Nearly Optimal First-Order Methods for Convex Optimization under Gradient Norm Measure: an Adaptive Regularization Approach

In the development of first-order methods for smooth (resp., composite) convex optimization problems, where functions with Lipschitz continuous gradients are minimized, gradient mapping) norm becomes a fundamental optimality measure. Under this measure, fixed iteration algorithm optimal complexity case is known, while determining number to obtain desired accuracy requires prior knowledge distance from initial point solution set. paper, we report an adaptive regularization approach, which attains nearly without knowing To further faster convergence adaptively, secondly apply approach construct method that Hölderian error bound condition (or equivalently, ?ojasiewicz property), covers moderately wide classes applications. The proposed respect mapping norm.