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. In this talk, I will discuss some general mathematical principles allowing for efficient optimization in over-parameterized non-linear systems, a setting that includes deep neural networks. I will discuss that optimization problems corresponding to these systems are not convex, even locally, but instead satisfy the Polyak-Lojasiewicz (PL) condition on most of the parameter space, allowing for efficient optimization by gradient descent or SGD. I will connect the PL condition of these systems to the condition number associated with the tangent kernel and show how a non-linear theory for those systems parallels classical analyses of over-parameterized linear equations. As a separate related development, I will discuss a perspective on the remarkable recently discovered phenomenon of transition to linearity (constancy of NTK) in certain classes of large neural networks. I will show how this transition to linearity results from the scaling of the Hessian with the size of the network controlled by certain functional norms. Combining these ideas, I will show how the transition to linearity can be used to demonstrate the PL condition and convergence for a general class of wide neural networks. Finally, I will comment on systems that are ''almost'' over-parameterized, which appears to be common in practice.

Based on joint work with Chaoyue Liu and Libin Zhu