State-Dependent Temperature Control for Langevin Diffusions
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AbstractWe 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.
All Author(s) ListXuefeng Gao, Zuo Quan Xu, Xun Yu Zhou
Journal nameSIAM Journal on Control and Optimization
Year2022
Volume Number60
Issue Number3
Pages1250 - 1268
ISSN0363-0129
eISSN1095-7138
LanguagesEnglish-United States