Despite their immense promise in performing a variety of learning tasks, a theoretical understanding of the effectiveness and limitations of Deep Neural Networks (DNNs) has so far eluded practitioners. This is partly due to the inability to determine the closed forms of the learned functions, making it harder to assess their precise dependence on the training data and to study their generalization properties on unseen datasets. Recent work has shown that randomly initialized DNNs in the infinite width limit converge to kernel machines relying on a Neural Tangent Kernel (NTK) with known closed form. These results suggest, and experimental evidence corroborates, that empirical kernel machines can also act as surrogates for finite width DNNs. The high computational cost of assembling the full NTK, however, makes this approach infeasible in practice, motivating the need for low-cost approximations. In the current work, we study the performance of the Conjugate Kernel (CK), an efficient approximation to the NTK that has been observed to yield fairly similar results. For the regression problem of smooth functions and classification using logistic regression, we show that the CK performance is only marginally worse than that of the NTK and, in certain cases, is shown to be superior. In particular, we establish bounds for the relative test losses, verify them with numerical tests, and identify the regularity of the kernel as the key determinant of performance. In addition to providing a theoretical grounding for using CKs instead of NTKs, our framework provides insights into understanding the robustness of the various approximants and suggests a recipe for improving DNN accuracy inexpensively. We present a demonstration of this on the foundation model GPT-2 by comparing its performance on a classification task using a conventional approach and our prescription.

通过研究神经切线核在代替有限宽度深度神经网络中的性能表现，揭示了核的正则性是性能的关键决定因素，从而提出了一种廉价改进深度神经网络准确性的方法。这一理论框架不仅为使用共轭核代替神经切线核提供了理论基础，而且揭示了不同近似方法的稳健性，为提高深度神经网络的准确性提供了建议。

基于神经网络的回归的高效核替代模型