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zai-org
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RPC-Bench

@@ -461,7 +461,7 @@ $$
 \sum _ { l = 1 } ^ { L } \| \mathbf { W } _ { l } \| _ { F } ^ { 2 } \geq \sum _ { l = 1 } ^ { L - 2 } \| \mathbf { W } _ { l } \| _ { F } ^ { 2 } + 2 \sum _ { j = 1 } ^ { m } | w _ { L , j } | \ \| \mathbf { w } _ { L - 1 , j } \| _ { 2 } ,
 $$
-where the equality is achieved with the scaling choice αj =  |wL,j |kwL−1,jk2  is used. Since the scaling operation does not change the right-hand side of the inequality, we can set $\| \mathbf { w } _ { L - 1 , j } \| _ { 2 } = 1 , \forall j$ . Therefore, the right-hand side becomes $\| \mathbf { w } _ { L } \| _ { 1 }$ .
 Now, let us consider a modified version of the problem, where the unit norm equality constraint is relaxed as $\| \mathbf { w } _ { L - 1 , j } \| _ { 2 } \leq 1$ . Let us also assume that for a certain index $j$ , we obtain $\| \mathbf { w } _ { L - 1 , j } \| _ { 2 } <$ 1 with $w _ { L , j } \neq 0$ as an optimal solution. This shows that the unit norm inequality constraint is not active for ${ \bf w } _ { L - 1 , j }$ , and hence removing the constraint for ${ \bf w } _ { L - 1 , j }$ will not change the optimal solution. However, when we remove the constraint, $\| \mathbf { w } _ { L - 1 , j } \| _ { 2 } \to \bar { \infty }$ reduces the objective value since it yields $w _ { L , j } = 0$ . Therefore, we have a contradiction, which proves that all the constraints that correspond to a nonzero $w _ { L , j }$ must be active for an optimal solution. This also shows that replacing $\| \mathbf { w } _ { L - 1 , j } \| _ { 2 } = 1$ with $\| \tilde { \mathbf { w } } _ { L - 1 , j } \| _ { 2 } \leq 1$ does not change the solution to the problem.

 \sum _ { l = 1 } ^ { L } \| \mathbf { W } _ { l } \| _ { F } ^ { 2 } \geq \sum _ { l = 1 } ^ { L - 2 } \| \mathbf { W } _ { l } \| _ { F } ^ { 2 } + 2 \sum _ { j = 1 } ^ { m } | w _ { L , j } | \ \| \mathbf { w } _ { L - 1 , j } \| _ { 2 } ,
 $$
+where the equality is achieved with the scaling choice αj =  |wL,j |kwL−1,jk2  is used. Since the scaling operation does not change the right-hand side of the inequality, we can set $\| \mathbf { w } _ { L - 1 , j } \| _ { 2 } = 1 , \forall j$ . Therefore, the right-hand side becomes $\| \mathbf { w } _ { L } \| _ { 1 }$ .
 Now, let us consider a modified version of the problem, where the unit norm equality constraint is relaxed as $\| \mathbf { w } _ { L - 1 , j } \| _ { 2 } \leq 1$ . Let us also assume that for a certain index $j$ , we obtain $\| \mathbf { w } _ { L - 1 , j } \| _ { 2 } <$ 1 with $w _ { L , j } \neq 0$ as an optimal solution. This shows that the unit norm inequality constraint is not active for ${ \bf w } _ { L - 1 , j }$ , and hence removing the constraint for ${ \bf w } _ { L - 1 , j }$ will not change the optimal solution. However, when we remove the constraint, $\| \mathbf { w } _ { L - 1 , j } \| _ { 2 } \to \bar { \infty }$ reduces the objective value since it yields $w _ { L , j } = 0$ . Therefore, we have a contradiction, which proves that all the constraints that correspond to a nonzero $w _ { L , j }$ must be active for an optimal solution. This also shows that replacing $\| \mathbf { w } _ { L - 1 , j } \| _ { 2 } = 1$ with $\| \tilde { \mathbf { w } } _ { L - 1 , j } \| _ { 2 } \leq 1$ does not change the solution to the problem.