mlsanityData to 14 Oct 2026

A Test-Function Approach to Incremental Stability

arXiv 2507.00695 ↗First posted 1 Jul 2025cs.LG cs.SY eess.SYcited by 1

On the map: Deep reinforcement learning methods → RL theory and sample complexity

Abstract (arXiv)

This paper presents a novel framework for analyzing Incremental-Input-to-State Stability (δISS) based on the idea of using rewards as "test functions." Whereas control theory traditionally deals with Lyapunov functions that satisfy a time-decrease condition, reinforcement learning (RL) value functions are constructed by exponentially decaying a Lipschitz reward function that may be non-smooth and unbounded on both sides. Thus, these RL-style value functions cannot be directly understood as Lyapunov certificates. We develop a new equivalence between a variant of incremental input-to-state stability of a closed-loop system under given a policy, and the regularity of RL-style value functions under adversarial selection of a H"older-continuous reward function. This result highlights that the regularity of value functions, and their connection to incremental stability, can be understood in a way that is distinct from the traditional Lyapunov-based approach to certifying stability in control theory.

Key facts. Cited by 1 paper in the map (data through 14 Oct 2026). Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods. Builds on 6 of its 27 references that are in the map. First posted on arXiv 1 Jul 2025.

Builds on

6 of 27 references.

References · 6

  1. 2016Generative Adversarial Imitation Learningcited by 1,597
  2. 2019Model-Based Reinforcement Learning for Ataricited by 502
  3. 2021Of Moments and Matching: A Game-Theoretic Framework for Closing the Imitation Gapcited by 81
  4. 2023Provable Guarantees for Generative Behavior Cloning: Bridging Low-Level Stability and High-Level Behaviorcited by 33
  5. 2025The Pitfalls of Imitation Learning when Actions are Continuouscited by 26
  6. 2022TaSIL: Taylor Series Imitation Learningcited by 21

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 1.

1 paper.

  1. 2026Fast Rates for Inverse Reinforcement Learningcited by 1

Similar, not cited

Close in wording, no citation either way

  1. 2025Certifying Stability of Reinforcement Learning Policies using Generalized Lyapunov Functions
  2. 2024On the continuity and smoothness of the value function in reinforcement learning and optimal control
  3. 2026Reinforcement Learning for Control with Probabilistic Stability Guarantee: A Finite-Sample Approach
  4. 2024Steady-State Error Compensation for Reinforcement Learning with Quadratic Rewards
  5. 2024Taming "data-hungry" reinforcement learning? Stability in continuous state-action spaces
  6. 2024Decomposing Control Lyapunov Functions for Efficient Reinforcement Learning
  7. 2024Enhancing Robustness in Deep Reinforcement Learning: A Lyapunov Exponent Approach
  8. 2026Certified Set Convergence for Piecewise Affine Systems via Neural Lyapunov Functions ↗