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md/train/-_FVMKvxVCQo1/-_FVMKvxVCQo1.md

@@ -230,7 +230,7 @@ where $T _ { \mathrm { m i n } }$ was set to a constant 50 to avoid stopping too
 12 $( \alpha _ { \bullet } , \mathbf { v _ { \bullet } } , \varphi _ { \bullet } ( \cdot ) )  \mathrm { B A S E } ( \mathbf { X _ { \bullet } } , \mathbf { Y _ { \bullet } } , \mathbf { W _ { \bullet } } )$   
 13 $\mathrm { I N S E R T } \big ( S , \big ( \mathbf { v _ { * } } , \varphi _ { \bullet } ( \cdot ) , \mathbf { X _ { \bullet } } , \mathbf { Y _ { \bullet } } , \bullet , j \big ) , \gamma ( \mathbf { v _ { * } } , \varphi _ { \bullet } , \mathbf { X _ { \bullet } } , \mathbf { Y _ { \bullet } } , \mathbf { W _ { \bullet } } ) - \gamma ( \mathbf { v } _ { j } , \varphi _ { j } , \mathbf { X _ { \bullet } } , \mathbf { Y _ { \bullet } } , \mathbf { W _ { \bullet } } ) \big )$ . key = edge improvement over parent edge 1 1 + γ(h1, W)   
 14 α = 2 log 1 − γ(h1, W) . standard coefficient of the full tree classifier ${ \mathfrak { h } } _ { 1 }$ (14) 2   
-15 return α, H, l, r

+15 return α, H, l, r

 
 Figure 2. The pseudocode of the Hamming tree base learner. $N$ is the number of inner nodes. The algorithm returns a list of base classifiers ${ \mathfrak H }$ , two index lists l and $\mathfrak { r }$ , and the base coefficient $\alpha$ . The tree classifier is then defined by (14).   
 Table 1. Test error percentages on mid-sized benchmark data sets.   

md/train/Sy-tszZRZ/Sy-tszZRZ.md

@@ -136,7 +136,7 @@ Proof. $\dim ( { \mathcal S } ) \ = \ \operatorname { r a n k } ( \sigma _ { S ^
 
 In the remainder of the proof of Theorem 1, we combine Lemmas 3 and 4 to construct a recurrence $R ( l , d )$ that bounds the number of regions within a given region of dimension $d$ . Simplifying this recurrence yields the expression in Theorem 1. We formalize this idea and complete the proof of Theorem 1 in Appendix D.
 
-As a side note, Theorem 1 can be further tightened if the weight matrices are known to have small rank. The bound from Lemma 3 can be rewritten as j=0 Pmin{rank(W l),dim(S)} nl
 if we do not relax $\mathrm { r a n k } ( W ^ { l } )$ to $n _ { l }$ in the proof. The term $\mathrm { r a n k } ( W ^ { l } )$ follows through the proof of Theorem 1 and the index set $J$ in the theorem becomes $\{ ( j _ { 1 } , \dotsc , j _ { L } ) \in \mathbb { Z } ^ { L } : 0 \leq j _ { l } \leq \operatorname* { m i n } \{ n _ { 0 } , n _ { 1 } - j _ { 1 } , \dotsc , n _ { l - 1 } -$ $j _ { l - 1 } , \mathrm { r a n k } ( W ^ { l } ) \} \forall l \geq 1 \}$ .
+As a side note, Theorem 1 can be further tightened if the weight matrices are known to have small rank. The bound from Lemma 3 can be rewritten as j=0 Pmin{rank(W l),dim(S)} nl
 if we do not relax $\mathrm { r a n k } ( W ^ { l } )$ to $n _ { l }$ in the proof. The term $\mathrm { r a n k } ( W ^ { l } )$ follows through the proof of Theorem 1 and the index set $J$ in the theorem becomes $\{ ( j _ { 1 } , \dotsc , j _ { L } ) \in \mathbb { Z } ^ { L } : 0 \leq j _ { l } \leq \operatorname* { m i n } \{ n _ { 0 } , n _ { 1 } - j _ { 1 } , \dotsc , n _ { l - 1 } -$ $j _ { l - 1 } , \mathrm { r a n k } ( W ^ { l } ) \} \forall l \geq 1 \}$ .
 
 A key insight from Lemmas 3 and 4 is that the dimensions of the regions are non-increasing as we move through the layers partitioning it. In other words, if at any layer the dimension of a region becomes small, then that region will not be able to be further partitioned into a large number of regions. For instance, if the dimension of a region falls to zero, then that region will never be further partitioned. This suggests that if we want to have many regions, we need to keep dimensions high. We use this idea in the next section to construct a DNN with many regions.
 
@@ -419,7 +419,7 @@ Proof. As illustrated in Figure 1, the partitioning can be viewed as a sequentia
 
 With this process in mind, we recursively bound the number of subregions within a region. More precisely, we construct a recurrence $R ( l , d )$ to be an upper bound to the maximal number of regions obtained from partitioning a region of dimension $d$ with layers $l , l + 1 , \ldots , L$ . The base case of the recurrence is given by Lemma 3: $\begin{array} { r } { R ( L , d ) = \sum _ { j = 0 } ^ { \operatorname* { m i n } \{ n _ { L } , d \} } \binom { n _ { L } } { j } } \end{array}$ . Based on Lemma 4, we can write the recurrence by grouping together regions with the same activation set size $\begin{array} { r } { R ( l , d ) = \sum _ { j = 0 } ^ { n _ { l } } { N _ { n _ { l } , d , j } \bar { R } ( l ^ { ' } + 1 , \bar { \operatorname* { m i n } } \{ j , d \} ) } } \end{array}$ for all $l = 1 , \ldots , L - 1$ . Here, $N _ { n _ { l } , d , j }$ $| S ^ { l } |$ represents the , as follows: maximum number of regions with $| S ^ { l } | = j$ obtained by partitioning a space of dimension $d$ with $n _ { l }$ hyperplanes. We bound this value next.
 
-For each $j$ , there are at most $\binom { n _ { l } } { j }$ regions with $| S ^ { l } | = j$ , as they can be viewed as subsets of $n _ { l }$ neurons of size $j$ . In total, Lemma 3 states that there are at most Pmin{nl,d}j=0 nlj 
 regions. If we allow these regions to have the highest $| S ^ { l } |$ possible, for each $j$ from 0 to $\operatorname* { m i n } \{ n _ { l } , d \}$ we have at most $\textstyle { \binom { n _ { l } } { n _ { l } - j } } = { \binom { n _ { l } } { j } }$ regions with $\vert S ^ { l } \vert = n _ { l } - j$ .
+For each $j$ , there are at most $\binom { n _ { l } } { j }$ regions with $| S ^ { l } | = j$ , as they can be viewed as subsets of $n _ { l }$ neurons of size $j$ . In total, Lemma 3 states that there are at most Pmin{nl,d}j=0 nlj 
 regions. If we allow these regions to have the highest $| S ^ { l } |$ possible, for each $j$ from 0 to $\operatorname* { m i n } \{ n _ { l } , d \}$ we have at most $\textstyle { \binom { n _ { l } } { n _ { l } - j } } = { \binom { n _ { l } } { j } }$ regions with $\vert S ^ { l } \vert = n _ { l } - j$ .
 
 Therefore, we can write the recurrence as
 

md/train/URc7gYBcjVn/URc7gYBcjVn.md

@@ -364,7 +364,7 @@ stant 305 $\omega _ { M }$ such that, for all $k \in [ K ] \colon \mathbb { E }
 
 The first statement stems from the fact that each bucket is quantized using StoVoQ which is unbiased. The second statement is more challenging; proof is postponed to Appendix A.6. We stress that this result differs from Theorem 2, which corresponds to the distortion of a source with distribution $q$ .
 
-Convergence results. Theorem 4 proves that our compression method satisfies the assumptions needed to obtain fast convergence rate, for DoStoVoQ-SGD, and for its variants DoStoVoQ-(VR)- DIANA. Consider a Smooth and Strongly Convex (SSC) function $\textstyle F = \sum _ { k = 1 } ^ { K } f _ { k }$ , with condition number $\kappa > 1$ . We measure the complexity of the algorithm by the number of iterations $t$ required to obtain a model $\theta _ { t }$ such that $\mathbb { E } [ F ( \theta _ { t } ) ] - \operatorname* { m i n } _ { \mathbb { R } ^ { D } } F \leq \epsilon .$ The result of VR-DIANA [16], which provides a complexity of $O _ { \kappa \to \infty }$  $\kappa ( 1 + \omega _ { M } / K ) \log ( \epsilon ^ { - 1 } ) )$ [16, Corollary 2], applies to DoStoVoQVR-DIANA.
+Convergence results. Theorem 4 proves that our compression method satisfies the assumptions needed to obtain fast convergence rate, for DoStoVoQ-SGD, and for its variants DoStoVoQ-(VR)- DIANA. Consider a Smooth and Strongly Convex (SSC) function $\textstyle F = \sum _ { k = 1 } ^ { K } f _ { k }$ , with condition number $\kappa > 1$ . We measure the complexity of the algorithm by the number of iterations $t$ required to obtain a model $\theta _ { t }$ such that $\mathbb { E } [ F ( \theta _ { t } ) ] - \operatorname* { m i n } _ { \mathbb { R } ^ { D } } F \leq \epsilon .$ The result of VR-DIANA [16], which provides a complexity of $O _ { \kappa \to \infty }$  $\kappa ( 1 + \omega _ { M } / K ) \log ( \epsilon ^ { - 1 } ) )$ [16, Corollary 2], applies to DoStoVoQVR-DIANA.
 
 Convergence rates for DoStoVoQ-DIANA (without VR), and on non-convex optimization problems can be obtained from Horváth et al. [16, Corollary 1,3,4]. As in the strongly-convex case, complexities increase by a factor depending on $( 1 + \omega _ { M } / K )$ w.r.t. uncompressed algorithm. Intuitively, the impact on the optimization complexity of a high compression is mitigated by the number of workers, which supports the use of independent and unbiased compressors when the number of workers is large and high compression factors are required.
 

md/train/rylDfnCqF7/rylDfnCqF7.md

@@ -68,7 +68,7 @@ $$
 \begin{array} { r l } { \mathcal { L } ( \mathbf { x } ; \theta , \phi ) = } & { { } \underbrace { \log p _ { \theta } ( \mathbf { x } ) } _ { \mathrm { ~ \forall ~ \theta ~ \equiv ~ \phi ~ , ~ \dots ~ } } - \underbrace { D _ { \mathrm { K L } } ( q _ { \phi } ( \mathbf { z } | \mathbf { x } ) | | p _ { \theta } ( \mathbf { z } | \mathbf { x } ) ) } _ { \mathrm { ~ \forall ~ \theta ~ \equiv ~ \phi ~ , ~ \dots ~ } } } \end{array}
 $$
 
-3.2 OBSERVATIONS ON SYNTHETIC DATAWith this view, the only goal of approximate posterior $q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ is to match model posterior $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ , while the optimization of $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ is influenced by two forces, one of which is the ideal objecAs a synthetic dataset we use discrete sequenctive marginal data likelihood, and the other is $D _ { \mathrm { K L } } ( q _ { \phi } ( \mathbf { z } | \mathbf { x } ) \| p _ { \theta } ( \mathbf { z } | \mathbf { x } ) )$ pse has been fo, which drives $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ mos tosevewards $q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ t modeling tasks. To mimic the diversity of the real data distribution, we sample discrete. Ideally if the approximate posterior is perfect, the second force will vanish, with $\nabla _ { \pmb \theta } D _ { \mathrm { K L } } \big ( q _ { \phi } ( \mathbf { z } | \mathbf { x } ) | p _ { \pmb \theta } ( \mathbf { z } | \mathbf { x } ) \big ) = 0$ odel, wwhen $q _ { \phi } ( \mathbf { z } | \mathbf { x } ) = p _ { \theta } ( \mathbf { z } | \mathbf { x } )$ xture of Gaussian prior — f. At the start of training, $\mathbf { z }$ ur-mand $\mathbf { x }$ es ofare a Gaussian mixture as a prionearly independent under both $q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ an  and $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ enerator. More details about this sy as we show in Section 3.2, i.e. all $\mathbf { x }$ hetic dataset suffer from and experiment details can be found in Appendix B.1.model collapse in the beginning. Then the only component in the training objective that possibly causes dependence between $\mathbf { z }$ and $\mathbf { x }$ under $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ is $\log p _ { \pmb { \theta } } ( \mathbf { x } )$ 2. However, this pressure may be overwhelmed by the KL term when $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ and $q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ start to diverge but $\mathbf { z }$ and $\mathbf { x }$ remain independent under $q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ . We hypothesize that, in practice, training drives $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ and $q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ to t he prior in order to bring them into alignment, while locking into model parameters that capture the distribution of $\mathbf { x }$ while ignoring $\mathbf { z }$ x,          x,✓ . Critically, posterior collapse is a local optimum; once a set of paXrameters that achieves these goals are reached, gradient optimization fails to make further progress, E [even if better overall models that make use of $\mathbf { z }$ = [zipto describe $\mathbf { x }$ i|x)], exist.
+3.2 OBSERVATIONS ON SYNTHETIC DATAWith this view, the only goal of approximate posterior $q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ is to match model posterior $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ , while the optimization of $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ is influenced by two forces, one of which is the ideal objecAs a synthetic dataset we use discrete sequenctive marginal data likelihood, and the other is $D _ { \mathrm { K L } } ( q _ { \phi } ( \mathbf { z } | \mathbf { x } ) \| p _ { \theta } ( \mathbf { z } | \mathbf { x } ) )$ pse has been fo, which drives $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ mos tosevewards $q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ t modeling tasks. To mimic the diversity of the real data distribution, we sample discrete. Ideally if the approximate posterior is perfect, the second force will vanish, with $\nabla _ { \pmb \theta } D _ { \mathrm { K L } } \big ( q _ { \phi } ( \mathbf { z } | \mathbf { x } ) | p _ { \pmb \theta } ( \mathbf { z } | \mathbf { x } ) \big ) = 0$ odel, wwhen $q _ { \phi } ( \mathbf { z } | \mathbf { x } ) = p _ { \theta } ( \mathbf { z } | \mathbf { x } )$ xture of Gaussian prior — f. At the start of training, $\mathbf { z }$ ur-mand $\mathbf { x }$ es ofare a Gaussian mixture as a prionearly independent under both $q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ an  and $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ enerator. More details about this sy as we show in Section 3.2, i.e. all $\mathbf { x }$ hetic dataset suffer from and experiment details can be found in Appendix B.1.model collapse in the beginning. Then the only component in the training objective that possibly causes dependence between $\mathbf { z }$ and $\mathbf { x }$ under $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ is $\log p _ { \pmb { \theta } } ( \mathbf { x } )$ 2. However, this pressure may be overwhelmed by the KL term when $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ and $q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ start to diverge but $\mathbf { z }$ and $\mathbf { x }$ remain independent under $q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ . We hypothesize that, in practice, training drives $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ and $q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ to t he prior in order to bring them into alignment, while locking into model parameters that capture the distribution of $\mathbf { x }$ while ignoring $\mathbf { z }$ x,          x,✓ . Critically, posterior collapse is a local optimum; once a set of paXrameters that achieves these goals are reached, gradient optimization fails to make further progress, E [even if better overall models that make use of $\mathbf { z }$ = [zipto describe $\mathbf { x }$ i|x)], exist.
 
 Next we visualize the posterior mean space by training a basic VAE with a scalar latent variable on C           a relatively simple synthetic dataset to examine our hypothesis.
 
@@ -76,7 +76,7 @@ Next we visualize the posterior mean space by training a basic VAE with a scalar
 
 second stage, the points starts to spread along the µx,✓ axis. This phenomenon implies that for someAs a synthetic dataset we use discrete sequence data since posterior collapse has been found the most data points p✓(z|x) moves far away from the prior p(z), and confirms that log p✓(x) is able to helpsevere in text modeling tasks. Details on this synthetic dataset and experiment are in Appendix B.1.
 
-that q(z x) fails to catch up to p✓(z x) and these points are still in an inference collapsed state.We train a basic VAE with a scalar latent variable, LSTM encoder, and LSTM decoder on our synthetic dataset. We sample 500 data points from the validation set and show them on the posterior This LSTM decoder has less capacity than the one used for creating the dataset since in real world modelmean space plots at four different training stages from initialization to convergence in Figure 2. The capacity is usually insufficient to exactly model thmean of the approximate posterior distribution $\mu _ { \mathbf { x } , \phi }$ pirical distribution.is from the output of the inference network, and $\mu _ { \mathbf { x } , \theta }$ can be approximated by discretization of the true model posterior $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ (see Appendix A).
+that q(z x) fails to catch up to p✓(z x) and these points are still in an inference collapsed state.We train a basic VAE with a scalar latent variable, LSTM encoder, and LSTM decoder on our synthetic dataset. We sample 500 data points from the validation set and show them on the posterior This LSTM decoder has less capacity than the one used for creating the dataset since in real world modelmean space plots at four different training stages from initialization to convergence in Figure 2. The capacity is usually insufficient to exactly model thmean of the approximate posterior distribution $\mu _ { \mathbf { x } , \phi }$ pirical distribution.is from the output of the inference network, and $\mu _ { \mathbf { x } , \theta }$ can be approximated by discretization of the true model posterior $p _ { \pmb { \theta } } ( \mathbf { z } | \mathbf { x } )$ (see Appendix A).
 
 <table><tr><td>Algorithm1 VAE training with controlled aggressive inference network optimization.</td></tr><tr><td>1:0, ← Initialize parameters</td></tr><tr><td>2:aggressive ←TRUE</td></tr><tr><td>3: repeat</td></tr><tr><td>4: if aggressive then</td></tr><tr><td>5: repeat</td></tr><tr><td>[aggressive updates] X←Randomdata minibatch</td></tr><tr><td>Compute gradients gΦ ← VL(X;0,Φ)</td></tr><tr><td>Update Φ using gradients gΦ</td></tr><tr><td>until convergence</td></tr><tr><td>9: 10:</td></tr><tr><td>X←Randomdata minibatch Compute gradients ge ← VθL(X; 0,Φ)</td></tr><tr><td>11: 12: Update θ using gradients ge</td></tr><tr><td>13: else</td></tr><tr><td>14: X←Randomdataminibatch</td></tr><tr><td>15: Compute gradients g0,𝜙 ← V,θL(X;0,𝜙)</td></tr><tr><td>16: Update 0,Φ using ge,𝜙</td></tr><tr><td>end if</td></tr><tr><td>17:</td></tr><tr><td>18: Update aggressive as discussed in Section 4.2 19: until convergence</td></tr></table>