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The Neural Tangent Kernel (NTK) has recently attracted intense study, as itdescribes the evolution of an over-parameterized Neural Network (NN) trained bygradient descent. However, it is now well-known that gradient descent is notalways a good optimizer for NNs, which can partially explain the unsatisfactorypractical performance of the NTK regression estimator. In this paper, weintroduce the Weighted Neural Tangent Kernel (WNTK), a generalized and improvedtool, which can capture an over-parameterized NN's training dynamics underdifferent optimizers. Theoretically, in the infinite-width limit, we prove: i)the stability of the WNTK at initialization and during training, and ii) theequivalence between the WNTK regression estimator and the corresponding NNestimator with different learning rates on different parameters. With theproposed weight update algorithm, both empirical and analytical WNTKsoutperform the corresponding NTKs in numerical experiments.

Weighted Neural Tangent Kernel: A Generalized and Improved Network-Induced Kernel