Adaptive Optimization in the $\infty$-Width LimitDownload PDF

Published: 01 Feb 2023, Last Modified: 02 Mar 2023ICLR 2023 posterReaders: Everyone
Keywords: Infinite width, neural tangent kernels, feature learning, theory, adaptive optimization, tensor programs
TL;DR: We derive the infinite width limits of neural networks trained with adaptive optimizers
Abstract: Recent works have developed detailed understanding of large neural networks' behaviors via their infinite-width limits, e.g., the neural tangent kernel (NTK) and the feature learning ($\mu$) limits. These theories were developed for stochastic gradient descent. Yet, in practice, all large NN are trained using Adam or other adaptive gradient optimizers (AGO), which are not covered by such previous works. Here, we close this gap via the Tensor Programs framework. Specifically, for deep MLPs, we derive the NTK and $\mu$ parametrizations as well as their infinite-width limits. We find 1) The NTK limit of AGO, in contrast to that of SGD, now depends nonlinearly on the loss derivative but nevertheless still fails to learn features; 2) this is fixed by the $\mu$ limit of AGO (as in the case of SGD). To obtain these results, we extend the Tensor Programs language with a new instruction that allows one to express the gradient processing done by AGOs.
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