  Motivated by the pressing need for efficient optimization in online
recommender systems, we revisit the cascading bandit model proposed by Kveton
et al. (2015). While Thompson sampling (TS) algorithms have been shown to be
empirically superior to Upper Confidence Bound (UCB) algorithms for cascading
bandits, theoretical guarantees are only known for the latter. In this paper,
we first provide a problem-dependent upper bound on the regret of a TS
algorithm with Beta-Bernoulli updates; this upper bound is tighter than a
recent derivation under a more general setting by Huyuk and Tekin (2019). Next,
we design and analyze another TS algorithm with Gaussian updates, TS-Cascade.
TS-Cascade achieves the state-of-the-art regret bound for cascading bandits.
Complementarily, we consider a linear generalization of the cascading bandit
model, which allows efficient learning in large cascading bandit problem
instances. We introduce and analyze a TS algorithm, which enjoys a regret bound
that depends on the dimension of the linear model but not the number of items.
Finally, by using information-theoretic techniques and judiciously constructing
cascading bandit instances, we derive a nearly matching regret lower bound for
the standard model. Our paper establishes the first theoretical guarantees on
TS algorithms for stochastic combinatorial bandit problem model with partial
feedback. Numerical experiments demonstrate the superiority of the proposed TS
algorithms compared to existing UCB-based ones.
