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Thoughts on Massively Scalable Gaussian Processes

Andrew Gordon Wilson, Christoph Dann, Hannes Nickisch

arXiv:1511.01870v1cs.LGcs.AIstat.MEstat.ML

TL;DR

Gaussian processes face steep computational costs on large datasets, while scalable approximations often restrict the number of inducing points. MSGP extends KISS-GP with structure-exploiting methods and reports exceptional scalability across training points, inducing points, and test points compared with state-of-the-art alternatives.

  • Problem

    Standard Gaussian-process inference and learning cost O(n^3), and scalable inducing-point approaches can require m ≪ n, limiting predictive performance, representational power, and kernel learning.

    Source: §1 · p. 2

  • Method

    MSGP extends KISS-GP with constant-time test predictions, circulant approximations, BTTB structure, and projections.

    Source: §1 · p. 2, §7 · p. 18

  • Results

    MSGP is exceptionally scalable in training points, inducing points, and test points compared with state-of-the-art alternatives.

    Source: §7 · p. 18

  • Takeaways & Limitations

    Handling large inducing-point sets is important for retaining accuracy in scalable Gaussian-process methods and enabling large-scale kernel learning.

    Source: §7 · p. 18

    The Kronecker covariance decomposition assumes regularly spaced multidimensional inputs and a stationary product kernel.

    Source: §5.2 · p. 7

Abstract

from arXiv · show

We introduce a framework and early results for massively scalable Gaussian processes (MSGP), significantly extending the KISS-GP approach of Wilson and Nickisch (2015). The MSGP framework enables the use of Gaussian processes (GPs) on billions of datapoints, without requiring distributed inference, or severe assumptions. In particular, MSGP reduces the standard $O(n^3)$ complexity of GP learning and inference to $O(n)$, and the standard $O(n^2)$ complexity per test point prediction to $O(1)$. MSGP involves 1) decomposing covariance matrices as Kronecker products of Toeplitz matrices approximated by circulant matrices. This multi-level circulant approximation allows one to unify the orthogonal computational benefits of fast Kronecker and Toeplitz approaches, and is significantly faster than either approach in isolation; 2) local kernel interpolation and inducing points to allow for arbitrarily located data inputs, and $O(1)$ test time predictions; 3) exploiting block-Toeplitz Toeplitz-block structure (BTTB), which enables fast inference and learning when multidimensional Kronecker structure is not present; and 4) projections of the input space to flexibly model correlated inputs and high dimensional data. The ability to handle many ($m \approx n$) inducing points allows for near-exact accuracy and large scale kernel learning.

1 Introduction

Large-scale Gaussian processes are constrained by the cost of standard inference and by the small inducing-point sets required by many scalable approximations. MSGP extends KISS-GP with near-exact linear-time inference and learning, constant-time test predictions, and additional structure-exploiting methods.

1 Introduction

  • Large datasets challenge Gaussian processes because standard methods require O(n^3) computation and O(n^2) storage, while many scalable approximations require m ≪ n.The restricted inducing-point count can affect predictive performance, representational power, and kernel learning.Source: §1 · p. 2
  • MSGP provides near-exact O(n) inference and learning and O(1) mean and variance predictions per test point.Standard GPs and KISS-GP require O(n) predictive-mean and O(n^2) predictive-variance work per test point.Source: §1 · p. 2
  • Circulant approximations integrate Kronecker and Toeplitz structure and enable fast log-determinant evaluations for kernel learning.The paper also reports increased speed over Toeplitz methods for one-dimensional predictors.Source: §1 · p. 2
  • BTTB structure supports fast inference and learning when multidimensional Kronecker structure is absent.Source: §1 · p. 2
  • Projections help address the limitation of Kronecker methods to low-dimensional input spaces.Source: §1 · p. 2

2 Gaussian Processes

The section defines the Gaussian-process model, its covariance notation, and the marginal likelihood used for kernel learning. It also identifies the computational costs that make standard inference and learning difficult at scale.

2 Gaussian Processes

  • A Gaussian process assigns a joint Gaussian distribution to function values, with mean and covariance determined by the process mean and kernel.The covariance entries are k_θ(x_i, x_j), and the kernel is parameterized by θ.Source: §2 · p. 3
  • The marginal likelihood is optimized with respect to θ to perform kernel learning.Source: §2 · p. 3
  • Standard Cholesky-based inference and learning require O(n^3) operations and O(n^2) storage; prediction costs O(n) for the mean and O(n^2) for the variance per test point.Source: §2 · p. 3

3 Structure Exploiting Inference

Structure-exploiting inference uses covariance structure to accelerate Gaussian-process computation, but its usefulness depends on the input grid and dimensionality. The section reviews Kronecker and Toeplitz structure as complementary cases.

3 Structure Exploiting Inference

  • Structure-exploiting approaches can provide exact predictive accuracy and impressive scalability, but severe grid restrictions make them inapplicable to many problems.Source: §3 · p. 3

3.1 Kronecker Structure

  • For multidimensional rectilinear grids with a product kernel, Kronecker structure factors the covariance into per-dimension matrices for efficient inference and learning.The approach offers no efficiency gain for one-dimensional inputs.Source: §3.1 · p. 4

3.2 Toeplitz Structure

  • Stationary kernels on regularly spaced one-dimensional grids yield Toeplitz covariance matrices, enabling O(n log n) inference but leaving log determinants and single-point predictive variances at O(n^2).The section presents Toeplitz methods as complementary to Kronecker methods for multidimensional grids.Source: §3.2 · p. 4

4 KISS-GP

KISS-GP combines inducing points, local kernel interpolation, and structure-exploiting algebra to scale Gaussian-process inference and learning. Its sparse interpolation and large inducing-point sets support near-exact performance without requiring the data inputs themselves to lie on a grid.

4 KISS-GP

  • KISS-GP approximates training-to-inducing cross-covariances with interpolation weights and inducing-point covariances, yielding an approximate training covariance.Source: §4 · pp. 4–5
  • Local cubic interpolation uses only 4^d nonzero entries per row of the interpolation matrix for d-dimensional inputs.Source: §4 · p. 5
  • KISS-GP can scale inference and learning as O(n), accommodate large inducing-point sets, and support near-exact performance and large-scale kernel learning.The approach places inducing points on a rectilinear grid and exploits Toeplitz or Kronecker structure without requiring grid-aligned data inputs.Source: §4 · p. 5
  • Linear conjugate gradients use matrix-vector products for inference, while fast structure-exploiting eigendecompositions approximate eigenvalues for marginal-likelihood evaluation.The cited KISS-GP account reports convergence in j ≪ n iterations, with j depending on conditioning rather than n.Source: §4 · p. 5

5 Massively Scalable Gaussian Processes

MSGP extends KISS-GP with near-exact linear-time inference and learning, constant-time test predictions, circulant and BTTB structure, and input projections. Its circulant comparison reports Whittle approximation error below 1% for grid sizes above 1000.

5 Massively Scalable Gaussian Processes

  • MSGP extends KISS-GP with near-exact O(n) inference and learning, O(1) test predictions, and structure-exploiting methods for broader settings.Its extensions include circulant log-determinant approximations, BTTB structure, and input projections.Source: §5 · pp. 5–6

5.1.1 Predictive Mean

  • O(1) operations per test point follow by precomputing the interpolated mean contribution and multiplying by sparse test-input interpolation weights.The mean approximation uses structured kernel interpolation, with four nonzero weights per row under local cubic interpolation.Source: §5.1.1 · p. 6

5.1.2 Predictive Variance

  • O(1) test-point variance computation uses a stochastic estimate of grid explained variance, followed by multiplication with sparse interpolation weights.The estimate avoids solving a size-n linear system separately for every grid entry.Source: §5.1.2 · p. 7

5.2 Circulant Approximation

  • Circulant approximation unifies Kronecker and Toeplitz benefits, scales Toeplitz log-determinant evaluations, and removes the need for grid-located data inputs.The nested Toeplitz structure otherwise does not make the required full eigendecompositions efficient.Source: §5.2 · pp. 7–8
  • Below 1% relative error for m > 1000, the Whittle approximation remains accurate across tested covariance functions, lengthscales, and noise variances.The reported relative error decays with grid size.Source: §5.2 · p. 9

5.3 BCCB Approximation for BTTB

  • BTTB structure supports fast inference and learning for multivariate grid data without requiring a factorizing covariance function.Dimension-wise circulant embedding yields BCCB structure and carries over the Toeplitz case's computational and approximation benefits.Source: §5.3 · p. 10

5.4 Projections

  • Input projections preserve Kronecker structure while capturing correlations across input dimensions and supporting data with more than five dimensions.The projection matrix is learned in a supervised manner through the Gaussian process marginal likelihood.Source: §5.4 · pp. 10–11

6 Experiments

The preliminary experiments test MSGP's runtime and predictive accuracy in one-dimensional stress tests and high-dimensional projection tasks. They report scalable training and prediction, accurate fast predictions, and projection recovery that remains competitive through about D = 40.

6.1 Stress tests

  • The stress tests use randomly located one-dimensional inputs and regularly spaced inducing points, then assess runtime, prediction accuracy, and scalability.The authors choose one dimension to expose circulant-approximation scalability before moving to higher-dimensional problems.Source: §6 · p. 11, §6.1 · p. 12
  • At n = 10^7, MSGP takes the same training time with m = 10^3 as with m = 10^6, and can use m = 10^6 while overtaking alternatives using m = 10^3.The authors associate this large inducing-point capacity with near-exact accuracy and retaining structure for large-scale kernel learning.Source: §6.1 · p. 12
  • The authors describe MSGP and BDGP as complementary approaches that could be combined, noting BDGP's minibatch optimization of variational and GP hyperparameters.They also note that BDGP's stochastic optimization can take time to converge.Source: §6.1 · p. 12
  • MSGP prediction runtime is practically independent of m and n, and for fixed m is much faster than alternatives whose costs depend at least quadratically on them.Local interpolation also improves runtime over MSGP's standard predictive equations using Kronecker and Toeplitz algebra.Source: §6.1 · p. 12, Figure 3 · p. 14
  • Fast local-interpolation mean predictions are essentially indistinguishable from standard MSGP means, while fast variances have reasonable accuracy and improve with more inducing points and stochastic-estimator samples.Overall fast-prediction error is reported as much less than average data variability.Source: §6.1 · p. 13, Figure 4 · p. 15

6.2 Projections

  • Through about D = 40, learned projections have extremely low subspace and SMAE errors and are competitive in SMAE with an exact GP on the true subspace; at D = 100, they still outperform an exact GP on observed inputs.The projection experiment averages 30 runs per D and reports two standard errors.Source: §6.2 · p. 17

7 Discussion

The discussion presents MSGP as an extension of KISS-GP and structure-exploiting methods, with components for scalable predictions, covariance structure, and projections. The paper reports scalability across training points, inducing points, and test points, and highlights large inducing-point counts for accuracy and kernel learning.

7 Discussion

  • MSGP combines O(1) test predictions, circulant log-determinant approximations, BTTB inference, and input projections to extend KISS-GP and structure-exploiting approaches.The circulant approximations unify Toeplitz and Kronecker structure and extend scalable kernel learning to D = 1; BTTB supports exact multidimensional inference without Kronecker decompositions.Source: §7 · p. 18
  • The reported comparisons show MSGP scales across training points, inducing points, and test points; handling large m is presented as important for accuracy and large-scale kernel learning.The paper characterizes MSGP as exceptionally scalable relative to state-of-the-art alternatives.Source: §7 · p. 18
  • The document describes its contribution as substantial developments to the SKI and KISS-GP frameworks introduced by Wilson and Nickisch (2015).Source: §7 · p. 18

A.1 Derivatives For Normal Projections

The passage states a definition for the orthonormal projection matrix Q, but supplies no further derivative procedure in this excerpt.

A.1 Derivatives For Normal Projections

  • The passage defines Q as diag(p)P in R^(d×D), identifying it as an orthonormal projection matrix.Source: §A.1 · p. 21

A.2 Derivatives For Orthonormal Projections

The passage defines an orthonormal projection using P and specifies an eigenvalue-decomposition construction and componentwise division convention.

A.2 Derivatives For Orthonormal Projections

  • The orthonormal projection matrix is defined as Q = (PP^T)^(-1/2)P, so that QQ^T = I.Source: §A.2 · p. 21
  • The passage specifies an eigenvalue decomposition PP^T = V F V^T and defines S using V and a diagonal matrix of square roots of eigenvalues.Source: §A.2 · p. 21
  • Division in A is component-wise.Source: §A.2 · p. 21

A.3 Circulant Determinant Approximation Benchmark

This section presents additional benchmarks of different circulant approximations.

A.3 Circulant Determinant Approximation Benchmark

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