Faster Acceleration for Steepest Descent

Recent advances (Sherman, 2017; Sidford and Tian, 2018; Cohen et al., 2021) have overcome the fundamental barrier of dimension dependence in the iteration complexity of solving ℓ_∞ regression with first-order methods. Yet it remains unclear to what extent such acceleration can be achieved for general ℓp smooth functions. In this paper, we propose a new accelerated first-order method for convex optimization under non-Euclidean smoothness assumptions. In contrast to standard acceleration techniques, our approach uses primal-dual iterate sequences taken with respect to textitdiffering norms, which are then coupled using an textitimplicitly determined interpolation parameter. For ℓp norm smooth problems in d dimensions, our method provides an iteration complexity improvement of up to O(d^1-frac2p) in terms of calls to a first-order oracle, thereby allowing us to circumvent long-standing barriers in accelerated non-Euclidean steepest descent.