Title: Contrastive Predictive Coding Done Right for Mutual Information Estimation
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
Keywords: information estimation, contrastive predictive coding, representation learning, noise contrastive estimation, density ratio estimation

Abstract: The InfoNCE objective, originally introduced for contrastive representation learning, has become a popular choice for mutual information (MI) estimation, despite its indirect connection to MI. In this paper, we demonstrate why InfoNCE should not be regarded as a valid MI estimator, and we introduce a simple modification, which we refer to as *InfoNCE-anchor*, for accurate MI estimation. 
Our modification introduces an auxiliary \emph{anchor} class, enabling consistent density ratio estimation and yielding a plug-in MI estimator with significantly reduced bias.
Beyond this, we generalize our framework using proper scoring rules, which recover InfoNCE-anchor as a special case when the log score is employed. 
This formulation unifies a broad spectrum of contrastive objectives, including NCE, InfoNCE, and $f$-divergence variants, under a single principled framework. 
Empirically, we find that InfoNCE-anchor with the log score achieves the most accurate MI estimates; however, in self-supervised representation learning experiments, we find that the anchor does not improve the downstream task performance.
These findings corroborate that contrastive representation learning benefits not from accurate MI estimation per se, but from the learning of structured density ratios.

TL;DR: We propose a simple yet fundamental fix to InfoNCE for accurate and principled mutual information estimation.