Optimization-Based Separations for Neural Networks
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Depth separation results propose a possible theoretical explanation for the benefits of deep neural networks over shallower architectures, establishing that the former possess superior approximation capabilities. However, there are no known results in which the deeper architecture leverages this advantage into a provable optimization guarantee. We prove that when the data are generated by a distribution with radial symmetry which satisfies some mild assumptions, gradient descent can efficiently learn ball indicator functions using a depth 2 neural network with two layers of sigmoidal activations, and where the hidden layer is held fixed throughout training. Since it is known that ball indicators are hard to approximate with respect to a certain heavy-tailed distribution when using depth 2 networks with a single layer of non-linearities (Safran and Shamir, 2017), this establishes what is to the best of our knowledge, the first optimization-based separation result where the approximation benefits of the stronger architecture provably manifest in practice. Our proof technique relies on a random features approach which reduces the problem to learning with a single neuron, where new tools are required to show the convergence of gradient descent when the distribution of the data is heavy-tailed.
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