  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.
