# Your Learned Constraint is Secretly a Backward Reachable Tube

arXiv: https://arxiv.org/abs/2501.15618 · HTML: https://mlsanity.com/paper/2501.15618/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 41 references that are in the map.
- First posted on arXiv 26 Jan 2025.

## Builds on (most cited first, 25 of 25)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Generative Adversarial Imitation Learning](https://mlsanity.com/paper/1606.03476/) (2016)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [Responsive Safety in Reinforcement Learning by PID Lagrangian Methods](https://mlsanity.com/paper/2007.03964/) (2020)
- [Inverse Reward Design](https://mlsanity.com/paper/1711.02827/) (2017)
- [Hamilton-Jacobi Reachability: A Brief Overview and Recent Advances](https://mlsanity.com/paper/1709.07523/) (2017)
- [Tianshou: a Highly Modularized Deep Reinforcement Learning Library](https://mlsanity.com/paper/2107.14171/) (2021)
- [DeepReach: A Deep Learning Approach to High-Dimensional Reachability](https://mlsanity.com/paper/2011.02082/) (2020)
- [Of Moments and Matching: A Game-Theoretic Framework for Closing the Imitation Gap](https://mlsanity.com/paper/2103.03236/) (2021)
- [Reach-Avoid Problems with Time-Varying Dynamics, Targets and Constraints](https://mlsanity.com/paper/1410.6445/) (2014)
- [Inverse Reinforcement Learning without Reinforcement Learning](https://mlsanity.com/paper/2303.14623/) (2023)
- [Maximum Likelihood Constraint Inference for Inverse Reinforcement Learning](https://mlsanity.com/paper/1909.05477/) (2019)
- [Sequence Model Imitation Learning with Unobserved Contexts](https://mlsanity.com/paper/2208.02225/) (2022)
- [Hybrid Inverse Reinforcement Learning](https://mlsanity.com/paper/2402.08848/) (2024)
- [ISAACS: Iterative Soft Adversarial Actor-Critic for Safety](https://mlsanity.com/paper/2212.03228/) (2022)
- [What Went Wrong? Closing the Sim-to-Real Gap via Differentiable Causal Discovery](https://mlsanity.com/paper/2306.15864/) (2023)
- [Learning Safety Constraints from Demonstrations with Unknown Rewards](https://mlsanity.com/paper/2305.16147/) (2023)
- [Diffusing States and Matching Scores: A New Framework for Imitation Learning](https://mlsanity.com/paper/2410.13855/) (2024)
- [Maximum Likelihood Constraint Inference from Stochastic Demonstrations](https://mlsanity.com/paper/2102.12554/) (2021)
- [Learning Shared Safety Constraints from Multi-task Demonstrations](https://mlsanity.com/paper/2309.00711/) (2023)
- [A Comprehensive Survey on Inverse Constrained Reinforcement Learning: Definitions, Progress and Challenges](https://mlsanity.com/paper/2409.07569/) (2024)
- [EvIL: Evolution Strategies for Generalisable Imitation Learning](https://mlsanity.com/paper/2406.11905/) (2024)
- [CALIPSO: A Differentiable Solver for Trajectory Optimization with Conic and Complementarity Constraints](https://mlsanity.com/paper/2205.09255/) (2022)
- [Learning Covariances for Estimation with Constrained Bilevel Optimization](https://mlsanity.com/paper/2309.09718/) (2023)
- [Learning Observation Models with Incremental Non-Differentiable Graph Optimizers in the Loop for Robotics State Estimation](https://mlsanity.com/paper/2309.02525/) (2023)

## Built on by (most cited first, 3 of 3)

- [Learning Neural Control Barrier Functions from Expert Demonstrations using Inverse Constraint Learning](https://mlsanity.com/paper/2510.21560/) (2025)
- [Multi-Objective Constraint Inference using Inverse reinforcement learning](https://mlsanity.com/paper/2605.06951/) (2026)
- [Instance-Dependent Regret for CMDPs with Step-Wise Constraints](https://mlsanity.com/paper/2610.02520/) (2026)

## Abstract (arXiv)

Inverse Constraint Learning (ICL) is the problem of inferring constraints from safe (i.e., constraint-satisfying) demonstrations. The hope is that these inferred constraints can then be used downstream to search for safe policies for new tasks and, potentially, under different dynamics. Our paper explores the question of what mathematical entity ICL recovers. Somewhat surprisingly, we show that both in theory and in practice, ICL recovers the set of states where failure is inevitable, rather than the set of states where failure has already happened. In the language of safe control, this means we recover a backwards reachable tube (BRT) rather than a failure set. In contrast to the failure set, the BRT depends on the dynamics of the data collection system. We discuss the implications of the dynamics-conditionedness of the recovered constraint on both the sample-efficiency of policy search and the transferability of learned constraints.