MCMC practitioners may be familiar with the Wang-Landau algorithm, which is widely used in Physics. This algorithm divides the sample space into “boxes”. Given a target distribution, the algorithm then samples proportionally to the target in each box, while aiming at spending a pre-defined proportion of the sample in each box. (Usually these predefined proportions are uniform.)
This strategy can help move faster between modes of a distribution, by forcing the sample to visit often the space between modes.
The most sophisticated versions of this algorithm combine a decreasing stochastic schedule and the so-called flat histogram criterion: whenever the proportions of the sample in each box are close enough to the desired frequencies, the stochastic schedule decreases. A decreasing schedule is necessary for diminishing adaptation to hold.
Until now, it was unknown whether the flat histogram is necessarily reached in finite time, and hence whether the schedule ever starts decreasing.