A Theoretical Analysis of Deep Q-Learning (1901.00137v3)

Abstract: Despite the great empirical success of deep reinforcement learning, its theoretical foundation is less well understood. In this work, we make the first attempt to theoretically understand the deep Q-network (DQN) algorithm (Mnih et al., 2015) from both algorithmic and statistical perspectives. In specific, we focus on a slight simplification of DQN that fully captures its key features. Under mild assumptions, we establish the algorithmic and statistical rates of convergence for the action-value functions of the iterative policy sequence obtained by DQN. In particular, the statistical error characterizes the bias and variance that arise from approximating the action-value function using deep neural network, while the algorithmic error converges to zero at a geometric rate. As a byproduct, our analysis provides justifications for the techniques of experience replay and target network, which are crucial to the empirical success of DQN. Furthermore, as a simple extension of DQN, we propose the Minimax-DQN algorithm for zero-sum Markov game with two players. Borrowing the analysis of DQN, we also quantify the difference between the policies obtained by Minimax-DQN and the Nash equilibrium of the Markov game in terms of both the algorithmic and statistical rates of convergence.

A Theoretical Analysis of Deep Q-Learning

The paper "A Theoretical Analysis of Deep Q-Learning" addresses crucial gaps in our understanding of the Deep Q-Network (DQN) algorithm by analyzing it through the lenses of algorithmic and statistical convergence. Despite the empirical success of deep reinforcement learning, the theoretical underpinnings remain inadequately explored, particularly in scenarios involving complex, non-linear function approximators such as deep neural networks.

Core Contributions

1. Convergence Rates: The paper delineates the algorithmic and statistical rates of convergence for the iterative policy sequences generated by a simplified form of the DQN algorithm. The authors focus on a version of DQN suitable for theoretical analysis while maintaining the essence of experience replay and target networks.
2. Experience Replay and Target Network: The analysis provides justification for using experience replay and target networks. Experience replay reduces variance in gradient estimates by considering i.i.d. samples rather than correlated trajectories. The target network stabilizes training by minimizing a bias inherent in the mean-squared BeLLMan error while holding predicted targets constant over several updates.
3. Minimax-DQN: Extending DQN to zero-sum Markov games, the authors propose the Minimax-DQN algorithm. This variant efficiently addresses two-player scenarios, including calculating Nash equilibria within the game framework, enhancing applicability to complex strategic environments.

Theoretical Insights

• Statistical Error Characterization: The paper highlights how the statistical error reflects the intrinsic approximation bias of neural networks along with sample variance, achieving geometric convergence to the optimal action-value function.
• Complexity and Capacity Relations: Detailed analysis reveals that the error rate is sensitive to the network architecture, with a direct correlation between the depth, width, and sparsity of ReLU networks and the convergence rate.

Implications and Future Directions

This paper's findings emphasize the necessity for a unified theoretical framework that accommodates both linearized and fully non-linear discrete decision-making processes. As AI continues to advance into strategic decision-making arenas like games and simulations, the need for robust theoretical foundations becomes imperative. Beyond classical applications, future research could extend these insights to realms such as continuous control domains, which pose additional complexities and challenges.

Moreover, a deeper exploration of non-convex optimization landscapes in network training through paradigm shifts (such as Neural Tangent Kernel or Lottery Ticket Hypothesis) could bridge the gap between optimization guarantees and empirical successes. Integrating insights from this paper with parallel advances in over-parametrized model theory may eventually yield comprehensive, theoretically grounded design principles for scalable, reliable reinforcement learning algorithms.

In summary, this paper lays a comprehensive groundwork for understanding DQN by dissecting its mechanisms and reinforcing its empirical practices with theoretical justifications, a commendable stride towards addressing the long-standing challenge of bridging the empirical-theoretical divide in reinforcement learning.