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md/train/66H4g_OHdnl/66H4g_OHdnl.md

@@ -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 $\| \mathbf { w } _ { L } \| _ { 1 }$ .
+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.
 

md/train/ujmgfuxSLrO/ujmgfuxSLrO.md

@@ -48,7 +48,7 @@ $$
 \mathbf { Y } ^ { l } = \left\{ \begin{array} { l l } { \mathcal { F } \left( \mathbf { X } , \mathbf { W } ^ { l } , \mathbf { b } ^ { l } , g ^ { l } \right) , } & { l = 1 } \\ { \mathcal { F } \left( \mathcal { H } \left( \mathbf { X } , \mathbf { Y } ^ { l - 1 } \right) , \mathbf { W } ^ { l } , \mathbf { b } ^ { l } , g ^ { l } \right) , } & { \mathrm { O t h e r w i s e } } \end{array} \right.
 $$
 
-where $\mathbf { W } ^ { l } = \left\{ \mathbf { W } _ { 1 } ^ { l } , \cdot \cdot \cdot , \mathbf { W } _ { g ^ { l } } ^ { l } \right\}$ and $\mathbf { b } ^ { l } = \left\{ \mathbf { b } _ { 1 } ^ { l } , \cdot \cdot \cdot , \mathbf { b } _ { g ^ { l } } ^ { l } \right\}$ are the learnable weights and biases of group linear transformation $\mathcal { F }$ with $g ^ { l }$ groups at the $l$ -th layer. Briefly, the $\mathcal { F }$ function takes the input $\mathbf { X }$ or $\mathcal { H } \left( \mathbf { X } , \mathbf { Y } ^ { l - 1 } \right) \Big )$ and splits into $g ^ { l }$ non-overlapping groups such that $\mathbf { X } = \left\{ \mathbf { X } _ { 1 } , \cdots , \mathbf { X } _ { g ^ { l } } \right\}$ . The function $\mathcal { F }$ then linearly transforms each $\mathbf { X } _ { i }$ with weights $\mathbf { W } _ { i } ^ { l }$ and bias $\mathbf { b } _ { i } ^ { l }$ to produce output $\mathbf { Y } _ { i } ^ { l } = \mathbf { X } _ { i } \mathbf { W } _ { i } ^ { l } + \mathbf { b } _ { i } ^ { l }$ . The outputs of each group $\mathbf { Y } _ { i } ^ { l }$ are then concatenated to produce the output $\mathbf { Y } ^ { l }$ . The function $\mathcal { H }$ first shuffles the output of each group in $\mathbf { Y } ^ { l - 1 }$ and then combines it with the input $\mathbf { X }$ using the input mixer connection of Mehta et al. (2020) to avoid vanishing gradient problems. Figure 2 visualizes the expansion phase in the DeLighT transformation with group linear transformation, feature shuffling, and the input mixer connection.
+where $\mathbf { W } ^ { l } = \left\{ \mathbf { W } _ { 1 } ^ { l } , \cdot \cdot \cdot , \mathbf { W } _ { g ^ { l } } ^ { l } \right\}$ and $\mathbf { b } ^ { l } = \left\{ \mathbf { b } _ { 1 } ^ { l } , \cdot \cdot \cdot , \mathbf { b } _ { g ^ { l } } ^ { l } \right\}$ are the learnable weights and biases of group linear transformation $\mathcal { F }$ with $g ^ { l }$ groups at the $l$ -th layer. Briefly, the $\mathcal { F }$ function takes the input $\mathbf { X }$ or $\mathcal { H } \left( \mathbf { X } , \mathbf { Y } ^ { l - 1 } \right) \Big )$ and splits into $g ^ { l }$ non-overlapping groups such that $\mathbf { X } = \left\{ \mathbf { X } _ { 1 } , \cdots , \mathbf { X } _ { g ^ { l } } \right\}$ . The function $\mathcal { F }$ then linearly transforms each $\mathbf { X } _ { i }$ with weights $\mathbf { W } _ { i } ^ { l }$ and bias $\mathbf { b } _ { i } ^ { l }$ to produce output $\mathbf { Y } _ { i } ^ { l } = \mathbf { X } _ { i } \mathbf { W } _ { i } ^ { l } + \mathbf { b } _ { i } ^ { l }$ . The outputs of each group $\mathbf { Y } _ { i } ^ { l }$ are then concatenated to produce the output $\mathbf { Y } ^ { l }$ . The function $\mathcal { H }$ first shuffles the output of each group in $\mathbf { Y } ^ { l - 1 }$ and then combines it with the input $\mathbf { X }$ using the input mixer connection of Mehta et al. (2020) to avoid vanishing gradient problems. Figure 2 visualizes the expansion phase in the DeLighT transformation with group linear transformation, feature shuffling, and the input mixer connection.
 
 The number of groups at the $l$ -th GLT in DeLighT transformation are computed as:
 

parse/dev/XByg4kotW5/XByg4kotW5.md

@@ -136,7 +136,7 @@ Now that we have a better sense of what policy RCSL will converge to with infini
 
 Theorem 2 (Reduction of RCSL to SL). Consider any function $f : S  \mathbb { R }$ such that the following two assumptions hold:
 
-1. Bounded occupancy mismatch: ⇡fP(s)  Cf for all s.
+1. Bounded occupancy mismatch: ⇡fP(s)  Cf for all s.
 
 2. Return coverage: $P _ { \beta } ( g = f ( s ) | s ) \ge \alpha _ { f }$ for all $s$