Review for NeurIPS paper: A Contour Stochastic Gradient Langevin Dynamics Algorithm for Simulations of Multi-modal Distributions

My main concern is that using a flattened surrogate energy in this fashion is suitable for most sampling situations. The main reason is, by construction our iterates are not following the true distribution particularly closely; for example a plot of the samples obtained in the synthetic experiments (figs 2c--d) would look quite different from the original. While this does allow the algorithm to bounce out of local optima, the deviance from the true energy would make samples obtained after convergence to not be super useful. For point estimation situations, we might be able to get away with these samples for cases where the multiple modes of the real energy are sort of symmetric (as in the synthetic Gaussian experiments); it seems that even if we use a'flattened' energy (can be thought of as lower peaks with higher elevation between them), the original distribution's symmetry would be essentially preserved and the mean / other point estimates would be close enough. But flattening energies with skewed distribution of modes might not be as accurate, as the flattened version might have a mean closer to the'center' of the space, but the original would be closer to one of the modes near the periphery (am visualizing a simple 2-d space).