Ridge normal equations and effect

Problem

Derive the minimizer of

$\lVert X\mathbf{w}-\mathbf{y}\rVert_2^2+\lambda\lVert\mathbf{w}\rVert_2^2$

for $\lambda>0$ . State how increasing $\lambda$ typically changes coefficient magnitude and training fit.

Reveal answer or reference solution

$\mathbf{w}=(X^\top X+\lambda I)^{-1}X^\top\mathbf{y}.$

Increasing $\lambda$ shrinks coefficients and usually worsens training fit while potentially improving generalization.

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Regression Practice