# Variance-Reduced Gradient Estimation via Noise-Reuse in Online Evolution Strategies

arXiv: https://arxiv.org/abs/2304.12180 · HTML: https://mlsanity.com/paper/2304.12180/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 46 references that are in the map.
- First posted on arXiv 21 Apr 2023.

## Builds on (most cited first, 28 of 28)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Training Deep Nets with Sublinear Memory Cost](https://mlsanity.com/paper/1604.06174/) (2016)
- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Evolution Strategies as a Scalable Alternative to Reinforcement Learning](https://mlsanity.com/paper/1703.03864/) (2017)
- [Dataset Distillation by Matching Training Trajectories](https://mlsanity.com/paper/2203.11932/) (2022)
- [The CMA Evolution Strategy: A Tutorial](https://mlsanity.com/paper/1604.00772/) (2016)
- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [Forward and Reverse Gradient-Based Hyperparameter Optimization](https://mlsanity.com/paper/1703.01785/) (2017)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [Understanding and correcting pathologies in the training of learned optimizers](https://mlsanity.com/paper/1810.10180/) (2018)
- [Gradients are Not All You Need](https://mlsanity.com/paper/2111.05803/) (2021)
- [Understanding Short-Horizon Bias in Stochastic Meta-Optimization](https://mlsanity.com/paper/1803.02021/) (2018)
- [PIPPS: Flexible Model-Based Policy Search Robust to the Curse of Chaos](https://mlsanity.com/paper/1902.01240/) (2019)
- [Unbiased Gradient Estimation in Unrolled Computation Graphs with Persistent Evolution Strategies](https://mlsanity.com/paper/2112.13835/) (2021)
- [Structured Evolution with Compact Architectures for Scalable Policy Optimization](https://mlsanity.com/paper/1804.02395/) (2018)
- [Unbiased Online Recurrent Optimization](https://mlsanity.com/paper/1702.05043/) (2017)
- [Unbiasing Truncated Backpropagation Through Time](https://mlsanity.com/paper/1705.08209/) (2017)
- [Practical tradeoffs between memory, compute, and performance in learned optimizers](https://mlsanity.com/paper/2203.11860/) (2022)
- [A Closer Look at Learned Optimization: Stability, Robustness, and Inductive Biases](https://mlsanity.com/paper/2209.11208/) (2022)
- [Approximating Real-Time Recurrent Learning with Random Kronecker Factors](https://mlsanity.com/paper/1805.10842/) (2018)
- [Optimal Kronecker-Sum Approximation of Real Time Recurrent Learning](https://mlsanity.com/paper/1902.03993/) (2019)
- [Generalizing Gaussian Smoothing for Random Search](https://mlsanity.com/paper/2211.14721/) (2022)
- [Convergence of Online Adaptive and Recurrent Optimization Algorithms](https://mlsanity.com/paper/2005.05645/) (2020)

## Built on by (most cited first, 5 of 5)

- [Evolution Strategies at the Hyperscale](https://mlsanity.com/paper/2511.16652/) (2025)
- [μLO: Compute-Efficient Meta-Generalization of Learned Optimizers](https://mlsanity.com/paper/2406.00153/) (2024)
- [Stein Variational Evolution Strategies](https://mlsanity.com/paper/2410.10390/) (2024)
- [Applications of fractional calculus in learned optimization](https://mlsanity.com/paper/2411.14855/) (2024)
- [Efficient Long-Horizon Learning for Learned Optimization](https://mlsanity.com/paper/2607.06772/) (2026)

## Abstract (arXiv)

Unrolled computation graphs are prevalent throughout machine learning but present challenges to automatic differentiation (AD) gradient estimation methods when their loss functions exhibit extreme local sensitivtiy, discontinuity, or blackbox characteristics. In such scenarios, online evolution strategies methods are a more capable alternative, while being more parallelizable than vanilla evolution strategies (ES) by interleaving partial unrolls and gradient updates. In this work, we propose a general class of unbiased online evolution strategies methods. We analytically and empirically characterize the variance of this class of gradient estimators and identify the one with the least variance, which we term Noise-Reuse Evolution Strategies (NRES). Experimentally, we show NRES results in faster convergence than existing AD and ES methods in terms of wall-clock time and number of unroll steps across a variety of applications, including learning dynamical systems, meta-training learned optimizers, and reinforcement learning.