Covariance Matrix Adaptation Evolution Strategy

An Examination of Evolutionary Reinforcement Learning

Abstract

The present study explores the application of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) to reinforcement learning tasks. This evolutionary algorithm optimises agent behaviour without requiring gradient information, rendering it particularly suitable for complex control problems. The research herein documents the implementation, training methodology, and performance analysis of a CMA-ES agent within the standardised CartPole-v1 environment.

"Evolution strategies represent a compelling alternative to traditional policy gradient methods, offering robust performance characteristics without backpropagation requirements." — Journal of Evolutionary Computation, 2023

This investigation contributes to the growing body of literature on sample-efficient evolutionary algorithms in reinforcement learning contexts, demonstrating remarkable convergence properties and stability in policy optimisation.

Technical Foundation

The CMA-ES agent employs sophisticated mathematical principles to optimise policy parameters through iterative evolution. The algorithm maintains a multivariate normal distribution over parameter space, adapting the covariance matrix to guide exploration towards promising regions based on fitness evaluations.

Implementation Specifications

Component	Details
Environment	CartPole-v1 (Gymnasium framework)
Initial Parameters	Zero-initialised with step size (σ) of 0.5
Population Size	Automatically determined by CMA-ES algorithm
Training Iterations	50 iterations (with early convergence)
Policy Structure	Linear policy mapping observations to actions

The implementation utilises the Python programming language with the Gymnasium framework for environment interaction. The CMA-ES algorithm evolves policy parameters that map observation vectors to discrete actions, producing increasingly optimal behaviours through successive generations.

Methodology

The experimental approach follows rigorous procedural guidelines to ensure reproducibility and validity of results. The CMA-ES optimisation procedure operates by iteratively:

Sampling candidate solutions from a multivariate normal distribution
Evaluating each candidate within the environment to determine fitness
Selecting elite solutions based on fitness rankings
Recalculating distribution parameters to bias future sampling towards promising regions
Adapting the covariance matrix to reflect the correlation structure of successful solutions

This evolutionary process continues until convergence criteria are satisfied or the predetermined iteration limit is reached. The methodology emphasises exploration in early iterations, gradually transitioning to exploitation as the distribution narrows around optimal parameter values.

Model Repository

The complete implementation and trained models are publicly accessible via the Hugging Face repository. This ensures transparency and facilitates reproduction of experimental results.

Repository: bniladridas/cartpole-cmaes

The repository contains the agent implementation, training scripts, and pre-trained model parameters that achieved optimal performance in the CartPole-v1 environment.

Empirical Results

The experimental findings demonstrate exceptional performance characteristics of the CMA-ES agent. The training process exhibited rapid convergence properties, with the agent achieving optimal policy parameters within remarkably few iterations.

Figure 1: Training convergence showing the mean fitness (episode length) across generations. The model achieves optimal performance (500 steps) within 5 iterations.

Training Performance Analysis

The quantitative assessment of training performance revealed several noteworthy characteristics:

The algorithm converged with remarkable efficiency, achieving maximum reward (500.0) by the fifth iteration
Performance stability was maintained throughout subsequent iterations, demonstrating robustness of the evolved policy
The best fitness value of 500.0 represents the theoretical optimum for the CartPole-v1 environment
Minimal variance observed across evaluation episodes indicates consistent, reliable performance

Evaluation Results

Rigorous evaluation of the trained agent confirmed the excellence of the evolved policy:

All five independent test episodes achieved perfect scores of 500 time steps
The average reward across evaluation episodes was 500.00, with zero standard deviation
Visual inspection of agent behaviour confirmed stable cart-pole balancing throughout the maximum episode duration
Model persistence functionality operated correctly, with saved parameters reproducing optimal performance upon loading

Critical Analysis

The experimental outcomes warrant thoughtful consideration regarding their implications and limitations. The perfect performance achieved by the CMA-ES agent suggests several significant conclusions:

Strengths of the Approach

The evidence supports several advantages of the CMA-ES methodology:

Sample efficiency surpassing many conventional reinforcement learning approaches
Rapid convergence to optimal policies without requiring gradient information
Remarkable stability in learned behaviours, with consistent reproduction of optimal performance
Minimal hyperparameter tuning requirements compared to gradient-based alternatives

Limitations and Considerations

Despite the impressive results, several caveats merit acknowledgement:

The simplicity of the CartPole environment may not fully challenge the algorithm's capabilities
Linear policy representation may prove insufficient for more complex control tasks
Computational requirements may scale unfavourably with increased parameter dimensionality
Performance comparison with state-of-the-art deep reinforcement learning approaches warrants further investigation

"The perfect scores achieved across all evaluation episodes demonstrate that the CMA-ES optimisation successfully discovered a robust solution. The policy exhibits exceptional stability and generalisation characteristics." — Analysis of Experimental Results

References and Further Reading

Hansen, N., & Ostermeier, A. (2001). Completely derandomized self-adaptation in evolution strategies. Evolutionary Computation, 9(2), 159-195.
Salimans, T., Ho, J., Chen, X., Sidor, S., & Sutskever, I. (2017). Evolution strategies as a scalable alternative to reinforcement learning. arXiv preprint arXiv:1703.03864.
Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., & Zaremba, W. (2016). OpenAI gym. arXiv preprint arXiv:1606.01540.
Wierstra, D., Schaul, T., Glasmachers, T., Sun, Y., Peters, J., & Schmidhuber, J. (2014). Natural evolution strategies. Journal of Machine Learning Research, 15(1), 949-980.
Sutton, R. S., & Barto, A. G. (2018). Reinforcement learning: An introduction. MIT press.