Transportation-Inequalities, Lyapunov Stability and Sampling for Dynamical Systems on Continuous State Space
This work addresses challenges in reinforcement learning and controls by providing new concentration bounds for dynamical systems, though it appears incremental in its functional analytic approach.
The paper tackles the problem of deriving exponential concentration inequalities for discrete-time random dynamical systems with unbounded state spaces, achieving results that apply to unbounded observables without requiring reversibility or exact system knowledge.
We study the concentration phenomenon for discrete-time random dynamical systems with an unbounded state space. We develop a heuristic approach towards obtaining exponential concentration inequalities for dynamical systems using an entirely functional analytic framework. We also show that existence of exponential-type Lyapunov function, compared to the purely deterministic setting, not only implies stability but also exponential concentration inequalities for sampling from the stationary distribution, via \emph{transport-entropy inequality} (T-E). These results have significant impact in \emph{reinforcement learning} (RL) and \emph{controls}, leading to exponential concentration inequalities even for unbounded observables, while neither assuming reversibility nor exact knowledge of random dynamical system (assumptions at heart of concentration inequalities in statistical mechanics and Markov diffusion processes).