TY - JOUR
T1 - Langevin dynamics based algorithm e-THεO POULA for stochastic optimization problems with discontinuous stochastic gradient
AU - Lim, Dong-Young
AU - Neufeld, Ariel
AU - Sabanis, Sotirios
AU - Zhang, Ying
PY - 2024/9/11
Y1 - 2024/9/11
N2 - We introduce a new Langevin dynamics based algorithm, called the extended tamed hybrid ε-order polygonal unadjusted Langevin algorithm (e-THεO POULA), to solve optimization problems with discontinuous stochastic gradients, which naturally appear in real-world applications such as quantile estimation, vector quantization, conditional value at risk (CVaR) minimization, and regularized optimization problems involving rectified linear unit (ReLU) neural networks. We demonstrate both theoretically and numerically the applicability of the e-THεO 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 nonasymptotic error bounds for e-THεO POULA in Wasserstein distances and provide a nonasymptotic estimate for the expected excess risk, which can be controlled to be arbitrarily small. Three key applications in finance and insurance are provided, namely, multiperiod portfolio optimization, transfer learning in multiperiod portfolio optimization, and insurance claim prediction, which involve neural networks with (Leaky)-ReLU activation functions. Numerical experiments conducted using real-world data sets illustrate the superior empirical performance of e-THεO POULA compared with SGLD (stochastic gradient Langevin dynamics), TUSLA (tamed unadjusted stochastic Langevin algorithm), adaptive moment estimation, and Adaptive Moment Estimation with a Strongly Non-Convex Decaying Learning Rate in terms of model accuracy.
AB - We introduce a new Langevin dynamics based algorithm, called the extended tamed hybrid ε-order polygonal unadjusted Langevin algorithm (e-THεO POULA), to solve optimization problems with discontinuous stochastic gradients, which naturally appear in real-world applications such as quantile estimation, vector quantization, conditional value at risk (CVaR) minimization, and regularized optimization problems involving rectified linear unit (ReLU) neural networks. We demonstrate both theoretically and numerically the applicability of the e-THεO 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 nonasymptotic error bounds for e-THεO POULA in Wasserstein distances and provide a nonasymptotic estimate for the expected excess risk, which can be controlled to be arbitrarily small. Three key applications in finance and insurance are provided, namely, multiperiod portfolio optimization, transfer learning in multiperiod portfolio optimization, and insurance claim prediction, which involve neural networks with (Leaky)-ReLU activation functions. Numerical experiments conducted using real-world data sets illustrate the superior empirical performance of e-THεO POULA compared with SGLD (stochastic gradient Langevin dynamics), TUSLA (tamed unadjusted stochastic Langevin algorithm), adaptive moment estimation, and Adaptive Moment Estimation with a Strongly Non-Convex Decaying Learning Rate in terms of model accuracy.
U2 - 10.1287/moor.2022.0307
DO - 10.1287/moor.2022.0307
M3 - Article
SN - 0364-765X
JO - Mathematics of Operations Research
JF - Mathematics of Operations Research
ER -