提出新方法分析非平稳在线学习中的动态遗憾，适用于逻辑回归和Adam优化器。
Dynamic Regret via Discounted-to-Dynamic Reduction with Applications to Curved Losses and Adam Optimizer
- 基于折扣到动态的简化框架，统一推导FTRL方法的动态遗憾边界。
- 首次获得在线逻辑回归的动态遗憾新上界，证明对非凸非光滑问题仍有效。
- 可细化分析Adam优化器参数影响，适配研究自适应优化器的学者。
我们研究非平稳在线学习中的动态遗憾最小化，重点聚焦于带正则化的追随领先者（FTRL）方法。尽管FTRL在处理曲率损失及理解自适应优化器（如Adam）中至关重要，但现有动态遗憾分析对其研究不足。为此，我们基于折扣到动态的归约方法，提出一种模块化方式以获得与FTRL相关问题的动态遗憾上界。具体地，我们关注两类典型曲率损失：线性回归与逻辑回归。该方法不仅简化了在线线性回归最优动态遗憾的已有证明，还为在线逻辑回归首次提供了新的动态遗憾保证。超越在线凸优化范畴，我们将该归约应用于分析Adam优化器，在随机、非凸、非光滑设定下获得了最优收敛速率。此外，该框架支持对两个折扣参数(β₁, β₂)的精细化分析，推动了剪裁版与无剪裁版Adam优化器的新结果。
原文摘要 · Abstract (English)
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.
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