Inference via Interpolation: Contrastive Representations Provably Enable Planning and Inference

Benjamin Eysenbach, Vivek Myers, Russ Salakhutdinov, Sergey Levine

Keywords: contrastive learning, prediction, planning, inference, time-series

TL;DR: While prediction and planning over time series data is challenging, these problems have closed form solutions in terms of temporal contrastive representations

Abstract: Given time series data, how can we answer questions like ``what will happen in the future?'' and ``how did we get here?'' These sorts of probabilistic inference questions are challenging when observations are high-dimensional. In this paper, we show how these questions can have compact, closed form solutions in terms of learned representations. The key idea is to apply a variant of contrastive learning to time series data. Prior work already shows that the representations learned by contrastive learning encode a probability ratio. By extending prior work to show that the marginal distribution over representations is Gaussian, we can then prove that joint distribution of representations is also Gaussian. Taken together, these results show that representations learned via temporal contrastive learning follow a Gauss-Markov chain, a graphical model where inference (e.g., prediction, planning) over representations corresponds to inverting a low-dimensional matrix. In one special case, inferring intermediate representations will be equivalent to interpolating between the learned representations. We validate our theory using numerical simulations on tasks up to 46-dimensions.

Primary Area: Probabilistic methods (for example: variational inference, Gaussian processes)

Submission Number: 12940