# A Note on Monte Carlo Integration in High Dimensions

arXiv: https://arxiv.org/abs/2206.09036 · HTML: https://mlsanity.com/paper/2206.09036/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jun 2022.

## Built on by (most cited first, 5 of 5)

- [Fast Ergodic Search with Kernel Functions](https://mlsanity.com/paper/2403.01536/) (2024)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [Monte Carlo and quasi-Monte Carlo integration for likelihood functions](https://mlsanity.com/paper/2506.21733/) (2025)
- [Distributional Computational Graphs: Error Bounds](https://mlsanity.com/paper/2601.16250/) (2026)
- [Understanding the geometry of deep learning with decision boundary volume](https://mlsanity.com/paper/2603.14768/) (2026)

## Abstract (arXiv)

Monte Carlo integration is a commonly used technique to compute intractable integrals and is typically thought to perform poorly for very high-dimensional integrals. To show that this is not always the case, we examine Monte Carlo integration using techniques from the high-dimensional statistics literature by allowing the dimension of the integral to increase. In doing so, we derive non-asymptotic bounds for the relative and absolute error of the approximation for some general classes of functions through concentration inequalities. We provide concrete examples in which the magnitude of the number of points sampled needed to guarantee a consistent estimate vary between polynomial to exponential, and show that in theory arbitrarily fast or slow rates are possible. This demonstrates that the behaviour of Monte Carlo integration in high dimensions is not uniform. Through our methods we also obtain non-asymptotic confidence intervals for Monte Carlo estimate which are valid regardless of the number of points sampled.