Stochastic gradient descent and fast relaxation to thermodynamic equilibrium: A stochastic control approach

We study the convergence to equilibrium of an underdamped Langevin equation that is controlled by a linear feedback force. Specifically, we are interested in sampling possibly multimodal invariant probability distribution system at small noise (or low temperature), for which dynamics can easily get trapped inside metastable subsets phase space. follow Chen et al. [J. Math. Phys. 56, 113302 (2015)] and consider simulated high temperature, with control playing role friction balances additional so as restore original measure lower temperature. discuss different limits temperature ratio goes infinity prove limit dynamics. It turns out that, depending on whether (“target”) or higher (“simulation”) fixed, converges either overdamped deterministic gradient flow. This implies (a) ergodic large separation do not commute general (b) it possible accelerate speed making larger larger. implications these observations from perspective stochastic optimization algorithms enhanced schemes molecular