  Discrete integration in a high dimensional space of n variables poses
fundamental challenges. The WISH algorithm reduces the intractable discrete
integration problem into n optimization queries subject to randomized
constraints, obtaining a constant approximation guarantee. The optimization
queries are expensive, which limits the applicability of WISH. We propose
AdaWISH, which is able to obtain the same guarantee but accesses only a small
subset of queries of WISH. For example, when the number of function values is
bounded by a constant, AdaWISH issues only O(log n) queries. The key idea is to
query adaptively, taking advantage of the shape of the weight function being
integrated. In general, we prove that AdaWISH has a regret of only O(log n)
relative to an idealistic oracle that issues queries at data-dependent optimal
points. Experimentally, AdaWISH gives precise estimates for discrete
integration problems, of the same quality as that of WISH and better than
several competing approaches, on a variety of probabilistic inference
benchmarks. At the same time, it saves substantially on the number of
optimization queries compared to WISH. On a suite of UAI inference challenge
benchmarks, it saves 81.5% of WISH queries while retaining the quality of
results.
