258 research outputs found
The Simulator: Understanding Adaptive Sampling in the Moderate-Confidence Regime
We propose a novel technique for analyzing adaptive sampling called the {\em
Simulator}. Our approach differs from the existing methods by considering not
how much information could be gathered by any fixed sampling strategy, but how
difficult it is to distinguish a good sampling strategy from a bad one given
the limited amount of data collected up to any given time. This change of
perspective allows us to match the strength of both Fano and change-of-measure
techniques, without succumbing to the limitations of either method. For
concreteness, we apply our techniques to a structured multi-arm bandit problem
in the fixed-confidence pure exploration setting, where we show that the
constraints on the means imply a substantial gap between the
moderate-confidence sample complexity, and the asymptotic sample complexity as
found in the literature. We also prove the first instance-based
lower bounds for the top-k problem which incorporate the appropriate
log-factors. Moreover, our lower bounds zero-in on the number of times each
\emph{individual} arm needs to be pulled, uncovering new phenomena which are
drowned out in the aggregate sample complexity. Our new analysis inspires a
simple and near-optimal algorithm for the best-arm and top-k identification,
the first {\em practical} algorithm of its kind for the latter problem which
removes extraneous log factors, and outperforms the state-of-the-art in
experiments
Query Complexity of Derivative-Free Optimization
This paper provides lower bounds on the convergence rate of Derivative Free
Optimization (DFO) with noisy function evaluations, exposing a fundamental and
unavoidable gap between the performance of algorithms with access to gradients
and those with access to only function evaluations. However, there are
situations in which DFO is unavoidable, and for such situations we propose a
new DFO algorithm that is proved to be near optimal for the class of strongly
convex objective functions. A distinctive feature of the algorithm is that it
uses only Boolean-valued function comparisons, rather than function
evaluations. This makes the algorithm useful in an even wider range of
applications, such as optimization based on paired comparisons from human
subjects, for example. We also show that regardless of whether DFO is based on
noisy function evaluations or Boolean-valued function comparisons, the
convergence rate is the same
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