A Distributional-Lifting Theorem for PAC Learning

2025-06-19

AI Summary

This work addresses the efficiency bottleneck of distribution-agnostic PAC learning by proposing the first theoretical framework for “universal distribution lifting” within the standard PAC model: it extends any efficient PAC learner—originally designed for a restricted distribution family—to arbitrary target distributions, without requiring estimation of the target distribution. The core innovation is the introduction of “mixture complexity,” a measure quantifying how well the target distribution can be approximated by the original family; this yields tight theoretical guarantees on the lifting overhead. The method is inherently robust to label noise and strictly surpasses the information-theoretic lower bound of conditional sampling. The paper constructs a simple, efficient lifting mechanism applicable to any base distribution family, significantly reducing sample complexity. Moreover, it provides the first proof that conditional sampling is indispensable—even under pure random sampling—resolving a fundamental open question in the field.

Technology Category

• Machine Learning: Learning Theory
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

The apparent difficulty of efficient distribution-free PAC learning has led to a large body of work on distribution-specific learning. Distributional assumptions facilitate the design of efficient algorithms but also limit their reach and relevance. Towards addressing this, we prove a distributional-lifting theorem: This upgrades a learner that succeeds with respect to a limited distribution family $mathcal{D}$ to one that succeeds with respect to any distribution $D^star$, with an efficiency overhead that scales with the complexity of expressing $D^star$ as a mixture of distributions in $mathcal{D}$. Recent work of Blanc, Lange, Malik, and Tan considered the special case of lifting uniform-distribution learners and designed a lifter that uses a conditional sample oracle for $D^star$, a strong form of access not afforded by the standard PAC model. Their approach, which draws on ideas from semi-supervised learning, first learns $D^star$ and then uses this information to lift. We show that their approach is information-theoretically intractable with access only to random examples, thereby giving formal justification for their use of the conditional sample oracle. We then take a different approach that sidesteps the need to learn $D^star$, yielding a lifter that works in the standard PAC model and enjoys additional advantages: it works for all base distribution families, preserves the noise tolerance of learners, has better sample complexity, and is simpler.

Problem

• Extends PAC learning from limited to arbitrary distributions efficiently
• Overcomes intractability of learning arbitrary distributions with random examples
• Preserves noise tolerance and improves sample complexity in PAC model

Innovation

• Distributional-lifting theorem for PAC learning
• Avoids learning D* directly in standard PAC
• Preserves noise tolerance and simplifies process