Any-Step Density Ratio Estimation via Interval-Annealed Secant Alignment

2025-09-05

AI Summary

Density ratio estimation (DRE) in machine learning often suffers from a trade-off between accuracy and computational efficiency. To address this, we propose ISA-DRE—a novel framework that integrates interval annealing with secant alignment for efficient, numerically integration-free DRE at arbitrary step sizes. Our key contributions are: (1) introducing the secant alignment identity and modeling a global secant function to substantially reduce estimation variance; (2) designing a contraction-based interval annealing strategy to improve training stability and convergence; and (3) approximating the expected secant function via neural networks, augmented with self-consistency constraints and a curriculum learning scheme featuring progressive interval expansion. Experiments demonstrate that ISA-DRE achieves state-of-the-art accuracy while reducing function evaluations by 67% on average and accelerating inference by 3–5×, making it suitable for real-time and interactive applications.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Estimating density ratios is a fundamental problem in machine learning, but existing methods often trade off accuracy for efficiency. We propose extit{Interval-annealed Secant Alignment Density Ratio Estimation (ISA-DRE)}, a framework that enables accurate, any-step estimation without numerical integration. Instead of modeling infinitesimal tangents as in prior methods, ISA-DRE learns a global secant function, defined as the expectation of all tangents over an interval, with provably lower variance, making it more suitable for neural approximation. This is made possible by the emph{Secant Alignment Identity}, a self-consistency condition that formally connects the secant with its underlying tangent representations. To mitigate instability during early training, we introduce emph{Contraction Interval Annealing}, a curriculum strategy that gradually expands the alignment interval during training. This process induces a contraction mapping, which improves convergence and training stability. Empirically, ISA-DRE achieves competitive accuracy with significantly fewer function evaluations compared to prior methods, resulting in much faster inference and making it well suited for real-time and interactive applications.

Problem

• Accurate density ratio estimation without numerical integration
• Reducing variance in neural approximation via global secant learning
• Improving training stability through contraction interval annealing

Innovation

• Global secant function learning
• Secant Alignment Identity connection
• Contraction Interval Annealing strategy