A Dynamical View on Optimization Algorithms of Overparameterized Neural Networks

• URL: http://arxiv.org/abs/2010.13165v2
• Date: Wed, 10 Mar 2021 17:28:13 GMT
• Title: A Dynamical View on Optimization Algorithms of Overparameterized Neural Networks
• Authors: Zhiqi Bu, Shiyun Xu, Kan Chen
• Abstract summary: We consider a broad class of optimization algorithms that are commonly used in practice. As a consequence, we can leverage the convergence behavior of neural networks. We believe our approach can also be extended to other optimization algorithms and network theory.
• Score: 23.038631072178735
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: When equipped with efficient optimization algorithms, the over-parameterized neural networks have demonstrated high level of performance even though the loss function is non-convex and non-smooth. While many works have been focusing on understanding the loss dynamics by training neural networks with the gradient descent (GD), in this work, we consider a broad class of optimization algorithms that are commonly used in practice. For example, we show from a dynamical system perspective that the Heavy Ball (HB) method can converge to global minimum on mean squared error (MSE) at a linear rate (similar to GD); however, the Nesterov accelerated gradient descent (NAG) may only converges to global minimum sublinearly. Our results rely on the connection between neural tangent kernel (NTK) and finite over-parameterized neural networks with ReLU activation, which leads to analyzing the limiting ordinary differential equations (ODE) for optimization algorithms. We show that, optimizing the non-convex loss over the weights corresponds to optimizing some strongly convex loss over the prediction error. As a consequence, we can leverage the classical convex optimization theory to understand the convergence behavior of neural networks. We believe our approach can also be extended to other optimization algorithms and network architectures.

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