A Distributional-Lifting Theorem for PAC Learning

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 D to one that succeeds with respect to any distribution D^, with an efficiency overhead that scales with the complexity of expressing D^ as a mixture of distributions in 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^, 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^ 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^, 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.

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