# High-Dimensional Simulation Optimization via Brownian Fields and Sparse Grids

arXiv: https://arxiv.org/abs/2107.08595 · HTML: https://mlsanity.com/paper/2107.08595/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 49 references that are in the map.
- First posted on arXiv 19 Jul 2021.

## Builds on (most cited first, 15 of 15)

- [When Gaussian Process Meets Big Data: A Review of Scalable GPs](https://mlsanity.com/paper/1807.01065/) (2018)
- [Bayesian Optimization in a Billion Dimensions via Random Embeddings](https://mlsanity.com/paper/1301.1942/) (2013)
- [High Dimensional Bayesian Optimisation and Bandits via Additive Models](https://mlsanity.com/paper/1503.01673/) (2015)
- [High-Dimensional Bayesian Optimization via Additive Models with Overlapping Groups](https://mlsanity.com/paper/1802.07028/) (2018)
- [Bandit optimisation of functions in the Mat'ern kernel RKHS](https://mlsanity.com/paper/2001.10396/) (2020)
- [Global optimization of Lipschitz functions](https://mlsanity.com/paper/1703.02628/) (2017)
- [On the choice of the low-dimensional domain for global optimization via random embeddings](https://mlsanity.com/paper/1704.05318/) (2017)
- [Simulation optimization: A review of algorithms and applications](https://mlsanity.com/paper/1706.08591/) (2017)
- [High-Dimensional Bayesian Optimization via Nested Riemannian Manifolds](https://mlsanity.com/paper/2010.10904/) (2020)
- [A theoretical framework for calibration in computer models: parametrization, estimation and convergence properties](https://mlsanity.com/paper/1508.07155/) (2015)
- [BdryGP: a new Gaussian process model for incorporating boundary information](https://mlsanity.com/paper/1908.08868/) (2019)
- [Continuum-Armed Bandits: A Function Space Perspective](https://mlsanity.com/paper/2010.08007/) (2020)
- [Fast prediction of deterministic functions using sparse grid experimental designs](https://mlsanity.com/paper/1402.6350/) (2014)
- [Surrogate-Based Simulation Optimization](https://mlsanity.com/paper/2105.03893/) (2021)
- [Generalization Guarantees for Sparse Kernel Approximation with Entropic Optimal Features](https://mlsanity.com/paper/2002.04195/) (2020)

## Abstract (arXiv)

High-dimensional simulation optimization is notoriously challenging. We propose a new sampling algorithm that converges to a global optimal solution and suffers minimally from the curse of dimensionality. The algorithm consists of two stages. First, we take samples following a sparse grid experimental design and approximate the response surface via kernel ridge regression with a Brownian field kernel. Second, we follow the expected improvement strategy -- with critical modifications that boost the algorithm's sample efficiency -- to iteratively sample from the next level of the sparse grid. Under mild conditions on the smoothness of the response surface and the simulation noise, we establish upper bounds on the convergence rate for both noise-free and noisy simulation samples. These upper bounds deteriorate only slightly in the dimension of the feasible set, and they can be improved if the objective function is known to be of a higher-order smoothness. Extensive numerical experiments demonstrate that the proposed algorithm dramatically outperforms typical alternatives in practice.