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

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

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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 noisy data distribution via

Suppose at timestep

, the cumulative noise schedule is

. For the noisy input

, the model predicts the noise vector:

Compute the estimated score vector

.

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Question 3 MCQ · 3.0 marks

During inference with a Classifier-Free Guidance (CFG) diffusion model, you apply a guidance scale of

. At a particular denoising timestep, the model produces: • Unconditional output:

• Conditional output (given text prompt

):

Using the classifier-free guidance formula

compute the guided prediction

.

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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 doubles

.

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It increases 10-fold.

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It increases by a factor of

.

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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-probabilities: • Beam A:

(ending in token 'hello') • Beam B:

(ending in token 'hi') At step

, the model outputs the following top log-probabilities: • Given 'hello': 'world' (

), 'there' (

) • Given 'hi': 'there' (

), 'you' (

) Using standard beam search with additive log-probabilities, which two paths form the new beams after step

?

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('hello', 'world') and ('hello', 'there')

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('hello', 'world') and ('hi', 'there')

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('hi', 'there') and ('hi', 'you')

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('hello', 'there') and ('hi', 'you')

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Question 6 MCQ · 3.0 marks

DPO eliminates the explicit reward model by defining an implicit reward

based on the optimal policy

and reference policy

. Given: •

•

•

(KL penalty coefficient) Calculate the implicit reward value assigned to this response.

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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 + previously generated tokens). • The **Action Space** is the vocabulary

. Suppose the vocabulary size is

and we generate a sequence of length

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

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Output dim: 25; Trajectory space:

.

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Output dim: 32000; Trajectory space:

.

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Output dim: 32000; Trajectory space:

.

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Output dim: 1; 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?

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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 baseline term?

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Question 10 MCQ · 3.0 marks

DDIMs are faster than DDPM mainly because:

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They remove the neural network entirely.

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The sampling process is deterministic when

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They require more noise steps.

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They require additional forward passes.

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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?

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The state

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

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The reward

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

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The trajectory

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

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The policy

corresponds to the LM’s conditional distribution

.

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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:

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Sinusoidal positional encodings are learned parameters updated during training.

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

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The sinusoidal frequencies grow geometrically with the dimension index

.

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The encoding scheme ensures that relative positions can be represented through linear functions of the embeddings.

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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

.

Calculate the raw attention scores (before Softmax) scaled by the factor

. What is the value of the element at position

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

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Question 14 NAT · 3.0 marks

In a decoder-only transformer (GPT-style), we compute attention scores

. Before applying softmax, we add a causal mask

. Let the raw scores for a sequence

be:

The mask

sets future positions to

. Calculate the probability

after applying Softmax to the second row.

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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 preference model. For a given prompt

, you have a preferred completion

and a non-preferred completion

. The RM predicts the following scalar rewards:

Compute the negative log-likelihood loss for this comparison:

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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. Compute the discretized state transition parameter

. Formula:

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