Fast learning rates for plug-in classifiers

TL;DR

This paper demonstrates that plug-in classifiers can achieve super-fast convergence rates under the margin assumption, surpassing previous conjectures and establishing optimality bounds.

Contribution

It disproves two conjectures by showing plug-in classifiers can attain super-fast rates and provides minimax lower bounds confirming these rates are optimal.

Findings

Plug-in classifiers can achieve rates faster than $n^{-1}$.

Super-fast convergence rates are possible under the margin assumption.

Minimax lower bounds confirm the optimality of these rates.

Abstract

It has been recently shown that, under the margin (or low noise) assumption, there exist classifiers attaining fast rates of convergence of the excess Bayes risk, that is, rates faster than $n^{-1/2}$. The work on this subject has suggested the following two conjectures: (i) the best achievable fast rate is of the order $n^{-1}$, and (ii) the plug-in classifiers generally converge more slowly than the classifiers based on empirical risk minimization. We show that both conjectures are not correct. In particular, we construct plug-in classifiers that can achieve not only fast, but also super-fast rates, that is, rates faster than $n^{-1}$. We establish minimax lower bounds showing that the obtained rates cannot be improved.