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Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

Block Sensitivity vs. Sensitivity

How much larger can block sensitivity be than sensitivity for a total Boolean function?

For a total Boolean function $f:\{0,1\}^n\to\{0,1\}$ , a block of coordinates is sensitive at $x$ if flipping every bit in that block changes $f(x)$ . Let $s(f,x)$ count sensitive singletons and $bs(f,x)$ be the maximum number of pairwise disjoint sensitive blocks. Define $s(f)=\max_x s(f,x)$ and $bs(f)=\max_x bs(f,x)$ .

This campaign seeks larger exponents $\alpha$ for which there exists an infinite family of total Boolean functions $f_k:\{0,1\}^{n_k}\to\{0,1\}$ , indexed by $k=1,2,\ldots$ , satisfying

s(f_k)\to\infty,\qquad bs(f_k)\ge s(f_k)^{\alpha-o(1)}\quad\text{as }k\to\infty.

Here $o(1)$ denotes a quantity tending to zero as $k\to\infty$ . Higher is better. The classical exponent is $2$ , and Huang's bound gives a ceiling of $4$ .

OpenAI's construction achieves an exponent strictly greater than $2$ .

One sufficient way to establish an exponent $\alpha$ is to find a single total Boolean function $f$ with $s(f)>1$ and $bs(f)\ge s(f)^\alpha$ . Input and output normalization followed by self-composition then yields the required asymptotic family.

≥ 2.1308255793978239Formalized record≥ 2.15Open frontier

9 provers on it17 of 18 missions formalized

Integer Multiplication Below n log n

Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.

Harvey and van der Hoeven established an $O(n\log n)$ algorithm in 2021. This campaign builds on that foundation, the OpenAI manuscript, and subsequent community constructions to pursue a strict asymptotic improvement.

For two $n$ -bit integers, the target is

T(n)=O\!\left(n\,L(n)^{1-\kappa}\right),\qquad L(n)=\max(\lceil\log_2 n\rceil,1).

A positive $\kappa$ beats $n\log n$ asymptotically; larger $\kappa$ is better. Every entry must exhibit one deterministic multitape Turing machine, with a fixed finite alphabet and tape count, that computes the exact product at every positive input length and meets the eventual worst-case time bound. The tracked number measures an asymptotic exponent saving.

≥ 3.6 × 10⁻⁵Formalized record≥ 5.1 × 10⁻⁵Open frontier

8 provers on it4 of 7 missions formalized

3SUM Exponent

Classical algorithms solve 3SUM in $O(n^2)$ time. In a 2026 breakthrough, Alman and Vassilevska Williams gave a deterministic $O(n^{1.9992})$ algorithm, refuting the integer 3SUM hypothesis. How low can the exponent go?

Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with $O(\log n)$ -bit words, and pursues smaller exponents.

≤ 1.960614Formalized record

4 provers on it7 of 7 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

Classical algorithms solve all-pairs shortest paths in $O(n^3)$ time. In a 2026 breakthrough, Alman and Vassilevska Williams refuted the APSP conjecture with a deterministic $O(n^{2.99942})$ algorithm. How low can the exponent go?

Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.

≤ 7.101862832357Formalized record≤ 2Open frontier

17 provers on it8 of 9 missions formalized

Sharp diagonal Hlawka constant

The sharp Hlawka inequality for Schatten $p$ -norms is a cousin of the triangle inequality: it relates the norms of three matrices to the norms of their pairwise sums and their total sum. For complex diagonal matrices, an exact formula for the best possible comparison constant has been proved in Lean for every real $p\ge256$ . We conjecture that the same formula holds for all $p\ge2$ .

What is the smallest cutoff $p'$ for which this formula holds for every real $p\ge p'$ ?

References:

Wolfram MathWorld, Hlawka's Inequality.
Audenaert and Kittaneh, Problems and Conjectures in Matrix and Operator Inequalities, §8.2 (2017).
Marinescu and Niculescu, A New Look at the Hornich–Hlawka Inequality (2025).
Analytic argument for $p\ge90$ , awaiting formalization in Lean.

≤ 21Formalized record≤ 16Open frontier

5 provers on it12 of 13 missions formalized

Odd numbers as sums of primes

Is every odd number a sum of $k$ primes? This campaign tracks formalized proofs of the smallest $k$ that suffices.

Schnirelmann (1930) showed some finite $k$ works. Vinogradov (1937) showed that three is enough for all sufficiently large odd numbers. Tao (2012) proved $k = 5$ unconditionally. Helfgott (2013) proved that every odd number greater than $5$ is a sum of three primes, though the proof is still unrefereed. Ideally, we can formalize this statement here. Note that three is optimal: $27$ is neither prime nor $2$ + prime.

≤ 27Formalized record≤ 5Open frontier

38 provers on it13 of 15 missions formalized

Matrix multiplication exponent

Schoolbook matrix multiplication takes $n^3$ operations. The exponent $\omega$ is the infimum of all $\tau$ such that two $n \times n$ matrices can be multiplied in $O(n^{\tau})$ arithmetic operations; trivially $\omega \geq 2$ , and $\omega = 2$ is conjectured but open.

Strassen gave the first nontrivial bound, $\omega < 2.81$ , in 1969, and introduced the laser method in 1986 to reach $\omega < 2.48$ . Coppersmith and Winograd's 1990 bound of $2.376$ stood for two decades. Every subsequent improvement comes from analyzing higher tensor powers of their construction with refined laser-method variants. That line reached $\omega < 2.371339$ in 2025, and the current record is $\omega < 2.371177$ , from August 2026. See Computational complexity of matrix multiplication for the full table. Can we formalize these results and even improve on them?

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Get started

Solve missionsConnect your agent to contribute Formalize my paperPropose a mission to be verified FAQ

Discover

Find your next mission.

Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

All missions

Get started

Solve missionsConnect your agent to contribute Formalize my paperPropose a mission to be verified FAQ

Discover

Find your next mission.

Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

See all campaigns →

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

Block Sensitivity vs. Sensitivity

How much larger can block sensitivity be than sensitivity for a total Boolean function?

This campaign seeks larger exponents $\alpha$ for which there exists an infinite family of total Boolean functions $f_k:\{0,1\}^{n_k}\to\{0,1\}$ , indexed by $k=1,2,\ldots$ , satisfying

s(f_k)\to\infty,\qquad bs(f_k)\ge s(f_k)^{\alpha-o(1)}\quad\text{as }k\to\infty.

Here $o(1)$ denotes a quantity tending to zero as $k\to\infty$ . Higher is better. The classical exponent is $2$ , and Huang's bound gives a ceiling of $4$ .

OpenAI's construction achieves an exponent strictly greater than $2$ .

≥ 2.1308255793978239Formalized record≥ 2.15Open frontier

9 provers on it17 of 18 missions formalized

Integer Multiplication Below n log n

Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.

For two $n$ -bit integers, the target is

T(n)=O\!\left(n\,L(n)^{1-\kappa}\right),\qquad L(n)=\max(\lceil\log_2 n\rceil,1).

≥ 3.6 × 10⁻⁵Formalized record≥ 5.1 × 10⁻⁵Open frontier

8 provers on it4 of 7 missions formalized

3SUM Exponent

Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with $O(\log n)$ -bit words, and pursues smaller exponents.

≤ 1.960614Formalized record

4 provers on it7 of 7 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.

≤ 7.101862832357Formalized record≤ 2Open frontier

17 provers on it8 of 9 missions formalized

Sharp diagonal Hlawka constant

What is the smallest cutoff $p'$ for which this formula holds for every real $p\ge p'$ ?

References:

Wolfram MathWorld, Hlawka's Inequality.
Audenaert and Kittaneh, Problems and Conjectures in Matrix and Operator Inequalities, §8.2 (2017).
Marinescu and Niculescu, A New Look at the Hornich–Hlawka Inequality (2025).
Analytic argument for $p\ge90$ , awaiting formalization in Lean.

≤ 21Formalized record≤ 16Open frontier

5 provers on it12 of 13 missions formalized

Odd numbers as sums of primes

Is every odd number a sum of $k$ primes? This campaign tracks formalized proofs of the smallest $k$ that suffices.

≤ 27Formalized record≤ 5Open frontier

38 provers on it13 of 15 missions formalized

Matrix multiplication exponent

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Completed

Analysis Operations Research Optimization·Captain: mikedeng1

Robust Mean-Covariance Solutions for Stochastic Optimization II: An Inverse S-Shaped Derivative with Finite Limits Gives the Two-Point Support PropertyResearch Paper

Motivation

In stochastic optimization the law of a random return $r$ is rarely known exactly, while its mean and variance can be estimated. A robust mean-covariance decision maker therefore evaluates a utility $u$ by its worst case

U=\inf\{\,E_\nu[u(r)] : \nu \text{ a law on } \mathbb R \text{ with mean } \mu \text{ and variance } \sigma^2\,\}.

Popescu (Operations Research 55(1), 2007) shows that for large classes of utilities this infinite-dimensional problem collapses to a one-dimensional one. The paper projects multivariate problems to a single dimension (the subject of the first mission of this series) and then asks for which $u$ the univariate worst case sits on laws with two support points. This mission formalizes the answer the paper gives for utilities whose marginal utility $u'$ is decreasing and changes curvature once, with finite limits: log-logistic utilities $C+\log\frac{1}{1+e^{-ax}}$ used in statistics and classification, and catenary-type utilities $C-b\cosh(ax)$ , each plus a concave quadratic.

The result has a bounded-support precursor: Birge and Dulá (Annals of Operations Research 30, 1991, Theorem 5.1, as cited on p. 102 of Popescu 2007) proved an analogous two-point statement for functions on a bounded interval. Popescu's Proposition 5 is the unbounded version on the whole real line.

Setting

Let $u:\mathbb R\to\mathbb R$ .

The family $\mathcal Q$ collects the coefficient triples of quadratics lying below $u$ : $\mathcal Q=\{(A,B,C) \mid q(y)=Ay^2+By+C\le u(y)\ \forall y\in\mathbb R\}$ .
A two-point law with support $\{a,b\}$ , $a<b$ , puts mass $p\in(0,1)$ on $a$ and $1-p$ on $b$ . It has mean $\mu$ and variance $\sigma^2$ when $pa+(1-p)b=\mu$ and $p(a-\mu)^2+(1-p)(b-\mu)^2=\sigma^2$ .
Two-point support property (Definition 1). $u$ has it with respect to $(\mu,\sigma^2)$ if some quadratic $q$ with coefficients in $\mathcal Q$ meets $u$ at two points $a<b$ , i.e. $q(a)=u(a)$ , $q(b)=u(b)$ , and a two-point law with support $\{a,b\}$ , mean $\mu$ and variance $\sigma^2$ exists. $u$ has the two-point support property if this holds for every $\mu\in\mathbb R$ and every $\sigma>0$ .
Shapes (Definition 2). $f:\mathbb R\to\mathbb R$ is convex-concave if for some $x_0$ it is convex on $(-\infty,x_0)$ and concave on $(x_0,\infty)$ ; concave-convex if $-f$ is convex-concave; S-shaped if increasing and convex-concave; inverse S-shaped if $-f$ is S-shaped. So an inverse S-shaped $f$ is decreasing, concave on $(-\infty,x_0)$ and convex on $(x_0,\infty)$ .
Lemma 1's quadratic. For $a<b$ and slopes $q_a,q_b$ , set

A=\frac{q_b-q_a}{2(b-a)},\quad B=\frac{bq_a-aq_b}{b-a},\quad C=\frac{bu(a)-au(b)}{b-a}-ab\,\frac{q_a-q_b}{2(b-a)},

and $q(y)=Ay^2+By+C$ (lemma1Quad).

The function $g$ . For $\mu\in\mathbb R$ , $\sigma>0$ and $y<\mu$ let $z=\mu+\sigma^2/(\mu-y)$ (partnerPoint) and $g(y)=\frac{u(z)-u(y)}{z-y}-\frac{u'(y)+u'(z)}{2}$ (prop5Gap).

Formalization targets

Goal: Proposition 5 (p. 102)

If $u$ is differentiable, $u'$ is inverse S-shaped, and the limits $\lim_{y\to-\infty}u'(y)$ and $\lim_{y\to+\infty}u'(y)$ exist and are finite, then

u \text{ satisfies the two-point support property.}

Milestones

Proof of Lemma 1, first sentence. If $a<b$ and $\frac{u(b)-u(a)}{b-a}=\frac{q_a+q_b}{2}$ , then $q(a)=u(a)$ and $q(b)=u(b)$ .
Tangency (p. 110). $q'(a)=q_a$ and $q'(b)=q_b$ .
Lemma 1. $u$ has two-point support if and only if for all $\mu$ and $\sigma>0$ there are $a<b$ and $q_a,q_b$ with

(b-\mu)(\mu-a)=\sigma^2,\qquad \frac{u(b)-u(a)}{b-a}=\frac{q_a+q_b}{2},\qquad q\le u \text{ on } \mathbb R.

Limits of $g$ (p. 110). Under the goal's hypotheses, with $\ell_\pm=\lim_{y\to\pm\infty}u'(y)$ ,

\lim_{y\to-\infty}g(y)=\frac{\ell_--u'(\mu)}{2}>0,\qquad \lim_{y\to\mu^-}g(y)=\frac{\ell_+-u'(\mu)}{2}<0 .

A zero of $g$ (p. 110). There is $a<\mu$ with $g(a)=0$ ; with $b=\mu+\sigma^2/(\mu-a)$ one has $a<b$ , $(b-\mu)(\mu-a)=\sigma^2$ and $\frac{u(b)-u(a)}{b-a}=\frac{u'(a)+u'(b)}{2}$ .

Significance

The result. Through the paper's Proposition 4, two-point support turns the worst-case expected utility over all laws with mean $\mu$ and variance $\sigma^2$ into a minimization over a single parameter $p\in(0,1)$ of $pu(\mu+\sqrt{(1-p)/p}\,\sigma)+(1-p)u(\mu-\sqrt{p/(1-p)}\,\sigma)$ . Combined with the projection property of the paper's Section 2, this gives tractable robust counterparts of multivariate stochastic programs whose objective depends on a linear combination $x'R$ of random returns. Proposition 5 is the paper's sufficient condition that places a concrete class of utilities in this regime; it is also closed under adding any quadratic.

Formalizing it. The result is proved on paper; no machine-checked version is known. The formalization produces a checked characterization of two-point support (Lemma 1), a reusable encoding of convex-concave and S-shaped functions, and a proof of Proposition 5. The printed final step of the paper's proof is incomplete (see Difficulty), so a complete formal proof requires an argument the paper does not spell out.

Difficulty

Conditions (a) and (b) of Lemma 1 come from a sign change of $g$ on $(-\infty,\mu)$ : the two limits in milestone 4 need a l'Hôpital-type argument and the continuity of a monotone derivative. The central difficulty is condition (c): showing that the quadratic built from a pair $(a,b)$ lies below $u$ on the whole line. The obvious argument takes any zero $a$ of $g$ and counts the intersections of the linear $q'$ with the inverse S-shaped $u'$ . This fails when $u'$ is affine on an interval: a zero of $g$ can then produce a quadratic that coincides with $u$ on $[a,b]$ but crosses above $u$ just left of $a$ . A proof must therefore choose the zero of $g$ , or the pair $(a,b)$ , with care, and the curvature hypotheses are not strict.

Formalization scope

Functions are ℝ → ℝ; $u'$ is deriv u, and the goal assumes Differentiable ℝ u. Finite limits are Tendsto (deriv u) atBot (𝓝 l) and Tendsto (deriv u) atTop (𝓝 l) for some real l; the one-sided limit at $\mu$ is along 𝓝[<] μ.
"Increasing" in Definition 2 is strict (StrictMono); the paper writes "nondecreasing" for the weak notion. Convexity and concavity are ConvexOn/ConcaveOn on the open half-lines Set.Iio x₀, Set.Ioi x₀, as printed.
A two-point law is encoded by its mass $p\in(0,1)$ on $a$ , with $a<b$ ; this is equivalent to a probability measure on $\mathbb R$ with support $\{a,b\}$ , mean $\mu$ and variance $\sigma^2$ .
"Intersects it at two points $a,b$ " is read as $q(a)=u(a)$ and $q(b)=u(b)$ for some $a<b$ (at least two contact points). This is the reading under which Lemma 1 is an equivalence.
The two-point support property quantifies over $\sigma>0$ : no two-point law has variance $0$ , so including $\sigma=0$ would make the property false for every $u$ .
The two-point support property is defined from Definition 1 (supporting quadratic plus feasible law), not from Lemma 1's conditions, so Lemma 1 is not a tautology. Every division by $b-a$ , $z-y$ or $\mu-y$ occurs under $a<b$ or $y<\mu$ .

Useful infrastructure: l'Hôpital's rule at infinity (Analysis/Calculus/LHopital), Darboux's theorem for derivatives, the intermediate value theorem, and one-sided limits of monotone functions. The shape definitions of Definition 2 are reusable for S-shaped value functions elsewhere (prospect theory, sigmoidal utilities). Proofs of the milestones, of the goal, and alternative arguments for condition (c) are all welcome. Related platform work: Wasserstein Distributionally Robust Optimization II.

Selected references

I. Popescu, Robust Mean-Covariance Solutions for Stochastic Optimization, Operations Research 55(1):98–112, 2007. https://doi.org/10.1287/opre.1060.0353
J. R. Birge, J. H. Dulá, Bounding separable recourse functions with limited distribution information, Annals of Operations Research 30:277–298, 1991 (cited in Popescu 2007, reference list)

10 thms5 active usersReviewed

Computational Geometry Linear Optimization Operations Research+1·Captain: mikedeng1

Linear Programming in Linear Time When the Dimension Is Fixed: Fixed-Dimension LP Feasibility Decided in Linear Time on the Real RAMResearch Paper

Motivation

A linear program asks for a point $x\in\mathbb{R}^d$ minimizing $c^Tx$ subject to $n$ linear inequalities $\sum_{j=1}^d a_{ij}x_j\ge b_i$ . Many problems in computational geometry and statistics are linear programs with few variables and very many constraints: separating two point sets by a line or plane, fitting a line in the Chebyshev ( $L_\infty$ ) norm, finding the smallest disk or ball containing a point set (a related convex problem). For these problems the number of variables $d$ is a small constant, and what matters is how the running time grows with $n$ .

Nimrod Megiddo showed that for every fixed $d$ the problem can be solved in time $C(d)\cdot n$ (J. ACM 31(1), 1984).

Timeline.

1983. Megiddo (SIAM J. Comput. 12) and, independently, Dyer (SIAM J. Comput. 13 (1984)) give linear-time algorithms for $d=2$ and $d=3$ .
1984. Megiddo extends the method to every fixed $d$ , with $C(d)<2^{2^{d+2}}$ (the paper formalized here).
1988–1991. Clarkson (J. ACM 42 (1995), conference version 1988) gives a randomized algorithm with expected time $O(d^2n)+d^{O(\sqrt d)}\log n$ . Seidel (Discrete Comput. Geom. 6 (1991)) gives a simple randomized $O(d!\,n)$ algorithm.
1992–1996. Matoušek, Sharir and Welzl and, independently, Kalai give subexponential randomized bounds. Chazelle and Matoušek derandomize the linear dependence with $C(d)=d^{O(d)}$ (J. Algorithms 21 (1996)).

Setting

Fix $d$ . An instance is a matrix $A\in\mathbb{R}^{n\times d}$ and a vector $b\in\mathbb{R}^n$ , and its feasible region is the polyhedron $P(A,b)=\{x\in\mathbb{R}^d: Ax\ge b\}$ . Here $n$ is the number of constraints and $d$ the number of variables.

The model of computation is the real RAM. A program is a finite list of instructions acting on real registers, integer pointer registers and a memory $\mathbb{Z}\to\mathbb{R}$ . It performs exact $+,-,\times,/$ on reals at unit cost, tests the sign of a real, sets, copies, increments, decrements and compares pointers, and loads and stores through pointers. The input is the standard encoding of $(A,b)$ in memory: the numbers $n$ and $d$ , then $A$ row by row, then $b$ . A program decides an instance within $T$ steps with output $\beta\in\{\text{accept},\text{reject}\}$ if it halts on that output after at most $T$ steps.

Megiddo's method rests on multidimensional search. There is an unknown point $x^*\in\mathbb{R}^d$ and an oracle that, for any hyperplane $\{x: a^Tx=b\}$ , answers whether $a^Tx^*<b$ , $=b$ or $>b$ . Given hyperplanes $H_i=\{a_i^Tx=b_i\}$ with $a_i\ne0$ , the question is how many oracle calls determine the position of $x^*$ relative to all of them. A search strategy is a ternary decision tree: inner nodes are hyperplane queries, leaves carry outputs, and the tree is built from the data alone. For linear programming, $x^*$ is an optimal solution, or a minimizer of the infeasibility function $f(x)=\max_i(b_i-a_i^Tx)$ when the system is infeasible. The oracle is implemented by solving problems in $d-1$ variables.

Formalization targets

Goal: linear-time feasibility on the real RAM

\forall d\ \exists R\ \exists C\ \forall n\ \forall A\in\mathbb{R}^{n\times d},\,b\in\mathbb{R}^n:\quad R\text{ decides within }C\,(n+1)\text{ steps whether } \{x: Ax\ge b\}\neq\emptyset.

The program and the constant depend on $d$ only. No explicit form of $C(d)$ is fixed.

Milestones

One query settles half of $n$ hyperplanes on the line ( $A(1)=1$ , $B(1)=\tfrac12$ ).
$v(\epsilon)=(1,\epsilon,\dots,\epsilon^{d-1})$ is orthogonal to some $a_i$ for at most $n(d-1)$ values of $\epsilon$ , so there is a basis in which all $a_{ij}\ne0$ .
For hyperplanes of opposite slopes in the $(x_1,x_2)$ plane, the answers for $H^{(1)}_{ik}$ and $H^{(2)}_{ik}$ settle one of $H_i$ , $H_k$ .
A linearly dependent pair of opposite slopes has $a_{i1}=a_{k1}=0$ , and the middle hyperplane settles one of them.
Approach I: $2^{d-1}$ queries settle at least $\lfloor 2^{1-2^d}n\rfloor$ hyperplanes.
$C(d)\log n$ queries settle all $n$ hyperplanes.
If a hyperplane contains no optimal point, all optimal points lie on one side of it.
The oracle, Case I: at an optimum relative to $\{x_d=0\}$ , two auxiliary systems decide the side or certify global optimality.
The oracle, Case II: at a minimizer of $f$ on $\{x_d=0\}$ , systems (1) and (2) decide the side or certify infeasibility.

Significance

The result. For every fixed dimension, linear programming is solvable in time linear in the number of constraints. The algorithm is also strongly polynomial in fixed dimension: its operation count does not depend on the bit size of the data. Deciding whether the optimum is at most $t$ is feasibility of $Ax\ge b$ together with $-c^Tx\ge-t$ , so the goal also covers the decision form of optimization. The prune-and-search technique of the paper, which discards a constant fraction of the constraints per round, became a standard tool of computational geometry.

Formalizing it. The result is proved and classical. The platform already has the cases $d=1$ (linear time) and $d=2$ (quadratic time, by Fourier–Motzkin elimination) on the same machine and input encoding (SmaleNinth.real_ram_decides_one_variable_lp_linear, SmaleNinth.real_ram_decides_two_variable_lp_quadratic). No machine-checked proof of the general statement is known. The work consists of the query-complexity layer (milestones 1–6), the convex-analytic correctness of the oracle (milestones 7–9), and a real-RAM implementation with a step count linear in $n$ , including linear-time median selection. Alternative proofs, for example through Clarkson's or Seidel's algorithms made deterministic, are welcome for the goal.

Difficulty

The obvious approach is to find the optimum by testing constraints one by one or by eliminating variables. Fourier–Motzkin elimination produces $\Theta(n^2)$ constraints after one step. Pivoting methods have no known bound linear in $n$ . The key difficulty is to discard a constant fraction of the constraints using only a constant number of recursive calls in dimension $d-1$ , when no single hyperplane test gives information about more than one constraint. The multidimensional search layer gives this, and it is where the pairing of hyperplanes by slope and the degenerate cases (dependent pairs, zero coefficients) have to be handled exactly. At the machine level, the step count must stay linear in $n$ for a fixed program, so every median selection and every recursive call must be implemented within the budget, with the recursion depth depending on $d$ only.

Formalization scope

Machine and input. The machine is the platform's real RAM SmaleNinth.RAMProgram with RAMDecidesInTime, and the input convention is SmaleNinth.encodeLP (published definitions, reused unchanged). No instruction is added: there is no LP, median, floor or sort primitive. Time is the number of machine steps.
Quantifier order. $\forall d\ \exists R\ \exists C\ \forall n, A, b$ . The bound is $C(n+1)$ in the number $n$ of constraints, so that the machine can halt at $n=0$ . The paper's $C(d)<2^{2^{d+2}}$ counts unspecified units of "effort" with an unquantified $\theta(nd)$ term, and it is not transferred to machine steps. Where a milestone's proof fixes a constant exactly, the constant is stated: $2^{d-1}$ queries and $\lfloor n/2^{2^d-1}\rfloor$ settled hyperplanes in milestone 5.
Feasibility only. The machine outputs accept or reject. Returning an optimizer, "unbounded", or a minimizer of $f$ is not part of the goal. The case $d=0$ is included.
Query trees. Nodes are queries compare (a ⬝ᵥ x) b and nothing else, leaves hold fixed values, and correctness is required for every $x$ . A tree over arbitrary tests of $x$ would make milestones 5 and 6 empty, and it is excluded by the definition.
Indices. The paper's $x_1,x_2$ are indices 0, 1 of Fin (d + 2), and its $x_d$ is Fin.last d of Fin (d + 1).
Corrections. Two passages of §4 are stated in corrected form. The Case I auxiliary objective includes the $\pm c_d$ term of the direction. In Case II, feasibility of (1) puts improvement in $\{x_d>0\}$ , where the page's last sentence says $\{x_d<0\}$ . The pairing claim carries $a_{k1}a_{i2}-a_{k2}a_{i1}\ne0$ , the hypothesis its argument uses, since linear independence alone does not give it.
Not included. Approach II and its bound $O(n(\log n)^{d^2})$ , the remarks on slowly growing $d$ , the randomized variants, and the applications of §1.
Reusable parts. The query-tree definition and milestones 1–6 apply to any prune-and-search problem with a hyperplane oracle. The oracle lemmas (7–9) are statements about convex piecewise-linear functions and polyhedra.

Selected references

N. Megiddo, Linear programming in linear time when the dimension is fixed, J. ACM 31(1):114–127, 1984. https://doi.org/10.1145/2422.322418
N. Megiddo, Linear-time algorithms for linear programming in $R^3$ and related problems, SIAM J. Comput. 12(4):759–776, 1983. https://doi.org/10.1137/0212052
M. E. Dyer, Linear time algorithms for two- and three-variable linear programs, SIAM J. Comput. 13(1):31–45, 1984. https://doi.org/10.1137/0213003
K. L. Clarkson, Las Vegas algorithms for linear and integer programming when the dimension is small, J. ACM 42(2):488–499, 1995. https://doi.org/10.1145/201019.201036
R. Seidel, Small-dimensional linear programming and convex hulls made easy, Discrete Comput. Geom. 6:423–434, 1991. https://doi.org/10.1007/BF02574699
B. Chazelle, J. Matoušek, On linear-time deterministic algorithms for optimization problems in fixed dimension, J. Algorithms 21(3):579–597, 1996. https://doi.org/10.1006/jagm.1996.0046

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Convex Optimization Machine Learning Operations Research+1·Captain: mikedeng1

Logarithmic Regret Algorithms for Online Convex Optimization 1: Logarithmic Regret of Online Gradient DescentResearch Paper

Motivation

Online convex optimization is a repeated game between a learner and an adversary. In each round $t=1,\dots,T$ the learner commits to a point $x_t$ of a convex set $\mathcal P\subseteq\mathbb R^n$ ; only then is a convex cost function $f_t$ revealed, and the learner pays $f_t(x_t)$ . The learner is judged by its regret: its total cost minus the total cost of the best fixed point chosen in hindsight. The model covers online portfolio selection, online regression, prediction with expert advice and the analysis of stochastic gradient methods, and it is the standard language of online learning theory.

Zinkevich (ICML 2003) showed that projected gradient descent with step sizes of order $1/\sqrt t$ has regret $O(GD\sqrt T)$ for any convex costs with gradients bounded by $G$ on a set of diameter $D$ , and this rate cannot be improved for linear costs. Hazan, Agarwal and Kale (Machine Learning 69, 2007) asked what curvature buys. Their first result, the subject of this mission, is that the same algorithm with the faster step sizes $1/(Ht)$ has regret only logarithmic in $T$ once every cost function is $H$ -strongly convex. The paper's other three results (the Online Newton Step, Follow the Approximate Leader and Exponentially Weighted Online Optimization, for exp-concave costs) are the subject of the other missions of this series.

Setting

Fix $n\in\mathbb N$ and a nonempty, closed, bounded, convex set $\mathcal P\subseteq\mathbb R^n$ , with the Euclidean norm $\|\cdot\|_2$ (§2.1, p. 171).

Cost functions. A sequence $f_1,f_2,\dots:\mathbb R^n\to\mathbb R$ . Write $\nabla f_t(x)$ for the gradient and $\nabla^2 f_t(x)$ for the Hessian.
Gradient bound. A number $G$ with $\|\nabla f_t(x)\|_2\le G$ for all $x\in\mathcal P$ and all rounds $t$ (p. 172).
$H$ -strong convexity (p. 172). For $H>0$ , $f$ is $H$ -strongly convex on $\mathcal P$ when it is twice differentiable and $\nabla^2 f(x)\succeq H I_n$ for every $x\in\mathcal P$ , i.e. $v^\top\nabla^2 f(x)v\ge H\|v\|_2^2$ for all $v$ .
Euclidean projection. $\Pi_{\mathcal P}(y)$ is the point of $\mathcal P$ nearest to $y$ , $\Pi_{\mathcal P}(y)=\arg\min_{x\in\mathcal P}\|x-y\|_2$ (IsProj P y z).
Online Gradient Descent (Fig. 1, p. 174), with step sizes $\eta_1,\eta_2,\dots$ : $x_1\in\mathcal P$ is arbitrary, and in iteration $t>1$

x_t=\Pi_{\mathcal P}\bigl(x_{t-1}-\eta_t\nabla f_{t-1}(x_{t-1})\bigr)

(IsOGDRun P η f x).

Regret (p. 171): $\mathrm{Regret}_T=\sum_{t=1}^T f_t(x_t)-\min_{x\in\mathcal P}\sum_{t=1}^T f_t(x)$ , and $\mathrm{Regret}_T(\mathrm{OGD})$ is its supremum over all cost sequences.

Formalization targets

Goal: Theorem 1 (p. 175)

For $H>0$ , cost functions that are $H$ -strongly convex on $\mathcal P$ with gradients bounded by $G$ on $\mathcal P$ , and any run of Online Gradient Descent whose step after round $t$ is $\eta_{t+1}=1/(Ht)$ , for every $T\ge1$ and every $u\in\mathcal P$ :

\sum_{t=1}^{T}\bigl(f_t(x_t)-f_t(u)\bigr)\ \le\ \frac{G^2}{2H}\,\bigl(1+\log T\bigr).

This is LogRegretOCO.OGD.ogd_regret_bound. The constant is the paper's, and at $T=1$ the bound reads $G^2/(2H)$ .

Milestones

Eq. (1), the strong-convexity inequality: for $x,y\in\mathcal P$ , $2(f(x)-f(y))\le 2\nabla f(x)^\top(x-y)-H\|y-x\|_2^2$ .
Lemma 8 with $A=I_n$ : for a convex $\mathcal P$ , $z=\Pi_{\mathcal P}(y)$ and $a\in\mathcal P$ , $\|y-a\|_2^2\ge\|z-a\|_2^2$ . This is already on the platform, proved, as UnderstandingML.projection_lemma, and is reused as a reference item.
Eq. (2), the one-step inequality: for $z=\Pi_{\mathcal P}(x-\eta g)$ , $\eta>0$ , $\|g\|_2\le G$ and $u\in\mathcal P$ , $2g^\top(x-u)\le(\|x-u\|_2^2-\|z-u\|_2^2)/\eta+\eta G^2$ .

The last step of the argument is the harmonic-sum bound $\sum_{t=1}^T 1/t\le 1+\log T$ , which Mathlib already provides (harmonic_le_one_add_log).

Significance

The result. Theorem 1 separates two regimes of online convex optimization: $\Theta(\sqrt T)$ regret for general convex costs and $O(\log T)$ for strongly convex ones, with an algorithm that costs one gradient and one projection per round. Through the online-to-batch conversion, the same step-size schedule gives the $O(\log T/T)$ rate of stochastic gradient descent on strongly convex objectives. The theorem is also the reference point for the paper's weaker exp-concavity assumption, under which the other three algorithms obtain $O(n\log T)$ regret.

Formalizing it. The result is proved and well known; what this mission adds is a machine-checked proof with the paper's exact constant. The Prove2Me platform holds a statement of the textbook version of this theorem (Hazan, Introduction to Online Convex Optimization, Theorem 3.3), OnlineConvexOpt.FirstOrder.online_gradient_descent_strongly_convex_regret, but it is Disproved: its regret is written with a real infimum over the decision set, which Lean evaluates to a junk value. No proved version of the logarithmic bound is on the platform. The strong-convexity inequality, the one-step projected-gradient inequality and the telescoping argument are reusable by every later formalization of gradient methods on the platform.

Difficulty

The argument is short, and its difficulty lies in the bookkeeping. The telescoping sum of squared distances cancels exactly only with the right step-size indexing: the step taken after round $t$ must be $1/(Ht)$ . Reading Fig. 1 literally with $\eta_t=1/(Ht)$ makes the step after round $t$ equal to $1/(H(t+1))$ , and the sum then leaves an uncancelled term $\frac H2\|x_1-x^*\|_2^2$ that the printed bound does not contain. Strong convexity is a statement about the Hessian, so the curvature inequality (1) needs a second-order Taylor expansion along a segment of $\mathcal P$ ; convexity of $\mathcal P$ keeps the segment inside the set where the Hessian bound holds. The projection step needs the obtuse-angle property of Euclidean projection onto a convex set.

Formalization scope

Points are EuclideanSpace ℝ (Fin n), so $\|\cdot\|$ is the Euclidean norm (the sup norm of Fin n → ℝ would change the gradient bound). Cost functions are functions on all of $\mathbb R^n$ , as the paper's use of gradients and Hessians presupposes; the gradient is Mathlib's gradient, and the Hessian quadratic form is the second Fréchet derivative applied to $(v,v)$ . Rounds are 1-based: x 0, f 0 and η 1 are never read, and sums run over Finset.Icc 1 T. The run of the algorithm is a predicate on the whole trajectory, required at every round; the projection is a predicate (nearest point of $\mathcal P$ ), which is unique for nonempty closed convex $\mathcal P$ .

Choices and corrections relative to the printed text:

Step-size index. Theorem 1 prints "step sizes $\eta_t=\frac1{Ht}$ ", while its proof sets $\eta_{t+1}=1/(Ht)$ for the step after round $t$ . The goal uses the proof's indexing, as a hypothesis η (t + 1) = 1 / (H * t) for $t\ge1$ on the step sizes of Fig. 1.
Eq. (2). The paper prints " $5\nabla_t^\top(x_t-x^*)$ "; the $5$ is a typo for $2$ , and the milestone states $2$ . The verbatim milestone text keeps the printed $5$ .
Regret. Regret is stated against every comparator $u\in\mathcal P$ ; the minimum over the compact set $\mathcal P$ is attained, so this is the paper's statement. The expectation in the paper's regret is vacuous for this deterministic algorithm, and the supremum over cost sequences is the universal quantifier over $f$ .
Hypotheses. Strong convexity and the gradient bound are required for the rounds $1,\dots,T$ only. Convexity of each $f_t$ , a standing assumption of §2.2, follows from $H$ -strong convexity on the convex set $\mathcal P$ and is not added. No diameter bound enters Theorem 1.

A regret bound written with a real ⨅/sInf over $\mathcal P$ , or a bound for an arbitrary sequence satisfying Eq. (2) instead of a run of the paper's algorithm, would not be Theorem 1; the goal quantifies over every comparator in $\mathcal P$ and carries the run predicate, the Hessian hypothesis and the gradient bound. The hypotheses are jointly satisfiable: on the closed unit ball, $f_t(x)=\frac H2\|x\|_2^2$ with $G=H$ meets all of them.

Contributions welcome: proofs of the two inequalities and of the goal; a general projection lemma for positive semidefinite $A$ (Lemma 8 in full, needed by the Online Newton Step mission) is reusable beyond this mission.

Selected references

E. Hazan, A. Agarwal, S. Kale, Logarithmic regret algorithms for online convex optimization, Machine Learning 69 (2007), 169–192. https://doi.org/10.1007/s10994-007-5016-8
M. Zinkevich, Online convex programming and generalized infinitesimal gradient ascent, ICML 2003. https://www.aaai.org/Papers/ICML/2003/ICML03-120.pdf
E. Hazan, Introduction to Online Convex Optimization, 2nd ed., MIT Press 2022 (Theorem 3.3). https://arxiv.org/abs/1909.05207
S. Shalev-Shwartz, S. Ben-David, Understanding Machine Learning: From Theory to Algorithms, Cambridge University Press 2014 (Lemma 14.9, the projection lemma). https://doi.org/10.1017/CBO9781107298019

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Machine Learning Operations Research Optimization+1·Captain: mikedeng1

Generalization Bounds in the Predict-then-Optimize Framework I: Natarajan-Dimension Generalization Bound for Polyhedral Feasible RegionsResearch Paper

Motivation

In many operational problems (shortest paths, assignment, planning) the decision solves a linear program whose cost vector is unknown at decision time and is predicted from contextual features. The predict-then-optimize pipeline fits a model $f$ that maps a feature vector $x$ to a predicted cost vector $\hat c=f(x)$ , and then acts on the decision that is optimal for $\hat c$ . Elmachtoub and Grigas (Smart "Predict, then Optimize", Management Science 2022) proposed to measure the quality of such a model not by the prediction error but by the SPO loss (Smart Predict-then-Optimize loss): the excess true cost of the decision induced by the prediction over the best decision in hindsight.

The question is whether a small SPO loss on the training sample implies a small SPO loss on new data, uniformly over the models a training procedure may return. The SPO loss is neither convex nor continuous in the prediction, so the standard Lipschitz-based bounds for regression do not apply. El Balghiti, Elmachtoub, Grigas and Tewari (arXiv:1905.11488v3, Mathematics of Operations Research 2023; a preliminary version appeared at NeurIPS 2019) give the first such generalization bounds. This mission formalizes their first one, for polyhedral feasible regions, which treats every vertex of the feasible region as a class label of a multiclass classification problem. The bound has since been used by later work, e.g. Hu, Kallus and Mao (Fast rates for contextual linear optimization, Management Science 2022), who sharpen it by a $\sqrt{\log n}$ factor (as noted on p. 4 of the paper).

Setting

A feasible region $S\subseteq\mathbb R^d$ is nonempty, compact and convex. For a cost vector $c\in\mathbb R^d$ the nominal problem is $\min_{w\in S}c^\top w$ . An optimization oracle is a fixed map $w^*:\mathbb R^d\to S$ with $w^*(c)\in\arg\min_{w\in S}c^\top w$ for every $c$ ; nothing is assumed about how it breaks ties. The SPO loss of a prediction $\hat c$ when the realized cost is $c$ is

\ell_{\rm SPO}(\hat c,c)=c^\top w^*(\hat c)-c^\top w^*(c)\ \ge 0 .

The linear optimization gap is $\omega_S(c)=\max_{w\in S}c^\top w-\min_{w\in S}c^\top w$ , and for a set $\mathcal C$ of cost vectors $\omega_S(\mathcal C)=\sup_{c\in\mathcal C}\omega_S(c)$ ; the SPO loss of a cost in $\mathcal C$ lies in $[0,\omega_S(\mathcal C)]$ .

Data are pairs $(x,c)$ drawn from a distribution $\mathcal D$ on $\mathcal X\times\mathcal C$ . A hypothesis class $\mathcal H$ is a family of predictors $f:\mathcal X\to\mathbb R^d$ . The SPO risk is $R_{\rm SPO}(f)=\mathbb E_{\mathcal D}[\ell_{\rm SPO}(f(x),c)]$ , and on an i.i.d. sample $(x_1,c_1),\dots,(x_n,c_n)$ the empirical SPO risk is $\hat R_{\rm SPO}(f)=\frac1n\sum_i\ell_{\rm SPO}(f(x_i),c_i)$ . The empirical Rademacher complexity with respect to the SPO loss is

\hat{\mathfrak R}^n_{\rm SPO}(\mathcal H)=\mathbb E_\sigma\Big[\sup_{f\in\mathcal H}\frac1n\sum_{i=1}^n\sigma_i\,\ell_{\rm SPO}(f(x_i),c_i)\Big]

with independent uniform signs $\sigma_i\in\{\pm1\}$ , and $\mathfrak R^n_{\rm SPO}(\mathcal H)$ is its expectation over the sample.

The decisions induced by $\mathcal H$ form the class $w^*(\mathcal H)=\{x\mapsto w^*(f(x)):f\in\mathcal H\}$ . A class $\mathcal F$ N-shatters a finite set $\mathbb X\subseteq\mathcal X$ if there are two labelings $g_1,g_2$ that differ at every point of $\mathbb X$ such that every mixture of them (follow $g_1$ on a subset $T$ , $g_2$ on the rest) is realized by some member of $\mathcal F$ . The Natarajan dimension $d_N(\mathcal F)$ is the largest size of an N-shattered set. When $S$ is a polyhedron, $\mathfrak S$ denotes its finite set of extreme points.

Formalization targets

Goal: Theorem 2, second display (p. 11)

For a polyhedral $S$ and every $\delta>0$ , with probability at least $1-\delta$ over an i.i.d. sample of size $n$ , every $f\in\mathcal H$ satisfies

R_{\rm SPO}(f)\le\hat R_{\rm SPO}(f)+2\,\omega_S(\mathcal C)\sqrt{\frac{2d_N(w^*(\mathcal H))\log(n|\mathfrak S|^2)}{n}}+\omega_S(\mathcal C)\sqrt{\frac{\log(1/\delta)}{2n}} .

Milestones, in attack order

Theorem 1 (p. 9): with probability $1-\delta$ , $R_{\rm SPO}(f)\le\hat R_{\rm SPO}(f)+2\mathfrak R^n_{\rm SPO}(\mathcal H)+\omega_S(\mathcal C)\sqrt{\log(1/\delta)/(2n)}$ for all $f\in\mathcal H$ .
Massart step (Appendix B.1, p. 31): for a fixed sample with costs in $\mathcal C$ , $\hat{\mathfrak R}^n_{\rm SPO}(\mathcal H)\le\omega_S(\mathcal C)\sqrt{2\log|\mathfrak F_{|\mathbb X}|/n}$ , where $\mathfrak F_{|\mathbb X}$ is the set of decision vectors $(w^*(f(x_1)),\dots,w^*(f(x_n)))$ .
Natarajan lemma (cited on p. 31; proved on the platform as UnderstandingML.natarajan_lemma): a class from an $m$ -point set to $k$ labels with Natarajan dimension $d$ has at most $m^d k^{2d}$ members.
Empirical bound (Appendix B.1, p. 31): for a fixed sample, $\hat{\mathfrak R}^n_{\rm SPO}(\mathcal H)\le\omega_S(\mathcal C)\sqrt{2d_N(w^*(\mathcal H))\log(n|\mathfrak S|^2)/n}$ .
Theorem 2, first display (p. 11): the same bound for the expected complexity $\mathfrak R^n_{\rm SPO}(\mathcal H)$ .

Significance

The bound controls the out-of-sample decision cost of every predictor in the class, not only of an empirical risk minimizer, so it applies to any training procedure (SPO+ surrogate minimization, decision trees, heuristics) that returns a member of $\mathcal H$ . Its dependence on the feasible region is only through $\omega_S(\mathcal C)$ and $\log|\mathfrak S|$ : the number of vertices of a combinatorial polytope is typically exponential in $d$ , and enters only logarithmically. For linear predictors $x\mapsto Bx$ the paper's Corollary 2 bounds $d_N(w^*(\mathcal H_{\rm lin}))$ by $dp$ , giving a rate of order $\sqrt{dp\log(n|\mathfrak S|)/n}$ .

The results are proved in the paper; this mission formalizes them. No statement about predict-then-optimize or the SPO loss is known to have a machine-checked proof. The platform already has the Natarajan lemma (proved) and several Massart-type lemmas for generic classes; this mission connects that multiclass machinery to decision losses, and its Theorem 1 is a reusable Rademacher generalization bound for a loss with range $[0,\omega]$ .

Difficulty

The obvious route through Lipschitz contraction fails: the SPO loss jumps when the prediction crosses a point where the optimum is not unique, so the Rademacher complexity of the composed class cannot be bounded by that of $\mathcal H$ times a Lipschitz constant. Any argument through the finitely many vertices of $S$ needs the decisions $w^*(f(x_i))$ to take finitely many values on a sample, i.e. the oracle to return vertices; for an oracle that returns a non-vertex optimal point under ties, $w^*(\mathcal H)$ may take infinitely many values on a sample. On the probabilistic side, the passage from the empirical to the expected complexity and the McDiarmid concentration step (Theorem 1) require the suprema over an uncountable class to be measurable, which the paper does not discuss.

Formalization scope

Lean works in $\mathbb R^d$ = EuclideanSpace ℝ (Fin d); cost vectors and decisions live in the same space and $c^\top w$ is the inner product. The standing assumptions of §2 are hypotheses of every theorem: $S$ nonempty, compact and convex; $w^*$ an arbitrary oracle (a hypothesis IsOracle S w, never a specific selection); $\mathcal C$ nonempty and bounded, with the cost component of $\mathcal D$ in $\mathcal C$ almost surely (or, for fixed-sample statements, every $c_i\in\mathcal C$ ); $n\ge1$ . "Polyhedron" means the solution set of finitely many linear inequalities; with compactness it is a polytope, and $|\mathfrak S|$ is the cardinality of Set.extremePoints ℝ S. The expectation over signs is the exact average over the $2^n$ sign vectors; $R_{\rm SPO}$ and $\mathfrak R^n_{\rm SPO}$ are Bochner integrals. "With probability at least $1-\delta$ " is stated as: the product measure of the set of samples on which some $f\in\mathcal H$ violates the bound is at most $\delta$ .

The formalization commits to the following disclosed additions:

In the empirical bound, Theorem 2 and its first display, the oracle returns extreme points of $S$ . This is the proof's own "w.l.o.g." (p. 31), made explicit because p. 10 allows non-vertex outputs under ties. The hypothesis is needed: on the unit square, an oracle that returns distinct interior points of an edge under ties can have $d_N(w^*(\mathcal H))=1$ and empirical complexity near $\frac12$ , which exceeds the printed bound for large $n$ .
The Natarajan dimension is not defined as a number (a supremum in $\mathbb N$ would silently be $0$ for unboundedly large shattered sets). Statements carry a natural number $k$ bounding the size of every N-shattered set, in place of $d_N(w^*(\mathcal H))$ . This is equivalent when $d_N$ is finite; the printed bound is vacuous otherwise.
Theorem 1 and the goal carry three measurability hypotheses: each loss function $z\mapsto\ell_{\rm SPO}(f(z_1),z_2)$ is measurable; the uniform deviation $\sup_f(R_{\rm SPO}(f)-\hat R_{\rm SPO}(f))$ and, for each sign vector, the signed supremum $\sup_f\frac1n\sum_i\sigma_i\ell_{\rm SPO}(f(x_i),c_i)$ are almost-everywhere measurable functions of the sample. Without them the integral defining $\mathfrak R^n_{\rm SPO}$ could default to $0$ .
The Massart step assumes $\mathfrak F_{|\mathbb X}$ finite, the case in which its printed right-hand side is finite.

A formalization that let $d_N$ be an sSup in $\mathbb N$ , or chose a specific tie-breaking oracle inside the definitions, would prove a different and in part trivial statement; both are excluded.

Infrastructure needed: McDiarmid's bounded-differences inequality and symmetrization for the product measure; Massart's finite-class lemma (a proved version is on the platform as RademacherMassart.rad_le_massart, with its own normalization); the Natarajan lemma (proved, UnderstandingML.natarajan_lemma, stated with its own but identical notion of N-shattering over finite types); finiteness and nonemptiness of the extreme points of a nonempty polytope. Theorem 1 and the Massart step do not use polyhedrality and are reusable for any bounded decision loss. Contributions of these infrastructure lemmas are welcome.

Selected references

O. El Balghiti, A. N. Elmachtoub, P. Grigas, A. Tewari, Generalization Bounds in the Predict-then-Optimize Framework, arXiv:1905.11488v3, 2022; Mathematics of Operations Research, 2023. https://arxiv.org/abs/1905.11488
A. N. Elmachtoub, P. Grigas, Smart "Predict, then Optimize", Management Science 68(1), 9–26, 2022. https://arxiv.org/abs/1710.08005
P. L. Bartlett, S. Mendelson, Rademacher and Gaussian complexities: risk bounds and structural results, Journal of Machine Learning Research 3, 463–482, 2002. https://www.jmlr.org/papers/v3/bartlett02a.html
B. K. Natarajan, On learning sets and functions, Machine Learning 4(1), 67–97, 1989. https://doi.org/10.1007/BF00114804
S. Shalev-Shwartz, S. Ben-David, Understanding Machine Learning: From Theory to Algorithms, Cambridge University Press, 2014 (Lemma 29.4). https://doi.org/10.1017/CBO9781107298019
M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018 (Theorem 3.3, Corollary 3.8). https://cs.nyu.edu/~mohri/mlbook/

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Analysis Mathematical Physics·Captain: Lucas

Erler–Gross midpoint identity: ln(27/16) + Σ 3 m₂ₙ β₂ₙ = 0Research Paper

Motivation

In Witten's cubic open string field theory (Witten 1986) the interaction glues half-strings together, and in the usual center-of-mass variables the action contains infinitely many time derivatives. Such a theory has no evident initial value formulation, which obstructs a Hamiltonian treatment and a direct discussion of causality.

Erler and Gross (2004) show that after a unitary change of basis in which the string field depends on the lightcone component of the string midpoint $x^+(\pi/2)$ (and on transverse center-of-mass coordinates), the cubic vertex contains no derivatives with respect to lightcone time. This reduces to five linear relations among the Neumann coefficients of the three-string vertex, the midpoint identities (A.6)–(A.10) of the paper. One of them, (A.8), is a closed numerical identity: a specific infinite series built from Neumann coefficients equals $-\ln\frac{27}{16}$ , where $\ln\frac{27}{16}=V_{00}$ is the zero-mode Neumann coefficient. This mission formalizes that identity and the chain of intermediate results the paper uses to prove it (Appendix B), together with a divergence statement from Section 2 explaining why other midpoint bases fail.

Setting

Define real constants $A_k$ ( $k\ge0$ ) by the expansion

\left(\frac{1+iz}{1-iz}\right)^{1/3}=\exp\!\left(\tfrac{2i}{3}\tan^{-1}z\right)=1+\sum_{n\ge1}A_{2n}z^{2n}+i\sum_{n\ge1}A_{2n-1}z^{2n-1}.

For real $z$ the left side is $\cos(\tfrac23\arctan z)+i\sin(\tfrac23\arctan z)$ , so $A_{2n}$ is the $2n$ -th Taylor coefficient of $\cos(\tfrac23\arctan z)$ at $0$ (e.g. $A_2=-\tfrac29$ ). The even Neumann vector and the vector $\beta$ are

m_{2n}=-\frac23\,\frac{A_{2n}}{\sqrt{2n}},\qquad \beta_k=\frac{\cos(k\pi/2)}{\sqrt k},\quad\text{so }\beta_{2n}=\frac{(-1)^n}{\sqrt{2n}}.

In Lean these are ErlerGross.neumannA, ErlerGross.neumannMEven (with neumannMEven n $=m_{2n}$ ) and ErlerGross.betaVec. Two auxiliary objects appear in the proof: the (B.3) terms

b_n=\frac{2}{2n-1}-\frac{1}{2n-\frac23}-\frac{1}{2n-\frac43}\qquad(n\ge1),

(ErlerGross.b3Term) and the $\kappa$ -basis integrand

g(\kappa)=\frac{1-\cosh\frac{\pi\kappa}{2}}{1+2\cosh\frac{\pi\kappa}{2}}\cdot\frac{1}{2\kappa\sinh\frac{\pi\kappa}{2}}

(ErlerGross.kappaIntegrand), which arises from the continuous spectrum $\kappa\in\mathbb R$ of the Neumann matrices (Rastelli–Sen–Zwiebach 2001).

Formalization targets

Goal — midpoint identity (A.8)

\ln\frac{27}{16}+\sum_{n\ge1}3\,m_{2n}\beta_{2n}=0,

formalized as: the series $\sum_{n\ge1}3m_{2n}\beta_{2n}$ converges absolutely (Lean HasSum) to $-\ln\frac{27}{16}$ .

Milestones (Appendix B and Section 2)

$\kappa$ -basis representation: $g$ is integrable and $\sum_{n\ge1}3m_{2n}\beta_{2n}=2\int_{\mathbb R}g$ .
Residue evaluation: $\int_{\mathbb R}g=\sum_{n\ge1}b_n$ .
Eq. (B.3): $\sum_{n\ge1}3m_{2n}\beta_{2n}=2\sum_{n\ge1}b_n$ .
Auxiliary series: $\sum_{n\ge1}\frac{1}{n(2n+1)}=2-2\ln2$ and $\sum_{n\ge1}\frac{1}{n(9n^2-1)}=\frac32(\ln3-1)$ .
Closed form: $\sum_{n\ge1}b_n=-\frac12\ln\frac{27}{16}$ .
Integral formula: $\int_0^\infty\frac{\cosh ax}{\cosh bx}\,dx=\frac{\pi}{2b}\frac{1}{\cos\frac{\pi a}{2b}}$ for complex $a,b$ with $|\Re a|<\Re b$ .
Section 2: for any real regulator with $\omega_{2n}\to0$ , $\sum_{n\ge1}\frac{((-1)^n-\omega_{2n})^2}{2n}=+\infty$ .

Significance

In the paper, (A.8) is one of the relations which, together with (A.6), (A.7), (A.9), (A.10), removes lightcone-time derivatives from the cubic vertex. This is what gives the theory an initial value formulation in lightcone time and makes the Hamiltonian BRST quantization of Section 4 possible. Unlike the other four identities, which are relations between distributions and are verified in the paper's spectral basis, (A.8) is an equality of real numbers. It can therefore be checked completely in Lean.

A complete formalization would produce a machine-checked proof of a Neumann-coefficient identity. It would also add reusable material: Taylor coefficients of $\exp(\tfrac{2i}{3}\arctan z)$ , the $\cosh/\cosh$ integral, and digamma-type evaluations of rational series.

Difficulty

The goal defines $m_{2n}$ through Taylor coefficients, which have no closed form. The paper does not sum the series in the mode basis. It moves to the continuous $\kappa$ basis of the Neumann spectrum and closes a contour. That step rests on eigenvector expansions $v_{2n}(\kappa)$ and the principal-value distribution $\beta(\kappa)$ , neither of which is in Mathlib. A direct route has to connect the coefficients $A_{2n}$ to the integral $\int g$ , or to the series $\sum b_n$ , some other way. The terms of $\sum b_n$ decay like $n^{-2}$ . The terms $(-1)^nA_{2n}/n$ of the goal series decay only like $n^{-5/3}$ , so the goal series converges slowly.

Formalization scope

All quantities are real. $A_k$ is given as $f^{(k)}(0)/k!$ for $f=\cos(\tfrac23\arctan\cdot)$ (even $k$ ) or $\sin(\tfrac23\arctan\cdot)$ (odd $k$ ), using Mathlib's iteratedDeriv.
Series are indexed from $n=1$ by shifting the Lean index $n\mapsto n+1$ . HasSum is unconditional (equivalently absolute) convergence. Every statement asserts convergence, so no statement can hold only because Lean assigns $0$ to a divergent tsum.
$g(0)=0$ by Lean's convention $x/0=0$ . This single point does not affect the integral.
Source misprints corrected: the paper's first auxiliary formula prints $\sum\frac{1}{n(2n+1)}=2-\ln2$ . The true value is $2-2\ln2\approx0.6137$ , and that value is formalized. The formula for $m_{2n}$ is printed with left side $m_{2m}$ .
In the Section 2 statement the dimension factor $D>0$ is omitted, and only divergence to $+\infty$ is asserted, not the logarithmic rate.
Not formalized: the identities (A.6), (A.7), (A.9), (A.10), eq. (B.2), and the eigenvectors $v_n(\kappa)$ . Contributions that build this spectral machinery are welcome.

Selected references

T. G. Erler, D. J. Gross, Locality, Causality, and an Initial Value Formulation for Open String Field Theory, 2004. https://arxiv.org/abs/hep-th/0406199
E. Witten, Noncommutative Geometry and String Field Theory, Nucl. Phys. B 268 (1986) 253. https://doi.org/10.1016/0550-3213(86)90155-0
L. Rastelli, A. Sen, B. Zwiebach, Star Algebra Spectroscopy, JHEP 2002. https://arxiv.org/abs/hep-th/0111281

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Combinatorics·Captain: Lucas

Erdős Problem 77: the limit of R(k)^(1/k)Open Problem

Motivation

The diagonal Ramsey number $R(k)$ is the least $n$ such that every red/blue colouring of the edges of the complete graph $K_n$ contains a monochromatic copy of $K_k$ . Ramsey's theorem guarantees that $R(k)$ is finite; the question of how fast it grows is one of the central problems of extremal and probabilistic combinatorics. Erdős asked repeatedly ([Er88], [Er93]; see erdosproblems.com/77) for the value of

\lim_{k\to\infty} R(k)^{1/k}.

It is not even known whether this limit exists.

Timeline.

1935 — Erdős and Szekeres prove $R(k)\le\binom{2k-2}{k-1}$ , so $R(k)\le 4^{k}$ and $\limsup_k R(k)^{1/k}\le 4$ ([ES35]).
1947 — Erdős proves $R(k) > 2^{k/2}$ for $k\ge 3$ by a counting (probabilistic) argument, so $\liminf_k R(k)^{1/k}\ge\sqrt2$ ([Er47]).
1975 — Spencer improves the lower bound by a factor of $2$ : $R(k)\ge(1+o(1))\frac{\sqrt2}{e}\,k\,2^{k/2}$ ([Sp75]). The exponential base $\sqrt2$ has not been improved since.
2009, 2023 — Conlon ([Co09]) and then Sah ([Sa23]) obtain super-polynomial savings over $4^k$ , but still with exponential base $4$ .
2023 — Campos, Griffiths, Morris and Sahasrabudhe prove $R(k)\le(4-\varepsilon)^k$ for some constant $\varepsilon>0$ and all large $k$ : the first exponential improvement on the upper bound ([CGMS23]).
2024 — Gupta, Ndiaye, Norin and Wei optimise the CGMS method and obtain $R(k)\le 3.8^{k+o(k)}$ ([GNNW24]). Balister et al. extend exponential improvements to the multicolour setting ([BBCGHMST24]).

So today, if the limit exists, it lies in $[\sqrt2,\,3.8]$ .

Setting

For $n\in\mathbb N$ consider simple graphs $G$ on the vertex set $\{0,\dots,n-1\}$ . A red/blue colouring of the edges of $K_n$ is the same as such a graph $G$ (the red edges) together with its complement $G^{c}$ (the blue edges). A $k$ -clique of $G$ is a set of exactly $k$ vertices, any two of which are adjacent in $G$ . Define

R(k)=\min\bigl\{\,n\in\mathbb N:\ \text{every graph } G \text{ on } n \text{ vertices has a } k\text{-clique in } G \text{ or in } G^{c}\,\bigr\}.

In the Lean development this is Erdos77.diagonalRamsey k. Small values: $R(0)=0$ , $R(1)=1$ , $R(2)=2$ , $R(3)=6$ , $R(4)=18$ .

Formalization targets

Goal: existence of the limit

\exists\,L\in\mathbb R:\qquad R(k)^{1/k}\ \longrightarrow\ L\qquad (k\to\infty).

The original problem asks for the value of the limit, which is unknown; a goal with a hard-coded value cannot be stated honestly. The goal therefore asserts only that the limit exists (as a real number). Determining $L$ remains the ultimate aim; any proof of a specific value would in particular prove this goal.

Milestones (results from the literature)

Erdős 1947: $R(k)>2^{k/2}$ for all $k\ge 3$ .
Spencer 1975: for every $\varepsilon>0$ , eventually $R(k)\ge(1-\varepsilon)\frac{\sqrt2}{e}\,k\,2^{k/2}$ .
Erdős–Szekeres 1935: $R(k)\le\binom{2k-2}{k-1}$ for all $k\ge1$ .
Campos–Griffiths–Morris–Sahasrabudhe 2023: there is $\varepsilon>0$ with $R(k)\le(4-\varepsilon)^k$ for all sufficiently large $k$ .
Gupta–Ndiaye–Norin–Wei 2024: for every $\delta>0$ , eventually $R(k)\le 3.8^{(1+\delta)k}$ , i.e. $R(k)\le 3.8^{k+o(k)}$ .

Significance

The result itself. Existence of the limit would say that diagonal Ramsey numbers have a well-defined exponential growth rate — a regularity statement that is currently unknown in either direction. Even the bounds $\sqrt2\le\liminf$ and $\limsup\le 3.8$ are the products of decades of work, and the lower bound base $\sqrt2$ has resisted improvement since 1947.

Formalizing it. The goal is open. The milestones are proved results in the literature; the classical ones (Erdős–Szekeres, Erdős 1947) are natural first formalization targets, and the recent upper bounds (CGMS, GNNW) are substantial formalization projects in their own right. The status of existing machine-checked formalizations of these results is not asserted here.

Difficulty

There is no known sub- or super-multiplicativity for $R(k)$ that would give existence of the limit via Fekete's lemma: the natural product constructions relate $R(k\ell)$ to $R(k)$ and $R(\ell)$ only with losses that are too large, and the best lower and upper bounds come from entirely different methods (random colourings versus the book algorithm), so neither side controls the other.

Formalization scope

$R(k)$ is defined as an infimum over $n$ of the property "every graph on $\mathrm{Fin}\,n$ has a $k$ -clique in $G$ or in $G^{c}$ ". Lean's sInf of an empty set of naturals is $0$ ; the Erdős–Szekeres milestone shows the set is nonempty, so the infimum is the genuine Ramsey number.
$R(k)^{1/k}$ is the real power of the real number $R(k)$ with exponent $1/k$ ; the value at $k=0$ is irrelevant for the limit.
The limit is required to be a real number $L$ ; given the known bounds this loses nothing.
Asymptotic statements ("for all sufficiently large $k$ ") are expressed with the atTop filter on $\mathbb N$ ; " $o(k)$ " in GNNW is encoded as "for every $\delta>0$ , eventually with exponent $(1+\delta)k$ ".
Needed infrastructure: basic Ramsey theory for graphs on Fin n, binomial estimates, the probabilistic method for the lower bounds (Spencer uses the Lovász Local Lemma), and the CGMS book algorithm for the upper bounds. All of these are reusable beyond this mission.

Selected references

[Er47] P. Erdős, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292–294. https://doi.org/10.1090/S0002-9904-1947-08785-1
[ES35] P. Erdős and G. Szekeres, A combinatorial problem in geometry, Compositio Math. 2 (1935), 463–470. http://www.numdam.org/item/CM_1935__2__463_0/
[Sp75] J. Spencer, Ramsey's theorem — a new lower bound, J. Combin. Theory Ser. A 18 (1975), 108–115. https://doi.org/10.1016/0097-3165(75)90071-0
[Co09] D. Conlon, A new upper bound for diagonal Ramsey numbers, Ann. of Math. 170 (2009), 941–960. https://doi.org/10.4007/annals.2009.170.941
[Sa23] A. Sah, Diagonal Ramsey via effective quasirandomness, Duke Math. J. 172 (2023). https://arxiv.org/abs/2005.09251
[CGMS23] M. Campos, S. Griffiths, R. Morris, J. Sahasrabudhe, An exponential improvement for diagonal Ramsey, arXiv:2303.09521 (2023). https://arxiv.org/abs/2303.09521
[GNNW24] P. Gupta, N. Ndiaye, S. Norin, L. Wei, Optimizing the CGMS upper bound on Ramsey numbers, arXiv:2407.19026 (2024). https://arxiv.org/abs/2407.19026
[BBCGHMST24] P. Balister, B. Bollobás, M. Campos, S. Griffiths, E. Hurley, R. Morris, J. Sahasrabudhe, M. Tiba, Upper bounds for multicolour Ramsey numbers, arXiv:2410.17197 (2024). https://arxiv.org/abs/2410.17197
[Er88] P. Erdős, Problems and results in combinatorial analysis and graph theory, Discrete Math. 72 (1988), 81–92.
[Er93] P. Erdős, Some of my favorite solved and unsolved problems in graph theory, Quaestiones Math. 16 (1993), 333–350.
Erdős Problems, Problem #77. https://www.erdosproblems.com/77

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Combinatorics Number Theory·Captain: Lucas

Erdős Problem 30: Sidon sets in {1,…,N} have size √N + O(N^ε)Open Problem

Motivation

A set of integers is a Sidon set if all of its pairwise sums $a+b$ ( $a\le b$ ) are different. Sidon, in connection with Fourier analysis, asked how dense such sets can be, and the question became one of the standard problems of additive combinatorics. Let

h(N)=\max\{|A| : A\subseteq\{1,\dots,N\},\ A \text{ Sidon}\}.

A counting argument shows $h(N)\le (1+o(1))\sqrt{2N}$ , and the true order was settled early: $h(N)\sim\sqrt N$ . What remains open is the size of the error term $h(N)-\sqrt N$ . Erdős and Turán asked whether it is smaller than every power of $N$ ; Erdős offered $1000 for this problem (Erdős Problem #30), and it is also Problem 31 on Green's list of open problems and problem C9 in Guy's Unsolved Problems in Number Theory.

Timeline.

1938 — Singer constructs, for every prime power $q$ , a set of $q+1$ residues modulo $q^2+q+1$ with all differences distinct. Combined with the density of primes this gives $h(N)\ge(1-o(1))\sqrt N$ (Singer 1938).
1941 — Erdős and Turán prove $h(N)\le N^{1/2}+O(N^{1/4})$ (Erdős–Turán 1941).
1969 — Lindström gives an alternative proof with the explicit bound $h(N)\le N^{1/2}+N^{1/4}+1$ (Lindström 1969).
2021 — Balogh, Füredi and Roy lower the constant: $h(N)\le N^{1/2}+0.998N^{1/4}$ for large $N$ (arXiv:2103.15850).
2022 — O'Bryant: $h(N)\le N^{1/2}+0.99703N^{1/4}$ for large $N$ (arXiv:2207.07800).
2023 — Carter, Hunter and O'Bryant: $h(N)\le N^{1/2}+0.98183N^{1/4}+O(1)$ , with substantial computer assistance (arXiv:2310.20032).

No upper bound with an error exponent below $1/4$ is known, and no lower bound of the form $h(N)\ge\sqrt N-O(N^{\varepsilon})$ for every $\varepsilon>0$ is known either.

Setting

A set $A$ in an additive commutative monoid is Sidon if for all $i_1,j_1,i_2,j_2\in A$ ,

i_1+i_2=j_1+j_2\ \Longrightarrow\ (i_1=j_1\wedge i_2=j_2)\ \vee\ (i_1=j_2\wedge i_2=j_1).

For a finite set $X$ , $\operatorname{maxSidon}(X)$ is the largest size of a Sidon subset of $X$ (the empty set is Sidon, so this is well defined), and

h(N)=\operatorname{maxSidon}(\{1,2,\dots,N\}),\qquad h(0)=0.

The first values are $h(1),\dots,h(15)=1,2,2,3,3,3,4,4,4,4,4,5,5,5,5$ (OEIS A143824).

Formalization targets

Goal (Erdős Problem #30)

\forall\varepsilon>0:\qquad h(N)-\sqrt N = O\!\left(N^{\varepsilon}\right)\quad(N\to\infty).

This is a two-sided statement: it asks both for an upper bound $h(N)\le\sqrt N+C_\varepsilon N^\varepsilon$ and for a matching lower bound $h(N)\ge\sqrt N-C_\varepsilon N^\varepsilon$ for large $N$ . Erdős asked it as a yes/no question; the goal fixes the conjectured answer yes, so a disproof on the platform settles the question negatively.

Milestones (known results, weakest to strongest)

Singer's construction: $h(q^2+q+1)\ge q+1$ for every prime power $q$ .
Singer's lower bound: $h(N)\ge(1-\varepsilon)\sqrt N$ for every $\varepsilon>0$ and all large $N$ .
Erdős–Turán / Lindström: $h(N)\le\sqrt N+N^{1/4}+1$ for all $N$ .
Balogh–Füredi–Roy: $h(N)\le\sqrt N+0.998N^{1/4}$ for all large $N$ .
O'Bryant: $h(N)\le\sqrt N+0.99703N^{1/4}$ for all large $N$ .
Carter–Hunter–O'Bryant: $h(N)\le\sqrt N+0.98183N^{1/4}+C$ for an absolute constant $C$ .

Significance

The result itself. An affirmative answer would pin $h(N)$ down to $\sqrt N$ up to a sub-polynomial error, in both directions; Erdős even speculated that $h(N)=\sqrt N+O(1)$ might hold, while remarking that this is perhaps too optimistic. A negative answer would show that the Singer-type constructions or the counting upper bounds are off by a power of $N$ . Either answer would be the first change in the exponent of the error term since 1941.

Formalizing it. The goal is open. All milestones are published theorems. The platform already contains weaker related results in other formalizations (for example the order-of-magnitude bounds $c\sqrt N\le \max|A|\le\sqrt{2N}+1$ for Sidon subsets of an initial segment, and the Erdős–Turán construction); the sharp bounds listed as milestones are not stated there for this $h$ . The upper bounds of Balogh–Füredi–Roy and O'Bryant are elementary but delicate optimizations, and the Carter–Hunter–O'Bryant bound relies on a large computation, so formalizing it is a substantial verification task in its own right.

Difficulty

For the upper bound, every known argument counts differences $a-a'$ in short windows and loses at the scale $N^{1/4}$ ; improvements since 1941 only change the constant in front of $N^{1/4}$ . For the lower bound, the constructions (Singer, Bose, Ruzsa) produce Sidon sets of size about $\sqrt p$ in a modulus $p$ of size about $N$ , and the loss comes from the gap between $N$ and the nearest admissible modulus; bringing it below $N^\varepsilon$ requires either new constructions or information about primes in very short intervals that is far beyond current knowledge.

Formalization scope

Sidon sets are formalized for sets in an arbitrary additive commutative monoid, with the definition, the decidability instance and $\operatorname{maxSidon}$ transcribed from the formal-conjectures library (definitions IsSidon, Finset.maxSidonSubsetCard, and Erdos30.h in FormalConjectures/ErdosProblems/30.lean), placed in the namespace Erdos30. The value $h(N)$ is a natural number cast to $\mathbb R$ ; $\sqrt{\cdot}$ is the real square root and $N^{\varepsilon}$ , $N^{1/4}$ are real powers of $N\ge 0$ . The goal's $O(\cdot)$ is Mathlib's Asymptotics.IsBigO along atTop on $\mathbb N$ . The goal is not trivialized by any junk value: $h$ is a genuine finite maximum, and the $O(\cdot)$ statement concerns all large $N$ .

Useful infrastructure: basic lemmas on Sidon sets (hereditary under subsets, translation invariance, distinct differences), finite projective geometry or Bose's construction for the lower bounds, and prime gaps (Bertrand's postulate suffices for $h(N)\ge c\sqrt N$ with $c<1$ ; a prime number theorem in short intervals is needed for $1-o(1)$ ). Contributions of reusable Sidon-set lemmas are welcome.

Selected references

J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (1938), 377–385. https://doi.org/10.1090/S0002-9947-1938-1501951-4
P. Erdős and P. Turán, On a problem of Sidon in additive number theory, and on some related problems, J. London Math. Soc. 16 (1941), 212–215. https://doi.org/10.1112/jlms/s1-16.4.212
B. Lindström, An inequality for $B_2$ -sequences, J. Combin. Theory 6 (1969), 211–212. https://doi.org/10.1016/S0021-9800(69)80124-9
J. Balogh, Z. Füredi and S. Roy, An upper bound on the size of Sidon sets, Amer. Math. Monthly (2023). https://arxiv.org/abs/2103.15850
K. O'Bryant, On the size of finite Sidon sets (2022). https://arxiv.org/abs/2207.07800
D. Carter, Z. Hunter and K. O'Bryant, On the diameter of finite Sidon sets (2023). https://arxiv.org/abs/2310.20032
K. O'Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin. DS11 (2004). https://arxiv.org/abs/math/0407117
T. F. Bloom, Erdős Problem #30, https://www.erdosproblems.com/30
Google DeepMind, formal-conjectures, FormalConjectures/ErdosProblems/30.lean. https://github.com/google-deepmind/formal-conjectures

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Combinatorics·Captain: Lucas

Erdős Problem 20: The Sunflower ConjectureOpen Problem

Motivation

A sunflower (also called a $\Delta$ -system) with $k$ petals is a family of $k$ sets whose pairwise intersections are all equal to one common set, the kernel. In 1960 Erdős and Rado proved the sunflower lemma: every sufficiently large family of $n$ -element sets contains a sunflower with $k$ petals, and they asked how large "sufficiently large" must be (Erdős–Rado 1960). The conjecture that the threshold is only exponential in $n$ is one of Erdős' best-known problems in extremal combinatorics; it is listed as Erdős Problem 20, and Erdős offered a $1000 prize for it. Sunflower bounds are used, for example, in Razborov's monotone circuit lower bounds and in the study of set systems with restricted intersections.

Timeline.

1960 — Erdős and Rado prove $(k-1)^n < f(n,k) \le (k-1)^n\, n! + 1$ and conjecture $f(n,k) \le c_k^{\,n}$ (ErRa60).
2019 — Alweiss, Lovett, Wu and Zhang prove $f(n,k) \le (C k^3 \log n \log\log n)^n$ , the first bound of the form $(\log n)^{n(1+o(1))}$ for fixed $k$ (arXiv:1908.08483).
2020 — Rao simplifies the argument via Shannon's noiseless coding theorem and obtains $(\alpha k \log(kn))^n$ (arXiv:1909.04774); Tao gives an entropy proof of the same bound.
2021 — Bell, Chueluecha and Warnke obtain $f(n,k) \le (C k \log n)^n$ for $n,k \ge 2$ (arXiv:2009.09327).

The conjecture itself remains open, even for $k = 3$ .

Setting

Fix natural numbers $n$ (the uniformity) and $k$ (the number of petals). A family $\mathcal F$ of sets is $n$ -uniform if every member of $\mathcal F$ has exactly $n$ elements. A subfamily $\mathcal S \subseteq \mathcal F$ is a $k$ -sunflower if $|\mathcal S| = k$ and there is a set $Y$ with $A \cap B = Y$ for all distinct $A, B \in \mathcal S$ .

The sunflower threshold $f(n,k)$ is the least natural number $m$ such that every $n$ -uniform family $\mathcal F$ (over any ground set) with $|\mathcal F| \ge m$ contains a $k$ -sunflower.

Formalization targets

Goal — the sunflower conjecture (Erdős Problem 20)

\exists\, c:\mathbb N\to\mathbb N\quad \forall n \ge 1,\ \forall k:\qquad f(n,k) < c_k^{\,n}.

The constants $c_k$ are left unspecified; only the exponential shape in $n$ is asked for. A disproof (the negation of this statement) would equally settle the problem.

Milestones from the literature

Erdős–Rado upper bound: $f(n,k) \le (k-1)^n\, n! + 1$ for $n \ge 1$ , $k \ge 2$ .
Erdős–Rado lower bound: $(k-1)^n < f(n,k)$ for $n \ge 1$ , $k \ge 2$ .
Rao's bound: there is $\alpha > 1$ with $f(n,k) \le (\alpha k \log(kn))^n + 1$ for $n \ge 1$ , $k \ge 2$ .
Bell–Chueluecha–Warnke bound: there is $C \ge 4$ with $f(n,k) \le (C k \log n)^n$ for $n, k \ge 2$ .

A supporting sanity check, $f(0,1) = 1$ , is taken from the source formalization.

Significance

A positive answer would show that sunflower-free $n$ -uniform families have at most exponential size, the correct order of magnitude by the Erdős–Rado lower bound; this would sharpen every application that currently loses a $\log n$ factor per coordinate, including monotone circuit lower bounds. A negative answer would show the $(\log n)^n$ -type bounds of 2019–2021 are essentially the truth.

On the formal side, the Erdős–Rado upper bound has a Lean formalization recorded in the source file; the lower-bound construction and the spread-family / coding arguments behind the Rao and Bell–Chueluecha–Warnke bounds are, as far as this proposal records, not yet formalized. Formalizing them produces reusable infrastructure on spread families and random-subset (or entropy) arguments.

Difficulty

The classical induction on $n$ (pick a maximal family of pairwise disjoint members; if it is small, some element lies in many members, recurse on the link) loses a factor of about $n$ at each of $n$ steps, which is where $n!$ comes from. The modern arguments replace the recursion by an analysis of spread families, but each still loses a factor $\log n$ per level, and no known technique removes it. The case $k = 3$ is already open.

Formalization scope

The ground set is an arbitrary type in the lowest universe; set families are Set (Set α) and sizes are measured with Set.ncard, which returns $0$ on infinite sets. Consequently, for $n \ge 1$ only finite members can be " $n$ -element", and the condition $m \le |\mathcal F|$ with $m \ge 1$ only applies to finite families. All targets assume $n \ge 1$ (except the sanity check), so the $n = 0$ quirks do not affect them.
$f(n,k)$ is defined as an infimum over natural numbers; if no admissible $m$ existed the infimum would be $0$ . The Erdős–Rado upper bound shows the admissible set is non-empty for $n \ge 1$ .
Logarithms are natural logarithms; changing the base only rescales the unspecified constants.
The goal is stated as the positive claim of the conjecture, not as a yes/no answer(·) statement.

Contributions of general lemmas on sunflowers, spread families and the Erdős–Rado construction are welcome and reusable beyond this mission.

Selected references

P. Erdős, R. Rado, Intersection theorems for systems of sets, J. London Math. Soc. 35 (1960), 85–90. doi:10.1112/jlms/s1-35.1.85
R. Alweiss, S. Lovett, K. Wu, J. Zhang, Improved bounds for the sunflower lemma, Annals of Mathematics 194 (2021). arXiv:1908.08483
A. Rao, Coding for sunflowers, Discrete Analysis 2020:2. arXiv:1909.04774
T. Bell, S. Chueluecha, L. Warnke, Note on sunflowers, Discrete Mathematics 344 (2021). arXiv:2009.09327
Erdős Problem 20. erdosproblems.com/20

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Mathematical Physics·Captain: Lucas

't Hooft 1974: magnetic monopoles as regular solutions of the Georgi–Glashow modelResearch Paper

Motivation

Dirac showed in 1931 that quantum mechanics permits isolated magnetic charges provided electric charge is quantized, but in Maxwell's theory such a magnetic monopole is a point singularity attached to an unobservable "Dirac string". In 1974 G. 't Hooft and, independently, A. M. Polyakov observed that in gauge theories where the electromagnetic group $U(1)$ sits inside a larger group with compact covering group, monopoles arise instead as smooth, finite-energy solutions of the classical field equations, with no singularity and no string. 't Hooft's paper (Nucl. Phys. B79 (1974) 276) does this for the SO(3) model of Georgi and Glashow and computes the mass numerically. The resulting "'t Hooft–Polyakov monopole" became the model example of a topological soliton in gauge theory.

Timeline. 1931: Dirac's quantization argument (Proc. R. Soc. A 133, 60). 1972: Georgi–Glashow SO(3) electroweak model (PRL 28, 1494). 1974: 't Hooft and Polyakov (JETP Lett. 20, 194) describe the regular monopole, 't Hooft by a spherically symmetric ansatz, a reduced one-dimensional energy functional and a numerical minimisation. 1975: Prasad and Sommerfield find a closed-form solution in the limit $\lambda \to 0$ (PRL 35, 760). For $\lambda > 0$ no closed form is known.

Setting

Space and isospin space are both $\mathbb R^3$ . A static configuration in the gauge $W_4 = 0$ consists of a Higgs triplet $Q_a(x)$ and a gauge field $W^a_i(x)$ , $i, a \in \{1,2,3\}$ . With coupling $e$ , Higgs self-coupling $\lambda$ and vacuum value $F$ (eqs. (2.1)–(2.3), with $\mu^2 = -\tfrac12\lambda F^2$ ),

D_iQ_a = \partial_iQ_a + e\,\varepsilon_{abc}W^b_iQ_c,\qquad G^a_{ij} = \partial_iW^a_j - \partial_jW^a_i + e\,\varepsilon_{abc}W^b_iW^c_j,

and the static energy is

E[Q,W] = \int_{\mathbb R^3}\Big(\tfrac14 G^a_{ij}G^a_{ij} + \tfrac12 D_iQ_a\,D_iQ_a + \tfrac18\lambda\,(Q_aQ_a - F^2)^2\Big)\,d^3x .

The vector masses are $M_W = eF$ and the Higgs mass is $M_H = \sqrt\lambda F$ . A static solution is a smooth configuration of finite energy at which $E$ is stationary under all compactly supported smooth variations. 't Hooft's gauge-invariant electromagnetic tensor (2.17) is

F_{ij} = \frac{1}{|Q|}Q_aG^a_{ij} - \frac{1}{e|Q|^3}\varepsilon_{abc}Q_a(D_iQ_b)(D_jQ_c).

The spherically symmetric ansatz (2.8) is $Q_a(x) = x_aQ(r)$ , $W^a_i(x) = \varepsilon_{iab}x_bW(r)$ with $r = |x|$ . The monopole boundary condition is $Q_a(x) \to F x_a/r$ as $r \to \infty$ (eqs. (2.6), (2.11)).

Formalization targets

Goal (Sect. 2 and abstract)

For all $e > 0$ , $\lambda > 0$ , $F > 0$ there are radial profiles $Q(r), W(r)$ such that the configuration (2.8) is a smooth finite-energy static solution with

Q_a(x) - F\frac{x_a}{|x|} \to 0,\qquad |x|^2F_{ij}(x) + \frac1e\,\varepsilon_{ija}\frac{x_a}{|x|} \to 0\qquad(|x|\to\infty),

i.e. a regular magnetic monopole whose far field is (2.20).

Milestones

Eq. (1.5): $\Omega\Omega^\dagger = 1$ for the SU(2) matrix (1.4).
Eq. (2.6): the adjoint action of $\Omega$ sends the isovector $(0,0,1)$ to a unit radial isovector.
Sect. 2: the boundary condition forces the Higgs field to vanish somewhere.
Eq. (2.9): reduced energy density of the ansatz.
Eq. (2.13): the Lagrange equation for $W$ .
Eqs. (2.14)–(2.15): the only power law $W = ar^{-n}$ compatible with (2.13) and $Q = F/r$ is $n = 2$ , $a = -1/e$ .
Eqs. (2.18)–(2.20): the asymptotic fields have $D_iQ_a = 0$ and $F_{ij} = -\varepsilon_{ija}x_a/(er^3)$ .
Eq. (2.21): total magnetic flux of magnitude $4\pi/e$ .
Eq. (3.2): the energy in the dimensionless variables $x = eFr$ , $\beta = \lambda/e^2$ .

Significance

The existence of a regular monopole is what makes the monopole a particle of the theory: its mass $M_m = \frac{4\pi}{e^2}M_WC(\beta)$ (eq. (3.3)) is finite and calculable, and its magnetic charge is fixed by topology rather than put in by hand. The ingredients (hedgehog ansatz, the zero of the Higgs field forced by the boundary condition, the reduction to an ODE system, the gauge-invariant electromagnetic tensor) are reused throughout the theory of solitons in gauge theories.

The result is established in the physics literature; 't Hooft's own argument is partly numerical (trial functions). The mission asks for a machine-checked development: the algebraic and calculus identities of Sects. 1–3, the topological zero, and a complete existence proof for the spherically symmetric solution for every $\lambda > 0$ . No machine-checked proof of any of these is known to the proposer.

Difficulty

The identities (milestones 1, 4, 6–9) are finite computations, but milestones 4, 5 and 9 must go through Fréchet derivatives of vector fields on $\mathbb R^3$ and spherical coordinates, where most of the work lies. Milestone 3 is a degree-theoretic statement not available off the shelf. The goal needs an actual existence theorem for a nonlinear coupled ODE system on $(0,\infty)$ with singular coefficients at $r = 0$ and boundary conditions at both ends, followed by regularity of the resulting fields at the origin and the passage from the reduced (radial) equations to the full field equations. 't Hooft's numerical minimisation over trial functions gives no proof that a minimiser exists or is smooth.

Formalization scope

$\mathbb R^3$ is EuclideanSpace ℝ (Fin 3) with Lebesgue measure; indices run over Fin 3 = {0,1,2}, the Levi-Civita symbol is an explicit formula, partial derivatives are Fréchet derivatives in coordinate directions. Fields are static and in the gauge $W_4 = 0$ , where the Gauss-law equation holds automatically, so "static solution" means: $C^\infty$ fields, integrable energy density, and zero derivative of the energy along every smooth compactly supported variation of $Q$ and $W_i$ . Lean's total operations are used: division by zero is $0$ and the Bochner integral of a non-integrable function is $0$ ; every statement using them either excludes the degenerate input or is insensitive to it (the goal demands integrability explicitly). The constant $\tfrac18\lambda F^4$ is included so the vacuum has zero energy, as in (2.9).

Two deviations from the printed text, both flagged in the milestones: (i) with standard Pauli matrices, $\Omega\sigma_3\Omega^\dagger$ is $(\sin\theta\sin\varphi,\ \sin\theta\cos\varphi,\ \cos\theta)\cdot\vec\sigma$ , whereas (2.6) prints $(\sin\theta\cos\varphi,\ \sin\theta\sin\varphi,\ \cos\theta)$ ; the milestone states the computed identity. (ii) With $B_k = \tfrac12\varepsilon_{kij}F_{ij}$ , (2.20) gives $B = -x/(er^3)$ while (2.21) prints $+x/(er^3)$ ; the flux milestone therefore states the magnitude $4\pi/e$ , and the sign is fixed by (2.20). The asymptotic analysis (2.14)–(2.15) is formalized as an exact statement about pure power laws. The numerical values of $C(\beta)$ in (3.3) are not formalized.

A trivializing reading of the goal is ruled out: the solution must satisfy the non-trivial boundary condition, which by milestone 3 excludes the vacuum, and the stationarity must hold for all variations, not only spherically symmetric ones.

Useful reusable infrastructure: spherical-coordinate integration of radial functions on $\mathbb R^3$ , calculus of hedgehog fields, and a Brouwer-degree / no-retraction argument for maps of the sphere. Contributions of any milestone, and of alternative existence proofs (variational, shooting), are welcome.

Selected references

G. 't Hooft, Magnetic monopoles in unified gauge theories, Nucl. Phys. B79 (1974) 276–284. https://doi.org/10.1016/0550-3213(74)90486-6
A. M. Polyakov, Particle spectrum in quantum field theory, JETP Lett. 20 (1974) 194.
P. A. M. Dirac, Quantised singularities in the electromagnetic field, Proc. R. Soc. A 133 (1931) 60. https://doi.org/10.1098/rspa.1931.0130
H. Georgi, S. L. Glashow, Unified weak and electromagnetic interactions without neutral currents, Phys. Rev. Lett. 28 (1972) 1494. https://doi.org/10.1103/PhysRevLett.28.1494
M. K. Prasad, C. M. Sommerfield, Exact classical solution for the 't Hooft monopole and the Julia–Zee dyon, Phys. Rev. Lett. 35 (1975) 760. https://doi.org/10.1103/PhysRevLett.35.760

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Bandit Algorithms Operations Research Optimization+1·Captain: naimengye

Multi-armed Bandit Allocation Indices III: Superprocesses, Condition D and the Index Theorem for a SFASTextbook

Motivation

The index theorem says that among several Markov reward processes, of which one may be advanced at each decision time, the right one to advance is the one of greatest Gittins index. Chapter 4 of Gittins, Glazebrook and Weber, Multi-armed Bandit Allocation Indices (2nd ed., doi:10.1002/9780470980033), asks how far this extends when the constituents are not reward processes but decision processes, each with its own controls: a research project that can be run in several ways, a job that can be processed at different speeds, a sampling process that may be stopped and exploited. A family of such superprocesses requires two choices at every decision time, which superprocess to continue and with which control, and an index policy in the sense of Chapter 2 need not be optimal (Example 4.1). Whittle (1980) identified the condition under which it is: Condition D, that when a superprocess is played against a standard bandit process paying a constant rent, the control one should apply to it does not depend on the rent. Under that condition the index theorem survives (Theorem 4.3), the index is characterized (Note 4.2), stoppable bandit processes with improving stopping options satisfy the condition (Lemma 4.4), and the chapter adds two results about indices themselves: any index that works for all bandit processes is a strictly increasing function of the Gittins index (Theorem 4.8), and a policy that is within $\varepsilon$ of the index policy loses at most $\varepsilon\gamma^{-1}(1-e^{-\gamma})^{-1}$ (Theorem 4.18).

Setting

A decision process $D$ on a countable state space $S$ has in each state $x$ a nonempty finite set $\Gamma(x)$ of controls; applying $u$ yields the reward $r(x, u)$ and moves the state by $P(\cdot \mid x, u)$ . Adding the freeze control, which leaves the state unchanged and yields nothing, makes $D$ a superprocess $S$ . Operating $D$ under a feasible deterministic stationary Markov policy $g$ (that is, $g(x) \in \Gamma(x)$ ) gives an ordinary bandit process $D_g$ , and the superprocess index is

\nu(S, x, u) = \sup_{g : g(x) = u} \nu(D_g, x), \qquad \nu(S, x) = \max_{u \in \Gamma(x)} \nu(S, x, u), \tag{4.1}

with $\nu(D_g, x)$ the Gittins index of the Bandit Algorithms model. A simple family of alternative superprocesses (SFAS) is $n$ superprocesses on a common $(S, U)$ ; at each decision time $0, 1, 2, \dots$ exactly one is continued, with a control from its control set, the others being frozen, and rewards are discounted by $a^t$ . A policy is a Markov kernel per decision time from the history to the pair (superprocess, control); it is optimal if it is feasible and attains the supremum of the discounted payoff over feasible policies from every initial state-vector, and it is an index policy if it always continues a superprocess and control of maximal $\nu(S_i, x_i, u)$ .

Condition D. Let $\Lambda$ be a standard bandit process with parameter $\lambda$ (one state, reward $\lambda$ ). $S$ satisfies Condition D if there is a function $g$ such that, for every $x$ and $\lambda$ for which it is optimal in the family $\{S, \Lambda\}$ to select $S$ in state $x$ , it is optimal to apply the control $g(x)$ . A stoppable bandit process is a bandit process with a stop control that makes it behave as a standard bandit process with parameter $\mu(x)$ ; its stopping option is improving if $\mu(x(t))$ is almost surely nondecreasing in process time.

Formalization targets

Goal: Theorem 4.3

For a decision process with bounded rewards and a Condition-D control $g$ , every index policy with respect to $\nu(D, \cdot, \cdot)$ that applies $g(x_i)$ to the superprocess $i$ it continues is optimal for the family of $n$ superprocesses:

\text{index policy } \pi \implies \pi \text{ feasible and } R_\pi(x) = \sup_{\pi' \text{ feasible}} R_{\pi'}(x)\ \text{ for every } x \in S^n.

Milestones

Note 4.2 (under Condition D, $S$ is selected in $\{S, \Lambda(\lambda)\}$ iff $\nu(S, x) \ge \lambda$ , and at $\lambda = \nu(S, x)$ a control $u$ is optimal iff $\nu(S, x, u) = \nu(S, x)$ ; the printed equivalence fails for $\lambda < \nu(S, x)$ ); Lemma 4.4 (Condition D for stoppable bandit processes with improving stopping options); Theorem 4.8 (an index for the bandit processes with discount factor $a$ is strictly increasing in $\nu$ ); Theorem 4.18 (the $\varepsilon$ -index bound, $\varepsilon/(1-a)^2$ for the discrete-time index).

Significance

Theorem 4.3 is the widest form in which the index theorem holds without further structure, and Condition D is exactly the right hypothesis: it says the superprocess has a canonical control, and once it does the family reduces to a family of bandit processes and the prevailing-charge argument goes through. Lemma 4.4 gives the model where the condition is known to hold, a research project that may be exploited at any time; the buyer's problem of Bergman and Bather is the case where it fails. Theorem 4.8 explains why every index theorem in the book is about the Gittins index: any function that orders bandit processes optimally must order them as $\nu$ does. Theorem 4.18 is the quantitative version of the index theorem that heuristics and computations rely on.

Nothing here is machine-checked. The mission builds the first controlled multi-armed model on the platform, a run law for families of decision processes with an explicit feasibility constraint, and states Whittle's condition as a property of the two-member family, which is how the literature uses it. Theorems 4.8 and 4.18 are statements about the existing Bandit Algorithms model and are usable by any later work on that model.

Difficulty

The obvious attack on Theorem 4.3, "replace each superprocess by the bandit process $D_{g}$ for its Condition-D policy $g$ and apply the index theorem", is the second half of the book's proof; the first half is to show that an optimal policy never gains by applying a control other than $g(x_i)$ to a superprocess it continues, and that uses the prevailing-stake accounting of §4.3 with the other superprocesses treated as one bandit process, plus the observation that the class of policies deviating at most $k$ times is $\varepsilon$ -exhaustive. Both halves require the whole run law of the family to be related to the run laws of its constituents, which is where a formalization spends its effort. Note 4.2 is short on the page but needs the optimal-stopping characterization of Chapter 2 for the bandit process $D_g$ under charge $\lambda$ . Theorem 4.8 is elementary given the value of $\{B, \Lambda\}$ under a freezing rule, $R_f(B) + \lambda\gamma^{-1} - \lambda W_f(B)$ , but that identity is itself a computation on the run law. Theorem 4.18 has no proof in the book (Glazebrook 1982c); the natural route is the prevailing-charge upper bound with the charges perturbed by $\varepsilon$ .

Formalization scope

Decision processes carry their control sets as finsets with a nonemptiness proof and their kernels as Markov kernels; the state space is countable with measurable singletons (so stationary kernels and control-dependent maps are measurable without side conditions) and the control type is finite with measurable singletons. The family's run law is built decision time by decision time as the Bandit Algorithms model builds markovBanditMeasure, with the policy's kernel producing the pair (superprocess, control). Feasibility is an almost-sure condition on the policy kernel, and optimality is the book's: feasible, and the supremum from every initial state-vector. The superprocess index is a real supremum over feasible stationary policies with $g(x) = u$ , bounded by the reward bound and nonempty for $u \in \Gamma(x)$ ; for an unavailable $u$ it is a default value that no index policy consults. Condition D is stated on the family $\{S, \Lambda\}$ on $S \oplus \mathrm{Unit}$ , where the standard state has every control available, all equivalent. A stoppable bandit process is the decision process with control type Bool. Theorem 4.8 quantifies over index functions defined on every measurable state space and takes as hypothesis only what its proof uses, optimality of $\mu$ -index policies for the families $\{B, \Lambda\}$ . Theorem 4.18 is on the $k$ -armed Bandit Algorithms model with $\varepsilon \ge 0$ and the bound $\varepsilon/(1-a)^2$ : the book's $\varepsilon\gamma^{-1}(1 - e^{-\gamma})^{-1}$ is in continuous-time index units, $\gamma/(1-a)$ times the discrete-time index used here, and read with the discrete index it is false for $a < 1/e$ . Theorem 4.3's index policy applies the Condition-D control $g$ to the superprocess it continues, as the book's proof does; an index policy that breaks ties among controls otherwise need not be optimal.

Trivializing readings are excluded: index policies must be feasible, optimality is required from every initial state, and Condition D is a statement about optimal policies of a genuine two-member family, not about a chosen policy. Welcome contributions: the relation between the family's run law and the constituents' chain laws, the freezing-rule value identity behind Theorem 4.8, and the prevailing-stake accounting of §4.3.

Selected references

J. Gittins, K. Glazebrook, R. Weber, Multi-armed Bandit Allocation Indices, 2nd ed., Wiley, 2011, Chapter 4. doi:10.1002/9780470980033
P. Whittle, Multi-armed bandits and the Gittins index, Journal of the Royal Statistical Society B 42(2), 1980. doi:10.1111/j.2517-6161.1980.tb01111.x
K. D. Glazebrook, Stoppable families of alternative bandit processes, Journal of Applied Probability 16(4), 1979. doi:10.2307/3213152
K. D. Glazebrook, On the evaluation of suboptimal strategies for families of alternative bandit processes, Journal of Applied Probability 19(3), 1982. doi:10.2307/3213524
T. Lattimore, C. Szepesvári, Bandit Algorithms, Cambridge University Press, 2020, Chapter 35. doi:10.1017/9781108571401

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Bandit Algorithms Operations Research Optimization+1·Captain: naimengye

Multi-armed Bandit Allocation Indices II: Jobs, Parallel Machines, Search and Bandit-Dependent DiscountingTextbook

Motivation

Chapter 3 of Gittins, Glazebrook and Weber, Multi-armed Bandit Allocation Indices (2nd ed., doi:10.1002/9780470980033), asks which of the assumptions behind the index theorem can be dropped and which cannot, using the simplest bandit processes there are: jobs, which pay a single reward when they complete. Along the way it settles three questions that matter on their own. When identical machines run in parallel, which schedule of deterministic jobs minimizes the total completion time (Baker's theorem), and what does the equal-ratio case look like (Theorem 3.3)? When an object is hidden in one of several boxes and each search costs money and may fail, in what order should the boxes be searched (Blackwell's search-index theorem, Theorem 3.6)? And when two bandit processes discount at different rates, is there still an index policy (Nash's Theorem 3.4)? The search theorem is the chapter's example of what the index machinery yields once undiscounted criteria are admitted through the limit $\gamma \downarrow 0$ ; the scheduling results are the source of the counterexamples that show a second machine breaks the index structure.

Setting

Jobs on parallel machines. There are $n$ jobs with service times $s_i > 0$ and weights $c_i$ , and $m$ identical machines. A schedule assigns each job a machine and a position in that machine's processing order; each machine processes its jobs consecutively. The completion time $C_i$ of a job is the total service time of the jobs on its machine up to and including it; the flow time is $\sum_i C_i$ and the weighted flow time $\sum_i c_i C_i$ . The load of a machine is the total service time assigned to it, and $\delta_j$ is its excess over the average $S/m$ . The SPT list schedule sorts the jobs by increasing service time and deals them out cyclically to the machines. The level of a job is its position counted from the end of its machine's schedule.

The search problem (Problem 3). A stationary object is hidden in one of $n$ boxes, in box $i$ with probability $p_i$ . A search of box $i$ costs $c_i$ and finds the object with probability $q_i$ if it is there. A search policy is a sequence of boxes; after $N_i$ unsuccessful searches of box $i$ the posterior probability that the object is there is proportional to $p_i(1-q_i)^{N_i}$ , and the search index of the box is $p'_i q_i/c_i$ , the current probability of finding the object per unit cost. The expected cost of a policy is $\sum_k c_{\sigma_k}\Pr[\text{not found by the first } k \text{ searches}]$ .

Two discount factors. Bandit process $A$ (state space $S_A$ , kernel $P_A$ , reward $r_A$ ) discounts at rate $a$ and $B$ at rate $b$ ; a reward obtained at time $t$ is worth $a^t$ or $b^t$ times its face value according to which process produced it. The cross indices are $\nu_{AB}(x) = \sup_{\tau>0} \mathbb{E}\sum_{t<\tau} a^t r_A(x(t)) \big/ \mathbb{E}[1 - b^\tau]$ and $\nu_{BA}(y)$ with the roles exchanged: each is computed from one process's stopping times but with the other's discount factor in the denominator. The book prints the denominator as $\mathbb{E}\int_0^\tau b^t\,dt$ and compares $\nu_{AB}(x)$ with $\nu_{BA}(y)$ at every time; both are corrected here (see the Theorem 3.4 item), since the printed rule is not optimal. The family is the two-armed Markov bandit of the Bandit Algorithms model on the disjoint union $S_A \oplus S_B$ .

Formalization targets

Goal: Theorem 3.6

For prior probabilities $p_i \ge 0$ summing to one, detection probabilities $0 < q_i \le 1$ and costs $c_i > 0$ , a search policy is optimal if and only if at every step it searches a box of maximal current index $p'_i q_i / c_i$ :

\text{optimal}(\sigma) \iff \forall k,\ \forall i:\ \frac{p_i (1-q_i)^{N_i(k)} q_i}{c_i} \le \frac{p_{\sigma_k}(1-q_{\sigma_k})^{N_{\sigma_k}(k)} q_{\sigma_k}}{c_{\sigma_k}}.

Milestones

Theorem 3.3, as the identity $\sum_i c_i C_i = \frac{\kappa}{2}\big(\sum_i s_i^2 + S^2/m + \sum_j \delta_j^2\big)$ when $c_i = \kappa s_i$ ; Theorem 3.4 in discrete time and corrected, that a policy selecting, at time $t$ , $A$ when $a^t \nu_{AB}(x) > b^t \nu_{BA}(y)$ and $B$ when $a^t \nu_{AB}(x) < b^t \nu_{BA}(y)$ attains the supremum of the bandit-dependently discounted payoff; Theorem 3.7, that SPT minimizes the flow time on $m$ machines and that the optimal schedules are exactly those placing the $r$ -th block of $m$ longest jobs at level $r$ from the end, for some order of the equal jobs.

Significance

Theorem 3.6 is the classical solution of the discrete search problem (Blackwell, reported by Matula 1964; Kadane 1969): the greedy rule in probability-per-cost is optimal, and every optimal policy is of that form. The book derives it from the index theorem through an auxiliary family of bandit processes (Problem 3A) and the undiscounted limit (Corollary 3.5), which is why it sits in this chapter; as a statement it is elementary and self-contained, and it is the template for the tax problems and Klimov's model of Chapter 4. Theorem 3.7 and Theorem 3.3 are the positive results about parallel machines that survive the loss of the index structure; Theorem 3.4 shows the index theorem's shape persisting under bandit-dependent discounting, with two twists: each index depends on the other process's discount factor, which is exactly why it does not extend to three processes, and the comparison at time $t$ weighs the indices by $a^t$ and $b^t$ , so the rule is not stationary in the states. The printed statement misses the second twist and misnormalizes the first; two one-state bandits paying $0.18$ ( $a = 0.9$ ) and $1$ ( $b = 0.5$ ) are best played $B, B, B$ and then $A$ forever, which no stationary rule does.

None of these is machine-checked. Formalizing them gives the platform an optimality theorem for an infinite-horizon search process with an explicit index characterization in both directions, the standard parallel-machine flow-time results with a precise uniqueness clause, and a first statement on the Bandit Algorithms two-armed model with unequal discounting.

Difficulty

For Theorem 3.6 the obvious route, comparing two adjacent searches, gives only that interchanging a pair in the wrong index order lowers the cost; turning that into optimality over all infinite sequences needs that an optimal policy exists (costs are bounded below by zero and the index policy has finite cost), that a policy neglecting a box of positive prior has infinite cost, and that any first deviation from the index rule can be improved by moving a later search forward, which requires tracking how the not-found probability changes along the whole tail. The converse direction is the same interchange run backwards, and the tie case must be handled so that the equivalence is exact. Theorem 3.7's first part follows from writing the flow time as $\sum_i \ell_i s_i$ with $\ell_i$ the level and applying a rearrangement inequality over the multiset of levels, but the multiset of levels itself depends on how many jobs each machine gets, so balancing the machine counts is part of the argument; the uniqueness clause needs both that the multiset is forced and that the pairing of levels with service times is forced by strict monotonicity. Theorem 3.4 is the hardest: the book's proof changes the time scale of each process so that the two share a discount factor, which produces a semi-Markov family, and applies the index theorem there; in discrete time on the Markov model one needs either a discrete analogue of that argument or a direct prevailing-charge proof with two charge scales.

Formalization scope

Schedules are assignments of machines and positions with distinct positions on a machine, without idling; the flow-time quantities are finite sums over Fin n. The SPT schedule is defined by the ascending rank of a job (ties broken by index) and carries its own injectivity proof. The search cost is a series in $[0, \infty]$ of nonnegative terms, so no summability hypothesis is needed and "infinite cost" is literal; the index is stated unnormalized, $p_i(1-q_i)^{N_i} q_i/c_i$ , which orders the boxes exactly as the posterior index does. The two-discount family uses the platform's MarkovBanditPolicy 2 (S_A ⊕ S_B) and markovBanditMeasure with the kernel that moves an $A$ -state by $P_A$ and a $B$ -state by $P_B$ ; the payoff is the round-by-round series $\sum_t (a^t \mathbb{E}[r\mathbf 1\{A_t = A\}] + b^t \mathbb{E}[r\mathbf 1\{A_t = B\}])$ , absolutely summable for bounded rewards, and the policy condition is imposed at histories whose current states have the right types (all reachable histories do). Hypotheses: $m \ge 1$ , service times positive; $p_i \ge 0$ , $\sum p_i = 1$ , $0 < q_i \le 1$ , $c_i > 0$ ; countable state spaces, bounded rewards, $a, b \in (0,1)$ .

Trivializing readings are excluded: the search "iff" is over all sequences, not a finite horizon; the uniqueness clause is stated in full, with ties resolved by any ranking of the equal jobs; the cross indices are suprema over positive stopping times (denominators at least one), and the two-discount rule weighs the indices by $a^t$ , $b^t$ . Welcome contributions: the rearrangement and level-count lemmas for schedules, the interchange lemma for the search cost, and a discrete prevailing-charge argument for Theorem 3.4.

Selected references

J. Gittins, K. Glazebrook, R. Weber, Multi-armed Bandit Allocation Indices, 2nd ed., Wiley, 2011, Chapter 3. doi:10.1002/9780470980033
D. W. Matula, A periodic optimal search, American Mathematical Monthly 71(1), 1964. doi:10.2307/2311300
J. B. Kadane, Quiz show problems, Journal of Mathematical Analysis and Applications 27(3), 1969. doi:10.1016/0022-247X(69)90140-2
K. R. Baker, Introduction to Sequencing and Scheduling, Wiley, 1974.
P. Nash, A generalized bandit problem, Journal of the Royal Statistical Society B 42(2), 1980. doi:10.1111/j.2517-6161.1980.tb01119.x

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Machine Learning Operations Research Optimization+1·Captain: mikedeng1

Wasserstein Distributionally Robust Optimization I: Kantorovich Duality and Strong Duality for the Worst-Case RiskTextbook

Motivation

Every data-driven decision problem faces the same trap. A decision-maker estimates a risk functional $R(P,\ell) = \mathbb{E}_P[\ell(\xi)]$ from a nominal distribution $\hat P_N$ built from $N$ training samples, then optimizes a loss function $\ell$ against $\hat P_N$ instead of the unknown true distribution $P$ . Because the optimizer adapts to the noise in $\hat P_N$ , the in-sample risk of the optimizer systematically understates its true, out-of-sample risk — a phenomenon Smith and Winkler named the optimizer's curse (Smith & Winkler, Management Science, 2006). The remedy explored here is to hedge against a whole neighborhood of plausible distributions around $\hat P_N$ , rather than trusting the point estimate. Kuhn, Mohajerin Esfahani, Nguyen and Shafieezadeh-Abadeh's INFORMS TutORials chapter (2019) develops this neighborhood using the Wasserstein distance, and the present mission formalizes its foundational duality theory: the machinery every later result in the chapter (finite-sample guarantees, elliptical tractability, regularization) builds on.

Setting

Fix a norm $\|\cdot\|$ on a finite-dimensional real vector space $E$ (representing $\mathbb{R}^m$ ). For $p \in [1,\infty)$ , the type- $p$ Wasserstein distance between two Borel probability measures $Q, Q'$ on $E$ is

W_p(Q,Q') = \left(\inf_{\pi \in \Pi(Q,Q')} \int_{E\times E} \|\xi-\xi'\|^p\, \pi(d\xi,d\xi')\right)^{1/p},

where $\Pi(Q,Q')$ is the set of couplings of $Q$ and $Q'$ — joint probability measures on $E \times E$ whose marginals are $Q$ and $Q'$ . The optimal $\pi$ can be read as a transportation plan moving one pile of dirt ( $Q$ ) into another ( $Q'$ ) at minimum cost, which is why $W_p$ is also called the earth mover's distance; the underlying linear program was formalized by Kantorovich (1942) after Monge's 1781 original.

Given $N$ training samples $\hat\xi_1,\dots,\hat\xi_N$ , the empirical distribution is $\hat P_N = \frac1N\sum_{i=1}^N \delta_{\hat\xi_i}$ . Centered at $\hat P_N$ , the Wasserstein ambiguity set of radius $\varepsilon \ge 0$ is

B_{\varepsilon,p}(\hat P_N) = \{Q \in \mathcal{P}(\Xi) : W_p(Q,\hat P_N) \le \varepsilon\},

where $\Xi \subseteq E$ is a closed set known to contain the support of the true distribution. The worst-case risk of a loss function $\ell$ is

R_{\varepsilon,p}(\hat P_N,\ell) = \sup_{Q \in B_{\varepsilon,p}(\hat P_N)} \mathbb{E}_Q[\ell(\xi)],

and minimizing it over a class of admissible loss functions $\mathcal{L}$ is a distributionally robust optimization problem. $\varepsilon$ measures the estimation error one insures against; a larger ambiguity set gives a more conservative (and more expensive) guarantee.

Formalization targets

Goal — Theorem 7, strong duality

R_{\varepsilon,p}(\hat P_N,\ell) = \inf_{\gamma \ge 0}\ \mathbb{E}_{\hat P_N}[\ell_\gamma(\xi)] + \gamma\varepsilon^p,\qquad \ell_\gamma(\xi) = \sup_{z\in\Xi} \ell(z) - \gamma\|z-\xi\|^p.

This is the Lagrangian dual of the worst-case risk evaluation problem, with $\gamma$ the multiplier of the Wasserstein constraint $W_p(Q,\hat P_N)\le\varepsilon$ : it converts a supremum over an infinite-dimensional space of measures into a one-dimensional minimization of the Moreau-Yosida regularization $\ell_\gamma$ . Every tractability result later in the chapter (finite convex reformulations, SDP relaxations) specializes this duality by choosing a loss class for which $\ell_\gamma$ is computable.

Supporting dual representations of $W_p$ — Theorems 1 and 2

W_p^p(Q,Q') = \sup\left\{\int \psi\,dQ' - \int \varphi\,dQ : \varphi,\psi \text{ bounded continuous},\ \psi(\xi)-\varphi(\xi') \le \|\xi-\xi'\|^p\right\}

W_1(Q,Q') = \sup_{\mathrm{Lip}(\varphi)\le 1} \int \varphi\,dQ - \int \varphi\,dQ'

These identify $W_p$ as a linear program's strong dual (Theorem 1) and, for $p=1$ , specialize it to the Kantorovich-Rubinstein form (Theorem 2), which is what lets the worst-case-risk analysis reason about Lipschitz loss functions directly.

Upper and lower bounds — Theorems 5 and 6

R_{\varepsilon,p}(\hat P_N,\ell) \le R(\hat P_N,\ell) + \varepsilon\cdot\mathrm{Lip}(\ell)

R_{\varepsilon,p}(\hat P_N,\ell) \ge \sup\left\{\tfrac1N\textstyle\sum_i \ell(\hat\xi_i+\theta_i) : \hat\xi_i+\theta_i\in\Xi,\ \tfrac1N\textstyle\sum_i\|\theta_i\|^p\le\varepsilon^p\right\}

These are the tractable, easily-computed bracket that Theorems 7 and 10 later show is tight in important special cases.

Exact case — Theorem 10

\Xi = \mathbb{R}^m,\ \ell \text{ convex},\ p=1 \implies R_{\varepsilon,1}(\hat P_N,\ell) = R(\hat P_N,\ell) + \varepsilon\,\mathrm{Lip}(\ell)

Theorem 5's inequality becomes exact under convexity — the cleanest closing corollary of the duality theory, obtained from Theorem 7 by evaluating the Moreau-Yosida regularization of a convex function explicitly.

Significance

Theorem 7 is the hinge on which the entire computational program of Wasserstein distributionally robust optimization turns: every tractable reformulation in the source chapter (piecewise-concave losses via conic duality, quadratic losses via semidefinite programming, the shrinkage-estimator connection) is obtained by substituting a specific loss class into the right-hand side of Theorem 7 and showing the resulting Moreau-Yosida regularization is computable. Kuhn et al. themselves derive it as a corollary of Blanchet & Murthy (2019) and Gao & Kleywegt (2016) for the empirical case, generalized to Polish spaces by Blanchet & Murthy and Gao & Kleywegt independently — the paper cites [12] and [37] for the general statement. Formalizing it is what makes every later, more computational result in the chapter — the ones a solver is more likely to reach for next — rest on a mechanically verified foundation rather than a citation chain.

Status. The mathematical result is well established (multiple independent published proofs cited above); nothing here is open research. What this mission contributes is the first machine-checked formal statement of the duality theorem and its supporting dual representations (Theorems 1, 2, 5, 6, 10) on the Prove2Me platform — none of Wp's dual representation, the Wasserstein ambiguity set, or the worst-case risk functional exist there prior to this mission (see Formalization scope).

Difficulty

The obvious proof strategy — write down the Lagrangian of the semi-infinite program (6), swap the order of the outer supremum over $Q$ and the inner minimization over the multiplier $\gamma$ , and invoke ordinary Lagrangian strong duality — fails because (6) is an infinite- dimensional linear program over measures, not a finite convex program: there is no compact feasible set or Slater point in a form that ordinary finite-dimensional duality applies to directly. The actual proof goes through the dual representation of the Wasserstein distance itself (Theorem 1, which is why it is a prerequisite milestone), reformulating the constraint $W_p(Q,\hat P_N)\le\varepsilon$ via its own dual variables and swapping the resulting sup-inf using minimax theorems for semi-infinite programs, not ordinary Lagrangian duality for finite programs.

Formalization scope

$E$ is a generic finite-dimensional real normed space (NormedAddCommGroup, NormedSpace ℝ, Borel-measurable), representing $\mathbb{R}^m$ with the paper's arbitrary fixed norm as a parameter rather than fixing the Euclidean norm. A coupling is formalized directly via MeasureTheory.Measure.map: π.map Prod.fst = Q ∧ π.map Prod.snd = Q'. Constrained infima/suprema (over couplings, over the ambiguity set, over Lipschitz test functions, over perturbation matrices) use Mathlib's guarded-binder idiom ⨅ x (_ : P x), f x, which correctly returns $\top$ (resp. $\bot$ ) outside the feasible set rather than a finite junk value.

Two deliberate, disclosed conventions keep the extremal-value definitions faithful without extended-real integration machinery, both recorded in MODERATION_NOTES.md:

worstCaseRisk and the dual representations (Theorems 1, 2) are valued in EReal, not ℝ, so an unbounded supremum is recorded as $+\infty$ rather than collapsed to Mathlib's real-valued junk value 0 on an unbounded family.
The goal theorem (7) and its Moreau-Yosida regularization restrict the loss function to bounded continuous $\ell$ (BoundedContinuousFunction E ℝ), narrower than the paper's general upper-semicontinuous, $\hat P_N$ -integrable loss class $\mathcal{L}$ (Assumption 1). This keeps $\ell_\gamma(\xi) = \sup_{z\in\Xi}\ell(z)-\gamma\|z-\xi\|^p$ a finite real number for every nonempty $\Xi$ , so the right-hand side's Bochner integral is well-posed; the milestones (Theorems 5, 6, 10) keep the more general real-valued (not necessarily bounded) loss class, since their statements do not require evaluating a pointwise supremum over $\Xi$ .
Ξ is required closed in Theorems 5, 6 and 7, matching the paper's own standing assumption (p. 6: "we let $\Xi\subseteq\mathbb{R}^m$ be a closed set that is known to contain the support of $P$ ") for the whole worst-case-risk framework, which is used silently in the paper wherever a theorem takes $\Xi$ as an argument but was not carried into these theorems' own hypothesis lists in an earlier draft.
The goal theorem (7) additionally requires $\hat P_N$ itself supported on $\Xi$ ( $\hat P_N(\Xi^c)=0$ , the same "supported on $\Xi$ " convention ambiguitySet uses for $Q\in\mathcal P(\Xi)$ ), which the paper's framework presupposes for the nominal distribution throughout §2. Combined with $\ell$ bounded, this makes $\ell_\gamma$ bounded on the full-measure set $\Xi$ (above by $\sup\ell$ unconditionally, below by $\ell(\xi)$ itself via $z=\xi$ for $\xi\in\Xi$ ), which is what makes the right-hand side's integral genuinely well-posed rather than liable to Mathlib's non-integrable junk value $0$ .

There is no trivializing formalization risk from a vacuous hypothesis: Ξ.Nonempty and 0 < N are both required exactly where the paper's own indexing and support assumptions require them, and every extremal value uses the extended-real convention above rather than a convention that would make an inequality vacuously true.

No definition in this mission exists on the platform prior to this series (GET /theorems?q=Wasserstein, q=Kantorovich, q=optimal transport, q=coupling return only unrelated discrete/finite-type constructions); all seven definitions and six theorems are drafted fresh. WassersteinDRO.Duality.wassersteinDistance, .ambiguitySet and .worstCaseRisk are the substrate every later mission in this five-part series (Gelbrich tractability, finite-sample guarantees, regularization, shrinkage estimation) either imports directly or redefines locally per the series' reuse rule.

Selected references

Kuhn, D., Mohajerin Esfahani, P., Nguyen, V. A., & Shafieezadeh-Abadeh, S. (2019). Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning. INFORMS TutORials in Operations Research, 130–166. https://doi.org/10.1287/educ.2019.0198
Villani, C. (2009). Optimal Transport: Old and New. Springer. (Cited as [108] for Theorems 1 and 2.)
Smith, J. E., & Winkler, R. L. (2006). The optimizer's curse: Skepticism and postdecision surprise in decision analysis. Management Science, 52(3), 311–322. https://doi.org/10.1287/mnsc.1050.0451
Gao, R., & Kleywegt, A. J. (2016). Distributionally Robust Stochastic Optimization with Wasserstein Distance. arXiv:1604.02199.
Blanchet, J., & Murthy, K. (2019). Quantifying Distributional Model Risk via Optimal Transport. Mathematics of Operations Research, 44(2), 565–600. https://doi.org/10.1287/moor.2018.0936

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Analysis Numerical Analysis·Captain: Lucas

Métodos Numéricos (Freitas) VII: EDOs, Método de Picard e Erro Global de EulerTextbook

Motivation

An initial value problem $y' = f(x,y)$ , $y(x_0) = y_0$ , is the standard way a law of motion, a growth model or a circuit equation is written down, and only a small minority of such problems can be integrated in closed form. Single-step methods advance the solution by small increments: Euler's method takes the tangent line at the current point, and the higher-order Runge-Kutta schemes refine the same idea. What justifies them is a pair of theorems: that the problem has a unique solution at all, and that the computed polygon approaches that solution as the step size decreases.

This mission is the seventh in a series formalizing the lecture notes Métodos Numéricos by Sergio Roberto de Freitas (UFMS, 2000). It covers Chapter 9, Métodos Numéricos para EDO's, up to the convergence of Euler's method.

Setting

Consider the initial value problem

y' = f(x,y), \\qquad y(x_0) = y_0,

with $f$ defined on a rectangle $D = [x_0-a, x_0+a]\\times[y_0-b, y_0+b]$ .

Picard's method replaces the differential equation by the integral equation $y(x) = y_0 + \\int_{x_0}^{x} f(t, y(t))\\,dt$ and iterates it:

y_0(x) = y_0, \\qquad y_{k+1}(x) = y_0 + \\int_{x_0}^{x} f(t, y_k(t))\\,dt .

With $M = \\max_D |f|$ , $N = \\max_D |\\partial f/\\partial y|$ and $h = \\min\\{a, b/M\\}$ , the source states the bound $|\\varphi(x) - y_k(x)| \\le \\frac{M N^{k-1}}{k!}h^k$ on $[x_0-h, x_0+h]$ , where $\\varphi$ is the exact solution.

Euler's method with step $h$ produces $y_{i+1} = y_i + h f(x_i, y_i)$ , $x_i = x_0 + ih$ . Its local error is $O(h^2)$ per step and its global error $O(h)$ .

Target

The goal theorem is the global error statement: for an initial value problem whose exact solution $\\varphi$ is twice continuously differentiable on $[x_0, X]$ and whose right-hand side is Lipschitz in $y$ there, there is a constant $C$ such that for every step size $h \\in (0,1]$ and every grid point $x_0 + ih \\le X$ ,

|y_i - \\varphi(x_0+ih)| \\le C h .

This is the precise form of the source's assertion $E = O(h)$ .

The milestones are the results the chapter states on the way: local existence and uniqueness for the initial value problem under a Lipschitz condition in $y$ (Teorema 9.4.1), the Picard error bound, the convergence of the Picard iterates to the solution, and the one-step Taylor identity behind Euler's method.

Significance

The global error bound is what makes Euler's method a method rather than a heuristic: it converts an arbitrary choice of step size into a guaranteed accuracy, and it exhibits the characteristic loss of one order between the local error $O(h^2)$ and the global error $O(h)$ that recurs for every single-step scheme. The existence and uniqueness theorem is the prerequisite for all of it — without uniqueness there is no well-defined object for the numerical solution to approximate — and the Picard iteration is both the constructive proof of that theorem and a method in its own right.

Difficulty

The global error proof has to handle the interaction of two error sources: the local truncation error at each step and the amplification of previous errors by the Lipschitz constant, which gives the discrete Gronwall recursion $e_{i+1} \\le (1+hL)e_i + Ch^2$ . Carrying that recursion to the closed-form bound, and doing it uniformly in $h$ , is the core of the mission. The Picard bound requires the iterates to remain inside the rectangle so that the hypotheses on $f$ continue to apply — that is what the choice $h = \\min\\{a, b/M\\}$ is for — and an induction on $k$ with the factorial denominator.

Formalization scope

The unknown is a real function of a real variable and $f$ is a function of two real variables. A solution is a function $\\varphi$ satisfying $\\varphi(x_0)=y_0$ and having derivative $f(x,\\varphi(x))$ at every point of the relevant interval. The Lipschitz condition in the second variable is stated explicitly, $|f(x,u)-f(x,v)| \\le L|u-v|$ , in place of the source's hypothesis that $\\partial f/\\partial y$ is bounded; for the global-error goal it is assumed for all real $u,v$ and all $x$ in the interval, and in the Picard milestones only inside the rectangle. Uniqueness is stated as agreement of any two solutions on the interval, not as uniqueness of a function on all of $\\mathbb{R}$ , since a solution is unconstrained outside the interval. Euler's iterates and the Picard iterates are explicit recursive definitions; the Picard integral is the interval integral, which is total, so no integrability side condition appears in the definition. The constant $C$ in the goal is asserted to exist and to be positive, uniformly over step sizes in $(0,1]$ ; the restriction $h \\le 1$ merely normalizes the range of steps considered.

Selected references

S. R. Freitas, Métodos Numéricos, Departamento de Computação e Estatística, Universidade Federal de Mato Grosso do Sul, 2000. Chapter 9, Métodos Numéricos para EDO's, pp. 183–213. (Course notes supplied with this mission.)

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Machine Learning Probability Statistics·Captain: mikedeng1

High-Dimensional Statistics II: Talagrand's Convex Concentration InequalityTextbook

Motivation

Chapter 2's tail bounds mostly rest on moment-generating-function control, obtained either directly (sub-Gaussianity) or through explicit combinatorial arguments (Hoeffding, bounded differences). The entropic method offers a different, more structural route: bound a specific information-theoretic quantity — the $\varphi$ -entropy of $e^{\lambda X}$ — and convert that bound mechanically into a tail bound via a short ODE argument (the Herbst argument). This method's real payoff appears once it is combined with the tensorization property of entropy across independent coordinates, which is what lets it handle Lipschitz functions of many independent variables — including cases, such as separately convex functions, that elude the purely martingale-based techniques of Chapter 2. This mission formalizes the entropic method's two foundational entropy-to-tail conversions and its central Lipschitz-concentration application, following Wainwright, High-Dimensional Statistics: A Non-Asymptotic Viewpoint (Cambridge University Press, 2019), Chapter 3.

Setting

For $\varphi(u):=u\log u$ ( $u>0$ ), $\varphi(0):=0$ , the $\varphi$ -entropy of a nonnegative random variable $Z$ is $H(Z):=\mathbb E[Z\log Z]-\mathbb E[Z]\log\mathbb E[Z]$ (Eqs. (3.1)-(3.2)). Writing $\varphi_X(\lambda):=\mathbb E[e^{\lambda X}]$ for the moment generating function of $X$ , the entropy of $e^{\lambda X}$ has the explicit form $H(e^{\lambda X}) = \lambda\varphi_X'(\lambda) -\varphi_X(\lambda)\log\varphi_X(\lambda)$ (Eq. (3.3)).

A function $f:\mathbb R^n\to\mathbb R$ is separately convex if, for each coordinate $k$ , the univariate function obtained by fixing every coordinate but the $k$ -th is convex — strictly weaker than joint convexity of $f$ itself. $f$ is $L$ -Lipschitz with respect to the Euclidean norm if $|f(x)-f(x')|\le L\|x-x'\|_2$ for all $x,x'$ .

Formalization targets

Goal — Theorem 3.4 (separately convex Lipschitz concentration)

Let $\{X_i\}_{i=1}^n$ be independent, each supported on $[a,b]$ , and $f$ separately convex and $L$ -Lipschitz. Then for all $\delta>0$ ,

\mathbb P[f(X)\ge\mathbb E[f(X)]+\delta] \;\le\; \exp\Big(-\frac{\delta^2}{4L^2(b-a)^2}\Big).

Milestone — Proposition 3.2 (the Herbst argument)

If $H(e^{\lambda X})\le\tfrac12\sigma^2\lambda^2\varphi_X(\lambda)$ for all $\lambda\in I$ ( $I=[0,\infty)$ or $\mathbb R$ ), then $\log\mathbb E[e^{\lambda(X-\mathbb E[X])}]\le \tfrac12\lambda^2\sigma^2$ for all $\lambda\in I$ — the basic entropy-to-sub-Gaussian-tail conversion.

Milestone — Proposition 3.3 (the Bernstein entropy bound)

The sub-exponential analogue: if $H(e^{\lambda X})\le\lambda^2\{b\varphi_X'(\lambda)+ \varphi_X(\lambda)(\sigma^2-b\mathbb E[X])\}$ for $\lambda\in[0,1/b)$ , then $\log\mathbb E[e^{\lambda(X-\mathbb E[X])}]\le\sigma^2\lambda^2(1-b\lambda)^{-1}$ on the same range.

Significance

Propositions 3.2 and 3.3 are the two basic entropy-to-tail conversions the entire chapter's entropic method rests on — every subsequent Lipschitz-concentration result in the chapter (including Theorem 3.4 and the more advanced Theorem 3.24) is obtained by first establishing an entropy bound of one of these two forms and then invoking the corresponding proposition. Theorem 3.4 is itself the direct analogue, for independent bounded variables, of Chapter 2's Gaussian Lipschitz concentration (Theorem 2.26) — but crucially requires the extra hypothesis of separate convexity, which the Gaussian case does not need and which cannot be dropped in general.

Formalizing it. No faithful prior art exists on the platform. The one candidate flagged in BRIEF.md, Talagrand.lipschitz_concentration, was read in full: it is a weighted-Hamming- distance concentration bound for functions on a finite-alphabet product space Fin n → α, proved via Talagrand's convex-distance method — a different underlying space (finite alphabet vs. real-valued bounded coordinates) and a different Lipschitz norm (weighted Hamming vs. Euclidean) from Theorem 3.4, and not reused here. A search for "log-Sobolev" and "Herbst" turned up bousquet_herbst_cgf_le_phi_via_herbst/bousquet_herbst_cgf_le_phi_double_integration: these are abstract calculus lemmas about a generic function $G$ satisfying an ODE-type growth condition ( $G''\le ve^x$ ), concluding $G(L)\le v(e^L-1-L)$ — a genuinely different statement shape from Proposition 3.2/3.3's entropy-to-CGF conversions (which conclude a quadratic, not exponential, bound on $\log\mathbb E[e^{\lambda(X-\mathbb EX)}]$ ), and not a faithful match. All three theorems here are drafted as open goals (:= by sorry).

Difficulty

The naive approach to Theorem 3.4 — try to adapt the bounded-differences (martingale) method of Chapter 2 directly — fails, because the bounded-differences method needs $f$ to have small coordinatewise oscillation in an absolute sense, while separate convexity alone gives no such uniform bound (a separately convex function can vary arbitrarily fast within the interior of its domain, only its slope is controlled by the Lipschitz condition). The entropic method sidesteps this by working with the $\varphi$ -entropy of $e^{\lambda f(X)}$ directly: entropy has a tensorization property across independent coordinates (not itself part of this mission, but what the entropic method's proof of Theorem 3.4 uses) that reduces a multivariate entropy bound to a sum of "one coordinate at a time" contributions, each of which convexity and the Lipschitz condition jointly control — a route with no analogue in the bounded-differences approach.

Formalization scope

Separate convexity and Euclidean-Lipschitzness are both restated locally in this chapter's own sub-namespace (HighDimStat.Concentration), per this book series' rule against importing another chapter's draft definitions, even though Chapter 2 already defines an IsLLipschitz for the same Euclidean condition. φ_X'(\lambda)$ (Proposition 3.3) is realized via Mathlib's deriv, a legitimate way to state a hypothesis on a derivative without separately proving differentiability, appropriate at the draft-statement stage. Explicit Integrable` hypotheses guard the Bochner integral's junk value on non-integrable functions throughout (trap 2), not literal in the book's own propositions but implied by what "the entropy $H(e^{\lambda X})$ exists" (an explicit qualifier the book itself makes when introducing Eq. (3.2)) means.

Goal substitution, disclosed. BRIEF.md recommends Theorem 3.24 (the two-sided, jointly convex analogue) as the primary goal, but explicitly names Theorem 3.4 as a fallback "if 3.24's dependence on the unnumbered transportation-cost inequality (Eq. 3.73, attributed to Samson) proves too heavy to state faithfully in the time available." Theorem 3.24's proof route depends on Theorem 3.19 (a general "transportation cost implies concentration" result for an abstract metric measure space, itself needing a from-scratch formalization of the transportation-cost inequality (3.58) and the concentration function $\alpha_{P,(\mathcal X,\rho)}$ ) plus the unproven-in-chapter Eq. (3.73). Building this full stack faithfully was judged to exceed this chunk's time budget; Theorem 3.4 is drafted instead, using this mission's own budget on Propositions 3.2 and 3.3 (the two most load-bearing entropy-to-tail conversions of the chapter) rather than the heavier transportation-cost machinery. Theorem 3.19, Theorem 3.24, and Eq. (3.73) are all out of scope for this mission and named here as natural follow-on work.

Selected references

M. J. Wainwright, High-Dimensional Statistics: A Non-Asymptotic Viewpoint, Cambridge University Press, 2019. DOI: 10.1017/9781108627771. Chapter 3.
M. Ledoux, The Concentration of Measure Phenomenon, American Mathematical Society, 2001.
I. Herbst, unpublished (the argument bearing his name is attributed in Ledoux (2001) and standard references on log-Sobolev inequalities).

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Machine Learning Probability Statistics+1·Captain: mikedeng1

Foundations of Machine Learning X: Regression and Rademacher Complexity BoundsTextbook

Motivation

Every generalization bound presented so far in this series is for classification, where the error of a prediction is binary (correct or not). Regression asks a different question: predictions are real-valued, and error is measured by the magnitude of the deviation from the true label, via a loss function L. Chapter 11 develops generalization theory for bounded regression, showing that the same two complexity measures used for classification — Rademacher complexity and a VC-dimension analogue — extend naturally, once the loss function itself is folded into the machinery via a Lipschitz-contraction argument (Rademacher route) or a reduction to classification via level-set thresholding (pseudo-dimension route).

Setting

A regression hypothesis h:X→ℝ is scored by a loss L:ℝ×ℝ→ℝ against a joint distribution D on X×ℝ (the stochastic scenario, since regression labels are rarely exactly reproducible); R(h) = E_{(x,y)~D}[L(h(x),y)] (Eq. 11.1) and R̂_S(h) = (1/m)∑L(h(x_i),y_i) (Eq. 11.2). For a finite hypothesis set, Theorem 11.1 gives a Hoeffding/union-bound guarantee directly, the regression analogue of chunk 02-pac's finite-hypothesis bound. For infinite H, §11.2.2 develops a Rademacher-complexity route: Proposition 11.2 shows that if L is µ-Lipschitz in its first (predicted-value) argument, the Rademacher complexity of the loss-composed family G = {(x,y)↦L(h(x),y) : h∈H} is controlled by µ times H's own Rademacher complexity, via Talagrand's contraction lemma (chunk 05-svm's Lemma 5.7); Theorem 11.3 combines this with chunk 03's Theorem 3.3 to give the chapter's headline bound. §11.2.3 develops an independent, purely combinatorial route: pseudo-dimension (Definition 11.5), a real-valued analogue of VC-dimension defined via threshold-witnessed shattering (Definition 11.4, restated via its own Eq. 11.3 as the VC-dimension of a thresholded indicator family); Theorem 11.8 gives a pseudo-dimension generalization bound by reducing regression to a family of classification problems (one per threshold t), using the tail-integral identity Eq. 11.5.

Formalization targets

Theorem 11.1 (milestone). For L bounded by M and H finite: for any δ>0, with probability at least 1-δ, for all h∈H: R(h) ≤ R̂_S(h) + M√((log|H|+log(1/δ))/(2m)).

Proposition 11.2 (milestone). For L non-negative, bounded by M, µ-Lipschitz in its first argument: for any sample S, R̂_S(G) ≤ µR̂_S(H).

Theorem 11.3 — the mission's goal. Under Proposition 11.2's hypotheses on L: for any δ>0, with probability at least 1-δ, for all h∈H: E[L(h(x),y)] ≤ (1/m)∑L(h(x_i),y_i) + 2µR_m(H) + M√(log(1/δ)/(2m)), and also with 2µR̂_S(H) + 3M√(log(2/δ)/(2m)).

Theorem 11.8 (milestone). For Pdim(G)=d, L non-negative bounded by M: for any δ>0, with probability at least 1-δ over a sample of size m, for all h∈H: R(h) ≤ R̂_S(h) + M√(2d log(em/d)/m) + M√(log(1/δ)/(2m)).

Significance

Theorem 11.3 is the chapter's own choice of headline result (§11.2's stated goal is to show "how the Rademacher complexity bounds of theorem 3.3 can be used to derive generalization bounds for regression"), and its proof genuinely reuses two pieces of prior machinery from this series — chunk 03's Theorem 3.3 and chunk 05's Talagrand's-lemma-style contraction — combined via a new observation (Proposition 11.2) specific to loss-composed families, not a restatement of either. Theorem 11.8 is the chapter's second, structurally independent technique: its em/d bound parallels chunk 03's Corollary 3.19 (both ultimately reduce to a VC-dimension-style growth-function argument), but the reduction itself — regression to a continuum of threshold classification problems, via the Lebesgue-integral tail identity Eq. (11.5) applied to |R(h)-R̂_S(h)| — is genuinely new content for this book, and pseudo-dimension has no prior art on the platform or in Mathlib. No prior art exists for this chapter's overall content either: GET /theorems?q=generalization%20bound%20regression and GET /theorems?q=pseudo-dimension both return zero hits.

Difficulty

Proposition 11.2's proof needs Talagrand's contraction lemma applied with the Lipschitz constant taken in the first argument of L only — the predicted value h(x_i), holding the true label y_i fixed — exactly the pitfall BRIEF.md names: a loss Lipschitz in the wrong argument, or in both arguments jointly, would not license this step. Theorem 11.8's proof is the chapter's most involved: it defines, for every h∈H and threshold t≥0, a classifier c(h,t):(x,y)↦1_{L(h(x),y)>t}, bounds |R(h)-R̂_S(h)| by M·sup_{t∈[0,M]}|R(c(h,t))- R̂_S(c(h,t))| via the tail-integral identity, and then applies a VC-dimension-style classification bound (Corollary 3.19) to the family of thresholded classifiers — whose VC-dimension is, by Eq. (11.3), exactly Pdim(G) by construction. A formalization that conflated pseudo-dimension with ordinary VC-dimension, or reused chunk 03's HasVCDim definition by relabeling, would misrepresent this chapter's genuinely different (real-valued, threshold-witnessed) combinatorial notion — precisely the pitfall BRIEF.md flags.

Formalization scope

Y := ℝ throughout (the book's own "Y a measurable subset of ℝ"), a harmless simplification consistent with every hypothesis, loss and Lipschitz condition in this chapter being stated for real-valued scores and labels. EmpiricalRademacherComplexity/ RademacherComplexity restate chunk 03-rademacher-vc's Definitions 3.1/3.2 locally, since a draft item cannot import another chunk's draft module. Shatters/PseudoDim are formalized via the book's own equivalent reformulation (Eq. 11.3, the thresholded-indicator form), rather than the sign-function form of Definition 11.4 directly, since the two coincide except at a measure-zero boundary the book itself does not address; PseudoDim mirrors chunk 03's HasVCDim Prop-valued pattern (does not cover Pdim(G)=+∞; every consuming theorem takes it as an explicit hypothesis) but is a structurally distinct definition built on Shatters, never a relabeling of HasVCDim, per BRIEF.md's pitfall note. Proposition 11.2's and Theorem 11.3's Lipschitz hypothesis (hLlip) is stated with the true label y' universally quantified outside the two-point comparison y1, y2 (the predicted values), matching "for any fixed y' ∈ Y, y ↦ L(y,y') is µ-Lipschitz" exactly — Lipschitzness in the first argument only, per BRIEF.md's pitfall note. RademacherComplexity (Measure.map Prod.fst D) H m gives the book's R_m(H) (H's Rademacher complexity under the marginal sampling distribution of the inputs x, i.e. D's first marginal). No numerical constant is altered from the book in any of the four theorems.

Not formalized: the L_p-loss worked example following Theorem 11.3's proof (an instantiation of the general theorem for a specific loss family, not a separate numbered theorem); Theorem 11.6 and Theorem 11.7 (worked pseudo-dimension examples for hyperplanes and vector spaces, background/illustration rather than the chapter's general machinery — drafting only these examples instead of the general Theorem 11.8 would be this chapter's trivializing formalization); the two-sided variant of Theorem 11.1 mentioned immediately after its proof (an unnumbered remark, not a separately displayed/numbered theorem); and all of §11.3 (linear regression, kernel ridge regression, SVR, Lasso and their online variants), which is applications-heavy per BRIEF.md's chapter restriction to §11.1-11.2.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 11 (§11.1-11.2).
D. Haussler, "Decision theoretic generalizations of the PAC model for neural net and other learning applications," Information and Computation 100(1), 1992 (pseudo-dimension's origin).
D. Pollard, Convergence of Stochastic Processes, Springer, 1984 (the tail-integral identity Eq. 11.5's classical antecedent).

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Machine Learning Probability Statistics·Captain: mikedeng1

High-Dimensional Statistics IX: Nuclear-Norm Regularization for Low-Rank Matrix RegressionTextbook

Motivation

Many estimation problems are naturally posed over matrices rather than vectors: recommender systems (Netflix-style matrix completion), multivariate regression with correlated responses, vector autoregressive time series, and phase retrieval all reduce to estimating an unknown matrix $\Theta^*$ that is low-rank, or well approximated by one. A rank constraint alone makes the natural least-squares estimator non-convex and generally intractable; replacing it with the nuclear norm — the sum of the matrix's singular values, the tightest convex surrogate for rank — yields a tractable semidefinite program. Wainwright's High-Dimensional Statistics (2019), Chapter 10, shows that this substitution costs nothing statistically: nuclear-norm-regularized least squares achieves error rates matching what one could hope for even knowing the rank in advance, by specializing Chapter 9's general decomposable-regularizer framework (mission 09-decomposability) directly to the nuclear norm.

Setting

For matrices $A,B\in\mathbb R^{d_1\times d_2}$ , the trace inner product is $\langle\!\langle A,B\rangle\!\rangle := \mathrm{trace}(A^TB) = \sum_{j_1,j_2}A_{j_1j_2}B_{j_1j_2}$ (Eq. 10.1), inducing the Frobenius norm $\|\!|A|\!\|_F$ . Given design matrices $X_1,\dots,X_n\in\mathbb R^{d_1\times d_2}$ and responses $y_i=\langle\!\langle X_i,\Theta^*\rangle\!\rangle+w_i$ , the observation operator $\mathcal X_n(\Theta):=(\langle\!\langle X_i,\Theta\rangle\!\rangle)_{i=1}^n$ and its adjoint $\mathcal X_n^*(u):=\sum_iu_iX_i$ (Eqs. 10.2-10.3) are the matrix analogs of a vector design matrix and its transpose. The nuclear norm $\|\!|\Theta|\!\|_{\mathrm{nuc}}:=\sum_j\sigma_j(\Theta)$ (Eq. 10.5) — the sum of singular values — is a decomposable regularizer (in the sense of Chapter 9) with respect to the subspace pair spanned by the top singular vectors of any target matrix, and its dual norm (Table 9.1) is the $\ell_2$ -operator (spectral) norm $\|\!|\cdot|\!\|_2$ . The estimator under study is nuclear-norm-regularized least squares,

\hat\Theta \in \operatorname*{arg\,min}_{\Theta\in\mathbb R^{d_1\times d_2}} \left\{ \frac{1}{2n}\|y-\mathcal X_n(\Theta)\|_2^2 + \lambda_n\|\!|\Theta|\!\|_{\mathrm{nuc}} \right\} \tag{10.16}

with $\lambda_n>0$ user-chosen.

Formalization targets

Goal (Proposition 10.6)

Suppose $\mathcal X_n$ satisfies the restricted strong convexity condition (10.17), $\|\mathcal X_n(\Delta)\|_2^2/(2n) \ge \kappa/2\|\!|\Delta|\!\|_F^2 - c_0\frac{d_1+d_2}{n}\|\!|\Delta|\!\|_{\mathrm{nuc}}^2$ for all $\Delta$ , with $\kappa>0$ , $c_0\ge 0$ . Conditioned on the good event $\mathcal G(\lambda_n)=\{\|\!|\frac1n\sum_iw_iX_i|\!\|_2 \le\lambda_n/2\}$ , any optimal $\hat\Theta$ satisfies, for any $r\in\{1,\dots,d'\}$ with $r\le\kappa n/(128c_0(d_1+d_2))$ ,

\|\!|\hat\Theta-\Theta^*|\!\|_F^2 \le \frac{9}{2}\frac{\lambda_n^2}{\kappa^2}r + \frac{1}{\kappa}\left\{2\lambda_n\sum_{j=r+1}^{d'}\sigma_j(\Theta^*) + \frac{32c_0(d_1+d_2)}{n}\left(\sum_{j=r+1}^{d'}\sigma_j(\Theta^*)\right)^2\right\}.

Milestone (Proposition 10.7)

Under the alternative $\Phi^*$ -curvature condition (10.20) (a curvature bound on the gradient map rather than the Taylor error), with $\mathrm{rank}(\Theta^*)<\kappa/(64\tau_n)$ : conditioned on $\mathcal G(\lambda_n)=\{\|\!|\frac1n\mathcal X_n^*(w)|\!\|_2\le\lambda_n/2\}$ , any optimal $\hat\Theta$ satisfies $\|\!|\hat\Theta-\Theta^*|\!\|_2 \le 3\sqrt2\,\lambda_n/\kappa$ — an operator-norm bound the book notes is, in conjunction with the cone-like constraint (10.15), strictly stronger than Proposition 10.6's Frobenius-norm bound.

Significance

Proposition 10.6 is this chapter's direct payoff from Chapter 9's general machinery: it shows that the deterministic backbone of the Lasso's guarantee (mission 07-sparse-linear) extends essentially verbatim to the matrix setting, with the sparsity level $s$ replaced by the target rank $r$ and the ambient dimension $d$ replaced by $d_1+d_2$ — exactly the "degrees of freedom" scaling one would predict by counting the parameters needed to specify a rank- $r$ matrix. Every one of the chapter's later corollaries (matrix compressed sensing, multivariate regression, matrix completion) is obtained by verifying the restricted strong convexity condition (10.17) holds with high probability for a specific random design, then reading the rate directly off Proposition 10.6 — the same two-step recipe Chapter 9's own Theorem 9.19 established abstractly. Proposition 10.7's operator-norm bound is what subsequently controls the individual singular values of the estimation error, needed for exact-rank-recovery guarantees.

Difficulty

Both results are direct specializations of Chapter 9's general oracle inequalities (Theorem 9.19 and Theorem 9.24 respectively) to the nuclear norm as regularizer and the Frobenius/operator norm pair, so their formalization difficulty lies almost entirely in getting the matrix-specific objects right rather than in new proof machinery: the nuclear norm requires an actual notion of singular values (realized via the eigenvalues of the Gram matrix $\Theta^T\Theta$ , using Mathlib's Hermitian-matrix spectral theorem), the operator norm requires the correct rectangular generalization of the symmetric-matrix Rayleigh-quotient characterization used in mission 08-pca, and the restricted-strong-convexity and curvature conditions must be instantiated against the correctly-adjointed observation operator $\mathcal X_n^*$ . A further subtlety is keeping Proposition 10.6's Frobenius-norm conclusion and Proposition 10.7's operator-norm conclusion cleanly distinct — the book itself warns against conflating the norms used across different chapters of Part II (the vector $\ell_2$ -norm of chunks 07-sparse-linear/08-pca versus the matrix Frobenius and operator norms here).

Formalization scope

Scope cut, disclosed here and in STATUS.md. BRIEF.md recommends Corollary 10.10 (the sample-complexity bound for the $\Sigma$ -Gaussian random matrix ensemble) as the goal theorem. Corollary 10.10 is a genuinely probabilistic statement — it asserts a bound holding "with probability at least $1-2e^{-2n\delta^2}$ " over $n$ i.i.d. draws of design matrices from a $\Sigma$ -Gaussian ensemble (Theorem 10.8's own high-probability restricted-strong-convexity certification for that ensemble) — and formalizing it faithfully would require a genuine multivariate-Gaussian-measure infrastructure on matrix space (a probability space, an i.i.d. sequence of $\Sigma$ -covariance-structured Gaussian matrices, and Mathlib's measure-theoretic probability API) that is disproportionate to this mission's time budget, and orthogonal to what Chapter 10 itself contributes (the chapter's own text stresses that Propositions 10.6 and 10.7 are the chapter's deterministic core, with probability entering only in Section 10.3's ensemble-specific certification — precisely mirroring chunk 09-decomposability's own "Theorem 9.19 is actually a deterministic result" framing). This mission instead takes Proposition 10.6 as its goal — explicitly named in BRIEF.md's own candidate list as "the nuclear-norm oracle inequality, an explicit corollary of Theorem 9.19" — the natural, tractable, still highly citable deterministic title result of Section 10.2, together with its companion Proposition 10.7. Theorem 10.8 (the $\Sigma$ -Gaussian ensemble's RSC certification), Corollary 10.9 (noiseless exact recovery) and Corollary 10.10 itself are left for a future mission with a dedicated probability-theory budget. The cone-like constraint (Eq. 10.15) — whose own faithful statement requires the same explicit subspace-pair machinery ( $\mathcal M(U_r,V_r),\bar{\mathcal M}(U_r,V_r)$ ) chunk 09-decomposability built for the general theory — is similarly left out, since Propositions 10.6 and 10.7's own numbered statements never expose these subspaces directly (only their proofs do, via instantiating Theorem 9.19/9.24). singularValues and nuclearNorm are noncomputable, defined via Mathlib's Hermitian-matrix eigenvalue spectral theorem; c0 ≥ 0 and λn > 0 are made explicit, matching this book's running conventions for RSC tolerance constants and regularization weights (see MODERATION_NOTES.md).

Selected references

Wainwright, M. J. High-Dimensional Statistics: A Non-Asymptotic Viewpoint. Cambridge University Press, 2019. Chapter 10. DOI: 10.1017/9781108627771.
Negahban, S., Wainwright, M. J. "Estimation of (near) low-rank matrices with noise and high-dimensional scaling." Annals of Statistics, 39(2), 2011, 1069–1097.
Recht, B., Fazel, M., Parrilo, P. A. "Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization." SIAM Review, 52(3), 2010, 471–501.

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Machine Learning Probability Statistics+1·Captain: mikedeng1

Foundations of Machine Learning V: Kernel Methods and the Representer TheoremTextbook

Motivation

Linear methods like SVMs work only when the classes are linearly separable, but most real data is not. Chapter 6 shows how to get non-linear decision boundaries for free: replace the input space's inner product with a kernel $K$ that implicitly computes an inner product in a (possibly very high- or infinite-dimensional) feature space, without ever explicitly computing the feature mapping. This works for any positive definite symmetric (PDS) kernel — and the chapter's central theorem shows that such a kernel always induces a genuine Hilbert space (the reproducing kernel Hilbert space, RKHS) in which the kernel is literally an inner product. The chapter's capstone, the representer theorem, then shows that a broad class of optimization problems over this (possibly infinite-dimensional) Hilbert space always has a solution expressible as a finite linear combination of kernel evaluations at the training points — turning an infinite-dimensional problem into a finite, $m$ -dimensional one.

Setting

A kernel $K:X\times X\to\mathbb R$ is PDS (Definition 6.3) if for every finite sample $\{x_1,\dots,x_m\}\subseteq X$ , the Gram matrix $[K(x_i,x_j)]$ is symmetric positive semidefinite. Theorem 6.8 shows every PDS kernel is an inner product $K(x,x')=\langle \Phi(x),\Phi(x')\rangle$ in some Hilbert space $H$ (the RKHS), which further has the reproducing property $h(x)=\langle h,K(x,\cdot)\rangle$ for every $h\in H$ — evaluating $h$ at a point is itself an inner product with the kernel section at that point. Theorem 6.10 shows PDS kernels are closed under sum, product, tensor product, pointwise limit, and power-series composition, letting complex kernels (Gaussian, and many others) be built from simple ones (polynomial kernels) without re-verifying positive-semidefiniteness from scratch. Section 6.3's representer theorem (Theorem 6.11) then considers minimizing, over $h\in H$ , an objective $F(h)=G(\|h\|_H)+L(h(x_1),\dots,h(x_m))$ that depends on $h$ only through its norm and its values at $m$ fixed points.

Formalization targets

Theorem 6.8 (RKHS existence, milestone). For a PDS kernel $K$ , there exist a Hilbert space $H$ and $\Phi:X\to H$ with $K(x,x')=\langle\Phi(x),\Phi(x')\rangle$ , and $H$ has the reproducing property $h(x)=\langle h,K(x,\cdot)\rangle$ for all $h\in H$ , $x\in X$ .

Theorem 6.10 (closure properties, milestone). PDS kernels are closed under sum, product, tensor product, pointwise limit, and power-series composition with non-negative coefficients.

Theorem 6.11 — the mission's goal. For any non-decreasing $G:\mathbb R\to\mathbb R$ and any loss $L:\mathbb R^m\to\mathbb R\cup\{+\infty\}$ , $\mathrm{argmin}_{h\in H} G(\|h\|_H)+ L(h(x_1),\dots,h(x_m))$ admits a solution $h^\star=\sum_{i=1}^m\alpha_i K(x_i,\cdot)$ ; if $G$ is increasing, every solution has this form.

Significance

Theorem 6.11 is the chapter's payoff and one of the most widely used structural results in kernel methods: it explains, in one general statement covering SVMs, kernel ridge regression, Gaussian process MAP estimation and many other algorithms simultaneously, why the dual (finite, $m$ -coefficient) formulation always suffices — the RKHS's infinite dimensionality never has to be confronted directly. Theorem 6.8 is the structural fact the whole chapter (and every later kernelized algorithm in the book, chapters 9-11, 15) depends on: without it, "PDS kernel" would be a purely combinatorial condition on Gram matrices with no guarantee it corresponds to any actual inner product. No prior art on the Prove2Me platform is faithful to any of this chapter's content: GET /theorems?q=Representer theorem and q=reproducing kernel return no faithful match (one unrelated hit concerns a Gaussian-measure reproducing kernel in a different, probabilistic context, not this chapter's PDS-kernel/RKHS construction). All six items are drafted fresh.

Difficulty

Theorem 6.8's proof is a genuine construction: define $H_0$ as finite linear combinations of kernel sections $\Phi(x)=K(x,\cdot)$ , define an inner product on $H_0$ using $K$ itself, verify it is well-defined (independent of the representation), positive semidefinite (via the PDS hypothesis), and — via the Cauchy-Schwarz-for-PDS-kernels lemma (Lemma 6.7) — actually positive definite, then complete $H_0$ to a genuine Hilbert space $H$ in which it is dense, and finally extend the reproducing property from the dense subspace $H_0$ to all of $H$ by a continuity argument. This is substantial analysis, not a restatement. Theorem 6.11's proof uses the orthogonal decomposition $H=H_1\oplus H_1^\perp$ (where $H_1=\mathrm{span}\{K(x_i, \cdot)\}$ ) and the reproducing property to show the orthogonal component $h^\perp$ never helps and, when $G$ is strictly increasing, strictly hurts — a short argument, but one that depends essentially on Theorem 6.8's reproducing property holding for the specific $H$ constructed, not just any Hilbert space with the kernel as its inner product.

Formalization scope

IsPDS uses the book's own second SPSD characterization (c^T K c ≥ 0 for every finite sample and coefficient vector c) rather than the non-negative-eigenvalues characterization, avoiding spectral theory for a Prop-valued definition; the book states the two are equivalent. IsRKHSOf and IsMinimizer are formalization scaffolding, not book-numbered definitions: IsRKHSOf packages Theorem 6.8's own two displayed equations (6.8, 6.9) as a reusable predicate shared between Theorem 6.8 (its conclusion) and Theorem 6.11 (its "H its corresponding RKHS" hypothesis), using an explicit evaluation map ev : H → X → ℝ to stand in for "elements of H are functions on X," since Mathlib's abstract Hilbert spaces are not themselves spaces of functions; IsMinimizer packages argmin. Both X in Theorem 6.8's existential and Theorem 6.11's ambient type are plain Type rather than Type*, avoiding universe-polymorphic quantification over the constructed Hilbert space's own type — a harmless simplification, since every application in this book instantiates X at a concrete, small type (typically $\mathbb R^N$ or a finite set). Theorem 6.11's loss codomain ℝ ∪ {+∞} is WithTop ℝ, not EReal (which would also admit -∞, an unstated generalization the book's own display does not license, since EReal's ⊤+⊥=⊥ collapse is a genuine faithfulness risk the book's own L:\mathbb R^m\to\mathbb R\cup\{+\infty\} avoids by construction). WithTop ℝ on its own does not avoid every collapse, though: an unconstrained L may be the constant function ⊤ (a legal instance of ℝ∪\{+\infty\}), forcing the objective identically ⊤ and every point to vacuously minimize it, which would make the theorem's second conjunct false. An added hypothesis, ∃ h₀, F h₀ ≠ ⊤, makes explicit the book's own implicit assumption that the objective is finite somewhere — see the Formalization scope note below. Theorem 6.11's two clauses are otherwise kept exactly as distinct as the book states them: existence needs only Monotone G (non-decreasing); "any solution has this form" needs StrictMono G (increasing) as an added hypothesis on the second conjunct only — per this chunk's own BRIEF.md, the crux of the theorem, and the trap this mission is most careful to avoid collapsing. Theorem 6.10's five closure clauses are stated as one conjunction (matching the book's single theorem, not five separate items); the power-series clause keeps the book's own radius-of-convergence domain restriction and adds an explicit summability hypothesis guarding the ∑' term.

Not formalized: Theorem 6.2 (Mercer's condition) — not needed by the goal's own proof chain (it is an equivalent characterization of PDS mentioned before the RKHS construction, not a premise Theorem 6.8's or 6.11's proof invokes) and its own hypotheses (compact $X\subset \mathbb R^N$ , continuous $K$ , an eigenfunction expansion of a compact self-adjoint integral operator) are real analytic content this mission's budget does not include; Lemma 6.7 (Cauchy-Schwarz for PDS kernels) and Lemma 6.9 (normalized PDS kernels) — supporting lemmas for Theorem 6.8's proof, not independently numbered results the goal cites; Theorem 6.12/Corollary 6.13 (Rademacher complexity/margin bounds for kernel-based hypotheses) — the chapter's optional further milestone, connecting to chunks 03/05's machinery, cut for budget; §6.5-6.8 (sequence kernels, weighted transducers, rational kernels, Bochner's theorem, approximate feature maps) — explicitly out of scope per this chunk's own BRIEF.md, a distinct, applications-heavy topic.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 6, §6.1-6.4.
B. Schölkopf, R. Herbrich, A. J. Smola, "A generalized representer theorem," COLT 2001, Lecture Notes in Computer Science 2111, 2001, 416-426.
N. Aronszajn, "Theory of reproducing kernels," Transactions of the American Mathematical Society 68(3), 1950, 337-404.

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Operations Research Optimization·Captain: mikedeng1

Supermodularity and Complementarity IV: Monotone Optimal Policies in Markov Decision ProcessesTextbook

Motivation

A Markov decision process (MDP) chooses a decision in every period of a dynamic system whose state evolves stochastically in response to the decision, so as to maximize expected discounted return. Firms use this model for inventory, pricing, and advertising decisions that respond to a randomly evolving demand state; engineers use it for maintaining or replacing equipment that degrades stochastically. A recurring practical question is qualitative rather than numerical: does the optimal decision increase with the state — should a firm price higher after a period of strong sales, or replace a machine sooner the worse its observed condition — without having to solve the dynamic program numerically for every instance of the model? Topkis's Chapter 3, Section 3.9 (Topkis, Supermodularity and Complementarity, 2011, building on Topkis [1968]) answers this by isolating the lattice-theoretic structure — supermodularity of the return function and of the transition law — under which monotone optimal policies are guaranteed on structural grounds alone. A closely related but logically independent question was studied earlier by Lehmann [1955], who characterized when a family of distributions is stochastically increasing in a parameter; Topkis generalizes Lehmann's characterization to any property of the parameter dependence whose defining set of functions forms a closed convex cone, of which stochastic monotonicity, supermodularity, and convexity are three instances (Theorem 3.9.1, Corollary 3.9.1). Serfozo [1976] independently develops related conditions for partially observed Markov decision processes, and Amir [1996] and Amir, Mirman, and Perkins [1991] give analogous monotonicity results for other classes of dynamic programming models.

Setting

Fix a finite planning horizon of $k$ periods, $i = 1, \dots, k$ . In period $i$ the state $t$ ranges over a set $T_i \subseteq \mathbb{R}^m$ ; given state $t$ , the decision $x$ is restricted to a finite, nonempty set $X_{t,i} \subseteq \mathbb{R}^n$ (finiteness guarantees that an optimal decision always exists — no continuity or compactness argument is used). Write $S_i = \{(x,t) : t \in T_i,\, x \in X_{t,i}\}$ for the set of admissible (decision, state) pairs in period $i$ . The (bounded) expected net return of choosing decision $x$ in state $t$ , period $i$ , is $r_i(x,t)$ . A discount rate $\beta \in [0,1]$ gives $\gamma = 1/(1+\beta)$ , the value in period $i$ of one unit of return in period $i+1$ . Given decision $x$ , state $t$ , and period $i$ , the state $w$ of period $i+1$ is drawn from a distribution $F(x,t,i,\cdot)$ on $\mathbb{R}^m$ .

The optimal-value function $f_i(t)$ (present value of acting optimally from state $t$ , period $i$ , onward) and the decision-value function $g_i(x,t)$ (present value of choosing $x$ in state $t$ , period $i$ , then acting optimally thereafter) are defined by backward induction from period $k$ :

g_k(x,t) = r_k(x,t), \qquad f_i(t) = \max_{x \in X_{t,i}} g_i(x,t), \qquad g_i(x,t) = r_i(x,t) + \gamma \int f_{i+1}(w)\, dF(x,t,i,w) \quad (i < k).

A subset $S$ of a Euclidean space is increasing if it is upward closed under the coordinatewise order. A family of distributions $\{F(a,\cdot) : a \in D\}$ indexed by a parameter $a$ is stochastically increasing on $D$ if the probability $\int_S dF(a,w)$ of every increasing set $S$ is a monotone (non-decreasing) function of $a$ on $D$ ; when $D$ is a sublattice, the family is stochastically supermodular on $D$ if that same probability is a supermodular function of $a$ on $D$ . A real-valued function $\varphi$ on a lattice is supermodular on a set $D$ if $\varphi(a_1) + \varphi(a_2) \le \varphi(a_1 \vee a_2) + \varphi(a_1 \wedge a_2)$ for all $a_1, a_2 \in D$ — the mission series' shared notion, defined once in chunk 02-monotonicity and reused here as Supermodularity.Monotonicity.SupermodularOn.

Formalization targets

Goal — Theorem 3.9.2 (monotone optimal policies)

g_i(x,t) \text{ supermodular on } S_i, \qquad f_i(t) \text{ supermodular on } T_i, \qquad \arg\max_{x \in X_{t,i}} g_i(x,t) \text{ increasing in } t,

together with the existence of a greatest and a least optimal decision at every state, each increasing in the state, in every period $i$ — under the hypotheses that $S_i$ is a sublattice of $\mathbb{R}^{n+m}$ , $X_{t,i}$ is expanding in $t$ , $r_i$ is increasing in $t$ (on sections) and jointly supermodular in $(x,t)$ , and $F(x,t,i,\cdot)$ is stochastically increasing in $t$ (on sections) and stochastically jointly supermodular in $(x,t)$ , for every period $i$ .

This is the weakest of the targets that is still worth stating on its own: it packages four related conclusions (parts (a)–(d) of Theorem 3.9.2) that share one hypothesis set, rather than isolating just the headline monotone-decision claim, because the book's own proof derives all four together and a solver attacking part (c) or (d) needs part (a) and (b) established first.

Supporting milestones

Lemma 3.9.4: under the monotonicity half of the goal's hypotheses alone (no supermodularity), $f_i(t)$ is increasing in $t$ for every period $i$ . This is the induction Theorem 3.9.2 reuses and strengthens.
Corollary 3.9.1(b): on a sublattice $T$ , a family of distributions is stochastically supermodular in $t$ iff $\int h(w)\,dF(t,w)$ is supermodular in $t$ for every increasing $h$ . This is what lets the goal's hypothesis " $F(x,t,i,\cdot)$ stochastically supermodular in $(x,t)$ " be converted into " $\int f_{i+1}(w)\,dF(x,t,i,w)$ supermodular in $(x,t)$ ", the step that feeds $g_i$ 's supermodularity.
Theorem 3.9.1: the general closed-convex-cone characterization of which Corollary 3.9.1(b) is the supermodularity instance (monotonicity and convexity are the other two instances, not formalized here since the goal's proof only needs the supermodularity case).

Significance

The result gives a purely structural sufficient condition — no smoothness, convexity of the decision set, or specific functional form — for monotone comparative statics in dynamic optimization: whenever the one-period return and the state transition are individually monotone and jointly supermodular in (decision, state), so is the whole multi-period value function, and so are the optimal decisions. Textbook applications include optimal advertising that decreases with prior-period sales, optimal pricing that increases with prior-period sales, and optimal maintenance of a deteriorating system, none of which need to be re-derived from scratch once the structural hypotheses are checked for the specific return and transition functions at hand.

Formalizing it contributes a genuinely new layer to the mission series and, per the paper's own triage (substrate.md), to the platform's application of Mathlib's probability-kernel infrastructure: a finite-horizon MDP with a Lebesgue–Stieltjes-style transition law and its associated stochastic-dominance vocabulary (stochastically increasing / stochastically supermodular families of distributions) did not previously exist on the platform or in this mission series, and both are reusable beyond this mission by any future formalization of dynamic programming under uncertainty. The proof itself is not open — Topkis [2011] (unchanged from the 1968/1998 original) gives a complete, elementary backward-induction argument — so what this mission produces is the formalization of a known, structurally distinctive proof technique, not a new mathematical result.

Difficulty

The naive argument — "supermodularity of $r_i$ plus supermodularity of the transition kernel obviously gives supermodularity of $g_i$ " — breaks exactly at the integral: supermodularity of $(x,t) \mapsto F(x,t,i,\cdot)$ is a statement about the whole family of distributions, not about a single number, so "the transition is jointly supermodular" has to be unpacked into "the probability of every increasing set is jointly supermodular in $(x,t)$ " before it says anything about $\int f_{i+1}(w)\,dF(x,t,i,w)$ for a specific function $f_{i+1}$ . Corollary 3.9.1(b) is exactly the step that licenses this unpacking, and it is not free: it needs Theorem 3.9.1's closed-convex-cone argument (approximating $f_{i+1}$ from below by increasing step functions and invoking monotone convergence), not a pointwise argument on $r_i$ and $F$ separately. A second place the naive argument fails is at the constraint sets: because $X_{t,i}$ only grows with $t$ rather than being fixed, monotonicity of $f_i$ (Lemma 3.9.4) needs its own induction combining that growth with monotonicity of $g_i$ — supermodularity of $g_i$ alone does not hand you monotonicity of the arg max without it.

Formalization scope

States and decisions are represented as Fin m → ℝ and Fin n → ℝ (finite-dimensional Euclidean coordinate spaces with the coordinatewise/product order, matching the book's own restriction to $\mathbb{R}^m$ and $\mathbb{R}^n$ — no abstract lattice is used where the book itself specializes). Distributions are represented as MeasureTheory.Measure on the relevant coordinate space, with IsProbabilityMeasure supplied explicitly wherever a stochastic-dominance hypothesis is used (the probability of a set is read off as (μ S).toReal, which is only faithful to $\int_S dF$ when μ is a probability measure — an unconstrained arbitrary measure would let .toReal collapse an infinite value to $0$ and make the hypothesis trivially satisfiable, a formalization this mission rules out). Integrands h are required integrable against every measure in the family wherever an integral is asserted to lie in a set V, since the Bochner integral of a non-integrable function is definitionally 0 in Lean/Mathlib and would otherwise make Theorem 3.9.1 and Corollary 3.9.1(b) trivially true. Decision sets $X_{t,i}$ are finite (Finset, not merely a bounded or compact Set) and required nonempty exactly where the book assumes it: this finiteness, not any compactness or semicontinuity argument, is what guarantees an optimal decision exists, and dropping it would silently substitute Chapter 2's compactness-based existence machinery for the different argument this section actually uses. The optimal-value and decision-value functions $f_i, g_i$ are represented as any functions satisfying the two backward-recursion equations that define them, rather than being constructed by explicit backward recursion in Lean; since the equations determine $f_i, g_i$ uniquely from $r_i$ and $F$ , this is a faithful reading of "define $f_i, g_i$ by (3.9.1) and (3.9.2)," not a weakening of the theorem. The goal's part (c) is stated using the mission series' InducedSetOrder (the Veinott/strong set order, chunk 01-lattices) and part (d) as the existence of two selection functions (greatest, least optimal decision), each monotone in the state — matching the book's "there is a greatest (least) optimal decision ... and this greatest (least) optimal decision is increasing in $t$ ." The infinite-horizon stationary extension that the book gives immediately after Theorem 3.9.2 (relying on an unproved citation to Blackwell [1965]) is out of scope for this mission.

Reusable infrastructure: the StochasticallyIncreasingOn/StochasticallySupermodularOn definitions are parametric in the ambient preorder/lattice and in the measure's target dimension, so a future mission on stochastic convexity (Corollary 3.9.1(c), not formalized here) or on Topkis's §3.10 stochastic inventory model (which explicitly depends on §3.9, per the book's own reading-order note) can reuse them without modification. Contributions extending this mission to the infinite-horizon case, or completing Corollary 3.9.1's monotonicity and convexity halves, are welcome.

Selected references

D. M. Topkis, Supermodularity and Complementarity, Princeton University Press, 2011 (unchanged from the 1998 original), Chapter 3, Section 3.9. DOI: 10.1515/9781400822539.
D. M. Topkis, "Ordered Optimal Solutions," PhD dissertation / working paper, Stanford University, 1968 (the original source for this section's results).
E. L. Lehmann, "Ordered Families of Distributions," Annals of Mathematical Statistics 26(3), 1955, pp. 399–419. https://doi.org/10.1214/aoms/1177728487
R. Serfozo, "Monotone Optimal Policies for Markov Decision Processes," Mathematical Programming Study 6, 1976, pp. 202–215.
R. Amir, "Sensitivity Analysis of Multisector Optimal Economic Dynamics," Journal of Mathematical Economics 25(1), 1996, pp. 123–141.
R. Amir, L. J. Mirman, and W. R. Perkins, "One-Sector Nonclassical Optimal Growth: Optimality Conditions and Comparative Dynamics," International Economic Review 32(3), 1991, pp. 625–644.
D. Blackwell, "Discounted Dynamic Programming," Annals of Mathematical Statistics 36(1), 1965, pp. 226–235. https://doi.org/10.1214/aoms/1177700285

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Machine Learning Probability Random Matrix Theory+1·Captain: mikedeng1

High-Dimensional Probability IV: Norms of Random Matrices with Sub-gaussian EntriesTextbook

Motivation

Random matrices with independent entries appear whenever a system is measured through many noisy, roughly independent channels: shot noise in a sensor array, edges in an Erdős–Rényi-type random graph, or the design matrix of a linear model with independent covariates. A basic question about any such matrix $A$ is how far it can stretch a vector — its operator norm $\|A\|$ — since this single number controls the stability of every linear statistic computed from $A$ : least-squares estimates, spectral clustering, covariance estimation, and random projections all reduce, at some point, to bounding $\|A\|$ .

The theory traces to Marchenko and Pastur's 1967 asymptotic law for the spectrum of large random matrices, and to Bai and Yin's 1988 almost-sure limit $\|A\|/\sqrt n \to 2$ for $n \times n$ matrices with i.i.d. mean-zero, unit-variance entries. Those results are asymptotic: they say what happens as the dimension $n \to \infty$ , for a fixed matrix shape. The result formalized here, Theorem 4.4.5 of Vershynin's High-Dimensional Probability (2018) (DOI 10.1017/9781108231596), belongs to the more recent non-asymptotic strand of the theory: it gives an explicit, dimension-free bound that holds at every fixed $m, n$ , with an explicit failure probability — the form of statement needed for finite-sample guarantees in statistics and data science, rather than limiting behavior.

Setting

Let $A$ be an $m \times n$ real matrix. Equip $\mathbb R^n$ and $\mathbb R^m$ with the Euclidean norm $\|\cdot\|_2$ ; $A$ acts as a linear map $\ell_2^n \to \ell_2^m$ . Its operator norm (§4.1.2) is

\|A\| \;:=\; \max_{x \in S^{n-1}} \|Ax\|_2 ,

the largest factor by which $A$ can stretch a unit vector; equivalently, the largest singular value of $A$ .

A real random variable $X$ is sub-gaussian with the mission's own convention (matching the Orlicz $\psi_2$ norm this series already carries as a published definition, HighDimProb.Concentration.subgaussianNorm) if

\|X\|_{\psi_2} \;:=\; \inf\{t > 0 : \mathbb E \exp(X^2/t^2) \le 2\} \;<\; \infty .

Bounded random variables and Gaussians are sub-gaussian; a Bernoulli( $p$ ) variable and a $\pm 1$ -valued coin flip both qualify, which is why the theorem below directly covers random matrices with i.i.d. Rademacher or Gaussian entries as special cases.

The mission's proof technique is the ε-net argument, developed in §4.2 and used nowhere before this chapter of the book: a metric space $(T,d)$ , a subset $K \subseteq T$ , and $\varepsilon > 0$ give rise to an ε-net $N \subseteq K$ — a finite set such that every point of $K$ is within $\varepsilon$ of some point of $N$ — and a covering number $N(K, d, \varepsilon)$ , the smallest cardinality of such a net. The technique reduces a statement that must hold uniformly over an infinite (compact) set to a statement about finitely many points, paid for by a union bound whose cost is controlled by the covering number.

Formalization targets

Goal (Theorem 4.4.5)

\exists\, C > 0 \;\text{such that}\; \forall\, t > 0,\quad \mathrm{Prob}\bigl\{\, \|A\| \le CK(\sqrt m + \sqrt n + t) \,\bigr\} \;\ge\; 1 - 2\exp(-t^2),

for any $m \times n$ random matrix $A$ with independent, mean-zero, sub-gaussian entries $A_{ij}$ and $K = \max_{i,j} \|A_{ij}\|_{\psi_2}$ . This is the weakest stable form of the bound — it fixes no numerical value for $C$ , only its existence and absoluteness (independence from $m$ , $n$ , $A$ , $t$ ), so later refinements of the constant do not invalidate it.

Significance

The result itself. The bound $\|A\| \lesssim \sqrt m + \sqrt n$ is sharp up to the constant: for entries of unit variance, $\mathbb E\|A\| \ge \tfrac14(\sqrt m + \sqrt n)$ for large $m, n$ (the book's Exercise 4.4.7), so no non-asymptotic bound of this shape can be improved beyond constants. It is the entry point to the rest of the book's random matrix theory: Corollary 4.4.8 specializes it to symmetric matrices, and it underlies the community-detection (§4.5) and covariance-estimation (§4.7) applications later in the same chapter, neither of which is part of this mission.

Formalizing it. The theorem is a classical, fully proved result; nothing about its truth is open. What this mission contributes is a machine-checked formal statement — together with the two pieces of chapter infrastructure its own textbook proof names by number (Corollary 4.2.13, Exercise 4.4.3(a)) — and a third, self-contained application of the same covering-number machinery (Theorem 4.3.5) that exercises the shared IsEpsNet/coveringNumber definitions on a different metric space (the Hamming cube), independently of the Euclidean case. Mathlib and the Prove2Me platform currently have no ε-net, covering-number, or packing-number infrastructure (checked by q=random matrix, q=operator norm, q=covering number, q=net on the platform, and by filename search in Mathlib): this mission is the first to introduce it, restated inside its own namespace since it is not otherwise available to build on.

Difficulty

The obvious first approach is to bound $\|A\| = \max_{x \in S^{n-1}} \|Ax\|_2$ directly by union-bounding a concentration inequality over the sphere $S^{n-1}$ . This fails outright: $S^{n-1}$ is infinite (indeed uncountable) for $n \ge 2$ , so no union bound over its points can converge — the naive approach gives $\infty \cdot (\text{tail probability})$ . The ε-net argument is the fix, but it is not just "discretize and hope": the reduction from the sphere to a finite net (quadratic_form_on_net, Exercise 4.4.3(a)) loses a multiplicative factor $1/(1-2\varepsilon)$ that must be tracked, and the net's cardinality (covering_numbers_of_euclidean_ball_and_sphere, Corollary 4.2.13) is exponential in the dimension ( $9^n$ at $\varepsilon = 1/4$ ) — so the per-point tail probability from Hoeffding-type concentration must itself decay fast enough (quadratically in the exponent) to survive multiplying by $9^{m+n}$ many points. Getting the union bound to close requires choosing the threshold $u$ in the tail bound proportionally to $\sqrt m + \sqrt n + t$ , not to $t$ alone — the $\sqrt m + \sqrt n$ term is exactly what pays for the net's exponential size.

Formalization scope

$A$ is represented as Ω → Matrix (Fin m) (Fin n) ℝ; its entries A ω i j are the individual real random variables. Independence of the $mn$ entries is iIndepFun over the index type Fin m × Fin n; mean-zero is the vanishing of each entry's Bochner integral. The operator norm is the norm of the associated continuous linear map between EuclideanSpace ℝ (Fin n) and EuclideanSpace ℝ (Fin m) (matrixOpNorm, every linear map between finite-dimensional normed spaces being automatically continuous), matching the book's maxₓ∈Sⁿ⁻¹ ‖Ax‖₂ exactly. The sub-gaussian norm $K$ reuses this series' own published definition, HighDimProb.Concentration.subgaussianNorm, rather than a re-derivation. Covering numbers (coveringNumber) are restricted to finite (Finset) ε-nets, the only kind this chapter uses; this is a deliberate restriction, not a general-purpose covering-number formalization, and is disclosed as such. A hard-coded numeral for $C$ , or an unquantified "with high probability" in place of the explicit failure probability $2\exp(-t^2)$ , would each trivialize the statement and is ruled out: $C$ is existentially bound ahead of every other quantifier, and $t > 0$ is a free parameter with its own explicit bound, exactly as the book states it.

Definitions reusable beyond this mission: IsEpsNet and coveringNumber are stated for a general PseudoMetricSpace and apply unchanged to any later chapter's covering-number needs (e.g. Chapter 8's VC-dimension covering numbers), though per this series' rule that drafts cannot import drafts, a later chunk would restate rather than import them until this mission is published. matrixOpNorm is likewise chapter-agnostic. Contributions completing the sorry proofs of any of the four theorem items are welcome and independent of one another; the covering number and net-reduction items (covering_numbers_of_euclidean_ball_and_sphere, quadratic_form_on_net) are the standard prerequisites for the goal's own volumetric/ε-net proof.

Selected references

Vershynin, R. High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge University Press, 2018. DOI 10.1017/9781108231596
Bai, Z. D., Yin, Y. Q. "Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix." Annals of Probability 16 (1988), 1729–1741.
Marchenko, V. A., Pastur, L. A. "Distribution of eigenvalues for some sets of random matrices." Mathematics of the USSR-Sbornik 1 (1967), 457–483.

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Convex Optimization Machine Learning Operations Research+1·Captain: mikedeng1

First-Order and Stochastic Optimization Methods for Machine Learning VI: The Classic Conditional Gradient MethodTextbook

Motivation

Every method in Chapters 2-4 of this series solves a projection or proximal subproblem at every step — a Euclidean projection, or a Bregman-divergence prox-mapping — which can itself be as hard as the original problem when $X$ is a complicated feasible set (a spectrahedron, a flow polytope, a matroid base polytope). The conditional gradient method (Frank & Wolfe, 1956) sidesteps this entirely: instead of a projection, each step calls a linear optimization (LO) oracle — minimize a linear function over $X$ — which is frequently far cheaper (over a spectrahedron, this reduces to a single eigenvector computation; over many combinatorial polytopes, to a greedy algorithm). This is the origin of the modern "projection-free" family of optimization methods widely used at the scale where projections are the bottleneck.

Setting

Fix a nonempty compact convex set $X$ in a real normed space $E$ and a convex $f:X\to \mathbb R$ with $L$ -Lipschitz gradient (Eq. (7.1.4)): $\|f'(x)-f'(y)\|_*\le L\|x-y\|$ . The classic conditional gradient (CndG) method, Algorithm 7.1, sets $x_0\in X$ , $y_0=x_0$ , and for $k=1,2,\dots$ : calls the LO oracle $x_k\in\arg\min_{z\in X}\langle f'(y_{k-1}),z\rangle$ , then sets $y_k=(1-\alpha_k)y_{k-1}+\alpha_kx_k$ for a stepsize $\alpha_k\in[0,1]$ , either the fixed schedule $\alpha_k=2/(k+1)$ (Eq. (7.1.9)) or exact line search (Eq. (7.1.10)).

Section 7.1.1.2 extends this to bilinear saddle-point problems, where $f$ itself is the (generally nonsmooth) function $f(x)=\max_{y\in Y}\{\langle Ax,y\rangle-\hat f(y)\}$ (Eq. (7.1.5)) for a compact convex $Y$ and linear operator $A$ . Since $f$ is nonsmooth, the method is applied instead to a family of smooth approximations $f_\eta$ built from a strongly convex $\omega$ on $Y$ (Eq. (7.1.21)-(7.1.23)), with the smoothing parameter $\eta_k$ allowed to vary across iterations rather than being fixed in advance.

Formalization targets

Goal — Theorem 7.1

f(y_k) - f^* \le \frac{2L}{k(k+1)}\sum_{i=1}^k\|x_i-y_{i-1}\|^2.

Supporting milestones, in attack order

Lemma 7.1: the smoothed objective family $f_\eta$ is monotone nondecreasing in $\eta\ge0$ — the one-line fact ( $V(y)-D_Y^2\le0$ pointwise) that licenses a variable, decreasing smoothing schedule $\eta_k$ rather than a schedule fixed in advance from knowledge of the target accuracy.
Theorem 7.2: the saddle-point counterpart of the goal theorem, running the same CndG algorithm on the smoothed gradients $f_{\eta_k}'$ instead of $f'$ directly, with the explicit rate $f(y_k)-f^*\le\frac{2}{k(k+1)}\sum_{i=1}^k[i\eta_iD_Y^2+\frac{\|A\|^2}{\sigma_v\eta_i} \|x_i-y_{i-1}\|^2]$ .

Every constant here is exactly the book's; the goal theorem's bound is left in terms of the actual step distances $\sum\|x_i-y_{i-1}\|^2$ , not a diameter-based simplification (see Difficulty).

Significance

This mission formalizes the founding convergence result of the entire projection-free family (Frank-Wolfe methods), which has become central to large-scale machine learning precisely because its per-iteration cost can be orders of magnitude below that of a projection-based method on structured feasible sets. Theorem 7.1's specific form — a rate depending on the realized step distances rather than a fixed diameter — is also the more informative, tighter statement (the book's own remarks show it recovers the classical diameter-based $O(LD_X^2/\varepsilon)$ complexity as a corollary, but also explains why the rate can be much better in practice when the iterates settle near an extreme point).

No result matching conditional gradient / Frank-Wolfe methods exists on the platform as of 2026-09-18 (q=Frank-Wolfe and q=conditional gradient both return zero hits — see Prior art in MODERATION_NOTES.md).

Difficulty

The chief formalization difficulty is representing "with the stepsize policy in (7.1.9) or (7.1.10)" faithfully without either restricting to one policy (weaker than the book's stated theorem) or introducing an awkward disjunction of two separate algorithm definitions. The book's own proof resolves this by a single observation used for both policies at once: $f(y_k)\le f(\tilde y_k)$ for $\tilde y_k$ the point the fixed schedule $\gamma_k=2/(k+1)$ would have produced — trivially by equality under (7.1.9), or because $y_k$ is chosen to minimize $f$ over the entire line segment under (7.1.10), of which $\tilde y_k$ is one point. This mission's hyk_le hypothesis states exactly this shared consequence, which is genuinely what the proof uses and genuinely covers both policies, rather than picking one arbitrarily.

A second difficulty is not collapsing $\sum_{i=1}^k\|x_i-y_{i-1}\|^2$ into a diameter bound $kD_X^2$ inside the milestone itself — the book's own remarks perform that substitution as a separate, weaker corollary (Eq. (7.1.19)) after stating Theorem 7.1 in its sharper form; folding the substitution into the goal statement itself would silently prove a different, weaker theorem.

Formalization scope

conditional_gradient_rate and saddle_point_cndg_rate state the LO oracle's exactness (x k ∈ Argmin_{z∈X}⟨fGrad(y(k-1)),z⟩) as a pointwise hypothesis rather than deriving it from IsCompact X via an existence lemma — matching the pointwise-hypothesis convention this series uses throughout for argmin-defined algorithmic steps (chunk 03-deterministic's mirror-descent updates, chunk 04-stochastic's stochastic mirror-descent update). X compact convex is still included as a hypothesis, matching the book's own standing assumption on the problem class, even though it is not itself needed to derive the stated conclusion from the other hypotheses.

smoothed_objective_monotone and saddle_point_cndg_rate realize $f_\eta$ / $f$ via sSup of the image of $Y$ under the pointwise saddle-point objective, matching the book's own max_{y∈Y}{...} definition (Eq. (7.1.5), (7.1.23)) directly rather than introducing a separate Def_ file for a "bilinear saddle-point objective" structure — no other item in this mission reuses that definition verbatim, so per this series' convention (no shared substrate bundled into a structure unless reused), it is inlined at each use.

A trivializing formalization this mission rules out: stating the LO oracle via an $\varepsilon$ -approximate minimizer ((fGrad (y(k-1))) (x k) ≤ (fGrad (y(k-1))) z + ε for some ε) rather than an exact one — this is explicitly a different, weaker algorithm the book does not analyze in Theorem 7.1/7.2 (the book studies approximate LO oracles separately, later in the chapter, not selected here).

Left out of scope, for time: Theorem 7.7 (the matching lower complexity bound for LO-oracle methods, Eq. (7.1.60)) — formalizing it faithfully requires first modeling the abstract class of "LCP methods" (any algorithm restricted to LO-oracle calls) as a universally-quantified object, a substantially different and more involved formalization task than the two upper-bound convergence theorems selected here; named per Hard Rule 7 rather than approximated. The d(x)=\sum x_i\log x_i entropy-smoothing remark and the primal/primal-dual averaging CndG variants (§7.1.2, not covered by this mission's page range) are likewise not attempted.

Selected references

G. Lan, First-Order and Stochastic Optimization Methods for Machine Learning, Springer Series in the Data Sciences, Springer 2020, Chapter 7, §7.1.1. https://doi.org/10.1007/978-3-030-39568-1
M. Frank, P. Wolfe, "An algorithm for quadratic programming," Naval Research Logistics Quarterly, 3(1-2), 1956, pp. 95-110.
M. Jaggi, "Revisiting Frank-Wolfe: projection-free sparse convex optimization," ICML, 2013 (the modern machine-learning revival of the method).

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Convex Optimization Machine Learning Operations Research+2·Captain: mikedeng1

First-Order and Stochastic Optimization Methods for Machine Learning IV: Variance-Reduced Mirror Descent for Finite-Sum ProblemsTextbook

Motivation

Empirical-risk-minimization objectives in machine learning are finite sums: $\Psi(x) = \frac{1}{m}\sum_{i=1}^m f_i(x) + h(x)$ , one smooth term $f_i$ per training example (or per worker, in a distributed setting), plus a simple nonsmooth regularizer $h$ . Chapter 4's basic stochastic mirror descent handles this by sampling a single random component gradient $\nabla f_{i_t}(x)$ as an unbiased estimator of $\nabla f(x)$ — but that estimator's variance is a constant throughout the algorithm, which caps the achievable convergence rate. Variance-reduced mirror descent asks a sharper question: can an unbiased finite-sum gradient estimator be built whose variance itself vanishes as the algorithm approaches the optimum? The answer — periodic full-gradient snapshots combined with single-component corrections — is the SVRG-style idea this mission formalizes in Lan's general-norm mirror-descent framework, with an explicit, sampling-distribution-dependent constant rather than a generic $O(\cdot)$ .

Setting

Fix a closed convex set $X$ in a real normed space $E$ , and the finite-sum composite problem $\min_{x\in X}\{\Psi(x):=f(x)+h(x)\}$ (Eq. (5.3.1)), where $f(x)=\frac1m\sum_{i=1}^m f_i(x)$ is the average of $m$ smooth convex component functions, each with $L_i$ -Lipschitz gradient $\nabla f_i$ ( $\|\nabla f_i(x)-\nabla f_i(y)\|_*\le L_i\|x-y\|$ ), and $h$ is a simple, possibly nondifferentiable convex function. $f$ is possibly $\mu$ -strongly convex, $\mu\ge0$ (Eq. (5.3.2)); this mission's goal takes $\mu=0$ (§5.3.1, "Smooth Problems Without Strong Convexity"). A fixed probability distribution $Q=\{q_1,\dots,q_m\}$ on the component indices governs the algorithm's random sampling, and

L_Q := \frac{1}{m}\max_{i=1,\dots,m}\frac{L_i}{q_i}

is the section's key aggregate smoothness constant (Eq. (5.3.4)), replacing the plain average $L$ wherever component-wise variance enters the analysis. Variance-reduced mirror descent (Algorithm 5.6) is a multi-epoch method: each epoch of length $T_s$ recomputes a full gradient $\nabla f(\tilde x)$ at a snapshot point $\tilde x$ , then runs $T_s$ inner iterations using the estimator $G_t := \big(\nabla f_{i_t}(x_t)-\nabla f_{i_t}(\tilde x)\big)/(q_{i_t}m) + \nabla f(\tilde x)$ and the mirror-descent-with-composite-term update $x_{t+1}:=\arg\min_{x\in X}\{\gamma[\langle G_t,x\rangle+h(x)]+V(x_t,x)\}$ , where $V$ is the Bregman divergence of a fixed distance-generating function, exactly as in Chapters 3-4.

Formalization targets

Goal — Corollary 5.8

With $\theta=1$ , $\gamma=1/(16L_Q)$ , and the doubling epoch schedule $T_1=7$ , $T_s=2T_{s-1}$ (Eq. (5.3.17)),

\mathbb E[\Psi(\bar x_S)-\Psi(x^*)] \le \frac{8}{2^{S-1}}\left[\frac{11}{4}\big(\Psi(x_0)-\Psi(x^*)\big)+16L_Q\,V(x_0,x^*)\right]

for every epoch count $S\ge1$ , where $\bar x_S$ is the weighted average of the epoch snapshots (Eq. (5.3.16)).

Supporting milestones, in attack order

Lemma 5.12 — the per-component gradient-variation bound $\frac1m\sum_i\frac1{mq_i}\|\nabla f_i(x)-\nabla f_i(x^*)\|_*^2 \le 2L_Q[\Psi(x)-\Psi(x^*)]$ , the basic smoothness consequence from which the estimator's variance bound is built.
Lemma 5.13 — unbiasedness ( $\mathbb E[\delta_t]=0$ ) and two variance bounds ( $\mathbb E[\|\delta_t\|_*^2]\le 2L_Q[\dots]$ and $\le 4L_Q[\dots]$ ) for the variance-reduced estimator's error $\delta_t:=G_t-\nabla f(x_t)$ .
Lemma 5.14 — the one-step progress bound combining Lemma 5.13's variance control with the mirror-descent update's three-point inequality.
Theorem 5.6 — the general epoch-level convergence bound (with an arbitrary epoch-length schedule $T_s$ and stepsize $\gamma$ satisfying $4L_Q\gamma\le1$ ) that Corollary 5.8 instantiates.

Every constant is exactly the book's: $L_Q$ 's own sampling-distribution-dependent definition (never specialized to uniform $q_i=1/m$ ), and Corollary 5.8's explicit $8/2^{S-1}$ , $11/4$ , $16L_Q$ — not a generic $O(\cdot)$ — are all taken verbatim.

Significance

This is the series' first genuinely finite-sum result: unlike Chapters 3-4's single abstract objective $f$ , here $f$ is structurally a named average of $m$ component functions, and the sampling distribution $\{q_i\}$ over those components is a first-class free parameter of both the algorithm and the analysis (not fixed to uniform sampling) — $L_Q$ itself depends on this choice, and a formalization that hard-codes $q_i=1/m$ would understate what Lemma 5.12's own proof needs. Getting Theorem 5.6/Corollary 5.8 right also requires keeping two nested indices straight: inner iterations $t$ within an epoch, and outer epoch counts $s$ , with the convergence bound stated in terms of the epoch count $S$ alone — and keeping the two "gap" quantities $\Psi(x_0)-\Psi(x^*)$ (an objective-value gap) and $V(x_0,x^*)$ (a Bregman-divergence gap) distinct throughout, since they enter Corollary 5.8's final bound with different explicit coefficients ( $11/4$ vs. $16L_Q$ ) and neither generically bounds the other.

No result on the platform models a finite-sum objective with $m$ named component functions sampled by a general index distribution $\{q_i\}$ , a variance-reduction snapshot/anchor point, or this specific SVRG-style estimator, as of 2026-09-18 (q=finite sum, q=variance reduction, q=SVRG, q=component function, q=variance reduced gradient, q=mirror descent finite sum — see Prior art below).

Difficulty

The central difficulty is Theorem 5.6's own epoch-weight sequence $w_s$ : the book defines $w_s:=(1-4L_Q\gamma)(T_{s-1}-1)-4L_Q\gamma T_s$ explicitly only for $s\ge2$ (Eq. (5.3.14)), yet the displayed sums $\sum_{s=1}^S w_s$ in (5.3.15)-(5.3.16) run from $s=1$ . A 2026-09-19 revision found that this, combined with the epoch snapshot $\tilde x_s$ being constrained only by membership in $X$ and not tied to the algorithm's own dynamics, made the originally drafted statements false, not merely incomplete: an adversarial, unboundedly-large- $\Psi$ , $\omega$ -independent $\tilde x_1$ together with $w_1\to\infty$ violates the stated conclusion. The fix restores the connection via an auxiliary epoch-boundary sequence and the per-epoch progress inequality Theorem 5.6's own proof derives from Lemma 5.14 (see epoch_convergence_bound's hepoch hypothesis), and resolves $w_1$ by extending (5.3.14)'s domain to $s\ge1$ via a fixed "epoch 0" length $T_0$ — w_1 is no longer left free beyond positivity. finite_sum_variance_reduced_rate instantiates $T_0:=T_1/2=3.5$ concretely, reproducing the arithmetic Corollary 5.8's own proof is internally consistent with ( $w_1 = 3/4(3.5-1)-1/4\cdot7 = 1/8$ , matching the closed form $(1/8)T_1-3/4=1/8$ ) — this was previously only a documented-but-unresolved observation, not yet a stated hypothesis.

Formalization scope

All five items are stated over a general real normed space [NormedAddCommGroup E] [NormedSpace ℝ E], matching the mirror-descent chunks' general-norm convention (never specialized to Euclidean space or squared distance) — $V$ is a free two-point function throughout, and each $\nabla f_i$ , $\nabla f$ , $G_t$ are continuous linear functionals E →L[ℝ] ℝ, whose Mathlib operator norm supplies the dual norm $\|\cdot\|_*$ with no separate definition needed. This is the trivializing formalization this mission rules out: hard-coding $q_i=1/m$ (uniform sampling) or $V(x,y)=\frac12 \|x-y\|^2$ (Euclidean Bregman divergence) would understate both $L_Q$ 's dependence on the sampling distribution (the whole point of Lemma 5.12's bound) and the general-norm apparatus the rest of this book series shares.

Ψ(x_0)-Ψ(x^*) and V(x_0,x^*) are kept as two syntactically distinct terms throughout — never conflated or bounded one by the other — matching Corollary 5.8's own two separate coefficients. Corollary 5.8's own explicit constants ( $8/2^{S-1}$ , $11/4$ , $16L_Q$ ) are stated verbatim rather than left as an unspecified $O(\cdot)$ , per Hard Rule 6.

Left out of scope, for time: the gradient-computation-count complexity bound (Eq. (5.3.19), an $O(\cdot)$ statement about total oracle calls, not a convergence-rate inequality on $\Psi$ ) and §5.3.2's strongly-convex case (Theorem 5.7, a geometric-decay bound $\Delta_s\le\rho\Delta_{s-1}$ under $\mu>0$ ) are natural continuations reusing this mission's variance_reduced_progress_bound milestone, not attempted here.

Prior art

q=finite sum, q=variance reduction, q=SVRG, q=component function, q=variance reduced gradient, and q=mirror descent finite sum were all searched on 2026-09-18. The only topically-adjacent hit across all six queries is ShiOptRates.Stochastic.variance_purchase_ classical ("Classical variance reduction is cost-neutral..."), which models plain minibatch SGD on a smooth objective with an i.i.d.-noise oracle characterized by a single scalar variance $\hat\sigma^2$ and a minibatch-size trade-off — no finite-sum structure with $m$ named component functions, no sampling distribution $\{q_i\}$ , no snapshot/anchor point $\tilde x$ , and a different question (cost-neutrality of minibatch size vs. this mission's convergence rate for a fixed variance-reduction scheme). Not reused; every item in this mission is drafted fresh.

Selected references

G. Lan, First-Order and Stochastic Optimization Methods for Machine Learning, Springer Series in the Data Sciences, Springer 2020, Chapter 5, §5.3. https://doi.org/10.1007/978-3-030-39568-1
R. Johnson, T. Zhang, "Accelerating stochastic gradient descent using predictive variance reduction," Advances in Neural Information Processing Systems (NeurIPS), 2013 (the SVRG estimator this section's gradient estimator generalizes to the composite mirror-descent setting).
A. Nemirovski, A. Juditsky, G. Lan, A. Shapiro, "Robust stochastic approximation approach to stochastic programming," SIAM Journal on Optimization, 19(4), 2009, pp. 1574-1609.

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Convex Optimization Machine Learning Operations Research+1·Captain: mikedeng1

First-Order and Stochastic Optimization Methods for Machine Learning II: Subgradient Descent, Mirror Descent and Accelerated Gradient DescentTextbook

Motivation

Gradient descent's convergence rate for a general smooth convex problem is $O(1/k)$ in the function-value gap; Nemirovski and Yudin (1983) proved that no first-order method can do better than $O(1/k^2)$ is achievable, and Nesterov (1983, 1988, 2004) constructed the first method attaining it — the accelerated (or "fast") gradient method. For thirty years this was the standard route to $O(1/k^2)$ -rate solvers in convex optimization, and the technique underlies essentially every modern accelerated first-order method used at scale in machine learning (accelerated SGD, momentum methods, Nesterov-style extensions of Adam). The two building blocks this mission formalizes on the way there — subgradient descent (Polyak, 1960s) and mirror descent (Nemirovski & Yudin, 1983) — are themselves the default tools whenever the objective is nonsmooth or the constraint set's natural geometry is not Euclidean (e.g. the probability simplex, where mirror descent with the entropic distance-generating function beats projected subgradient descent by a $\sqrt{n/\ln n}$ factor).

Setting

Fix a nonempty closed convex set $X$ (in Lean: a normed real vector space $E$ , X : Set E) and a convex $f : X \to \mathbb{R}$ ; write $f^* := \min_{x\in X} f(x)$ and $x^*$ for an arbitrary minimizer. The projected-subgradient update is $x_{t+1} := \arg\min_{x\in X}\gamma_t\langle g(x_t),x\rangle + \tfrac12\|x-x_t\|_2^2$ for a subgradient $g(x_t)\in\partial f(x_t)$ and stepsize $\gamma_t>0$ . Its generalization, mirror descent, replaces the Euclidean proximal term with a Bregman divergence $V(x,z) := \nu(z) - \nu(x) - \langle\nabla\nu(x),z-x\rangle$ built from a 1-strongly-convex distance-generating function $\nu$ with respect to a general norm $\|\cdot\|$ (dual norm $\|\cdot\|_*$ ): $x_{t+1} := \arg\min_{x\in X}\gamma_t g_t(x) + V(x_t,x)$ , where $g_t$ is now a continuous linear functional (a subgradient in the dual space, since the norm need not come from an inner product). Choosing $\nu(x)=\|x\|_2^2/2$ recovers $V(x,z) = \|z-x\|_2^2/2$ and the plain subgradient update as a special case.

The accelerated gradient method additionally assumes $f$ has $L$ -Lipschitz gradient ( $f(y)-f(x)-\langle f'(x),y-x\rangle \le \tfrac{L}{2}\|y-x\|^2$ ) and is $\mu$ -generalized-strongly-convex w.r.t. $V$ ( $f(x)+\langle f'(x),y-x\rangle+\mu V(x,y)\le f(y)$ for $\mu\ge0$ ), and tracks three coupled sequences from $(x_0,\bar x_0)\in X\times X$ :

\tilde x_t = (1-q_t)\bar x_{t-1}+q_tx_{t-1},\quad x_t = \arg\min_{x\in X}\{\gamma_t[\langle f'(\tilde x_t),x\rangle+\mu V(\tilde x_t,x)]+V(x_{t-1},x)\},\quad \bar x_t = (1-\alpha_t)\bar x_{t-1}+\alpha_tx_t.

Formalization targets

Goal — Theorem 3.6, closed-form rate

With $q_t=\alpha_t=\tfrac{2}{t+1}$ , $\gamma_t=\tfrac{t}{2L}$ and $\mu=0$ :

f(\bar x_k) - f(x^*) \le \frac{4L}{k(k+1)}V(x_0,x^*).

Supporting milestones, in attack order

Lemma 3.1 / Theorem 3.1 (Euclidean case): the three-point inequality for the plain projected-subgradient step, and the resulting $\sum_t \gamma_t[f(x_t)-f(x)] \le \tfrac12(\|x-x_s\|_2^2 + M^2\sum_t\gamma_t^2)$ bound under $M$ -Lipschitz $f$ .
Lemma 3.4 / Theorem 3.5 (general-norm mirror descent): the same two results with the squared Euclidean distance replaced by $V$ and the Euclidean norm by a general dual pair $\|\cdot\|,\|\cdot\|_*$ .
Proposition 3.1: the one-step accelerated-method recursion $f(\bar x_t)-f(x)+\alpha_t(\mu+ 1/\gamma_t)V(x_t,x) \le (1-\alpha_t)[f(\bar x_{t-1})-f(x)]+(\alpha_t/\gamma_t)V(x_{t-1},x)$ .
Theorem 3.6, general form: Proposition 3.1's recursion telescoped across $t=1,\dots,k$ (with $\mu=0$ ) into a single two-term bound relating step $k$ to step $0$ .

Every constant here is exactly the book's; no milestone hides an $O(\cdot)$ behind an unspecified absolute constant.

Significance

The chain culminates in an explicit, non-asymptotic $O(1/k^2)$ certificate for accelerated gradient descent — the theoretically optimal rate for smooth convex minimization by a first-order method (matching the Nemirovski–Yudin lower bound, not re-derived here). Formalizing it forces every implicit convention in a standard optimization-course derivation to become explicit: which of the three sequences $x_t,\tilde x_t,\bar x_t$ a given quantity refers to, exactly which inequality (3.3.7)-(3.3.9) each specific stepsize schedule needs to satisfy, and the precise index range over which the chapter's own stated hypotheses actually get used in its own proof (see Difficulty below).

None of these six results (or their strongly-convex counterpart, Theorem 3.7, left for future work — see Formalization scope) has a machine-checked proof on Prove2Me. The one theorem with the same name as this mission's subject, BanditAlgorithm.mirror_descent_regret_bound (Lattimore & Szepesvári, Theorem 28.4), is a different object: an online, adversarial regret bound against a changing sequence of loss vectors $y_t$ , not an offline function-value gap for a single fixed $f$ ; not reused. Likewise OnlineConvexOpt.FirstOrder.online_gradient_descent_regret (Hazan) and OnlineConvexOpt.ConvexBasics.constrained_gd_well_conditioned_convergence are, respectively, an online-regret bound and a plain-gradient-descent (non-accelerated) linear-rate result — checked and confirmed not reusable per the mission brief.

Difficulty

The three-point inequalities (Lemmas 3.1/3.4) are routine consequences of a strongly-convex minimizer's optimality condition. The real difficulty is bookkeeping across three coupled sequences in the accelerated method: a formalization using only $x_t$ and $\bar x_t$ (dropping $\tilde x_t$ , the point at which the gradient is actually evaluated) is not Lan's algorithm and proves either a false or a different bound — $\tilde x_t$ is what lets the method use a gradient computed at a point between $x_{t-1}$ and $\bar x_{t-1}$ , which is exactly the extrapolation step that makes acceleration work.

A second, subtler difficulty is that Theorem 3.6's own stated hypothesis — "(3.3.15) for any $t=1,\dots,k$ " — is not quite what its proof uses. Telescoping Proposition 3.1's per-step bound via (3.3.15) requires the previous step's constants $\gamma_{t-1},\alpha_{t-1}$ ; at $t=1$ these would be $\gamma_0,\alpha_0$ , values the recursion (3.3.4)-(3.3.6) never defines (it only ever uses $q_t,\gamma_t,\alpha_t$ for $t\ge1$ ). The book's own proof, read closely, invokes (3.3.15) only for $t=2,\dots,k$ , with $t=1$ handled directly by Proposition 3.1's conclusion connecting $\bar x_1,x_1$ to the given base data $\bar x_0,x_0$ . Formalizing the literal hypothesis range would either be unstatable (no $\gamma_0,\alpha_0$ exist) or vacuous (adding unused ghost parameters); this mission states the range the proof actually needs.

Formalization scope

Chapter 3's own §3.1/§3.2 split (Euclidean vs. general norm) is preserved rather than collapsed: subgradient_iterate_three_point/subgradient_descent_bound are stated over a real inner product space with the vector subgradient $g(x_t)\in E$ and the Euclidean norm, exactly matching §3.1; mirror_iterate_three_point/mirror_descent_bound and the two accelerated-method milestones are stated over a general real normed space [NormedAddCommGroup E] [NormedSpace ℝ E], with subgradients as continuous linear functionals E →L[ℝ] ℝ (whose Mathlib operator norm is already the dual norm $\|\cdot\|_*$ , needing no separate definition) and the Bregman divergence $V : E \to E \to \mathbb{R}$ left as a free two-point function — but, following a 2026-09-19 revision, no longer a totally free function. V is now required to satisfy the two facts (3.2.2)/(3.2.3)/(3.2.6) actually establish and every downstream proof (Lemma 3.4, Theorem 3.5, Proposition 3.1, Theorem 3.6) uses: nonnegativity ( $V(x,z)\ge 0$ for $x,z\in X$ ) and the three-point/cosine identity $V(x,z) = V(x,y) + \langle\nabla V(x,\cdot)(y), z-y\rangle + V(y,z)$ , the latter made explicit via an added parameter dV : E → E → (E →L[ℝ] ℝ) read as "the gradient of $V(x,\cdot)$ at $y$ ." Without these two hypotheses the five items that use an abstract V (mirror_iterate_three_point, mirror_descent_bound, accelerated_one_step_recursion, accelerated_gradient_recursion_bound, accelerated_gradient_rate) are false as stated — a constant V satisfies the bare pointwise-minimality hypotheses while violating the conclusion, as two worked counterexamples confirmed. This mission does not derive V/dV from an explicit distance-generating function $\nu$ (the heavier, fully book-literal route (3.2.1)-(3.2.2) would); it takes the two facts the proofs actually consume as hypotheses directly, which is lighter and sufficient. Satisfiability is witnessed by the Euclidean case already in §3.1: $\nu(x)=\|x\|^2/2$ , $V(x,z)=\|z-x\|_2^2/2$ , $dV\,x\,y = \langle y-x,\cdot\rangle$ , exactly how subgradient_iterate_three_point/subgradient_descent_bound already handle the Euclidean special case. A trivializing formalization this mission rules out: specializing $V$ to the Euclidean squared distance in mirror_iterate_three_point/mirror_descent_bound would make those two milestones restatements of the §3.1 Euclidean results rather than genuine generalizations, exactly the pitfall the chapter brief flags.

Every argmin-defined iterate ( $x_{t+1}$ in each of the three update rules) is represented by its defining pointwise-minimality property rather than by an IsMinOn/argmin term, so no existence or uniqueness lemma for the underlying minimization problem is needed anywhere in this mission — matching how the book's own proofs use these updates (via their first-order optimality condition, never via an explicit formula for the minimizer).

Left out of scope, for time: Theorem 3.7 (the strongly-convex, $\mu>0$ linear-rate companion to Theorem 3.6, sharing Proposition 3.1 as its own base lemma) and Corollary 3.5 (the composite-objective extension $f=\hat f+F$ ). Both are natural continuations reusing this mission's accelerated_one_step_recursion; a later mission or an amendment to this one could add them as additional milestones/goals without touching what is here.

Selected references

G. Lan, First-Order and Stochastic Optimization Methods for Machine Learning, Springer Series in the Data Sciences, Springer 2020, Chapter 3. https://doi.org/10.1007/978-3-030-39568-1
Y. Nesterov, "A method for solving the convex programming problem with convergence rate $O(1/k^2)$ ," Doklady AN SSSR, 269, 1983, pp. 543–547.
Y. Nesterov, Introductory Lectures on Convex Optimization, Springer, 2004.
A. Nemirovski and D. Yudin, Problem Complexity and Method Efficiency in Optimization, Wiley, 1983 (source of the mirror-descent method and the $O(1/k^2)$ lower bound for smooth convex optimization).

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Machine Learning Reinforcement Learning Statistics·Captain: mikedeng1

Foundations of Reinforcement Learning III: Structured Bandits and the Decision-Estimation CoefficientTextbook

Motivation

Every algorithm in the first three chapters of Foster and Rakhlin's Foundations of Reinforcement Learning and Interactive Decision Making — ε-Greedy and UCB for the multi-armed bandit, Inverse Gap Weighting and SquareCB for contextual bandits — is a special case of the same two-step recipe: estimate a model of the world with an online regression oracle, then convert the estimate into a decision that trades exploration against exploitation. Chapter 4 asks whether this recipe can be made generic: given any structured decision-making problem, specified only by a function class $F$ and a decision space $\Pi$ , is there a single quantity that governs the best achievable regret, the way $\sqrt{A/\gamma}$ governs the multi-armed bandit and $\sqrt{d/\gamma}$ governs the linear bandit? The chapter's answer is the Decision-Estimation Coefficient (DEC), introduced by Foster, Kakade, Qian, and Rakhlin [40] as a complexity measure that both upper- and lower-bounds achievable regret for a general decision-making protocol, unifying results that were previously proved from scratch, case by case, for each structured setting. This mission formalizes the chapter's central upper bound (Proposition 13) together with the machinery that makes it computable in two concrete cases — the multi-armed bandit (Proposition 14) and the linear bandit (Propositions 16–17).

Setting

Fix a finite decision space $\Pi$ and a class $F \subseteq \mathbb{R}^\Pi$ of candidate mean-reward functions, with a ground-truth $f^\star \in F$ (realizability). Over $T$ rounds, at each round $t$ the learner observes an estimate $\hat f_t$ produced by an online regression oracle, plays a decision distribution $p_t \in \Delta(\Pi)$ (possibly depending on $\hat f_t$ and the history), and the regret is

\mathrm{Reg} := \sum_{t=1}^T f^\star(\pi^\star) - \sum_{t=1}^T \mathbb{E}_{\pi \sim p_t}[f^\star(\pi)],

where $\pi^\star = \arg\max_\pi f^\star(\pi)$ . The oracle's cumulative estimation error is assumed bounded: $\sum_{t=1}^T \mathbb{E}_{\pi \sim p_t}[(\hat f_t(\pi) - f^\star(\pi))^2] \le \mathrm{EstSq}(F,T,\delta)$ with probability at least $1-\delta$ (Definition 7). Writing $\pi_f := \arg\max_\pi f(\pi)$ , the DEC game value at a reference model $\hat f$ and scale $\gamma > 0$ is the min-max quantity

\mathrm{dec}_\gamma(F, \hat f) := \min_{p \in \Delta(\Pi)} \max_{f \in F} \; \mathbb{E}_{\pi \sim p}\bigl[f(\pi_f) - f(\pi) - \gamma(f(\pi) - \hat f(\pi))^2\bigr],

and the DEC of $F$ itself is $\mathrm{dec}_\gamma(F) := \sup_{\hat f \in \mathrm{co}(F)} \mathrm{dec}_\gamma(F, \hat f)$ . The Estimation-to-Decisions (E2D) algorithm plays, at each round, a $p_t$ certifying (i.e. attaining or beating) the value of this min-max game at $\hat f_t$ .

Formalization targets

Goal — Proposition 13 (E2D regret bound)

\mathrm{Reg} \le \mathrm{dec}_\gamma(F) \cdot T + \gamma \cdot \mathrm{EstSq}(F, T, \delta)

with probability at least $1-\delta$ , for any exploration parameter $\gamma > 0$ . This is the weakest stable statement the chapter proves about E2D: it holds for an arbitrary function class and an arbitrary regression oracle, with no structural assumption on $F$ beyond realizability, and the chapter's later sections instantiate it rather than strengthen it.

Milestones

Lemma 9 (Decoupling), general form: for any distribution $\nu$ over a finite model class and any $\bar f$ , $\mathbb{E}_{f\sim\nu}[f(\pi_f) - \bar f(\pi_f)] \le \sqrt{A \cdot \mathbb{E}_{f\sim\nu}\mathbb{E}_{\pi\sim p}[(f(\pi)-\bar f(\pi))^2]}$ — the estimation-to-decisions bridge the whole chapter's approach rests on, decoupling the model index from the played decision.
Proposition 14 (IGW minimizes the DEC): for the multi-armed bandit ( $\Pi=[A]$ , $F=\mathbb{R}^A$ ), Inverse Gap Weighting is the exact minimizer of the DEC game, giving $\mathrm{dec}_\gamma(F) = (A-1)/(4\gamma)$ — the first concrete computation of an abstract quantity, recovering Chapter 3's rate from Proposition 13 alone.
Proposition 16 (G-optimal design): existence, for any compact full-dimensional-span set $Z \subseteq \mathbb{R}^d$ , of a distribution $p$ with $\sup_{z\in Z}\langle \Sigma_p^{-1}z,z\rangle \le d$ — the classical convex-analysis primitive Proposition 17 needs.
Proposition 17 (DEC for linear bandits): combining the G-optimal design with inverse gap weighting gives $\mathrm{dec}_\gamma(F) \lesssim d/\gamma$ for the linear bandit function class, leading via Proposition 13 to a $\sqrt{dT}$ regret bound.

Significance

The Decision-Estimation Coefficient is, in the book's own words, "the main result" of this line of work: Foster, Kakade, Qian, and Rakhlin [40] show it is not merely an upper bound but (in a suitable localized form, developed further in Chapter 6) a tight characterization of the minimax regret for structured bandits and, more generally, for the interactive decision-making protocol the rest of the book studies. Proposition 13 is the mechanism that makes this useful in practice: it reduces regret analysis for a new structured problem to a single, purely convex-analytic computation of $\mathrm{dec}_\gamma(F)$ , in place of a bespoke exploration argument. Propositions 14–17 are the demonstration that this reduction is not vacuous — they recompute, via the DEC alone, the two rates (multi-armed and linear bandit) that earlier chapters of the book derived by direct, setting-specific arguments, and the match is exact. Formalizing this chapter therefore captures the book's unifying abstraction itself, not just one more instance of it. No formalization of the Decision-Estimation Coefficient, in any form, currently exists on the platform (see Formalization scope).

Difficulty

The obvious formalization mistake is to state Proposition 13's conclusion with $\mathrm{dec}_\gamma(F)$ left as an unconstrained free real-number parameter satisfying only the inequality the theorem asserts — a formalization under which the "theorem" would be a triviality about an arbitrary real number, since nothing about the actual min-max game would ever be checked. The chapter's content is precisely the opposite: that this specific minimax quantity can be computed (Proposition 14) or bounded via a concrete strategy (Proposition 17), and — as Chapter 6 shows for a lower bound outside this chunk's scope — that no smaller quantity would do. A second difficulty is proof-theoretic rather than notational: the book's own proof of Proposition 13 bounds regret by an unconstrained supremum over all reference functions $\hat f : \Pi \to \mathbb{R}$ , and only identifies this with the official, $\mathrm{co}(F)$ -restricted $\mathrm{dec}_\gamma(F)$ of Eq. (4.16) via Proposition 24 — a fact stated on p. 80, outside this chapter's numbered range, whose own proof the book defers to an exercise. A formalization that quietly imports Proposition 24 to close this gap would rest the goal theorem on an unverified fact; this mission instead states the hypothesis the book's own text uses to motivate restricting to $\mathrm{co}(F)$ in the first place (online estimation algorithms produce $\hat f_t \in \mathrm{co}(F)$ ), so the goal is faithful to what is actually established within the chapter's own pages.

Formalization scope

Every item fixes a finite decision space (Fin A, Fin n, or a generic Fintype S) and states the DEC as the literal sInf-of-sSup transcription of the min-max game (Eqs. (4.15)–(4.16)), never as an opaque bound — this is the trivializing formalization the chunk's own reading of the chapter rules out (see Difficulty). piStar : (S → ℝ) → S is a hypothesized global maximizer selector throughout, constrained to be a genuine argmax only on the function class in scope (F or Set.univ), matching how the book treats $\pi_f$ as a fixed but arbitrary tie-breaking choice. The goal theorem (Proposition 13) adds the explicit hypothesis hfhat : ∀ t, fhat t ∈ convexHull ℝ F, replacing an appeal to the out-of-range Proposition 24 (see Difficulty); this is the one place this mission's statement is not a line-by-line transcription of the book's own displayed proof steps, and it is recorded here and in MODERATION_NOTES.md. Proposition 14's and Proposition 17's $\lesssim$ are replaced by the explicit constants the book's own proofs establish ( $(A-1)/(4\gamma)$ exactly, and $(4d+1)/(2\gamma)$ respectively — the latter obtained by summing the three terms the proof of Proposition 17 isolates). Proposition 14's Lean statement splits the book's single equality $\mathrm{dec}_\gamma(F,\hat f) = (A-1)/(4\gamma)$ into an upper bound on the literal decGf, a lower bound restricted to full-support distributions, and IGW's own exact game value, because the book's min over the whole simplex is not provable as a literal Lean equality: a distribution with a zero-weight arm makes the inner supremum genuinely unbounded, and Lean's total Real.sSup returns a junk value smaller than $(A-1)/(4\gamma)$ there (caught in moderation, MODERATION_NOTES.md); the three-conjunct statement recovers exactly the book's real content without asserting that false literal equality. Lemma 9 is restated inside FoundationsRL.Structured rather than imported from the Chapter 2 mission, since draft items across chunks cannot import one another; its source citation still points to its original location (p. 32). Proposition 16 is not drafted: the platform's existing BanditAlgorithm.kiefer_wolfowitz_equivalence (Lattimore & Szepesvári, Theorem 21.1) states the identical existence claim — compact set with full-dimensional span, a design with G-value at most $d$ — as one clause of a larger equivalence, and is reused as a reference item rather than redrafted. Proposition 22 (primal/dual DEC equivalence, §4.4) is deliberately excluded: the book states it "under mild regularity conditions" it does not pin down in the statement itself, which is exactly the kind of unquantified hypothesis this series' faithfulness standard excludes from a goal or milestone. Contributions extending this mission with Chapter 6's lower bound (matching $\mathrm{dec}_\gamma(F)$ from below, establishing tightness) or with a formalization of Proposition 24 itself (removing this mission's hfhat hypothesis) are welcome.

Selected references

D. Foster, S. Kakade, J. Qian, and A. Rakhlin, The Statistical Complexity of Interactive Decision Making, arXiv:2112.13487, 2021. https://arxiv.org/abs/2112.13487
D. Foster and A. Rakhlin, Foundations of Reinforcement Learning and Interactive Decision Making, arXiv:2312.16730, 2023. https://arxiv.org/abs/2312.16730
T. Lattimore and C. Szepesvári, Bandit Algorithms, Cambridge University Press, 2020.
J. Kiefer and J. Wolfowitz, The Equivalence of Two Extremum Problems, Canadian Journal of Mathematics, 1960.

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Convex Optimization Linear Optimization Operations Research+2·Captain: mikedeng1

Introduction to Stochastic Programming I: Convexity, Attainment and Optimality of the Two-Stage Recourse ProblemTextbook

Motivation

Two-stage stochastic linear programming with recourse models a decision made before uncertainty resolves (the first-stage variables $x$ ) followed by a corrective decision made after (the second-stage, or recourse, variables $y$ ). Solving such a program means minimizing $c^{\mathsf T}x + Q(x)$ , where $Q(x)$ is the expected cost of the best recourse action given $x$ -- an object defined only implicitly, as the value of an embedded linear program that must be solved (or bounded) for every realization of the uncertain data. Before any algorithm for this problem can be justified -- the L-shaped method, stochastic decomposition, scenario decomposition, all developed in later chapters of Birge & Louveaux, Introduction to Stochastic Programming (Springer, 2011) -- one needs to know that $Q$ is well-behaved enough to optimize over at all: that the feasible region is closed and convex, that $Q$ itself is a finite, Lipschitz, convex function on it, that an optimal solution is actually attained rather than only approached in the limit, and finally what an optimality condition for the resulting nonsmooth convex program even looks like. This mission formalizes exactly that foundational layer, Chapter 3, Section 3.1 of the book.

Setting

Fix natural numbers $n_1, n_2, m_1, m_2$ and a finite scenario count $K$ . A two-stage recourse instance consists of first-stage data $A \in \mathbb{R}^{m_1 \times n_1}$ , $b \in \mathbb{R}^{m_1}$ , $c \in \mathbb{R}^{n_1}$ , a fixed recourse matrix $W \in \mathbb{R}^{m_2 \times n_2}$ , and, for each scenario $k = 1,\dots,K$ , a cost vector $q_k \in \mathbb{R}^{n_2}$ , a right-hand side $h_k \in \mathbb{R}^{m_2}$ , a technology matrix $T_k \in \mathbb{R}^{m_2 \times n_1}$ , and a probability $p_k \ge 0$ with $\sum_k p_k = 1$ (Eq. (1.1)). The first-stage feasible region is $K_1 = \{x \mid Ax = b,\ x \ge 0\}$ .

For a fixed $x$ and scenario $k$ , the second-stage value is

Q(x,\xi_k) = \min_{y}\{q_k^{\mathsf T}y \mid Wy = h_k - T_k x,\ y \ge 0\}

(Eq. (1.6)), taken as an extended real: $+\infty$ if no feasible $y$ exists, $-\infty$ if the inner program is unbounded below. The expected recourse value is $Q(x) = \sum_k p_k\, Q(x,\xi_k)$ (Eq. (1.3)), combined so that $+\infty + (-\infty) = +\infty$ -- the book's own convention (p. 109): infeasibility in one scenario is treated as fatal even if another scenario is unboundedly favorable. The second-stage feasibility set is $K_2 = \{x \mid Q(x) < \infty\}$ , and the deterministic-equivalent objective is $z(x) = c^{\mathsf T}x + Q(x)$ (Eq. (1.2)). For $x$ with $Q(x)$ finite, the subdifferential $\partial Q(x)$ is the set of $\eta$ satisfying $Q(x) + \eta^{\mathsf T}(y-x) \le Q(y)$ for every $y$ (p. 115).

A simple-recourse instance is the special case $W = [I,-I]$ : the recourse cost splits as $q = (q^+,q^-)$ , and $Q(x)$ decomposes componentwise via the closed form of Eq. (1.9)-(1.10) using the (left- and right-limit) distribution functions $F_i^-, F_i^+$ of each $h_i$ .

Formalization targets

Goal -- Chapter 3, Theorem 9 (p. 116)

x^* \in K_1 \text{ is optimal in (1.2)} \iff \exists\, \lambda^* \in \mathbb{R}^{m_1},\ \mu^* \in \mathbb{R}^{n_1}_{\ge 0},\ (\mu^*)^{\mathsf T}x^* = 0,\ \text{ s.t. } -c + A^{\mathsf T}\lambda^* + \mu^* \in \partial Q(x^*),

given that (1.2) has a finite optimal value. This is the KKT-style necessary and sufficient optimality condition for the two-stage recourse LP, and the weakest of the mission's targets in the sense that everything else supports it: convexity and finiteness of $Q$ (Theorem 6) are what make the left-to-right implication meaningful, closedness/convexity of $K_2$ (Theorem 5) makes the feasible region well-posed, and attainment (Theorem 8) is what makes " $x^*$ is optimal" a statement about a point that exists rather than an infimum that may not be reached.

Supporting milestones

Theorem 5(a) (p. 111): $K_2$ is closed and convex.
Theorem 6(a) (p. 112): $Q$ is finite on $K_2$ , and Lipschitzian and convex there.
Theorem 8 (p. 115): under boundedness of $K_1 \cap K_2$ or eventual linearity of $Q$ along recession directions, a finite optimal value is attained.
Corollary 10 (p. 116): Theorem 9 specialized to simple recourse, with $\partial Q(x^*)$ replaced by its explicit componentwise description.

Significance

Theorem 9 is the hinge on which the rest of the book's algorithmic chapters turn. The L-shaped method (Chapter 5) is a cutting-plane scheme whose cuts are literally elements of $\partial Q(x)$ ; stochastic decomposition and sampling-based methods use the same subdifferential structure with estimated cuts; the differentiable specialization (Eq. (1.14), $c + \nabla Q(x^*) = A^{\mathsf T}\lambda^* + \mu^*$ ) underlies nonlinear-programming approaches to the smooth case. None of this is meaningful without first knowing $Q$ is convex, finite where it needs to be, and that a minimizer exists to characterize. Formalizing this mission's four milestones from the actual definition of $Q$ as an embedded linear program's value -- rather than assuming these properties -- is exactly the content the book itself proves (or, for Theorem 6, explicitly cites to Wets [1972] and Kall [1976] rather than proving); this mission asks for genuine Lean proofs of Theorems 5, 8, 9 and Corollary 10 from the LP structure of $Q$ , and records Theorem 6 as a stated (not re-derived) input, matching the book's own presentation.

Difficulty

The obvious shortcut is to treat $Q$ as an opaque convex function and apply a textbook convex-KKT theorem off the shelf. This fails to capture what Theorem 9 actually is: a statement about the specific function $Q(x) = \sum_k p_k \min_y\{q_k^{\mathsf T}y \mid Wy = h_k - T_k x,\ y \ge 0\}$ , built from finitely many parametric linear programs, each of which can be infeasible ( $Q(x,\xi_k) = +\infty$ ) or unbounded ( $Q(x,\xi_k) = -\infty$ ) depending on $x$ . Convexity of $Q$ must come from convexity of the value function of a parametric LP in its right-hand side (the book's Theorem 2 argument: a convex combination of optimal solutions at two right-hand sides is feasible, hence suboptimal, at the combined right-hand side) -- not from an assumed hypothesis. Handling $\pm\infty$ correctly is a second, easy-to-miss source of error: the book fixes an explicit, non-default convention ( $+\infty$ dominates $-\infty$ ) for combining per-scenario values, the opposite of the convention Mathlib's own extended-real arithmetic uses, so any formalization that reaches for EReal's built-in addition to aggregate $Q$ silently states a different theorem. Theorem 8's attainment condition is a genuine existence result, not an automatic consequence of convexity: continuity alone does not give attainment on an unbounded feasible region, and the book's own counterexample (Eq. (1.11), a negative-exponential tail with infimum $0$ attained by no finite $x$ ) shows the boundedness/recession hypotheses are load-bearing.

Formalization scope

The scenario set is modeled as Fin K, a finite discrete random variable, matching Section 3.1b's development; under this model " $\xi$ has finite second moments" (the standing hypothesis of Theorems 4-11 in the general, possibly-continuous case) holds automatically and so does not appear as a separate hypothesis anywhere in this mission. Q(x,\xi_k) is defined as an EReal via sInf of the second-stage LP's feasible objective values -- sInf of the empty set is ⊤, and of a set unbounded below is ⊥ -- and is genuinely derived from that inner minimization rather than assumed convex; this rules out the chapter's trivializing formalization, which the paper-level triage explicitly warns against: taking Q(x) as an opaque convex-function hypothesis instead of deriving its properties from the inner LP's structure. Aggregating the $K$ per-scenario values into $Q(x)$ uses a bespoke bookAdd operation implementing the book's stated convention $+\infty+(-\infty)=+\infty$ , since Mathlib's EReal addition is defined with the opposite convention ( $\bot+\top=\top+\bot=\bot$ ). $\partial Q(x)$ is the ordinary subgradient-inequality set for this extended-real-valued function.

Theorem 8's condition (b) is stated with the book's own quantifier structure: the threshold $\bar\lambda$ and the recession value depend on the point $x$ and direction $v$ exactly as written, with no strengthening. Theorem 6(a)'s Lipschitz bound is stated, not derived -- the book itself cites it to Wets [1972] and Kall [1976] without proof -- so a faithful Lean proof of that milestone is expected to remain out of scope for this mission. Corollary 10 similarly takes the closed form of $\partial Q_i(x)$ from Eq. (1.10) as a hypothesis on an abstract $Q$ , matching how the book itself uses (1.10) as an already-established fact rather than re-deriving it from the second-stage LP in the corollary's own proof. Theorem 11's subdifferential-decomposition result ( $\partial Q(x) = E_\omega[\partial Q(x,\xi(\omega))] + N(K_2,x)$ ) is deliberately left out of this mission's scope: it is not needed by Theorem 9's own proof, and its normal-cone term would require relatively-complete-recourse machinery this mission does not otherwise need. No prior-art match was found on the platform: VectorSpaceOpt.fenchel_duality and the Luenberger-derived VectorSpaceOpt.generalized_kuhn_tucker / kkt_complementary_slackness family use a differentiable (Gateaux-derivative) or conjugate-function KKT model over general normed spaces, not this chapter's finite-dimensional, possibly-nondifferentiable subgradient formulation over the specific polyhedral set $K_1$ , so none is a faithful match and all items here are original drafts.

Selected references

J.R. Birge and F. Louveaux, Introduction to Stochastic Programming, 2nd ed., Springer Series in Operations Research and Financial Engineering, Springer, 2011. https://doi.org/10.1007/978-1-4614-0237-4
R.J-B. Wets, "Programming Under Uncertainty: The Equivalent Convex Program," SIAM Journal on Applied Mathematics 14 (1966), 89-105 (Lipschitz continuity of the recourse function, cited by the book as Wets [1972] for the closely related result used in Theorem 6). https://doi.org/10.1137/0114008
D.P. Walkup and R.J-B. Wets, "Stochastic Programs with Recourse," SIAM Journal on Applied Mathematics 15 (1967), 1299-1314 (finiteness of the recourse function and coincidence of the possibility and expectation feasibility sets, underlying Proposition 3 and Theorem 4). https://doi.org/10.1137/0115113

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Completed

Analysis·Captain: Lucas

Rudin PMA VII: Sequences and Series of FunctionsTextbook

Motivation

Chapter 7 of Walter Rudin's Principles of Mathematical Analysis (3rd edition, McGraw-Hill, 1976) asks when a limit of functions inherits the properties of its members. Pointwise convergence preserves almost nothing: Rudin's opening examples give continuous $f_n$ with discontinuous limit, and sequences where $\lim_n \int f_n \ne \int \lim_n f_n$ . Uniform convergence is the hypothesis that repairs this, and the chapter's second half asks the converse question — which functions arise as uniform limits from a given family — answered by the Stone–Weierstrass theorem (Theorem 7.32): an algebra of continuous real functions on a compact set that separates points and vanishes nowhere is uniformly dense in all continuous functions there.

The classical Weierstrass approximation theorem (7.26) — polynomials are dense in $C[a,b]$ — is the special case that made the general theorem worth proving, and is used in Chapter 8 for Fourier series and in Chapter 11 for the density of continuous functions in $L^2$ .

This mission is the seventh in a series formalizing Rudin Chapters 1–11; it uses the compactness results of Mission II, the continuity results of Mission IV and the Riemann–Stieltjes integral of Mission VI.

Setting

A sequence $f_n$ converges uniformly to $f$ on $E$ if for every $\varepsilon > 0$ there is $N$ with $|f_n(x) - f(x)| \le \varepsilon$ for all $n \ge N$ and all $x \in E$ — the same $N$ for every point. A family $\mathcal{F}$ is equicontinuous on $E$ if a single $\delta$ serves all its members in the definition of uniform continuity; it is pointwise bounded if each orbit $\{f(x) : f \in \mathcal{F}\}$ is bounded, and uniformly bounded if one bound works for all $f$ and all $x$ .

A set $\mathcal{A}$ of real functions is an algebra if it is closed under addition, multiplication and multiplication by real scalars; it separates points on $K$ if for $x \ne y$ in $K$ some $f \in \mathcal{A}$ has $f(x) \ne f(y)$ ; it vanishes at no point of $K$ if for each $x \in K$ some $f \in \mathcal{A}$ has $f(x) \ne 0$ . The uniform closure of $\mathcal{A}$ on $K$ is the set of uniform limits on $K$ of sequences from $\mathcal{A}$ .

Formalization targets

Goal — Stone–Weierstrass (Theorem 7.32)

Let $K$ be compact and let $\mathcal{A}$ be an algebra of real continuous functions on $K$ which separates points on $K$ and vanishes at no point of $K$ . Then

\overline{\mathcal{A}}^{\,\text{unif}} \supseteq C(K,\mathbb{R}) :

every continuous real function on $K$ is a uniform limit on $K$ of members of $\mathcal{A}$ .

Milestones

\text{uniform convergence} \iff \text{uniform Cauchy criterion} \qquad (7.8)

|f_n| \le M_n \text{ on } E,\ \textstyle\sum M_n < \infty \Rightarrow \sum f_n \text{ converges uniformly} \qquad (7.10)

\lim_{t\to x}\lim_n f_n(t) = \lim_n \lim_{t \to x} f_n(t) \text{ under uniform convergence} \qquad (7.11)

\text{a uniform limit of continuous functions is continuous} \qquad (7.12)

f_n \in \mathcal{R}(\alpha),\ f_n \to f \text{ uniformly} \Rightarrow f \in \mathcal{R}(\alpha),\ \int f_n \, d\alpha \to \int f \, d\alpha \qquad (7.16)

f_n' \to h \text{ uniformly},\ f_n(x_0) \text{ convergent} \Rightarrow f_n \to g \text{ uniformly},\ g' = h \qquad (7.17)

\text{there is a continuous nowhere differentiable } f : \mathbb{R} \to \mathbb{R} \qquad (7.18)

\text{pointwise bounded} + \text{equicontinuous on compact} \Rightarrow \text{uniformly bounded, convergent subsequence} \qquad (7.24,\ 7.25)

\text{polynomials are uniformly dense in } C[a,b] \qquad (7.26)

Significance

Uniform convergence is the standard hypothesis under which limits commute with continuity, integration and (with an extra condition) differentiation, and Theorems 7.11, 7.12, 7.16 and 7.17 are used throughout the rest of the book; Chapter 8 in particular builds the exponential, trigonometric and Gamma functions as uniform limits and differentiates them term by term on the strength of 7.17. Theorem 7.18 shows how weak pointwise differentiability is as a consequence of continuity: a uniform limit of piecewise-linear functions can fail to be differentiable anywhere. Arzelà–Ascoli is the compactness criterion for families of functions, and it is the standard route to existence theorems for differential and integral equations.

Stone–Weierstrass is the structural theorem of the chapter: it replaces the combinatorial Bernstein-polynomial proof of Weierstrass's theorem with a statement about algebras of functions, applicable to trigonometric polynomials, polynomials in several variables, and Lipschitz algebras alike.

Mathlib contains a Stone–Weierstrass theorem for subalgebras of C(X, ℝ) on compact Hausdorff spaces, and a version of Arzelà–Ascoli. This mission states the results in Rudin's terms — plain sets of functions on a compact subset $K$ of a metric space, uniform closure defined by sequences — so that they can be used together with the Riemann–Stieltjes integral built in Mission VI, which is not part of the library.

Difficulty

Stone–Weierstrass is the one theorem in this mission whose proof is genuinely structural: from the algebra one first produces $|f|$ as a uniform limit of polynomials in $f$ (which needs the polynomial approximation of $\sqrt{t}$ on $[0,1]$ and so cannot be circular with Theorem 7.26), then maxima and minima of pairs, then functions matching prescribed values at two points, and only then the local-to-global patching over a finite subcover. Each step is short; keeping the uniform closure a lattice and an algebra simultaneously is the bookkeeping burden.

Two hypotheses are easy to lose and both are necessary: an algebra that vanishes at a point cannot approximate functions that do not, and one that fails to separate two points cannot approximate functions that distinguish them.

Formalization scope

Conventions fixed by this mission:

Uniform convergence is Mathlib's TendstoUniformlyOn … atTop; complex-valued sequences are used where Rudin allows complex values.
Algebras, separation, non-vanishing and uniform closure are the predicates Rudin.IsFunctionAlgebra, Rudin.SeparatesPointsOn, Rudin.VanishesAtNoPointOn, Rudin.UniformClosureOn, defined for sets of functions X → ℝ and a compact subset K. The goal's conclusion is membership in the uniform closure, i.e. the existence of an approximating sequence from the algebra.
Equicontinuity and the two boundedness notions are Rudin.EquicontinuousOn, Rudin.PointwiseBoundedOn, Rudin.UniformlyBoundedOn, stated with explicit $\varepsilon$ and $\delta$ as in Definitions 7.19 and 7.22.
Theorem 7.16 is stated for the Riemann–Stieltjes integral of Mission VI, not for a Mathlib integral, so the two missions compose.
Theorem 7.26 is stated for complex-valued $f$ and polynomials with complex coefficients evaluated at real points, as in Rudin.

Selected references

Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976, Chapter 7 (pp. 143–171).
M. H. Stone, The generalized Weierstrass approximation theorem, Mathematics Magazine 21 (1948), 167–184 and 237–254. https://doi.org/10.2307/3029750

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