State-Dependent Temperature Control for Langevin Diffusions
This addresses the sensitivity issue in temperature control for optimization algorithms, but it is incremental as it builds on existing Langevin diffusion methods.
The paper tackles the problem of temperature control in Langevin diffusions for non-convex optimization by proposing a stochastic relaxed control formulation that smooths out bang-bang control, and demonstrates its performance in a numerical experiment comparing it to other algorithms for finding a global optimum.
We study the temperature control problem for Langevin diffusions in the context of non-convex 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 HJB partial differential equation. We carry out a numerical experiment on a one-dimensional baseline example, in which the HJB equation can be easily solved, to compare the performance of the algorithm with three other available algorithms in search of a global optimum.