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：我们考虑对抗性线性上下文赌博机设置，允许与K个臂各自相关的损失函数随时间无限制变化。假设d维上下文来自固定的已知分布，在T轮博弈中最坏情况下的期望遗憾度量为$\tilde O(\sqrt{Kd T})$。在额外假设上下文密度为对数凹函数的条件下，我们获得了关于学习者损失累积二阶矩$V_T$的量级为$\tilde O(K\sqrt{d V_T})$的二阶界限，以及关于最优策略累积损失$L_T^*$的量级为$\tilde O(K\sqrt{d L_T^*})$的紧密相关的一阶界限。由于$V_T$或$L_T^*$可能显著小于T，因此当环境相对温和时，这些界限优于最坏情况下的遗憾值。我们的结果通过使用概率单纯形上的截断连续指数加权算法获得，并通过利用该算法与无上下文线性赌博机设置的新颖联系进行分析。