# Asymptotic Bayes risk for Gaussian mixture in a semi-supervised setting

## TL;DR

This paper analytically quantifies the performance gain in semi-supervised learning over supervised learning for Gaussian mixture models in high dimensions, using advanced mathematical tools from statistical physics.

## Contribution

It provides the first rigorous analysis of the asymptotic Bayes risk gap in semi-supervised Gaussian mixture models, leveraging recent high-dimensional inference theories.

## Key findings

- Quantifies the accuracy improvement due to unlabeled data.
- Provides explicit formulas for the Bayes risk gap.
- Demonstrates the impact of unlabeled data in high-dimensional settings.

## Abstract

Semi-supervised learning (SSL) uses unlabeled data for training and has been shown to greatly improve performance when compared to a supervised approach on the labeled data available. This claim depends both on the amount of labeled data available and on the algorithm used. In this paper, we compute analytically the gap between the best fully-supervised approach using only labeled data and the best semi-supervised approach using both labeled and unlabeled data. We quantify the best possible increase in performance obtained thanks to the unlabeled data, i.e. we compute the accuracy increase due to the information contained in the unlabeled data. Our work deals with a simple high-dimensional Gaussian mixture model for the data in a Bayesian setting. Our rigorous analysis builds on recent theoretical breakthroughs in high-dimensional inference and a large body of mathematical tools from statistical physics initially developed for spin glasses.