Importance Weighting Without Importance Weights: An Efficient Algorithm for Combinatorial Semi-Bandits

Gergely Neu, Gábor Bartók; 17(154):1−21, 2016.

Abstract

We propose a sample-efficient alternative for importance weighting for situations where one only has sample access to the probability distribution that generates the observations. Our new method, called Geometric Resampling (GR), is described and analyzed in the context of online combinatorial optimization under semi-bandit feedback, where a learner sequentially selects its actions from a combinatorial decision set so as to minimize its cumulative loss. In particular, we show that the well-known Follow-the-Perturbed-Leader (FPL) prediction method coupled with Geometric Resampling yields the first computationally efficient reduction from offline to online optimization in this setting. We provide a thorough theoretical analysis for the resulting algorithm, showing that its performance is on par with previous, inefficient solutions. Our main contribution is showing that, despite the relatively large variance induced by the GR procedure, our performance guarantees hold with high probability rather than only in expectation. As a side result, we also improve the best known regret bounds for FPL in online combinatorial optimization with full feedback, closing the perceived performance gap between FPL and exponential weights in this setting. (A preliminary version of this paper was published as Neu and Bartók (2013). Parts of this work were completed while Gergely Neu was with the SequeL team at INRIA Lille -- Nord Europe, France and Gábor Bartók was with the Department of Computer Science at ETH Zürich.)

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