Stabilizing Temporal Difference Learning via Implicit Stochastic Recursion (2505.01361v2)

Abstract: Temporal difference (TD) learning is a foundational algorithm in reinforcement learning (RL). For nearly forty years, TD learning has served as a workhorse for applied RL as well as a building block for more complex and specialized algorithms. However, despite its widespread use, TD procedures are generally sensitive to step size specification. A poor choice of step size can dramatically increase variance and slow convergence in both on-policy and off-policy evaluation tasks. In practice, researchers use trial and error to identify stable step sizes, but these approaches tend to be ad hoc and inefficient. As an alternative, we propose implicit TD algorithms that reformulate TD updates into fixed point equations. Such updates are more stable and less sensitive to step size without sacrificing computational efficiency. Moreover, we derive asymptotic convergence guarantees and finite-time error bounds for our proposed implicit TD algorithms, which include implicit TD(0), TD($\lambda$), and TD with gradient correction (TDC). Our results show that implicit TD algorithms are applicable to a much broader range of step sizes, and thus provide a robust and versatile framework for policy evaluation and value approximation in modern RL tasks. We demonstrate these benefits empirically through extensive numerical examples spanning both on-policy and off-policy tasks.

Stabilizing Temporal Difference Learning via Implicit Stochastic Approximation

Temporal Difference (TD) learning has been a cornerstone in reinforcement learning (RL), utilized extensively for policy evaluation and value prediction tasks. Despite its widespread application, TD learning poses challenges, particularly its sensitivity to step size, which can impact the convergence and stability of RL algorithms. This paper by Kim et al. addresses these challenges by proposing implicit TD algorithms, leveraging implicit stochastic approximation techniques to mitigate step size sensitivity and enhance stability without sacrificing computational efficiency.

Approach and Theoretical Insights

The proposed implicit TD algorithms reformulate standard TD updates into fixed-point equations, introducing implicit recursion that stabilizes the learning process. The core idea is inspired by implicit stochastic gradient descent (SGD), where updates are made with random step sizes inversely proportional to feature norms. This approach constrains updates naturally, ensuring robustness even in ill-conditioned environments.

The paper provides rigorous theoretical analyses, demonstrating asymptotic convergence and finite-time error bounds of implicit TD algorithms. Notably, implicit TD(0) and TD($\lambda$) algorithms exhibit convergence guarantees under a wide range of step size selections, a significant improvement over traditional TD methods which require meticulously chosen step sizes to avoid divergence.

Theoretical justifications are backed by Lyapunov function-based analysis and extend established results in stochastic approximation theory to the implicit context. This ensures robustness and highlights the minimal computational overhead associated with these implicit updates, making them practical for large-scale RL applications.

Numerical Experiments

Empirical evaluations in synthetic and real-world environments underscore the practical benefits of these algorithms. In tasks such as random walks, Markov reward processes, and continuous-domain control problems, implicit TD algorithms consistently outperform standard TD methods, demonstrating lower bias and variance in value function approximation. Moreover, they remain stable even when larger learning rates are utilized, which aids in faster initial learning phases.

Implications and Future Directions

The paper’s contributions have significant implications for the field of RL, offering a robust alternative to existing TD methods. Implicit TD algorithms can enhance the numerical stability of RL systems, allowing for broader applicability and easier implementation without the burden of extensive hyperparameter tuning.

Future research can explore extensions to other RL paradigms, such as Q-learning or SARSA with implicit approaches, potentially improving stability in off-policy learning settings. Additionally, integrating implicit updates into deep RL frameworks could provide scalable solutions for complex environments and tasks.

Conclusion

Kim et al. have provided a comprehensive paper on addressing TD learning's inherent instability through implicit stochastic approximation. Their work not only advances theoretical understanding and guarantees for TD methods but also establishes practical, efficient, and stable algorithms for real-world RL applications. These developments pave the way for more reliable and adaptable reinforcement learning systems capable of handling diverse and intricate tasks.