# No-Regret Constrained Bayesian Optimization of Noisy and Expensive Hybrid Models using Differentiable Quantile Function Approximations

arXiv: https://arxiv.org/abs/2305.03824 · HTML: https://mlsanity.com/paper/2305.03824/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 89 references that are in the map.
- First posted on arXiv 5 May 2023.

## Builds on (most cited first, 26 of 26)

- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [Fast Differentiable Sorting and Ranking](https://mlsanity.com/paper/2002.08871/) (2020)
- [On Information Gain and Regret Bounds in Gaussian Process Bandits](https://mlsanity.com/paper/2009.06966/) (2020)
- [Bayesian Optimization with Safety Constraints: Safe and Automatic Parameter Tuning in Robotics](https://mlsanity.com/paper/1602.04450/) (2016)
- [Practical Multi-fidelity Bayesian Optimization for Hyperparameter Tuning](https://mlsanity.com/paper/1903.04703/) (2019)
- [Scalable Constrained Bayesian Optimization](https://mlsanity.com/paper/2002.08526/) (2020)
- [A Literature Survey of Benchmark Functions For Global Optimization Problems](https://mlsanity.com/paper/1308.4008/) (2013)
- [The reparameterization trick for acquisition functions](https://mlsanity.com/paper/1712.00424/) (2017)
- [Constrained Efficient Global Optimization of Expensive Black-box Functions](https://mlsanity.com/paper/2211.00162/) (2022)
- [Bayesian Optimization of Composite Functions](https://mlsanity.com/paper/1906.01537/) (2019)
- [Bayesian Optimization with High-Dimensional Outputs](https://mlsanity.com/paper/2106.12997/) (2021)
- [Bayesian optimization under mixed constraints with a slack-variable augmented Lagrangian](https://mlsanity.com/paper/1605.09466/) (2016)
- [Stochastic Optimization Using a Trust-Region Method and Random Models](https://mlsanity.com/paper/1504.04231/) (2015)
- [Thinking inside the box: A tutorial on grey-box Bayesian optimization](https://mlsanity.com/paper/2201.00272/) (2022)
- [Performance-oriented model learning for data-driven MPC design](https://mlsanity.com/paper/1904.10839/) (2019)
- [A Data-Driven Automatic Tuning Method for MPC under Uncertainty using Constrained Bayesian Optimization](https://mlsanity.com/paper/2011.11841/) (2020)
- [COBALT: COnstrained Bayesian optimizAtion of computationaLly expensive grey-box models exploiting derivaTive information](https://mlsanity.com/paper/2105.04114/) (2021)
- [Global Optimization of Gaussian processes](https://mlsanity.com/paper/2005.10902/) (2020)
- [Upper Trust Bound Feasibility Criterion for Mixed Constrained Bayesian Optimization with Application to Aircraft Design](https://mlsanity.com/paper/2005.05067/) (2020)
- [Improving the Flexibility and Robustness of Model-Based Derivative-Free Optimization Solvers](https://mlsanity.com/paper/1804.00154/) (2018)
- [A Stochastic-Gradient-based Interior-Point Algorithm for Solving Smooth Bound-Constrained Optimization Problems](https://mlsanity.com/paper/2304.14907/) (2023)
- [Regret Bounds for Gaussian-Process Optimization in Large Domains](https://mlsanity.com/paper/2104.14113/) (2021)
- [Safe Bayesian Optimization using Interior-Point Methods -- Applied to Personalized Insulin Dose Guidance](https://mlsanity.com/paper/2205.02327/) (2022)
- [Derivative-free optimization methods](https://mlsanity.com/paper/1904.11585/) (2019)
- [Data-Driven Scenario Optimization for Automated Controller Tuning with Probabilistic Performance Guarantees](https://mlsanity.com/paper/2011.07445/) (2020)

## Built on by (most cited first, 6 of 6)

- [Bayesian optimization as a flexible and efficient design framework for sustainable process systems](https://mlsanity.com/paper/2401.16373/) (2024)
- [BONSAI: Structure-exploiting robust Bayesian optimization for networked black-box systems under uncertainty](https://mlsanity.com/paper/2510.03893/) (2025)
- [Multi-Objective Bayesian Optimization for Networked Black-Box Systems: A Path to Greener Profits and Smarter Designs](https://mlsanity.com/paper/2502.14121/) (2025)
- [Bayesian Optimization of Expensive Nested Grey-Box Functions](https://mlsanity.com/paper/2306.05150/) (2023)
- [Model-free Reinforcement Learning for Model-based Control: Towards Safe, Interpretable and Sample-efficient Agents](https://mlsanity.com/paper/2507.13491/) (2025)
- [BITS for GAPS: Bayesian Information-Theoretic Sampling for hierarchical GAussian Process Surrogates](https://mlsanity.com/paper/2511.16815/) (2025)

## Abstract (arXiv)

This paper investigates the problem of efficient constrained global optimization of hybrid models that are a composition of a known white-box function and an expensive multi-output black-box function subject to noisy observations, which often arises in real-world science and engineering applications. We propose a novel method, Constrained Upper Quantile Bound (CUQB), to solve such problems that directly exploits the composite structure of the objective and constraint functions that we show leads substantially improved sampling efficiency. CUQB is a conceptually simple, deterministic approach that avoid constraint approximations used by previous methods. Although the CUQB acquisition function is not available in closed form, we propose a novel differentiable sample average approximation that enables it to be efficiently maximized. We further derive bounds on the cumulative regret and constraint violation under a non-parametric Bayesian representation of the black-box function. Since these bounds depend sublinearly on the number of iterations under some regularity assumptions, we establis bounds on the convergence rate to the optimal solution of the original constrained problem. In contrast to most existing methods, CUQB further incorporates a simple infeasibility detection scheme, which we prove triggers in a finite number of iterations when the original problem is infeasible (with high probability given the Bayesian model). Numerical experiments on several test problems, including environmental model calibration and real-time optimization of a reactor system, show that CUQB significantly outperforms traditional Bayesian optimization in both constrained and unconstrained cases. Furthermore, compared to other state-of-the-art methods that exploit composite structure, CUQB achieves competitive empirical performance while also providing substantially improved theoretical guarantees.