# Learning Curves for Gaussian Process Regression: Approximations and Bounds

@article{Sollich2002LearningCF, title={Learning Curves for Gaussian Process Regression: Approximations and Bounds}, author={Peter Sollich and Anason S. Halees}, journal={Neural Computation}, year={2002}, volume={14}, pages={1393-1428} }

We consider the problem of calculating learning curves (i.e., average generalization performance) of gaussian processes used for regression. On the basis of a simple expression for the generalization error, in terms of the eigenvalue decomposition of the covariance function, we derive a number of approximation schemes. We identify where these become exact and compare with existing bounds on learning curves; the new approximations, which can be used for any input space dimension, generally get…

## 42 Citations

Asymptotic analysis of the learning curve for Gaussian process regression

- Mathematics, Computer ScienceMachine Learning
- 2014

The main result is the proof of a theorem giving the generalization error for a large class of correlation kernels and for any dimension when the number of observations is large.

Regularity dependence of the rate of convergence of the learning curve for Gaussian process regression

- Mathematics, Computer Science
- 2012

The presented proof generalizes previous ones that were limited to more specific kernels or to small dimensions (one or two) and can be used to build an optimal strategy for resources allocation.

Learning curves for multi-task Gaussian process regression

- Computer ScienceNIPS
- 2012

It is demonstrated that when learning many tasks, the learning curves separate into an initial phase, where the Bayes error on each task is reduced down to a plateau value by "collective learning" even though most tasks have not seen examples, and a final decay that occurs once the number of examples is proportional to thenumber of tasks.

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

- Computer Science, MathematicsArXiv
- 2019

A novel bound is derived 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 and it is demonstrated that the extension of the bound to an average learning bound outperforms existing approaches.

Continuous-Space Gaussian Process Regression and Generalized Wiener Filtering with Application to Learning Curves

- Computer Science, MathematicsSCIA
- 2013

The general continuous-space Gaussian process regression equations are presented and their close connection with Wiener filtering is discussed and the results to estimation of learning curves as functions of training set size and input dimensionality are applied.

Generalization Errors and Learning Curves for Regression with Multi-task Gaussian Processes

- Computer ScienceNIPS
- 2009

The asymmetric two-tasks case, where a secondary task is to help the learning of a primary task, is analyzed, and bounds on the generalization error and the learning curve of the primary task are given.

Exact learning curves for Gaussian process regression on large random graphs

- Computer ScienceNIPS
- 2010

It is shown that for discrete input domains, where similarity between input points is characterised in terms of a graph, accurate predictions can be obtained and should in fact become exact for large graphs drawn from a broad range of random graph ensembles with arbitrary degree distributions.

Learning curves for Gaussian process regression with power-law priors and targets

- Computer ScienceArXiv
- 2021

It is shown that the generalization error of kernel ridge regression (KRR) has the same asymptotics as well as that of Gaussian process regression (GPR) when the eigenspectrum of the prior and the eigenexpansion coefficients of the target function decay with rate β.

Learning Curves for Gaussian Processes via Numerical Cubature Integration

- Computer ScienceICANN
- 2011

An approach where the recursion equations for the generalization error are approximately solved using numerical cubature integration methods where the eigenfunction expansion of the covariance function does not need to be known.

Replica theory for learning curves for Gaussian processes on random graphs

- Computer Science
- 2012

It is shown that replica techniques can be used to obtain exact performance predictions in the limit of large graphs, after first rewriting the average error in terms of a graphical model.

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