Density Ratio Estimation with Conditional Probability Paths

Hanlin Yu, Arto Klami, Aapo Hyvarinen, Anna Korba, Omar Chehab

Abstract: Density ratio estimation in high dimensions can be reframed as integrating a certain quantity, the time score, over probability paths which interpolate between the two densities. In practice, the time score has to be estimated based on samples from the two densities. However, existing methods for this problem remain computationally expensive and can yield inaccurate estimates. Inspired by recent advances in generative modeling, we introduce a novel framework for time score estimation, based on a conditioning variable. Choosing the conditioning variable judiciously enables a closed-form objective function. We demonstrate that, compared to previous approaches, our approach results in faster learning of the time score and competitive or better estimation accuracies of the density ratio on challenging tasks. Furthermore, we establish theoretical guarantees on the error of the estimated density ratio.

Lay Summary:

One of the classical machine learning tasks is learning a probability density from a collection of observations (e.g. images). The density associates each observation with positive number, a probability, indicating how typical he observation is in the context of the data. This is a highly challenging problem because we need to associate such probabilities for all possible observations, not just the ones available in the data. Density estimation is needed within several machine learning methods, and the topic remains a highly active area of research in the field. Advances in accuracy or computational efficiency will help several applications, for instance generative AI.

We make contributions for an active line of density estimation research, for methods that solve the problem by contrasting the density of interest with some simpler density, learning the ratio of the associated probabilities. We identify both theoretical and practical limitations in recently proposed methods. We then present a new method that has theoretical guarantees on how accurately we can estimate the density, while also being faster to compute. Our paper focuses on the theory and experimental evaluation on relatively simple benchmark problems, while paving the way for larger scale applications.

Link To Code: https://github.com/ksnxr/dre-prob-paths

Primary Area: Probabilistic Methods

Keywords: score matching, density ratio estimation, probability paths

Submission Number: 6415