GTU 3170724 Machine Learning (ML) Winter 2025 Solved Paper & Solutions

Machine learning is the ability of a computer to learn from data and get better at a task without being programmed step by step.

Human learning is the way a person gains knowledge and skill by study, by being taught, or from experience.

Reinforcement learning is a type of machine learning in which an agent (the learner) learns by trial and error. The agent acts in an environment. It gets a reward for a good action and a penalty for a bad one. Over time it learns the actions that give the highest total reward.

Example: a robot learning to walk. Each step without falling earns a reward. Each fall earns a penalty.

Key features:

1. No labelled data. Nobody tells the agent the right answer. It learns only from rewards.
2. Trial and error. The agent must try actions to find out which ones are good.
3. Delayed reward. A reward can come many steps after the action that caused it, as in a game of chess.
4. Exploration and exploitation. The agent has to balance trying new actions (exploration) with repeating actions that already worked (exploitation).
5. Sequence of decisions. Each action changes the next situation, so the decisions depend on one another.

Data pre-processing techniques

Real data is often incomplete, noisy and measured on different scales. Pre-processing prepares it for a model. The main techniques are:

1. Data cleaning: fill in or remove missing values and correct wrong or noisy values.
2. Data integration: combine data from several sources into one data set.
3. Data transformation: bring values to a common scale (normalisation) and convert text labels into numbers (encoding).
4. Data reduction: make the data smaller without losing important information. Dimensionality reduction and feature selection are two ways to do this.

Dimensionality reduction

A feature is one input column of the data set. The number of features is called the dimension of the data.

Dimensionality reduction builds a small set of new features from the original ones and keeps most of the information.

• A common method is Principal Component Analysis (PCA). It combines related features into a few new features called principal components.
• Example: 100 features of an image can be reduced to 10 principal components.
• Benefits: faster training, less storage, and data that can be plotted in 2D or 3D.

Feature selection

Feature selection keeps some of the original features and drops the rest. No new feature is created.

• It removes irrelevant features, which have no link with the output. Example: roll number when predicting marks.
• It removes redundant features, which repeat the same information. Example: height in cm and height in inches.
• Approaches: filter (score each feature with a statistical measure), wrapper (train the model on different subsets and keep the best) and embedded (the model selects features while it trains).

Under-fitting happens when a model is too simple to learn the pattern in the training data. It gives poor results on the training data and also on new data.

Example: fitting a straight line to data that follows a curve.

Major causes:

1. The model is too simple for the data.
2. Too few features, or features not related to the output.
3. Too little training: not enough data, or training stopped too early.
4. Too much regularisation, which stops the model from fitting the data.

No. Machine learning cannot solve all problems. It works well only when three things are true: there is a pattern to learn, enough good data is available, and the rules are too hard to write by hand.

Machine learning is not suitable in these cases:

1. No data or poor data. A model learns only from data. With too little or wrong data, the result is poor.
2. A simple rule or formula already exists. To calculate salary or to sort numbers, a normal program is exact, faster and cheaper.
3. Human judgement is needed. Tasks that need ethics, emotions or common sense cannot be left to a model.
4. Every decision must be explained. Many models cannot give a reason for their output.
5. The situation is new. A model cannot predict well for cases that never appeared in its training data.

It is suitable for tasks such as spam detection and face recognition, where a lot of past data exists.

How success is evaluated

An unsupervised model has no labelled answers, so its output cannot be marked right or wrong. For clustering, success means good-quality clusters:

• points inside one cluster are close to each other (cohesion),
• different clusters are far apart (separation).

Popular measures

1. Silhouette coefficient
2. Sum of squared errors (SSE) inside the clusters
3. Dunn index and Davies-Bouldin index
4. Purity and Rand index, used when the true classes are known

1. Silhouette coefficient

For one data point:

• $a$ = average distance to the other points in its own cluster
• $b$ = average distance to the points in the nearest other cluster

The value lies between $-1$ and $+1$. A value near $+1$ means the point is well placed. A value near $0$ means it lies on the border of two clusters. A negative value means it is probably in the wrong cluster.

Example: for a point, $a = 2$ and $b = 8$.

The value is close to $+1$, so the point is well clustered. The average of $s$ over all points gives the score of the whole model.

2. Purity

Purity is used when the true class of each point is known. Each cluster is given the class that appears most often in it.

Example: 20 points are grouped into two clusters.

A purity of 1 means every cluster contains only one class.

A predictive model predicts an unknown or future output from known inputs. It learns from past data in which the output is already known, so it belongs to supervised learning.

Example: a bank wants to know whether a new customer will repay a loan.

• Inputs: income, age, past loans.
• The model is trained on old customers whose result is known.
• For a new customer, it predicts "will repay" or "will not repay".

If the output is a class, it is classification. If the output is a number, such as house price, it is regression.

Supervised learning trains a model on labelled data, which means the correct output of each training example is known. It has two types.

1. Classification

The model puts each input into one of a fixed set of classes.

Sample application: email spam detection. The model is trained on emails already marked as spam or not spam. For a new email it predicts one of the two classes. Another application is disease diagnosis: disease present or not present.

2. Regression

The model predicts a numeric value.

Sample application: house price prediction. The model is trained on past sales with area, location and price. For a new house it predicts the price as a number. Another application is predicting tomorrow's temperature.

Definition

The Naive Bayes classifier is a supervised classification method based on Bayes' theorem. It is called "naive" because it assumes that all features are independent of each other for a given class.

Bayes' theorem

• $P(c)$: prior probability of class $c$
• $P(x \mid c)$: likelihood, the probability of seeing data $x$ in class $c$
• $P(c \mid x)$: posterior probability of class $c$ for the given data

For features $x_1, x_2, \dots, x_n$ the classifier calculates, for every class,

and selects the class with the highest value.

Steps

1. Calculate the prior probability of each class from the training data.
2. Calculate the likelihood of each feature value for each class.
3. For new data, multiply the prior and the likelihoods for each class.
4. Choose the class with the highest result.

Example from practical life: spam filter

Training data: 10 emails, of which 4 are spam and 6 are not spam. The word "offer" appears in 3 of the 4 spam emails and in 1 of the 6 other emails.

A new email contains the word "offer".

Since $0.30 > 0.10$, the email is classified as spam.

Uses, strengths and limits

• Uses: spam filtering, sentiment analysis of reviews, document classification, medical diagnosis.
• Strengths: simple, fast, and works with a small amount of training data.
• Limits: features are rarely independent in real data. If a feature value never appears with a class in training, its probability becomes zero. Laplace smoothing is used to correct this.

In a curvilinear relationship the graph is a curve, so the slope is not constant. The slope at a point is the slope of the tangent at that point.

• Curve linear positive slope: the curve rises from left to right. As $x$ increases, $y$ also increases. The slope is positive at every point, but its value changes.
• Curve linear negative slope: the curve falls from left to right. As $x$ increases, $y$ decreases. The slope is negative at every point, and its value changes.

A Bayesian belief network is a graph that shows how a set of variables depend on each other through probability. It has two parts:

1. A directed acyclic graph. Each node is a variable. An arrow from A to B means that A directly affects B. The graph has no cycles.
2. A conditional probability table for each node, which gives the probability of that node for every combination of values of its parent nodes.

Unlike Naive Bayes, it allows some variables to depend on each other.

Where they are used: medical diagnosis (from symptoms to disease), fault finding in machines, spam filtering, weather prediction and risk analysis.

Can they solve all types of problems? No.

• The structure of the graph must be known or learned, and that is difficult.
• Every probability table needs a lot of data.
• The calculation becomes very heavy when there are many variables.
• Relationships that form a cycle cannot be shown.

Definition

Support Vector Machine (SVM) is a supervised learning algorithm used mainly for classification. It finds the boundary that separates two classes with the largest possible gap between them.

Important terms

• Hyperplane: the decision boundary that separates the classes. For two features it is a line: $w \cdot x + b = 0$.
• Margin: the distance between the hyperplane and the nearest data points of each class.
• Support vectors: the data points nearest to the hyperplane. They alone decide its position.

How it works

Many lines can separate the two classes. SVM chooses the one with the maximum margin, because a wide margin gives fewer errors on new data. The margin equals $\dfrac{2}{\lVert w \rVert}$, so SVM makes $\lVert w \rVert$ as small as possible while keeping every training point on the correct side:

Soft margin

Real data has noise. A soft margin allows a few points to fall on the wrong side. The parameter $C$ controls how many such errors are allowed.

Kernel trick

When the classes cannot be separated by a straight line, a kernel function maps the data to a higher dimension where a hyperplane can separate them. Common kernels are linear, polynomial and radial basis function (RBF).

Strengths, limits and uses

• Strengths: works well with many features, gives good accuracy on small and medium data sets, and is less likely to over-fit.
• Limits: slow on very large data sets, and choosing the right kernel is difficult.
• Uses: text classification, face detection, handwriting recognition.

Deep learning is a part of machine learning that uses artificial neural networks with many hidden layers. The word "deep" refers to the number of layers.

• It learns useful features directly from raw data such as images, sound and text.
• Each layer learns something more complex than the layer before it.
• It needs a large amount of data and high computing power.

Example: face recognition. The first layers learn simple edges. The middle layers learn parts of a face such as eyes and nose. The last layers recognise the whole face.

Density-based methods are clustering methods that treat a cluster as a dense region of data points. Clusters are separated by regions with few points, and points in such thin regions are treated as noise.

The best-known method is DBSCAN (Density-Based Spatial Clustering of Applications with Noise). It uses two parameters:

• Eps ($\varepsilon$): the radius of the neighbourhood around a point.
• MinPts: the minimum number of points needed inside that radius to call the region dense.

Each point is one of three types:

1. Core point: has at least MinPts points within its $\varepsilon$ radius.
2. Border point: has fewer than MinPts points, but lies within the radius of a core point.
3. Noise point: is neither a core point nor a border point.

Working: start from a core point and add all points within its radius. Repeat from every new core point until the cluster stops growing.

Advantages: it finds clusters of any shape, needs no number of clusters, and handles noise.

Limitations: choosing $\varepsilon$ and MinPts is difficult, and it works poorly when clusters have different densities.

Learning process of an ANN

An artificial neural network (ANN) is made of layers of neurons joined by links. Every link has a number called a synaptic weight, which shows how strongly one neuron affects the next. Learning means finding the weights that give the correct output.

1. Initialise: give every weight a small random value.
2. Forward pass: feed the input. Each neuron calculates the weighted sum of its inputs, adds a bias and applies an activation function. The result moves forward to the output layer.
3. Find the error: compare the output $o$ with the target $t$.

4. Backward pass: send the error back through the network and change each weight a little so that the error becomes smaller.

Here $\eta$ is the learning rate, which sets the size of each change.

5. Repeat steps 2 to 4 for all training examples, many times, until the error is small. One full pass over the training data is called an epoch.

The challenge in assigning weights

A network has a very large number of weights, and their correct values are not known in advance. They cannot be set by hand.

For the output layer the error is known directly, because the target is given. For a hidden neuron there is no target value. So it is not clear how much each hidden weight is responsible for the final error.

Example: a network with 3 inputs, 4 hidden neurons and 1 output has $3 \times 4 + 4 \times 1 = 16$ weights. If the output is 0.8 and the target is 1, which of the 16 weights should change, and by how much?

How the challenge is addressed

The backpropagation algorithm with gradient descent solves it.

• It first calculates the error at the output layer.
• Using the chain rule of differentiation, it passes this error backwards, layer by layer. Each weight gets its own share of the error, called the gradient.
• Each weight is then moved a small step in the direction that reduces the error.

Starting with random weights and choosing a suitable learning rate help the network learn properly.