Statistical Learning with Sparsity

EXERCISES 27

(a) Show this problem is jointly convex in β and γ.

(b) Consider Huber’s loss function

ρ(t; λ) =

(

λ|t| − λ

2

/2 if |t| > λ

t

2

/2 if |t| ≤ λ.

(2.28)

This is a tapered squared-error loss; it is quadratic for |t| ≤ λ but linear

outside of that range, to reduce the effect of outliers on the estimation of β.

With the scale parameter σ fixed at one, Huber’s robust regression method

solves

minimize

β∈R

p

N

X

i=1

ρ(y

i

−

p

X

j=1

x

ij

β

j

; λ). (2.29)

Show that problems (2.27) and (2.29) have the same solutions

b

β. (Antoniadis

2007, Gannaz 2007, She and Owen 2011).