Asymptotic and Finite Sample Analysis of Nonexpansive Stochastic Approximations with Markovian Noise (2409.19546v4)

Abstract: Stochastic approximation is an important class of algorithms, and a large body of previous analysis focuses on stochastic approximations driven by contractive operators, which is not applicable in some important reinforcement learning settings. This work instead investigates stochastic approximations with merely nonexpansive operators. In particular, we study nonexpansive stochastic approximations with Markovian noise, providing both asymptotic and finite sample analysis. Key to our analysis are a few novel bounds of noise terms resulting from the Poisson equation. As an application, we prove, for the first time, that the classical tabular average reward temporal difference learning converges to a sample path dependent fixed point.

• The paper provides the first almost sure convergence proof for Tabular Average Reward TD learning under mild assumptions.
• It extends stochastic approximation techniques with SKM iterations to robustly handle Markovian and additive noise in TD updates.
• The analysis leverages Poisson’s equation for noise decomposition, addressing long-standing stability and convergence challenges in AR-RL.

Almost Sure Convergence of Average Reward Temporal Difference Learning

The paper "Almost Sure Convergence of Average Reward Temporal Difference Learning" presents a significant theoretical advancement in reinforcement learning (RL). This work addresses the convergence properties of Tabular Average Reward Temporal Difference (TD) learning, a cornerstone algorithm in average reward reinforcement learning (AR-RL).

Key Contributions

1. Almost Sure Convergence Proof: The paper provides the first proof that, under mild conditions, the iterates of the Tabular Average Reward TD learning algorithm converge almost surely to a sample-path-dependent fixed point. This result resolves a long-standing theoretical gap that has persisted since the algorithm's inception over 25 years ago.
2. Stochastic Krasnoselskii-Mann (SKM) Iterations: To achieve the convergence proof, the authors extend recent results in stochastic approximation, particularly those concerning Stochastic Krasnoselskii-Mann (SKM) iterations. They adapt these results to handle situations with Markovian and additive noise, a significant step forward given that existing SKM results typically assume i.i.d. noise or deterministic settings.
3. Poisson's Equation and Stochastic Approximation: The analysis leverages Poisson's equation for Markov chains to decompose the noise in the TD updates. This decomposition is a crucial technique for managing the complexity introduced by the Tabular TD algorithm's asynchronous nature.

Methodology

The authors consider the algorithm's iterative updates: $J_{t+1} = J_t + \beta_{t+1} (R_{t+1} - J_t),$

$$v_{t+1}(S_t) = v_t(S_t) + \alpha_{t+1} (R_{t+1} - J_t + v_t(S_{t+1}) - v_t(S_t)),$$

where $\{S_t, R_t\}$ is a sequence of states and rewards from a Markov Decision Process (MDP) with a fixed policy, $J_t$ is the average reward estimate, and $v_t$ is the differential value estimate.

Challenges Addressed

1. Stability and Convergence: Analyzing the stability and convergence of Tabular Average Reward TD is challenging due to its non-expansive nature and the lack of a discount factor, which complicates the application of traditional ODE-based techniques for stochastic approximation analysis.
2. Markovian Noise: The sequence of state-action pairs $\{Y_t = (S_t, A_t)\}$ forms a Markov chain rather than an i.i.d. sequence, which further complicates the noise analysis in TD updates.

Extensions and Implications

• Broader Applicability:

By extending the analysis to Markovian noise settings, the results pave the way for applying SKM-based approaches to other RL algorithms that operate under similar conditions.

• Linear Function Approximation:

Although the results in this paper do not directly address linear function approximation, they lay a foundation for future work. The analysis highlights the difficulties in extending the results from tabular to function approximation settings, particularly where the feature matrix $\Phi$ does not satisfy the properties assumed in prior work.

• Future Research Directions:

The work opens several avenues for further investigation, such as exploring the $L^p$ convergence of SKM iterations, deriving central limit theorems or laws of iterated logarithms for these iterates, and extending these results to two-timescale settings. Additionally, developing finite sample analyses for these techniques remains an open and significant challenge.

Conclusion

This paper marks a decisive step in understanding the convergence properties of fundamental algorithms in AR-RL. By leveraging advanced techniques in stochastic approximation and providing the first almost sure convergence proof for Tabular Average Reward TD learning, the authors resolve a long-standing open question. This contribution not only enriches the theoretical foundation of RL but also suggests promising new directions for future research and algorithm development.