Non-asymptotic analysis of SGLD with discontinuous stochastic gradients: a new route to portfolio CVaR minimization

Abstract

Conditional-Value-at-Risk (CVaR) optimization is a fundamental tool in portfolio selection, yet its nonsmooth loss structure poses a challenge for the existing theory of stochastic gradient methods including associated Langevin algorithms. In particular, the positive-part loss and the softmax portfolio parametrization lead to stochastic gradients that need not be globally Lipschitz continuous, but rather discontinuous, where available non-convex stochastic gradient Langevin dynamics (SGLD) guarantees often exhibit an unfavorable exponential dependence on the problem dimension. In this article, we develop a non-asymptotic analysis of SGLD for a regularized CVaR portfolio optimization problem. We exploit the geometric structure of the CVaR objective to establish linear growth, convexity-at-infinity, and one-sided Lipschitz properties of the stochastic gradient without imposing density assumptions on the asset losses that are typical in previous works. Crucially, the constants appearing in the aforementioned structural conditions are shown to be independent of the number of assets. Moreover, we establish a convergence bound in Wasserstein-2 distance with rate $\mathcal{O}(λ^{1/4})$ and a finite-time guarantee for the expected excess risk. Our analysis, together with the dimension-independent constants in the structural conditions, leads to an excess risk bound that depends linearly on the number of assets. Numerical experiments with simulated Gaussian portfolios exhibit the predicted dimension dependence. Computational comparisons with alternative stochastic optimization methods further illustrate the practical efficiency of SGLD for the optimization problem under consideration. Finally, we extend the framework to a broader class of non-convex stochastic optimization problems with discontinuous stochastic gradients.