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# On the Convergence of Gradient Flow on Multi-layer Linear Models

*Proceedings of the 40th International Conference on Machine Learning*, PMLR 202:24850-24887, 2023.

#### Abstract

In this paper, we analyze the convergence of gradient flow on a multi-layer linear model with a loss function of the form $f(W_1W_2\cdots W_L)$. We show that when $f$ satisfies the gradient dominance property, proper weight initialization leads to exponential convergence of the gradient flow to a global minimum of the loss. Moreover, the convergence rate depends on two trajectory-specific quantities that are controlled by the weight initialization: the

*imbalance matrices*, which measure the difference between the weights of adjacent layers, and the least singular value of the*weight product*$W=W_1W_2\cdots W_L$. Our analysis exploits the fact that the gradient of the overparameterized loss can be written as the composition of the non-overparametrized gradient with a time-varying (weight-dependent) linear operator whose smallest eigenvalue controls the convergence rate. The key challenge we address is to derive a uniform lower bound for this time-varying eigenvalue that lead to improved rates for several multi-layer network models studied in the literature.