arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2202.04732cs.LGmath.OCstat.ML
收藏

基于最小选择原理的运输在线学习

Online Learning to Transport via the Minimal Selection Principle

University of Chicago(芝加哥大学);University of British Columbia(不列颠哥伦比亚大学)

Wenxuan Guo, YoonHaeng Hur, Tengyuan Liang, Christopher Ryan

摘要:

基于运筹学中的鲁棒动态资源分配问题,本文研究了决策变量为概率测度的无穷维对象的\textit{Online Learning to Transport}(OLT)问题。我们通过一种称为最小选择原理的见解,在在线学习、最优运输和偏微分方程之间建立联系,该原理最初由\citet{Ambrosio_2005}在Wasserstein梯度流设置中研究。这使我们能够将标准的在线学习框架无缝地扩展到无限维环境。基于我们的框架,我们推导出一种新的方法,称为\textit{minimal selection or exploration (MSoE) algorithm}解决OLT问题,使用平均场近似和离散化技术。在位移凸设置,支撑我们的方法的主要理论信息是,随着时间的推移,最小化运输成本(通过最小选择原则),确保最佳的累积遗憾上限。在算法方面,我们的MSoE算法适用于位移凸设置之外,使得最优传输的数学理论实际上与动态资源分配中常见的非凸设置相关。

英文摘要:

Motivated by robust dynamic resource allocation in operations research, we study the \textit{Online Learning to Transport} (OLT) problem where the decision variable is a probability measure, an infinite-dimensional object. We draw connections between online learning, optimal transport, and partial differential equations through an insight called the minimal selection principle, originally studied in the Wasserstein gradient flow setting by \citet{Ambrosio_2005}. This allows us to extend the standard online learning framework to the infinite-dimensional setting seamlessly. Based on our framework, we derive a novel method called the \textit{minimal selection or exploration (MSoE) algorithm} to solve OLT problems using mean-field approximation and discretization techniques. In the displacement convex setting, the main theoretical message underpinning our approach is that minimizing transport cost over time (via the minimal selection principle) ensures optimal cumulative regret upper bounds. On the algorithmic side, our MSoE algorithm applies beyond the displacement convex setting, making the mathematical theory of optimal transport practically relevant to non-convex settings common in dynamic resource allocation.

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。

↑