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：随机梯度方法已成为一类流行且强大的优化方法，旨在最小化目标函数。其优势在于通过近似梯度而非使用完整的雅可比矩阵进行计算。与此相关的一个研究方向是将该方法应用于无穷维问题，其中自然可采用希尔伯特空间框架。然而，在更一般的设定（例如自然框架为Banach空间的情形）下考虑此问题的研究仍然有限。本文旨在通过引入一种新颖的随机方法——随机最速下降法（SSD）来解决这一问题。SSD将遵循随机梯度下降法的思想，利用Riesz表示定理来识别梯度和导数。我们选择此类方法的原因在于它天然适用于Banach空间框架，近期研究（如PDE约束的形状优化）已展现出该框架的优势。我们在温和假设下建立了该方法的收敛性理论。此外，我们通过两个数值算例（即$p$-拉普拉斯方程和最优控制问题）展示了该方法的性能，并在这些应用中验证了我们的假设。