We analyse adversarial bandit convex optimisation with an adversary that is restricted to playing functions of the form $f(x) = g(\langle x, \theta\rangle)$ for convex $g : \mathbb R \to \mathbb R$ and $\theta \in \mathbb R^d$. We provide a short information-theoretic proof that the minimax regret is at most $O(d\sqrt{n} \log(\operatorname{diam}\mathcal K))$ where $n$ is the number of interactions, $d$ the dimension and $\operatorname{diam}(\mathcal K)$ is the diameter of the constraint set. Hence, this class of functions is at most logarithmically harder than the linear case.

翻译：我们用一个对手来分析对抗性土匪的优化, 该对手仅限于为 convex $g:\mathbb R\to\mathbb R$ 和 $\theta\ in\mathbb R ⁇ d$。 我们提供了简短的信息- 理论证明, 迷你马克思的悔恨最多为 $O( d\ sqrt{n}\log( operatorname{ diam\ mathcal K) $, 其中, $n是互动的数量, $d$ 维度和 $\ opatorname{diam} (\mathcal K) 是约束设置的直径。 因此, 这个函数的类别比线性案例要困难得多 。