This paper investigates the application of mini-batch gradient descent to semiflows. Given a loss function, we introduce a continuous version of mini-batch gradient descent by randomly selecting sub-loss functions over time, defining a piecewise flow. We prove that, under suitable assumptions on the gradient flow, the mini-batch descent flow trajectory closely approximates the original gradient flow trajectory on average. Additionally, we propose a randomized minimizing movement scheme that also approximates the gradient flow of the loss function. We illustrate the versatility of this approach across various problems, including constrained optimization, sparse inversion, and domain decomposition. Finally, we validate our results with several numerical examples.

翻译：本文研究了小批量梯度下降在半流中的应用。针对给定的损失函数，我们通过随时间随机选择子损失函数，引入了一种连续版本的小批量梯度下降方法，从而定义了一个分段流。我们证明，在梯度流满足适当假设的条件下，小批量下降流的轨迹在平均意义上能够紧密逼近原始梯度流的轨迹。此外，我们提出了一种随机化最小化移动格式，该格式同样能够逼近损失函数的梯度流。我们通过多种问题展示了该方法的普适性，包括约束优化、稀疏反演和区域分解。最后，我们通过若干数值算例验证了所得结果。