Exact Lower Bounds for the Agnostic Probably-Approximately-Correct (PAC) Machine Learning Model

TL;DR

This paper derives exact non-asymptotic and asymptotic lower bounds on the minimax expected excess risk in the agnostic PAC classification model, revealing fundamental limits and optimal algorithms.

Contribution

It provides the first exact non-asymptotic lower bound on minimax EER and characterizes minimax algorithms as symmetric voting procedures, advancing theoretical understanding of PAC learning.

Findings

Exact non-asymptotic lower bound on minimax EER derived

Asymptotic lower bound of c_infinity/√ν established

Improved bounds on tail probability of excess risk obtained

Abstract

We provide an exact non-asymptotic lower bound on the minimax expected excess risk (EER) in the agnostic probably-ap\-proximately-correct (PAC) machine learning classification model and identify minimax learning algorithms as certain maximally symmetric and minimally randomized "voting" procedures. Based on this result, an exact asymptotic lower bound on the minimax EER is provided. This bound is of the simple form $c_\infty/\sqrt{\nu}$ as $\nu\to\infty$, where $c_\infty=0.16997\dots$ is a universal constant, $\nu=m/d$, $m$ is the size of the training sample, and $d$ is the Vapnik--Chervonenkis dimension of the hypothesis class. It is shown that the differences between these asymptotic and non-asymptotic bounds, as well as the differences between these two bounds and the maximum EER of any learning algorithms that minimize the empirical risk, are asymptotically negligible, and all these differences are due to ties in the mentioned "voting" procedures. A few easy to compute non-asymptotic lower bounds on the minimax EER are also obtained, which are shown to be close to the exact asymptotic lower bound $c_\infty/\sqrt{\nu}$ even for rather small values of the ratio $\nu=m/d$. As an application of these results, we substantially improve existing lower bounds on the tail probability of the excess risk. Among the tools used are Bayes estimation and apparently new identities and inequalities for binomial distributions.