Stochastic gradient methods have been a popular and powerful choice of optimization methods, aimed at minimizing functions. Their advantage lies in the fact that that one approximates the gradient as opposed to using the full Jacobian matrix. One research direction, related to this, has been on the application to infinite-dimensional problems, where one may naturally have a Hilbert space framework. However, there has been limited work done on considering this in a more general setup, such as where the natural framework is that of a Banach space. This article aims to address this by the introduction of a novel stochastic method, the stochastic steepest descent method (SSD). The SSD will follow the spirit of stochastic gradient descent, which utilizes Riesz representation to identify gradients and derivatives. Our choice for using such a method is that it naturally allows one to adopt a Banach space setting, for which recent applications have exploited the benefit of this, such as in PDE-constrained shape optimization. We provide a convergence theory related to this under mild assumptions. Furthermore, we demonstrate the performance of this method on a couple of numerical applications, namely a $p$-Laplacian and an optimal control problem. Our assumptions are verified in these applications.

翻译：随机梯度方法作为一种流行的优化方法，在函数最小化问题中具有强大优势。其核心在于用梯度近似替代完整的雅可比矩阵。相关研究方向之一涉及无限维问题的应用，这类问题通常具有希尔伯特空间框架。然而，在更一般的框架（如巴拿赫空间）下研究该方法的成果较为有限。本文旨在通过引入新型随机方法——随机最速下降法(SSD)来解决这一问题。SSD遵循随机梯度下降的思想，利用里斯表示定理来识别梯度与导数。选择该方法的原因在于其天然适用于巴拿赫空间设定，而近期应用（如PDE约束形状优化）已充分体现了该设定优势。我们在温和假设下证明了相关收敛理论，并通过两个数值案例（p-拉普拉斯问题与最优控制问题）验证了方法性能，同时验证了假设条件在应用中的有效性。