Why does the LASSO's L1 penalty produce sparse models by setting some coefficients exactly to zero?

The LASSO (Least Absolute Shrinkage and Selection Operator) uses an L1 penalty -- the sum of absolute values of coefficients -- rather than the L2 squared penalty of ridge. This seemingly small change has a profound geometric consequence: the L1 constraint region has corners at the axes, and the OLS solution is more likely to touch these corners, setting some coefficients exactly to zero.\n\nObjective Function:\n\nLASSO minimizes: sum of (y_i - X_i x beta)^2 + lambda x sum of |beta_j|\n\nGeometric Insight:\n\n`mermaid\ngraph LR\n subgraph \"Ridge (L2)\"\n A[\"Circular constraint
No corners on axes
All coefficients nonzero\"]\n end\n subgraph \"LASSO (L1)\" \n B[\"Diamond constraint
Corners on axes
Some coefficients = 0\"]\n end\n A --> C[\"Shrinks but keeps all variables\"]\n B --> D[\"Shrinks AND selects variables\"]\n`\n\nImagine the OLS contours (ellipses of equal RSS) expanding outward from the unconstrained minimum. For ridge, they hit the circular boundary, which is smooth everywhere -- the contact point almost never falls exactly on an axis. For LASSO, the diamond has sharp corners on the axes, and the expanding ellipse frequently touches a corner first, zeroing out that dimension's coefficient.\n\nWorked Example:\n\nPortfolio manager Svetlana at Ashworth Investments uses 15 macroeconomic variables to predict equity risk premium. Most are noisy or redundant.\n\nOLS with 15 variables: R-squared = 0.28, adjusted R-squared = 0.11 (many insignificant coefficients).\n\nLASSO with lambda chosen by 10-fold cross-validation:\n\n| Variable | OLS Coefficient | LASSO Coefficient |\n|---|---|---|\n| Term Spread | 0.42 | 0.31 |\n| Credit Spread | 0.38 | 0.25 |\n| Earnings Yield | 0.29 | 0.18 |\n| Inflation Surprise | -0.15 | -0.08 |\n| Industrial Production | 0.22 | 0.00 |\n| Consumer Confidence | 0.08 | 0.00 |\n| Trade Balance | -0.04 | 0.00 |\n| ... (8 more) | various | 0.00 |\n\nLASSO keeps 4 variables and zeros out 11. The sparse model has cross-validated R-squared of 0.16 -- lower than OLS in-sample but dramatically better out-of-sample.\n\nLASSO vs. Ridge Summary:\n- LASSO produces interpretable sparse models (built-in variable selection)\n- Ridge is better when all variables contribute signal and are correlated\n- LASSO struggles when predictors are highly correlated (tends to pick one arbitrarily)\n- LASSO's solution path is piecewise linear, making it computationally efficient\n\nCFA Exam Tips:\n- LASSO = L1 = sparsity = variable selection\n- Ridge = L2 = shrinkage without selection\n- Both require standardized predictors and cross-validation for lambda\n\nMaster regularization concepts in our CFA Quantitative Methods course.