Stability Verification in Stochastic Control Systems via Neural Network Supermartingales
This addresses a critical gap in verifying stability for stochastic control systems, particularly relevant for learning algorithms with neural network policies, and is incremental by building on ranking supermartingales instead of classical Lyapunov functions.
The paper tackles the problem of formally verifying almost-sure asymptotic stability in discrete-time nonlinear stochastic control systems, which is an open issue, by introducing a method using neural network ranking supermartingales that guarantees stability and provides bounds on stabilization time, validated experimentally on reinforcement learning environments.
We consider the problem of formally verifying almost-sure (a.s.) asymptotic stability in discrete-time nonlinear stochastic control systems. While verifying stability in deterministic control systems is extensively studied in the literature, verifying stability in stochastic control systems is an open problem. The few existing works on this topic either consider only specialized forms of stochasticity or make restrictive assumptions on the system, rendering them inapplicable to learning algorithms with neural network policies. In this work, we present an approach for general nonlinear stochastic control problems with two novel aspects: (a) instead of classical stochastic extensions of Lyapunov functions, we use ranking supermartingales (RSMs) to certify a.s.~asymptotic stability, and (b) we present a method for learning neural network RSMs. We prove that our approach guarantees a.s.~asymptotic stability of the system and provides the first method to obtain bounds on the stabilization time, which stochastic Lyapunov functions do not. Finally, we validate our approach experimentally on a set of nonlinear stochastic reinforcement learning environments with neural network policies.