What can linearized neural networks actually say about generalization?
This work addresses the gap between theoretical NTK predictions and practical neural network performance, providing insights for researchers in deep learning theory and algorithm design.
The authors investigated the practical validity of linear approximations (neural tangent kernel) for neural networks by systematically comparing their behavior across tasks, finding that these approximations can rank task learning complexity but do not always outperform networks, with the gap depending on architecture, dataset size, and task, revealing a new implicit bias due to kernel evolution during training.
For certain infinitely-wide neural networks, the neural tangent kernel (NTK) theory fully characterizes generalization, but for the networks used in practice, the empirical NTK only provides a rough first-order approximation. Still, a growing body of work keeps leveraging this approximation to successfully analyze important deep learning phenomena and design algorithms for new applications. In our work, we provide strong empirical evidence to determine the practical validity of such approximation by conducting a systematic comparison of the behavior of different neural networks and their linear approximations on different tasks. We show that the linear approximations can indeed rank the learning complexity of certain tasks for neural networks, even when they achieve very different performances. However, in contrast to what was previously reported, we discover that neural networks do not always perform better than their kernel approximations, and reveal that the performance gap heavily depends on architecture, dataset size and training task. We discover that networks overfit to these tasks mostly due to the evolution of their kernel during training, thus, revealing a new type of implicit bias.