  In this paper we develop a Stochastic Gradient Langevin Dynamics (SGLD)
algorithm tailored for solving a certain class of non-convex distributionally
robust optimisation problems. By deriving non-asymptotic convergence bounds, we
build an algorithm which for any prescribed accuracy $\varepsilon>0$ outputs an
estimator whose expected excess risk is at most $\varepsilon$. As a concrete
application, we employ our robust SGLD algorithm to solve the (regularised)
distributionally robust Mean-CVaR portfolio optimisation problem using real
financial data. We empirically demonstrate that the trading strategy obtained
by our robust SGLD algorithm outperforms the trading strategy obtained when
solving the corresponding non-robust Mean-CVaR portfolio optimisation problem
using, e.g., a classical SGLD algorithm. This highlights the practical
relevance of incorporating model uncertainty when optimising portfolios in real
financial markets.
