Dynamic Regret via Discounted-to-Dynamic Reduction with Applications to Curved Losses and Adam Optimizer

TL;DR

This paper introduces a modular reduction approach to analyze dynamic regret in non-stationary online learning, with applications to curved losses like linear and logistic regression, and to Adam optimizers.

Contribution

It provides a unified framework for dynamic regret bounds of FTRL methods, simplifying proofs and extending guarantees to new settings including Adam optimizers.

Findings

Optimal dynamic regret bounds for online linear regression.

New dynamic regret guarantees for online logistic regression.

Enhanced analysis of Adam optimizer variants with discount parameters.

Abstract

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 $(\beta_1,\beta_2)$, leading to new results for both clipped and clip-free variants of Adam optimizers.