# Sparse Gaussian Process Regression (SGPR)¶

## Overview¶

In this notebook, we’ll overview how to use SGPR in which the inducing point locations are learned.

[1]:

import math
import torch
import gpytorch
import tqdm.notebook as tqdm
from matplotlib import pyplot as plt

# Make plots inline
%matplotlib inline


For this example notebook, we’ll be using the elevators UCI dataset used in the paper. Running the next cell downloads a copy of the dataset that has already been scaled and normalized appropriately. For this notebook, we’ll simply be splitting the data using the first 80% of the data as training and the last 20% as testing.

Note: Running the next cell will attempt to download a ~400 KB dataset file to the current directory.

[2]:

import urllib.request
import os
from math import floor

# this is for running the notebook in our testing framework
smoke_test = ('CI' in os.environ)

if not smoke_test and not os.path.isfile('../elevators.mat'):

if smoke_test:  # this is for running the notebook in our testing framework
X, y = torch.randn(1000, 3), torch.randn(1000)
else:
X = data[:, :-1]
X = X - X.min(0)[0]
X = 2 * (X / X.max(0)[0]) - 1
y = data[:, -1]

train_n = int(floor(0.8 * len(X)))
train_x = X[:train_n, :].contiguous()
train_y = y[:train_n].contiguous()

test_x = X[train_n:, :].contiguous()
test_y = y[train_n:].contiguous()

if torch.cuda.is_available():
train_x, train_y, test_x, test_y = train_x.cuda(), train_y.cuda(), test_x.cuda(), test_y.cuda()

[3]:

X.size()

[3]:

torch.Size([16599, 18])


## Defining the SGPR Model¶

We now define the GP model. For more details on the use of GP models, see our simpler examples. This model constructs a base scaled RBF kernel, and then simply wraps it in an InducingPointKernel. Other than this, everything should look the same as in the simple GP models.

[4]:

from gpytorch.means import ConstantMean
from gpytorch.kernels import ScaleKernel, RBFKernel, InducingPointKernel
from gpytorch.distributions import MultivariateNormal

class GPRegressionModel(gpytorch.models.ExactGP):
def __init__(self, train_x, train_y, likelihood):
super(GPRegressionModel, self).__init__(train_x, train_y, likelihood)
self.mean_module = ConstantMean()
self.base_covar_module = ScaleKernel(RBFKernel())
self.covar_module = InducingPointKernel(self.base_covar_module, inducing_points=train_x[:500, :].clone(), likelihood=likelihood)

def forward(self, x):
mean_x = self.mean_module(x)
covar_x = self.covar_module(x)
return MultivariateNormal(mean_x, covar_x)

[5]:

likelihood = gpytorch.likelihoods.GaussianLikelihood()
model = GPRegressionModel(train_x, train_y, likelihood)

if torch.cuda.is_available():
model = model.cuda()
likelihood = likelihood.cuda()


### Training the model¶

[6]:

training_iterations = 2 if smoke_test else 100

# Find optimal model hyperparameters
model.train()
likelihood.train()

# "Loss" for GPs - the marginal log likelihood
mll = gpytorch.mlls.ExactMarginalLogLikelihood(likelihood, model)

def train():
iterator = tqdm.tqdm(range(training_iterations), desc="Train")

for i in iterator:
# Get output from model
output = model(train_x)
# Calc loss and backprop derivatives
loss = -mll(output, train_y)
loss.backward()
iterator.set_postfix(loss=loss.item())
optimizer.step()
torch.cuda.empty_cache()

%time train()

CPU times: user 2.7 s, sys: 852 ms, total: 3.55 s
Wall time: 3.58 s


### Making Predictions¶

[7]:

model.eval()
likelihood.eval()

Test MAE: 0.07258129864931107

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