On stabilizing reinforcement learning without Lyapunov functions

Reinforcement learning remains one of the major directions of the contemporary development of control engineering and machine learning. Nice intuition, flexible settings, ease of application are among the many perks of this methodology. From the standpoint of machine learning, the main strength of a reinforcement learning agent is its ability to ``capture" (learn) the optimal behavior in the given environment. Typically, the agent is built on neural networks and it is their approximation abilities that give rise to the above belief. From the standpoint of control engineering, however, reinforcement learning has serious deficiencies. The most significant one is the lack of stability guarantee of the agent-environment closed loop. A great deal of research was and is being made towards stabilizing reinforcement learning. Speaking of stability, the celebrated Lyapunov theory is the de facto tool. It is thus no wonder that so many techniques of stabilizing reinforcement learning rely on the Lyapunov theory in one way or another. In control theory, there is an intricate connection between a stabilizing controller and a Lyapunov function. Employing such a pair seems thus quite attractive to design stabilizing reinforcement learning. However, computation of a Lyapunov function is generally a cumbersome process. In this note, we show how to construct a stabilizing reinforcement learning agent that does not employ such a function at all. We only assume that a Lyapunov function exists, which is a natural thing to do if the given system (read: environment) is stabilizable, but we do not need to compute one.

On stochastic stabilization via non-smooth control Lyapunov functions

Control Lyapunov function is a central tool in stabilization. It generalizes an abstract energy function -- a Lyapunov function -- to the case of controlled systems. It is a known fact that most control Lyapunov functions are non-smooth -- so is the case in non-holonomic systems, like wheeled robots and cars. Frameworks for stabilization using non-smooth control Lyapunov functions exist, like Dini aiming and steepest descent. This work generalizes the related results to the stochastic case. As the groundwork, sampled control scheme is chosen in which control actions are computed at discrete moments in time using discrete measurements of the system state. In such a setup, special attention should be paid to the sample-to-sample behavior of the control Lyapunov function. A particular challenge here is a random noise acting on the system. The central result of this work is a theorem that states, roughly, that if there is a, generally non-smooth, control Lyapunov function, the given stochastic dynamical system can be practically stabilized in the sample-and-hold mode meaning that the control actions are held constant within sampling time steps. A particular control method chosen is based on Moreau-Yosida regularization, in other words, inf-convolution of the control Lyapunov function, but the overall framework is extendable to further control schemes. It is assumed that the system noise be bounded almost surely, although the case of unbounded noise is briefly addressed.

A comment on stabilizing reinforcement learning

This is a short comment on the paper "Asymptotically Stable Adaptive-Optimal Control Algorithm With Saturating Actuators and Relaxed Persistence of Excitation" by Vamvoudakis et al. The question of stability of reinforcement learning (RL) agents remains hard and the said work suggested an on-policy approach with a suitable stability property using a technique from adaptive control - a robustifying term to be added to the action. However, there is an issue with this approach to stabilizing RL, which we will explain in this note. Furthermore, Vamvoudakis et al. seems to have made a fallacious assumption on the Hamiltonian under a generic policy. To provide a positive result, we will not only indicate this mistake, but show critic neural network weight convergence under a stochastic, continuous-time environment, provided certain conditions on the behavior policy hold.