The TCS Formalization Drive

Every guarantee in the preceding chapters assumes a fixed distribution and i.i.d. sampling. On-line learning drops both assumptions: an algorithm processes one example at a time, in an adversarial (worst-case) sequence, and is judged by regret against the best fixed comparator in hindsight rather than by generalization error. This chapter develops the…

Weak learning — a base classifier only slightly better than random guessing — is easy to come by; strong learning, in the PAC sense of Chapter 2, is not. Boosting is the technique that turns the first into the second: combine many weak classifiers, each trained on a reweighted version of the sample that emphasizes previously misclassified points, into a…

A matching cut is a vertex bipartition whose crossing edges form a matching. The property was introduced under the name decomposability and has links to graph algorithms, stable cutsets in line graphs, and several graph-labeling problems. An Open Problem Garden question asked whether sufficiently large girth forces a matching cut once average degree is…

Randomness appears to enlarge efficient computation, but the Sipser–Gács–Lautemann theorem places every bounded-error probabilistic polynomial-time language at the second level of the polynomial hierarchy, giving one of complexity theory’s foundational limits on the power of randomization.

The pancake problem was posed in 1975: a stack of $n$ pancakes of distinct sizes is to be sorted, smallest on top, by repeatedly inserting a spatula and flipping the pancakes above it. Mathematically the stack is a permutation of $\llbracket 1;n\rrbracket$ and a move is a prefix reversal. Two questions arise: the diameter $f(n)$ of the resulting pancake network (t…

Many two-party quantum protocols, such as oblivious transfer and password-based identification, start the same way: Alice sends Bob $n$ qubits $|x\rangle_\theta$ encoding random bits $x$ in random BB84 bases $\theta\in\{+,\times\}^n$. Their security proofs work only against a restricted Bob, for example one with bounded quantum memory (Damgård, Fehr, Salvail,…

Certify $\alpha\ge2.006470828$ by specializing OpenAI's general seed theorem to $d=20$ and $M=9^{21}$.

Many quantum algorithms end in a measurement that succeeds only with some probability $a$. Two questions follow: how to raise that probability, and how to learn its value. G. Brassard, P. Høyer, M. Mosca and A. Tapp answered both in Quantum Amplitude Amplification and Estimation (arXiv:quant-ph/0005055, 2000; Contemporary Mathematics 305, 2002). Amplitude…

In stochastic matching a platform knows, for every potential pair, the probability that the pair would succeed, but learns whether it does only by trying it, and a successful trial is binding. Kidney exchange, online dating and labour markets are the standard examples: a crossmatch test, a date or an interview is a probe whose outcome is random, and a…

A firm sells several products, each in limited supply. Customers arrive one at a time, and each product can be offered at one of a few prices. This is the setting of airline fare classes, hotel rate plans and e-commerce discounting. Because future demand is unknown, the firm must decide online whether to sell a unit now at a low price or keep it for a…

Online matching and its revenue-management variants model a firm that allocates limited inventory to customers who arrive one at a time, without knowing who comes next. Classical results in this area assume each item has a single price, or weight. For a single price and large inventories, the BALANCE algorithm of Kalyanasundaram and Pruhs (2000) and its…

Online scheduling asks for decisions before the full job set is known. In the online-list model, jobs are revealed one at a time and each must be assigned to a machine before the next job is revealed; the scheduler may choose the processing order after the list is complete. When processing durations are random and machines are unrelated, the same job may…

In multi-agent scheduling, several decision makers share one processing resource, and each one cares only about the completion of its own jobs. Agnetis, Mirchandani, Pacciarelli and Pacifici (Oper. Res. 52(2), 2004) set up the two-agent version of this problem and classified its complexity for most combinations of classical objectives. Typical settings are…

Differential privacy (Dwork, McSherry, Nissim and Smith, 2006) requires a randomized algorithm's output distribution to change little when one individual's record is changed. PAC learning asks an algorithm to output, from i.i.d. labeled examples, a hypothesis whose error is small with high probability. Kasiviswanathan, Lee, Nissim, Raskhodnikova and Smith (a…

Split Euclidean space into finitely many measurable cells and record, for each cell, its Gaussian first moment, the integral of $x$ against the standard Gaussian measure over that cell. How large can the sum of the squared lengths of these vectors be? The propeller conjecture predicts that, with at least three cells in dimension at least two, the answer is …

Many learning tasks run on data about people: medical records, purchase histories, survey answers. A learner whose output reveals too much about any one training example is unusable on such data, however accurate it is. Differential privacy (Dwork, McSherry, Nissim and Smith, 2006) makes "reveals too much" precise: changing one individual's record may…

Automatic program analyses for heap-manipulating code have to infer, for each procedure, a precondition describing the memory the procedure needs. Calcagno, Distefano, O'Hearn and Yang (J. ACM 2011) do this with bi-abduction: at a call site the analysis asks for an anti-frame $M$, the missing memory, and a frame, the memory left untouched, so that the current…

Biometric readings, typed passphrases and physical measurements are secrets that are never reproduced exactly: a second reading $w'$ of the same fingerprint differs from the enrolled reading $w$ in a few positions. Dodis, Ostrovsky, Reyzin and Smith (arXiv:cs/0602007, SIAM J. Comput. 38(1), 2008) introduced the secure sketch, public helper data that lets anyone…

In the on-line prediction model studied by Littlestone and Warmuth (Inform. and Comput. 108, 1994), a learner receives instances one at a time, predicts a binary label for each, and is then told the correct label. No probabilistic assumption is made: performance is measured by the number of mistakes on a worst-case sequence. When every label is given by…

Cryptographic keys must be uniformly random and reproducible exactly. Many secrets people actually possess are neither: a biometric reading (an iris code, a fingerprint template), a long passphrase recalled with typos, or the correlated bits two parties obtain from a quantum channel are noisy, so two readings of the same secret differ in a few positions,…

Grover's algorithm (Grover 1996) finds a marked item among $N$ with $O(\sqrt N)$ queries, a quadratic speed-up over any classical algorithm. Whether quantum computers can do substantially better on search, and so solve NP-type problems by brute force in polynomial time, was settled in the negative by Bennett, Bernstein, Brassard and Vazirani (arXiv:quant-ph/9701…