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md/train/SyxZOsA9tX/SyxZOsA9tX.md

@@ -247,7 +247,7 @@ Input: ${ \pmb v } ^ { ( 0 ) } , P , { \pmb r } , \gamma , k , T$
 9:   
 10: else   
 11: $\mathbf { \tilde { \alpha } } _ { 1 } ^ { ( t ) } = 1 , \alpha _ { i } ^ { ( t ) } = 0$ for i 6= 1   
-12: v(t) = max rπ + γPπv(t−1)
   
+12: v(t) = max rπ + γPπv(t−1)
   
 13: end if   
 14: end if   
 15: end for   

md/train/ZdJQ8KekIyd/ZdJQ8KekIyd.md

@@ -269,7 +269,7 @@ Figure 2: The data-generating process for the observational data $\left\{ X ^ {
 214 Gibbs sampling is a well-known MCMC algorithm that allows one to sample posterior distributions.   
 215 For convenience, we introduce the following notations. Let parameters $\pmb { \theta } = \{ \theta _ { u } | \forall U \in U , \forall u \}$   
 216 and ⇠ = n ⇠ (paV ,uV )V | $\bigstar \bigstar = \Big \{ \xi _ { V } ^ { ( p a _ { V } , u _ { V } ) } | \forall V \in V , \forall p a _ { V } , u _ { V } \Big \}$ . The set $\bar { U } = \left\{ U ^ { ( n ) } \right\} _ { n = 1 } ^ { N }$ are exogenous variables   
-217 affecting N observations V¯ =  V (n) N n=1; we use u¯ to represent their realizations. Our blocked   
+217 affecting N observations V¯ =  V (n) N n=1; we use u¯ to represent their realizations. Our blocked   
 218 Gibbs sampler works by iteratively drawing values from the conditional distributions of variables as   
 219 follows $\lVert \hat { 2 2 } \rVert$ . Detailed derivations of complete conditional distributions are shown in Appendix F.   
 220 Sampling $P \left( \bar { \pmb { u } } | \bar { \pmb { v } } , \pmb { \theta } , \pmb { \xi } \right)$ . Exogenous variables $U ^ { ( n ) }$ , $n = 1 , \ldots , N$ , are mutually independent   
@@ -351,7 +351,7 @@ $$
 260 observations $\underline { { \| 2 0 \| } } , \boxed { 4 7 } , \boxed { 3 7 } , \boxed { 8 } , \boxed { 4 6 } $ . As the number of observational data $N$ grows (to infinite), the $1 0 0 \%$   
 261 credible interval $[ l _ { 0 } , r _ { 0 } ]$ eventually converges to the optimal asymptotic bound $[ l , r ]$ in Eq. $( 6 )$ [11].
 
-Let  ✓(t) T be $T$ samples drawn from $P \left( \theta _ { \mathrm { c t f } } \mid \bar { \mathbf { v } } \right)$ . One could compute the $1 0 0 ( 1 - \alpha ) \%$ credible interval for $\bar { \theta _ { \mathrm { c t f } } }$ using the following consistent estimators $\pmb { \| 3 9 \| }$ :
+Let  ✓(t) T be $T$ samples drawn from $P \left( \theta _ { \mathrm { c t f } } \mid \bar { \mathbf { v } } \right)$ . One could compute the $1 0 0 ( 1 - \alpha ) \%$ credible interval for $\bar { \theta _ { \mathrm { c t f } } }$ using the following consistent estimators $\pmb { \| 3 9 \| }$ :
 
 $$
 \hat { l } _ { \alpha } ( T ) = \theta ^ { ( \lceil ( \alpha / 2 ) T \rceil ) } , \hat { r } _ { \alpha } ( T ) = \theta ^ { ( \lceil ( 1 - \alpha / 2 ) T \rceil ) } ,
@@ -417,7 +417,7 @@ generated SCM. Fig. 4a shows samples drawn from the posterior distribution of th
 311 unobserved confounding between $X$ and $Y$ has been acknowledged in [5]. For binary $X , Y , Z$ , [2]   
 312 derived closed-form, sharp bounds over $P ( y _ { x } )$ (labelled as opt). We collect $N = 1 0 ^ { 5 }$ observational   
 313 samples $\bar { \cal V } = \{ X ^ { ( n ) } , Y ^ { ( n ) } , Z ^ { ( n ) } \} _ { n = 1 } ^ { N }$ from a randomly generated SCM instance. Fig. $4  { \mathbf { b } }$ shows   
-314 samples drawn from the posterior distribution of  $P ( Y _ { x = 0 } = 1 ) \mid \bar { V } )$ . As a baseline, we also include   
+314 samples drawn from the posterior distribution of  $P ( Y _ { x = 0 } = 1 ) \mid \bar { V } )$ . As a baseline, we also include   
 315 the optimal bound opt, and posterior samples obtained from the Gibbs sampler of $\mathbb { \ m }$ , which utilizes   
 316 the canonical partitions of exogenous domains in $\pmb { \Vert 2 \Vert }$ $( b p )$ . The analysis reveals that our algorithm   
 317 derives the valid bound over the actual probability $P ( Y _ { x = 0 } = 1 ) = 0 . 3 9 5 4$ ; the $1 0 0 \%$ credible   

md/train/rkgKBhA5Y7/rkgKBhA5Y7.md

@@ -370,7 +370,7 @@ $\mathfrak { L } _ { d } [ R _ { \mathrm { M S E } } ( w + s d ) ] - R _ { \math
 
 # A.7 INCLUDING HIGH LEARNING RATE ITERATES INTO SWA
 
-As discussed in Mandt et al. (2017), under certain assumptions SGD samples from a Gaussian distribution centered at the optimum of the loss $w _ { 0 }$ with covariance proportional to the learning rate. Suppose then that we have $n$ weights sampled at learning rate $\eta _ { 1 }$ , $w _ { i } ^ { ( 1 ) } \stackrel { i i d } { \sim } \mathcal { N } ( w _ { 0 } , \eta _ { 1 } \Sigma )$ and $m$ weights sampled with the higher learning rate $\eta _ { 2 }$ , $w _ { j } ^ { ( 2 ) } \stackrel { i i d } { \sim } \mathcal { N } ( w _ { 0 } , \eta _ { 2 } \Sigma )$ . For the SWA estimator $\begin{array} { r } { \hat { w } _ { \mathrm { S W A } } = \frac { 1 } { n } \sum _ { i } w _ { i } ^ { ( 1 ) } , \mathbb { E } [ \| \hat { w } _ { \mathrm { S W A } } - w _ { 0 } \| ^ { 2 } ] = \mathrm { t r } ( \mathrm { C o v } ( \hat { w } _ { \mathrm { S W A } } ) ) = \frac { \eta _ { 1 } } { n } \mathrm { t r } ( \Sigma ) } \end{array}$ . But if we include the high variance points in the average, as in fast-SWA, wˆfSWA = 1n+m  $\begin{array} { r } { \hat { w } _ { \mathrm { f S W A } } = \frac { 1 } { n + m } \big ( \sum _ { i } w _ { i } ^ { ( 1 ) } + \sum _ { j } w _ { j } ^ { ( 2 ) } \big ) } \end{array}$ , then $\begin{array} { r } { \mathbb { E } [ \| \hat { w } _ { \mathrm { f S W A } } - w _ { 0 } \| ^ { 2 } ] = \frac { n \eta _ { 1 } + m \eta _ { 2 } } { ( n + m ) ^ { 2 } } \mathrm { t r } ( \Sigma ) } \end{array}$ . If $\begin{array} { r } { \frac { n \eta _ { 1 } + m \eta _ { 2 } } { ( n + m ) ^ { 2 } } < \frac { \eta _ { 1 } } { n } } \end{array}$ then including the high learning rate points decreases the MSE of the estimator for $\begin{array} { r } { m > n \bigl ( \frac { \eta _ { 2 } } { \eta _ { 1 } } - 2 \bigr ) } \end{array}$ . If we include enough points, we will still improve the estimate.
+As discussed in Mandt et al. (2017), under certain assumptions SGD samples from a Gaussian distribution centered at the optimum of the loss $w _ { 0 }$ with covariance proportional to the learning rate. Suppose then that we have $n$ weights sampled at learning rate $\eta _ { 1 }$ , $w _ { i } ^ { ( 1 ) } \stackrel { i i d } { \sim } \mathcal { N } ( w _ { 0 } , \eta _ { 1 } \Sigma )$ and $m$ weights sampled with the higher learning rate $\eta _ { 2 }$ , $w _ { j } ^ { ( 2 ) } \stackrel { i i d } { \sim } \mathcal { N } ( w _ { 0 } , \eta _ { 2 } \Sigma )$ . For the SWA estimator $\begin{array} { r } { \hat { w } _ { \mathrm { S W A } } = \frac { 1 } { n } \sum _ { i } w _ { i } ^ { ( 1 ) } , \mathbb { E } [ \| \hat { w } _ { \mathrm { S W A } } - w _ { 0 } \| ^ { 2 } ] = \mathrm { t r } ( \mathrm { C o v } ( \hat { w } _ { \mathrm { S W A } } ) ) = \frac { \eta _ { 1 } } { n } \mathrm { t r } ( \Sigma ) } \end{array}$ . But if we include the high variance points in the average, as in fast-SWA, wˆfSWA = 1n+m  $\begin{array} { r } { \hat { w } _ { \mathrm { f S W A } } = \frac { 1 } { n + m } \big ( \sum _ { i } w _ { i } ^ { ( 1 ) } + \sum _ { j } w _ { j } ^ { ( 2 ) } \big ) } \end{array}$ , then $\begin{array} { r } { \mathbb { E } [ \| \hat { w } _ { \mathrm { f S W A } } - w _ { 0 } \| ^ { 2 } ] = \frac { n \eta _ { 1 } + m \eta _ { 2 } } { ( n + m ) ^ { 2 } } \mathrm { t r } ( \Sigma ) } \end{array}$ . If $\begin{array} { r } { \frac { n \eta _ { 1 } + m \eta _ { 2 } } { ( n + m ) ^ { 2 } } < \frac { \eta _ { 1 } } { n } } \end{array}$ then including the high learning rate points decreases the MSE of the estimator for $\begin{array} { r } { m > n \bigl ( \frac { \eta _ { 2 } } { \eta _ { 1 } } - 2 \bigr ) } \end{array}$ . If we include enough points, we will still improve the estimate.
 
 # A.8 NETWORK ARCHITECTURES
 

md/train/tGZu6DlbreV/tGZu6DlbreV.md

@@ -155,7 +155,7 @@ For more detailed analysis of the EM algorithm for optimizing $\mathcal { O } (
 
 Datasets. We choose four datasets for evaluation, including FB15k-237 (Toutanova & Chen, 2015), WN18RR (Dettmers et al., 2018), Kinship and UMLS (Kok & Domingos, 2007). For Kinship and UMLS, there are no standard data splits, so we randomly sample $30 \%$ of all the triplets for training, $20 \%$ for validation, and the rest $50 \%$ for testing. The detailed statistics are summarized in the App. D.
 
-Compared Algorithms. We compare the following algorithms in experiment:  Rule learning methods. For traditional statistical relational learning methods, we choose Markov logic networks (Richardson & Domingos, 2006), boosted relational dependency networks (Natarajan et al., 2010) and path ranking (Lao & Cohen, 2010). We also consider neural logic programming methods, including NeuralLP (Yang et al., 2017), DRUM (Sadeghian et al., 2019) and NLIL (Yang & Song, 2020). In addition, we compare against CTP (Minervini et al., 2020), a differentiable method based on neural theorem provers. Besides, we consider three reinforcement learning methods, which are MINERVA (Das et al., 2018), MultiHopKG (Lin et al., 2018) and M-Walk (Shen et al., 2018).  Other methods. We also compare with some embedding methods, including TransE (Bordes et al., 2013), DistMult (Yang et al., 2015), ComplEx (Trouillon et al., 2016), ComplEx-N3 (Lacroix et al., 2018), ConvE (Dettmers et al., 2018), TuckER (Balazevic et al., 2019) and RotatE (Sun et al., 2019).  RNNLogic. For RNNLogic, we consider two model variants. The first variant assigns a constant score to different grounding paths in the reasoning predictor, i.e., $\phi _ { w } ( p a t h ) = 1 $ in Eq. (4), and we denote this variant as w/o emb.. The second variant leverages entity embeddings and relation embeddings to compute the path score $\phi _ { w } ( p a t h )$ , and we denote the variant as with emb..
+Compared Algorithms. We compare the following algorithms in experiment:  Rule learning methods. For traditional statistical relational learning methods, we choose Markov logic networks (Richardson & Domingos, 2006), boosted relational dependency networks (Natarajan et al., 2010) and path ranking (Lao & Cohen, 2010). We also consider neural logic programming methods, including NeuralLP (Yang et al., 2017), DRUM (Sadeghian et al., 2019) and NLIL (Yang & Song, 2020). In addition, we compare against CTP (Minervini et al., 2020), a differentiable method based on neural theorem provers. Besides, we consider three reinforcement learning methods, which are MINERVA (Das et al., 2018), MultiHopKG (Lin et al., 2018) and M-Walk (Shen et al., 2018).  Other methods. We also compare with some embedding methods, including TransE (Bordes et al., 2013), DistMult (Yang et al., 2015), ComplEx (Trouillon et al., 2016), ComplEx-N3 (Lacroix et al., 2018), ConvE (Dettmers et al., 2018), TuckER (Balazevic et al., 2019) and RotatE (Sun et al., 2019).  RNNLogic. For RNNLogic, we consider two model variants. The first variant assigns a constant score to different grounding paths in the reasoning predictor, i.e., $\phi _ { w } ( p a t h ) = 1 $ in Eq. (4), and we denote this variant as w/o emb.. The second variant leverages entity embeddings and relation embeddings to compute the path score $\phi _ { w } ( p a t h )$ , and we denote the variant as with emb..
 
 Table 1: Results of reasoning on FB15k-237 and WN18RR. $\mathrm { H @ } k$ is in $\%$ . $[ ^ { * } ]$ means the numbers are taken from the original papers. [†] means we rerun the methods with the same evaluation process.