Complex Dynamics in Simple Neural Networks: Understanding Gradient Flow in Phase Retrieval
This addresses a fundamental issue in non-convex optimization for machine learning, providing insights into when gradient-based methods can avoid poor local minima, though it is incremental as it focuses on a specific problem.
The paper tackles the problem of gradient flow dynamics getting trapped in spurious minima during phase retrieval from random measurements, finding that above a critical measurement-to-dimension ratio, these minima become unstable, allowing the algorithm to escape and find the global minimum, as shown through analytical and numerical experiments.
Despite the widespread use of gradient-based algorithms for optimizing high-dimensional non-convex functions, understanding their ability of finding good minima instead of being trapped in spurious ones remains to a large extent an open problem. Here we focus on gradient flow dynamics for phase retrieval from random measurements. When the ratio of the number of measurements over the input dimension is small the dynamics remains trapped in spurious minima with large basins of attraction. We find analytically that above a critical ratio those critical points become unstable developing a negative direction toward the signal. By numerical experiments we show that in this regime the gradient flow algorithm is not trapped; it drifts away from the spurious critical points along the unstable direction and succeeds in finding the global minimum. Using tools from statistical physics we characterize this phenomenon, which is related to a BBP-type transition in the Hessian of the spurious minima.