  Understanding the asymptotic behavior of gradient-descent training of deep
neural networks is essential for revealing inductive biases and improving
network performance. We derive the infinite-time training limit of a
mathematically tractable class of deep nonlinear neural networks, gated linear
networks (GLNs), and generalize these results to gated networks described by
general homogeneous polynomials. We study the implications of our results,
focusing first on two-layer GLNs. We then apply our theoretical predictions to
GLNs trained on MNIST and show how architectural constraints and the implicit
bias of gradient descent affect performance. Finally, we show that our theory
captures a substantial portion of the inductive bias of ReLU networks. By
making the inductive bias explicit, our framework is poised to inform the
development of more efficient, biologically plausible, and robust learning
algorithms.
