In this blog, we shall discuss on Gaussian Process Regression, the basic concepts, how it can be implemented with python from scratch and also using the GPy library. Then we shall demonstrate an application of GPR in Bayesian optimiation. The problems appeared in this coursera course on Bayesian methods for Machine Learning by UCSanDiego HSE and also in this Machine learning course provided at UBC.
Gaussian Process
A GP is a Gaussian distribution over functions, that takes two parameters, namely the mean (m) and the kernel function K (to ensure smoothness). In this article, we shall implement non-linear regression with GP.
Given training data points (X,y) we want to learn a non-linear function f:R^d -> R (here X is d-dimensional), s.t., y = f(x).
Then use the function f to predict the value of y for unseen data points Xtest, along with the confidence of prediction.
The following figure describes the basic concepts of a GP and how it can be used for regression. We need to use the conditional expectation and variance formula (given the data) to compute the posterior distribution for the GP.
Noiseless Gaussian Process Regression
Let's follow the steps below to get some intuition on noiseless GP:
Generate 10 data points (these points will serve as training datapoints) with negligible noise (corresponds to noiseless GP regression). Use the following python function with default noise variance.
import numpy as np
def generate_noisy_points(n=10, noise_variance=1e-6):
np.random.seed(777)
X = np.random.uniform(-3., 3., (n, 1))
y = np.sin(X) + np.random.randn(n, 1) * noise_variance**0.5
return X, yPlot the points with the following code.
import matplotlib.pylab as plt
X, y = generate_noisy_points()
plt.plot(X, y, 'x')
plt.show()
Now, let's implement the algorithm for GP regression, the one shown in the above figure. First lets generate 100 test data points.
Xtest, ytest = generate_noisy_points(100)
Xtest.sort(axis=0)Draw 10 function samples from the GP prior distribution using the following python code.
n = len(Xtest)
K = kernel(Xtest, Xtest)
L = np.linalg.cholesky(K + noise_var*np.eye(n))
f_prior = np.dot(L, np.random.normal(size=(n, n_samples)))The following animation shows the samples drawn from the GP prior.
Next, let's compute the GP posterior given the original (training) 10 data points, using the following python code.
N, n = len(X), len(Xtest)
K = kernel(X, X)
L = np.linalg.cholesky(K + noise_var*np.eye(N))
K_ = kernel(Xtest, Xtest)
Lk = np.linalg.solve(L, kernel(X, Xtest))
mu = np.dot(Lk.T, np.linalg.solve(L, y))
L = np.linalg.cholesky(K_ + noise_var*np.eye(n) - np.dot(Lk.T, Lk))
f_post = mu.reshape(-1,1) + np.dot(L, np.random.normal(size=(n,n_samples)))The following animation shows 10 function samples drawn from the GP posterior istribution. As expected, we get nearly zero uncertainty in the prediction of the points that are present in the training dataset and the variance increase as we move further from the points.
The kernel function used here is Gaussian squared exponential kernel, can be implemented with the following python code snippet.
def kernel(x, y, l2):
sqdist = np.sum(x**2,1).reshape(-1,1) + \
np.sum(y**2,1) - 2*np.dot(x, y.T)
return np.exp(-.5 * (1/l2) * sqdist)Now, let's predict with the Gaussian Process Regression model, using the following python function:
def posterior(X, Xtest, l2=0.1, noise_var=1e-6):
N, n = len(X), len(Xtest)
K = kernel(X, X, l2)
L = np.linalg.cholesky(K + noise_var*np.eye(N))
Lk = np.linalg.solve(L, kernel(X, Xtest, l2))
# compute the mean at our test points.
mu = np.dot(Lk.T, np.linalg.solve(L, y))
# compute the variance at our test points.
K_ = kernel(Xtest, Xtest, l2)
sd = np.sqrt(np.diag(K_) - np.sum(Lk**2, axis=0))
return (mu, sd)Use the above function to predict the mean and standard deviation at x=1.
mu, sd = posterior(X, np.array([[1]]))
print(mu, sd)
# [[0.85437633]] [0.26444361]The following figure shows the predicted values along with the associated 3 s.d. confidence.
As can be seen, the highest confidence (corresponds to zero confidence interval) is again at the training data points. The blue curve represents the original function, the red one being the predicted function with GP and the red "+" points are the training data points.
Noisy Gaussian Regression
Now let’s increase the noise variance to implement the noisy version of GP.
The following animation shows how the predictions and the confidence intervals change as noise variance is increased: the predictions become less and less uncertain, as expected.
Next, let's see how varying the kernel parameter l changes the confidence interval, in the following animation.
Gaussian processes and Bayesian optimization
Now, let's learn how to use GPy and GPyOpt libraries to deal with gaussian processes. These libraries provide quite simple and inuitive interfaces for training and inference, and we will try to get familiar with them in a few tasks. The following figure shows the basic concepts required for GP regression again.
Gaussian processes Regression with GPy (documentation)
Again, let's start with a simple regression problem, for which we will try to fit a Gaussian Process with RBF kernel.
Create RBF kernel with variance sigma_f and length-scale parameter l for 1D samples and compute value of the kernel between points, using the following code snippet.
import GPy
import GPyOpt
import seaborn as sns
sigma_f, l = 1.5, 2
kernel = GPy.kern.RBF(1, sigma_f, l)
sns.heatmap(kernel.K(X, X))
plt.show()The following figure shows how the kernel heatmap looks like (we have 10 points in the training data, so the computed kernel is a 10X10 matrix.
Let's fit a GP on the training data points. Use kernel from previous task. As shown in the code below, use GPy.models.GPRegression class to predict mean and vairance at position 𝑥=1, e.g.
X, y = generate_noisy_points(noise_variance=0.01)
model = GPy.models.GPRegression(X,y,kernel)
mean, variance = model.predict(np.array([[1]]))
print(mean, variance)
# 0.47161301004863576 1.1778512693257484Let's see the parameters of the model and plot the model
print(model)
model.plot()
plt.show()
# Name : GP regression
# Objective : 11.945995014694255
# Number of Parameters : 3
# Number of Optimization Parameters : 3
# Updates : True
# Parameters:
# GP_regression. | value | constraints |
# priors
# rbf.variance | 0.5884024388364221 | +ve |
# rbf.lengthscale | 1.565659066396689 | +ve |
# Gaussian_noise.variance | 1.0 | +ve |
Observe that the model didn't fit the data quite well. Let's try to fit kernel and noise parameters automatically.
model.optimize()
print(model)
# Name : GP regression
# Objective : 0.691713117288967
# Number of Parameters : 3
# Number of Optimization Parameters : 3
# Updates : True
# Parameters:
# GP_regression. | value | constraints | # priors
# rbf.variance | 0.5088590289246014 | +ve |
# rbf.lengthscale | 1.1019439281553658 | +ve |
# Gaussian_noise.variance | 0.0030183424485056066 | +ve | Now plot the model to obtain a figure like the following one.
As can be seen from the above figure, the process generates outputs just right.
Differentiating noise and signal with GP
Generate two datasets: sinusoid wihout noise (with the function generate_points() and noise variance 0) and samples from gaussian noise (with the function generate_noise() define below).
def generate_noise(n=10, noise_variance=0.01):
np.random.seed(777)
X = np.random.uniform(-3., 3., (n, 1))
y = np.random.randn(n, 1) * noise_variance**0.5
return X, y
X, y = generate_noise(noise_variance=10)Optimize kernel parameters compute the optimal values of noise component for the noise.
model = GPy.models.GPRegression(X,y,kernel)
model.optimize()
print(model)
# Name : GP regression
# Objective : 26.895319516885678
# Number of Parameters : 3
# Number of Optimization Parameters : 3
# Updates : True
# Parameters:
# GP_regression. | value | constraints |
# priors
# rbf.variance | 4.326712527380182 | +ve |
# rbf.lengthscale | 0.613701590417825 | +ve |
# Gaussian_noise.variance | 9.006031601676087 | +ve | Now optimize kernel parameters compute the optimal values of noise component for the signal without noise.
X, y = generate_noisy_points(noise_variance=0)
model = GPy.models.GPRegression(X,y,kernel)
model.optimize()
print(model)
# Name : GP regression
# Objective : -22.60291145140328
# Number of Parameters : 3
# Number of Optimization Parameters : 3
# Updates : True
# Parameters:
# GP_regression. | value | constraints | # priors
# rbf.variance | 2.498516381000242 | +ve |
# rbf.lengthscale | 2.4529513517426444 | +ve |
# Gaussian_noise.variance | 5.634716625480888e-16 | +ve | As can be seen from above, the GP detects the noise correctly with a high value of Gaussian_noise.variance output parameter.
Sparse GP
Now let's consider the speed of GP. Let's generate a dataset of 3000 points and measure the time that is consumed for prediction of mean and variance for each point.
Then let's try to use inducing inputs and find the optimal number of points according to quality-time tradeoff.
For the sparse model with inducing points, you should use GPy.models.SparseGPRegression class.
The number of inducing inputs can be set with parameter num_inducing and optimize their positions and values with .optimize() call.
Let's first create a dataset of 1000 points and fit GPRegression. Measure time for predicting mean and variance at position 𝑥=1.
import time
X, y = generate_noisy_points(1000)
start = time.time()
model = GPy.models.GPRegression(X,y,kernel)
mean, variance = model.predict(np.array([[1]]))
time_gp = time.time()-start
print(mean, variance, time_gp)
# [[0.84157973]] [[1.08298092e-06]] 7.353320360183716 Then fit SparseGPRegression with 10 inducing inputs and repeat the experiment. Let's find speedup as a ratio between consumed time without and with inducing inputs.
start = time.time()
model = GPy.models.SparseGPRegression(X,y,kernel,num_inducing=10)
model.optimize()
mean, variance = model.predict(np.array([[1]]))
time_sgp = time.time()-start
print(mean, variance, time_sgp)
# [[0.84159203]] [[1.11154212e-06]] 0.8615052700042725As can be seen, there is a speedup of more than 8 with sparse GP using only the inducing points.
time_gp / time_sgp
# 8.535432824627119The model is shown in the next figure.
Bayesian optimization
Bayesian Optimization is used when there is no explicit objective function and it's expensive to evaluate the objective function.
As shown in the next figure, a GP is used along with an acquisition (utility) function to choose the next point to sample, where it's more likely to find the maximum value in an unknown objective function.
A GP is constructed from the points already sampled and the next point is sampled from the region where the GP posterior has higher mean (to exploit) and larger variance (to explore), which is determined by the maximum value of the acquisition function (which is a function of GP posterior mean and variance).
To choose the next point to be sampled, the above process is repeated.
The next couple of figures show the basic concepts of Bayesian optimization using GP, the algorithm, how it works, along with a few popular acquisition functions.
Bayesian Optimization with GPyOpt (documentation)
Paramerter Tuning for XGBoost Regressor
Let's now try to find optimal hyperparameters to XGBoost model using Bayesian optimization with GP, with the diabetes dataset (from sklearn) as input. Let's first load the dataset with the following python code snippet:
from sklearn import datasets
from xgboost import XGBRegressor
from sklearn.model_selection import cross_val_score
dataset = sklearn.datasets.load_diabetes()
X = dataset['data']
y = dataset['target']We will use cross-validation score to estimate accuracy and our goal will be to tune: max_depth, learning_rate, n_estimators parameters. First, we have to define optimization function and domains, as shown in the code below.
# Score. Optimizer will try to find minimum, so we will add a "-" sign.
def f(parameters):
parameters = parameters[0]
score = -cross_val_score(
XGBRegressor(learning_rate=parameters[0],
max_depth=int(parameters[2]),
n_estimators=int(parameters[3]),
gamma=int(parameters[1]),
min_child_weight = parameters[4]),
X, y, scoring='neg_root_mean_squared_error'
).mean()
score = np.array(score)
return score
# Bounds (define continuous variables first, then discrete!)
bounds = [
{'name': 'learning_rate',
'type': 'continuous',
'domain': (0, 1)},
{'name': 'gamma',
'type': 'continuous',
'domain': (0, 5)},
{'name': 'max_depth',
'type': 'discrete',
'domain': (1, 50)},
{'name': 'n_estimators',
'type': 'discrete',
'domain': (1, 300)},
{'name': 'min_child_weight',
'type': 'discrete',
'domain': (1, 10)}
]Let's find the baseline RMSE with default XGBoost parameters is . Let's see if we can do better.
baseline = -cross_val_score(
XGBRegressor(), X, y, scoring='neg_root_mean_squared_error'
).mean()
baseline
# 64.90693011829266Now, run the Bayesian optimization with GPyOpt and plot convergence, as in the next code snippet:
optimizer = GPyOpt.methods.BayesianOptimization(
f=f, domain=bounds,
acquisition_type ='MPI',
acquisition_par = 0.1,
exact_eval=True
)
max_iter = 50
max_time = 60
optimizer.run_optimization(max_iter, max_time)
optimizer.plot_convergence()
Extract the best values of the parameters and compute the RMSE / gain obtained with Bayesian Optimization, using the following code.
optimizer.X[np.argmin(optimizer.Y)]
# array([2.01515532e-01, 1.35401092e+00, 1.00000000e+00,
# 3.00000000e+02, 1.00000000e+00])
print('RMSE:', np.min(optimizer.Y),
'Gain:', baseline/np.min(optimizer.Y)*100)
# RMSE: 57.6844355488563 Gain: 112.52069904249859As can be seen, we were able to get 12% boost without tuning parameters by hand.
Paramerter Tuning for SVR
Now, let's tune a Support Vector Regressor model with Bayesian Optimization and find the optimal values for three parameters: C, epsilon and gamma.
Let's use range (1e-5, 1000) for C, (1e-5, 10) for epsilon and gamma.
Let's use MPI as an acquisition function with weight 0.1.
from sklearn.svm import SVR
# Bounds (define continuous variables first, then discrete!)
bounds = [
{'name': 'C',
'type': 'continuous',
'domain': (1e-5, 1000)},
{'name': 'epsilon',
'type': 'continuous',
'domain': (1e-5, 10)},
{'name': 'gamma',
'type': 'continuous',
'domain': (1e-5, 10)}
]
# Score. Optimizer will try to find minimum, so we will add a "-" sign.
def f(parameters):
parameters = parameters[0]
score = -cross_val_score(
SVR(C = parameters[0],
epsilon = parameters[1],
gamma = parameters[2]),
X, y, scoring='neg_root_mean_squared_error'
).mean()
score = np.array(score)
return score
optimizer = GPyOpt.methods.BayesianOptimization(
f=f, domain=bounds,
acquisition_type ='MPI',
acquisition_par = 0.1,
exact_eval=True
)
max_iter = 50*4
max_time = 60*4
optimizer.run_optimization(max_iter, max_time)
baseline = -cross_val_score(
SVR(), X, y, scoring='neg_root_mean_squared_error'
).mean()
print(baseline)
# 70.44352670586173
print(optimizer.X[np.argmin(optimizer.Y)])
# [126.64337652 8.49323372 8.59189135]
print('RMSE:', np.min(optimizer.Y),
'Gain:', baseline/np.min(optimizer.Y)*100)
# RMSE: 54.02576574389976 Gain: 130.38876124364006
best_epsilon = optimizer.X[np.argmin(optimizer.Y)][1]
optimizer.plot_convergence()
For the model above the boost in RMSE that was obtained after tuning hyperparameters was 30%.
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