We analyze algorithms for approximating a function $f(x) = \Phi x$ mapping $\Re^d$ to $\Re^d$ using deep linear neural networks, i.e. that learn a function $h$ parameterized by matrices $\Theta_1,...,\Theta_L$ and defined by $h(x) = \Theta_L \Theta_{L-1} ... \Theta_1 x$. We focus on algorithms that learn through gradient descent on the population quadratic loss in the case that the distribution over the inputs is isotropic. We provide polynomial bounds on the number of iterations for gradient descent to approximate the optimum, in the case where the initial hypothesis $\Theta_1 = ... = \Theta_L = I$ has loss bounded by a small enough constant. On the other hand, we show that gradient descent fails to converge for $\Phi$ whose distance from the identity is a larger constant, and we show that some forms of regularization toward the identity in each layer do not help. If $\Phi$ is symmetric positive definite, we show that an algorithm that initializes $\Theta_i = I$ learns an $\epsilon$-approximation of $f$ using a number of updates polynomial in $L$, the condition number of $\Phi$, and $\log(d/\epsilon)$. In contrast, we show that if the target $\Phi$ is symmetric and has a negative eigenvalue, then all members of a class of algorithms that perform gradient descent with identity initialization, and optionally regularize toward the identity in each layer, fail to converge. We analyze an algorithm for the case that $\Phi$ satisfies $u^{\top} \Phi u > 0$ for all $u$, but may not be symmetric. This algorithm uses two regularizers: one that maintains the invariant $u^{\top} \Theta_L \Theta_{L-1} ... \Theta_1 u > 0$ for all $u$, and another that "balances" $\Theta_1 ... \Theta_L$ so that they have the same singular values.

本文介绍了使用梯度下降算法在深度线性神经网络中学习的近似函数的算法，并分析了在输入分布为各向同性时，使用多项式约束的梯度下降算法能多快地逼近最小二乘矩阵，以及针对不同情况下的正则化方法如何对收敛时效产生影响。此外，文章还探讨了对称正定矩阵条件下的算法，以及对于某些不对称矩阵的处理方式。

深度残差网络通过身份初始化高效学习正定线性变换的梯度下降