Mean-field Langevin dynamics (MLFD) is a class of interacting particle methods that tackle convex optimization over probability measures on a manifold, which are scalable, versatile, and enjoy computational guarantees. However, some important problems -- such as risk minimization for infinite width two-layer neural networks, or sparse deconvolution -- are originally defined over the set of signed, rather than probability, measures. In this paper, we investigate how to extend the MFLD framework to convex optimization problems over signed measures. Among two known reductions from signed to probability measures -- the lifting and the bilevel approaches -- we show that the bilevel reduction leads to stronger guarantees and faster rates (at the price of a higher per-iteration complexity). In particular, we investigate the convergence rate of MFLD applied to the bilevel reduction in the low-noise regime and obtain two results. First, this dynamics is amenable to an annealing schedule, adapted from Suzuki et al. (2023), that results in improved convergence rates to a fixed multiplicative accuracy. Second, we investigate the problem of learning a single neuron with the bilevel approach and obtain local exponential convergence rates that depend polynomially on the dimension and noise level (to compare with the exponential dependence that would result from prior analyses).

翻译：平均场朗之万动力学（MFLD）是一类在流形上处理概率测度凸优化问题的交互粒子方法，具有可扩展性、通用性，并享有计算保证。然而，一些重要问题——例如无限宽度双层神经网络的风险最小化，或稀疏反卷积——最初是在符号测度（而非概率测度）集合上定义的。本文研究了如何将MFLD框架扩展到符号测度的凸优化问题。在从符号测度到概率测度的两种已知约化方法——提升法与双层法中，我们证明双层法能带来更强的保证和更快的收敛速率（代价是更高的单次迭代复杂度）。具体而言，我们研究了在低噪声区域中应用于双层约化的MFLD收敛速率，并获得两项结果。首先，该动力学适用于从Suzuki等人（2023）改进的退火调度策略，从而提升了达到固定乘法精度的收敛速率。其次，我们研究了基于双层法的单神经元学习问题，获得了局部指数收敛速率，该速率对维度和噪声水平具有多项式依赖性（相较于先前分析可能产生的指数依赖性而言）。