We analyze speed of convergence to global optimum for gradient descent
training a deep linear neural network (parameterized as $x \mapsto W_N W_{N-1}
\cdots W_1 x$) by minimizing the $\ell_2$ loss over whitened data. Convergence
at a linear rate is guaranteed when the following hold: (i) dimensions of
hidden layers are at least the minimum of the input and output dimensions; (ii)
weight matrices at initialization are approximately balanced; and (iii) the
initial loss is smaller than the loss of any rank-deficient solution. The
assumptions on initialization (conditions (ii) and (iii)) are necessary, in the
sense that violating any one of them may lead to convergence failure. Moreover,
in the important case of output dimension 1, i.e. scalar regression, they are
met, and thus convergence to global optimum holds, with constant probability
under a random initialization scheme. Our results significantly extend previous
analyses, e.g., of deep linear residual networks (Bartlett et al., 2018).

本文研究在白化数据上，通过梯度下降来训练深度线性神经网络收敛到全局最优点的速度。当隐藏层数的维度不小于输入输出维度的最小值，并且初始化的权重矩阵大致平衡且初始损失小于任何秩缺失解时，可保证线性收敛。此外，在输出维度为 1 的情况下，即标量回归，这些条件是满足的，并且在随机初始化方案下具有恒定的概率达到全局最优解。