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We provide new gradient-based methods for efficiently solving a broad classof ill-conditioned optimization problems. We consider the problem of minimizinga function $f : \mathbb{R}^d \rightarrow \mathbb{R}$ which is implicitlydecomposable as the sum of $m$ unknown non-interacting smooth, strongly convexfunctions and provide a method which solves this problem with a number ofgradient evaluations that scales (up to logarithmic factors) as the product ofthe square-root of the condition numbers of the components. This complexitybound (which we prove is nearly optimal) can improve almost exponentially onthat of accelerated gradient methods, which grow as the square root of thecondition number of $f$. Additionally, we provide efficient methods for solvingstochastic, quadratic variants of this multiscale optimization problem. Ratherthan learn the decomposition of $f$ (which would be prohibitively expensive),our methods apply a clean recursive "Big-Step-Little-Step" interleaving ofstandard methods. The resulting algorithms use $\tilde{\mathcal{O}}(d m)$space, are numerically stable, and open the door to a more fine-grainedunderstanding of the complexity of convex optimization beyond condition number.

Big-Step-Little-Step: Efficient Gradient Methods for Objectives with Multiple Scales