Guide To GPyTorch: A Python Library For Gaussian Process Models

GPyTorch is a PyTorch-based library designed for implementing Gaussian processes. It was introduced by Jacob R. Gardner, Geoff Pleiss, David Bindel, Kilian Q. Weinberger and Andrew Gordon Wilson - researchers at Cornel University (research paper).

Before going into the details of GPyTorch, let us first understand what a Gaussian process means, in short.

Gaussian Process

In probability theory and statistics, the Gaussian process refers to a stochastic process i.e. a collection of random variables indexed by time or space in such a way that each finite collection of the random variables has a multivariate normal distribution (every finite linear combination of the variables is normally distributed).

You might have heard about statistical inference techniques such as Bayesian inference using which one can represent uncertainty over numeric values like the outcome of a dice roll or the height of a person. Gaussian process instead is a probability distribution over possible functions. Find a detailed description of the Gaussian process here.

Overview of GPyTorch

GPyTorch enables easy creation of flexible, scalable and modular Gaussian process models. It is implemented using PyTorch. It performs GP inference via Blackbox Matrix-Matrix multiplication (BBMM).

Pros of GPyTorch

Scalability: It enables training of GPs with millions of data points
Modular design: It has the capability of easily integrating GPs with deep neural networks
Speed: It can utilize state-of-the-art inference algorithms (such as SKI/KISS-GP, stochastic Lanczos expansions, LOVE, SKIP, stochastic variational deep kernel learning) and hardware acceleration using GPUs

Practical implementation

Here’s a demonstration of training an RBF kernel Gaussian process on the following function:

y = sin(2x) + E …(i)

E ~ (0, 0.04)

(where 0 is mean of the normal distribution and 0.04 is the variance)

The code has been implemented in Google colab with Python 3.7.10 and GPyTorch 1.4.0 versions. Step-wise explanation of the code is as follows:

Install the GPyTorch library

!pip install gpytorch

Import required libraries

 import math
 import torch
 import gpytorch
 from matplotlib import pyplot as plt
 %matplotlib inline #for visualization plots to appear at the frontend

Prepare training data

 # Choose regularly spaced 100 points from the interval [0,1] 
 x_train = torch.linspace(0, 1, 100)
 # Compute label as sin(2*pi*x) with Gaussian noise as described by eq.(i) above
 y_train = torch.sin(x_train * (2 * math.pi)) + torch.randn(x_train.size()) * math.sqrt(0.04)

Define the GP model

We have used exact inference - the simplest form of GP model

 class ExactGPModel(gpytorch.models.ExactGP):
     def __init__(self, x_train, y_train, likelihood):
         super(ExactGPModel, self).__init__(x_train, y_train, likelihood)
         self.mean_module = gpytorch.means.ConstantMean() #prior mean
         #covariance 
         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)

For most of the GP regression models, following objects should be constructed:

A ‘GP Model’ which handles most of the inference
A ‘Likelihood’
A ‘Mean’ defining prior mean of GP
A ‘Kernel’ defining covariance of GP
A Multivariate Normal Distribution

The two methods defined above are components of the Exact (non-variational) GP model.

The _init_ method takes a likelihood and the training data. It then constructs objects like mean module and kernel module required for the ‘forward’ method of the model. The ‘forward’ method takes in some data x. It returns a multivariate normal distribution with prior mean and covariance computed at x.

Initialize likelihood

lkh = gpytorch.likelihoods.GaussianLikelihood()

Initialize the GP model

model = ExactGPModel(x_train, y_train, lkh)

Find optimal hyperparameters of the model

 model.train()
 lkh.train()

Output:

 GaussianLikelihood(
    (noise_covar): HomoskedasticNoise(
                 (raw_noise_constraint): GreaterThan(1.000E-04)
               )
            )

Use Adam optimization algorithm

opt = torch.optim.Adam(model.parameters(), lr=0.1)

Define loss for GP

l = gpytorch.mlls.ExactMarginalLogLikelihood(lkh, model)

Compute loss, length scale (i.e. length of twists and turns in the function) and noise for each iteration of the GP

 for i in range(20):
     # Zero gradients from previous iteration
     opt.zero_grad()
     # Store output from the model
     op = model(x_train)
     # Compute loss and backprop gradients
     loss = -l(op, y_train)
     loss.backward()  #back propagation
 #torch.autograd.backward() calculates sum of gradients of given tensors
 #Print the loss, length scale and noise for 20 iterations
     print('Iter %d/%d - Loss: %.3f   lengthscale: %.3f   noise:  
     %.3f' % (
         i + 1, 20, loss.item(),  #iteration number
         model.covar_module.base_kernel.lengthscale.item(), #length scale
         model.likelihood.noise.item() #noise
     ))

Make prediction with the model

 #Evaluation (predictive posterior) mode
 model.eval()
 lkh.eval()

Make predictions by feeding model through likelihood

 with torch.no_grad(), gpytorch.settings.fast_pred_var():
     x_test = torch.linspace(0, 1, 51) 
#equally spaced 51 test points in [0,1]
     observed_pred = likelihood(model(x_test))

Plot the fitted model

 with torch.no_grad():  #disable gradient computation
     # Initialize plot
     fig, axis = plt.subplots(1, 1, figsize=(4, 3))
     # Upper and lower confidence bounds
     lower, upper = observed_pred.confidence_region()
     # Plot training data as black stars
     axis.plot(x_train.numpy(), y_train.numpy(), 'k*')
     # Plot predictive means as blue line
     axis.plot(x_test.numpy(), observed_pred.mean.numpy(), 'b')
     # Shade between the lower and upper confidence bounds
     #fill the area showing confidence
     axis.fill_between(x_test.numpy(), lower.numpy(), upper.numpy(),  
     alpha=0.5)
     axis.set_ylim([-3, 3])  #Y-direction limits
     axis.legend(['Observed Data', 'Mean', 'Confidence'])

Output plot:

Code source
Google colab notebook of the above implementation

References

To get a detailed understanding of the GPyTorch library, refer to the following web links: