Choose linear regression or decision tree appropriately

You are given 100,000 independent and identically distributed rows with features $x_1$ (0–100), $x_2$, $x_3$, and $y$. The true data-generating process is unknown to you, but it is piecewise linear with a hinge at $x_1 = 50$ and an interaction: $y \approx 3x_1 + 20 \cdot I[x_1 > 50] + 2 \cdot x_2 x_3 + \varepsilon$, with heteroskedastic noise $\mathrm{Var}(\varepsilon \mid x_1) = 0.01(1 + x_1)$. Outline an analysis to decide between linear regression and a decision tree. Specify: (1) the feature engineering and linearity tests you would run (for example, a spline basis for $x_1$ and an $x_2:x_3$ interaction), and how you would check residual diagnostics for heteroskedasticity; (2) a fair comparison protocol (CV split, identical preprocessing) and metrics; (3) how you would enforce monotonicity or interaction constraints in a tree-based model to reflect domain knowledge; (4) which model you expect to generalize better here and why, including bias–variance reasoning and how you would quantify it with learning curves.

Overview: This question evaluates model selection and diagnostic skills in supervised learning, specifically assessing feature engineering, interaction detection, handling heteroskedastic residuals, incorporation of monotonicity or interaction constraints in tree-based models, and fair cross-validation-based comparison between linear and tree approaches.