平均场情形下的神经网络函数逼近：熵正则化与受控McKean-Vlasov动力学
Function approximation by neural nets in the mean-field regime: Entropic regularization and controlled McKean-Vlasov dynamics
AI总结：
该研究将平均场极限下两层随机权重神经网络的函数逼近问题转化为路径空间自由能泛函最小化问题，刻画了对应Föllmer漂移的最优解，并验证了平均场朗之万扩散在对数凹正则化下的逼近有效性。
摘要：
我们研究采用随机权重的两层神经网络的函数逼近问题，其中随机权重在Kullback-Leibler散度意义下“近似高斯”。我们的研究设定为平均场极限，即隐藏层中的有限神经元群体被替换为连续集合。我们证明该问题可以表述为在权重上概率测度的（有限长度）路径空间上对自由能泛函进行全局最小化。该泛函在终端测度的$L^2$逼近风险与路径相对于各向同性布朗运动先验的KL散度之间进行权衡。我们刻画了唯一的全局最小解，并研究了能够实现该最小解的权重上概率测度空间中的动力学。特别地，我们证明最优路径空间测度对应于Föllmer漂移——这是与经典Schrödinger桥问题密切相关的McKean-Vlasov最优控制问题的解。虽然Föllmer漂移通常无法得到闭式解，从而限制了其潜在的算法实用性，但我们阐明了平均场朗之万扩散在熵正则化的各种条件下作为有限时间逼近的可行性。具体而言，我们证明当正则化使得最小化密度为对数凹时，它能够紧密追踪Föllmer漂移。
英文摘要：
We consider the problem of function approximation by two-layer neural nets with random weights that are "nearly Gaussian" in the sense of Kullback-Leibler divergence. Our setting is the mean-field limit, where the finite population of neurons in the hidden layer is replaced by a continuous ensemble. We show that the problem can be phrased as global minimization of a free energy functional on the space of (finite-length) paths over probability measures on the weights. This functional trades off the $L^2$ approximation risk of the terminal measure against the KL divergence of the path with respect to an isotropic Brownian motion prior. We characterize the unique global minimizer and examine the dynamics in the space of probability measures over weights that can achieve it. In particular, we show that the optimal path-space measure corresponds to the Föllmer drift, the solution to a McKean-Vlasov optimal control problem closely related to the classic Schrödinger bridge problem. While the Föllmer drift cannot in general be obtained in closed form, thus limiting its potential algorithmic utility, we illustrate the viability of the mean-field Langevin diffusion as a finite-time approximation under various conditions on entropic regularization. Specifically, we show that it closely tracks the Föllmer drift when the regularization is such that the minimizing density is log-concave.
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