This work considers minimizing a sum of convex functions, each with potentially different structure ranging from nonsmooth to smooth, Lipschitz to non-Lipschitz. Nesterov’s universal fast gradient method (Nesterov, Math Program 152:381–404, 2015) provides an optimal black-box first-order method for minimizing a single function that takes advantage of any continuity structure present without requiring prior knowledge. In this paper, we show that this landmark method (without modification) further adapts to heterogeneous sums. For example, it minimizes the sum of a nonsmooth M-Lipschitz function and an L-smooth function at a rate of \( O(M^2/\epsilon ^2 + \sqrt{L/\epsilon }) \) without knowledge of M, L, or even that the objective was a sum of two terms. This rate is precisely the sum of the optimal convergence rates for each term’s individual complexity class. More generally, we show sums of varied Hölder smooth functions introduce no new complexities and require at most as many iterations as is needed to minimize each summand separately. Extensions to strongly convex and growth/error bounds are also provided.

这项工作考虑最小化凸函数的总和，每个凸函数具有潜在的不同结构，从非光滑到光滑，从利普希茨到非利普希茨。Nesterov 的通用快速梯度方法（Nesterov，Math Program 152：381–404，2015）提供了一种最佳的黑盒一阶方法，用于最小化单个函数，该方法利用现有的任何连续性结构，而无需先验知识。在本文中，我们证明了这种具有里程碑意义的方法（未经修改）进一步适应异构和。例如，它在不知情的情况下以\( O(M^2/\epsilon ^2 + \sqrt{L/\epsilon }) \) 的速率最小化非平滑M -Lipschitz 函数和L -平滑函数的总和中号、中号，甚至目标是两项之和。该速率恰好是每个项的单独复杂性类别的最佳收敛速率的总和。更一般地说，我们展示了各种 Hölder 平滑函数的总和，不会引入新的复杂性，并且最多需要与分别最小化每个被加数所需的次数相同的迭代。还提供了对强凸边界和增长/误差边界的扩展。