ML Algorithm Implementation and Numerical Considerations Questions

Practical implementation details and algorithmic concerns when building machine learning systems. Topics include implementing gradient descent and variants such as stochastic and mini batch gradient descent, numerical stability and precision issues, vectorized matrix operations, efficient use of linear algebra libraries, feature normalization and standardization, distance metrics, algorithmic complexity, sorting and ranking techniques, back propagation implementation details, convergence criteria, initialization strategies, and performance trade offs for memory and compute. Also covers debugging model training, avoiding numerical overflow or underflow, and engineering considerations for productionizing ML algorithms.

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Implement K-Means clustering in Python using numpy with vectorized distance computations. Your function should accept initial centroids, compute assignments, update centroids until convergence or max iterations, and include a convergence threshold. Discuss initialization (k-means++) and how to handle empty clusters numerically.

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Implement mini-batch gradient descent with classical momentum in Python using numpy. Function signature should accept X, y, batch_size, learning_rate, momentum (mu), num_epochs, and initial weights. Use vectorized batch gradients, update a velocity vector, and return final weights and training loss history. Discuss numerical issues if momentum becomes large (mu close to 1) and possible mitigations.

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A custom AdamW optimizer implementation diverges (NaNs or exploding loss) for certain learning rates on a small CNN. Describe unit tests and stress tests to reproduce and debug: test bias-correction math, zero gradients, sparse gradients, extreme LR, and compare against a reference implementation. List possible coding bugs (incorrect bias correction, wrong weight-decay timing, in-place ops) and how to detect them.

MediumTechnical

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Implement a numerically stable sigmoid and its derivative in Python/numpy to avoid overflow for inputs with large magnitude. Show the algebraic trick: compute sigmoid differently for x>=0 and x<0, and express derivative in terms of the sigmoid value to reuse computation and minimize operations.

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Explain ZCA whitening (zero-phase component analysis): what it does to features and provide high-level pseudocode to compute the ZCA whitening matrix given data X. Discuss numerical stability when inverting the covariance (adding eps to eigenvalues) and how to apply or update the transform in a streaming setting with running covariance estimates.

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