Theoretical Foundations of Machine Learning Questions

Covers the mathematical and theoretical building blocks that underpin modern machine learning and artificial intelligence. Key areas include probability theory and Bayesian reasoning such as conditional probability, Bayes theorem, expectation and variance, and probabilistic inference; linear algebra and matrix analysis including eigenvalues, eigenvectors, matrix decompositions, matrix norms, rank, and geometric intuitions; optimization and calculus topics such as gradient descent, stochastic optimization, convexity, Lagrange multipliers, partial derivatives, the chain rule, and properties of optimization landscapes; and related theoretical themes such as information theory and approximation concepts. Candidates should be able to connect these foundations to algorithm behavior, model expressivity, convergence properties, and practical design decisions.

MediumTechnical

77 practiced

Formulate the Karush–Kuhn–Tucker (KKT) conditions for an optimization problem with inequality constraints g_i(x) ≤ 0. Explain stationarity, primal and dual feasibility, and complementary slackness with a simple illustrative problem and derivation of multipliers.

MediumTechnical

72 practiced

Derive the maximum likelihood estimators for the mean μ and variance σ^2 of a Gaussian distribution given i.i.d. samples. Then derive the MAP estimator using a conjugate Normal-Inverse-Gamma prior and discuss how the prior affects estimates, especially for small n.

MediumTechnical

71 practiced

Assume f is μ-strongly convex and L-smooth. Derive the convergence rate for gradient descent with step size η=1/L and show that f(x_t)−f(x*) ≤ (1 − μ/L)^t (f(x_0)−f(x*)). Outline the main inequalities used in the derivation.

MediumTechnical

95 practiced

Define VC-dimension formally. Compute (and justify) the VC-dimension of linear classifiers in R^d (with and without bias term). Discuss how VC-dimension informs sample complexity and how margin assumptions can improve effective capacity bounds.

HardTechnical

68 practiced

Derive an upper bound on the empirical Rademacher complexity for the class of linear predictors H = {x ↦ w·x : ||w||_2 ≤ B} over a dataset {x_i}_{i=1}^n. Express the bound in terms of B and the empirical covariance or norms of x_i, and explain how this leads to a generalization bound.

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