An Analytical Formula of Population Gradient for two-layered ReLU network and its Applications in Convergence and Critical Point Analysis

Yuandong Tian
Proceedings of the 34th International Conference on Machine Learning, PMLR 70:3404-3413, 2017.

Abstract

In this paper, we explore theoretical properties of training a two-layered 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 (“out-of-plane“) are not isolated and form manifolds, and characterize in-plane critical-point-free regions for two-ReLU 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 upper-bounded 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 symmetric-breaking (SSB) in physics. We assume no independence of ReLU activations. Simulation verifies our findings.

Cite this Paper

BibTeX
@InProceedings{pmlr-v70-tian17a, title = {An Analytical Formula of Population Gradient for two-layered {R}e{LU} network and its Applications in Convergence and Critical Point Analysis}, author = {Yuandong Tian}, booktitle = {Proceedings of the 34th International Conference on Machine Learning}, pages = {3404--3413}, year = {2017}, editor = {Precup, Doina and Teh, Yee Whye}, volume = {70}, series = {Proceedings of Machine Learning Research}, month = {06--11 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v70/tian17a/tian17a.pdf}, url = {https://proceedings.mlr.press/v70/tian17a.html}, abstract = {In this paper, we explore theoretical properties of training a two-layered 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 (“out-of-plane“) are not isolated and form manifolds, and characterize in-plane critical-point-free regions for two-ReLU 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 upper-bounded 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 symmetric-breaking (SSB) in physics. We assume no independence of ReLU activations. Simulation verifies our findings.} }
Endnote
%0 Conference Paper %T An Analytical Formula of Population Gradient for two-layered ReLU network and its Applications in Convergence and Critical Point Analysis %A Yuandong Tian %B Proceedings of the 34th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2017 %E Doina Precup %E Yee Whye Teh %F pmlr-v70-tian17a %I PMLR %P 3404--3413 %U https://proceedings.mlr.press/v70/tian17a.html %V 70 %X In this paper, we explore theoretical properties of training a two-layered 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 (“out-of-plane“) are not isolated and form manifolds, and characterize in-plane critical-point-free regions for two-ReLU 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 upper-bounded 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 symmetric-breaking (SSB) in physics. We assume no independence of ReLU activations. Simulation verifies our findings.
APA
Tian, Y.. (2017). An Analytical Formula of Population Gradient for two-layered ReLU network and its Applications in Convergence and Critical Point Analysis. Proceedings of the 34th International Conference on Machine Learning, in Proceedings of Machine Learning Research 70:3404-3413 Available from https://proceedings.mlr.press/v70/tian17a.html.

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