Towards Efficient and Optimal Covariance-Adaptive Algorithms for Combinatorial Semi-Bandits
This addresses the need for efficient and optimal algorithms in combinatorial decision-making problems, particularly in scenarios where semi-bandit feedback outperforms bandit feedback, representing a domain-specific incremental improvement.
The paper tackles the problem of stochastic combinatorial semi-bandits by designing covariance-adaptive algorithms (OLS-UCB-C and COS-V) that leverage online estimations of covariance structure to achieve improved gap-free regret, with COS-V being the first sampling-based algorithm to achieve T^1/2 gap-free regret (up to poly-logs).
We address the problem of stochastic combinatorial semi-bandits, where a player selects among P actions from the power set of a set containing d base items. Adaptivity to the problem's structure is essential in order to obtain optimal regret upper bounds. As estimating the coefficients of a covariance matrix can be manageable in practice, leveraging them should improve the regret. We design "optimistic" covariance-adaptive algorithms relying on online estimations of the covariance structure, called OLS-UCB-C and COS-V (only the variances for the latter). They both yields improved gap-free regret. Although COS-V can be slightly suboptimal, it improves on computational complexity by taking inspiration from ThompsonSampling approaches. It is the first sampling-based algorithm satisfying a T^1/2 gap-free regret (up to poly-logs). We also show that in some cases, our approach efficiently leverages the semi-bandit feedback and outperforms bandit feedback approaches, not only in exponential regimes where P >> d but also when P <= d, which is not covered by existing analyses.