# Variational Bounds for Perceptron Learning from Structured Data

arXiv: https://arxiv.org/abs/2608.04882 · HTML: https://mlsanity.com/paper/2608.04882/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 56 references that are in the map.
- First posted on arXiv 5 Aug 2026.

## Builds on (most cited first, 3 of 3)

- [Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning](https://mlsanity.com/paper/1810.01075/) (2018)
- [Fundamental limits of overparametrized shallow neural networks for supervised learning](https://mlsanity.com/paper/2307.05635/) (2023)
- [Characterizing Finite-Dimensional Posterior Marginals in High-Dimensional GLMs via Leave-One-Out](https://mlsanity.com/paper/2601.00091/) (2025)

## Abstract (arXiv)

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.