  Influence maximization, adaptive routing, and dynamic spectrum allocation all
require choosing the right action from a large set of alternatives. Thanks to
the advances in combinatorial optimization, these and many similar problems can
be efficiently solved given an environment with known stochasticity. In this
paper, we take this one step further and focus on combinatorial optimization in
unknown environments. We consider a very general learning framework called
combinatorial multi-armed bandit with probabilistically triggered arms and a
very powerful Bayesian algorithm called Combinatorial Thompson Sampling (CTS).
Under the semi-bandit feedback model and assuming access to an oracle without
knowing the expected base arm outcomes beforehand, we show that when the
expected reward is Lipschitz continuous in the expected base arm outcomes CTS
achieves $O(\sum_{i =1}^m\log T/(p_i\Delta_i))$ regret and
$O(\max\{\mathbb{E}[m\sqrt{T\log T/p^*}],\mathbb{E}[m^2/p^*]\})$ Bayesian
regret, where $m$ denotes the number of base arms, $p_i$ and $\Delta_i$ denote
the minimum non-zero triggering probability and the minimum suboptimality gap
of base arm $i$ respectively, $T$ denotes the time horizon, and $p^*$ denotes
the overall minimum non-zero triggering probability. We also show that when the
expected reward satisfies the triggering probability modulated Lipschitz
continuity, CTS achieves $O(\max\{m\sqrt{T\log T},m^2\})$ Bayesian regret, and
when triggering probabilities are non-zero for all base arms, CTS achieves
$O(1/p^*\log(1/p^*))$ regret independent of the time horizon. Finally, we
numerically compare CTS with algorithms based on upper confidence bounds in
several networking problems and show that CTS outperforms these algorithms by
at least an order of magnitude in majority of the cases.
