A Minimum Variance Path Principle for Accurate and Stable Score-Based Density Ratio Estimation

TL;DR

This paper introduces the MVP principle to minimize path variance in score-based density ratio estimation, leading to more accurate and stable results by optimizing the entire training objective.

Contribution

It derives a closed-form expression for path variance, enabling data-adaptive, low-variance path learning with a flexible model, improving estimator stability and accuracy.

Findings

Achieves state-of-the-art results on benchmarks.

Provides a tractable optimization for path variance.

Demonstrates improved stability in density ratio estimation.

Abstract

Score-based methods are powerful across machine learning, but they face a paradox: theoretically path-independent, yet practically path-dependent. We resolve this by proving that practical training objectives differ from the ideal, ground-truth objective by a crucial, overlooked term: the path variance of the score function. We propose the MVP (**M**imum **V**ariance **P**ath) Principle to minimize this path variance. Our key contribution is deriving a closed-form expression for the variance, making optimization tractable. By parameterizing the path with a flexible Kumaraswamy Mixture Model, our method learns data-adaptive, low-variance paths without heuristic manual selection. This principled optimization of the complete objective yields more accurate and stable estimators, establishing new state-of-the-art results on challenging benchmarks and providing a general framework for optimizing score-based interpolation. Our code can be found in https://github.com/Hoemr/OpenDRE.git.