  Black-box functions are broadly used to model complex problems that provide
no explicit information but the input and output. Despite existing studies of
black-box function optimization, the solution set satisfying an inequality with
a black-box function plays a more significant role than only one optimum in
many practical situations. Covering as much as possible of the solution set
through limited evaluations to the black-box objective function is defined as
the Black-Box Coverage (BBC) problem in this paper. We formalized this problem
in a sample-based search paradigm and constructed a coverage criterion with
Confusion Matrix Analysis. Further, we propose LAMBDA (Latent-Action
Monte-Carlo Beam Search with Density Adaption) to solve BBC problems. LAMBDA
can focus around the solution set quickly by recursively partitioning the
search space into accepted and rejected sub-spaces. Compared with La-MCTS,
LAMBDA introduces density information to overcome the sampling bias of
optimization and obtain more exploration. Benchmarking shows, LAMBDA achieved
state-of-the-art performance among all baselines and was at most 33x faster to
get 95% coverage than Random Search. Experiments also demonstrate that LAMBDA
has a promising future in the verification of autonomous systems in virtual
tests.
