We introduce a new Langevin dynamics based algorithm, called e-TH$\varepsilon$O POULA, to solve optimization problems with discontinuous stochastic gradients which naturally appear in real-world applications such as quantile estimation, vector quantization, CVaR minimization, and regularized optimization problems involving ReLU neural networks. We demonstrate both theoretically and numerically the applicability of the e-TH$\varepsilon$O POULA algorithm. More precisely, under the conditions that the stochastic gradient is locally Lipschitz in average and satisfies a certain convexity at infinity condition, we establish non-asymptotic error bounds for e-TH$\varepsilon$O POULA in Wasserstein distances and provide a non-asymptotic estimate for the expected excess risk, which can be controlled to be arbitrarily small. Three key applications in finance and insurance are provided, namely, multi-period portfolio optimization, transfer learning in multi-period portfolio optimization, and insurance claim prediction, which involve neural networks with (Leaky)-ReLU activation functions. Numerical experiments conducted using real-world datasets illustrate the superior empirical performance of e-TH$\varepsilon$O POULA compared to SGLD, TUSLA, ADAM, and AMSGrad in terms of model accuracy.

翻译：我们提出了一种新型基于朗之万动力学的算法，命名为e-TH$\varepsilon$O POULA，用于求解实际应用中自然出现的含不连续随机梯度的优化问题，如分位数估计、向量量化、CVaR最小化以及涉及ReLU神经网络的正则化优化问题。我们从理论和数值两方面论证了e-TH$\varepsilon$O POULA算法的适用性。具体而言，在随机梯度满足局部Lipschitz平均性及无穷远处凸性条件的假设下，我们建立了该算法在Wasserstein距离下的非渐近误差界，并给出了期望超额风险的非渐近估计，该风险可通过算法控制达到任意小值。本文在金融与保险领域提供了三个关键应用案例，包括多期投资组合优化、多期投资组合优化中的迁移学习以及保险理赔预测，这些场景涉及使用(Leaky)-ReLU激活函数的神经网络。基于真实数据集的数值实验结果表明，e-TH$\varepsilon$O POULA在模型精度上相较于SGLD、TUSLA、ADAM及AMSGrad具有更优的实证性能。