Extending Provably Correct Reinforcement Learning to Continuous Action Spaces

Core Concepts

This work extends provably correct reinforcement learning algorithms for low-rank Markov Decision Processes (MDPs) to settings with continuous action spaces, without requiring discretization of the action space.

Abstract

The paper discusses the limitations of existing provably correct (PAC) reinforcement learning algorithms for low-rank MDPs, which exhibit a super-linear dependence on the size of the action space |A|. This makes them unsuitable for applications with large or continuous action spaces. To address this, the authors propose two strategies: Utilizing smoothness of error functions: The authors provide a generalization of the importance sampling lemma used in these analyses, which can be applied to α-smooth functions. This allows bounding errors under any policy, not just uniform exploration, when the transition function errors are smooth in the actions. Using smoothed policies: The authors show that if the policies being evaluated have a uniformly bounded density ratio, then the importance sampling lemma can be applied without the |A| dependence. They further provide a way to extend this to unrestricted policies when the transition function and reward function are Hölder continuous. As a case study, the authors apply these techniques to the FLAMBE algorithm, a seminal PAC RL algorithm for low-rank MDPs. They provide two PAC bounds for FLAMBE in the continuous action setting - one for restricted policies with bounded density ratio, and one for unrestricted policies under additional smoothness assumptions. The key insights are that leveraging smoothness in the problem parameters can significantly improve the sample complexity scaling with action space size, and that restricting to policies with bounded density ratio provides a simple way to avoid the |A| dependence in certain parts of the analysis.

Stats

The number of trajectories collected by the FLAMBE algorithm in the continuous action setting is: For restricted policies: ˜ O(H^2 * K^(5+4τ) * d^(7+4τ) / ε^(10+8τ) * L^(9+8τ)κ) For unrestricted policies: ˜ O(H^2 * d^(7+4τ+(4τ+5)σ) / ε^(10+8τ+(4τ+5)σ) * L^(9+8τ)κ * L^(4τ+5)σ) Where: τ = m/αE, with αE the smoothness order of transition errors κ = m/(m + αE) σ = m/min(αT, αR), with αT the smoothness order of true transitions and αR the smoothness order of rewards

Quotes

None

Key Insights Distilled From

Low-Rank MDPs with Continuous Action Spaces

by Andrew Benne... at arxiv.org 04-03-2024

https://arxiv.org/pdf/2311.03564.pdf

Deeper Inquiries

How can the smoothness assumptions on the transition function and reward function be verified or justified in practical applications

In practical applications, the smoothness assumptions on the transition function and reward function can be verified or justified through various methods: Empirical Analysis: One approach is to analyze the behavior of the transition and reward functions empirically. By collecting data and observing the changes in these functions with varying actions or states, one can infer the level of smoothness exhibited. Theoretical Justification: For certain types of problems, the smoothness assumptions can be theoretically justified based on the underlying dynamics of the system. Mathematical models and domain knowledge can provide insights into the expected smoothness properties. Functional Approximation: Another method is to use function approximation techniques to model the transition and reward functions. By fitting these functions to data and evaluating the smoothness of the approximations, one can indirectly verify the smoothness assumptions. Expert Consultation: Seeking input from domain experts or researchers familiar with the specific application domain can also help in validating the smoothness assumptions. Experts can provide insights into the expected behavior of the functions based on their knowledge and experience.

Can the techniques developed in this work be extended to other PAC RL algorithms for low-rank MDPs beyond FLAMBE

The techniques developed in this work can be extended to other PAC RL algorithms for low-rank MDPs beyond FLAMBE by adapting the smoothness assumptions and analysis framework to suit the specific algorithms. Here are some ways to extend these techniques: Generalization of Smoothness Assumptions: The smoothness assumptions on transition and reward functions can be generalized to accommodate different algorithms. By defining appropriate smoothness criteria tailored to the specific characteristics of each algorithm, the techniques can be extended. Algorithm-Specific Analysis: Each PAC RL algorithm may have unique requirements and characteristics. Adapting the analysis framework to account for the specific features of different algorithms can enable the extension of the techniques to a broader range of low-rank MDP algorithms. Comparative Studies: Conducting comparative studies between FLAMBE and other PAC RL algorithms can help identify commonalities and differences in the smoothness assumptions and analysis methods. This comparative analysis can guide the extension of techniques to other algorithms. Collaborative Research: Collaborating with researchers working on different PAC RL algorithms can facilitate the transfer of techniques and methodologies. By sharing insights and collaborating on research projects, the techniques developed in this work can be applied to a wider range of algorithms.

What are some potential challenges or limitations in implementing the continuous action version of the FLAMBE algorithm, particularly the elliptical planner component

Implementing the continuous action version of the FLAMBE algorithm, particularly the elliptical planner component, may face some challenges and limitations: Optimization Complexity: Handling continuous actions in the elliptical planner may introduce computational challenges due to the need for non-trivial optimization procedures. Grid search or concave embeddings may help, but they can be computationally expensive for high-dimensional action spaces. Model Complexity: The complexity of the model may increase with continuous actions, leading to higher computational requirements for training and inference. Managing the complexity of the model while ensuring accuracy and efficiency can be a significant challenge. Smoothness Verification: Verifying the smoothness assumptions on transition and reward functions in practical applications can be challenging. Ensuring that the functions exhibit the required smoothness properties may require advanced mathematical analysis and empirical validation. Algorithm Robustness: Ensuring the robustness of the algorithm to variations in smoothness levels and action space dimensions is crucial. Adapting the algorithm to handle different levels of smoothness and action space complexities without sacrificing performance can be a challenging task. Real-World Applications: Applying the continuous action FLAMBE algorithm to real-world scenarios with complex environments and continuous action spaces may pose additional challenges. Ensuring the algorithm's effectiveness and scalability in practical applications is essential.