Convergence Rates of Zeroth Order Gradient Descent for ?ojasiewicz Functions

We prove convergence rates of Zeroth -order Gradient Descent (ZGD) algorithms for Lojasiewicz functions. Our results show that for smooth Lojasiewicz functions with Lojasiewicz exponent larger than 0.5 and smaller than 1, the functions values can converge much faster than the (zeroth -order) gradient descent trajectory. Similar results hold for convex nonsmooth Lojasiewicz functions.