更快的凸优化：具有有效通用屏障的模拟退火算法
Faster Convex Optimization: Simulated Annealing with an Efficient Universal Barrier
摘要：
本文探讨了两种截然不同的凸优化方法之间令人惊讶的等价性。本文证明了模拟随机游动算法在某种意义上与熵泛障碍函数的中心路径内点算法是等价的。这种连接表现出几个好处。首先，我们能够改善现有的凸优化的时间复杂度下的成员预言模型。我们利用Nesterov和Nemirovskii开发的工具，基于中心路径跟踪内点算法，改进了Kalai和Vempala的随机算法的分析。我们能够收紧模拟退火的温度时间表，从而提高运行时间，在某些情况下减少了尺寸的平方根。其次，我们得到了一个有效的随机内点方法，它具有一个有效的可计算的通用障碍，适用于任何由成员预言机描述的凸集。以前，有效计算的障碍是已知的，只有特定的凸集。
英文摘要：
This paper explores a surprising equivalence between two seemingly-distinct convex optimization methods. We show that simulated annealing, a well-studied random walk algorithms, is directly equivalent, in a certain sense, to the central path interior point algorithm for the the entropic universal barrier function. This connection exhibits several benefits. First, we are able improve the state of the art time complexity for convex optimization under the membership oracle model. We improve the analysis of the randomized algorithm of Kalai and Vempala by utilizing tools developed by Nesterov and Nemirovskii that underly the central path following interior point algorithm. We are able to tighten the temperature schedule for simulated annealing which gives an improved running time, reducing by square root of the dimension in certain instances. Second, we get an efficient randomized interior point method with an efficiently computable universal barrier for any convex set described by a membership oracle. Previously, efficiently computable barriers were known only for particular convex sets.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
