On the Theory of Implicit Deep Learning: Global Convergence with Implicit Layers

• URL: http://arxiv.org/abs/2102.07346v2
• Date: Thu, 18 Feb 2021 18:39:14 GMT
• Title: On the Theory of Implicit Deep Learning: Global Convergence with Implicit Layers
• Authors: Kenji Kawaguchi
• Abstract summary: A deep equilibrium model uses implicit numerical sequence which are implicitly defined through an equilibrium point sequence of computation. We prove a relation between the dynamics of the deep implicit and the dynamics of the dynamics method of a shallow layer of trust.
• Score: 6.548580592686076
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A deep equilibrium model uses implicit layers, which are implicitly defined through an equilibrium point of an infinite sequence of computation. It avoids any explicit computation of the infinite sequence by finding an equilibrium point directly via root-finding and by computing gradients via implicit differentiation. In this paper, we analyze the gradient dynamics of deep equilibrium models with nonlinearity only on weight matrices and non-convex objective functions of weights for regression and classification. Despite non-convexity, convergence to global optimum at a linear rate is guaranteed without any assumption on the width of the models, allowing the width to be smaller than the output dimension and the number of data points. Moreover, we prove a relation between the gradient dynamics of the deep implicit layer and the dynamics of trust region Newton method of a shallow explicit layer. This mathematically proven relation along with our numerical observation suggests the importance of understanding implicit bias of implicit layers and an open problem on the topic. Our proofs deal with implicit layers, weight tying and nonlinearity on weights, and differ from those in the related literature.

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