Self-Regularity of Non-Negative Output Weights for Overparameterized Two-Layer Neural Networks

• URL: http://arxiv.org/abs/2103.01887v1
• Date: Tue, 2 Mar 2021 17:36:03 GMT
• Title: Self-Regularity of Non-Negative Output Weights for Overparameterized Two-Layer Neural Networks
• Authors: David Gamarnik, Eren C. K{\i}z{\i}lda\u{g}, and Ilias Zadik
• Abstract summary: We consider the problem of finding a two-layer neural network with sigmoid, rectified linear unit (ReLU) We then leverage our bounds to establish guarantees for such networks through emphfat-shattering dimension Notably, our bounds also have good sample complexity (polynomials in $d$ with a low degree)
• Score: 16.64116123743938
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the problem of finding a two-layer neural network with sigmoid, rectified linear unit (ReLU), or binary step activation functions that "fits" a training data set as accurately as possible as quantified by the training error; and study the following question: \emph{does a low training error guarantee that the norm of the output layer (outer norm) itself is small?} We answer affirmatively this question for the case of non-negative output weights. Using a simple covering number argument, we establish that under quite mild distributional assumptions on the input/label pairs; any such network achieving a small training error on polynomially many data necessarily has a well-controlled outer norm. Notably, our results (a) have a polynomial (in $d$) sample complexity, (b) are independent of the number of hidden units (which can potentially be very high), (c) are oblivious to the training algorithm; and (d) require quite mild assumptions on the data (in particular the input vector $X\in\mathbb{R}^d$ need not have independent coordinates). We then leverage our bounds to establish generalization guarantees for such networks through \emph{fat-shattering dimension}, a scale-sensitive measure of the complexity class that the network architectures we investigate belong to. Notably, our generalization bounds also have good sample complexity (polynomials in $d$ with a low degree), and are in fact near-linear for some important cases of interest.

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