  Recent contrastive representation learning methods rely on estimating mutual
information (MI) between multiple views of an underlying context. E.g., we can
derive multiple views of a given image by applying data augmentation, or we can
split a sequence into views comprising the past and future of some step in the
sequence. Contrastive lower bounds on MI are easy to optimize, but have a
strong underestimation bias when estimating large amounts of MI. We propose
decomposing the full MI estimation problem into a sum of smaller estimation
problems by splitting one of the views into progressively more informed
subviews and by applying the chain rule on MI between the decomposed views.
This expression contains a sum of unconditional and conditional MI terms, each
measuring modest chunks of the total MI, which facilitates approximation via
contrastive bounds. To maximize the sum, we formulate a contrastive lower bound
on the conditional MI which can be approximated efficiently. We refer to our
general approach as Decomposed Estimation of Mutual Information (DEMI). We show
that DEMI can capture a larger amount of MI than standard non-decomposed
contrastive bounds in a synthetic setting, and learns better representations in
a vision domain and for dialogue generation.
