  Density-ratio estimation via classification is a cornerstone of unsupervised
learning. It has provided the foundation for state-of-the-art methods in
representation learning and generative modelling, with the number of use-cases
continuing to proliferate. However, it suffers from a critical limitation: it
fails to accurately estimate ratios p/q for which the two densities differ
significantly. Empirically, we find this occurs whenever the KL divergence
between p and q exceeds tens of nats. To resolve this limitation, we introduce
a new framework, telescoping density-ratio estimation (TRE), that enables the
estimation of ratios between highly dissimilar densities in high-dimensional
spaces. Our experiments demonstrate that TRE can yield substantial improvements
over existing single-ratio methods for mutual information estimation,
representation learning and energy-based modelling.
