A case where a spindly two-layer linear network whips any neural network with a fully connected input layer

• URL: http://arxiv.org/abs/2010.08625v1
• Date: Fri, 16 Oct 2020 20:49:58 GMT
• Title: A case where a spindly two-layer linear network whips any neural network with a fully connected input layer
• Authors: Manfred K. Warmuth, Wojciech Kot{\l}owski, Ehsan Amid
• Abstract summary: We show that a sparse input layer is needed to sample efficiently learn sparse targets with gradient descent. Surprisingly the same type of problem can be solved drastically more efficient by a simple 2-layer linear neural network.
• Score: 24.132345589750592
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: It was conjectured that any neural network of any structure and arbitrary differentiable transfer functions at the nodes cannot learn the following problem sample efficiently when trained with gradient descent: The instances are the rows of a $d$-dimensional Hadamard matrix and the target is one of the features, i.e. very sparse. We essentially prove this conjecture: We show that after receiving a random training set of size $k < d$, the expected square loss is still $1-\frac{k}{(d-1)}$. The only requirement needed is that the input layer is fully connected and the initial weight vectors of the input nodes are chosen from a rotation invariant distribution. Surprisingly the same type of problem can be solved drastically more efficient by a simple 2-layer linear neural network in which the $d$ inputs are connected to the output node by chains of length 2 (Now the input layer has only one edge per input). When such a network is trained by gradient descent, then it has been shown that its expected square loss is $\frac{\log d}{k}$. Our lower bounds essentially show that a sparse input layer is needed to sample efficiently learn sparse targets with gradient descent when the number of examples is less than the number of input features.

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