Information theoretic limits of learning a sparse rule

TL;DR

This paper investigates the fundamental limits of learning sparse signals in high-dimensional generalized linear models, revealing phase transitions in the minimum mean-square error and extending the all-or-nothing phenomenon beyond linear regression.

Contribution

It derives a variational formula for mutual information and MMSE in sparse high-dimensional models, demonstrating phase transitions and extending the all-or-nothing phenomenon to perceptron learning.

Findings

MMSE exhibits phase transitions with respect to sampling rate.

All-or-nothing phenomenon occurs in generalized linear models.

Results extend high-dimensional learning theory beyond linear regression.

Abstract

We consider generalized linear models in regimes where the number of nonzero components of the signal and accessible data points are sublinear with respect to the size of the signal. We prove a variational formula for the asymptotic mutual information per sample when the system size grows to infinity. This result allows us to derive an expression for the minimum mean-square error (MMSE) of the Bayesian estimator when the signal entries have a discrete distribution with finite support. We find that, for such signals and suitable vanishing scalings of the sparsity and sampling rate, the MMSE is nonincreasing piecewise constant. In specific instances the MMSE even displays an all-or-nothing phase transition, that is, the MMSE sharply jumps from its maximum value to zero at a critical sampling rate. The all-or-nothing phenomenon has previously been shown to occur in high-dimensional linear regression. Our analysis goes beyond the linear case and applies to learning the weights of a perceptron with general activation function in a teacher-student scenario. In particular, we discuss an all-or-nothing phenomenon for the generalization error with a sublinear set of training examples.