We analyze algorithms for approximating a function f(x)=Φx mapping ℜd to ℜd using deep linear neural networks, that is, that learn a function h parameterized by matrices Θ1,…,ΘL and defined by h(x)=ΘLΘL-1…Θ1x. 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 least-squares matrix Φ, in the case where the initial hypothesis Θ1=…=ΘL=I has excess loss bounded by a small enough constant. We also show that gradient descent fails to converge for Φ 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 Φ is symmetric positive definite, we show that an algorithm that initializes Θi=I learns an ε-approximation of f using a number of updates polynomial in L, the condition number of Φ, and log(d/ε). In contrast, we show that if the least-squares matrix Φ 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 Φ satisfies u⊤Φu>0 for all u but may not be symmetric. This algorithm uses two regularizers: one that maintains the invariant u⊤ΘLΘL-1…Θ1u>0 for all u and the other that “balances” Θ1,…,ΘL so that they have the same singular values.
