Error Exponent in Agnostic PAC Learning

TL;DR

This paper introduces an analysis of PAC learning using the error exponent from Information Theory, providing improved bounds on the exponential decay of error probability in agnostic binary classification.

Contribution

It establishes a new distribution-dependent error exponent for agnostic PAC learning, showing it can match realizable learning under certain stability conditions.

Findings

Improved error exponent bounds under stability assumptions

Agnostic learning can have the same error exponent as realizable learning

Application of error exponent analysis to knowledge distillation

Abstract

Statistical learning theory and the Probably Approximately Correct (PAC) criterion are the common approach to mathematical learning theory. PAC is widely used to analyze learning problems and algorithms, and have been studied thoroughly. Uniform worst case bounds on the convergence rate have been well established using, e.g., VC theory or Radamacher complexity. However, in a typical scenario the performance could be much better. In this paper, we consider PAC learning using a somewhat different tradeoff, the error exponent - a well established analysis method in Information Theory - which describes the exponential behavior of the probability that the risk will exceed a certain threshold as function of the sample size. We focus on binary classification and find, under some stability assumptions, an improved distribution dependent error exponent for a wide range of problems, establishing the exponential behavior of the PAC error probability in agnostic learning. Interestingly, under these assumptions, agnostic learning may have the same error exponent as realizable learning. The error exponent criterion can be applied to analyze knowledge distillation, a problem that so far lacks a theoretical analysis.