State-Dependent Temperature Control for Langevin Diffusions
Xuefeng Gao, Zuo Quan Xu, Xun Yu Zhou
Abstract
We study the temperature control problem for Langevin diffusions in the context of nonconvex optimization. The classical optimal control of such a problem is of the bang-bang type, which is overly sensitive to errors. A remedy is to allow the diffusions to explore other temperature values and hence smooth out the bang-bang control. We accomplish this by a stochastic relaxed control formulation incorporating randomization of the temperature control and regularizing its entropy. We derive a state-dependent, truncated exponential distribution, which can be used to sample temperatures in a Langevin algorithm, in terms of the solution to an Hamilton--Jacobi--Bellman partial differential equation. We carry out a numerical experiment on a one-dimensional baseline example, in which the Hamilton--Jacobi--Bellman equation can be easily solved, to compare the performance of the algorithm with three other available algorithms in search of a global optimum.