Going beyond stochastic gradient descent (SGD), what new phenomena emerge in wide neural networks trained by adaptive optimizers like Adam? Here we show: The same dichotomy between feature learning and kernel behaviors (as in SGD) holds for general optimizers as well, including Adam -- albeit with a nonlinear notion of "kernel." We derive the corresponding "neural tangent" and "maximal update" limits for any architecture. Two foundational advances underlie the above results: 1) A new Tensor Program language, NEXORT, that can express how adaptive optimizers process gradients into updates. 2) The introduction of bra-ket notation to drastically simplify expressions and calculations in Tensor Programs. This work summarizes and generalizes all previous results in the Tensor Programs series of papers.

翻译：超越随机梯度下降（SGD），在自适应优化器（如Adam）训练的宽神经网络中会出现哪些新现象？我们在此表明：特征学习与核行为之间的相同二分法（如同SGD中那样）也适用于一般优化器——包括Adam——尽管此时"核"的概念是非线性的。我们推导了任意架构下的相应"神经正切"和"最大更新"极限。以下两项基础性进展为上述结果奠定了基础：1）一种新的张量规划语言NEXORT，能够表达自适应优化器如何将梯度处理为更新；2）引入狄拉克符号，极大简化了张量规划中的表达式与计算。本工作总结并泛化了张量规划系列论文中的所有先前结果。