Top-k Combinatorial Bandits with Full-Bandit Feedback
This addresses a specific challenge in bandit algorithms for combinatorial settings where feedback is limited, though it appears incremental as it generalizes existing methods.
The paper tackles the problem of top-k combinatorial bandits with full-bandit feedback, where only the sum of rewards is observed, by proposing the CSAR algorithm with an efficient sampling scheme using Hadamard matrices to estimate individual arm rewards. The algorithm outperforms other methods in experiments and achieves tight sample complexity bounds for small k.
Top-k Combinatorial Bandits generalize multi-armed bandits, where at each round any subset of $k$ out of $n$ arms may be chosen and the sum of the rewards is gained. We address the full-bandit feedback, in which the agent observes only the sum of rewards, in contrast to the semi-bandit feedback, in which the agent observes also the individual arms' rewards. We present the Combinatorial Successive Accepts and Rejects (CSAR) algorithm, which generalizes SAR (Bubeck et al, 2013) for top-k combinatorial bandits. Our main contribution is an efficient sampling scheme that uses Hadamard matrices in order to estimate accurately the individual arms' expected rewards. We discuss two variants of the algorithm, the first minimizes the sample complexity and the second minimizes the regret. We also prove a lower bound on sample complexity, which is tight for $k=O(1)$. Finally, we run experiments and show that our algorithm outperforms other methods.