Density Ratio Estimation with Conditional Probability Paths

This paper introduces a new method for density ratio estimation called conditional time score matching (CTSM). CTSM estimates the time score along a probability path connecting two densities instead of directly estimating the ratio of two densities. By conditioning on additional variables, the authors propose an easy-to-estimate objective and a faster variant. They further provide the theoretical guarantee of error bounds and then show strong empirical results, especially in high-dimensional settings.

update after rebuttal:

I'm maintaining my score.

The authors tackle the problem of estimating the ratio of two probability densities, improving upon speed and accuracy of prior work. They also establish a theorem on a guarantee of the error in the estimated density ratio. The method applies the "marginalization trick" (a la flow matching) to make the learning objective tractable. Essentially a latent variable is introduced to the stochastic interpolant between the test and target densities, and object modified to match the expectation of score matching marginalizing out thelatent. The distributions are chosen in a clever way similar to previous literature so that the objective is tractable (so called Conditional Time Score Matching objective). The authors also introduce a variant where a multivariate test and target distributions are broken up into n terms in an autoregressive fashion and the terms matched separately. A novel contribution is the design of a weighting function, time score normalization, to stabilize training. This could become a standard statistical method.

This paper proposes the conditional time score matching (CTSM) objective, which is a variant of the time score matching (TSM) objective proposed in the recent density ratio estimation literature. The CTSM objective provides a principled way to detour the computational drawback of TSM, which requires two automatic differentiation steps. The idea is to introduce a conditioning variable z, and the mathematical trick is essentially same as that of denoising score matching. The authors provide a vector variant that can be useful in practice. They also analyze the quality of the estimated density $\hat{p}_1(x)$, where the integration of the time score is approximated by a discretization, assuming that $p_0(x)$ is given. The experiments support that CTSM can perform as well as TSM with less computation. Overall, it is a well-written paper with well-executed analyses and experiments.

update after rebuttal

I thank the authors' response. I will keep my positive score. For the final version, it'd be helpful if the authors further comment on the choice of KL divergence for the analysis and its limitation.

This paper proposes the conditional time score matching (CTSM) objective, which is a variant of the time score matching (TSM) objective proposed in the recent density ratio estimation literature. The CTSM objective provides a principled way to detour the computational drawback of TSM, which requires two automatic differentiation steps. The idea is to introduce a conditioning variable and apply the "marginalization trick" (a la flow matching) to make the learning objective tractable.

All reviewers are unanimously positive about this paper: the density ratio estimation problem is important, the methodology is clever and sound, and the paper is well-written with with well-executed analyses and experiments. Based on the reviews and my own reading, I happily recommend acceptance.