mm 3. (20%) Proximal operator for the group lasso regularizer. In this exercise we derive the proximal operator for the group lasso regularizer. We will be using the notion of subgradient to make the derivation more solid. Recall that the proximal operator of the function =1 ||0j || at a particular point , where ; is the jth row of 0, is the solution to the following optimization problem minimize ||0;ll + p 10,11 +3/10 - ll . m j=1 1 , Notice that the problem is separable over so it suffices to consider the simpler problem minimize pl|0; || + 5|0; ; 112. all +. , (1) (a) Give an expression for the subdifferential of the function || 03 ||. Recall that subdifferential is the set of subgradients, so you need to list all possible subgradients. Hint. The function is only non-differentiable when j = 0; when the function is differentiable, the only element in the subdifferential is the gradient; otherwise it is some convex set. (b) A necessary and sufficient condition for optimality is that 0 is an element of the subdifferential. Use this condition and the expression of the subdifferential to derive the solution of (1)