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

arXiv: https://arxiv.org/abs/2602.08372 · HTML: https://mlsanity.com/paper/2602.08372/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 65 of its 95 references that are in the map.
- First posted on arXiv 9 Feb 2026.

## Builds on (most cited first, 30 of 65)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019)
- [Adaptive Gradient Methods with Dynamic Bound of Learning Rate](https://mlsanity.com/paper/1902.09843/) (2019)
- [A Simple Convergence Proof of Adam and Adagrad](https://mlsanity.com/paper/2003.02395/) (2020)
- [Adaptive Online Learning in Dynamic Environments](https://mlsanity.com/paper/1810.10815/) (2018)
- [A Sufficient Condition for Convergences of Adam and RMSProp](https://mlsanity.com/paper/1811.09358/) (2018)
- [Adam Can Converge Without Any Modification On Update Rules](https://mlsanity.com/paper/2208.09632/) (2022)
- [Strongly Adaptive Online Learning](https://mlsanity.com/paper/1502.07073/) (2015)
- [Convergence of Adam Under Relaxed Assumptions](https://mlsanity.com/paper/2304.13972/) (2023)
- [A Second-order Bound with Excess Losses](https://mlsanity.com/paper/1402.2044/) (2014)
- [Scale-Free Online Learning](https://mlsanity.com/paper/1601.01974/) (2016)
- [Adapting to Misspecification in Contextual Bandits](https://mlsanity.com/paper/2107.05745/) (2021)
- [Adaptivity and Non-stationarity: Problem-dependent Dynamic Regret for Online Convex Optimization](https://mlsanity.com/paper/2112.14368/) (2021)
- [Relative Loss Bounds for On-line Density Estimation with the Exponential Family of Distributions](https://mlsanity.com/paper/1301.6677/) (2013)
- [Impossible Tuning Made Possible: A New Expert Algorithm and Its Applications](https://mlsanity.com/paper/2102.01046/) (2021)
- [A Novel Convergence Analysis for Algorithms of the Adam Family](https://mlsanity.com/paper/2112.03459/) (2021)
- [Parameter-free Mirror Descent](https://mlsanity.com/paper/2203.00444/) (2022)
- [Optimal Stochastic Non-smooth Non-convex Optimization through Online-to-Non-convex Conversion](https://mlsanity.com/paper/2302.03775/) (2023)
- [Dynamical Models and Tracking Regret in Online Convex Programming](https://mlsanity.com/paper/1301.1254/) (2013)
- [Logistic Regression: The Importance of Being Improper](https://mlsanity.com/paper/1803.09349/) (2018)
- [Bandit Convex Optimization in Non-stationary Environments](https://mlsanity.com/paper/1907.12340/) (2019)
- [Optimal Dynamic Regret in Exp-Concave Online Learning](https://mlsanity.com/paper/2104.11824/) (2021)
- [In Search of Adam's Secret Sauce](https://mlsanity.com/paper/2505.21829/) (2025)
- [A gradient sampling method with complexity guarantees for Lipschitz functions in high and low dimensions](https://mlsanity.com/paper/2112.06969/) (2021)
- [Closing the Gap Between the Upper Bound and the Lower Bound of Adam's Iteration Complexity](https://mlsanity.com/paper/2310.17998/) (2023)
- [No-Regret Learning in Time-Varying Zero-Sum Games](https://mlsanity.com/paper/2201.12736/) (2022)

## Built on by (most cited first, 4 of 4)

- [Online Learning with Gradient-Variation Interval Regret](https://mlsanity.com/paper/2606.03831/) (2026)
- [Adam Converges in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2606.22326/) (2026)
- [Minimax Optimality of Shampoo for Nonstationary Stochastic Optimization](https://mlsanity.com/paper/2610.16923/) (2026)
- [Dimension-Optimal Dynamic Regret in Exp-Concave Online Learning](https://mlsanity.com/paper/2610.20187/) (2026)

## Abstract (arXiv)

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 (β1,β2), leading to new results for both clipped and clip-free variants of Adam optimizers.