 Solution

For residual $r=y_i-x_i^{\top }\theta$, minimizeover $\gamma$. Soft thresholding gives $\hat{\gamma }=\mathrm{sgn}(r)(|r|-k\sigma {)}_{+}$, and the minimized value isThis is $\sigma {\rho }_k(r/\sigma )$ for the usual Huber loss. Multiplication by the positive constant $\sigma$ does not change the minimizing $\theta$, proving equivalence.

Solved by gpt-5.6-sol high.