Convergence of adaptive biasing potential methods for diffusions
Résumé
We prove the consistency of an adaptive importance sampling strategy based on biasing the potential energy function V of a diffusion process dX 0 t = −∇V (X 0 t)dt + dW t ; for the sake of simplicity, periodic boundary conditions are assumed, so that X 0 t lives on the flat d-dimensional torus. The goal is to sample its invariant distribution µ = Z −1 exp −V (x) dx. The bias V t −V , where V t is the new (random and time-dependent) potential function, acts only on some coordinates of the system, and is designed to flatten the corresponding empirical occupation measure of the diffusion X in the large time regime. The diffusion process writes dX t = −∇V t (X t)dt + dW t , where the bias V t − V is function of the key quantity µ t : a probability occupation measure which depends on the past of the process, i.e. on (X s) s∈[0,t]. We are thus dealing with a self-interacting diffusion. In this note, we prove that when t goes to infinity, µ t almost surely converges to µ. Moreover, the approach is justified by the convergence of the bias to a limit which has an intepretation in terms of a free energy. The main argument is a change of variables, which formally validates the consistency of the approach. The convergence is then rigorously proven adapting the ODE method from stochastic approximation .
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