This paper investigates the application of mini-batch gradient descent to semiflows (gradient flows). Given a loss function (potential), 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 potential generating the semiflow, the \textit{mini-batch descent flow} trajectory closely approximates the original semiflow trajectory on average. In addition, we study a randomized minimizing movement scheme that also approximates the semiflow of the full 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（梯度流）中的应用。给定一个损失函数（势函数），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，定义了一个分段流。我们证明，在生成半流的势函数满足适当假设的条件下，\textit{小批量下降流}的轨迹在平均意义上能紧密逼近原始半流轨迹。此外，我们研究了一种随机化最小化运动格式，该格式同样能逼近完整损失函数的半流。我们展示了该方法在多种问题中的适用性，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。