Minimax Regret for Cascading Bandits

Cascading bandits model the task of learning to rank $K$ out of $L$ items over $n$ rounds of partial feedback. For this model, the minimax (i.e., gap-free) regret is poorly understood; in particular, the best known lower and upper bounds are $\Omega(\sqrt{nL/K})$ and $\tilde{O}(\sqrt{nLK})$, respectively. We improve the lower bound to $\Omega(\sqrt{nL})$ and show CascadeKL-UCB (which ranks items by their KL-UCB indices) attains it up to log terms. Surprisingly, we also show CascadeUCB1 (which ranks via UCB1) can suffer suboptimal $\Omega(\sqrt{nLK})$ regret. This sharply contrasts with standard $L$-armed bandits, where the corresponding algorithms both achieve the minimax regret $\sqrt{nL}$ (up to log terms), and the main advantage of KL-UCB is only to improve constants in the gap-dependent bounds. In essence, this contrast occurs because Pinsker's inequality is tight for hard problems in the $L$-armed case but loose (by a factor of $K$) in the cascading case.