Rate-Distortion Bounds on Bayes Risk in Supervised Learning

We present an information-theoretic framework for bounding the number of labeled samples needed to train a classifier in a parametric Bayesian setting. We derive bounds on the average $Lp$ distance between the learned classifier and the true maximum a posteriori classifier, which are well-established surrogates for the excess classification error due to imperfect learning. We provide lower and upper bounds on the rate-distortion function, using $Lp$ loss as the distortion measure, of a maximum a priori classifier in terms of the differential entropy of the posterior distribution and a quantity called the interpolation dimension, which characterizes the complexity of the parametric distribution family. In addition to expressing the information content of a classifier in terms of lossy compression, the rate-distortion function also expresses the minimum number of bits a learning machine needs to extract from training data to learn a classifier to within a specified $Lp$ tolerance. We use results from universal source coding to express the information content in the training data in terms of the Fisher information of the parametric family and the number of training samples available. The result is a framework for computing lower bounds on the Bayes $Lp$ risk. This framework complements the well-known probably approximately correct (PAC) framework, which provides minimax risk bounds involving the Vapnik-Chervonenkis dimension or Rademacher complexity. Whereas the PAC framework provides upper bounds the risk for the worst-case data distribution, the proposed rate-distortion framework lower bounds the risk averaged over the data distribution. We evaluate the bounds for a variety of data models, including categorical, multinomial, and Gaussian models. In each case the bounds are provably tight orderwise, and in two cases we prove that the bounds are tight up to multiplicative constants.