Almost Sure Convergence of Linear Temporal Difference Learning with Arbitrary Features (2409.12135v2)

Abstract: Temporal difference (TD) learning with linear function approximation, abbreviated as linear TD, is a classic and powerful prediction algorithm in reinforcement learning. While it is well understood that linear TD converges almost surely to a unique point, this convergence traditionally requires the assumption that the features used by the approximator are linearly independent. However, this linear independence assumption does not hold in many practical scenarios. This work is the first to establish the almost sure convergence of linear TD without requiring linearly independent features. In fact, we do not make any assumptions on the features. We prove that the approximated value function converges to a unique point and the weight iterates converge to a set. We also establish a notion of local stability of the weight iterates. Importantly, we do not need to introduce any other additional assumptions and do not need to make any modification to the linear TD algorithm. Key to our analysis is a novel characterization of bounded invariant sets of the mean ODE of linear TD.

• The paper proves almost sure convergence of linear TD learning without requiring linearly independent features, broadening its applicability in practical RL scenarios.
• It employs ODE analysis to characterize TD fixed points and bounded invariant sets, ensuring algorithmic stability even in unconstrained feature spaces.
• The study links TD updates to stochastic approximation theory, demonstrating local stability and convergence through properties of the underlying Markov chain.

Insights on Almost Sure Convergence of Linear Temporal Difference Learning with Arbitrary Features

The paper by Wang and Zhang addresses a significant limitation in the classical analysis of Linear Temporal Difference (TD) learning, a pivotal algorithm in Reinforcement Learning (RL). Previously, the convergence analysis of linear TD relied heavily on the assumption that the features used must be linearly independent. This paper extends the foundational work by removing this assumption, thereby broadening the applicability of linear TD learning, particularly in real-world scenarios where the feature sets are not necessarily linearly independent.

Main Contributions

1. Convergence Proof Without Linear Independence: The authors prove the almost sure convergence of linear TD learning without assuming that the features are linearly independent. This is a major extension as linear dependency of features is common when dealing with large state spaces encoded by neural networks or in continual learning settings where features evolve over time.
2. TD Fixed Points and Mean ODE Analysis: Wang and Zhang explore the properties of TD fixed points and present significant results concerning the solutions of the associated Ordinary Differential Equation (ODE) – a continuous counterpart of the discrete TD updates. They prove that even when features are not linearly independent, the averaged value function iterates towards a set that contains TD fixed points.
3. Bounded Invariant Set Characterization: The research further dives into the ODE method and characterizes the bounded invariant sets. They show that for the unconstrained feature space, TD iterates converge to a bounded invariant set of solutions to the ODE, which is crucial for analyzing the stability of the learning algorithm.
4. Connections to Stochastic Approximation: The paper strengthens the results by linking the TD algorithm's updates to stochastic approximation theory. This involves showing that the process involves a series of updates that converge based on predefined stochastic properties such as the irreducibility of the underlying Markov Chain.
5. Local Stability: An interesting insight is the established notion of local stability of weight iterates. The authors prove that any convergent subsequence of the iterates converges to a TD fixed point, which reaffirms the theoretical stability under weaker assumptions.

Implications and Future Directions

The elimination of the linear independence requirement significantly aligns theoretical RL research with practical applications. It supports the use of function approximation in environments where the state or observation features rendered do not exhibit linear independence, such as in the case of neural network-based feature extraction.

The implications of these findings are vast, allowing for more robust application in RL tasks like autonomous driving, robotics, and complex simulation environments where prior linear constraints restrict practical implementation. The groundwork laid by this paper could inspire further research into the convergence properties of more complex and nonlinear RL algorithms, such as actor-critic methods or policies utilizing deep neural networks.

Future research could leverage these theoretical advancements to explore convergence properties in overparametrized neural networks, where feature redundancy and nonlinear dependencies inherently exist. Moreover, the techniques and insights can be pivotal in developing more resilient algorithms that provide reliable performance without meticulous manual tuning of feature sets or the simplification of complex state representations.

In summary, Wang and Zhang's work on the almost sure convergence of linear TD under arbitrary features marks an important milestone in RL, alleviating constraints that have historically bridged a gap between theory and applied RL systems.