# The CMA Evolution Strategy: A Tutorial

arXiv: https://arxiv.org/abs/1604.00772 · HTML: https://mlsanity.com/paper/1604.00772/

## Key facts

- Cited by 377 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Apr 2016.

## Built on by (most cited first, 30 of 377)

- [Black-Box Tuning for Language-Model-as-a-Service](https://mlsanity.com/paper/2201.03514/) (2022)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [Maximum Entropy RL (Provably) Solves Some Robust RL Problems](https://mlsanity.com/paper/2103.06257/) (2021)
- [Tree-Structured Parzen Estimator: Understanding Its Algorithm Components and Their Roles for Better Empirical Performance](https://mlsanity.com/paper/2304.11127/) (2023)
- [Advances and Challenges in Foundation Agents: From Brain-Inspired Intelligence to Evolutionary, Collaborative, and Safe Systems](https://mlsanity.com/paper/2504.01990/) (2025)
- [Paint by Word](https://mlsanity.com/paper/2103.10951/) (2021)
- [AutoFlow: Learning a Better Training Set for Optical Flow](https://mlsanity.com/paper/2104.14544/) (2021)
- [Constrained Graphic Layout Generation via Latent Optimization](https://mlsanity.com/paper/2108.00871/) (2021)
- [Diffusion Models for Black-Box Optimization](https://mlsanity.com/paper/2306.07180/) (2023)
- [Test-Time Model Adaptation with Only Forward Passes](https://mlsanity.com/paper/2404.01650/) (2024)
- [Differentiable Quality Diversity](https://mlsanity.com/paper/2106.03894/) (2021)
- [Auditing Privacy Defenses in Federated Learning via Generative Gradient Leakage](https://mlsanity.com/paper/2203.15696/) (2022)
- [InstructZero: Efficient Instruction Optimization for Black-Box Large Language Models](https://mlsanity.com/paper/2306.03082/) (2023)
- [A Survey on Model-based Reinforcement Learning](https://mlsanity.com/paper/2206.09328/) (2022)
- [MarioGPT: Open-Ended Text2Level Generation through Large Language Models](https://mlsanity.com/paper/2302.05981/) (2023)
- [Fast and Efficient Locomotion via Learned Gait Transitions](https://mlsanity.com/paper/2104.04644/) (2021)
- [Unbiased Gradient Estimation in Unrolled Computation Graphs with Persistent Evolution Strategies](https://mlsanity.com/paper/2112.13835/) (2021)
- [COMBO: Compositional World Models for Embodied Multi-Agent Cooperation](https://mlsanity.com/paper/2404.10775/) (2024)
- [Training of Physical Neural Networks](https://mlsanity.com/paper/2406.03372/) (2024)
- [Approximate Differentiable Rendering with Algebraic Surfaces](https://mlsanity.com/paper/2207.10606/) (2022)
- [What Drives Success in Physical Planning with Joint-Embedding Predictive World Models?](https://mlsanity.com/paper/2512.24497/) (2025)
- [LassoBench: A High-Dimensional Hyperparameter Optimization Benchmark Suite for Lasso](https://mlsanity.com/paper/2111.02790/) (2021)
- [Efficient Differentiable Simulation of Articulated Bodies](https://mlsanity.com/paper/2109.07719/) (2021)
- [Differentiable Simulation of Soft Multi-body Systems](https://mlsanity.com/paper/2205.01758/) (2022)
- [Approximating Gradients for Differentiable Quality Diversity in Reinforcement Learning](https://mlsanity.com/paper/2202.03666/) (2022)
- [On the Importance of Environments in Human-Robot Coordination](https://mlsanity.com/paper/2106.10853/) (2021)
- [Monte Carlo Tree Search based Variable Selection for High Dimensional Bayesian Optimization](https://mlsanity.com/paper/2210.01628/) (2022)
- [World Models and Predictive Coding for Cognitive and Developmental Robotics: Frontiers and Challenges](https://mlsanity.com/paper/2301.05832/) (2023)
- [Illuminating Diverse Neural Cellular Automata for Level Generation](https://mlsanity.com/paper/2109.05489/) (2021)
- [CPG-RL: Learning Central Pattern Generators for Quadruped Locomotion](https://mlsanity.com/paper/2211.00458/) (2022)

## Abstract (arXiv)

This tutorial introduces the CMA Evolution Strategy (ES), where CMA stands for Covariance Matrix Adaptation. The CMA-ES is a stochastic, or randomized, method for real-parameter (continuous domain) optimization of non-linear, non-convex functions. We try to motivate and derive the algorithm from intuitive concepts and from requirements of non-linear, non-convex search in continuous domain.
