Geometric Analysis of Non-Convex Optimization Landscapes for Robust M-Estimation of Location

In this paper, we study the classic problem of robust M-estimation of a location parameter. This problem involves minimizing a finite sum of non-convex loss functions. We investigate the geometric structure of the empirical non-convex objective. Under certain assumptions, we prove that the optimization landscape can be characterized by two favorable regions: a strong convex region within a ball centered at the minimum and a one-point strong convex region outside a ball centered at the minimum. Utilizing these results, we establish conditions under which the typically non-convex estimation problem possesses a unique global minimum that is close to the ground truth. By exploiting the favorable landscape properties, numerical methods such as gradient descent can achieve global convergence to the unique optimum from any starting point. Our theoretical conclusions are supported by numerical experiments.