da5002_2025T3_ET_FN.pdf

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

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

Question 2 MCQ · 3.0 marks

In the context of Denoising Diffusion Probabilistic Models (DDPM), the trained noise predictor [[IMAGE:ef8a71bf9c941dbd_2_2]] approximates the score function of the data distribution via [[IMAGE:ef8a71bf9c941dbd_2_3]] Consider a specific timestep [[IMAGE:ef8a71bf9c941dbd_2_4]] where the cumulative noise variance schedule is given by [[IMAGE:ef8a71bf9c941dbd_2_5]] . At this timestep, for a specific input [[IMAGE:ef8a71bf9c941dbd_2_6]] , the model predicts a noise vector: [[IMAGE:ef8a71bf9c941dbd_2_7]] Calculate the estimated score vector, [[IMAGE:ef8a71bf9c941dbd_2_8]] .

[[IMAGE:ef8a71bf9c941dbd_2_9]]
[[IMAGE:ef8a71bf9c941dbd_2_10]]
[[IMAGE:ef8a71bf9c941dbd_2_11]]
[[IMAGE:ef8a71bf9c941dbd_2_12]]

A published solution is not available for this question yet.

Question 3 MCQ · 3.0 marks

You are performing inference using a Guided Diffusion model with Classifier-Free Guidance (CFG) with a guidance scale [[IMAGE:ef8a71bf9c941dbd_3_13]] . At a specific step, the model outputs: [[IMAGE:ef8a71bf9c941dbd_3_14]] • Unconditional prediction: [[IMAGE:ef8a71bf9c941dbd_3_16]] • Conditional prediction (given class [[IMAGE:ef8a71bf9c941dbd_3_15]] ): Using the standard CFG formula, calculate the final guided noise vector [[IMAGE:ef8a71bf9c941dbd_3_17]] .

[[IMAGE:ef8a71bf9c941dbd_3_18]]
[[IMAGE:ef8a71bf9c941dbd_3_19]]
[[IMAGE:ef8a71bf9c941dbd_3_20]]
[[IMAGE:ef8a71bf9c941dbd_3_21]]

A published solution is not available for this question yet.

Question 4 MCQ · 3.0 marks

In a Deep Transformer with Residual connections, the output of layer [[IMAGE:ef8a71bf9c941dbd_4_22]] is [[IMAGE:ef8a71bf9c941dbd_4_23]] . Assume that at initialization, the function [[IMAGE:ef8a71bf9c941dbd_4_24]] (Attention + FFN) outputs values with variance roughly [[IMAGE:ef8a71bf9c941dbd_4_25]] . After 10 layers, assuming independence, approximately how much has the signal variance increased relative to the input variance [[IMAGE:ef8a71bf9c941dbd_4_26]] ? (Approximation: [[IMAGE:ef8a71bf9c941dbd_4_27]] ).

It remains the same [[IMAGE:ef8a71bf9c941dbd_4_28]] .
It increases by a factor of [[IMAGE:ef8a71bf9c941dbd_4_29]] .
It doubles [[IMAGE:ef8a71bf9c941dbd_4_30]] .
It increases 10-fold.

A published solution is not available for this question yet.

Question 5 MCQ · 3.0 marks

You are performing Beam Search with width [[IMAGE:ef8a71bf9c941dbd_4_31]] . At step [[IMAGE:ef8a71bf9c941dbd_4_32]] , you have two active beams with accumulated log-probs: Beam A: [[IMAGE:ef8a71bf9c941dbd_4_33]] (ending in token 'cat') Beam B: [[IMAGE:ef8a71bf9c941dbd_4_34]] (ending in token 'dog') At step [[IMAGE:ef8a71bf9c941dbd_4_35]] , 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?

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

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Question 6 MCQ · 3.0 marks

[[IMAGE:ef8a71bf9c941dbd_5_36]]

[[IMAGE:ef8a71bf9c941dbd_5_37]]
[[IMAGE:ef8a71bf9c941dbd_5_38]]
[[IMAGE:ef8a71bf9c941dbd_5_39]]
Undefined

A published solution is not available for this question yet.

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 [[IMAGE:ef8a71bf9c941dbd_5_40]] . If the vocabulary size is [[IMAGE:ef8a71bf9c941dbd_5_41]] and we generate a sequence of length [[IMAGE:ef8a71bf9c941dbd_5_42]] . What is the dimension of the policy output distribution at a single timestep, and what is the dimensionality of the trajectory space?

Output dim: [[IMAGE:ef8a71bf9c941dbd_5_43]] ; Trajectory space: [[IMAGE:ef8a71bf9c941dbd_5_44]] .
Output dim: [[IMAGE:ef8a71bf9c941dbd_5_45]] ; Trajectory space: [[IMAGE:ef8a71bf9c941dbd_5_46]] .
Output dim: 1; Trajectory space: [[IMAGE:ef8a71bf9c941dbd_5_47]] .
Output dim: 50,000; Trajectory space: [[IMAGE:ef8a71bf9c941dbd_5_48]] .

A published solution is not available for this question yet.

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?

[[IMAGE:ef8a71bf9c941dbd_6_49]]
[[IMAGE:ef8a71bf9c941dbd_6_50]]
[[IMAGE:ef8a71bf9c941dbd_6_51]]
[[IMAGE:ef8a71bf9c941dbd_6_52]]

A published solution is not available for this question yet.

Question 9 MCQ · 3.0 marks

The Policy Gradient theorem states [[IMAGE:ef8a71bf9c941dbd_6_53]] . If we introduce a baseline [[IMAGE:ef8a71bf9c941dbd_6_54]] that depends only on state, the term becomes [[IMAGE:ef8a71bf9c941dbd_6_55]] . What is the expected value of the following baseline term? [[IMAGE:ef8a71bf9c941dbd_6_56]]

[[IMAGE:ef8a71bf9c941dbd_6_57]]
[[IMAGE:ef8a71bf9c941dbd_6_58]]
[[IMAGE:ef8a71bf9c941dbd_6_59]]
[[IMAGE:ef8a71bf9c941dbd_6_60]]

A published solution is not available for this question yet.

Question 10 MCQ · 3.0 marks

DDIMs are faster than DDPM mainly because:

They remove the neural network entirely.
They require more noise steps.
They require additional forward passes.
The sampling process is deterministic when [[IMAGE:ef8a71bf9c941dbd_7_61]] .

A published solution is not available for this question yet.

Question 11 MSQ · 4.0 marks

An auto-regressive language model predicts the next token [[IMAGE:ef8a71bf9c941dbd_7_62]] given the history [[IMAGE:ef8a71bf9c941dbd_7_63]] . In the RL interpretation used for alignment, which of the following correspondences between LM objects and RL objects are correct?

The state [[IMAGE:ef8a71bf9c941dbd_7_64]] corresponds to the current context: prompt plus all tokens generated so far.
The action [[IMAGE:ef8a71bf9c941dbd_7_65]] corresponds to selecting the next token from the vocabulary.
The trajectory [[IMAGE:ef8a71bf9c941dbd_7_66]] corresponds to the full sequence of (prompt, generated tokens) over time.
The reward [[IMAGE:ef8a71bf9c941dbd_7_67]] is necessarily the log-likelihood of the token under the pretraining objective.

A published solution is not available for this question yet.

Question 12 MSQ · 4.0 marks

The sinusoidal positional encoding (PE) for Transformers is defined as: [[IMAGE:ef8a71bf9c941dbd_8_68]] Based on the properties of sinusoidal positional encodings, identify all correct statements:

Positional encodings allow the model to incorporate order information that attention alone cannot provide.
The sinusoidal frequencies grow geometrically with the dimension index [[IMAGE:ef8a71bf9c941dbd_8_69]] .
Sinusoidal positional encodings are learned parameters updated during training.
Adding positional encodings to token embeddings guarantees each position has a unique representation.

A published solution is not available for this question yet.

Question 13 NAT · 3.0 marks

Consider a single-head self-attention mechanism with [[IMAGE:ef8a71bf9c941dbd_8_70]] . You are given the following Query ( [[IMAGE:ef8a71bf9c941dbd_8_71]] ) and Key ( [[IMAGE:ef8a71bf9c941dbd_8_72]] ) matrices for a sequence length of [[IMAGE:ef8a71bf9c941dbd_8_73]] : [[IMAGE:ef8a71bf9c941dbd_8_74]] Compute the raw attention scores (before Softmax) and scale them by the factor [[IMAGE:ef8a71bf9c941dbd_8_75]] . What is the value of the element at position [[IMAGE:ef8a71bf9c941dbd_8_76]] (i.e., row 1, column 2) of the scaled score matrix?

A published solution is not available for this question yet.

Question 14 NAT · 3.0 marks

You have a Transformer with model dimension [[IMAGE:ef8a71bf9c941dbd_9_77]] . You employ [[IMAGE:ef8a71bf9c941dbd_9_78]] parallel attention heads. The output of the heads is concatenated and projected by a matrix [[IMAGE:ef8a71bf9c941dbd_9_79]] . If the input batch size is [[IMAGE:ef8a71bf9c941dbd_9_80]] and sequence length is [[IMAGE:ef8a71bf9c941dbd_9_81]] , what is the total number of floating point elements in the final output tensor of the Multi-Head Attention block (before the residual connection)?

A published solution is not available for this question yet.

Question 15 NAT · 3.0 marks

In RLHF (Reinforcement Learning from Human Feedback), the reward used to update the language model [[IMAGE:ef8a71bf9c941dbd_9_82]] is defined as [[IMAGE:ef8a71bf9c941dbd_9_83]] Given: • Reward model score [[IMAGE:ef8a71bf9c941dbd_9_84]] • [[IMAGE:ef8a71bf9c941dbd_9_85]] • [[IMAGE:ef8a71bf9c941dbd_9_86]] • KL coefficient [[IMAGE:ef8a71bf9c941dbd_9_87]] Calculate the total reward. (Use [[IMAGE:ef8a71bf9c941dbd_9_88]] .)

A published solution is not available for this question yet.

Question 16 NAT · 3.0 marks

You are training a Reward Model (RM) using the Bradley-Terry model. For a specific prompt [[IMAGE:ef8a71bf9c941dbd_10_89]] , you have a winning completion [[IMAGE:ef8a71bf9c941dbd_10_90]] and a losing completion [[IMAGE:ef8a71bf9c941dbd_10_91]] . The RM currently predicts the scalar rewards as: [[IMAGE:ef8a71bf9c941dbd_10_92]] Calculate the negative log-likelihood loss for this pair: [[IMAGE:ef8a71bf9c941dbd_10_93]] .

A published solution is not available for this question yet.

Question 17 NAT · 3.0 marks

A continuous time State-Space Model (SSM) has a scalar state parameter [[IMAGE:ef8a71bf9c941dbd_10_94]] (representing decay). You discretize this system with a step size [[IMAGE:ef8a71bf9c941dbd_10_95]] using the Zero-Order Hold (ZOH) method (or approximation thereom). Calculate the discretized state transition parameter [[IMAGE:ef8a71bf9c941dbd_10_96]] . Formula: [[IMAGE:ef8a71bf9c941dbd_10_97]] .

A published solution is not available for this question yet.