# LAMBDA: Covering the Solution Set of Black-Box Inequality by Search Space Quantization

arXiv: https://arxiv.org/abs/2203.13708 · HTML: https://mlsanity.com/paper/2203.13708/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 23 references that are in the map.
- First posted on arXiv 25 Mar 2022.

## Builds on (most cited first, 6 of 6)

- [Billion-scale similarity search with GPUs](https://mlsanity.com/paper/1702.08734/) (2017)
- [BOHB: Robust and Efficient Hyperparameter Optimization at Scale](https://mlsanity.com/paper/1807.01774/) (2018)
- [Evaluation: from precision, recall and F-measure to ROC, informedness, markedness and correlation](https://mlsanity.com/paper/2010.16061/) (2020)
- [Scalable Global Optimization via Local Bayesian Optimization](https://mlsanity.com/paper/1910.01739/) (2019)
- [Learning Search Space Partition for Black-box Optimization using Monte Carlo Tree Search](https://mlsanity.com/paper/2007.00708/) (2020)
- [Scenarios for Development, Test and Validation of Automated Vehicles](https://mlsanity.com/paper/1801.08598/) (2018)

## Abstract (arXiv)

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.