Langevin 扩散的州依赖温度控制 (State-Dependent Temperature Control for Langevin Diffusions)

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.

翻译：我们从非电流优化的角度研究Langevin扩散的温度控制问题。这一问题的典型最佳控制方式是爆炸性爆炸型,对错误过于敏感。一种补救措施是允许扩散探索其他温度值,从而平滑爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性的爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸性爆炸。