Sparse Gaussian Process Regression (SGPR)

Overview

In this notebook, we’ll overview how to use SGPR, the method of http://proceedings.mlr.press/v5/titsias09a/titsias09a.pdf in which the inducing point locations are learned.

[1]:
import math
import torch
import gpytorch
from matplotlib import pyplot as plt

# Make plots inline
%matplotlib inline

Loading Data

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.path
from scipy.io import loadmat
from math import floor

if not os.path.isfile('elevators.mat'):
    print('Downloading \'elevators\' UCI dataset...')
    urllib.request.urlretrieve('https://drive.google.com/uc?export=download&id=1jhWL3YUHvXIaftia4qeAyDwVxo6j1alk', 'elevators.mat')

data = torch.Tensor(loadmat('elevators.mat')['data'])
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().cuda()
train_y = y[:train_n].contiguous().cuda()

test_x = X[train_n:, :].contiguous().cuda()
test_y = y[train_n:].contiguous().cuda()
[3]:
X.size()
[3]:
torch.Size([16599, 18])

Defining the GP 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.

[14]:
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, :], likelihood=likelihood)

    def forward(self, x):
        mean_x = self.mean_module(x)
        covar_x = self.covar_module(x)
        return MultivariateNormal(mean_x, covar_x)
[15]:
likelihood = gpytorch.likelihoods.GaussianLikelihood().cuda()
model = GPRegressionModel(train_x, train_y, likelihood).cuda()

Training the model

[16]:
# Find optimal model hyperparameters
model.train()
likelihood.train()

# Use the adam optimizer
optimizer = torch.optim.SGD(model.parameters(), lr=0.1)

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

training_iterations = 25
def train():
    for i in range(training_iterations):
        # Zero backprop gradients
        optimizer.zero_grad()
        # Get output from model
        output = model(train_x)
        # Calc loss and backprop derivatives
        loss = -mll(output, train_y)
        loss.backward()
        print('Iter %d/%d - Loss: %.3f' % (i + 1, training_iterations, loss.item()))
        optimizer.step()
        torch.cuda.empty_cache()

# See dkl_mnist.ipynb for explanation of this flag
with gpytorch.settings.use_toeplitz(True):
    %time train()
Iter 1/25 - Loss: 0.796
Iter 2/25 - Loss: 0.786
Iter 3/25 - Loss: 0.773
Iter 4/25 - Loss: 0.762
Iter 5/25 - Loss: 0.748
Iter 6/25 - Loss: 0.735
Iter 7/25 - Loss: 0.724
Iter 8/25 - Loss: 0.711
Iter 9/25 - Loss: 0.698
Iter 10/25 - Loss: 0.685
Iter 11/25 - Loss: 0.670
Iter 12/25 - Loss: 0.657
Iter 13/25 - Loss: 0.645
Iter 14/25 - Loss: 0.631
Iter 15/25 - Loss: 0.617
Iter 16/25 - Loss: 0.602
Iter 17/25 - Loss: 0.588
Iter 18/25 - Loss: 0.574
Iter 19/25 - Loss: 0.561
Iter 20/25 - Loss: 0.545
Iter 21/25 - Loss: 0.529
Iter 22/25 - Loss: 0.515
Iter 23/25 - Loss: 0.500
Iter 24/25 - Loss: 0.484
Iter 25/25 - Loss: 0.470
CPU times: user 10.2 s, sys: 13.2 s, total: 23.4 s
Wall time: 3.29 s

Making Predictions

The next cell makes predictions with SKIP. We use the same max_root_decomposition size, and we also demonstrate increasing the max preconditioner size. Increasing the preconditioner size on this dataset is not necessary, but can make a big difference in final test performance, and is often preferable to increasing the number of CG iterations if you can afford the space.

[17]:
model.eval()
likelihood.eval()
with gpytorch.settings.max_preconditioner_size(10), torch.no_grad():
    with gpytorch.settings.use_toeplitz(False), gpytorch.settings.max_root_decomposition_size(30), gpytorch.settings.fast_pred_var():
        preds = model(test_x)
[18]:
print('Test MAE: {}'.format(torch.mean(torch.abs(preds.mean - test_y))))
Test MAE: 0.0909833088517189
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