  The analysis of gradient descent-type methods typically relies on the
Lipschitz continuity of the objective gradient. This generally requires an
expensive hyperparameter tuning process to appropriately calibrate a stepsize
for a given problem. In this work we introduce a local first-order smoothness
oracle (LFSO) which generalizes the Lipschitz continuous gradients smoothness
condition and is applicable to any twice-differentiable function. We show that
this oracle can encode all relevant problem information for tuning stepsizes
for a suitably modified gradient descent method and give global and local
convergence results. We also show that LFSOs in this modified first-order
method can yield global linear convergence rates for non-strongly convex
problems with extremely flat minima, and thus improve over the lower bound on
rates achievable by general (accelerated) first-order methods.
