We analyze algorithms for approximating a function f (x) = x mapping d to d using deep linear neural networks, that is, that learn a function h parameterized by matrices 1 , . . . , L and defined by h(x) = L L−1 . . . 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 least-squares matrix , in the case where the initial hypothesis 1 = . . . = L = I has excess loss bounded by a small enough constant. We also show that gradient descent fails to converge for 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 is symmetric positive definite, we show that an algorithm that initializes i = I learns anapproximation of f using a number of updates polynomial in L, the condition number of , and log(d/ ). In contrast, we show that if the least-squares matrix 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 satisfies u u > 0 for all u but may not be symmetric. This algorithm uses two regularizers: one that maintains the invariant u L L−1 . . . 1 u > 0 for all u and the other that "balances" 1 , . . . , L so that they have the same singular values.
More recently, REF proved convergence results for gradient descent on such problems, assuming the inputs are isotropic and the target linear mapping is symmetric and positive definite.
Gradient Descent with Identity Initialization Efficiently Learns Positive-Definite Linear Transformations by Deep Residual Networks
Neural Computation
Neural Computation
Computer Science Mathematics Medicine