I started with gradient descent because it felt more natural to code up, then circled back to the normal equation.
Start by clarifying the problem and assumptions (e.g., whether to include an intercept, data shape). Then implement the LinearRegression class with fit and predict methods, using the normal equation for closed-form and gradient descent for iterative optimization. Discuss trade-offs between the two methods and validate with a simple example.
Pro tip: Mention numerical stability: for the normal equation, use np.linalg.solve instead of explicitly computing the inverse, and consider adding a small regularization term if X^T X is ill-conditioned. For gradient descent, feature scaling can dramatically improve convergence.
Ask about input data format, whether to include an intercept term, and any constraints (e.g., time complexity, memory). Confirm that both methods should be implemented in the same class with a flag to choose the solver.
Use the normal equation: theta = (X^T X)^(-1) X^T y. Handle the intercept by adding a column of ones to X. Use np.linalg.solve for numerical stability.
Initialize weights (zeros or small random), compute gradients of MSE loss, and update weights iteratively. Include configurable learning rate and number of iterations. Optionally add convergence check.
Given new X, add intercept column if needed, then return X @ theta. Ensure consistent preprocessing between fit and predict.
Compare computational complexity: normal equation O(n^3) due to matrix inversion, gradient descent O(k n d) per iteration. Mention when to use each (small vs large datasets, feature scaling). Validate with a simple synthetic dataset.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by defining the normal equation and gradient descent, then compare their computational and numerical properties. Discuss when the normal equation is preferable (small to medium datasets, low dimensionality, well-conditioned matrices) and when gradient descent is better (large datasets, high dimensionality, ill-conditioned matrices). Highlight tradeoffs in terms of time complexity, memory, and numerical stability.
Pro tip: Mention that while the normal equation is exact and avoids hyperparameter tuning, its O(n^3) inversion cost and O(n^2) memory make it impractical for high-dimensional data; gradient descent scales better but requires careful tuning and may converge to approximate solutions.
Briefly explain the closed-form normal equation (θ = (XᵀX)⁻¹Xᵀy) and gradient descent as an iterative optimization algorithm.
State that the normal equation has O(n^3) time complexity due to matrix inversion and O(n^2) memory, while gradient descent is O(kn^2) per iteration (k iterations) and can handle larger datasets with mini-batch.
Explain that the normal equation can be unstable if XᵀX is ill-conditioned (near singular), leading to large errors; gradient descent is more robust but may converge slowly or get stuck.
For small n (features) and moderate m (samples), normal equation is fast and exact; for large m or n, gradient descent (especially stochastic) is preferred due to scalability.
Conclude that the choice depends on the problem: use normal equation for simplicity and exactness when n is small and XᵀX is well-conditioned; otherwise use gradient descent with regularization or iterative solvers like conjugate gradient.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by explaining the mathematical formulation of L2 regularization and its effect on the loss function, then describe how to modify the gradient computation and update rule. Finally, discuss implementation details like scaling, hyperparameter tuning, and trade-offs in a production setting.
Pro tip: Mention that L2 regularization is equivalent to adding a Gaussian prior on the weights in a Bayesian framework, and that it can be implemented efficiently by augmenting the design matrix—this shows depth and practical awareness.
Write the modified loss function: J(w) = MSE(w) + λ * ||w||^2 (excluding the bias term). Explain that λ controls the strength of regularization.
Compute the gradient: ∇J(w) = (2/n) * X^T (Xw - y) + 2λw. Show that the update becomes w := w - α * (∇MSE + 2λw), which shrinks weights toward zero.
Describe how to add the regularization term in code, e.g., by adding λ * np.sum(w**2) to the loss and 2λw to the gradient. Mention that the bias term is typically not regularized.
Emphasize that features should be standardized so that regularization penalizes all weights equally. Discuss using cross-validation to tune λ.
Compare L2 with L1 (Lasso) and Elastic Net, and explain when L2 is preferred (e.g., when all features are useful and we want to avoid overfitting).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Covered MSE and R-squared, talked about train vs validation error gaps.
Start by defining the metrics used to evaluate linear regression fit, such as R-squared, adjusted R-squared, RMSE, and residual plots. Then explain how to diagnose underfitting and overfitting by comparing training and validation performance, and discuss techniques to address each issue. Finally, tie your answer to Uber's scale and product context by mentioning the importance of generalization and business impact.
Pro tip: Emphasize that no single metric tells the whole story—combine quantitative metrics with residual analysis and domain knowledge. Also, mention that at Uber's scale, even small improvements in model fit can have significant business impact, so it's crucial to balance bias-variance trade-off with interpretability and operational constraints.
List common metrics for linear regression: R-squared, adjusted R-squared, RMSE, MAE, and residual plots. Explain what each indicates about fit quality.
Describe signs of underfitting: high bias, poor performance on both training and validation sets, low R-squared, and systematic patterns in residuals. Mention causes like overly simple models or insufficient features.
Describe signs of overfitting: high variance, excellent training performance but poor validation performance, large gap between training and validation errors, and complex models with many features relative to data points.
Explain how to use train/validation/test splits, cross-validation (e.g., k-fold), and learning curves to diagnose underfitting vs. overfitting.
Discuss remedies: for underfitting, add features, use polynomial terms, or more complex models; for overfitting, use regularization (L1/L2), reduce features, gather more data, or simplify the model. Emphasize iterative improvement.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.