Homework 4 Basic optimization

ISYE 6740 Homework 4
Total 100 points + 15 bonus points.
1. Basic optimization. (40 points.)
Consider a simplified logistic regression problem. Given m training samples (x
i
, yi
), i = 1, . . . , m.
The data x
i ∈ R (note that we only have one feature for each sample), and y
i ∈ {0, 1}. To fit a
logistic regression model for classification, we solve the following optimization problem, where θ ∈ R
is a parameter we aim to find:
max
θ
`(θ), (1)
where the log-likelhood function
`(θ) = Xm
i=1

− log(1 + exp{−θxi
}) + (y
i − 1)θxi
    
.
(a) (10 points) Show step-by-step mathematical derivation for the gradient of the cost function `(θ)
in (1).
(b) (10 points) Write a pseudo-code for performing gradient descent to find the optimizer θ
∗
. This
is essentially what the training procedure does. (pseudo-code means you will write down the steps
of the algorithm, not necessarily any specific programming language.)
(c) (10 points) Write the pseudo-code for performing the stochastic gradient descent algorithm to
solve the training of logistic regression problem (1). Please explain the difference between gradient
descent and stochastic gradient descent for training logistic regression.
(d) (10 points) We will show that the training problem in basic logistic regression problem
is concave. Derive the Hessian matrix of `(θ) and based on this, show the training problem (1)
is concave (note that in this case, since we only have one feature, the Hessian matrix is just a
scalar). Explain why the problem can be solved efficiently and gradient descent will achieve a
unique global optimizer, as we discussed in class.
2. Comparing Bayes, logistic, and KNN classifiers. (60 points)
In lectures, we learn three different classifiers. This question is to implement and compare them.
Python users, please feel free to use Scikit-learn, which is a commonly-used and powerful Python library
with various machine learning tools. But you can also use other similar libraries in other languages of
your choice to perform the tasks.
Part One (Divorce classification/prediction). (30 points)
This dataset is about participants who completed the personal information form and a divorce predictors scale.
The data is a modified version of the publicly available at
https://archive.ics.uci.edu/ml/datasets/Divorce+Predictors+data+set (by injecting noise so
you will not get the exactly same results as on UCI website). The dataset marriage.csv is contained
in the homework folder. There are 170 participants and 54 attributes (or predictor variables) that are
all real-valued. The last column of the CSV file is label y (1 means “divorce”, 0 means “no divorce”).
Each column is for one feature (predictor variable), and each row is a sample (participant). A detailed
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explanation for each feature (predictor variable) can be found at the website link above. Our goal is
to build a classifier using training data, such that given a test sample, we can classify (or essentially
predict) whether its label is 0 (“no divorce”) or 1 (“divorce”).
Build three classifiers using (Naive Bayes, Logistic Regression, KNN). Use the first 80% data for
training and the remaining 20% for testing. If you use scikit-learn you can use train test split to split
the dataset.
Remark: Please note that, here, for Naive Bayes, this means that we have to estimate the variance for
each individual feature from training data. When estimating the variance, if the variance is zero to
close to zero (meaning that there is very little variability in the feature), you can set the variance to
be a small number, e.g.,  = 10−3
. We do not want to have include zero or nearly variance in Naive
Bayes. This tip holds for both Part One and Part Two of this question.
(a) (15 points) Report testing accuracy for each of the three classifiers. Comment on their performance: which performs the best and make a guess why they perform the best in this setting.
(b) (15 points) Now perform PCA to project the data into two-dimensional space. Plot the data
points and decision boundary of each classifier. Comment on the difference between the decision
boundary for the three classifiers. Please clearly represent the data points with different labels
using different colors.
Part Two (Handwritten digits classification). (30 points) Repeat the above using the MNIST
Data in our previous homework. Here, give “digit” 6 label y = 1, and give “digit” 2 label y = 0. All
the pixels in each image will be the feature (predictor variables) for that sample (i.e., image). Our goal
is to build classifier to such that given a new test sample, we can tell is it a 2 or a 6. Using the first
80% of the samples for training and remaining 20% for testing.
(a) (15 points) Report testing accuracy for each of the three classifiers. Comment on their performance: which performs the best and make a guess why they perform the best in this setting.
(b) (15 points) Now perform PCA to project the data into two-dimensional space. Plot the data
points and decision boundary of each classifier. Comment on the difference between the decision
boundary for the three classifiers. Please clearly represent the data points with different labels
using different colors.
3. Naive Bayes for spam filtering. (15 points)
In this problem, we will use the Naive Bayes algorithm to fit a spam filter by hand. This will enhance your understanding to Bayes classifier and build intuition. This question does not involve any
programming but only derivation and hand calculation.
Spam filters are used in all email services to classify received emails as “Spam” or “Not Spam”. A
simple approach involves maintaining a vocabulary of words that commonly occur in “Spam” emails
and classifying an email as “Spam” if the number of words from the dictionary that are present in the
email is over a certain threshold. We are given the vocabulary consists of 15 words
V = {secret, offer, low, price, valued, customer, today, dollar, million, sports, is, for, play, healthy, pizza}.
We will use Vi to represent the ith word in V . As our training dataset, we are also given 3 example
spam messages,
• million dollar offer
• secret offer today
• secret is secret
and 4 example non-spam messages
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• low price for valued customer
• play secret sports today
• sports is healthy
• low price pizza
Recall that the Naive Bayes classifier assumes the probability of an input depends on its input feature.
The feature for each sample is defined as x
(i) = [x
(i)
1
, x
(i)
2
, . . . , x
(i)
d
]
T
, i = 1, . . . , m and the class of the
ith sample is y
(i)
. In our case the length of the input vector is d = 15, which is equal to the number
of words in the vocabulary V . Each entry x
(i)
j
is equal to the number of times word Vj occurs in the
i-th message.
(a) (5 points) Calculate class prior P(y = 0) and P(y = 1) from the training data, where y = 0
corresponds to spam messages, and y = 1 corresponds to non-spam messages. Note that these
class prior essentially corresponds to the frequency of each class in the training sample. Write
down the feature vectors for each spam and non-spam messages.
(b) (5 points) In the Naive Bayes model, assuming the keywords are independent of each other (this
is a simplification), the likelihood of a sentence with its feature vector x given a class c is given
by
P(x|y = c) = Y
d
k=1
θ
xk
c,k, c = {0, 1}
where 0 ≤ θc,k ≤ 1 is the probability of word k appearing in class c, which satisfies
θ0,k = 1 − θ1,k, ∀k.
Given this, the complete log-likelihood function for our training data is given by
`(θ0,1, . . . , θ0,d, θ1,1, . . . , θ1,d) = Xm
i=1
X
d
k=1
x
(i)
k
log θy(i),k
(In this example, m = 7.) Calculate the maximum likelihood estimates of θ0,1, θ0,7, θ1,1, θ1,15 by
maximizing the log-likelihood function above. (Hint: We are solving a constrained maximization
problem: you can introduce Lagrangian multipliers, or directly substitute the θ0,k = 1 − θ1,k into
the objective function so you do not need to worry about the constraint.)
(c) (5 points) Given a test message “today is secret”, using the Naive Bayes classier that you have
trained in Part (a)-(b), to calculate the posterior and decide whether it is spam or not spam.
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