Derive Linear Regression Solution
Company: Meta
Role: Machine Learning Engineer
Category: Machine Learning
Difficulty: medium
Interview Round: Technical Screen
Given training pairs `(x_i, y_i)` for a one-dimensional linear regression model without bias, `y_hat = w * x`, derive the mean squared error objective, solve for the optimal closed-form value of `w`, and implement a function that estimates `w` from data.
Also explain what happens in the degenerate case where all `x_i` values are zero, and compare the closed-form solution with gradient descent.
Quick Answer: This question evaluates understanding of one-dimensional linear regression estimation, the statistical derivation of the mean squared error objective and closed-form parameter estimation, as well as recognition of degenerate data cases and comparisons between analytical and iterative optimization methods.