One-dimensional ridge estimate

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Problem

With no intercept, fit $\hat{y}=wx$ to $(1,1)$ and $(2,3)$ by minimizing

J(w)=\frac{1}{2}\sum_i(wx_i-y_i)^2+\frac{\lambda}{2}w^2

with $\lambda=2$ . Derive the minimizer and compare it with the unregularized estimate.

Reveal answer or reference solution

The derivative is

J'(w)=(1^2+2^2+2)w-(1\cdot1+2\cdot3)=7w-7,

so the ridge estimate is $w=1$ . The unregularized estimate is $\frac{7}{5}=1.4$ . The penalty shrinks the coefficient toward zero. At $w=1$ , the data term is $\frac{1}{2}$ , the penalty is $1$ , and $J=\frac{3}{2}$ .

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One-dimensional ridge estimate

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