Machine Learning and Statistical Modeling Discussion

This interview focuses on a comprehensive discussion of fundamental and advanced topics in machine learning and statistical modeling. Be prepared to explain concepts, compare techniques, derive relationships, and discuss practical implications across various scenarios.

Core Machine Learning Concepts

• Class Imbalance: Discuss techniques used at the data, loss, model, and evaluation levels when dealing with severe class imbalance in supervised learning problems.
• Loss Functions Comparison: Compare MAE, MSE, and Huber loss. Explain when each is preferable.
• Custom Loss Function Design: Describe how you would design a loss function that penalizes overestimation more than underestimation, or vice versa.
• Bias-Variance Trade-off: Explain the bias-variance trade-off and illustrate how it appears across different model classes.

Advanced Regression and Estimation

• Quantile Regression: Define quantile regression, explain its objective function, and discuss its advantages over mean regression in specific scenarios.
• Linear Regression with Measurement Noise: Analyze the effect on estimated coefficients in linear regression if the observed feature X_obs = X + U, where U is independent measurement noise.
• Maximum Likelihood Estimation (MLE): Provide examples where maximum likelihood estimation is not consistent.
• Covariance Matrix Estimation: Given x_i is distributed as N(0, I + beta v v^T) with ||v|| = 1, explain how to estimate v and determine the leading-order dependence of the estimation error on sample size n, dimension d, and signal strength beta.
• Two-Stage Regression: Explain how residualization or innovations can be used in a two-stage regression pipeline when predictors are strongly correlated.

Deep Learning and Sequence Models

• Generative Adversarial Networks (GANs): Explain the adversarial objective in GAN training and discuss common stability issues encountered.
• RNNs vs. Transformers: Compare Recurrent Neural Networks (RNNs) and Transformers for sequence modeling, highlighting their strengths and weaknesses.

Dimensionality Reduction and Linear Algebra

• Orthogonal Regression and PCA: Show why one-dimensional orthogonal regression is closely related to Principal Component Analysis (PCA).
• Power Method and Sparse PCA: Explain how the power method recovers the top eigenvector. Discuss why sparse PCA is significantly harder than ordinary PCA.

Statistical Inference and Probability

• Game Analysis (Luck vs. Skill): Given only win/loss outcomes among n players, describe how you would test whether the game is mostly luck versus skill.
• Minimax Inequality: Discuss whether it is always true that min over y of E_X[f(X, y)] is less than or equal to E_{X,Y}[f(X, Y)]. Explain what changes if X and Y are independent.