# Scalable Multitask GP Regression (w/ KISS-GP)¶

This notebook demonstrates how to perform a scalable multitask regression.

It does everything that the standard multitask GP example does, but using the SKI scalable GP approximation. This can be used on much larger datasets (up to 100,000+ data points).

:

import math
import torch
import gpytorch
from matplotlib import pyplot as plt

%matplotlib inline


## Set up training data¶

In the next cell, we set up the training data for this example. We’ll be using 1000 regularly spaced points on [0,1] which we evaluate the function on and add Gaussian noise to get the training labels.

We’ll have two functions - a sine function (y1) and a cosine function (y2).

For MTGPs, our train_targets will actually have two dimensions: with the second dimension corresponding to the different tasks.

:

train_x = torch.linspace(0, 1, 1000)

train_y = torch.stack([
torch.sin(train_x * (2 * math.pi)) + torch.randn(train_x.size()) * 0.2,
torch.cos(train_x * (2 * math.pi)) + torch.randn(train_x.size()) * 0.2,
], -1)


### Set up the model¶

The model should be somewhat similar to the ExactGP model in the simple regression example.

The differences:

1. We’re going to wrap ConstantMean with a MultitaskMean. This makes sure we have a mean function for each task.
2. Rather than just using a RBFKernel, we’re using that in conjunction with a MultitaskKernel. This gives us the covariance function described in the introduction.
3. We’re using a MultitaskMultivariateNormal and MultitaskGaussianLikelihood. This allows us to deal with the predictions/outputs in a nice way. For example, when we call MultitaskMultivariateNormal.mean, we get a n x num_tasks matrix back.

In addition, we’re going to wrap the RBFKernel in a GridInterpolationKernel. This approximates the kernel using SKI, which makes GP regression very scalable.

You may also notice that we don’t use a ScaleKernel, since the IndexKernel will do some scaling for us. (This way we’re not overparameterizing the kernel.)

:

class MultitaskGPModel(gpytorch.models.ExactGP):
def __init__(self, train_x, train_y, likelihood):

# SKI requires a grid size hyperparameter. This util can help with that
grid_size = gpytorch.utils.grid.choose_grid_size(train_x)

)
gpytorch.kernels.GridInterpolationKernel(
gpytorch.kernels.RBFKernel(), grid_size=grid_size, num_dims=1,
)

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



### Train the model hyperparameters¶

:

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

{'params': model.parameters()},  # Includes GaussianLikelihood parameters
], lr=0.1)

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

n_iter = 50
for i in range(n_iter):
output = model(train_x)
loss = -mll(output, train_y)
loss.backward()
print('Iter %d/%d - Loss: %.3f' % (i + 1, n_iter, loss.item()))
optimizer.step()

Iter 1/50 - Loss: 404.388
Iter 2/50 - Loss: 352.338
Iter 3/50 - Loss: 300.768
Iter 4/50 - Loss: 249.549
Iter 5/50 - Loss: 198.132
Iter 6/50 - Loss: 145.569
Iter 7/50 - Loss: 91.422
Iter 8/50 - Loss: 38.138
Iter 9/50 - Loss: -11.845
Iter 10/50 - Loss: -61.598
Iter 11/50 - Loss: -110.397
Iter 12/50 - Loss: -157.206
Iter 13/50 - Loss: -203.842
Iter 14/50 - Loss: -249.601
Iter 15/50 - Loss: -292.015
Iter 16/50 - Loss: -340.180
Iter 17/50 - Loss: -384.880
Iter 18/50 - Loss: -427.049
Iter 19/50 - Loss: -471.487
Iter 20/50 - Loss: -516.107
Iter 21/50 - Loss: -553.546
Iter 22/50 - Loss: -600.415
Iter 23/50 - Loss: -634.620
Iter 24/50 - Loss: -676.436
Iter 25/50 - Loss: -715.459
Iter 26/50 - Loss: -754.357
Iter 27/50 - Loss: -792.451
Iter 28/50 - Loss: -826.312
Iter 29/50 - Loss: -854.712
Iter 30/50 - Loss: -887.396
Iter 31/50 - Loss: -918.314
Iter 32/50 - Loss: -945.657
Iter 33/50 - Loss: -971.053
Iter 34/50 - Loss: -992.324
Iter 35/50 - Loss: -1014.124
Iter 36/50 - Loss: -1035.698
Iter 37/50 - Loss: -1052.513
Iter 38/50 - Loss: -1065.371
Iter 39/50 - Loss: -1074.025
Iter 40/50 - Loss: -1083.747
Iter 41/50 - Loss: -1090.505
Iter 42/50 - Loss: -1092.604
Iter 43/50 - Loss: -1097.734
Iter 44/50 - Loss: -1096.438
Iter 45/50 - Loss: -1098.831
Iter 46/50 - Loss: -1098.317
Iter 47/50 - Loss: -1094.847
Iter 48/50 - Loss: -1092.078
Iter 49/50 - Loss: -1088.565
Iter 50/50 - Loss: -1082.591


### Make predictions with the model¶

:

# Set into eval mode
model.eval()
likelihood.eval()

# Initialize plots
f, (y1_ax, y2_ax) = plt.subplots(1, 2, figsize=(8, 3))
# Test points every 0.02 in [0,1]

# Make predictions
test_x = torch.linspace(0, 1, 51)
observed_pred = likelihood(model(test_x))
# Get mean
mean = observed_pred.mean
# Get lower and upper confidence bounds
lower, upper = observed_pred.confidence_region()

# This contains predictions for both tasks, flattened out
# The first half of the predictions is for the first task
# The second half is for the second task

# Plot training data as black stars
y1_ax.plot(train_x.detach().numpy(), train_y[:, 0].detach().numpy(), 'k*')
# Predictive mean as blue line
y1_ax.plot(test_x.numpy(), mean[:, 0].numpy(), 'b')
y1_ax.fill_between(test_x.numpy(), lower[:, 0].numpy(), upper[:, 0].numpy(), alpha=0.5)
y1_ax.set_ylim([-3, 3])
y1_ax.legend(['Observed Data', 'Mean', 'Confidence'])
y1_ax.set_title('Observed Values (Likelihood)')

# Plot training data as black stars
y2_ax.plot(train_x.detach().numpy(), train_y[:, 1].detach().numpy(), 'k*')
# Predictive mean as blue line
y2_ax.plot(test_x.numpy(), mean[:, 1].numpy(), 'b') [ ]: