Regularized Q-learning through Robust Averaging

Abstract: We propose a new Q-learning variant, called 2RA Q-learning, that addresses some weaknesses of existing Q-learning methods in a principled manner. One such weakness is an underlying estimation bias which cannot be controlled and often results in poor performance. We propose a distributionally robust estimator for the maximum expected value term, which allows us to precisely control the level of estimation bias introduced. The distributionally robust estimator admits a closed-form solution such that the proposed algorithm has a computational cost per iteration comparable to Watkins' Q-learning. For the tabular case, we show that 2RA Q-learning converges to the optimal policy and analyze its asymptotic mean-squared error. Lastly, we conduct numerical experiments for various settings, which corroborate our theoretical findings and indicate that 2RA Q-learning often performs better than existing methods.

• The paper introduces 2RA Q-learning, which reduces estimation bias through robust averaging and a distributionally robust regularization mechanism.
• It employs two key parameters to control bias, stabilizing the learning process and achieving faster convergence to optimal policies.
• Numerical experiments in synthetic settings and OpenAI gym environments demonstrate its practical efficiency and superiority over traditional Q-learning variants.

Understanding 2RA Q-Learning: An Advanced Q-Learning Variant

Introduction to 2RA Q-Learning

In the world of Reinforcement Learning (RL), Q-learning represents a cornerstone method for learning optimal policies in Markov Decision Processes (MDPs). However, classic Q-learning, particularly Watkins' Q-learning, is known for its susceptibility to estimation biases, which can negatively affect the performance of the learned policies. To tackle the challenges associated with estimation biases—both overestimation and underestimation—this paper introduces a novel Q-learning variant: Regularized Q-learning through Robust Averaging (2RA Q-learning).

Estimation Bias in Q-Learning

Estimation bias occurs when the expected value of the max operator, commonly used in standard Q-learning updates, deviates significantly due to inherent algorithmic estimation errors. This deviation either overestimates or underestimates the true state-action values, leading to suboptimal policy performance.

• Overestimation Bias: Traditional Q-learning tends to estimate higher than the actual values, which can skew the learning process towards non-optimal policies.
• Underestimation Bias: Double Q-learning and similar methods were developed to address overestimation but often end up underestimating the values, which equally harms the learning outcomes.

Addressing the Bias: Key Features of 2RA Q-Learning

The 2RA Q-learning tackles estimation biases in a novel way by introducing robust averaging and regularization mechanisms into the Q-learning framework:

• Robust Averaging: This technique incorporates averaging over multiple estimates of the Q-values, which helps in reducing the variance and stabilizes the learning process.
• Regularization: By incorporating a distributionally robust estimator, 2RA Q-learning controls the extent of regularization, helping mitigate the problem of over and underestimation observed in traditional methods.
• Parameter Control: The method introduces two parameters, $\rho$ and $N$. $\rho$ dictates the level of robustness/regularization in the estimation process, whereas $N$ decides the number of state-action pairs sampled for averaging.

Theoretical Insights and Practical Implications

• Convergence Guarantee: The proposed 2RA Q-learning method is proven to converge to the optimal policy under specified conditions. This theoretical guarantee parallels that of traditional Q-learning but with improved performance against bias.
• Computational Efficiency: The computational cost per iteration for 2RA Q-learning is comparable to Watkins' Q-learning, making it a practical alternative in terms of implementation and runtime.
• Bias Reduction: The dual influence of $\rho$ and $N$ helps in fine-tuning the estimation bias, allowing a smoother and potentially faster convergence to the optimal policy, as demonstrated through various numerical experiments.

Numerical Experiments and Results

The efficacy of 2RA Q-learning has been exemplified through numerical experiments in both synthetic and practical settings using the OpenAI gym environment. These experiments displayed that 2RA Q-learning often outperforms traditional Q-learning variants by achieving better performance metrics.

• Synthetic Environments: In controlled experiments, 2RA Q-learning consistently showed lower estimation errors and faster convergence towards optimal policies compared to other Q-learning methods.
• OpenAI Gym Tests: Practical tests further validated the superiority of 2RA Q-learning over existing alternatives, highlighting its robustness across different interactive environments.

Speculations on Future Developments

The promising results of 2RA Q-learning suggest several directions for future research:

• Extension to Complex Models: Exploring how 2RA Q-learning performs in more complex environments, including continuous spaces and large state-action spaces.
• Integration with Deep Learning: Adapting 2RA Q-learning to deep reinforcement learning frameworks to handle high-dimensional sensory inputs more effectively.
• Further Bias Control Techniques: Innovating additional mechanisms to control biases in value estimation, enhancing the reliability and efficiency of reinforcement learning algorithms.

Conclusion

The introduction of 2RA Q-learning marks a significant step forward in addressing the critical problem of estimation bias in Q-learning. By merging robust averaging and regularization into the Q-learning framework, it provides a more stable and reliable method for learning optimal policies in MDPs, poised to advance both the theoretical and practical aspects of reinforcement learning.