Efficient factoring of GP likelihoods

Inducing variables

Sparse Gaussian processes are ones where you approximate the target posterior by summarising the data with a short list of inducing points that have nearly the same similar posterior density, by some metric. These have been invented in various forms by various people, but most of the differences we can ignore. Based on citations, thethe Right Way is the version of Titsias (2009), which we use in all the subsequent things as a building-block.

An explanation, based on Hensman, Fusi, and Lawrence (2013) summarizing Titsias (2009):

We consider an output vector \(\mathbf{y},\) where each entry \(y_{i}\) is a noisy observation of the function \(f,\) i.e. \(y_{i}=f(\mathbf{x}_{i})+\varepsilon_{i}\) for all the points \(\mathbf{X}=\left\{\mathbf{x}_{i}\right\}_{i=1}^{n}.\) We assume the noises \(\varepsilon_{i}\sim\mathcal{N}(0,1/\beta)\) to be independent. We introduce a Gaussian process prior over \(f(\cdot).\) Now let the vector \(\mathbf{f}=\{f(x_{i})\}_{i=1}^{n}\) contain values of the function evaluated at the inputs. We introduce a set of inducing variables, defined similarly: let the vector \(\mathbf{u}=\{f(\mathbf{z}_{i})\}_{i=1}^{m}\). Standard properties of Gaussian processes allow us to write \[ \begin{aligned} p(\mathbf{y} \mid \mathbf{f}) &=\mathcal{N}\left(\mathbf{y} \mid \mathbf{f}, \beta^{-1} \mathbf{I}\right) \\ p(\mathbf{f} \mid \mathbf{u}) &=\mathcal{N}\left(\mathbf{f} \mid \mathbf{K}_{n m} \mathbf{K}_{m m}^{-1} \mathbf{u}, \tilde{\mathbf{K}}\right) \\ p(\mathbf{u}) &=\mathcal{N}\left(\mathbf{u} \mid \mathbf{0}, \mathbf{K}_{m m}\right) \end{aligned} \] where \(\mathbf{K}_{m m}\) is the covariance function evaluated between all the inducing points and \(\mathbf{K}_{n m}\) is the covariance function between all inducing points and training points and we have defined with \(\tilde{\mathbf{K}}=\mathbf{K}_{n n}-\) \(\mathbf{K}_{n m} \mathbf{K}_{m m}^{-1} \mathbf{K}_{m n}\)

Quoting them:

We first apply Jensen’s inequality on the conditional probability \(p(\mathbf{y} \mid \mathbf{u})\) \[ \begin{aligned} \log p(\mathbf{y} \mid \mathbf{u}) &=\log \langle p(\mathbf{y} \mid \mathbf{f})\rangle_{p(\mathbf{f} \mid \mathbf{u})} \\ & \geq\langle\log p(\mathbf{y} \mid \mathbf{f})\rangle_{p(\mathbf{f} \mid \mathbf{u})} \triangleq \mathcal{L}_{1} \end{aligned} \] where \(\langle\cdot\rangle_{p(x)}\) denotes an expectation under \(p(x).\) For Gaussian noise taking the expectation inside the log is tractable, but it results in an expression containing \(\mathbf{K}_{n n}^{-1},\) which has a computational complexity of \(\mathcal{O}\left(n^{3}\right).\) Bringing the expectation outside the log gives a lower bound, \(\mathcal{L}_{1},\) which can be computed with has complexity \(\mathcal{O}\left(m^{3}\right).\) Further, when \(p(\mathbf{y} \mid \mathbf{f})\) factorises across the data, \[ p(\mathbf{y} \mid \mathbf{f})=\prod_{i=1}^{n} p\left(y_{i} \mid f_{i}\right) \] then this lower bound can be shown to be separable across y giving \[ \exp \left(\mathcal{L}_{1}\right)=\prod_{i=1}^{n} \mathcal{N}\left(y_{i} \mid \mu_{i}, \beta^{-1}\right) \exp \left(-\frac{1}{2} \beta \tilde{k}_{i, i}\right) \] where \(\boldsymbol{\mu}=\mathbf{K}_{n m} \mathbf{K}_{m m}^{-1} \mathbf{u}\) and \(\tilde{k}_{i, i}\) is the \(i\) th diagonal element of \(\widetilde{\mathbf{K}}\). Note that the difference between our bound and the original log likelihood is given by the Kullback Leibler (KL) divergence between the posterior over the mapping function given the data and the inducing variables and the posterior of the mapping function given the inducing variables only, \[ \mathrm{KL}(p(\mathbf{f} \mid \mathbf{u}) \| p(\mathbf{f} \mid \mathbf{u}, \mathbf{y})) \] This KL divergence is minimized when there are \(m=n\) inducing variables and they are placed at the training data locations. This means that \(\mathbf{u}=\mathbf{f}\), \(\mathbf{K}_{m m}=\mathbf{K}_{n m}=\mathbf{K}_{n n}\) meaning that \(\tilde{\mathbf{K}}=\mathbf{0} .\) In this case we recover \(\exp \left(\mathcal{L}_{1}\right)=p(\mathbf{y} \mid \mathbf{f})\) and the bound becomes equality because \(p(\mathbf{f} \mid \mathbf{u})\) is degenerate. However, since \(m=n\) and that there would be no computational or storage advantage from the representation. When \(m<n\) the bound can be maximised with respect to \(\mathbf{Z}\) (which are variational parameters). This minimises the KL divergence and ensures that \(\mathbf{Z}\) are distributed amongst the training data \(\mathbf{X}\) such that all \(\tilde{k}_{i, i}\) are small. In practice this means that the expectations in (1) are only taken across a narrow domain (\(\tilde{k}_{i, i}\) is the marginal variance of \(p\left(f_{i} \mid \mathbf{u}\right)\) ), keeping Jensen’s bound tight.

…we recover the bound of Titsias (2009) by marginalising the inducing variables, \[ \begin{aligned} \log p(\mathbf{y} \mid \mathbf{X}) &=\log \int p(\mathbf{y} \mid \mathbf{u}) p(\mathbf{u}) \mathrm{d} \mathbf{u} \\ & \geq \log \int \exp \left\{\mathcal{L}_{1}\right\} p(\mathbf{u}) \mathrm{d} \mathbf{u} \triangleq \mathcal{L}_{2} \end{aligned} \] which with some linear algebraic manipulation leads to \[ \mathcal{L}_{2}=\log \mathcal{N}\left(\mathbf{y} \mid \mathbf{0}, \mathbf{K}_{n m} \mathbf{K}_{m m}^{-1} \mathbf{K}_{m n}+\beta^{-1} \mathbf{I}\right)-\frac{1}{2} \beta \operatorname{tr}(\widetilde{\mathbf{K}}) \] matching the result of Titsias, with the implicit approximating distribution \(q(\mathbf{u})\) having precision \[ \mathbf{\Lambda}=\beta \mathbf{K}_{m m}^{-1} \mathbf{K}_{m n} \mathbf{K}_{n m} \mathbf{K}_{m m}^{-1}+\mathbf{K}_{m m}^{-1} \] and mean \[ \hat{\mathbf{u}}=\beta \mathbf{\Lambda}^{-1} \mathbf{K}_{m m}^{-1} \mathbf{K}_{m n} \mathbf{y} \]

Spectral and rank sparsity

Loosely speaking, where the functions can be represented in a small number of basis functions. See, for, example (Adam et al. 2020; Zammit-Mangion and Cressie 2021).

SVI for Gaussian processes

As seen in Hensman, Fusi, and Lawrence (2013);Salimbeni and Deisenroth (2017).

Low rank methods

Represent the GP in terms of a controlled budget of basis functions. See low-rank Gaussian processes.

Vecchia factorisation

Approximate the precision matrix by one with a sparse cholesky factorisation. See Vecchia factorization.

Latent Gaussian Process models

The Edwin V. Bonilla, Krauth, and Dezfouli (2019) set up for Latent Gaussian Process models (“LGPMs”) goes as follows:

We are learning a mapping \(\boldsymbol{f}:\mathbb{R}^D\to\mathbb{R}^P\) from data. The dataset looks like \(\mathcal{D}=\left\{\mathbf{x}_{n}, \mathbf{y}_{n}\right\}_{n=1}^{N}\equiv \left\{\mathbf{x}, \mathbf{y}\right\}.\) \(\mathbf{x}_{n}\in \mathbb{R}^D\) is an input vector and \(\mathbf{y}_{n}\in\mathbb{R}^P\) is an output. We decree that the mapping from inputs to outputs, may be expressed by \(Q\) underlying latent functions \(\left\{f_{j}\right\}_{j=1}^{Q}.\) We assume that the \(Q\) latent functions \(\left\{f_{j}\right\}\) are drawn from (a priori) independent zero-mean Gaussian processes.

\[ \begin{aligned} p\left(f_{j} \mid \boldsymbol{\theta}_{j}\right) & \sim \mathcal{G} \mathcal{P}\left(0, \kappa_{j}\left(\cdot, \cdot ; \boldsymbol{\theta}_{j}\right)\right), \quad j=1, \ldots Q, \quad \text { and } \\ p(\mathbf{f} \mid \boldsymbol{\theta}) &=\prod_{j=1}^{Q} p\left(\mathbf{f}_{\cdot j} \mid \boldsymbol{\theta}_{j}\right) \\ &=\prod_{j=1}^{Q} \mathcal{N}\left(\mathbf{f}_{\cdot j} ; \mathbf{0}, \mathbf{K}_{\mathbf{x x}}^{j}\right). \end{aligned} \] Here \(\mathbf{f}\) is the set of all latent function values; \(\mathbf{f}_{\cdot j}=\left\{f_{j}\left(\mathbf{x}_{n}\right)\right\}_{n=1}^{N}\) denotes the values of latent function \(j\). The Gram matrix is \(\mathbf{K}_{\mathrm{xx}}^{j}\), induced by a covariance kernel, \(\kappa_{j}\left(\cdot, \cdot ; \boldsymbol{\theta}_{j}\right)\). The parameters of all kernel functions we call \(\boldsymbol{\theta}=\left\{\boldsymbol{\theta}_{j}\right\}.\) Our observation model can have various likelihoods; We call the corresponding parameter \(\boldsymbol{\phi}\). We assume that our multi-dimensional observations \(\left\{\mathbf{y}_{n}\right\}\) are i.i.d. given the latent functions \(\left\{\mathbf{f}_{n}\right\},\) so that \[ p(\mathbf{y} \mid \mathbf{f}, \boldsymbol{\phi})=\prod_{n=1}^{N} p\left(\mathbf{y}_{n} \mid \mathbf{f}_{n \cdot}, \boldsymbol{\phi}\right) \] \(\mathbf{f}_{n\cdot}=\{f_{j}(\boldsymbol{x}_n)\}_{j=1}^{q}\) is the set of latent \(\boldsymbol{f}\) values upon which \(\mathbf{y}_{n}\) depends.

There are several factorizations to note here

1. The prior is factored into latent functions per-coordinate
2. the conditional likelihood is factored over observations (i.e. nosie in independent)

If we further factorise the variational approximation in some way this will work out nicely, e.g. into Gaussian mixtures. This is going to work out well for us when we try to devise a system of inference later to minimise the ELBO. TBC.

For now, though, let us examine exactly tractable inference

• Implementing a scalable multi-output GP model with exact inference
• Scaling multi-output Gaussian process models with exact inference

References