让温度随自身密度变化，提升非凸优化中找全局最优的效率。
Mean-Field Langevin Diffusions with Density-dependent Temperature
- 温度随扩散过程自身密度动态调整，实现自适应扰动。
- 密度随时间演化收敛至含朗伯W函数的稳态分布。
- 数值实验显示能同时加快收敛速度并提高精度。
在非凸优化背景下，本文将朗之万扩散的温度设为自身密度的函数。其原理在于，密度可部分反映待优化非凸函数所施加的景观特征，因此密度依赖的温度能提供位置相关的随机扰动，更好地响应局部极小值的位置与深度。由于朗之万动力学受自身密度调节，形成了非标准的诺米茨基型平均场随机微分方程，区别于传统的麦凯恩-弗拉索夫方程。基于Wasserstein次微分微积分，我们首先证明对应的（非线性）福克-普兰克方程存在唯一解；随后通过特雷维桑的叠加原理，从福克-普兰克解构造出该SDE的弱解。当时间趋于无穷时，诱导密度收敛至一个不变分布，其表达式可用朗伯W函数显式写出。数值示例表明，密度依赖温度可同时提升全局极小值估计的准确性和收敛速率。
原文摘要 · Abstract (English)
In the context of non-convex optimization, we let the temperature of a Langevin diffusion to depend on the diffusion's own density function. The rationale is that the induced density captures to some extent the landscape imposed by the non-convex function to be minimized, such that a density-dependent temperature provides location-wise random perturbation that may better react to, for instance, the location and depth of local minimizers. As the Langevin dynamics is now self-regulated by its own density, it forms a mean-field stochastic differential equation (SDE) of the Nemytskii type, distinct from the standard McKean-Vlasov equations. Relying on Wasserstein subdifferential calculus, we first show that the corresponding (nonlinear) Fokker-Planck equation has a unique solution. Next, a weak solution to the SDE is constructed from the solution to the Fokker-Planck equation, by Trevisan's superposition principle. As time goes to infinity, we further show that the induced density converges to an invariant distribution, which admits an explicit formula in terms of the Lambert $W$ function. A numerical example suggests that the density-dependent temperature can simultaneously improve the accuracy of and rate of convergence to the estimate of global minimizers.
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