We consider the adversarial linear contextual bandit setting, which allows for the loss functions associated with each of $K$ arms to change over time without restriction. Assuming the $d$-dimensional contexts are drawn from a fixed known distribution, the worst-case expected regret over the course of $T$ rounds is known to scale as $\tilde O(\sqrt{Kd T})$. Under the additional assumption that the density of the contexts is log-concave, we obtain a second-order bound of order $\tilde O(K\sqrt{d V_T})$ in terms of the cumulative second moment of the learner's losses $V_T$, and a closely related first-order bound of order $\tilde O(K\sqrt{d L_T^*})$ in terms of the cumulative loss of the best policy $L_T^*$. Since $V_T$ or $L_T^*$ may be significantly smaller than $T$, these improve over the worst-case regret whenever the environment is relatively benign. Our results are obtained using a truncated version of the continuous exponential weights algorithm over the probability simplex, which we analyse by exploiting a novel connection to the linear bandit setting without contexts.

翻译：考虑对抗性线性上下文赌博机设定，允许与$K$个臂相关的损失函数随时间无限制变化。假设$d$维上下文来自已知固定分布，$T$轮中最坏情况下期望遗憾的阶为$\tilde O(\sqrt{Kd T})$。在额外假设上下文密度为对数凹的条件下，我们得到关于学习者损失累积二阶矩$V_T$的$\tilde O(K\sqrt{d V_T})$阶二阶界限，以及关于最优策略累积损失$L_T^*$的$\tilde O(K\sqrt{d L_T^*})$阶一阶界限。由于$V_T$或$L_T^*$可能显著小于$T$，当环境相对温和时，这些界限优于最坏情况遗憾。我们的结果通过概率单纯形上的截断连续指数权重算法获得，并利用与无上下文线性赌博机的新联系进行分析。