An Analytical Formula of Population Gradient for twolayered ReLU network and its Applications in Convergence and Critical Point Analysis
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Proceedings of the 34th International Conference on Machine Learning, PMLR 70:34043413, 2017.
Abstract
In this paper, we explore theoretical properties of training a twolayered ReLU network $g(\mathbf{x}; \mathbf{w}) = \sum_{j=1}^K \sigma(\mathbf{w}_j^\top\mathbf{x})$ with centered $d$dimensional spherical Gaussian input $\mathbf{x}$ ($\sigma$=ReLU). We train our network with gradient descent on $\mathbf{w}$ to mimic the output of a teacher network with the same architecture and fixed parameters $\mathbf{w}^*$. We show that its population gradient has an analytical formula, leading to interesting theoretical analysis of critical points and convergence behaviors. First, we prove that critical points outside the hyperplane spanned by the teacher parameters (“outofplane“) are not isolated and form manifolds, and characterize inplane criticalpointfree regions for twoReLU case. On the other hand, convergence to $\mathbf{w}^*$ for one ReLU node is guaranteed with at least $(1\epsilon)/2$ probability, if weights are initialized randomly with standard deviation upperbounded by $O(\epsilon/\sqrt{d})$, in accordance with empirical practice. For network with many ReLU nodes, we prove that an infinitesimal perturbation of weight initialization results in convergence towards $\mathbf{w}^*$ (or its permutation), a phenomenon known as spontaneous symmetricbreaking (SSB) in physics. We assume no independence of ReLU activations. Simulation verifies our findings.
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