A Finite Time Analysis of Temporal Difference Learning With Linear Function Approximation (1806.02450v2)

Abstract: Temporal difference learning (TD) is a simple iterative algorithm used to estimate the value function corresponding to a given policy in a Markov decision process. Although TD is one of the most widely used algorithms in reinforcement learning, its theoretical analysis has proved challenging and few guarantees on its statistical efficiency are available. In this work, we provide a simple and explicit finite time analysis of temporal difference learning with linear function approximation. Except for a few key insights, our analysis mirrors standard techniques for analyzing stochastic gradient descent algorithms, and therefore inherits the simplicity and elegance of that literature. Final sections of the paper show how all of our main results extend to the study of TD learning with eligibility traces, known as TD($\lambda$), and to Q-learning applied in high-dimensional optimal stopping problems.

Overview of Finite Time Analysis of Temporal Difference Learning with Linear Function Approximation

This paper presents a rigorous finite-time analysis of Temporal Difference Learning (TD) with linear function approximation, a cornerstone algorithm in reinforcement learning, primarily used for estimating value functions in Markov Decision Processes (MDP). While TD has been a pivotal tool within the domain, its theoretical understanding, especially in finite time, has been limited. The authors address this gap by providing explicit finite-time performance guarantees for TD algorithms in various settings and extensions, notably including TD with eligibility traces (TD($\lambda$)) and Q-learning for optimal stopping problems.

Key Contributions

1. Non-Asymptotic Analysis Framework: The paper introduces a finite-time analysis framework similar to Stochastic Gradient Descent (SGD), allowing theoretical insights into the convergence of TD learning. The analysis extends the asymptotic convergence proofs by incorporating key properties of SGD to understand both bias and variance of TD estimates over finite iterations.
2. Empirical and Theoretical Implications: By studying a projected variant of TD under different observational models (i.i.d. and Markov chain noise), the authors provide theoretical bounds and predictions about data efficiency and convergence rates. The paper details analysis extensions for Markov noise, augmenting classical TD with eligibility traces in TD($\lambda$), and delivering guarantees for Q-learning in high-dimensional stopping problems.
3. Comprehensive Coverage of TD Variants: Significant contributions include coverage of both basic TD(0) and its variant TD($\lambda$), further extending to Q-learning for dealing with optimal stopping problems, thus widening the applicability of their results across broader computational tasks.
4. Impacts on Algorithm Choice and Design: Results imply critical insights for practitioners on selecting appropriate discount factors and eligibility trace parameters, understanding how these choices influence the trade-off between convergence speed and final approximation accuracy in embedded TD algorithms.

Numerical Results and Methodological Insights

The paper provides quantitative insights into expected convergence rates, demonstrating:

• Constant vs. Decaying Step Sizes: TD with constant step sizes reaches a level of convergence that shows fast geometric decay concerning the initial error, conditioned by the feature matrix and variance of updates.
• Robust Step-Size with Iterative Averaging: A robust convergence result without reliance on feature-conditioning highlights the efficacy of iterate averaging—convergence in the order of $O(1/\sqrt{T})$, drawing parallels to analyses of SGD.
• Markov Noise Considerations: Inmarked dependency on observation models, the variance of Markov processes in the TD paradigm inherits a scaling with mixing time, which directly impacts practical learning convergence rates and choices in non-i.i.d. settings.

Speculation on Future AI Developments

The findings presented in the paper have broader implications for the development of algorithms within AI, contributing theoretically to more sophisticated, reliable, and scalable RL algorithms. As the domain seeks more robust generalization across unseen environments, understanding these deterministic factors in convergence becomes critical. The paper suggests pathways for ensuring more ‘principled’ improvements to TD learning techniques through deeper synergy with optimization insights—providing groundwork for advancing more stable, hybrid SD-TD strategies.

In conclusion, this research provides crucial finite-time performance understanding for TD algorithms, fostering both theoretical advancements and practical guidelines for reinforcement learning. The work paves the way for enhancing algorithm efficiency and reliability, which is essential as AI systems increasingly undertake decision-making in complex, dynamic environments.