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

Which of the following statements about the KL divergence is true?

Consider a Generative Adversarial Network (GAN) trained to generate realistic images of cars. Which of the following statements are true?

Consider a GAN where the training dataset contains three times as many car images as bike images. Suppose the generator has learned to produce only high-quality car images that are indistinguishable from real ones. The discriminator is trained to output the probability that an input image is real. What is the optimal output of the discriminator when given a car image? Enter the answer correct to two decimal places.

A Gaussian Mixture Model has two components,

and

, with prior probabilities

. The components are 1D Gaussians with parameters

and

. For a data point

, calculate the responsibility (posterior probability) of component

for this point, i.e.,

. Provide the answer to three decimal places.

Consider a VAE where we gradually increase the weight of the KL divergence term in the ELBO loss function (i.e., increase the

parameter in

-VAE where the loss is

). Assume a standard normal prior,

. As

gradually increases from 0, what happens to the learned latent representations?

For a fixed input size and embedding dimension

, consider standard multi-head attention where the embedding dimension is evenly split across heads. Select the correct statements from the following:

In the context of the Transformer model's encoder-decoder architecture, which of the following statements are accurate?

A sequence of length

is represented by

where each row of

represents a token. The attention module uses

Thus

Each row of

and

correspond to the representation of a single input token. Use

and scaling by

. Based on the above data, answer the given subquestions.

Let the scaled scores for query

be defined as . Compute the

norm of this vector. Enter the answer correct to two decimal places.

A sequence of length

is represented by

where each row of

represents a token. The attention module uses

Thus

Each row of

and

correspond to the representation of a single input token. Use

and scaling by

. Based on the above data, answer the given subquestions.

Compute the attention output vector for the second token.

A decoder has sequence of length

and pre-mask scaled score matrix

Causal masking is applied to the scores exactly as in autoregressive transformers. The masking uses

for the mask matrix

.Use

Based on the above data, answer the given subquestions.

Which of the following is the masked score vector for the third query?

A decoder has sequence of length

and pre-mask scaled score matrix

Causal masking is applied to the scores exactly as in autoregressive transformers. The masking uses

for the mask matrix

.Use

Based on the above data, answer the given subquestions.

For token 4, compute the attention probability assigned jointly to positions 1 and 2. Use the masked attention weights. Enter the answer correct to three decimal places.

Based on the above data, answer the given subquestions.

Find the mean and variance of

.

Based on the above data, answer the given subquestions.

Compute the second coordinate of the final layer-normalized output. Enter the answer correct to three decimal places.

A language model is treated as a policy that generates a response of length 4. The rewards are

with discount factor

. Suppose the state-value estimates are

The sampled log-policy gradient scalars

are given as

Based on the above data, answer the given subquestions.

Compute the full discounted return

Enter the answer correct to three decimal places.

A language model is treated as a policy that generates a response of length 4. The rewards are

with discount factor

. Suppose the state-value estimates are

The sampled log-policy gradient scalars

are given as

Based on the above data, answer the given subquestions.

Compute the reward-to-go

. Enter the answer correct to two decimal places.

A language model is treated as a policy that generates a response of length 4. The rewards are

with discount factor

. Suppose the state-value estimates are

The sampled log-policy gradient scalars

are given as

Based on the above data, answer the given subquestions.

Compute the gradient estimate

where

is the advantage estimate given by

Enter the answer correct to three decimal places.

Based on the above data, answer the given subquestions.

Compute the importance sampling ratio

for time step

. Enter the answer correct to one decimal place.

Based on the above data, answer the given subquestions.

Compute the clipped surrogate contribution at time 0. Enter the answer correct to one decimal place.

Based on the above data, answer the given subquestions.

Compute the clipped surrogate contribution at time 1. Enter the answer correct to two decimal places.

Based on the above data, answer the given subquestions.

Compute the clipped surrogate contribution at time 2. Enter the answer correct to one decimal place.

Based on the above data, answer the given subquestions.

Compute the total PPO clipped objective contribution over the three time steps. Enter the answer correct to two decimal places.