This work studies online zero-order optimization of convex and Lipschitz functions. We present a novel gradient estimator based on two function evaluation and randomization on the $\ell_1$-sphere. Considering different geometries of feasible sets and Lipschitz assumptions we analyse online mirror descent algorithm with our estimator in place of the usual gradient. We consider two types of assumptions on the noise of the zero-order oracle: canceling noise and adversarial noise. We provide an anytime and completely data-driven algorithm, which is adaptive to all parameters of the problem. In the case of canceling noise that was previously studied in the literature, our guarantees are either comparable or better than state-of-the-art bounds obtained by~\citet{duchi2015} and \citet{Shamir17} for non-adaptive algorithms. Our analysis is based on deriving a new Poincar\'e type inequality for the uniform measure on the $\ell_1$-sphere with explicit constants, which may be of independent interest.
翻译:这项工作研究在网上对 convex 和 Lipschitz 函数进行零顺序优化。 我们根据对$\ ell_ 1$- sphere 的两种功能评估和随机化, 提出了一个新的梯度估计器。 考虑到各种可行的组合和Lipschitz 假设的不同地理分布, 我们用我们的估计器来分析在线反镜下位算法, 以取代通常的梯度。 我们考虑对零顺序或断层的噪音的两种假设: 取消噪音和对立噪音。 我们提供了一种随时完全的数据驱动算法, 它可以适应问题的所有参数。 在以前研究过的取消噪音的情况下, 我们的保证比citet{duchi2015} 和\citet{Shamir17} 所获得的非适应性算法的最先进的界限类似或更好。 我们的分析基于对$\ ell_ 1$- sphe 和直线常数的统一计量法的一种新的 Poincar\ 类型不平等, 可能具有独立的兴趣。