arXiv：cs.LG· Tharrshann Jayan Logarajah, Jeremias Knoblauch, Louis Sharrock·· 3 小时前
过参数化加权 MMD 的优化景观研究
On the Optimisation Landscape of Overparameterised Weighted MMD
AI 导读
研究针对加权原子测度上最大均值差异（MMD）的优化景观，证明在一维高斯核下，粒子数足够多时每个局部极小值的 MMD 可任意小，且 MMD 有界远离零的驻点在无权重梯度流下不稳定。当目标分布为有限支撑时，超出显式粒子阈值后不存在虚假局部极小值，所有虚假驻点均不稳定。
正文
Abstract:We study the optimisation landscape of the maximum mean discrepancy (MMD) over weighted atomic measures. For the one-dimensional Gaussian kernel, we show that, for any compactly supported target distribution, sufficiently many particles ensure that every local minimum has arbitrarily small MMD. Moreover, any stationary point whose MMD remains bounded away from zero is unstable under an unweighted gradient flow over the particle locations. These results hold both when the weights are constrained to the simplex, and when they are unconstrained but optimised at each step. When the target is itself finitely supported, we further strengthen this result: beyond an explicit particle threshold, there are no spurious local minima, and every spurious stationary point is unstable. To our knowledge, these are the first finite-particle overparameterisation guarantees of their kind for weighted MMD optimisation.
| Subjects: | Machine Learning (stat.ML); Machine Learning (cs.LG); Optimization and Control (math.OC) |
| Cite as: | arXiv:2610.19861 [stat.ML] |
| (or arXiv:2610.19861v1 [stat.ML] for this version) | |
| https://doi.org/10.48550/arXiv.2610.19861 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Tharrshann Logarajah [view email]
[v1]
Wed, 14 Oct 2026 09:24:51 UTC (72 KB)
来源：arXiv：cs.LG · arxiv.org