# Further Understanding of a Local Gaussian Process Approximation: Characterising Convergence in the Finite Regime

arXiv: https://arxiv.org/abs/2404.06200 · HTML: https://mlsanity.com/paper/2404.06200/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 27 references that are in the map.
- First posted on arXiv 9 Apr 2024.

## Builds on (most cited first, 7 of 7)

- [When Gaussian Process Meets Big Data: A Review of Scalable GPs](https://mlsanity.com/paper/1807.01065/) (2018)
- [Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets](https://mlsanity.com/paper/1406.7343/) (2014)
- [Local Gaussian process approximation for large computer experiments](https://mlsanity.com/paper/1303.0383/) (2013)
- [Convergence of Sparse Variational Inference in Gaussian Processes Regression](https://mlsanity.com/paper/2008.00323/) (2020)
- [Sparse within Sparse Gaussian Processes using Neighbor Information](https://mlsanity.com/paper/2011.05041/) (2020)
- [Improved Convergence Rates for Sparse Approximation Methods in Kernel-Based Learning](https://mlsanity.com/paper/2202.04005/) (2022)
- [Leveraging Locality and Robustness to Achieve Massively Scalable Gaussian Process Regression](https://mlsanity.com/paper/2306.14731/) (2023)

## Abstract (arXiv)

We show that common choices of kernel functions for a highly accurate and massively scalable nearest-neighbour based GP regression model (GPnn: citeGPnn) exhibit gradual convergence to asymptotic behaviour as dataset-size n increases. For isotropic kernels such as Mat'ern and squared-exponential, an upper bound on the predictive MSE can be obtained as O(n^-fracpd) for input dimension d, p dictated by the kernel (and d>p) and fixed number of nearest-neighbours m with minimal assumptions on the input distribution. Similar bounds can be found under model misspecification and combined to give overall rates of convergence of both MSE and an important calibration metric. We show that lower bounds on n can be given in terms of m, l, p, d, a tolerance varepsilon and a probability δ. When m is chosen to be O(n^fracpp+d) minimax optimal rates of convergence are attained. Finally, we demonstrate empirical performance and show that in many cases convergence occurs faster than the upper bounds given here.