Randomized algorithms and PAC bounds for inverse reinforcement learning in continuous spaces (2405.15509v1)

Abstract: This work studies discrete-time discounted Markov decision processes with continuous state and action spaces and addresses the inverse problem of inferring a cost function from observed optimal behavior. We first consider the case in which we have access to the entire expert policy and characterize the set of solutions to the inverse problem by using occupation measures, linear duality, and complementary slackness conditions. To avoid trivial solutions and ill-posedness, we introduce a natural linear normalization constraint. This results in an infinite-dimensional linear feasibility problem, prompting a thorough analysis of its properties. Next, we use linear function approximators and adopt a randomized approach, namely the scenario approach and related probabilistic feasibility guarantees, to derive epsilon-optimal solutions for the inverse problem. We further discuss the sample complexity for a desired approximation accuracy. Finally, we deal with the more realistic case where we only have access to a finite set of expert demonstrations and a generative model and provide bounds on the error made when working with samples.

• The paper presents a randomized, PAC-based formulation that overcomes traditional ill-posedness in continuous inverse reinforcement learning.
• It employs occupation measures, duality, and complementary slackness to recast the IRL problem into an infinite-dimensional linear feasibility framework amenable to finite approximations.
• The scenario-based method yields ε-optimal solutions, providing practical sample complexity bounds and robust performance guarantees for real-world applications.

Overview of Inverse Reinforcement Learning in Continuous State and Action Spaces

This paper addresses the challenges of inverse reinforcement learning (IRL) within the context of discrete-time discounted Markov decision processes (MDPs) that have continuous state and action spaces. This setting is particularly significant due to its applicability in domains like autonomous driving and robotics. The authors employ a range of mathematical tools including occupation measures, linear duality, and complementary slackness to characterize the solutions to the IRL problem. They introduce an infinite-dimensional linear feasibility problem and address its practical tractability by using linear function approximators and scenario-based probabilistic solutions.

In-depth Analysis of the Approach

Full Knowledge of Expert Policy

The initial part of the work considers scenarios where the entire policy of the expert is known. By leveraging the linear programming (LP) formulation of continuous MDPs, the authors characterize the set of solutions to the IRL problem using occupation measures. Occupation measures provide an elegant way to express the time-average spent in various states under a specific policy. Using complementary slackness conditions and linear duality, the authors establish that the set of feasible solutions can be viewed as a convex cone.

Avoiding Ill-posedness

To counter the ill-posedness and triviality issues typically associated with IRL, a normalization constraint is introduced, resulting in an infinite-dimensional linear feasibility problem. This constraint helps in disregarding degenerate solutions like constant cost functions which do not provide meaningful information about the task. Approximations are then employed to handle the infinite-dimensionality: the decision variables and constraints from an infinite-dimensional function space are tightened to a finite-dimensional subspace and then approximated by a finite subset (relaxation).

Utilizing Scenarios for Epsilon-Optimal Solutions

To find $\varepsilon$-optimal solutions, the paper adopts a randomized scenario approach. This method involves sampling from the continuous state and action spaces to create a finite but probabilistically sound representation of the feasible set. The paper elaborates on the sample complexity required to achieve a desired level of approximation, ensuring that solutions are within $\varepsilon$-accuracy with high probability.

Practical Case: Finite Set of Expert Demonstrations

In a more realistic setting where only a finite set of expert demonstrations is available, the authors extend their methodology to provide practical bounds on the error induced due to sampling. This scenario is addressed by leveraging a generative model that supplies successive state transitions given current state-action pairs. Error bounds are derived to quantify the accuracy of the recovered cost function.

Strong Results and Theoretical Insights

The theoretical contributions are significant, offering:

• Characterization of Inverse Feasibility: The IRL problem is reformulated into a linear program that characterizes the set of all inverse feasible cost functions.
• Normalization Constraint: Helps in avoiding trivial solutions and makes the IRL problem computationally tractable.
• Probabilistic Guarantees: The scenario-based approach ensures solutions are $\varepsilon$-optimal with high probability, offering robust performance guarantees.
• Sample Complexity: Detailed analysis and bounds are provided for the sample complexity required to ensure the accuracy of the solution.

Future Implications and Speculations

The framework set forth in this paper opens new avenues for the application of IRL in continuous spaces, especially in complex systems such as robotics and autonomous vehicles. Future research could explore:

• High-Dimensional State Spaces: Exploring techniques to further handle the curse of dimensionality to make the approach scalable to high-dimensional MDPs.
• More Complex Models and Dynamics: Extending the work to incorporate non-Lipschitz continuous dynamics while ensuring theoretical guarantees.
• Adaptive Sampling: Investigating adaptive sampling methods that can provide more information-efficient ways to approximate the inverse feasible set.

Conclusion

This paper makes a substantial contribution to the field of inverse reinforcement learning by tackling the challenge of continuous state and action spaces. Through thoughtful introduction of normalization constraints and robust probabilistic methods, it provides both practical solutions and theoretical guarantees. This work paves the way for more advanced and tractable applications of IRL in real-world complex systems.