Commenced in January 2007
Paper Count: 30528
Convergence Analysis of Training Two-Hidden-Layer Partially Over-Parameterized ReLU Networks via Gradient Descent
Authors: Zhifeng Kong
Abstract:Over-parameterized neural networks have attracted a great deal of attention in recent deep learning theory research, as they challenge the classic perspective of over-fitting when the model has excessive parameters and have gained empirical success in various settings. While a number of theoretical works have been presented to demystify properties of such models, the convergence properties of such models are still far from being thoroughly understood. In this work, we study the convergence properties of training two-hidden-layer partially over-parameterized fully connected networks with the Rectified Linear Unit activation via gradient descent. To our knowledge, this is the first theoretical work to understand convergence properties of deep over-parameterized networks without the equally-wide-hidden-layer assumption and other unrealistic assumptions. We provide a probabilistic lower bound of the widths of hidden layers and proved linear convergence rate of gradient descent. We also conducted experiments on synthetic and real-world datasets to validate our theory. Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 90
 Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016.
 Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. arXiv preprint arXiv:1605.07146, 2016.
 Levent Sagun, Utku Evci, V Ugur Guney, Yann Dauphin, and Leon Bottou. Empirical analysis of the hessian of over-parametrized neural networks. arXiv preprint arXiv:1706.04454, 2017.
 Ji Xu, Daniel J Hsu, and Arian Maleki. Benefits of over-parameterization with em. In Advances in Neural Information Processing Systems, pages 10662–10672, 2018.
 Yuanzhi Li and Yingyu Liang. Learning overparameterized neural networks via stochastic gradient descent on structured data. In Advances in Neural Information Processing Systems, pages 8157–8166, 2018.
 Simon S Du, Xiyu Zhai, Barnabas Poczos, and Aarti Singh. Gradient descent provably optimizes over-parameterized neural networks. arXiv preprint arXiv:1810.02054, 2018.
 Lili Su and Pengkun Yang. On learning over-parameterized neural networks: A functional approximation prospective. arXiv preprint arXiv:1905.10826, 2019.
 Xiaoxia Wu, Simon S Du, and Rachel Ward. Global convergence of adaptive gradient methods for an over-parameterized neural network. arXiv preprint arXiv:1902.07111, 2019.
 Samet Oymak and Mahdi Soltanolkotabi. Towards moderate overparameterization: global convergence guarantees for training shallow neural networks. arXiv preprint arXiv:1902.04674, 2019.
 Difan Zou, Yuan Cao, Dongruo Zhou, and Quanquan Gu. Stochastic gradient descent optimizes over-parameterized deep relu networks. arXiv preprint arXiv:1811.08888, 2018.
 Zeyuan Allen-Zhu, Yuanzhi Li, and Yingyu Liang. Learning and generalization in overparameterized neural networks, going beyond two layers. arXiv preprint arXiv:1811.04918, 2018.
 Zeyuan Allen-Zhu, Yuanzhi Li, and Zhao Song. A convergence theory for deep learning via over-parameterization. arXiv preprint arXiv:1811.03962, 2018.
 Difan Zou and Quanquan Gu. An improved analysis of training over-parameterized deep neural networks. arXiv preprint arXiv:1906.04688, 2019.
 Yann LeCun, L´eon Bottou, Yoshua Bengio, Patrick Haffner, et al. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
 Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747, 2017.
 Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.
 Alon Brutzkus, Amir Globerson, Eran Malach, and Shai Shalev-Shwartz. Sgd learns over-parameterized networks that provably generalize on linearly separable data. arXiv preprint arXiv:1710.10174, 2017.
 Arthur Jacot, Franck Gabriel, and Cl´ement Hongler. Neural tangent kernel: Convergence and generalization in neural networks. In Advances in neural information processing systems, pages 8571–8580, 2018.
 Bo Xie, Yingyu Liang, and Le Song. Diverse neural network learns true target functions. arXiv preprint arXiv:1611.03131, 2016.
 Russell Tsuchida, Farbod Roosta-Khorasani, and Marcus Gallagher. Invariance of weight distributions in rectified mlps. arXiv preprint arXiv:1711.09090, 2017.
 Kenji Kawaguchi. Deep learning without poor local minima. In Advances in neural information processing systems, pages 586–594, 2016.
 Moritz Hardt and Tengyu Ma. Identity matters in deep learning. arXiv preprint arXiv:1611.04231, 2016.
 Yi Zhou and Yingbin Liang. Critical points of neural networks: Analytical forms and landscape properties. arXiv preprint arXiv:1710.11205, 2017.
 Sanjeev Arora, Nadav Cohen, Noah Golowich, and Wei Hu. A convergence analysis of gradient descent for deep linear neural networks. arXiv preprint arXiv:1810.02281, 2018.
 Sanjeev Arora, Nadav Cohen, and Elad Hazan. On the optimization of deep networks: Implicit acceleration by overparameterization. arXiv preprint arXiv:1802.06509, 2018.
 Peter L Bartlett, David P Helmbold, and Philip M Long. Gradient descent with identity initialization efficiently learns positive-definite linear transformations by deep residual networks. Neural computation, 31(3):477–502, 2019.
 Samet Oymak and Mahdi Soltanolkotabi. Overparameterized nonlinear learning: Gradient descent takes the shortest path? arXiv preprint arXiv:1812.10004, 2018.
 Robert Devaney. An introduction to chaotic dynamical systems. CRC Press, 2018.