Langevin Dynamics Based Algorithm e-THεO POULA for Stochastic Optimization Problems with Discontinuous Stochastic Gradient

Dongyoung Lim, Ariel Neufeld, Ying Zhang, Sotirios Sabanis

Abstract: We introduce a new Langevin dynamics based algorithm, called the extendedtamed hybrid ε-order polygonal unadjusted Langevin algorithm (e-THεO POULA), to solveoptimization problems with discontinuous stochastic gradients, which naturally appear inreal-world applications such as quantile estimation, vector quantization, conditional valueat risk (CVaR) minimization, and regularized optimization problems involving rectified lin-ear unit (ReLU) neural networks. We demonstrate both theoretically and numerically theapplicability of the e-THεO POULA algorithm. More precisely, under the conditions thatthe stochastic gradient is locally Lipschitz in average and satisfies a certain convexity at infin-ity condition, we establish nonasymptotic error bounds for e-THεO POULA in Wassersteindistances and provide a nonasymptotic estimate for the expected excess risk, which can becontrolled to be arbitrarily small. Three key applications in finance and insurance are pro-vided, namely, multiperiod portfolio optimization, transfer learning in multiperiod portfoliooptimization, and insurance claim prediction, which involve neural networks with (Leaky)-ReLU activation functions. Numerical experiments conducted using real-world data setsillustrate the superior empirical performance of e-THεO POULA compared with SGLD (sto-chastic gradient Langevin dynamics), TUSLA (tamed unadjusted stochastic Langevin algo-rithm), adaptive moment estimation, and Adaptive Moment Estimation with a StronglyNon-Convex Decaying Learning Rate in terms of model accuracy.