Linear bandits with polylogarithmic minimax regret

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 log³(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 λₘᵢₙ ( Vₜ ) = Ω (√λₘₐₓ(Vₜ ) ) for the design matrix Vₜ 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(1t), leading to the logarithmic scaling of the cumulative regret.