通过折扣到动态的还原实现动态后悔，应用于曲面损失和Adam优化器
Dynamic Regret via Discounted-to-Dynamic Reduction with Applications to Curved Losses and Adam Optimizer
- National Key Laboratory for Novel Software Technology（新型软件技术国家实验室）
- School of Artificial Intelligence（人工智能学院）
- University of Washington（华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过折扣到动态的还原方法，改进了FTRL相关问题的动态后悔界，并应用于曲面损失和Adam优化器，获得了非凸非光滑设置下的最优收敛速率。
AI中文摘要：
我们研究非平稳在线学习中的动态后悔最小化，主要关注跟随正则化领导者（FTRL）方法。FTRL对于曲面损失和理解自适应优化器如Adam至关重要，但现有的动态后悔分析对FTRL的研究较少。为解决这一问题，我们基于折扣到动态的还原，提出一种模块化的方法来获得FTRL相关问题的动态后悔界。具体而言，我们专注于两种代表性的曲面损失：线性回归和逻辑回归。我们的方法不仅简化了现有在线线性回归最优动态后悔的证明，还为在线逻辑回归提供了新的动态后悔保证。除了在线凸优化之外，我们将该还原应用于分析Adam优化器，获得了在随机、非凸和非光滑设置中的最优收敛速率。该还原还使Adam的处理更加细致，通过两个折扣参数（β₁，β₂），导致了对裁剪和无裁剪Adam优化器变体的新结果。
英文摘要：
We study dynamic regret minimization in non-stationary online learning, with a primary focus on follow-the-regularized-leader (FTRL) methods. FTRL is important for curved losses and for understanding adaptive optimizers such as Adam, yet existing dynamic regret analyses are less explored for FTRL. To address this, we build on the discounted-to-dynamic reduction and present a modular way to obtain dynamic regret bounds of FTRL-related problems. Specifically, we focus on two representative curved losses: linear regression and logistic regression. Our method not only simplifies existing proofs for the optimal dynamic regret of online linear regression, but also yields new dynamic regret guarantees for online logistic regression. Beyond online convex optimization, we apply the reduction to analyze the Adam optimizers, obtaining optimal convergence rates in stochastic, non-convex, and non-smooth settings. The reduction also enables a more detailed treatment of Adam with two discount parameters $(β_1,β_2)$, leading to new results for both clipped and clip-free variants of Adam optimizers.
