Error Exponent in Agnostic PAC Learning

Statistical learning theory and the Probably Ap-proximately Correct (PAC) criterion are the common approach to mathematical learning theory. PAC is widely used to ana-lyze 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. Inter-estingly, 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.