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We give a polynomial-time algorithm for learning neural networks with one layer of sigmoids feeding into any Lipschitz, monotone activation function (e.g., sigmoid or ReLU). We make no assumptions on the structure of the network, and the algorithm succeeds with respect to {\em any} distribution on the unit ball in n dimensions (hidden weight vectors also have unit norm). This is the first assumption-free, provably efficient algorithm for learning neural networks with two nonlinear layers. Our algorithm-- Alphatron-- is a simple, iterative update rule that combines isotonic regression with kernel methods. It outputs a hypothesis that yields efficient oracle access to interpretable features. It also suggests a new approach to Boolean learning problems via real-valued conditional-mean functions, sidestepping traditional hardness results from computational learning theory. Along these lines, we subsume and improve many longstanding results for PAC learning Boolean functions to the more general, real-valued setting of {\em probabilistic concepts}, a model that (unlike PAC learning) requires non-i.i.d. noise-tolerance.
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A Distributional-Lifting Theorem for Learning
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.
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- Award ID(s):
- 2310818
- PAR ID:
- 10677375
- Publisher / Repository:
- 38th Annual Conference on Learning Theory (COLT 2025)
- Date Published:
- Format(s):
- Medium: X
- Location:
- Lyon, France
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Conference Paper:
- The DOI is not currently available.
