QEP Regression with Uncertain Inputs¶
Introduction¶
In this notebook, we’re going to demonstrate one way of dealing with uncertainty in our training data. Let’s say that we’re collecting training data that models the following function.
\[\begin{split}\begin{align}
y &= \sin(2\pi x) + \epsilon \\
\epsilon &\sim \mathcal{Q}(0, 0.2)
\end{align}\end{split}\]
However, now assume that we’re a bit uncertain about our features. In particular, we’re going to assume that every x_i value is not a point but a distribution instead. E.g.
\[x_i \sim \mathcal{Q}(\mu_i, \sigma_i).\]
Using a distributional kernel to deal with uncertain inputs¶
Rather than using a variational method (see the QEP Regression with Uncertian Inputs tutorial in the variational examples), if we explicitly know the type of uncertainty in our inputs we can pass that into our kernel.
More specifically, assuming Gaussian inputs, we will compute the symmetrized KL divergence between the Q-exponential inputs.
[1]:
import math
import torch
import tqdm
import qpytorch
from matplotlib import pyplot as plt
%matplotlib inline
%load_ext autoreload
%autoreload 2
[2]:
# Training data is 100 points in [0,1] inclusive regularly spaced
train_x_mean = torch.linspace(0, 1, 20)
# We'll assume the variance shrinks the closer we get to 1
train_x_stdv = torch.linspace(0.03, 0.01, 20)
# True function is sin(2*pi*x) with Gaussian noise
train_y = torch.sin(train_x_mean * (2 * math.pi)) + torch.randn(train_x_mean.size()) * 0.2
To effectively pass in the training distributional data, we will need to stack the mean and log variances.
[3]:
train_x_distributional = torch.stack((train_x_mean, (train_x_stdv**2).log()), dim=1)
[4]:
f, ax = plt.subplots(1, 1, figsize=(8, 3))
ax.errorbar(train_x_mean, train_y, xerr=(train_x_stdv * 2), fmt="k*", label="Train Data")
ax.legend()
[4]:
<matplotlib.legend.Legend at 0x124cb7b20>
We train the hyperparameters of the resulting distributional GP via type-II gradient descent, as is standard in many settings. We could also do fully Bayesian inference.
[5]:
from qpytorch.models import ExactQEP
from qpytorch.kernels import QExponentialSymmetrizedKLKernel, ScaleKernel
from qpytorch.means import ConstantMean
POWER =1.0
class ExactQEPModel(ExactQEP):
def __init__(self, train_x, train_y, likelihood):
super(ExactQEPModel, self).__init__(train_x, train_y, likelihood)
self.power = torch.tensor(POWER)
self.mean_module = ConstantMean()
self.covar_module = ScaleKernel(QExponentialSymmetrizedKLKernel())
def forward(self, x):
mean_x = self.mean_module(x)
covar_x = self.covar_module(x)
return qpytorch.distributions.MultivariateQExponential(mean_x, covar_x, power=self.power)
# initialize likelihood and model
likelihood = qpytorch.likelihoods.QExponentialLikelihood(power=torch.tensor(POWER))
model = ExactQEPModel(train_x_distributional, train_y, likelihood)
[6]:
# this is for running the notebook in our testing framework
import os
smoke_test = ('CI' in os.environ)
training_iter = 2 if smoke_test else 500
# Find optimal model hyperparameters
model.train()
likelihood.train()
# Use the adam optimizer
optimizer = torch.optim.Adam(model.parameters(), lr=0.25) # Includes QExponentialLikelihood parameters
# "Loss" for QEPs - the marginal log likelihood
mll = qpytorch.mlls.ExactMarginalLogLikelihood(likelihood, model)
for i in range(training_iter):
# Zero gradients from previous iteration
optimizer.zero_grad()
# Output from model
output = model(train_x_distributional)
# 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()
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Now, we test predictions. For simplicity, we will assume a fixed variance of \(0.01.\)
[7]:
# 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(), qpytorch.settings.fast_pred_var():
test_x = torch.linspace(0, 1, 51)
test_x_distributional = torch.stack((test_x, (1e-3 * torch.ones_like(test_x)).log()), dim=1)
observed_pred = likelihood(model(test_x_distributional))
with torch.no_grad():
# Initialize plot
f, ax = plt.subplots(1, 1, figsize=(8, 3))
# Get upper and lower confidence bounds
lower, upper = observed_pred.confidence_region(rescale=True)
# Plot training data as black stars
ax.errorbar(train_x_mean.numpy(), train_y.numpy(), xerr=train_x_stdv, fmt='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'])
As a final note, we’ve made it very easy to extend the distributional kernel class by exposing a generic DistributionalInputKernel class that takes as input any distance function over probability distributions.
[ ]: