Statistically Efficient, Polynomial Time Algorithms for Combinatorial Semi Bandits

• URL: http://arxiv.org/abs/2002.07258v2
• Date: Wed, 13 Jan 2021 17:12:58 GMT
• Title: Statistically Efficient, Polynomial Time Algorithms for Combinatorial Semi Bandits
• Authors: Thibaut Cuvelier and Richard Combes and Eric Gourdin
• Abstract summary: We consider semi-bandits over a set of arms $cal X subset 0,1d$ where rewards are uncorrelated across items. For this problem, the algorithm ESCB yields the smallest known regret bound $R(T) = cal OBig( d (ln m)2 (ln T) over Delta_min Big)$, but it has computational complexity $cal O(|cal X|)$ which is typically exponential in $d$, and cannot
• Score: 3.9919781077062497
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider combinatorial semi-bandits over a set of arms ${\cal X} \subset \{0,1\}^d$ where rewards are uncorrelated across items. For this problem, the algorithm ESCB yields the smallest known regret bound $R(T) = {\cal O}\Big( {d (\ln m)^2 (\ln T) \over \Delta_{\min} }\Big)$, but it has computational complexity ${\cal O}(|{\cal X}|)$ which is typically exponential in $d$, and cannot be used in large dimensions. We propose the first algorithm which is both computationally and statistically efficient for this problem with regret $R(T) = {\cal O} \Big({d (\ln m)^2 (\ln T)\over \Delta_{\min} }\Big)$ and computational complexity ${\cal O}(T {\bf poly}(d))$. Our approach involves carefully designing an approximate version of ESCB with the same regret guarantees, showing that this approximate algorithm can be implemented in time ${\cal O}(T {\bf poly}(d))$ by repeatedly maximizing a linear function over ${\cal X}$ subject to a linear budget constraint, and showing how to solve this maximization problems efficiently.

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