  We consider first order gradient methods for effectively optimizing a
composite objective in the form of a sum of smooth and, potentially, non-smooth
functions. We present accelerated and adaptive gradient methods, called FLAG
and FLARE, which can offer the best of both worlds. They can achieve the
optimal convergence rate by attaining the optimal first-order oracle complexity
for smooth convex optimization. Additionally, they can adaptively and
non-uniformly re-scale the gradient direction to adapt to the limited curvature
available and conform to the geometry of the domain. We show theoretically and
empirically that, through the compounding effects of acceleration and
adaptivity, FLAG and FLARE can be highly effective for many data fitting and
machine learning applications.
