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Lower Complexity Bounds for Minimizing Regularized Functions

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 $ν\in [0, 1]$ 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.