A Generative Conditional Distribution Equality Testing Framework and Its Minimax Analysis

2025-09-22

AI Summary

This paper addresses the problem of testing equality of conditional distributions under covariate shift. To tackle this challenge, we propose a general framework that leverages generative modeling and sample splitting to reduce the conditional distribution test to an unconditional one. Building upon this framework, we introduce two novel nonparametric tests: a generative permutation test and a classification accuracy test. Both achieve the optimal minimax lower bound under nonparametric settings and are provably statistically consistent. Our theoretical analysis integrates shifted Rademacher complexity, function approximation theory, and convergence rates of conditional generative models. Extensive experiments on synthetic and real-world datasets demonstrate the effectiveness of our methods and their substantial improvement over existing baselines.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

In this paper, we propose a general framework for testing the equality of the conditional distributions in a two-sample problem. This problem is most relevant to transfer learning under covariate shift. Our framework is built on neural network-based generative methods and sample splitting techniques by transforming the conditional distribution testing problem into an unconditional one. We introduce two special tests: the generative permutation-based conditional distribution equality test and the generative classification accuracy-based conditional distribution equality test. Theoretically, we establish a minimax lower bound for statistical inference in testing the equality of two conditional distributions under certain smoothness conditions. We demonstrate that the generative permutation-based conditional distribution equality test and its modified version can attain this lower bound precisely or up to some iterated logarithmic factor. Moreover, we prove the testing consistency of the generative classification accuracy-based conditional distribution equality test. We also establish the convergence rate for the learned conditional generator by deriving new results related to the recently-developed offset Rademacher complexity and approximation properties using neural networks. Empirically, we conduct numerical studies including synthetic datasets and two real-world datasets, demonstrating the effectiveness of our approach.

Problem

• Tests equality of conditional distributions in two-sample problems
• Addresses transfer learning under covariate shift conditions
• Transforms conditional distribution testing into unconditional testing

Innovation

• Transforms conditional distribution testing into unconditional testing
• Uses neural network-based generative methods with sample splitting
• Develops permutation-based and classification accuracy-based tests