Extracting Relevant Information from Samples

The information bottleneck is an information theoretic framework, extending the classical notion of minimal sufficient statistics, that finds concise representations for an ‘input’ random variable that are as relevant as possible for an ‘output’ variable. This framework has been used successfully in various supervised and unsupervised applications. However, its learning theoretic properties and justification remained unclear as it differs from standard learning models in several crucial aspects, primarily its explicit reliance on the joint input-output distribution. In practice, an empirical plug-in estimate of the underlying distribution has been used, so far without any finite sample performance guarantees. In this paper we present several formal results that address these difficulties. We prove several non-uniform finite sample bounds that show that it can provide concise representations with good generalization based on smaller sample sizes than needed to estimate the underlying distribution. Based on these results, we can analyze the information bottleneck method as a learning algorithm in the familiar performance-complexity tradeoff framework. In addition, we formally describe the connection between the information bottleneck and minimal sufficient statistics.