A Test-Function Approach to Incremental Stability

2025-07-01

AI Summary

Conventional Lyapunov-based methods struggle to verify incremental input-to-state stability (δISS) in reinforcement learning (RL) due to the inherent non-smoothness and unboundedness of value functions. Method: This paper introduces a novel analytical framework that employs the reward function—as opposed to a Lyapunov function—as the test function. By constructing reward functions exhibiting exponential-decay Lipschitz continuity and adversarial Hölder continuity, we establish a rigorous equivalence between δISS and RL-style value function regularity. Contribution/Results: Our approach overcomes classical Lyapunov requirements—namely smoothness, boundedness, and positive definiteness—enabling δISS certification under bidirectionally unbounded and non-smooth rewards. Experiments demonstrate that δISS can be verified directly on policy-closed-loop systems without explicit Lyapunov construction, offering a new paradigm for stability analysis of learning-based control systems.

Technology Category

• Intelligent Robots: Learning & Optimization for ROB
• Reasoning under Uncertainty: Sequential Decision Making
• Machine Learning: Reinforcement Learning

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

This paper presents a novel framework for analyzing Incremental-Input-to-State Stability ($δ$ISS) 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 Hölder-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.

Problem

• Analyzing incremental stability using RL-style test functions
• Connecting value function regularity to δISS stability
• Developing non-Lyapunov stability certification framework

Innovation

• Uses rewards as test functions for stability
• Links RL value functions to incremental stability
• Develops new equivalence beyond Lyapunov methods