In recent years, interest in gradient-based optimization over Riemannian manifolds has surged. However, a significant challenge lies in the reliance on hyperparameters, especially the learning rate, which requires meticulous tuning by practitioners to ensure convergence at a suitable rate. In this work, we introduce innovative learning-rate-free algorithms for stochastic optimization over Riemannian manifolds, eliminating the need for hand-tuning and providing a more robust and user-friendly approach. We establish high probability convergence guarantees that are optimal, up to logarithmic factors, compared to the best-known optimally tuned rate in the deterministic setting. Our approach is validated through numerical experiments, demonstrating competitive performance against learning-rate-dependent algorithms.

翻译：近年来，基于梯度的黎曼流形优化研究兴趣激增。然而，一个重大挑战在于对超参数（尤其是学习率）的依赖，这需要实践者进行精细调参以确保以合适的速率收敛。本文针对黎曼流形上的随机优化问题，提出了创新的免学习率算法，无需人工调参，提供了更鲁棒且用户友好的方法。我们建立了高概率收敛保证，与确定性环境下已知最优调参速率相比，该保证在忽略对数因子意义下达到最优。数值实验验证了所提方法的有效性，结果表明其性能与依赖学习率的算法具有竞争力。