AI: Sequence to Sequence

Derive the backpropagation algorithm for a multi-layer perceptron by applying the chain rule recursively on a computational graph.

Analyze the vanishing and exploding gradient problems by examining the product of Jacobians during backpropagation.

Derive the Xavier and He weight initialization schemes from the principle of preserving activation variance during forward and backward passes.

Derive the update rules for SGD with Momentum, RMSprop, and Adam, and analyze how each component addresses the challenges of deep learning optimization.

Derive the Bayesian interpretation of L1 (Laplace prior) and L2 (Gaussian prior) regularization as Maximum A Posteriori (MAP) estimation.

Explain dropout as a form of ensemble approximation and derive the necessary scaling of activations at inference time.

Derive the forward and backward pass equations for a Batch Normalization layer and explain its role in mitigating internal covariate shift.

Derive the 2D convolution operation and explain how parameter sharing and sparse connectivity lead to translation equivariance.

Calculate the output dimensions and receptive field of a convolutional network based on kernel size, stride, padding, and dilation.

Derive the backpropagation equations for a convolutional layer with respect to its inputs and filters.

Analyze the architectural evolution from LeNet to VGG, focusing on the impact of depth and small-receptive-field filters.

Derive how residual connections in ResNet architectures combat the degradation problem and enable the training of much deeper networks.

Explain the core architectural components of the Vision Transformer (ViT), including patch embedding and the application of self-attention to image patches.

Formulate the forward pass of a vanilla Recurrent Neural Network (RNN) and derive the Backpropagation Through Time (BPTT) algorithm.

Analyze the mathematical reasons why vanilla RNNs are susceptible to vanishing and exploding gradients over long sequences.

Derive the forward pass equations for a Long Short-Term Memory (LSTM) cell, explaining how the input, forget, and output gates mitigate long-range dependency issues.

Derive the Gated Recurrent Unit (GRU) equations and compare its architecture and complexity to the LSTM cell.

Formulate the encoder-decoder (seq2seq) architecture for sequence transduction tasks and explain its fixed-context bottleneck limitation.

Explain the mechanism of teacher forcing during training and analyze the resulting exposure bias problem during inference.

Motivate attention as a solution to the fixed-bottleneck problem in seq2seq models, allowing for dynamic context.

Derive scaled dot-product attention and justify the scaling factor by analyzing the variance of dot products.

Formulate multi-head self-attention and explain how it allows the model to jointly attend to information from different representation subspaces.

Describe the full architecture of a Transformer encoder block, explaining the roles of self-attention, feed-forward networks, residual connections, and layer normalization.

Describe the Transformer decoder block, detailing the purpose of masked self-attention and encoder-decoder cross-attention.

Derive sinusoidal positional encodings and explain their property of representing relative positions via linear transformations.

Derive rotary positional encodings (RoPE) and explain their geometric interpretation of encoding absolute positions via rotation matrices to preserve relative positions.

Analyze the O(N^2) computational complexity of self-attention and explain the core principle of I/O-aware optimizations like FlashAttention.

Explain the core principle of self-supervised learning, where a pretext task is used to learn representations from unlabeled data.

Derive the word2vec (skip-gram) objective with negative sampling as an efficient approximation to the full softmax.

Contrast the predictive learning of word2vec with the matrix factorization approach of GloVe, which is based on global co-occurrence statistics.

Derive the Masked Language Modeling (MLM) objective used in BERT and explain how it enables the learning of deep bidirectional representations.

Contrast static embeddings (word2vec, GloVe) with contextual embeddings (BERT, ELMo) and their implications for ambiguity resolution.

Derive the InfoNCE loss function and explain its role in contrastive learning frameworks like SimCLR.

Distinguish between discriminative and generative modeling paradigms and survey the major families of generative models.

Derive the Evidence Lower Bound (ELBO) for Variational Autoencoders (VAEs) and interpret its reconstruction and regularization terms.

Explain the reparameterization trick and derive how it enables gradient-based optimization of the stochastic VAE objective.

Analyze the posterior collapse phenomenon in VAEs and discuss common mitigation strategies like KL annealing.

Derive the minimax objective for Generative Adversarial Networks (GANs) and prove its equivalence to minimizing the Jensen-Shannon divergence at optimality.

Analyze common GAN training instabilities (mode collapse, vanishing gradients) and derive how the Wasserstein GAN (WGAN) objective provides more stable gradients.

Derive the autoregressive likelihood factorization and explain its application in models like PixelCNN via masked convolutions.

Derive the forward diffusion process (noising) and the closed-form expression for sampling x_t at any timestep t given x_0.

Derive the reverse process (denoising) as a variational inference problem and formulate the corresponding variational lower bound.

Show how the VLB can be simplified to a noise-prediction objective (MSE between predicted and true noise), as used in DDPMs.

Explain the connection between denoising diffusion models and score-based generative models via score matching.

Derive the DDIM update rule as a non-Markovian, deterministic generative process that allows for accelerated sampling.

Derive classifier-free guidance and analyze its role in trading off sample quality and diversity.

Explain the architecture of Latent Diffusion Models (e.g., Stable Diffusion) and the role of the pretrained VAE in enabling diffusion in a compressed latent space.

Compare the architectural design and typical use cases of encoder-only (BERT-style), decoder-only (GPT-style), and encoder-decoder (T5-style) Transformers.

Explain common subword tokenization algorithms (e.g., Byte-Pair Encoding) and analyze their impact on model performance and handling of rare words.

State and interpret the Chinchilla scaling laws for training compute-optimal large language models.

Describe emergent abilities and in-context learning in LLMs as phenomena observed at scale.

Explain Mixture-of-Experts (MoE) layers and derive the routing and load-balancing losses used in their training.

Explain how sliding window attention and grouped-query attention improve inference efficiency for long sequences.

Derive Position Interpolation (PI) and explain how it enables fine-tuning models to handle longer context lengths than they were trained on.

Frame language model generation as a sequential decision-making problem, defining states, actions, and policies within the Markov Decision Process (MDP) framework.

Derive the vanilla policy gradient theorem (REINFORCE algorithm) from first principles.

Explain supervised fine-tuning (SFT) and instruction tuning for adapting a pretrained LLM to specific downstream tasks or conversational styles.

Derive the reward modeling objective from pairwise human preference data using the Bradley-Terry model.

Derive the clipped surrogate objective of Proximal Policy Optimization (PPO) and explain how it enables stable policy updates.

Formulate Reinforcement Learning from Human Feedback (RLHF), applying PPO with a KL-divergence penalty against a reference policy to align LLMs.

Derive the Direct Preference Optimization (DPO) objective and explain its relationship to the RLHF objective, offering a more direct and stable alignment method.

Derive Low-Rank Adaptation (LoRA) and explain its mechanism for parameter-efficient fine-tuning by updating low-rank decomposition matrices of the original weights.

Derive the contrastive learning objective of CLIP and explain how it enables zero-shot image classification by aligning image and text representations in a shared space.

Explain the end-to-end architecture of a latent text-to-image diffusion model like Stable Diffusion, detailing the roles of the text encoder, U-Net, and VAE decoder.

Describe the LLaVA architecture, explaining how a vision encoder's outputs are projected and fed into a pretrained LLM for visual instruction following.

Explain the Flamingo architecture, focusing on how its gated cross-attention layers are inserted into a frozen LLM to condition generation on visual inputs.

Explain the two-stage training strategy and Q-Former architecture of BLIP-2 for bridging the modality gap between vision encoders and LLMs.

Describe the core components of a modern speech-to-text model like Whisper, including its encoder-decoder structure and multitask training.

Explain how spatiotemporal tokenization (e.g., patching) can be used to adapt Transformer architectures for video understanding, as in Sora-style models.

Analyze Retrieval-Augmented Generation (RAG) as a framework for combining the parametric knowledge of an LLM with non-parametric external knowledge.

Describe the LLM-as-agent paradigm, including the use of tools, function calling, and reasoning frameworks like ReAct (Reason + Act).

Compare chain-of-thought prompting with self-consistency decoding as methods to improve multi-step reasoning.

Explain the core motivation for State Space Models (SSMs) as an alternative to Transformers, focusing on their linear-time complexity and RNN-like properties.

Derive the selective scan mechanism (SSM) in Mamba, including the continuous-to-discrete state-space transformation and the parallel scan algorithm.

Analyze the conceptual lineage from the perceptron to modern generative AI, synthesizing the key architectural and objective function innovations into a coherent narrative.