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md/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.
 

md/train/B8fp0LVMHa/B8fp0LVMHa.md

@@ -103,7 +103,7 @@ The bounds in equation 8 capture the main intuitions about the interplay between
 # Algorithm 1: Test-Time Policy $\pi _ { \mathrm { t e s t } }$
 
 Function TestEnsemble(values): return $\lambda \cdot \operatorname* { m i n } ( v a l u e s ) + ( 1 - \lambda ) \cdot \operatorname* { m a x } ( v a l u e s )$   
-Function $\pi _ { \mathrm { t e s t } } \left( s \right)$ : $\{ a _ { i } \} ^ { N } \sim \mu ( a | s )$ return arg $\operatorname* { m a x } _ { \{ a _ { i } \} ^ { N } }$ TestEnsemble[Qi(s, a) for all i]

+Function $\pi _ { \mathrm { t e s t } } \left( s \right)$ : $\{ a _ { i } \} ^ { N } \sim \mu ( a | s )$ return arg $\operatorname* { m a x } _ { \{ a _ { i } \} ^ { N } }$ TestEnsemble[Qi(s, a) for all i]

 
 close to $\pi _ { \underline { { \mu } } } ^ { * }$ . The two variants of $\Delta ( s , N )$ , based on $Q _ { \mu } ^ { * }$ or $Q _ { \mu } ^ { N }$ , suggest an intriguing notion of difficulty for an offline RL problem. If we could estimate either of these $\mathrm { Q }$ -value functions, then for a desired set of states (such as initial states) we could plot $\Delta ( s , N )$ as decreasing function of $N$ . The rate at which this function decreases could serve as an intuitive notion of difficulty for a given offline offline RL problem which consists of an MDP and a given behavior policy. While we leave theoretical investigations of this measure for future work, our empirical results in Section 5 demonstrate that the effective value of $N$ may be surprisingly small.
 

md/train/H1ebTsActm/H1ebTsActm.md

@@ -679,7 +679,7 @@ $$
 \begin{array} { r l r } {  { \| f - R _ { K } ( f ) \| _ { L ^ { r } } ^ { r } \lesssim \sum _ { K < \| k \| _ { 1 } \leq K ^ { * } } \| p _ { k } - G _ { k } ( p _ { k } ) \| _ { L ^ { r } } ^ { r } + \sum _ { K ^ { * } < \| k \| _ { 1 } } \| p _ { k } \| _ { L ^ { r } } ^ { r } } } \\ & { } & { \lesssim \sum _ { K < \| k \| _ { 1 } \leq K ^ { * } } [ \| p _ { k } \| _ { p } 2 ^ { \delta \| k \| _ { 1 } } n _ { k } ^ { - \delta } ] ^ { r } + \sum _ { K ^ { * } < \| k \| _ { 1 } } [ 2 ^ { \delta \| k \| _ { 1 } } \| p _ { k } \| _ { L ^ { p } } ] ^ { r } . } \end{array}
 $$
 
-In the following, we require an upper bound of k+d−1d−1 
. Hence, we evaluate this quantity beforehand. This can be upper bounded by the Stering’s formula as
+In the following, we require an upper bound of k+d−1d−1 
. Hence, we evaluate this quantity beforehand. This can be upper bounded by the Stering’s formula as
 
 $$
 \binom { k + d - 1 } { d - 1 } \leq \frac { \sqrt { 2 } e } { 2 \pi } \underbrace { \left( 1 + \frac { d - 1 } { k } \right) ^ { k } \left( 1 + \frac { k } { d - 1 } \right) ^ { d - 1 } } _ { = D _ { k , d } } \leq D _ { k , d } .
@@ -925,7 +925,7 @@ $$
 \leq \sum _ { k = 1 } ^ { L } ( B W ) ^ { L - k } \delta ( B \vee 1 ) ^ { k - 1 } ( W + 1 ) ^ { k } \leq \delta L ( B \vee 1 ) ^ { L - 1 } ( W + 1 ) ^ { L } .
 $$
 
-Thus, for a fixed sparsity pattern (the locations of non-zero parameters), the covering number is bounded by $\left( \delta / [ L ( B \vee 1 ) ^ { L - 1 } ( W + 1 ) ^ { L } ] \right) ^ { - S }$ . There are the number of configurations of the sparsity pattern is bounded b y (W+1)LS 
 ≤ (W + 1)LS. Thus, the covering number of the whole space $\Phi$ is bounded as
+Thus, for a fixed sparsity pattern (the locations of non-zero parameters), the covering number is bounded by $\left( \delta / [ L ( B \vee 1 ) ^ { L - 1 } ( W + 1 ) ^ { L } ] \right) ^ { - S }$ . There are the number of configurations of the sparsity pattern is bounded b y (W+1)LS 
 ≤ (W + 1)LS. Thus, the covering number of the whole space $\Phi$ is bounded as
 
 $$
 ( W + 1 ) ^ { L S } \left\{ \delta / [ L ( B \vee 1 ) ^ { L - 1 } ( W + 1 ) ^ { L } ] \right\} ^ { - S } = [ \delta ^ { - 1 } L ( B \vee 1 ) ^ { L - 1 } ( W + 1 ) ^ { 2 L } ] ^ { S } ,

md/train/RUQ1zwZR8_/RUQ1zwZR8_.md

@@ -73,14 +73,14 @@ Let $m$ be the number of users in a round and let $\gamma \in [ 0 , 1 ]$ denote
 
 # Algorithm 1 DPFedAvg-M with adaptive clipping
 
-function $\mathrm { T r a i n } ( m , \gamma , \eta _ { c } , \eta _ { s } , \eta _ { C } , z , \sigma _ { b } , \beta )$ Initialize model $\theta ^ { 0 }$ , clipping bound $C ^ { 0 }$ z∆ ← z−2 − (2σb)−2
−1/2 for each round $t = 0 , 1 , 2 , \ldots { \dot { \mathbf { c } } }$ o ${ \mathcal { Q } } ^ { t } \gets$ (sample $m$ users uniformly) for each user $i \in \mathcal { Q } ^ { t }$ in parallel do $\begin{array} { r l } & { \quad ( \Delta _ { i } ^ { t } , b _ { i } ^ { t } ) \gets \mathrm { ~ F e d A v g } ( i , \theta ^ { t } , \eta _ { c } , C ^ { t } ) } \\ & { \sigma _ { \Delta } \gets z _ { \Delta } C ^ { t } } \\ & { \tilde { \Delta } ^ { t } = \frac { 1 } { m } \left( \sum _ { i \in \mathcal { Q } ^ { t } } \Delta _ { i } ^ { t } + \mathcal { N } ( 0 , I \sigma _ { \Delta } ^ { 2 } ) \right) } \\ & { \bar { \Delta } ^ { t } = \beta \bar { \Delta } ^ { t - 1 } + ( 1 - \beta ) \tilde { \Delta } ^ { t } } \\ & { \theta ^ { t + 1 } \gets \theta ^ { t } + \eta _ { s } \bar { \Delta } ^ { t } } \\ & { \tilde { b } ^ { t } = \frac { 1 } { m } \left( \sum _ { i \in \mathcal { Q } ^ { t } } b _ { i } ^ { t } + \mathcal { N } ( O , \sigma _ { b } ^ { 2 } ) \right) } \end{array}$
+function $\mathrm { T r a i n } ( m , \gamma , \eta _ { c } , \eta _ { s } , \eta _ { C } , z , \sigma _ { b } , \beta )$ Initialize model $\theta ^ { 0 }$ , clipping bound $C ^ { 0 }$ z∆ ← z−2 − (2σb)−2
−1/2 for each round $t = 0 , 1 , 2 , \ldots { \dot { \mathbf { c } } }$ o ${ \mathcal { Q } } ^ { t } \gets$ (sample $m$ users uniformly) for each user $i \in \mathcal { Q } ^ { t }$ in parallel do $\begin{array} { r l } & { \quad ( \Delta _ { i } ^ { t } , b _ { i } ^ { t } ) \gets \mathrm { ~ F e d A v g } ( i , \theta ^ { t } , \eta _ { c } , C ^ { t } ) } \\ & { \sigma _ { \Delta } \gets z _ { \Delta } C ^ { t } } \\ & { \tilde { \Delta } ^ { t } = \frac { 1 } { m } \left( \sum _ { i \in \mathcal { Q } ^ { t } } \Delta _ { i } ^ { t } + \mathcal { N } ( 0 , I \sigma _ { \Delta } ^ { 2 } ) \right) } \\ & { \bar { \Delta } ^ { t } = \beta \bar { \Delta } ^ { t - 1 } + ( 1 - \beta ) \tilde { \Delta } ^ { t } } \\ & { \theta ^ { t + 1 } \gets \theta ^ { t } + \eta _ { s } \bar { \Delta } ^ { t } } \\ & { \tilde { b } ^ { t } = \frac { 1 } { m } \left( \sum _ { i \in \mathcal { Q } ^ { t } } b _ { i } ^ { t } + \mathcal { N } ( O , \sigma _ { b } ^ { 2 } ) \right) } \end{array}$
 
 $C ^ { t + 1 } \gets C ^ { t } \cdot \exp \left( - \eta _ { C } ( \tilde { b } ^ { t } - \gamma ) \right)$ function FedAvg(i, θ0, η, C) θ ← θ0 $\mathcal { G } $ (user i’s local data split into batches) for batch $g \in { \mathcal { G } }$ do $\begin{array} { r l } & { \quad \theta \gets \theta - \eta \nabla \ell ( \theta ; g ) } \\ & { \Delta \gets \theta - \theta ^ { 0 } } \\ & { b \gets \mathbb { I } _ { | | \Delta | | \leq C } } \\ & { \Delta ^ { \prime } \gets \Delta \cdot \operatorname* { m i n } \left( 1 , \frac { C } { | | \Delta | | } \right) } \\ & { \mathrm { ~ r e t u r n } \left( \Delta ^ { \prime } , b \right) } \end{array}$
 
 ![](images/d1f5d3ef5dfde7726538c717f8da8f80339ce3990f0c0907a44973a4dd5dbc83.jpg)  
 Figure 2: Evolution of the quantile estimate on data drawn from log-normal distributions. The three plots use data drawn from the exponential of $\mathcal { N } ( 0 . 0 , 1 . 0 )$ , $\mathcal { N } ( \bar { 0 . 0 } , 0 . 1 )$ , and $\mathcal { N } ( \log { 1 0 } , 1 . 0 )$ , respectively. Curves are shown for each of five quantiles: (0.1, 0.3, 0.5, 0.7, 0.9), and the dashed lines show the true value at each quantile. Hyperparameters are as discussed in the text and used in the experiments of Section 3: $\eta _ { C } = 0 . 2 , C ^ { 0 } \stackrel { \sim } { = } 0 . \dot { 1 } , m = 1 0 0 , \sigma _ { b } = m / 2 0$ . After an initial phase of exponential growth, the true quantile is fairly closely tracked. A smaller value of $\eta _ { C }$ would allow more accurate tracking at the cost of slower convergence, but since the quantile value is only used as a heuristic for clipping, a small amount of noise is tolerable. The entire sequence of values estimated for each target quantile satisfy $( 0 . 0 3 4 , n ^ { - 1 . 1 } )$ -differential privacy using RDP composition across the 200 rounds assuming fixed-size samples of $m = 1 0 0$ out of a total population of $\bar { n } = 1 0 ^ { 6 }$ [29].
 
-Theorem 1. One step of $D P$ -FedAvg with adaptive clipping using $\sigma _ { b }$ noise standard deviation on the clipped counts $\sum b _ { i } ^ { t }$ and $z _ { \Delta }$ noise multiplier on the vector sums $\sum \Delta _ { i } ^ { t }$ is equivalent (so far as privacy accounting is concerned) to one step of non-adaptive $D P$ -FedAvg with noise multiplier $z$ if we set z∆ = z−2 − (2σb)−2
−1/2.
+Theorem 1. One step of $D P$ -FedAvg with adaptive clipping using $\sigma _ { b }$ noise standard deviation on the clipped counts $\sum b _ { i } ^ { t }$ and $z _ { \Delta }$ noise multiplier on the vector sums $\sum \Delta _ { i } ^ { t }$ is equivalent (so far as privacy accounting is concerned) to one step of non-adaptive $D P$ -FedAvg with noise multiplier $z$ if we set z∆ = z−2 − (2σb)−2
−1/2.
 
 Proof. We make a conceptual change to the algorithm that does not change the behavior or privacy properties but allows us to analyze each step as if it were a single private Gaussian sum. Instead of sending $( \Delta _ { i } ^ { t } , b _ { i } ^ { t } )$ , each user sends $( \hat { \Delta } _ { i } ^ { t } , \hat { b } _ { i } ^ { t } ) \triangleq \big ( \Delta _ { i } ^ { t } / \sigma _ { \Delta } , ( b _ { i } ^ { t } - \scriptscriptstyle 1 / 2 ) / \sigma _ { b } \big )$ . The server adds noise with covariance $I$ and averages, then reverses the transformation so $\begin{array} { r } { \tilde { \Delta } ^ { t } = \frac { \sigma _ { \Delta } } { m } \biggl ( \sum _ { i \in \mathcal { Q } ^ { t } } \hat { \Delta } _ { i } ^ { t } + \mathcal { N } ( 0 , I ) \biggr ) } \end{array}$ and $\begin{array} { r } { \tilde { b } ^ { t } = \frac { \sigma _ { b } } { m } \left( \sum _ { i \in \mathcal { Q } ^ { t } } \hat { b } _ { i } ^ { t } + \mathcal { N } ( 0 , 1 ) \right) + 1 / 2 } \end{array}$ . Noting that $\vert \vert ( \hat { \Delta } _ { i } ^ { t } , \hat { b } _ { i } ^ { t } ) \vert \vert \leq S \triangleq \left( ( C ^ { t } / \sigma _ { \Delta } ) ^ { 2 } + \left( 1 / 2 \sigma _ { b } \right) ^ { 2 } \right) ^ { 1 / 2 }$ , it is clear that the two Gaussian sum queries of Algorithm 1 are equivalent to pre- and post-processing of a single query with sensitivity $S$ and covariance $I$ , or noise multiplier $z = { \overset { \cdot } { 1 } } / s = \overset { \cdot } { \left( z _ { \Delta } ^ { - 2 } + ( 2 \sigma _ { b } ) ^ { - 2 } \right) ^ { - 1 / 2 } }$ Rearranging yields the result.
 

md/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)
 

md/train/bXehDYUjjXi/bXehDYUjjXi.md

@@ -202,7 +202,7 @@ $$
 
 which we refer to as the discrete-time ELBO (DT-ELBO). The reconstruction error of the stochastic layer can be seen as some form of finite difference approximation to differentiation, which gives rise to $\nabla \cdot \boldsymbol { \mu }$ in the CT-ELBO in the infinitesimal limit (as $\Delta t$ approaches 0). The regularization of $| | a | | ^ { 2 }$ pops up when we compare the difference between $\tilde { \mu } _ { i }$ and $\hat { \mu } _ { i }$ using the Gaussian reparameterization to compute the reconstruction error. We formalize this idea in the following theorem.
 
-Theorem 5 (Consistency). Assume $\mu$ , , $\sigma ^ { - 2 }$ , $^ { a , }$ , $| | a | | ^ { 2 }$ and their derivatives up to the fourth order are all bounded and continuous, and that $\sigma$ is non-singular. Then $\mathcal { E } ^ { L } \to \mathcal { E } ^ { \infty }$ as $L \to \infty$ .
+Theorem 5 (Consistency). Assume $\mu$ , , $\sigma ^ { - 2 }$ , $^ { a , }$ , $| | a | | ^ { 2 }$ and their derivatives up to the fourth order are all bounded and continuous, and that $\sigma$ is non-singular. Then $\mathcal { E } ^ { L } \to \mathcal { E } ^ { \infty }$ as $L \to \infty$ .
 
 This theorem tells us that the CT-ELBO we derive for continuous-time diffusion models is not that different from the traditional ELBO, and that maximizing the CT-ELBO can be seen as training an infinitely deep hierarchical VAE. We present the proof in Appendix F, which formalizes the above intuition, using Taylor’s theorem to control the polynomial approximation error, which will go to 0 as the step size $\Delta t$ vanishes when the number of layers $L$ increases to infinity.
 

md/train/cVwc7IHWEWi/cVwc7IHWEWi.md

@@ -42,7 +42,7 @@ Notation. For a positive integer $n$ , let $[ n ] : = \{ 1 , 2 , . . . , n \}$ .
 
 No-regret learning in games. We consider a game $G$ with $m \in \mathbb { N }$ players, where player $i \in [ m ]$ has action space $A _ { i }$ with $n _ { i } : = | A _ { i } |$ actions. We may assume that $\mathcal { A } _ { i } = [ n _ { i } ]$ for each player $i$ . The joint action space is $\mathcal { A } : = \mathcal { A } _ { 1 } \times \dots \times \mathcal { A } _ { m }$ . The specification of the game $G$ is completed by a collection of loss functions $\mathcal { L } _ { 1 } , \ldots , \mathcal { L } _ { m } : \mathcal { A }  [ 0 , 1 ]$ . For an action profile $a = ( a _ { 1 } , \ldots , a _ { m } ) \in \mathcal { A }$ and $i \in [ m ]$ , $\mathcal { L } _ { i } ( a )$ is the loss player $i$ experiences when each player $i ^ { \prime } \in [ m ]$ plays $a _ { i ^ { \prime } }$ . A mixed strategy $x _ { i } \in \Delta ( \mathcal { A } _ { i } )$ for player $i$ is a distribution over $A _ { i }$ , with the probability of playing action $j \in \mathcal A _ { i }$ given by $x _ { i } ( j )$ . Given a mixed strategy profile $\boldsymbol { x } = ( x _ { 1 } , \dots , x _ { m } )$ (or an action profile $\boldsymbol { a } = ( a _ { 1 } , \dots , a _ { m } ) )$ and a player $i \in [ m ]$ we let $x _ { - i }$ (or $a _ { - i }$ , respectively) denote the profile after removing the ith mixed strategy $x _ { i }$ (or the $i$ th action $a _ { i }$ , respectively).
 
-The $m$ players play the game $G$ for a total of $T$ rounds. At the beginning of each round $t \in [ T ]$ , each player $i$ chooses a mixed strategy $x _ { i } ^ { ( t ) } \in \Delta ( \mathcal { A } _ { i } )$ . The loss vector of player $i$ , denoted $\ell _ { i } ^ { ( t ) } \in [ 0 , 1 ] ^ { n _ { i } }$ , is defined as $\ell _ { i } ^ { ( t ) } ( j ) = \mathbb { E } _ { a _ { - i } \sim x _ { - i } ^ { ( t ) } } [ \mathcal { L } _ { i } ( \dot { j } , a _ { - i } ) ]$ . As a matter of convention, set $\ell _ { i } ^ { ( 0 ) } = \mathbf { 0 }$ to be the all-zeros vector. We consider the full-information setting in this paper, meaning that player $i$ observes its full loss vector $\ell _ { i } ^ { ( t ) }$ for each round $t$ . Finally, player $i$ experiences a loss of $\langle \ell _ { i } ^ { ( t ) } , x _ { i } ^ { ( t ) } \rangle$ . The goal of each player $i$ is to minimize its regret, defined as: $\begin{array} { r } { \mathrm { R e g } _ { i , T } : = \sum _ { t \in [ T ] } \langle x _ { i } ^ { ( t ) } , \ell _ { i } ^ { ( t ) } \rangle - \operatorname* { m i n } _ { j \in [ n _ { i } ] } \sum _ { t \in [ T ] } \ell _ { i } ^ { ( t ) } ( j ) } \end{array}$ .
+The $m$ players play the game $G$ for a total of $T$ rounds. At the beginning of each round $t \in [ T ]$ , each player $i$ chooses a mixed strategy $x _ { i } ^ { ( t ) } \in \Delta ( \mathcal { A } _ { i } )$ . The loss vector of player $i$ , denoted $\ell _ { i } ^ { ( t ) } \in [ 0 , 1 ] ^ { n _ { i } }$ , is defined as $\ell _ { i } ^ { ( t ) } ( j ) = \mathbb { E } _ { a _ { - i } \sim x _ { - i } ^ { ( t ) } } [ \mathcal { L } _ { i } ( \dot { j } , a _ { - i } ) ]$ . As a matter of convention, set $\ell _ { i } ^ { ( 0 ) } = \mathbf { 0 }$ to be the all-zeros vector. We consider the full-information setting in this paper, meaning that player $i$ observes its full loss vector $\ell _ { i } ^ { ( t ) }$ for each round $t$ . Finally, player $i$ experiences a loss of $\langle \ell _ { i } ^ { ( t ) } , x _ { i } ^ { ( t ) } \rangle$ . The goal of each player $i$ is to minimize its regret, defined as: $\begin{array} { r } { \mathrm { R e g } _ { i , T } : = \sum _ { t \in [ T ] } \langle x _ { i } ^ { ( t ) } , \ell _ { i } ^ { ( t ) } \rangle - \operatorname* { m i n } _ { j \in [ n _ { i } ] } \sum _ { t \in [ T ] } \ell _ { i } ^ { ( t ) } ( j ) } \end{array}$ .
 
 Optimistic hedge. The Optimistic Hedge algorithm chooses mixed strategies for player $i \in [ m ]$ as follows: at time $t = 1$ , it sets $x _ { i } ^ { ( 1 ) } = ( 1 / n _ { i } , \ldots , 1 / n _ { i } )$ to be the uniform distribution on $\mathbf { \mathcal { A } } _ { i }$ . Then for all $t < T$ , player $i$ ’s strategy at iteration $t + 1$ is defined as follows, for $j \in [ n _ { i } ]$ :
 
@@ -52,7 +52,7 @@ $$
 
 Optimistic Hedge is a modification of Hedge, which performs the updates $x _ { i } ^ { ( t + 1 ) } ( j ) : =$ $\begin{array} { r l } & { \frac { x _ { i } ^ { ( t ) } ( j ) \cdot \exp ( - \eta \cdot \ell _ { i } ^ { ( t ) } ( j ) ) } { \sum _ { k \in [ n _ { i } ] } x _ { i } ^ { ( t ) } ( k ) \cdot \exp ( - \eta \cdot \ell _ { i } ^ { ( t ) } ( k ) ) } } \end{array}$ . The update (1) modifies the Hedge update by replacing the loss vector $\ell _ { i } ^ { ( t ) }$ 2 i   with a predictor of the following iteration’s loss vector, $\ell _ { i } ^ { ( t ) } + ( \ell _ { i } ^ { ( t ) } - \ell _ { i } ^ { ( t - 1 ) } )$ . Hedge corresponds to FTRL with a negative entropy regularizer (see, e.g., [Bub15]), whereas Optimistic Hedge corresponds to Optimistic FTRL with a negative entropy regularizer [RS13b, RS13a].
 
-Distributions $\pmb { \& }$ divergences. For distributions $P , Q$ on a finite domain $[ n ]$ , the $K L$ divergence between $P , Q$ is $\begin{array} { r } { \mathrm { K L } ( P ; Q ) = \sum _ { j = 1 } ^ { n } P ( j ) \cdot \log \left( \frac { P ( j ) } { Q ( j ) } \right) } \end{array}$ . The chi-squared divergence between $P , Q$ is $\begin{array} { r } { \chi ^ { 2 } ( P ; Q ) = \sum _ { j = 1 } ^ { n } Q ( j ) \cdot \left( \frac { P ( j ) } { Q ( j ) } \right) ^ { 2 } - 1 = \sum _ { j = 1 } ^ { n } \frac { ( P ( j ) - Q ( j ) ) ^ { 2 } } { Q ( j ) } } \end{array}$ (P (j)Q(j))2 . For a distribution P on [n] and a vector $v \in \mathbb { R } ^ { n }$ , we write $\begin{array} { r } { \mathrm { V a r } _ { P } \left( v \right) : = \sum _ { j = 1 } ^ { n } P ( j ) \cdot \left( v ( j ) - \sum _ { k = 1 } ^ { n } P ( k ) v ( k ) \right) ^ { 2 } } \end{array}$ . Also define $\begin{array} { r } { \| v \| _ { P } : = \sqrt { \sum _ { j = 1 } ^ { n } P ( j ) \cdot v ( j ) ^ { 2 } } } \end{array}$ . If further $P$ has full support, then define $\begin{array} { r } { \left\| v \right\| _ { P } ^ { \star } = \sqrt { \sum _ { j = 1 } ^ { n } \frac { v ( j ) ^ { 2 } } { P ( j ) } } } \end{array}$ The above notations will often be used when $P$ is the mixed strategy profile $x _ { i }$ for some player $i$ and $v$ is a loss vector $\ell _ { i }$ ; in such a case the norms $\| v \| _ { P }$ and $\| v \| _ { P } ^ { \star }$ are often called local norms.
+Distributions $\pmb { \& }$ divergences. For distributions $P , Q$ on a finite domain $[ n ]$ , the $K L$ divergence between $P , Q$ is $\begin{array} { r } { \mathrm { K L } ( P ; Q ) = \sum _ { j = 1 } ^ { n } P ( j ) \cdot \log \left( \frac { P ( j ) } { Q ( j ) } \right) } \end{array}$ . The chi-squared divergence between $P , Q$ is $\begin{array} { r } { \chi ^ { 2 } ( P ; Q ) = \sum _ { j = 1 } ^ { n } Q ( j ) \cdot \left( \frac { P ( j ) } { Q ( j ) } \right) ^ { 2 } - 1 = \sum _ { j = 1 } ^ { n } \frac { ( P ( j ) - Q ( j ) ) ^ { 2 } } { Q ( j ) } } \end{array}$ (P (j)Q(j))2 . For a distribution P on [n] and a vector $v \in \mathbb { R } ^ { n }$ , we write $\begin{array} { r } { \mathrm { V a r } _ { P } \left( v \right) : = \sum _ { j = 1 } ^ { n } P ( j ) \cdot \left( v ( j ) - \sum _ { k = 1 } ^ { n } P ( k ) v ( k ) \right) ^ { 2 } } \end{array}$ . Also define $\begin{array} { r } { \| v \| _ { P } : = \sqrt { \sum _ { j = 1 } ^ { n } P ( j ) \cdot v ( j ) ^ { 2 } } } \end{array}$ . If further $P$ has full support, then define $\begin{array} { r } { \left\| v \right\| _ { P } ^ { \star } = \sqrt { \sum _ { j = 1 } ^ { n } \frac { v ( j ) ^ { 2 } } { P ( j ) } } } \end{array}$ The above notations will often be used when $P$ is the mixed strategy profile $x _ { i }$ for some player $i$ and $v$ is a loss vector $\ell _ { i }$ ; in such a case the norms $\| v \| _ { P }$ and $\| v \| _ { P } ^ { \star }$ are often called local norms.
 
 # 3 Results
 
@@ -94,7 +94,7 @@ $$
 \mathrm { R e g } _ { i , T } \leq \frac { \log n _ { i } } { \eta } + \sum _ { t = 1 } ^ { T } \left. x _ { i } ^ { ( t ) } - \tilde { x } _ { i } ^ { ( t ) } \right. _ { x _ { i } ^ { ( t ) } } ^ { \times } \sqrt { \mathrm { V a r } _ { x _ { i } ^ { ( t ) } } \left( \ell _ { i } ^ { ( t ) } - \ell _ { i } ^ { ( t - 1 ) } \right) } - \frac { 1 } { \eta } \sum _ { t = 1 } ^ { T } \mathrm { K L } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } ) - \frac { 1 } { \eta } \sum _ { t = 1 } ^ { T } \mathrm { K L } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } ) ,
 $$
 
-2) thand $\mathrm { K L } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } )$ $\mathrm { K L } ( x _ { i } ^ { ( t ) } ; \tilde { x } _ { i } ^ { ( t - 1 ) } )$ may be lower bounded bytively. Note it is a standard $( 1 / 2 - O ( \eta ) ) \cdot \chi ^ { 2 } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } )$ $( 1 / 2 - O ( \eta ) ) \cdot \chi ^ { 2 } ( x _ { i } ^ { ( t ) } ; \tilde { x } _ { i } ^ { ( t - 1 ) } )$ fact that the KL divergence between two distributions is upper bounded by the chi-squared distribution between them; by contrast, Lemma A.2 can exploit that $x _ { i } ^ { ( t ) }$ , $\tilde { x } _ { i } ^ { ( t ) }$ t) and x˜ (t1)i a re close to each other between lower bo $x _ { i } ^ { ( t ) }$ i and x˜ $\tilde { x } _ { i } ^ { ( t - 1 ) }$ hat the , leadi $\chi ^ { 2 }$ -divergenco the term $\chi ^ { 2 } ( x _ { i } ^ { ( t ) } ; \tilde { x } _ { i } ^ { ( t - 1 ) } )$ $\left( 1 - O ( \eta ) \right) \cdot \eta ^ { 2 } \cdot \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( \ell _ { i } ^ { ( t - 1 ) } \right)$ $\begin{array} { r } { \frac { ( 1 - C \eta ) \eta } { 2 } \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( \ell _ { i } ^ { ( t - 1 ) } \right) } \end{array}$ being subtracted in (4). The $\chi ^ { 2 }$ -divergence $\chi ^ { 2 } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } )$ , as well as the term $\left\| \boldsymbol { x } _ { i } ^ { ( t ) } - \tilde { \boldsymbol { x } } _ { i } ^ { ( t ) } \right\| _ { \boldsymbol { x } _ { i } ^ { ( t ) } } ^ { \star }$ in (5) are bounded in a similar manner to obtain (4).
+2) thand $\mathrm { K L } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } )$ $\mathrm { K L } ( x _ { i } ^ { ( t ) } ; \tilde { x } _ { i } ^ { ( t - 1 ) } )$ may be lower bounded bytively. Note it is a standard $( 1 / 2 - O ( \eta ) ) \cdot \chi ^ { 2 } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } )$ $( 1 / 2 - O ( \eta ) ) \cdot \chi ^ { 2 } ( x _ { i } ^ { ( t ) } ; \tilde { x } _ { i } ^ { ( t - 1 ) } )$ fact that the KL divergence between two distributions is upper bounded by the chi-squared distribution between them; by contrast, Lemma A.2 can exploit that $x _ { i } ^ { ( t ) }$ , $\tilde { x } _ { i } ^ { ( t ) }$ t) and x˜ (t1)i a re close to each other between lower bo $x _ { i } ^ { ( t ) }$ i and x˜ $\tilde { x } _ { i } ^ { ( t - 1 ) }$ hat the , leadi $\chi ^ { 2 }$ -divergenco the term $\chi ^ { 2 } ( x _ { i } ^ { ( t ) } ; \tilde { x } _ { i } ^ { ( t - 1 ) } )$ $\left( 1 - O ( \eta ) \right) \cdot \eta ^ { 2 } \cdot \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( \ell _ { i } ^ { ( t - 1 ) } \right)$ $\begin{array} { r } { \frac { ( 1 - C \eta ) \eta } { 2 } \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( \ell _ { i } ^ { ( t - 1 ) } \right) } \end{array}$ being subtracted in (4). The $\chi ^ { 2 }$ -divergence $\chi ^ { 2 } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } )$ , as well as the term $\left\| \boldsymbol { x } _ { i } ^ { ( t ) } - \tilde { \boldsymbol { x } } _ { i } ^ { ( t ) } \right\| _ { \boldsymbol { x } _ { i } ^ { ( t ) } } ^ { \star }$ in (5) are bounded in a similar manner to obtain (4).
 
 # 4.2 Finite differences
 
@@ -190,7 +190,7 @@ $$
 \sum _ { t = 1 } ^ { T - h - 1 } \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( ( \operatorname { D } _ { h + 1 } \ell _ { i } ) ^ { ( t ) } \right) \leq O ( 1 / H ) \cdot \sum _ { t = 1 } ^ { T - h } \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( ( \operatorname { D } _ { h } \ell _ { i } ) ^ { ( t ) } \right) + \tilde { O } \left( \mu ^ { 2 h } \right) ,
 $$
 
-where $\tilde { O }$ hides factors polynomial in $\log T$ . The validity of (12) for $h = 0$ implies Lemma 4.2. On implies that the other hand, as long we choose the value $\begin{array} { r } { \sum _ { t = 1 } ^ { T - H } \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( \left( \operatorname { D } _ { H } \ell _ { i } \right) ^ { ( t ) } \right) \leq O ( \mu ^ { 2 H } ) } \end{array}$ $\mu$ in (12) to satisfy . This gives that (12) holds for $\mu \geq m \eta H ^ { \Omega ( 1 ) }$ , then Lemma 4.4 $h = H - 1$ . To show that (12) holds for all $H - 1 > h \geq 0$ , we use downwards induction; fix any $h$ , and assume that (12) has been shown for all $h ^ { \prime }$ satisfying $h < h ^ { \prime } \leq H - 1$ . Our main tool in the inductive step is to apply Lemma 4.7 below. To state it, for $\zeta > 0 , n \in \mathbb { N }$ , we say that a sequence of distributions $P ^ { ( 1 ) } , \ldots , P ^ { ( T ) } \in \Delta ^ { n }$ is $\zeta$ -consecutively close if for each $1 \leq t < T$ , it holds that max $\left\{ \left\| \frac { P ^ { ( t ) } } { P ^ { ( t + 1 ) } } \right\| _ { \infty } , \left\| \frac { P ^ { ( t + 1 ) } } { P ^ { ( t ) } } \right\| _ { \infty } \right\} \leq 1 + \zeta$ . 4 Lemma 4.7 shows that given a sequence of vectors for which the variances of its second-order finite differences are bounded by the variances of its first-order finite differences, a similar relationship holds between its first- and zeroth-order finite differences.
+where $\tilde { O }$ hides factors polynomial in $\log T$ . The validity of (12) for $h = 0$ implies Lemma 4.2. On implies that the other hand, as long we choose the value $\begin{array} { r } { \sum _ { t = 1 } ^ { T - H } \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( \left( \operatorname { D } _ { H } \ell _ { i } \right) ^ { ( t ) } \right) \leq O ( \mu ^ { 2 H } ) } \end{array}$ $\mu$ in (12) to satisfy . This gives that (12) holds for $\mu \geq m \eta H ^ { \Omega ( 1 ) }$ , then Lemma 4.4 $h = H - 1$ . To show that (12) holds for all $H - 1 > h \geq 0$ , we use downwards induction; fix any $h$ , and assume that (12) has been shown for all $h ^ { \prime }$ satisfying $h < h ^ { \prime } \leq H - 1$ . Our main tool in the inductive step is to apply Lemma 4.7 below. To state it, for $\zeta > 0 , n \in \mathbb { N }$ , we say that a sequence of distributions $P ^ { ( 1 ) } , \ldots , P ^ { ( T ) } \in \Delta ^ { n }$ is $\zeta$ -consecutively close if for each $1 \leq t < T$ , it holds that max $\left\{ \left\| \frac { P ^ { ( t ) } } { P ^ { ( t + 1 ) } } \right\| _ { \infty } , \left\| \frac { P ^ { ( t + 1 ) } } { P ^ { ( t ) } } \right\| _ { \infty } \right\} \leq 1 + \zeta$ . 4 Lemma 4.7 shows that given a sequence of vectors for which the variances of its second-order finite differences are bounded by the variances of its first-order finite differences, a similar relationship holds between its first- and zeroth-order finite differences.
 
 Lemma 4.7. There is a sufficiently large constant $C _ { 0 } > 1$ so that the following holds. For any $M , \zeta , \alpha > 0$ and $n \in \mathbb N$ , suppose that $P ^ { ( 1 ) } , \ldots , P ^ { ( T ) } \in \Delta ^ { n }$ and $Z ^ { ( 1 ) } , \ldots , Z ^ { ( \breve { T } ) } \in [ - M , M ] ^ { \breve { n } }$ satisfy the following conditions:
 
@@ -216,7 +216,7 @@ $$
 \begin{array} { r } { ( \mathrm { D } _ { h } ^ { \circ } L ) ^ { ( t ) } = \left\{ \begin{array} { l l } { \left( \mathrm { D } _ { h - 1 } ^ { \circ } L \right) ^ { ( t + 1 ) } - \left( \mathrm { D } _ { h - 1 } ^ { \circ } L \right) ^ { ( t ) } } & { : 0 \le t \le S - 2 } \\ { \left( \mathrm { D } _ { h - 1 } ^ { \circ } L \right) ^ { ( 1 ) } - \left( \mathrm { D } _ { h - 1 } ^ { \circ } L \right) ^ { ( T ) } } & { : t = S - 1 . } \end{array} \right. } \end{array}
 $$
 
-Circular finite differences for a sequence $L ^ { ( 0 ) } , \ldots , L ^ { ( S - 1 ) }$ are defined similarly to finite differences (Definition 4.1) except that unlike for finite differences, where (Dh L) (Sh) , $\left( \mathrm { D } _ { h } \cal L \right) ^ { ( S - h ) } , . . . , \left( \mathrm { D } _ { h } \cal L \right) ^ { ( S - 1 ) }$ are not defined, (Dh L) (Sh) , $\big ( \mathrm { D } _ { h } ^ { \circ } L \big ) ^ { ( S - h ) } , \ldots , \big ( \mathrm { D } _ { h } ^ { \circ } L \big ) ^ { ( S - 1 ) }$ are defined by “wrapping around” back to the beginning of the sequence. The above-described reductions, which are worked out in detail in Section C.2, allow us to reduce proving Lemma 4.7 to proving the following simpler lemma:
+Circular finite differences for a sequence $L ^ { ( 0 ) } , \ldots , L ^ { ( S - 1 ) }$ are defined similarly to finite differences (Definition 4.1) except that unlike for finite differences, where (Dh L) (Sh) , $\left( \mathrm { D } _ { h } \cal L \right) ^ { ( S - h ) } , . . . , \left( \mathrm { D } _ { h } \cal L \right) ^ { ( S - 1 ) }$ are not defined, (Dh L) (Sh) , $\big ( \mathrm { D } _ { h } ^ { \circ } L \big ) ^ { ( S - h ) } , \ldots , \big ( \mathrm { D } _ { h } ^ { \circ } L \big ) ^ { ( S - 1 ) }$ are defined by “wrapping around” back to the beginning of the sequence. The above-described reductions, which are worked out in detail in Section C.2, allow us to reduce proving Lemma 4.7 to proving the following simpler lemma:
 
 Lemma 4.8. Suppose $\mu \in \mathbb { R } , \alpha > 0$ , and $W ^ { ( 0 ) } , \ldots , W ^ { ( S - 1 ) } \in \mathbb { R }$ is a sequence of reals satisfying
 

md/train/rkgK3oC5Fm/rkgK3oC5Fm.md

@@ -340,7 +340,7 @@ Details of the recognition model used in the CVAE baseline. We show the layer wi
 
 <table><tr><td>Layer</td><td>Type</td><td>Size</td><td>Activation</td><td>Input</td><td>Output</td></tr><tr><td>In1</td><td>Input</td><td></td><td></td><td>x,y</td><td>ConV1,1</td></tr><tr><td>ConV1,1</td><td>Conv2D</td><td>128</td><td>ReLU</td><td>In1</td><td>ConV1,2</td></tr><tr><td>ConV1,2</td><td>Conv2D</td><td>128</td><td>ReLU</td><td>ConV1,1</td><td>MaxPool1</td></tr><tr><td>MaxPool1</td><td>Max Pooling</td><td>2×2</td><td></td><td>ConV1,2</td><td>ConV2,1</td></tr><tr><td>ConV2,1</td><td>Conv2D</td><td>256</td><td>ReLU</td><td>MaxPool1</td><td>ConV2,2</td></tr><tr><td>ConV2,2</td><td>Conv2D</td><td>256</td><td>ReLU</td><td>ConV2,1</td><td>MaxPool2</td></tr><tr><td>MaxPool2</td><td>Max Pooling</td><td>2×2</td><td></td><td>ConV2,2</td><td>Conv3,1</td></tr><tr><td>ConV3,1</td><td>Conv2D</td><td>512</td><td>ReLU</td><td>MaxPool2</td><td>MaxPool2</td></tr><tr><td>MaxPool3</td><td>Max Pooling</td><td>2×2</td><td></td><td>ConV3,1</td><td>ConV4,1</td></tr><tr><td>ConV4,1</td><td>Conv2D</td><td>512</td><td>ReLU</td><td>MaxPool3</td><td>MaxPool4</td></tr><tr><td>MaxPool4</td><td>Max Pooling</td><td>2×2</td><td></td><td>ConV4,1</td><td>Flatten</td></tr><tr><td>Flatten</td><td></td><td></td><td></td><td>MaxPool4</td><td>Dense1</td></tr><tr><td>Dense1</td><td>Fully Connected</td><td>1024</td><td>ReLU</td><td>Flatten</td><td>Dense2</td></tr><tr><td>Dense2</td><td>Fully Connected</td><td>1024</td><td>ReLU</td><td>Dense1</td><td>Out</td></tr><tr><td>Out</td><td>Fully Connected</td><td>1</td><td></td><td>Dense2</td><td></td></tr></table>
 
-Table 9: Details our discriminator model. The final output Out provides the synthetic likelihoods  D(x,ˆy)1−D(x,ˆy) 
.
+Table 9: Details our discriminator model. The final output Out provides the synthetic likelihoods  D(x,ˆy)1−D(x,ˆy) 
.
 
 <table><tr><td>Layer</td><td>Type</td><td>Size</td><td>Activation</td><td>Input</td><td>Output</td></tr><tr><td>In1</td><td>Input</td><td></td><td></td><td>x,y</td><td>ConV1,1</td></tr><tr><td>ConV1,1</td><td>Conv2D</td><td>128</td><td>ReLU</td><td>In1</td><td>ConV1,2</td></tr><tr><td>ConV1,2</td><td>Conv2D</td><td>128</td><td>ReLU</td><td>ConV1,1</td><td>MaxPool1</td></tr><tr><td>MaxPool1</td><td>Max Pooling</td><td>2×2</td><td></td><td>ConV1,2</td><td>ConV2,1</td></tr><tr><td>Conv2,1</td><td>Conv2D</td><td>128</td><td>ReLU</td><td>MaxPool1</td><td>ConV2,2</td></tr><tr><td>ConV2,2</td><td>Conv2D</td><td>128</td><td>ReLU</td><td>ConV2,1</td><td>MaxPool2</td></tr><tr><td>MaxPool2</td><td>Max Pooling</td><td>2×2</td><td></td><td>ConV2,2</td><td>ConV3,1</td></tr><tr><td>ConV3,1</td><td>Conv2D</td><td>128</td><td>ReLU</td><td>MaxPool2</td><td>ConV4,1</td></tr><tr><td>ConV4,1</td><td>Conv2D</td><td>128</td><td>ReLU</td><td>ConV3,1</td><td>UpSamp1</td></tr><tr><td>UpSamp1</td><td>Up Sampling</td><td>2×2</td><td></td><td>ConV4,1</td><td>ConV5,1</td></tr><tr><td>ConV5,1 UpSamp2</td><td>Conv2D</td><td>128</td><td>ReLU</td><td>UpSamp1</td><td>UpSamp2</td></tr><tr><td></td><td>Up Sampling</td><td>2×2</td><td></td><td>ConV3,2</td><td>ConV4,1</td></tr><tr><td>ConV6,1</td><td>Conv2D</td><td>32</td><td></td><td>UpSamp2</td><td>21</td></tr><tr><td>ConV6,2</td><td>Conv2D</td><td>32</td><td></td><td>UpSamp2</td><td>2</td></tr><tr><td>ConV6,3</td><td>Conv2D</td><td>32</td><td></td><td>UpSamp2</td><td>23</td></tr></table>