Tight Bounds on the Smallest Eigenvalue of the Neural Tangent Kernel for Deep ReLU Networks
This work provides theoretical guarantees for the global convergence and generalization of deep neural networks, which is crucial for researchers and practitioners relying on NTK theory.
This paper establishes tight bounds on the smallest eigenvalue of the Neural Tangent Kernel (NTK) for deep ReLU networks, addressing a gap in existing literature that primarily focused on two-layer networks or assumed non-zero eigenvalues for multi-layer networks. The bounds are provided for both infinite-width and finite-width settings, with the latter accommodating general architectures having at least one wide layer.
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.