2022/02/09 by Doikov, Nikita · 1 citation
#FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2202.04545
In this paper, we establish lower bounds for the oracle complexity of the first-order methods minimizing regularized convex functions. We consider the composite representation of the objective. The smooth part has Hölder continuous gradient of degree ν∈ [0, 1] and is accessible by a black-box local oracle. The composite part is a power of a norm. We prove that the best possible rate for the first-order methods in the large-scale setting for Euclidean norms is of the order O(k- p(1 + 3ν) / (2(p - 1 - ν))) for the functional residual, where k is the iteration counter and p is the power of regularization. Our formulation covers several cases, including computation of the Cubically regularized Newton step by the first-order gradient methods, in which case the rate becomes O(k-6). It can be achieved by the Fast Gradient Method. Thus, our result proves the latter rate to be optimal. We also discover lower complexity bounds for non-Euclidean norms.