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：我们提出了一种新的基于朗之万动力学的算法，称为e-TH$\varepsilon$O POULA，用于求解具有间断随机梯度的优化问题。这类问题在现实应用中自然出现，例如分位数估计、矢量量化、条件风险价值（CVaR）最小化，以及涉及ReLU神经网络的正则化优化问题。我们从理论和数值上证明了e-TH$\varepsilon$O POULA算法的适用性。更准确地说，在随机梯度局部平均Lipschitz连续且满足某种无穷远处凸性条件的假设下，我们建立了e-TH$\varepsilon$O POULA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，该估计可以被控制到任意小。本文提供了在金融和保险领域的三个关键应用，即多阶段投资组合优化、多阶段投资组合优化中的迁移学习，以及保险索赔预测，这些应用都涉及使用（Leaky）-ReLU激活函数的神经网络。使用真实世界数据集进行的数值实验表明，在模型精度方面，e-TH$\varepsilon$O POULA相较于SGLD、TUSLA、ADAM和AMSGrad具有更优越的实证性能。