(15 points) Consider a regularized empirical risk minimization problem minw?n1??i=1n??i?(wTXi?)+?g(w) for some ?0, where ?i? 's and g are convex functions. Let ?? be an optimal solution of its dual problem max???n1??i=1n??i??(??i?)??g?(?n1?X?), where ?i?? and g? are the conjugate of ?i? and g, respectively. Assume that g? is differentiable at solutionspile.com

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(15 points) Consider a regularized empirical risk minimization problem minw?n1??i=1n??i?(wTXi?)+?g(w) for some ?>0, where ?i? 's and g are convex functions. Let ?? be an optimal solution of its dual problem max???n1??i=1n??i??(??i?)??g?(?n1?X?), where ?i?? and g? are the conjugate of ?i? and g, respectively. Assume that g? is differentiable at n?1?X??. Show that w?=?g?(n?1?X??) is an optimal solution of (2) (provide your detailed arguments). solutionspile.com

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