Mathematics of learning

TL;DR

This paper investigates the convergence of learning algorithms through the spectral analysis of random matrices, linking it to the roots of random polynomials and the behavior of harmonic means, with implications for understanding stability and tail behavior.

Contribution

It introduces a novel connection between learning algorithm convergence, random matrix eigenvalues, and the roots of random polynomials, extending to stable law convergence analysis.

Findings

Eigenvalue analysis of random matrices related to learning algorithms

Distribution of roots of random polynomials in [0,1]

Asymptotic behavior of harmonic means and stable laws

Abstract

We study the convergence properties of a pair of learning algorithms (learning with and without memory). This leads us to study the dominant eigenvalue of a class of random matrices. This turns out to be related to the roots of the derivative of random polynomials (generated by picking their roots uniformly at random in the interval [0, 1], although our results extend to other distributions). This, in turn, requires the study of the statistical behavior of the harmonic mean of random variables as above, which leads us to delicate question of the rate of convergence to stable laws and tail estimates for stable laws. The reader can find the proofs of most of the results announced here in the paper entitled "Harmonic mean, random polynomials, and random matrices", by the same authors.