Statistical Efficiency of Thompson Sampling for Combinatorial Semi-Bandits
This addresses a theoretical gap in bandit algorithms for researchers, providing efficient solutions with proven regret bounds, though it is incremental as it builds on existing Thompson Sampling methods.
The paper tackles the open problem of achieving optimal asymptotic regret in combinatorial semi-bandits for families like mutually independent and multivariate sub-Gaussian outcomes, by proposing tight analyses of Combinatorial Thompson Sampling variants, offering a computationally efficient alternative to non-efficient optimal policies.
We investigate stochastic combinatorial multi-armed bandit with semi-bandit feedback (CMAB). In CMAB, the question of the existence of an efficient policy with an optimal asymptotic regret (up to a factor poly-logarithmic with the action size) is still open for many families of distributions, including mutually independent outcomes, and more generally the multivariate sub-Gaussian family. We propose to answer the above question for these two families by analyzing variants of the Combinatorial Thompson Sampling policy (CTS). For mutually independent outcomes in $[0,1]$, we propose a tight analysis of CTS using Beta priors. We then look at the more general setting of multivariate sub-Gaussian outcomes and propose a tight analysis of CTS using Gaussian priors. This last result gives us an alternative to the Efficient Sampling for Combinatorial Bandit policy (ESCB), which, although optimal, is not computationally efficient.