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 $\lambda_{\min} ( V_t ) = \Omega
(\sqrt{\lambda_{\max}(V_t ) })$ 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.

研究线性随机赌博机的噪声模型，介绍一种基于加权最小二乘估计的算法，能够最小化后悔度，通过几何论证独立于噪声模型，能够紧密控制每个时间步骤的期望后悔度为 O (1/t)，从而导致了累积后悔度的对数缩放。