Hi everyone,
When using gp.models.SparseGPRegression, I found that the covariance matrices during predictions are always the same for all target variables, even when the targets have different value ranges. This was noticeable when plotting mean predictions and standard deviations for each target, where the standard deviations remained identical across targets, which seemed suspicious (also given the distinct target ranges).
However, when switching to gp.models.VariationalSparseGP, the covariance matrices differed for each target variable, which appears correct and aligns with the expected behavior.
I initially encountered this issue while working with a real-life dataset, and after testing various datasets with different numbers of target variables and features, I noticed that my issue remains. Below, I’m sharing a simple toy example that illustrates this.
Why does gp.models.SparseGPRegression produce the same covariance matrices for different targets, while gp.models.VariationalSparseGP generates varying ones? Is there something specific in the model setup or approximation method that explains this behaviour? Any insights or explanations would be greatly appreciated!
Thanks in advance for your help!
import numpy as np
import pyro
import pyro.contrib.gp as gp
import torch
from sklearn.preprocessing import StandardScaler
from torch.optim import Adam
class SparseGPModel:
def __init__(
self,
kernel_variance=1.0,
kernel_lengthscale=1.0,
inducing_points_divisor=5,
model_variant="VSGP",
):
"""
Initialize the SparseGPModel class.
"""
self.kernel_variance = torch.tensor(kernel_variance, dtype=torch.float64)
self.kernel_lengthscale = torch.tensor(kernel_lengthscale, dtype=torch.float64)
self.inducing_points_divisor = inducing_points_divisor
self.model_variant = model_variant
self.scaler = StandardScaler()
self.model = None
self.inducing_points = None
self.Xu = None
def construct_model(self, X: torch.Tensor, y: torch.Tensor):
"""
Construct the Gaussian Process Regression model.
Args:
X (torch.Tensor): Input features.
y (torch.Tensor): Target variables.
"""
# Define the RBF kernel
kernel = gp.kernels.RBF(
input_dim=X.shape[1], variance=self.kernel_variance, lengthscale=self.kernel_lengthscale
)
# Select inducing points
n_inducing_points = int(len(X) / self.inducing_points_divisor)
random_indices = torch.randint(0, len(self.inducing_points), (n_inducing_points,))
self.Xu = torch.nn.Parameter(self.inducing_points[random_indices])
# Construct the model based on the selected variant
if self.model_variant == "VFE":
# Sparse Gaussian Process Regression using VFE approximation
model = gp.models.SparseGPRegression(X, y, kernel, self.Xu, approx="VFE")
else:
# Variational Sparse Gaussian Process Regression
model = gp.models.VariationalSparseGP(X, y, kernel, self.Xu, gp.likelihoods.Gaussian())
return model
def predict(self, X: np.ndarray) -> np.ndarray:
"""
Predict the mean and standard deviation for the given input features.
Returns concatenated mean and standard deviation of predictions.
"""
# Scale the input features
X = self.scaler.transform(X)
# Get predictions
if self.model_variant == "VFE":
mean, cov = self.model(
torch.tensor(X, dtype=torch.float64), full_cov=True, noiseless=False
)
else:
mean, cov = self.model(torch.tensor(X, dtype=torch.float64), full_cov=True)
# Initialize an empty tensor to hold the standard deviations
std = torch.empty(cov.shape[0], cov.shape[1])
# Iterate over each set of covariance matrices
for i in range(cov.shape[0]):
# Compute the standard deviation from the diagonal elements
std[i] = torch.sqrt(torch.diagonal(cov[i]))
return np.concatenate([mean.detach().numpy(), std.detach().numpy()], axis=0).T
def train(self, X_train, y_train, epochs=100):
"""
Train the GP model.
Args:
X_train (np.ndarray): Training features.
y_train (np.ndarray): Training target variables.
epochs (int): Number of training epochs.
"""
# Standardize the input features
self.scaler.fit(X_train)
X_train = self.scaler.transform(X_train)
# Convert the dataset to torch tensors
X_train = torch.tensor(X_train, dtype=torch.float64)
y_train = torch.tensor(y_train.T, dtype=torch.float64)
# Set inducing points as part of the model
self.inducing_points = X_train.clone()
# Construct the GP model
self.model = self.construct_model(X_train, y_train)
# Define the optimizer
optimizer = Adam(self.model.parameters(), lr=0.01)
loss_fn = pyro.infer.Trace_ELBO().differentiable_loss
# Training loop
for epoch in range(epochs):
optimizer.zero_grad()
loss = loss_fn(self.model.model, self.model.guide) # Calculate the loss
loss.backward() # Backpropagate the loss
optimizer.step() # Update the model parameters
# Generate toy dataset with one feature and two target variables
np.random.seed(42)
X = np.random.rand(280, 1) # One feature
y1 = X.reshape(-1) * 2 + 0.2
y2 = 3 * X.reshape(-1) - 0.01
y = np.stack([y1, y2], axis=1)
# Train and predict using the VariationalSparseGP variant
gp_model_vsgp = SparseGPModel(model_variant="SVGP")
gp_model_vsgp.train(X, y, epochs=100)
predictions_vsgp = gp_model_vsgp.predict(X)
print("Predictions using VariationalSparseGP (Mean and Standard Deviations):")
print(predictions_vsgp)
# Train and predict using the SparseGPRegression variant
gp_model_sgp = SparseGPModel(model_variant="VFE")
gp_model_sgp.train(X, y, epochs=100)
predictions_sgp = gp_model_sgp.predict(X)
print("Predictions using SparseGPRegression (Mean and Standard Deviations):")
print(predictions_sgp)