Mathematical Foundations of Generative AI · End Term · May 2026 · FN

Mathematical Foundations of Generative AI · End Term · May 2026 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 NAT · 2.0 marks

An agent receives rewards

over an episode (

) with the discount factor equal to 0.5. Compute the full discounted return

for the trajectory

. Enter your answer correct to three decimal places.

A published solution is not available for this question yet.

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Question 3 MSQ · 3.0 marks

6406537372413.

Both TRPO and PPO aim to prevent the updated policy from deviating too far from

.

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PPO always produces a smaller policy update than TRPO for the same

and

.

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When

is within

, the PPO clip is inactive and

reduces to the unclipped surrogate

.

6406537372416.

Computing the PPO objective requires solving a constrained optimisation subproblem at each step, making it more expensive than TRPO.

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Question 4 MSQ · 3.0 marks

In Proximal Policy Optimization (PPO), the clipped surrogate objective for a single state-action pair is:

Which of the following statements are correct?

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When

and

, the clipped objective prevents the gradient from further increasing

, effectively imposing a trust region.

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When

and

, the clipped objective prevents the gradient from further decreasing

.

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The clipping mechanism is equivalent to adding an explicit KL-divergence penalty

to the objective.

6406537372420.

If

, then

regardless of the sign of

.

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Question 5 MSQ · 3.0 marks

In multi-head self-attention with

heads, the queries, keys, and values for each head have dimension

where

is the model dimension. Which of the following correctly explain why this choice makes the implementation efficient?

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Choosing

ensures that every head learns the same attention pattern while reducing the number of parameters.

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The total parameters across

query projections

equal

, the same as a single-head projection

. The same holds for key and value projections.

6406537372423.

Since

, all

heads can be computed as a single batched matrix multiplication, enabling parallel execution across heads on a GPU.

6406537372424.

Using

, the total computational cost across all

heads is reduced by a factor of

compared to single-head attention with

.

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Question 6 NAT · 3.0 marks

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Question 9 MSQ · 2.0 marks

Which of the following statements about Denoising Diffusion Implicit Models (DDIMs) are correct? Select all that apply.

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The forward process is non-Markovian, unlike the Markovian forward process in DDPM.

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DDIM achieves a tighter ELBO than DDPM.

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DDIM requires a separate training procedure from DDPM.

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When

for all

, the DDIM sampling process is deterministic.

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Question 10 MSQ · 2.0 marks

Consider two DDPMs at the same timestep

. For Model A,

, while for Model B,

. Recall that

Which of the following statements are correct?

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The sample

of Model A is more strongly influenced by the original sample

than that of Model B.

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The sample

of Model B is closer to

than that of Model A.

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As

decreases, the contribution of

to

increases.

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Model B produces a noisier representation at timestep

than Model A.

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Question 11 MSQ · 2.0 marks

Which of the following statements correctly describe the forward diffusion process in a DDPM?

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The forward process gradually corrupts

by adding Gaussian noise according to a predefined noise schedule.

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The parameters of the forward process are learned using the training data.

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The forward process requires a neural network to predict the noise added at every timestep.

6406537372450.

Given

,

can be sampled directly without simulating all intermediate timesteps.

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Question 12 MSQ · 2.0 marks

You are training a GAN with generator

,

, to generate handwritten digit images spanning all 10 digit classes. After training, you suspect that

has not converged to

and that the generator is suffering from mode collapse. Which of the following could be indicators of this problem? Select all that apply.

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Samples

for many different

all resemble the digit "3".

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The discriminator loss

drops close to zero while the generator loss

remains persistently high.

6406537372457.

The discriminator loss remains persistently high while the generator loss drops to zero.

6406537372458.

The generator loss and discriminator loss both decrease monotonically to zero, indicating the adversarial objective

has reached a global saddle point.

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Question 13 MSQ · 5.0 marks

6406537372434.

: both actions are equally good relative to the policy average in state B, so no policy gradient update will change

.

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In state C, action

has a higher advantage than action

; so a policy gradient update increases

and decreases

.

6406537372436.

In state C, action

has a higher advantage than action

; so a policy gradient update decreases

and increases

.

6406537372437.

If a sampled trajectory from state

yields the discounted return

, then the estimated advantage

; so the policy gradient update will decrease

.

6406537372438.

If a sampled trajectory from state

yields the discounted return

, then the estimated advantage

; so the policy gradient update will increase

.

6406537372439.

The expected return obtained by taking

in state

must be negative.

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Question 14 NAT · 3.0 marks

A transformer processes a sequence of

tokens with model dimension

using

attention heads, each with

. The input embedding matrix is:

Head 1 uses projections:

Head 2 uses projections:

Use

for scaling. All biases are zero. Based on the above data, answer the given subquestions.

Compute

,

,

for Head 1 and the scaled score matrix , and enter the value of the

entry of

. Enter the answer correct to two decimal places.

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Question 15 NAT · 3.0 marks

A transformer processes a sequence of

tokens with model dimension

using

attention heads, each with

. The input embedding matrix is:

Head 1 uses projections:

Head 2 uses projections:

Use

for scaling. All biases are zero. Based on the above data, answer the given subquestions.

Causal masking is applied on

. Compute the attention weights for the third query and enter the value of the largest weight among the three, correct to two decimal places.

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Question 16 NAT · 3.0 marks

A transformer processes a sequence of

tokens with model dimension

using

attention heads, each with

. The input embedding matrix is:

Head 1 uses projections:

Head 2 uses projections:

Use

for scaling. All biases are zero. Based on the above data, answer the given subquestions.

Using causal masking, compute the attention outputs for the third token from both heads and concatenate them as

Enter the fourth entry of the third row of

. Enter the answer correct to two decimal places.

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Question 17 NAT · 3.0 marks

In classifier-free guidance, a single diffusion model is jointly trained for both conditional and unconditional generation. During training, the conditioning information

is randomly replaced with a null token

with probability

. During sampling at timestep

, the model produces two noise predictions: 1. Conditional prediction:

(model receives conditioning

) 2. Unconditional prediction:

(model receives null token) The classifier-free guided noise prediction is defined as :

where

is the guidance scale. This follows from the guided score formulation:

The predicted clean data point is recovered via . Following are the values given:

Based on the above data, answer the given subquestions.

Compute the predicted clean data point

. Enter the answer correct to one decimal place.

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Question 18 MCQ · 2.0 marks

In classifier-free guidance, a single diffusion model is jointly trained for both conditional and unconditional generation. During training, the conditioning information

is randomly replaced with a null token

with probability

. During sampling at timestep

, the model produces two noise predictions: 1. Conditional prediction:

(model receives conditioning

) 2. Unconditional prediction:

(model receives null token) The classifier-free guided noise prediction is defined as :

where

is the guidance scale. This follows from the guided score formulation:

The predicted clean data point is recovered via . Following are the values given:

Based on the above data, answer the given subquestions.

If the guidance scale is set to

, which of the following correctly describes the effect on

and

?

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reduces to the conditional prediction

alone, giving

.

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reduces to the unconditional prediction

alone, giving

.

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reduces to the conditional prediction

alone, giving

.

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remains unchanged from

case because the guidance scale only affects the output projection layer of the U-Net.

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Question 19 NAT · 4.0 marks

Consider a 1-dimensional VAE with: 1. Encoder:

,

,

2. Decoder:

, with fixed decoder variance

Use the reparameterization trick to express the latent sample

. The reconstruction loss

for a single sample is given by

. Following are the values given:

, input

, sampled noise

. Compute

. Enter the answer correct to three decimal places.

A published solution is not available for this question yet.

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