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Sharpness is an almost generic assumption in continuous optimization thatbounds the distance from minima by objective function suboptimality. It leadsto the acceleration of first-order methods via restarts. However, sharpnessinvolves problem-specific constants that are typically unknown, and previousrestart schemes reduce convergence rates. Moreover, such schemes arechallenging to apply in the presence of noise or approximate model classes(e.g., in compressive imaging or learning problems), and typically assume thatthe first-order method used produces feasible iterates. We consider theassumption of approximate sharpness, a generalization of sharpness thatincorporates an unknown constant perturbation to the objective function error.This constant offers greater robustness (e.g., with respect to noise orrelaxation of model classes) for finding approximate minimizers. By employing anew type of search over the unknown constants, we design a restart scheme thatapplies to general first-order methods and does not require the first-ordermethod to produce feasible iterates. Our scheme maintains the same convergencerate as when assuming knowledge of the constants. The rates of convergence weobtain for various first-order methods either match the optimal rates orimprove on previously established rates for a wide range of problems. Weshowcase our restart scheme on several examples and point to futureapplications and developments of our framework and theory.

Restarts subject to approximate sharpness: A parameter-free and optimal scheme for first-order methods