# Kernelized Normalizing Constant Estimation: Bridging Bayesian Quadrature and Bayesian Optimization

arXiv: https://arxiv.org/abs/2401.05716 · HTML: https://mlsanity.com/paper/2401.05716/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 38 references that are in the map.
- First posted on arXiv 11 Jan 2024.

## Builds on (most cited first, 22 of 22)

- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [On Information Gain and Regret Bounds in Gaussian Process Bandits](https://mlsanity.com/paper/2009.06966/) (2020)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization](https://mlsanity.com/paper/1706.00090/) (2017)
- [Sharp convergence rates for Langevin dynamics in the nonconvex setting](https://mlsanity.com/paper/1805.01648/) (2018)
- [Sampling Can Be Faster Than Optimization](https://mlsanity.com/paper/1811.08413/) (2018)
- [Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo](https://mlsanity.com/paper/2202.05214/) (2022)
- [Control functionals for Monte Carlo integration](https://mlsanity.com/paper/1410.2392/) (2014)
- [Optimal Order Simple Regret for Gaussian Process Bandits](https://mlsanity.com/paper/2108.09262/) (2021)
- [Non-parametric Models for Non-negative Functions](https://mlsanity.com/paper/2007.03926/) (2020)
- [On Lower Bounds for Standard and Robust Gaussian Process Bandit Optimization](https://mlsanity.com/paper/2008.08757/) (2020)
- [Convergence Guarantees for Gaussian Process Means With Misspecified Likelihoods and Smoothness](https://mlsanity.com/paper/2001.10818/) (2020)
- [Convergence Guarantees for Adaptive Bayesian Quadrature Methods](https://mlsanity.com/paper/1905.10271/) (2019)
- [Estimating Normalizing Constants for Log-Concave Distributions: Algorithms and Lower Bounds](https://mlsanity.com/paper/1911.03043/) (2019)
- [Poisson intensity estimation with reproducing kernels](https://mlsanity.com/paper/1610.08623/) (2016)
- [Fisher information lower bounds for sampling](https://mlsanity.com/paper/2210.02482/) (2022)
- [Convergence Rates for Non-Log-Concave Sampling and Log-Partition Estimation](https://mlsanity.com/paper/2303.03237/) (2023)
- [Normalizing constants of log-concave densities](https://mlsanity.com/paper/1707.00460/) (2017)
- [On Average-Case Error Bounds for Kernel-Based Bayesian Quadrature](https://mlsanity.com/paper/2202.10615/) (2022)
- [Invariant Priors for Bayesian Quadrature](https://mlsanity.com/paper/2112.01578/) (2021)

## Built on by (most cited first, 2 of 2)

- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [Active Learning for Gaussian Process Regression Under Self-Induced Boltzmann Weights](https://mlsanity.com/paper/2605.10654/) (2026)

## Abstract (arXiv)

In this paper, we study the problem of estimating the normalizing constant int e-λ f(x)dx through queries to the black-box function f, where f belongs to a reproducing kernel Hilbert space (RKHS), and λ is a problem parameter. We show that to estimate the normalizing constant within a small relative error, the level of difficulty depends on the value of λ: When λ approaches zero, the problem is similar to Bayesian quadrature (BQ), while when λ approaches infinity, the problem is similar to Bayesian optimization (BO). More generally, the problem varies between BQ and BO. We find that this pattern holds true even when the function evaluations are noisy, bringing new aspects to this topic. Our findings are supported by both algorithm-independent lower bounds and algorithmic upper bounds, as well as simulation studies conducted on a variety of benchmark functions.