  Short autocorrelation times are essential for a reliable error assessment in
Monte Carlo simulations of lattice systems. In many interesting scenarios, the
decay of autocorrelations in the Markov chain is prohibitively slow. Generative
samplers can provide statistically independent field configurations, thereby
potentially ameliorating these issues. In this work, the applicability of
neural samplers to this problem is investigated. Specifically, we work with a
generative adversarial network (GAN). We propose to address difficulties
regarding its statistical exactness through the implementation of an
overrelaxation step, by searching the latent space of the trained generator
network. This procedure can be incorporated into a standard Monte Carlo
algorithm, which then permits a sensible assessment of ergodicity and balance
based on consistency checks. Numerical results for real, scalar $\phi^4$-theory
in two dimensions are presented. We achieve a significant reduction of
autocorrelations while accurately reproducing the correct statistics. We
discuss possible improvements to the approach as well as potential solutions to
persisting issues.
