Multi-label Contrastive Predictive Coding

Variational mutual information (MI) estimators are widely used in unsupervised representation learning methods such as contrastive predictive coding (CPC). A lower bound on MI can be obtained from a multi-class classification problem, where a critic attempts to distinguish a positive sample drawn from the underlying joint distribution from $(m-1)$ negative samples drawn from a suitable proposal distribution. Using this approach, MI estimates are bounded above by $\log m$, and could thus severely underestimate unless $m$ is very large. To overcome this limitation, we introduce a novel estimator based on a multi-label classification problem, where the critic needs to jointly identify multiple positive samples at the same time. We show that using the same amount of negative samples, multi-label CPC is able to exceed the $\log m$ bound, while still being a valid lower bound of mutual information. We demonstrate that the proposed approach is able to lead to better mutual information estimation, gain empirical improvements in unsupervised representation learning, and beat a current state-of-the-art knowledge distillation method over 10 out of 13 tasks.