Langevin dynamics based algorithm e-TH$\varepsilon$O POULA for stochastic optimization problems with discontinuous stochastic gradient

TL;DR

This paper introduces e-THεO POULA, a Langevin dynamics-based algorithm for stochastic optimization with discontinuous gradients, providing theoretical guarantees and demonstrating superior empirical performance in finance and insurance applications.

Contribution

The paper proposes a novel Langevin dynamics algorithm for discontinuous stochastic gradients, with non-asymptotic error bounds and practical applications in finance and insurance.

Findings

Non-asymptotic error bounds in Wasserstein distances.

Arbitrarily small expected excess risk achievable.

Superior empirical performance over existing methods.

Abstract

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.