Extensions of Robbins-Siegmund Theorem with Applications in Reinforcement Learning (2509.26442v1)

Abstract: The Robbins-Siegmund theorem establishes the convergence of stochastic processes that are almost supermartingales and is foundational for analyzing a wide range of stochastic iterative algorithms in stochastic approximation and reinforcement learning (RL). However, its original form has a significant limitation as it requires the zero-order term to be summable. In many important RL applications, this summable condition, however, cannot be met. This limitation motivates us to extend the Robbins-Siegmund theorem for almost supermartingales where the zero-order term is not summable but only square summable. Particularly, we introduce a novel and mild assumption on the increments of the stochastic processes. This together with the square summable condition enables an almost sure convergence to a bounded set. Additionally, we further provide almost sure convergence rates, high probability concentration bounds, and $L^p$ convergence rates. We then apply the new results in stochastic approximation and RL. Notably, we obtain the first almost sure convergence rate, the first high probability concentration bound, and the first $L^p$ convergence rate for $Q$-learning with linear function approximation.

• The paper introduces an extension of the Robbins-Siegmund Theorem that relaxes the summability requirement to square summability, enabling almost sure convergence in stochastic approximation.
• It establishes both asymptotic and nonasymptotic convergence metrics including high probability concentration bounds and L^p rates, which improve analysis of RL algorithms.
• The research validates its theoretical advances by demonstrating stable convergence in linear Q-learning, addressing previous limitations in reinforcement learning stability.

Extensions of Robbins-Siegmund Theorem with Applications in Reinforcement Learning

The paper "Extensions of Robbins-Siegmund Theorem with Applications in Reinforcement Learning" (2509.26442) addresses a critical limitation in the original Robbins-Siegmund Theorem, which is foundational to analyzing stochastic processes in stochastic approximation and reinforcement learning (RL). The traditional theorem necessitates the summability of the zero-order term, limiting its applicability to important RL applications. This research extends the theorem to scenarios where this zero-order term is only square summable, introducing novel convergence results applicable to stochastic approximation and RL, specifically benefiting $Q$-learning with linear function approximation.

Theoretical Contributions

Extensions to Robbins-Siegmund Theorem

The paper presents extensions to the Robbins-Siegmund Theorem to handle almost supermartingales where the zero-order term is not summable but only square summable. Key contributions include:

• New Assumptions: Introducing a novel assumption focused on the increments of the stochastic processes, paired with the square summability condition, to ensure almost sure convergence to a bounded set.
• Convergence Metrics: Providing almost sure convergence rates, high probability concentration bounds, and $L^p$ convergence rates for these processes, which were not attainable with the original theorem.

Asymptotic and Nonasymptotic Results

The research distinguishes between asymptotic and nonasymptotic extensions, offering insights into convergence behaviors without the stringent requirements of summability:

• Asymptotic Convergence: The theorem is extended to show almost sure convergence to a bounded set under revised conditions.
• Nonasymptotic Rates: It further introduces nonasymptotic convergence rates with almost sure convergence, concentration bounds, and $L^p$ norms, critical for evaluating algorithm performance in finite settings.

Practical Applications in Reinforcement Learning

Stochastic Approximation

The paper applies its theoretical findings to stochastic approximation algorithms characterized by time-inhomogeneous Markovian noise:

• Algorithm Analysis: By leveraging the new convergence rates, the paper provides novel analyses for stochastic approximation algorithms, overcoming limitations posed by previous assumptions about time-homogeneity.
• Integration Techniques: The work combines advanced analytical techniques, enabling the new theorem to address complexities associated with noise and asymptotic biases in stochastic approximation.

Linear $Q$-Learning

The extended theorem's applicability is demonstrated with linear $Q$-learning, overcoming historical stability concerns in RL:

• Stable Convergence: The paper achieves reliable convergence for linear $Q$-learning using the extended theorem, providing the first almost sure convergence rate, concentration bound, and $L^p$ convergence rate with explicit numerical guarantees.
• Practical Impact: This result fundamentally shifts previous considerations around the instability of linear $Q$-learning, offering a generalized and robust theoretical foundation without the need for algorithmic modifications.

Conclusion

The extension of the Robbins-Siegmund Theorem presented in this work significantly broadens the scope of its applicability in stochastic processes, particularly in reinforcement learning scenarios previously inaccessible due to stringent limitations. This research provides a comprehensive suite of mathematical tools for analyzing convergence properties of RL algorithms in realistic settings, offering promising directions for future research, such as adapting these findings to non-linear approximation and more complex noise models.