# OptEMA: Adaptive Exponential Moving Average for Stochastic Optimization with Zero-Noise Optimality

arXiv: https://arxiv.org/abs/2603.09923 · HTML: https://mlsanity.com/paper/2603.09923/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 29 references that are in the map.
- First posted on arXiv 10 Mar 2026.

## Builds on (most cited first, 22 of 22)

- [Adafactor: Adaptive Learning Rates with Sublinear Memory Cost](https://mlsanity.com/paper/1804.04235/) (2018)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [Symbolic Discovery of Optimization Algorithms](https://mlsanity.com/paper/2302.06675/) (2023)
- [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)
- [AdaGrad stepsizes: Sharp convergence over nonconvex landscapes](https://mlsanity.com/paper/1806.01811/) (2018)
- [On the Convergence of A Class of Adam-Type Algorithms for Non-Convex Optimization](https://mlsanity.com/paper/1808.02941/) (2018)
- [Mini-batch Stochastic Approximation Methods for Nonconvex Stochastic Composite Optimization](https://mlsanity.com/paper/1308.6594/) (2013)
- [Adam Can Converge Without Any Modification On Update Rules](https://mlsanity.com/paper/2208.09632/) (2022)
- [Learning-Rate-Free Learning by D-Adaptation](https://mlsanity.com/paper/2301.07733/) (2023)
- [DoG is SGD's Best Friend: A Parameter-Free Dynamic Step Size Schedule](https://mlsanity.com/paper/2302.12022/) (2023)
- [MARS: Unleashing the Power of Variance Reduction for Training Large Models](https://mlsanity.com/paper/2411.10438/) (2024)
- [SUPER-ADAM: Faster and Universal Framework of Adaptive Gradients](https://mlsanity.com/paper/2106.08208/) (2021)
- [Convergence of Muon with Newton-Schulz](https://mlsanity.com/paper/2601.19156/) (2026)
- [On the Convergence of Muon and Beyond](https://mlsanity.com/paper/2509.15816/) (2025)
- [Towards Practical Adam: Non-Convexity, Convergence Theory, and Mini-Batch Acceleration](https://mlsanity.com/paper/2101.05471/) (2021)
- [Convergence of Adam for Non-convex Objectives: Relaxed Hyperparameters and Non-ergodic Case](https://mlsanity.com/paper/2307.11782/) (2023)
- [Adaptive Variance Reduction for Stochastic Optimization under Weaker Assumptions](https://mlsanity.com/paper/2406.01959/) (2024)
- [META-STORM: Generalized Fully-Adaptive Variance Reduced SGD for Unbounded Functions](https://mlsanity.com/paper/2209.14853/) (2022)

## Abstract (arXiv)

Exponential moving averages (EMAs) are a central component of widely used adaptive optimizers such as Adam. However, existing analyses of Adam-style methods often yield suboptimal guarantees in the zero-noise regime, rely on open-loop parameter schedules, or require prior knowledge of smoothness constants. Motivated by these limitations, we introduce OptEMA and analyze two complementary variants: OptEMA-M, which applies an adaptive, decreasing EMA coefficient to the first moment with a fixed second-moment decay, and OptEMA-V, which swaps these roles. At the heart of these variants is a Corrected AdaGrad-Norm coefficient schedule. This formulation renders OptEMA algorithmically closed-loop and Lipschitz-free, meaning its effective stepsizes are trajectory-dependent and require no parameterization via the Lipschitz constant. Under lower-boundedness, unbiasedness, bounded variance, average smoothness, and a bounded stochastic-gradient condition used to control the adaptive normalizers, we prove that both variants achieve the unified noise-adaptive rate tildeO left(T-1/2+σ1/2T-1/4right) for the averaged gradient norm. In the zero-noise regime, these bounds automatically reduce to the nearly optimal deterministic rate widetildeO(T-1/2) without manual hyperparameter retuning.