ArXiv

Variational Bounds for Perceptron Learning from Structured Data

2026-08-05 · 14:08 UTC ·Francesco Camilli, Pierluigi Contucci, Federica Gerace... ·1 min read
Authors
Francesco Camilli, Pierluigi Contucci, Federica Gerace...
Categories
cs.LG, cond-mat.dis-nn, math-ph, stat.ML
arXiv
https://arxiv.org/abs/2608.04882v1
PDF
https://arxiv.org/pdf/2608.04882v1

Brief

Camilli et al. propose a variational framework for continuous-spin perceptrons trained on Gaussian mixtures that admits concave utilities and log‑concave separable priors. Combining interpolation with log‑concavity and concentration bounds, they obtain lower/upper minimax variational bounds for the limiting quenched pressure; when the two variational optimizations commute those bounds match and yield fixed‑point equations to compute ground‑state energy, training loss, and generalization error.

Why it matters

Introduces a variational approach for a finite-temperature continuous-spin perceptron trained on a Gaussian mixture, supporting a broad class of concave utilities and log-concave separable prior measures (Authors: Francesco Camilli, Pierluigi Contucci, Federica Gerace, Emanuele Mingione; arXiv:2608.04882v1; published 2026-08-05).

Key details

  • Derives lower and upper minimax variational bounds for the limiting quenched pressure using the interpolation method plus log-concavity and concentration estimates; the two bounds differ only by the order of optimization of two variational parameters and coincide when those optimizations commute, identifying the model solution.
  • The variational potential yields stationarity (fixed-point) equations and provides a unified route to compute ground-state energy, training loss, and generalization error; full paper is 51 pages with 10 figures (PDF available).
Source evidence

Abstract

We introduce a variational approach to a finite-temperature continuous-spin perceptron trained on a Gaussian mixture. The model allows for a broad class of concave utilities and log-concave separable prior measures on the spins. By combining the interpolation method with log-concavity and concentration estimates, we derive lower and upper minimax variational bounds for the limiting quenched pressure. Remarkably, the two bounds differ only in the order of optimization of two variational parameters, while all remaining extrema are controlled by the concave--convex structure of the variational potential. Whenever the two optimizations commute, the two bounds match and identify the solution of the model. The same potential yields the fixed-point equations as stationarity conditions and provides a unified route to the computation of the ground-state energy, training loss, and generalization error.

Comment: 51 pages, 10 figures