Linear bandits with polylogarithmic minimax regret
This work addresses a specific noise model in bandit optimization, offering a significant improvement in regret bounds for scenarios with vanishing noise, though it is incremental in the broader context of bandit algorithms.
The paper tackles the problem of linear stochastic bandits with a noise model where the subgaussian noise parameter decreases linearly as actions approach the unknown vector, achieving a minimax regret scaling as log^3(T) in time horizon T, in contrast to the typical square root scaling.
We study a noise model for linear stochastic bandits for which the subgaussian noise parameter vanishes linearly as we select actions on the unit sphere closer and closer to the unknown vector. We introduce an algorithm for this problem that exhibits a minimax regret scaling as $\log^3(T)$ in the time horizon $T$, in stark contrast the square root scaling of this regret for typical bandit algorithms. Our strategy, based on weighted least-squares estimation, achieves the eigenvalue relation $λ_{\min} ( V_t ) = Ω(\sqrt{λ_{\max}(V_t ) })$ for the design matrix $V_t$ at each time step $t$ through geometrical arguments that are independent of the noise model and might be of independent interest. This allows us to tightly control the expected regret in each time step to be of the order $O(\frac1{t})$, leading to the logarithmic scaling of the cumulative regret.