High-Probability Sample Complexity Bounds for Off-Policy Policy Evaluation with Linear Function Approximation

Core Concepts

This paper establishes the first high-probability sample complexity bounds for two widely-used policy evaluation algorithms - the temporal difference (TD) learning algorithm in the on-policy setting, and the two-timescale linear TD with gradient correction (TDC) algorithm in the off-policy setting. The bounds match the minimax-optimal dependence on the target accuracy level and provide explicit dependence on problem-related parameters.

Abstract

The paper investigates the sample complexities required to guarantee a predefined estimation error of the best linear coefficients for policy evaluation in discounted infinite horizon Markov decision processes.
In the on-policy setting, where observations are generated from the target policy, the authors establish the first sample complexity bound with high-probability convergence guarantee for the temporal difference (TD) learning algorithm. The bound matches the minimax-optimal dependence on the tolerance level and exhibits an explicit dependence on problem-related quantities, including the choice of the feature map and the problem dimension.
In the off-policy setting, where samples are drawn from a behavior policy potentially different from the target policy, the authors establish a high-probability sample complexity bound for the two-timescale linear TD with gradient correction (TDC) algorithm. This is the first bound to achieve high-probability convergence guarantee with non-varying stepsizes and without using projection steps or batched updates. The bound also provides explicit dependence on problem-related parameters.
The authors also provide minimax lower bounds to assess the tightness of their upper bounds, showing that the performances of the TD and TDC learning algorithms cannot be further improved in the minimax sense other than a factor of 1/(1-γ) - the effective horizon.

Stats

The paper does not contain any explicit numerical data or statistics. It focuses on establishing theoretical sample complexity bounds for policy evaluation algorithms.

Quotes

"This paper is concerned with the problem of policy evaluation with linear function approximation in discounted infinite horizon Markov decision processes."
"We establish the first sample complexity bound with high-probability convergence guarantee that attains the optimal dependence on the tolerance level."
"Our sample complexity bound also provides an explicit dependence on the salient parameters."

Key Insights Distilled From

High-probability sample complexities for policy evaluation with linear function approximation

by Gen Li,Weich... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2305.19001.pdf

Deeper Inquiries

How can the analysis techniques developed in this paper be extended to other reinforcement learning algorithms beyond policy evaluation, such as policy optimization

The analysis techniques developed in this paper can be extended to other reinforcement learning algorithms beyond policy evaluation, such as policy optimization, by adapting the framework to suit the specific characteristics of the algorithm in question. For policy optimization, the focus shifts from evaluating a given policy to finding the optimal policy. This involves maximizing the expected cumulative reward by adjusting the policy parameters.
To apply the analysis techniques to policy optimization, one would need to modify the objective function and the update rules to reflect the optimization goal. The key idea would be to formulate the optimization problem in terms of estimating the optimal policy parameters, similar to how the best linear coefficients were estimated in policy evaluation. The sample complexities and convergence guarantees would need to be redefined to account for the optimization process rather than evaluation.
By extending the analysis techniques to policy optimization, researchers can gain insights into the sample complexity required to find the optimal policy, the convergence properties of the optimization algorithm, and the trade-offs between exploration and exploitation in reinforcement learning tasks.

What are the implications of the minimax lower bounds established in this work for the design of more efficient reinforcement learning algorithms

The minimax lower bounds established in this work have significant implications for the design of more efficient reinforcement learning algorithms. These lower bounds provide a theoretical limit on the performance of algorithms in terms of sample complexity and accuracy. By showing that certain levels of accuracy cannot be achieved with fewer samples, the lower bounds highlight the inherent challenges and limitations of reinforcement learning tasks.
For algorithm designers, the minimax lower bounds serve as a benchmark for evaluating the performance of their algorithms. If an algorithm can achieve the accuracy levels specified in the lower bounds within the sample complexity constraints, it indicates that the algorithm is performing optimally. On the other hand, if an algorithm falls short of these bounds, it suggests that there is room for improvement in terms of efficiency and effectiveness.
The lower bounds also guide the development of new algorithms by setting realistic expectations for their performance. Researchers can use the insights from the lower bounds to identify areas where algorithm improvements are needed and to prioritize research efforts in reinforcement learning.

Can the high-probability sample complexity bounds be further improved by incorporating additional structural assumptions on the Markov decision process or the feature representation

The high-probability sample complexity bounds established in this work could potentially be further improved by incorporating additional structural assumptions on the Markov decision process or the feature representation. By leveraging the specific characteristics of the problem domain, researchers can tailor the analysis techniques to exploit the underlying structure and reduce the sample complexity required for accurate estimation.
One approach to improving the sample complexity bounds is to consider domain-specific properties of the Markov decision process, such as sparsity, symmetry, or smoothness. By incorporating these structural assumptions into the analysis, researchers can derive tighter bounds that reflect the intrinsic properties of the problem.
Similarly, by optimizing the feature representation to capture the most relevant information for the task, researchers can reduce the dimensionality of the problem and improve the efficiency of the learning process. Feature engineering techniques that highlight important features and reduce noise in the data can lead to more accurate estimations with fewer samples.
Overall, incorporating additional structural assumptions on the Markov decision process and the feature representation can help refine the high-probability sample complexity bounds and enhance the performance of reinforcement learning algorithms.