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md/train/3RMnfrH_Fi8eU/3RMnfrH_Fi8eU.md

@@ -97,10 +97,10 @@ We see that our method significantly outperforms the other two in nearly all cas
 ![](images/134eaaa4b5fbdd071df8f60b4e320dcb8c1d7dd039cf1d1237f5fb76aa788dc9.jpg)
 
 ![](images/28ea8908bc1ea044fc80942d93bc29b2cc82bdbf3c33aa97fd86095eaef7791f.jpg)  
-!
&#"&-)%&(&%" \* '\* $^ { \prime = 1 2 8 }$ "\$('\* $: = 3 2$ ""\$(' "& $_ { ; = 9 6 }$ "('\$(' " $_ { : = 2 5 6 }$
+!
&#"&-)%&(&%" \* '\* $^ { \prime = 1 2 8 }$ "\$('\* $: = 3 2$ ""\$(' "& $_ { ; = 9 6 }$ "('\$(' " $_ { : = 2 5 6 }$
 
 ![](images/149f8e0328907b567859084864c2be98638b893cea5769fefb16f134df06533a.jpg)  
-!
&#"&-)%&(&'\* %" \* $^ { - 7 }$ "\$('\* $\mathtt { \Gamma } _ { \mathtt { = } } 3 2$ ""\$(' "& $_ { ; = 9 6 }$ "('\$(' " $_ { : = 2 5 6 }$   
+!
&#"&-)%&(&'\* %" \* $^ { - 7 }$ "\$('\* $\mathtt { \Gamma } _ { \mathtt { = } } 3 2$ ""\$(' "& $_ { ; = 9 6 }$ "('\$(' " $_ { : = 2 5 6 }$   
 Figure 3: Speed comparison with respect to size of input image (top), kernel size (middle) and minibatch size (bottom)
 
 We next ran experiments with parameter configurations typical of those used in different layers of a large convolutional network. The time taken by the different methods are given in milliseconds. The top row is a 4-tuple $( k , n , f , f ^ { \prime } )$ indicating the width of the kernel, width of the input image, number of input feature maps and number of output feature maps. All kernels and input images are square, of size $k \times k$ and $n \times n$ respectively. All configurations have minibatches of size 128. The first configuration represents the first layer, which is why we did not report times for the updateGradInput operation. For each configuration, the best-performing method is highlighted in bold.

md/train/H1xFWgrFPS/H1xFWgrFPS.md

@@ -159,14 +159,14 @@ Table 2: Identity preserving performance on three prediction tasks.
 
 <table><tr><td rowspan="2"></td><td colspan="2">CelebA:Smiling</td><td colspan="2">CelebA:Young</td><td colspan="2"> Xray:Cardiomegaly</td></tr><tr><td>xGEM</td><td>Ours</td><td>xGEM</td><td>Ours</td><td>xGEM</td><td>Ours</td></tr><tr><td>Latent Space Closeness</td><td>88.2</td><td>88.0</td><td>89.5</td><td>81.6</td><td>2.2</td><td>27.9</td></tr><tr><td>Face Verification Accuracy</td><td>0.0</td><td>85.3</td><td>0.0</td><td>72.2</td><td>1</td><td>1</td></tr></table>
 
-For population-level analysis, we plotted the average heart size of $\mathbf { x } _ { \delta }$ vs the condition used for generation $( f ( \mathbf { x } ) + \delta )$ in Figure 4 (a). The plot shows a positive correlation between the heart size and the response of the classifier $f ( \mathbf { x } )$ , which agrees with the definition of Cardiomegaly. To better understand the results, we divided the population into two groups, the first group $( \mathbf { x } ^ { h } ; f ( \mathbf { x } ^ { h } ) < 0 . 1 )$ consists of real images of healthy $\mathbf { X }$ -rays, and the second group $( \mathbf { x } ^ { c }$ $\mathfrak { x } ^ { c } ; f ( \mathbf { x } ^ { c } ) > 0 . 9 )$ contains real images of abnormal $\mathbf { X }$ -rays positive for Cardiomegaly. For $\mathbf { x } ^ { h }$ we generated counterfactual as $\mathbf { x } _ { \delta } ^ { c }$ such that $f ( \mathbf { x } _ { \delta } ^ { c } ) ~ > ~ 0 . 9$ . Similarly, counterfactuals for $\mathbf { x } ^ { c }$ are derived as $\mathbf { x } _ { \delta } ^ { h }$ such that $f ( \mathbf { x } _ { \delta } ^ { h } ) ~ <$ 0.1. In Figure 4 (b), we show the distribution of heart size in the four groups. We reported the dependent t-test statistics for paired samples $\mathbf { \Delta x } ^ { h }$ and $\mathbf { x } _ { \delta } ^ { c }$ 
, $\mathbf { \Psi } _ { \mathbf { X } } ^ { c }$ and $\mathbf { x } _ { \delta } ^ { h }$ 
. A significant $\mathsf { p }$ -value $\ll$ 0.001 rejected the null hypothesis (i.e., that the two groups have similar distributions). We also reported the independent two-sample t-test statistics for healthy  $\mathbf { \bar { x } } ^ { h }$ and $\mathbf { x } _ { \delta } ^ { h }$ , p-value $> 0 . 0 1$ 
 and abnormal $\mathbf { x } ^ { c }$ and $\mathbf { x } _ { \delta } ^ { c }$ , p-value $< 0 . 0 1 \AA \AA ,$ ) populations. Given higher p-values, we cannot reject the null hypothesis of identical average distributions with high confidence. Our model derived explanations successfully captured the change in heart size while generating counterfactual explanations.
+For population-level analysis, we plotted the average heart size of $\mathbf { x } _ { \delta }$ vs the condition used for generation $( f ( \mathbf { x } ) + \delta )$ in Figure 4 (a). The plot shows a positive correlation between the heart size and the response of the classifier $f ( \mathbf { x } )$ , which agrees with the definition of Cardiomegaly. To better understand the results, we divided the population into two groups, the first group $( \mathbf { x } ^ { h } ; f ( \mathbf { x } ^ { h } ) < 0 . 1 )$ consists of real images of healthy $\mathbf { X }$ -rays, and the second group $( \mathbf { x } ^ { c }$ $\mathfrak { x } ^ { c } ; f ( \mathbf { x } ^ { c } ) > 0 . 9 )$ contains real images of abnormal $\mathbf { X }$ -rays positive for Cardiomegaly. For $\mathbf { x } ^ { h }$ we generated counterfactual as $\mathbf { x } _ { \delta } ^ { c }$ such that $f ( \mathbf { x } _ { \delta } ^ { c } ) ~ > ~ 0 . 9$ . Similarly, counterfactuals for $\mathbf { x } ^ { c }$ are derived as $\mathbf { x } _ { \delta } ^ { h }$ such that $f ( \mathbf { x } _ { \delta } ^ { h } ) ~ <$ 0.1. In Figure 4 (b), we show the distribution of heart size in the four groups. We reported the dependent t-test statistics for paired samples $\mathbf { \Delta x } ^ { h }$ and $\mathbf { x } _ { \delta } ^ { c }$ 
, $\mathbf { \Psi } _ { \mathbf { X } } ^ { c }$ and $\mathbf { x } _ { \delta } ^ { h }$ 
. A significant $\mathsf { p }$ -value $\ll$ 0.001 rejected the null hypothesis (i.e., that the two groups have similar distributions). We also reported the independent two-sample t-test statistics for healthy  $\mathbf { \bar { x } } ^ { h }$ and $\mathbf { x } _ { \delta } ^ { h }$ , p-value $> 0 . 0 1$ 
 and abnormal $\mathbf { x } ^ { c }$ and $\mathbf { x } _ { \delta } ^ { c }$ , p-value $< 0 . 0 1 \AA \AA ,$ ) populations. Given higher p-values, we cannot reject the null hypothesis of identical average distributions with high confidence. Our model derived explanations successfully captured the change in heart size while generating counterfactual explanations.
 
 ![](images/d39888c553e45320ac15dc7ec307e7d51106beb70913ede77cb9227604f3c00a.jpg)  
 Figure 4: Cardiomegaly disease is associated with large heart size. In (a) we show the positive correlation between the heart size and the response of the classifier $f ( \mathbf { x } )$ . (b) Comparison of the distribution of the heart size in the four groups. (c) Plot to show the drop in accuracy of the classifier as we perturb the most relevant pixels (relevance calculated from saliency map) in the image.
 
 # 4.3 SALIENCY MAP
 
-Saliency maps show the importance of each pixel of an image in the context of classification. Our method is not designed to produce saliency maps as a continuous score for every feature of the input. We extract an approximate saliency map by quantifying the regions that changed the most when comparing explanations at the opposing ends of the classification spectrum. For each query image, we generated two visual explanations corresponding to the two extremes of the decision boundary  $\bar { \ b { f } } ( \mathbf { x } _ { \delta } ) = 0$ and $f ( \mathbf { x } _ { \delta } ) = 1 $ 
. The absolute difference between these explanations is our saliency map. Figure 5 shows the saliency map obtain from our method and its comparison with popular gradient based methods. We restricted the saliency maps obtained from different methods to have positive values and normalize them to range [0,1]. Subjective, the saliency maps produced by our method are very localized and are comparable to the other methods.
+Saliency maps show the importance of each pixel of an image in the context of classification. Our method is not designed to produce saliency maps as a continuous score for every feature of the input. We extract an approximate saliency map by quantifying the regions that changed the most when comparing explanations at the opposing ends of the classification spectrum. For each query image, we generated two visual explanations corresponding to the two extremes of the decision boundary  $\bar { \ b { f } } ( \mathbf { x } _ { \delta } ) = 0$ and $f ( \mathbf { x } _ { \delta } ) = 1 $ 
. The absolute difference between these explanations is our saliency map. Figure 5 shows the saliency map obtain from our method and its comparison with popular gradient based methods. We restricted the saliency maps obtained from different methods to have positive values and normalize them to range [0,1]. Subjective, the saliency maps produced by our method are very localized and are comparable to the other methods.
 
 We adapted the metric introduced in (Samek et al., 2016) to compare the different saliency maps. In an iterative procedure, we progressively replace a percentage of the most relevant pixels in an image (as given by the saliency map) with random values sampled from a uniform distribution. We observe the corresponding change in the classification performance as shown in Figure 4 (c). All the methods experienced a drop in the accuracy of the classifier with increase in the fraction of perturb pixels. The saliency maps produced by our model is significantly better than random maps and are comparable to the other saliency map methods. It should be noted that, there are many ways to quantify important regions in a image, using the series of explanations generated by our method. We didn’t optimize to find the best saliency map and showed results for one such method.
 

md/train/J28lNO4p3ki/J28lNO4p3ki.md

@@ -169,7 +169,7 @@ Remark 3 (The Optimal Batch Sizes and Local Updates Trade-off). The parameter $\
 10: end for   
 11: Return: $\scriptstyle { \bar { x } } _ { a }$ where $a \sim \mathcal { U } \{ 1 , . . . , T \}$ .
 
-Remark 4 (The Sub-Optimal Batch Sizes and Local Updates Trade-off). From our proof (Theorem $\left| \overline { { \mathbf { C . 1 0 } } } \right|$ included in the supplemental material), we can see that STEM requires $\bar { \tilde { O } } ( \operatorname* { m a x } \big \{ ( b \cdot$ $I ) \epsilon ^ { - 1 } , K ^ { - 1 } \epsilon ^ { - 3 / 2 } \rbrace )$ samples and $\tilde { \mathcal { O } } \big ( \operatorname* { m a x } \big \{ \epsilon ^ { - 1 } , ( b \cdot I ) ^ { - 1 } K ^ { - 1 } \epsilon ^ { - 3 / 2 } \big \} \big )$  and communication rounds. According to the above expressions, if $b \cdot I$ increases beyond $\mathcal { O } ( K ^ { - 1 } \epsilon ^ { - 1 / 2 } )$ , then the sample complexity will increase from the optimal $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ ; otherwise, the optimal sample complexity $\tilde { \mathcal O } ( \epsilon ^ { - 3 / 2 } )$ is maintained. On the other hand, if $b \cdot I$ decreases beyond $\mathcal { O } ( K ^ { - 1 } \epsilon ^ { - 1 / 2 } )$ , the communication complexity increases from $\tilde { \mathcal { O } } ( \epsilon ^ { - 1 } )$ . For instance, if we choose $b = \mathcal { O } ( 1 )$ and $I = { \mathcal { O } } ( 1 )$ the communication complexity becomes $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ while the optimal sample complexity $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ is maintained. This trade-off is illustrated in Figure $\boxed { 1 \mathrm { a } }$ where we maintain the optimal sample complexity, while changing $b$ and $I$ to generate the trade-off surface. □
+Remark 4 (The Sub-Optimal Batch Sizes and Local Updates Trade-off). From our proof (Theorem $\left| \overline { { \mathbf { C . 1 0 } } } \right|$ included in the supplemental material), we can see that STEM requires $\bar { \tilde { O } } ( \operatorname* { m a x } \big \{ ( b \cdot$ $I ) \epsilon ^ { - 1 } , K ^ { - 1 } \epsilon ^ { - 3 / 2 } \rbrace )$ samples and $\tilde { \mathcal { O } } \big ( \operatorname* { m a x } \big \{ \epsilon ^ { - 1 } , ( b \cdot I ) ^ { - 1 } K ^ { - 1 } \epsilon ^ { - 3 / 2 } \big \} \big )$  and communication rounds. According to the above expressions, if $b \cdot I$ increases beyond $\mathcal { O } ( K ^ { - 1 } \epsilon ^ { - 1 / 2 } )$ , then the sample complexity will increase from the optimal $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ ; otherwise, the optimal sample complexity $\tilde { \mathcal O } ( \epsilon ^ { - 3 / 2 } )$ is maintained. On the other hand, if $b \cdot I$ decreases beyond $\mathcal { O } ( K ^ { - 1 } \epsilon ^ { - 1 / 2 } )$ , the communication complexity increases from $\tilde { \mathcal { O } } ( \epsilon ^ { - 1 } )$ . For instance, if we choose $b = \mathcal { O } ( 1 )$ and $I = { \mathcal { O } } ( 1 )$ the communication complexity becomes $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ while the optimal sample complexity $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ is maintained. This trade-off is illustrated in Figure $\boxed { 1 \mathrm { a } }$ where we maintain the optimal sample complexity, while changing $b$ and $I$ to generate the trade-off surface. □
 
 Remark 5 (Data Heterogeneity). The term $\begin{array} { r } { \tilde { \mathcal { O } } \biggl ( \frac { \zeta ^ { 2 } } { K ^ { 2 \nu / 3 } T ^ { 1 - \nu / 3 } } \biggr ) } \end{array}$ in the gradient bound $\textcircled{4}$ captures the effect of the heterogeneity of data across WNs, where $\zeta$ is the parameter characterizing the intra-node variance and has been defined in Assumption $\bigstar$ (ii). Highly heterogeneous data with large $\zeta ^ { 2 }$ can adversely impact the performance of STEM. Note that such a dependency on $\zeta$ also appears in other existing FL algorithms, such as $[ \bigcirc , \bigcirc , \bigcirc , \bigcirc , \bigcirc ]$ . However, there is one special case of STEM that does not depend on the parameter $\zeta$ . This is the case where $I = 1$ , i.e., the minibatch SGD counterpart of STEM where only a single local iteration is performed between two communication rounds. We have the following corollary. □
 

md/train/Z2vksUFuVst/Z2vksUFuVst.md

@@ -145,7 +145,7 @@ $$
 \begin{array} { r l } & { F _ { Z ^ { \pi } ( s , a ) } ^ { - 1 } ( \tau ) \geq F _ { \tilde { Z } ^ { \pi } ( s , a ) } ^ { - 1 } ( \tau ) + ( 1 - \gamma ) ^ { - 1 } \underset { s ^ { \prime } , a ^ { \prime } } { \operatorname* { m i n } } \{ c ( s ^ { \prime } , a ^ { \prime } ) - \Delta ( s ^ { \prime } , a ^ { \prime } ) \} , } \\ & { F _ { Z ^ { \pi } ( s , a ) } ^ { - 1 } ( \tau ) \leq F _ { \tilde { Z } ^ { \pi } ( s , a ) } ^ { - 1 } ( \tau ) + ( 1 - \gamma ) ^ { - 1 } \underset { s ^ { \prime } , a ^ { \prime } } { \operatorname* { m a x } } \{ c ( s ^ { \prime } , a ^ { \prime } ) - \Delta ( s ^ { \prime } , a ^ { \prime } ) \} } \end{array}
 $$
 
-for all sufficie $s \in \mathcal { D }$ $a \in { \mathcal { A } } ,$ $\tau \in [ 0 , 1 ]$ $\begin{array} { r } { \Delta ( s , a ) = \frac { 1 } { \zeta } \sqrt { \frac { 5 | S | } { n ( s , a ) } \log \frac { 4 | S | | A | } { \delta } } } \end{array}$ rmore, for . $\alpha$ ntly large (i.e., ↵  maxs,a{ p·(s,a)p1c (s,a) } $F _ { Z ^ { \pi } ( s , a ) } ^ { - 1 } ( \tau ) \geq F _ { \tilde { Z } ^ { \pi } ( s , a ) } ^ { - 1 } ( \tau ) .$
+for all sufficie $s \in \mathcal { D }$ $a \in { \mathcal { A } } ,$ $\tau \in [ 0 , 1 ]$ $\begin{array} { r } { \Delta ( s , a ) = \frac { 1 } { \zeta } \sqrt { \frac { 5 | S | } { n ( s , a ) } \log \frac { 4 | S | | A | } { \delta } } } \end{array}$ rmore, for . $\alpha$ ntly large (i.e., ↵  maxs,a{ p·(s,a)p1c (s,a) } $F _ { Z ^ { \pi } ( s , a ) } ^ { - 1 } ( \tau ) \geq F _ { \tilde { Z } ^ { \pi } ( s , a ) } ^ { - 1 } ( \tau ) .$
 
 We give a proof in Appendix $\boxed { \mathbf { A . 2 } }$ The first inequality says that the quantile estimates computed by CDE form a lower bound on the true quantiles; this bound is not vacuous as long as $\alpha$ satisfies the given condition. Furthermore, the second inequality states that this lower bound is tight.
 

md/train/vlcVTDaufN/vlcVTDaufN.md

@@ -130,7 +130,7 @@ a given $\beta$ and given batch samples $( x , y )$ , we can then define loss as
 
 To illustrate gradient descent over a combinatorial loss, we first focus on a simple image recognition problem. Consider a photo of a group of people with a caption listing each of the persons in the picture, but missing the ”from left to right” part. Given a collection of such labeled photos, can a model learn to recognize individual faces? Similarly, consider a shopping cart and a printout from the register. Given a collection of unordered shopping carts together with matching receipts, can a model learn to recognize individual shopping items? These are example of a weakly-supervised learning where the goal is to learn to classify previously unseen feature vectors, but a training sample is a bag of feature vectors accompanied by a bag of correct labels, instead of a feature-vector and a correct label. We are not told which class belongs to which sample, which prevents us from directly using the standard cross-entropy loss.
 
-More formally, consider a $d$ -class classification problem, and a model $F ( x _ { j } ; \beta )$ that for sample $x _ { j }$ returns a $d$ -dimensional vector of class probabilities, $p _ { j }$ , with $\mathbf { \Delta } _ { p _ { j } ^ { c } } ^ { p _ { j } ^ { c } }$ denoting the predicted conditional probability of class $c$ given feature vector $x _ { j }$ . Let $y _ { j }$ denote a $d$ -dimensional, one-hot representation of the true class label of sample $x _ { j }$ , with $y _ { j } ^ { c } = 1$ if sample $j$ is of class $c$ , and zero otherwise. In weakly supervised learning involving bags of size $b$ , we are given a tuple of $b$ feature vectors, $\boldsymbol { X } = \left( \boldsymbol { x } _ { j } \right) _ { j = 1 } ^ { b }$ , and a tuple of permuted labels Y = yσ(i)
bi=1 as one-hot-vectors, for some permutation $\sigma$ ; we will refer to the $j$ -th element of the tuple $Y$ as $\bar { Y } _ { j }$ . The permutation $\sigma$ is unknown, thus using a loss $\ell ( p _ { j } , Y _ { j } ) = \ell ( p _ { j } , y _ { \sigma ( i ) } )$ to compare predicted distribution over classes for sample $j$ with one-hot representation of $j$ -th element in the randomly ordered set of true classes $Y _ { j }$ makes no sense, since most likely $i \neq j ; Y _ { j } = y _ { \sigma ( i ) }$ is the class for some other sample $i$ , not for sample $j$ .
+More formally, consider a $d$ -class classification problem, and a model $F ( x _ { j } ; \beta )$ that for sample $x _ { j }$ returns a $d$ -dimensional vector of class probabilities, $p _ { j }$ , with $\mathbf { \Delta } _ { p _ { j } ^ { c } } ^ { p _ { j } ^ { c } }$ denoting the predicted conditional probability of class $c$ given feature vector $x _ { j }$ . Let $y _ { j }$ denote a $d$ -dimensional, one-hot representation of the true class label of sample $x _ { j }$ , with $y _ { j } ^ { c } = 1$ if sample $j$ is of class $c$ , and zero otherwise. In weakly supervised learning involving bags of size $b$ , we are given a tuple of $b$ feature vectors, $\boldsymbol { X } = \left( \boldsymbol { x } _ { j } \right) _ { j = 1 } ^ { b }$ , and a tuple of permuted labels Y = yσ(i)
bi=1 as one-hot-vectors, for some permutation $\sigma$ ; we will refer to the $j$ -th element of the tuple $Y$ as $\bar { Y } _ { j }$ . The permutation $\sigma$ is unknown, thus using a loss $\ell ( p _ { j } , Y _ { j } ) = \ell ( p _ { j } , y _ { \sigma ( i ) } )$ to compare predicted distribution over classes for sample $j$ with one-hot representation of $j$ -th element in the randomly ordered set of true classes $Y _ { j }$ makes no sense, since most likely $i \neq j ; Y _ { j } = y _ { \sigma ( i ) }$ is the class for some other sample $i$ , not for sample $j$ .
 
 While the permutation is unknown, with repeated presentation of bags of samples and bags of corresponding labels, we do have some information connecting the feature vector to classes. Intuitively, we can try to match model’s outputs for feature vectors in the bag to the class labels using the information in the probability distribution $p _ { j }$ over classes provided by the model for each feature vector $x _ { j }$ . That is, we can aim to find permutation $\hat { \sigma }$ optimal in the average loss sense $\begin{array} { r } { \operatorname* { m i n } _ { \hat { \sigma } } \sum _ { j = 1 } ^ { b } \ell ( p _ { j } , \hat { \sigma } ( \bar { Y } ) _ { j } ) , } \end{array}$ . If the class conditional probabilities $p _ { j }$ resulting from the model perfectly match the one-hot vectors, the optimal $\hat { \sigma }$ will be the inverse of the permutation $\sigma$ , that is, $\hat { \sigma } ( Y ) _ { j } = y _ { j }$ .