Source code for

# Copyright (c) 2017-2019 Uber Technologies, Inc.
# SPDX-License-Identifier: Apache-2.0

import torch
from torch.distributions import constraints
from torch.nn import Parameter

import pyro
import pyro.distributions as dist
from import GPModel
from pyro.nn.module import PyroParam, pyro_method

[docs]class SparseGPRegression(GPModel): """ Sparse Gaussian Process Regression model. In :class:`.GPRegression` model, when the number of input data :math:`X` is large, the covariance matrix :math:`k(X, X)` will require a lot of computational steps to compute its inverse (for log likelihood and for prediction). By introducing an additional inducing-input parameter :math:`X_u`, we can reduce computational cost by approximate :math:`k(X, X)` by a low-rank Nystr\u00F6m approximation :math:`Q` (see reference [1]), where .. math:: Q = k(X, X_u) k(X_u,X_u)^{-1} k(X_u, X). Given inputs :math:`X`, their noisy observations :math:`y`, and the inducing-input parameters :math:`X_u`, the model takes the form: .. math:: u & \\sim \\mathcal{GP}(0, k(X_u, X_u)),\\\\ f & \\sim q(f \\mid X, X_u) = \\mathbb{E}_{p(u)}q(f\\mid X, X_u, u),\\\\ y & \\sim f + \\epsilon, where :math:`\\epsilon` is Gaussian noise and the conditional distribution :math:`q(f\\mid X, X_u, u)` is an approximation of .. math:: p(f\\mid X, X_u, u) = \\mathcal{N}(m, k(X, X) - Q), whose terms :math:`m` and :math:`k(X, X) - Q` is derived from the joint multivariate normal distribution: .. math:: [f, u] \\sim \\mathcal{GP}(0, k([X, X_u], [X, X_u])). This class implements three approximation methods: + Deterministic Training Conditional (DTC): .. math:: q(f\\mid X, X_u, u) = \\mathcal{N}(m, 0), which in turns will imply .. math:: f \\sim \\mathcal{N}(0, Q). + Fully Independent Training Conditional (FITC): .. math:: q(f\\mid X, X_u, u) = \\mathcal{N}(m, diag(k(X, X) - Q)), which in turns will correct the diagonal part of the approximation in DTC: .. math:: f \\sim \\mathcal{N}(0, Q + diag(k(X, X) - Q)). + Variational Free Energy (VFE), which is similar to DTC but has an additional `trace_term` in the model's log likelihood. This additional term makes "VFE" equivalent to the variational approach in :class:`.VariationalSparseGP` (see reference [2]). .. note:: This model has :math:`\\mathcal{O}(NM^2)` complexity for training, :math:`\\mathcal{O}(NM^2)` complexity for testing. Here, :math:`N` is the number of train inputs, :math:`M` is the number of inducing inputs. References: [1] `A Unifying View of Sparse Approximate Gaussian Process Regression`, Joaquin Qui\u00F1onero-Candela, Carl E. Rasmussen [2] `Variational learning of inducing variables in sparse Gaussian processes`, Michalis Titsias :param torch.Tensor X: A input data for training. Its first dimension is the number of data points. :param torch.Tensor y: An output data for training. Its last dimension is the number of data points. :param kernel: A Pyro kernel object, which is the covariance function :math:`k`. :param torch.Tensor Xu: Initial values for inducing points, which are parameters of our model. :param torch.Tensor noise: Variance of Gaussian noise of this model. :param callable mean_function: An optional mean function :math:`m` of this Gaussian process. By default, we use zero mean. :param str approx: One of approximation methods: "DTC", "FITC", and "VFE" (default). :param float jitter: A small positive term which is added into the diagonal part of a covariance matrix to help stablize its Cholesky decomposition. :param str name: Name of this model. """ def __init__( self, X, y, kernel, Xu, noise=None, mean_function=None, approx=None, jitter=1e-6 ): assert isinstance( X, torch.Tensor ), "X needs to be a torch Tensor instead of a {}".format(type(X)) if y is not None: assert isinstance( y, torch.Tensor ), "y needs to be a torch Tensor instead of a {}".format(type(y)) assert isinstance( Xu, torch.Tensor ), "Xu needs to be a torch Tensor instead of a {}".format(type(Xu)) super().__init__(X, y, kernel, mean_function, jitter) self.Xu = Parameter(Xu) noise = self.X.new_tensor(1.0) if noise is None else noise self.noise = PyroParam(noise, constraints.positive) if approx is None: self.approx = "VFE" elif approx in ["DTC", "FITC", "VFE"]: self.approx = approx else: raise ValueError( "The sparse approximation method should be one of " "'DTC', 'FITC', 'VFE'." )
[docs] @pyro_method def model(self): self.set_mode("model") # W = (inv(Luu) @ Kuf).T # Qff = Kfu @ inv(Kuu) @ Kuf = W @ W.T # Fomulas for each approximation method are # DTC: y_cov = Qff + noise, trace_term = 0 # FITC: y_cov = Qff + diag(Kff - Qff) + noise, trace_term = 0 # VFE: y_cov = Qff + noise, trace_term = tr(Kff-Qff) / noise # y_cov = W @ W.T + D # trace_term is added into log_prob N = self.X.size(0) M = self.Xu.size(0) Kuu = self.kernel(self.Xu).contiguous() Kuu.view(-1)[:: M + 1] += self.jitter # add jitter to the diagonal Luu = torch.linalg.cholesky(Kuu) Kuf = self.kernel(self.Xu, self.X) W = torch.linalg.solve_triangular(Luu, Kuf, upper=False).t() D = self.noise.expand(N) if self.approx == "FITC" or self.approx == "VFE": Kffdiag = self.kernel(self.X, diag=True) Qffdiag = W.pow(2).sum(dim=-1) if self.approx == "FITC": D = D + Kffdiag - Qffdiag else: # approx = "VFE" trace_term = (Kffdiag - Qffdiag).sum() / self.noise trace_term = trace_term.clamp(min=0) zero_loc = self.X.new_zeros(N) f_loc = zero_loc + self.mean_function(self.X) if self.y is None: f_var = D + W.pow(2).sum(dim=-1) return f_loc, f_var else: if self.approx == "VFE": pyro.factor(self._pyro_get_fullname("trace_term"), -trace_term / 2.0) return pyro.sample( self._pyro_get_fullname("y"), dist.LowRankMultivariateNormal(f_loc, W, D) .expand_by(self.y.shape[:-1]) .to_event(self.y.dim() - 1), obs=self.y, )
[docs] @pyro_method def guide(self): self.set_mode("guide") self._load_pyro_samples()
[docs] def forward(self, Xnew, full_cov=False, noiseless=True): r""" Computes the mean and covariance matrix (or variance) of Gaussian Process posterior on a test input data :math:`X_{new}`: .. math:: p(f^* \mid X_{new}, X, y, k, X_u, \epsilon) = \mathcal{N}(loc, cov). .. note:: The noise parameter ``noise`` (:math:`\epsilon`), the inducing-point parameter ``Xu``, together with kernel's parameters have been learned from a training procedure (MCMC or SVI). :param torch.Tensor Xnew: A input data for testing. Note that ``Xnew.shape[1:]`` must be the same as ``self.X.shape[1:]``. :param bool full_cov: A flag to decide if we want to predict full covariance matrix or just variance. :param bool noiseless: A flag to decide if we want to include noise in the prediction output or not. :returns: loc and covariance matrix (or variance) of :math:`p(f^*(X_{new}))` :rtype: tuple(torch.Tensor, torch.Tensor) """ self._check_Xnew_shape(Xnew) self.set_mode("guide") # W = inv(Luu) @ Kuf # Ws = inv(Luu) @ Kus # D as in self.model() # K = I + W @ inv(D) @ W.T = L @ L.T # S = inv[Kuu + Kuf @ inv(D) @ Kfu] # = inv(Luu).T @ inv[I + inv(Luu)@ Kuf @ inv(D)@ Kfu @ inv(Luu).T] @ inv(Luu) # = inv(Luu).T @ inv[I + W @ inv(D) @ W.T] @ inv(Luu) # = inv(Luu).T @ inv(K) @ inv(Luu) # = inv(Luu).T @ inv(L).T @ inv(L) @ inv(Luu) # loc = Ksu @ S @ Kuf @ inv(D) @ y = Ws.T @ inv(L).T @ inv(L) @ W @ inv(D) @ y # cov = Kss - Ksu @ inv(Kuu) @ Kus + Ksu @ S @ Kus # = kss - Ksu @ inv(Kuu) @ Kus + Ws.T @ inv(L).T @ inv(L) @ Ws N = self.X.size(0) M = self.Xu.size(0) # TODO: cache these calculations to get faster inference Kuu = self.kernel(self.Xu).contiguous() Kuu.view(-1)[:: M + 1] += self.jitter # add jitter to the diagonal Luu = torch.linalg.cholesky(Kuu) Kuf = self.kernel(self.Xu, self.X) W = torch.linalg.solve_triangular(Luu, Kuf, upper=False) D = self.noise.expand(N) if self.approx == "FITC": Kffdiag = self.kernel(self.X, diag=True) Qffdiag = W.pow(2).sum(dim=0) D = D + Kffdiag - Qffdiag W_Dinv = W / D K = W_Dinv.matmul(W.t()).contiguous() K.view(-1)[:: M + 1] += 1 # add identity matrix to K L = torch.linalg.cholesky(K) # get y_residual and convert it into 2D tensor for packing y_residual = self.y - self.mean_function(self.X) y_2D = y_residual.reshape(-1, N).t() W_Dinv_y = W_Dinv.matmul(y_2D) # End caching ---------- Kus = self.kernel(self.Xu, Xnew) Ws = torch.linalg.solve_triangular(Luu, Kus, upper=False) pack =, Ws), dim=1) Linv_pack = torch.linalg.solve_triangular(L, pack, upper=False) # unpack Linv_W_Dinv_y = Linv_pack[:, : W_Dinv_y.shape[1]] Linv_Ws = Linv_pack[:, W_Dinv_y.shape[1] :] C = Xnew.size(0) loc_shape = self.y.shape[:-1] + (C,) loc = Linv_W_Dinv_y.t().matmul(Linv_Ws).reshape(loc_shape) if full_cov: Kss = self.kernel(Xnew).contiguous() if not noiseless: Kss.view(-1)[:: C + 1] += self.noise # add noise to the diagonal Qss = Ws.t().matmul(Ws) cov = Kss - Qss + Linv_Ws.t().matmul(Linv_Ws) cov_shape = self.y.shape[:-1] + (C, C) cov = cov.expand(cov_shape) else: Kssdiag = self.kernel(Xnew, diag=True) if not noiseless: Kssdiag = Kssdiag + self.noise Qssdiag = Ws.pow(2).sum(dim=0) cov = Kssdiag - Qssdiag + Linv_Ws.pow(2).sum(dim=0) cov_shape = self.y.shape[:-1] + (C,) cov = cov.expand(cov_shape) return loc + self.mean_function(Xnew), cov