(SEM VII) THEORY EXAMINATION 2024-25 ARTIFICIAL INTELLIGENCE

ARTIFICIAL INTELLIGENCE (KDS071) – COMPLETE SOLVED PAPER

Time: 3 Hours  Max Marks: 100                                                      Instructions: Attempt all sections

SECTION A (2 × 10 = 20 Marks)

Attempt all questions in brief

a) Examples of convex optimization problems

Linear Programming (LP)                                                      Least Squares Regression

Support Vector Machines (SVM)                                           Lasso and Ridge Regression

Quadratic Programming

b) Karush–Kuhn–Tucker (KKT) conditions                             For constrained optimization:

Stationarity                                                                            Primal feasibility

Dual feasibility                                                                       Complementary slackness

c) Basic steps of mirror descent algorithm                            Initialize parameter

Compute gradient                                                                 Map to dual space

Perform gradient step                                                           Map back using mirror function

d) Proximal gradient methods

They solve non-smooth convex problems by combining:  Gradient descent (smooth part)

Proximal operator (non-smooth part)

e) Monotone operators

An operator T is monotone if:                           (Tx−Ty)T(x−y)≥0(Tx - Ty)^T (x - y) \ge 0(Tx−Ty)T(x−y)≥0

Used in convex optimization and variational inequalities.

f) Intuition behind Douglas–Rachford splitting

It solves problems with two complex constraints by splitting them into simpler subproblems and iteratively combining their solutions.

g) Langevin dynamics

A stochastic optimization method that adds Gaussian noise to gradient descent to escape local minima and sample from probability distributions.

h) Purpose of averaging methods in stochastic optimization

Reduces variance

Improves convergence stability                          Produces smoother trajectories

i) Supervised vs Unsupervised learning

j) Feature extraction in machine learning

Transforming raw data into meaningful numerical features to improve model accuracy and efficiency.

SECTION B (10 × 3 = 30 Marks)

Attempt any three

a) Duality in optimization (with example)

Primal problem minimizes cost; dual maximizes constraints.
Example:
Minimize x2x^2x2 subject to x≥1x ≥ 1x≥1.
Dual formulation provides bounds and optimality checks.

b) ODE interpretations and optimization algorithms

Gradient descent can be viewed as a discretized Ordinary Differential Equation (ODE), explaining convergence behavior and stability.

c) Augmented Lagrangian method (quadratic constraint)

Adds penalty term to Lagrangian:

L(x,λ)=f(x)+λg(x)+ρ2g(x)2L(x,λ)=f(x)+λg(x)+\frac{ρ}{2}g(x)^2L(x,λ)=f(x)+λg(x)+2ρ​g(x)2

Improves convergence for constrained problems.

d) Polyak–Juditsky averaging

Averages all previous iterates:       xˉk=1k∑i=1kxi\bar{x}_k=\frac{1}{k}\sum_{i=1}^k x_ixˉk​=k1​i=1∑k​xi​

Reduces noise in stochastic gradients.

e) Application domains of machine learning

Healthcare (diagnosis)                    Finance (fraud detection)

NLP (chatbots)                                Computer Vision

Autonomous vehicles                     Recommendation systems

SECTION C (10 × 5 = 50 Marks)

Attempt one from each question

Q3(a) Apply KKT conditions to a quadratic programming problem

Minimize:                                          f(x)=x2f(x)=x^2f(x)=x2

Subject to:                                         x≥1x ≥ 1x≥1

Steps:

Lagrangian: L=x2+λ(1−x)L=x^2+λ(1-x)L=x2+λ(1−x)

Apply KKT conditions                        Optimal solution: x = 1

Q4(b) Gradient descent for f(x)=x2+4x+4f(x)=x^2+4x+4f(x)=x2+4x+4

Gradient:                                          f′(x)=2x+4f'(x)=2x+4f′(x)=2x+4

Set gradient = 0 → x = −2              Minimum value: f(−2)=0

Q5(a) Augmented Lagrangian vs Standard Lagrangian

Q6(a) Polyak–Juditsky averaging application

Used in SGD                                      Improves convergence speed

Reduces oscillations                          Common in deep learning optimizers

Q7(a) Bayesian classifier use case

Email Spam Detection                    Uses Bayes’ theorem

Computes probability of spam given words

Fast, interpretable, efficient for text data

Q7(b) PCA – real-world optimization scenario

Image Compression                         High-dimensional pixels reduced

Retains maximum variance                Faster computation

Less storage and noise