  The success of deep learning is due, to a large extent, to the remarkable
effectiveness of gradient-based optimization methods applied to large neural
networks. The purpose of this work is to propose a modern view and a general
mathematical framework for loss landscapes and efficient optimization in
over-parameterized machine learning models and systems of non-linear equations,
a setting that includes over-parameterized deep neural networks. Our starting
observation is that optimization problems corresponding to such systems are
generally not convex, even locally. We argue that instead they satisfy PL$^*$,
a variant of the Polyak-Lojasiewicz condition on most (but not all) of the
parameter space, which guarantees both the existence of solutions and efficient
optimization by (stochastic) gradient descent (SGD/GD). The PL$^*$ condition of
these systems is closely related to the condition number of the tangent kernel
associated to a non-linear system showing how a PL$^*$-based non-linear theory
parallels classical analyses of over-parameterized linear equations. We show
that wide neural networks satisfy the PL$^*$ condition, which explains the
(S)GD convergence to a global minimum. Finally we propose a relaxation of the
PL$^*$ condition applicable to "almost" over-parameterized systems.
