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Gradient descent efficiently learns positive definite deep linear residual networks

Peter L. Bartlett
David P. Helmbold
Philip M. Long
ICML (2018)


We analyze algorithms that aim to approximate a function $f(x) = \Phi x$ mapping $\Re^d$ to $\Re^d$ using deep linear neural networks, i.e.\ a function $h$ parameterized by matrices $\Theta_1,...,\Theta_L$ 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 $\Phi$ is symmetric and has a negative eigenvalue, that 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 that maintain the invariants that $u^{\top} \Theta_L \Theta_{L-1} ... \Theta_1 u > 0$ for all $u$, and that ``balance'' $\Theta_1 ... \Theta_L$ so that they have the same singular values.