We introduce a data-driven approach to analyze the performance of continuous optimization algorithms using generalization guarantees from statistical learning theory. We study classical and learned optimizers to solve families of parametric optimization problems. We build generalization guarantees for classical optimizers, using a sample convergence bound, and for learned optimizers, using the Probably Approximately Correct (PAC)-Bayes framework. To train learned optimizers, we use a gradient-based algorithm to directly minimize the PAC-Bayes upper bound. Numerical experiments in signal processing, control, and meta-learning showcase the ability of our framework to provide strong generalization guarantees for both classical and learned optimizers given a fixed budget of iterations. For classical optimizers, our bounds are much tighter than those that worst-case guarantees provide. For learned optimizers, our bounds outperform the empirical outcomes observed in their non-learned counterparts.

翻译：本文提出一种基于统计学习理论泛化保证的数据驱动方法，用于分析连续优化算法的性能。我们研究经典优化器与学习型优化器在求解参数化优化问题族时的表现。针对经典优化器，我们利用样本收敛界构建其泛化保证；针对学习型优化器，则采用概率近似正确（PAC）-贝叶斯框架建立保证。为训练学习型优化器，我们采用基于梯度的算法直接最小化PAC-贝叶斯上界。在信号处理、控制与元学习领域的数值实验表明，在给定有限迭代次数的条件下，本框架能为经典及学习型优化器提供强泛化保证。对于经典优化器，所得边界远优于最坏情况保证所提供的界限；对于学习型优化器，其边界表现优于非学习型对照方法中观测到的经验结果。