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Optimizing Percentile Criterion in Offline Reinforcement Learning with Value-at-Risk Bellman Operator


Core Concepts
The core message of this paper is to propose a novel Value-at-Risk (VaR) based dynamic programming framework for optimizing a tight lower bound on the percentile criterion in offline reinforcement learning, without explicitly constructing ambiguity sets. The authors show that the VaR Bellman operator implicitly constructs smaller ambiguity sets compared to Bayesian credible regions, leading to less conservative robust policies.
Abstract
The paper addresses the challenge of computing robust policies for high-stakes decision-making problems with limited data in reinforcement learning. The authors focus on the percentile criterion, which aims to optimize the policy for the worst α-percentile transition probability model. The key insights are: Existing work uses Robust Markov Decision Processes (RMDPs) with Bayesian credible regions (BCR) as ambiguity sets to approximately solve the non-convex percentile criterion. However, the BCR ambiguity sets are often unnecessarily large, resulting in overly conservative policies. The authors propose a novel Value-at-Risk (VaR) based dynamic programming framework to optimize a lower bound on the percentile criterion without explicitly constructing ambiguity sets. They show that the VaR Bellman operator is a valid contraction mapping that optimizes a tighter lower bound on the percentile criterion compared to RMDPs with BCR ambiguity sets. The authors theoretically analyze the performance of the VaR framework and show that the ambiguity sets implicitly constructed by the VaR Bellman operator tend to be smaller than the BCR ambiguity sets, especially as the number of states increases. The authors provide a Generalized VaR Value Iteration algorithm and analyze its error bounds. They also empirically demonstrate the efficacy of the VaR framework in three domains, showing that it outperforms various baseline methods in terms of robust performance.
Stats
The number of states in the Riverswim MDP is 5. The number of states in the Population Growth MDP is 50. The number of states in the Inventory Management MDP is 30.
Quotes
None.

Key Insights Distilled From

by Elita A. Lob... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.05055.pdf
Percentile Criterion Optimization in Offline Reinforcement Learning

Deeper Inquiries

What are the potential limitations of the VaR framework, and how could it be extended to handle more complex forms of uncertainty, such as correlations between transition probabilities across states and actions

The potential limitations of the VaR framework lie in its inability to consider correlations in the uncertainty of transition probabilities across states and actions. This limitation can lead to suboptimal policies, especially in scenarios where these correlations play a significant role in decision-making. To address this limitation, the VaR framework could be extended by incorporating a Conditional Value at Risk (CVaR) Bellman operator. The CVaR operator is convex and provides a lower bound on the Value at Risk measure, taking into account correlations in uncertainty. By integrating the CVaR operator into the VaR framework, it can handle more complex forms of uncertainty and provide more robust policies in the presence of correlated transition probabilities.

How could the VaR framework be adapted to handle continuous state-action spaces, and what are the challenges in doing so

Adapting the VaR framework to handle continuous state-action spaces involves addressing the challenges posed by the increased complexity and dimensionality of the problem. One approach could be to use function approximation techniques, such as neural networks, to represent the value function in continuous spaces. By training the neural network on sampled data from the continuous space, the VaR framework can learn to approximate the VaR Bellman operator for continuous state-action spaces. However, challenges such as convergence issues, computational complexity, and the curse of dimensionality need to be carefully addressed in this adaptation process. Additionally, techniques like discretization or dimensionality reduction may be employed to make the problem more tractable.

Given that the Soft-Robust method optimizes the mean of expected returns and does not provide probabilistic guarantees, how could the strengths of the VaR framework and the Soft-Robust method be combined to achieve a balance between robustness and performance

To achieve a balance between robustness and performance, the strengths of the VaR framework and the Soft-Robust method can be combined in a hybrid approach. One way to do this is to use the VaR framework to optimize the robustness of the policy while incorporating the mean of expected returns as a secondary objective. This can be achieved by introducing a regularization term in the VaR optimization that penalizes deviations from the mean of expected returns. By optimizing for both robustness (through VaR) and mean performance (through Soft-Robust), the hybrid approach can strike a balance between the two objectives, ensuring both probabilistic guarantees and optimization of the mean returns. Additionally, techniques like multi-objective optimization or ensemble methods can be employed to integrate the strengths of both methods effectively.
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