In this notebook, we demonstrate many of the design features of GPyTorch using the simplest example, training an RBF kernel Gaussian process on a simple function. We'll be modeling the function
\begin{align*} y &= \sin(2\pi x) + \epsilon \\ \epsilon &\sim \mathcal{N}(0, 0.2) \end{align*}with 11 training examples, and testing on 51 test examples.
Note: this notebook is not necessarily intended to teach the mathematical background of Gaussian processes, but rather how to train a simple one and make predictions in GPyTorch. For a mathematical treatment, Chapter 2 of Gaussian Processes for Machine Learning provides a very thorough introduction to GP regression (this entire text is highly recommended): http://www.gaussianprocess.org/gpml/chapters/RW2.pdf
In [7]:
import math
import torch
import gpytorch
from matplotlib import pyplot as plt
%matplotlib inline
%load_ext autoreload
%autoreload 2
In [8]:
# Training data is 11 points in [0,1] inclusive regularly spaced
train_x = torch.linspace(0, 1, 100)
# True function is sin(2*pi*x) with Gaussian noise
train_y = torch.sin(train_x * (2 * math.pi)) + torch.randn(train_x.size()) * 0.2
The next cell demonstrates the most critical features of a user-defined Gaussian process model in GPyTorch. Building a GP model in GPyTorch is different in a number of ways.
First in contrast to many existing GP packages, we do not provide full GP models for the user. Rather, we provide the tools necessary to quickly construct one. This is because we believe, analogous to building a neural network in standard PyTorch, it is important to have the flexibility to include whatever components are necessary. As can be seen in more complicated examples, like the CIFAR10 Deep Kernel Learning example which combines deep learning and Gaussian processes, this allows the user great flexibility in designing custom models.
For most GP regression models, you will need to construct the following GPyTorch objects:
gpytorch.models.ExactGP) - This handles most of the inference.gpytorch.likelihoods.GaussianLikelihood) - This is the most common likelihood used for GP regression.gpytorch.means.ConstantMean() is a good place to start.gpytorch.kernels.ScaleKernel(gpytorch.kernels.RBFKernel()) is a good place to start.gpytorch.distributions.MultivariateNormal) - This is the object used to represent multivariate normal distributions.The components of a user built (Exact, i.e. non-variational) GP model in GPyTorch are, broadly speaking:
An __init__ method that takes the training data and a likelihood, and constructs whatever objects are necessary for the model's forward method. This will most commonly include things like a mean module and a kernel module.
A forward method that takes in some $n \times d$ data x and returns a MultivariateNormal with the prior mean and covariance evaluated at x. In other words, we return the vector $\mu(x)$ and the $n \times n$ matrix $K_{xx}$ representing the prior mean and covariance matrix of the GP.
This specification leaves a large amount of flexibility when defining a model. For example, to compose two kernels via addition, you can either add the kernel modules directly:
self.covar_module = ScaleKernel(RBFKernel() + WhiteNoiseKernel())
Or you can add the outputs of the kernel in the forward method:
covar_x = self.rbf_kernel_module(x) + self.white_noise_module(x)
In [9]:
# We will use the simplest form of GP model, exact inference
class ExactGPModel(gpytorch.models.ExactGP):
def __init__(self, train_x, train_y, likelihood):
super(ExactGPModel, self).__init__(train_x, train_y, likelihood)
self.mean_module = gpytorch.means.ConstantMean()
self.covar_module = gpytorch.kernels.ScaleKernel(gpytorch.kernels.RBFKernel())
def forward(self, x):
mean_x = self.mean_module(x)
covar_x = self.covar_module(x)
return gpytorch.distributions.MultivariateNormal(mean_x, covar_x)
# initialize likelihood and model
likelihood = gpytorch.likelihoods.GaussianLikelihood()
model = ExactGPModel(train_x, train_y, likelihood)
In the next cell, we handle using Type-II MLE to train the hyperparameters of the Gaussian process.
The most obvious difference here compared to many other GP implementations is that, as in standard PyTorch, the core training loop is written by the user. In GPyTorch, we make use of the standard PyTorch optimizers as from torch.optim, and all trainable parameters of the model should be of type torch.nn.Parameter. Because GP models directly extend torch.nn.Module, calls to methods like model.parameters() or model.named_parameters() function as you might expect coming from PyTorch.
In most cases, the boilerplate code below will work well. It has the same basic components as the standard PyTorch training loop:
However, defining custom training loops allows for greater flexibility. For example, it is easy to save the parameters at each step of training, or use different learning rates for different parameters (which may be useful in deep kernel learning for example).
In [10]:
# Find optimal model hyperparameters
model.train()
likelihood.train()
# Use the adam optimizer
optimizer = torch.optim.Adam([
{'params': model.parameters()}, # Includes GaussianLikelihood parameters
], lr=0.1)
# "Loss" for GPs - the marginal log likelihood
mll = gpytorch.mlls.ExactMarginalLogLikelihood(likelihood, model)
training_iter = 50
for i in range(training_iter):
# Zero gradients from previous iteration
optimizer.zero_grad()
# Output from model
output = model(train_x)
# Calc loss and backprop gradients
loss = -mll(output, train_y)
loss.backward()
print('Iter %d/%d - Loss: %.3f lengthscale: %.3f noise: %.3f' % (
i + 1, training_iter, loss.item(),
model.covar_module.base_kernel.lengthscale.item(),
model.likelihood.noise.item()
))
optimizer.step()
In the next cell, we make predictions with the model. To do this, we simply put the model and likelihood in eval mode, and call both modules on the test data.
Just as a user defined GP model returns a MultivariateNormal containing the prior mean and covariance from forward, a trained GP model in eval mode returns a MultivariateNormal containing the posterior mean and covariance. Thus, getting the predictive mean and variance, and then sampling functions from the GP at the given test points could be accomplished with calls like:
f_preds = model(test_x)
y_preds = likelihood(model(test_x))
f_mean = f_preds.mean
f_var = f_preds.variance
f_covar = f_preds.covariance_matrix
f_samples = f_preds.sample(sample_shape=torch.Size(1000,))
The gpytorch.settings.fast_pred_var context is not needed, but here we are giving a preview of using one of our cool features, getting faster predictive distributions using LOVE.
In [11]:
# Get into evaluation (predictive posterior) mode
model.eval()
likelihood.eval()
# Test points are regularly spaced along [0,1]
# Make predictions by feeding model through likelihood
with torch.no_grad(), gpytorch.settings.fast_pred_var():
test_x = torch.linspace(0, 1, 51)
observed_pred = likelihood(model(test_x))
In [12]:
with torch.no_grad():
# Initialize plot
f, ax = plt.subplots(1, 1, figsize=(4, 3))
# Get upper and lower confidence bounds
lower, upper = observed_pred.confidence_region()
# Plot training data as black stars
ax.plot(train_x.numpy(), train_y.numpy(), 'k*')
# Plot predictive means as blue line
ax.plot(test_x.numpy(), observed_pred.mean.numpy(), 'b')
# Shade between the lower and upper confidence bounds
ax.fill_between(test_x.numpy(), lower.numpy(), upper.numpy(), alpha=0.5)
ax.set_ylim([-3, 3])
ax.legend(['Observed Data', 'Mean', 'Confidence'])
In [ ]: