Variational Bounds for Perceptron Learning from Structured Data

This paper studies a finite-temperature continuous-spin perceptron trained on Gaussian-mixture data. It supports a broad class of concave utilities and log-concave separable priors over the spins. Using interpolation, log-concavity, and concentration estimates, the authors derive lower and upper minimax variational bounds for the limiting quenched pressure. The bounds differ only in the order of optimizing two variational parameters. When those optimizations commute, the bounds coincide and characterize the model. The resulting potential also supplies stationarity equations and a unified route to computing ground-state energy, training loss, and generalization error.