Depth creates no more spurious local minima

We show that for any convex differentiable loss, a deep linear network has no spurious local minima as long as it is true for the two layer case. This reduction greatly simplifies the study on the existence of spurious local minima in deep linear networks. When applied to the quadratic loss, our result immediately implies the powerful result in [Kawaguchi 2016]. Further, with the work in [Zhou and Liang 2018], we can remove all the assumptions in [Kawaguchi 2016]. This property holds for more general "multi-tower" linear networks too. Our proof builds on [Laurent and von Brecht 2018] and develops a new perturbation argument to show that any spurious local minimum must have full rank, a structural property which can be useful more generally.