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parse/train/B1eyO1BFPr/B1eyO1BFPr.md

@@ -456,7 +456,7 @@ Note that in our experiments of large-batch SGD, either with the default large-b
 
 # A.4.2 IMAGENET
 
-ResNet-50 training is limited to 90 passes over the data in total, and the data is disjointly partitioned and is re-shuffled globally every epoch. All competing methods access the same total number of data samples (i.e. gradients) regardless of the number of local steps. We adopt the large-batch learning schemes as in Goyal et al. (2017) below. We linearly scale the learning rate based on Number of $\mathrm { G P U s } \ \times \ \frac { 0 . 1 } { 2 5 6 } \ \times \ \mathrm { \bar { \it B } _ { g l o b } } )$ where 0.1 and 256 is the base learning rate and mini-batch size 256 respectively for standard single GPU training. The local mini-batch size is set to 128. For learning rate scaling, we perform gradual warmup for the first 5 epochs, and decay the scaled learning rate by the factor of 10 when local models have access 30, 60, 80 epochs of training samples respectively.
+ResNet-50 training is limited to 90 passes over the data in total, and the data is disjointly partitioned and is re-shuffled globally every epoch. All competing methods access the same total number of data samples (i.e. gradients) regardless of the number of local steps. We adopt the large-batch learning schemes as in Goyal et al. (2017) below. We linearly scale the learning rate based on Number of $\mathrm { G P U s } \ \times \ \frac { 0 . 1 } { 2 5 6 } \ \times \ \mathrm { \bar { \it B } _ { g l o b } } )$ where 0.1 and 256 is the base learning rate and mini-batch size 256 respectively for standard single GPU training. The local mini-batch size is set to 128. For learning rate scaling, we perform gradual warmup for the first 5 epochs, and decay the scaled learning rate by the factor of 10 when local models have access 30, 60, 80 epochs of training samples respectively.
 
 # A.5 SYSTEM PERFORMANCE EVALUATION
 
@@ -850,7 +850,7 @@ input: the initial model ${ \pmb w } _ { ( 0 ) } \in \mathbb { R } ^ { d }$ ; tr
 9: end if   
 10: 11: for s $h : = 1 , \ldots , H _ { ( t ) }$ doch from $\mathcal { T } _ { ( t ) + h - 1 } ^ { k }$   
 12: compute the gradient $\begin{array} { r } { \pmb { g } _ { ( t ) + h - 1 } ^ { k } : = \frac { 1 } { B _ { \mathrm { l o c } } } \sum _ { i \in \mathcal { T } _ { ( t ) + h - 1 } ^ { k } } \nabla f _ { i } \left( \pmb { w } _ { ( t ) + h - 1 } ^ { k } \right) . } \end{array}$ . ™ can involve weight decay and momentum.   
-13: update the local model wkptqh : wptqh
1 
 γptqgkptqh
1 .   
+13: update the local model wkptqh : wptqh
1 
 γptqgkptqh
1 .   
 14: end for   
 15: get model difference $\Delta _ { ( t ) } ^ { k } : = { \pmb w } _ { ( t ) } ^ { k } - { \pmb w } _ { ( t ) + H } ^ { k } + { \pmb e } _ { ( t ) }$ .   
 16: compress the model difference: $\begin{array} { r } { \pmb { s } _ { ( t ) } ^ { k } = \mathrm { s i g n } ( \Delta _ { ( t ) } ^ { k } ) } \end{array}$ and $\begin{array} { r } { p _ { ( t ) } ^ { k } = \frac { \left\| \Delta _ { ( t ) } ^ { k } \right\| _ { 1 } } { d } } \end{array}$   
@@ -885,7 +885,7 @@ $$
 \begin{array} { r l } & { w _ { [ ( t ) + l ] + H } ^ { k } : = w _ { [ ( t ) + l ] } ^ { k } - \displaystyle \sum _ { h = 1 } ^ { H } \frac { \gamma _ { [ ( t ) ] } } { B _ { \operatorname* { l o c } } } \cdot \sum _ { \nabla _ { [ ( t ) + l ] + h - 1 } } \nabla f _ { i } \big ( w _ { [ ( t ) + l ] + h - 1 } ^ { k } } \\ & { } \\ & { w _ { [ ( t ) + l + 1 ] } ^ { k } : = w _ { [ ( t ) + l ] } ^ { k } - \displaystyle \frac { K _ { i } } { K _ { i } } \sum _ { k = 1 } ^ { K _ { i } } \big ( w _ { [ ( t ) + l ] } ^ { k } - w _ { [ ( t ) + l ] + H } ^ { k } \big ) } \\ & { \qquad w _ { [ ( t + 1 ) ] } ^ { k } : = w _ { [ ( t ) ] } ^ { k } - \displaystyle \frac { K } { K } \sum _ { k = 1 } ^ { K } \big ( w _ { [ ( t ) ] } ^ { k } - w _ { [ ( t ) + H ^ { k } ] } ^ { k } \big ) } \end{array}
 $$
 
-where ${ \pmb w } _ { [ ( t ) + l ] + H } ^ { k }$ indicates the model after $l$ k1 block update steps and $H$ local update steps, and $K _ { i }$ is rptqlsHthe number of GPUs on the GPU-block $i$ . The definition of $\gamma _ { [ ( t ) ] }$ and $\mathcal { T } _ { [ ( t ) + l ] + h - 1 } ^ { k }$ follows a similar scheme.
+where ${ \pmb w } _ { [ ( t ) + l ] + H } ^ { k }$ indicates the model after $l$ k1 block update steps and $H$ local update steps, and $K _ { i }$ is rptqlsHthe number of GPUs on the GPU-block $i$ . The definition of $\gamma _ { [ ( t ) ] }$ and $\mathcal { T } _ { [ ( t ) + l ] + h - 1 } ^ { k }$ follows a similar scheme.
 
 As the number of devices grows to the thousands (Goyal et al., 2017; You et al., 2017b), the difference between ‘within’ and ‘between’ block communication efficiency becomes more drastic. Thus, the performance benefits of our adaptive scheme compared to flat & large mini-batch SGD will be even more pronounced.
 

parse/train/ahYIlRBeCFw/ahYIlRBeCFw.md

@@ -100,13 +100,13 @@ $$
 \operatorname* { m i n } _ { x \in \mathcal C , z \in \mathcal Z } f ( x ) \ge \operatorname* { m i n } _ { x \in \mathcal C , z \in \mathcal Z } \mathbf W ^ { ( L ) } \mathbf D ^ { ( L - 1 ) } z ^ { ( L - 1 ) } + \mathrm { c o n s t . }
 $$
 
-To enforce the split neurons at layer $L - 1$ , we use a Lagrange function with $\beta ^ { ( L - 1 ) \top } \mathbf { S } ^ { ( L - 1 ) }$ multiplied on z(L1):
+To enforce the split neurons at layer $L - 1$ , we use a Lagrange function with $\beta ^ { ( L - 1 ) \top } \mathbf { S } ^ { ( L - 1 ) }$ multiplied on z(L1):
 
 $$
 \begin{array} { l } { \displaystyle \operatorname* { m i n } _ { x \in \mathcal { C } , z \in \mathcal { Z } } f ( x ) \geq \operatorname* { m i n } _ { \bar { x } \in \mathcal { C } _ { \mathcal { C } } ^ { ( L - 2 ) } \in \mathcal { L } ^ { ( L - 1 ) } \geq 0 } \operatorname* { m a x } _ { 0 ^ { ( L - 1 ) } \geq 0 } \mathbf { W } ^ { ( L ) } \mathbf { D } ^ { ( L - 1 ) } z ^ { ( L - 1 ) } + \beta ^ { ( L - 1 ) ^ { \top } } \mathbf { S } ^ { ( L - 1 ) } z ^ { ( L - 1 ) } + \mathrm { c o n s t } } \\ { \displaystyle \qquad \geq \operatorname* { m a x } _ { \beta ^ { ( L - 1 ) } \geq 0 } \operatorname* { m i n } _ { \bar { x } \in \mathcal { Z } _ { \mathcal { C } } ^ { ( L - 2 ) } \in \mathcal { Z } ( L - 2 ) } \left( \mathbf { W } ^ { ( L ) } \mathbf { D } ^ { ( L - 1 ) } + \beta ^ { ( L - 1 ) ^ { \top } } \mathbf { S } ^ { ( L - 1 ) } \right) z ^ { ( L - 1 ) } + \mathrm { c o n s t } } \end{array}
 $$
 
-The first inequality is due to the definition of the Lagrange function: we remove the constraint 2 Zduality. Due to the design of S(L1), neuron split z (L1)j  0 has a negative multiplier (L1)j $z ^ { ( L - 1 ) } \in \mathcal { Z } ^ { ( \bar { L } - 1 ) }$ and use a multiplier to replace this constraint. The second inequality is due to weak and split $z _ { j } ^ { ( L - 1 ) } < 0$ has a positive multiplier $\beta _ { j } ^ { ( L - 1 ) }$ . Any $\beta ^ { ( L - 1 ) } \geq 0$ yields a lower bound for the constrained optimization problem. Then we substitute $z ^ { ( L - 1 ) }$ with $\mathbf { W } ^ { ( L - 1 ) } \hat { z } ^ { ( L - 2 ) } + \mathbf { b } ^ { ( L - 1 ) }$ for next layer:
+The first inequality is due to the definition of the Lagrange function: we remove the constraint 2 Zduality. Due to the design of S(L1), neuron split z (L1)j  0 has a negative multiplier (L1)j $z ^ { ( L - 1 ) } \in \mathcal { Z } ^ { ( \bar { L } - 1 ) }$ and use a multiplier to replace this constraint. The second inequality is due to weak and split $z _ { j } ^ { ( L - 1 ) } < 0$ has a positive multiplier $\beta _ { j } ^ { ( L - 1 ) }$ . Any $\beta ^ { ( L - 1 ) } \geq 0$ yields a lower bound for the constrained optimization problem. Then we substitute $z ^ { ( L - 1 ) }$ with $\mathbf { W } ^ { ( L - 1 ) } \hat { z } ^ { ( L - 2 ) } + \mathbf { b } ^ { ( L - 1 ) }$ for next layer:
 
 $$
 \operatorname* { m i n } _ { x \in \mathcal { C } , z \in \mathcal { Z } } f ( x ) \geq \operatorname* { m a x } _ { \beta ^ { ( L - 1 ) } \geq 0 } \operatorname* { m i n } _ { \tilde { x } ^ { ( L - 2 ) } \in \mathcal { C } ( L - 2 ) } \left( \mathbf { W } ^ { ( L ) } \mathbf { D } ^ { ( L - 1 ) } + \beta ^ { ( L - 1 ) ^ { \top } } \mathbf { S } ^ { ( L - 1 ) } \right) \mathbf { W } ^ { ( L - 1 ) } \hat { z } ^ { ( L - 2 ) } + \mathrm { c o n s t }
@@ -124,7 +124,7 @@ $$
 \operatorname* { m i n } _ { x \in \mathcal { C } , z \in \mathcal { Z } } f ( x ) \ge \operatorname* { m a x } _ { \beta \ge 0 } \operatorname* { m i n } _ { x \in \mathcal { C } } { \mathbf { A } ^ { ( 0 ) } } x + \mathrm { c o n s t } ,
 $$
 
-where  := ⇥ (1)> (2)> (L1)>⇤> concatenates all $\beta ^ { ( i ) }$ vectors. Following the above idea, we present the main theorem in Theorem $3 . 1$ (proof is given in Appendix $\mathbf { A } )$ .
+where  := ⇥ (1)> (2)> (L1)>⇤> concatenates all $\beta ^ { ( i ) }$ vectors. Following the above idea, we present the main theorem in Theorem $3 . 1$ (proof is given in Appendix $\mathbf { A } )$ .
 
 Theorem 3.1 ( $\beta$ -CROWN bound). Given an $L$ -layer NN $f ( \boldsymbol { x } ) : \mathbb { R } ^ { d _ { 0 } }  \mathbb { R }$ with weights $\mathbf { W } ^ { ( i ) }$ , biases $\mathbf { b } ^ { ( i ) }$ , pre-ReLU bounds $\mathbf { l } ^ { ( i ) } \le z ^ { ( i ) } \le \mathbf { u } ^ { ( i ) }$ ( $1 \leq i \leq L )$ ), input bounds $\mathcal { C }$ , split constraints $\mathcal { Z }$ . We have:
 
@@ -158,7 +158,7 @@ $\operatorname* { m i n } f ( x ) : = z ^ { ( L ) } ( x )$ s.t.
 Network and Input Bounds: $z ^ { ( i ) } = \mathbf { W } ^ { ( i ) } \hat { z } ^ { ( i - 1 ) } + \mathbf { b } ^ { ( i ) } ; \hat { z } ^ { ( 0 ) } \geq x _ { 0 } - \epsilon ; \hat { z } ^ { ( 0 ) } \leq x _ { 0 } + \epsilon ;$   
 Stable ReLUs: $\hat { z } _ { j } ^ { ( i ) } = z _ { j } ^ { ( i ) }$ (if $1 _ { j } ^ { ( i ) } \geq 0$ ); $\hat { z } _ { j } ^ { ( i ) } = 0$ (if $\begin{array} { r } { \mathbf { u } _ { j } ^ { ( i ) } \le 0 . } \end{array}$ );   
 Unstable: $\begin{array} { r } { \hat { z } _ { j } ^ { ( i ) } \ge 0 , \hat { z } _ { j } ^ { ( i ) } \ge z _ { j } ^ { ( i ) } , \hat { z } _ { j } ^ { ( i ) } \le \frac { \mathbf { u } _ { j } ^ { ( i ) } } { \mathbf { u } _ { j } ^ { ( i ) } - \mathbf { 1 } _ { j } ^ { ( i ) } } \left( z _ { j } ^ { ( i ) } - \mathbf { 1 } _ { j } ^ { ( i ) } \right) ( \mathrm { i f } \mathbf { l } _ { j } ^ { ( i ) } < 0 < \mathbf { u } _ { j } ^ { ( i ) } , j \notin \mathcal { Z } ^ { + ( i ) } \cup \mathcal { Z } ^ { - ( i ) } ) } \end{array}$   
-Neuron Split Constraints: $\hat { z } _ { j } ^ { ( i ) } = z _ { j } ^ { ( i ) } , z _ { j } ^ { ( i ) } \geq 0$ (if $j \in \mathcal { Z } ^ { + ( i ) }$ ); ˆz (i)j = 0, z(i)j < 0 (if j 2 Z (i) )
+Neuron Split Constraints: $\hat { z } _ { j } ^ { ( i ) } = z _ { j } ^ { ( i ) } , z _ { j } ^ { ( i ) } \geq 0$ (if $j \in \mathcal { Z } ^ { + ( i ) }$ ); ˆz (i)j = 0, z(i)j < 0 (if j 2 Z (i) )
 
 Compared to the formulation in $\mathbb { \lVert \rVert }$ , we have neuron split constraints. Many BaB based complete verifiers $\mathbb { B } \mathbb { B } \mathbb { 3 } \mathbb { I }$ use an LP solver for Eq. $\boxed { 1 0 }$ as the incomplete verifier. We first show that it is possible to derive Theorem $3 . 1$ from the dual of this LP, leading to Theorem 3.2:
 
@@ -194,7 +194,7 @@ $$
 \operatorname* { m i n } _ { x \in \mathcal { C } , z \in \mathcal { Z } } f ( x ) \ge \operatorname* { m a x } _ { 0 \le \hat { \alpha } \le 1 , \hat { \beta } \ge 0 } g ( \hat { \alpha } , \hat { \beta } ) .
 $$
 
-This formulation is non-convex and has a large number of variables. Since any $0 \leq \hat { \alpha } \leq 1$ , ${ \hat { \beta } } \geq 0$ leads to a valid lower bound, the non-convexity does not affect soundness. When intermediate layer bounds are also allowed to be tightened during optimization, we can outperform the LP verifier for Eq. $\textcircled { 1 0 }$ using fixed intermediate layer bounds. Typically, in many previous works $\boxed { 8 } \boxed { 2 3 } \boxed { 6 }$ , when the LP formulation Eq. 10 is formed, intermediate layer bounds are pre-computed with bound propagation procedures $\boxed { 8 } \boxed { 2 3 }$ , which are far from optimal. To estimate the dimension of this problem, we denote the number of unstable neurons at layer $i$ as $s _ { i } : = \operatorname { T r } ( | \mathbf { S } ^ { ( i ) } | )$ . Each neuron in layer $i$ is associated with $2 \times \textstyle \sum _ { k = 1 } ^ { i - 1 } s _ { k }$ variables $\mathbf { { \boldsymbol { \alpha } } ^ { \prime } }$ . Suppose each hidden layer has $d$ neurons $( s _ { i } = { O } ( d ) )$ , then ↵ˆ has 2 ⇥ PL1i=1 d $\begin{array} { r } { 2 \times \sum _ { i = 1 } ^ { L - 1 } d _ { i } \sum _ { k = 1 } ^ { i - 1 } s _ { k } = O ( L ^ { 2 } d ^ { 2 } ) } \end{array}$ variables in total. This can be too large for efficientintermediate neurons of the same layer, leading to a $\mathbf { { \boldsymbol { \alpha } } ^ { \prime } }$ $\beta ^ { \prime }$ total number of $O ( L ^ { 2 } d )$ variables to optimize. Note that a weaker form of joint optimization was also discussed in $\lVert \boldsymbol { \mathsf { \overline { { 4 5 } } } } \rVert$ without $\beta$ , and a detailed analysis can be found in Appendix B.2.
+This formulation is non-convex and has a large number of variables. Since any $0 \leq \hat { \alpha } \leq 1$ , ${ \hat { \beta } } \geq 0$ leads to a valid lower bound, the non-convexity does not affect soundness. When intermediate layer bounds are also allowed to be tightened during optimization, we can outperform the LP verifier for Eq. $\textcircled { 1 0 }$ using fixed intermediate layer bounds. Typically, in many previous works $\boxed { 8 } \boxed { 2 3 } \boxed { 6 }$ , when the LP formulation Eq. 10 is formed, intermediate layer bounds are pre-computed with bound propagation procedures $\boxed { 8 } \boxed { 2 3 }$ , which are far from optimal. To estimate the dimension of this problem, we denote the number of unstable neurons at layer $i$ as $s _ { i } : = \operatorname { T r } ( | \mathbf { S } ^ { ( i ) } | )$ . Each neuron in layer $i$ is associated with $2 \times \textstyle \sum _ { k = 1 } ^ { i - 1 } s _ { k }$ variables $\mathbf { { \boldsymbol { \alpha } } ^ { \prime } }$ . Suppose each hidden layer has $d$ neurons $( s _ { i } = { O } ( d ) )$ , then ↵ˆ has 2 ⇥ PL1i=1 d $\begin{array} { r } { 2 \times \sum _ { i = 1 } ^ { L - 1 } d _ { i } \sum _ { k = 1 } ^ { i - 1 } s _ { k } = O ( L ^ { 2 } d ^ { 2 } ) } \end{array}$ variables in total. This can be too large for efficientintermediate neurons of the same layer, leading to a $\mathbf { { \boldsymbol { \alpha } } ^ { \prime } }$ $\beta ^ { \prime }$ total number of $O ( L ^ { 2 } d )$ variables to optimize. Note that a weaker form of joint optimization was also discussed in $\lVert \boldsymbol { \mathsf { \overline { { 4 5 } } } } \rVert$ without $\beta$ , and a detailed analysis can be found in Appendix B.2.
 
 # 3.4 $\beta$ -CROWN with Branch and Bound ( $\beta$ -CROWN BaB)