Tight bounds on the smallest Eigenvalue of the neural tangent kernel for deep ReLU networks

conference paper Tight bounds on the smallest Eigenvalue of the neural tangent kernel for deep ReLU networks published yes Quynh Nguyen author Marco Mondelli author 27EB676C-8706-11E9-9510-7717E66974250000-0002-3242-7020 Guido Montufar author MaMo department ICML: International Conference on Machine Learning Prix Lopez-Loretta 2019 - Marco Mondelli project A recent line of work has analyzed the theoretical properties of deep neural networks via the Neural Tangent Kernel (NTK). In particular, the smallest eigenvalue of the NTK has been related to the memorization capacity, the global convergence of gradient descent algorithms and the generalization of deep nets. However, existing results either provide bounds in the two-layer setting or assume that the spectrum of the NTK matrices is bounded away from 0 for multi-layer networks. In this paper, we provide tight bounds on the smallest eigenvalue of NTK matrices for deep ReLU nets, both in the limiting case of infinite widths and for finite widths. In the finite-width setting, the network architectures we consider are fairly general: we require the existence of a wide layer with roughly order of N neurons, N being the number of data samples; and the scaling of the remaining layer widths is arbitrary (up to logarithmic factors). To obtain our results, we analyze various quantities of independent interest: we give lower bounds on the smallest singular value of hidden feature matrices, and upper bounds on the Lipschitz constant of input-output feature maps. https://research-explorer.ista.ac.at/download/13146/13155/2021_PMLR_Nguyen.pdf application/pdfno https://creativecommons.org/licenses/by/4.0/ ML Research Press2021Virtual eng Proceedings of the 38th International Conference on Machine Learning 2640-3498 9781713845065 2012.11654 1398119-8129 Nguyen, Q., Mondelli, M., & Montufar, G. (2021). Tight bounds on the smallest Eigenvalue of the neural tangent kernel for deep ReLU networks. In Proceedings of the 38th International Conference on Machine Learning (Vol. 139, pp. 8119–8129). Virtual: ML Research Press. Q. Nguyen, M. Mondelli, and G. Montufar, “Tight bounds on the smallest Eigenvalue of the neural tangent kernel for deep ReLU networks,” in Proceedings of the 38th International Conference on Machine Learning, Virtual, 2021, vol. 139, pp. 8119–8129. Nguyen, Quynh, Marco Mondelli, and Guido Montufar. “Tight Bounds on the Smallest Eigenvalue of the Neural Tangent Kernel for Deep ReLU Networks.” In Proceedings of the 38th International Conference on Machine Learning, 139:8119–29. ML Research Press, 2021. Nguyen, Quynh, et al. “Tight Bounds on the Smallest Eigenvalue of the Neural Tangent Kernel for Deep ReLU Networks.” Proceedings of the 38th International Conference on Machine Learning, vol. 139, ML Research Press, 2021, pp. 8119–29. Nguyen Q, Mondelli M, Montufar G. Tight bounds on the smallest Eigenvalue of the neural tangent kernel for deep ReLU networks. In: Proceedings of the 38th International Conference on Machine Learning. Vol 139. ML Research Press; 2021:8119-8129. Q. Nguyen, M. Mondelli, G. Montufar, in:, Proceedings of the 38th International Conference on Machine Learning, ML Research Press, 2021, pp. 8119–8129. Nguyen Q, Mondelli M, Montufar G. 2021. Tight bounds on the smallest Eigenvalue of the neural tangent kernel for deep ReLU networks. Proceedings of the 38th International Conference on Machine Learning. ICML: International Conference on Machine Learning vol. 139, 8119–8129. 131462023-06-18T22:00:48Z2025-07-10T11:50:36Z