Variance-Aware Sparse Linear Bandits

Understand

It is well-known that for sparse linear bandits, when ignoring the dependency on sparsity which is much smaller than the ambient dimension, the worst-case minimax regret is $\widetilde{\Theta}\left(\sqrt{dT}\right)$ where $d$ is the ambient dimension and $T$ is the number of rounds.

• On the other hand, in the benign setting where there is no noise and the action set is the unit sphere, one can use divide-and-conquer to achieve $\widetilde{\mathcal O}(1)$ regret, which is (nearly) independent of $d$ and $T$.
• In this paper, we present the first variance-aware regret guarantee for sparse linear bandits: $\widetilde{\mathcal O}\left(\sqrt{d\sum_{t=1}^T \sigma_t^2} + 1\right)$, where $\sigma_t^2$ is the variance of the noise at the $t$-th round.
• This bound naturally interpolates the regret bounds for the worst-case constant-variance regime (i.e., $\sigma_t \equiv \Omega(1)$) and the benign deterministic regimes (i.e., $\sigma_t \equiv 0$).