  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 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 that sums of varied H\"older smooth functions
introduce no new complexities and require at most as many iterations as is
needed for minimizing each summand separately. Extensions to strongly convex
and H\"older growth settings as well as simple matching lower bounds are also
provided.
