# Bayesian Optimization for Cascade-type Multi-stage Processes

arXiv: https://arxiv.org/abs/2111.08330 · HTML: https://mlsanity.com/paper/2111.08330/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 43 references that are in the map.
- First posted on arXiv 16 Nov 2021.

## Builds on (most cited first, 12 of 12)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Batch Bayesian Optimization via Local Penalization](https://mlsanity.com/paper/1505.08052/) (2015)
- [Information Directed Sampling and Bandits with Heteroscedastic Noise](https://mlsanity.com/paper/1801.09667/) (2018)
- [Multi-objective Bayesian Optimization using Pareto-frontier Entropy](https://mlsanity.com/paper/1906.00127/) (2019)
- [GLASSES: Relieving The Myopia Of Bayesian Optimisation](https://mlsanity.com/paper/1510.06299/) (2015)
- [BINOCULARS for Efficient, Nonmyopic Sequential Experimental Design](https://mlsanity.com/paper/1909.04568/) (2019)
- [Sequential- and Parallel- Constrained Max-value Entropy Search via Information Lower Bound](https://mlsanity.com/paper/2102.09788/) (2021)
- [Multi-fidelity Bayesian Optimization with Max-value Entropy Search and its parallelization](https://mlsanity.com/paper/1901.08275/) (2019)
- [Bayesian optimisation under uncertain inputs](https://mlsanity.com/paper/1902.07908/) (2019)
- [Mean-Variance Analysis in Bayesian Optimization under Uncertainty](https://mlsanity.com/paper/2009.08166/) (2020)
- [Active learning for distributionally robust level-set estimation](https://mlsanity.com/paper/2102.04000/) (2021)

## Built on by (most cited first, 17 of 17)

- [Active Learning for Optimal Intervention Design in Causal Models](https://mlsanity.com/paper/2209.04744/) (2022)
- [Model-based Causal Bayesian Optimization](https://mlsanity.com/paper/2211.10257/) (2022)
- [Randomized Gaussian Process Upper Confidence Bound with Tighter Bayesian Regret Bounds](https://mlsanity.com/paper/2302.01511/) (2023)
- [Posterior Sampling-Based Bayesian Optimization with Tighter Bayesian Regret Bounds](https://mlsanity.com/paper/2311.03760/) (2023)
- [Bayesian Optimization of Function Networks with Partial Evaluations](https://mlsanity.com/paper/2311.02146/) (2023)
- [Bounding Box-based Multi-objective Bayesian Optimization of Risk Measures under Input Uncertainty](https://mlsanity.com/paper/2301.11588/) (2023)
- [Regret Analysis for Randomized Gaussian Process Upper Confidence Bound](https://mlsanity.com/paper/2409.00979/) (2024)
- [Regret Bounds for Noise-Free Cascaded Kernelized Bandits](https://mlsanity.com/paper/2211.05430/) (2022)
- [Active Learning for Level Set Estimation Using Randomized Straddle Algorithms](https://mlsanity.com/paper/2408.03144/) (2024)
- [Distributionally Robust Active Learning for Gaussian Process Regression](https://mlsanity.com/paper/2502.16870/) (2025)
- [On Regret Bounds of Thompson Sampling for Bayesian Optimization](https://mlsanity.com/paper/2603.09276/) (2026)
- [Validation of Composite Systems by Discrepancy Propagation](https://mlsanity.com/paper/2210.12061/) (2022)
- [Randomized Kriging Believer for Parallel Bayesian Optimization with Regret Bounds](https://mlsanity.com/paper/2603.01470/) (2026)
- [Multi-stage Bayesian optimisation for dynamic decision-making in self-driving labs](https://mlsanity.com/paper/2512.15483/) (2025)
- [Reducing Hyperparameter Tuning Costs in ML, Vision and Language Model Training Pipelines via Memoization-Awareness](https://mlsanity.com/paper/2411.03731/) (2024)
- [Fast Bayesian Optimization of Function Networks with Partial Evaluations](https://mlsanity.com/paper/2506.11456/) (2025)
- [Towards Regret Guarantees for One-Step Lookahead Bayesian Optimization](https://mlsanity.com/paper/2606.00956/) (2026)

## Abstract (arXiv)

Complex processes in science and engineering are often formulated as multistage decision-making problems. In this paper, we consider a type of multistage decision-making process called a cascade process. A cascade process is a multistage process in which the output of one stage is used as an input for the subsequent stage. When the cost of each stage is expensive, it is difficult to search for the optimal controllable parameters for each stage exhaustively. To address this problem, we formulate the optimization of the cascade process as an extension of the Bayesian optimization framework and propose two types of acquisition functions based on credible intervals and expected improvement. We investigate the theoretical properties of the proposed acquisition functions and demonstrate their effectiveness through numerical experiments. In addition, we consider an extension called suspension setting in which we are allowed to suspend the cascade process at the middle of the multistage decision-making process that often arises in practical problems. We apply the proposed method in a test problem involving a solar cell simulator, which was the motivation for this study.