Tensor methods for finding approximate stationary points of convex functions

In this paper, we consider the problem of finding epsilon-approximate stationary points of convex functions that are p-times differentiable with nu-Holder continuous pth derivatives. We present tensor methods with and without acceleration. Specifically, we show that the non-accelerated schemes take at most O(epsilon(-1/(p+nu-1))) iterations to reduce the norm of the gradient of the objective below given epsilon is an element of (0, 1). For accelerated tensor schemes, we establish improved complexity bounds of O(epsilon(-(p+nu)/[(p+nu-1)(p+nu+1)])) and O (vertical bar log(epsilon)vertical bar epsilon(-1/(p+nu))), when the Holder parameter nu is an element of [0, 1] is known. For the case in which nu is unknown, we obtain a bound of O (epsilon(-(p+1)/[(p+nu-1)(p+2)])) for a universal accelerated scheme. Finally, we also obtain a lower complexity bound of O (epsilon(-2/[3(p+nu)-2])) for finding epsilon-approximate stationary points using p-order tensor methods.