Ensuring Unique Solution for Bellman Equation with Two-Discount-Factor Surrogate Rewards for LTL Objectives

Core Concepts

The Bellman equation for the two-discount-factor surrogate reward used for LTL objectives may have multiple solutions when one of the discount factors is set to 1, leading to inaccurate policy evaluation. A sufficient condition to ensure the Bellman equation has a unique solution equal to the value function is that the solution for states in rejecting bottom strongly connected components (BSCCs) is set to 0.

Abstract

The content discusses the uniqueness of the solution to the Bellman equation when using a two-discount-factor surrogate reward approach for planning problems with linear temporal logic (LTL) objectives on Markov decision processes (MDPs).
Key highlights:

The two-discount-factor surrogate reward approach is commonly used to translate LTL objectives into a form suitable for reinforcement learning. It assigns a constant positive reward to "good" states and a constant discount factor.
Previous works have allowed setting one of the discount factors to 1, but this can lead to the Bellman equation having multiple solutions, which can mislead the reinforcement learning algorithm to converge to a suboptimal policy.
The authors demonstrate this issue with a concrete example, where the Bellman equation has multiple solutions when one discount factor is 1.
The authors propose a sufficient condition to ensure the Bellman equation has a unique solution equal to the value function: the solution for states in rejecting bottom strongly connected components (BSCCs) must be set to 0.
The authors prove this condition is sufficient by showing that when one discount factor is 1, the solution can be separated into states with discounting and states without discounting, and the unique solution can be derived for each part.

The content provides a thorough analysis of the uniqueness issue in the Bellman equation for the two-discount-factor surrogate reward approach and proposes a solution to ensure the value function is the unique fixed point of the Bellman operator, which is crucial for the convergence of reinforcement learning algorithms.

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Key Insights Distilled From

On the Uniqueness of Solution for the Bellman Equation of LTL Objectives

by Zetong Xuan,... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.05074.pdf

Deeper Inquiries

What are the implications of the proposed sufficient condition on the practical implementation of reinforcement learning algorithms for LTL objectives

The proposed sufficient condition has significant implications for the practical implementation of reinforcement learning algorithms for LTL objectives. By ensuring that the solutions for states within rejecting BSCCs are set to zero, the uniqueness of the solution to the Bellman equation is guaranteed. This uniqueness is crucial for accurate evaluation of the expected return and the convergence to optimal policies in reinforcement learning. Without this condition, the Bellman equation may have multiple solutions, leading to inaccurate evaluations and potentially suboptimal policies. Therefore, the proposed condition enhances the reliability and effectiveness of reinforcement learning algorithms for LTL objectives.

How would the solution change if the discount factors are allowed to vary over time instead of being constant

If the discount factors are allowed to vary over time instead of being constant, the solution to the Bellman equation would become more dynamic and adaptive. By incorporating time-varying discount factors, the reinforcement learning algorithm can adjust the importance placed on immediate rewards versus future rewards based on the evolving circumstances. This adaptability can potentially lead to more flexible and responsive decision-making, allowing the algorithm to optimize its policy in a more nuanced and context-sensitive manner. However, the dynamic nature of varying discount factors may introduce additional complexity in the optimization process and require sophisticated algorithms to handle the changing reward structures effectively.

Can the proposed approach be extended to handle more general reward structures beyond the two-discount-factor surrogate reward

The proposed approach can be extended to handle more general reward structures beyond the two-discount-factor surrogate reward. By incorporating different types of rewards, such as sparse rewards, dense rewards, or shaped rewards, the reinforcement learning algorithm can adapt to a wider range of scenarios and objectives. The key lies in formulating the Bellman equation and the value function to accommodate the specific characteristics of the reward structure being used. This extension would enable the algorithm to learn and optimize policies for a diverse set of tasks and environments, enhancing its versatility and applicability in various real-world applications.