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Started with gradient descent which they accepted, but then they pushed for a closed-form solution.
Start by explaining the standard closed-form solution (normal equations) and its computational complexity, then discuss why it becomes impractical for large datasets. Transition to iterative methods like gradient descent and stochastic variants, emphasizing their scalability and trade-offs. Conclude with practical considerations for choosing an approach in a software engineering context.
Pro tip: Mention that for very large datasets, stochastic gradient descent (SGD) and its variants (e.g., Adam) are often preferred due to their ability to handle streaming data and avoid memory issues, but be prepared to discuss convergence and tuning challenges.
Describe how coefficients are estimated using the normal equations: β = (X^T X)^{-1} X^T y. Mention that this minimizes the sum of squared residuals.
Highlight that computing the inverse of X^T X is O(n^3) in the number of features, and forming X^T X requires O(n^2) memory, which is infeasible when n is large.
Explain gradient descent: iteratively update coefficients in the direction of the negative gradient of the loss function. Mention batch, mini-batch, and stochastic variants.
For large number of observations (n >> p), SGD or mini-batch gradient descent is efficient because each update uses a small subset of data, reducing memory and computation per iteration.
Discuss convergence criteria, learning rate tuning, regularization (e.g., L1/L2), and the use of libraries like scikit-learn or TensorFlow that implement these methods.
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