FixedNoiseGaussianLikelihood results in negative variance

[mgarort]

Hi,

I am trying to train a most likely heteroscedastic GP (from "Most likely heteroscedastic GP regression", Kersting et al. 2007). To this end I am using the likelihood FixedNoiseGaussianLikelihood. I am setting the noise to positive values r.

(Pdb) print(r)
tensor([0.0086, 0.0071, 0.0071, 0.0069, 0.0067, 0.0067, 0.0065, 0.0065, 0.0065,
        0.0065, 0.0065, 0.0065, 0.0065, 0.0066, 0.0066, 0.0066, 0.0070, 0.0076,
        0.0076, 0.0090, 0.0107, 0.0110, 0.0117, 0.0122, 0.0130, 0.0135, 0.0140,
        0.0154, 0.0202, 0.0208, 0.0218, 0.0226, 0.0229, 0.0265, 0.0270, 0.0280,
        0.0282, 0.0285, 0.0287, 0.0289, 0.0290, 0.0290, 0.0289, 0.0288, 0.0287,
        0.0276, 0.0269, 0.0241, 0.0219, 0.0218, 0.0191, 0.0177, 0.0175, 0.0166,
        0.0164, 0.0160, 0.0146, 0.0145, 0.0131, 0.0120, 0.0112, 0.0105, 0.0100,
        0.0094, 0.0082, 0.0081, 0.0079, 0.0072, 0.0072, 0.0070, 0.0070, 0.0059,
        0.0058, 0.0057, 0.0057, 0.0056, 0.0052, 0.0051, 0.0049, 0.0049, 0.0048,
        0.0048, 0.0045, 0.0045, 0.0044, 0.0043, 0.0043, 0.0042, 0.0042, 0.0042,
        0.0042, 0.0042, 0.0042, 0.0042, 0.0042, 0.0042, 0.0045, 0.0046, 0.0046,
        0.0047, 0.0050, 0.0051, 0.0059, 0.0067, 0.0086, 0.0104, 0.0115, 0.0161,
        0.0213, 0.0226, 0.0278, 0.0399, 0.0413, 0.0418, 0.0463, 0.0567, 0.0592,
        0.2299, 0.2421, 0.2920, 0.3486, 0.5690, 0.7409, 0.8167, 1.3840, 1.4557,
        1.3335, 1.2206, 1.1017, 0.8272])

lik_3 = FixedNoiseGaussianLikelihood(noise=r, learn_additional_noise=False) 
GP3 = ExactGPModel(self.train_x,self.train_y,lik_3)
GP3, lik_3 = train_a_GP(GP3,self.train_x,self.train_y,lik_3,self.training_iter)

where train_a_GP is simply the following training function (copied from a GPytorch regression tutorial):

def train_a_GP(model, train_x, train_y, likelihood, training_iter):
    # train GP_model for training_iter iterations
    model.train()
    likelihood.train()

    # Use the adam optimizer
    optimizer = torch.optim.Adam([
        {'params': model.parameters()},  # Includes GaussianLikelihood parameters
    ], lr=0.1)

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

    for i in range(training_iter):
        # Zero gradients from previous iteration
        optimizer.zero_grad()
        # Output from model
        output = model(train_x)
        # Calc loss and backprop gradients
        loss = -mll(output, train_y)
        loss.backward()
        print('Iter %d/%d - Loss: %.3f   lengthscale: %.3f' % (
            i + 1, training_iter, loss.item(),
            model.covar_module.base_kernel.lengthscale.item(),
        ))
        optimizer.step()

        model.eval()
        likelihood.eval()
    return model, likelihood

However when I try to obtain predictions, the variance of the MultivariateNormal returned seems to be negative.

GP3.eval()
lik_3.eval()
with torch.no_grad(), gpytorch.settings.fast_pred_var():
    train_pred = lik_3(GP3(self.train_x),noise=r)

(Pdb) train_pred.variance
tensor([-0.1621, -0.1650, -0.1650, -0.1652, -0.1653, -0.1653, -0.1651, -0.1651,
        -0.1650, -0.1650, -0.1648, -0.1648, -0.1647, -0.1644, -0.1642, -0.1642,
        -0.1631, -0.1619, -0.1618, -0.1586, -0.1548, -0.1542, -0.1528, -0.1515,
        -0.1497, -0.1487, -0.1475, -0.1444, -0.1334, -0.1320, -0.1295, -0.1279,
        -0.1271, -0.1192, -0.1182, -0.1162, -0.1159, -0.1155, -0.1152, -0.1150,
        -0.1152, -0.1159, -0.1160, -0.1167, -0.1168, -0.1206, -0.1226, -0.1303,
        -0.1358, -0.1360, -0.1424, -0.1454, -0.1460, -0.1479, -0.1483, -0.1493,
        -0.1522, -0.1524, -0.1552, -0.1574, -0.1590, -0.1602, -0.1611, -0.1622,
        -0.1642, -0.1645, -0.1648, -0.1660, -0.1660, -0.1662, -0.1663, -0.1680,
        -0.1682, -0.1683, -0.1684, -0.1684, -0.1691, -0.1693, -0.1696, -0.1696,
        -0.1697, -0.1698, -0.1701, -0.1702, -0.1703, -0.1704, -0.1705, -0.1706,
        -0.1707, -0.1707, -0.1707, -0.1707, -0.1707, -0.1708, -0.1707, -0.1707,
        -0.1705, -0.1704, -0.1703, -0.1702, -0.1700, -0.1699, -0.1690, -0.1681,
        -0.1660, -0.1638, -0.1626, -0.1569, -0.1504, -0.1486, -0.1416, -0.1242,
        -0.1221, -0.1214, -0.1145, -0.0983, -0.0943,  0.2122,  0.2350,  0.3277,
         0.4312,  0.8048,  1.0528,  1.1486,  1.5613,  1.5205,  1.2962,  1.1957,
         1.1100,  0.8646])

What am I doing wrong? Any help would be greatly appreciated.

Thanks a lot!

Miguel

[jacobrgardner]

Hi @mgarort --

Nothing immediately jumps out at me as wrong. Would you possibly be able to provide example data for which this happens, as well as the definition ExactGPModel if it differs from our basic examples?

[mgarort]

Hi @jacobrgardner

Thanks for the quick reply. Here's a simple script that reproduces the error, together with the 3 files needed (x, y and noise r).

reproduce_negative_variance.zip

[Balandat]

If it helps, you can also take a look at our convenience wrapper model for observed variances in BoTorch: https://github.com/pytorch/botorch/blob/master/botorch/models/gp_regression.py#L118-L127

There is also a PR open for the most likely heteroskedastic GP fitting (which I'll probably get to in the near future): pytorch/botorch#250. Note that this uses a full heteroskedastic GP, where the noise model is itself a GP (rather than fixed variances).

[mgarort]

Hi @Balandat

Thanks! I'll take a look at the wrapper model in BoTorch

I think I have modelled the noise correctly according to the most likely heteroscedastic GP (although of course there's always room for error!). The vector r with variances that is passed to GP3 has been produced by a previous GP that models the noise called GP2 (following the notation convention on section 4 of the paper, "Optimization"). If anything, I think I am most likely to have made an incorrect assumption about the way GPytorch incorporates the heteroscedastic noise r into the model...

Best,

Miguel

[mgarort]

@Balandat @jacobrgardner

Here's my full attempt at an implementation of the most likely heteroscedastic GP.

import torch
import gpytorch
from gpytorch.means import ConstantMean
from gpytorch.kernels import ScaleKernel, RBFKernel
from gpytorch.distributions import MultivariateNormal
from gpytorch.likelihoods import GaussianLikelihood, FixedNoiseGaussianLikelihood
from gpytorch.mlls import ExactMarginalLogLikelihood
from gpytorch.constraints import Positive

def train_a_GP(model, train_x, train_y, likelihood, training_iter):
    """
    Simple utility function to train a Gaussian process (GP) model with Adam (following the examples on the docs).

    :param model: GP model
    :param train_x: tensor with training features X
    :param train_y: tensor with training targets Y
    :param likelihood: likelihood function
    :param training_iter: number of iterations to train
    :return: trained GP model, trained likelihood
    """
    # train GP_model for training_iter iterations
    model.train()
    likelihood.train()

    # Use the adam optimizer
    optimizer = torch.optim.Adam([
        {'params': model.parameters()},  # Includes GaussianLikelihood parameters
    ], lr=0.1)

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

    for i in range(training_iter):
        # Zero gradients from previous iteration
        optimizer.zero_grad()
        # Output from model
        output = model(train_x)
        # Calc loss and backprop gradients
        loss = -mll(output, train_y)
        loss.backward()
        print('Iter %d/%d - Loss: %.3f   lengthscale: %.3f' % (
            i + 1, training_iter, loss.item(),
            model.covar_module.base_kernel.lengthscale.item(),
        ))
        optimizer.step()

        model.eval()
        likelihood.eval()
    return model, likelihood


class ExactGPModel(gpytorch.models.ExactGP):
    """
    Exact Gaussian process model (following the examples in the docs).
    """
    def __init__(self, train_x, train_y, likelihood):
        """
        Initializer function. Specifies the mean and the covariance functions.

        :param train_x: tensor with training features X
        :param train_y: tensor with training targets Y
        :param likelihood: likelihood function
        :return: None
        """
        super(ExactGPModel, self).__init__(train_x, train_y, likelihood)
        self.mean_module = ConstantMean()
        self.covar_module = ScaleKernel(RBFKernel())

    def forward(self, x):
        """
        Forward method to evaluate GP.

        :param x: tensor with features X on which to evaluate the GP.
        :return: MultivariateNormal
        """
        mean_x = self.mean_module(x)
        covar_x = self.covar_module(x)
        return MultivariateNormal(mean_x, covar_x)


class hetGPModel():
    """
    Most likely heteroscedastic GP model.
    """
    def __init__(self,train_x,train_y,training_iter=100,het_fitting_iter=10,var_estimator_n=50):
        """
        Initializer function.

        :param train_x: tensor with training features X
        :param train_y: tensor with training targets Y
        :param training_iter: number of iterations to train GP1, GP2 and GP3
        :param het_fitting_iter: number of iterations to run the pseudo expectation maximization (EM) algorithm while refining GP3
        :param var_estimator_n: number of samples to estimate the variance at each training point
        :return: None
        """
        self.train_x = train_x
        self.train_y = train_y
        self.training_iter = training_iter
        self.var_estimator_n = var_estimator_n
        self.het_fitting_iter = het_fitting_iter
        self.final_GP = None
        self.final_lik = None
        self.final_r = None

    def predict(self,x):
        """
        Predict method to evaluate GP.

        :param x: tensor with features X on which to evaluate the GP.
        :return: MultivariateNormal
        """
        if self.final_GP is None:
            raise RuntimeError('hetGPModel needs to be trained before using it')
        return self.final_GP(x)

    def train_model(self):
        """
        Train most likely heteroscedastic GP, in which one GP predicts the mean and another GP predicts the variance. This function
        corresponds to section '4. Optimization' in the original most likely heteroscedastic GP paper (Kersting et al. 2007).

        :return: None
        """
        # train self.GP1 if self.is_GP1_trained == False, and then set it to True. Otherwise ignore
        lik_1 = GaussianLikelihood()
        GP1 = ExactGPModel(self.train_x,self.train_y,lik_1)
        GP1, lik_1 = train_a_GP(GP1,self.train_x,self.train_y,lik_1,self.training_iter)
        for i in range(self.het_fitting_iter):
            # estimate the noise levels z
            z = torch.log(self.get_r_hat(GP1,lik_1))
            # fit the noise z at train_x
            lik_2 = GaussianLikelihood()
            GP2 = ExactGPModel(self.train_x,z,lik_2)
            GP2, lik_2 = train_a_GP(GP2,self.train_x,z,lik_2,self.training_iter)
            # create a heteroscedastic GP
            with torch.no_grad(), gpytorch.settings.fast_pred_var():
                r_pred = lik_2(GP2(self.train_x))
            r = torch.exp(r_pred.mean)
            lik_3 = FixedNoiseGaussianLikelihood(noise=r, learn_additional_noise=False) 
            GP3 = ExactGPModel(self.train_x,self.train_y,lik_3)
            GP3, lik_3 = train_a_GP(GP3,self.train_x,self.train_y,lik_3,self.training_iter)
            GP1 = GP3
            lik_1 = lik_3
            
        self.final_GP = GP3
        self.final_lik = lik_3
        self.final_r = r


    def get_r_hat(self,GP,likelihood):
        """
        Estimate variance at each training point.

        :param GP: GP model that predicts the mean in the heteroscedastic GP model.
        :return: tensor r_hat with the estimated variances
        """
        with torch.no_grad(), gpytorch.settings.fast_pred_var():
            train_pred = likelihood(GP(self.train_x))
        r_hat = torch.sum(0.5*(self.train_y.reshape(1,-1) - train_pred.sample_n(self.var_estimator_n))**2,dim=0)/self.var_estimator_n
        return r_hat

[mgarort]

Hi @Balandat and @jacobrgardner

I have tried to rewrite my heteroscedastic GP using the wrapper FixedNoiseGP from Botorch instead of FixedNoiseGaussianLikelihood and I obtain the same result (negative variance).

My code for the heteroscedastic GP with FixedNoiseGP is the following.

Thanks a lot in advance.

import torch
import gpytorch
from gpytorch.means import ConstantMean
from gpytorch.kernels import ScaleKernel, RBFKernel
from gpytorch.distributions import MultivariateNormal
from gpytorch.likelihoods import GaussianLikelihood, FixedNoiseGaussianLikelihood
from gpytorch.mlls import ExactMarginalLogLikelihood
from gpytorch.constraints import Positive

from botorch.models.gp_regression import FixedNoiseGP

def train_a_GP(model, train_x, train_y, likelihood, training_iter):
    """
    Simple utility function to train a Gaussian process (GP) model with Adam (following the examples on the docs).

    :param model: GP model
    :param train_x: tensor with training features X
    :param train_y: tensor with training targets Y
    :param likelihood: likelihood function
    :param training_iter: number of iterations to train
    :return: trained GP model, trained likelihood
    """
    # train GP_model for training_iter iterations
    model.train()
    likelihood.train()

    # Use the adam optimizer
    optimizer = torch.optim.Adam([
        {'params': model.parameters()},  # Includes GaussianLikelihood parameters
    ], lr=0.1)

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

    for i in range(training_iter):
        # Zero gradients from previous iteration
        optimizer.zero_grad()
        # Output from model
        output = model(train_x)
        # Calc loss and backprop gradients
        loss = -mll(output, train_y)
        loss.backward()
        print('Iter %d/%d - Loss: %.3f   lengthscale: %.3f' % (
            i + 1, training_iter, loss.item(),
            model.covar_module.base_kernel.lengthscale.item(),
        ))
        optimizer.step()

        model.eval()
        likelihood.eval()
    return model, likelihood


class ExactGPModel(gpytorch.models.ExactGP):
    """
    Exact Gaussian process model (following the examples in the docs).
    """
    def __init__(self, train_x, train_y, likelihood):
        """
        Initializer function. Specifies the mean and the covariance functions.

        :param train_x: tensor with training features X
        :param train_y: tensor with training targets Y
        :param likelihood: likelihood function
        :return: None
        """
        super(ExactGPModel, self).__init__(train_x, train_y, likelihood)
        self.mean_module = ConstantMean()
        self.covar_module = ScaleKernel(RBFKernel())

    def forward(self, x):
        """
        Forward method to evaluate GP.

        :param x: tensor with features X on which to evaluate the GP.
        :return: MultivariateNormal
        """
        mean_x = self.mean_module(x)
        covar_x = self.covar_module(x)
        return MultivariateNormal(mean_x, covar_x)


class hetGPModel():
    """
    Most likely heteroscedastic GP model.
    """
    def __init__(self,train_x,train_y,training_iter=100,het_fitting_iter=10,var_estimator_n=50):
        """
        Initializer function.

        :param train_x: tensor with training features X
        :param train_y: tensor with training targets Y
        :param training_iter: number of iterations to train GP1, GP2 and GP3
        :param het_fitting_iter: number of iterations to run the pseudo expectation maximization (EM) algorithm while refining GP3
        :param var_estimator_n: number of samples to estimate the variance at each training point
        :return: None
        """
        self.train_x = train_x
        self.train_y = train_y
        self.training_iter = training_iter
        self.var_estimator_n = var_estimator_n
        self.het_fitting_iter = het_fitting_iter
        self.final_GP = None
        self.final_lik = None
        self.final_r_func = None

    def predict(self,x):
        """
        Predict method to evaluate GP.

        :param x: tensor with features X on which to evaluate the GP.
        :return: MultivariateNormal
        """
        if self.final_GP is None:
            raise RuntimeError('hetGPModel needs to be trained before using it')
        return self.final_GP(x)

    def train_model(self):
        """
        Train most likely heteroscedastic GP, in which one GP predicts the mean and another GP predicts the variance. This function
        corresponds to section '4. Optimization' in the original most likely heteroscedastic GP paper (Kersting et al. 2007).

        :return: None
        """
        # train self.GP1 if self.is_GP1_trained == False, and then set it to True. Otherwise ignore
        lik_1 = GaussianLikelihood()
        GP1 = ExactGPModel(self.train_x,self.train_y,lik_1)
        GP1, lik_1 = train_a_GP(GP1,self.train_x,self.train_y,lik_1,self.training_iter)
        for i in range(self.het_fitting_iter):
            # estimate the noise levels z
            z = torch.log(self.get_r_hat(GP1,lik_1))
            # fit the noise z at train_x
            lik_2 = GaussianLikelihood()
            GP2 = ExactGPModel(self.train_x,z,lik_2)
            GP2, lik_2 = train_a_GP(GP2,self.train_x,z,lik_2,self.training_iter)
            # create a heteroscedastic GP
            with torch.no_grad(), gpytorch.settings.fast_pred_var():
                r_func_pred = lik_2(GP2(self.train_x))
            r_func = torch.exp(r_func_pred.mean)
            lik_3 = GaussianLikelihood()
            #import pdb; pdb.set_trace()
            GP3 = FixedNoiseGP(self.train_x.reshape(-1,1),self.train_y.reshape(-1,1),r_func.reshape(-1,1))
            GP3, lik_3 = train_a_GP(GP3,self.train_x,self.train_y,lik_3,self.training_iter)
            GP1 = GP3
            lik_1 = lik_3
            
        self.final_GP = GP3
        self.final_lik = lik_3
        self.final_r_func = r_func


    def get_r_hat(self,GP,likelihood):
        """
        Estimate variance at each training point.

        :param GP: GP model that predicts the mean in the heteroscedastic GP model.
        :return: tensor r_hat with the estimated variances
        """
        with torch.no_grad(), gpytorch.settings.fast_pred_var():
            train_pred = likelihood(GP(self.train_x))
        r_hat = torch.sum(0.5*(self.train_y.reshape(1,-1) - train_pred.sample_n(self.var_estimator_n))**2,dim=0)/self.var_estimator_n
        return r_hat

[stanbiryukov]

Hey there - I had this problem a few weeks ago and solved it by adding a constraint to the the likelihood:
likelihood=gpytorch.likelihoods.GaussianLikelihood(noise_constraint=gpytorch.constraints.GreaterThan(1e-3)).to(device)

or for fixednoise:
gpytorch.likelihoods.FixedNoiseGaussianLikelihood(noise=noise, noise_constraint=gpytorch.constraints.GreaterThan(1e-3))

[mgarort]

Hi @stanbiryukov

Thanks for the tip. Unfortunately I've tried FixedNoiseGaussianLikelihood(noise=noise, noise_constraint=gpytorch.constraints.GreaterThan(1e-3)) and variances are still negative :/

[jacobrgardner]

@mgarort I need sample data that produces this issue. When I run your code with toy data:

train_x = torch.linspace(0, 6, 100)
train_y = torch.sin(train_x) + 0.01 * torch.randn(100)

I get positive variances.

[mgarort]

Hi @jacobrgardner

I attached sample data and a script that results in negative variance in a zip a few posts back, but I guess it has ended up hidden under the subsequent posts!

Here's the zip again. You just need to run the script in the same folder as x.txt, y.txt and r.txt
reproduce_negative_variance.zip

Thanks,

Miguel

[jacobrgardner]

@mgarort Okay, I'm pretty sure I know what's going on here. It's actually pretty technical.

Basically, for fast predictive variances we decompose (K+\sigma^{2} I)^{-1} in a way that is fine because the added noise doesn't change the eigenvalue clustering, it only shifts the whole spectrum. In the heteroscedastic noise setting, this is violated in the sense that adding an arbitrary diagonal component does change the eigenvalue clustering. Turning off fast predictive variances gives positive variances.

To work around this, we could instead decompose K, and then use a QR decomposition to effectively get a root for K^{-1}. This will take a bit to implement. For now, is turning off fast_pred_var a feasible work around, or do you anticipate having too much data?

[jacobrgardner]

cc/ @gpleiss on this one actually, since we've discussed decomposing K instead of K+\sigma^{2} I before for LOVE, but this is the first time we've had any motivation for it at all.