  Cascading bandits is a natural and popular model that frames the task of
learning to rank from Bernoulli click feedback in a bandit setting. For the
case of unstructured rewards, we prove matching upper and lower bounds for the
problem-independent (i.e., gap-free) regret, both of which strictly improve the
best known. A key observation is that the hard instances of this problem are
those with small mean rewards, i.e., the small click-through rates that are
most relevant in practice. Based on this, and the fact that small mean implies
small variance for Bernoullis, our key technical result shows that
variance-aware confidence sets derived from the Bernstein and Chernoff bounds
lead to optimal algorithms (up to log terms), whereas Hoeffding-based
algorithms suffer order-wise suboptimal regret. This sharply contrasts with the
standard (non-cascading) bandit setting, where the variance-aware algorithms
only improve constants. In light of this and as an additional contribution, we
propose a variance-aware algorithm for the structured case of linear rewards
and show its regret strictly improves the state-of-the-art.
