Paper ID: 2403.02418

From Zero to Hero: How local curvature at artless initial conditions leads away from bad minima

Tony Bonnaire, Giulio Biroli, Chiara Cammarota

We provide an analytical study of the evolution of the Hessian during gradient descent dynamics, and relate a transition in its spectral properties to the ability of finding good minima. We focus on the phase retrieval problem as a case study for complex loss landscapes. We first characterize the high-dimensional limit where both the number $M$ and the dimension $N$ of the data are going to infinity at fixed signal-to-noise ratio $\alpha = M/N$. For small $\alpha$, the Hessian is uninformative with respect to the signal. For $\alpha$ larger than a critical value, the Hessian displays at short-times a downward direction pointing towards good minima. While descending, a transition in the spectrum takes place: the direction is lost and the system gets trapped in bad minima. Hence, the local landscape is benign and informative at first, before gradient descent brings the system into a uninformative maze. Through both theoretical analysis and numerical experiments, we show that this dynamical transition plays a crucial role for finite (even very large) $N$: it allows the system to recover the signal well before the algorithmic threshold corresponding to the $N\rightarrow\infty$ limit. Our analysis sheds light on this new mechanism that facilitates gradient descent dynamics in finite dimensions, and highlights the importance of a good initialization based on spectral properties for optimization in complex high-dimensional landscapes.

Submitted: Mar 4, 2024