## AbstractIn many applications a key step is estimating some unknown quantity X from a sequence of trials, each having expected value X. Optimal algorithms are known when the task is to estimate X within a multiplicative factor of e, for an e<1 given in advance. In this paper we consider <i>anytime</i> approximation algorithms, i.e., algorithms that must give a reliable approximation after each trial, and whose approximations have to be increasingly accurate as the number of trials grows. We give an anytime algorithm for this task when the only a-priori known property of X is its range, and show that it is asymptotically optimal in some cases, in the sense that no correct anytime algorithm can give asymptotically better approximations. The key ingredient is a new large deviation bound for the supremum of the deviations in an infinite sequence of trials, which can be seen as a non-limit analog of the classical Law of the Iterated Logarithm.
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