Sharpness is an almost generic assumption in continuous optimization that bounds the distance from minima by objective function suboptimality. It facilitates the acceleration of first-order methods through restarts. However, sharpness involves problem-specific constants that are typically unknown, and restart schemes typically reduce convergence rates. Moreover, these schemes are challenging to apply in the presence of noise or with approximate model classes (e.g., in compressive imaging or learning problems), and they generally assume that the first-order method used produces feasible iterates. We consider the assumption of approximate sharpness, a generalization of sharpness that incorporates an unknown constant perturbation to the objective function error. This constant offers greater robustness (e.g., with respect to noise or relaxation of model classes) for finding approximate minimizers. By employing a new type of search over the unknown constants, we design a restart scheme that applies to general first-order methods and does not require the first-order method to produce feasible iterates. Our scheme maintains the same convergence rate as when the constants are known. The convergence rates we achieve for various first-order methods match the optimal rates or improve on previously established rates for a wide range of problems. We showcase our restart scheme in several examples and highlight potential future applications and developments of our framework and theory.

翻译：锐度是连续优化中一个几乎通用的假设，它通过目标函数次优性来界定与极小点的距离。该假设通过重启机制促进一阶方法的加速。然而，锐度涉及通常未知的问题特定常数，且重启方案通常会降低收敛速率。此外，这些方案在存在噪声或使用近似模型类（例如在压缩成像或学习问题中）时难以应用，并且通常假设所采用的一阶方法能产生可行迭代点。我们考虑近似锐度假设——这是对锐度概念的推广，其中包含对目标函数误差的未知常数扰动。该常数为寻找近似极小点提供了更强的鲁棒性（例如对噪声或模型类松弛的鲁棒性）。通过对未知常数进行新型搜索，我们设计了一种适用于通用一阶方法的重启方案，且不要求一阶方法必须产生可行迭代点。我们的方案在常数未知时仍能保持与已知常数时间等的收敛速率。我们为多种一阶方法实现的收敛速率，在广泛问题范围内达到了最优速率或改进了先前建立的速率。我们在若干示例中展示了重启方案的应用，并展望了本框架及理论潜在的未来应用与发展方向。