Modeling Bellman-Error with Logistic Distribution with Applications in Reinforcement Learning

Background and Research Objectives

Reinforcement Learning (RL) has recently become a dynamic and transformative field within artificial intelligence, aiming to maximize cumulative rewards through the interaction between agents and the environment. However, the application of RL faces challenges in optimizing the Bellman Error. This error is particularly crucial in deep Q-learning and related algorithms, where traditional methods mainly use Mean-Squared Bellman Error (MSELoss) as the standard loss function. However, the assumption that the Bellman Error follows a normal distribution may oversimplify the complex characteristics of RL applications. Therefore, this paper revisits the distribution of Bellman Error in RL training and finds that it more likely follows a Logistic Distribution.

Source and Authors of the Paper

The paper titled “Modeling Bellman-error with Logistic Distribution with Applications in Reinforcement Learning” is jointly written by Outongyi Lv and Bingxin Zhou from the Institute of Natural Sciences and the School of Mathematical Sciences at Shanghai Jiao Tong University, and Lin F. Yang from the Department of Electrical and Computer Engineering at the University of California, Los Angeles. The paper was published in the journal “Neural Networks” on May 15, 2024.

Research Content and Methods

This study mainly focuses on the following aspects:

Research Process and Experimental Design

1. Analysis of Distribution Characteristics: Initially, through numerical experiments, it is demonstrated that in RL training environments, the Bellman Error tends to follow a Logistic Distribution rather than the traditionally assumed normal distribution. Consequently, this paper proposes using Logistic Maximum Likelihood Function (L-Loss) to replace MSELoss.
2. Kolmogorov-Smirnov Test: To verify the accuracy of the Logistic Distribution fitting the Bellman Error, the study uses the Kolmogorov-Smirnov test to compare the fit between Logistic Distribution and normal distribution, with results showing that the Logistic Distribution has a better fit.
3. Study on the Relationship Between Reward Proportional Scaling and Distribution: The paper theoretically establishes the explicit relationship between Bellman Error distribution and Proportional Reward Scaling, a common RL performance enhancement technique.
4. Trade-off Analysis of Sampling Accuracy: The study delves into the trade-off between sample accuracy when approximating the Logistic Distribution, using the bias-variance decomposition method to balance the use of computational resources.

Sample and Algorithm Design

Extensive numerical experiments were conducted in ten online and nine offline RL environments to test the performance improvement obtained by integrating Logistic Distribution corrections into various benchmark RL methods. The experiments demonstrated that using L-Loss significantly enhances the performance of these algorithms compared to MSELoss. Additionally, to obtain the true characteristics of the Logistic Distribution, the study analyzed the Bellman Error under initializations of both Logistic and normal distributions, finding that the former better represents the Bellman Error.

Experimental Results and Conclusion

Experimental Results

1. Distribution Fitting Results:
   • Detailed numerical experiments show that the Logistic Distribution offers higher accuracy in fitting the Bellman Error in various environments. The Kolmogorov-Smirnov test specifically indicates a significant advantage of the Logistic Distribution over the normal distribution.
2. Performance Comparison:
   • In numerous RL environments, integrating L-Loss into different benchmark RL methods helps enhance overall performance, including deep Q-learning, conservative Q-learning, and others.
3. Theoretical Validation:
   • The study reveals the intrinsic connection between Bellman Error and reward proportional scaling, providing guidance for selecting the optimal scaling factors while emphasizing the risk of excessive scaling.

Conclusion

The paper theoretically and experimentally proves that the Bellman Error is more accurately modeled by the Logistic Distribution hypothesis, laying an important foundation for the optimization and understanding of future RL algorithms. By replacing the traditional MSELoss, L-Loss can bring significant performance improvements in practical applications.

Research Significance and Value

1. Scientific Value: This research overturns the previous common belief that Bellman Error follows a normal distribution, providing a new theoretical basis for the design and performance improvement of RL optimization methods.
2. Application Value: The research results can be directly applied to various RL algorithms, enhancing model stability and optimization effects by introducing the Logistic Distribution into the loss function.

Research Highlights

1. Discovery of Logistic Distribution Characteristics: For the first time, the paper proposes that Bellman Error follows a Logistic Distribution and verifies this through numerical experiments.
2. Optimization of RL Methods: By adjusting the loss function, the study significantly improves the optimization effects of various RL algorithms.
3. Theoretical Innovation: It establishes a clear connection between Bellman Error distribution and reward proportional scaling, providing theoretical support for reward adjustment.

Other Valuable Information

The experimental section of the paper also thoroughly discusses sampling strategies in RL training. By precisely setting the training batch size, it ensures sampling error reaches an optimal level, thereby improving training efficiency.

By revisiting the type of distribution for Bellman Error, this study demonstrates the feasibility and superiority of this new approach from theory to practice. Whether in theoretical innovation or practical optimization, this study reveals new directions and possibilities for future reinforcement learning research.