We study the implicit regularization imposed by gradient descent for learning multi-layer homogeneous functions including feed-forward fully connected and convolutional deep neural networks with linear, ReLU or Leaky ReLU activation. We rigorously prove that gradient flow (i.e. gradient descent with infinitesimal step size) effectively enforces the differences between squared norms across different layers to remain invariant without any explicit regularization. This result implies that if the weights are initially small, gradient flow automatically balances the magnitudes of all layers. Using a discretization argument, we analyze gradient descent with positive step size for the non-convex low-rank asymmetric matrix factorization problem without any regularization. Inspired by our findings for gradient flow, we prove that gradient descent with step sizes $\eta_t = O\left(t^{-\left( \frac12+\delta\right)} \right)$ ($0<\delta\le\frac12$) automatically balances two low-rank factors and converges to a bounded global optimum. Furthermore, for rank-$1$ asymmetric matrix factorization we give a finer analysis showing gradient descent with constant step size converges to the global minimum at a globally linear rate. We believe that the idea of examining the invariance imposed by first order algorithms in learning homogeneous models could serve as a fundamental building block for studying optimization for learning deep models.

证明了通过梯度下降（以及正步长）学习多层同质函数时，该算法的梯度流可以有效地强制不同层之间的平方范数差异保持不变，从而自动平衡所有层的大小，由此可以深入研究利用一阶算法来优化学习深度模型的基本方法。

学习深度同质模型中的算法正则化：层次自动平衡