We introduce a computationally efficient algorithm for zeroth-order bandit convex optimisation and prove that in the adversarial setting its regret is at most $d^{3.5} \sqrt{n} \mathrm{polylog}(n, d)$ with high probability where $d$ is the dimension and $n$ is the time horizon. In the stochastic setting the bound improves to $M d^{2} \sqrt{n} \mathrm{polylog}(n, d)$ where $M \in [d^{-1/2}, d^{-1 / 4}]$ is a constant that depends on the geometry of the constraint set and the desired computational properties.

翻译：我们提出了一种计算高效的零阶老虎机凸优化算法，并证明在对抗性设置下，其高概率遗憾上界为 $d^{3.5} \sqrt{n} \mathrm{polylog}(n, d)$，其中 $d$ 为维度，$n$ 为时间范围。在随机性设置中，该上界改进为 $M d^{2} \sqrt{n} \mathrm{polylog}(n, d)$，其中 $M \in [d^{-1/2}, d^{-1 / 4}]$ 是一个取决于约束集几何性质与期望计算特性的常数。