An Optimal Anytime Estimator Algorithm
Departament de Llenguatges i Sistemes Informatics, UPC, Barcelona, Spain.
In 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.