The committee machine: Computational to statistical gaps in learning a two-layers neural network

TL;DR

This paper rigorously analyzes the statistical and computational limits of learning a two-layer neural network, revealing a significant gap where optimal learning is possible but computationally infeasible.

Contribution

It provides a rigorous justification of physics-inspired methods and introduces an AMP algorithm for the committee machine, highlighting a computational gap in learning.

Findings

Optimal generalization error is information-theoretically achievable.

AMP algorithm fails in certain regimes despite the possibility of low error.

Large computational gap identified in learning regimes.

Abstract

Heuristic tools from statistical physics have been used in the past to locate the phase transitions and compute the optimal learning and generalization errors in the teacher-student scenario in multi-layer neural networks. In this contribution, we provide a rigorous justification of these approaches for a two-layers neural network model called the committee machine. We also introduce a version of the approximate message passing (AMP) algorithm for the committee machine that allows to perform optimal learning in polynomial time for a large set of parameters. We find that there are regimes in which a low generalization error is information-theoretically achievable while the AMP algorithm fails to deliver it, strongly suggesting that no efficient algorithm exists for those cases, and unveiling a large computational gap.