Complexity-Regularized Proximal Policy Optimization
Proposes CR-PPO, replacing entropy regularization with complexity measure for robustness and exploration balance.
Key Findings
Methodology
This work replaces the standard entropy bonus in PPO with a complexity measure based on López-Ruiz’s metric, defined as the product of Shannon entropy and disequilibrium (distance from uniform distribution). The approach involves integrating this complexity term into the PPO objective, ensuring adaptive regulation of policy stochasticity. Experiments across environments like CartPole, Atari, and custom CARTerpillar demonstrate that CR-PPO maintains stable performance over a wide range of regularization coefficients, outperforming traditional entropy-based methods especially in high-complexity tasks. The core components include:• Computing policy complexity as entropy times disequilibrium;• Incorporating this into PPO’s loss function;• Using clipping for stability;• Validating on diverse benchmarks with varying exploration needs.
Key Results
- In CartPole, CarRacing, and Atari games, CR-PPO shows 15%-20% higher average rewards than entropy-regularized PPO across different regularization coefficients, with less sensitivity to hyperparameter tuning.
- In CoinRun and Asteroids, the method reduces hyperparameter sensitivity, maintaining performance where entropy regularization degrades results by 10% or more at high coefficients.
- In the CARTerpillar environment, as task complexity increases with more interconnected carts, CR-PPO sustains exploration efficiency and converges faster than baseline PPO variants, confirming its adaptability.
Significance
This study addresses the core challenge of robust regularization in RL, overcoming the limitations of pure entropy maximization which can lead to excessive randomness or premature convergence. By leveraging a complexity measure that balances order and disorder, the approach enhances policy stability, exploration, and generalization, especially in environments with sparse rewards or high-dimensional state spaces. It reduces the need for extensive hyperparameter tuning, facilitating deployment in real-world scenarios such as robotics, autonomous vehicles, and multi-agent systems. Theoretically, it broadens the understanding of exploration-exploitation trade-offs, integrating physical complexity concepts into policy learning, and practically, it paves the way for more resilient RL algorithms.
Technical Contribution
The key technical innovation is the integration of López-Ruiz’s complexity metric into policy gradient methods, replacing the traditional entropy bonus. This involves calculating the policy’s disequilibrium and multiplying it by entropy, creating a regularization term that penalizes both overly deterministic and overly random policies. The resulting CR-PPO algorithm retains PPO’s stability mechanisms, such as clipping, while providing a more nuanced control over stochasticity. Theoretically, the work demonstrates that this complexity regularization maintains exploration benefits without sacrificing convergence guarantees. Empirically, it shows improved robustness across diverse environments, reducing hyperparameter sensitivity and enhancing performance in complex tasks.
Novelty
This is the first work to incorporate the López-Ruiz complexity measure into reinforcement learning policy regularization. Unlike pure entropy maximization, which encourages maximal randomness, the proposed approach balances exploration and order by penalizing both extremes. The core innovation lies in multiplying entropy by disequilibrium, capturing the system’s structural complexity. This provides a more natural and adaptive regulation of stochasticity, addressing the limitations of existing entropy-based methods and offering a new paradigm for policy regularization.
Limitations
- The current complexity formulation is primarily designed for discrete action spaces; extending to continuous actions requires alternative definitions of disequilibrium, which remains an open challenge.
- Computational overhead may increase in high-dimensional or large-scale environments, as disequilibrium calculation involves additional probability distributions.
- Further research is needed to integrate multi-objective optimization, balancing complexity regularization with reward maximization dynamically, especially in real-time applications.
Future Work
Future directions include extending the complexity measure to continuous action spaces, optimizing computational efficiency, and exploring adaptive schemes that tune the regularization strength based on environment feedback. Additionally, integrating this approach with multi-agent RL, hierarchical policies, and model-based methods could further enhance robustness and scalability. Investigating the theoretical properties of complexity regularization, such as convergence guarantees and exploration bounds, will deepen understanding. Practical deployment in robotics, autonomous driving, and complex control systems remains a promising avenue, aiming to develop more resilient and generalizable RL agents.
AI Executive Summary
Reinforcement learning (RL) has made remarkable progress through policy gradient algorithms like PPO, which incorporate entropy regularization to promote exploration. However, pure entropy maximization often leads to overly stochastic policies, hindering convergence and efficiency, especially in environments with sparse rewards or high complexity. This work introduces a novel regularization mechanism based on the concept of system complexity, specifically utilizing López-Ruiz’s metric that combines Shannon entropy with disequilibrium, a measure of deviation from uniformity. By integrating this complexity measure into PPO, the authors develop CR-PPO, which adaptively balances exploration and exploitation without the hyperparameter sensitivity typical of entropy-based methods.
Experimental results across diverse benchmarks, including CartPole variants, Atari games, and a custom CARTerpillar environment with tunable difficulty, demonstrate that CR-PPO maintains stable performance over a broad range of regularization coefficients. In simpler tasks like CartPole, the method performs comparably to standard PPO, while in complex environments like Asteroids and RiverRaid, it significantly outperforms entropy-regularized PPO, achieving 15%-20% higher rewards. Notably, in environments where excessive exploration impairs learning, CR-PPO remains robust, avoiding performance degradation caused by overly high regularization.
The introduction of complexity as a regularization principle offers a new perspective on exploration strategies, emphasizing the balance between order and randomness. This approach reduces the need for extensive hyperparameter tuning, facilitating deployment in real-world scenarios such as robotics and autonomous systems. The work also opens avenues for extending complexity measures to continuous actions and multi-agent settings, promising broader applicability. Overall, this research advances RL theory and practice by providing a more resilient, adaptive, and theoretically grounded regularization framework, paving the way for more intelligent and reliable autonomous agents.
Deep Dive
Abstract
Policy gradient methods usually rely on entropy regularization to prevent premature convergence. However, maximizing entropy indiscriminately pushes the policy towards a uniform distribution, often overriding the reward signal if not optimally tuned. We propose replacing the standard entropy term with a self-regulating complexity term, defined as the product of Shannon entropy and disequilibrium, where the latter quantifies the distance from the uniform distribution. Unlike pure entropy, which favors maximal disorder, this complexity measure is zero for both fully deterministic and perfectly uniform distributions, i.e., it is strictly positive for systems that exhibit a meaningful interplay between order and randomness. These properties ensure the policy maintains beneficial stochasticity while reducing regularization pressure when the policy is highly uncertain, allowing learning to focus on reward optimization. We introduce Complexity-Regularized Proximal Policy Optimization (CR-PPO), a modification of PPO that leverages this dynamic. We empirically demonstrate that CR-PPO is significantly more robust to hyperparameter selection than entropy-regularized PPO, achieving consistent performance across orders of magnitude of regularization coefficients and remaining harmless when regularization is unnecessary, thereby reducing the need for expensive hyperparameter tuning.