We develop an algorithm for parameter-free stochastic convex optimization (SCO) whose rate of convergence is only a double-logarithmic factor larger than the optimal rate for the corresponding known-parameter setting. In contrast, the best previously known rates for parameter-free SCO are based on online parameter-free regret bounds, which contain unavoidable excess logarithmic terms compared to their known-parameter counterparts. Our algorithm is conceptually simple, has high-probability guarantees, and is also partially adaptive to unknown gradient norms, smoothness, and strong convexity. At the heart of our results is a novel parameter-free certificate for SGD step size choice, and a time-uniform concentration result that assumes no a-priori bounds on SGD iterates.

翻译：我们提出了一种用于无参数随机凸优化（SCO）的算法，其收敛速度仅比相应已知参数设置下的最优速度高出一个双对数因子。相比之下，先前已知的无参数SCO最优速度基于在线无参数遗憾界，这些界与已知参数设定相比包含不可避免的额外对数项。我们的算法概念简单，具有高概率保证，并且对未知梯度范数、光滑性和强凸性具有部分自适应性。我们结果的核心是一项新颖的、无参数化的SGD步长选择证书，以及一个时间均匀的集中结果，该结果不需要对SGD迭代过程预设先验界限。