  Often in the analysis of first-order methods for both smooth and nonsmooth
optimization, assuming the existence of a growth/error bound or KL condition
facilitates much stronger convergence analysis. Hence separate analysis is
typically needed for the general case and for the growth bounded cases. We give
meta-theorems for deriving general convergence rates from those assuming a
growth lower bound. Applying this simple but conceptually powerful tool to the
proximal point, subgradient, bundle, dual averaging, gradient descent,
Frank-Wolfe, and universal accelerated methods immediately recovers their known
convergence rates for general convex optimization problems from their
specialized rates. New convergence results follow for bundle methods, dual
averaging, and Frank-Wolfe. Our results can lift any rate based on H\"older
continuous gradients and H\"older growth bounds. Moreover, our theory provides
simple proofs of optimal convergence lower bounds under H\"older growth from
textbook examples without growth bounds.
