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