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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, thealgorithm 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 computationalcomplexity ${\cal O}(|{\cal X}|)$ which is typically exponential in $d$, andcannot be used in large dimensions. We propose the first algorithm which isboth computationally and statistically efficient for this problem with regret$R(T) = {\cal O} \Big({d (\ln m)^2 (\ln T)\over \Delta_{\min} }\Big)$ andcomputational complexity ${\cal O}(T {\bf poly}(d))$. Our approach involvescarefully designing an approximate version of ESCB with the same regretguarantees, 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 thismaximization problems efficiently.

Statistically Efficient, Polynomial Time Algorithms for Combinatorial Semi Bandits