insight - Inverse Reinforcement Learning - # Multi-Planning Horizon Inverse Reinforcement Learning

Learning Reward Functions and Discount Factors from Experts with Multiple Planning Horizons

Q: How can the proposed algorithms be extended to handle settings where both the reward function and the discount factors vary across experts?

The proposed algorithms, MPLP-IRL and MPMCE-IRL, can be extended to accommodate settings where both the reward function and the discount factors vary across experts by modifying the underlying optimization framework. In the current formulation, the algorithms assume a shared global reward function with distinct discount factors for each expert. To generalize this to allow for varying reward functions, we can introduce a separate reward function for each expert while maintaining the structure of the optimization problem. This can be achieved by defining a set of reward functions ( r_k ) for each expert ( k ) and incorporating these into the optimization objectives. The optimization problem would then need to maximize the expected return for each expert's policy under its corresponding reward function and discount factor. The constraints would also need to be adjusted to ensure that each expert's policy remains optimal under its specific reward function and discount factor. Additionally, the algorithms could leverage techniques from multi-task learning, where the relationships between the different reward functions are modeled to capture shared structures or commonalities. This would allow the algorithms to learn from the collective behavior of the experts while still accommodating individual differences in both reward functions and discount factors.

Q: What are the theoretical limits on the identifiability of the reward function and discount factors as the number of experts increases?

The theoretical limits on the identifiability of the reward function and discount factors in the context of inverse reinforcement learning (IRL) are influenced by the rank conditions of the linear systems derived from the expert demonstrations. As the number of experts increases, the complexity of the linear system also increases, which can lead to a more constrained feasible solution space. According to Proposition 2 in the context provided, the identifiability of the reward function and discount factors is contingent upon the rank of the matrix formed by the transition dynamics and the expert policies. Specifically, if the rank of the augmented matrix ( \Phi|b ) exceeds the rank of ( \Phi ), it indicates that there is no reward function that can reconstruct the optimal policies of the experts. Conversely, if the ranks are equal, a unique reward function (up to a constant) exists that can reconstruct the expert policies. As the number of experts increases, the number of constraints in the linear system grows, which can lead to a situation where the feasible solution set becomes smaller. This implies that while having more experts can provide more information, it can also complicate the identifiability of the true reward function and discount factors. In highly heterogeneous settings, however, the increased number of experts can enhance identifiability, as the diversity in expert behaviors can help delineate the underlying reward structures more clearly.

Q: Can the proposed algorithms be adapted to handle continuous state and action spaces, or high-dimensional domains?

Yes, the proposed algorithms can be adapted to handle continuous state and action spaces, as well as high-dimensional domains, though this requires significant modifications to the existing formulations. The current algorithms, MPLP-IRL and MPMCE-IRL, are primarily designed for discrete state and action spaces, which simplifies the optimization process through linear programming techniques. To extend these algorithms to continuous domains, one approach is to utilize function approximation methods, such as neural networks, to represent the reward functions and policies. This would allow the algorithms to generalize across continuous state and action spaces by learning a parameterized representation of the reward function that can capture complex relationships in high-dimensional spaces. Additionally, techniques from maximum entropy IRL, such as those used in MCE-IRL, can be employed to handle continuous state-action spaces. This involves formulating the problem in terms of expected feature counts and using optimization techniques that are suitable for continuous variables, such as gradient-based methods or evolutionary algorithms. Moreover, the algorithms could incorporate sampling methods to estimate the visitation counts and optimize the reward functions over continuous spaces. This would involve discretizing the state and action spaces during the learning process or using Monte Carlo methods to approximate the necessary expectations. In summary, while adapting the proposed algorithms to continuous and high-dimensional domains presents challenges, leveraging function approximation, maximum entropy principles, and sampling techniques can facilitate this extension, allowing for effective learning in more complex environments.

Core Concepts

We develop algorithms to jointly learn a global reward function and agent-specific discount factors from expert demonstrations with different planning horizons.

Abstract

The paper studies an inverse reinforcement learning (IRL) problem where experts are planning under a shared reward function but with different, unknown planning horizons. Without the knowledge of discount factors, the reward function has a larger feasible solution set, making it harder for existing IRL approaches to identify the reward function.

To address this challenge, the authors develop two algorithms:

Multi-Planning Horizon LP-IRL (MPLP-IRL):
- Extends the linear programming IRL (LP-IRL) approach to handle multiple planning horizons.
- Avoids undesirable solutions by maximizing the minimal non-zero difference of Q-functions over states where expert policies are distinguishable.
- Performs a bi-level optimization to jointly learn the reward function and discount factors.
Multi-Planning Horizon MCE-IRL (MPMCE-IRL):
- Extends the max causal entropy IRL (MCE-IRL) approach to handle multiple planning horizons.
- Shows that strong duality does not hold between the MPMCE-IRL problem and its Lagrangian dual, making the inference less tractable.
- Proposes a bi-level optimization approach to jointly learn the reward function and discount factors.

The authors provide theoretical analyses on the identifiability of the reward function and discount factors. They show that with a sufficiently large number of experts, both the reward function and discount factors become identifiable.

Experiments on three domains demonstrate that the learned reward functions generalize well to similar tasks, and the algorithms converge quickly compared to exhaustive grid search.

Customize Summary

Rewrite with AI

Generate Citations

Translate Source

To Another Language

Generate MindMap

from source content

Visit Source

arxiv.org

Stats

The value function V^(γ^, r^)(s_0) of expert policies π^* under the true reward r^*.
The value function V^(r̂, γ̂)(s_0) of reconstructed optimal policies π under the learned reward function r̂.

Quotes

None

Key Insights Distilled From

Inverse Reinforcement Learning with Multiple Planning Horizons

by Jiayu Yao, W... at arxiv.org 09-27-2024

https://arxiv.org/pdf/2409.18051.pdf

Inverse Reinforcement Learning with Multiple Planning Horizons

Deeper Inquiries

How can the proposed algorithms be extended to handle settings where both the reward function and the discount factors vary across experts?

The proposed algorithms, MPLP-IRL and MPMCE-IRL, can be extended to accommodate settings where both the reward function and the discount factors vary across experts by modifying the underlying optimization framework. In the current formulation, the algorithms assume a shared global reward function with distinct discount factors for each expert. To generalize this to allow for varying reward functions, we can introduce a separate reward function for each expert while maintaining the structure of the optimization problem.
This can be achieved by defining a set of reward functions ( r_k ) for each expert ( k ) and incorporating these into the optimization objectives. The optimization problem would then need to maximize the expected return for each expert's policy under its corresponding reward function and discount factor. The constraints would also need to be adjusted to ensure that each expert's policy remains optimal under its specific reward function and discount factor.
Additionally, the algorithms could leverage techniques from multi-task learning, where the relationships between the different reward functions are modeled to capture shared structures or commonalities. This would allow the algorithms to learn from the collective behavior of the experts while still accommodating individual differences in both reward functions and discount factors.

What are the theoretical limits on the identifiability of the reward function and discount factors as the number of experts increases?

The theoretical limits on the identifiability of the reward function and discount factors in the context of inverse reinforcement learning (IRL) are influenced by the rank conditions of the linear systems derived from the expert demonstrations. As the number of experts increases, the complexity of the linear system also increases, which can lead to a more constrained feasible solution space.
According to Proposition 2 in the context provided, the identifiability of the reward function and discount factors is contingent upon the rank of the matrix formed by the transition dynamics and the expert policies. Specifically, if the rank of the augmented matrix ( \Phi|b ) exceeds the rank of ( \Phi ), it indicates that there is no reward function that can reconstruct the optimal policies of the experts. Conversely, if the ranks are equal, a unique reward function (up to a constant) exists that can reconstruct the expert policies.
As the number of experts increases, the number of constraints in the linear system grows, which can lead to a situation where the feasible solution set becomes smaller. This implies that while having more experts can provide more information, it can also complicate the identifiability of the true reward function and discount factors. In highly heterogeneous settings, however, the increased number of experts can enhance identifiability, as the diversity in expert behaviors can help delineate the underlying reward structures more clearly.

Can the proposed algorithms be adapted to handle continuous state and action spaces, or high-dimensional domains?

Yes, the proposed algorithms can be adapted to handle continuous state and action spaces, as well as high-dimensional domains, though this requires significant modifications to the existing formulations. The current algorithms, MPLP-IRL and MPMCE-IRL, are primarily designed for discrete state and action spaces, which simplifies the optimization process through linear programming techniques.
To extend these algorithms to continuous domains, one approach is to utilize function approximation methods, such as neural networks, to represent the reward functions and policies. This would allow the algorithms to generalize across continuous state and action spaces by learning a parameterized representation of the reward function that can capture complex relationships in high-dimensional spaces.
Additionally, techniques from maximum entropy IRL, such as those used in MCE-IRL, can be employed to handle continuous state-action spaces. This involves formulating the problem in terms of expected feature counts and using optimization techniques that are suitable for continuous variables, such as gradient-based methods or evolutionary algorithms.
Moreover, the algorithms could incorporate sampling methods to estimate the visitation counts and optimize the reward functions over continuous spaces. This would involve discretizing the state and action spaces during the learning process or using Monte Carlo methods to approximate the necessary expectations.
In summary, while adapting the proposed algorithms to continuous and high-dimensional domains presents challenges, leveraging function approximation, maximum entropy principles, and sampling techniques can facilitate this extension, allowing for effective learning in more complex environments.