Mathematical Foundations of Generative AI · End Term · Sep 2025 · FN

Mathematical Foundations of Generative AI · End Term · Sep 2025 FN

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Questions and published explanations below are available without starting a test. Some questions may not have a published solution yet.

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Question 2 MCQ · 3.0 marks

In the context of Denoising Diffusion Probabilistic Models (DDPM), the trained noise predictor

approximates the score function of the data distribution via

Consider a specific timestep

where the cumulative noise variance schedule is given by

. At this timestep, for a specific input

, the model predicts a noise vector:

Calculate the estimated score vector,

.

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6406536024993.

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Question 3 MCQ · 3.0 marks

You are performing inference using a Guided Diffusion model with Classifier-Free Guidance (CFG) with a guidance scale

. At a specific step, the model outputs:

• Unconditional prediction:

• Conditional prediction (given class

): Using the standard CFG formula, calculate the final guided noise vector

.

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6406536024995.

6406536024996.

6406536024997.

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Question 4 MCQ · 3.0 marks

In a Deep Transformer with Residual connections, the output of layer

is

. Assume that at initialization, the function

(Attention + FFN) outputs values with variance roughly

. After 10 layers, assuming independence, approximately how much has the signal variance increased relative to the input variance

? (Approximation:

).

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It remains the same

.

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It increases by a factor of

.

6406536025000.

It doubles

.

6406536025001.

It increases 10-fold.

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Question 5 MCQ · 3.0 marks

You are performing Beam Search with width

. At step

, you have two active beams with accumulated log-probs: Beam A:

(ending in token 'cat') Beam B:

(ending in token 'dog') At step

, the model gives the following top log-probs: Given 'cat': 'sits' (-0.5), 'naps' (-1.0) Given 'dog': 'barks' (-0.1), 'runs' (-0.4) Which two paths form the new beams?

6406536025002.

('cat', 'sits') and ('dog', 'barks')

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('cat', 'sits') and ('cat', 'naps')

6406536025004.

('dog', 'barks') and ('dog', 'runs')

6406536025005.

('cat', 'naps') and ('dog', 'runs')

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Question 6 MCQ · 3.0 marks

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Undefined

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Question 7 MCQ · 3.0 marks

When treating an Auto-Regressive Language Model as an RL policy: • The **State** is the context (prompt + generated tokens so far). • The **Action Space** is the vocabulary

. If the vocabulary size is

and we generate a sequence of length

. What is the dimension of the policy output distribution at a single timestep, and what is the dimensionality of the trajectory space?

6406536025010.

Output dim:

; Trajectory space:

.

6406536025011.

Output dim:

; Trajectory space:

.

6406536025012.

Output dim: 1; Trajectory space:

.

6406536025013.

Output dim: 50,000; Trajectory space:

.

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Question 8 MCQ · 3.0 marks

Trust Region Policy Optimization (TRPO) maximizes the surrogate objective subject to a constraint on the KL divergence between the old and new policies. What is the nature of this constraint?

6406536025014.

6406536025015.

6406536025016.

6406536025017.

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Question 9 MCQ · 3.0 marks

The Policy Gradient theorem states

. If we introduce a baseline

that depends only on state, the term becomes

. What is the expected value of the following baseline term?

6406536025018.

6406536025019.

6406536025020.

6406536025021.

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Question 10 MCQ · 3.0 marks

DDIMs are faster than DDPM mainly because:

6406536025022.

They remove the neural network entirely.

6406536025023.

They require more noise steps.

6406536025024.

They require additional forward passes.

6406536025025.

The sampling process is deterministic when

.

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Question 11 MSQ · 4.0 marks

An auto-regressive language model predicts the next token

given the history

. In the RL interpretation used for alignment, which of the following correspondences between LM objects and RL objects are correct?

6406536025026.

The state

corresponds to the current context: prompt plus all tokens generated so far.

6406536025027.

The action

corresponds to selecting the next token from the vocabulary.

6406536025028.

The trajectory

corresponds to the full sequence of (prompt, generated tokens) over time.

6406536025029.

The reward

is necessarily the log-likelihood of the token under the pretraining objective.

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Question 12 MSQ · 4.0 marks

The sinusoidal positional encoding (PE) for Transformers is defined as:

Based on the properties of sinusoidal positional encodings, identify all correct statements:

6406536025030.

Positional encodings allow the model to incorporate order information that attention alone cannot provide.

6406536025031.

The sinusoidal frequencies grow geometrically with the dimension index

.

6406536025032.

Sinusoidal positional encodings are learned parameters updated during training.

6406536025033.

Adding positional encodings to token embeddings guarantees each position has a unique representation.

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Question 13 NAT · 3.0 marks

Consider a single-head self-attention mechanism with

. You are given the following Query (

) and Key (

) matrices for a sequence length of

:

Compute the raw attention scores (before Softmax) and scale them by the factor

. What is the value of the element at position

(i.e., row 1, column 2) of the scaled score matrix?

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Question 14 NAT · 3.0 marks

You have a Transformer with model dimension

. You employ

parallel attention heads. The output of the heads is concatenated and projected by a matrix

. If the input batch size is

and sequence length is

, what is the total number of floating point elements in the final output tensor of the Multi-Head Attention block (before the residual connection)?

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Question 15 NAT · 3.0 marks

In RLHF (Reinforcement Learning from Human Feedback), the reward used to update the language model

is defined as

Given: • Reward model score

•

•

• KL coefficient

Calculate the total reward. (Use

.)

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Question 16 NAT · 3.0 marks

You are training a Reward Model (RM) using the Bradley-Terry model. For a specific prompt

, you have a winning completion

and a losing completion

. The RM currently predicts the scalar rewards as:

Calculate the negative log-likelihood loss for this pair:

.

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Question 17 NAT · 3.0 marks

A continuous time State-Space Model (SSM) has a scalar state parameter

(representing decay). You discretize this system with a step size

using the Zero-Order Hold (ZOH) method (or approximation thereom). Calculate the discretized state transition parameter

. Formula:

.

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