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COMP3670/6670 Programming Assignment 3 - Linear

Regression

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In [ ]:

The following section provides some helper functions.

import numpy as np

import matplotlib.pyplot as plt

In [ ]:

Task 0: Warming Up

The following code block visualises the difference between different methods of performing

linear regression.

## GENERAL FUNCTIONS - DO NOT MODIFY ##

def lr_mle(X, y):

# maximum likelihood (least squares) for linear regression

XtX = np.dot(X.T, X)

Xty = np.dot(X.T, y)

theta = np.linalg.solve(XtX, Xty)

return theta

def lr_map(X, y, alpha=0.1):

# maximum a-posteriori (regularised least squares) for linear regression

N, D = X.shape[0], X.shape[1]

XtX = np.dot(X.T, X) + np.diag(alpha*N*np.ones(D))

Xty = np.dot(X.T, y)

theta = np.linalg.solve(XtX, Xty)

return theta

def lr_bayes(X, y, alpha=0.1, noise_var=0.01):

# exact posterior for Bayesian linear regression

N, D = X.shape[0], X.shape[1]

XtX = np.dot(X.T, X) + np.diag(alpha*N*np.ones(D))

Xty = np.dot(X.T, y)

mean = np.linalg.solve(XtX, Xty)

# note: calling inv directly is not ideal

cov = np.linalg.inv(XtX) * noise_var

return mean, cov

def predict_point(X, theta):

# predict given parameter estimate

return np.dot(X, theta)

def predict_bayes(X, theta_mean, theta_cov):

# predict gien parameter posterior

mean = np.dot(X, theta_mean)

cov = np.dot(X, np.dot(theta_cov, X.T))

return mean, cov

def add_bias_col(x):

# add an all-one column

n = x.shape[0]

return np.hstack([x, np.ones([n, 1])])

## END GENERAL FUNCTIONS ##

2 9

In [ ]:

Task 1: What makes a good regression?

As can be seen from the visualisation above, the regressed line seems to be far from the

# load data

data = np.loadtxt("./data/ass3_data1_train.txt")

x_train, y_train = data[:, 0][:, None], data[:, 1][:, None]

data = np.loadtxt("./data/ass3_data1_valid.txt")

x_valid, y_valid = data[:, 0][:, None], data[:, 1][:, None]

# some data for visualisation

N_plot = 100

x_plot = np.linspace(-2.5, 2.5, N_plot).reshape([N_plot, 1])

# add one col to the inputs

x_train_with_bias = add_bias_col(x_train)

x_plot_with_bias = add_bias_col(x_plot)

# MLE = least squares

theta_mle = lr_mle(x_train_with_bias, y_train)

f_mle = predict_point(x_plot_with_bias, theta_mle)

# MAP = regularised least squares

alpha = 0.1

theta_map = lr_map(x_train_with_bias, y_train, alpha)

f_map = predict_point(x_plot_with_bias, theta_map)

# exact Bayesian

theta_mean, theta_cov = lr_bayes(x_train_with_bias, y_train, alpha)

f_bayes_mean, f_bayes_cov = predict_bayes(

x_plot_with_bias, theta_mean, theta_cov)

# plot utility

def plot(x, y, x_plot, f_mle, f_map, f_bayes_mean, f_bayes_cov):

# plot utility

plt.figure(figsize=(6, 4))

plt.plot(x, y, '+g', label='train data', ms=12)

if f_mle is not None:

plt.plot(x_plot, f_mle, '-k', label='mle')

if f_map is not None:

plt.plot(x_plot, f_map, '--k', label="map", zorder=10)

if f_bayes_mean is not None:

plt.plot(x_plot, f_bayes_mean, '-r', label="bayes", lw=3)

f_std = np.sqrt(np.diag(f_bayes_cov))

upper = f_bayes_mean[:, 0] + 2*f_std

lower = f_bayes_mean[:, 0] - 2*f_std

plt.fill_between(x_plot[:, 0], upper, lower, color='r', alpha=

plt.legend()

plt.xlabel('x')

plt.ylabel('y')

plt.ylim([-3, 3])

# plot the training data and predictions

plot(x_train, y_train, x_plot, f_mle, f_map, f_bayes_mean, f_bayes_cov

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datapoints. Are there any ways we can improve the regression?

Task 1.1

Explain why the above linear regression fails.

-----Your answer here-----

Task 1.2

What kind of features would lead to a better result? Why?

-----Your answer here-----

Task1.3

Implement featurise function that takes raw datapoints as the input and output a

reasonable design matrix Φ according to the method you mentioned in Task 1.2.

In [ ]:

Task 2: Estimating noise variance through the marginal

likelihood

One commonly asked question in Bayesian linear regression is how can we define the noise

level of the target. In previous questions, we set the noise variance in lr_bayes to be 0.01

def featurise(x):

# TODO: Try to come up with proper features

features = add_bias_col(x) # change this!

return features

x_train_feat = featurise(x_train)

x_valid_feat = featurise(x_valid)

x_plot_feat = featurise(x_plot)

# repeat but now with features

# MLE

theta_mle = lr_mle(x_train_feat, y_train)

f_mle = predict_point(x_plot_feat, theta_mle)

# MAP

alpha = 0.1

theta_map = lr_map(x_train_feat, y_train, alpha)

f_map = predict_point(x_plot_feat, theta_map)

# exact Bayesian

theta_mean, theta_cov = lr_bayes(x_train_feat, y_train, alpha)

f_bayes_mean, f_bayes_cov = predict_bayes(

x_plot_feat, theta_mean, theta_cov)

plot(x_train, y_train, x_plot, f_mle, f_map, f_bayes_mean, f_bayes_cov

4 9

- a fixed constant. But intuitively, after we have observed some datapoints, the noise level

can actually be inferred or estimated. This tasks is designed for you to investigate the

marginal likelihood (a.k.a. model evidence) and how we can use this to pick the noise

variance.

Task 2.1

Implement the negative log marginal likelihood, given the noise level of the likelihood,

training inputs and outputs, and the prior variance. We can pick prior_var using the

same procedure, but assume prior_var = 0.5 for this exercise. The form of the

marginal likelihood is provided in Week 7's lecture slides.

In [ ]:

Task 2.2

Select the most appropriate noise level that minimises the negative log marginal likelihood. In

practice, we can do this mimimisation by gradient descent, but for this exercise, we assume

we have access to a predefined set of potential noise levels and just need to pick one.

In [ ]:

Task 2.3

We visualise the predictions using the estimated noise variance, and compare to those when

the noise is very large or very small. Based on these graphs and the negative log marginal

likelihood corresponding to these noise levels, explain why finding a proper noise level of the

likelihood is important.

# 2a

def negative_log_marginal_likelihood(noise_var, x, y, prior_var=0.5):

# TODO: implement this

return 0

# 2.2

# a predefined list

potential_noise_vars = np.logspace(-4, 1.5, 50)

## YOUR CODE HERE! ##

noise_var_estimated = potential_noise_vars[0] # change this!

#####################

5 9

In [ ]:

-----Your answer here-----

** Task 2.4 - Optional **

The naive implementation of the negative log marginal likelihood above would require the

inverse of an N by N matrix, which is of time complexity . This is computationally

intractable for a large dataset (large N). Can we speed this up?

Θ(?3)

In [ ]:

Task 3: Regularisation

In machine learning, regularisation is an important technique to reduce overfitting.

Regularisation also provides better generalisation in general. This task aims to show how

regularisation affects the parameter estimates.

Task 3.1

Implement L1 , L2 . Both functions take the weight as input, output the regularisation

value and the gradient of the regularisation term (NOT THE GRADIENT OF THE ENTIRE

?

# fit with the estimated noise variance

N = x_train_feat.shape[0]

prior_var = 0.5

alpha = noise_var_estimated / prior_var / N

theta_mean, theta_cov = lr_bayes(x_train_feat, y_train, alpha, noise_var_estimated

f_bayes_mean, f_bayes_cov = predict_bayes(

x_plot_feat, theta_mean, theta_cov)

plot(x_train, y_train, x_plot, None, None, f_bayes_mean, f_bayes_cov)

# fit with a very large noise

noise_var = 5

alpha = noise_var / prior_var / N

theta_mean, theta_cov = lr_bayes(x_train_feat, y_train, alpha, noise_var

f_bayes_mean, f_bayes_cov = predict_bayes(

x_plot_feat, theta_mean, theta_cov)

plot(x_train, y_train, x_plot, None, None, f_bayes_mean, f_bayes_cov)

# fit with a very small noise

noise_var = 0.00001

alpha = noise_var / prior_var / N

theta_mean, theta_cov = lr_bayes(x_train_feat, y_train, alpha, noise_var

f_bayes_mean, f_bayes_cov = predict_bayes(

x_plot_feat, theta_mean, theta_cov)

plot(x_train, y_train, x_plot, None, None, f_bayes_mean, f_bayes_cov)

data = np.loadtxt("./data/ass3_data1_train_large.txt")

x_large, y_large = data[:, 0][:, None], data[:, 1][:, None]

x_large_feat = featurise(x_large)

def negative_log_marginal_likelihood_v2(noise_var, x, y, prior_var=0.5

# TODO: implement this

return 0

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OBJECTIVE FUNCTION).

In [ ]:

Task 3.2

We now run gradient descent and plot the predictions. Comment on the results.

def L1(theta):

# TODO: implement this

return 0, np.zeros_like(theta) # change this

def L2(theta):

# TODO: implement this

return 0, np.zeros_like(theta) # change this

def data_fit(theta, x, y):

diff = y - np.dot(x, theta) # N x 1

f = np.mean(diff**2) # 1 x 1

df = - 2 * np.dot(diff.T, x).T / x.shape[0]

return f, df

def objective(theta, x, y, alpha, l2=True):

reg_func = L2 if l2 else L1

reg, dreg = reg_func(theta)

fit, dfit = data_fit(theta, x, y)

obj = fit + alpha * reg

dobj = dfit + alpha * dreg

return obj, dobj

7 9

In [ ]:

-----Your answer here-----

D = x_train_feat.shape[1]

theta_l2_sgd_init = np.random.randn(D, 1)

theta_l2_sgd = theta_l2_sgd_init

no_iters = 2000

learning_rate = 0.1

alpha = 0.1

l2 = True

for i in range(no_iters):

obj, dobj = objective(theta_l2_sgd, x_train_feat, y_train, alpha,

theta_l2_sgd -= learning_rate * dobj

if i % 100 == 0:

print(i, obj)

f_l2_sgd = predict_point(x_plot_feat, theta_l2_sgd)

theta_l1_sgd = theta_l2_sgd_init

l2 = False

for i in range(no_iters):

obj, dobj = objective(theta_l1_sgd, x_train_feat, y_train, alpha,

theta_l1_sgd -= learning_rate * dobj

if i % 100 == 0:

print(i, obj)

f_l1_sgd = predict_point(x_plot_feat, theta_l1_sgd)

# Without any regularisation

theta_noreg_sgd = theta_l2_sgd_init

for i in range(no_iters):

obj, dobj = objective(theta_noreg_sgd, x_train_feat, y_train, 0, l2

theta_noreg_sgd -= learning_rate * dobj

if i % 100 == 0:

print(i, obj)

f_noreg_sgd = predict_point(x_plot_feat, theta_noreg_sgd)

theta_map = lr_map(x_train_feat, y_train, alpha)

f_map = predict_point(x_plot_feat, theta_map)

# plot utility

plt.figure(figsize=(6, 4))

plt.plot(x_train, y_train, '+g', label='train data', ms=12)

plt.plot(x_plot, f_l2_sgd, '-k', lw=3, label='sgd l2')

plt.plot(x_plot, f_l1_sgd, '-r', lw=2, label='sgd l1')

plt.plot(x_plot, f_map, '--o', label="map", zorder=10, lw=1)

plt.plot(x_plot, f_noreg_sgd, '-x', label="no reg", lw=2)

plt.legend()

plt.xlabel('x')

plt.ylabel('y')

plt.ylim([-3, 3])

plt.show()

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