We provide a novel computer-assisted technique for systematically analyzing first-order methods for optimization. In contrast with previous works, the approach is particularly suited for handling sublinear convergence rates and stochastic oracles. The technique relies on semidefinite programming and potential functions. It allows simultaneously obtaining worst-case guarantees on the behavior of those algorithms, and assisting in choosing appropriate parameters for tuning their worst-case performances. The technique also benefits from comfortable tightness guarantees, meaning that unsatisfactory results can be improved only by changing the setting. We use the approach for analyzing deterministic and stochastic first-order methods under different assumptions on the nature of the stochastic noise. Among others, we treat unstructured noise with bounded variance, different noise models arising in over-parametrized expectation minimization problems, and randomized block-coordinate descent schemes.

中文翻译：

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随机一阶方法：通过势函数进行非渐近和计算机辅助分析

我们提供了一种新颖的计算机辅助技术，用于系统地分析一阶优化方法。与之前的工作相比，该方法特别适合处理次线性收敛率和随机预言。该技术依赖于半定规划和势函数。它允许同时获得对这些算法行为的最坏情况保证，并协助选择适当的参数来调整它们的最坏情况性能。该技术还受益于舒适的密封性保证，这意味着只有通过更改设置才能改善不令人满意的结果。我们使用该方法在对随机噪声性质的不同假设下分析确定性和随机一阶方法。其中，