Adaptive generative moment matching networks for improved learning of dependence structures

An adaptive bandwidth selection procedure for the mixture kernel in the maximum mean discrepancy (MMD) for fitting generative moment matching networks (GMMNs) is introduced, and its ability to improve the learning of copula random number generators is demonstrated. Based on the relative error of the training loss, the number of kernels is increased during training; additionally, the relative error of the validation loss is used as an early stopping criterion. While training time of such adaptively trained GMMNs (AGMMNs) is similar to that of GMMNs, training performance is increased significantly in comparison to GMMNs, which is assessed and shown based on validation MMD trajectories, samples and validation MMD values. Superiority of AGMMNs over GMMNs, as well as typical parametric copula models, is demonstrated in terms of three applications. First, convergence rates of quasi-random versus pseudo-random samples from high-dimensional copulas are investigated for three functionals of interest and in dimensions as large as 100 for the first time. Second, replicated validation MMDs, as well as Monte Carlo and quasi-Monte Carlo applications based on the expected payoff of a basked call option and the risk measure expected shortfall as functionals are used to demonstrate the improved training of AGMMNs over GMMNs for a copula model fitted to the standardized residuals of the 50 constituents of the S&P 500 index after deGARCHing. Last, both the latter dataset and 50 constituents of the FTSE~100 are used to demonstrate that the improved training of AGMMNs over GMMNs and in comparison to the fitting of classical parametric copula models indeed also translates to an improved model prediction.