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

FLAG n' FLARE: Fast Linearly-Coupled Adaptive Gradient Methods