Dynamic Regret via Discounted-to-Dynamic Reduction with Applications to Curved Losses and Adam Optimizer
This work addresses dynamic regret analysis for FTRL and Adam, which is important for adaptive optimization in machine learning, but it appears incremental as it builds on existing reduction techniques to extend results.
The paper tackles dynamic regret minimization in non-stationary online learning for FTRL methods, particularly for curved losses like linear and logistic regression, and applies this to analyze Adam optimizers, achieving optimal convergence rates in various settings and new results for Adam variants.
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.