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We study the dynamics of gradient descent on objective functions of the form$f(\prod_{i=1}^{k} w_i)$ (with respect to scalar parameters $w_1,\ldots,w_k$),which arise in the context of training depth-$k$ linear neural networks. Weprove that for standard random initializations, and under mild assumptions on$f$, the number of iterations required for convergence scales exponentiallywith the depth $k$. We also show empirically that this phenomenon can occur inhigher dimensions, where each $w_i$ is a matrix. This highlights a potentialobstacle in understanding the convergence of gradient-based methods for deeplinear neural networks, where $k$ is large.

Exponential Convergence Time of Gradient Descent for One-Dimensional Deep Linear Neural Networks