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A Framework for Evaluating Approximation Methods for Gaussian Process Regression
Krzysztof Chalupka, Christopher K. I. Williams, Iain Murray
TL;DR
Gaussian process regression approximations lack clear evidence about their relative merits and useful operating conditions. This paper evaluates four approximation algorithms across four prediction problems using prediction quality versus compute time, finding that simple Subset of Data or Hybrid methods are leading choices when hyperparameter learning dominates computation.
Problem
Existing GP approximation methods lack clear comparisons of their relative merits and the situations in which they are most useful, despite exact GPR requiring O(n^2) space and O(n^3) time.
Method
The paper empirically evaluates four approximation algorithms across four prediction problems using prediction quality versus compute time and comparisons with standard baselines such as SoD and FITC.
Results
When hyperparameter learning dominates computation time, Subset of Data or Hybrid are leading choices; FITC provides better accuracy for a given test time on the considered datasets.
Takeaways & Limitations
Future evaluations should compare error-time curves with standard approximations such as SoD and FITC while considering hyperparameter learning, training, testing, and other relevant factors.
Takeaways & Limitations
Iterative GPR methods are usually applied only to mean prediction, while variance prediction would require solving a new linear system for each test point and lacks a demonstrated well-scaled method.
Abstract
from arXiv · showhide
Gaussian process (GP) predictors are an important component of many Bayesian approaches to machine learning. However, even a straightforward implementation of Gaussian process regression (GPR) requires O(n^2) space and O(n^3) time for a dataset of n examples. Several approximation methods have been proposed, but there is a lack of understanding of the relative merits of the different approximations, and in what situations they are most useful. We recommend assessing the quality of the predictions obtained as a function of the compute time taken, and comparing to standard baselines (e.g., Subset of Data and FITC). We empirically investigate four different approximation algorithms on four different prediction problems, and make our code available to encourage future comparisons.
1. Approximation algorithms for Gaussian Process Regression (GPR)
GPR computes predictive distributions and learns kernel hyperparameters, but its full implementation is expensive, motivating approximations that trade prediction quality against computation across distinct phases and settings.
- Full GPR: GPR combines a Gaussian-process prior with Gaussian observations to produce predictive means and variances for latent function values at test inputs.The kernel matrix, test-input covariance vector, targets, and noise variance determine the predictive quantities.
- Full GPR: Hyperparameter learning maximizes the log marginal likelihood, whose value and gradient require O(n^3) computation and can dominate fixed-hyperparameter training.Training computes quantities independent of the test input, while testing computes only the remaining test-input-dependent operations.
- Computational phases and complexity: Full GPR requires O(n^2) storage for the kernel matrix, whereas special stationary grid cases can admit analytic matrix operations.The computational burden therefore depends on both the dataset and exploitable covariance structure.
- Approximation families: Approximation families include Subset of Data, inducing-point, Local, and iterative fast-matrix-vector-multiplication methods, each changing how computations scale or use data.SoD replaces n with a subset size m; FITC uses a rank-m plus diagonal approximation, while Local methods partition nearby data.
- Inducing-point and hybrid methods: The Hybrid method learns hyperparameters with SoD and predicts with FITC, reducing hyperparameter-learning cost from O(m^2n) to O(m^3) while potentially reducing FITC performance for fixed m.Its motivation is to combine SoD’s cheaper learning phase with FITC’s potentially stronger predictions.
- Local GPR: Recursive Projection Clustering was introduced because Farthest Point Clustering produced unequal cluster sizes that limited Local GPR speedups.Partitioning choices can therefore affect both computational efficiency and predictive behavior.
- Iterative methods and IFGT: Iterative methods require careful termination and remain difficult to deploy routinely because runtimes depend on hyperparameters and variance prediction is not straightforward.Fast MVMs are still important for reducing the O(n^2) cost and memory burden of explicitly constructing the kernel matrix.
- Comparing approximation methods: For fixed m, FITC and Local may improve predictions over SoD, but their O(m^2n) training cost can allow SoD to use a larger m at equal training time.The paper therefore compares approximation methods using error-time trade-offs rather than subset size alone.
2. A Basis for Comparing Approximations
The paper frames approximation comparison as a prediction-quality-versus-compute problem complicated by multiple cost definitions, hyperparameter choices, and problem characteristics. It therefore evaluates methods under realistic hyperparameter optimization and considers kernel, dimensionality, noise, and usability factors.
- Comparing methods: Approximation quality should be compared against predictive error at a given computational cost, with cost defined separately for hyperparameter learning, training, and testing.Possible cost measures include CPU time, flops, or other operation counts.
- Comparing methods: Error metrics reduce predictive mean and variance functions to one number, so visualizing these functions can clarify differences for low-dimensional problems.Visualization complements, rather than replaces, quantitative error comparisons.
- Hyperparameters: Real-world comparisons should let approximate methods determine their own hyperparameters because known appropriate values are uncommon.This is more realistic than reusing full-GPR hyperparameters, but it changes both the approximation and the hyperparameters.
- Experimental setup: The experiments measure computational cost as CPU time in seconds using Matlab implementations, except for IFGT, which uses Figtree C++ code with Matlab wrappers.The core GPR calculations and several approximation implementations derive from the gpml toolbox.
- Factors affecting suitability: Appropriateness depends on the regression problem, model, and approximation, including function frequency, input dimensionality, irrelevant dimensions, noise, kernel choice, and practical usability.Usability includes numerical robustness, clear tweak-parameter settings, and efficiency across hyperparameter settings.
3. Experiments
The experiments compare approximate GP methods across four datasets using prediction quality against training and test-time costs. Results show strong method- and dataset-dependence, with SoD, FITC, Hybrid, Local, and iterative approaches each exhibiting distinct trade-offs.
- Experimental setup: The study evaluates SMSE and MSLL on synth2, synth8, chem, and sarcos datasets, separating hyperparameter-training time from test time.SMSE measures normalized squared error, whereas MSLL also incorporates predictive variances.
- Results for iterative methods: 50 iterations were insufficient for meaningful convergence in the reported iterative-method experiment, while SoD achieved better SMSE than the iterative methods.A separate synth8 setting showed that DD could outperform CG and SoD in a smaller-time regime, demonstrating dataset and hyperparameter dependence.
- Results for IFGT: Useful IFGT matrix-vector multiplication speedups over direct implementation were obtained only for synth2 across the four datasets.The timing varied with kernel lengthscale and dataset.
- Comparison of SoD, FITC, Hybrid and Local GPR: SoD outperformed FITC for equal hyperparameter-training time on all datasets, whereas FITC was superior for equal test time.The pattern matches their computational scalings: FITC has more expensive training but better predictive performance at test time.
- Comparison of SoD, FITC, Hybrid and Local GPR: Hybrid generally improved performance over SoD for similar hyperparameter-training time, while remaining inferior to FITC at test time for the same subset size.Its faster hyperparameter learning can permit larger subset sizes.
- Comparison of SoD, FITC, Hybrid and Local GPR: Local runtimes were non-monotonic in m; despite overhead and occasional error increases, Local often performed better in the time regimes where it operated.For small m, other methods could be faster than Local for every tested m.
- Dataset-specific results: On synth2, all methods approached the noise-level SMSE of 10^-6 for sufficiently large m, while synth8 remained difficult with SMSE far above its 10^-3 noise level.FPC improved SoD and FITC on synth2, whereas random inducing-point selection slightly improved them on synth8.
- Comparison with Prediction using the Generative Hyperparameters: Learned and fixed hyperparameters generally agreed for SoD and FITC, but differed noticeably for SoD at m ≤128 on synth2 and m ≤512 on synth8.For FITC on synth8, learned hyperparameters slightly improved MSLL by around 0.05 nats and SMSE by up to 0.05.
4. Future directions
The paper identifies open questions around scaling approximation methods, especially how to balance dataset size, approximation controls, and computation. It also reports that Local GPR can outperform alternatives in some time ranges, despite implementation overhead at small m.
- Local GPR can sometimes make better predictions than other methods for some ranges of available computer time.Its performance advantage depends on the available compute-time regime.
- Local GPR has unusual small-m runtime scaling because the implementation tracks thousands of small matrices.More careful, lower-level programming than the Matlab implementation might reduce this overhead.
- Random-selection SoD will eventually outperform the other considered approximations as dataset size tends to infinity because it has no n-dependence.The paper notes that other approximations such as FITC could also be run on a subset.
- Future work should jointly choose dataset size, approximation control m, inducing-point locations, and hyperparameter-training effort.The paper identifies this balancing problem as an open research area for methods with multiple control parameters.
5. Conclusions
The paper recommends evaluating GP approximations by prediction quality versus compute time and comparing them with SoD and FITC. Across the tested datasets, simple SoD or Hybrid methods are strongest for hyperparameter learning, while FITC often gives better test-time accuracy and Local can win in selected regimes.
- Future evaluations should compare prediction-quality-versus-compute-time curves with standard SoD and FITC baselines.The authors made their data and code available to facilitate such comparisons.
- SoD and Hybrid dominate FITC for hyperparameter learning on the considered datasets.This conclusion concerns the datasets and evaluation setting studied in the paper.
- FITC gives better accuracy than SoD and Hybrid for a given test time in the reported comparisons.The conclusion is explicitly limited to the duration for which FITC was run.
- Local produces more varied results but can outperform other methods for some problems and cluster sizes.The paper also notes that iterative methods can be comparable with simpler approaches, while IFGT sped up only synth2 among the tested datasets.
- When hyperparameter learning dominates computation time, SoD or Hybrid is the leading contender among the evaluated methods.The conclusion follows the reported trade-offs between hyperparameter-learning time and test-time accuracy.