Non-asymptotic analysis of SGLD with discontinuous stochastic gradients: a new route to portfolio CVaR minimization
Ding et al.
For regularised conditional value at risk, stochastic gradient Langevin dynamics converges in Wasserstein-2 distance at rate \(\mathcal{O}(\lambda^{1/4})\) in the step size, with excess risk linear in the number of assets.
Source published · In the Daily Brief of 4 November 2026
Filed under Quantitative finance, Optimization
Conditional value at risk has a positive-part loss, and with a softmax portfolio parametrisation the stochastic gradients can be discontinuous. Existing guarantees for non-convex Langevin methods then often depend exponentially on dimension.
The authors use the geometry of the objective to establish linear growth, convexity at infinity and one-sided Lipschitz conditions, without density assumptions on asset losses and with constants independent of the number of assets. They prove a Wasserstein-2 bound with rate \(\mathcal{O}(\lambda^{1/4})\) for step size \(\lambda\), a finite-time excess risk guarantee, and an excess risk bound linear in the number of assets. Experiments on simulated Gaussian portfolios show the predicted dependence on dimension.