Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 218 / 1 / a

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- Solution a

Solution

Put

s = x^{T} x > 0

and

z = x^{T} Y

. With objective

∥ Y - x β ∥_{2}^{2} + λ β^{2}

, ridge regression gives

β = \frac{z}{s + λ} .

For the duplicated design, the objective depends on

β_{1} + β_{2}

through the loss and symmetry makes the minimum-penalty decomposition equal:

β_{1} = β_{2} = \frac{z}{2 s + λ}, β_{1} + β_{2} = \frac{2 z}{2 s + λ} .

Duplicating a predictor therefore halves its effective ridge penalty.

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