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

We investigate the optimization dynamics of gradient descent in a non-convex and high-dimensional setting, with a focus on the phase retrieval problem as a case study for complex loss landscapes. We first study 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 $\alpha = M/N$. By analyzing how the local curvature changes during optimization, we uncover that for intermediate $\alpha$, the Hessian displays a downward direction pointing towards good minima in the first regime of the descent, before being trapped in bad minima at the end. Hence, the local landscape is benign and informative at first, before gradient descent brings the system into a uninformative maze. The transition between the two regimes is associated to a BBP-type threshold in the time-dependent Hessian. Through both theoretical analysis and numerical experiments, we show that in practical cases, i.e. for finite but even very large $N$, successful optimization via gradient descent in phase retrieval is achieved by falling towards the good minima before reaching the bad ones. This mechanism explains why successful recovery is obtained well before the algorithmic transition corresponding to the high-dimensional limit. Technically, this is associated to strong logarithmic corrections of the algorithmic transition at large $N$ with respect to the one expected in the $N\to\infty$ limit. Our analysis sheds light on such a new mechanism that facilitate gradient descent dynamics in finite large dimensions, also highlighting the importance of good initialization of spectral properties for optimization in complex high-dimensional landscapes.