NTR Labs Webinar

Sequential Decision Making

 with Gaussian Processes

Viacheslav Borovitskiy (Slava)

Sequential Decision Making

Gaussian processes — gold standard in uncertainty estimation

Gaussian Processes

Definition. A Gaussian process is random function $f : X \times \Omega \to \R$ such that for any $x_1,..,x_n$, the vector $f(x_1),..,f(x_n)$ is multivariate Gaussian.

The distribution of a Gaussian process is characterized by

  • a mean function $m(x) = \E(f(x))$,
  • a kernel (covariance) function $k(x, x') = \Cov(f(x), f(x'))$,

Notation: $f \~ \f{GP}(m, k)$.

The kernel $k$ must be positive (semi-)definite.

Gaussian Process Regression


  • a prior Gaussian Process $\f{GP}(m, k)$
  • and data $(x_1, y_1), .., (x_n, y_n) \in X \x \R$,

giving the posterior Gaussian process $\f{GP}(\hat{m}, \hat{k})$.

The functions $\hat{m}$ and $\hat{k}$ may be explicitly expressed in terms of $m$ and $k$.

An Application: Bayesian Optimization

Goal: minimize unknown function $\phi$ in as few evaluations as possible.

  1. Build GP posterior $f \given \v{y}$ using data $(x_1,\phi(x_1)),..,(x_n,\phi(x_n))$.
  2. Choose $$ \htmlClass{fragment}{ x_{n+1} = \argmax_{x\in\c{X}} \underbrace{\alpha_{f\given\v{y}}(x).}_{\text{acquisition function}} } $$ For instance, expected improvement $$ \alpha_{f\given\v{y}}(x) = \E_{f\given\v{y}} \max(0, {\displaystyle\min_{i=1,..,n}} \phi(x_i) - f(x)). $$

Bayesian Optimization 1/15

Bayesian Optimization 2/15

Bayesian Optimization 3/15

Bayesian Optimization 4/15

Bayesian Optimization 5/15

Bayesian Optimization 6/15

Bayesian Optimization 7/15

Bayesian Optimization 8/15

Bayesian Optimization 9/15

Bayesian Optimization 10/15

Bayesian Optimization 11/15

Bayesian Optimization 12/15

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Bayesian Optimization 15/15

Bayesian Optimization



  • geostatistics,
  • robotics,
  • more...

Efficient Algorithms

Algorithms Connected to Gaussian Processes

And hyperparameter optimization! Complexity: $O(N^3 + M^3)$ for $N$ data points and $M$ sampling nodes.

Efficient Conditioning

Idea. Approximate the posterior process $\f{GP}(\hat{m}, \hat{k})$ with a simpler process belonging to some parametric family $\f{GP}(m_{\theta}, k_{\theta})$.

$O(N^3)$ turns into $O(N K^2)$ where $K$ controls accuracy

Titsias (2009, AISTATS)

Efficient Sampling From Priors

Priors are usually assumed stationary.

Idea. Approximate with a process of form

$$ \htmlData{class=yoyo}{ \tilde{f}(x) = \sum_{l=1}^L w_l \phi_l(x) \qquad w_l \overset{\textrm{i.i.d.}}{\sim} N(0, 1), } $$

which is easy to sample from.

$O(M^3)$ turns into $O(M L)$ where $L$ controls accuracy


Efficient Sampling From Posteriors

Idea. From a parametric family $\f{GP}(m_{\theta}, k_{\theta})$ build another family $\f{GP}(\tilde{m}_{\theta}, \tilde{k}_{\theta})$ which is easy to sample from.

$O(N^3 + M^3)$ turns into $O(N K^2 + M L)$ where $K$ and $L$ control accuracy

Wilson, Borovitskiy, Terenin, Mostowsky & Deisenroth (2020, ICML)

Geometry-aware Priors
on Manifolds and Graphs

Euclidean Priors: Matérn Gaussian Processes

$$ \htmlData{class=fragment fade-out,fragment-index=9}{ \footnotesize \mathclap{ k_{\nu, \kappa, \sigma^2}(x,x') = \sigma^2 \frac{2^{1-\nu}}{\Gamma(\nu)} \del{\sqrt{2\nu} \frac{\norm{x-x'}}{\kappa}}^\nu K_\nu \del{\sqrt{2\nu} \frac{\norm{x-x'}}{\kappa}} } } \htmlData{class=fragment d-print-none,fragment-index=9}{ \footnotesize \mathclap{ k_{\infty, \kappa, \sigma^2}(x,x') = \sigma^2 \exp\del{-\frac{\norm{x-x'}^2}{2\kappa^2}} } } $$ $\sigma^2$: variance $\kappa$: length scale $\nu$: smoothness
$\nu\to\infty$: Gaussian kernel (RBF)

$\nu = 1/2$

$\nu = 3/2$

$\nu = 5/2$

$\nu = \infty$

Gaussian Processes on
Non-Euclidean Domains

Non-Euclidean Domains



How should Matérn kernels generalize to this setting?

Explicit Formula for Compact Riemannian Manifolds

$$ k_{\nu, \kappa, \sigma^2}(x,x') = \frac{\sigma^2}{C_\nu} \sum_{n=0}^\infty \del{\frac{2\nu}{\kappa^2} - \lambda_n}^{-\nu-\frac{d}{2}} f_n(x) f_n(x') $$

$\lambda_n, f_n$: eigenvalues and eigenfunctions of the Laplace–Beltrami


Borovitskiy, Terenin, Mostowsky & Deisenroth (2020, NeurIPS)

Explicit Formula for Finite Undirected Graphs

$$ k_{\nu, \kappa, \sigma^2}(i, j) = \frac{\sigma^2}{C_{\nu}} \sum_{n=0}^{\abs{V}-1} \del{\frac{2\nu}{\kappa^2} + \mathbf{\lambda_n}}^{-\nu} \mathbf{f_n}(i)\mathbf{f_n}(j) $$

$\lambda_n, \mathbf{f_n}$: eigenvalues and eigenvectors of the Laplacian matrix


Borovitskiy et al. (2021, AISTATS)

Graph Matérn kernels: examples

$k_{5/2}(\htmlStyle{color:rgb(255, 19, 0)!important}{\circ},\.)$

An Application: Modeling Dynamical Systems with Uncertainty

$$ \htmlData{fragment-index=0,class=fragment}{ x_0 } \qquad \htmlData{fragment-index=1,class=fragment}{ x_1 = x_0 + f(x_0)\Delta t } \qquad \htmlData{fragment-index=2,class=fragment}{ x_2 = x_1 + f(x_1)\Delta t } \qquad \htmlData{fragment-index=3,class=fragment}{ .. } $$

Pendulum Dynamics: GP on a cylinder

(a) Ground truth

(b) Gaussian process

Borovitskiy, Terenin, Mostowsky & Deisenroth (2020, NeurIPS)

Traffic Speed Extrapolation on a Road Network

(a) Prediction

(b) Standard Deviation

Borovitskiy et al. (2021, AISTATS)

Wind Speed Modeling: Vector Fields on Manifolds

Hutchinson, Terenin, Borovitskiy, Takao, Teh, Deisenroth (2021, NeurIPS)

Thank you!

Thank you!

Special thanks: Ekaterina Noskova for the wonderful Bayesian optimization animations. viacheslav.borovitskiy@gmail.com                       https://vab.im


J. Wilson, V. Borovitskiy, A. Terenin, P. Mostowsky, M. P. Deisenroth. Efficiently sampling functions from Gaussian process posteriors. International Conference on Machine Learning, 2020.

V. Borovitskiy, A. Terenin, P. Mostowsky, M. P. Deisenroth. Matérn Gaussian Processes on Riemannian Manifolds.
In Neural Information Processing Systems (NeurIPS) 2020.

V. Borovitskiy, I. Azangulov, A. Terenin, P. Mostowsky, M. P. Deisenroth. Matérn Gaussian Processes on Graphs.
In International Conference on Artificial Intelligence and Statistics (AISTATS) 2021.

J. Wilson, V. Borovitskiy, A. Terenin, P. Mostowsky, M. P. Deisenroth. Pathwise Conditioning of Gaussian Processes. Journal of Machine Learning Research, 2021.

N. Jaquier, V. Borovitskiy, A. Smolensky, A. Terenin, T. Asfour, L. Rozo. Geometry-aware Bayesian Optimization in Robotics using Riemannian Matérn Kernels. To appear in Conference on Robot Learning (CoRL), 2021.

M. Hutchinson, A. Terenin, V. Borovitskiy, S. Takao, Y. W. Teh, M. P. Deisenroth. Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge-Independent Projected Kernels. To appear in NeurIPS 2021.

viacheslav.borovitskiy@gmail.com                       https://vab.im