Katkat
Back

Explain linear regression and Transformer fundamentals

ImcData Scientist · Onsite
MediumData ScientistOnsiteMachine Learning

Last updated

This question tests your ability to explain the theoretical foundations of linear regression, including its probabilistic interpretation, and your technical depth regarding the mechanics of the Transformer architecture.

Part 1: Linear Regression

Suppose you are training a linear regression model using Ordinary Least Squares (OLS) on a dataset with $n$ observations and $p$ features.

  1. Explain the probabilistic assumptions required for OLS to be the Best Linear Unbiased Estimator (BLUE) under the Gauss-Markov theorem.
  2. If the features exhibit high multicollinearity, how does this affect the variance of the coefficient estimates, and why does this complicate the interpretability of individual feature impacts?

Part 2: Transformer Fundamentals

Transformers have become the standard for sequence modeling due to their self-attention mechanism.

  1. Describe the mathematical process of Scaled Dot-Product Attention. Why is the scaling factor of 1/sqrt(d_k) necessary for training stability?
  2. Transformers are inherently permutation-invariant. Explain why positional encodings are necessary and discuss the trade-offs between using additive sinusoidal encodings versus learned positional embeddings in terms of sequence length generalization.
  3. Analyze the computational complexity of the self-attention mechanism in terms of sequence length $L$ and embedding dimension $d$. If we were to scale this model to handle sequences of 100,000 tokens, what are the primary architectural bottlenecks?

Hints

Hint 1

Recall the assumptions regarding the error term distribution and the properties of the design matrix X.

Hint 2

Consider the gradients during backpropagation when the dot product of large-magnitude vectors grows too large.

Hint 3

Think about the quadratic relationship between the attention matrix size and the sequence length during the computation of the QK^T product.

Follow-up questions

  • How would you implement L2 regularization (Ridge) to mitigate the issues caused by multicollinearity in the linear regression model?
  • If you were to replace the self-attention mechanism with a linear complexity variant (e.g., Sparse Attention or State Space Models), what potential trade-offs would you face regarding performance on downstream tasks?
  • How does the concept of multi-head attention allow the model to capture different types of dependencies, and how does this affect the model's total parameter count?