Fast Adaptive Tensor Methods Under Local Smoothness

Abstract

A new, fast adaptive regularization methods is proposed and analyzed under local Lipschitz smoothness of the $p$-th order tensor. For nonconvex problems, it achieves the optimal $\mathcal{O}\!\left(|\log(ε)|ε^{-(p+1)/p}\right)$ complexity to obtain first-order $ε$-stationary points and in the convex case, it yields $\mathcal{O}\!\left(|\log(ε)|ε^{-1/p}\right)$ iterations to drive the optimality gap below $ε$, thus matching the complexity bounds of standard tensor methods under global Lipschitz smoothness yp to logarithmic terms. The proposed algorithm follows the line of standard tensor methods with an appropriately chosen regularization and suitable modifications. Initial numerical experiments and comparisons for some nonconvex regression problems are made with the standard adaptive cubic regularization where we showcase some potential of the proposed method.