Asymptotic Learning Curve and Renormalizable Condition in Statistical Learning Theory

TL;DR

This paper introduces a renormalizable condition in statistical learning theory that guarantees universal asymptotic learning curves, connecting concepts from Bayesian statistics and physics, and explores cases where this universality does not hold.

Contribution

It defines a renormalizable condition ensuring universal learning curves and analyzes nonrenormalizable cases with different asymptotic behaviors.

Findings

Renormalizable condition guarantees universal asymptotic law.

Universal law holds even for unrealizable and singular true distributions.

Nonrenormalizable cases exhibit different asymptotic behaviors.

Abstract

Bayes statistics and statistical physics have the common mathematical structure, where the log likelihood function corresponds to the random Hamiltonian. Recently, it was discovered that the asymptotic learning curves in Bayes estimation are subject to a universal law, even if the log likelihood function can not be approximated by any quadratic form. However, it is left unknown what mathematical property ensures such a universal law. In this paper, we define a renormalizable condition of the statistical estimation problem, and show that, under such a condition, the asymptotic learning curves are ensured to be subject to the universal law, even if the true distribution is unrealizable and singular for a statistical model. Also we study a nonrenormalizable case, in which the learning curves have the different asymptotic behaviors from the universal law.