In an online decision problem, one makes a sequence of decisions without knowledge of the future. Each period, one pays a cost based on the decision and observed state. We give a simple approach for doing nearly as well as the best single decision, where the best is chosen with the benefit of hindsight. A natural idea is to follow the leader, i.e. each period choose the decision which has done best so far. We show that by slightly perturbing the totals and then choosing the best decision, the expected performance is nearly as good as the best decision in hindsight. Our approach, which is very much like Hannan's original game-theoretic approach from the 1950s, yields guarantees competitive with the more modern exponential weighting algorithms like Weighted Majority.More importantly, these follow-the-leader style algorithms extend naturally to a large class of structured online problems for which the exponential algorithms are inefficient.
We describe a slightly sub-exponential time algorithm for learning parity functions in the presence of random classification noise. This results in a polynomial-time algorithm for the case of parity functions that depend on only the first O(log n log log n) bits of input. This is the first known instance of an efficient noise-tolerant algorithm for a concept class that is provably not learnable in the Statistical Query model of Kearns [7]. Thus, we demonstrate that the set of problems learnable in the statistical query model is a strict subset of those problems learnable in the presence of noise in the PAC model. In coding-theory terms, what we give is a poly(n)-time algorithm for decoding linear k × n codes in the presence of random noise for the case of k = clog n log log n for some c > 0. (The case of k ---O(log n) is trivial since one can just individually check each of the 2 k possible messages and choose the one that yields the closest codeword.) A natural extension of the statistical query model is to allow queries about statistical properties that involve t-tuples of examples (as opposed to single examples). The second result of this paper is to show that any class of functions learnable (strongly or weakly) with t-wise queries for t = O(log n) is also weakly learnable with standard unary queries. Hence this natural extension to the statistical query model does not increase the set of weakly learnable functions.
We give the first algorithm that (under distributional assumptions) efficiently learns halfspaces in the notoriously difficult agnostic framework of Kearns, Schapire, & Sellie, where a learner is given access to labeled examples drawn from a distribution, without restriction on the labels (e.g. adversarial noise). The algorithm constructs a hypothesis whose error rate on future examples is within an additive ǫ of the optimal halfspace, in time poly(n) for any constant ǫ > 0, under the uniform distribution over {−1, 1} n or the unit sphere in R n , as well as under any log-concave distribution over R n. It also agnostically learns Boolean disjunctions in time 2 ˜ O(√ n) with respect to any distribution. The new algorithm, essentially L 1 polynomial regression, is a noise-tolerant arbitrary-distribution generalization of the "low-degree" Fourier algorithm of Linial, Mansour, & Nisan. We also give a new algorithm for PAC learning halfspaces under the uniform distribution on the unit sphere with the current best bounds on tolerable rate of "malicious noise."
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