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Computer Science > Machine Learning

arXiv:2106.10165 (cs)
[Submitted on 18 Jun 2021 (v1), last revised 24 Aug 2021 (this
version, v2)]

Title:The Principles of Deep Learning Theory

Authors:Daniel A. Roberts, Sho Yaida, Boris Hanin
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    Abstract: This book develops an effective theory approach to
    understanding deep neural networks of practical relevance.
    Beginning from a first-principles component-level picture of
    networks, we explain how to determine an accurate description of
    the output of trained networks by solving layer-to-layer
    iteration equations and nonlinear learning dynamics. A main
    result is that the predictions of networks are described by
    nearly-Gaussian distributions, with the depth-to-width aspect
    ratio of the network controlling the deviations from the
    infinite-width Gaussian description. We explain how these
    effectively-deep networks learn nontrivial representations from
    training and more broadly analyze the mechanism of representation
    learning for nonlinear models. From a nearly-kernel-methods
    perspective, we find that the dependence of such models'
    predictions on the underlying learning algorithm can be expressed
    in a simple and universal way. To obtain these results, we
    develop the notion of representation group flow (RG flow) to
    characterize the propagation of signals through the network. By
    tuning networks to criticality, we give a practical solution to
    the exploding and vanishing gradient problem. We further explain
    how RG flow leads to near-universal behavior and lets us
    categorize networks built from different activation functions
    into universality classes. Altogether, we show that the
    depth-to-width ratio governs the effective model complexity of
    the ensemble of trained networks. By using information-theoretic
    techniques, we estimate the optimal aspect ratio at which we
    expect the network to be practically most useful and show how
    residual connections can be used to push this scale to arbitrary
    depths. With these tools, we can learn in detail about the
    inductive bias of architectures, hyperparameters, and optimizers.

Comments:      471 pages, to be published by Cambridge University
               Press; v2: hyperlinks fixed, index added
               Machine Learning (cs.LG); Artificial Intelligence
Subjects:      (cs.AI); High Energy Physics - Theory (hep-th);
               Machine Learning (stat.ML)
Report number: MIT-CTP/5306
Cite as:       arXiv:2106.10165 [cs.LG]
               (or arXiv:2106.10165v2 [cs.LG] for this version)
               https://doi.org/10.48550/arXiv.2106.10165
               Focus to learn more
               arXiv-issued DOI via DataCite

Submission history

From: Sho Yaida [view email]
[v1] Fri, 18 Jun 2021 15:00:00 UTC (706 KB)
[v2] Tue, 24 Aug 2021 17:12:56 UTC (718 KB)
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