# Posterior Variance Analysis of Gaussian Processes with Application to Average Learning Curves

arXiv: https://arxiv.org/abs/1906.01404 · HTML: https://mlsanity.com/paper/1906.01404/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2019.

## Built on by (most cited first, 12 of 12)

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- [Networked Online Learning for Control of Safety-Critical Resource-Constrained Systems based on Gaussian Processes](https://mlsanity.com/paper/2202.11491/) (2022)
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- [Global Safe Sequential Learning via Efficient Knowledge Transfer](https://mlsanity.com/paper/2402.14402/) (2024)
- [PolicyGuard: Towards Test-time and Step-level Adversary (Backdoor) Defense for Reinforcement Learning Agent](https://mlsanity.com/paper/2606.12896/) (2026)
- [Excess risk analysis for epistemic uncertainty with application to variational inference](https://mlsanity.com/paper/2206.01606/) (2022)
- [OptEx: Expediting First-Order Optimization with Approximately Parallelized Iterations](https://mlsanity.com/paper/2402.11427/) (2024)
- [Estimating Mixture Distributions via Stochastic Mirror Descent](https://mlsanity.com/paper/2605.24929/) (2026)

## Abstract (arXiv)

The posterior variance of Gaussian processes is a valuable measure of the learning error which is exploited in various applications such as safe reinforcement learning and control design. However, suitable analysis of the posterior variance which captures its behavior for finite and infinite number of training data is missing. This paper derives a novel bound for the posterior variance function which requires only local information because it depends only on the number of training samples in the proximity of a considered test point. Furthermore, we prove sufficient conditions which ensure the convergence of the posterior variance to zero. Finally, we demonstrate that the extension of our bound to an average learning bound outperforms existing approaches.