Born Discrete, Made Smooth: Variational Formulation of Shallow Neural Networks
Although neural networks are remarkably effective, their underlying optimization principles remain theoretically elusive, often characterized by non-convex landscapes and stochastic heuristics. In this work, we propose a paradigm shift by replacing the discrete training problem of shallow neural networks with a well-posed continuum variational surrogate. We identify a family of $λ$-convex functionals over parameter densities in weighted Sobolev spaces and prove that these variational problems are globally well-posed, stable, and exhibit unexpected almost $C^3$ regularity. Unlike existing Wasserstein-based or Mean-Field approaches, which often face limited regularity and discretization cha...
| Authors | Matej Benko, Pierre Bousquet, Iwona Chlebicka et al. |
|---|---|
| Primary category | stat.ML |
| Published | 2 July 2026 |
| Last updated at source | 2 July 2026 |
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