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Offline Inverse RL: New Solution Concepts and Provably Efficient Algorithms

Filippo Lazzati, Mirco Mutti, Alberto Maria Metelli

TL;DR

This work tackles offline inverse reinforcement learning (IRL), where only pre-collected trajectories are available and exploration is limited. It reframes IRL around a feasible reward set that is robust to incomplete state-action coverage, introducing the new set \mathcal{R}_{p,\pi^E} and its inclusion-monotone sub-/super-variants. Two algorithms, IRLO and PIRLO, are proposed to estimate and manipulate these sets with PAC guarantees; PIRLO additionally imposes pessimism to ensure inclusion monotonicity, enabling a reward-sanity check mechanism. The framework relies on two distance measures and Hausdorff-type set distances to quantify learnability under offline data and partial coverage, and it demonstrates a fundamental limitation: learning the exact feasible set from expert data alone is impossible without sufficient coverage. Overall, the paper provides a principled, computable offline IRL toolkit and highlights the practical and theoretical challenges posed by offline data collection.

Abstract

Inverse reinforcement learning (IRL) aims to recover the reward function of an expert agent from demonstrations of behavior. It is well-known that the IRL problem is fundamentally ill-posed, i.e., many reward functions can explain the demonstrations. For this reason, IRL has been recently reframed in terms of estimating the feasible reward set (Metelli et al., 2021), thus, postponing the selection of a single reward. However, so far, the available formulations and algorithmic solutions have been proposed and analyzed mainly for the online setting, where the learner can interact with the environment and query the expert at will. This is clearly unrealistic in most practical applications, where the availability of an offline dataset is a much more common scenario. In this paper, we introduce a novel notion of feasible reward set capturing the opportunities and limitations of the offline setting and we analyze the complexity of its estimation. This requires the introduction an original learning framework that copes with the intrinsic difficulty of the setting, for which the data coverage is not under control. Then, we propose two computationally and statistically efficient algorithms, IRLO and PIRLO, for addressing the problem. In particular, the latter adopts a specific form of pessimism to enforce the novel desirable property of inclusion monotonicity of the delivered feasible set. With this work, we aim to provide a panorama of the challenges of the offline IRL problem and how they can be fruitfully addressed.

Offline Inverse RL: New Solution Concepts and Provably Efficient Algorithms

TL;DR

This work tackles offline inverse reinforcement learning (IRL), where only pre-collected trajectories are available and exploration is limited. It reframes IRL around a feasible reward set that is robust to incomplete state-action coverage, introducing the new set \mathcal{R}_{p,\pi^E} and its inclusion-monotone sub-/super-variants. Two algorithms, IRLO and PIRLO, are proposed to estimate and manipulate these sets with PAC guarantees; PIRLO additionally imposes pessimism to ensure inclusion monotonicity, enabling a reward-sanity check mechanism. The framework relies on two distance measures and Hausdorff-type set distances to quantify learnability under offline data and partial coverage, and it demonstrates a fundamental limitation: learning the exact feasible set from expert data alone is impossible without sufficient coverage. Overall, the paper provides a principled, computable offline IRL toolkit and highlights the practical and theoretical challenges posed by offline data collection.

Abstract

Inverse reinforcement learning (IRL) aims to recover the reward function of an expert agent from demonstrations of behavior. It is well-known that the IRL problem is fundamentally ill-posed, i.e., many reward functions can explain the demonstrations. For this reason, IRL has been recently reframed in terms of estimating the feasible reward set (Metelli et al., 2021), thus, postponing the selection of a single reward. However, so far, the available formulations and algorithmic solutions have been proposed and analyzed mainly for the online setting, where the learner can interact with the environment and query the expert at will. This is clearly unrealistic in most practical applications, where the availability of an offline dataset is a much more common scenario. In this paper, we introduce a novel notion of feasible reward set capturing the opportunities and limitations of the offline setting and we analyze the complexity of its estimation. This requires the introduction an original learning framework that copes with the intrinsic difficulty of the setting, for which the data coverage is not under control. Then, we propose two computationally and statistically efficient algorithms, IRLO and PIRLO, for addressing the problem. In particular, the latter adopts a specific form of pessimism to enforce the novel desirable property of inclusion monotonicity of the delivered feasible set. With this work, we aim to provide a panorama of the challenges of the offline IRL problem and how they can be fruitfully addressed.
Paper Structure (48 sections, 54 theorems, 247 equations, 1 figure, 2 tables, 2 algorithms)

This paper contains 48 sections, 54 theorems, 247 equations, 1 figure, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Solution Concepts for Offline IRL
  4. PAC Framework
  5. Inverse Reinforcement Learning for Offline data (IRLO)
  6. Pessimistic Inverse Reinforcement Learning for Offline data (PIRLO)
  7. Reward Sanity Check with PIRLO
  8. A Bitter Lesson
  9. Conclusion
  10. Related Works
  11. Additional Related Works
  12. A Framework for the "old" Feasible Set
  13. On the Learnability of the Feasible Set
  14. Further considerations
  15. About the need of two datasets
  16. ...and 33 more sections

Key Result

Theorem 3.1

In the setting of Definition def: new fs, the feasible reward set $\mathcal{R}_{p,\pi^E}$ satisfies:

Figures (1)

  • Figure 1: $\mathfrak{R}$ = set of all rewards, $\mathcal{R}$ = true feasible set, $\widehat{\mathcal{R}}^{\cap}$ and $\widehat{\mathcal{R}}^{\cup}$ = examples of inclusion monotonic estimated feasible set (i.e., $\widehat{\mathcal{R}}^{\cap} \subseteq \mathcal{R} \subseteq \widehat{\mathcal{R}}^{\cup}$), $\widehat{\mathcal{R}}$ = example of inclusion non-monotonic estimated feasible set (i.e., $\widehat{\mathcal{R}}\not\subseteq \mathcal{R}$ and $\mathcal{R}\not\subseteq \widehat{\mathcal{R}}$).

Theorems & Definitions (104)

  • Definition 3.1: "Old" Feasible Set $\overline{\mathcal{R}}_{p,\pi^E}$, metelli2021provably
  • Definition 3.2: Feasible Set $\mathcal{R}_{p,\pi^E}$
  • Theorem 3.1
  • Definition 3.3: Sub- and Super-Feasible Sets
  • Theorem 3.2
  • Definition 4.1: Semimetrics $d$ and $d_\infty$ between rewards
  • Proposition 1
  • Proposition 2
  • Definition 4.2: Hausdorff distance, rockafellar1998variational
  • Definition 4.3: $(\epsilon,\delta)$-PAC Algorithm
  • ...and 94 more