On the Power of Overparametrization in Neural Networks with Quadratic Activation
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Proceedings of the 35th International Conference on Machine Learning, PMLR 80:13291338, 2018.
Abstract
We provide new theoretical insights on why overparametrization is effective in learning neural networks. For a $k$ hidden node shallow network with quadratic activation and $n$ training data points, we show as long as $ k \ge \sqrt{2n}$, overparametrization enables local search algorithms to find a globally optimal solution for general smooth and convex loss functions. Further, despite that the number of parameters may exceed the sample size, using theory of Rademacher complexity, we show with weight decay, the solution also generalizes well if the data is sampled from a regular distribution such as Gaussian. To prove when $k\ge \sqrt{2n}$, the loss function has benign landscape properties, we adopt an idea from smoothed analysis, which may have other applications in studying loss surfaces of neural networks.
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