Abstract
Stochastic optimization powers the scalability of modern artificial intelligence, spanning machine learning, deep learning, reinforcement learning, and large language model training. Yet, existing theory remains largely confined to Hilbert spaces, relying on inner-product frameworks and orthogonality. This paradigm fails to capture non-Euclidean settings, such as mirror descent on simplices, Bregman proximal methods for sparse learning, natural gradient descent in information geometry, or Kullback--Leibler-regularized language model training. Unlike Euclidean-based Hilbert-space methods, this approach embraces general Banach spaces. This work introduces a pioneering Banach--Bregman framework for stochastic iterations, establishing Bregman geometry as a foundation for next-generation optimization. It (i) provides a unified template via Bregman projections and Bregman--Fejer monotonicity, encompassing stochastic approximation, mirror descent, natural gradient, adaptive methods, and mirror-prox; (ii) establishes super-relaxations ($\lambda > 2$) in non-Hilbert settings, enabling flexible geometries and elucidating their acceleration effect; and (iii) delivers convergence theorems spanning almost-sure boundedness to geometric rates, validated on synthetic and real-world tasks. Empirical studies across machine learning (UCI benchmarks), deep learning (e.g., Transformer training), reinforcement learning (actor--critic), and large language models (WikiText-2 with distilGPT-2) show up to 20% faster convergence, reduced variance, and enhanced accuracy over classical baselines. These results position Banach--Bregman geometry as a cornerstone unifying optimization theory and practice across core AI paradigms.
Abstract (translated)
随机优化为现代人工智能的可扩展性提供了动力，涵盖了机器学习、深度学习、强化学习和大型语言模型训练等多个领域。然而，现有的理论主要局限于希尔伯特空间，并依赖于内积框架和正交性。这种范式无法捕捉非欧几里得场景下的情况，比如单纯形上的镜像下降法、稀疏学习的布雷格曼近似方法、信息几何中的自然梯度下降或带Kullback-Leibler正则化的语言模型训练。不同于基于欧氏空间的希尔伯特空间方法，这种方法采用了一般巴拿赫空间。这项工作引入了一个开创性的巴拿赫-布雷格曼框架用于随机迭代，确立了布雷格曼几何作为下一代优化的基础。它（i）通过布雷格曼投影和布雷格曼-Féjer单调性提供了一个统一的模板，涵盖了随机近似、镜像下降法、自然梯度、自适应方法和镜像-普罗克；（ii）在非希尔伯特空间设置中建立了超松弛化($\lambda > 2$)，使得灵活的几何结构成为可能，并阐明了它们加速效果；以及（iii）提供了涵盖几乎必然有界性到几何速率的收敛定理，通过合成和现实世界任务进行了验证。跨机器学习（UCI基准测试）、深度学习（例如Transformer训练）、强化学习（演员-评论家方法）和大型语言模型（使用distilGPT-2的WikiText-2）进行的经验研究表明，在经典基线之上实现了高达20%更快的收敛速度，减少了方差，并提高了精度。这些结果将巴拿赫-布雷格曼几何确立为在核心AI范式中统一优化理论和实践的一个基石。
URL
https://arxiv.org/abs/2509.14216
