Mathematics of Deep Learning Seminar: Mikhail Belkin

Title: The Polyak-Lojasiewicz condition as a framework for over-parameterized optimization and its application to deep learning

Abstract: 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 to 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 systems which are ”almost” over-parameterized, which appears to be common in practice.

Based on joint work with Chaoyue Liu and Libin Zhu