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+parse/train/URc7gYBcjVn/URc7gYBcjVn_origin.pdf filter=lfs diff=lfs merge=lfs -text +parse/train/URc7gYBcjVn/URc7gYBcjVn_span.pdf filter=lfs diff=lfs merge=lfs -text +parse/train/URc7gYBcjVn/URc7gYBcjVn_layout.pdf filter=lfs diff=lfs merge=lfs -text +parse/train/Hyl9xxHYPr/Hyl9xxHYPr_layout.pdf filter=lfs diff=lfs merge=lfs -text +parse/train/Hyl9xxHYPr/Hyl9xxHYPr_span.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/parse/dev/4p6_5HBWPCw/4p6_5HBWPCw.md b/parse/dev/4p6_5HBWPCw/4p6_5HBWPCw.md new file mode 100644 index 0000000000000000000000000000000000000000..31e0939b813c7b9d773ee6d2c0585f9a98c6399c --- /dev/null +++ b/parse/dev/4p6_5HBWPCw/4p6_5HBWPCw.md @@ -0,0 +1,471 @@ +# GRAPH-LESS NEURAL NETWORKS: TEACHING OLD MLPS NEW TRICKS VIA DISTILLATION + +Shichang Zhang∗ University of California, Los Angeles shichang@cs.ucla.edu + +Yozen Liu +Snap Inc. +yliu2@snap.com + +# Yizhou Sun + +University of California, Los Angeles yzsun@cs.ucla.edu + +Neil Shah +Snap Inc. +nshah@snap.com + +# ABSTRACT + +Graph Neural Networks (GNNs) are popular for graph machine learning and have shown great results on wide node classification tasks. Yet, they are less popular for practical deployments in the industry owing to their scalability challenges incurred by data dependency. Namely, GNN inference depends on neighbor nodes multiple hops away from the target, and fetching them burdens latency-constrained applications. Existing inference acceleration methods like pruning and quantization can speed up GNNs by reducing Multiplication-and-ACcumulation (MAC) operations, but the improvements are limited given the data dependency is not resolved. Conversely, multi-layer perceptrons (MLPs) have no graph dependency and infer much faster than GNNs, even though they are less accurate than GNNs for node classification in general. Motivated by these complementary strengths and weaknesses, we bring GNNs and MLPs together via knowledge distillation (KD). Our work shows that the performance of MLPs can be improved by large margins with GNN KD. We call the distilled MLPs Graph-less Neural Networks (GLNNs) as they have no inference graph dependency. We show that GLNNs with competitive accuracy infer faster than GNNs by $1 4 6 \times - 2 7 3 \times$ and faster than other acceleration methods by $1 4 \times - 2 7 \times$ . Under a production setting involving both transductive and inductive predictions across 7 datasets, GLNN accuracies improve over stand-alone MLPs by $1 2 . 3 6 \%$ on average and match GNNs on 6/7 datasets. Comprehensive analysis shows when and why GLNNs can achieve competitive accuracies to GNNs and suggests GLNN as a handy choice for latency-constrained applications. + +# 1 INTRODUCTION + +Graph Neural Networks (GNNs) have recently become very popular for graph machine learning (GML) research and have shown great results on node classification tasks (Kipf & Welling, 2016; Hamilton et al., 2017; Velickovi ˇ c et al. ´ , 2017) like product prediction on co-purchasing graphs and paper category prediction on citation graphs. However, for large-scale industrial applications, MLPs remain the major workhorse, despite common (implicit) underlying graphs and suitability for GML formalisms. One reason for this academic-industrial gap is the challenges in scalability and deployment brought by data dependency in GNNs (Zhang et al., 2020; Jia et al., 2020), which makes GNNs hard to deploy for latency-constrained applications that require fast inference. + +Neighborhood fetching caused by graph dependency is one of the major sources of GNN latency. Inference on a target node necessitates fetching topology and features of many neighbor nodes, especially on small-world graphs (detailed discussion in Section 4). Common inference acceleration techniques like pruning (Zhou et al., 2021) and quantization (Tailor et al., 2021; Zhao et al., 2020) can speed up GNNs to some extent by reducing Multiplication-and-ACcumulation (MAC) operations. + +However, their improvements are limited given the graph dependency is not resolved. Unlike GNNs, MLPs have no dependency on graph data and are easier to deploy than GNNs. They also enjoy the auxiliary benefit of sidestepping the cold-start problem that often happens during the online prediction of relational data (Wei et al., 2020), meaning MLPs can infer reasonably even when neighbor information of a new encountered node is not immediately available. On the other hand, this lack of graph dependency typically hurts for relational learning tasks, limiting MLP performance on GML tasks compared to GNNs. We thus ask: can we bridge the two worlds, enjoying the low-latency, dependency-free nature of MLPs and the graph context-awareness of GNNs at the same time? + +Present work. Our key finding is that it is possible to distill knowledge from GNNs to MLPs without losing significant performance, but reducing the inference time drastically for node classification. The knowledge distillation (KD) can be done offline, coupled with model training. In other words, we can shift considerable work from the latency-constrained inference step, where time reduction in milliseconds makes a huge difference, to the less time-sensitive training step, where time cost in hours or days is often tolerable. We call our approach Graph-less Neural Network (GLNN). Specifically, GLNN is a modeling paradigm involving KD from a GNN teacher to a student MLP; the resulting GLNN is an MLP optimized through KD, so it enjoys the benefits of graph contextawareness in training but has no graph dependency in inference. Regarding speed, GLNNs have superior efficiency and are $1 4 6 \times - 2 7 3 \times$ faster than GNNs and $1 4 \times - 2 7 \times$ faster than other inference acceleration methods. Regarding performance, under a production setting involving both transductive and inductive predictions on 7 datasets, GLNN accuracies improve over MLPs by $1 2 . 3 6 \%$ on average and match GNNs on $6 / 7$ datasets. We comprehensively study when and why GLNNs can achieve competitive results as GNNs. Our analysis suggests the critical factors for such great performance are large MLP sizes and high mutual information between node features and labels. Our observations align with recent results in vision and language, which posit that large enough (or slightly modified) MLPs can achieve similar results as CNNs and Transformers (Liu et al., 2021; Tolstikhin et al., 2021; Melas-Kyriazi, 2021; Touvron et al., 2021; Ding et al., 2021). Our core contributions are as follows: + +• We propose GLNN, which eliminates neighbor-fetching latency in GNN inference via KD to MLP. • We show GLNNs has competitive performance as GNNs, while enjoying $1 4 6 \times - 2 7 3 \times$ faster inference than vanilla GNNs and $1 4 \times - 2 7 \times$ faster inference than other inference acceleration methods. • We study GLNN properties comprehensively by investigating their performance under different settings, how they work as regularizers, their inductive bias, expressiveness, and limitations. + +# 2 RELATED WORK + +Graph Neural Networks. The early GNNs generalize convolution nets to graphs (Bruna et al., 2014; Defferrard et al., 2017) and later simplified to message-passing neural net (MPNN) by GCN (Kipf & Welling, 2016). Most GNNs after can be put as MPNNs. For example, GAT employs attention (Velickovi ˇ c et al. ´ , 2017), PPNP employs personalized PageRank (Klicpera et al., 2019), GCNII and DeeperGCN employ residual connections and dense connections (Chen et al., 2020; Li et al., 2019). + +Inference Acceleration. Inference acceleration have been proposed by hardware improvements (Chen et al., 2016; Judd et al., 2016) and algorithmic improvements through pruning (Han et al., 2015), quantization (Gupta et al., 2015). For GNNs, pruning (Zhou et al., 2021) and quantizing GNN parameters (Zhao et al., 2020) have been studied. These approaches speed up GNN inference to a certain extent but do not eliminate the neighbor-fetching latency. In contrast, our cross-model KD solves this issue. Concurrently, Graph-MLP also tries to bypass GNN neighbor fetching (Hu et al., 2021) by training an MLP with a neighbor contrastive loss, but it only considers transductive but not the more practical inductive setting. Some sampling works focus on speed up GNN training (Zou et al., 2019; Chen et al., 2018), which are complementary to our goal on inference acceleration. + +GNN distillation. Existing GNN KD works try to distill large GNNs to smaller GNNs. LSP (Yang et al., 2021b) and TinyGNN (Yan et al., 2020) do KD while preserving local information. Their students are GNNs with fewer parameters but not necessarily fewer layers. Thus, both designs still require latency-inducing fetching. GFKD (Deng & Zhang, 2021) does graph-level KD via graph generation. In GFKD, data instances are independent graphs, whereas we focus on dependent nodes within a graph. GraphSAIL (Xu et al., 2020) uses KD to learn students work well on new data while preserving performance on old data. CPF (Yang et al., 2021a) combines KD and label propagation (LP). The student in CPF is not a GNN, but it is still heavily graph-dependent as it uses LP. + +![](images/3c59c6aff8764b30e7d812c76a7e7b24c6d87080f7daff930791e286b7e0e843.jpg) +Figure 1: The number of fetches and the inference time of GNNs are both magnitudes more than MLPs and grow exponentially as functions of the number of layers. Left: neighbors need to be fetched for two GNN layers. Middle: the total number of fetches for inference. Right: the total inference time. (Inductive inference for 10 random nodes on OGB Products (Hu et al., 2020)) + +# 3 PRELIMINARIES + +Notations. For GML tasks, the input is usually a graph and its node features, which we write as $\mathcal { G } = ( \nu , \mathcal { E } )$ , with $\nu$ stands for all nodes, and $\mathcal { E }$ stands for all edges. Let $N$ denote the total number of nodes. We use $\pmb { X } \in \mathbb { R } ^ { N \times D }$ to represent node features, with row $\mathbf { \boldsymbol { x } } _ { v }$ being the $D$ -dimensional feature of node $v \in \mathcal V$ . We represent edges with an adjacency matrix $\pmb { A }$ , with $A _ { u , v } = 1$ if edge $( u , v ) \in \mathcal { E }$ , and 0 otherwise. For node classification, one of the most important GML applications, the prediction targets are $\pmb { Y } \in \mathbb { R } ^ { N \times K }$ , where row $\mathbf { \Delta } _ { \mathbf { \mathcal { Y } } _ { v } }$ is a $K$ -dim one-hot vector for node $v$ . For a given $\mathcal { G }$ , usually a small portion of nodes will be labeled, which we mark using superscript $L$ , i.e. $\mathcal { V } ^ { \tilde { L } }$ , $X ^ { L }$ , and $\mathbf { \nabla } _ { \mathbf { Y } } \breve { L }$ . The majority of nodes will be unlabeled, and we mark using the superscript $U$ , i.e. $\mathcal { V } ^ { U }$ , $X ^ { U }$ , and $\mathbf { \nabla } _ { \mathbf { Y } } U$ . + +Graph Neural Networks. Most GNNs fit under the message-passing framework, where the representation $h _ { v }$ of each node $v$ is updated iteratively in each layer by collecting messages from its neighbors denoted as $\mathcal { N } ( v )$ . For the $l$ -th layer, $\mathbf { \Sigma } _ { h _ { v } ^ { ( l ) } }$ is obtained from the previous layer representation $ { \boldsymbol { h } } _ { u } ^ { ( l - 1 ) }$ $\mathbf { \mathcal { h } } _ { u } ^ { ( 0 ) } = \mathbf { \mathcal { x } } _ { u } )$ ) via an aggregation operation AGGR followed by an UPDATE operation as + +$$ +\begin{array} { r } { \pmb { h } _ { N ( v ) } ^ { ( l ) } = \mathrm { A G G R } ( \{ \pmb { h } _ { u } ^ { ( l - 1 ) } : u \in \mathcal { N } ( v ) \} ) \qquad \mathrm { a n d } \qquad \pmb { h } _ { v } ^ { ( l ) } = \mathrm { U P D A T E } ( \pmb { h } _ { N ( v ) } ^ { ( l ) } , \pmb { h } _ { v } ^ { ( l - 1 ) } ) } \end{array} +$$ + +# 4 MOTIVATION + +GNNs have considerable inference latency due to graph dependency. One more GNN layer means fetching one more hop of neighbors. To infer a node with a $L$ -layer GNN on a graph with average degree $R$ requires $\mathcal { O } ( R ^ { L } )$ fetches. $R$ can be large for real-world graphs, e.g. 208 for the Twitter (Ching et al., 2015). Also, as layer fetching must be done sequentially, the total latency explodes quickly as $L$ increases. Figure 1 shows the dependency added by each GNN layer and the exponential explosion of inference time. In contrast, the MLP inference time is much smaller and grows linearly. This marked gap contributes greatly to the practicality of MLPs in industrial applications over GNNs. + +The node-fetching latency is exacerbated by two factors: firstly, newer GNN architectures are getting deeper from 64 layers (Chen et al., 2020) to even 1001 layers (Li et al., 2021). Secondly, industrialscale graphs are frequently too large to fit into the memory of a single machine (Jin et al., 2022), necessitating sharding of the graph out of the main memory. For example, Twitter has 288M monthly active users (nodes) and an estimated 60B followers (edges) as of 3/2015. Facebook has 1.39B active users with more than 400B edges as of 12/2014 (Ching et al., 2015). Even when stored in a sparse-matrix-friendly format (often COO or CSR), these graphs are on the order of TBs and are constantly growing. Moving away from in-memory storage results in even slower neighbor-fetching. + +MLPs, on the other hand, lack the means to exploit graph topology, which hurts their performance for node classification. For example, test accuracy on Products is 78.61 for GraphSAGE compared to 62.47 for an equal-sized MLP. Nonetheless, recent results in vision and language posit that large (or slightly modified) MLPs can achieve similar results as CNNs and Transformers (Liu et al., 2021). We thus also ask: Can we bridge the best of GNNs and MLPs to get high-accuracy and low-latency models? This motivates us to do cross-model KD from GNNs to MLPs. + +![](images/fe13a4358c6462ce72c20699e317599db71d44b7281ec5d0c4ff85854d7ca559.jpg) +Figure 2: The GLNN framework: In offline training, a trained GNN teacher is applied on the graph for soft targets. Then, a student MLP is trained on node features guided by the soft targets. The distilled MLP, now GLNN, is deployed for online predictions. Since graph dependency is eliminated for inference, GLNNs infer much faster than GNNs, and hence the name “Graph-less Neural Network.” + +# 5 GRAPH-LESS NEURAL NETWORKS + +We introduce GLNN and answer exploration questions of its properties: 1) How do GLNNs compare to MLPs and GNNs? 2) Can GLNNs work well under both transductive and inductive settings? 3) How do GLNNs compare to other inference acceleration methods? 4) How do GLNNs benefit from KD? 5) Do GLNNs have sufficient model expressiveness? 6) When will GLNNs fail to work? + +# 5.1 THE GLNN FRAMEWORK + +The idea of GLNN is straightforward, yet as we will see, extremely effective. In short, we train a “boosted” MLP via KD from a teacher GNN. KD was introduced in Hinton et al. (2015), where knowledge was transferred from a cumbersome teacher to a simpler student. In our case, we generate soft targets $z _ { v }$ for each node $v$ with a teacher GNN. Then we train a student MLP with both true labels $\mathbf { \Delta } _ { \mathbf { \mathcal { Y } } _ { v } }$ and $z _ { v }$ . The objective is as Equation 1, with $\lambda$ being a weight parameter, $\mathcal { L } _ { l a b e l }$ being the cross-entropy between $\mathbf { \nabla } _ { \mathbf { y } _ { v } }$ and student predictions $\hat { y } _ { v }$ , $\mathcal { L } _ { t e a c h e r }$ being the KL-divergence. + +$$ +\mathcal { L } = \lambda \Sigma _ { v \in \mathcal { V } ^ { L } } \mathcal { L } _ { l a b e l } ( \hat { y } _ { v } , y _ { v } ) + ( 1 - \lambda ) \Sigma _ { v \in \mathcal { V } } \mathcal { L } _ { t e a c h e r } ( \hat { y } _ { v } , z _ { v } ) +$$ + +The model after KD, i.e. GLNN, is essentially a MLP. Therefore, GLNNs have no graph dependency during inference and are as fast as MLPs. On the other hand, through offline KD, GLNN parameters are optimized to predict and generalize as well as GNNs, with the added benefit of faster inference and easier deployment. In Figure 2, we show the offline KD and online inference steps of GLNNs. + +# 5.2 EXPERIMENT SETTINGS + +Datasets. We consider all five datasets used in the CPF paper (Yang et al., 2021a), i.e. Cora, Citeseer, Pubmed, A-computer, and A-photo. To fully evaluate our method, we also include two more larger OGB datasets (Hu et al., 2020), i.e. Arxiv and Products. + +Model Architectures. For consistent results, we use GraphSAGE (Hamilton et al., 2017) with GCN aggregation as the teacher. We conduct ablation studies of other GNN teachers like GCN (Kipf & Welling, 2016), GAT (Velickovi ˇ c et al. ´ , 2017) and, APPNP (Klicpera et al., 2019) in Section 6. + +Evaluation Protocol. For all experiments in this section, we report the average and standard deviation over ten runs with different random seeds. Model performance is measured as accuracy, and results are reported on test data with the best model selected using validation data. + +Transductive vs. Inductive. Given ${ \mathcal { G } } , X$ , and $Y ^ { L }$ , we consider node classification under two settings: transductive $( t r a n )$ and inductive $( i n d )$ . For ind, we hold out some test data for inductive evaluation only. We first select inductive nodes $\mathcal { V } _ { i n d } ^ { U } \subset \mathcal { V } ^ { U }$ , which partitions $\mathcal { V } ^ { U }$ into the disjoint inductive subset and observed subset, i.e. $\mathcal { V } ^ { U } = \mathcal { V } _ { o b s } ^ { U } \sqcup \mathcal { V } _ { i n d } ^ { U }$ . Then we hold out $v \in \mathcal { V } _ { i n d } ^ { U }$ and all edges connected to $v \in \mathcal { V } _ { i n d } ^ { U }$ obs ind, which leads to two disjoint graphs $\mathcal { G } = \mathcal { G } _ { o b s } \sqcup \mathcal { G } _ { i n d }$ ind with no shared nodes or + +Table 1: GLNNs outperform MLPs by large margins and match GNNs on 5 of 7 datasets under the transductive setting. $\Delta _ { M L P }$ $( \Delta _ { G N N } )$ represents difference between the GLNN and a trained MLP (GNN). Results show accuracy (higher is better); $\Delta _ { G N N } 2 0$ indicates GLNN outperforms GNN. + +
DatasetsSAGEMLPGLNN△MLP△GNN
Cora80.52 ± 1.7759.22 ±1.3180.54 ± 1.3521.32 (36.00%)0.02 (0.02%)
Citeseer70.33 ± 1.9759.61 ± 2.8871.77 ± 2.0112.16 (20.40%)1.44 (2.05%)
Pubmed75.39 ± 2.0967.55 ± 2.3175.42 ± 2.317.87 (11.65%)0.03 (0.04%)
A-computer82.97 ± 2.1667.80 ± 1.0683.03 ± 1.8715.23 (22.46%)0.06 (0.07%)
A-photo90.90 ± 0.8478.77 ± 1.7492.11 ± 1.0813.34 (16.94%)1.21 (1.33%)
Arxiv70.92 ± 0.1756.05 ± 0.4663.46 ± 0.457.41 (13.24%)-7.46 (-10.52%)
Products78.61 ± 0.4962.47 ± 0.1068.86 ± 0.466.39 (10.23%)-9.75 (-12.4%)
+ +Table 2: Enlarged GLNNs match the performance of GNNs on the OGB datasets. For Arxiv, we use MLPw4 (GLNNw4). For Products, we use MLPw8 (GLNNw8). + +
DatasetsSAGEMLP+GLNN+△MLP△GNN
Arxiv70.92 ± 0.1755.31± 0.4772.15 ± 0.2716.85 (30.46%)0.51 (0.71%)
Products78.61 ± 0.4964.50 ± 0.4577.65 ± 0.4813.14 (20.38%)-0.97 (-1.23%)
+ +els are partitioned into three disjoint sets, i.e. . Concretely, the input/output of both setting $X = X ^ { L } \sqcup X _ { o b s } ^ { U } \sqcup X _ { i n d } ^ { U }$ $\pmb { Y } = \pmb { Y } ^ { L } \sqcup \pmb { Y } _ { o b s } ^ { U } \sqcup \pmb { Y } _ { i n d } ^ { U }$ + +• ind: train on • tran: train on $\mathcal { G } _ { o b s }$ ${ \mathcal { G } } , X$ , $X ^ { L }$ , and , $X _ { o b s } ^ { U }$ $\mathbf { \nabla } _ { \mathbf { Y } ^ { L } }$ ; evaluate on , and $Y ^ { L }$ ; evaluate on $( X ^ { U } , Y ^ { U } )$ $( X _ { i n d } ^ { U } , Y _ { i n d } ^ { U } )$ ; $\mathrm { K D }$ uses $z _ { v }$ v ; KD uses for $v \in \mathcal V$ $z _ { v }$ . for + +Note that for tran, all the nodes in the graph including the validation and test nodes are used to generate $_ { z }$ . A discussion of this choice along with other experiment details are in Appendix A. + +# 5.3 HOW DO GLNNS COMPARE TO MLPS AND GNNS? + +We start by comparing GLNNs to MLPs and GNNs with the same number of layers and hidden dimensions. We first consider the standard transductive setting, so our results in Table 1 are directly comparable to results reported in previous literature like Yang et al. (2021a) and Hu et al. (2020). + +As shown in Table 1, the performance of all GLNNs improve over MLPs by large margins. On smaller datasets (first 5 rows), GLNNs can even outperform the teacher GNNs. In other words, for each task, with the same parameter budget, there exists a set of MLP parameters that has GNN-competitive performance (detailed discussion in Sections 5.6 and 5.7). For the larger OGB datasets (last 2 rows), the GLNN performance is improved over MLPs but still worse than the teacher GNNs. However, as we show in Table 2, this gap can be mitigated by increasing MLP size to MLPwi1. In Figure 3 (right), we visualize the trade-off between prediction accuracy and model inference time with different model sizes. We show that gradually increasing GLNN size pushes its performance to be close to SAGE. On the other hand, when we reduce the number of layers of $\mathrm { S A G E } ^ { 2 }$ , the accuracy quickly drops to be worse than GLNNs. A detailed discussion of the rationale for increasing MLP sizes is in Appendix B. + +# 5.4 CAN GLNNS WORK WELL UNDER BOTH TRANSDUCTIVE AND INDUCTIVE SETTINGS? + +Although transductive is the commonly studied setting for node classification, it does not encompass prediction on unseen nodes. Therefore, it may not be the best way to evaluate a deployed model, which must often generate predictions for new data points as well as reliably maintain performance on old ones. Thus, to better understand the effectiveness of GLNN, we also consider their performance under a realistic production setting, which contains both transductive and inductive predictions. + +To evaluate a model inductively, we hold out some test nodes from training to form an inductive set, i.e. $\mathcal { V } ^ { U } = \mathcal { V } _ { o b s } ^ { U } \sqcup \mathcal { V } _ { i n d } ^ { U }$ . In production, a model might be re-trained periodically, e.g. weekly. The obs hold-out nodes in $V _ { i n d } ^ { U }$ represent new nodes entered the graph between two trainings. $V _ { i n d } ^ { U }$ is usually small compared to $V _ { o b s } ^ { U }$ – e.g. Graham (2012) estimates $5- 7 \%$ for the fastest-growing tech startups. In our case, to mitigate randomness and better evaluate generalizability, we use V Uind containing 20% of the test data. We also evaluate on standard transductive prediction on $V _ { o b s } ^ { U }$ containing the otherrved unlabeled nodes $80 \%$ of the test data, representing thee inference is commonly redone on existing nodes in real-world cases. We report both results and a interpolated production (prod) results in Table 3. The prod results paint a clearer picture of model generalization as well as accuracy in production. See Section 6 for an ablation study of different inductive split rates other than 20-80. + +![](images/56094e9645d24171cda661a02619bd1120d8d877a8d7ebb8ed27856fa4567de6.jpg) +Figure 3: Enlarged MLPs (GLNNs) can match GNN accuracy, but infer dramatically faster. Plots are under the same setting as Figure 1. Left: inference time of MLPs vs. GNN (SAGE) for different model sizes. Right: model accuracy vs. inference time. Note: time axes are log-scaled. + +Table 3: GLNNs match GNN performance on a production setting with both inductive and transductive predictions. We uProducts. ind results on $V _ { i n d } ^ { U }$ LP for the 5 C, tran results on $V _ { o b s } ^ { U }$ atasets, MLPw4 for Arxiv, and MLPw8 for, and the interpolated prod results are reported. + +
DatasetsEvalSAGEMLP/MLP+GLNN/GLNN+△MLP△GNN
Coraprod79.2958.9878.2819.30 (32.72%)-1.01 (-1.28%)
ind81.33 ± 2.1959.09 ± 2.9673.82 ± 1.9314.73 (24.93%)-7.51 (-9.23%)
tran78.78 ± 1.9258.95 ± 1.6679.39 ± 1.6420.44 (34.66%)0.61 (0.77%)
Citeseerprod68.3859.8169.279.46 (15.82%)0.89 (1.30%)
ind69.75 ± 3.5960.06 ± 5.0069.25 ± 2.259.19 (15.30%)-0.5 (-0.7%)
tran68.04 ± 3.3459.75 ± 2.4869.28 ± 3.129.63 (15.93%)1.24 (1.82%)
Pubmedprod74.8866.8074.717.91 (11.83%)-0.17 (-0.22%)
ind75.26 ± 2.5766.85 ± 2.9674.30 ± 2.617.45 (11.83%)-0.96 (-1.27%)
tran74.78 ± 2.2266.79 ± 2.9074.81 ± 2.398.02 (12.01%)0.03 (0.04%)
A-computerprod82.1467.3882.2914.90 (22.12%)0.15 (0.19%)
ind82.08 ± 1.7967.84 ± 1.7880.92 ± 1.3613.08 (19.28%)-1.16 (-1.41%)
tran82.15 ± 1.5567.27 ± 1.3682.63 ± 1.4015.36 (22.79%)0.48 (0.58%)
A-photoprod91.0879.2592.3813.13 (16.57%)1.30 (1.42%)
ind91.50 ± 0.7979.44 ± 1.7291.18 ± 0.8111.74 (14.78%)-0.32 (-0.35%)
tran90.80 ± 0.7779.20 ± 1.6492.68 ± 0.5613.48 (17.01%)1.70 (1.87%)
Arxivprod70.7355.3065.099.79 (17.70%)-5.64 (-7.97%)
ind70.64 ± 0.6755.40 ± 0.5660.48 ± 0.464.3 (7.76%)-10.94 (-15.49%)
tran70.75± 0.2755.28 ± 0.4971.46 ± 0.3311.16 (20.18%)-4.31 (-6.09%)
Productsprod76.6063.7275.7712.05 (18.91%)-0.83 (-1.09%)
ind76.89 ± 0.5363.70 ± 0.6675.16 ± 0.3411.44 (17.96%)-1.73 (-2.25%)
tran76.53 ±0.5563.73 ± 0.6975.92 ± 0.6112.20 (19.15%)-0.61 (-0.79%)
+ +In Table 3, we see that GLNNs can still improve over MLP by large margins for inductive predictions. On 6/7 datasets, the GLNN prod performance are competitive to GNNs, which supports deploying GLNN as a much faster model with no or only slight performance loss. On the Arxiv dataset, the GLNN performance is notably less than GNNs – we hypothesize this is due to Arxiv having a particularly challenging data split which causes distribution shift between test nodes and training + +Table 4: While other inference acceleration methods speed up SAGE, they are considerably slower than GLNNs. Numbers (in ms) are inductive inference time on 10 randomly chosen nodes. + +
DatasetsSAGEQSAGEPSAGENeighbor SampleGLNN+
Arxiv489.49433.90 (1.13x)465.43 (1.05x)91.03 (5.37x)3.34 (146.55x)
Products2071.301946.49 (1.06x)2001.46 (1.04x)107.71 (19.23x)7.56 (273.98x)
+ +nodes, which is hard for GLNNs to capture without utilizing neighbor information like GNNs. +However, we note that GLNN performance is substantially improved over MLP. + +# 5.5 HOW DO GLNNS COMPARE TO OTHER INFERENCE ACCELERATION METHODS? + +Common techniques of inference acceleration include pruning and quantization. These approaches can reduce model parameters and Multiplication-and-ACcumulation (MACs) operations. Still, they don’t eliminate neighbor-fetching latency. Therefore, their speed gain on GNNs is less significant than on NNs. For GNNs, neighbor sampling is also used to reduce the fetching latency. We show an explicit speed comparison between vanilla SAGE, quantized SAGE from FP32 to INT8 (QSAGE), SAGE with $50 \%$ weights pruned (PSAGE), inference neighbor sampling with fan-out 15, and GLNN in Table 4. With the same setting as Figure 1, we see that GLNN is considerably faster. + +Two other kinds of methods considered as inference acceleration are GNN-to-GNN KD like TinyGNN (Yan et al., 2020) and Graph Augmented-MLPs (GA-MLPs) like SGC (Wu et al., 2019) or SIGN (Frasca et al., 2020). Inference of GNN-to-GNN KD is likely to be slower than a GNN-Li with the same $i$ as the student, since there will usually be some extra overheads like the Peer-Aware Module (PAM) in TinyGNN. GA-MLPs precompute augmented node features and apply MLPs to them. With precomputation, their inference time will be the same as MLPs for dimension-preserving augmentation (SGC) and the same as enlarged MLPwi for augmentation involves concatenation (SIGN). Thus, for both kinds of approaches, it is sufficient to compare GLNN with GNN-Li and MLPwi, which we have already shown in Figure 3 (left). We see that GNN-Lis are much slower than MLPs. For GA-MLPs, since full pre-computation cannot be done for inductive nodes, GA-MLPs still need to fetch neighbor nodes. This makes them much slower than MLPwi in the inductive setting, and even slower than pruned GNNs and TinyGNN as shown in Zhou et al. (2021). + +# 5.6 HOW DOES GLNN BENEFIT FROM DISTILLATION? + +We showed that GNNs are markedly better than MLPs on node classification tasks. But, with KD, GLNNs can often become competitive to GNNs. This indicates that there exist suitable MLP parameters which can well approximate the ideal prediction function from node features to labels. However, these parameters can be difficult to learn through standard stochastic gradient descent. We hypothesize that KD helps to find them through regularization and transfer of inductive bias. + +First, we show that KD can help to regularize the student model. From loss curves of a directly trained MLP and the GLNN in Figure 4, we see the gap between training and validation loss is visibly larger for MLPs than GLNNs, and MLPs show obvious overfitting trends. Second, we analyze the inductive bias that makes GNNs powerful on node classification, which suggests that node inferences should be influenced by the graph topology. Whereas MLPs have less inductive bias. Similar difference exists between Transformers (Vaswani et al., 2017) and MLPs. Liu et al. (2021) shows that the inductive bias in Transformers can be mitigated by a simple gate on large MLPs. For node classification, we hypothesize that KD helps to mitigate the inductive bias, so GLNNs can perform competitively. Soft labels from GNN teachers are heavily influenced by the graph topology due to inductive bias. They maintain nonzero probabilities on classes other than the ground truth provided by labels, which can be useful for the student to learn to complement the missing inductive bias in MLPs. To evaluate this hypothesis quantitatively, we define the cut loss $\mathcal { L } _ { c u t } \in [ 0 , 1 ]$ in Equation 2 to measure the consistency between model predictions and graph topology (details in Appendix C): + +$$ +\mathcal { L } _ { c u t } = \frac { T r ( \hat { Y } ^ { T } A \hat { Y } ) } { T r ( \hat { Y } ^ { T } D \hat { Y } ) } +$$ + +Here $\hat { \pmb { Y } } \in [ 0 , 1 ] ^ { N \times K }$ is the soft classification probability output by the model, $\pmb { A }$ and $_ { D }$ are the adjacency and degree matrices. When $\mathcal { L } _ { c u t }$ is close to $^ { 1 }$ , it means the predictions and the graph topology are very consistent. In our experiment, we observe that the average $\mathcal { L } _ { c u t }$ for SAGE over five CPF datasets is 0.9221, which means high consistency. The same $\mathcal { L } _ { c u t }$ for MLPs is only 0.7644, but for GLNNs it is 0.8986. This shows that the GLNN predictions indeed benefit from the graph topology knowledge contained in the teacher outputs (the full table of $\mathcal { L } _ { c u t }$ values in Appendix C). + +![](images/e13d36ad0fd34e672832534415bdedeb0f31097113d6e61931f935801ee4e5d2.jpg) +Figure 4: Loss curves on CPF datasets show GLNN distillation can help to regularize the training. Here the training loss of GLNN is on hard labels, only corresponding to the first term in Equation 1. + +# 5.7 DO GLNNS HAVE ENOUGH MODEL EXPRESSIVENESS? + +Intuitively, the addition of neighbor information makes GNNs more powerful than MLPs when classifying nodes. Thus, a natural question regarding KD from GNNs to MLPs is whether MLPs are expressive enough to represent graph data as well as GNNs. Many recent works studied GNN model expressiveness (Xu et al., 2018; Chen et al., 2021). The latter analyzed GNNs and GA-MLPs for node classification and characterized expressiveness as the number of equivalence classes of rooted graphs induced by the model (formal definitions in Appendix D). The conclusion is that GNNs are more powerful than GA-MLPs, but in most real-world cases their expressiveness is indistinguishable. + +We adopt the analysis framework from Chen et al. (2021) and show in Appendix D that the number of equivalence classes induced by GNNs and MLPs are |X |+m−2m−1 2L−1 a nd $| \mathcal { X } |$ respectively. Here $m$ denotes the max node degree, $L$ denotes the number of GNN layers, and $\mathcal { X }$ denotes the set of all possible node features. The former is apparently larger which concludes that GNNs are more expressive. Empirically, however, the gap makes little difference when $| \mathcal { X } |$ is large. In real applications, node features can be high dimensional like bag-of-words, or even word embeddings, thus making $| \mathcal { X } |$ enormous. Like for bag-of-words, $| \mathcal { X } |$ is in the order of $\mathcal { O } ( p ^ { D } )$ , where $D$ is the vocabulary size, and $p$ is the max word frequency. The expressiveness of a L-layer GNN is lower bounded by |X |+m−22L−1 $\left( \stackrel { | \mathcal { X } | + m - 2 } { m - 1 } \right) ^ { 2 ^ { L } - 1 } = \mathcal { O } ( p ^ { D ( m - 1 ) ( 2 ^ { L } - 1 ) } )$ , but empirically, both MLPs and GNNs should have enough expressiveness given $D$ is usually hundreds or bigger (see Table 5). + +# 5.8 WHEN WILL GLNNS FAIL TO WORK? + +As discussed in Section 5.7 and Appendix D, the goal of GML node classification is to fit a function $f$ on the rooted graph $\mathcal { G } ^ { [ i ] }$ and label $\mathbf { \nabla } _ { \mathbf { \psi } _ { 3 } } \mathbf { \psi } _ { 2 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \qquad \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 }$ . From the information theoretic perspective, fitting $f$ by minimizing the commonly used cross-entropy loss is equivalent to maximizing the mutual information (MI), $I ( \mathcal G ^ { [ i ] } ; \boldsymbol y _ { i } )$ as shown in Qin et al. (2020). If we consider $\mathcal { G } ^ { [ i ] }$ as a joint distribution of two random variables $X ^ { [ i ] }$ and ${ \mathcal { E } } ^ { [ i ] }$ representing the node features and edges in $\mathcal { G } ^ { [ i ] }$ respectively, we have + +$$ +I ( \mathcal { G } ^ { [ i ] } ; \pmb { y } _ { i } ) = I ( \pmb { X } ^ { [ i ] } , \pmb { \mathcal { E } } ^ { [ i ] } ; \pmb { y } _ { i } ) = I ( \pmb { \mathcal { E } } ^ { [ i ] } ; \pmb { y } _ { i } ) + I ( \pmb { X } ^ { [ i ] } ; \pmb { y } _ { i } | \pmb { \mathcal { E } } ^ { [ i ] } ) +$$ + +$I ( \mathcal { E } ^ { [ i ] } ; y _ { i } )$ only depends on edges and labels, thus MLPs can only maximize $I ( X ^ { [ i ] } ; y _ { i } | \mathcal { E } ^ { [ i ] } )$ . In the extreme case, $I ( X ^ { [ i ] } ; y _ { i } | \mathcal { E } ^ { [ i ] } )$ can be zero when $y ^ { [ i ] }$ is conditionally independent from $X ^ { [ i ] }$ given $\mathcal { E } ^ { [ i ] }$ . For example, when every node is labeled by its degree or whether it forms a triangle. Then MLPs won’t be able to fit meaningful functions, and neither will GLNNs. However, such cases are typically rare, and unexpected in practical settings our work is mainly concerned with. For real GML tasks, node features and structural roles are often highly correlated (Lerique et al., 2020), hence MLPs can achieve reasonable results even only based on node features, and thus GLNNs can potentially achieve much better results. We study the failure case of GLNNs by creating a low MI scenario in Section 6. + +![](images/5b5ba4cf66e93dda6d93901e3acfc6306b2eb1a932c1b44ff6ba4c6b436be96e.jpg) +Figure 5: Left: Node feature noise. GLNN has comparable performance to GNNs only when nodes are less noisy. Adding more noise decreases GLNN performance faster than GNNs. Middle: Inductive split rate. Altering the inductive:transductive ratio in the production setting doesn’t affect the accuracy much. Right: Teacher GNN architecture. GLNNs can learn from different GNN teachers to improve over MLPs and achieve comparable results. Accuracies are averaged over five CPF datasets. + +# 6 ABLATION STUDIES + +In this section, we do ablation studies of GLNNs on node feature noise, inductive split rates, and teacher GNN architecture. Reported results are test accuracies averaged over five datasets in CPF. More experiments can be found in Appendix including advanced GNN teachers (Appendix F), GA-MLP student (Appendix G), and non-homogeneous data (Appendix I). + +Noisy node features. Following Section 5.8, we investigate failure cases of GLNN by adding different levels of Gaussian noise to node features to decrease their mutual information with labels. Specifically, we replace $\boldsymbol { X }$ with $\tilde { \cal X } = ( 1 - \alpha ) { \cal X } + \alpha \epsilon$ . $\epsilon$ is an isotropic Gaussian independent from $\boldsymbol { X }$ , and $\alpha \in [ 0 , 1 ]$ denotes the noise level. We show the inductive performance of MLP, GNN, and GLNN under different noise levels in Figure 5 (left). We see that as $\alpha$ increases, the accuracy of MLPs and GLNNs decrease faster than GNNs, while the performance of GLNNs and GNNs are still comparable for small $\alpha \mathbf { s }$ . When $\alpha$ reaches 1, $\tilde { X }$ and $\mathbf { Y }$ will become independent corresponding to the extreme case discussed in Section 5.8. A more detailed discussion is in Appendix J. + +Inductive split rate. In Section 5.4, we use a 20-80 split of the test data for inductive evaluation. In Figure 5 (middle), we show the results under different split rates (More detailed plots in Appendix H). We see that as the inductive portion increase, GNN and MLP performance stays roughly the same, and the GLNN inductive performance drops slightly. We only consider rates up to 50-50 since having $50 \%$ or even more inductive nodes is highly atypical in practice. When a large amount of new data are encountered, practitioners can opt to retrain the model on all the data before deployment. + +Teacher GNN architecture. We used SAGE to represent GNNs so far. In Figure 5 (right), we show results with other various GNN teachers, e.g. GCN, GAT, and APPNP. We see that GLNNs can learn from different teachers and improve over MLPs. The performance is similar for all four teachers, with the GLNN distilled from APPNP very slightly worse than others. In fact, a similar phenomenon has been observed in Yang et al. (2021a) as well, i.e. APPNP benefits the student the least. One possible reason is that the first step of APPNP is to utilize the node’s own feature for prediction (prior to propagating over the graph), which is very similar to what the student MLP is doing, and thus provides less additional information to MLPs than other teachers. + +# 7 CONCLUSION AND FUTURE WORK + +In this paper, we explored whether we can bridge the best of GNNs and MLPs to achieve accurate and fast GML models for deployment. We found that KD from GNNs to MLPs helps to eliminate inference graph dependency, which results in GLNNs that are $1 4 6 \times - 2 7 3 \times$ faster than GNNs while enjoying competitive performance. We do a comprehensive study of GLNN properties. The promising results on 7 datasets across different domains show that GLNNs can be a handy choice for deploying latencyconstraint models. In our experiments, the current version of GLNNs on the Arxiv dataset doesn’t show competitive inductive performance. 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Learned low precision graph neural networks, 2020. + +Hongkuan Zhou, Ajitesh Srivastava, Hanqing Zeng, Rajgopal Kannan, and Viktor K. Prasanna. Accelerating large scale real-time GNN inference using channel pruning. CoRR, abs/2105.04528, 2021. URL https://arxiv.org/abs/2105.04528. + +Difan Zou, Ziniu Hu, Yewen Wang, Song Jiang, Yizhou Sun, and Quanquan Gu. Layer-dependent importance sampling for training deep and large graph convolutional networks. arXiv preprint arXiv:1911.07323, 2019. + +# A DETAILED EXPERIMENT SETTINGS + +# A.1 DATASETS + +Here we provide a detailed description of the datasets we used to support our argument. Out of these datasets, 4 of them are citation graphs. Cora, Citeseer, Pubmed, ogbn-arxiv with the node features being descriptions of the papers, either bag-of-word vector, TF-IDF vector, or word embedding vectors. + +In Table 5, we provided the basic statistics of these datasets. + +Table 5: Dataset Statistics. + +
Dataset#Nodes#Edges#Features# Classes
Cora2,4855,0691,4337
Citeseer2,1103,6683,7036
Pubmed19,71744,3245003
A-computer13,381245,77876710
A-photo7,487119,0437458
Arxiv169,3431,166,24312840
Products2,449,02961,859,14010047
+ +For all datasets, we follow the setting in the original paper to split the data. Specifically, for the five smaller datasets from the CPF paper, we use the CPF splitting strategy and each random seed corresponds to a different split. For the OGB datasets, we follow the OGB official splits based on time and popularity for Arxiv and Products respectively. + +# A.2 MODEL HYPERPARAMETERS + +The hyperparameters of GNN models on each dataset are taken from the best hyperparameters provided by the CPF paper and the OGB official examples. For the student MLPs and GLNN s, unless otherwise specified with -wi or $- \mathbf { L } i$ , we set the number of layers and the hidden dimension of each layer to be the same as the teacher GNN, so their total number of parameters stays the same as the teacher GNN. + +Table 6: Hyperparameters for GNNs on five datasets from the CPF paper. + +
SAGEGCNGATAPPNP
#layers2222
hidden dim128646464
learning rate0.010.010.010.01
weight decay0.00050.0010.010.01
dropout00.80.60.5
fan out5,511-
attention heads1=81
power iterations1=110
+ +Table 7: Hyperparameters for GraphSAGE on OGB datasets. + +
DatasetArxivProducts
# layers33
hidden dim256256
learning rate0.010.003
weight decay00
dropout0.20.5
normalizationbatchbatch
fan out[5,10,15][5,10,15]
+ +For GLNN s we do a hyperparameter search of learning rate from [0.01, 0.005, 0.001], weight decay from [0, 0.001, 0.002, 0.005, 0.01], and dropout from [0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6] + +# A.3 KNOWLEDGE DISTILLATION + +We use the distillation method proposed in Hinton et al. (2015) as in Equation 1, the hard labels are found to be helpful, so nonzero $\lambda s$ was suggested. In our case, we did a little tuning for $\lambda$ but didn’t find nonzero $\lambda s$ to be very helpful. Therefore, we report all of our results with $\lambda = 0$ , i.e. only the second term involving soft labels is effective. More careful tuning of $\lambda$ should further improve the results since the searching space is strictly larger. We implemented a weighted version in our code, and we leave the choice of $\lambda$ as future work. + +# A.4 THE TRANSDUCTIVE SETTING AND THE INDUCTIVE SETTING + +Given ${ \mathcal { G } } , X$ , and $Y ^ { L }$ , the goal of node classification can be divided into two different settings, i.e. transductive and inductive. In real applications, the former can correspond to predict missing attributes of a user based on the user profile and other existing users, and the latter can correspond to predict labels of some new nodes that are only seen during inference time. To create the inductive setting on a given dataset, we hold out some nodes along with edges connected to these nodes during data. Using notation defined above, we pick the inductive nodes training and use them for inductive evaluation only. These nodes and edges are picked from the test $\mathcal { V } _ { i n d } ^ { U } \subset \mathcal { V } ^ { U }$ , which partitions $\mathcal { V } ^ { U }$ into the disjoint inductive subset and observed subset, i.e. $\mathcal { V } ^ { U } = \mathcal { V } _ { o b s } ^ { U } \sqcup \mathcal { V } _ { i n d } ^ { U }$ V ind . Then we can take , $\mathcal { V } _ { i n d } ^ { U }$ furth, and raph, so we end up with. We show the input and $\mathcal { G } = \mathcal { G } _ { o b s } \sqcup \mathcal { G } _ { i n d }$ $\pmb { X } = \pmb { X } ^ { L } \sqcup \pmb { X } _ { o b s } ^ { U } \sqcup \overbrace { \pmb { X } _ { i n d } ^ { U } }$ $\pmb { Y } = \pmb { Y } ^ { L } \sqcup \pmb { Y } _ { o b s } ^ { U } \sqcup \pmb { Y } _ { i n d } ^ { U }$ output of both settings using the notations below. + +We visualize the difference between the inductive setting and the transductive setting in Figure 6. + +![](images/2a239fb19d94400d2ccb7f18def7895440e8208ed97573eb1cf684c4025f00ac.jpg) +Figure 6: The transductive setting and inductive setting illustrated by a 2-layer GNN. The middle shows the original graph used for training. The left shows the transductive setting, where the test node is in red and within the graph. The right shows the inductive setting, where the test node is an unseen new node. + +# A.5 CHOOSING SOFT TARGETS UNDER THE TRANSDUCTIVE SETTING + +For the transductive setting in Section 5.3, all the nodes in the graph, including the validation and test nodes, are used for the soft target generation. It seems less practical compared to the inductive case, but it is a necessary step to develop our argument. We now discuss the rationale behind this choice. + +Firstly, the transductive setting is the most common setting for graph data and it was used in most GNN architecture works and GNN acceleration works we mentioned in related work. Therefore, to avoid any confusion and for a fair comparison with numbers from previous literature, we start our experiments with exactly the same input and output as the standard transductive setting. Under this setting, the inputs to GNNs include all the node features and the graph structure, so GLNN is set to be able to access the same input. As GLNN includes a teacher training step and a distillation step, the soft labels of all the nodes are intermediate outputs produced by the teacher training step, and thus used for the second distillation step for the best GLNN performance. This transductive setting can boil down to a sanity check when the student is sufficiently large. Therefore, we separate the setting to be GLNN and $\mathsf { G L N N + }$ and report the results in Table 1 and Table 2 separately. In Table 1, we are checking how well GLNNs can perform compared to GNNs under the equal-parameter constraint. The results can be interpreted as given a fixed parameter budget, whether there exists one set of parameters (one instantiation of the MLP) that can achieve competitive results as the GNN. Only when this holds, should we further investigate the more interesting and challenging inductive case as in Section 5.4. + +Secondly, the task we focus on is node classification, which in many cases is considered as semisupervised learning with very scarce labels. For example, Pubmed only uses 60 labeled nodes (20 per class) out of 20K nodes for training. Rather than design an advanced model that can do few-shot learning, our goal here is to leverage as much data as possible to simplify the model for more efficient inference. We thus utilize the soft pseudo-labels on all the unlabelled nodes for the best GLNN performance. In reality, when there is a large amount of separate unlabeled data, these unlabeled data can be used for GLNN distillation training and a different set of labeled data can be used for evaluation. In our case, we mimic this scenario in the inductive setting in Section 5.4. + +# A.6 IMPLEMENTATION AND HARDWARD DETAILS + +The experiments on both baselines and our approach are implemented using PyTorch, the DGL (Wang et al., 2019) library for GNN algorithms, and Adam (Kingma & Ba, 2015) for optimization. We run all experiments on a machine with 80 Intel(R) Xeon(R) E5-2698 v4 $@$ 2.20GHz CPUs, and a single NVIDIA V100 GPU with 16GB RAM. + +# B SPACE AND TIME COMPLEXITY OF GNNS VS. MLPS + +Compared to MLP and GNN, GLNN provides a handy tool for users to trade-off between model accuracy and time complexity, which does not directly focus on space complexity. Given the space and time complexity are related, we now provide a more detailed discussion regarding these two complexities in our experiments. + +In Table 1, the model comparison was between equal-sized MLPs (GLNNs) and GNNs. While fixing parameter budget to control space complexity is a standard approach when comparing models, it is not completely fair for cross-model comparison especially for MLPs vs. GNNs. To do inference with GNNs, the graph needs to be loaded in the memory either entirely or batch by batch, and may use much larger space than the model parameters. Thus, the actual space complexity of GNNs is much higher than equal-sized MLPs. From the time complexity perspective, the major inference latency of GNNs comes from the data dependency as shown in Section 4. Under the same setting as Figure 1, we show in Figure 3 Left that even a 5-layer MLP with 8 times wider hidden layers still runs much faster than a single-layer SAGE. Another example of cross-model comparison is Transformers vs. RNNs. Large Transformers can have more parameters than RNNs because of the attention mechanism, but they are also faster than RNNs in general, which is an important consideration in the context of inference time minimization. + +In Table 1, we saw that for equal-sized comparison, GLNNs are not as accurate as GNNs on the OGB datasets. Following the discussion above and given the GLNNs used in Table 1 are relatively small (3 layers and 256 hidden dimensions) for millions of nodes in the OGB datasets, we ask whether this gap can be mitigated by increasing the MLP and thus GLNN sizes. The answer is yes as shown in Table 2. + +# C CONSISTENCY MEASURE OF MODEL PREDICTIONS AND GRAPH TOPOLOGY BASED ON MIN-CUT + +We introduce a metric to measure the consistency between model predictions and graph topology based on the min-cut problem in Section 5.6. The $K$ -way normalized min-cut problem, or simply min-cut, partitions $N$ nodes in $\nu$ into $K$ disjoint subsets by removing the minimum volume of edges. According to Dhillon et al. (2004), the min-cut problem can be expressed as + +$$ +\operatorname* { m a x } \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \frac { C _ { k } ^ { T } A C _ { k } } { C _ { k } ^ { T } D C _ { k } } +$$ + +with $C$ being the node assignment matrix that partitions $\nu$ , i.e. $C _ { i , j } = 1$ if node $i$ is assigned to class $j$ . $\pmb { A }$ being the adjacency matrix and $_ { D }$ being the degree matrix. This quantity we try to maximize here tells us whether the assignment is consistent with the graph topology. The bigger it is, the less edges need to be removed, and the assignment is more consistent with existing connections in the graph. In Bianchi et al. (2019), the authors show that when replacing the hard assignments $C \in \{ 0 , 1 \} ^ { N \times K }$ with a soft classification probability $\hat { \pmb Y } \in [ 0 , 1 ] ^ { N \times K }$ , a cut loss $\mathcal { L } _ { c u t }$ in Equation 2 can become a good approximation of Equation 4 and be used as the measuring metric. + +Table 8: GLNN predictions are much more consistent with the graph topology than MLPs. We show the $\mathcal { L } _ { c u t }$ values of GNNs, MLPs, and GLNN s on five CPF datasets. GLNN $\mathcal { L } _ { c u t }$ values become pretty close to the high $\mathcal { L } _ { c u t }$ values of GNNs, which were closely related to the GNN inductive bias. + +
DatasetsSAGEMLPGLNN
Cora0.93470.70260.8852
Citeseer0.94850.76930.9339
Pubmed0.96050.94550.9701
A-computer0.90030.69760.8638
A-photo0.86640.70690.8398
Average0.92210.76440.8986
+ +# D EXPRESSIVENESS OF GNNS VS. MLPS IN TERMS OF EQUIVALENCE CLASSES OF ROOTED GRAPHS + +In Chen et al. (2021), the expressiveness of GNNs and GA-MLPs were theoretically quantified in terms of induced equivalence classes of rooted graphs. We adopt their framework and perform a similar analysis for GNNs vs. MLPs. We first define rooted graphs. + +Definition 1 (Rooted Graph). A rooted graph, denoted as $\mathcal { G } ^ { [ i ] }$ is a graph with one node i in $\mathcal { G } ^ { [ i ] }$ designated as the root. GNNs, GA-MLPs, and MLPs can all be considered as functions on rooted graphs. The goal of a node-level task on node $i$ with label $\mathbf { \nabla } _ { \mathbf { \psi } _ { 3 } } \psi _ { i }$ is to fit a function to the input-output pairs $( \mathcal G ^ { [ i ] } , y _ { i } )$ . + +We denote the space of rooted graphs as $\mathcal { E }$ . Following Chen et al. (2021), the expressive power of a model on graph data is evaluated by its ability to approximate functions on $\mathcal { E }$ . This is further characterized as the number of induced equivalence classes of rooted graphs on $\mathcal { E }$ , with the equivalence relation defined as the following. Given a family of functions $\mathcal { F }$ on $\mathcal { E }$ , we define an equivalence relation $\simeq _ { \varepsilon , F }$ among all rooted graphs such that $\forall \mathcal { G } ^ { [ i ] } , \mathcal { G } ^ { \prime [ j ] } \in \mathcal { E } , \mathcal { G } ^ { [ i ] } \simeq _ { \mathcal { E } , \mathcal { F } } \mathcal { G } ^ { \prime [ j ] }$ if and only if $\forall f \in \mathcal { F } , f ( \mathcal { G } ^ { [ i ] } ) = f ( \mathcal { G } ^ { \prime [ j ] } )$ . We now give a proposition to characterize the GNN expressive power (proof in Appendix E). + +Proposition 1. With $\mathcal { X }$ denotes the set of all possible node features and assuming $| \mathcal { X } | \ge 2$ , with $m$ denotes the maximum node degree and assuming $m \geq 3$ , the total number of equivalence classes of rooted graphs induced by an $L$ -layer GNN is lower bounded by $\binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 }$ + +As shown in Proposition 1, the expressive power of GNNs grows doubly-exponentially in the number of layers $L$ , which means it grows linearly in $L$ after taking $\log ( \log ( \cdot ) )$ . The expressive power GA-MLPs only grows exponentially in $L$ as shown in Chen et al. (2021). Under this framework, the expressive power of MLPs, which corresponds to a 0-layer GA-MLP, is $| \mathcal { X } |$ . Since the former is much larger than the latter, the conclusion will be GNNs are much more expressive than MLPs. The gap between these two numbers indeed exists, but empirically this gap will only make a difference when $| \mathcal { X } |$ is small. As in Chen et al. (2021), both the lower bound proof and the constructed examples showing GNNs are more powerful than GA-MLPs assumed $| \mathcal { X } | = 2$ . In real applications and datasets considered in this work, the node features can be high dimensional vectors like bag-of-words, which makes $| \mathcal { X } |$ enormous. Thus, this gap doesn’t matter much empirically. + +# E PROOF OF THE PROPOSITION 1 + +To prove Proposition 1, we first define rooted aggregation trees, which is similar to but different from rooted graphs. + +Definition 2 (Rooted Aggregation Tree). The depth-K rooted aggregation tree of a rooted graph $\mathcal { G } ^ { [ i ] }$ is a depth-K rooted tree with $a$ (possibly many-to-one) mapping from every node in the tree to some node in $\mathcal { G } ^ { [ i ] }$ , where (i) the root of the tree is mapped to node $i$ , and (ii) the children of every node $j$ in the tree are mapped to the neighbors of the node in $\mathcal { G } ^ { [ i ] }$ to which $j$ is mapped. + +A rooted aggregation tree can be obtained by unrolling the neighborhood aggregation steps in the GNNs. An illustration of rooted graphs and rooted aggregation trees can be found in Chen et al. (2021) Figure 4. We denote the set of all rooted aggregation trees of depth $\mathrm { L }$ using $\mathcal { T } _ { L }$ . Then we use $\mathcal { T } _ { L , \mathcal { X } , m }$ to denote a subset of $\mathcal { T } _ { L }$ , where the node features belong to $\mathcal { X }$ , and all the nodes have exactly degree $m$ ( $m$ children), and at least two nodes out of these m nodes have different features. In other words, a node can’t have all identical children. With rooted aggregation trees defined, we are ready to prove Proposition 1. The proof is adapted from the proof of Lemma 3 in Chen et al. (2021). + +Proof. Since the number of equivalence classes on $\mathcal { E }$ induced by the family of all depth-L GNNs consists of all rooted graphs that share the same rooted aggregation tree of depth- $\mathrm { . L }$ (Chen et al., 2021), the lower bound problem in Proposition 1 can be reduced to lower bound $| \mathcal { T } _ { L } |$ , which can be further reduced to lower bound the subset $| \mathcal { T } _ { L , \mathcal { X } , m } |$ . We now show $\begin{array} { r } { | \mathcal { T } _ { L , \mathcal { X } , m } | \geq \binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 } } \end{array}$ inductively. + +When $L = 1$ , the root of the tree can have $| \mathcal { X } |$ different choices. For the children nodes, we $m$ $| \mathcal { X } |$ are allowed. This leads to . $\binom { | x | + m - 1 } { m }$ cases. Therefore, $\mathcal { T } _ { L + 1 , \mathcal { X } , m } = | \mathcal { X } | \binom { | \mathcal { X } | + m - 1 } { m } \ge \binom { | \mathcal { X } | + m - 2 } { m - 1 }$ + +Assuming the statement holds for $L$ , we show it holds for $L + 1$ by constructing trees in $\tau _ { L + 1 , x , m }$ from $T , T ^ { \prime } \in \mathcal { T } _ { L , x , m }$ . We do this by assigning node features in $\mathcal { X }$ to the $m$ children of each leaf node in $T$ and $T ^ { \prime }$ . First note that when $T$ and $T ^ { \prime }$ are two non-isomorphic trees, two depth- $\mathrm { . L } { + } 1$ trees constructed from $T$ and $T ^ { \prime }$ will be different no matter how the node features are assigned. Now we consider all the trees can be constructed from $T$ by assign node features of children to leaf nodes. + +We first consider all paths from the root to leaves in $T$ . Each path consists of a sequence of nodes where the node features form a one-to-one mapping to an $\mathrm { L }$ -tuple $\tau \in \{ ( x _ { 1 } , \ldots , x _ { L } ) : x _ { i } \in \mathcal { X } \}$ . Leaf nodes are called node under $\tau$ if the path from the root to it corresponds to $\tau$ . The children of nodes under different $\tau \mathbf { S }$ are always distinguishable, and thus any assignments lead to distinct rooted aggregation trees of depth $L + 1$ . The assignment of children of nodes under the same $\tau$ , on the other hand, could be overcounted. Therefore, to lower bound $\tau _ { L + 1 , \mathcal { X } , m }$ , we only consider a special way of assignments to avoid over counting, which is that children of all nodes under the same $\tau$ are assigned the same set of features. + +Since we assumed that at least two nodes of $T$ have different features, there are at least $2 ^ { L }$ different $\tau \mathbf { S }$ corresponding to the path from the root to leaves. For a leaf node $j$ under a fixed $\tau$ , one of its children needs to have the same feature as $j$ ’s parent node. This restriction is due to the definition of rooted aggregation trees. Therefore, we only pick features for the other $m - 1$ nodes, which will be $\binom { | \mathcal { X } | + m - 2 } { m - 1 }$ cases for each $j$ . Then through this construction, the total number of depth- $\mathrm { . L } { + } 1$ trees from T can be lower bounded by |X |+m−2m−1 2L. $T \in \mathcal { T } _ { L , \mathcal { X } , m }$ , so we derive $\begin{array} { r } { \mathcal { T } _ { L + 1 , \mathcal { X } , m } \geq \binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } } \mathcal { T } _ { L , \mathcal { X } , m } } \end{array}$ hav, and $\begin{array} { r } { \mathcal { T } _ { L , \mathcal { X } , m } \geq \binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { \sum _ { l = 1 } ^ { L } 2 ^ { l } } = } \end{array}$ $\binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 }$ + +# F ADVANCED GNN ARCHITECTURES AS THE TEACHER + +In our experiment, SAGE teacher is used throughout to avoid influence by model architecture. Some other GNNs like GCN are also considered in the ablation studies, but they are not the best known architecture for a specific dataset. To show GLNN has stronger performance given a stronger teacher, we consider the best teacher we can access on Products. We take $\mathrm { M L P { + } C S }$ Huang et al. (2021) from the OGB leaderboard as a new teacher, which has reported accuracy $8 4 . 1 8 \%$ and ranks 8 on the leadarboard as of Nov 2021. We choose $_ { \mathrm { M L P + C S } }$ instead of the other top 7 because the others either rely on raw text (additional info to the given node feature), or require a large GPU with ${ > } 1 6 \mathrm { G B }$ memory, which we don’t have access to. Also, their improvement is not super significant compared to $\mathrm { M L P { + } C S }$ , i.e. $84 \%$ to $86 \%$ . The result with $\mathrm { M L P { + } C S }$ teacher is shown in Table 9. We see that with the new teacher, performance of ${ \mathrm { G L N N } } +$ improves to be even better than SAGE $( 7 8 . 6 1 \% )$ , which shows GLNN can get stronger given a stronger teacher. + +Table 9: ${ \mathrm { G L N N } } +$ with $\mathrm { M L P { + } C S }$ teacher on Products + +
MLP+C&SMLP+GLNN+
Acc84.1864.5082.94
+ +# G GLNN WITH FEATURE AUGMENTATION FROM ONE-HOP NEIGHBORS + +In our main experiment, the inductive performance of GLNN on the Arxiv dataset is less desirable than others. We thus consider augment the node features with their one-hop neighbors to include more graph information. This can be seen as a middle ground between pure GLNNs and GNNs. For this new experiment, we follow the setting in Table 3 but with two new approaches. We explain the setting of these two approaches below. + +1. 1-hop GA-MLP: firstly, for each node $v$ , we collect features of its 1-hop neighbors $u$ to augment the raw feature of $v$ , i.e. $x _ { v } \tilde { x } _ { v }$ , like in SGC. Then we train an MLP on the graph with $\tilde { x } _ { v }$ . Note if $v$ is in the observed graph but $u$ is in the inductive (unobserved during training) part, then $v$ doesn’t collect features from $u$ . + +2. 1-hop GA-GLNN: Go through the same feature augmentation step as 1-hop GA-MLP. Then train an MLP with distillation from teacher GNN. + +3. In summary, we compare 5 different models in the table below + +(a) SAGE: single model on $x _ { v }$ +(b) MLP: single model on $x _ { v }$ +(c) GLNN: SAGE teacher and MLP student on $x _ { v }$ +(d) 1-hop GA-MLP: single model on $\tilde { x } _ { v }$ +(e) 1-hop GA-GLNN: SAGE teacher on $x _ { v }$ , MLP student on $\tilde { x } _ { v }$ + +We show in the table below, with 1-hop neighbor features, performance of GLNN improves a lot. This is expected as we also observe significant improvement from MLP to 1-hop GA-MLP. However, we indeed see 1-hop GA-GLNN (68.83) can further improve from 1-hop GA-MLP (66.62) and nearly match the teacher (70.64). + +Table 10: GLNN with feature augmentation from one-hop neighbor on Arxiv + +
EvalSAGEMLPGLNN1-hop GA-MLP1-hop GA-GLNN
Arxivind70.6455.4060.4866.6268.83
tran70.7555.2871.4666.6769.82
+ +As we have shown in Figure 3, the 1-Layer GNN in our case is roughly 4 times slower than GLNN (29.31ms vs. $7 . 5 6 \mathrm { m s } ,$ ), which should be a good approximation for the speed comparison between 1-hop GA-MLP/GA-GLNN and GLNN. This result is practically beneficial, as it gives practitioners more flexibility about how much accuracy they want to trade for less inference time. + +This section is a continuation of the ablation study of inductive split rate in Section 6. It generalizes Figure 5 Middle to more split rates (from 10:90 to 90:10), and explicitly show the inductive and transductive performance on each dataset. For better visualization, the training data label rate is also reduced from 20 per class to 5 per class in the following plots. + +![](images/46024ef2ae15c38b05bcf61172d3fc0a7188e2c13b24b720ed673291135bb45b.jpg) +Figure 7: Model inductive performance comparison between MLP, GNN(SAGE), and GLNN under different inductive split rate in the production setting. + +![](images/f66528cb30229f3d636a468c436934c7c9c543d8c492f108574b73bb3899b35d.jpg) +Figure 8: Model transductive performance comparison between MLP, GNN(SAGE), and GLNN under different inductive split rate in the production setting. + +# I GLNN UNDER NODE FEATURE HETEROGENEITY AND NON-HOMOPHILY + +Besides the 7 datasets used in the main experiments, we consider 4 more datasets from Ivanov & Prokhorenkova (2021) and Lim et al. (2021) to further evaluate GLNN. + +The House_class and VK_class datasets are from Ivanov & Prokhorenkova (2021). The node features of these two graphs are based on tabular data, which have different types, scales, and meanings as the opposite of the bag-of-word node features in Cora and etc. Some basic statistics of the datasets are shown in the following table. + +Table 11: Statistics of dataset with heterogeneous node features + +
Dataset# Nodes#Edges# Features#Classes
House_class20,640182,14665
VK_class54,028213,644147
+ +We apply the GLNN on House_class and VK_class using the best BGNN model from Ivanov & Prokhorenkova (2021) as the teacher. The comparison is shown in the following table. Ivanov & Prokhorenkova (2021) also includes GAT, GCN, AGNN, and APPNP as baselines, whose performance on these two datasets are quite similar (difference $< 0 . 0 2 5$ ). We compare with these baselines by including the best result among the 4 GNN models and refer it as GNN in the table below, i.e. GNN $=$ max(GAT, GCN, AGNN, APPNP). From the table, we see that GLNN can improve from MLP, outperform GNN and LightGBM, and become competitive to the teacher BGNN. + +Table 12: GLNN on datasets with heterogeneous node features. Numbers other than GLNN are taken from Ivanov & Prokhorenkova (2021) + +
DatasetLightGBMGNNsBGNNMLPGLNN
House_class0.550.6250.6820.5340.672
VK_class0.570.5770.6830.5670.641
+ +We further pick the non-homophilous Penn94 and Pokec datasets from Lim et al. (2021). Some basic statistics of the datasets are shown in the following table. + +Table 13: Statistics of non-homophilous datasets + +
Dataset# Nodes#Edges# Features# Classes
Penn9441,5361,590,65552
Pokec1,632,80330,622,564652
+ +Using the GCN teacher, we see that the performance of GLNN is improved over MLP and becomes competitive to the teacher GCN on $\mathtt { P e n n 9 4 }$ . However, on Pokec, the simple LINK model can achieve very good performance, and it is better than most GNNs reported in Lim et al. (2021). LINK is a purely structural model which does not use node features at all. This shows that the Pokec dataset corresponds to the setting we discussed in Sec 5.8 (limitations of GLNN) – if the node labels can be largely determined by only the graph structure, then GLNN will struggle. We observe that GLNN is not as good as LINK owing to this limitation. However, we still see that for most of the non-homophilous datasets, MLPs already work quite well on them, and we can use GLNN for the other ones like Penn94. + +Table 14: GLNN on non-homophilous datasets. Numbers other than GLNN are taken from Lim et al. (2021) + +
DatasetLINKGCNMLPGLNN
Penn9480.7982.4773.6181.69
Pokec80.5475.4562.3761.32
+ +# J MODEL COMPARISON WITH NOISY NODE FEATURES + +In Section 6, we conducted an ablation study to compare model performance with noisy node features, and the result is shown in the left plot in Figure 5. There are two subtle points in this plot. (1) The performance of GNN is still relatively high for high noisy features, even when $\alpha = 1$ and the features are completely random. (2) For completely random features, the performance of GLNN is still higher than MLP. We now discuss and explain them in more detail. + +GNN Performance on Random Features. GNN still performs well because nodes with the same labels are likely to be connected and GNN can overfit the training data. We explain the detail through a toy example. Suppose there is a 4-clique containing nodes A, B, C, D in the graph with only a single edge D-E connects this clique to other graph nodes. Suppose A, B, C, D all have iid random Gaussian raw features and the same class label c. Let’s pick A to be the inductive test node and assume E and the triangle formed by B, C, D to be in the training graph. Let’s consider a simple example for 1-layer GCN and break down message passing into feature aggregation and nonlinear transformation. During training, GNN can overfit the data by learning a nonlinear transformation which maps the aggregated features of B, C, D to class c. The aggregated features of B and C will just be the average of the raw features of B, C, D. Although E is also involved in D’s feature aggregation step, the aggregated features of D will also be very close to this average. Then when test on A, the aggregated feature of A will likely be classified to the same class c by the overfitted nonlinear transformation because it is the average of raw node features of A, B, C, D. In this case, GNN can actually correctly classify A because of the overfitting. For GNNs with more layers and graphs with more neighbor nodes, the conclusion may be generalized.This is roughly sort of a “majority vote” process. For a test node A, if many nodes, which A collects features from, have the same class label and appear in the training graph, then A will be classified as this class by an overfitted classifier. + +GLNN and MLP Performance on Random Features. The gap between MLP and GLNN is due to imbalanced datasets. The GLNN can learn the imbalance from soft labels, whereas MLPs can only access uniformly picked training nodes. We explain more detail using the A-computer dataset as an example, for which the gap between MLP and GLNN is obvious. The task is 10-class classification. With random node features $( \alpha { = } 1 )$ ), the inductive accuracy for MLP is 0.0652 and 0.2538 for GLNN. If the data labels are uniform, then both models should give an accuracy around 0.1. However, the labels on the inductive dataset are actually imbalanced. We show the results in Figure 9. The hist on the left is the label distribution of the inductive test set. In particular, class 4 takes about $40 \%$ . However, given this imbalance, the standard train-test split selects training nodes uniformly among labels. In this case, 20 nodes per class. Therefore, the predictions of MLP on random features are expected to be relatively uniform because the 200 nodes we train it on are uniform. This gives the hist shown in the middle, where the largest class takes about $1 7 . 5 \%$ . Finally, for GLNN, we train it on all the 200 training nodes with hard labels, plus soft labels of other nodes in the observed graph $\mathcal { G } _ { o b s }$ (see Section 5.2). Since these extra nodes are selected randomly, whose label distribution is actually similar to the label distribution on the whole data and the distribution on the inductive test set. Therefore, we get the GLNN predictions hist on the right. Although for each node, we can’t assign a prediction correlated to its feature, on average the distribution is very close to the true label distribution on the inductive test set and has a much higher expectation. In fact, if the prediction distribution is exactly the true distribution on the inductive test set, the expectation will be 0.2169. GLNN actually does even a bit better by putting its bet more on the largest class. + +![](images/403a9c5a1b32a03dcf520caf81b8a2a56b3a306859c74e50c516bf26d61ec307.jpg) +Figure 9: Inductive (predicted) label distribution on the A-computer dataset. Left: true labels. Middle: predicted labels by MLP. Right: predicted labels by GLNN. \ No newline at end of file diff --git a/parse/dev/4p6_5HBWPCw/4p6_5HBWPCw_content_list.json b/parse/dev/4p6_5HBWPCw/4p6_5HBWPCw_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..746c1b11cb43af48edeedab05876d34b9f9e0793 --- /dev/null +++ b/parse/dev/4p6_5HBWPCw/4p6_5HBWPCw_content_list.json @@ -0,0 +1,2473 @@ +[ + { + "type": "text", + "text": "GRAPH-LESS NEURAL NETWORKS: TEACHING OLD MLPS NEW TRICKS VIA DISTILLATION ", + "text_level": 1, + "bbox": [ + 176, + 99, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Shichang Zhang∗ University of California, Los Angeles shichang@cs.ucla.edu ", + "bbox": [ + 184, + 170, + 433, + 212 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yozen Liu \nSnap Inc. \nyliu2@snap.com ", + "bbox": [ + 552, + 170, + 692, + 212 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yizhou Sun ", + "text_level": 1, + "bbox": [ + 184, + 234, + 266, + 246 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "University of California, Los Angeles yzsun@cs.ucla.edu ", + "bbox": [ + 184, + 248, + 431, + 275 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Neil Shah \nSnap Inc. \nnshah@snap.com ", + "bbox": [ + 560, + 233, + 700, + 275 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 313, + 544, + 327 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Graph Neural Networks (GNNs) are popular for graph machine learning and have shown great results on wide node classification tasks. Yet, they are less popular for practical deployments in the industry owing to their scalability challenges incurred by data dependency. Namely, GNN inference depends on neighbor nodes multiple hops away from the target, and fetching them burdens latency-constrained applications. Existing inference acceleration methods like pruning and quantization can speed up GNNs by reducing Multiplication-and-ACcumulation (MAC) operations, but the improvements are limited given the data dependency is not resolved. Conversely, multi-layer perceptrons (MLPs) have no graph dependency and infer much faster than GNNs, even though they are less accurate than GNNs for node classification in general. Motivated by these complementary strengths and weaknesses, we bring GNNs and MLPs together via knowledge distillation (KD). Our work shows that the performance of MLPs can be improved by large margins with GNN KD. We call the distilled MLPs Graph-less Neural Networks (GLNNs) as they have no inference graph dependency. We show that GLNNs with competitive accuracy infer faster than GNNs by $1 4 6 \\times - 2 7 3 \\times$ and faster than other acceleration methods by $1 4 \\times - 2 7 \\times$ . Under a production setting involving both transductive and inductive predictions across 7 datasets, GLNN accuracies improve over stand-alone MLPs by $1 2 . 3 6 \\%$ on average and match GNNs on 6/7 datasets. Comprehensive analysis shows when and why GLNNs can achieve competitive accuracies to GNNs and suggests GLNN as a handy choice for latency-constrained applications. ", + "bbox": [ + 233, + 344, + 766, + 636 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 665, + 334, + 681 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Graph Neural Networks (GNNs) have recently become very popular for graph machine learning (GML) research and have shown great results on node classification tasks (Kipf & Welling, 2016; Hamilton et al., 2017; Velickovi ˇ c et al. ´ , 2017) like product prediction on co-purchasing graphs and paper category prediction on citation graphs. However, for large-scale industrial applications, MLPs remain the major workhorse, despite common (implicit) underlying graphs and suitability for GML formalisms. One reason for this academic-industrial gap is the challenges in scalability and deployment brought by data dependency in GNNs (Zhang et al., 2020; Jia et al., 2020), which makes GNNs hard to deploy for latency-constrained applications that require fast inference. ", + "bbox": [ + 174, + 696, + 825, + 809 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Neighborhood fetching caused by graph dependency is one of the major sources of GNN latency. Inference on a target node necessitates fetching topology and features of many neighbor nodes, especially on small-world graphs (detailed discussion in Section 4). Common inference acceleration techniques like pruning (Zhou et al., 2021) and quantization (Tailor et al., 2021; Zhao et al., 2020) can speed up GNNs to some extent by reducing Multiplication-and-ACcumulation (MAC) operations. ", + "bbox": [ + 174, + 815, + 825, + 886 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, their improvements are limited given the graph dependency is not resolved. Unlike GNNs, MLPs have no dependency on graph data and are easier to deploy than GNNs. They also enjoy the auxiliary benefit of sidestepping the cold-start problem that often happens during the online prediction of relational data (Wei et al., 2020), meaning MLPs can infer reasonably even when neighbor information of a new encountered node is not immediately available. On the other hand, this lack of graph dependency typically hurts for relational learning tasks, limiting MLP performance on GML tasks compared to GNNs. We thus ask: can we bridge the two worlds, enjoying the low-latency, dependency-free nature of MLPs and the graph context-awareness of GNNs at the same time? ", + "bbox": [ + 174, + 103, + 825, + 215 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Present work. Our key finding is that it is possible to distill knowledge from GNNs to MLPs without losing significant performance, but reducing the inference time drastically for node classification. The knowledge distillation (KD) can be done offline, coupled with model training. In other words, we can shift considerable work from the latency-constrained inference step, where time reduction in milliseconds makes a huge difference, to the less time-sensitive training step, where time cost in hours or days is often tolerable. We call our approach Graph-less Neural Network (GLNN). Specifically, GLNN is a modeling paradigm involving KD from a GNN teacher to a student MLP; the resulting GLNN is an MLP optimized through KD, so it enjoys the benefits of graph contextawareness in training but has no graph dependency in inference. Regarding speed, GLNNs have superior efficiency and are $1 4 6 \\times - 2 7 3 \\times$ faster than GNNs and $1 4 \\times - 2 7 \\times$ faster than other inference acceleration methods. Regarding performance, under a production setting involving both transductive and inductive predictions on 7 datasets, GLNN accuracies improve over MLPs by $1 2 . 3 6 \\%$ on average and match GNNs on $6 / 7$ datasets. We comprehensively study when and why GLNNs can achieve competitive results as GNNs. Our analysis suggests the critical factors for such great performance are large MLP sizes and high mutual information between node features and labels. Our observations align with recent results in vision and language, which posit that large enough (or slightly modified) MLPs can achieve similar results as CNNs and Transformers (Liu et al., 2021; Tolstikhin et al., 2021; Melas-Kyriazi, 2021; Touvron et al., 2021; Ding et al., 2021). Our core contributions are as follows: ", + "bbox": [ + 174, + 222, + 826, + 472 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We propose GLNN, which eliminates neighbor-fetching latency in GNN inference via KD to MLP. • We show GLNNs has competitive performance as GNNs, while enjoying $1 4 6 \\times - 2 7 3 \\times$ faster inference than vanilla GNNs and $1 4 \\times - 2 7 \\times$ faster inference than other inference acceleration methods. • We study GLNN properties comprehensively by investigating their performance under different settings, how they work as regularizers, their inductive bias, expressiveness, and limitations. ", + "bbox": [ + 174, + 478, + 825, + 549 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 570, + 344, + 587 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Graph Neural Networks. The early GNNs generalize convolution nets to graphs (Bruna et al., 2014; Defferrard et al., 2017) and later simplified to message-passing neural net (MPNN) by GCN (Kipf & Welling, 2016). Most GNNs after can be put as MPNNs. For example, GAT employs attention (Velickovi ˇ c et al. ´ , 2017), PPNP employs personalized PageRank (Klicpera et al., 2019), GCNII and DeeperGCN employ residual connections and dense connections (Chen et al., 2020; Li et al., 2019). ", + "bbox": [ + 174, + 603, + 825, + 672 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Inference Acceleration. Inference acceleration have been proposed by hardware improvements (Chen et al., 2016; Judd et al., 2016) and algorithmic improvements through pruning (Han et al., 2015), quantization (Gupta et al., 2015). For GNNs, pruning (Zhou et al., 2021) and quantizing GNN parameters (Zhao et al., 2020) have been studied. These approaches speed up GNN inference to a certain extent but do not eliminate the neighbor-fetching latency. In contrast, our cross-model KD solves this issue. Concurrently, Graph-MLP also tries to bypass GNN neighbor fetching (Hu et al., 2021) by training an MLP with a neighbor contrastive loss, but it only considers transductive but not the more practical inductive setting. Some sampling works focus on speed up GNN training (Zou et al., 2019; Chen et al., 2018), which are complementary to our goal on inference acceleration. ", + "bbox": [ + 174, + 680, + 825, + 805 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "GNN distillation. Existing GNN KD works try to distill large GNNs to smaller GNNs. LSP (Yang et al., 2021b) and TinyGNN (Yan et al., 2020) do KD while preserving local information. Their students are GNNs with fewer parameters but not necessarily fewer layers. Thus, both designs still require latency-inducing fetching. GFKD (Deng & Zhang, 2021) does graph-level KD via graph generation. In GFKD, data instances are independent graphs, whereas we focus on dependent nodes within a graph. GraphSAIL (Xu et al., 2020) uses KD to learn students work well on new data while preserving performance on old data. CPF (Yang et al., 2021a) combines KD and label propagation (LP). The student in CPF is not a GNN, but it is still heavily graph-dependent as it uses LP. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/3c59c6aff8764b30e7d812c76a7e7b24c6d87080f7daff930791e286b7e0e843.jpg", + "image_caption": [ + "Figure 1: The number of fetches and the inference time of GNNs are both magnitudes more than MLPs and grow exponentially as functions of the number of layers. Left: neighbors need to be fetched for two GNN layers. Middle: the total number of fetches for inference. Right: the total inference time. (Inductive inference for 10 random nodes on OGB Products (Hu et al., 2020)) " + ], + "image_footnote": [], + "bbox": [ + 222, + 102, + 771, + 247 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 176, + 329, + 339, + 344 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Notations. For GML tasks, the input is usually a graph and its node features, which we write as $\\mathcal { G } = ( \\nu , \\mathcal { E } )$ , with $\\nu$ stands for all nodes, and $\\mathcal { E }$ stands for all edges. Let $N$ denote the total number of nodes. We use $\\pmb { X } \\in \\mathbb { R } ^ { N \\times D }$ to represent node features, with row $\\mathbf { \\boldsymbol { x } } _ { v }$ being the $D$ -dimensional feature of node $v \\in \\mathcal V$ . We represent edges with an adjacency matrix $\\pmb { A }$ , with $A _ { u , v } = 1$ if edge $( u , v ) \\in \\mathcal { E }$ , and 0 otherwise. For node classification, one of the most important GML applications, the prediction targets are $\\pmb { Y } \\in \\mathbb { R } ^ { N \\times K }$ , where row $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { Y } } _ { v } }$ is a $K$ -dim one-hot vector for node $v$ . For a given $\\mathcal { G }$ , usually a small portion of nodes will be labeled, which we mark using superscript $L$ , i.e. $\\mathcal { V } ^ { \\tilde { L } }$ , $X ^ { L }$ , and $\\mathbf { \\nabla } _ { \\mathbf { Y } } \\breve { L }$ . The majority of nodes will be unlabeled, and we mark using the superscript $U$ , i.e. $\\mathcal { V } ^ { U }$ , $X ^ { U }$ , and $\\mathbf { \\nabla } _ { \\mathbf { Y } } U$ . ", + "bbox": [ + 173, + 359, + 826, + 472 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Graph Neural Networks. Most GNNs fit under the message-passing framework, where the representation $h _ { v }$ of each node $v$ is updated iteratively in each layer by collecting messages from its neighbors denoted as $\\mathcal { N } ( v )$ . For the $l$ -th layer, $\\mathbf { \\Sigma } _ { h _ { v } ^ { ( l ) } }$ is obtained from the previous layer representation $ { \\boldsymbol { h } } _ { u } ^ { ( l - 1 ) }$ $\\mathbf { \\mathcal { h } } _ { u } ^ { ( 0 ) } = \\mathbf { \\mathcal { x } } _ { u } )$ ) via an aggregation operation AGGR followed by an UPDATE operation as ", + "bbox": [ + 174, + 477, + 826, + 542 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/4bc240481ba9c1fa2eea593b231ecd03030f7cea35038b6d250adb06797d22f4.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { h } _ { N ( v ) } ^ { ( l ) } = \\mathrm { A G G R } ( \\{ \\pmb { h } _ { u } ^ { ( l - 1 ) } : u \\in \\mathcal { N } ( v ) \\} ) \\qquad \\mathrm { a n d } \\qquad \\pmb { h } _ { v } ^ { ( l ) } = \\mathrm { U P D A T E } ( \\pmb { h } _ { N ( v ) } ^ { ( l ) } , \\pmb { h } _ { v } ^ { ( l - 1 ) } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 209, + 546, + 785, + 570 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4 MOTIVATION ", + "text_level": 1, + "bbox": [ + 174, + 585, + 315, + 602 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "GNNs have considerable inference latency due to graph dependency. One more GNN layer means fetching one more hop of neighbors. To infer a node with a $L$ -layer GNN on a graph with average degree $R$ requires $\\mathcal { O } ( R ^ { L } )$ fetches. $R$ can be large for real-world graphs, e.g. 208 for the Twitter (Ching et al., 2015). Also, as layer fetching must be done sequentially, the total latency explodes quickly as $L$ increases. Figure 1 shows the dependency added by each GNN layer and the exponential explosion of inference time. In contrast, the MLP inference time is much smaller and grows linearly. This marked gap contributes greatly to the practicality of MLPs in industrial applications over GNNs. ", + "bbox": [ + 173, + 616, + 826, + 715 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The node-fetching latency is exacerbated by two factors: firstly, newer GNN architectures are getting deeper from 64 layers (Chen et al., 2020) to even 1001 layers (Li et al., 2021). Secondly, industrialscale graphs are frequently too large to fit into the memory of a single machine (Jin et al., 2022), necessitating sharding of the graph out of the main memory. For example, Twitter has 288M monthly active users (nodes) and an estimated 60B followers (edges) as of 3/2015. Facebook has 1.39B active users with more than 400B edges as of 12/2014 (Ching et al., 2015). Even when stored in a sparse-matrix-friendly format (often COO or CSR), these graphs are on the order of TBs and are constantly growing. Moving away from in-memory storage results in even slower neighbor-fetching. ", + "bbox": [ + 173, + 720, + 826, + 834 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "MLPs, on the other hand, lack the means to exploit graph topology, which hurts their performance for node classification. For example, test accuracy on Products is 78.61 for GraphSAGE compared to 62.47 for an equal-sized MLP. Nonetheless, recent results in vision and language posit that large (or slightly modified) MLPs can achieve similar results as CNNs and Transformers (Liu et al., 2021). We thus also ask: Can we bridge the best of GNNs and MLPs to get high-accuracy and low-latency models? This motivates us to do cross-model KD from GNNs to MLPs. ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/fe13a4358c6462ce72c20699e317599db71d44b7281ec5d0c4ff85854d7ca559.jpg", + "image_caption": [ + "Figure 2: The GLNN framework: In offline training, a trained GNN teacher is applied on the graph for soft targets. Then, a student MLP is trained on node features guided by the soft targets. The distilled MLP, now GLNN, is deployed for online predictions. Since graph dependency is eliminated for inference, GLNNs infer much faster than GNNs, and hence the name “Graph-less Neural Network.” " + ], + "image_footnote": [], + "bbox": [ + 173, + 98, + 820, + 253 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "5 GRAPH-LESS NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 333, + 495, + 349 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We introduce GLNN and answer exploration questions of its properties: 1) How do GLNNs compare to MLPs and GNNs? 2) Can GLNNs work well under both transductive and inductive settings? 3) How do GLNNs compare to other inference acceleration methods? 4) How do GLNNs benefit from KD? 5) Do GLNNs have sufficient model expressiveness? 6) When will GLNNs fail to work? ", + "bbox": [ + 173, + 366, + 825, + 421 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "5.1 THE GLNN FRAMEWORK ", + "text_level": 1, + "bbox": [ + 174, + 440, + 390, + 454 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The idea of GLNN is straightforward, yet as we will see, extremely effective. In short, we train a “boosted” MLP via KD from a teacher GNN. KD was introduced in Hinton et al. (2015), where knowledge was transferred from a cumbersome teacher to a simpler student. In our case, we generate soft targets $z _ { v }$ for each node $v$ with a teacher GNN. Then we train a student MLP with both true labels $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { Y } } _ { v } }$ and $z _ { v }$ . The objective is as Equation 1, with $\\lambda$ being a weight parameter, $\\mathcal { L } _ { l a b e l }$ being the cross-entropy between $\\mathbf { \\nabla } _ { \\mathbf { y } _ { v } }$ and student predictions $\\hat { y } _ { v }$ , $\\mathcal { L } _ { t e a c h e r }$ being the KL-divergence. ", + "bbox": [ + 173, + 467, + 825, + 551 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/af326a8c346f30af403e9dc1e4dbd3fe4b6f2d694fbafce9361c3dd60840254a.jpg", + "text": "$$\n\\mathcal { L } = \\lambda \\Sigma _ { v \\in \\mathcal { V } ^ { L } } \\mathcal { L } _ { l a b e l } ( \\hat { y } _ { v } , y _ { v } ) + ( 1 - \\lambda ) \\Sigma _ { v \\in \\mathcal { V } } \\mathcal { L } _ { t e a c h e r } ( \\hat { y } _ { v } , z _ { v } )\n$$", + "text_format": "latex", + "bbox": [ + 295, + 571, + 700, + 590 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The model after KD, i.e. GLNN, is essentially a MLP. Therefore, GLNNs have no graph dependency during inference and are as fast as MLPs. On the other hand, through offline KD, GLNN parameters are optimized to predict and generalize as well as GNNs, with the added benefit of faster inference and easier deployment. In Figure 2, we show the offline KD and online inference steps of GLNNs. ", + "bbox": [ + 174, + 602, + 825, + 659 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "5.2 EXPERIMENT SETTINGS ", + "text_level": 1, + "bbox": [ + 174, + 676, + 382, + 691 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Datasets. We consider all five datasets used in the CPF paper (Yang et al., 2021a), i.e. Cora, Citeseer, Pubmed, A-computer, and A-photo. To fully evaluate our method, we also include two more larger OGB datasets (Hu et al., 2020), i.e. Arxiv and Products. ", + "bbox": [ + 174, + 703, + 825, + 746 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Model Architectures. For consistent results, we use GraphSAGE (Hamilton et al., 2017) with GCN aggregation as the teacher. We conduct ablation studies of other GNN teachers like GCN (Kipf & Welling, 2016), GAT (Velickovi ˇ c et al. ´ , 2017) and, APPNP (Klicpera et al., 2019) in Section 6. ", + "bbox": [ + 173, + 752, + 823, + 795 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Evaluation Protocol. For all experiments in this section, we report the average and standard deviation over ten runs with different random seeds. Model performance is measured as accuracy, and results are reported on test data with the best model selected using validation data. ", + "bbox": [ + 174, + 801, + 821, + 844 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Transductive vs. Inductive. Given ${ \\mathcal { G } } , X$ , and $Y ^ { L }$ , we consider node classification under two settings: transductive $( t r a n )$ and inductive $( i n d )$ . For ind, we hold out some test data for inductive evaluation only. We first select inductive nodes $\\mathcal { V } _ { i n d } ^ { U } \\subset \\mathcal { V } ^ { U }$ , which partitions $\\mathcal { V } ^ { U }$ into the disjoint inductive subset and observed subset, i.e. $\\mathcal { V } ^ { U } = \\mathcal { V } _ { o b s } ^ { U } \\sqcup \\mathcal { V } _ { i n d } ^ { U }$ . Then we hold out $v \\in \\mathcal { V } _ { i n d } ^ { U }$ and all edges connected to $v \\in \\mathcal { V } _ { i n d } ^ { U }$ obs ind, which leads to two disjoint graphs $\\mathcal { G } = \\mathcal { G } _ { o b s } \\sqcup \\mathcal { G } _ { i n d }$ ind with no shared nodes or ", + "bbox": [ + 173, + 849, + 825, + 925 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/7931cb574eb9e40028be345ce26f3087cf810557f9904d4f1fc8852cfd022ec3.jpg", + "table_caption": [ + "Table 1: GLNNs outperform MLPs by large margins and match GNNs on 5 of 7 datasets under the transductive setting. $\\Delta _ { M L P }$ $( \\Delta _ { G N N } )$ represents difference between the GLNN and a trained MLP (GNN). Results show accuracy (higher is better); $\\Delta _ { G N N } 2 0$ indicates GLNN outperforms GNN. " + ], + "table_footnote": [], + "table_body": "
DatasetsSAGEMLPGLNN△MLP△GNN
Cora80.52 ± 1.7759.22 ±1.3180.54 ± 1.3521.32 (36.00%)0.02 (0.02%)
Citeseer70.33 ± 1.9759.61 ± 2.8871.77 ± 2.0112.16 (20.40%)1.44 (2.05%)
Pubmed75.39 ± 2.0967.55 ± 2.3175.42 ± 2.317.87 (11.65%)0.03 (0.04%)
A-computer82.97 ± 2.1667.80 ± 1.0683.03 ± 1.8715.23 (22.46%)0.06 (0.07%)
A-photo90.90 ± 0.8478.77 ± 1.7492.11 ± 1.0813.34 (16.94%)1.21 (1.33%)
Arxiv70.92 ± 0.1756.05 ± 0.4663.46 ± 0.457.41 (13.24%)-7.46 (-10.52%)
Products78.61 ± 0.4962.47 ± 0.1068.86 ± 0.466.39 (10.23%)-9.75 (-12.4%)
", + "bbox": [ + 173, + 145, + 825, + 271 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/618cbe9609dadcbc5b9f8ff792a545f059adc05ef870ebb449d8f99b90319b99.jpg", + "table_caption": [ + "Table 2: Enlarged GLNNs match the performance of GNNs on the OGB datasets. For Arxiv, we use MLPw4 (GLNNw4). For Products, we use MLPw8 (GLNNw8). " + ], + "table_footnote": [], + "table_body": "
DatasetsSAGEMLP+GLNN+△MLP△GNN
Arxiv70.92 ± 0.1755.31± 0.4772.15 ± 0.2716.85 (30.46%)0.51 (0.71%)
Products78.61 ± 0.4964.50 ± 0.4577.65 ± 0.4813.14 (20.38%)-0.97 (-1.23%)
", + "bbox": [ + 173, + 310, + 825, + 371 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "els are partitioned into three disjoint sets, i.e. . Concretely, the input/output of both setting $X = X ^ { L } \\sqcup X _ { o b s } ^ { U } \\sqcup X _ { i n d } ^ { U }$ $\\pmb { Y } = \\pmb { Y } ^ { L } \\sqcup \\pmb { Y } _ { o b s } ^ { U } \\sqcup \\pmb { Y } _ { i n d } ^ { U }$ ", + "bbox": [ + 176, + 387, + 823, + 417 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• ind: train on • tran: train on $\\mathcal { G } _ { o b s }$ ${ \\mathcal { G } } , X$ , $X ^ { L }$ , and , $X _ { o b s } ^ { U }$ $\\mathbf { \\nabla } _ { \\mathbf { Y } ^ { L } }$ ; evaluate on , and $Y ^ { L }$ ; evaluate on $( X ^ { U } , Y ^ { U } )$ $( X _ { i n d } ^ { U } , Y _ { i n d } ^ { U } )$ ; $\\mathrm { K D }$ uses $z _ { v }$ v ; KD uses for $v \\in \\mathcal V$ $z _ { v }$ . for ", + "bbox": [ + 169, + 424, + 748, + 455 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Note that for tran, all the nodes in the graph including the validation and test nodes are used to generate $_ { z }$ . A discussion of this choice along with other experiment details are in Appendix A. ", + "bbox": [ + 173, + 460, + 826, + 489 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5.3 HOW DO GLNNS COMPARE TO MLPS AND GNNS? ", + "text_level": 1, + "bbox": [ + 173, + 506, + 566, + 521 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We start by comparing GLNNs to MLPs and GNNs with the same number of layers and hidden dimensions. We first consider the standard transductive setting, so our results in Table 1 are directly comparable to results reported in previous literature like Yang et al. (2021a) and Hu et al. (2020). ", + "bbox": [ + 174, + 532, + 825, + 575 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As shown in Table 1, the performance of all GLNNs improve over MLPs by large margins. On smaller datasets (first 5 rows), GLNNs can even outperform the teacher GNNs. In other words, for each task, with the same parameter budget, there exists a set of MLP parameters that has GNN-competitive performance (detailed discussion in Sections 5.6 and 5.7). For the larger OGB datasets (last 2 rows), the GLNN performance is improved over MLPs but still worse than the teacher GNNs. However, as we show in Table 2, this gap can be mitigated by increasing MLP size to MLPwi1. In Figure 3 (right), we visualize the trade-off between prediction accuracy and model inference time with different model sizes. We show that gradually increasing GLNN size pushes its performance to be close to SAGE. On the other hand, when we reduce the number of layers of $\\mathrm { S A G E } ^ { 2 }$ , the accuracy quickly drops to be worse than GLNNs. A detailed discussion of the rationale for increasing MLP sizes is in Appendix B. ", + "bbox": [ + 174, + 582, + 825, + 720 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5.4 CAN GLNNS WORK WELL UNDER BOTH TRANSDUCTIVE AND INDUCTIVE SETTINGS? ", + "text_level": 1, + "bbox": [ + 173, + 739, + 799, + 752 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Although transductive is the commonly studied setting for node classification, it does not encompass prediction on unseen nodes. Therefore, it may not be the best way to evaluate a deployed model, which must often generate predictions for new data points as well as reliably maintain performance on old ones. Thus, to better understand the effectiveness of GLNN, we also consider their performance under a realistic production setting, which contains both transductive and inductive predictions. ", + "bbox": [ + 174, + 763, + 825, + 834 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To evaluate a model inductively, we hold out some test nodes from training to form an inductive set, i.e. $\\mathcal { V } ^ { U } = \\mathcal { V } _ { o b s } ^ { U } \\sqcup \\mathcal { V } _ { i n d } ^ { U }$ . In production, a model might be re-trained periodically, e.g. weekly. The obs hold-out nodes in $V _ { i n d } ^ { U }$ represent new nodes entered the graph between two trainings. $V _ { i n d } ^ { U }$ is usually small compared to $V _ { o b s } ^ { U }$ – e.g. Graham (2012) estimates $5- 7 \\%$ for the fastest-growing tech startups. In our case, to mitigate randomness and better evaluate generalizability, we use V Uind containing 20% of the test data. We also evaluate on standard transductive prediction on $V _ { o b s } ^ { U }$ containing the otherrved unlabeled nodes $80 \\%$ of the test data, representing thee inference is commonly redone on existing nodes in real-world cases. We report both results and a interpolated production (prod) results in Table 3. The prod results paint a clearer picture of model generalization as well as accuracy in production. See Section 6 for an ablation study of different inductive split rates other than 20-80. ", + "bbox": [ + 176, + 840, + 825, + 886 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/56094e9645d24171cda661a02619bd1120d8d877a8d7ebb8ed27856fa4567de6.jpg", + "image_caption": [ + "Figure 3: Enlarged MLPs (GLNNs) can match GNN accuracy, but infer dramatically faster. Plots are under the same setting as Figure 1. Left: inference time of MLPs vs. GNN (SAGE) for different model sizes. Right: model accuracy vs. inference time. Note: time axes are log-scaled. " + ], + "image_footnote": [], + "bbox": [ + 228, + 106, + 767, + 289 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/48643f2b7590c7ec7d45551a8acf5a14bd95329b32a6aee2b04294fcbff0de8b.jpg", + "table_caption": [ + "Table 3: GLNNs match GNN performance on a production setting with both inductive and transductive predictions. We uProducts. ind results on $V _ { i n d } ^ { U }$ LP for the 5 C, tran results on $V _ { o b s } ^ { U }$ atasets, MLPw4 for Arxiv, and MLPw8 for, and the interpolated prod results are reported. " + ], + "table_footnote": [], + "table_body": "
DatasetsEvalSAGEMLP/MLP+GLNN/GLNN+△MLP△GNN
Coraprod79.2958.9878.2819.30 (32.72%)-1.01 (-1.28%)
ind81.33 ± 2.1959.09 ± 2.9673.82 ± 1.9314.73 (24.93%)-7.51 (-9.23%)
tran78.78 ± 1.9258.95 ± 1.6679.39 ± 1.6420.44 (34.66%)0.61 (0.77%)
Citeseerprod68.3859.8169.279.46 (15.82%)0.89 (1.30%)
ind69.75 ± 3.5960.06 ± 5.0069.25 ± 2.259.19 (15.30%)-0.5 (-0.7%)
tran68.04 ± 3.3459.75 ± 2.4869.28 ± 3.129.63 (15.93%)1.24 (1.82%)
Pubmedprod74.8866.8074.717.91 (11.83%)-0.17 (-0.22%)
ind75.26 ± 2.5766.85 ± 2.9674.30 ± 2.617.45 (11.83%)-0.96 (-1.27%)
tran74.78 ± 2.2266.79 ± 2.9074.81 ± 2.398.02 (12.01%)0.03 (0.04%)
A-computerprod82.1467.3882.2914.90 (22.12%)0.15 (0.19%)
ind82.08 ± 1.7967.84 ± 1.7880.92 ± 1.3613.08 (19.28%)-1.16 (-1.41%)
tran82.15 ± 1.5567.27 ± 1.3682.63 ± 1.4015.36 (22.79%)0.48 (0.58%)
A-photoprod91.0879.2592.3813.13 (16.57%)1.30 (1.42%)
ind91.50 ± 0.7979.44 ± 1.7291.18 ± 0.8111.74 (14.78%)-0.32 (-0.35%)
tran90.80 ± 0.7779.20 ± 1.6492.68 ± 0.5613.48 (17.01%)1.70 (1.87%)
Arxivprod70.7355.3065.099.79 (17.70%)-5.64 (-7.97%)
ind70.64 ± 0.6755.40 ± 0.5660.48 ± 0.464.3 (7.76%)-10.94 (-15.49%)
tran70.75± 0.2755.28 ± 0.4971.46 ± 0.3311.16 (20.18%)-4.31 (-6.09%)
Productsprod76.6063.7275.7712.05 (18.91%)-0.83 (-1.09%)
ind76.89 ± 0.5363.70 ± 0.6675.16 ± 0.3411.44 (17.96%)-1.73 (-2.25%)
tran76.53 ±0.5563.73 ± 0.6975.92 ± 0.6112.20 (19.15%)-0.61 (-0.79%)
", + "bbox": [ + 171, + 414, + 826, + 729 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 746, + 825, + 848 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In Table 3, we see that GLNNs can still improve over MLP by large margins for inductive predictions. On 6/7 datasets, the GLNN prod performance are competitive to GNNs, which supports deploying GLNN as a much faster model with no or only slight performance loss. On the Arxiv dataset, the GLNN performance is notably less than GNNs – we hypothesize this is due to Arxiv having a particularly challenging data split which causes distribution shift between test nodes and training ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/d36420376c5e091858f4677ce8bf4e738b5d977ca47d51e96ed2ebb598008531.jpg", + "table_caption": [ + "Table 4: While other inference acceleration methods speed up SAGE, they are considerably slower than GLNNs. Numbers (in ms) are inductive inference time on 10 randomly chosen nodes. " + ], + "table_footnote": [], + "table_body": "
DatasetsSAGEQSAGEPSAGENeighbor SampleGLNN+
Arxiv489.49433.90 (1.13x)465.43 (1.05x)91.03 (5.37x)3.34 (146.55x)
Products2071.301946.49 (1.06x)2001.46 (1.04x)107.71 (19.23x)7.56 (273.98x)
", + "bbox": [ + 173, + 132, + 825, + 189 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "nodes, which is hard for GLNNs to capture without utilizing neighbor information like GNNs. \nHowever, we note that GLNN performance is substantially improved over MLP. ", + "bbox": [ + 173, + 207, + 823, + 234 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.5 HOW DO GLNNS COMPARE TO OTHER INFERENCE ACCELERATION METHODS? ", + "text_level": 1, + "bbox": [ + 173, + 252, + 751, + 265 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Common techniques of inference acceleration include pruning and quantization. These approaches can reduce model parameters and Multiplication-and-ACcumulation (MACs) operations. Still, they don’t eliminate neighbor-fetching latency. Therefore, their speed gain on GNNs is less significant than on NNs. For GNNs, neighbor sampling is also used to reduce the fetching latency. We show an explicit speed comparison between vanilla SAGE, quantized SAGE from FP32 to INT8 (QSAGE), SAGE with $50 \\%$ weights pruned (PSAGE), inference neighbor sampling with fan-out 15, and GLNN in Table 4. With the same setting as Figure 1, we see that GLNN is considerably faster. ", + "bbox": [ + 174, + 276, + 825, + 375 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Two other kinds of methods considered as inference acceleration are GNN-to-GNN KD like TinyGNN (Yan et al., 2020) and Graph Augmented-MLPs (GA-MLPs) like SGC (Wu et al., 2019) or SIGN (Frasca et al., 2020). Inference of GNN-to-GNN KD is likely to be slower than a GNN-Li with the same $i$ as the student, since there will usually be some extra overheads like the Peer-Aware Module (PAM) in TinyGNN. GA-MLPs precompute augmented node features and apply MLPs to them. With precomputation, their inference time will be the same as MLPs for dimension-preserving augmentation (SGC) and the same as enlarged MLPwi for augmentation involves concatenation (SIGN). Thus, for both kinds of approaches, it is sufficient to compare GLNN with GNN-Li and MLPwi, which we have already shown in Figure 3 (left). We see that GNN-Lis are much slower than MLPs. For GA-MLPs, since full pre-computation cannot be done for inductive nodes, GA-MLPs still need to fetch neighbor nodes. This makes them much slower than MLPwi in the inductive setting, and even slower than pruned GNNs and TinyGNN as shown in Zhou et al. (2021). ", + "bbox": [ + 173, + 381, + 825, + 547 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.6 HOW DOES GLNN BENEFIT FROM DISTILLATION? ", + "text_level": 1, + "bbox": [ + 174, + 564, + 555, + 579 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We showed that GNNs are markedly better than MLPs on node classification tasks. But, with KD, GLNNs can often become competitive to GNNs. This indicates that there exist suitable MLP parameters which can well approximate the ideal prediction function from node features to labels. However, these parameters can be difficult to learn through standard stochastic gradient descent. We hypothesize that KD helps to find them through regularization and transfer of inductive bias. ", + "bbox": [ + 174, + 590, + 825, + 660 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "First, we show that KD can help to regularize the student model. From loss curves of a directly trained MLP and the GLNN in Figure 4, we see the gap between training and validation loss is visibly larger for MLPs than GLNNs, and MLPs show obvious overfitting trends. Second, we analyze the inductive bias that makes GNNs powerful on node classification, which suggests that node inferences should be influenced by the graph topology. Whereas MLPs have less inductive bias. Similar difference exists between Transformers (Vaswani et al., 2017) and MLPs. Liu et al. (2021) shows that the inductive bias in Transformers can be mitigated by a simple gate on large MLPs. For node classification, we hypothesize that KD helps to mitigate the inductive bias, so GLNNs can perform competitively. Soft labels from GNN teachers are heavily influenced by the graph topology due to inductive bias. They maintain nonzero probabilities on classes other than the ground truth provided by labels, which can be useful for the student to learn to complement the missing inductive bias in MLPs. To evaluate this hypothesis quantitatively, we define the cut loss $\\mathcal { L } _ { c u t } \\in [ 0 , 1 ]$ in Equation 2 to measure the consistency between model predictions and graph topology (details in Appendix C): ", + "bbox": [ + 174, + 667, + 825, + 848 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/56de8c84db91bba1c82eb94a8d760a0587d8ff7a98d5859a4187e038006e4f10.jpg", + "text": "$$\n\\mathcal { L } _ { c u t } = \\frac { T r ( \\hat { Y } ^ { T } A \\hat { Y } ) } { T r ( \\hat { Y } ^ { T } D \\hat { Y } ) }\n$$", + "text_format": "latex", + "bbox": [ + 424, + 851, + 573, + 890 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Here $\\hat { \\pmb { Y } } \\in [ 0 , 1 ] ^ { N \\times K }$ is the soft classification probability output by the model, $\\pmb { A }$ and $_ { D }$ are the adjacency and degree matrices. When $\\mathcal { L } _ { c u t }$ is close to $^ { 1 }$ , it means the predictions and the graph topology are very consistent. In our experiment, we observe that the average $\\mathcal { L } _ { c u t }$ for SAGE over five CPF datasets is 0.9221, which means high consistency. The same $\\mathcal { L } _ { c u t }$ for MLPs is only 0.7644, but for GLNNs it is 0.8986. This shows that the GLNN predictions indeed benefit from the graph topology knowledge contained in the teacher outputs (the full table of $\\mathcal { L } _ { c u t }$ values in Appendix C). ", + "bbox": [ + 174, + 895, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/e13d36ad0fd34e672832534415bdedeb0f31097113d6e61931f935801ee4e5d2.jpg", + "image_caption": [ + "Figure 4: Loss curves on CPF datasets show GLNN distillation can help to regularize the training. Here the training loss of GLNN is on hard labels, only corresponding to the first term in Equation 1. " + ], + "image_footnote": [], + "bbox": [ + 173, + 104, + 821, + 227 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 281, + 825, + 337 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.7 DO GLNNS HAVE ENOUGH MODEL EXPRESSIVENESS? ", + "text_level": 1, + "bbox": [ + 173, + 356, + 586, + 369 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Intuitively, the addition of neighbor information makes GNNs more powerful than MLPs when classifying nodes. Thus, a natural question regarding KD from GNNs to MLPs is whether MLPs are expressive enough to represent graph data as well as GNNs. Many recent works studied GNN model expressiveness (Xu et al., 2018; Chen et al., 2021). The latter analyzed GNNs and GA-MLPs for node classification and characterized expressiveness as the number of equivalence classes of rooted graphs induced by the model (formal definitions in Appendix D). The conclusion is that GNNs are more powerful than GA-MLPs, but in most real-world cases their expressiveness is indistinguishable. ", + "bbox": [ + 173, + 382, + 825, + 479 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We adopt the analysis framework from Chen et al. (2021) and show in Appendix D that the number of equivalence classes induced by GNNs and MLPs are \u0000|X |+m−2m−1 \u00012L−1 a nd $| \\mathcal { X } |$ respectively. Here $m$ denotes the max node degree, $L$ denotes the number of GNN layers, and $\\mathcal { X }$ denotes the set of all possible node features. The former is apparently larger which concludes that GNNs are more expressive. Empirically, however, the gap makes little difference when $| \\mathcal { X } |$ is large. In real applications, node features can be high dimensional like bag-of-words, or even word embeddings, thus making $| \\mathcal { X } |$ enormous. Like for bag-of-words, $| \\mathcal { X } |$ is in the order of $\\mathcal { O } ( p ^ { D } )$ , where $D$ is the vocabulary size, and $p$ is the max word frequency. The expressiveness of a L-layer GNN is lower bounded by \u0000|X |+m−2\u00012L−1 $\\left( \\stackrel { | \\mathcal { X } | + m - 2 } { m - 1 } \\right) ^ { 2 ^ { L } - 1 } = \\mathcal { O } ( p ^ { D ( m - 1 ) ( 2 ^ { L } - 1 ) } )$ , but empirically, both MLPs and GNNs should have enough expressiveness given $D$ is usually hundreds or bigger (see Table 5). ", + "bbox": [ + 173, + 486, + 825, + 643 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.8 WHEN WILL GLNNS FAIL TO WORK? ", + "text_level": 1, + "bbox": [ + 176, + 661, + 468, + 676 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As discussed in Section 5.7 and Appendix D, the goal of GML node classification is to fit a function $f$ on the rooted graph $\\mathcal { G } ^ { [ i ] }$ and label $\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\mathbf { \\psi } _ { 2 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\qquad \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 }$ . From the information theoretic perspective, fitting $f$ by minimizing the commonly used cross-entropy loss is equivalent to maximizing the mutual information (MI), $I ( \\mathcal G ^ { [ i ] } ; \\boldsymbol y _ { i } )$ as shown in Qin et al. (2020). If we consider $\\mathcal { G } ^ { [ i ] }$ as a joint distribution of two random variables $X ^ { [ i ] }$ and ${ \\mathcal { E } } ^ { [ i ] }$ representing the node features and edges in $\\mathcal { G } ^ { [ i ] }$ respectively, we have ", + "bbox": [ + 174, + 688, + 825, + 763 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/cabc0b009bcf5d43058de368843e6996201b8abfced99c292e08c327b5a83b2a.jpg", + "text": "$$\nI ( \\mathcal { G } ^ { [ i ] } ; \\pmb { y } _ { i } ) = I ( \\pmb { X } ^ { [ i ] } , \\pmb { \\mathcal { E } } ^ { [ i ] } ; \\pmb { y } _ { i } ) = I ( \\pmb { \\mathcal { E } } ^ { [ i ] } ; \\pmb { y } _ { i } ) + I ( \\pmb { X } ^ { [ i ] } ; \\pmb { y } _ { i } | \\pmb { \\mathcal { E } } ^ { [ i ] } )\n$$", + "text_format": "latex", + "bbox": [ + 295, + 771, + 700, + 791 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "$I ( \\mathcal { E } ^ { [ i ] } ; y _ { i } )$ only depends on edges and labels, thus MLPs can only maximize $I ( X ^ { [ i ] } ; y _ { i } | \\mathcal { E } ^ { [ i ] } )$ . In the extreme case, $I ( X ^ { [ i ] } ; y _ { i } | \\mathcal { E } ^ { [ i ] } )$ can be zero when $y ^ { [ i ] }$ is conditionally independent from $X ^ { [ i ] }$ given $\\mathcal { E } ^ { [ i ] }$ . For example, when every node is labeled by its degree or whether it forms a triangle. Then MLPs won’t be able to fit meaningful functions, and neither will GLNNs. However, such cases are typically rare, and unexpected in practical settings our work is mainly concerned with. For real GML tasks, node features and structural roles are often highly correlated (Lerique et al., 2020), hence MLPs can achieve reasonable results even only based on node features, and thus GLNNs can potentially achieve much better results. We study the failure case of GLNNs by creating a low MI scenario in Section 6. ", + "bbox": [ + 173, + 808, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/5b5ba4cf66e93dda6d93901e3acfc6306b2eb1a932c1b44ff6ba4c6b436be96e.jpg", + "image_caption": [ + "Figure 5: Left: Node feature noise. GLNN has comparable performance to GNNs only when nodes are less noisy. Adding more noise decreases GLNN performance faster than GNNs. Middle: Inductive split rate. Altering the inductive:transductive ratio in the production setting doesn’t affect the accuracy much. Right: Teacher GNN architecture. GLNNs can learn from different GNN teachers to improve over MLPs and achieve comparable results. Accuracies are averaged over five CPF datasets. " + ], + "image_footnote": [], + "bbox": [ + 179, + 103, + 820, + 236 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 ABLATION STUDIES ", + "text_level": 1, + "bbox": [ + 176, + 333, + 372, + 348 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this section, we do ablation studies of GLNNs on node feature noise, inductive split rates, and teacher GNN architecture. Reported results are test accuracies averaged over five datasets in CPF. More experiments can be found in Appendix including advanced GNN teachers (Appendix F), GA-MLP student (Appendix G), and non-homogeneous data (Appendix I). ", + "bbox": [ + 174, + 366, + 825, + 422 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Noisy node features. Following Section 5.8, we investigate failure cases of GLNN by adding different levels of Gaussian noise to node features to decrease their mutual information with labels. Specifically, we replace $\\boldsymbol { X }$ with $\\tilde { \\cal X } = ( 1 - \\alpha ) { \\cal X } + \\alpha \\epsilon$ . $\\epsilon$ is an isotropic Gaussian independent from $\\boldsymbol { X }$ , and $\\alpha \\in [ 0 , 1 ]$ denotes the noise level. We show the inductive performance of MLP, GNN, and GLNN under different noise levels in Figure 5 (left). We see that as $\\alpha$ increases, the accuracy of MLPs and GLNNs decrease faster than GNNs, while the performance of GLNNs and GNNs are still comparable for small $\\alpha \\mathbf { s }$ . When $\\alpha$ reaches 1, $\\tilde { X }$ and $\\mathbf { Y }$ will become independent corresponding to the extreme case discussed in Section 5.8. A more detailed discussion is in Appendix J. ", + "bbox": [ + 174, + 429, + 825, + 545 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Inductive split rate. In Section 5.4, we use a 20-80 split of the test data for inductive evaluation. In Figure 5 (middle), we show the results under different split rates (More detailed plots in Appendix H). We see that as the inductive portion increase, GNN and MLP performance stays roughly the same, and the GLNN inductive performance drops slightly. We only consider rates up to 50-50 since having $50 \\%$ or even more inductive nodes is highly atypical in practice. When a large amount of new data are encountered, practitioners can opt to retrain the model on all the data before deployment. ", + "bbox": [ + 174, + 551, + 825, + 636 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Teacher GNN architecture. We used SAGE to represent GNNs so far. In Figure 5 (right), we show results with other various GNN teachers, e.g. GCN, GAT, and APPNP. We see that GLNNs can learn from different teachers and improve over MLPs. The performance is similar for all four teachers, with the GLNN distilled from APPNP very slightly worse than others. In fact, a similar phenomenon has been observed in Yang et al. (2021a) as well, i.e. APPNP benefits the student the least. One possible reason is that the first step of APPNP is to utilize the node’s own feature for prediction (prior to propagating over the graph), which is very similar to what the student MLP is doing, and thus provides less additional information to MLPs than other teachers. ", + "bbox": [ + 174, + 642, + 825, + 753 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSION AND FUTURE WORK ", + "text_level": 1, + "bbox": [ + 176, + 777, + 493, + 794 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we explored whether we can bridge the best of GNNs and MLPs to achieve accurate and fast GML models for deployment. We found that KD from GNNs to MLPs helps to eliminate inference graph dependency, which results in GLNNs that are $1 4 6 \\times - 2 7 3 \\times$ faster than GNNs while enjoying competitive performance. We do a comprehensive study of GLNN properties. The promising results on 7 datasets across different domains show that GLNNs can be a handy choice for deploying latencyconstraint models. In our experiments, the current version of GLNNs on the Arxiv dataset doesn’t show competitive inductive performance. More advanced distillation techniques can potentially improve the GLNN performance, and we leave this investigation as future work. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 102, + 287, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Filippo Maria Bianchi, Daniele Grattarola, and Cesare Alippi. Mincut pooling in graph neural networks. CoRR, abs/1907.00481, 2019. URL http://arxiv.org/abs/1907.00481. ", + "bbox": [ + 171, + 126, + 825, + 155 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. 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", + "bbox": [ + 174, + 849, + 826, + 892 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A DETAILED EXPERIMENT SETTINGS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 501, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 DATASETS ", + "text_level": 1, + "bbox": [ + 176, + 133, + 290, + 148 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Here we provide a detailed description of the datasets we used to support our argument. Out of these datasets, 4 of them are citation graphs. Cora, Citeseer, Pubmed, ogbn-arxiv with the node features being descriptions of the papers, either bag-of-word vector, TF-IDF vector, or word embedding vectors. ", + "bbox": [ + 174, + 160, + 825, + 217 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In Table 5, we provided the basic statistics of these datasets. ", + "bbox": [ + 174, + 223, + 566, + 238 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/3af418bb4154021d5d6d31f9d53bc3a9c0dc59e64e2d2e42bf237c9c2f750a7e.jpg", + "table_caption": [ + "Table 5: Dataset Statistics. " + ], + "table_footnote": [], + "table_body": "
Dataset#Nodes#Edges#Features# Classes
Cora2,4855,0691,4337
Citeseer2,1103,6683,7036
Pubmed19,71744,3245003
A-computer13,381245,77876710
A-photo7,487119,0437458
Arxiv169,3431,166,24312840
Products2,449,02961,859,14010047
", + "bbox": [ + 272, + 270, + 720, + 397 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For all datasets, we follow the setting in the original paper to split the data. Specifically, for the five smaller datasets from the CPF paper, we use the CPF splitting strategy and each random seed corresponds to a different split. For the OGB datasets, we follow the OGB official splits based on time and popularity for Arxiv and Products respectively. ", + "bbox": [ + 176, + 412, + 823, + 468 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.2 MODEL HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 176, + 487, + 418, + 501 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The hyperparameters of GNN models on each dataset are taken from the best hyperparameters provided by the CPF paper and the OGB official examples. For the student MLPs and GLNN s, unless otherwise specified with -wi or $- \\mathbf { L } i$ , we set the number of layers and the hidden dimension of each layer to be the same as the teacher GNN, so their total number of parameters stays the same as the teacher GNN. ", + "bbox": [ + 174, + 512, + 825, + 583 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/022791f2522d89f04b9b698dd7a78165dc34649c3ca90c9c4e684e558de4ee63.jpg", + "table_caption": [ + "Table 6: Hyperparameters for GNNs on five datasets from the CPF paper. " + ], + "table_footnote": [], + "table_body": "
SAGEGCNGATAPPNP
#layers2222
hidden dim128646464
learning rate0.010.010.010.01
weight decay0.00050.0010.010.01
dropout00.80.60.5
fan out5,511-
attention heads1=81
power iterations1=110
", + "bbox": [ + 320, + 612, + 671, + 753 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/1542923fcb3a1d43a58ce9a9e1cd9197eec9f69cc9a6ea0584e4dc4822a3adcb.jpg", + "table_caption": [ + "Table 7: Hyperparameters for GraphSAGE on OGB datasets. " + ], + "table_footnote": [], + "table_body": "
DatasetArxivProducts
# layers33
hidden dim256256
learning rate0.010.003
weight decay00
dropout0.20.5
normalizationbatchbatch
fan out[5,10,15][5,10,15]
", + "bbox": [ + 356, + 794, + 635, + 921 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For GLNN s we do a hyperparameter search of learning rate from [0.01, 0.005, 0.001], weight decay from [0, 0.001, 0.002, 0.005, 0.01], and dropout from [0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6] ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.3 KNOWLEDGE DISTILLATION ", + "text_level": 1, + "bbox": [ + 176, + 148, + 415, + 164 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We use the distillation method proposed in Hinton et al. (2015) as in Equation 1, the hard labels are found to be helpful, so nonzero $\\lambda s$ was suggested. In our case, we did a little tuning for $\\lambda$ but didn’t find nonzero $\\lambda s$ to be very helpful. Therefore, we report all of our results with $\\lambda = 0$ , i.e. only the second term involving soft labels is effective. More careful tuning of $\\lambda$ should further improve the results since the searching space is strictly larger. We implemented a weighted version in our code, and we leave the choice of $\\lambda$ as future work. ", + "bbox": [ + 174, + 174, + 825, + 258 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.4 THE TRANSDUCTIVE SETTING AND THE INDUCTIVE SETTING ", + "text_level": 1, + "bbox": [ + 174, + 275, + 647, + 290 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Given ${ \\mathcal { G } } , X$ , and $Y ^ { L }$ , the goal of node classification can be divided into two different settings, i.e. transductive and inductive. In real applications, the former can correspond to predict missing attributes of a user based on the user profile and other existing users, and the latter can correspond to predict labels of some new nodes that are only seen during inference time. To create the inductive setting on a given dataset, we hold out some nodes along with edges connected to these nodes during data. Using notation defined above, we pick the inductive nodes training and use them for inductive evaluation only. These nodes and edges are picked from the test $\\mathcal { V } _ { i n d } ^ { U } \\subset \\mathcal { V } ^ { U }$ , which partitions $\\mathcal { V } ^ { U }$ into the disjoint inductive subset and observed subset, i.e. $\\mathcal { V } ^ { U } = \\mathcal { V } _ { o b s } ^ { U } \\sqcup \\mathcal { V } _ { i n d } ^ { U }$ V ind . Then we can take , $\\mathcal { V } _ { i n d } ^ { U }$ furth, and raph, so we end up with. We show the input and $\\mathcal { G } = \\mathcal { G } _ { o b s } \\sqcup \\mathcal { G } _ { i n d }$ $\\pmb { X } = \\pmb { X } ^ { L } \\sqcup \\pmb { X } _ { o b s } ^ { U } \\sqcup \\overbrace { \\pmb { X } _ { i n d } ^ { U } }$ $\\pmb { Y } = \\pmb { Y } ^ { L } \\sqcup \\pmb { Y } _ { o b s } ^ { U } \\sqcup \\pmb { Y } _ { i n d } ^ { U }$ output of both settings using the notations below. ", + "bbox": [ + 173, + 301, + 825, + 459 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We visualize the difference between the inductive setting and the transductive setting in Figure 6. ", + "bbox": [ + 171, + 465, + 808, + 481 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/2a239fb19d94400d2ccb7f18def7895440e8208ed97573eb1cf684c4025f00ac.jpg", + "image_caption": [ + "Figure 6: The transductive setting and inductive setting illustrated by a 2-layer GNN. The middle shows the original graph used for training. The left shows the transductive setting, where the test node is in red and within the graph. The right shows the inductive setting, where the test node is an unseen new node. " + ], + "image_footnote": [], + "bbox": [ + 176, + 496, + 826, + 604 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.5 CHOOSING SOFT TARGETS UNDER THE TRANSDUCTIVE SETTING ", + "text_level": 1, + "bbox": [ + 176, + 696, + 663, + 710 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For the transductive setting in Section 5.3, all the nodes in the graph, including the validation and test nodes, are used for the soft target generation. It seems less practical compared to the inductive case, but it is a necessary step to develop our argument. We now discuss the rationale behind this choice. ", + "bbox": [ + 176, + 722, + 825, + 763 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Firstly, the transductive setting is the most common setting for graph data and it was used in most GNN architecture works and GNN acceleration works we mentioned in related work. Therefore, to avoid any confusion and for a fair comparison with numbers from previous literature, we start our experiments with exactly the same input and output as the standard transductive setting. Under this setting, the inputs to GNNs include all the node features and the graph structure, so GLNN is set to be able to access the same input. As GLNN includes a teacher training step and a distillation step, the soft labels of all the nodes are intermediate outputs produced by the teacher training step, and thus used for the second distillation step for the best GLNN performance. This transductive setting can boil down to a sanity check when the student is sufficiently large. Therefore, we separate the setting to be GLNN and $\\mathsf { G L N N + }$ and report the results in Table 1 and Table 2 separately. In Table 1, we are checking how well GLNNs can perform compared to GNNs under the equal-parameter constraint. The results can be interpreted as given a fixed parameter budget, whether there exists one set of parameters (one instantiation of the MLP) that can achieve competitive results as the GNN. Only when this holds, should we further investigate the more interesting and challenging inductive case as in Section 5.4. ", + "bbox": [ + 174, + 770, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Secondly, the task we focus on is node classification, which in many cases is considered as semisupervised learning with very scarce labels. For example, Pubmed only uses 60 labeled nodes (20 per class) out of 20K nodes for training. Rather than design an advanced model that can do few-shot learning, our goal here is to leverage as much data as possible to simplify the model for more efficient inference. We thus utilize the soft pseudo-labels on all the unlabelled nodes for the best GLNN performance. In reality, when there is a large amount of separate unlabeled data, these unlabeled data can be used for GLNN distillation training and a different set of labeled data can be used for evaluation. In our case, we mimic this scenario in the inductive setting in Section 5.4. ", + "bbox": [ + 174, + 166, + 825, + 277 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.6 IMPLEMENTATION AND HARDWARD DETAILS ", + "text_level": 1, + "bbox": [ + 174, + 294, + 532, + 308 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The experiments on both baselines and our approach are implemented using PyTorch, the DGL (Wang et al., 2019) library for GNN algorithms, and Adam (Kingma & Ba, 2015) for optimization. We run all experiments on a machine with 80 Intel(R) Xeon(R) E5-2698 v4 $@$ 2.20GHz CPUs, and a single NVIDIA V100 GPU with 16GB RAM. ", + "bbox": [ + 174, + 320, + 825, + 375 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B SPACE AND TIME COMPLEXITY OF GNNS VS. MLPS ", + "text_level": 1, + "bbox": [ + 176, + 395, + 648, + 411 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Compared to MLP and GNN, GLNN provides a handy tool for users to trade-off between model accuracy and time complexity, which does not directly focus on space complexity. Given the space and time complexity are related, we now provide a more detailed discussion regarding these two complexities in our experiments. ", + "bbox": [ + 174, + 425, + 825, + 481 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In Table 1, the model comparison was between equal-sized MLPs (GLNNs) and GNNs. While fixing parameter budget to control space complexity is a standard approach when comparing models, it is not completely fair for cross-model comparison especially for MLPs vs. GNNs. To do inference with GNNs, the graph needs to be loaded in the memory either entirely or batch by batch, and may use much larger space than the model parameters. Thus, the actual space complexity of GNNs is much higher than equal-sized MLPs. From the time complexity perspective, the major inference latency of GNNs comes from the data dependency as shown in Section 4. Under the same setting as Figure 1, we show in Figure 3 Left that even a 5-layer MLP with 8 times wider hidden layers still runs much faster than a single-layer SAGE. Another example of cross-model comparison is Transformers vs. RNNs. Large Transformers can have more parameters than RNNs because of the attention mechanism, but they are also faster than RNNs in general, which is an important consideration in the context of inference time minimization. ", + "bbox": [ + 174, + 489, + 825, + 655 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In Table 1, we saw that for equal-sized comparison, GLNNs are not as accurate as GNNs on the OGB datasets. Following the discussion above and given the GLNNs used in Table 1 are relatively small (3 layers and 256 hidden dimensions) for millions of nodes in the OGB datasets, we ask whether this gap can be mitigated by increasing the MLP and thus GLNN sizes. The answer is yes as shown in Table 2. ", + "bbox": [ + 174, + 662, + 825, + 732 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C CONSISTENCY MEASURE OF MODEL PREDICTIONS AND GRAPH TOPOLOGY BASED ON MIN-CUT ", + "text_level": 1, + "bbox": [ + 176, + 751, + 740, + 785 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We introduce a metric to measure the consistency between model predictions and graph topology based on the min-cut problem in Section 5.6. The $K$ -way normalized min-cut problem, or simply min-cut, partitions $N$ nodes in $\\nu$ into $K$ disjoint subsets by removing the minimum volume of edges. According to Dhillon et al. (2004), the min-cut problem can be expressed as ", + "bbox": [ + 174, + 800, + 825, + 856 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/baabbe24cd148f6dbd2618e3af9afe60d0917f844b53b0d83dffb22427c8249b.jpg", + "text": "$$\n\\operatorname* { m a x } \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\frac { C _ { k } ^ { T } A C _ { k } } { C _ { k } ^ { T } D C _ { k } }\n$$", + "text_format": "latex", + "bbox": [ + 380, + 858, + 617, + 922 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "with $C$ being the node assignment matrix that partitions $\\nu$ , i.e. $C _ { i , j } = 1$ if node $i$ is assigned to class $j$ . $\\pmb { A }$ being the adjacency matrix and $_ { D }$ being the degree matrix. This quantity we try to maximize here tells us whether the assignment is consistent with the graph topology. The bigger it is, the less edges need to be removed, and the assignment is more consistent with existing connections in the graph. In Bianchi et al. (2019), the authors show that when replacing the hard assignments $C \\in \\{ 0 , 1 \\} ^ { N \\times K }$ with a soft classification probability $\\hat { \\pmb Y } \\in [ 0 , 1 ] ^ { N \\times K }$ , a cut loss $\\mathcal { L } _ { c u t }$ in Equation 2 can become a good approximation of Equation 4 and be used as the measuring metric. ", + "bbox": [ + 173, + 103, + 825, + 204 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/6f56e9f82af720b815340b0bc2a5f76144d5b01fc88d7011474f114b1ef06c00.jpg", + "table_caption": [ + "Table 8: GLNN predictions are much more consistent with the graph topology than MLPs. We show the $\\mathcal { L } _ { c u t }$ values of GNNs, MLPs, and GLNN s on five CPF datasets. GLNN $\\mathcal { L } _ { c u t }$ values become pretty close to the high $\\mathcal { L } _ { c u t }$ values of GNNs, which were closely related to the GNN inductive bias. " + ], + "table_footnote": [], + "table_body": "
DatasetsSAGEMLPGLNN
Cora0.93470.70260.8852
Citeseer0.94850.76930.9339
Pubmed0.96050.94550.9701
A-computer0.90030.69760.8638
A-photo0.86640.70690.8398
Average0.92210.76440.8986
", + "bbox": [ + 338, + 268, + 653, + 390 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "D EXPRESSIVENESS OF GNNS VS. MLPS IN TERMS OF EQUIVALENCE CLASSES OF ROOTED GRAPHS ", + "text_level": 1, + "bbox": [ + 173, + 431, + 769, + 465 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In Chen et al. (2021), the expressiveness of GNNs and GA-MLPs were theoretically quantified in terms of induced equivalence classes of rooted graphs. We adopt their framework and perform a similar analysis for GNNs vs. MLPs. We first define rooted graphs. ", + "bbox": [ + 174, + 486, + 823, + 529 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Definition 1 (Rooted Graph). A rooted graph, denoted as $\\mathcal { G } ^ { [ i ] }$ is a graph with one node i in $\\mathcal { G } ^ { [ i ] }$ designated as the root. GNNs, GA-MLPs, and MLPs can all be considered as functions on rooted graphs. The goal of a node-level task on node $i$ with label $\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\psi _ { i }$ is to fit a function to the input-output pairs $( \\mathcal G ^ { [ i ] } , y _ { i } )$ . ", + "bbox": [ + 174, + 536, + 825, + 594 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We denote the space of rooted graphs as $\\mathcal { E }$ . Following Chen et al. (2021), the expressive power of a model on graph data is evaluated by its ability to approximate functions on $\\mathcal { E }$ . This is further characterized as the number of induced equivalence classes of rooted graphs on $\\mathcal { E }$ , with the equivalence relation defined as the following. Given a family of functions $\\mathcal { F }$ on $\\mathcal { E }$ , we define an equivalence relation $\\simeq _ { \\varepsilon , F }$ among all rooted graphs such that $\\forall \\mathcal { G } ^ { [ i ] } , \\mathcal { G } ^ { \\prime [ j ] } \\in \\mathcal { E } , \\mathcal { G } ^ { [ i ] } \\simeq _ { \\mathcal { E } , \\mathcal { F } } \\mathcal { G } ^ { \\prime [ j ] }$ if and only if $\\forall f \\in \\mathcal { F } , f ( \\mathcal { G } ^ { [ i ] } ) = f ( \\mathcal { G } ^ { \\prime [ j ] } )$ . We now give a proposition to characterize the GNN expressive power (proof in Appendix E). ", + "bbox": [ + 173, + 609, + 825, + 712 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proposition 1. With $\\mathcal { X }$ denotes the set of all possible node features and assuming $| \\mathcal { X } | \\ge 2$ , with $m$ denotes the maximum node degree and assuming $m \\geq 3$ , the total number of equivalence classes of rooted graphs induced by an $L$ -layer GNN is lower bounded by $\\binom { | \\mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 }$ ", + "bbox": [ + 173, + 718, + 825, + 770 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "As shown in Proposition 1, the expressive power of GNNs grows doubly-exponentially in the number of layers $L$ , which means it grows linearly in $L$ after taking $\\log ( \\log ( \\cdot ) )$ . The expressive power GA-MLPs only grows exponentially in $L$ as shown in Chen et al. (2021). Under this framework, the expressive power of MLPs, which corresponds to a 0-layer GA-MLP, is $| \\mathcal { X } |$ . Since the former is much larger than the latter, the conclusion will be GNNs are much more expressive than MLPs. The gap between these two numbers indeed exists, but empirically this gap will only make a difference when $| \\mathcal { X } |$ is small. As in Chen et al. (2021), both the lower bound proof and the constructed examples showing GNNs are more powerful than GA-MLPs assumed $| \\mathcal { X } | = 2$ . In real applications and datasets considered in this work, the node features can be high dimensional vectors like bag-of-words, which makes $| \\mathcal { X } |$ enormous. Thus, this gap doesn’t matter much empirically. ", + "bbox": [ + 173, + 784, + 825, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E PROOF OF THE PROPOSITION 1 ", + "text_level": 1, + "bbox": [ + 174, + 102, + 464, + 118 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "To prove Proposition 1, we first define rooted aggregation trees, which is similar to but different from rooted graphs. ", + "bbox": [ + 173, + 143, + 823, + 171 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Definition 2 (Rooted Aggregation Tree). The depth-K rooted aggregation tree of a rooted graph $\\mathcal { G } ^ { [ i ] }$ is a depth-K rooted tree with $a$ (possibly many-to-one) mapping from every node in the tree to some node in $\\mathcal { G } ^ { [ i ] }$ , where (i) the root of the tree is mapped to node $i$ , and (ii) the children of every node $j$ in the tree are mapped to the neighbors of the node in $\\mathcal { G } ^ { [ i ] }$ to which $j$ is mapped. ", + "bbox": [ + 174, + 184, + 825, + 241 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A rooted aggregation tree can be obtained by unrolling the neighborhood aggregation steps in the GNNs. An illustration of rooted graphs and rooted aggregation trees can be found in Chen et al. (2021) Figure 4. We denote the set of all rooted aggregation trees of depth $\\mathrm { L }$ using $\\mathcal { T } _ { L }$ . Then we use $\\mathcal { T } _ { L , \\mathcal { X } , m }$ to denote a subset of $\\mathcal { T } _ { L }$ , where the node features belong to $\\mathcal { X }$ , and all the nodes have exactly degree $m$ ( $m$ children), and at least two nodes out of these m nodes have different features. In other words, a node can’t have all identical children. With rooted aggregation trees defined, we are ready to prove Proposition 1. The proof is adapted from the proof of Lemma 3 in Chen et al. (2021). ", + "bbox": [ + 173, + 258, + 825, + 358 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. Since the number of equivalence classes on $\\mathcal { E }$ induced by the family of all depth-L GNNs consists of all rooted graphs that share the same rooted aggregation tree of depth- $\\mathrm { . L }$ (Chen et al., 2021), the lower bound problem in Proposition 1 can be reduced to lower bound $| \\mathcal { T } _ { L } |$ , which can be further reduced to lower bound the subset $| \\mathcal { T } _ { L , \\mathcal { X } , m } |$ . We now show $\\begin{array} { r } { | \\mathcal { T } _ { L , \\mathcal { X } , m } | \\geq \\binom { | \\mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 } } \\end{array}$ inductively. ", + "bbox": [ + 173, + 419, + 826, + 500 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "When $L = 1$ , the root of the tree can have $| \\mathcal { X } |$ different choices. For the children nodes, we $m$ $| \\mathcal { X } |$ are allowed. This leads to . $\\binom { | x | + m - 1 } { m }$ cases. Therefore, $\\mathcal { T } _ { L + 1 , \\mathcal { X } , m } = | \\mathcal { X } | \\binom { | \\mathcal { X } | + m - 1 } { m } \\ge \\binom { | \\mathcal { X } | + m - 2 } { m - 1 }$ ", + "bbox": [ + 174, + 503, + 825, + 556 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Assuming the statement holds for $L$ , we show it holds for $L + 1$ by constructing trees in $\\tau _ { L + 1 , x , m }$ from $T , T ^ { \\prime } \\in \\mathcal { T } _ { L , x , m }$ . We do this by assigning node features in $\\mathcal { X }$ to the $m$ children of each leaf node in $T$ and $T ^ { \\prime }$ . First note that when $T$ and $T ^ { \\prime }$ are two non-isomorphic trees, two depth- $\\mathrm { . L } { + } 1$ trees constructed from $T$ and $T ^ { \\prime }$ will be different no matter how the node features are assigned. Now we consider all the trees can be constructed from $T$ by assign node features of children to leaf nodes. ", + "bbox": [ + 174, + 561, + 825, + 632 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We first consider all paths from the root to leaves in $T$ . Each path consists of a sequence of nodes where the node features form a one-to-one mapping to an $\\mathrm { L }$ -tuple $\\tau \\in \\{ ( x _ { 1 } , \\ldots , x _ { L } ) : x _ { i } \\in \\mathcal { X } \\}$ . Leaf nodes are called node under $\\tau$ if the path from the root to it corresponds to $\\tau$ . The children of nodes under different $\\tau \\mathbf { S }$ are always distinguishable, and thus any assignments lead to distinct rooted aggregation trees of depth $L + 1$ . The assignment of children of nodes under the same $\\tau$ , on the other hand, could be overcounted. Therefore, to lower bound $\\tau _ { L + 1 , \\mathcal { X } , m }$ , we only consider a special way of assignments to avoid over counting, which is that children of all nodes under the same $\\tau$ are assigned the same set of features. ", + "bbox": [ + 173, + 638, + 825, + 751 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Since we assumed that at least two nodes of $T$ have different features, there are at least $2 ^ { L }$ different $\\tau \\mathbf { S }$ corresponding to the path from the root to leaves. For a leaf node $j$ under a fixed $\\tau$ , one of its children needs to have the same feature as $j$ ’s parent node. This restriction is due to the definition of rooted aggregation trees. Therefore, we only pick features for the other $m - 1$ nodes, which will be $\\binom { | \\mathcal { X } | + m - 2 } { m - 1 }$ cases for each $j$ . Then through this construction, the total number of depth- $\\mathrm { . L } { + } 1$ trees from T can be lower bounded by \u0000|X |+m−2m−1 \u00012L. $T \\in \\mathcal { T } _ { L , \\mathcal { X } , m }$ , so we derive $\\begin{array} { r } { \\mathcal { T } _ { L + 1 , \\mathcal { X } , m } \\geq \\binom { | \\mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } } \\mathcal { T } _ { L , \\mathcal { X } , m } } \\end{array}$ hav, and $\\begin{array} { r } { \\mathcal { T } _ { L , \\mathcal { X } , m } \\geq \\binom { | \\mathcal { X } | + m - 2 } { m - 1 } ^ { \\sum _ { l = 1 } ^ { L } 2 ^ { l } } = } \\end{array}$ $\\binom { | \\mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 }$ ", + "bbox": [ + 173, + 756, + 825, + 904 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "F ADVANCED GNN ARCHITECTURES AS THE TEACHER ", + "text_level": 1, + "bbox": [ + 173, + 102, + 645, + 118 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In our experiment, SAGE teacher is used throughout to avoid influence by model architecture. Some other GNNs like GCN are also considered in the ablation studies, but they are not the best known architecture for a specific dataset. To show GLNN has stronger performance given a stronger teacher, we consider the best teacher we can access on Products. We take $\\mathrm { M L P { + } C S }$ Huang et al. (2021) from the OGB leaderboard as a new teacher, which has reported accuracy $8 4 . 1 8 \\%$ and ranks 8 on the leadarboard as of Nov 2021. We choose $_ { \\mathrm { M L P + C S } }$ instead of the other top 7 because the others either rely on raw text (additional info to the given node feature), or require a large GPU with ${ > } 1 6 \\mathrm { G B }$ memory, which we don’t have access to. Also, their improvement is not super significant compared to $\\mathrm { M L P { + } C S }$ , i.e. $84 \\%$ to $86 \\%$ . The result with $\\mathrm { M L P { + } C S }$ teacher is shown in Table 9. We see that with the new teacher, performance of ${ \\mathrm { G L N N } } +$ improves to be even better than SAGE $( 7 8 . 6 1 \\% )$ , which shows GLNN can get stronger given a stronger teacher. ", + "bbox": [ + 173, + 132, + 825, + 286 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/b301cddd40a502492ef559bca7c1201681d50bea2e75223fe84d6bd56955145c.jpg", + "table_caption": [ + "Table 9: ${ \\mathrm { G L N N } } +$ with $\\mathrm { M L P { + } C S }$ teacher on Products " + ], + "table_footnote": [], + "table_body": "
MLP+C&SMLP+GLNN+
Acc84.1864.5082.94
", + "bbox": [ + 367, + 316, + 625, + 362 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "G GLNN WITH FEATURE AUGMENTATION FROM ONE-HOP NEIGHBORS ", + "text_level": 1, + "bbox": [ + 168, + 390, + 759, + 405 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In our main experiment, the inductive performance of GLNN on the Arxiv dataset is less desirable than others. We thus consider augment the node features with their one-hop neighbors to include more graph information. This can be seen as a middle ground between pure GLNNs and GNNs. For this new experiment, we follow the setting in Table 3 but with two new approaches. We explain the setting of these two approaches below. ", + "bbox": [ + 174, + 420, + 825, + 491 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "1. 1-hop GA-MLP: firstly, for each node $v$ , we collect features of its 1-hop neighbors $u$ to augment the raw feature of $v$ , i.e. $x _ { v } \\tilde { x } _ { v }$ , like in SGC. Then we train an MLP on the graph with $\\tilde { x } _ { v }$ . Note if $v$ is in the observed graph but $u$ is in the inductive (unobserved during training) part, then $v$ doesn’t collect features from $u$ . ", + "bbox": [ + 214, + 503, + 823, + 559 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "2. 1-hop GA-GLNN: Go through the same feature augmentation step as 1-hop GA-MLP. Then train an MLP with distillation from teacher GNN. ", + "bbox": [ + 207, + 564, + 823, + 592 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "3. In summary, we compare 5 different models in the table below ", + "bbox": [ + 210, + 597, + 643, + 612 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "(a) SAGE: single model on $x _ { v }$ \n(b) MLP: single model on $x _ { v }$ \n(c) GLNN: SAGE teacher and MLP student on $x _ { v }$ \n(d) 1-hop GA-MLP: single model on $\\tilde { x } _ { v }$ \n(e) 1-hop GA-GLNN: SAGE teacher on $x _ { v }$ , MLP student on $\\tilde { x } _ { v }$ ", + "bbox": [ + 238, + 616, + 660, + 695 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We show in the table below, with 1-hop neighbor features, performance of GLNN improves a lot. This is expected as we also observe significant improvement from MLP to 1-hop GA-MLP. However, we indeed see 1-hop GA-GLNN (68.83) can further improve from 1-hop GA-MLP (66.62) and nearly match the teacher (70.64). ", + "bbox": [ + 176, + 708, + 823, + 765 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/4d9c60ce7f02eccca496494587a83e4e1d7b75581009d319d05958e80d040ad5.jpg", + "table_caption": [ + "Table 10: GLNN with feature augmentation from one-hop neighbor on Arxiv " + ], + "table_footnote": [], + "table_body": "
EvalSAGEMLPGLNN1-hop GA-MLP1-hop GA-GLNN
Arxivind70.6455.4060.4866.6268.83
tran70.7555.2871.4666.6769.82
", + "bbox": [ + 225, + 795, + 766, + 853 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "As we have shown in Figure 3, the 1-Layer GNN in our case is roughly 4 times slower than GLNN (29.31ms vs. $7 . 5 6 \\mathrm { m s } ,$ ), which should be a good approximation for the speed comparison between 1-hop GA-MLP/GA-GLNN and GLNN. This result is practically beneficial, as it gives practitioners more flexibility about how much accuracy they want to trade for less inference time. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "This section is a continuation of the ablation study of inductive split rate in Section 6. It generalizes Figure 5 Middle to more split rates (from 10:90 to 90:10), and explicitly show the inductive and transductive performance on each dataset. For better visualization, the training data label rate is also reduced from 20 per class to 5 per class in the following plots. ", + "bbox": [ + 173, + 138, + 825, + 195 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/46024ef2ae15c38b05bcf61172d3fc0a7188e2c13b24b720ed673291135bb45b.jpg", + "image_caption": [ + "Figure 7: Model inductive performance comparison between MLP, GNN(SAGE), and GLNN under different inductive split rate in the production setting. " + ], + "image_footnote": [], + "bbox": [ + 210, + 217, + 785, + 501 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/f66528cb30229f3d636a468c436934c7c9c543d8c492f108574b73bb3899b35d.jpg", + "image_caption": [ + "Figure 8: Model transductive performance comparison between MLP, GNN(SAGE), and GLNN under different inductive split rate in the production setting. " + ], + "image_footnote": [], + "bbox": [ + 209, + 579, + 787, + 863 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "I GLNN UNDER NODE FEATURE HETEROGENEITY AND NON-HOMOPHILY ", + "text_level": 1, + "bbox": [ + 171, + 102, + 790, + 118 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Besides the 7 datasets used in the main experiments, we consider 4 more datasets from Ivanov & Prokhorenkova (2021) and Lim et al. (2021) to further evaluate GLNN. ", + "bbox": [ + 174, + 133, + 823, + 162 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The House_class and VK_class datasets are from Ivanov & Prokhorenkova (2021). The node features of these two graphs are based on tabular data, which have different types, scales, and meanings as the opposite of the bag-of-word node features in Cora and etc. Some basic statistics of the datasets are shown in the following table. ", + "bbox": [ + 174, + 170, + 825, + 227 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/4772ecdb71b8496f5e9eb2fa90327aba44f0066da19aff5489bae5c1691aecd4.jpg", + "table_caption": [ + "Table 11: Statistics of dataset with heterogeneous node features " + ], + "table_footnote": [], + "table_body": "
Dataset# Nodes#Edges# Features#Classes
House_class20,640182,14665
VK_class54,028213,644147
", + "bbox": [ + 281, + 257, + 709, + 318 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We apply the GLNN on House_class and VK_class using the best BGNN model from Ivanov & Prokhorenkova (2021) as the teacher. The comparison is shown in the following table. Ivanov & Prokhorenkova (2021) also includes GAT, GCN, AGNN, and APPNP as baselines, whose performance on these two datasets are quite similar (difference $< 0 . 0 2 5$ ). We compare with these baselines by including the best result among the 4 GNN models and refer it as GNN in the table below, i.e. GNN $=$ max(GAT, GCN, AGNN, APPNP). From the table, we see that GLNN can improve from MLP, outperform GNN and LightGBM, and become competitive to the teacher BGNN. ", + "bbox": [ + 173, + 332, + 825, + 430 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/9032ba8bd479a1b4997c0e4e77d354f21137f2694dd6360ab612d6d48a879ad3.jpg", + "table_caption": [ + "Table 12: GLNN on datasets with heterogeneous node features. Numbers other than GLNN are taken from Ivanov & Prokhorenkova (2021) " + ], + "table_footnote": [], + "table_body": "
DatasetLightGBMGNNsBGNNMLPGLNN
House_class0.550.6250.6820.5340.672
VK_class0.570.5770.6830.5670.641
", + "bbox": [ + 264, + 476, + 723, + 535 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We further pick the non-homophilous Penn94 and Pokec datasets from Lim et al. (2021). Some basic statistics of the datasets are shown in the following table. ", + "bbox": [ + 174, + 550, + 825, + 579 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/11ffbde4f449318037249a7417142a9d32e9ca1f15d808b135d322091926e27f.jpg", + "table_caption": [ + "Table 13: Statistics of non-homophilous datasets " + ], + "table_footnote": [], + "table_body": "
Dataset# Nodes#Edges# Features# Classes
Penn9441,5361,590,65552
Pokec1,632,80330,622,564652
", + "bbox": [ + 290, + 611, + 700, + 670 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Using the GCN teacher, we see that the performance of GLNN is improved over MLP and becomes competitive to the teacher GCN on $\\mathtt { P e n n 9 4 }$ . However, on Pokec, the simple LINK model can achieve very good performance, and it is better than most GNNs reported in Lim et al. (2021). LINK is a purely structural model which does not use node features at all. This shows that the Pokec dataset corresponds to the setting we discussed in Sec 5.8 (limitations of GLNN) – if the node labels can be largely determined by only the graph structure, then GLNN will struggle. We observe that GLNN is not as good as LINK owing to this limitation. However, we still see that for most of the non-homophilous datasets, MLPs already work quite well on them, and we can use GLNN for the other ones like Penn94. ", + "bbox": [ + 173, + 684, + 825, + 810 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/6346a480b97623096c4c0a3398fa52071b30e5375568b9c59b4a6757b36cb2ab.jpg", + "table_caption": [ + "Table 14: GLNN on non-homophilous datasets. Numbers other than GLNN are taken from Lim et al. (2021) " + ], + "table_footnote": [], + "table_body": "
DatasetLINKGCNMLPGLNN
Penn9480.7982.4773.6181.69
Pokec80.5475.4562.3761.32
", + "bbox": [ + 343, + 853, + 648, + 912 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "J MODEL COMPARISON WITH NOISY NODE FEATURES ", + "text_level": 1, + "bbox": [ + 174, + 102, + 642, + 118 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "In Section 6, we conducted an ablation study to compare model performance with noisy node features, and the result is shown in the left plot in Figure 5. There are two subtle points in this plot. (1) The performance of GNN is still relatively high for high noisy features, even when $\\alpha = 1$ and the features are completely random. (2) For completely random features, the performance of GLNN is still higher than MLP. We now discuss and explain them in more detail. ", + "bbox": [ + 174, + 133, + 825, + 203 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "GNN Performance on Random Features. GNN still performs well because nodes with the same labels are likely to be connected and GNN can overfit the training data. We explain the detail through a toy example. Suppose there is a 4-clique containing nodes A, B, C, D in the graph with only a single edge D-E connects this clique to other graph nodes. Suppose A, B, C, D all have iid random Gaussian raw features and the same class label c. Let’s pick A to be the inductive test node and assume E and the triangle formed by B, C, D to be in the training graph. Let’s consider a simple example for 1-layer GCN and break down message passing into feature aggregation and nonlinear transformation. During training, GNN can overfit the data by learning a nonlinear transformation which maps the aggregated features of B, C, D to class c. The aggregated features of B and C will just be the average of the raw features of B, C, D. Although E is also involved in D’s feature aggregation step, the aggregated features of D will also be very close to this average. Then when test on A, the aggregated feature of A will likely be classified to the same class c by the overfitted nonlinear transformation because it is the average of raw node features of A, B, C, D. In this case, GNN can actually correctly classify A because of the overfitting. For GNNs with more layers and graphs with more neighbor nodes, the conclusion may be generalized.This is roughly sort of a “majority vote” process. For a test node A, if many nodes, which A collects features from, have the same class label and appear in the training graph, then A will be classified as this class by an overfitted classifier. ", + "bbox": [ + 173, + 210, + 825, + 446 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "GLNN and MLP Performance on Random Features. The gap between MLP and GLNN is due to imbalanced datasets. The GLNN can learn the imbalance from soft labels, whereas MLPs can only access uniformly picked training nodes. We explain more detail using the A-computer dataset as an example, for which the gap between MLP and GLNN is obvious. The task is 10-class classification. With random node features $( \\alpha { = } 1 )$ ), the inductive accuracy for MLP is 0.0652 and 0.2538 for GLNN. If the data labels are uniform, then both models should give an accuracy around 0.1. However, the labels on the inductive dataset are actually imbalanced. We show the results in Figure 9. The hist on the left is the label distribution of the inductive test set. In particular, class 4 takes about $40 \\%$ . However, given this imbalance, the standard train-test split selects training nodes uniformly among labels. In this case, 20 nodes per class. Therefore, the predictions of MLP on random features are expected to be relatively uniform because the 200 nodes we train it on are uniform. This gives the hist shown in the middle, where the largest class takes about $1 7 . 5 \\%$ . Finally, for GLNN, we train it on all the 200 training nodes with hard labels, plus soft labels of other nodes in the observed graph $\\mathcal { G } _ { o b s }$ (see Section 5.2). Since these extra nodes are selected randomly, whose label distribution is actually similar to the label distribution on the whole data and the distribution on the inductive test set. Therefore, we get the GLNN predictions hist on the right. Although for each node, we can’t assign a prediction correlated to its feature, on average the distribution is very close to the true label distribution on the inductive test set and has a much higher expectation. In fact, if the prediction distribution is exactly the true distribution on the inductive test set, the expectation will be 0.2169. GLNN actually does even a bit better by putting its bet more on the largest class. ", + "bbox": [ + 173, + 454, + 826, + 731 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/403a9c5a1b32a03dcf520caf81b8a2a56b3a306859c74e50c516bf26d61ec307.jpg", + "image_caption": [ + "Figure 9: Inductive (predicted) label distribution on the A-computer dataset. Left: true labels. Middle: predicted labels by MLP. Right: predicted labels by GLNN. 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Namely, GNN inference depends on neighbor nodes", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 318, + 470, + 329 + ], + "spans": [ + { + "bbox": [ + 142, + 318, + 470, + 329 + ], + "score": 1.0, + "content": "multiple hops away from the target, and fetching them burdens latency-constrained", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 329, + 470, + 341 + ], + "spans": [ + { + "bbox": [ + 141, + 329, + 470, + 341 + ], + "score": 1.0, + "content": "applications. 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Motivated by these complementary strengths and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 393, + 470, + 407 + ], + "spans": [ + { + "bbox": [ + 141, + 393, + 470, + 407 + ], + "score": 1.0, + "content": "weaknesses, we bring GNNs and MLPs together via knowledge distillation (KD).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 405, + 469, + 418 + ], + "spans": [ + { + "bbox": [ + 141, + 405, + 469, + 418 + ], + "score": 1.0, + "content": "Our work shows that the performance of MLPs can be improved by large margins", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 415, + 469, + 429 + ], + "spans": [ + { + "bbox": [ + 141, + 415, + 469, + 429 + ], + "score": 1.0, + "content": "with GNN KD. 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One reason for this academic-industrial gap is the challenges in scalability and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 631 + ], + "score": 1.0, + "content": "deployment brought by data dependency in GNNs (Zhang et al., 2020; Jia et al., 2020), which makes", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 629, + 447, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 447, + 642 + ], + "score": 1.0, + "content": "GNNs hard to deploy for latency-constrained applications that require fast inference.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 552, + 506, + 642 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 646, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 646, + 507, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 507, + 660 + ], + "score": 1.0, + "content": "Neighborhood fetching caused by graph dependency is one of the major sources of GNN latency.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 657, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 506, + 670 + ], + "score": 1.0, + "content": "Inference on a target node necessitates fetching topology and features of many neighbor nodes,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "especially on small-world graphs (detailed discussion in Section 4). Common inference acceleration", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 680, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 692 + ], + "score": 1.0, + "content": "techniques like pruning (Zhou et al., 2021) and quantization (Tailor et al., 2021; Zhao et al., 2020)", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 690, + 507, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 507, + 704 + ], + "score": 1.0, + "content": "can speed up GNNs to some extent by reducing Multiplication-and-ACcumulation (MAC) operations.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 646, + 507, + 704 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 507, + 96 + ], + "score": 1.0, + "content": "However, their improvements are limited given the graph dependency is not resolved. Unlike GNNs,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "score": 1.0, + "content": "MLPs have no dependency on graph data and are easier to deploy than GNNs. They also enjoy", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "the auxiliary benefit of sidestepping the cold-start problem that often happens during the online", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "prediction of relational data (Wei et al., 2020), meaning MLPs can infer reasonably even when", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "neighbor information of a new encountered node is not immediately available. On the other hand, this", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 104, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "lack of graph dependency typically hurts for relational learning tasks, limiting MLP performance on", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 507, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 507, + 162 + ], + "score": 1.0, + "content": "GML tasks compared to GNNs. We thus ask: can we bridge the two worlds, enjoying the low-latency,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 483, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 483, + 172 + ], + "score": 1.0, + "content": "dependency-free nature of MLPs and the graph context-awareness of GNNs at the same time?", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 176, + 506, + 374 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "Present work. Our key finding is that it is possible to distill knowledge from GNNs to MLPs without", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 507, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 507, + 200 + ], + "score": 1.0, + "content": "losing significant performance, but reducing the inference time drastically for node classification.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "The knowledge distillation (KD) can be done offline, coupled with model training. In other words,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "we can shift considerable work from the latency-constrained inference step, where time reduction", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "in milliseconds makes a huge difference, to the less time-sensitive training step, where time cost", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "in hours or days is often tolerable. We call our approach Graph-less Neural Network (GLNN).", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 507, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 507, + 255 + ], + "score": 1.0, + "content": "Specifically, GLNN is a modeling paradigm involving KD from a GNN teacher to a student MLP;", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 506, + 265 + ], + "score": 1.0, + "content": "the resulting GLNN is an MLP optimized through KD, so it enjoys the benefits of graph context-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "awareness in training but has no graph dependency in inference. Regarding speed, GLNNs have", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 218, + 287 + ], + "score": 1.0, + "content": "superior efficiency and are", + "type": "text" + }, + { + "bbox": [ + 218, + 275, + 264, + 285 + ], + "score": 0.84, + "content": "1 4 6 \\times - 2 7 3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 275, + 357, + 287 + ], + "score": 1.0, + "content": "faster than GNNs and", + "type": "text" + }, + { + "bbox": [ + 358, + 275, + 394, + 285 + ], + "score": 0.89, + "content": "1 4 \\times - 2 7 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 275, + 506, + 287 + ], + "score": 1.0, + "content": "faster than other inference", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 298 + ], + "score": 1.0, + "content": "acceleration methods. Regarding performance, under a production setting involving both transductive", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 295, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 428, + 311 + ], + "score": 1.0, + "content": "and inductive predictions on 7 datasets, GLNN accuracies improve over MLPs by", + "type": "text" + }, + { + "bbox": [ + 428, + 297, + 459, + 307 + ], + "score": 0.88, + "content": "1 2 . 3 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 295, + 506, + 311 + ], + "score": 1.0, + "content": "on average", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 192, + 321 + ], + "score": 1.0, + "content": "and match GNNs on", + "type": "text" + }, + { + "bbox": [ + 193, + 308, + 208, + 318 + ], + "score": 0.29, + "content": "6 / 7", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "datasets. We comprehensively study when and why GLNNs can achieve", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 319, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 506, + 331 + ], + "score": 1.0, + "content": "competitive results as GNNs. Our analysis suggests the critical factors for such great performance", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "are large MLP sizes and high mutual information between node features and labels. Our observations", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "align with recent results in vision and language, which posit that large enough (or slightly modified)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "score": 1.0, + "content": "MLPs can achieve similar results as CNNs and Transformers (Liu et al., 2021; Tolstikhin et al., 2021;", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 507, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 507, + 375 + ], + "score": 1.0, + "content": "Melas-Kyriazi, 2021; Touvron et al., 2021; Ding et al., 2021). Our core contributions are as follows:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "• We propose GLNN, which eliminates neighbor-fetching latency in GNN inference via KD to MLP.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 388, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 396, + 404 + ], + "score": 1.0, + "content": "• We show GLNNs has competitive performance as GNNs, while enjoying", + "type": "text" + }, + { + "bbox": [ + 396, + 390, + 442, + 401 + ], + "score": 0.73, + "content": "1 4 6 \\times - 2 7 3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 388, + 506, + 404 + ], + "score": 1.0, + "content": "faster inference", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 115, + 401, + 482, + 414 + ], + "spans": [ + { + "bbox": [ + 115, + 401, + 210, + 414 + ], + "score": 1.0, + "content": "than vanilla GNNs and", + "type": "text" + }, + { + "bbox": [ + 210, + 402, + 247, + 412 + ], + "score": 0.88, + "content": "1 4 \\times - 2 7 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 401, + 482, + 414 + ], + "score": 1.0, + "content": "faster inference than other inference acceleration methods.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 109, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 109, + 411, + 506, + 425 + ], + "score": 1.0, + "content": "• We study GLNN properties comprehensively by investigating their performance under different", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 115, + 423, + 485, + 437 + ], + "spans": [ + { + "bbox": [ + 115, + 423, + 485, + 437 + ], + "score": 1.0, + "content": "settings, how they work as regularizers, their inductive bias, expressiveness, and limitations.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 452, + 211, + 465 + ], + "lines": [ + { + "bbox": [ + 104, + 451, + 213, + 468 + ], + "spans": [ + { + "bbox": [ + 104, + 451, + 213, + 468 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "score": 1.0, + "content": "Graph Neural Networks. The early GNNs generalize convolution nets to graphs (Bruna et al., 2014;", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "Defferrard et al., 2017) and later simplified to message-passing neural net (MPNN) by GCN (Kipf", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "& Welling, 2016). Most GNNs after can be put as MPNNs. For example, GAT employs attention", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "(Velickovi ˇ c et al. ´ , 2017), PPNP employs personalized PageRank (Klicpera et al., 2019), GCNII and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "DeeperGCN employ residual connections and dense connections (Chen et al., 2020; Li et al., 2019).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "Inference Acceleration. Inference acceleration have been proposed by hardware improvements", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 506, + 563 + ], + "score": 1.0, + "content": "(Chen et al., 2016; Judd et al., 2016) and algorithmic improvements through pruning (Han et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "2015), quantization (Gupta et al., 2015). For GNNs, pruning (Zhou et al., 2021) and quantizing GNN", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "parameters (Zhao et al., 2020) have been studied. These approaches speed up GNN inference to a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "certain extent but do not eliminate the neighbor-fetching latency. In contrast, our cross-model KD", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "solves this issue. Concurrently, Graph-MLP also tries to bypass GNN neighbor fetching (Hu et al.,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "2021) by training an MLP with a neighbor contrastive loss, but it only considers transductive but not", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 616, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 504, + 628 + ], + "score": 1.0, + "content": "the more practical inductive setting. Some sampling works focus on speed up GNN training (Zou", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 626, + 489, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 489, + 640 + ], + "score": 1.0, + "content": "et al., 2019; Chen et al., 2018), which are complementary to our goal on inference acceleration.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "score": 1.0, + "content": "GNN distillation. Existing GNN KD works try to distill large GNNs to smaller GNNs. LSP (Yang", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "et al., 2021b) and TinyGNN (Yan et al., 2020) do KD while preserving local information. Their", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 666, + 504, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 504, + 677 + ], + "score": 1.0, + "content": "students are GNNs with fewer parameters but not necessarily fewer layers. Thus, both designs still", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "require latency-inducing fetching. GFKD (Deng & Zhang, 2021) does graph-level KD via graph", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "generation. In GFKD, data instances are independent graphs, whereas we focus on dependent nodes", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "within a graph. GraphSAIL (Xu et al., 2020) uses KD to learn students work well on new data while", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "preserving performance on old data. CPF (Yang et al., 2021a) combines KD and label propagation", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 720, + 471, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 471, + 732 + ], + "score": 1.0, + "content": "(LP). The student in CPF is not a GNN, but it is still heavily graph-dependent as it uses LP.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 49.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 507, + 96 + ], + "score": 1.0, + "content": "However, their improvements are limited given the graph dependency is not resolved. Unlike GNNs,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "score": 1.0, + "content": "MLPs have no dependency on graph data and are easier to deploy than GNNs. They also enjoy", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "the auxiliary benefit of sidestepping the cold-start problem that often happens during the online", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "prediction of relational data (Wei et al., 2020), meaning MLPs can infer reasonably even when", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "neighbor information of a new encountered node is not immediately available. On the other hand, this", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 104, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "lack of graph dependency typically hurts for relational learning tasks, limiting MLP performance on", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 507, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 507, + 162 + ], + "score": 1.0, + "content": "GML tasks compared to GNNs. We thus ask: can we bridge the two worlds, enjoying the low-latency,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 483, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 483, + 172 + ], + "score": 1.0, + "content": "dependency-free nature of MLPs and the graph context-awareness of GNNs at the same time?", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 104, + 82, + 507, + 172 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 176, + 506, + 374 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "Present work. Our key finding is that it is possible to distill knowledge from GNNs to MLPs without", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 507, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 507, + 200 + ], + "score": 1.0, + "content": "losing significant performance, but reducing the inference time drastically for node classification.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "The knowledge distillation (KD) can be done offline, coupled with model training. In other words,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "we can shift considerable work from the latency-constrained inference step, where time reduction", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "in milliseconds makes a huge difference, to the less time-sensitive training step, where time cost", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "in hours or days is often tolerable. We call our approach Graph-less Neural Network (GLNN).", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 507, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 507, + 255 + ], + "score": 1.0, + "content": "Specifically, GLNN is a modeling paradigm involving KD from a GNN teacher to a student MLP;", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 506, + 265 + ], + "score": 1.0, + "content": "the resulting GLNN is an MLP optimized through KD, so it enjoys the benefits of graph context-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "awareness in training but has no graph dependency in inference. Regarding speed, GLNNs have", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 218, + 287 + ], + "score": 1.0, + "content": "superior efficiency and are", + "type": "text" + }, + { + "bbox": [ + 218, + 275, + 264, + 285 + ], + "score": 0.84, + "content": "1 4 6 \\times - 2 7 3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 275, + 357, + 287 + ], + "score": 1.0, + "content": "faster than GNNs and", + "type": "text" + }, + { + "bbox": [ + 358, + 275, + 394, + 285 + ], + "score": 0.89, + "content": "1 4 \\times - 2 7 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 275, + 506, + 287 + ], + "score": 1.0, + "content": "faster than other inference", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 298 + ], + "score": 1.0, + "content": "acceleration methods. Regarding performance, under a production setting involving both transductive", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 295, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 428, + 311 + ], + "score": 1.0, + "content": "and inductive predictions on 7 datasets, GLNN accuracies improve over MLPs by", + "type": "text" + }, + { + "bbox": [ + 428, + 297, + 459, + 307 + ], + "score": 0.88, + "content": "1 2 . 3 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 295, + 506, + 311 + ], + "score": 1.0, + "content": "on average", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 192, + 321 + ], + "score": 1.0, + "content": "and match GNNs on", + "type": "text" + }, + { + "bbox": [ + 193, + 308, + 208, + 318 + ], + "score": 0.29, + "content": "6 / 7", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "datasets. We comprehensively study when and why GLNNs can achieve", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 319, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 506, + 331 + ], + "score": 1.0, + "content": "competitive results as GNNs. Our analysis suggests the critical factors for such great performance", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "are large MLP sizes and high mutual information between node features and labels. Our observations", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "align with recent results in vision and language, which posit that large enough (or slightly modified)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "score": 1.0, + "content": "MLPs can achieve similar results as CNNs and Transformers (Liu et al., 2021; Tolstikhin et al., 2021;", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 507, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 507, + 375 + ], + "score": 1.0, + "content": "Melas-Kyriazi, 2021; Touvron et al., 2021; Ding et al., 2021). Our core contributions are as follows:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 176, + 507, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "• We propose GLNN, which eliminates neighbor-fetching latency in GNN inference via KD to MLP.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 388, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 396, + 404 + ], + "score": 1.0, + "content": "• We show GLNNs has competitive performance as GNNs, while enjoying", + "type": "text" + }, + { + "bbox": [ + 396, + 390, + 442, + 401 + ], + "score": 0.73, + "content": "1 4 6 \\times - 2 7 3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 388, + 506, + 404 + ], + "score": 1.0, + "content": "faster inference", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 115, + 401, + 482, + 414 + ], + "spans": [ + { + "bbox": [ + 115, + 401, + 210, + 414 + ], + "score": 1.0, + "content": "than vanilla GNNs and", + "type": "text" + }, + { + "bbox": [ + 210, + 402, + 247, + 412 + ], + "score": 0.88, + "content": "1 4 \\times - 2 7 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 401, + 482, + 414 + ], + "score": 1.0, + "content": "faster inference than other inference acceleration methods.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 109, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 109, + 411, + 506, + 425 + ], + "score": 1.0, + "content": "• We study GLNN properties comprehensively by investigating their performance under different", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 115, + 423, + 485, + 437 + ], + "spans": [ + { + "bbox": [ + 115, + 423, + 485, + 437 + ], + "score": 1.0, + "content": "settings, how they work as regularizers, their inductive bias, expressiveness, and limitations.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 379, + 506, + 437 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 452, + 211, + 465 + ], + "lines": [ + { + "bbox": [ + 104, + 451, + 213, + 468 + ], + "spans": [ + { + "bbox": [ + 104, + 451, + 213, + 468 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "score": 1.0, + "content": "Graph Neural Networks. The early GNNs generalize convolution nets to graphs (Bruna et al., 2014;", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "Defferrard et al., 2017) and later simplified to message-passing neural net (MPNN) by GCN (Kipf", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "& Welling, 2016). Most GNNs after can be put as MPNNs. For example, GAT employs attention", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "(Velickovi ˇ c et al. ´ , 2017), PPNP employs personalized PageRank (Klicpera et al., 2019), GCNII and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "DeeperGCN employ residual connections and dense connections (Chen et al., 2020; Li et al., 2019).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 477, + 506, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "Inference Acceleration. Inference acceleration have been proposed by hardware improvements", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 506, + 563 + ], + "score": 1.0, + "content": "(Chen et al., 2016; Judd et al., 2016) and algorithmic improvements through pruning (Han et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "2015), quantization (Gupta et al., 2015). For GNNs, pruning (Zhou et al., 2021) and quantizing GNN", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "parameters (Zhao et al., 2020) have been studied. These approaches speed up GNN inference to a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "certain extent but do not eliminate the neighbor-fetching latency. In contrast, our cross-model KD", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "solves this issue. Concurrently, Graph-MLP also tries to bypass GNN neighbor fetching (Hu et al.,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "2021) by training an MLP with a neighbor contrastive loss, but it only considers transductive but not", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 616, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 504, + 628 + ], + "score": 1.0, + "content": "the more practical inductive setting. Some sampling works focus on speed up GNN training (Zou", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 626, + 489, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 489, + 640 + ], + "score": 1.0, + "content": "et al., 2019; Chen et al., 2018), which are complementary to our goal on inference acceleration.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 538, + 506, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "score": 1.0, + "content": "GNN distillation. Existing GNN KD works try to distill large GNNs to smaller GNNs. LSP (Yang", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "et al., 2021b) and TinyGNN (Yan et al., 2020) do KD while preserving local information. Their", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 666, + 504, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 504, + 677 + ], + "score": 1.0, + "content": "students are GNNs with fewer parameters but not necessarily fewer layers. Thus, both designs still", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "require latency-inducing fetching. GFKD (Deng & Zhang, 2021) does graph-level KD via graph", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "generation. In GFKD, data instances are independent graphs, whereas we focus on dependent nodes", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "within a graph. GraphSAIL (Xu et al., 2020) uses KD to learn students work well on new data while", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "preserving performance on old data. CPF (Yang et al., 2021a) combines KD and label propagation", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 720, + 471, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 471, + 732 + ], + "score": 1.0, + "content": "(LP). The student in CPF is not a GNN, but it is still heavily graph-dependent as it uses LP.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 49.5, + "bbox_fs": [ + 105, + 642, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 136, + 81, + 472, + 196 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 136, + 81, + 472, + 196 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 136, + 81, + 472, + 196 + ], + "spans": [ + { + "bbox": [ + 136, + 81, + 472, + 196 + ], + "score": 0.968, + "type": "image", + "image_path": "3c59c6aff8764b30e7d812c76a7e7b24c6d87080f7daff930791e286b7e0e843.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 136, + 81, + 472, + 119.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 136, + 119.33333333333334, + 472, + 157.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 136, + 157.66666666666669, + 472, + 196.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 204, + 505, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "Figure 1: The number of fetches and the inference time of GNNs are both magnitudes more than", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "MLPs and grow exponentially as functions of the number of layers. Left: neighbors need to be", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "fetched for two GNN layers. Middle: the total number of fetches for inference. Right: the total", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 237, + 495, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 495, + 249 + ], + "score": 1.0, + "content": "inference time. (Inductive inference for 10 random nodes on OGB Products (Hu et al., 2020))", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 108, + 261, + 208, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 209, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 209, + 276 + ], + "score": 1.0, + "content": "3 PRELIMINARIES", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 285, + 506, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "score": 1.0, + "content": "Notations. For GML tasks, the input is usually a graph and its node features, which we write as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 296, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 107, + 297, + 153, + 309 + ], + "score": 0.92, + "content": "\\mathcal { G } = ( \\nu , \\mathcal { E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 296, + 176, + 308 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 176, + 297, + 185, + 307 + ], + "score": 0.8, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 296, + 283, + 308 + ], + "score": 1.0, + "content": "stands for all nodes, and", + "type": "text" + }, + { + "bbox": [ + 283, + 297, + 290, + 307 + ], + "score": 0.79, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 296, + 388, + 308 + ], + "score": 1.0, + "content": "stands for all edges. Let", + "type": "text" + }, + { + "bbox": [ + 388, + 297, + 398, + 307 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 296, + 506, + 308 + ], + "score": 1.0, + "content": "denote the total number of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 304, + 507, + 321 + ], + "spans": [ + { + "bbox": [ + 104, + 304, + 166, + 321 + ], + "score": 1.0, + "content": "nodes. We use", + "type": "text" + }, + { + "bbox": [ + 166, + 307, + 216, + 317 + ], + "score": 0.92, + "content": "\\pmb { X } \\in \\mathbb { R } ^ { N \\times D }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 304, + 361, + 321 + ], + "score": 1.0, + "content": "to represent node features, with row", + "type": "text" + }, + { + "bbox": [ + 361, + 309, + 373, + 318 + ], + "score": 0.87, + "content": "\\mathbf { \\boldsymbol { x } } _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 304, + 413, + 321 + ], + "score": 1.0, + "content": "being the", + "type": "text" + }, + { + "bbox": [ + 414, + 308, + 423, + 317 + ], + "score": 0.79, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 304, + 507, + 321 + ], + "score": 1.0, + "content": "-dimensional feature", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 140, + 331 + ], + "score": 1.0, + "content": "of node", + "type": "text" + }, + { + "bbox": [ + 140, + 318, + 165, + 329 + ], + "score": 0.9, + "content": "v \\in \\mathcal V", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 318, + 355, + 331 + ], + "score": 1.0, + "content": ". We represent edges with an adjacency matrix", + "type": "text" + }, + { + "bbox": [ + 356, + 319, + 365, + 328 + ], + "score": 0.72, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 318, + 389, + 331 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 389, + 318, + 428, + 331 + ], + "score": 0.93, + "content": "A _ { u , v } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 318, + 460, + 331 + ], + "score": 1.0, + "content": "if edge", + "type": "text" + }, + { + "bbox": [ + 461, + 318, + 502, + 330 + ], + "score": 0.93, + "content": "( u , v ) \\in \\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 318, + 506, + 331 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "and 0 otherwise. For node classification, one of the most important GML applications, the prediction", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 337, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 104, + 337, + 150, + 353 + ], + "score": 1.0, + "content": "targets are", + "type": "text" + }, + { + "bbox": [ + 151, + 339, + 200, + 351 + ], + "score": 0.91, + "content": "\\pmb { Y } \\in \\mathbb { R } ^ { N \\times K }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 337, + 249, + 353 + ], + "score": 1.0, + "content": ", where row", + "type": "text" + }, + { + "bbox": [ + 249, + 342, + 261, + 352 + ], + "score": 0.87, + "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { Y } } _ { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 337, + 278, + 353 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 278, + 341, + 289, + 350 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 337, + 405, + 353 + ], + "score": 1.0, + "content": "-dim one-hot vector for node", + "type": "text" + }, + { + "bbox": [ + 405, + 342, + 411, + 350 + ], + "score": 0.73, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 337, + 462, + 353 + ], + "score": 1.0, + "content": ". For a given", + "type": "text" + }, + { + "bbox": [ + 463, + 341, + 470, + 351 + ], + "score": 0.82, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 337, + 506, + 353 + ], + "score": 1.0, + "content": ", usually", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 103, + 349, + 507, + 364 + ], + "spans": [ + { + "bbox": [ + 103, + 349, + 404, + 364 + ], + "score": 1.0, + "content": "a small portion of nodes will be labeled, which we mark using superscript", + "type": "text" + }, + { + "bbox": [ + 405, + 351, + 411, + 361 + ], + "score": 0.5, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 349, + 430, + 364 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 430, + 351, + 444, + 361 + ], + "score": 0.82, + "content": "\\mathcal { V } ^ { \\tilde { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 349, + 448, + 364 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 449, + 351, + 466, + 361 + ], + "score": 0.81, + "content": "X ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 349, + 486, + 364 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 487, + 350, + 503, + 361 + ], + "score": 0.88, + "content": "\\mathbf { \\nabla } _ { \\mathbf { Y } } \\breve { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 349, + 507, + 364 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 360, + 507, + 375 + ], + "spans": [ + { + "bbox": [ + 104, + 360, + 403, + 375 + ], + "score": 1.0, + "content": "The majority of nodes will be unlabeled, and we mark using the superscript", + "type": "text" + }, + { + "bbox": [ + 403, + 362, + 410, + 372 + ], + "score": 0.41, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 360, + 429, + 375 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 429, + 362, + 443, + 373 + ], + "score": 0.79, + "content": "\\mathcal { V } ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 360, + 447, + 375 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 448, + 362, + 465, + 373 + ], + "score": 0.77, + "content": "X ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 360, + 486, + 375 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 486, + 362, + 503, + 372 + ], + "score": 0.87, + "content": "\\mathbf { \\nabla } _ { \\mathbf { Y } } U", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 360, + 507, + 375 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 378, + 506, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 393 + ], + "score": 1.0, + "content": "Graph Neural Networks. Most GNNs fit under the message-passing framework, where the rep-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 154, + 403 + ], + "score": 1.0, + "content": "resentation", + "type": "text" + }, + { + "bbox": [ + 154, + 390, + 166, + 401 + ], + "score": 0.88, + "content": "h _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 390, + 223, + 403 + ], + "score": 1.0, + "content": "of each node", + "type": "text" + }, + { + "bbox": [ + 223, + 392, + 229, + 400 + ], + "score": 0.78, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "is updated iteratively in each layer by collecting messages from its", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 401, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 191, + 417 + ], + "score": 1.0, + "content": "neighbors denoted as", + "type": "text" + }, + { + "bbox": [ + 191, + 403, + 215, + 416 + ], + "score": 0.92, + "content": "\\mathcal { N } ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 402, + 249, + 417 + ], + "score": 1.0, + "content": ". For the", + "type": "text" + }, + { + "bbox": [ + 250, + 404, + 254, + 414 + ], + "score": 0.56, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 402, + 290, + 417 + ], + "score": 1.0, + "content": "-th layer,", + "type": "text" + }, + { + "bbox": [ + 290, + 401, + 307, + 415 + ], + "score": 0.92, + "content": "\\mathbf { \\Sigma } _ { h _ { v } ^ { ( l ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 402, + 506, + 417 + ], + "score": 1.0, + "content": "is obtained from the previous layer representation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 411, + 492, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 133, + 428 + ], + "score": 0.63, + "content": " { \\boldsymbol { h } } _ { u } ^ { ( l - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 411, + 138, + 434 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 138, + 415, + 183, + 429 + ], + "score": 0.58, + "content": "\\mathbf { \\mathcal { h } } _ { u } ^ { ( 0 ) } = \\mathbf { \\mathcal { x } } _ { u } )", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 411, + 492, + 434 + ], + "score": 1.0, + "content": ") via an aggregation operation AGGR followed by an UPDATE operation as", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 433, + 481, + 452 + ], + "lines": [ + { + "bbox": [ + 128, + 433, + 481, + 452 + ], + "spans": [ + { + "bbox": [ + 128, + 433, + 481, + 452 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\pmb { h } _ { N ( v ) } ^ { ( l ) } = \\mathrm { A G G R } ( \\{ \\pmb { h } _ { u } ^ { ( l - 1 ) } : u \\in \\mathcal { N } ( v ) \\} ) \\qquad \\mathrm { a n d } \\qquad \\pmb { h } _ { v } ^ { ( l ) } = \\mathrm { U P D A T E } ( \\pmb { h } _ { N ( v ) } ^ { ( l ) } , \\pmb { h } _ { v } ^ { ( l - 1 ) } ) } \\end{array}", + "type": "interline_equation", + "image_path": "4bc240481ba9c1fa2eea593b231ecd03030f7cea35038b6d250adb06797d22f4.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 128, + 433, + 481, + 452 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 464, + 193, + 477 + ], + "lines": [ + { + "bbox": [ + 104, + 463, + 194, + 480 + ], + "spans": [ + { + "bbox": [ + 104, + 463, + 194, + 480 + ], + "score": 1.0, + "content": "4 MOTIVATION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 506, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "GNNs have considerable inference latency due to graph dependency. One more GNN layer means", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 348, + 514 + ], + "score": 1.0, + "content": "fetching one more hop of neighbors. To infer a node with a", + "type": "text" + }, + { + "bbox": [ + 348, + 501, + 356, + 510 + ], + "score": 0.76, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "-layer GNN on a graph with average", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 136, + 524 + ], + "score": 1.0, + "content": "degree", + "type": "text" + }, + { + "bbox": [ + 136, + 512, + 145, + 521 + ], + "score": 0.81, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 510, + 182, + 524 + ], + "score": 1.0, + "content": "requires", + "type": "text" + }, + { + "bbox": [ + 183, + 511, + 213, + 523 + ], + "score": 0.93, + "content": "\\mathcal { O } ( R ^ { L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 510, + 250, + 524 + ], + "score": 1.0, + "content": "fetches.", + "type": "text" + }, + { + "bbox": [ + 251, + 512, + 260, + 522 + ], + "score": 0.67, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "can be large for real-world graphs, e.g. 208 for the Twitter", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "(Ching et al., 2015). Also, as layer fetching must be done sequentially, the total latency explodes", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 148, + 546 + ], + "score": 1.0, + "content": "quickly as", + "type": "text" + }, + { + "bbox": [ + 149, + 534, + 156, + 543 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "increases. Figure 1 shows the dependency added by each GNN layer and the exponential", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "explosion of inference time. In contrast, the MLP inference time is much smaller and grows linearly.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 554, + 507, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 507, + 568 + ], + "score": 1.0, + "content": "This marked gap contributes greatly to the practicality of MLPs in industrial applications over GNNs.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 506, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 585 + ], + "score": 1.0, + "content": "The node-fetching latency is exacerbated by two factors: firstly, newer GNN architectures are getting", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "deeper from 64 layers (Chen et al., 2020) to even 1001 layers (Li et al., 2021). Secondly, industrial-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "scale graphs are frequently too large to fit into the memory of a single machine (Jin et al., 2022),", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "necessitating sharding of the graph out of the main memory. For example, Twitter has 288M monthly", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 617, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 627 + ], + "score": 1.0, + "content": "active users (nodes) and an estimated 60B followers (edges) as of 3/2015. Facebook has 1.39B", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "active users with more than 400B edges as of 12/2014 (Ching et al., 2015). Even when stored in a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "sparse-matrix-friendly format (often COO or CSR), these graphs are on the order of TBs and are", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 648, + 507, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 507, + 662 + ], + "score": 1.0, + "content": "constantly growing. Moving away from in-memory storage results in even slower neighbor-fetching.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "MLPs, on the other hand, lack the means to exploit graph topology, which hurts their performance for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "node classification. For example, test accuracy on Products is 78.61 for GraphSAGE compared to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "score": 1.0, + "content": "62.47 for an equal-sized MLP. Nonetheless, recent results in vision and language posit that large (or", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "slightly modified) MLPs can achieve similar results as CNNs and Transformers (Liu et al., 2021).", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "We thus also ask: Can we bridge the best of GNNs and MLPs to get high-accuracy and low-latency", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 395, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 395, + 732 + ], + "score": 1.0, + "content": "models? This motivates us to do cross-model KD from GNNs to MLPs.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 136, + 81, + 472, + 196 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 136, + 81, + 472, + 196 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 136, + 81, + 472, + 196 + ], + "spans": [ + { + "bbox": [ + 136, + 81, + 472, + 196 + ], + "score": 0.968, + "type": "image", + "image_path": "3c59c6aff8764b30e7d812c76a7e7b24c6d87080f7daff930791e286b7e0e843.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 136, + 81, + 472, + 119.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 136, + 119.33333333333334, + 472, + 157.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 136, + 157.66666666666669, + 472, + 196.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 204, + 505, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "Figure 1: The number of fetches and the inference time of GNNs are both magnitudes more than", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "MLPs and grow exponentially as functions of the number of layers. Left: neighbors need to be", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "fetched for two GNN layers. Middle: the total number of fetches for inference. Right: the total", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 237, + 495, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 495, + 249 + ], + "score": 1.0, + "content": "inference time. (Inductive inference for 10 random nodes on OGB Products (Hu et al., 2020))", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 108, + 261, + 208, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 209, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 209, + 276 + ], + "score": 1.0, + "content": "3 PRELIMINARIES", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 285, + 506, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "score": 1.0, + "content": "Notations. For GML tasks, the input is usually a graph and its node features, which we write as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 296, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 107, + 297, + 153, + 309 + ], + "score": 0.92, + "content": "\\mathcal { G } = ( \\nu , \\mathcal { E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 296, + 176, + 308 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 176, + 297, + 185, + 307 + ], + "score": 0.8, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 296, + 283, + 308 + ], + "score": 1.0, + "content": "stands for all nodes, and", + "type": "text" + }, + { + "bbox": [ + 283, + 297, + 290, + 307 + ], + "score": 0.79, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 296, + 388, + 308 + ], + "score": 1.0, + "content": "stands for all edges. Let", + "type": "text" + }, + { + "bbox": [ + 388, + 297, + 398, + 307 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 296, + 506, + 308 + ], + "score": 1.0, + "content": "denote the total number of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 304, + 507, + 321 + ], + "spans": [ + { + "bbox": [ + 104, + 304, + 166, + 321 + ], + "score": 1.0, + "content": "nodes. We use", + "type": "text" + }, + { + "bbox": [ + 166, + 307, + 216, + 317 + ], + "score": 0.92, + "content": "\\pmb { X } \\in \\mathbb { R } ^ { N \\times D }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 304, + 361, + 321 + ], + "score": 1.0, + "content": "to represent node features, with row", + "type": "text" + }, + { + "bbox": [ + 361, + 309, + 373, + 318 + ], + "score": 0.87, + "content": "\\mathbf { \\boldsymbol { x } } _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 304, + 413, + 321 + ], + "score": 1.0, + "content": "being the", + "type": "text" + }, + { + "bbox": [ + 414, + 308, + 423, + 317 + ], + "score": 0.79, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 304, + 507, + 321 + ], + "score": 1.0, + "content": "-dimensional feature", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 140, + 331 + ], + "score": 1.0, + "content": "of node", + "type": "text" + }, + { + "bbox": [ + 140, + 318, + 165, + 329 + ], + "score": 0.9, + "content": "v \\in \\mathcal V", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 318, + 355, + 331 + ], + "score": 1.0, + "content": ". 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To infer a node with a", + "type": "text" + }, + { + "bbox": [ + 348, + 501, + 356, + 510 + ], + "score": 0.76, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "-layer GNN on a graph with average", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 136, + 524 + ], + "score": 1.0, + "content": "degree", + "type": "text" + }, + { + "bbox": [ + 136, + 512, + 145, + 521 + ], + "score": 0.81, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 510, + 182, + 524 + ], + "score": 1.0, + "content": "requires", + "type": "text" + }, + { + "bbox": [ + 183, + 511, + 213, + 523 + ], + "score": 0.93, + "content": "\\mathcal { O } ( R ^ { L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 510, + 250, + 524 + ], + "score": 1.0, + "content": "fetches.", + "type": "text" + }, + { + "bbox": [ + 251, + 512, + 260, + 522 + ], + "score": 0.67, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "can be large for real-world graphs, e.g. 208 for the Twitter", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "(Ching et al., 2015). Also, as layer fetching must be done sequentially, the total latency explodes", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 148, + 546 + ], + "score": 1.0, + "content": "quickly as", + "type": "text" + }, + { + "bbox": [ + 149, + 534, + 156, + 543 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "increases. Figure 1 shows the dependency added by each GNN layer and the exponential", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "explosion of inference time. In contrast, the MLP inference time is much smaller and grows linearly.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 554, + 507, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 507, + 568 + ], + "score": 1.0, + "content": "This marked gap contributes greatly to the practicality of MLPs in industrial applications over GNNs.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 489, + 507, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 506, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 585 + ], + "score": 1.0, + "content": "The node-fetching latency is exacerbated by two factors: firstly, newer GNN architectures are getting", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "deeper from 64 layers (Chen et al., 2020) to even 1001 layers (Li et al., 2021). Secondly, industrial-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "scale graphs are frequently too large to fit into the memory of a single machine (Jin et al., 2022),", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "necessitating sharding of the graph out of the main memory. For example, Twitter has 288M monthly", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 617, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 627 + ], + "score": 1.0, + "content": "active users (nodes) and an estimated 60B followers (edges) as of 3/2015. Facebook has 1.39B", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "active users with more than 400B edges as of 12/2014 (Ching et al., 2015). Even when stored in a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "sparse-matrix-friendly format (often COO or CSR), these graphs are on the order of TBs and are", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 648, + 507, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 507, + 662 + ], + "score": 1.0, + "content": "constantly growing. Moving away from in-memory storage results in even slower neighbor-fetching.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 570, + 507, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "MLPs, on the other hand, lack the means to exploit graph topology, which hurts their performance for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "node classification. For example, test accuracy on Products is 78.61 for GraphSAGE compared to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "score": 1.0, + "content": "62.47 for an equal-sized MLP. Nonetheless, recent results in vision and language posit that large (or", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "slightly modified) MLPs can achieve similar results as CNNs and Transformers (Liu et al., 2021).", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "We thus also ask: Can we bridge the best of GNNs and MLPs to get high-accuracy and low-latency", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 395, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 395, + 732 + ], + "score": 1.0, + "content": "models? This motivates us to do cross-model KD from GNNs to MLPs.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 666, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 78, + 502, + 201 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 78, + 502, + 201 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 78, + 502, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 502, + 201 + ], + "score": 0.969, + "type": "image", + "image_path": "fe13a4358c6462ce72c20699e317599db71d44b7281ec5d0c4ff85854d7ca559.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 78, + 502, + 119.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 119.0, + 502, + 160.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 160.0, + 502, + 201.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 208, + 506, + 252 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 207, + 506, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 220 + ], + "score": 1.0, + "content": "Figure 2: The GLNN framework: In offline training, a trained GNN teacher is applied on the graph for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "soft targets. Then, a student MLP is trained on node features guided by the soft targets. The distilled", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 229, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 104, + 229, + 506, + 243 + ], + "score": 1.0, + "content": "MLP, now GLNN, is deployed for online predictions. Since graph dependency is eliminated for", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "inference, GLNNs infer much faster than GNNs, and hence the name “Graph-less Neural Network.”", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 107, + 264, + 303, + 277 + ], + "lines": [ + { + "bbox": [ + 104, + 262, + 304, + 279 + ], + "spans": [ + { + "bbox": [ + 104, + 262, + 304, + 279 + ], + "score": 1.0, + "content": "5 GRAPH-LESS NEURAL NETWORKS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 290, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "We introduce GLNN and answer exploration questions of its properties: 1) How do GLNNs compare", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "to MLPs and GNNs? 2) Can GLNNs work well under both transductive and inductive settings? 3)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "How do GLNNs compare to other inference acceleration methods? 4) How do GLNNs benefit from", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 323, + 478, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 478, + 335 + ], + "score": 1.0, + "content": "KD? 5) Do GLNNs have sufficient model expressiveness? 6) When will GLNNs fail to work?", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 107, + 349, + 239, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 240, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 240, + 362 + ], + "score": 1.0, + "content": "5.1 THE GLNN FRAMEWORK", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "The idea of GLNN is straightforward, yet as we will see, extremely effective. In short, we train a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 381, + 504, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 504, + 393 + ], + "score": 1.0, + "content": "“boosted” MLP via KD from a teacher GNN. KD was introduced in Hinton et al. (2015), where", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "knowledge was transferred from a cumbersome teacher to a simpler student. In our case, we generate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 404, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 154, + 415 + ], + "score": 1.0, + "content": "soft targets", + "type": "text" + }, + { + "bbox": [ + 154, + 405, + 165, + 414 + ], + "score": 0.86, + "content": "z _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 404, + 226, + 415 + ], + "score": 1.0, + "content": "for each node", + "type": "text" + }, + { + "bbox": [ + 226, + 405, + 233, + 413 + ], + "score": 0.74, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 404, + 505, + 415 + ], + "score": 1.0, + "content": "with a teacher GNN. Then we train a student MLP with both true", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 413, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 132, + 427 + ], + "score": 1.0, + "content": "labels", + "type": "text" + }, + { + "bbox": [ + 132, + 416, + 144, + 426 + ], + "score": 0.86, + "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { Y } } _ { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 413, + 162, + 427 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 162, + 416, + 174, + 425 + ], + "score": 0.87, + "content": "z _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 413, + 324, + 427 + ], + "score": 1.0, + "content": ". The objective is as Equation 1, with", + "type": "text" + }, + { + "bbox": [ + 325, + 415, + 331, + 424 + ], + "score": 0.81, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 413, + 439, + 427 + ], + "score": 1.0, + "content": "being a weight parameter,", + "type": "text" + }, + { + "bbox": [ + 439, + 415, + 464, + 426 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { l a b e l }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 413, + 505, + 427 + ], + "score": 1.0, + "content": "being the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 425, + 463, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 199, + 438 + ], + "score": 1.0, + "content": "cross-entropy between", + "type": "text" + }, + { + "bbox": [ + 199, + 427, + 210, + 437 + ], + "score": 0.86, + "content": "\\mathbf { \\nabla } _ { \\mathbf { y } _ { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 425, + 307, + 438 + ], + "score": 1.0, + "content": "and student predictions", + "type": "text" + }, + { + "bbox": [ + 307, + 426, + 318, + 437 + ], + "score": 0.84, + "content": "\\hat { y } _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 425, + 322, + 438 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 322, + 425, + 358, + 437 + ], + "score": 0.83, + "content": "\\mathcal { L } _ { t e a c h e r }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 425, + 463, + 438 + ], + "score": 1.0, + "content": "being the KL-divergence.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 453, + 429, + 468 + ], + "lines": [ + { + "bbox": [ + 181, + 453, + 429, + 468 + ], + "spans": [ + { + "bbox": [ + 181, + 453, + 429, + 468 + ], + "score": 0.88, + "content": "\\mathcal { L } = \\lambda \\Sigma _ { v \\in \\mathcal { V } ^ { L } } \\mathcal { L } _ { l a b e l } ( \\hat { y } _ { v } , y _ { v } ) + ( 1 - \\lambda ) \\Sigma _ { v \\in \\mathcal { V } } \\mathcal { L } _ { t e a c h e r } ( \\hat { y } _ { v } , z _ { v } )", + "type": "interline_equation", + "image_path": "af326a8c346f30af403e9dc1e4dbd3fe4b6f2d694fbafce9361c3dd60840254a.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 181, + 453, + 429, + 468 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 477, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 476, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 491 + ], + "score": 1.0, + "content": "The model after KD, i.e. GLNN, is essentially a MLP. Therefore, GLNNs have no graph dependency", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "during inference and are as fast as MLPs. On the other hand, through offline KD, GLNN parameters", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "are optimized to predict and generalize as well as GNNs, with the added benefit of faster inference", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 510, + 500, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 500, + 523 + ], + "score": 1.0, + "content": "and easier deployment. In Figure 2, we show the offline KD and online inference steps of GLNNs.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 107, + 536, + 234, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 235, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 235, + 549 + ], + "score": 1.0, + "content": "5.2 EXPERIMENT SETTINGS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 557, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 507, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 507, + 571 + ], + "score": 1.0, + "content": "Datasets. We consider all five datasets used in the CPF paper (Yang et al., 2021a), i.e. Cora,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "Citeseer, Pubmed, A-computer, and A-photo. To fully evaluate our method, we also", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 579, + 451, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 451, + 592 + ], + "score": 1.0, + "content": "include two more larger OGB datasets (Hu et al., 2020), i.e. Arxiv and Products.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 504, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "Model Architectures. For consistent results, we use GraphSAGE (Hamilton et al., 2017) with GCN", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "aggregation as the teacher. We conduct ablation studies of other GNN teachers like GCN (Kipf &", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 617, + 487, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 487, + 631 + ], + "score": 1.0, + "content": "Welling, 2016), GAT (Velickovi ˇ c et al. ´ , 2017) and, APPNP (Klicpera et al., 2019) in Section 6.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 635, + 503, + 669 + ], + "lines": [ + { + "bbox": [ + 105, + 636, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 648 + ], + "score": 1.0, + "content": "Evaluation Protocol. For all experiments in this section, we report the average and standard deviation", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "over ten runs with different random seeds. Model performance is measured as accuracy, and results", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 657, + 408, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 408, + 670 + ], + "score": 1.0, + "content": "are reported on test data with the best model selected using validation data.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 673, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 673, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 259, + 686 + ], + "score": 1.0, + "content": "Transductive vs. Inductive. Given", + "type": "text" + }, + { + "bbox": [ + 259, + 674, + 283, + 685 + ], + "score": 0.29, + "content": "{ \\mathcal { G } } , X", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 673, + 307, + 686 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 307, + 674, + 322, + 685 + ], + "score": 0.88, + "content": "Y ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 673, + 506, + 686 + ], + "score": 1.0, + "content": ", we consider node classification under two", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 685, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 685, + 195, + 696 + ], + "score": 1.0, + "content": "settings: transductive", + "type": "text" + }, + { + "bbox": [ + 196, + 686, + 220, + 696 + ], + "score": 0.29, + "content": "( t r a n )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 685, + 279, + 696 + ], + "score": 1.0, + "content": "and inductive", + "type": "text" + }, + { + "bbox": [ + 279, + 686, + 300, + 696 + ], + "score": 0.29, + "content": "( i n d )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 685, + 505, + 696 + ], + "score": 1.0, + "content": ". 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We first select inductive nodes", + "type": "text" + }, + { + "bbox": [ + 302, + 696, + 349, + 708 + ], + "score": 0.9, + "content": "\\mathcal { V } _ { i n d } ^ { U } \\subset \\mathcal { V } ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 691, + 422, + 714 + ], + "score": 1.0, + "content": ", which partitions", + "type": "text" + }, + { + "bbox": [ + 422, + 695, + 437, + 706 + ], + "score": 0.89, + "content": "\\mathcal { V } ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 691, + 509, + 714 + ], + "score": 1.0, + "content": "into the disjoint", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 101, + 701, + 509, + 727 + ], + "spans": [ + { + "bbox": [ + 101, + 701, + 268, + 727 + ], + "score": 1.0, + "content": "inductive subset and observed subset, i.e.", + "type": "text" + }, + { + "bbox": [ + 269, + 708, + 340, + 721 + ], + "score": 0.91, + "content": "\\mathcal { V } ^ { U } = \\mathcal { V } _ { o b s } ^ { U } \\sqcup \\mathcal { V } _ { i n d } ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 701, + 414, + 727 + ], + "score": 1.0, + "content": ". Then we hold out", + "type": "text" + }, + { + "bbox": [ + 415, + 707, + 452, + 721 + ], + "score": 0.92, + "content": "v \\in \\mathcal { V } _ { i n d } ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 701, + 509, + 727 + ], + "score": 1.0, + "content": "and all edges", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 102, + 714, + 506, + 738 + ], + "spans": [ + { + "bbox": [ + 102, + 714, + 159, + 738 + ], + "score": 1.0, + "content": "connected to", + "type": "text" + }, + { + "bbox": [ + 160, + 720, + 197, + 733 + ], + "score": 0.92, + "content": "v \\in \\mathcal { V } _ { i n d } ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 714, + 339, + 738 + ], + "score": 1.0, + "content": "obs ind, which leads to two disjoint graphs", + "type": "text" + }, + { + "bbox": [ + 339, + 721, + 407, + 732 + ], + "score": 0.91, + "content": "\\mathcal { G } = \\mathcal { G } _ { o b s } \\sqcup \\mathcal { G } _ { i n d }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 714, + 506, + 738 + ], + "score": 1.0, + "content": "ind with no shared nodes or", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 78, + 502, + 201 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 78, + 502, + 201 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 78, + 502, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 502, + 201 + ], + "score": 0.969, + "type": "image", + "image_path": "fe13a4358c6462ce72c20699e317599db71d44b7281ec5d0c4ff85854d7ca559.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 78, + 502, + 119.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 119.0, + 502, + 160.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 160.0, + 502, + 201.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 208, + 506, + 252 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 207, + 506, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 220 + ], + "score": 1.0, + "content": "Figure 2: The GLNN framework: In offline training, a trained GNN teacher is applied on the graph for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "soft targets. Then, a student MLP is trained on node features guided by the soft targets. The distilled", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 229, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 104, + 229, + 506, + 243 + ], + "score": 1.0, + "content": "MLP, now GLNN, is deployed for online predictions. Since graph dependency is eliminated for", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "inference, GLNNs infer much faster than GNNs, and hence the name “Graph-less Neural Network.”", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 107, + 264, + 303, + 277 + ], + "lines": [ + { + "bbox": [ + 104, + 262, + 304, + 279 + ], + "spans": [ + { + "bbox": [ + 104, + 262, + 304, + 279 + ], + "score": 1.0, + "content": "5 GRAPH-LESS NEURAL NETWORKS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 290, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "We introduce GLNN and answer exploration questions of its properties: 1) How do GLNNs compare", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "to MLPs and GNNs? 2) Can GLNNs work well under both transductive and inductive settings? 3)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "How do GLNNs compare to other inference acceleration methods? 4) How do GLNNs benefit from", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 323, + 478, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 478, + 335 + ], + "score": 1.0, + "content": "KD? 5) Do GLNNs have sufficient model expressiveness? 6) When will GLNNs fail to work?", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 290, + 506, + 335 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 349, + 239, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 240, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 240, + 362 + ], + "score": 1.0, + "content": "5.1 THE GLNN FRAMEWORK", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "The idea of GLNN is straightforward, yet as we will see, extremely effective. In short, we train a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 381, + 504, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 504, + 393 + ], + "score": 1.0, + "content": "“boosted” MLP via KD from a teacher GNN. KD was introduced in Hinton et al. (2015), where", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "knowledge was transferred from a cumbersome teacher to a simpler student. In our case, we generate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 404, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 154, + 415 + ], + "score": 1.0, + "content": "soft targets", + "type": "text" + }, + { + "bbox": [ + 154, + 405, + 165, + 414 + ], + "score": 0.86, + "content": "z _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 404, + 226, + 415 + ], + "score": 1.0, + "content": "for each node", + "type": "text" + }, + { + "bbox": [ + 226, + 405, + 233, + 413 + ], + "score": 0.74, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 404, + 505, + 415 + ], + "score": 1.0, + "content": "with a teacher GNN. 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The objective is as Equation 1, with", + "type": "text" + }, + { + "bbox": [ + 325, + 415, + 331, + 424 + ], + "score": 0.81, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 413, + 439, + 427 + ], + "score": 1.0, + "content": "being a weight parameter,", + "type": "text" + }, + { + "bbox": [ + 439, + 415, + 464, + 426 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { l a b e l }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 413, + 505, + 427 + ], + "score": 1.0, + "content": "being the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 425, + 463, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 199, + 438 + ], + "score": 1.0, + "content": "cross-entropy between", + "type": "text" + }, + { + "bbox": [ + 199, + 427, + 210, + 437 + ], + "score": 0.86, + "content": "\\mathbf { \\nabla } _ { \\mathbf { y } _ { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 425, + 307, + 438 + ], + "score": 1.0, + "content": "and student predictions", + "type": "text" + }, + { + "bbox": [ + 307, + 426, + 318, + 437 + ], + "score": 0.84, + "content": "\\hat { y } _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 425, + 322, + 438 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 322, + 425, + 358, + 437 + ], + "score": 0.83, + "content": "\\mathcal { L } _ { t e a c h e r }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 425, + 463, + 438 + ], + "score": 1.0, + "content": "being the KL-divergence.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 370, + 506, + 438 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 453, + 429, + 468 + ], + "lines": [ + { + "bbox": [ + 181, + 453, + 429, + 468 + ], + "spans": [ + { + "bbox": [ + 181, + 453, + 429, + 468 + ], + "score": 0.88, + "content": "\\mathcal { L } = \\lambda \\Sigma _ { v \\in \\mathcal { V } ^ { L } } \\mathcal { L } _ { l a b e l } ( \\hat { y } _ { v } , y _ { v } ) + ( 1 - \\lambda ) \\Sigma _ { v \\in \\mathcal { V } } \\mathcal { L } _ { t e a c h e r } ( \\hat { y } _ { v } , z _ { v } )", + "type": "interline_equation", + "image_path": "af326a8c346f30af403e9dc1e4dbd3fe4b6f2d694fbafce9361c3dd60840254a.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 181, + 453, + 429, + 468 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 477, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 476, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 491 + ], + "score": 1.0, + "content": "The model after KD, i.e. GLNN, is essentially a MLP. Therefore, GLNNs have no graph dependency", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "during inference and are as fast as MLPs. On the other hand, through offline KD, GLNN parameters", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "are optimized to predict and generalize as well as GNNs, with the added benefit of faster inference", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 510, + 500, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 500, + 523 + ], + "score": 1.0, + "content": "and easier deployment. In Figure 2, we show the offline KD and online inference steps of GLNNs.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 476, + 505, + 523 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 536, + 234, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 235, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 235, + 549 + ], + "score": 1.0, + "content": "5.2 EXPERIMENT SETTINGS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 557, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 507, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 507, + 571 + ], + "score": 1.0, + "content": "Datasets. We consider all five datasets used in the CPF paper (Yang et al., 2021a), i.e. Cora,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "Citeseer, Pubmed, A-computer, and A-photo. To fully evaluate our method, we also", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 579, + 451, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 451, + 592 + ], + "score": 1.0, + "content": "include two more larger OGB datasets (Hu et al., 2020), i.e. Arxiv and Products.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 556, + 507, + 592 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 504, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "Model Architectures. For consistent results, we use GraphSAGE (Hamilton et al., 2017) with GCN", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "aggregation as the teacher. We conduct ablation studies of other GNN teachers like GCN (Kipf &", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 617, + 487, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 487, + 631 + ], + "score": 1.0, + "content": "Welling, 2016), GAT (Velickovi ˇ c et al. ´ , 2017) and, APPNP (Klicpera et al., 2019) in Section 6.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 596, + 506, + 631 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 635, + 503, + 669 + ], + "lines": [ + { + "bbox": [ + 105, + 636, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 648 + ], + "score": 1.0, + "content": "Evaluation Protocol. For all experiments in this section, we report the average and standard deviation", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "over ten runs with different random seeds. Model performance is measured as accuracy, and results", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 657, + 408, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 408, + 670 + ], + "score": 1.0, + "content": "are reported on test data with the best model selected using validation data.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 636, + 505, + 670 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 673, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 673, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 259, + 686 + ], + "score": 1.0, + "content": "Transductive vs. Inductive. Given", + "type": "text" + }, + { + "bbox": [ + 259, + 674, + 283, + 685 + ], + "score": 0.29, + "content": "{ \\mathcal { G } } , X", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 673, + 307, + 686 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 307, + 674, + 322, + 685 + ], + "score": 0.88, + "content": "Y ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 673, + 506, + 686 + ], + "score": 1.0, + "content": ", we consider node classification under two", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 685, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 685, + 195, + 696 + ], + "score": 1.0, + "content": "settings: transductive", + "type": "text" + }, + { + "bbox": [ + 196, + 686, + 220, + 696 + ], + "score": 0.29, + "content": "( t r a n )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 685, + 279, + 696 + ], + "score": 1.0, + "content": "and inductive", + "type": "text" + }, + { + "bbox": [ + 279, + 686, + 300, + 696 + ], + "score": 0.29, + "content": "( i n d )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 685, + 505, + 696 + ], + "score": 1.0, + "content": ". 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Then we hold out", + "type": "text" + }, + { + "bbox": [ + 415, + 707, + 452, + 721 + ], + "score": 0.92, + "content": "v \\in \\mathcal { V } _ { i n d } ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 701, + 509, + 727 + ], + "score": 1.0, + "content": "and all edges", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 102, + 714, + 506, + 738 + ], + "spans": [ + { + "bbox": [ + 102, + 714, + 159, + 738 + ], + "score": 1.0, + "content": "connected to", + "type": "text" + }, + { + "bbox": [ + 160, + 720, + 197, + 733 + ], + "score": 0.92, + "content": "v \\in \\mathcal { V } _ { i n d } ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 714, + 339, + 738 + ], + "score": 1.0, + "content": "obs ind, which leads to two disjoint graphs", + "type": "text" + }, + { + "bbox": [ + 339, + 721, + 407, + 732 + ], + "score": 0.91, + "content": "\\mathcal { G } = \\mathcal { G } _ { o b s } \\sqcup \\mathcal { G } _ { i n d }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 714, + 506, + 738 + ], + "score": 1.0, + "content": "ind with no shared nodes or", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 101, + 673, + 509, + 738 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 115, + 505, + 215 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 506, + 114 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "score": 1.0, + "content": "Table 1: GLNNs outperform MLPs by large margins and match GNNs on 5 of 7 datasets under the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 194, + 104 + ], + "score": 1.0, + "content": "transductive setting.", + "type": "text" + }, + { + "bbox": [ + 194, + 91, + 224, + 102 + ], + "score": 0.48, + "content": "\\Delta _ { M L P }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 91, + 262, + 103 + ], + "score": 0.42, + "content": "( \\Delta _ { G N N } )", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 91, + 506, + 104 + ], + "score": 1.0, + "content": "represents difference between the GLNN and a trained MLP", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 491, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 304, + 115 + ], + "score": 1.0, + "content": "(GNN). 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DatasetsSAGEMLPGLNN△MLP△GNN
Cora80.52 ± 1.7759.22 ±1.3180.54 ± 1.3521.32 (36.00%)0.02 (0.02%)
Citeseer70.33 ± 1.9759.61 ± 2.8871.77 ± 2.0112.16 (20.40%)1.44 (2.05%)
Pubmed75.39 ± 2.0967.55 ± 2.3175.42 ± 2.317.87 (11.65%)0.03 (0.04%)
A-computer82.97 ± 2.1667.80 ± 1.0683.03 ± 1.8715.23 (22.46%)0.06 (0.07%)
A-photo90.90 ± 0.8478.77 ± 1.7492.11 ± 1.0813.34 (16.94%)1.21 (1.33%)
Arxiv70.92 ± 0.1756.05 ± 0.4663.46 ± 0.457.41 (13.24%)-7.46 (-10.52%)
Products78.61 ± 0.4962.47 ± 0.1068.86 ± 0.466.39 (10.23%)-9.75 (-12.4%)
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DatasetsSAGEMLP+GLNN+△MLP△GNN
Arxiv70.92 ± 0.1755.31± 0.4772.15 ± 0.2716.85 (30.46%)0.51 (0.71%)
Products78.61 ± 0.4964.50 ± 0.4577.65 ± 0.4813.14 (20.38%)-0.97 (-1.23%)
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A discussion of this choice along with other experiment details are in Appendix A.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 106, + 401, + 347, + 413 + ], + "lines": [ + { + "bbox": [ + 106, + 401, + 347, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 347, + 414 + ], + "score": 1.0, + "content": "5.3 HOW DO GLNNS COMPARE TO MLPS AND GNNS?", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 422, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "We start by comparing GLNNs to MLPs and GNNs with the same number of layers and hidden", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "dimensions. We first consider the standard transductive setting, so our results in Table 1 are directly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 444, + 495, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 495, + 456 + ], + "score": 1.0, + "content": "comparable to results reported in previous literature like Yang et al. (2021a) and Hu et al. (2020).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 571 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "As shown in Table 1, the performance of all GLNNs improve over MLPs by large margins. On smaller", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "score": 1.0, + "content": "datasets (first 5 rows), GLNNs can even outperform the teacher GNNs. In other words, for each task,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "with the same parameter budget, there exists a set of MLP parameters that has GNN-competitive", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 493, + 507, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 507, + 507 + ], + "score": 1.0, + "content": "performance (detailed discussion in Sections 5.6 and 5.7). For the larger OGB datasets (last 2 rows),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "the GLNN performance is improved over MLPs but still worse than the teacher GNNs. However, as", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "we show in Table 2, this gap can be mitigated by increasing MLP size to MLPwi1. In Figure 3 (right),", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 526, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 538 + ], + "score": 1.0, + "content": "we visualize the trade-off between prediction accuracy and model inference time with different model", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "sizes. We show that gradually increasing GLNN size pushes its performance to be close to SAGE. 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DatasetsSAGEMLPGLNN△MLP△GNN
Cora80.52 ± 1.7759.22 ±1.3180.54 ± 1.3521.32 (36.00%)0.02 (0.02%)
Citeseer70.33 ± 1.9759.61 ± 2.8871.77 ± 2.0112.16 (20.40%)1.44 (2.05%)
Pubmed75.39 ± 2.0967.55 ± 2.3175.42 ± 2.317.87 (11.65%)0.03 (0.04%)
A-computer82.97 ± 2.1667.80 ± 1.0683.03 ± 1.8715.23 (22.46%)0.06 (0.07%)
A-photo90.90 ± 0.8478.77 ± 1.7492.11 ± 1.0813.34 (16.94%)1.21 (1.33%)
Arxiv70.92 ± 0.1756.05 ± 0.4663.46 ± 0.457.41 (13.24%)-7.46 (-10.52%)
Products78.61 ± 0.4962.47 ± 0.1068.86 ± 0.466.39 (10.23%)-9.75 (-12.4%)
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DatasetsSAGEMLP+GLNN+△MLP△GNN
Arxiv70.92 ± 0.1755.31± 0.4772.15 ± 0.2716.85 (30.46%)0.51 (0.71%)
Products78.61 ± 0.4964.50 ± 0.4577.65 ± 0.4813.14 (20.38%)-0.97 (-1.23%)
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A discussion of this choice along with other experiment details are in Appendix A.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 365, + 505, + 389 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 401, + 347, + 413 + ], + "lines": [ + { + "bbox": [ + 106, + 401, + 347, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 347, + 414 + ], + "score": 1.0, + "content": "5.3 HOW DO GLNNS COMPARE TO MLPS AND GNNS?", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 422, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "We start by comparing GLNNs to MLPs and GNNs with the same number of layers and hidden", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "dimensions. 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On smaller", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "score": 1.0, + "content": "datasets (first 5 rows), GLNNs can even outperform the teacher GNNs. In other words, for each task,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "with the same parameter budget, there exists a set of MLP parameters that has GNN-competitive", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 493, + 507, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 507, + 507 + ], + "score": 1.0, + "content": "performance (detailed discussion in Sections 5.6 and 5.7). For the larger OGB datasets (last 2 rows),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "the GLNN performance is improved over MLPs but still worse than the teacher GNNs. However, as", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "we show in Table 2, this gap can be mitigated by increasing MLP size to MLPwi1. In Figure 3 (right),", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 526, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 538 + ], + "score": 1.0, + "content": "we visualize the trade-off between prediction accuracy and model inference time with different model", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "sizes. We show that gradually increasing GLNN size pushes its performance to be close to SAGE. On", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 547, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 335, + 562 + ], + "score": 1.0, + "content": "the other hand, when we reduce the number of layers of", + "type": "text" + }, + { + "bbox": [ + 336, + 548, + 366, + 559 + ], + "score": 0.73, + "content": "\\mathrm { S A G E } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 547, + 506, + 562 + ], + "score": 1.0, + "content": ", the accuracy quickly drops to be", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 560, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 506, + 572 + ], + "score": 1.0, + "content": "worse than GLNNs. A detailed discussion of the rationale for increasing MLP sizes is in Appendix B.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 461, + 507, + 572 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 586, + 489, + 596 + ], + "lines": [ + { + "bbox": [ + 106, + 585, + 491, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 491, + 597 + ], + "score": 1.0, + "content": "5.4 CAN GLNNS WORK WELL UNDER BOTH TRANSDUCTIVE AND INDUCTIVE SETTINGS?", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 505, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 605, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 619 + ], + "score": 1.0, + "content": "Although transductive is the commonly studied setting for node classification, it does not encompass", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 617, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 629 + ], + "score": 1.0, + "content": "prediction on unseen nodes. Therefore, it may not be the best way to evaluate a deployed model,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 627, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 641 + ], + "score": 1.0, + "content": "which must often generate predictions for new data points as well as reliably maintain performance on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 638, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 652 + ], + "score": 1.0, + "content": "old ones. 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In production, a model might be re-trained periodically, e.g. weekly. The", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 102, + 683, + 510, + 708 + ], + "spans": [ + { + "bbox": [ + 102, + 683, + 178, + 708 + ], + "score": 1.0, + "content": "obs hold-out nodes in", + "type": "text" + }, + { + "bbox": [ + 178, + 690, + 198, + 702 + ], + "score": 0.89, + "content": "V _ { i n d } ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 683, + 444, + 708 + ], + "score": 1.0, + "content": "represent new nodes entered the graph between two trainings.", + "type": "text" + }, + { + "bbox": [ + 445, + 689, + 464, + 702 + ], + "score": 0.93, + "content": "V _ { i n d } ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 683, + 510, + 708 + ], + "score": 1.0, + "content": "is usually", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 101, + 585, + 510, + 610 + ], + "spans": [ + { + "bbox": [ + 101, + 585, + 183, + 610 + ], + "score": 1.0, + "content": "small compared to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 183, + 590, + 202, + 604 + ], + "score": 0.71, + "content": "V _ { o b s } ^ { U }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 203, + 585, + 333, + 610 + ], + "score": 1.0, + "content": "– e.g. Graham (2012) estimates", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 333, + 592, + 356, + 602 + ], + "score": 0.86, + "content": "5- 7 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 357, + 585, + 510, + 610 + ], + "score": 1.0, + "content": "for the fastest-growing tech startups.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 103, + 601, + 508, + 619 + ], + "spans": [ + { + "bbox": [ + 103, + 602, + 420, + 618 + ], + "score": 1.0, + "content": "In our case, to mitigate randomness and better evaluate generalizability, we use", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 422, + 601, + 508, + 619 + ], + "score": 1.0, + "content": "V Uind containing 20%", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 102, + 609, + 509, + 639 + ], + "spans": [ + { + "bbox": [ + 102, + 609, + 252, + 639 + ], + "score": 1.0, + "content": "of the test data. We also evaluate on standard transductive prediction on", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 252, + 615, + 271, + 628 + ], + "score": 0.91, + "content": "V _ { o b s } ^ { U }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 271, + 609, + 354, + 639 + ], + "score": 1.0, + "content": "containing the otherrved unlabeled nodes", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 354, + 616, + 374, + 627 + ], + "score": 0.87, + "content": "80 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 375, + 609, + 509, + 639 + ], + "score": 1.0, + "content": "of the test data, representing thee inference is commonly redone", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "on existing nodes in real-world cases. We report both results and a interpolated production (prod)", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "results in Table 3. The prod results paint a clearer picture of model generalization as well as accuracy", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "in production. See Section 6 for an ablation study of different inductive split rates other than 20-80.", + "type": "text", + "cross_page": true + } + ], + "index": 16 + } + ], + "index": 37, + "bbox_fs": [ + 102, + 665, + 510, + 708 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 140, + 84, + 470, + 229 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 140, + 84, + 470, + 229 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 140, + 84, + 470, + 229 + ], + "spans": [ + { + "bbox": [ + 140, + 84, + 470, + 229 + ], + "score": 0.967, + "type": "image", + "image_path": "56094e9645d24171cda661a02619bd1120d8d877a8d7ebb8ed27856fa4567de6.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 140, + 84, + 470, + 132.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 140, + 132.33333333333334, + 470, + 180.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 140, + 180.66666666666669, + 470, + 229.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 241, + 505, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "Figure 3: Enlarged MLPs (GLNNs) can match GNN accuracy, but infer dramatically faster. Plots are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "under the same setting as Figure 1. Left: inference time of MLPs vs. GNN (SAGE) for different", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 263, + 457, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 457, + 276 + ], + "score": 1.0, + "content": "model sizes. Right: model accuracy vs. inference time. Note: time axes are log-scaled.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 105, + 328, + 506, + 578 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 293, + 507, + 327 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 293, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 305 + ], + "score": 1.0, + "content": "Table 3: GLNNs match GNN performance on a production setting with both inductive and trans-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 101, + 303, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 101, + 303, + 216, + 333 + ], + "score": 1.0, + "content": "ductive predictions. 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DatasetsEvalSAGEMLP/MLP+GLNN/GLNN+△MLP△GNN
Coraprod79.2958.9878.2819.30 (32.72%)-1.01 (-1.28%)
ind81.33 ± 2.1959.09 ± 2.9673.82 ± 1.9314.73 (24.93%)-7.51 (-9.23%)
tran78.78 ± 1.9258.95 ± 1.6679.39 ± 1.6420.44 (34.66%)0.61 (0.77%)
Citeseerprod68.3859.8169.279.46 (15.82%)0.89 (1.30%)
ind69.75 ± 3.5960.06 ± 5.0069.25 ± 2.259.19 (15.30%)-0.5 (-0.7%)
tran68.04 ± 3.3459.75 ± 2.4869.28 ± 3.129.63 (15.93%)1.24 (1.82%)
Pubmedprod74.8866.8074.717.91 (11.83%)-0.17 (-0.22%)
ind75.26 ± 2.5766.85 ± 2.9674.30 ± 2.617.45 (11.83%)-0.96 (-1.27%)
tran74.78 ± 2.2266.79 ± 2.9074.81 ± 2.398.02 (12.01%)0.03 (0.04%)
A-computerprod82.1467.3882.2914.90 (22.12%)0.15 (0.19%)
ind82.08 ± 1.7967.84 ± 1.7880.92 ± 1.3613.08 (19.28%)-1.16 (-1.41%)
tran82.15 ± 1.5567.27 ± 1.3682.63 ± 1.4015.36 (22.79%)0.48 (0.58%)
A-photoprod91.0879.2592.3813.13 (16.57%)1.30 (1.42%)
ind91.50 ± 0.7979.44 ± 1.7291.18 ± 0.8111.74 (14.78%)-0.32 (-0.35%)
tran90.80 ± 0.7779.20 ± 1.6492.68 ± 0.5613.48 (17.01%)1.70 (1.87%)
Arxivprod70.7355.3065.099.79 (17.70%)-5.64 (-7.97%)
ind70.64 ± 0.6755.40 ± 0.5660.48 ± 0.464.3 (7.76%)-10.94 (-15.49%)
tran70.75± 0.2755.28 ± 0.4971.46 ± 0.3311.16 (20.18%)-4.31 (-6.09%)
Productsprod76.6063.7275.7712.05 (18.91%)-0.83 (-1.09%)
ind76.89 ± 0.5363.70 ± 0.6675.16 ± 0.3411.44 (17.96%)-1.73 (-2.25%)
tran76.53 ±0.5563.73 ± 0.6975.92 ± 0.6112.20 (19.15%)-0.61 (-0.79%)
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We also evaluate on standard transductive prediction on", + "type": "text" + }, + { + "bbox": [ + 252, + 615, + 271, + 628 + ], + "score": 0.91, + "content": "V _ { o b s } ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 609, + 354, + 639 + ], + "score": 1.0, + "content": "containing the otherrved unlabeled nodes", + "type": "text" + }, + { + "bbox": [ + 354, + 616, + 374, + 627 + ], + "score": 0.87, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 609, + 509, + 639 + ], + "score": 1.0, + "content": "of the test data, representing thee inference is commonly redone", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "on existing nodes in real-world cases. We report both results and a interpolated production (prod)", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "results in Table 3. The prod results paint a clearer picture of model generalization as well as accuracy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "in production. See Section 6 for an ablation study of different inductive split rates other than 20-80.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 507, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 507, + 690 + ], + "score": 1.0, + "content": "In Table 3, we see that GLNNs can still improve over MLP by large margins for inductive predictions.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 686, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 505, + 702 + ], + "score": 1.0, + "content": "On 6/7 datasets, the GLNN prod performance are competitive to GNNs, which supports deploying", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "GLNN as a much faster model with no or only slight performance loss. On the Arxiv dataset, the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "GLNN performance is notably less than GNNs – we hypothesize this is due to Arxiv having a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "particularly challenging data split which causes distribution shift between test nodes and training", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 140, + 84, + 470, + 229 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 140, + 84, + 470, + 229 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 140, + 84, + 470, + 229 + ], + "spans": [ + { + "bbox": [ + 140, + 84, + 470, + 229 + ], + "score": 0.967, + "type": "image", + "image_path": "56094e9645d24171cda661a02619bd1120d8d877a8d7ebb8ed27856fa4567de6.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 140, + 84, + 470, + 132.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 140, + 132.33333333333334, + 470, + 180.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 140, + 180.66666666666669, + 470, + 229.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 241, + 505, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "Figure 3: Enlarged MLPs (GLNNs) can match GNN accuracy, but infer dramatically faster. Plots are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "under the same setting as Figure 1. Left: inference time of MLPs vs. GNN (SAGE) for different", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 263, + 457, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 457, + 276 + ], + "score": 1.0, + "content": "model sizes. Right: model accuracy vs. inference time. Note: time axes are log-scaled.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 105, + 328, + 506, + 578 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 293, + 507, + 327 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 293, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 305 + ], + "score": 1.0, + "content": "Table 3: GLNNs match GNN performance on a production setting with both inductive and trans-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 101, + 303, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 101, + 303, + 216, + 333 + ], + "score": 1.0, + "content": "ductive predictions. We uProducts. ind results on", + "type": "text" + }, + { + "bbox": [ + 216, + 314, + 236, + 327 + ], + "score": 0.91, + "content": "V _ { i n d } ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 303, + 299, + 333 + ], + "score": 1.0, + "content": "LP for the 5 C, tran results on", + "type": "text" + }, + { + "bbox": [ + 299, + 314, + 317, + 327 + ], + "score": 0.91, + "content": "V _ { o b s } ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 303, + 506, + 333 + ], + "score": 1.0, + "content": "atasets, MLPw4 for Arxiv, and MLPw8 for, and the interpolated prod results are reported.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "table_body", + "bbox": [ + 105, + 328, + 506, + 578 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 328, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 506, + 578 + ], + "score": 0.986, + "html": "
DatasetsEvalSAGEMLP/MLP+GLNN/GLNN+△MLP△GNN
Coraprod79.2958.9878.2819.30 (32.72%)-1.01 (-1.28%)
ind81.33 ± 2.1959.09 ± 2.9673.82 ± 1.9314.73 (24.93%)-7.51 (-9.23%)
tran78.78 ± 1.9258.95 ± 1.6679.39 ± 1.6420.44 (34.66%)0.61 (0.77%)
Citeseerprod68.3859.8169.279.46 (15.82%)0.89 (1.30%)
ind69.75 ± 3.5960.06 ± 5.0069.25 ± 2.259.19 (15.30%)-0.5 (-0.7%)
tran68.04 ± 3.3459.75 ± 2.4869.28 ± 3.129.63 (15.93%)1.24 (1.82%)
Pubmedprod74.8866.8074.717.91 (11.83%)-0.17 (-0.22%)
ind75.26 ± 2.5766.85 ± 2.9674.30 ± 2.617.45 (11.83%)-0.96 (-1.27%)
tran74.78 ± 2.2266.79 ± 2.9074.81 ± 2.398.02 (12.01%)0.03 (0.04%)
A-computerprod82.1467.3882.2914.90 (22.12%)0.15 (0.19%)
ind82.08 ± 1.7967.84 ± 1.7880.92 ± 1.3613.08 (19.28%)-1.16 (-1.41%)
tran82.15 ± 1.5567.27 ± 1.3682.63 ± 1.4015.36 (22.79%)0.48 (0.58%)
A-photoprod91.0879.2592.3813.13 (16.57%)1.30 (1.42%)
ind91.50 ± 0.7979.44 ± 1.7291.18 ± 0.8111.74 (14.78%)-0.32 (-0.35%)
tran90.80 ± 0.7779.20 ± 1.6492.68 ± 0.5613.48 (17.01%)1.70 (1.87%)
Arxivprod70.7355.3065.099.79 (17.70%)-5.64 (-7.97%)
ind70.64 ± 0.6755.40 ± 0.5660.48 ± 0.464.3 (7.76%)-10.94 (-15.49%)
tran70.75± 0.2755.28 ± 0.4971.46 ± 0.3311.16 (20.18%)-4.31 (-6.09%)
Productsprod76.6063.7275.7712.05 (18.91%)-0.83 (-1.09%)
ind76.89 ± 0.5363.70 ± 0.6675.16 ± 0.3411.44 (17.96%)-1.73 (-2.25%)
tran76.53 ±0.5563.73 ± 0.6975.92 ± 0.6112.20 (19.15%)-0.61 (-0.79%)
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DatasetsSAGEQSAGEPSAGENeighbor SampleGLNN+
Arxiv489.49433.90 (1.13x)465.43 (1.05x)91.03 (5.37x)3.34 (146.55x)
Products2071.301946.49 (1.06x)2001.46 (1.04x)107.71 (19.23x)7.56 (273.98x)
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These approaches", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 104, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "can reduce model parameters and Multiplication-and-ACcumulation (MACs) operations. Still, they", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "don’t eliminate neighbor-fetching latency. Therefore, their speed gain on GNNs is less significant", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "than on NNs. For GNNs, neighbor sampling is also used to reduce the fetching latency. We show an", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "score": 1.0, + "content": "explicit speed comparison between vanilla SAGE, quantized SAGE from FP32 to INT8 (QSAGE),", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 275, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 154, + 287 + ], + "score": 1.0, + "content": "SAGE with", + "type": "text" + }, + { + "bbox": [ + 154, + 275, + 174, + 285 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 275, + 506, + 287 + ], + "score": 1.0, + "content": "weights pruned (PSAGE), inference neighbor sampling with fan-out 15, and GLNN", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 285, + 453, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 453, + 299 + ], + "score": 1.0, + "content": "in Table 4. With the same setting as Figure 1, we see that GLNN is considerably faster.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 505, + 434 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "Two other kinds of methods considered as inference acceleration are GNN-to-GNN KD like TinyGNN", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "(Yan et al., 2020) and Graph Augmented-MLPs (GA-MLPs) like SGC (Wu et al., 2019) or SIGN", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "(Frasca et al., 2020). Inference of GNN-to-GNN KD is likely to be slower than a GNN-Li with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 146, + 348 + ], + "score": 1.0, + "content": "the same", + "type": "text" + }, + { + "bbox": [ + 146, + 336, + 151, + 345 + ], + "score": 0.68, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "as the student, since there will usually be some extra overheads like the Peer-Aware", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "Module (PAM) in TinyGNN. GA-MLPs precompute augmented node features and apply MLPs to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "them. With precomputation, their inference time will be the same as MLPs for dimension-preserving", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 369, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 380 + ], + "score": 1.0, + "content": "augmentation (SGC) and the same as enlarged MLPwi for augmentation involves concatenation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "(SIGN). Thus, for both kinds of approaches, it is sufficient to compare GLNN with GNN-Li and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "MLPwi, which we have already shown in Figure 3 (left). We see that GNN-Lis are much slower than", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "MLPs. For GA-MLPs, since full pre-computation cannot be done for inductive nodes, GA-MLPs still", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 411, + 507, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 507, + 425 + ], + "score": 1.0, + "content": "need to fetch neighbor nodes. This makes them much slower than MLPwi in the inductive setting,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 423, + 436, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 436, + 434 + ], + "score": 1.0, + "content": "and even slower than pruned GNNs and TinyGNN as shown in Zhou et al. (2021).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 107, + 447, + 340, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 341, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 341, + 460 + ], + "score": 1.0, + "content": "5.6 HOW DOES GLNN BENEFIT FROM DISTILLATION?", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "We showed that GNNs are markedly better than MLPs on node classification tasks. But, with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "KD, GLNNs can often become competitive to GNNs. This indicates that there exist suitable MLP", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 490, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 502 + ], + "score": 1.0, + "content": "parameters which can well approximate the ideal prediction function from node features to labels.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "However, these parameters can be difficult to learn through standard stochastic gradient descent. We", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 512, + 477, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 477, + 525 + ], + "score": 1.0, + "content": "hypothesize that KD helps to find them through regularization and transfer of inductive bias.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 529, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "First, we show that KD can help to regularize the student model. From loss curves of a directly trained", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "MLP and the GLNN in Figure 4, we see the gap between training and validation loss is visibly larger", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "for MLPs than GLNNs, and MLPs show obvious overfitting trends. Second, we analyze the inductive", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "bias that makes GNNs powerful on node classification, which suggests that node inferences should be", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "influenced by the graph topology. Whereas MLPs have less inductive bias. Similar difference exists", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "between Transformers (Vaswani et al., 2017) and MLPs. Liu et al. (2021) shows that the inductive", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 594, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 608 + ], + "score": 1.0, + "content": "bias in Transformers can be mitigated by a simple gate on large MLPs. For node classification, we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "hypothesize that KD helps to mitigate the inductive bias, so GLNNs can perform competitively. Soft", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 615, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 630 + ], + "score": 1.0, + "content": "labels from GNN teachers are heavily influenced by the graph topology due to inductive bias. They", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "maintain nonzero probabilities on classes other than the ground truth provided by labels, which can", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "be useful for the student to learn to complement the missing inductive bias in MLPs. To evaluate", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 325, + 662 + ], + "score": 1.0, + "content": "this hypothesis quantitatively, we define the cut loss", + "type": "text" + }, + { + "bbox": [ + 326, + 649, + 380, + 661 + ], + "score": 0.95, + "content": "\\mathcal { L } _ { c u t } \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "in Equation 2 to measure the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 660, + 444, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 444, + 673 + ], + "score": 1.0, + "content": "consistency between model predictions and graph topology (details in Appendix C):", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 39 + }, + { + "type": "interline_equation", + "bbox": [ + 260, + 674, + 351, + 705 + ], + "lines": [ + { + "bbox": [ + 260, + 674, + 351, + 705 + ], + "spans": [ + { + "bbox": [ + 260, + 674, + 351, + 705 + ], + "score": 0.95, + "content": "\\mathcal { L } _ { c u t } = \\frac { T r ( \\hat { Y } ^ { T } A \\hat { Y } ) } { T r ( \\hat { Y } ^ { T } D \\hat { Y } ) }", + "type": "interline_equation", + "image_path": "56de8c84db91bba1c82eb94a8d760a0587d8ff7a98d5859a4187e038006e4f10.jpg" + } + ] + } + ], + "index": 46.5, + "virtual_lines": [ + { + "bbox": [ + 260, + 674, + 351, + 689.5 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 260, + 689.5, + 351, + 705.0 + ], + "spans": [], + "index": 47 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 706, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 129, + 723 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 129, + 708, + 195, + 722 + ], + "score": 0.92, + "content": "\\hat { \\pmb { Y } } \\in [ 0 , 1 ] ^ { N \\times K }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 706, + 432, + 723 + ], + "score": 1.0, + "content": "is the soft classification probability output by the model,", + "type": "text" + }, + { + "bbox": [ + 433, + 710, + 443, + 720 + ], + "score": 0.74, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 706, + 462, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 462, + 710, + 473, + 720 + ], + "score": 0.79, + "content": "_ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 706, + 506, + 723 + ], + "score": 1.0, + "content": "are the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 267, + 734 + ], + "score": 1.0, + "content": "adjacency and degree matrices. When", + "type": "text" + }, + { + "bbox": [ + 267, + 721, + 287, + 732 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { c u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 720, + 334, + 734 + ], + "score": 1.0, + "content": "is close to", + "type": "text" + }, + { + "bbox": [ + 334, + 721, + 340, + 730 + ], + "score": 0.49, + "content": "^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 720, + 505, + 734 + ], + "score": 1.0, + "content": ", it means the predictions and the graph", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 105, + 505, + 150 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 504, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 506, + 93 + ], + "score": 1.0, + "content": "Table 4: While other inference acceleration methods speed up SAGE, they are considerably slower", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 467, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 467, + 104 + ], + "score": 1.0, + "content": "than GLNNs. Numbers (in ms) are inductive inference time on 10 randomly chosen nodes.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 105, + 505, + 150 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 105, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 150 + ], + "score": 0.978, + "html": "
DatasetsSAGEQSAGEPSAGENeighbor SampleGLNN+
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These approaches", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 104, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "can reduce model parameters and Multiplication-and-ACcumulation (MACs) operations. Still, they", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "don’t eliminate neighbor-fetching latency. Therefore, their speed gain on GNNs is less significant", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "than on NNs. For GNNs, neighbor sampling is also used to reduce the fetching latency. We show an", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "score": 1.0, + "content": "explicit speed comparison between vanilla SAGE, quantized SAGE from FP32 to INT8 (QSAGE),", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 275, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 154, + 287 + ], + "score": 1.0, + "content": "SAGE with", + "type": "text" + }, + { + "bbox": [ + 154, + 275, + 174, + 285 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 275, + 506, + 287 + ], + "score": 1.0, + "content": "weights pruned (PSAGE), inference neighbor sampling with fan-out 15, and GLNN", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 285, + 453, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 453, + 299 + ], + "score": 1.0, + "content": "in Table 4. With the same setting as Figure 1, we see that GLNN is considerably faster.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 104, + 219, + 506, + 299 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 505, + 434 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "Two other kinds of methods considered as inference acceleration are GNN-to-GNN KD like TinyGNN", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "(Yan et al., 2020) and Graph Augmented-MLPs (GA-MLPs) like SGC (Wu et al., 2019) or SIGN", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "(Frasca et al., 2020). Inference of GNN-to-GNN KD is likely to be slower than a GNN-Li with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 146, + 348 + ], + "score": 1.0, + "content": "the same", + "type": "text" + }, + { + "bbox": [ + 146, + 336, + 151, + 345 + ], + "score": 0.68, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "as the student, since there will usually be some extra overheads like the Peer-Aware", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "Module (PAM) in TinyGNN. GA-MLPs precompute augmented node features and apply MLPs to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "them. With precomputation, their inference time will be the same as MLPs for dimension-preserving", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 369, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 380 + ], + "score": 1.0, + "content": "augmentation (SGC) and the same as enlarged MLPwi for augmentation involves concatenation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "(SIGN). Thus, for both kinds of approaches, it is sufficient to compare GLNN with GNN-Li and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "MLPwi, which we have already shown in Figure 3 (left). We see that GNN-Lis are much slower than", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "MLPs. For GA-MLPs, since full pre-computation cannot be done for inductive nodes, GA-MLPs still", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 411, + 507, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 507, + 425 + ], + "score": 1.0, + "content": "need to fetch neighbor nodes. This makes them much slower than MLPwi in the inductive setting,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 423, + 436, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 436, + 434 + ], + "score": 1.0, + "content": "and even slower than pruned GNNs and TinyGNN as shown in Zhou et al. (2021).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 302, + 507, + 434 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 447, + 340, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 341, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 341, + 460 + ], + "score": 1.0, + "content": "5.6 HOW DOES GLNN BENEFIT FROM DISTILLATION?", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "We showed that GNNs are markedly better than MLPs on node classification tasks. But, with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "KD, GLNNs can often become competitive to GNNs. This indicates that there exist suitable MLP", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 490, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 502 + ], + "score": 1.0, + "content": "parameters which can well approximate the ideal prediction function from node features to labels.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "However, these parameters can be difficult to learn through standard stochastic gradient descent. We", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 512, + 477, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 477, + 525 + ], + "score": 1.0, + "content": "hypothesize that KD helps to find them through regularization and transfer of inductive bias.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 468, + 506, + 525 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 529, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "First, we show that KD can help to regularize the student model. From loss curves of a directly trained", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "MLP and the GLNN in Figure 4, we see the gap between training and validation loss is visibly larger", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "for MLPs than GLNNs, and MLPs show obvious overfitting trends. Second, we analyze the inductive", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "bias that makes GNNs powerful on node classification, which suggests that node inferences should be", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "influenced by the graph topology. Whereas MLPs have less inductive bias. Similar difference exists", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "between Transformers (Vaswani et al., 2017) and MLPs. Liu et al. (2021) shows that the inductive", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 594, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 608 + ], + "score": 1.0, + "content": "bias in Transformers can be mitigated by a simple gate on large MLPs. For node classification, we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "hypothesize that KD helps to mitigate the inductive bias, so GLNNs can perform competitively. Soft", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 615, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 630 + ], + "score": 1.0, + "content": "labels from GNN teachers are heavily influenced by the graph topology due to inductive bias. They", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "maintain nonzero probabilities on classes other than the ground truth provided by labels, which can", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "be useful for the student to learn to complement the missing inductive bias in MLPs. To evaluate", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 325, + 662 + ], + "score": 1.0, + "content": "this hypothesis quantitatively, we define the cut loss", + "type": "text" + }, + { + "bbox": [ + 326, + 649, + 380, + 661 + ], + "score": 0.95, + "content": "\\mathcal { L } _ { c u t } \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "in Equation 2 to measure the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 660, + 444, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 444, + 673 + ], + "score": 1.0, + "content": "consistency between model predictions and graph topology (details in Appendix C):", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 528, + 506, + 673 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 260, + 674, + 351, + 705 + ], + "lines": [ + { + "bbox": [ + 260, + 674, + 351, + 705 + ], + "spans": [ + { + "bbox": [ + 260, + 674, + 351, + 705 + ], + "score": 0.95, + "content": "\\mathcal { L } _ { c u t } = \\frac { T r ( \\hat { Y } ^ { T } A \\hat { Y } ) } { T r ( \\hat { Y } ^ { T } D \\hat { Y } ) }", + "type": "interline_equation", + "image_path": "56de8c84db91bba1c82eb94a8d760a0587d8ff7a98d5859a4187e038006e4f10.jpg" + } + ] + } + ], + "index": 46.5, + "virtual_lines": [ + { + "bbox": [ + 260, + 674, + 351, + 689.5 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 260, + 689.5, + 351, + 705.0 + ], + "spans": [], + "index": 47 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 706, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 129, + 723 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 129, + 708, + 195, + 722 + ], + "score": 0.92, + "content": "\\hat { \\pmb { Y } } \\in [ 0 , 1 ] ^ { N \\times K }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 706, + 432, + 723 + ], + "score": 1.0, + "content": "is the soft classification probability output by the model,", + "type": "text" + }, + { + "bbox": [ + 433, + 710, + 443, + 720 + ], + "score": 0.74, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 706, + 462, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 462, + 710, + 473, + 720 + ], + "score": 0.79, + "content": "_ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 706, + 506, + 723 + ], + "score": 1.0, + "content": "are the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 267, + 734 + ], + "score": 1.0, + "content": "adjacency and degree matrices. When", + "type": "text" + }, + { + "bbox": [ + 267, + 721, + 287, + 732 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { c u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 720, + 334, + 734 + ], + "score": 1.0, + "content": "is close to", + "type": "text" + }, + { + "bbox": [ + 334, + 721, + 340, + 730 + ], + "score": 0.49, + "content": "^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 720, + 505, + 734 + ], + "score": 1.0, + "content": ", it means the predictions and the graph", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 406, + 235 + ], + "score": 1.0, + "content": "topology are very consistent. In our experiment, we observe that the average", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 407, + 224, + 426, + 234 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { c u t }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 426, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "for SAGE over five", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 367, + 247 + ], + "score": 1.0, + "content": "CPF datasets is 0.9221, which means high consistency. The same", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 367, + 234, + 387, + 245 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { c u t }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 387, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "for MLPs is only 0.7644, but", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 244, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 505, + 259 + ], + "score": 1.0, + "content": "for GLNNs it is 0.8986. This shows that the GLNN predictions indeed benefit from the graph topology", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 256, + 464, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 349, + 268 + ], + "score": 1.0, + "content": "knowledge contained in the teacher outputs (the full table of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 349, + 256, + 368, + 267 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { c u t }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 369, + 257, + 464, + 268 + ], + "score": 1.0, + "content": "values in Appendix C).", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 706, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 83, + 503, + 180 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 83, + 503, + 180 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 83, + 503, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 503, + 180 + ], + "score": 0.966, + "type": "image", + "image_path": "e13d36ad0fd34e672832534415bdedeb0f31097113d6e61931f935801ee4e5d2.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 83, + 503, + 115.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 115.33333333333334, + 503, + 147.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 147.66666666666669, + 503, + 180.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 186, + 504, + 210 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 185, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 201 + ], + "score": 1.0, + "content": "Figure 4: Loss curves on CPF datasets show GLNN distillation can help to regularize the training.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "Here the training loss of GLNN is on hard labels, only corresponding to the first term in Equation 1.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 223, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 105, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 406, + 235 + ], + "score": 1.0, + "content": "topology are very consistent. 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The same", + "type": "text" + }, + { + "bbox": [ + 367, + 234, + 387, + 245 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { c u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "for MLPs is only 0.7644, but", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 244, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 505, + 259 + ], + "score": 1.0, + "content": "for GLNNs it is 0.8986. This shows that the GLNN predictions indeed benefit from the graph topology", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 256, + 464, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 349, + 268 + ], + "score": 1.0, + "content": "knowledge contained in the teacher outputs (the full table of", + "type": "text" + }, + { + "bbox": [ + 349, + 256, + 368, + 267 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { c u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 257, + 464, + 268 + ], + "score": 1.0, + "content": "values in Appendix C).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 106, + 282, + 359, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 282, + 360, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 360, + 295 + ], + "score": 1.0, + "content": "5.7 DO GLNNS HAVE ENOUGH MODEL EXPRESSIVENESS?", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 303, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "Intuitively, the addition of neighbor information makes GNNs more powerful than MLPs when", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "classifying nodes. Thus, a natural question regarding KD from GNNs to MLPs is whether MLPs are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "expressive enough to represent graph data as well as GNNs. Many recent works studied GNN model", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 336, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 348 + ], + "score": 1.0, + "content": "expressiveness (Xu et al., 2018; Chen et al., 2021). The latter analyzed GNNs and GA-MLPs for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "node classification and characterized expressiveness as the number of equivalence classes of rooted", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "graphs induced by the model (formal definitions in Appendix D). The conclusion is that GNNs are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 370, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 506, + 381 + ], + "score": 1.0, + "content": "more powerful than GA-MLPs, but in most real-world cases their expressiveness is indistinguishable.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "We adopt the analysis framework from Chen et al. (2021) and show in Appendix D that the number", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 101, + 395, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 101, + 395, + 401, + 420 + ], + "score": 1.0, + "content": "of equivalence classes induced by GNNs and MLPs are \u0000|X |+m−2m−1 \u00012L−1 a", + "type": "text" + }, + { + "bbox": [ + 396, + 403, + 414, + 415 + ], + "score": 1.0, + "content": "nd", + "type": "text" + }, + { + "bbox": [ + 414, + 403, + 429, + 415 + ], + "score": 0.87, + "content": "| \\mathcal { X } |", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "respectively. Here", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 415, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 107, + 417, + 117, + 425 + ], + "score": 0.67, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 415, + 246, + 427 + ], + "score": 1.0, + "content": "denotes the max node degree,", + "type": "text" + }, + { + "bbox": [ + 247, + 415, + 255, + 425 + ], + "score": 0.68, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 415, + 428, + 427 + ], + "score": 1.0, + "content": "denotes the number of GNN layers, and", + "type": "text" + }, + { + "bbox": [ + 429, + 416, + 439, + 425 + ], + "score": 0.8, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 415, + 505, + 427 + ], + "score": 1.0, + "content": "denotes the set", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "of all possible node features. The former is apparently larger which concludes that GNNs are", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 437, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 423, + 449 + ], + "score": 1.0, + "content": "more expressive. Empirically, however, the gap makes little difference when", + "type": "text" + }, + { + "bbox": [ + 423, + 437, + 438, + 449 + ], + "score": 0.9, + "content": "| \\mathcal { X } |", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 437, + 506, + 449 + ], + "score": 1.0, + "content": "is large. In real", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 447, + 507, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 507, + 461 + ], + "score": 1.0, + "content": "applications, node features can be high dimensional like bag-of-words, or even word embeddings,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 458, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 160, + 471 + ], + "score": 1.0, + "content": "thus making", + "type": "text" + }, + { + "bbox": [ + 160, + 459, + 175, + 471 + ], + "score": 0.9, + "content": "| \\mathcal { X } |", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 459, + 320, + 471 + ], + "score": 1.0, + "content": "enormous. Like for bag-of-words,", + "type": "text" + }, + { + "bbox": [ + 320, + 459, + 335, + 471 + ], + "score": 0.91, + "content": "| \\mathcal { X } |", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 459, + 408, + 471 + ], + "score": 1.0, + "content": "is in the order of", + "type": "text" + }, + { + "bbox": [ + 408, + 458, + 437, + 471 + ], + "score": 0.93, + "content": "\\mathcal { O } ( p ^ { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 459, + 469, + 471 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 469, + 459, + 479, + 469 + ], + "score": 0.82, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 459, + 506, + 471 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 192, + 483 + ], + "score": 1.0, + "content": "vocabulary size, and", + "type": "text" + }, + { + "bbox": [ + 192, + 472, + 198, + 481 + ], + "score": 0.78, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "is the max word frequency. The expressiveness of a L-layer GNN is lower", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 102, + 479, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 102, + 479, + 227, + 500 + ], + "score": 1.0, + "content": "bounded by \u0000|X |+m−2\u00012L−1", + "type": "text" + }, + { + "bbox": [ + 158, + 481, + 312, + 500 + ], + "score": 0.93, + "content": "\\left( \\stackrel { | \\mathcal { X } | + m - 2 } { m - 1 } \\right) ^ { 2 ^ { L } - 1 } = \\mathcal { O } ( p ^ { D ( m - 1 ) ( 2 ^ { L } - 1 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 483, + 506, + 500 + ], + "score": 1.0, + "content": ", but empirically, both MLPs and GNNs should", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 498, + 430, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 245, + 512 + ], + "score": 1.0, + "content": "have enough expressiveness given", + "type": "text" + }, + { + "bbox": [ + 245, + 500, + 254, + 509 + ], + "score": 0.79, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 498, + 430, + 512 + ], + "score": 1.0, + "content": "is usually hundreds or bigger (see Table 5).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 108, + 524, + 287, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 524, + 288, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 288, + 537 + ], + "score": 1.0, + "content": "5.8 WHEN WILL GLNNS FAIL TO WORK?", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "score": 1.0, + "content": "As discussed in Section 5.7 and Appendix D, the goal of GML node classification is to fit a function", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 556, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 107, + 559, + 114, + 570 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 556, + 199, + 572 + ], + "score": 1.0, + "content": "on the rooted graph", + "type": "text" + }, + { + "bbox": [ + 199, + 557, + 214, + 569 + ], + "score": 0.9, + "content": "\\mathcal { G } ^ { [ i ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 556, + 256, + 572 + ], + "score": 1.0, + "content": "and label", + "type": "text" + }, + { + "bbox": [ + 256, + 559, + 267, + 570 + ], + "score": 0.83, + "content": "\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\mathbf { \\psi } _ { 2 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\qquad \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 556, + 483, + 572 + ], + "score": 1.0, + "content": ". From the information theoretic perspective, fitting", + "type": "text" + }, + { + "bbox": [ + 484, + 558, + 491, + 570 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 556, + 506, + 572 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "score": 1.0, + "content": "minimizing the commonly used cross-entropy loss is equivalent to maximizing the mutual information", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 131, + 594 + ], + "score": 1.0, + "content": "(MI),", + "type": "text" + }, + { + "bbox": [ + 131, + 579, + 173, + 593 + ], + "score": 0.92, + "content": "I ( \\mathcal G ^ { [ i ] } ; \\boldsymbol y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 579, + 367, + 594 + ], + "score": 1.0, + "content": "as shown in Qin et al. 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If we consider", + "type": "text" + }, + { + "bbox": [ + 367, + 579, + 383, + 591 + ], + "score": 0.9, + "content": "\\mathcal { G } ^ { [ i ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 579, + 506, + 594 + ], + "score": 1.0, + "content": "as a joint distribution of two", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 591, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 177, + 607 + ], + "score": 1.0, + "content": "random variables", + "type": "text" + }, + { + "bbox": [ + 177, + 592, + 196, + 603 + ], + "score": 0.9, + "content": "X ^ { [ i ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 591, + 213, + 607 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 214, + 592, + 229, + 604 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } ^ { [ i ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 591, + 402, + 607 + ], + "score": 1.0, + "content": "representing the node features and edges in", + "type": "text" + }, + { + "bbox": [ + 402, + 592, + 417, + 604 + ], + "score": 0.9, + "content": "\\mathcal { G } ^ { [ i ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 591, + 505, + 607 + ], + "score": 1.0, + "content": "respectively, we have", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 611, + 429, + 627 + ], + "lines": [ + { + "bbox": [ + 181, + 611, + 429, + 627 + ], + "spans": [ + { + "bbox": [ + 181, + 611, + 429, + 627 + ], + "score": 0.91, + "content": "I ( \\mathcal { G } ^ { [ i ] } ; \\pmb { y } _ { i } ) = I ( \\pmb { X } ^ { [ i ] } , \\pmb { \\mathcal { E } } ^ { [ i ] } ; \\pmb { y } _ { i } ) = I ( \\pmb { \\mathcal { E } } ^ { [ i ] } ; \\pmb { y } _ { i } ) + I ( \\pmb { X } ^ { [ i ] } ; \\pmb { y } _ { i } | \\pmb { \\mathcal { E } } ^ { [ i ] } )", + "type": "interline_equation", + "image_path": "cabc0b009bcf5d43058de368843e6996201b8abfced99c292e08c327b5a83b2a.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 181, + 611, + 429, + 627 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 640, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 148, + 653 + ], + "score": 0.92, + "content": "I ( \\mathcal { E } ^ { [ i ] } ; y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 639, + 413, + 654 + ], + "score": 1.0, + "content": "only depends on edges and labels, thus MLPs can only maximize", + "type": "text" + }, + { + "bbox": [ + 413, + 640, + 475, + 653 + ], + "score": 0.92, + "content": "I ( X ^ { [ i ] } ; y _ { i } | \\mathcal { E } ^ { [ i ] } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 639, + 506, + 654 + ], + "score": 1.0, + "content": ". 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For real GML tasks,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "node features and structural roles are often highly correlated (Lerique et al., 2020), hence MLPs can", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "achieve reasonable results even only based on node features, and thus GLNNs can potentially achieve", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "much better results. 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Thus, a natural question regarding KD from GNNs to MLPs is whether MLPs are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "expressive enough to represent graph data as well as GNNs. Many recent works studied GNN model", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 336, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 348 + ], + "score": 1.0, + "content": "expressiveness (Xu et al., 2018; Chen et al., 2021). The latter analyzed GNNs and GA-MLPs for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "node classification and characterized expressiveness as the number of equivalence classes of rooted", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "graphs induced by the model (formal definitions in Appendix D). The conclusion is that GNNs are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 370, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 506, + 381 + ], + "score": 1.0, + "content": "more powerful than GA-MLPs, but in most real-world cases their expressiveness is indistinguishable.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 303, + 506, + 381 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "We adopt the analysis framework from Chen et al. 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Here", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 415, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 107, + 417, + 117, + 425 + ], + "score": 0.67, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 415, + 246, + 427 + ], + "score": 1.0, + "content": "denotes the max node degree,", + "type": "text" + }, + { + "bbox": [ + 247, + 415, + 255, + 425 + ], + "score": 0.68, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 415, + 428, + 427 + ], + "score": 1.0, + "content": "denotes the number of GNN layers, and", + "type": "text" + }, + { + "bbox": [ + 429, + 416, + 439, + 425 + ], + "score": 0.8, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 415, + 505, + 427 + ], + "score": 1.0, + "content": "denotes the set", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "of all possible node features. The former is apparently larger which concludes that GNNs are", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 437, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 423, + 449 + ], + "score": 1.0, + "content": "more expressive. Empirically, however, the gap makes little difference when", + "type": "text" + }, + { + "bbox": [ + 423, + 437, + 438, + 449 + ], + "score": 0.9, + "content": "| \\mathcal { X } |", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 437, + 506, + 449 + ], + "score": 1.0, + "content": "is large. In real", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 447, + 507, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 507, + 461 + ], + "score": 1.0, + "content": "applications, node features can be high dimensional like bag-of-words, or even word embeddings,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 458, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 160, + 471 + ], + "score": 1.0, + "content": "thus making", + "type": "text" + }, + { + "bbox": [ + 160, + 459, + 175, + 471 + ], + "score": 0.9, + "content": "| \\mathcal { X } |", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 459, + 320, + 471 + ], + "score": 1.0, + "content": "enormous. Like for bag-of-words,", + "type": "text" + }, + { + "bbox": [ + 320, + 459, + 335, + 471 + ], + "score": 0.91, + "content": "| \\mathcal { X } |", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 459, + 408, + 471 + ], + "score": 1.0, + "content": "is in the order of", + "type": "text" + }, + { + "bbox": [ + 408, + 458, + 437, + 471 + ], + "score": 0.93, + "content": "\\mathcal { O } ( p ^ { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 459, + 469, + 471 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 469, + 459, + 479, + 469 + ], + "score": 0.82, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 459, + 506, + 471 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 192, + 483 + ], + "score": 1.0, + "content": "vocabulary size, and", + "type": "text" + }, + { + "bbox": [ + 192, + 472, + 198, + 481 + ], + "score": 0.78, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "is the max word frequency. 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From the information theoretic perspective, fitting", + "type": "text" + }, + { + "bbox": [ + 484, + 558, + 491, + 570 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 556, + 506, + 572 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "score": 1.0, + "content": "minimizing the commonly used cross-entropy loss is equivalent to maximizing the mutual information", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 131, + 594 + ], + "score": 1.0, + "content": "(MI),", + "type": "text" + }, + { + "bbox": [ + 131, + 579, + 173, + 593 + ], + "score": 0.92, + "content": "I ( \\mathcal G ^ { [ i ] } ; \\boldsymbol y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 579, + 367, + 594 + ], + "score": 1.0, + "content": "as shown in Qin et al. 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GLNN has comparable performance to GNNs only when nodes", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 207, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 219 + ], + "score": 1.0, + "content": "are less noisy. Adding more noise decreases GLNN performance faster than GNNs. Middle: Inductive", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 217, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 232 + ], + "score": 1.0, + "content": "split rate. Altering the inductive:transductive ratio in the production setting doesn’t affect the accuracy", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 228, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 506, + 243 + ], + "score": 1.0, + "content": "much. Right: Teacher GNN architecture. GLNNs can learn from different GNN teachers to improve", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 240, + 476, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 476, + 253 + ], + "score": 1.0, + "content": "over MLPs and achieve comparable results. Accuracies are averaged over five CPF datasets.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "title", + "bbox": [ + 108, + 264, + 228, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 262, + 229, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 229, + 279 + ], + "score": 1.0, + "content": "6 ABLATION STUDIES", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 290, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "In this section, we do ablation studies of GLNNs on node feature noise, inductive split rates, and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 302, + 507, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 507, + 314 + ], + "score": 1.0, + "content": "teacher GNN architecture. Reported results are test accuracies averaged over five datasets in CPF.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 312, + 507, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 507, + 326 + ], + "score": 1.0, + "content": "More experiments can be found in Appendix including advanced GNN teachers (Appendix F),", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 324, + 407, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 407, + 337 + ], + "score": 1.0, + "content": "GA-MLP student (Appendix G), and non-homogeneous data (Appendix I).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 340, + 505, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 354 + ], + "score": 1.0, + "content": "Noisy node features. Following Section 5.8, we investigate failure cases of GLNN by adding different", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "levels of Gaussian noise to node features to decrease their mutual information with labels. Specifically,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 153, + 377 + ], + "score": 1.0, + "content": "we replace", + "type": "text" + }, + { + "bbox": [ + 154, + 364, + 165, + 374 + ], + "score": 0.75, + "content": "\\boldsymbol { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 363, + 188, + 377 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 188, + 363, + 281, + 376 + ], + "score": 0.89, + "content": "\\tilde { \\cal X } = ( 1 - \\alpha ) { \\cal X } + \\alpha \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 363, + 287, + 377 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 287, + 366, + 293, + 374 + ], + "score": 0.52, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 363, + 472, + 377 + ], + "score": 1.0, + "content": "is an isotropic Gaussian independent from", + "type": "text" + }, + { + "bbox": [ + 472, + 364, + 483, + 374 + ], + "score": 0.75, + "content": "\\boldsymbol { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 363, + 505, + 377 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 107, + 375, + 147, + 387 + ], + "score": 0.92, + "content": "\\alpha \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "denotes the noise level. 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We see that as", + "type": "text" + }, + { + "bbox": [ + 348, + 388, + 356, + 396 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "increases, the accuracy of MLPs and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "GLNNs decrease faster than GNNs, while the performance of GLNNs and GNNs are still comparable", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 408, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 145, + 422 + ], + "score": 1.0, + "content": "for small", + "type": "text" + }, + { + "bbox": [ + 145, + 411, + 157, + 420 + ], + "score": 0.51, + "content": "\\alpha \\mathbf { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 408, + 187, + 422 + ], + "score": 1.0, + "content": ". 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A more detailed discussion is in Appendix J.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "Inductive split rate. In Section 5.4, we use a 20-80 split of the test data for inductive evaluation. 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We only consider rates up to 50-50 since having", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 126, + 492 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "or even more inductive nodes is highly atypical in practice. When a large amount of new data", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 492, + 478, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 478, + 505 + ], + "score": 1.0, + "content": "are encountered, practitioners can opt to retrain the model on all the data before deployment.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 509, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "Teacher GNN architecture. We used SAGE to represent GNNs so far. In Figure 5 (right), we show", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "results with other various GNN teachers, e.g. GCN, GAT, and APPNP. 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One", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "possible reason is that the first step of APPNP is to utilize the node’s own feature for prediction (prior", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "to propagating over the graph), which is very similar to what the student MLP is doing, and thus", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 586, + 370, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 370, + 598 + ], + "score": 1.0, + "content": "provides less additional information to MLPs than other teachers.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 108, + 616, + 302, + 629 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 304, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 304, + 631 + ], + "score": 1.0, + "content": "7 CONCLUSION AND FUTURE WORK", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "score": 1.0, + "content": "In this paper, we explored whether we can bridge the best of GNNs and MLPs to achieve accurate and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "fast GML models for deployment. We found that KD from GNNs to MLPs helps to eliminate inference", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 665, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 319, + 680 + ], + "score": 1.0, + "content": "graph dependency, which results in GLNNs that are", + "type": "text" + }, + { + "bbox": [ + 319, + 666, + 366, + 676 + ], + "score": 0.9, + "content": "1 4 6 \\times - 2 7 3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 665, + 506, + 680 + ], + "score": 1.0, + "content": "faster than GNNs while enjoying", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "competitive performance. We do a comprehensive study of GLNN properties. The promising results", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "score": 1.0, + "content": "on 7 datasets across different domains show that GLNNs can be a handy choice for deploying latency-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "constraint models. In our experiments, the current version of GLNNs on the Arxiv dataset doesn’t", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "show competitive inductive performance. 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We only consider rates up to 50-50 since having", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 126, + 492 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "or even more inductive nodes is highly atypical in practice. 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We see that GLNNs can learn", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "from different teachers and improve over MLPs. The performance is similar for all four teachers,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "with the GLNN distilled from APPNP very slightly worse than others. In fact, a similar phenomenon", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "has been observed in Yang et al. (2021a) as well, i.e. APPNP benefits the student the least. One", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "possible reason is that the first step of APPNP is to utilize the node’s own feature for prediction (prior", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "to propagating over the graph), which is very similar to what the student MLP is doing, and thus", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 586, + 370, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 370, + 598 + ], + "score": 1.0, + "content": "provides less additional information to MLPs than other teachers.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 508, + 506, + 598 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 616, + 302, + 629 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 304, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 304, + 631 + ], + "score": 1.0, + "content": "7 CONCLUSION AND FUTURE WORK", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "score": 1.0, + "content": "In this paper, we explored whether we can bridge the best of GNNs and MLPs to achieve accurate and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "fast GML models for deployment. We found that KD from GNNs to MLPs helps to eliminate inference", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 665, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 319, + 680 + ], + "score": 1.0, + "content": "graph dependency, which results in GLNNs that are", + "type": "text" + }, + { + "bbox": [ + 319, + 666, + 366, + 676 + ], + "score": 0.9, + "content": "1 4 6 \\times - 2 7 3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 665, + 506, + 680 + ], + "score": 1.0, + "content": "faster than GNNs while enjoying", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "competitive performance. We do a comprehensive study of GLNN properties. The promising results", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "score": 1.0, + "content": "on 7 datasets across different domains show that GLNNs can be a handy choice for deploying latency-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "constraint models. In our experiments, the current version of GLNNs on the Arxiv dataset doesn’t", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "show competitive inductive performance. 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Layer-dependent", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 116, + 685, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 116, + 685, + 505, + 697 + ], + "score": 1.0, + "content": "importance sampling for training deep and large graph convolutional networks. arXiv preprint", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 116, + 695, + 219, + 707 + ], + "spans": [ + { + "bbox": [ + 116, + 695, + 219, + 707 + ], + "score": 1.0, + "content": "arXiv:1911.07323, 2019.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44, + "bbox_fs": [ + 106, + 672, + 506, + 707 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 307, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 308, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 308, + 95 + ], + "score": 1.0, + "content": "A DETAILED EXPERIMENT SETTINGS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 108, + 106, + 178, + 118 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 179, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 179, + 118 + ], + "score": 1.0, + "content": "A.1 DATASETS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 127, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "Here we provide a detailed description of the datasets we used to support our argument. 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Dataset#Nodes#Edges#Features# Classes
Cora2,4855,0691,4337
Citeseer2,1103,6683,7036
Pubmed19,71744,3245003
A-computer13,381245,77876710
A-photo7,487119,0437458
Arxiv169,3431,166,24312840
Products2,449,02961,859,14010047
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SAGEGCNGATAPPNP
#layers2222
hidden dim128646464
learning rate0.010.010.010.01
weight decay0.00050.0010.010.01
dropout00.80.60.5
fan out5,511-
attention heads1=81
power iterations1=110
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DatasetArxivProducts
# layers33
hidden dim256256
learning rate0.010.003
weight decay00
dropout0.20.5
normalizationbatchbatch
fan out[5,10,15][5,10,15]
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Dataset#Nodes#Edges#Features# Classes
Cora2,4855,0691,4337
Citeseer2,1103,6683,7036
Pubmed19,71744,3245003
A-computer13,381245,77876710
A-photo7,487119,0437458
Arxiv169,3431,166,24312840
Products2,449,02961,859,14010047
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SAGEGCNGATAPPNP
#layers2222
hidden dim128646464
learning rate0.010.010.010.01
weight decay0.00050.0010.010.01
dropout00.80.60.5
fan out5,511-
attention heads1=81
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DatasetArxivProducts
# layers33
hidden dim256256
learning rate0.010.003
weight decay00
dropout0.20.5
normalizationbatchbatch
fan out[5,10,15][5,10,15]
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It seems less practical compared to the inductive case,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "but it is a necessary step to develop our argument. We now discuss the rationale behind this choice.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "Firstly, the transductive setting is the most common setting for graph data and it was used in most", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "GNN architecture works and GNN acceleration works we mentioned in related work. Therefore, to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "avoid any confusion and for a fair comparison with numbers from previous literature, we start our", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "experiments with exactly the same input and output as the standard transductive setting. Under this", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "setting, the inputs to GNNs include all the node features and the graph structure, so GLNN is set to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "be able to access the same input. As GLNN includes a teacher training step and a distillation step,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "the soft labels of all the nodes are intermediate outputs produced by the teacher training step, and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "thus used for the second distillation step for the best GLNN performance. This transductive setting", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "can boil down to a sanity check when the student is sufficiently large. Therefore, we separate the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 204, + 722 + ], + "score": 1.0, + "content": "setting to be GLNN and", + "type": "text" + }, + { + "bbox": [ + 205, + 710, + 238, + 720 + ], + "score": 0.71, + "content": "\\mathsf { G L N N + }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "and report the results in Table 1 and Table 2 separately. In Table", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "1, we are checking how well GLNNs can perform compared to GNNs under the equal-parameter", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 97 + ], + "score": 1.0, + "content": "For GLNN s we do a hyperparameter search of learning rate from [0.01, 0.005, 0.001], weight decay", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 441, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 441, + 106 + ], + "score": 1.0, + "content": "from [0, 0.001, 0.002, 0.005, 0.01], and dropout from [0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6]", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 505, + 106 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 118, + 254, + 130 + ], + "lines": [ + { + "bbox": [ + 107, + 119, + 255, + 131 + ], + "spans": [ + { + "bbox": [ + 107, + 119, + 255, + 131 + ], + "score": 1.0, + "content": "A.3 KNOWLEDGE DISTILLATION", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 138, + 505, + 205 + ], + "lines": [ + { + "bbox": [ + 106, + 139, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 505, + 151 + ], + "score": 1.0, + "content": "We use the distillation method proposed in Hinton et al. (2015) as in Equation 1, the hard labels are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 233, + 163 + ], + "score": 1.0, + "content": "found to be helpful, so nonzero", + "type": "text" + }, + { + "bbox": [ + 234, + 150, + 245, + 160 + ], + "score": 0.37, + "content": "\\lambda s", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 150, + 455, + 163 + ], + "score": 1.0, + "content": "was suggested. In our case, we did a little tuning for", + "type": "text" + }, + { + "bbox": [ + 456, + 150, + 463, + 160 + ], + "score": 0.75, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "but didn’t", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 161, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 160, + 173 + ], + "score": 1.0, + "content": "find nonzero", + "type": "text" + }, + { + "bbox": [ + 160, + 162, + 172, + 171 + ], + "score": 0.51, + "content": "\\lambda s", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 161, + 425, + 173 + ], + "score": 1.0, + "content": "to be very helpful. Therefore, we report all of our results with", + "type": "text" + }, + { + "bbox": [ + 425, + 161, + 450, + 171 + ], + "score": 0.9, + "content": "\\lambda = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 161, + 505, + 173 + ], + "score": 1.0, + "content": ", i.e. only the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 172, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 386, + 184 + ], + "score": 1.0, + "content": "second term involving soft labels is effective. More careful tuning of", + "type": "text" + }, + { + "bbox": [ + 387, + 172, + 394, + 182 + ], + "score": 0.77, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 172, + 505, + 184 + ], + "score": 1.0, + "content": "should further improve the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 183, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 506, + 195 + ], + "score": 1.0, + "content": "results since the searching space is strictly larger. We implemented a weighted version in our code,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 194, + 285, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 214, + 205 + ], + "score": 1.0, + "content": "and we leave the choice of", + "type": "text" + }, + { + "bbox": [ + 215, + 194, + 222, + 204 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 194, + 285, + 205 + ], + "score": 1.0, + "content": "as future work.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 139, + 506, + 205 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 218, + 396, + 230 + ], + "lines": [ + { + "bbox": [ + 106, + 218, + 398, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 398, + 231 + ], + "score": 1.0, + "content": "A.4 THE TRANSDUCTIVE SETTING AND THE INDUCTIVE SETTING", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 239, + 505, + 364 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 507, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 134, + 253 + ], + "score": 1.0, + "content": "Given", + "type": "text" + }, + { + "bbox": [ + 134, + 239, + 158, + 250 + ], + "score": 0.5, + "content": "{ \\mathcal { G } } , X", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 237, + 181, + 253 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 181, + 239, + 198, + 250 + ], + "score": 0.88, + "content": "Y ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 237, + 507, + 253 + ], + "score": 1.0, + "content": ", the goal of node classification can be divided into two different settings,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 248, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 104, + 248, + 506, + 264 + ], + "score": 1.0, + "content": "i.e. transductive and inductive. In real applications, the former can correspond to predict missing", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 262, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 505, + 273 + ], + "score": 1.0, + "content": "attributes of a user based on the user profile and other existing users, and the latter can correspond to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 104, + 271, + 506, + 285 + ], + "score": 1.0, + "content": "predict labels of some new nodes that are only seen during inference time. To create the inductive", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 282, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 296 + ], + "score": 1.0, + "content": "setting on a given dataset, we hold out some nodes along with edges connected to these nodes during", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 102, + 295, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 102, + 299, + 369, + 324 + ], + "score": 1.0, + "content": "data. Using notation defined above, we pick the inductive nodes", + "type": "text" + }, + { + "bbox": [ + 106, + 295, + 505, + 306 + ], + "score": 1.0, + "content": "training and use them for inductive evaluation only. These nodes and edges are picked from the test", + "type": "text" + }, + { + "bbox": [ + 370, + 304, + 416, + 317 + ], + "score": 0.91, + "content": "\\mathcal { V } _ { i n d } ^ { U } \\subset \\mathcal { V } ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 299, + 489, + 324 + ], + "score": 1.0, + "content": ", which partitions", + "type": "text" + }, + { + "bbox": [ + 489, + 304, + 504, + 315 + ], + "score": 0.88, + "content": "\\mathcal { V } ^ { U }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 102, + 312, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 102, + 312, + 347, + 336 + ], + "score": 1.0, + "content": "into the disjoint inductive subset and observed subset, i.e.", + "type": "text" + }, + { + "bbox": [ + 347, + 317, + 425, + 330 + ], + "score": 0.91, + "content": "\\mathcal { V } ^ { U } = \\mathcal { V } _ { o b s } ^ { U } \\sqcup \\mathcal { V } _ { i n d } ^ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 319, + 435, + 331 + ], + "score": 1.0, + "content": "V ind", + "type": "text" + }, + { + "bbox": [ + 426, + 317, + 505, + 330 + ], + "score": 1.0, + "content": ". 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We show the input and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 341, + 405, + 354 + ], + "spans": [ + { + "bbox": [ + 107, + 341, + 175, + 353 + ], + "score": 0.88, + "content": "\\mathcal { G } = \\mathcal { G } _ { o b s } \\sqcup \\mathcal { G } _ { i n d }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 341, + 285, + 354 + ], + "score": 0.9, + "content": "\\pmb { X } = \\pmb { X } ^ { L } \\sqcup \\pmb { X } _ { o b s } ^ { U } \\sqcup \\overbrace { \\pmb { X } _ { i n d } ^ { U } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 341, + 405, + 354 + ], + "score": 0.92, + "content": "\\pmb { Y } = \\pmb { Y } ^ { L } \\sqcup \\pmb { Y } _ { o b s } ^ { U } \\sqcup \\pmb { Y } _ { i n d } ^ { U }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 352, + 305, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 305, + 365 + ], + "score": 1.0, + "content": "output of both settings using the notations below.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5, + "bbox_fs": [ + 102, + 237, + 507, + 365 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 369, + 495, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 495, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 495, + 383 + ], + "score": 1.0, + "content": "We visualize the difference between the inductive setting and the transductive setting in Figure 6.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 367, + 495, + 383 + ] + }, + { + "type": "image", + "bbox": [ + 108, + 393, + 506, + 479 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 393, + 506, + 479 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 393, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 108, + 393, + 506, + 479 + ], + "score": 0.963, + "type": "image", + "image_path": "2a239fb19d94400d2ccb7f18def7895440e8208ed97573eb1cf684c4025f00ac.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 108, + 393, + 506, + 421.6666666666667 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 108, + 421.6666666666667, + 506, + 450.33333333333337 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 108, + 450.33333333333337, + 506, + 479.00000000000006 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 486, + 506, + 531 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "Figure 6: The transductive setting and inductive setting illustrated by a 2-layer GNN. The middle", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 498, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 506, + 510 + ], + "score": 1.0, + "content": "shows the original graph used for training. The left shows the transductive setting, where the test", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "score": 1.0, + "content": "node is in red and within the graph. 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It seems less practical compared to the inductive case,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "but it is a necessary step to develop our argument. We now discuss the rationale behind this choice.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 572, + 507, + 607 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "Firstly, the transductive setting is the most common setting for graph data and it was used in most", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "GNN architecture works and GNN acceleration works we mentioned in related work. Therefore, to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "avoid any confusion and for a fair comparison with numbers from previous literature, we start our", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "experiments with exactly the same input and output as the standard transductive setting. Under this", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "setting, the inputs to GNNs include all the node features and the graph structure, so GLNN is set to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "be able to access the same input. As GLNN includes a teacher training step and a distillation step,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "the soft labels of all the nodes are intermediate outputs produced by the teacher training step, and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "thus used for the second distillation step for the best GLNN performance. This transductive setting", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "can boil down to a sanity check when the student is sufficiently large. Therefore, we separate the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 204, + 722 + ], + "score": 1.0, + "content": "setting to be GLNN and", + "type": "text" + }, + { + "bbox": [ + 205, + 710, + 238, + 720 + ], + "score": 0.71, + "content": "\\mathsf { G L N N + }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "and report the results in Table 1 and Table 2 separately. In Table", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "1, we are checking how well GLNNs can perform compared to GNNs under the equal-parameter", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "constraint. The results can be interpreted as given a fixed parameter budget, whether there exists one", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "set of parameters (one instantiation of the MLP) that can achieve competitive results as the GNN.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "Only when this holds, should we further investigate the more interesting and challenging inductive", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 197, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 197, + 127 + ], + "score": 1.0, + "content": "case as in Section 5.4.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 610, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "constraint. The results can be interpreted as given a fixed parameter budget, whether there exists one", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "set of parameters (one instantiation of the MLP) that can achieve competitive results as the GNN.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "Only when this holds, should we further investigate the more interesting and challenging inductive", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 197, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 197, + 127 + ], + "score": 1.0, + "content": "case as in Section 5.4.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 220 + ], + "lines": [ + { + "bbox": [ + 106, + 132, + 507, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 507, + 144 + ], + "score": 1.0, + "content": "Secondly, the task we focus on is node classification, which in many cases is considered as semi-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "supervised learning with very scarce labels. For example, Pubmed only uses 60 labeled nodes (20", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 155, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 506, + 166 + ], + "score": 1.0, + "content": "per class) out of 20K nodes for training. Rather than design an advanced model that can do few-shot", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 166, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 506, + 177 + ], + "score": 1.0, + "content": "learning, our goal here is to leverage as much data as possible to simplify the model for more efficient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "inference. We thus utilize the soft pseudo-labels on all the unlabelled nodes for the best GLNN", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "performance. In reality, when there is a large amount of separate unlabeled data, these unlabeled", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "data can be used for GLNN distillation training and a different set of labeled data can be used for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 449, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 449, + 221 + ], + "score": 1.0, + "content": "evaluation. In our case, we mimic this scenario in the inductive setting in Section 5.4.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 107, + 233, + 326, + 244 + ], + "lines": [ + { + "bbox": [ + 105, + 233, + 327, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 327, + 246 + ], + "score": 1.0, + "content": "A.6 IMPLEMENTATION AND HARDWARD DETAILS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 252, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 267 + ], + "score": 1.0, + "content": "The experiments on both baselines and our approach are implemented using PyTorch, the DGL (Wang", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "et al., 2019) library for GNN algorithms, and Adam (Kingma & Ba, 2015) for optimization. We run", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 377, + 288 + ], + "score": 1.0, + "content": "all experiments on a machine with 80 Intel(R) Xeon(R) E5-2698 v4", + "type": "text" + }, + { + "bbox": [ + 378, + 276, + 387, + 286 + ], + "score": 0.54, + "content": "@", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "2.20GHz CPUs, and a single", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 287, + 264, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 264, + 297 + ], + "score": 1.0, + "content": "NVIDIA V100 GPU with 16GB RAM.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 108, + 313, + 397, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 399, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 399, + 328 + ], + "score": 1.0, + "content": "B SPACE AND TIME COMPLEXITY OF GNNS VS. MLPS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 337, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "Compared to MLP and GNN, GLNN provides a handy tool for users to trade-off between model", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "accuracy and time complexity, which does not directly focus on space complexity. Given the space", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "and time complexity are related, we now provide a more detailed discussion regarding these two", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 371, + 240, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 240, + 383 + ], + "score": 1.0, + "content": "complexities in our experiments.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 402 + ], + "score": 1.0, + "content": "In Table 1, the model comparison was between equal-sized MLPs (GLNNs) and GNNs. While fixing", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 399, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 411 + ], + "score": 1.0, + "content": "parameter budget to control space complexity is a standard approach when comparing models, it is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "not completely fair for cross-model comparison especially for MLPs vs. GNNs. To do inference with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 420, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 434 + ], + "score": 1.0, + "content": "GNNs, the graph needs to be loaded in the memory either entirely or batch by batch, and may use", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "much larger space than the model parameters. Thus, the actual space complexity of GNNs is much", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "higher than equal-sized MLPs. From the time complexity perspective, the major inference latency of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "GNNs comes from the data dependency as shown in Section 4. Under the same setting as Figure 1, we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "show in Figure 3 Left that even a 5-layer MLP with 8 times wider hidden layers still runs much faster", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 475, + 507, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 507, + 488 + ], + "score": 1.0, + "content": "than a single-layer SAGE. Another example of cross-model comparison is Transformers vs. RNNs.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 487, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 498 + ], + "score": 1.0, + "content": "Large Transformers can have more parameters than RNNs because of the attention mechanism, but", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "they are also faster than RNNs in general, which is an important consideration in the context of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 508, + 223, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 223, + 520 + ], + "score": 1.0, + "content": "inference time minimization.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "In Table 1, we saw that for equal-sized comparison, GLNNs are not as accurate as GNNs on the OGB", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "datasets. Following the discussion above and given the GLNNs used in Table 1 are relatively small (3", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "layers and 256 hidden dimensions) for millions of nodes in the OGB datasets, we ask whether this", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "score": 1.0, + "content": "gap can be mitigated by increasing the MLP and thus GLNN sizes. The answer is yes as shown in", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 568, + 141, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 141, + 581 + ], + "score": 1.0, + "content": "Table 2.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 108, + 595, + 453, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 456, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 456, + 610 + ], + "score": 1.0, + "content": "C CONSISTENCY MEASURE OF MODEL PREDICTIONS AND GRAPH", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 127, + 610, + 296, + 623 + ], + "spans": [ + { + "bbox": [ + 127, + 610, + 296, + 623 + ], + "score": 1.0, + "content": "TOPOLOGY BASED ON MIN-CUT", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 505, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "score": 1.0, + "content": "We introduce a metric to measure the consistency between model predictions and graph topology", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 311, + 658 + ], + "score": 1.0, + "content": "based on the min-cut problem in Section 5.6. 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While fixing", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 399, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 411 + ], + "score": 1.0, + "content": "parameter budget to control space complexity is a standard approach when comparing models, it is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "not completely fair for cross-model comparison especially for MLPs vs. GNNs. To do inference with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 420, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 434 + ], + "score": 1.0, + "content": "GNNs, the graph needs to be loaded in the memory either entirely or batch by batch, and may use", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "much larger space than the model parameters. Thus, the actual space complexity of GNNs is much", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "higher than equal-sized MLPs. From the time complexity perspective, the major inference latency of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "GNNs comes from the data dependency as shown in Section 4. Under the same setting as Figure 1, we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "show in Figure 3 Left that even a 5-layer MLP with 8 times wider hidden layers still runs much faster", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 475, + 507, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 507, + 488 + ], + "score": 1.0, + "content": "than a single-layer SAGE. 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DatasetsSAGEMLPGLNN
Cora0.93470.70260.8852
Citeseer0.94850.76930.9339
Pubmed0.96050.94550.9701
A-computer0.90030.69760.8638
A-photo0.86640.70690.8398
Average0.92210.76440.8986
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MLPS IN TERMS OF EQUIVALENCE", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 127, + 358, + 281, + 370 + ], + "spans": [ + { + "bbox": [ + 127, + 358, + 281, + 370 + ], + "score": 1.0, + "content": "CLASSES OF ROOTED GRAPHS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 504, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "In Chen et al. (2021), the expressiveness of GNNs and GA-MLPs were theoretically quantified in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "score": 1.0, + "content": "terms of induced equivalence classes of rooted graphs. We adopt their framework and perform a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 407, + 377, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 377, + 420 + ], + "score": 1.0, + "content": "similar analysis for GNNs vs. MLPs. We first define rooted graphs.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 104, + 425, + 504, + 438 + ], + "spans": [ + { + "bbox": [ + 104, + 425, + 347, + 438 + ], + "score": 1.0, + "content": "Definition 1 (Rooted Graph). A rooted graph, denoted as", + "type": "text" + }, + { + "bbox": [ + 347, + 425, + 363, + 437 + ], + "score": 0.89, + "content": "\\mathcal { G } ^ { [ i ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 425, + 488, + 438 + ], + "score": 1.0, + "content": "is a graph with one node i in", + "type": "text" + }, + { + "bbox": [ + 488, + 425, + 504, + 437 + ], + "score": 0.86, + "content": "\\mathcal { G } ^ { [ i ] }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 438, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 449 + ], + "score": 1.0, + "content": "designated as the root. GNNs, GA-MLPs, and MLPs can all be considered as functions on rooted", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 447, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 294, + 462 + ], + "score": 1.0, + "content": "graphs. The goal of a node-level task on node", + "type": "text" + }, + { + "bbox": [ + 294, + 450, + 299, + 458 + ], + "score": 0.33, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 447, + 342, + 462 + ], + "score": 1.0, + "content": "with label", + "type": "text" + }, + { + "bbox": [ + 343, + 450, + 353, + 460 + ], + "score": 0.82, + "content": "\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\psi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 447, + 506, + 462 + ], + "score": 1.0, + "content": "is to fit a function to the input-output", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 458, + 171, + 472 + ], + "spans": [ + { + "bbox": [ + 104, + 458, + 129, + 472 + ], + "score": 1.0, + "content": "pairs", + "type": "text" + }, + { + "bbox": [ + 129, + 459, + 165, + 471 + ], + "score": 0.86, + "content": "( \\mathcal G ^ { [ i ] } , y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 458, + 171, + 472 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 276, + 496 + ], + "score": 1.0, + "content": "We denote the space of rooted graphs as", + "type": "text" + }, + { + "bbox": [ + 277, + 484, + 284, + 493 + ], + "score": 0.69, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 482, + 506, + 496 + ], + "score": 1.0, + "content": ". Following Chen et al. (2021), the expressive power", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 432, + 506 + ], + "score": 1.0, + "content": "of a model on graph data is evaluated by its ability to approximate functions on", + "type": "text" + }, + { + "bbox": [ + 433, + 495, + 440, + 504 + ], + "score": 0.76, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 493, + 506, + 506 + ], + "score": 1.0, + "content": ". This is further", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 412, + 518 + ], + "score": 1.0, + "content": "characterized as the number of induced equivalence classes of rooted graphs on", + "type": "text" + }, + { + "bbox": [ + 413, + 506, + 420, + 515 + ], + "score": 0.79, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 505, + 505, + 518 + ], + "score": 1.0, + "content": ", with the equivalence", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 363, + 528 + ], + "score": 1.0, + "content": "relation defined as the following. Given a family of functions", + "type": "text" + }, + { + "bbox": [ + 363, + 516, + 373, + 526 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 515, + 387, + 528 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 388, + 516, + 395, + 526 + ], + "score": 0.8, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 515, + 506, + 528 + ], + "score": 1.0, + "content": ", we define an equivalence", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 526, + 507, + 541 + ], + "spans": [ + { + "bbox": [ + 104, + 526, + 140, + 541 + ], + "score": 1.0, + "content": "relation", + "type": "text" + }, + { + "bbox": [ + 141, + 529, + 164, + 540 + ], + "score": 0.88, + "content": "\\simeq _ { \\varepsilon , F }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 526, + 312, + 541 + ], + "score": 1.0, + "content": "among all rooted graphs such that", + "type": "text" + }, + { + "bbox": [ + 312, + 527, + 444, + 540 + ], + "score": 0.92, + "content": "\\forall \\mathcal { G } ^ { [ i ] } , \\mathcal { G } ^ { \\prime [ j ] } \\in \\mathcal { E } , \\mathcal { G } ^ { [ i ] } \\simeq _ { \\mathcal { E } , \\mathcal { F } } \\mathcal { G } ^ { \\prime [ j ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 526, + 507, + 541 + ], + "score": 1.0, + "content": "if and only if", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 539, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 217, + 553 + ], + "score": 0.9, + "content": "\\forall f \\in \\mathcal { F } , f ( \\mathcal { G } ^ { [ i ] } ) = f ( \\mathcal { G } ^ { \\prime [ j ] } )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 539, + 506, + 554 + ], + "score": 1.0, + "content": ". We now give a proposition to characterize the GNN expressive power", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 200, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 200, + 564 + ], + "score": 1.0, + "content": "(proof in Appendix E).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 569, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 504, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 191, + 582 + ], + "score": 1.0, + "content": "Proposition 1. With", + "type": "text" + }, + { + "bbox": [ + 191, + 570, + 201, + 579 + ], + "score": 0.62, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 569, + 437, + 582 + ], + "score": 1.0, + "content": "denotes the set of all possible node features and assuming", + "type": "text" + }, + { + "bbox": [ + 437, + 569, + 470, + 581 + ], + "score": 0.91, + "content": "| \\mathcal { X } | \\ge 2", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 569, + 494, + 582 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 494, + 571, + 504, + 579 + ], + "score": 0.58, + "content": "m", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 580, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 303, + 594 + ], + "score": 1.0, + "content": "denotes the maximum node degree and assuming", + "type": "text" + }, + { + "bbox": [ + 304, + 581, + 332, + 591 + ], + "score": 0.89, + "content": "m \\geq 3", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 580, + 506, + 594 + ], + "score": 1.0, + "content": ", the total number of equivalence classes of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 591, + 425, + 611 + ], + "spans": [ + { + "bbox": [ + 104, + 596, + 222, + 611 + ], + "score": 1.0, + "content": "rooted graphs induced by an", + "type": "text" + }, + { + "bbox": [ + 223, + 598, + 230, + 607 + ], + "score": 0.41, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 596, + 361, + 611 + ], + "score": 1.0, + "content": "-layer GNN is lower bounded by", + "type": "text" + }, + { + "bbox": [ + 362, + 591, + 425, + 611 + ], + "score": 0.93, + "content": "\\binom { | \\mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 }", + "type": "inline_equation" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "As shown in Proposition 1, the expressive power of GNNs grows doubly-exponentially in the number", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 146, + 646 + ], + "score": 1.0, + "content": "of layers", + "type": "text" + }, + { + "bbox": [ + 146, + 633, + 154, + 643 + ], + "score": 0.69, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 632, + 299, + 646 + ], + "score": 1.0, + "content": ", which means it grows linearly in", + "type": "text" + }, + { + "bbox": [ + 299, + 633, + 308, + 643 + ], + "score": 0.78, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 632, + 360, + 646 + ], + "score": 1.0, + "content": "after taking", + "type": "text" + }, + { + "bbox": [ + 360, + 633, + 405, + 645 + ], + "score": 0.9, + "content": "\\log ( \\log ( \\cdot ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 632, + 506, + 646 + ], + "score": 1.0, + "content": ". The expressive power", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 263, + 657 + ], + "score": 1.0, + "content": "GA-MLPs only grows exponentially in", + "type": "text" + }, + { + "bbox": [ + 263, + 644, + 272, + 654 + ], + "score": 0.76, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 642, + 506, + 657 + ], + "score": 1.0, + "content": "as shown in Chen et al. (2021). 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The", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "gap between these two numbers indeed exists, but empirically this gap will only make a difference", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 129, + 700 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 130, + 688, + 145, + 700 + ], + "score": 0.9, + "content": "| \\mathcal { X } |", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "is small. As in Chen et al. 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Cora0.93470.70260.8852
Citeseer0.94850.76930.9339
Pubmed0.96050.94550.9701
A-computer0.90030.69760.8638
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A rooted graph, denoted as", + "type": "text" + }, + { + "bbox": [ + 347, + 425, + 363, + 437 + ], + "score": 0.89, + "content": "\\mathcal { G } ^ { [ i ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 425, + 488, + 438 + ], + "score": 1.0, + "content": "is a graph with one node i in", + "type": "text" + }, + { + "bbox": [ + 488, + 425, + 504, + 437 + ], + "score": 0.86, + "content": "\\mathcal { G } ^ { [ i ] }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 438, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 449 + ], + "score": 1.0, + "content": "designated as the root. GNNs, GA-MLPs, and MLPs can all be considered as functions on rooted", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 447, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 294, + 462 + ], + "score": 1.0, + "content": "graphs. The goal of a node-level task on node", + "type": "text" + }, + { + "bbox": [ + 294, + 450, + 299, + 458 + ], + "score": 0.33, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 447, + 342, + 462 + ], + "score": 1.0, + "content": "with label", + "type": "text" + }, + { + "bbox": [ + 343, + 450, + 353, + 460 + ], + "score": 0.82, + "content": "\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\psi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 447, + 506, + 462 + ], + "score": 1.0, + "content": "is to fit a function to the input-output", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 458, + 171, + 472 + ], + "spans": [ + { + "bbox": [ + 104, + 458, + 129, + 472 + ], + "score": 1.0, + "content": "pairs", + "type": "text" + }, + { + "bbox": [ + 129, + 459, + 165, + 471 + ], + "score": 0.86, + "content": "( \\mathcal G ^ { [ i ] } , y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 458, + 171, + 472 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 425, + 506, + 472 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 276, + 496 + ], + "score": 1.0, + "content": "We denote the space of rooted graphs as", + "type": "text" + }, + { + "bbox": [ + 277, + 484, + 284, + 493 + ], + "score": 0.69, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 482, + 506, + 496 + ], + "score": 1.0, + "content": ". Following Chen et al. (2021), the expressive power", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 432, + 506 + ], + "score": 1.0, + "content": "of a model on graph data is evaluated by its ability to approximate functions on", + "type": "text" + }, + { + "bbox": [ + 433, + 495, + 440, + 504 + ], + "score": 0.76, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 493, + 506, + 506 + ], + "score": 1.0, + "content": ". This is further", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 412, + 518 + ], + "score": 1.0, + "content": "characterized as the number of induced equivalence classes of rooted graphs on", + "type": "text" + }, + { + "bbox": [ + 413, + 506, + 420, + 515 + ], + "score": 0.79, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 505, + 505, + 518 + ], + "score": 1.0, + "content": ", with the equivalence", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 363, + 528 + ], + "score": 1.0, + "content": "relation defined as the following. Given a family of functions", + "type": "text" + }, + { + "bbox": [ + 363, + 516, + 373, + 526 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 515, + 387, + 528 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 388, + 516, + 395, + 526 + ], + "score": 0.8, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 515, + 506, + 528 + ], + "score": 1.0, + "content": ", we define an equivalence", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 526, + 507, + 541 + ], + "spans": [ + { + "bbox": [ + 104, + 526, + 140, + 541 + ], + "score": 1.0, + "content": "relation", + "type": "text" + }, + { + "bbox": [ + 141, + 529, + 164, + 540 + ], + "score": 0.88, + "content": "\\simeq _ { \\varepsilon , F }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 526, + 312, + 541 + ], + "score": 1.0, + "content": "among all rooted graphs such that", + "type": "text" + }, + { + "bbox": [ + 312, + 527, + 444, + 540 + ], + "score": 0.92, + "content": "\\forall \\mathcal { G } ^ { [ i ] } , \\mathcal { G } ^ { \\prime [ j ] } \\in \\mathcal { E } , \\mathcal { G } ^ { [ i ] } \\simeq _ { \\mathcal { E } , \\mathcal { F } } \\mathcal { G } ^ { \\prime [ j ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 526, + 507, + 541 + ], + "score": 1.0, + "content": "if and only if", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 539, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 217, + 553 + ], + "score": 0.9, + "content": "\\forall f \\in \\mathcal { F } , f ( \\mathcal { G } ^ { [ i ] } ) = f ( \\mathcal { G } ^ { \\prime [ j ] } )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 539, + 506, + 554 + ], + "score": 1.0, + "content": ". We now give a proposition to characterize the GNN expressive power", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 200, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 200, + 564 + ], + "score": 1.0, + "content": "(proof in Appendix E).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 482, + 507, + 564 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 569, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 504, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 191, + 582 + ], + "score": 1.0, + "content": "Proposition 1. With", + "type": "text" + }, + { + "bbox": [ + 191, + 570, + 201, + 579 + ], + "score": 0.62, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 569, + 437, + 582 + ], + "score": 1.0, + "content": "denotes the set of all possible node features and assuming", + "type": "text" + }, + { + "bbox": [ + 437, + 569, + 470, + 581 + ], + "score": 0.91, + "content": "| \\mathcal { X } | \\ge 2", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 569, + 494, + 582 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 494, + 571, + 504, + 579 + ], + "score": 0.58, + "content": "m", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 580, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 303, + 594 + ], + "score": 1.0, + "content": "denotes the maximum node degree and assuming", + "type": "text" + }, + { + "bbox": [ + 304, + 581, + 332, + 591 + ], + "score": 0.89, + "content": "m \\geq 3", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 580, + 506, + 594 + ], + "score": 1.0, + "content": ", the total number of equivalence classes of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 591, + 425, + 611 + ], + "spans": [ + { + "bbox": [ + 104, + 596, + 222, + 611 + ], + "score": 1.0, + "content": "rooted graphs induced by an", + "type": "text" + }, + { + "bbox": [ + 223, + 598, + 230, + 607 + ], + "score": 0.41, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 596, + 361, + 611 + ], + "score": 1.0, + "content": "-layer GNN is lower bounded by", + "type": "text" + }, + { + "bbox": [ + 362, + 591, + 425, + 611 + ], + "score": 0.93, + "content": "\\binom { | \\mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 }", + "type": "inline_equation" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 569, + 506, + 611 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "As shown in Proposition 1, the expressive power of GNNs grows doubly-exponentially in the number", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 146, + 646 + ], + "score": 1.0, + "content": "of layers", + "type": "text" + }, + { + "bbox": [ + 146, + 633, + 154, + 643 + ], + "score": 0.69, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 632, + 299, + 646 + ], + "score": 1.0, + "content": ", which means it grows linearly in", + "type": "text" + }, + { + "bbox": [ + 299, + 633, + 308, + 643 + ], + "score": 0.78, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 632, + 360, + 646 + ], + "score": 1.0, + "content": "after taking", + "type": "text" + }, + { + "bbox": [ + 360, + 633, + 405, + 645 + ], + "score": 0.9, + "content": "\\log ( \\log ( \\cdot ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 632, + 506, + 646 + ], + "score": 1.0, + "content": ". The expressive power", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 642, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 263, + 657 + ], + "score": 1.0, + "content": "GA-MLPs only grows exponentially in", + "type": "text" + }, + { + "bbox": [ + 263, + 644, + 272, + 654 + ], + "score": 0.76, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 642, + 506, + 657 + ], + "score": 1.0, + "content": "as shown in Chen et al. (2021). Under this framework, the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 403, + 668 + ], + "score": 1.0, + "content": "expressive power of MLPs, which corresponds to a 0-layer GA-MLP, is", + "type": "text" + }, + { + "bbox": [ + 404, + 655, + 419, + 667 + ], + "score": 0.89, + "content": "| \\mathcal { X } |", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 654, + 506, + 668 + ], + "score": 1.0, + "content": ". Since the former is", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "much larger than the latter, the conclusion will be GNNs are much more expressive than MLPs. The", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "gap between these two numbers indeed exists, but empirically this gap will only make a difference", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 129, + 700 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 130, + 688, + 145, + 700 + ], + "score": 0.9, + "content": "| \\mathcal { X } |", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "is small. As in Chen et al. (2021), both the lower bound proof and the constructed examples", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 341, + 712 + ], + "score": 1.0, + "content": "showing GNNs are more powerful than GA-MLPs assumed", + "type": "text" + }, + { + "bbox": [ + 341, + 699, + 374, + 711 + ], + "score": 0.91, + "content": "| \\mathcal { X } | = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 699, + 505, + 712 + ], + "score": 1.0, + "content": ". In real applications and datasets", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "considered in this work, the node features can be high dimensional vectors like bag-of-words, which", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 388, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 134, + 733 + ], + "score": 1.0, + "content": "makes", + "type": "text" + }, + { + "bbox": [ + 135, + 721, + 150, + 732 + ], + "score": 0.9, + "content": "| \\mathcal { X } |", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 720, + 388, + 733 + ], + "score": 1.0, + "content": "enormous. Thus, this gap doesn’t matter much empirically.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 622, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 284, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 286, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 286, + 95 + ], + "score": 1.0, + "content": "E PROOF OF THE PROPOSITION 1", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 114, + 504, + 136 + ], + "lines": [ + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "score": 1.0, + "content": "To prove Proposition 1, we first define rooted aggregation trees, which is similar to but different from", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 124, + 165, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 165, + 137 + ], + "score": 1.0, + "content": "rooted graphs.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 146, + 505, + 191 + ], + "lines": [ + { + "bbox": [ + 104, + 145, + 504, + 159 + ], + "spans": [ + { + "bbox": [ + 104, + 145, + 488, + 159 + ], + "score": 1.0, + "content": "Definition 2 (Rooted Aggregation Tree). The depth-K rooted aggregation tree of a rooted graph", + "type": "text" + }, + { + "bbox": [ + 488, + 145, + 504, + 158 + ], + "score": 0.86, + "content": "\\mathcal { G } ^ { [ i ] }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 156, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 223, + 171 + ], + "score": 1.0, + "content": "is a depth-K rooted tree with", + "type": "text" + }, + { + "bbox": [ + 223, + 159, + 230, + 167 + ], + "score": 0.35, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 156, + 506, + 171 + ], + "score": 1.0, + "content": "(possibly many-to-one) mapping from every node in the tree to some", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 167, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 138, + 182 + ], + "score": 1.0, + "content": "node in", + "type": "text" + }, + { + "bbox": [ + 138, + 168, + 153, + 180 + ], + "score": 0.89, + "content": "\\mathcal { G } ^ { [ i ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 167, + 343, + 182 + ], + "score": 1.0, + "content": ", where (i) the root of the tree is mapped to node", + "type": "text" + }, + { + "bbox": [ + 344, + 170, + 348, + 178 + ], + "score": 0.3, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 167, + 487, + 182 + ], + "score": 1.0, + "content": ", and (ii) the children of every node", + "type": "text" + }, + { + "bbox": [ + 488, + 169, + 493, + 180 + ], + "score": 0.62, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 167, + 506, + 182 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 178, + 418, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 312, + 192 + ], + "score": 1.0, + "content": "the tree are mapped to the neighbors of the node in", + "type": "text" + }, + { + "bbox": [ + 313, + 178, + 327, + 190 + ], + "score": 0.89, + "content": "\\mathcal { G } ^ { [ i ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 178, + 365, + 192 + ], + "score": 1.0, + "content": "to which", + "type": "text" + }, + { + "bbox": [ + 366, + 180, + 372, + 191 + ], + "score": 0.71, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 178, + 418, + 192 + ], + "score": 1.0, + "content": "is mapped.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 205, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 205, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 219 + ], + "score": 1.0, + "content": "A rooted aggregation tree can be obtained by unrolling the neighborhood aggregation steps in the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 216, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 230 + ], + "score": 1.0, + "content": "GNNs. An illustration of rooted graphs and rooted aggregation trees can be found in Chen et al.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 402, + 240 + ], + "score": 1.0, + "content": "(2021) Figure 4. We denote the set of all rooted aggregation trees of depth", + "type": "text" + }, + { + "bbox": [ + 403, + 228, + 411, + 238 + ], + "score": 0.37, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 227, + 435, + 240 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 435, + 228, + 448, + 239 + ], + "score": 0.85, + "content": "\\mathcal { T } _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 227, + 506, + 240 + ], + "score": 1.0, + "content": ". Then we use", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 238, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 107, + 239, + 137, + 251 + ], + "score": 0.91, + "content": "\\mathcal { T } _ { L , \\mathcal { X } , m }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 238, + 221, + 252 + ], + "score": 1.0, + "content": "to denote a subset of", + "type": "text" + }, + { + "bbox": [ + 222, + 239, + 234, + 250 + ], + "score": 0.88, + "content": "\\mathcal { T } _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 238, + 372, + 252 + ], + "score": 1.0, + "content": ", where the node features belong to", + "type": "text" + }, + { + "bbox": [ + 372, + 239, + 382, + 249 + ], + "score": 0.79, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 238, + 506, + 252 + ], + "score": 1.0, + "content": ", and all the nodes have exactly", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 135, + 262 + ], + "score": 1.0, + "content": "degree", + "type": "text" + }, + { + "bbox": [ + 135, + 252, + 145, + 260 + ], + "score": 0.68, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 249, + 150, + 262 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 150, + 252, + 160, + 260 + ], + "score": 0.62, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "children), and at least two nodes out of these m nodes have different features. In other", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "score": 1.0, + "content": "words, a node can’t have all identical children. With rooted aggregation trees defined, we are ready to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 272, + 474, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 474, + 284 + ], + "score": 1.0, + "content": "prove Proposition 1. The proof is adapted from the proof of Lemma 3 in Chen et al. (2021).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 332, + 506, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 332, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 316, + 344 + ], + "score": 1.0, + "content": "Proof. Since the number of equivalence classes on", + "type": "text" + }, + { + "bbox": [ + 316, + 333, + 324, + 343 + ], + "score": 0.79, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 332, + 506, + 344 + ], + "score": 1.0, + "content": "induced by the family of all depth-L GNNs", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 443, + 356 + ], + "score": 1.0, + "content": "consists of all rooted graphs that share the same rooted aggregation tree of depth-", + "type": "text" + }, + { + "bbox": [ + 444, + 344, + 451, + 354 + ], + "score": 0.29, + "content": "\\mathrm { . L }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "(Chen et al.,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 353, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 438, + 367 + ], + "score": 1.0, + "content": "2021), the lower bound problem in Proposition 1 can be reduced to lower bound", + "type": "text" + }, + { + "bbox": [ + 439, + 355, + 456, + 367 + ], + "score": 0.9, + "content": "| \\mathcal { T } _ { L } |", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 353, + 506, + 367 + ], + "score": 1.0, + "content": ", which can", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 366, + 504, + 385 + ], + "spans": [ + { + "bbox": [ + 104, + 370, + 290, + 385 + ], + "score": 1.0, + "content": "be further reduced to lower bound the subset", + "type": "text" + }, + { + "bbox": [ + 290, + 371, + 327, + 384 + ], + "score": 0.95, + "content": "| \\mathcal { T } _ { L , \\mathcal { X } , m } |", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 370, + 391, + 385 + ], + "score": 1.0, + "content": ". We now show", + "type": "text" + }, + { + "bbox": [ + 391, + 366, + 504, + 385 + ], + "score": 0.93, + "content": "\\begin{array} { r } { | \\mathcal { T } _ { L , \\mathcal { X } , m } | \\geq \\binom { | \\mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 382, + 155, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 155, + 396 + ], + "score": 1.0, + "content": "inductively.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 399, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 134, + 414 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 134, + 401, + 164, + 411 + ], + "score": 0.9, + "content": "L = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 398, + 294, + 414 + ], + "score": 1.0, + "content": ", the root of the tree can have", + "type": "text" + }, + { + "bbox": [ + 294, + 400, + 309, + 412 + ], + "score": 0.9, + "content": "| \\mathcal { X } |", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 398, + 505, + 414 + ], + "score": 1.0, + "content": "different choices. For the children nodes, we", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 127, + 411, + 507, + 447 + ], + "spans": [ + { + "bbox": [ + 127, + 415, + 137, + 424 + ], + "score": 0.73, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 414, + 212, + 425 + ], + "score": 0.88, + "content": "| \\mathcal { X } |", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 411, + 387, + 447 + ], + "score": 1.0, + "content": "are allowed. This leads to .", + "type": "text" + }, + { + "bbox": [ + 387, + 412, + 430, + 427 + ], + "score": 0.91, + "content": "\\binom { | x | + m - 1 } { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 411, + 507, + 447 + ], + "score": 1.0, + "content": "cases. Therefore,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 425, + 277, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 277, + 442 + ], + "score": 0.87, + "content": "\\mathcal { T } _ { L + 1 , \\mathcal { X } , m } = | \\mathcal { X } | \\binom { | \\mathcal { X } | + m - 1 } { m } \\ge \\binom { | \\mathcal { X } | + m - 2 } { m - 1 }", + "type": "inline_equation" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 504, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 243, + 460 + ], + "score": 1.0, + "content": "Assuming the statement holds for", + "type": "text" + }, + { + "bbox": [ + 244, + 446, + 252, + 456 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 443, + 339, + 460 + ], + "score": 1.0, + "content": ", we show it holds for", + "type": "text" + }, + { + "bbox": [ + 340, + 446, + 365, + 456 + ], + "score": 0.9, + "content": "L + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 443, + 462, + 460 + ], + "score": 1.0, + "content": "by constructing trees in", + "type": "text" + }, + { + "bbox": [ + 462, + 446, + 504, + 458 + ], + "score": 0.91, + "content": "\\tau _ { L + 1 , x , m }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 129, + 470 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 129, + 457, + 195, + 468 + ], + "score": 0.9, + "content": "T , T ^ { \\prime } \\in \\mathcal { T } _ { L , x , m }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 456, + 369, + 470 + ], + "score": 1.0, + "content": ". 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First note that when", + "type": "text" + }, + { + "bbox": [ + 262, + 468, + 271, + 478 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 466, + 288, + 480 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 289, + 468, + 300, + 478 + ], + "score": 0.87, + "content": "T ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 466, + 464, + 480 + ], + "score": 1.0, + "content": "are two non-isomorphic trees, two depth-", + "type": "text" + }, + { + "bbox": [ + 465, + 468, + 483, + 478 + ], + "score": 0.81, + "content": "\\mathrm { . L } { + } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 466, + 505, + 480 + ], + "score": 1.0, + "content": "trees", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 477, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 177, + 493 + ], + "score": 1.0, + "content": "constructed from", + "type": "text" + }, + { + "bbox": [ + 177, + 479, + 186, + 489 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 477, + 204, + 493 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 204, + 479, + 215, + 489 + ], + "score": 0.86, + "content": "T ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 477, + 505, + 493 + ], + "score": 1.0, + "content": "will be different no matter how the node features are assigned. Now we", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 490, + 497, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 289, + 502 + ], + "score": 1.0, + "content": "consider all the trees can be constructed from", + "type": "text" + }, + { + "bbox": [ + 290, + 490, + 298, + 500 + ], + "score": 0.78, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 490, + 497, + 502 + ], + "score": 1.0, + "content": "by assign node features of children to leaf nodes.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 318, + 519 + ], + "score": 1.0, + "content": "We first consider all paths from the root to leaves in", + "type": "text" + }, + { + "bbox": [ + 319, + 507, + 327, + 516 + ], + "score": 0.74, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 506, + 505, + 519 + ], + "score": 1.0, + "content": ". Each path consists of a sequence of nodes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 345, + 531 + ], + "score": 1.0, + "content": "where the node features form a one-to-one mapping to an", + "type": "text" + }, + { + "bbox": [ + 346, + 518, + 354, + 528 + ], + "score": 0.26, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 517, + 379, + 531 + ], + "score": 1.0, + "content": "-tuple", + "type": "text" + }, + { + "bbox": [ + 379, + 518, + 503, + 530 + ], + "score": 0.91, + "content": "\\tau \\in \\{ ( x _ { 1 } , \\ldots , x _ { L } ) : x _ { i } \\in \\mathcal { X } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 517, + 506, + 531 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 243, + 541 + ], + "score": 1.0, + "content": "Leaf nodes are called node under", + "type": "text" + }, + { + "bbox": [ + 243, + 531, + 250, + 538 + ], + "score": 0.75, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 528, + 429, + 541 + ], + "score": 1.0, + "content": "if the path from the root to it corresponds to", + "type": "text" + }, + { + "bbox": [ + 429, + 531, + 436, + 538 + ], + "score": 0.77, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 528, + 506, + 541 + ], + "score": 1.0, + "content": ". The children of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 192, + 552 + ], + "score": 1.0, + "content": "nodes under different", + "type": "text" + }, + { + "bbox": [ + 193, + 541, + 204, + 550 + ], + "score": 0.72, + "content": "\\tau \\mathbf { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "are always distinguishable, and thus any assignments lead to distinct rooted", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 210, + 563 + ], + "score": 1.0, + "content": "aggregation trees of depth", + "type": "text" + }, + { + "bbox": [ + 210, + 551, + 235, + 561 + ], + "score": 0.9, + "content": "L + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 550, + 446, + 563 + ], + "score": 1.0, + "content": ". 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Therefore, to lower bound", + "type": "text" + }, + { + "bbox": [ + 325, + 561, + 366, + 573 + ], + "score": 0.92, + "content": "\\tau _ { L + 1 , \\mathcal { X } , m }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 561, + 506, + 574 + ], + "score": 1.0, + "content": ", we only consider a special way of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 573, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 446, + 585 + ], + "score": 1.0, + "content": "assignments to avoid over counting, which is that children of all nodes under the same", + "type": "text" + }, + { + "bbox": [ + 446, + 575, + 453, + 582 + ], + "score": 0.77, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 573, + 506, + 585 + ], + "score": 1.0, + "content": "are assigned", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 583, + 205, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 205, + 595 + ], + "score": 1.0, + "content": "the same set of features.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 599, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 106, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 284, + 612 + ], + "score": 1.0, + "content": "Since we assumed that at least two nodes of", + "type": "text" + }, + { + "bbox": [ + 284, + 601, + 292, + 610 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 599, + 455, + 612 + ], + "score": 1.0, + "content": "have different features, there are at least", + "type": "text" + }, + { + "bbox": [ + 455, + 599, + 467, + 610 + ], + "score": 0.87, + "content": "2 ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "different", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 107, + 613, + 118, + 622 + ], + "score": 0.6, + "content": "\\tau \\mathbf { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 611, + 389, + 623 + ], + "score": 1.0, + "content": "corresponding to the path from the root to leaves. 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Then through this construction, the total number of depth-", + "type": "text" + }, + { + "bbox": [ + 465, + 646, + 483, + 656 + ], + "score": 0.83, + "content": "\\mathrm { . L } { + } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 641, + 506, + 660 + ], + "score": 1.0, + "content": "trees", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 102, + 657, + 504, + 705 + ], + "spans": [ + { + "bbox": [ + 102, + 657, + 307, + 681 + ], + "score": 1.0, + "content": "from T can be lower bounded by \u0000|X |+m−2m−1 \u00012L.", + "type": "text" + }, + { + "bbox": [ + 106, + 684, + 157, + 696 + ], + "score": 0.89, + "content": "T \\in \\mathcal { T } _ { L , \\mathcal { X } , m }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 663, + 216, + 705 + ], + "score": 1.0, + "content": ", so we derive", + "type": "text" + }, + { + "bbox": [ + 216, + 678, + 354, + 698 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathcal { T } _ { L + 1 , \\mathcal { X } , m } \\geq \\binom { | \\mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } } \\mathcal { T } _ { L , \\mathcal { X } , m } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 663, + 375, + 705 + ], + "score": 1.0, + "content": "hav, and", + "type": "text" + }, + { + "bbox": [ + 375, + 677, + 504, + 698 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathcal { T } _ { L , \\mathcal { X } , m } \\geq \\binom { | \\mathcal { X } | + m - 2 } { m - 1 } ^ { \\sum _ { l = 1 } ^ { L } 2 ^ { l } } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 696, + 172, + 717 + ], + "spans": [ + { + "bbox": [ + 107, + 696, + 172, + 717 + ], + "score": 0.87, + "content": "\\binom { | \\mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 }", + "type": "inline_equation" + } + ], + "index": 41 + } + ], + "index": 38 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 721, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 496, + 723, + 504, + 730 + ], + "spans": [ + { + "bbox": [ + 496, + 723, + 504, + 730 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 284, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 286, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 286, + 95 + ], + "score": 1.0, + "content": "E PROOF OF THE PROPOSITION 1", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 114, + 504, + 136 + ], + "lines": [ + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "score": 1.0, + "content": "To prove Proposition 1, we first define rooted aggregation trees, which is similar to but different from", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 124, + 165, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 165, + 137 + ], + "score": 1.0, + "content": "rooted graphs.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 113, + 506, + 137 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 146, + 505, + 191 + ], + "lines": [ + { + "bbox": [ + 104, + 145, + 504, + 159 + ], + "spans": [ + { + "bbox": [ + 104, + 145, + 488, + 159 + ], + "score": 1.0, + "content": "Definition 2 (Rooted Aggregation Tree). The depth-K rooted aggregation tree of a rooted graph", + "type": "text" + }, + { + "bbox": [ + 488, + 145, + 504, + 158 + ], + "score": 0.86, + "content": "\\mathcal { G } ^ { [ i ] }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 156, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 223, + 171 + ], + "score": 1.0, + "content": "is a depth-K rooted tree with", + "type": "text" + }, + { + "bbox": [ + 223, + 159, + 230, + 167 + ], + "score": 0.35, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 156, + 506, + 171 + ], + "score": 1.0, + "content": "(possibly many-to-one) mapping from every node in the tree to some", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 167, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 138, + 182 + ], + "score": 1.0, + "content": "node in", + "type": "text" + }, + { + "bbox": [ + 138, + 168, + 153, + 180 + ], + "score": 0.89, + "content": "\\mathcal { G } ^ { [ i ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 167, + 343, + 182 + ], + "score": 1.0, + "content": ", where (i) the root of the tree is mapped to node", + "type": "text" + }, + { + "bbox": [ + 344, + 170, + 348, + 178 + ], + "score": 0.3, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 167, + 487, + 182 + ], + "score": 1.0, + "content": ", and (ii) the children of every node", + "type": "text" + }, + { + "bbox": [ + 488, + 169, + 493, + 180 + ], + "score": 0.62, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 167, + 506, + 182 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 178, + 418, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 312, + 192 + ], + "score": 1.0, + "content": "the tree are mapped to the neighbors of the node in", + "type": "text" + }, + { + "bbox": [ + 313, + 178, + 327, + 190 + ], + "score": 0.89, + "content": "\\mathcal { G } ^ { [ i ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 178, + 365, + 192 + ], + "score": 1.0, + "content": "to which", + "type": "text" + }, + { + "bbox": [ + 366, + 180, + 372, + 191 + ], + "score": 0.71, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 178, + 418, + 192 + ], + "score": 1.0, + "content": "is mapped.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 145, + 506, + 192 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 205, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 205, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 219 + ], + "score": 1.0, + "content": "A rooted aggregation tree can be obtained by unrolling the neighborhood aggregation steps in the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 216, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 230 + ], + "score": 1.0, + "content": "GNNs. An illustration of rooted graphs and rooted aggregation trees can be found in Chen et al.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 402, + 240 + ], + "score": 1.0, + "content": "(2021) Figure 4. We denote the set of all rooted aggregation trees of depth", + "type": "text" + }, + { + "bbox": [ + 403, + 228, + 411, + 238 + ], + "score": 0.37, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 227, + 435, + 240 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 435, + 228, + 448, + 239 + ], + "score": 0.85, + "content": "\\mathcal { T } _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 227, + 506, + 240 + ], + "score": 1.0, + "content": ". Then we use", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 238, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 107, + 239, + 137, + 251 + ], + "score": 0.91, + "content": "\\mathcal { T } _ { L , \\mathcal { X } , m }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 238, + 221, + 252 + ], + "score": 1.0, + "content": "to denote a subset of", + "type": "text" + }, + { + "bbox": [ + 222, + 239, + 234, + 250 + ], + "score": 0.88, + "content": "\\mathcal { T } _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 238, + 372, + 252 + ], + "score": 1.0, + "content": ", where the node features belong to", + "type": "text" + }, + { + "bbox": [ + 372, + 239, + 382, + 249 + ], + "score": 0.79, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 238, + 506, + 252 + ], + "score": 1.0, + "content": ", and all the nodes have exactly", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 135, + 262 + ], + "score": 1.0, + "content": "degree", + "type": "text" + }, + { + "bbox": [ + 135, + 252, + 145, + 260 + ], + "score": 0.68, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 249, + 150, + 262 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 150, + 252, + 160, + 260 + ], + "score": 0.62, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "children), and at least two nodes out of these m nodes have different features. In other", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "score": 1.0, + "content": "words, a node can’t have all identical children. With rooted aggregation trees defined, we are ready to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 272, + 474, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 474, + 284 + ], + "score": 1.0, + "content": "prove Proposition 1. The proof is adapted from the proof of Lemma 3 in Chen et al. (2021).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 205, + 506, + 284 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 332, + 506, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 332, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 316, + 344 + ], + "score": 1.0, + "content": "Proof. Since the number of equivalence classes on", + "type": "text" + }, + { + "bbox": [ + 316, + 333, + 324, + 343 + ], + "score": 0.79, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 332, + 506, + 344 + ], + "score": 1.0, + "content": "induced by the family of all depth-L GNNs", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 443, + 356 + ], + "score": 1.0, + "content": "consists of all rooted graphs that share the same rooted aggregation tree of depth-", + "type": "text" + }, + { + "bbox": [ + 444, + 344, + 451, + 354 + ], + "score": 0.29, + "content": "\\mathrm { . L }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "(Chen et al.,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 353, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 438, + 367 + ], + "score": 1.0, + "content": "2021), the lower bound problem in Proposition 1 can be reduced to lower bound", + "type": "text" + }, + { + "bbox": [ + 439, + 355, + 456, + 367 + ], + "score": 0.9, + "content": "| \\mathcal { T } _ { L } |", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 353, + 506, + 367 + ], + "score": 1.0, + "content": ", which can", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 366, + 504, + 385 + ], + "spans": [ + { + "bbox": [ + 104, + 370, + 290, + 385 + ], + "score": 1.0, + "content": "be further reduced to lower bound the subset", + "type": "text" + }, + { + "bbox": [ + 290, + 371, + 327, + 384 + ], + "score": 0.95, + "content": "| \\mathcal { T } _ { L , \\mathcal { X } , m } |", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 370, + 391, + 385 + ], + "score": 1.0, + "content": ". We now show", + "type": "text" + }, + { + "bbox": [ + 391, + 366, + 504, + 385 + ], + "score": 0.93, + "content": "\\begin{array} { r } { | \\mathcal { T } _ { L , \\mathcal { X } , m } | \\geq \\binom { | \\mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 382, + 155, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 155, + 396 + ], + "score": 1.0, + "content": "inductively.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 332, + 506, + 396 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 399, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 134, + 414 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 134, + 401, + 164, + 411 + ], + "score": 0.9, + "content": "L = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 398, + 294, + 414 + ], + "score": 1.0, + "content": ", the root of the tree can have", + "type": "text" + }, + { + "bbox": [ + 294, + 400, + 309, + 412 + ], + "score": 0.9, + "content": "| \\mathcal { X } |", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 398, + 505, + 414 + ], + "score": 1.0, + "content": "different choices. For the children nodes, we", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 127, + 411, + 507, + 447 + ], + "spans": [ + { + "bbox": [ + 127, + 415, + 137, + 424 + ], + "score": 0.73, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 414, + 212, + 425 + ], + "score": 0.88, + "content": "| \\mathcal { X } |", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 411, + 387, + 447 + ], + "score": 1.0, + "content": "are allowed. This leads to .", + "type": "text" + }, + { + "bbox": [ + 387, + 412, + 430, + 427 + ], + "score": 0.91, + "content": "\\binom { | x | + m - 1 } { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 411, + 507, + 447 + ], + "score": 1.0, + "content": "cases. Therefore,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 425, + 277, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 277, + 442 + ], + "score": 0.87, + "content": "\\mathcal { T } _ { L + 1 , \\mathcal { X } , m } = | \\mathcal { X } | \\binom { | \\mathcal { X } | + m - 1 } { m } \\ge \\binom { | \\mathcal { X } | + m - 2 } { m - 1 }", + "type": "inline_equation" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 398, + 507, + 447 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 504, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 243, + 460 + ], + "score": 1.0, + "content": "Assuming the statement holds for", + "type": "text" + }, + { + "bbox": [ + 244, + 446, + 252, + 456 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 443, + 339, + 460 + ], + "score": 1.0, + "content": ", we show it holds for", + "type": "text" + }, + { + "bbox": [ + 340, + 446, + 365, + 456 + ], + "score": 0.9, + "content": "L + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 443, + 462, + 460 + ], + "score": 1.0, + "content": "by constructing trees in", + "type": "text" + }, + { + "bbox": [ + 462, + 446, + 504, + 458 + ], + "score": 0.91, + "content": "\\tau _ { L + 1 , x , m }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 129, + 470 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 129, + 457, + 195, + 468 + ], + "score": 0.9, + "content": "T , T ^ { \\prime } \\in \\mathcal { T } _ { L , x , m }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 456, + 369, + 470 + ], + "score": 1.0, + "content": ". 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Some", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 506, + 129 + ], + "score": 1.0, + "content": "other GNNs like GCN are also considered in the ablation studies, but they are not the best known", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "score": 1.0, + "content": "architecture for a specific dataset. To show GLNN has stronger performance given a stronger teacher,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 383, + 151 + ], + "score": 1.0, + "content": "we consider the best teacher we can access on Products. 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We thus consider augment the node features with their one-hop neighbors to include", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 355, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 368 + ], + "score": 1.0, + "content": "more graph information. This can be seen as a middle ground between pure GLNNs and GNNs. For", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "this new experiment, we follow the setting in Table 3 but with two new approaches. We explain the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 378, + 262, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 262, + 390 + ], + "score": 1.0, + "content": "setting of these two approaches below.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 131, + 399, + 504, + 443 + ], + "lines": [ + { + "bbox": [ + 129, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 129, + 398, + 301, + 412 + ], + "score": 1.0, + "content": "1. 1-hop GA-MLP: firstly, for each node", + "type": "text" + }, + { + "bbox": [ + 302, + 402, + 308, + 409 + ], + "score": 0.74, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 398, + 486, + 412 + ], + "score": 1.0, + "content": ", we collect features of its 1-hop neighbors", + "type": "text" + }, + { + "bbox": [ + 486, + 402, + 493, + 409 + ], + "score": 0.73, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 398, + 505, + 412 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 410, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 141, + 410, + 256, + 422 + ], + "score": 1.0, + "content": "augment the raw feature of", + "type": "text" + }, + { + "bbox": [ + 256, + 412, + 262, + 420 + ], + "score": 0.71, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 410, + 283, + 422 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 283, + 411, + 322, + 421 + ], + "score": 0.92, + "content": "x _ { v } \\tilde { x } _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 410, + 506, + 422 + ], + "score": 1.0, + "content": ", like in SGC. 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Then", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 457, + 343, + 470 + ], + "spans": [ + { + "bbox": [ + 141, + 457, + 343, + 470 + ], + "score": 1.0, + "content": "train an MLP with distillation from teacher GNN.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 129, + 473, + 394, + 485 + ], + "lines": [ + { + "bbox": [ + 128, + 472, + 394, + 486 + ], + "spans": [ + { + "bbox": [ + 128, + 472, + 394, + 486 + ], + "score": 1.0, + "content": "3. 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However,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 584, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 506, + 597 + ], + "score": 1.0, + "content": "we indeed see 1-hop GA-GLNN (68.83) can further improve from 1-hop GA-MLP (66.62) and nearly", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 595, + 213, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 213, + 606 + ], + "score": 1.0, + "content": "match the teacher (70.64).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "table", + "bbox": [ + 138, + 630, + 469, + 676 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 146, + 616, + 464, + 628 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 145, + 614, + 465, + 631 + ], + "spans": [ + { + "bbox": [ + 145, + 614, + 465, + 631 + ], + "score": 1.0, + "content": "Table 10: GLNN with feature augmentation from one-hop neighbor on Arxiv", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "table_body", + "bbox": [ + 138, + 630, + 469, + 676 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 138, + 630, + 469, + 676 + ], + "spans": [ + { + "bbox": [ + 138, + 630, + 469, + 676 + ], + "score": 0.972, + "html": "
EvalSAGEMLPGLNN1-hop GA-MLP1-hop GA-GLNN
Arxivind70.6455.4060.4866.6268.83
tran70.7555.2871.4666.6769.82
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This result is practically beneficial, as it gives practitioners", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 721, + 445, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 445, + 733 + ], + "score": 1.0, + "content": "more flexibility about how much accuracy they want to trade for less inference time.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5 + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 81, + 395, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 397, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 397, + 96 + ], + "score": 1.0, + "content": "F ADVANCED GNN ARCHITECTURES AS THE TEACHER", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 505, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 107, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 107, + 505, + 118 + ], + "score": 1.0, + "content": "In our experiment, SAGE teacher is used throughout to avoid influence by model architecture. Some", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 506, + 129 + ], + "score": 1.0, + "content": "other GNNs like GCN are also considered in the ablation studies, but they are not the best known", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "score": 1.0, + "content": "architecture for a specific dataset. To show GLNN has stronger performance given a stronger teacher,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 383, + 151 + ], + "score": 1.0, + "content": "we consider the best teacher we can access on Products. We take", + "type": "text" + }, + { + "bbox": [ + 383, + 139, + 424, + 150 + ], + "score": 0.68, + "content": "\\mathrm { M L P { + } C S }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 139, + 506, + 151 + ], + "score": 1.0, + "content": "Huang et al. (2021)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 408, + 163 + ], + "score": 1.0, + "content": "from the OGB leaderboard as a new teacher, which has reported accuracy", + "type": "text" + }, + { + "bbox": [ + 409, + 150, + 442, + 161 + ], + "score": 0.89, + "content": "8 4 . 1 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 149, + 505, + 163 + ], + "score": 1.0, + "content": "and ranks 8 on", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 284, + 173 + ], + "score": 1.0, + "content": "the leadarboard as of Nov 2021. We choose", + "type": "text" + }, + { + "bbox": [ + 285, + 161, + 325, + 172 + ], + "score": 0.73, + "content": "_ { \\mathrm { M L P + C S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "instead of the other top 7 because the others", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 504, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 473, + 184 + ], + "score": 1.0, + "content": "either rely on raw text (additional info to the given node feature), or require a large GPU with", + "type": "text" + }, + { + "bbox": [ + 473, + 172, + 504, + 183 + ], + "score": 0.81, + "content": "{ > } 1 6 \\mathrm { G B }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 196 + ], + "score": 1.0, + "content": "memory, which we don’t have access to. Also, their improvement is not super significant compared to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 193, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 145, + 204 + ], + "score": 0.77, + "content": "\\mathrm { M L P { + } C S }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 193, + 166, + 206 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 166, + 194, + 186, + 204 + ], + "score": 0.88, + "content": "84 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 193, + 199, + 206 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 199, + 194, + 219, + 204 + ], + "score": 0.88, + "content": "86 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 193, + 285, + 206 + ], + "score": 1.0, + "content": ". The result with", + "type": "text" + }, + { + "bbox": [ + 285, + 194, + 325, + 204 + ], + "score": 0.81, + "content": "\\mathrm { M L P { + } C S }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 193, + 505, + 206 + ], + "score": 1.0, + "content": "teacher is shown in Table 9. We see that with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 239, + 217 + ], + "score": 1.0, + "content": "the new teacher, performance of", + "type": "text" + }, + { + "bbox": [ + 240, + 205, + 275, + 216 + ], + "score": 0.8, + "content": "{ \\mathrm { G L N N } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 205, + 435, + 217 + ], + "score": 1.0, + "content": "improves to be even better than SAGE", + "type": "text" + }, + { + "bbox": [ + 436, + 205, + 473, + 216 + ], + "score": 0.83, + "content": "( 7 8 . 6 1 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 205, + 505, + 217 + ], + "score": 1.0, + "content": ", which", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 329, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 329, + 228 + ], + "score": 1.0, + "content": "shows GLNN can get stronger given a stronger teacher.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 107, + 506, + 228 + ] + }, + { + "type": "table", + "bbox": [ + 225, + 251, + 383, + 287 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 192, + 237, + 418, + 249 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 192, + 236, + 418, + 250 + ], + "spans": [ + { + "bbox": [ + 192, + 236, + 227, + 250 + ], + "score": 1.0, + "content": "Table 9:", + "type": "text" + }, + { + "bbox": [ + 228, + 237, + 263, + 248 + ], + "score": 0.49, + "content": "{ \\mathrm { G L N N } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 236, + 283, + 250 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 284, + 238, + 324, + 248 + ], + "score": 0.83, + "content": "\\mathrm { M L P { + } C S }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 236, + 418, + 250 + ], + "score": 1.0, + "content": "teacher on Products", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "table_body", + "bbox": [ + 225, + 251, + 383, + 287 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 225, + 251, + 383, + 287 + ], + "spans": [ + { + "bbox": [ + 225, + 251, + 383, + 287 + ], + "score": 0.969, + "html": "
MLP+C&SMLP+GLNN+
Acc84.1864.5082.94
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This result is practically beneficial, as it gives practitioners", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 721, + 445, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 445, + 733 + ], + "score": 1.0, + "content": "more flexibility about how much accuracy they want to trade for less inference time.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 687, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 122 + ], + "score": 1.0, + "content": "This section is a continuation of the ablation study of inductive split rate in Section 6. 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(2021) to further evaluate GLNN.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 135, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 106, + 135, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 505, + 147 + ], + "score": 1.0, + "content": "The House_class and VK_class datasets are from Ivanov & Prokhorenkova (2021). The node", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 146, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 505, + 158 + ], + "score": 1.0, + "content": "features of these two graphs are based on tabular data, which have different types, scales, and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 157, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 506, + 168 + ], + "score": 1.0, + "content": "meanings as the opposite of the bag-of-word node features in Cora and etc. Some basic statistics of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 168, + 288, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 288, + 180 + ], + "score": 1.0, + "content": "the datasets are shown in the following table.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "table", + "bbox": [ + 172, + 204, + 434, + 252 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 177, + 191, + 433, + 203 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 177, + 189, + 434, + 204 + ], + "spans": [ + { + "bbox": [ + 177, + 189, + 434, + 204 + ], + "score": 1.0, + "content": "Table 11: Statistics of dataset with heterogeneous node features", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "table_body", + "bbox": [ + 172, + 204, + 434, + 252 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 172, + 204, + 434, + 252 + ], + "spans": [ + { + "bbox": [ + 172, + 204, + 434, + 252 + ], + "score": 0.975, + "html": "
Dataset# Nodes#Edges# Features#Classes
House_class20,640182,14665
VK_class54,028213,644147
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DatasetLightGBMGNNsBGNNMLPGLNN
House_class0.550.6250.6820.5340.672
VK_class0.570.5770.6830.5670.641
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Dataset# Nodes#Edges# Features# Classes
Penn9441,5361,590,65552
Pokec1,632,80330,622,564652
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DatasetLINKGCNMLPGLNN
Penn9480.7982.4773.6181.69
Pokec80.5475.4562.3761.32
", + "type": "table", + "image_path": "6346a480b97623096c4c0a3398fa52071b30e5375568b9c59b4a6757b36cb2ab.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 210, + 676, + 397, + 691.6666666666666 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 210, + 691.6666666666666, + 397, + 707.3333333333333 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 210, + 707.3333333333333, + 397, + 722.9999999999999 + ], + "spans": [], + "index": 42 + } + ] + } + ], + "index": 39.75 + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 105, + 81, + 484, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 81, + 485, + 95 + ], + "spans": [ + { + "bbox": [ + 104, + 81, + 485, + 95 + ], + "score": 1.0, + "content": "I GLNN UNDER NODE FEATURE HETEROGENEITY AND NON-HOMOPHILY", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 504, + 129 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 506, + 119 + ], + "score": 1.0, + "content": "Besides the 7 datasets used in the main experiments, we consider 4 more datasets from Ivanov &", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 390, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 390, + 129 + ], + "score": 1.0, + "content": "Prokhorenkova (2021) and Lim et al. (2021) to further evaluate GLNN.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 106, + 106, + 506, + 129 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 135, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 106, + 135, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 505, + 147 + ], + "score": 1.0, + "content": "The House_class and VK_class datasets are from Ivanov & Prokhorenkova (2021). The node", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 146, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 505, + 158 + ], + "score": 1.0, + "content": "features of these two graphs are based on tabular data, which have different types, scales, and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 157, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 506, + 168 + ], + "score": 1.0, + "content": "meanings as the opposite of the bag-of-word node features in Cora and etc. Some basic statistics of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 168, + 288, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 288, + 180 + ], + "score": 1.0, + "content": "the datasets are shown in the following table.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 135, + 506, + 180 + ] + }, + { + "type": "table", + "bbox": [ + 172, + 204, + 434, + 252 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 177, + 191, + 433, + 203 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 177, + 189, + 434, + 204 + ], + "spans": [ + { + "bbox": [ + 177, + 189, + 434, + 204 + ], + "score": 1.0, + "content": "Table 11: Statistics of dataset with heterogeneous node features", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "table_body", + "bbox": [ + 172, + 204, + 434, + 252 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 172, + 204, + 434, + 252 + ], + "spans": [ + { + "bbox": [ + 172, + 204, + 434, + 252 + ], + "score": 0.975, + "html": "
Dataset# Nodes#Edges# Features#Classes
House_class20,640182,14665
VK_class54,028213,644147
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DatasetLightGBMGNNsBGNNMLPGLNN
House_class0.550.6250.6820.5340.672
VK_class0.570.5770.6830.5670.641
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Dataset# Nodes#Edges# Features# Classes
Penn9441,5361,590,65552
Pokec1,632,80330,622,564652
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DatasetLINKGCNMLPGLNN
Penn9480.7982.4773.6181.69
Pokec80.5475.4562.3761.32
", + "type": "table", + "image_path": "6346a480b97623096c4c0a3398fa52071b30e5375568b9c59b4a6757b36cb2ab.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 210, + 676, + 397, + 691.6666666666666 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 210, + 691.6666666666666, + 397, + 707.3333333333333 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 210, + 707.3333333333333, + 397, + 722.9999999999999 + ], + "spans": [], + "index": 42 + } + ] + } + ], + "index": 39.75 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 393, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 80, + 395, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 80, + 395, + 96 + ], + "score": 1.0, + "content": "J MODEL COMPARISON WITH NOISY NODE FEATURES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 161 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "In Section 6, we conducted an ablation study to compare model performance with noisy node features,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "and the result is shown in the left plot in Figure 5. There are two subtle points in this plot. (1) The", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 414, + 141 + ], + "score": 1.0, + "content": "performance of GNN is still relatively high for high noisy features, even when", + "type": "text" + }, + { + "bbox": [ + 414, + 129, + 440, + 138 + ], + "score": 0.9, + "content": "\\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "and the features", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 139, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 505, + 151 + ], + "score": 1.0, + "content": "are completely random. (2) For completely random features, the performance of GLNN is still higher", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 347, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 347, + 163 + ], + "score": 1.0, + "content": "than MLP. We now discuss and explain them in more detail.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 167, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 106, + 166, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 506, + 180 + ], + "score": 1.0, + "content": "GNN Performance on Random Features. GNN still performs well because nodes with the same", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "labels are likely to be connected and GNN can overfit the training data. We explain the detail through", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 189, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 104, + 189, + 506, + 202 + ], + "score": 1.0, + "content": "a toy example. Suppose there is a 4-clique containing nodes A, B, C, D in the graph with only a single", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 200, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 213 + ], + "score": 1.0, + "content": "edge D-E connects this clique to other graph nodes. Suppose A, B, C, D all have iid random Gaussian", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "raw features and the same class label c. Let’s pick A to be the inductive test node and assume E and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "score": 1.0, + "content": "the triangle formed by B, C, D to be in the training graph. Let’s consider a simple example for 1-layer", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 247 + ], + "score": 1.0, + "content": "GCN and break down message passing into feature aggregation and nonlinear transformation. During", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "score": 1.0, + "content": "training, GNN can overfit the data by learning a nonlinear transformation which maps the aggregated", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "features of B, C, D to class c. The aggregated features of B and C will just be the average of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 506, + 279 + ], + "score": 1.0, + "content": "raw features of B, C, D. Although E is also involved in D’s feature aggregation step, the aggregated", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "score": 1.0, + "content": "features of D will also be very close to this average. Then when test on A, the aggregated feature of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 288, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 506, + 299 + ], + "score": 1.0, + "content": "A will likely be classified to the same class c by the overfitted nonlinear transformation because it is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "score": 1.0, + "content": "the average of raw node features of A, B, C, D. In this case, GNN can actually correctly classify A", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "because of the overfitting. For GNNs with more layers and graphs with more neighbor nodes, the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 321, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 506, + 333 + ], + "score": 1.0, + "content": "conclusion may be generalized.This is roughly sort of a “majority vote” process. For a test node A,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 330, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 104, + 330, + 506, + 346 + ], + "score": 1.0, + "content": "if many nodes, which A collects features from, have the same class label and appear in the training", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 343, + 387, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 387, + 355 + ], + "score": 1.0, + "content": "graph, then A will be classified as this class by an overfitted classifier.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 360, + 506, + 579 + ], + "lines": [ + { + "bbox": [ + 106, + 360, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 371 + ], + "score": 1.0, + "content": "GLNN and MLP Performance on Random Features. The gap between MLP and GLNN is due to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "imbalanced datasets. The GLNN can learn the imbalance from soft labels, whereas MLPs can only", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 104, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "access uniformly picked training nodes. We explain more detail using the A-computer dataset as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 392, + 507, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 507, + 404 + ], + "score": 1.0, + "content": "an example, for which the gap between MLP and GLNN is obvious. The task is 10-class classification.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 403, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 219, + 415 + ], + "score": 1.0, + "content": "With random node features", + "type": "text" + }, + { + "bbox": [ + 219, + 404, + 241, + 414 + ], + "score": 0.77, + "content": "( \\alpha { = } 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 403, + 506, + 415 + ], + "score": 1.0, + "content": "), the inductive accuracy for MLP is 0.0652 and 0.2538 for GLNN.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "If the data labels are uniform, then both models should give an accuracy around 0.1. However, the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "labels on the inductive dataset are actually imbalanced. We show the results in Figure 9. The hist", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 436, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 483, + 448 + ], + "score": 1.0, + "content": "on the left is the label distribution of the inductive test set. In particular, class 4 takes about", + "type": "text" + }, + { + "bbox": [ + 483, + 436, + 503, + 447 + ], + "score": 0.87, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 436, + 506, + 448 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 445, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 104, + 445, + 506, + 461 + ], + "score": 1.0, + "content": "However, given this imbalance, the standard train-test split selects training nodes uniformly among", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "labels. In this case, 20 nodes per class. Therefore, the predictions of MLP on random features are", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "expected to be relatively uniform because the 200 nodes we train it on are uniform. This gives the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 351, + 492 + ], + "score": 1.0, + "content": "hist shown in the middle, where the largest class takes about", + "type": "text" + }, + { + "bbox": [ + 351, + 480, + 378, + 491 + ], + "score": 0.88, + "content": "1 7 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 479, + 506, + 492 + ], + "score": 1.0, + "content": ". Finally, for GLNN, we train it", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 491, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 503 + ], + "score": 1.0, + "content": "on all the 200 training nodes with hard labels, plus soft labels of other nodes in the observed graph", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 501, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 126, + 513 + ], + "score": 0.9, + "content": "\\mathcal { G } _ { o b s }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 501, + 506, + 514 + ], + "score": 1.0, + "content": "(see Section 5.2). Since these extra nodes are selected randomly, whose label distribution is", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 513, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 524 + ], + "score": 1.0, + "content": "actually similar to the label distribution on the whole data and the distribution on the inductive test", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 524, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 536 + ], + "score": 1.0, + "content": "set. Therefore, we get the GLNN predictions hist on the right. Although for each node, we can’t", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 535, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 547 + ], + "score": 1.0, + "content": "assign a prediction correlated to its feature, on average the distribution is very close to the true label", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "distribution on the inductive test set and has a much higher expectation. In fact, if the prediction", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "distribution is exactly the true distribution on the inductive test set, the expectation will be 0.2169.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 567, + 431, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 431, + 580 + ], + "score": 1.0, + "content": "GLNN actually does even a bit better by putting its bet more on the largest class.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 32.5 + }, + { + "type": "image", + "bbox": [ + 127, + 600, + 481, + 692 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 127, + 600, + 481, + 692 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 600, + 481, + 692 + ], + "spans": [ + { + "bbox": [ + 127, + 600, + 481, + 692 + ], + "score": 0.968, + "type": "image", + "image_path": "403a9c5a1b32a03dcf520caf81b8a2a56b3a306859c74e50c516bf26d61ec307.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 127, + 600, + 481, + 630.6666666666666 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 127, + 630.6666666666666, + 481, + 661.3333333333333 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 127, + 661.3333333333333, + 481, + 691.9999999999999 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 699, + 505, + 723 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "Figure 9: Inductive (predicted) label distribution on the A-computer dataset. Left: true labels.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 711, + 383, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 383, + 724 + ], + "score": 1.0, + "content": "Middle: predicted labels by MLP. Right: predicted labels by GLNN.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + } + ], + "index": 45.25 + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 393, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 80, + 395, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 80, + 395, + 96 + ], + "score": 1.0, + "content": "J MODEL COMPARISON WITH NOISY NODE FEATURES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 161 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "In Section 6, we conducted an ablation study to compare model performance with noisy node features,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "and the result is shown in the left plot in Figure 5. There are two subtle points in this plot. (1) The", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 414, + 141 + ], + "score": 1.0, + "content": "performance of GNN is still relatively high for high noisy features, even when", + "type": "text" + }, + { + "bbox": [ + 414, + 129, + 440, + 138 + ], + "score": 0.9, + "content": "\\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "and the features", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 139, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 505, + 151 + ], + "score": 1.0, + "content": "are completely random. (2) For completely random features, the performance of GLNN is still higher", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 347, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 347, + 163 + ], + "score": 1.0, + "content": "than MLP. We now discuss and explain them in more detail.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 106, + 506, + 163 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 167, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 106, + 166, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 506, + 180 + ], + "score": 1.0, + "content": "GNN Performance on Random Features. GNN still performs well because nodes with the same", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "labels are likely to be connected and GNN can overfit the training data. We explain the detail through", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 189, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 104, + 189, + 506, + 202 + ], + "score": 1.0, + "content": "a toy example. Suppose there is a 4-clique containing nodes A, B, C, D in the graph with only a single", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 200, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 213 + ], + "score": 1.0, + "content": "edge D-E connects this clique to other graph nodes. Suppose A, B, C, D all have iid random Gaussian", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "raw features and the same class label c. Let’s pick A to be the inductive test node and assume E and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "score": 1.0, + "content": "the triangle formed by B, C, D to be in the training graph. Let’s consider a simple example for 1-layer", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 247 + ], + "score": 1.0, + "content": "GCN and break down message passing into feature aggregation and nonlinear transformation. During", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "score": 1.0, + "content": "training, GNN can overfit the data by learning a nonlinear transformation which maps the aggregated", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "features of B, C, D to class c. The aggregated features of B and C will just be the average of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 506, + 279 + ], + "score": 1.0, + "content": "raw features of B, C, D. Although E is also involved in D’s feature aggregation step, the aggregated", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "score": 1.0, + "content": "features of D will also be very close to this average. Then when test on A, the aggregated feature of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 288, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 506, + 299 + ], + "score": 1.0, + "content": "A will likely be classified to the same class c by the overfitted nonlinear transformation because it is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "score": 1.0, + "content": "the average of raw node features of A, B, C, D. In this case, GNN can actually correctly classify A", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "because of the overfitting. For GNNs with more layers and graphs with more neighbor nodes, the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 321, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 506, + 333 + ], + "score": 1.0, + "content": "conclusion may be generalized.This is roughly sort of a “majority vote” process. For a test node A,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 330, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 104, + 330, + 506, + 346 + ], + "score": 1.0, + "content": "if many nodes, which A collects features from, have the same class label and appear in the training", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 343, + 387, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 387, + 355 + ], + "score": 1.0, + "content": "graph, then A will be classified as this class by an overfitted classifier.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 166, + 506, + 355 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 360, + 506, + 579 + ], + "lines": [ + { + "bbox": [ + 106, + 360, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 371 + ], + "score": 1.0, + "content": "GLNN and MLP Performance on Random Features. The gap between MLP and GLNN is due to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "imbalanced datasets. The GLNN can learn the imbalance from soft labels, whereas MLPs can only", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 104, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "access uniformly picked training nodes. We explain more detail using the A-computer dataset as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 392, + 507, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 507, + 404 + ], + "score": 1.0, + "content": "an example, for which the gap between MLP and GLNN is obvious. The task is 10-class classification.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 403, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 219, + 415 + ], + "score": 1.0, + "content": "With random node features", + "type": "text" + }, + { + "bbox": [ + 219, + 404, + 241, + 414 + ], + "score": 0.77, + "content": "( \\alpha { = } 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 403, + 506, + 415 + ], + "score": 1.0, + "content": "), the inductive accuracy for MLP is 0.0652 and 0.2538 for GLNN.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "If the data labels are uniform, then both models should give an accuracy around 0.1. However, the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "labels on the inductive dataset are actually imbalanced. We show the results in Figure 9. The hist", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 436, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 483, + 448 + ], + "score": 1.0, + "content": "on the left is the label distribution of the inductive test set. In particular, class 4 takes about", + "type": "text" + }, + { + "bbox": [ + 483, + 436, + 503, + 447 + ], + "score": 0.87, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 436, + 506, + 448 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 445, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 104, + 445, + 506, + 461 + ], + "score": 1.0, + "content": "However, given this imbalance, the standard train-test split selects training nodes uniformly among", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "labels. In this case, 20 nodes per class. Therefore, the predictions of MLP on random features are", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "expected to be relatively uniform because the 200 nodes we train it on are uniform. This gives the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 351, + 492 + ], + "score": 1.0, + "content": "hist shown in the middle, where the largest class takes about", + "type": "text" + }, + { + "bbox": [ + 351, + 480, + 378, + 491 + ], + "score": 0.88, + "content": "1 7 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 479, + 506, + 492 + ], + "score": 1.0, + "content": ". Finally, for GLNN, we train it", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 491, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 503 + ], + "score": 1.0, + "content": "on all the 200 training nodes with hard labels, plus soft labels of other nodes in the observed graph", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 501, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 126, + 513 + ], + "score": 0.9, + "content": "\\mathcal { G } _ { o b s }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 501, + 506, + 514 + ], + "score": 1.0, + "content": "(see Section 5.2). Since these extra nodes are selected randomly, whose label distribution is", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 513, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 524 + ], + "score": 1.0, + "content": "actually similar to the label distribution on the whole data and the distribution on the inductive test", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 524, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 536 + ], + "score": 1.0, + "content": "set. Therefore, we get the GLNN predictions hist on the right. Although for each node, we can’t", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 535, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 547 + ], + "score": 1.0, + "content": "assign a prediction correlated to its feature, on average the distribution is very close to the true label", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "distribution on the inductive test set and has a much higher expectation. In fact, if the prediction", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "distribution is exactly the true distribution on the inductive test set, the expectation will be 0.2169.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 567, + 431, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 431, + 580 + ], + "score": 1.0, + "content": "GLNN actually does even a bit better by putting its bet more on the largest class.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 360, + 507, + 580 + ] + }, + { + "type": "image", + "bbox": [ + 127, + 600, + 481, + 692 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 127, + 600, + 481, + 692 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 600, + 481, + 692 + ], + "spans": [ + { + "bbox": [ + 127, + 600, + 481, + 692 + ], + "score": 0.968, + "type": "image", + "image_path": "403a9c5a1b32a03dcf520caf81b8a2a56b3a306859c74e50c516bf26d61ec307.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 127, + 600, + 481, + 630.6666666666666 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 127, + 630.6666666666666, + 481, + 661.3333333333333 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 127, + 661.3333333333333, + 481, + 691.9999999999999 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 699, + 505, + 723 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "Figure 9: Inductive (predicted) label distribution on the A-computer dataset. Left: true labels.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 711, + 383, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 383, + 724 + ], + "score": 1.0, + "content": "Middle: predicted labels by MLP. 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DatasetsSAGEMLPGLNN△MLP△GNN
Cora80.52 ± 1.7759.22 ±1.3180.54 ± 1.3521.32 (36.00%)0.02 (0.02%)
Citeseer70.33 ± 1.9759.61 ± 2.8871.77 ± 2.0112.16 (20.40%)1.44 (2.05%)
Pubmed75.39 ± 2.0967.55 ± 2.3175.42 ± 2.317.87 (11.65%)0.03 (0.04%)
A-computer82.97 ± 2.1667.80 ± 1.0683.03 ± 1.8715.23 (22.46%)0.06 (0.07%)
A-photo90.90 ± 0.8478.77 ± 1.7492.11 ± 1.0813.34 (16.94%)1.21 (1.33%)
Arxiv70.92 ± 0.1756.05 ± 0.4663.46 ± 0.457.41 (13.24%)-7.46 (-10.52%)
Products78.61 ± 0.4962.47 ± 0.1068.86 ± 0.466.39 (10.23%)-9.75 (-12.4%)
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DatasetsSAGEMLP+GLNN+△MLP△GNN
Arxiv70.92 ± 0.1755.31± 0.4772.15 ± 0.2716.85 (30.46%)0.51 (0.71%)
Products78.61 ± 0.4964.50 ± 0.4577.65 ± 0.4813.14 (20.38%)-0.97 (-1.23%)
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DatasetsEvalSAGEMLP/MLP+GLNN/GLNN+△MLP△GNN
Coraprod79.2958.9878.2819.30 (32.72%)-1.01 (-1.28%)
ind81.33 ± 2.1959.09 ± 2.9673.82 ± 1.9314.73 (24.93%)-7.51 (-9.23%)
tran78.78 ± 1.9258.95 ± 1.6679.39 ± 1.6420.44 (34.66%)0.61 (0.77%)
Citeseerprod68.3859.8169.279.46 (15.82%)0.89 (1.30%)
ind69.75 ± 3.5960.06 ± 5.0069.25 ± 2.259.19 (15.30%)-0.5 (-0.7%)
tran68.04 ± 3.3459.75 ± 2.4869.28 ± 3.129.63 (15.93%)1.24 (1.82%)
Pubmedprod74.8866.8074.717.91 (11.83%)-0.17 (-0.22%)
ind75.26 ± 2.5766.85 ± 2.9674.30 ± 2.617.45 (11.83%)-0.96 (-1.27%)
tran74.78 ± 2.2266.79 ± 2.9074.81 ± 2.398.02 (12.01%)0.03 (0.04%)
A-computerprod82.1467.3882.2914.90 (22.12%)0.15 (0.19%)
ind82.08 ± 1.7967.84 ± 1.7880.92 ± 1.3613.08 (19.28%)-1.16 (-1.41%)
tran82.15 ± 1.5567.27 ± 1.3682.63 ± 1.4015.36 (22.79%)0.48 (0.58%)
A-photoprod91.0879.2592.3813.13 (16.57%)1.30 (1.42%)
ind91.50 ± 0.7979.44 ± 1.7291.18 ± 0.8111.74 (14.78%)-0.32 (-0.35%)
tran90.80 ± 0.7779.20 ± 1.6492.68 ± 0.5613.48 (17.01%)1.70 (1.87%)
Arxivprod70.7355.3065.099.79 (17.70%)-5.64 (-7.97%)
ind70.64 ± 0.6755.40 ± 0.5660.48 ± 0.464.3 (7.76%)-10.94 (-15.49%)
tran70.75± 0.2755.28 ± 0.4971.46 ± 0.3311.16 (20.18%)-4.31 (-6.09%)
Productsprod76.6063.7275.7712.05 (18.91%)-0.83 (-1.09%)
ind76.89 ± 0.5363.70 ± 0.6675.16 ± 0.3411.44 (17.96%)-1.73 (-2.25%)
tran76.53 ±0.5563.73 ± 0.6975.92 ± 0.6112.20 (19.15%)-0.61 (-0.79%)
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Dataset#Nodes#Edges#Features# Classes
Cora2,4855,0691,4337
Citeseer2,1103,6683,7036
Pubmed19,71744,3245003
A-computer13,381245,77876710
A-photo7,487119,0437458
Arxiv169,3431,166,24312840
Products2,449,02961,859,14010047
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SAGEGCNGATAPPNP
#layers2222
hidden dim128646464
learning rate0.010.010.010.01
weight decay0.00050.0010.010.01
dropout00.80.60.5
fan out5,511-
attention heads1=81
power iterations1=110
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DatasetArxivProducts
# layers33
hidden dim256256
learning rate0.010.003
weight decay00
dropout0.20.5
normalizationbatchbatch
fan out[5,10,15][5,10,15]
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Pubmed0.96050.94550.9701
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Dataset# Nodes#Edges# Features#Classes
House_class20,640182,14665
VK_class54,028213,644147
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Dataset# Nodes#Edges# Features# Classes
Penn9441,5361,590,65552
Pokec1,632,80330,622,564652
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DatasetLightGBMGNNsBGNNMLPGLNN
House_class0.550.6250.6820.5340.672
VK_class0.570.5770.6830.5670.641
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DatasetLINKGCNMLPGLNN
Penn9480.7982.4773.6181.69
Pokec80.5475.4562.3761.32
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{ + "page_no": 20, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/dev/7r6kDq0mK_/7r6kDq0mK_.md b/parse/dev/7r6kDq0mK_/7r6kDq0mK_.md new file mode 100644 index 0000000000000000000000000000000000000000..5048b234a2e4badd405486a45724965ebcfb5493 --- /dev/null +++ b/parse/dev/7r6kDq0mK_/7r6kDq0mK_.md @@ -0,0 +1,396 @@ +# LATENT IMAGE ANIMATOR: LEARNING TO ANIMATE IMAGES VIA LATENT SPACE NAVIGATION + +Yaohui Wang, Di Yang, Francois Bremond & Antitza Dantcheva Inria, Universite C´ ote d’Azur ˆ {yaohui.wang,di.yang,francois.bremond,antitza.dantcheva}@inria.fr + +# ABSTRACT + +Due to the remarkable progress of deep generative models, animating images has become increasingly efficient, whereas associated results have become increasingly realistic. Current animation-approaches commonly exploit structure representation extracted from driving videos. Such structure representation is instrumental in transferring motion from driving videos to still images. However, such approaches fail in case the source image and driving video encompass large appearance variation. Moreover, the extraction of structure information requires additional modules that endow the animation-model with increased complexity. Deviating from such models, we here introduce the Latent Image Animator (LIA), a self-supervised autoencoder that evades need for structure representation. LIA is streamlined to animate images by linear navigation in the latent space. Specifically, motion in generated video is constructed by linear displacement of codes in the latent space. Towards this, we learn a set of orthogonal motion directions simultaneously, and use their linear combination, in order to represent any displacement in the latent space. Extensive quantitative and qualitative analysis suggests that our model systematically and significantly outperforms state-of-art methods on VoxCeleb, Taichi and TED-talk datasets w.r.t. generated quality. Source code and pre-trained models are publicly available1. + +![](images/e378522fe7bd0a0020ce31d5201a8ccda8a4e2a620417861db6f080cf4f0c4a2.jpg) +Figure 1: LIA animation examples. The two images of Marilyn Monroe and Emmanuel Macron are animated by LIA, which transfers motion of a driving video (smaller images on the top) from VoxCeleb dataset (Chung et al., 2018) onto the still images. LIA is able to successfully animate these two images without relying on any explicit structure representations, such as landmarks and region representations. + +# 1 INTRODUCTION + +In the series of science fiction books Harry Potter (Rowling et al., 2016; Rowling, 2019), wizards and witches were able to magically enchant portraits, bringing them to life. Remarkable progress of deep generative models has recently turned this vision into reality. This work examines the scenario where a framework animates a source image by motion representations learned from a driving video. Existing approaches for image animation are classically related to computer graphics (Cao et al., 2014; Thies et al., 2016; 2019; Zhao et al., 2018) or exploit motion labels (Wang et al., 2020b) and structure representations such as semantic maps (Pan et al., 2019; Wang et al., 2018; 2019), human keypoints (Jang et al., 2018; Yang et al., 2018; Walker et al., 2017; Chan et al., 2019; Zakharov et al., 2019; Wang et al., 2019; Siarohin et al., 2019), 3D meshes (Liu et al., 2019; Chen et al., 2021), and optical flows (Li et al., 2018; Ohnishi et al., 2018). We note that the ground truth of such structure representations has been computed a-priori for the purpose of supervised training, which poses constraints on applications, where such representations of unseen testing images might be fragmentary or difficult to access. + +Self-supervised motion transfer approaches (Wiles et al., 2018; Siarohin et al., 2019; 2021) accept raw videos as input and learn to reconstruct driving images by warping source image with predicted dense optical flow fields. While the need for domain knowledge or labeled ground truth data has been obviated, which improves performance on in-the-wild testing images, such methods entail necessity of explicit structure representations as motion guidance. Prior information such as keypoints (Siarohin et al., 2019; Wang et al., 2021a) or regions (Siarohin et al., 2021) are learned in an end-to-end training manner by additional networks as intermediate features, in order to predict target flow fields. Although online prediction of such representations is less tedious than the acquisition of ground truth labels, it still strains the complexity of networks. + +Deviating from such approaches, we here aim to fully eliminate the need of explicit structure representations by directly manipulating the latent space of a deep generative model. To the best of our knowledge, this constitutes a new direction in the context of image animation. Our work is motivated by interpretation of GANs (Shen et al., 2020; Goetschalckx et al., 2019; Jahanian et al., 2020; Voynov & Babenko, 2020), showcasing that latent spaces of StyleGAN (Karras et al., 2019; 2020b) and BigGAN (Brock et al., 2019) contain rich semantically meaningful directions. Given that walking along such directions, basic visual transformations such as zooming and rotation can be induced in generated results. As in image animation, we have that motion between source and driving images can be considered as higher-level transformation, a natural question here arises: can we discover a set of directions in the latent space that induces high-level motion transformations collaboratively? + +Towards answering this question, we introduce LIA, a novel Latent Image Animator constituting of an autoencoder for animating still images via latent space navigation. LIA seeks to animate a source image via linearly navigating associated source latent code along a learned path to reach a target latent code, which represents the high-level transformation for animating the source image. We introduce a Linear Motion Decomposition (LMD) approach aiming to represent a latent path via a linear combination of a set of learned motion directions and associated magnitudes. Specifically, we constrain the set as an orthogonal basis, where each vector indicates a basic visual transformation. By describing the whole motion space using such learned basis, LIA eliminates the requirement of explicit structure representations. + +In addition, we design LIA to disentangle motion and appearance within a single encoder-generator architecture. Deviating from existing methods using separate networks to learn disentangled features, LIA integrates both, latent motion code, as well as appearance features in a single encoder, which highly reduces the model complexity and simplifies training. + +We provide evaluation on multiple datasets including VoxCeleb (Chung et al., 2018), TaichiHD (Siarohin et al., 2019) and TED-talk (Siarohin et al., 2021). In addition, we show that LIA outperforms the state-of-the-art in preserving the facial structure in generated videos in the setting of one-shot image animation on unseen datasets such as FFHQ (Karras et al., 2019) and GermanPublicTV (Thies et al., 2020). + +# 2 RELATED WORK + +Video generation GAN-based video generation is aimed at mapping Gaussian noise to video, directly and in the absence of prior information (Vondrick et al., 2016; Saito et al., 2017; Tulyakov et al., 2018; Wang et al., 2020a; Wang, 2021). Approaches based on deep probabilistic models (Denton & Birodkar, 2017; Li & Mandt, 2018; Bhagat et al., 2020; Xie et al., 2020) were also proposed to tackle this problem, however only show results on toy datasets with low resolution. Recently, with the progress of GANs in photo-realistic image generation (Brock et al., 2019; Karras et al., 2019; 2020a), a series of works (Clark et al., 2019; Wang et al., 2021c) explored production of highresolution videos by incorporating the architecture of an image generator into video GANs, trained jointly with RNNs. Tian et al. (2021) directly leveraged the knowledge of a pre-trained StyleGAN to produce videos of resolution up to $1 0 2 4 \times \mathrm { i } 0 2 4$ . Unlike these approaches, which generate random videos based on noise vectors in an unconditional manner, in this paper, we focus on conditionally creating novel videos by transferring motion from driving videos to input images. + +![](images/adf402bc242a87960126961b2dd01a4fedaa5916e13c5407ab9a2e27d13beee5.jpg) +Figure 2: General pipeline. Our objective is to transfer motion via latent space navigation. The entire training pipeline consists of two steps. Firstly, we encode a source image $x _ { s }$ into a latent code $z _ { s r }$ . By linearly navigating $z _ { s r }$ along a path $w _ { r \to d }$ , we reach a target latent code $z _ { s d }$ . The latent paths are represented by a linear combination between a set of learned motion directions (e.g., $d _ { 1 }$ and $d _ { 2 }$ ), which is an orthogonal basis, and associated magnitudes. In the second step, we decode $z _ { s d }$ to a target dense optical flow field $\phi _ { s d }$ , which is used to warp $x _ { s }$ into the driving image $x _ { d }$ . While we train our model using images from the same video sequence, in the testing phase, $x _ { s }$ and $x _ { d }$ generally pertain to different identities. + +Latent space editing In an effort to control generated images, recent works explored the discovery of semantically meaningful directions in the latent space of pre-trained GANs, where linear navigation corresponds to desired image manipulation. Supervised (Shen et al., 2020; Jahanian et al., 2020; Goetschalckx et al., 2019) and unsupervised (Voynov & Babenko, 2020; Peebles et al., 2020; Shen & Zhou, 2021) approaches were proposed to edit semantics such as facial attributes, colors and basic visual transformations (e.g., rotation and zooming) in generated or inverted real images (Zhu et al., 2020; Abdal et al., 2020). In this work, as opposed to finding directions corresponding to individual visual transformations, we seek to learn a set of directions that cooperatively allows for high-level visual transformations that can be beneficial in image animation. + +Image animation Related approaches (Chan et al., 2019; Wang et al., 2018; Zakharov et al., 2019; Wang et al., 2019; Yang et al., 2020) in image animation required strong prior structure labels as motion guidance. In particular, Chan et al. (2019), Yang et al. (2020) and Wang et al. (2018) proposed to map representations such as human keypoints and facial landmarks to videos in the setting of image-to-image translation proposed by Isola et al. (2017). However, such approaches were only able to learn an individual model for a single identity. Transferring motion on new appearances requires retraining the entire model from scratch by using videos of target identities. Several recent works (Zakharov et al., 2019; Wang et al., 2019) explored meta learning in fine-tuning models on target identities. While only few images of target identities were required during inference time, it was still compulsory to input pre-computed structure representations in those approaches, which usually are hard to access in many real-world scenarios. Towards addressing this issue, very recent works (Siarohin et al., 2019; 2021; Wang et al., 2021b; Wiles et al., 2018) proposed to learn image animation in self-supervised manner, only relying on RGB videos for both, training and testing without any priors. They firstly predicted dense flow fields from input images, which were then utilized to warp source images, in order to obtain final generated results. Inference only required one image of a target identity without any fine-tuning step on pre-trained models. While no priors were required, state-of-the-art methods still followed the idea of using explicit structure representations. FOMM (Siarohin et al., 2019) proposed a first order motion approach to predict keypoints and local transformations online to generate flow fields. Siarohin et al. (2021) developed this idea to model articulated objects by replacing a keypoints predictor by a PCA-based region prediction module. Wang et al. (2021b) extended FOMM by predicting 3D keypoints for view-free generation. We note though that in all approaches, given that keypoints or regions are inadequately predicted, the quality of generated images drastically decreases. In contrast to such approaches, our method does not require any explicit structure representations. We dive into the latent space of the generator and self-learn to navigate motion codes in certain directions with the goal to reach target codes, which are then decoded to flow fields for warping. + +![](images/203fe2748a461292b0737c6c4a21c45cb5adc8b929922aec910a81a2c52ff5dd.jpg) +Figure 3: Overview of LIA. LIA is an autoencoder consisting of two networks, an encoder $E$ and a generator $G$ . In the latent space, we apply Linear Motion Decomposition (LMD) towards learning a motion dictionary $D _ { m }$ , which is an orthogonal basis where each vector represents a basic visual transformation. LIA takes two frames sampled from the same video sequence as source image $x _ { s }$ and driving image $x _ { d }$ respectively during training. Firstly, it encodes $x _ { s }$ into a source latent code $z _ { s r }$ and $x _ { d }$ into a magnitude vector $\bar { A _ { r \to d } }$ . Then, it linearly combines $A _ { r d }$ and a trainable $D _ { m }$ using LMD to obtain a latent path $w _ { r \to d }$ , which is used to navigate $z _ { s _ { } r }$ to a target code $z _ { s d }$ . Finally, $G$ decodes $z _ { s d }$ into a target dense flow field and warps $x _ { s }$ to an output image $x _ { s \to d }$ . The training objective is to reconstruct $x _ { d }$ using $x _ { s \to d }$ . + +# 3 METHOD + +Self-supervised image animation aims at learning to transfer motion from a subject of a driving video to a subject in a source image based on training with a large video dataset. In this work, we propose to model such motion transformation via latent space navigation. The general pipeline is illustrated in Fig. 2. Specifically, for training, our model takes in a pair of source and driving images, randomly sampled from one video sequence. These two images are encoded into a latent code which is used to represent motion transformation in the image space. The training objective is to reconstruct the driving image by combining source image with learned motion transformation. For testing, frames of a driving video are sequentially processed with the source image to animate the source subject. + +We provide an overview of the proposed model in Fig. 3. Our model is an autoencoder, consisting of two main networks, an encoder $\bar { E }$ and a generator $\bar { G }$ . In general, our model requires two steps to transfer motion. In the first step, $E$ encodes source and driving images $\boldsymbol { x } _ { s } , \boldsymbol { x } _ { d } \sim \dot { \mathcal { X } } \in \mathbb { R } ^ { 3 \times H \times W }$ into latent codes in the latent space. The source code is then navigated into a target code, which is used to represent target motion transformation, along a learned latent path. Based on proposed Linear Motion Decomposition (LMD), we represent such a path as a linear combination of a set of learned motion directions and associated magnitudes, which are learned from $x _ { d }$ . In the second step, once the target latent code is obtained, $G$ decodes it as a dense flow field $\phi _ { s \to d } \sim \Phi \in \mathbb { R } ^ { 2 \times H \times W }$ and uses $\bar { \phi _ { s \to d } }$ to warp $x _ { s }$ and then to obtain the output image. In the following, we proceed to discuss the two steps in detail. + +# 3.1 LATENT MOTION REPRESENTATION + +Given a source image $x _ { s }$ and a driving image $x _ { d }$ , our first step constitutes of learning a latent code $z _ { s \to d } \sim \mathcal { Z } \in \mathbb { R } ^ { N }$ to represent the motion transformation from $x _ { s }$ to $x _ { d }$ . Due to the uncertainty of two images, directly learning $z _ { s d }$ puts forward a high requirement on the model to capture a complex distribution of motion. Mathematically, it requires modeling directions and norms of the vector $z _ { s d }$ simultaneously, which is challenging. Therefore, instead of modeling motion transformation $x _ { s } \to x _ { d }$ , we assume there exists a reference image $x _ { r }$ and motion transfer can be modeled as $x _ { s } x _ { r } x _ { d }$ , where $z _ { s d }$ is learned in an indirect manner. We model $z _ { s d }$ as a target point in the latent space, which can be reached by taking linear walks from a starting point $z _ { s r }$ along a linear path $w _ { r \to d }$ (see Fig. 2), given by + +$$ +z _ { s d } = z _ { s r } + w _ { r d } , +$$ + +where $z _ { s r }$ and $w _ { r \to d }$ indicate the transformation $x _ { s } \to x _ { r }$ and $x _ { r } \to x _ { d }$ respectively. Both $z _ { s r }$ and $w _ { r \to d }$ are learned independently and $z _ { s r }$ is obtained by passing $x _ { s }$ through $E$ . + +We learn $w _ { r \to d }$ via Linear Motion Decomposition (LMD). Our idea is to learn a set of motion directions $D _ { m } = \{ \mathbf { d _ { 1 } } , . . . , \mathbf { d _ { M } } \}$ to represent any path in the latent space. We constrain $D _ { m }$ as an orthogonal basis, where each vector indicates a motion direction $\mathbf { d _ { i } }$ . We then combine each vector in the basis with a vector $A _ { r d } = \{ a _ { 1 } , . . . , a _ { M } \}$ , where $a _ { i }$ represents the magnitude of $\mathbf { d _ { i } }$ . Hence, any linear path in the latent space can be represented using a linear combination + +$$ +w _ { r d } = \sum _ { i = 1 } ^ { M } a _ { i } { \bf d _ { i } } , +$$ + +where $\mathbf { d } _ { \mathbf { i } } \in \mathbb { R } ^ { N }$ and $a _ { i } \in \mathbb { R }$ for all $i \in \{ 1 , . . . , M \}$ . Semantically, each $\mathbf { d _ { i } }$ should represent a basic visual transformation and $a _ { i }$ indicates the required steps to walk in $\mathbf { d _ { i } }$ towards achieving $w _ { r \to d }$ . Due to $D _ { m }$ entailing an orthogonal basis, any two directions $\mathbf { d } _ { \mathrm { i } } , \mathbf { d } _ { \mathrm { j } }$ follow the constrain + +$$ +< { \bf d _ { i } } , { \bf d _ { j } } > = \left\{ \begin{array} { c c } { { 0 } } & { { i \neq j } } \\ { { 1 } } & { { i = j . } } \end{array} \right. +$$ + +We implement $D _ { m }$ as a learnable matrix and apply the Gram-Schmidt process during each forward pass, in order to meet the requirement of orthogonality. $A _ { r d }$ is obtained by mapping $z _ { d r }$ , which is the output of $x _ { d }$ after $E$ , through a 5-layer MLP. The final formulation of latent motion representation for each $x _ { s }$ and $x _ { d }$ is thus given as + +$$ +z _ { s d } = z _ { s r } + \sum _ { i = 1 } ^ { M } a _ { i } { \bf d _ { i } } . +$$ + +# 3.2 LATENT CODE DRIVEN IMAGE ANIMATION + +Once we obtain $z _ { s d }$ , in our second step, we use $G$ to decode a flow field $\phi _ { s \to d }$ and warp $x _ { s }$ . Our $G$ consists of two components, a flow field generator $G _ { f }$ and a refinement network $G _ { r }$ (we provide details in App. A). + +Towards learning multi-scale features, $G$ is designed as a residual network containing $N$ models to produce a pyramid of flow fields $\phi _ { s \to d } = \{ \bar { \phi _ { i } } \bar \} _ { 1 } ^ { N }$ in different layers of $G _ { f }$ . Multi-scale source features $x _ { s } ^ { e n c } = \{ x _ { i } ^ { e n c } \} _ { 1 } ^ { N }$ are obtained from $E$ and are warped in $G _ { f }$ . + +However, as pointed out by Siarohin et al. (2019), only relying on $\phi _ { s \to d }$ to warp source features is insufficient to precisely reconstruct driving images due to the existing occlusions in some positions of $x _ { s }$ . In order to predict pixels in those positions, the network is required to inpaint the warped feature maps. Therefore, we predict multi-scale masks $\{ m _ { i } \} _ { 1 } ^ { N }$ along with $\{ \phi _ { i } \} _ { 1 } ^ { N }$ in $G _ { f }$ to mask out the regions required to be inpainted. In each residual module, we have + +$$ +x _ { i } ^ { \prime } = \mathcal { T } ( \phi _ { i } , x _ { i } ^ { e n c } ) \odot m _ { i } , +$$ + +where $\odot$ denotes the Hadamard product and $\tau$ denotes warping operation, whereas $\boldsymbol { x } _ { i } ^ { \prime }$ signifies the masked features. We generate both dense flow fields, as well as masks by letting each residual module output a 3-channel feature map in which the first two channels represent ${ \bar { \phi } } _ { i }$ and the last channel $m _ { i }$ . Based on an inpainted feature map $f ( x _ { i } ^ { \prime } )$ , as well as an upsampled image $g ( x _ { i - 1 } )$ provided by the previous module in $G _ { r }$ , the RGB image from each module is given by + +$$ +o _ { i } = f ( x _ { i } ^ { \prime } ) + g ( o _ { i - 1 } ) , +$$ + +where $f$ and $g$ denote the inpainting and upsampling layers, respectively. The output image $o _ { N }$ from the last module constitutes the final generated image $x _ { s \to d } = o _ { N }$ . + +# 3.3 LEARNING + +We train LIA in a self-supervised manner to reconstruct $x _ { d }$ using three losses, i.e., a reconstruction loss $\mathcal { L } _ { r e c o n }$ , a perceptual loss $\mathcal { L } _ { v g g }$ (Johnson et al., 2016) and an adversarial loss $\mathcal { L } _ { a d v }$ . We use $\mathcal { L } _ { r e c o n }$ to minimize the pixel-wise $\mathbb { L } _ { 1 }$ distance between $x _ { d }$ and $x _ { s \to d }$ , calculated as + +$$ +\mathcal { L } _ { r e c o n } ( x _ { s d } , x _ { d } ) = \mathbb { E } [ \| x _ { d } - x _ { s d } \| _ { 1 } ] . +$$ + +Towards minimizing the perceptual distance, we apply a VGG19-based $\mathcal { L } _ { v g g }$ on multi-scale feature maps between real and generated images, written as + +$$ +\mathcal { L } _ { v g g } ( x _ { s d } , x _ { d } ) = \mathbb { E } [ \sum _ { n } ^ { N } \| F _ { n } ( x _ { d } ) - F _ { n } ( x _ { s d } ) \| _ { 1 } ] , +$$ + +where $F _ { n }$ denotes the $n ^ { t h }$ layer in a pre-trained VGG19 (Simonyan & Zisserman, 2015). In practice, towards penalizing real and generated images in multi-scale images, we use a pyramid of four resolutions, namely $2 5 6 \times 2 5 6$ , ${ \bar { 1 } } 2 8 \times 1 2 8$ , $6 4 \times 6 4$ and $3 2 \times 3 2$ as inputs of VGG19. The final perceptual loss is the addition of perceptual losses in four resolutions. + +Further, towards generating photo-realistic results, we incorporate a non-saturating adversarial loss $\mathcal { L } _ { a d v }$ on $x _ { s \to d }$ , which is calculated as + +$$ +\mathcal { L } _ { a d v } ( x _ { s d } ) = \mathbb { E } _ { x _ { s d } \sim p _ { r e c } } [ - l o g ( D ( x _ { s d } ) ] , +$$ + +where $D$ is a discriminator, aimed at distinguishing reconstructed images from the original ones. Our full loss function is the combination of three losses with $\lambda$ as a balanced hyperparameter + +$$ +{ \mathcal { L } } ( x _ { s d } , x _ { d } ) = { \mathcal { L } } _ { r e c o n } ( x _ { s d } , x _ { d } ) + \lambda { \mathcal { L } } _ { v g g } ( x _ { s d } , x _ { d } ) + { \mathcal { L } } _ { a d v } ( x _ { s d } ) . +$$ + +# 3.4 INFERENCE + +In inference stage, given a driving video sequence $V _ { d } = \{ x _ { t } \} _ { 1 } ^ { T }$ , we aim to transfer motion from $V _ { d }$ to $x _ { s }$ , in order to generate a novel video $V _ { d s } = \{ x _ { t s } \} _ { 1 } ^ { T }$ . If $V _ { d }$ and $x _ { s }$ stem from the same video sequence, i.e., $x _ { s } ~ = ~ x _ { 1 }$ , our task comprises of reconstructing the entire original video sequence. Therefore, we construct the latent motion representation of each frame using absolute transfer, which follows the training process, given as + +$$ +z _ { s t } = z _ { s r } + w _ { r t } , t \in \{ 1 , . . . , T \} . +$$ + +However, in real world applications, interest is rather placed on the scenario, where motion transfer between $x _ { s }$ and $V _ { d }$ , the latter stemming from different identities, i.e., $x _ { s } \neq x _ { 1 }$ . Taking a talking head video as an example, in this setting, beyond identity, $x _ { 1 }$ and $x _ { s }$ might also differ in pose and expression. Therefore, we propose relative transfer to eliminate the motion impact of $w _ { r \to 1 }$ and involve motion of $w _ { r \to s }$ in the full generated video sequence. Owing to a linear representation of the latent path, we can easily represent $z _ { s t }$ for each frame as + +$$ +\begin{array} { r l } & { z _ { s t } = ( z _ { s r } + w _ { r s } ) + ( w _ { r t } - w _ { r 1 } ) } \\ & { \qquad = z _ { s s } + ( w _ { r t } - w _ { r 1 } ) , \ t \in \{ 1 , . . . , T \} . } \end{array} +$$ + +The first term in Eq. (12), $z _ { s s }$ indicates the reconstruction of $x _ { s }$ , while the second term $( w _ { r t } - w _ { r 1 } )$ represents the motion from $x _ { 1 }$ to $x _ { t }$ . This equation indicates that the original pose is preserved in $x _ { s }$ , at the same time motion is transferred from $V _ { d }$ . We note that in order to completely replicate the position and pose in $V _ { d }$ , it requires $x _ { s }$ and $x _ { 1 }$ to contain similar poses in relative motion transfer. + +# 4 EXPERIMENTS + +In this section, we firstly describe our experimental setup including implementation details and datasets. Secondly, we qualitatively demonstrate generated results based on testing datasets. Then, we provide quantitative evaluation w.r.t. image quality on (a) same-identity reconstruction, (b) crossvideo motion transfer, presenting (c) a user study. Next, we conduct an ablation study that demonstrates (d) the effectiveness of our proposed motion dictionary, as well as (e) associated size. Finally, we provide an in-depth analysis of our (f) latent codes and (g) motion dictionary to interpret their semantic meanings. + +Datasets Our model is trained on the datasets VoxCeleb, TaichiHD and TED-talk. We follow the pre-processing method in (Siarohin et al., 2019) to crop frames into $2 5 6 \times 2 5 6$ resolution for quantitative evaluation. + +Implementation details Our model is implemented in PyTorch (Paszke et al., 2019). All models are trained on four 16G NVIDIA V100 GPUs. The total batch size is 32 with 8 images per GPU. We use a learning rate of 0.002 to train our model with the Adam optimizer (Kingma & Ba, 2014). The dimension of all latent codes, as well as directions in $D _ { m }$ is set to be 512. In our loss function, we use $\lambda = 1 0$ in order to penalize more on the perceptual loss. It takes around 150 hours to fully train our framework. + +![](images/a37a7371bb03be1d7874e44a06664ca4dc3ef0bc6d060aab959b38245f9df2b2.jpg) +Figure 4: Qualitative results. Examples for same-dataset absolute motion transfer on TaichiHD (top-right) and TED-talk (bottom-right). On VoxCeleb (left), we demonstrate cross-dataset relative motion transfer. We successfully transfer motion between $x _ { 1 }$ and $x _ { t }$ from videos in VoxCeleb to $x _ { s }$ from FFHQ, the latter not being used for training. + +Evaluation metrics We evaluate our model w.r.t. (i) reconstruction faithfulness using $\mathcal { L } _ { 1 }$ , LPIPS, (ii) generated video quality using video FID, as well as (iii) semantic consistency using average keypoint distance (AKD), missing keypoint rate (MKR) and average euclidean distance (AED). Details are available in App. B.2. + +# 4.1 QUALITATIVE RESULTS + +Firstly, we evaluate the ability of LIA to generate realistic videos and compare related results with four state-of-the-art methods. For TaichiHD and TED-talk datasets, we conduct an experiment related to cross-video generation. Corresponding results (see Fig. 4) confirm that our method is able to correctly transfer motion on articulated human bodies, in the absence of explicit structure representations. For the VoxCeleb dataset, we conduct a cross-dataset generation-experiment, where we transfer motion from VoxCeleb to images of the FFHQ dataset. We observe that our method outperforms FOMM and MRAA w.r.t. image quality, as both approaches visibly deform the shape of the original faces. This is specifically notable in the case that source and driving images entail large pose variations. At the same time, LIA is able to successfully tackle this challenge and no similar deformations are visible. + +# 4.2 COMPARISON WITH STATE-OF-THE-ART METHODS + +We quantitatively compare our method with the state-of-the-art approaches X2Face, Monkey-Net, FOMM and MRAA on two tasks, namely (a) same-identity reconstruction and (b) cross-video motion transfer. Additionally, we conduct a (c) user study. + +(a) Same-identity reconstruction We here evaluate the reconstruction ability of our method. Specifically, we reconstruct each testing video by using the first frame as $x _ { s }$ and the remaining frames as $x _ { d }$ . Results on three datasets are reported in Table 1. Focusing on foregroundreconstruction, our method outperforms the other approaches w.r.t. all metrics. More results are presented in App. B.3, discussing background-reconstruction. + +(b) Cross-video motion transfer Next, we conduct experiments, where source images and driving videos stem from different video sequences. In this context, we mainly focus on evaluating talking head videos and explore two different cases. In the first case, we generate videos using the VoxCeleb testing set to conduct motion transfer. In the second case, source images are from an unseen dataset, namely the GermanPublicTV dataset, as we conduct cross-dataset motion transfer. In both experiments, we randomly construct source and driving pairs and transfer motion from driving videos to source images to generate a novel manipulated dataset. Since ground truth data for our generated videos is not available, we use video FID (as initialized by Wang et al. (2020a)) to compute the distance between generated and real data distributions. As shown in Tab. 2, our method outperforms all other approaches w.r.t. video FID, indicating the best generated video quality. + +Table 1: Same-identity reconstruction. Comparison with state-of-the-art methods on three datasets for same-identity reconstruction. + +
MethodVoxCelebTaichiHDTED-talks
L1AKDAEDLPIPSL1(AKD,MKR)AEDLPIPSL1(AKD,MKR)AEDLPIPS
X2Face0.0787.6870.405-0.080(17.654,0.109)-=-
Monkey-Net0.0491.8780.199-0.077(10.798,0.059)==
FOMM0.0461.3950.1410.1360.063(6.472,0.032)0.4950.1910.030(3.759,0.0090)0.4280.13
MRAA w/o bg0.0431.3070.1400.1270.063(5.626,0.025)0.4600.1890.029(3.126,0.0092)0.3960.12
Ours0.0411.3530.1380.1230.057(4.823, 0.020)0.4310.1800.027(3.141,0.0095)0.3990.11
+ +Table 2: Cross-video generation. We report video FID for both inner- and cross-dataset tasks on VoxCeleb and GermanPublicTV. + +
VoxCelebGermanPublicTV
FOMM0.323 0.456
MRAA 0.3080.454
Ours0.161 0.406
+ +Table 3: User study. We ask 20 human raters to conduct a subjective video quality evaluation. + +
VoxCeleb(%)TaichiHD(%)TED-talk(%)
Ours/FOMM92.9/7.164.5/35.571.4/28.6
Ours/MRAA89.7/10.360.7/39.954.8/45.2
+ +(c) User study We conduct a user study to evaluate video quality. Towards this, we displayed paired videos and asked 20 human raters ‘which clip is more realistic?’. Each video-pair contains a generated video from our method, as well as a video generated from FOMM or MRAA. Results suggest that our results are more realistic in comparison to FOMM and MRAA across all three datasets (see Tab. 3). Hence, the obtained human preference is in accordance with our quantitative evaluation. + +Table 4: Ablation study on motion dictionary. We conduct experiments on three datasets with and without $D _ { m } ^ { \phantom { \dagger } }$ and show reconstruction results. + +
VoxCelebTaichiHDTED-talks
MethodL1 LPIPSL1LPIPSL1LPIPS
wloDm0.049 0.1650.0620.1860.0310.12
Full0.041 0.1230.0570.1800.0280.11
+ +Table 5: Ablation study on $D _ { m }$ size. We conduct experiments on three datasets with 5 different $D _ { m }$ size and show reconstruction results. + +
VoxCelebTaichiHDTED-talks
ML1LPIPSL1LPIPSL1LPIPS
5 100.051 0.0430.15 0.130.070 0.0650.22 0.200.037 0.0360.15
200.0410.120.0570.180.0280.13 0.11
400.0420.120.0600.190.0300.12
1000.0410.120.0580.180.0280.11
+ +# 4.3 ABLATION STUDY + +We here analyze our proposed motion dictionary and focus on answering following two questions. + +(d) Is the motion dictionary $D _ { m }$ beneficial? We here explore the impact of proposed $D _ { m }$ , by training our model without $D _ { m }$ . Specifically, we output $w _ { r \to d }$ directly from MLP, without using LMD to learn an orthogonal basis. From the evaluation results reported in Tab. 4 and qualitative results in App. B.5, we observe that in the absence of $D _ { m }$ , model fails to generate high-quality images, which proves the effectiveness of $D _ { m }$ , consistently on all datasets. + +(e) How many directions are required in $D _ { m }$ ? Towards finding an effective size of $D _ { m }$ , we empirically test three different $M$ , viz. 5, 10, 20, 40 and 100. Quantitative results in Tab. 5 show that when using 20 directions, the model achieves the best reconstruction results. + +# 4.4 FURTHER ANALYSIS + +(f) Latent code analysis. While our method successfully transfers motion via latent space navigation, we here aim at answering the question — what does $x _ { r }$ represent? Towards answering this question, we proceed to visualize $x _ { r }$ . We firstly decode $z _ { s r }$ into a dense flow field $\phi _ { s r }$ , which is then used to warp $x _ { s }$ (we show details in App. B.4). Fig. 5 shows examples of $x _ { s }$ and $x _ { r }$ . Interestingly, we observe that $x _ { r }$ represents the canonical pose of $x _ { s }$ , regardless of original poses of the subjects. And for all datasets, reference images resemble each other $w . r . t .$ . pose and scale. As such reference images can be considered as a normalized form of $x _ { s }$ , learning transformations between $x _ { s }$ and $x _ { d }$ using $x _ { s } x _ { r } x _ { d }$ is considerably more efficient than $x _ { s } \to x _ { d }$ , once $x _ { r }$ is fixed. + +![](images/f4021d1e9c0a6b78d5507518e6f476c3eb214d4b2fd88cb1cbad903d30c7d449.jpg) +Figure 5: Visualization of reference images. Example source (top) and reference images (down) from VoxCeleb, TaichiHD and TED-talk datasets. Our network learns reference images of a consistently frontal pose, systematically for all input images of each dataset. + +![](images/833c41d15a70be2fc45b2ced129969f51412a6047f24acd5518917c81c6c810d.jpg) +Figure 6: Linear manipulation of four motion directions on the painting of Mona Lisa. Manipulated results indicate that $d _ { 6 }$ represents eye movement, $d _ { 8 }$ represents head nodding, whereas $\bar { d _ { 1 9 } }$ and $d _ { 7 }$ represent facial expressions. + +Noteworthy, we found the similar idea of learning a ‘reference image’ has also been explored by Siarohin et al. (2019) (FOMM) and Wiles et al. (2018) (X2Face). However, deviating from our visualized ‘reference image’, the ’reference image ’in FOMM refers to a non-visualized and abstract concept. In addition, LIA only requires a latent code $z _ { s r }$ , rather than the ’reference image’ for both, training and testing, which is contrast to X2Face. + +(g) Motion dictionary interpretation. Towards further interpretation of directions in $D _ { m }$ , we conduct linear manipulations on each $d _ { i }$ . Images pertained to manipulating four motion directions are depicted in Fig. 6. The results suggest that the directions in $\boldsymbol { D } _ { m } ^ { \bar { 2 } }$ are semantically meaningful, as they represent basic visual transformations such as head nodding $( d _ { 8 } )$ , eye movement $( d _ { 6 } ) ^ { \dag }$ and facial expressions $\cdot d _ { 1 9 }$ and $d _ { 7 }$ ). More results can be found on our project webpage2. + +# 5 CONCLUSIONS + +In this paper, we presented a novel self-supervised autoencoder LIA, aimed at animating images via latent space navigation. By the proposed Linear Motion Decomposition (LMD), we were able to formulate the task of transferring motion from driving videos to source images as learning linear transformations in the latent space. We evaluated proposed method on real-world videos and demonstrated that our approach is able to successfully animate still images, while eliminating the necessity of explicit structure representations. In addition, we showed that the incorporated motion dictionary is interpretable and contains directions pertaining to basic visual transformations. Both quantitative and qualitative evaluations showed that LIA outperforms state-of-art algorithms on all benchmarks. We postulate that LIA opens a new door in design of interpretable generative models for video generation. + +# ETHIC STATEMENT + +In this work, we aim to synthesize high-quality videos by transferring motion on still images. Our approach can be used for movie production, making video games, online education, generating synthetic data for other computer vision tasks, etc. We note that our framework mainly focuses on learning how to model motion distribution rather than directly model appearance, therefore it is not biased towards any specific gender, race, region, or social class. It works equally well irrespective of the difference in subjects. + +# REPRODUCIBILITY STATEMENT + +We assure that all the results shown in the paper and supplemental materials can be reproduced. We intend to open-source our code, as well as trained models. + +# ACKNOWLEDGEMENTS + +This work was granted access to the HPC resources of IDRIS under the allocation AD011011627R1. It was supported by the French Government, by the National Research Agency (ANR) under Grant ANR-18-CE92-0024, project RESPECT and through the 3IA Cote d’Azur Investments in the Future ˆ project managed by the National Research Agency (ANR) with the reference number ANR-19- P3IA-0002. + +# REFERENCES + +Rameen Abdal, Yipeng Qin, and Peter Wonka. Image2stylegan $^ { + + }$ : How to edit the embedded images? In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020. + +Brandon Amos, Bartosz Ludwiczuk, and Mahadev Satyanarayanan. 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In CVPR, 2018. + +Long Zhao, Xi Peng, Yu Tian, Mubbasir Kapadia, and Dimitris Metaxas. Learning to forecast and refine residual motion for image-to-video generation. In ECCV, 2018. + +Zhedong Zheng, Tao Ruan, Yunchao Wei, Yi Yang, and Tao Mei. Vehiclenet: Learning robust visual representation for vehicle re-identification. IEEE Transaction on Multimedia (TMM), 2020. + +Jiapeng Zhu, Yujun Shen, Deli Zhao, and Bolei Zhou. In-domain gan inversion for real image editing. In Proceedings of European Conference on Computer Vision (ECCV), 2020. + +# A DETAILS OF MODEL ARCHITECTURE + +We proceed to describe the model architecture in this section. Fig. 7 shows details of our $E$ . In each ResBlock in $E$ , spatial size of input feature maps are downsampled. We take feature maps of spatial sizes from $8 \times 8$ to $2 5 6 \times 2 5 6$ as our appearance features $x _ { i } ^ { e n c }$ . We use a 5-layer MLP to predict a magnitude vector $A _ { d r }$ from $z _ { d r }$ . Fig. 8 (a) shows the general architecture of our $G$ , which consists of two components, a flow field generator $G _ { f }$ and a refinement network $G _ { r }$ . We apply StyleConv (Upsample $\mathbf { + \ C o n v 3 \times 3 }$ ), which is proposed by StyleGAN2, in $G _ { f }$ . StyleConv takes latent representation $z _ { s t }$ as style code and generates flow field $\phi _ { i }$ and corresponding mask $m _ { i }$ . $G _ { r }$ uses UpConv (Conv1 $\times \ 1 +$ Upsample) to upsample and refine inpainted feature maps to target resolution. We show details pertaining to $G$ block in Fig. 8 (b). Each $G$ block is used to upsample $\times 2$ the previous resolution. We stack 6 blocks towards producing 256 resolution images. + +![](images/9325fffdbfebc0ee68f0be3ae9d39d9a1ea8e2cd0cc42b57b0799a99bd6ea9c8.jpg) +Figure 7: Encoder architecture. We show details of architecture of $E$ in (a) and ResBlock in (b). + +# B EXPERIMENTS + +We proceed to introduce details of datasets and evaluation metrics used in our experiments. + +# B.1 DATASETS + +VoxCeleb (Nagrani et al., 2019) consists of a large amount of interview videos of different celebrities. Following the process of FOMM (Siarohin et al., 2019), we extract frames and crop them into $2 5 6 \times 2 5 6$ resolution. In total, VoxCeleb contains a training set of 17928 videos and a test set of 495 videos. + +![](images/9bfd82d4cf08f228e79a896a311bfc79b0ee414695a25bedad52ccbc1330672a.jpg) +Figure 8: Generator architecture. We show details about architecture of $G$ in (a) and $G$ block in (b). + +TaiChiHD (Siarohin et al., 2019) consists of videos of full human bodies performing Tai Chi actions. We follow the original pre-processing of FOMM (Siarohin et al., 2019) and utilize its $2 5 6 \times 2 5 6$ version. TaiChiHD contains 1096 training videos and 115 testing videos. + +TED-talk is a new dataset proposed in MRAA (Siarohin et al., 2021). It comprises a number of TED-talk videos, where the main subjects have been cropped out. We resize the original version into $2 5 6 \times 2 5 6$ resolution to train our model. This dataset includes 1124 training videos and 130 testing videos. + +# B.2 EVALUATION METRICS + +We use five different metrics to evaluate our experimental results, namely $\mathcal { L } _ { 1 }$ , LPIPS, AKD, MKR and AED that quantify the reconstructed results. In addition, we compute video FID to evaluate video quality in motion transferring tasks. + +$\mathcal { L } _ { 1 }$ represents the mean absolute pixel difference between reconstructed and real videos. + +LPIPS (Zhang et al., 2018) aims at measuring the perceptual similarity between reconstructed and real images by leveraging the deep features from AlexNet (Krizhevsky et al., 2012). + +Video FID is a modified version of the original FID (Heusel et al., 2017). We here follow the same implementation as Wang et al. (2020a) and utilize a pre-trained ResNext101 (Hara et al., 2018) to extract spatio-temporal features to compute the distance between real and generated videos distributions. We take the first 100 frames of each video as input of the feature-extractor to compute the final scores. + +Average keypoint distance (AKD) and missing keypoint rate (MKR) evaluate the difference between keypoints of reconstructed and ground truth videos. We extract landmarks using the face alignment approach of (Bulat & Tzimiropoulos, 2017) and extract body poses for both TaiChiHD and TED-talks using OpenPose (Cao et al., 2019). AKD is computed as the average distance between corresponding keypoints, whereas MKR is the proportion of keypoints present in the groundtruth that are missing in a reconstructed video. + +Average Euclidean distance (AED) measures the ability of preserving identity in reconstructed video. We use a person re-identification pretrained model (Zheng et al., 2020) for measuring human bodies (TaichiHD and TED-talk) and OpenFace (Amos et al., 2016) for faces to extract identity embeddings from reconstructed and ground truth frame pairs, then we compute MSE of their difference for all pairs. + +# B.3 COMPARISON WITH FULL MRAA + +We show quantitative evaluation results with the full MRAA model in Tab. 6. We observe that our method achieves competitive results in reconstruction and keypoint evaluation. While we do not explicitly predict keypoints, w.r.t. the TaichiHD dataset, interestingly we outperform MRAA in both, AKD and MKR. Such results showcase the effectiveness of our proposed method on modeling articulated human structures. However, reconstruction evaluation cannot provide a completely fair comparison on how well the main subjects (e.g., faces and human bodies) are generated in videos. This is in particular the case for TaichiHD and TED-talk, where backgrounds have large contributions to the final scores. + +Table 6: Comparison with full MRAA. + +
MethodVoxCelebTaichiHDTED-talks
L1AKDAEDLPIPSL1(AKD,MKR)AEDLPIPSL1(AKD,MKR)AEDLPIPS
MRAA0.0411.3030.1350.1240.045(5.551,( ,0.025)0.4310.1780.027(3.107,0.0093)0.3790.11
Ours0.0411.3530.1380.1230.057(4.823,0.020)0.4310.1800.027(3.141,0.0095)0.3990.11
+ +# B.4 REFERENCE IMAGE GENERATION. + +To produce $x _ { r }$ , we use $G$ to decode $z _ { s r }$ into the flow field $\phi _ { s r }$ . Reference image $x _ { r }$ is obtained by warping $x _ { s }$ using $\phi _ { s r }$ . The entire process is shown in Fig. 9. + +![](images/762132d8adc241e8fde2ec32f40c10dbb27f4f06b3953c429f112db181bf10c2.jpg) +Figure 9: Reference image generation. + +# B.5 QUALITATIVE RESULTS ON EFFECTIVENESS OF USING MOTION DICTIONARY + +Fig. 10 illustrates the generated results on transferring motion from VoxCeleb to GermanPublicTV with and without motion dictionary. We observe that without the motion dictionary, appearance information is undesirably transferred from driving videos to generated videos. + +# B.6 LIMITATIONS + +For human body, one limitation of our method is dealing with body occlusion. We observe in Fig. 11 that in taichi videos, in case of occlusion cause by legs and arms, motion is not transferred successfully. In addition, in TED-talks, transferring hand motion is challenging, as hands are of small size, articulated and sometimes occluded by human bodies. + +![](images/70acaf34c5c7203b948fb2919b9b562110b48bb7fdf0186567c1baf79c82cd42.jpg) +Figure 10: Generated results with and without $D _ { m }$ . We observe that the disentanglement of appearance and motion is much better by using $D _ { m }$ . + +![](images/8df647813e53c9e3f9fda4ddf2534e1d09430b0019731ce68455cebf897027e0.jpg) +Figure 11: Failure cases. We observed that it is still challenging for LIA to handle arm-leg occlusion (Taichi) and hand motion (TED-talk). \ No newline at end of file diff --git a/parse/dev/7r6kDq0mK_/7r6kDq0mK__content_list.json b/parse/dev/7r6kDq0mK_/7r6kDq0mK__content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..31fe4ef4da104bb154ea9b7a3b5788462b3fc16e --- /dev/null +++ b/parse/dev/7r6kDq0mK_/7r6kDq0mK__content_list.json @@ -0,0 +1,1986 @@ +[ + { + "type": "text", + "text": "LATENT IMAGE ANIMATOR: LEARNING TO ANIMATE IMAGES VIA LATENT SPACE NAVIGATION ", + "text_level": 1, + "bbox": [ + 174, + 99, + 823, + 166 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yaohui Wang, Di Yang, Francois Bremond & Antitza Dantcheva Inria, Universite C´ ote d’Azur ˆ {yaohui.wang,di.yang,francois.bremond,antitza.dantcheva}@inria.fr ", + "bbox": [ + 184, + 190, + 756, + 229 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 265, + 544, + 280 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Due to the remarkable progress of deep generative models, animating images has become increasingly efficient, whereas associated results have become increasingly realistic. Current animation-approaches commonly exploit structure representation extracted from driving videos. Such structure representation is instrumental in transferring motion from driving videos to still images. However, such approaches fail in case the source image and driving video encompass large appearance variation. Moreover, the extraction of structure information requires additional modules that endow the animation-model with increased complexity. Deviating from such models, we here introduce the Latent Image Animator (LIA), a self-supervised autoencoder that evades need for structure representation. LIA is streamlined to animate images by linear navigation in the latent space. Specifically, motion in generated video is constructed by linear displacement of codes in the latent space. Towards this, we learn a set of orthogonal motion directions simultaneously, and use their linear combination, in order to represent any displacement in the latent space. Extensive quantitative and qualitative analysis suggests that our model systematically and significantly outperforms state-of-art methods on VoxCeleb, Taichi and TED-talk datasets w.r.t. generated quality. Source code and pre-trained models are publicly available1. ", + "bbox": [ + 233, + 297, + 764, + 527 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/e378522fe7bd0a0020ce31d5201a8ccda8a4e2a620417861db6f080cf4f0c4a2.jpg", + "image_caption": [ + "Figure 1: LIA animation examples. The two images of Marilyn Monroe and Emmanuel Macron are animated by LIA, which transfers motion of a driving video (smaller images on the top) from VoxCeleb dataset (Chung et al., 2018) onto the still images. LIA is able to successfully animate these two images without relying on any explicit structure representations, such as landmarks and region representations. " + ], + "image_footnote": [], + "bbox": [ + 176, + 558, + 823, + 794 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 102, + 336, + 117 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In the series of science fiction books Harry Potter (Rowling et al., 2016; Rowling, 2019), wizards and witches were able to magically enchant portraits, bringing them to life. Remarkable progress of deep generative models has recently turned this vision into reality. This work examines the scenario where a framework animates a source image by motion representations learned from a driving video. Existing approaches for image animation are classically related to computer graphics (Cao et al., 2014; Thies et al., 2016; 2019; Zhao et al., 2018) or exploit motion labels (Wang et al., 2020b) and structure representations such as semantic maps (Pan et al., 2019; Wang et al., 2018; 2019), human keypoints (Jang et al., 2018; Yang et al., 2018; Walker et al., 2017; Chan et al., 2019; Zakharov et al., 2019; Wang et al., 2019; Siarohin et al., 2019), 3D meshes (Liu et al., 2019; Chen et al., 2021), and optical flows (Li et al., 2018; Ohnishi et al., 2018). We note that the ground truth of such structure representations has been computed a-priori for the purpose of supervised training, which poses constraints on applications, where such representations of unseen testing images might be fragmentary or difficult to access. ", + "bbox": [ + 174, + 138, + 825, + 303 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Self-supervised motion transfer approaches (Wiles et al., 2018; Siarohin et al., 2019; 2021) accept raw videos as input and learn to reconstruct driving images by warping source image with predicted dense optical flow fields. While the need for domain knowledge or labeled ground truth data has been obviated, which improves performance on in-the-wild testing images, such methods entail necessity of explicit structure representations as motion guidance. Prior information such as keypoints (Siarohin et al., 2019; Wang et al., 2021a) or regions (Siarohin et al., 2021) are learned in an end-to-end training manner by additional networks as intermediate features, in order to predict target flow fields. Although online prediction of such representations is less tedious than the acquisition of ground truth labels, it still strains the complexity of networks. ", + "bbox": [ + 174, + 309, + 825, + 424 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Deviating from such approaches, we here aim to fully eliminate the need of explicit structure representations by directly manipulating the latent space of a deep generative model. To the best of our knowledge, this constitutes a new direction in the context of image animation. Our work is motivated by interpretation of GANs (Shen et al., 2020; Goetschalckx et al., 2019; Jahanian et al., 2020; Voynov & Babenko, 2020), showcasing that latent spaces of StyleGAN (Karras et al., 2019; 2020b) and BigGAN (Brock et al., 2019) contain rich semantically meaningful directions. Given that walking along such directions, basic visual transformations such as zooming and rotation can be induced in generated results. As in image animation, we have that motion between source and driving images can be considered as higher-level transformation, a natural question here arises: can we discover a set of directions in the latent space that induces high-level motion transformations collaboratively? ", + "bbox": [ + 174, + 431, + 825, + 570 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Towards answering this question, we introduce LIA, a novel Latent Image Animator constituting of an autoencoder for animating still images via latent space navigation. LIA seeks to animate a source image via linearly navigating associated source latent code along a learned path to reach a target latent code, which represents the high-level transformation for animating the source image. We introduce a Linear Motion Decomposition (LMD) approach aiming to represent a latent path via a linear combination of a set of learned motion directions and associated magnitudes. Specifically, we constrain the set as an orthogonal basis, where each vector indicates a basic visual transformation. By describing the whole motion space using such learned basis, LIA eliminates the requirement of explicit structure representations. ", + "bbox": [ + 174, + 577, + 825, + 691 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In addition, we design LIA to disentangle motion and appearance within a single encoder-generator architecture. Deviating from existing methods using separate networks to learn disentangled features, LIA integrates both, latent motion code, as well as appearance features in a single encoder, which highly reduces the model complexity and simplifies training. ", + "bbox": [ + 174, + 698, + 825, + 750 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We provide evaluation on multiple datasets including VoxCeleb (Chung et al., 2018), TaichiHD (Siarohin et al., 2019) and TED-talk (Siarohin et al., 2021). In addition, we show that LIA outperforms the state-of-the-art in preserving the facial structure in generated videos in the setting of one-shot image animation on unseen datasets such as FFHQ (Karras et al., 2019) and GermanPublicTV (Thies et al., 2020). ", + "bbox": [ + 174, + 756, + 825, + 820 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 842, + 339, + 857 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Video generation GAN-based video generation is aimed at mapping Gaussian noise to video, directly and in the absence of prior information (Vondrick et al., 2016; Saito et al., 2017; Tulyakov et al., 2018; Wang et al., 2020a; Wang, 2021). Approaches based on deep probabilistic models (Denton & Birodkar, 2017; Li & Mandt, 2018; Bhagat et al., 2020; Xie et al., 2020) were also proposed to tackle this problem, however only show results on toy datasets with low resolution. Recently, with the progress of GANs in photo-realistic image generation (Brock et al., 2019; Karras et al., 2019; 2020a), a series of works (Clark et al., 2019; Wang et al., 2021c) explored production of highresolution videos by incorporating the architecture of an image generator into video GANs, trained jointly with RNNs. Tian et al. (2021) directly leveraged the knowledge of a pre-trained StyleGAN to produce videos of resolution up to $1 0 2 4 \\times \\mathrm { i } 0 2 4$ . Unlike these approaches, which generate random videos based on noise vectors in an unconditional manner, in this paper, we focus on conditionally creating novel videos by transferring motion from driving videos to input images. ", + "bbox": [ + 174, + 878, + 823, + 917 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/adf402bc242a87960126961b2dd01a4fedaa5916e13c5407ab9a2e27d13beee5.jpg", + "image_caption": [ + "Figure 2: General pipeline. Our objective is to transfer motion via latent space navigation. The entire training pipeline consists of two steps. Firstly, we encode a source image $x _ { s }$ into a latent code $z _ { s r }$ . By linearly navigating $z _ { s r }$ along a path $w _ { r \\to d }$ , we reach a target latent code $z _ { s d }$ . The latent paths are represented by a linear combination between a set of learned motion directions (e.g., $d _ { 1 }$ and $d _ { 2 }$ ), which is an orthogonal basis, and associated magnitudes. In the second step, we decode $z _ { s d }$ to a target dense optical flow field $\\phi _ { s d }$ , which is used to warp $x _ { s }$ into the driving image $x _ { d }$ . While we train our model using images from the same video sequence, in the testing phase, $x _ { s }$ and $x _ { d }$ generally pertain to different identities. " + ], + "image_footnote": [], + "bbox": [ + 183, + 104, + 825, + 260 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 393, + 825, + 508 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Latent space editing In an effort to control generated images, recent works explored the discovery of semantically meaningful directions in the latent space of pre-trained GANs, where linear navigation corresponds to desired image manipulation. Supervised (Shen et al., 2020; Jahanian et al., 2020; Goetschalckx et al., 2019) and unsupervised (Voynov & Babenko, 2020; Peebles et al., 2020; Shen & Zhou, 2021) approaches were proposed to edit semantics such as facial attributes, colors and basic visual transformations (e.g., rotation and zooming) in generated or inverted real images (Zhu et al., 2020; Abdal et al., 2020). In this work, as opposed to finding directions corresponding to individual visual transformations, we seek to learn a set of directions that cooperatively allows for high-level visual transformations that can be beneficial in image animation. ", + "bbox": [ + 173, + 525, + 825, + 641 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Image animation Related approaches (Chan et al., 2019; Wang et al., 2018; Zakharov et al., 2019; Wang et al., 2019; Yang et al., 2020) in image animation required strong prior structure labels as motion guidance. In particular, Chan et al. (2019), Yang et al. (2020) and Wang et al. (2018) proposed to map representations such as human keypoints and facial landmarks to videos in the setting of image-to-image translation proposed by Isola et al. (2017). However, such approaches were only able to learn an individual model for a single identity. Transferring motion on new appearances requires retraining the entire model from scratch by using videos of target identities. Several recent works (Zakharov et al., 2019; Wang et al., 2019) explored meta learning in fine-tuning models on target identities. While only few images of target identities were required during inference time, it was still compulsory to input pre-computed structure representations in those approaches, which usually are hard to access in many real-world scenarios. Towards addressing this issue, very recent works (Siarohin et al., 2019; 2021; Wang et al., 2021b; Wiles et al., 2018) proposed to learn image animation in self-supervised manner, only relying on RGB videos for both, training and testing without any priors. They firstly predicted dense flow fields from input images, which were then utilized to warp source images, in order to obtain final generated results. Inference only required one image of a target identity without any fine-tuning step on pre-trained models. While no priors were required, state-of-the-art methods still followed the idea of using explicit structure representations. FOMM (Siarohin et al., 2019) proposed a first order motion approach to predict keypoints and local transformations online to generate flow fields. Siarohin et al. (2021) developed this idea to model articulated objects by replacing a keypoints predictor by a PCA-based region prediction module. Wang et al. (2021b) extended FOMM by predicting 3D keypoints for view-free generation. We note though that in all approaches, given that keypoints or regions are inadequately predicted, the quality of generated images drastically decreases. In contrast to such approaches, our method does not require any explicit structure representations. We dive into the latent space of the generator and self-learn to navigate motion codes in certain directions with the goal to reach target codes, which are then decoded to flow fields for warping. ", + "bbox": [ + 173, + 657, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/203fe2748a461292b0737c6c4a21c45cb5adc8b929922aec910a81a2c52ff5dd.jpg", + "image_caption": [ + "Figure 3: Overview of LIA. LIA is an autoencoder consisting of two networks, an encoder $E$ and a generator $G$ . In the latent space, we apply Linear Motion Decomposition (LMD) towards learning a motion dictionary $D _ { m }$ , which is an orthogonal basis where each vector represents a basic visual transformation. LIA takes two frames sampled from the same video sequence as source image $x _ { s }$ and driving image $x _ { d }$ respectively during training. Firstly, it encodes $x _ { s }$ into a source latent code $z _ { s r }$ and $x _ { d }$ into a magnitude vector $\\bar { A _ { r \\to d } }$ . Then, it linearly combines $A _ { r d }$ and a trainable $D _ { m }$ using LMD to obtain a latent path $w _ { r \\to d }$ , which is used to navigate $z _ { s _ { } r }$ to a target code $z _ { s d }$ . Finally, $G$ decodes $z _ { s d }$ into a target dense flow field and warps $x _ { s }$ to an output image $x _ { s \\to d }$ . The training objective is to reconstruct $x _ { d }$ using $x _ { s \\to d }$ . " + ], + "image_footnote": [], + "bbox": [ + 181, + 99, + 808, + 286 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 440, + 825, + 505 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 METHOD ", + "text_level": 1, + "bbox": [ + 174, + 526, + 281, + 541 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Self-supervised image animation aims at learning to transfer motion from a subject of a driving video to a subject in a source image based on training with a large video dataset. In this work, we propose to model such motion transformation via latent space navigation. The general pipeline is illustrated in Fig. 2. Specifically, for training, our model takes in a pair of source and driving images, randomly sampled from one video sequence. These two images are encoded into a latent code which is used to represent motion transformation in the image space. The training objective is to reconstruct the driving image by combining source image with learned motion transformation. For testing, frames of a driving video are sequentially processed with the source image to animate the source subject. ", + "bbox": [ + 174, + 563, + 823, + 665 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We provide an overview of the proposed model in Fig. 3. Our model is an autoencoder, consisting of two main networks, an encoder $\\bar { E }$ and a generator $\\bar { G }$ . In general, our model requires two steps to transfer motion. In the first step, $E$ encodes source and driving images $\\boldsymbol { x } _ { s } , \\boldsymbol { x } _ { d } \\sim \\dot { \\mathcal { X } } \\in \\mathbb { R } ^ { 3 \\times H \\times W }$ into latent codes in the latent space. The source code is then navigated into a target code, which is used to represent target motion transformation, along a learned latent path. Based on proposed Linear Motion Decomposition (LMD), we represent such a path as a linear combination of a set of learned motion directions and associated magnitudes, which are learned from $x _ { d }$ . In the second step, once the target latent code is obtained, $G$ decodes it as a dense flow field $\\phi _ { s \\to d } \\sim \\Phi \\in \\mathbb { R } ^ { 2 \\times H \\times W }$ and uses $\\bar { \\phi _ { s \\to d } }$ to warp $x _ { s }$ and then to obtain the output image. In the following, we proceed to discuss the two steps in detail. ", + "bbox": [ + 173, + 670, + 825, + 803 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 LATENT MOTION REPRESENTATION ", + "text_level": 1, + "bbox": [ + 176, + 820, + 457, + 833 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Given a source image $x _ { s }$ and a driving image $x _ { d }$ , our first step constitutes of learning a latent code $z _ { s \\to d } \\sim \\mathcal { Z } \\in \\mathbb { R } ^ { N }$ to represent the motion transformation from $x _ { s }$ to $x _ { d }$ . Due to the uncertainty of two images, directly learning $z _ { s d }$ puts forward a high requirement on the model to capture a complex distribution of motion. Mathematically, it requires modeling directions and norms of the vector $z _ { s d }$ simultaneously, which is challenging. Therefore, instead of modeling motion transformation $x _ { s } \\to x _ { d }$ , we assume there exists a reference image $x _ { r }$ and motion transfer can be modeled as $x _ { s } x _ { r } x _ { d }$ , where $z _ { s d }$ is learned in an indirect manner. We model $z _ { s d }$ as a target point in the latent space, which can be reached by taking linear walks from a starting point $z _ { s r }$ along a linear path $w _ { r \\to d }$ (see Fig. 2), given by ", + "bbox": [ + 174, + 844, + 823, + 922 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 143 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/df5faf4fb0dcd135a23a966fa3b1c603d561a212356fc77108d9539a75f1e93f.jpg", + "text": "$$\nz _ { s d } = z _ { s r } + w _ { r d } ,\n$$", + "text_format": "latex", + "bbox": [ + 419, + 147, + 575, + 161 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $z _ { s r }$ and $w _ { r \\to d }$ indicate the transformation $x _ { s } \\to x _ { r }$ and $x _ { r } \\to x _ { d }$ respectively. Both $z _ { s r }$ and $w _ { r \\to d }$ are learned independently and $z _ { s r }$ is obtained by passing $x _ { s }$ through $E$ . ", + "bbox": [ + 169, + 164, + 823, + 191 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We learn $w _ { r \\to d }$ via Linear Motion Decomposition (LMD). Our idea is to learn a set of motion directions $D _ { m } = \\{ \\mathbf { d _ { 1 } } , . . . , \\mathbf { d _ { M } } \\}$ to represent any path in the latent space. We constrain $D _ { m }$ as an orthogonal basis, where each vector indicates a motion direction $\\mathbf { d _ { i } }$ . We then combine each vector in the basis with a vector $A _ { r d } = \\{ a _ { 1 } , . . . , a _ { M } \\}$ , where $a _ { i }$ represents the magnitude of $\\mathbf { d _ { i } }$ . Hence, any linear path in the latent space can be represented using a linear combination ", + "bbox": [ + 173, + 195, + 825, + 262 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/ce1f4fd033c395dd8b54261361e0752d9b643c625285ae9ed34fbcd1dc76b91b.jpg", + "text": "$$\nw _ { r d } = \\sum _ { i = 1 } ^ { M } a _ { i } { \\bf d _ { i } } ,\n$$", + "text_format": "latex", + "bbox": [ + 436, + 263, + 560, + 308 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\mathbf { d } _ { \\mathbf { i } } \\in \\mathbb { R } ^ { N }$ and $a _ { i } \\in \\mathbb { R }$ for all $i \\in \\{ 1 , . . . , M \\}$ . Semantically, each $\\mathbf { d _ { i } }$ should represent a basic visual transformation and $a _ { i }$ indicates the required steps to walk in $\\mathbf { d _ { i } }$ towards achieving $w _ { r \\to d }$ . Due to $D _ { m }$ entailing an orthogonal basis, any two directions $\\mathbf { d } _ { \\mathrm { i } } , \\mathbf { d } _ { \\mathrm { j } }$ follow the constrain ", + "bbox": [ + 173, + 310, + 825, + 352 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/bfc2c5b59b76752fb48e656a15dc1ab4f70d40f6618ec59824266734f74db017.jpg", + "text": "$$\n< { \\bf d _ { i } } , { \\bf d _ { j } } > = \\left\\{ \\begin{array} { c c } { { 0 } } & { { i \\neq j } } \\\\ { { 1 } } & { { i = j . } } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 415, + 354, + 586, + 388 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We implement $D _ { m }$ as a learnable matrix and apply the Gram-Schmidt process during each forward pass, in order to meet the requirement of orthogonality. $A _ { r d }$ is obtained by mapping $z _ { d r }$ , which is the output of $x _ { d }$ after $E$ , through a 5-layer MLP. The final formulation of latent motion representation for each $x _ { s }$ and $x _ { d }$ is thus given as ", + "bbox": [ + 174, + 391, + 825, + 443 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/ab23dd7519062548f3ca731d261f453a3c7b9dbc7fa310e1f0a65e6770d781b7.jpg", + "text": "$$\nz _ { s d } = z _ { s r } + \\sum _ { i = 1 } ^ { M } a _ { i } { \\bf d _ { i } } .\n$$", + "text_format": "latex", + "bbox": [ + 411, + 445, + 586, + 489 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 LATENT CODE DRIVEN IMAGE ANIMATION ", + "text_level": 1, + "bbox": [ + 176, + 503, + 508, + 517 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Once we obtain $z _ { s d }$ , in our second step, we use $G$ to decode a flow field $\\phi _ { s \\to d }$ and warp $x _ { s }$ . Our $G$ consists of two components, a flow field generator $G _ { f }$ and a refinement network $G _ { r }$ (we provide details in App. A). ", + "bbox": [ + 176, + 527, + 823, + 568 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Towards learning multi-scale features, $G$ is designed as a residual network containing $N$ models to produce a pyramid of flow fields $\\phi _ { s \\to d } = \\{ \\bar { \\phi _ { i } } \\bar \\} _ { 1 } ^ { N }$ in different layers of $G _ { f }$ . Multi-scale source features $x _ { s } ^ { e n c } = \\{ x _ { i } ^ { e n c } \\} _ { 1 } ^ { N }$ are obtained from $E$ and are warped in $G _ { f }$ . ", + "bbox": [ + 173, + 571, + 825, + 618 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "However, as pointed out by Siarohin et al. (2019), only relying on $\\phi _ { s \\to d }$ to warp source features is insufficient to precisely reconstruct driving images due to the existing occlusions in some positions of $x _ { s }$ . In order to predict pixels in those positions, the network is required to inpaint the warped feature maps. Therefore, we predict multi-scale masks $\\{ m _ { i } \\} _ { 1 } ^ { N }$ along with $\\{ \\phi _ { i } \\} _ { 1 } ^ { N }$ in $G _ { f }$ to mask out the regions required to be inpainted. In each residual module, we have ", + "bbox": [ + 173, + 622, + 825, + 690 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/f1fed0d72e23b786d5fd7c6ab0d7b925e01797b6981cab28e1c7543451d43876.jpg", + "text": "$$\nx _ { i } ^ { \\prime } = \\mathcal { T } ( \\phi _ { i } , x _ { i } ^ { e n c } ) \\odot m _ { i } ,\n$$", + "text_format": "latex", + "bbox": [ + 416, + 693, + 578, + 710 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\odot$ denotes the Hadamard product and $\\tau$ denotes warping operation, whereas $\\boldsymbol { x } _ { i } ^ { \\prime }$ signifies the masked features. We generate both dense flow fields, as well as masks by letting each residual module output a 3-channel feature map in which the first two channels represent ${ \\bar { \\phi } } _ { i }$ and the last channel $m _ { i }$ . Based on an inpainted feature map $f ( x _ { i } ^ { \\prime } )$ , as well as an upsampled image $g ( x _ { i - 1 } )$ provided by the previous module in $G _ { r }$ , the RGB image from each module is given by ", + "bbox": [ + 173, + 714, + 825, + 779 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/601a95e6281a7fb5e685caccc2b4354ef2a527cee69d14b6f4fba9163185cdda.jpg", + "text": "$$\no _ { i } = f ( x _ { i } ^ { \\prime } ) + g ( o _ { i - 1 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 423, + 781, + 573, + 799 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $f$ and $g$ denote the inpainting and upsampling layers, respectively. The output image $o _ { N }$ from the last module constitutes the final generated image $x _ { s \\to d } = o _ { N }$ . ", + "bbox": [ + 171, + 801, + 825, + 829 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 842, + 290, + 856 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We train LIA in a self-supervised manner to reconstruct $x _ { d }$ using three losses, i.e., a reconstruction loss $\\mathcal { L } _ { r e c o n }$ , a perceptual loss $\\mathcal { L } _ { v g g }$ (Johnson et al., 2016) and an adversarial loss $\\mathcal { L } _ { a d v }$ . We use $\\mathcal { L } _ { r e c o n }$ to minimize the pixel-wise $\\mathbb { L } _ { 1 }$ distance between $x _ { d }$ and $x _ { s \\to d }$ , calculated as ", + "bbox": [ + 173, + 867, + 825, + 906 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/cbb3836f85ac8c8ad821df8ecc4c383d89ef22c12144dd5d156a4e64e232c07d.jpg", + "text": "$$\n\\mathcal { L } _ { r e c o n } ( x _ { s d } , x _ { d } ) = \\mathbb { E } [ \\| x _ { d } - x _ { s d } \\| _ { 1 } ] .\n$$", + "text_format": "latex", + "bbox": [ + 366, + 909, + 630, + 926 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Towards minimizing the perceptual distance, we apply a VGG19-based $\\mathcal { L } _ { v g g }$ on multi-scale feature maps between real and generated images, written as ", + "bbox": [ + 174, + 103, + 823, + 131 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/f7928c756c955cc141db6fcd558a53236b4a94a707d3bc71ec40d8ed94509e02.jpg", + "text": "$$\n\\mathcal { L } _ { v g g } ( x _ { s d } , x _ { d } ) = \\mathbb { E } [ \\sum _ { n } ^ { N } \\| F _ { n } ( x _ { d } ) - F _ { n } ( x _ { s d } ) \\| _ { 1 } ] ,\n$$", + "text_format": "latex", + "bbox": [ + 326, + 131, + 669, + 174 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $F _ { n }$ denotes the $n ^ { t h }$ layer in a pre-trained VGG19 (Simonyan & Zisserman, 2015). In practice, towards penalizing real and generated images in multi-scale images, we use a pyramid of four resolutions, namely $2 5 6 \\times 2 5 6$ , ${ \\bar { 1 } } 2 8 \\times 1 2 8$ , $6 4 \\times 6 4$ and $3 2 \\times 3 2$ as inputs of VGG19. The final perceptual loss is the addition of perceptual losses in four resolutions. ", + "bbox": [ + 173, + 176, + 825, + 229 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Further, towards generating photo-realistic results, we incorporate a non-saturating adversarial loss $\\mathcal { L } _ { a d v }$ on $x _ { s \\to d }$ , which is calculated as ", + "bbox": [ + 171, + 234, + 823, + 261 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/05bea7d6e6d22c447e74d62df9efb0cfa6ed3c06ed8234e0270924bc5b87f9da.jpg", + "text": "$$\n\\mathcal { L } _ { a d v } ( x _ { s d } ) = \\mathbb { E } _ { x _ { s d } \\sim p _ { r e c } } [ - l o g ( D ( x _ { s d } ) ] ,\n$$", + "text_format": "latex", + "bbox": [ + 348, + 261, + 647, + 279 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $D$ is a discriminator, aimed at distinguishing reconstructed images from the original ones. Our full loss function is the combination of three losses with $\\lambda$ as a balanced hyperparameter ", + "bbox": [ + 171, + 280, + 823, + 305 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/98a99248b9ea0369084c2ed86f66e8b66265c9eda359ab2a875e641b6da0b3c9.jpg", + "text": "$$\n{ \\mathcal { L } } ( x _ { s d } , x _ { d } ) = { \\mathcal { L } } _ { r e c o n } ( x _ { s d } , x _ { d } ) + \\lambda { \\mathcal { L } } _ { v g g } ( x _ { s d } , x _ { d } ) + { \\mathcal { L } } _ { a d v } ( x _ { s d } ) .\n$$", + "text_format": "latex", + "bbox": [ + 264, + 305, + 732, + 323 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.4 INFERENCE ", + "text_level": 1, + "bbox": [ + 174, + 335, + 295, + 349 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In inference stage, given a driving video sequence $V _ { d } = \\{ x _ { t } \\} _ { 1 } ^ { T }$ , we aim to transfer motion from $V _ { d }$ to $x _ { s }$ , in order to generate a novel video $V _ { d s } = \\{ x _ { t s } \\} _ { 1 } ^ { T }$ . If $V _ { d }$ and $x _ { s }$ stem from the same video sequence, i.e., $x _ { s } ~ = ~ x _ { 1 }$ , our task comprises of reconstructing the entire original video sequence. Therefore, we construct the latent motion representation of each frame using absolute transfer, which follows the training process, given as ", + "bbox": [ + 173, + 361, + 825, + 430 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/906b3d65e6df9ab02a1357d7fd666aa6dc6b2c37473057c00ea3397676e29cf5.jpg", + "text": "$$\nz _ { s t } = z _ { s r } + w _ { r t } , t \\in \\{ 1 , . . . , T \\} .\n$$", + "text_format": "latex", + "bbox": [ + 370, + 430, + 627, + 446 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "However, in real world applications, interest is rather placed on the scenario, where motion transfer between $x _ { s }$ and $V _ { d }$ , the latter stemming from different identities, i.e., $x _ { s } \\neq x _ { 1 }$ . Taking a talking head video as an example, in this setting, beyond identity, $x _ { 1 }$ and $x _ { s }$ might also differ in pose and expression. Therefore, we propose relative transfer to eliminate the motion impact of $w _ { r \\to 1 }$ and involve motion of $w _ { r \\to s }$ in the full generated video sequence. Owing to a linear representation of the latent path, we can easily represent $z _ { s t }$ for each frame as ", + "bbox": [ + 173, + 454, + 825, + 531 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/ffd2269604154178f057dda4081be44f9ef04e2bc5ecbbf0667ba81e6b26d159.jpg", + "text": "$$\n\\begin{array} { r l } & { z _ { s t } = ( z _ { s r } + w _ { r s } ) + ( w _ { r t } - w _ { r 1 } ) } \\\\ & { \\qquad = z _ { s s } + ( w _ { r t } - w _ { r 1 } ) , \\ t \\in \\{ 1 , . . . , T \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 334, + 532, + 663, + 569 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The first term in Eq. (12), $z _ { s s }$ indicates the reconstruction of $x _ { s }$ , while the second term $( w _ { r t } - w _ { r 1 } )$ represents the motion from $x _ { 1 }$ to $x _ { t }$ . This equation indicates that the original pose is preserved in $x _ { s }$ , at the same time motion is transferred from $V _ { d }$ . We note that in order to completely replicate the position and pose in $V _ { d }$ , it requires $x _ { s }$ and $x _ { 1 }$ to contain similar poses in relative motion transfer. ", + "bbox": [ + 173, + 569, + 825, + 633 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 655, + 326, + 670 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we firstly describe our experimental setup including implementation details and datasets. Secondly, we qualitatively demonstrate generated results based on testing datasets. Then, we provide quantitative evaluation w.r.t. image quality on (a) same-identity reconstruction, (b) crossvideo motion transfer, presenting (c) a user study. Next, we conduct an ablation study that demonstrates (d) the effectiveness of our proposed motion dictionary, as well as (e) associated size. Finally, we provide an in-depth analysis of our (f) latent codes and (g) motion dictionary to interpret their semantic meanings. ", + "bbox": [ + 173, + 690, + 825, + 781 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Datasets Our model is trained on the datasets VoxCeleb, TaichiHD and TED-talk. We follow the pre-processing method in (Siarohin et al., 2019) to crop frames into $2 5 6 \\times 2 5 6$ resolution for quantitative evaluation. ", + "bbox": [ + 173, + 794, + 821, + 833 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Implementation details Our model is implemented in PyTorch (Paszke et al., 2019). All models are trained on four 16G NVIDIA V100 GPUs. The total batch size is 32 with 8 images per GPU. We use a learning rate of 0.002 to train our model with the Adam optimizer (Kingma & Ba, 2014). The dimension of all latent codes, as well as directions in $D _ { m }$ is set to be 512. In our loss function, we use $\\lambda = 1 0$ in order to penalize more on the perceptual loss. It takes around 150 hours to fully train our framework. ", + "bbox": [ + 174, + 845, + 825, + 922 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/a37a7371bb03be1d7874e44a06664ca4dc3ef0bc6d060aab959b38245f9df2b2.jpg", + "image_caption": [ + "Figure 4: Qualitative results. Examples for same-dataset absolute motion transfer on TaichiHD (top-right) and TED-talk (bottom-right). On VoxCeleb (left), we demonstrate cross-dataset relative motion transfer. We successfully transfer motion between $x _ { 1 }$ and $x _ { t }$ from videos in VoxCeleb to $x _ { s }$ from FFHQ, the latter not being used for training. " + ], + "image_footnote": [], + "bbox": [ + 174, + 101, + 823, + 335 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Evaluation metrics We evaluate our model w.r.t. (i) reconstruction faithfulness using $\\mathcal { L } _ { 1 }$ , LPIPS, (ii) generated video quality using video FID, as well as (iii) semantic consistency using average keypoint distance (AKD), missing keypoint rate (MKR) and average euclidean distance (AED). Details are available in App. B.2. ", + "bbox": [ + 174, + 417, + 825, + 469 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 QUALITATIVE RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 487, + 377, + 501 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Firstly, we evaluate the ability of LIA to generate realistic videos and compare related results with four state-of-the-art methods. For TaichiHD and TED-talk datasets, we conduct an experiment related to cross-video generation. Corresponding results (see Fig. 4) confirm that our method is able to correctly transfer motion on articulated human bodies, in the absence of explicit structure representations. For the VoxCeleb dataset, we conduct a cross-dataset generation-experiment, where we transfer motion from VoxCeleb to images of the FFHQ dataset. We observe that our method outperforms FOMM and MRAA w.r.t. image quality, as both approaches visibly deform the shape of the original faces. This is specifically notable in the case that source and driving images entail large pose variations. At the same time, LIA is able to successfully tackle this challenge and no similar deformations are visible. ", + "bbox": [ + 174, + 513, + 825, + 640 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 COMPARISON WITH STATE-OF-THE-ART METHODS ", + "text_level": 1, + "bbox": [ + 174, + 659, + 566, + 672 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We quantitatively compare our method with the state-of-the-art approaches X2Face, Monkey-Net, FOMM and MRAA on two tasks, namely (a) same-identity reconstruction and (b) cross-video motion transfer. Additionally, we conduct a (c) user study. ", + "bbox": [ + 174, + 684, + 823, + 723 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "(a) Same-identity reconstruction We here evaluate the reconstruction ability of our method. Specifically, we reconstruct each testing video by using the first frame as $x _ { s }$ and the remaining frames as $x _ { d }$ . Results on three datasets are reported in Table 1. Focusing on foregroundreconstruction, our method outperforms the other approaches w.r.t. all metrics. More results are presented in App. B.3, discussing background-reconstruction. ", + "bbox": [ + 174, + 741, + 823, + 804 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "(b) Cross-video motion transfer Next, we conduct experiments, where source images and driving videos stem from different video sequences. In this context, we mainly focus on evaluating talking head videos and explore two different cases. In the first case, we generate videos using the VoxCeleb testing set to conduct motion transfer. In the second case, source images are from an unseen dataset, namely the GermanPublicTV dataset, as we conduct cross-dataset motion transfer. In both experiments, we randomly construct source and driving pairs and transfer motion from driving videos to source images to generate a novel manipulated dataset. Since ground truth data for our generated videos is not available, we use video FID (as initialized by Wang et al. (2020a)) to compute the distance between generated and real data distributions. As shown in Tab. 2, our method outperforms all other approaches w.r.t. video FID, indicating the best generated video quality. ", + "bbox": [ + 174, + 820, + 823, + 922 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/c2aa88b5ad472a0d7b3a5f9284ee27dd781681e2c7f2352efd69236a7987ccec.jpg", + "table_caption": [ + "Table 1: Same-identity reconstruction. Comparison with state-of-the-art methods on three datasets for same-identity reconstruction. " + ], + "table_footnote": [], + "table_body": "
MethodVoxCelebTaichiHDTED-talks
L1AKDAEDLPIPSL1(AKD,MKR)AEDLPIPSL1(AKD,MKR)AEDLPIPS
X2Face0.0787.6870.405-0.080(17.654,0.109)-=-
Monkey-Net0.0491.8780.199-0.077(10.798,0.059)==
FOMM0.0461.3950.1410.1360.063(6.472,0.032)0.4950.1910.030(3.759,0.0090)0.4280.13
MRAA w/o bg0.0431.3070.1400.1270.063(5.626,0.025)0.4600.1890.029(3.126,0.0092)0.3960.12
Ours0.0411.3530.1380.1230.057(4.823, 0.020)0.4310.1800.027(3.141,0.0095)0.3990.11
", + "bbox": [ + 178, + 143, + 821, + 219 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/ad274f8070fac87f953923f2ff42c22d786ec650ba57cc10b4221205e6b9fe26.jpg", + "table_caption": [ + "Table 2: Cross-video generation. We report video FID for both inner- and cross-dataset tasks on VoxCeleb and GermanPublicTV. " + ], + "table_footnote": [], + "table_body": "
VoxCelebGermanPublicTV
FOMM0.323 0.456
MRAA 0.3080.454
Ours0.161 0.406
", + "bbox": [ + 199, + 295, + 441, + 349 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/a132cc7dfc5907d4d643154a3c90f40b6d770be090a737eb77630cf2dfb3443d.jpg", + "table_caption": [ + "Table 3: User study. We ask 20 human raters to conduct a subjective video quality evaluation. " + ], + "table_footnote": [], + "table_body": "
VoxCeleb(%)TaichiHD(%)TED-talk(%)
Ours/FOMM92.9/7.164.5/35.571.4/28.6
Ours/MRAA89.7/10.360.7/39.954.8/45.2
", + "bbox": [ + 496, + 304, + 813, + 342 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 363, + 825, + 390 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(c) User study We conduct a user study to evaluate video quality. Towards this, we displayed paired videos and asked 20 human raters ‘which clip is more realistic?’. Each video-pair contains a generated video from our method, as well as a video generated from FOMM or MRAA. Results suggest that our results are more realistic in comparison to FOMM and MRAA across all three datasets (see Tab. 3). Hence, the obtained human preference is in accordance with our quantitative evaluation. ", + "bbox": [ + 174, + 404, + 825, + 481 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/f09a5c2b6459de9d6ea611e9c6cf079f6a6df9128278bcecbdc537633eda7d9a.jpg", + "table_caption": [ + "Table 4: Ablation study on motion dictionary. We conduct experiments on three datasets with and without $D _ { m } ^ { \\phantom { \\dagger } }$ and show reconstruction results. " + ], + "table_footnote": [], + "table_body": "
VoxCelebTaichiHDTED-talks
MethodL1 LPIPSL1LPIPSL1LPIPS
wloDm0.049 0.1650.0620.1860.0310.12
Full0.041 0.1230.0570.1800.0280.11
", + "bbox": [ + 179, + 546, + 464, + 604 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/e356340e7653eeceefae4355bd4672591605f3b6600bca605a5f9eb9c4ad26f3.jpg", + "table_caption": [ + "Table 5: Ablation study on $D _ { m }$ size. We conduct experiments on three datasets with 5 different $D _ { m }$ size and show reconstruction results. " + ], + "table_footnote": [], + "table_body": "
VoxCelebTaichiHDTED-talks
ML1LPIPSL1LPIPSL1LPIPS
5 100.051 0.0430.15 0.130.070 0.0650.22 0.200.037 0.0360.15
200.0410.120.0570.180.0280.13 0.11
400.0420.120.0600.190.0300.12
1000.0410.120.0580.180.0280.11
", + "bbox": [ + 522, + 532, + 785, + 617 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 ABLATION STUDY ", + "text_level": 1, + "bbox": [ + 174, + 633, + 339, + 647 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We here analyze our proposed motion dictionary and focus on answering following two questions. ", + "bbox": [ + 174, + 657, + 815, + 672 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(d) Is the motion dictionary $D _ { m }$ beneficial? We here explore the impact of proposed $D _ { m }$ , by training our model without $D _ { m }$ . Specifically, we output $w _ { r \\to d }$ directly from MLP, without using LMD to learn an orthogonal basis. From the evaluation results reported in Tab. 4 and qualitative results in App. B.5, we observe that in the absence of $D _ { m }$ , model fails to generate high-quality images, which proves the effectiveness of $D _ { m }$ , consistently on all datasets. ", + "bbox": [ + 173, + 685, + 823, + 751 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(e) How many directions are required in $D _ { m }$ ? Towards finding an effective size of $D _ { m }$ , we empirically test three different $M$ , viz. 5, 10, 20, 40 and 100. Quantitative results in Tab. 5 show that when using 20 directions, the model achieves the best reconstruction results. ", + "bbox": [ + 174, + 766, + 821, + 805 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.4 FURTHER ANALYSIS ", + "text_level": 1, + "bbox": [ + 174, + 821, + 354, + 835 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(f) Latent code analysis. While our method successfully transfers motion via latent space navigation, we here aim at answering the question — what does $x _ { r }$ represent? Towards answering this question, we proceed to visualize $x _ { r }$ . We firstly decode $z _ { s r }$ into a dense flow field $\\phi _ { s r }$ , which is then used to warp $x _ { s }$ (we show details in App. B.4). Fig. 5 shows examples of $x _ { s }$ and $x _ { r }$ . Interestingly, we observe that $x _ { r }$ represents the canonical pose of $x _ { s }$ , regardless of original poses of the subjects. And for all datasets, reference images resemble each other $w . r . t .$ . pose and scale. As such reference images can be considered as a normalized form of $x _ { s }$ , learning transformations between $x _ { s }$ and $x _ { d }$ using $x _ { s } x _ { r } x _ { d }$ is considerably more efficient than $x _ { s } \\to x _ { d }$ , once $x _ { r }$ is fixed. ", + "bbox": [ + 173, + 847, + 823, + 922 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/f4021d1e9c0a6b78d5507518e6f476c3eb214d4b2fd88cb1cbad903d30c7d449.jpg", + "image_caption": [ + "Figure 5: Visualization of reference images. Example source (top) and reference images (down) from VoxCeleb, TaichiHD and TED-talk datasets. Our network learns reference images of a consistently frontal pose, systematically for all input images of each dataset. " + ], + "image_footnote": [], + "bbox": [ + 174, + 102, + 823, + 267 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/833c41d15a70be2fc45b2ced129969f51412a6047f24acd5518917c81c6c810d.jpg", + "image_caption": [ + "Figure 6: Linear manipulation of four motion directions on the painting of Mona Lisa. Manipulated results indicate that $d _ { 6 }$ represents eye movement, $d _ { 8 }$ represents head nodding, whereas $\\bar { d _ { 1 9 } }$ and $d _ { 7 }$ represent facial expressions. " + ], + "image_footnote": [], + "bbox": [ + 181, + 333, + 816, + 464 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 531, + 821, + 558 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Noteworthy, we found the similar idea of learning a ‘reference image’ has also been explored by Siarohin et al. (2019) (FOMM) and Wiles et al. (2018) (X2Face). However, deviating from our visualized ‘reference image’, the ’reference image ’in FOMM refers to a non-visualized and abstract concept. In addition, LIA only requires a latent code $z _ { s r }$ , rather than the ’reference image’ for both, training and testing, which is contrast to X2Face. ", + "bbox": [ + 174, + 564, + 823, + 628 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(g) Motion dictionary interpretation. Towards further interpretation of directions in $D _ { m }$ , we conduct linear manipulations on each $d _ { i }$ . Images pertained to manipulating four motion directions are depicted in Fig. 6. The results suggest that the directions in $\\boldsymbol { D } _ { m } ^ { \\bar { 2 } }$ are semantically meaningful, as they represent basic visual transformations such as head nodding $( d _ { 8 } )$ , eye movement $( d _ { 6 } ) ^ { \\dag }$ and facial expressions $\\cdot d _ { 1 9 }$ and $d _ { 7 }$ ). More results can be found on our project webpage2. ", + "bbox": [ + 174, + 645, + 825, + 710 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 732, + 328, + 748 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we presented a novel self-supervised autoencoder LIA, aimed at animating images via latent space navigation. By the proposed Linear Motion Decomposition (LMD), we were able to formulate the task of transferring motion from driving videos to source images as learning linear transformations in the latent space. We evaluated proposed method on real-world videos and demonstrated that our approach is able to successfully animate still images, while eliminating the necessity of explicit structure representations. In addition, we showed that the incorporated motion dictionary is interpretable and contains directions pertaining to basic visual transformations. Both quantitative and qualitative evaluations showed that LIA outperforms state-of-art algorithms on all benchmarks. We postulate that LIA opens a new door in design of interpretable generative models for video generation. ", + "bbox": [ + 174, + 768, + 825, + 896 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ETHIC STATEMENT ", + "text_level": 1, + "bbox": [ + 176, + 102, + 333, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this work, we aim to synthesize high-quality videos by transferring motion on still images. Our approach can be used for movie production, making video games, online education, generating synthetic data for other computer vision tasks, etc. We note that our framework mainly focuses on learning how to model motion distribution rather than directly model appearance, therefore it is not biased towards any specific gender, race, region, or social class. It works equally well irrespective of the difference in subjects. ", + "bbox": [ + 174, + 138, + 825, + 215 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REPRODUCIBILITY STATEMENT ", + "text_level": 1, + "bbox": [ + 176, + 237, + 437, + 252 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We assure that all the results shown in the paper and supplemental materials can be reproduced. We intend to open-source our code, as well as trained models. ", + "bbox": [ + 174, + 273, + 825, + 300 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 323, + 367, + 338 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This work was granted access to the HPC resources of IDRIS under the allocation AD011011627R1. It was supported by the French Government, by the National Research Agency (ANR) under Grant ANR-18-CE92-0024, project RESPECT and through the 3IA Cote d’Azur Investments in the Future ˆ project managed by the National Research Agency (ANR) with the reference number ANR-19- P3IA-0002. ", + "bbox": [ + 174, + 358, + 825, + 422 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 445, + 285, + 460 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Rameen Abdal, Yipeng Qin, and Peter Wonka. Image2stylegan $^ { + + }$ : How to edit the embedded images? In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020. ", + "bbox": [ + 173, + 481, + 821, + 520 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Brandon Amos, Bartosz Ludwiczuk, and Mahadev Satyanarayanan. Openface: A general-purpose face recognition library with mobile applications. 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Learning to forecast and refine residual motion for image-to-video generation. In ECCV, 2018. ", + "bbox": [ + 174, + 387, + 825, + 416 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Zhedong Zheng, Tao Ruan, Yunchao Wei, Yi Yang, and Tao Mei. Vehiclenet: Learning robust visual representation for vehicle re-identification. IEEE Transaction on Multimedia (TMM), 2020. ", + "bbox": [ + 173, + 422, + 823, + 450 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Jiapeng Zhu, Yujun Shen, Deli Zhao, and Bolei Zhou. In-domain gan inversion for real image editing. In Proceedings of European Conference on Computer Vision (ECCV), 2020. ", + "bbox": [ + 173, + 458, + 823, + 486 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A DETAILS OF MODEL ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 176, + 103, + 511, + 117 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We proceed to describe the model architecture in this section. Fig. 7 shows details of our $E$ . In each ResBlock in $E$ , spatial size of input feature maps are downsampled. We take feature maps of spatial sizes from $8 \\times 8$ to $2 5 6 \\times 2 5 6$ as our appearance features $x _ { i } ^ { e n c }$ . We use a 5-layer MLP to predict a magnitude vector $A _ { d r }$ from $z _ { d r }$ . Fig. 8 (a) shows the general architecture of our $G$ , which consists of two components, a flow field generator $G _ { f }$ and a refinement network $G _ { r }$ . We apply StyleConv (Upsample $\\mathbf { + \\ C o n v 3 \\times 3 }$ ), which is proposed by StyleGAN2, in $G _ { f }$ . StyleConv takes latent representation $z _ { s t }$ as style code and generates flow field $\\phi _ { i }$ and corresponding mask $m _ { i }$ . $G _ { r }$ uses UpConv (Conv1 $\\times \\ 1 +$ Upsample) to upsample and refine inpainted feature maps to target resolution. We show details pertaining to $G$ block in Fig. 8 (b). Each $G$ block is used to upsample $\\times 2$ the previous resolution. We stack 6 blocks towards producing 256 resolution images. ", + "bbox": [ + 173, + 138, + 825, + 266 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/9325fffdbfebc0ee68f0be3ae9d39d9a1ea8e2cd0cc42b57b0799a99bd6ea9c8.jpg", + "image_caption": [ + "Figure 7: Encoder architecture. We show details of architecture of $E$ in (a) and ResBlock in (b). " + ], + "image_footnote": [], + "bbox": [ + 178, + 300, + 826, + 694 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 765, + 330, + 781 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We proceed to introduce details of datasets and evaluation metrics used in our experiments. ", + "bbox": [ + 171, + 801, + 769, + 816 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B.1 DATASETS ", + "text_level": 1, + "bbox": [ + 174, + 842, + 290, + 857 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "VoxCeleb (Nagrani et al., 2019) consists of a large amount of interview videos of different celebrities. Following the process of FOMM (Siarohin et al., 2019), we extract frames and crop them into $2 5 6 \\times 2 5 6$ resolution. In total, VoxCeleb contains a training set of 17928 videos and a test set of 495 videos. ", + "bbox": [ + 174, + 871, + 825, + 922 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/9bfd82d4cf08f228e79a896a311bfc79b0ee414695a25bedad52ccbc1330672a.jpg", + "image_caption": [ + "Figure 8: Generator architecture. We show details about architecture of $G$ in (a) and $G$ block in (b). " + ], + "image_footnote": [], + "bbox": [ + 183, + 99, + 821, + 411 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "TaiChiHD (Siarohin et al., 2019) consists of videos of full human bodies performing Tai Chi actions. We follow the original pre-processing of FOMM (Siarohin et al., 2019) and utilize its $2 5 6 \\times 2 5 6$ version. TaiChiHD contains 1096 training videos and 115 testing videos. ", + "bbox": [ + 173, + 468, + 823, + 508 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "TED-talk is a new dataset proposed in MRAA (Siarohin et al., 2021). It comprises a number of TED-talk videos, where the main subjects have been cropped out. We resize the original version into $2 5 6 \\times 2 5 6$ resolution to train our model. This dataset includes 1124 training videos and 130 testing videos. ", + "bbox": [ + 174, + 513, + 825, + 565 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B.2 EVALUATION METRICS ", + "text_level": 1, + "bbox": [ + 176, + 582, + 375, + 597 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We use five different metrics to evaluate our experimental results, namely $\\mathcal { L } _ { 1 }$ , LPIPS, AKD, MKR and AED that quantify the reconstructed results. In addition, we compute video FID to evaluate video quality in motion transferring tasks. ", + "bbox": [ + 176, + 608, + 823, + 647 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "$\\mathcal { L } _ { 1 }$ represents the mean absolute pixel difference between reconstructed and real videos. ", + "bbox": [ + 174, + 652, + 750, + 666 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "LPIPS (Zhang et al., 2018) aims at measuring the perceptual similarity between reconstructed and real images by leveraging the deep features from AlexNet (Krizhevsky et al., 2012). ", + "bbox": [ + 173, + 672, + 826, + 699 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Video FID is a modified version of the original FID (Heusel et al., 2017). We here follow the same implementation as Wang et al. (2020a) and utilize a pre-trained ResNext101 (Hara et al., 2018) to extract spatio-temporal features to compute the distance between real and generated videos distributions. We take the first 100 frames of each video as input of the feature-extractor to compute the final scores. ", + "bbox": [ + 174, + 705, + 825, + 770 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Average keypoint distance (AKD) and missing keypoint rate (MKR) evaluate the difference between keypoints of reconstructed and ground truth videos. We extract landmarks using the face alignment approach of (Bulat & Tzimiropoulos, 2017) and extract body poses for both TaiChiHD and TED-talks using OpenPose (Cao et al., 2019). AKD is computed as the average distance between corresponding keypoints, whereas MKR is the proportion of keypoints present in the groundtruth that are missing in a reconstructed video. ", + "bbox": [ + 173, + 776, + 823, + 853 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Average Euclidean distance (AED) measures the ability of preserving identity in reconstructed video. We use a person re-identification pretrained model (Zheng et al., 2020) for measuring human bodies (TaichiHD and TED-talk) and OpenFace (Amos et al., 2016) for faces to extract identity embeddings from reconstructed and ground truth frame pairs, then we compute MSE of their difference for all pairs. ", + "bbox": [ + 174, + 859, + 823, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B.3 COMPARISON WITH FULL MRAA ", + "text_level": 1, + "bbox": [ + 176, + 103, + 449, + 117 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We show quantitative evaluation results with the full MRAA model in Tab. 6. We observe that our method achieves competitive results in reconstruction and keypoint evaluation. While we do not explicitly predict keypoints, w.r.t. the TaichiHD dataset, interestingly we outperform MRAA in both, AKD and MKR. Such results showcase the effectiveness of our proposed method on modeling articulated human structures. However, reconstruction evaluation cannot provide a completely fair comparison on how well the main subjects (e.g., faces and human bodies) are generated in videos. This is in particular the case for TaichiHD and TED-talk, where backgrounds have large contributions to the final scores. ", + "bbox": [ + 173, + 127, + 825, + 231 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/a627194bf94f738dbeb4d2f9efd1eb6f668ed70266fedc4e6e71e0ba390a5374.jpg", + "table_caption": [ + "Table 6: Comparison with full MRAA. " + ], + "table_footnote": [], + "table_body": "
MethodVoxCelebTaichiHDTED-talks
L1AKDAEDLPIPSL1(AKD,MKR)AEDLPIPSL1(AKD,MKR)AEDLPIPS
MRAA0.0411.3030.1350.1240.045(5.551,( ,0.025)0.4310.1780.027(3.107,0.0093)0.3790.11
Ours0.0411.3530.1380.1230.057(4.823,0.020)0.4310.1800.027(3.141,0.0095)0.3990.11
", + "bbox": [ + 199, + 272, + 802, + 319 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B.4 REFERENCE IMAGE GENERATION. ", + "text_level": 1, + "bbox": [ + 178, + 352, + 449, + 364 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "To produce $x _ { r }$ , we use $G$ to decode $z _ { s r }$ into the flow field $\\phi _ { s r }$ . Reference image $x _ { r }$ is obtained by warping $x _ { s }$ using $\\phi _ { s r }$ . The entire process is shown in Fig. 9. ", + "bbox": [ + 174, + 376, + 825, + 402 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/762132d8adc241e8fde2ec32f40c10dbb27f4f06b3953c429f112db181bf10c2.jpg", + "image_caption": [ + "Figure 9: Reference image generation. " + ], + "image_footnote": [], + "bbox": [ + 191, + 415, + 802, + 534 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B.5 QUALITATIVE RESULTS ON EFFECTIVENESS OF USING MOTION DICTIONARY ", + "text_level": 1, + "bbox": [ + 176, + 593, + 738, + 607 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Fig. 10 illustrates the generated results on transferring motion from VoxCeleb to GermanPublicTV with and without motion dictionary. We observe that without the motion dictionary, appearance information is undesirably transferred from driving videos to generated videos. ", + "bbox": [ + 174, + 616, + 825, + 657 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B.6 LIMITATIONS ", + "text_level": 1, + "bbox": [ + 174, + 671, + 310, + 685 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "For human body, one limitation of our method is dealing with body occlusion. We observe in Fig. 11 that in taichi videos, in case of occlusion cause by legs and arms, motion is not transferred successfully. In addition, in TED-talks, transferring hand motion is challenging, as hands are of small size, articulated and sometimes occluded by human bodies. ", + "bbox": [ + 174, + 696, + 825, + 748 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/70acaf34c5c7203b948fb2919b9b562110b48bb7fdf0186567c1baf79c82cd42.jpg", + "image_caption": [ + "Figure 10: Generated results with and without $D _ { m }$ . We observe that the disentanglement of appearance and motion is much better by using $D _ { m }$ . " + ], + "image_footnote": [], + "bbox": [ + 176, + 127, + 823, + 518 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/8df647813e53c9e3f9fda4ddf2534e1d09430b0019731ce68455cebf897027e0.jpg", + "image_caption": [ + "Figure 11: Failure cases. We observed that it is still challenging for LIA to handle arm-leg occlusion (Taichi) and hand motion (TED-talk). 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The two images of Marilyn Monroe and Emmanuel Macron", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 658, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 505, + 671 + ], + "score": 1.0, + "content": "are animated by LIA, which transfers motion of a driving video (smaller images on the top) from", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "VoxCeleb dataset (Chung et al., 2018) onto the still images. LIA is able to successfully animate", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "these two images without relying on any explicit structure representations, such as landmarks and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 689, + 200, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 200, + 702 + ], + "score": 1.0, + "content": "region representations.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + } + ], + "index": 28.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 206, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 208, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 208, + 96 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 505, + 121 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 505, + 121 + ], + "score": 1.0, + "content": "In the series of science fiction books Harry Potter (Rowling et al., 2016; Rowling, 2019), wizards", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 119, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 506, + 133 + ], + "score": 1.0, + "content": "and witches were able to magically enchant portraits, bringing them to life. Remarkable progress", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "score": 1.0, + "content": "of deep generative models has recently turned this vision into reality. This work examines the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 140, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 506, + 152 + ], + "score": 1.0, + "content": "scenario where a framework animates a source image by motion representations learned from a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "score": 1.0, + "content": "driving video. Existing approaches for image animation are classically related to computer graphics", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 160, + 504, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 504, + 171 + ], + "score": 1.0, + "content": "(Cao et al., 2014; Thies et al., 2016; 2019; Zhao et al., 2018) or exploit motion labels (Wang et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "score": 1.0, + "content": "2020b) and structure representations such as semantic maps (Pan et al., 2019; Wang et al., 2018;", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 178, + 506, + 193 + ], + "spans": [ + { + "bbox": [ + 104, + 178, + 506, + 193 + ], + "score": 1.0, + "content": "2019), human keypoints (Jang et al., 2018; Yang et al., 2018; Walker et al., 2017; Chan et al., 2019;", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 189, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 506, + 202 + ], + "score": 1.0, + "content": "Zakharov et al., 2019; Wang et al., 2019; Siarohin et al., 2019), 3D meshes (Liu et al., 2019; Chen", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "score": 1.0, + "content": "et al., 2021), and optical flows (Li et al., 2018; Ohnishi et al., 2018). We note that the ground truth", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 208, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 506, + 224 + ], + "score": 1.0, + "content": "of such structure representations has been computed a-priori for the purpose of supervised training,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "which poses constraints on applications, where such representations of unseen testing images might", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 229, + 255, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 255, + 242 + ], + "score": 1.0, + "content": "be fragmentary or difficult to access.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "Self-supervised motion transfer approaches (Wiles et al., 2018; Siarohin et al., 2019; 2021) accept", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "raw videos as input and learn to reconstruct driving images by warping source image with pre-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "dicted dense optical flow fields. While the need for domain knowledge or labeled ground truth data", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 276, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 287 + ], + "score": 1.0, + "content": "has been obviated, which improves performance on in-the-wild testing images, such methods entail", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "necessity of explicit structure representations as motion guidance. Prior information such as key-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 295, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 309 + ], + "score": 1.0, + "content": "points (Siarohin et al., 2019; Wang et al., 2021a) or regions (Siarohin et al., 2021) are learned in an", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 306, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 317 + ], + "score": 1.0, + "content": "end-to-end training manner by additional networks as intermediate features, in order to predict target", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "flow fields. 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To the best of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 373 + ], + "score": 1.0, + "content": "our knowledge, this constitutes a new direction in the context of image animation. Our work is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 371, + 504, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 504, + 383 + ], + "score": 1.0, + "content": "motivated by interpretation of GANs (Shen et al., 2020; Goetschalckx et al., 2019; Jahanian et al.,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 380, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 394 + ], + "score": 1.0, + "content": "2020; Voynov & Babenko, 2020), showcasing that latent spaces of StyleGAN (Karras et al., 2019;", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 390, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 404 + ], + "score": 1.0, + "content": "2020b) and BigGAN (Brock et al., 2019) contain rich semantically meaningful directions. Given", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 400, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 414 + ], + "score": 1.0, + "content": "that walking along such directions, basic visual transformations such as zooming and rotation can", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "be induced in generated results. As in image animation, we have that motion between source and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "driving images can be considered as higher-level transformation, a natural question here arises: can", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 431, + 504, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 504, + 444 + ], + "score": 1.0, + "content": "we discover a set of directions in the latent space that induces high-level motion transformations", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 441, + 174, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 174, + 452 + ], + "score": 1.0, + "content": "collaboratively?", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 457, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 506, + 469 + ], + "score": 1.0, + "content": "Towards answering this question, we introduce LIA, a novel Latent Image Animator constituting of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "score": 1.0, + "content": "an autoencoder for animating still images via latent space navigation. LIA seeks to animate a source", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "score": 1.0, + "content": "image via linearly navigating associated source latent code along a learned path to reach a target", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 487, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 499 + ], + "score": 1.0, + "content": "latent code, which represents the high-level transformation for animating the source image. We", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "introduce a Linear Motion Decomposition (LMD) approach aiming to represent a latent path via a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "score": 1.0, + "content": "linear combination of a set of learned motion directions and associated magnitudes. Specifically, we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 518, + 504, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 504, + 529 + ], + "score": 1.0, + "content": "constrain the set as an orthogonal basis, where each vector indicates a basic visual transformation.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "score": 1.0, + "content": "By describing the whole motion space using such learned basis, LIA eliminates the requirement of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 538, + 241, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 241, + 549 + ], + "score": 1.0, + "content": "explicit structure representations.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 553, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "score": 1.0, + "content": "In addition, we design LIA to disentangle motion and appearance within a single encoder-generator", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 563, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 574 + ], + "score": 1.0, + "content": "architecture. Deviating from existing methods using separate networks to learn disentangled fea-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "score": 1.0, + "content": "tures, LIA integrates both, latent motion code, as well as appearance features in a single encoder,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 581, + 378, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 378, + 597 + ], + "score": 1.0, + "content": "which highly reduces the model complexity and simplifies training.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 506, + 612 + ], + "score": 1.0, + "content": "We provide evaluation on multiple datasets including VoxCeleb (Chung et al., 2018),", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 609, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 620 + ], + "score": 1.0, + "content": "TaichiHD (Siarohin et al., 2019) and TED-talk (Siarohin et al., 2021). In addition, we show that", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "LIA outperforms the state-of-the-art in preserving the facial structure in generated videos in the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 629, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 640 + ], + "score": 1.0, + "content": "setting of one-shot image animation on unseen datasets such as FFHQ (Karras et al., 2019) and", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 638, + 260, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 260, + 651 + ], + "score": 1.0, + "content": "GermanPublicTV (Thies et al., 2020).", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 49 + }, + { + "type": "title", + "bbox": [ + 108, + 667, + 208, + 679 + ], + "lines": [ + { + "bbox": [ + 104, + 665, + 211, + 682 + ], + "spans": [ + { + "bbox": [ + 104, + 665, + 211, + 682 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 52 + }, + { + "type": "text", + "bbox": [ + 107, + 696, + 504, + 727 + ], + "lines": [ + { + "bbox": [ + 105, + 695, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 505, + 708 + ], + "score": 1.0, + "content": "Video generation GAN-based video generation is aimed at mapping Gaussian noise to video, di-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 705, + 504, + 717 + ], + "spans": [ + { + "bbox": [ + 105, + 705, + 504, + 717 + ], + "score": 1.0, + "content": "rectly and in the absence of prior information (Vondrick et al., 2016; Saito et al., 2017; Tulyakov", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 715, + 505, + 728 + ], + "spans": [ + { + "bbox": [ + 105, + 715, + 505, + 728 + ], + "score": 1.0, + "content": "et al., 2018; Wang et al., 2020a; Wang, 2021). Approaches based on deep probabilistic models (Den-", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 54 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 206, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 208, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 208, + 96 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 505, + 121 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 505, + 121 + ], + "score": 1.0, + "content": "In the series of science fiction books Harry Potter (Rowling et al., 2016; Rowling, 2019), wizards", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 119, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 506, + 133 + ], + "score": 1.0, + "content": "and witches were able to magically enchant portraits, bringing them to life. Remarkable progress", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "score": 1.0, + "content": "of deep generative models has recently turned this vision into reality. This work examines the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 140, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 506, + 152 + ], + "score": 1.0, + "content": "scenario where a framework animates a source image by motion representations learned from a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "score": 1.0, + "content": "driving video. Existing approaches for image animation are classically related to computer graphics", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 160, + 504, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 504, + 171 + ], + "score": 1.0, + "content": "(Cao et al., 2014; Thies et al., 2016; 2019; Zhao et al., 2018) or exploit motion labels (Wang et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "score": 1.0, + "content": "2020b) and structure representations such as semantic maps (Pan et al., 2019; Wang et al., 2018;", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 178, + 506, + 193 + ], + "spans": [ + { + "bbox": [ + 104, + 178, + 506, + 193 + ], + "score": 1.0, + "content": "2019), human keypoints (Jang et al., 2018; Yang et al., 2018; Walker et al., 2017; Chan et al., 2019;", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 189, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 506, + 202 + ], + "score": 1.0, + "content": "Zakharov et al., 2019; Wang et al., 2019; Siarohin et al., 2019), 3D meshes (Liu et al., 2019; Chen", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "score": 1.0, + "content": "et al., 2021), and optical flows (Li et al., 2018; Ohnishi et al., 2018). We note that the ground truth", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 208, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 506, + 224 + ], + "score": 1.0, + "content": "of such structure representations has been computed a-priori for the purpose of supervised training,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "which poses constraints on applications, where such representations of unseen testing images might", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 229, + 255, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 255, + 242 + ], + "score": 1.0, + "content": "be fragmentary or difficult to access.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 110, + 506, + 242 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "Self-supervised motion transfer approaches (Wiles et al., 2018; Siarohin et al., 2019; 2021) accept", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "raw videos as input and learn to reconstruct driving images by warping source image with pre-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "dicted dense optical flow fields. While the need for domain knowledge or labeled ground truth data", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 276, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 287 + ], + "score": 1.0, + "content": "has been obviated, which improves performance on in-the-wild testing images, such methods entail", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "necessity of explicit structure representations as motion guidance. Prior information such as key-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 295, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 309 + ], + "score": 1.0, + "content": "points (Siarohin et al., 2019; Wang et al., 2021a) or regions (Siarohin et al., 2021) are learned in an", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 306, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 317 + ], + "score": 1.0, + "content": "end-to-end training manner by additional networks as intermediate features, in order to predict target", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "flow fields. Although online prediction of such representations is less tedious than the acquisition of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 326, + 354, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 354, + 338 + ], + "score": 1.0, + "content": "ground truth labels, it still strains the complexity of networks.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 245, + 506, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 354 + ], + "score": 1.0, + "content": "Deviating from such approaches, we here aim to fully eliminate the need of explicit structure rep-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 352, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 363 + ], + "score": 1.0, + "content": "resentations by directly manipulating the latent space of a deep generative model. To the best of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 373 + ], + "score": 1.0, + "content": "our knowledge, this constitutes a new direction in the context of image animation. Our work is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 371, + 504, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 504, + 383 + ], + "score": 1.0, + "content": "motivated by interpretation of GANs (Shen et al., 2020; Goetschalckx et al., 2019; Jahanian et al.,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 380, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 394 + ], + "score": 1.0, + "content": "2020; Voynov & Babenko, 2020), showcasing that latent spaces of StyleGAN (Karras et al., 2019;", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 390, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 404 + ], + "score": 1.0, + "content": "2020b) and BigGAN (Brock et al., 2019) contain rich semantically meaningful directions. Given", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 400, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 414 + ], + "score": 1.0, + "content": "that walking along such directions, basic visual transformations such as zooming and rotation can", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "be induced in generated results. As in image animation, we have that motion between source and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "driving images can be considered as higher-level transformation, a natural question here arises: can", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 431, + 504, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 504, + 444 + ], + "score": 1.0, + "content": "we discover a set of directions in the latent space that induces high-level motion transformations", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 441, + 174, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 174, + 452 + ], + "score": 1.0, + "content": "collaboratively?", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 340, + 506, + 452 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 457, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 506, + 469 + ], + "score": 1.0, + "content": "Towards answering this question, we introduce LIA, a novel Latent Image Animator constituting of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "score": 1.0, + "content": "an autoencoder for animating still images via latent space navigation. LIA seeks to animate a source", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "score": 1.0, + "content": "image via linearly navigating associated source latent code along a learned path to reach a target", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 487, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 499 + ], + "score": 1.0, + "content": "latent code, which represents the high-level transformation for animating the source image. We", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "introduce a Linear Motion Decomposition (LMD) approach aiming to represent a latent path via a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "score": 1.0, + "content": "linear combination of a set of learned motion directions and associated magnitudes. Specifically, we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 518, + 504, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 504, + 529 + ], + "score": 1.0, + "content": "constrain the set as an orthogonal basis, where each vector indicates a basic visual transformation.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "score": 1.0, + "content": "By describing the whole motion space using such learned basis, LIA eliminates the requirement of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 538, + 241, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 241, + 549 + ], + "score": 1.0, + "content": "explicit structure representations.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 457, + 506, + 549 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 553, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "score": 1.0, + "content": "In addition, we design LIA to disentangle motion and appearance within a single encoder-generator", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 563, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 574 + ], + "score": 1.0, + "content": "architecture. Deviating from existing methods using separate networks to learn disentangled fea-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "score": 1.0, + "content": "tures, LIA integrates both, latent motion code, as well as appearance features in a single encoder,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 581, + 378, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 378, + 597 + ], + "score": 1.0, + "content": "which highly reduces the model complexity and simplifies training.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 552, + 506, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 506, + 612 + ], + "score": 1.0, + "content": "We provide evaluation on multiple datasets including VoxCeleb (Chung et al., 2018),", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 609, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 620 + ], + "score": 1.0, + "content": "TaichiHD (Siarohin et al., 2019) and TED-talk (Siarohin et al., 2021). In addition, we show that", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "LIA outperforms the state-of-the-art in preserving the facial structure in generated videos in the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 629, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 640 + ], + "score": 1.0, + "content": "setting of one-shot image animation on unseen datasets such as FFHQ (Karras et al., 2019) and", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 638, + 260, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 260, + 651 + ], + "score": 1.0, + "content": "GermanPublicTV (Thies et al., 2020).", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 597, + 506, + 651 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 667, + 208, + 679 + ], + "lines": [ + { + "bbox": [ + 104, + 665, + 211, + 682 + ], + "spans": [ + { + "bbox": [ + 104, + 665, + 211, + 682 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 52 + }, + { + "type": "text", + "bbox": [ + 107, + 696, + 504, + 727 + ], + "lines": [ + { + "bbox": [ + 105, + 695, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 505, + 708 + ], + "score": 1.0, + "content": "Video generation GAN-based video generation is aimed at mapping Gaussian noise to video, di-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 705, + 504, + 717 + ], + "spans": [ + { + "bbox": [ + 105, + 705, + 504, + 717 + ], + "score": 1.0, + "content": "rectly and in the absence of prior information (Vondrick et al., 2016; Saito et al., 2017; Tulyakov", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 715, + 505, + 728 + ], + "spans": [ + { + "bbox": [ + 105, + 715, + 505, + 728 + ], + "score": 1.0, + "content": "et al., 2018; Wang et al., 2020a; Wang, 2021). Approaches based on deep probabilistic models (Den-", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "score": 1.0, + "content": "ton & Birodkar, 2017; Li & Mandt, 2018; Bhagat et al., 2020; Xie et al., 2020) were also proposed", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "to tackle this problem, however only show results on toy datasets with low resolution. 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Unlike these approaches, which generate random", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "videos based on noise vectors in an unconditional manner, in this paper, we focus on conditionally", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 392, + 434, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 434, + 405 + ], + "score": 1.0, + "content": "creating novel videos by transferring motion from driving videos to input images.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 104, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "Latent space editing In an effort to control generated images, recent works explored the discovery", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "of semantically meaningful directions in the latent space of pre-trained GANs, where linear navi-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 437, + 504, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 504, + 448 + ], + "score": 1.0, + "content": "gation corresponds to desired image manipulation. 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In this work, as opposed to finding directions corresponding to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "individual visual transformations, we seek to learn a set of directions that cooperatively allows for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 496, + 409, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 409, + 509 + ], + "score": 1.0, + "content": "high-level visual transformations that can be beneficial in image animation.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 415, + 505, + 509 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 521, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "Image animation Related approaches (Chan et al., 2019; Wang et al., 2018; Zakharov et al., 2019;", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 506, + 543 + ], + "score": 1.0, + "content": "Wang et al., 2019; Yang et al., 2020) in image animation required strong prior structure labels as mo-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "tion guidance. In particular, Chan et al. (2019), Yang et al. (2020) and Wang et al. (2018) proposed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "score": 1.0, + "content": "to map representations such as human keypoints and facial landmarks to videos in the setting of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "image-to-image translation proposed by Isola et al. (2017). However, such approaches were only", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "score": 1.0, + "content": "able to learn an individual model for a single identity. 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In contrast to such approaches, our method does", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "not require any explicit structure representations. We dive into the latent space of the generator and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "self-learn to navigate motion codes in certain directions with the goal to reach target codes, which", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 387, + 282, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 282, + 403 + ], + "score": 1.0, + "content": "are then decoded to flow fields for warping.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 107, + 417, + 172, + 429 + ], + "lines": [ + { + "bbox": [ + 104, + 415, + 174, + 433 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 174, + 433 + ], + "score": 1.0, + "content": "3 METHOD", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 446, + 504, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "Self-supervised image animation aims at learning to transfer motion from a subject of a driving video", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 455, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 104, + 455, + 506, + 470 + ], + "score": 1.0, + "content": "to a subject in a source image based on training with a large video dataset. In this work, we propose", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "to model such motion transformation via latent space navigation. The general pipeline is illustrated", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "score": 1.0, + "content": "in Fig. 2. Specifically, for training, our model takes in a pair of source and driving images, randomly", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "score": 1.0, + "content": "sampled from one video sequence. These two images are encoded into a latent code which is used", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "to represent motion transformation in the image space. The training objective is to reconstruct the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "score": 1.0, + "content": "driving image by combining source image with learned motion transformation. For testing, frames", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 516, + 498, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 498, + 529 + ], + "score": 1.0, + "content": "of a driving video are sequentially processed with the source image to animate the source subject.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "score": 1.0, + "content": "We provide an overview of the proposed model in Fig. 3. Our model is an autoencoder, consisting", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 244, + 555 + ], + "score": 1.0, + "content": "of two main networks, an encoder", + "type": "text" + }, + { + "bbox": [ + 245, + 542, + 254, + 552 + ], + "score": 0.82, + "content": "\\bar { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 542, + 318, + 555 + ], + "score": 1.0, + "content": "and a generator", + "type": "text" + }, + { + "bbox": [ + 319, + 542, + 328, + 552 + ], + "score": 0.76, + "content": "\\bar { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 542, + 506, + 555 + ], + "score": 1.0, + "content": ". In general, our model requires two steps to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 103, + 551, + 508, + 567 + ], + "spans": [ + { + "bbox": [ + 103, + 551, + 236, + 567 + ], + "score": 1.0, + "content": "transfer motion. In the first step,", + "type": "text" + }, + { + "bbox": [ + 236, + 554, + 245, + 563 + ], + "score": 0.79, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 551, + 386, + 567 + ], + "score": 1.0, + "content": "encodes source and driving images", + "type": "text" + }, + { + "bbox": [ + 387, + 552, + 486, + 565 + ], + "score": 0.9, + "content": "\\boldsymbol { x } _ { s } , \\boldsymbol { x } _ { d } \\sim \\dot { \\mathcal { X } } \\in \\mathbb { R } ^ { 3 \\times H \\times W }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 551, + 508, + 567 + ], + "score": 1.0, + "content": "into", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 563, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 575 + ], + "score": 1.0, + "content": "latent codes in the latent space. The source code is then navigated into a target code, which is used", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "score": 1.0, + "content": "to represent target motion transformation, along a learned latent path. Based on proposed Linear", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "Motion Decomposition (LMD), we represent such a path as a linear combination of a set of learned", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 388, + 606 + ], + "score": 1.0, + "content": "motion directions and associated magnitudes, which are learned from", + "type": "text" + }, + { + "bbox": [ + 389, + 595, + 400, + 604 + ], + "score": 0.83, + "content": "x _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 593, + 506, + 606 + ], + "score": 1.0, + "content": ". In the second step, once", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 103, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 103, + 604, + 244, + 618 + ], + "score": 1.0, + "content": "the target latent code is obtained,", + "type": "text" + }, + { + "bbox": [ + 245, + 605, + 254, + 615 + ], + "score": 0.79, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 604, + 385, + 618 + ], + "score": 1.0, + "content": "decodes it as a dense flow field", + "type": "text" + }, + { + "bbox": [ + 386, + 604, + 486, + 616 + ], + "score": 0.9, + "content": "\\phi _ { s \\to d } \\sim \\Phi \\in \\mathbb { R } ^ { 2 \\times H \\times W }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 126, + 628 + ], + "score": 1.0, + "content": "uses", + "type": "text" + }, + { + "bbox": [ + 126, + 615, + 150, + 626 + ], + "score": 0.91, + "content": "\\bar { \\phi _ { s \\to d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 615, + 184, + 628 + ], + "score": 1.0, + "content": "to warp", + "type": "text" + }, + { + "bbox": [ + 184, + 617, + 195, + 626 + ], + "score": 0.85, + "content": "x _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "and then to obtain the output image. 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LIA is an autoencoder consisting of two networks, an encoder", + "type": "text" + }, + { + "bbox": [ + 471, + 246, + 480, + 256 + ], + "score": 0.78, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 245, + 506, + 258 + ], + "score": 1.0, + "content": "and a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 255, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 104, + 255, + 147, + 271 + ], + "score": 1.0, + "content": "generator", + "type": "text" + }, + { + "bbox": [ + 147, + 257, + 156, + 266 + ], + "score": 0.77, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 255, + 506, + 271 + ], + "score": 1.0, + "content": ". 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These two images are encoded into a latent code which is used", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "to represent motion transformation in the image space. The training objective is to reconstruct the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "score": 1.0, + "content": "driving image by combining source image with learned motion transformation. 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Our model is an autoencoder, consisting", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 244, + 555 + ], + "score": 1.0, + "content": "of two main networks, an encoder", + "type": "text" + }, + { + "bbox": [ + 245, + 542, + 254, + 552 + ], + "score": 0.82, + "content": "\\bar { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 542, + 318, + 555 + ], + "score": 1.0, + "content": "and a generator", + "type": "text" + }, + { + "bbox": [ + 319, + 542, + 328, + 552 + ], + "score": 0.76, + "content": "\\bar { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 542, + 506, + 555 + ], + "score": 1.0, + "content": ". In general, our model requires two steps to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 103, + 551, + 508, + 567 + ], + "spans": [ + { + "bbox": [ + 103, + 551, + 236, + 567 + ], + "score": 1.0, + "content": "transfer motion. 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The source code is then navigated into a target code, which is used", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "score": 1.0, + "content": "to represent target motion transformation, along a learned latent path. Based on proposed Linear", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "Motion Decomposition (LMD), we represent such a path as a linear combination of a set of learned", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 388, + 606 + ], + "score": 1.0, + "content": "motion directions and associated magnitudes, which are learned from", + "type": "text" + }, + { + "bbox": [ + 389, + 595, + 400, + 604 + ], + "score": 0.83, + "content": "x _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 593, + 506, + 606 + ], + "score": 1.0, + "content": ". 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We note that in order to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 291, + 493 + ], + "score": 1.0, + "content": "completely replicate the position and pose in", + "type": "text" + }, + { + "bbox": [ + 291, + 481, + 303, + 492 + ], + "score": 0.88, + "content": "V _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 480, + 351, + 493 + ], + "score": 1.0, + "content": ", it requires", + "type": "text" + }, + { + "bbox": [ + 352, + 483, + 363, + 492 + ], + "score": 0.86, + "content": "x _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 480, + 382, + 493 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 382, + 483, + 393, + 492 + ], + "score": 0.86, + "content": "x _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "to contain similar poses in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 491, + 204, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 204, + 502 + ], + "score": 1.0, + "content": "relative motion transfer.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 107, + 519, + 200, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 201, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 201, + 533 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 547, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 505, + 560 + ], + "score": 1.0, + "content": "In this section, we firstly describe our experimental setup including implementation details and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 556, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 571 + ], + "score": 1.0, + "content": "datasets. Secondly, we qualitatively demonstrate generated results based on testing datasets. Then,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "we provide quantitative evaluation w.r.t. image quality on (a) same-identity reconstruction, (b) cross-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "video motion transfer, presenting (c) a user study. Next, we conduct an ablation study that demon-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "strates (d) the effectiveness of our proposed motion dictionary, as well as (e) associated size. Finally,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "we provide an in-depth analysis of our (f) latent codes and (g) motion dictionary to interpret their", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 606, + 188, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 188, + 622 + ], + "score": 1.0, + "content": "semantic meanings.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 629, + 503, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 629, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 505, + 640 + ], + "score": 1.0, + "content": "Datasets Our model is trained on the datasets VoxCeleb, TaichiHD and TED-talk. We follow", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 401, + 651 + ], + "score": 1.0, + "content": "the pre-processing method in (Siarohin et al., 2019) to crop frames into", + "type": "text" + }, + { + "bbox": [ + 402, + 640, + 446, + 650 + ], + "score": 0.89, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "resolution for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 650, + 201, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 201, + 660 + ], + "score": 1.0, + "content": "quantitative evaluation.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 670, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 670, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 682 + ], + "score": 1.0, + "content": "Implementation details Our model is implemented in PyTorch (Paszke et al., 2019). 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We note that in order to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 291, + 493 + ], + "score": 1.0, + "content": "completely replicate the position and pose in", + "type": "text" + }, + { + "bbox": [ + 291, + 481, + 303, + 492 + ], + "score": 0.88, + "content": "V _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 480, + 351, + 493 + ], + "score": 1.0, + "content": ", it requires", + "type": "text" + }, + { + "bbox": [ + 352, + 483, + 363, + 492 + ], + "score": 0.86, + "content": "x _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 480, + 382, + 493 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 382, + 483, + 393, + 492 + ], + "score": 0.86, + "content": "x _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "to contain similar poses in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 491, + 204, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 204, + 502 + ], + "score": 1.0, + "content": "relative motion transfer.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 449, + 506, + 502 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 519, + 200, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 201, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 201, + 533 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 547, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 505, + 560 + ], + "score": 1.0, + "content": "In this section, we firstly describe our experimental setup including implementation details and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 556, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 571 + ], + "score": 1.0, + "content": "datasets. Secondly, we qualitatively demonstrate generated results based on testing datasets. Then,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "we provide quantitative evaluation w.r.t. image quality on (a) same-identity reconstruction, (b) cross-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "video motion transfer, presenting (c) a user study. Next, we conduct an ablation study that demon-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "strates (d) the effectiveness of our proposed motion dictionary, as well as (e) associated size. 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We follow", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 401, + 651 + ], + "score": 1.0, + "content": "the pre-processing method in (Siarohin et al., 2019) to crop frames into", + "type": "text" + }, + { + "bbox": [ + 402, + 640, + 446, + 650 + ], + "score": 0.89, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "resolution for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 650, + 201, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 201, + 660 + ], + "score": 1.0, + "content": "quantitative evaluation.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 629, + 505, + 660 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 670, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 670, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 682 + ], + "score": 1.0, + "content": "Implementation details Our model is implemented in PyTorch (Paszke et al., 2019). All models", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 680, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 693 + ], + "score": 1.0, + "content": "are trained on four 16G NVIDIA V100 GPUs. 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Examples for same-dataset absolute motion transfer on TaichiHD", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "score": 1.0, + "content": "(top-right) and TED-talk (bottom-right). On VoxCeleb (left), we demonstrate cross-dataset relative", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 295, + 504, + 311 + ], + "spans": [ + { + "bbox": [ + 104, + 295, + 339, + 311 + ], + "score": 1.0, + "content": "motion transfer. We successfully transfer motion between", + "type": "text" + }, + { + "bbox": [ + 339, + 299, + 351, + 308 + ], + "score": 0.87, + "content": "x _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 295, + 369, + 311 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 369, + 299, + 379, + 308 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 295, + 492, + 311 + ], + "score": 1.0, + "content": "from videos in VoxCeleb to", + "type": "text" + }, + { + "bbox": [ + 493, + 299, + 504, + 308 + ], + "score": 0.83, + "content": "x _ { s }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 306, + 307, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 307, + 321 + ], + "score": 1.0, + "content": "from FFHQ, the latter not being used for training.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 331, + 505, + 372 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 457, + 343 + ], + "score": 1.0, + "content": "Evaluation metrics We evaluate our model w.r.t. 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For the VoxCeleb dataset, we conduct a cross-dataset generation-experiment, where we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "transfer motion from VoxCeleb to images of the FFHQ dataset. We observe that our method out-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "performs FOMM and MRAA w.r.t. image quality, as both approaches visibly deform the shape of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 475, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 490 + ], + "score": 1.0, + "content": "the original faces. This is specifically notable in the case that source and driving images entail large", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "pose variations. At the same time, LIA is able to successfully tackle this challenge and no similar", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 497, + 208, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 208, + 508 + ], + "score": 1.0, + "content": "deformations are visible.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 107, + 522, + 347, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 347, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 347, + 534 + ], + "score": 1.0, + "content": "4.2 COMPARISON WITH STATE-OF-THE-ART METHODS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 504, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "We quantitatively compare our method with the state-of-the-art approaches X2Face, Monkey-Net,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "FOMM and MRAA on two tasks, namely (a) same-identity reconstruction and (b) cross-video mo-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 562, + 328, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 328, + 574 + ], + "score": 1.0, + "content": "tion transfer. Additionally, we conduct a (c) user study.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 504, + 637 + ], + "lines": [ + { + "bbox": [ + 106, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 599 + ], + "score": 1.0, + "content": "(a) Same-identity reconstruction We here evaluate the reconstruction ability of our method.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 597, + 504, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 421, + 608 + ], + "score": 1.0, + "content": "Specifically, we reconstruct each testing video by using the first frame as", + "type": "text" + }, + { + "bbox": [ + 422, + 599, + 433, + 608 + ], + "score": 0.84, + "content": "x _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 597, + 504, + 608 + ], + "score": 1.0, + "content": "and the remain-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 166, + 619 + ], + "score": 1.0, + "content": "ing frames as", + "type": "text" + }, + { + "bbox": [ + 167, + 608, + 178, + 618 + ], + "score": 0.85, + "content": "x _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 606, + 506, + 619 + ], + "score": 1.0, + "content": ". Results on three datasets are reported in Table 1. Focusing on foreground-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "reconstruction, our method outperforms the other approaches w.r.t. all metrics. More results are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 626, + 356, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 356, + 640 + ], + "score": 1.0, + "content": "presented in App. B.3, discussing background-reconstruction.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 650, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "(b) Cross-video motion transfer Next, we conduct experiments, where source images and driv-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 660, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 674 + ], + "score": 1.0, + "content": "ing videos stem from different video sequences. In this context, we mainly focus on evaluating", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "talking head videos and explore two different cases. In the first case, we generate videos using the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "score": 1.0, + "content": "VoxCeleb testing set to conduct motion transfer. In the second case, source images are from an", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 690, + 504, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 504, + 702 + ], + "score": 1.0, + "content": "unseen dataset, namely the GermanPublicTV dataset, as we conduct cross-dataset motion transfer.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "In both experiments, we randomly construct source and driving pairs and transfer motion from driv-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 710, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 724 + ], + "score": 1.0, + "content": "ing videos to source images to generate a novel manipulated dataset. Since ground truth data for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "our generated videos is not available, we use video FID (as initialized by Wang et al. (2020a)) to", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 7 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 80, + 504, + 266 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 80, + 504, + 266 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 80, + 504, + 266 + ], + "spans": [ + { + "bbox": [ + 107, + 80, + 504, + 266 + ], + "score": 0.975, + "type": "image", + "image_path": "a37a7371bb03be1d7874e44a06664ca4dc3ef0bc6d060aab959b38245f9df2b2.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 80, + 504, + 142.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 142.0, + 504, + 204.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 204.0, + 504, + 266.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 277, + 505, + 318 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 277, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 505, + 289 + ], + "score": 1.0, + "content": "Figure 4: Qualitative results. Examples for same-dataset absolute motion transfer on TaichiHD", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "score": 1.0, + "content": "(top-right) and TED-talk (bottom-right). On VoxCeleb (left), we demonstrate cross-dataset relative", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 295, + 504, + 311 + ], + "spans": [ + { + "bbox": [ + 104, + 295, + 339, + 311 + ], + "score": 1.0, + "content": "motion transfer. We successfully transfer motion between", + "type": "text" + }, + { + "bbox": [ + 339, + 299, + 351, + 308 + ], + "score": 0.87, + "content": "x _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 295, + 369, + 311 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 369, + 299, + 379, + 308 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 295, + 492, + 311 + ], + "score": 1.0, + "content": "from videos in VoxCeleb to", + "type": "text" + }, + { + "bbox": [ + 493, + 299, + 504, + 308 + ], + "score": 0.83, + "content": "x _ { s }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 306, + 307, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 307, + 321 + ], + "score": 1.0, + "content": "from FFHQ, the latter not being used for training.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 331, + 505, + 372 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 457, + 343 + ], + "score": 1.0, + "content": "Evaluation metrics We evaluate our model w.r.t. 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For TaichiHD and TED-talk datasets, we conduct an experiment re-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "lated to cross-video generation. Corresponding results (see Fig. 4) confirm that our method is able", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "to correctly transfer motion on articulated human bodies, in the absence of explicit structure repre-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "sentations. For the VoxCeleb dataset, we conduct a cross-dataset generation-experiment, where we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "transfer motion from VoxCeleb to images of the FFHQ dataset. We observe that our method out-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "performs FOMM and MRAA w.r.t. image quality, as both approaches visibly deform the shape of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 475, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 490 + ], + "score": 1.0, + "content": "the original faces. This is specifically notable in the case that source and driving images entail large", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "pose variations. At the same time, LIA is able to successfully tackle this challenge and no similar", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 497, + 208, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 208, + 508 + ], + "score": 1.0, + "content": "deformations are visible.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 406, + 506, + 508 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 522, + 347, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 347, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 347, + 534 + ], + "score": 1.0, + "content": "4.2 COMPARISON WITH STATE-OF-THE-ART METHODS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 504, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "We quantitatively compare our method with the state-of-the-art approaches X2Face, Monkey-Net,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "FOMM and MRAA on two tasks, namely (a) same-identity reconstruction and (b) cross-video mo-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 562, + 328, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 328, + 574 + ], + "score": 1.0, + "content": "tion transfer. Additionally, we conduct a (c) user study.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 542, + 506, + 574 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 504, + 637 + ], + "lines": [ + { + "bbox": [ + 106, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 599 + ], + "score": 1.0, + "content": "(a) Same-identity reconstruction We here evaluate the reconstruction ability of our method.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 597, + 504, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 421, + 608 + ], + "score": 1.0, + "content": "Specifically, we reconstruct each testing video by using the first frame as", + "type": "text" + }, + { + "bbox": [ + 422, + 599, + 433, + 608 + ], + "score": 0.84, + "content": "x _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 597, + 504, + 608 + ], + "score": 1.0, + "content": "and the remain-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 166, + 619 + ], + "score": 1.0, + "content": "ing frames as", + "type": "text" + }, + { + "bbox": [ + 167, + 608, + 178, + 618 + ], + "score": 0.85, + "content": "x _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 606, + 506, + 619 + ], + "score": 1.0, + "content": ". Results on three datasets are reported in Table 1. Focusing on foreground-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "reconstruction, our method outperforms the other approaches w.r.t. all metrics. 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In this context, we mainly focus on evaluating", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "talking head videos and explore two different cases. In the first case, we generate videos using the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "score": 1.0, + "content": "VoxCeleb testing set to conduct motion transfer. In the second case, source images are from an", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 690, + 504, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 504, + 702 + ], + "score": 1.0, + "content": "unseen dataset, namely the GermanPublicTV dataset, as we conduct cross-dataset motion transfer.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "In both experiments, we randomly construct source and driving pairs and transfer motion from driv-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 710, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 724 + ], + "score": 1.0, + "content": "ing videos to source images to generate a novel manipulated dataset. Since ground truth data for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "our generated videos is not available, we use video FID (as initialized by Wang et al. (2020a)) to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "compute the distance between generated and real data distributions. 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Comparison with state-of-the-art methods on three datasets", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 89, + 239, + 101 + ], + "spans": [ + { + "bbox": [ + 105, + 89, + 239, + 101 + ], + "score": 1.0, + "content": "for same-identity reconstruction.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 109, + 114, + 503, + 174 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 114, + 503, + 174 + ], + "spans": [ + { + "bbox": [ + 109, + 114, + 503, + 174 + ], + "score": 0.978, + "html": "
MethodVoxCelebTaichiHDTED-talks
L1AKDAEDLPIPSL1(AKD,MKR)AEDLPIPSL1(AKD,MKR)AEDLPIPS
X2Face0.0787.6870.405-0.080(17.654,0.109)-=-
Monkey-Net0.0491.8780.199-0.077(10.798,0.059)==
FOMM0.0461.3950.1410.1360.063(6.472,0.032)0.4950.1910.030(3.759,0.0090)0.4280.13
MRAA w/o bg0.0431.3070.1400.1270.063(5.626,0.025)0.4600.1890.029(3.126,0.0092)0.3960.12
Ours0.0411.3530.1380.1230.057(4.823, 0.020)0.4310.1800.027(3.141,0.0095)0.3990.11
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VoxCelebGermanPublicTV
FOMM0.323 0.456
MRAA 0.3080.454
Ours0.161 0.406
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VoxCeleb(%)TaichiHD(%)TED-talk(%)
Ours/FOMM92.9/7.164.5/35.571.4/28.6
Ours/MRAA89.7/10.360.7/39.954.8/45.2
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VoxCelebTaichiHDTED-talks
MethodL1 LPIPSL1LPIPSL1LPIPS
wloDm0.049 0.1650.0620.1860.0310.12
Full0.041 0.1230.0570.1800.0280.11
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VoxCelebTaichiHDTED-talks
ML1LPIPSL1LPIPSL1LPIPS
5 100.051 0.0430.15 0.130.070 0.0650.22 0.200.037 0.0360.15
200.0410.120.0570.180.0280.13 0.11
400.0420.120.0600.190.0300.12
1000.0410.120.0580.180.0280.11
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Specifically, we output", + "type": "text" + }, + { + "bbox": [ + 339, + 556, + 364, + 565 + ], + "score": 0.9, + "content": "w _ { r \\to d }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 552, + 506, + 568 + ], + "score": 1.0, + "content": "directly from MLP, without using", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "LMD to learn an orthogonal basis. 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MethodVoxCelebTaichiHDTED-talks
L1AKDAEDLPIPSL1(AKD,MKR)AEDLPIPSL1(AKD,MKR)AEDLPIPS
X2Face0.0787.6870.405-0.080(17.654,0.109)-=-
Monkey-Net0.0491.8780.199-0.077(10.798,0.059)==
FOMM0.0461.3950.1410.1360.063(6.472,0.032)0.4950.1910.030(3.759,0.0090)0.4280.13
MRAA w/o bg0.0431.3070.1400.1270.063(5.626,0.025)0.4600.1890.029(3.126,0.0092)0.3960.12
Ours0.0411.3530.1380.1230.057(4.823, 0.020)0.4310.1800.027(3.141,0.0095)0.3990.11
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VoxCelebGermanPublicTV
FOMM0.323 0.456
MRAA 0.3080.454
Ours0.161 0.406
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VoxCeleb(%)TaichiHD(%)TED-talk(%)
Ours/FOMM92.9/7.164.5/35.571.4/28.6
Ours/MRAA89.7/10.360.7/39.954.8/45.2
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VoxCelebTaichiHDTED-talks
MethodL1 LPIPSL1LPIPSL1LPIPS
wloDm0.049 0.1650.0620.1860.0310.12
Full0.041 0.1230.0570.1800.0280.11
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VoxCelebTaichiHDTED-talks
ML1LPIPSL1LPIPSL1LPIPS
5 100.051 0.0430.15 0.130.070 0.0650.22 0.200.037 0.0360.15
200.0410.120.0570.180.0280.13 0.11
400.0420.120.0600.190.0300.12
1000.0410.120.0580.180.0280.11
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While our method successfully transfers motion via latent space navi-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 681, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 349, + 693 + ], + "score": 1.0, + "content": "gation, we here aim at answering the question — what does", + "type": "text" + }, + { + "bbox": [ + 350, + 683, + 361, + 692 + ], + "score": 0.84, + "content": "x _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 681, + 506, + 693 + ], + "score": 1.0, + "content": "represent? 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(2019) (FOMM) and Wiles et al. (2018) (X2Face). However, deviating from", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "our visualized ‘reference image’, the ’reference image ’in FOMM refers to a non-visualized and ab-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 476, + 504, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 345, + 489 + ], + "score": 1.0, + "content": "stract concept. 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We here follow the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 569, + 502, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 502, + 580 + ], + "score": 1.0, + "content": "same implementation as Wang et al. (2020a) and utilize a pre-trained ResNext101 (Hara et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 577, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 104, + 577, + 506, + 592 + ], + "score": 1.0, + "content": "2018) to extract spatio-temporal features to compute the distance between real and generated videos", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "score": 1.0, + "content": "distributions. We take the first 100 frames of each video as input of the feature-extractor to compute", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 598, + 172, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 172, + 611 + ], + "score": 1.0, + "content": "the final scores.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 558, + 506, + 611 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 504, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 627 + ], + "score": 1.0, + "content": "Average keypoint distance (AKD) and missing keypoint rate (MKR) evaluate the difference be-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 625, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 506, + 637 + ], + "score": 1.0, + "content": "tween keypoints of reconstructed and ground truth videos. We extract landmarks using the face", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "alignment approach of (Bulat & Tzimiropoulos, 2017) and extract body poses for both TaiChiHD", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 644, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 658 + ], + "score": 1.0, + "content": "and TED-talks using OpenPose (Cao et al., 2019). 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MethodVoxCelebTaichiHDTED-talks
L1AKDAEDLPIPSL1(AKD,MKR)AEDLPIPSL1(AKD,MKR)AEDLPIPS
MRAA0.0411.3030.1350.1240.045(5.551,( ,0.025)0.4310.1780.027(3.107,0.0093)0.3790.11
Ours0.0411.3530.1380.1230.057(4.823,0.020)0.4310.1800.027(3.141,0.0095)0.3990.11
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MethodVoxCelebTaichiHDTED-talks
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MethodVoxCelebTaichiHDTED-talks
L1AKDAEDLPIPSL1(AKD,MKR)AEDLPIPSL1(AKD,MKR)AEDLPIPS
X2Face0.0787.6870.405-0.080(17.654,0.109)-=-
Monkey-Net0.0491.8780.199-0.077(10.798,0.059)==
FOMM0.0461.3950.1410.1360.063(6.472,0.032)0.4950.1910.030(3.759,0.0090)0.4280.13
MRAA w/o bg0.0431.3070.1400.1270.063(5.626,0.025)0.4600.1890.029(3.126,0.0092)0.3960.12
Ours0.0411.3530.1380.1230.057(4.823, 0.020)0.4310.1800.027(3.141,0.0095)0.3990.11
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VoxCelebTaichiHDTED-talks
ML1LPIPSL1LPIPSL1LPIPS
5 100.051 0.0430.15 0.130.070 0.0650.22 0.200.037 0.0360.15
200.0410.120.0570.180.0280.13 0.11
400.0420.120.0600.190.0300.12
1000.0410.120.0580.180.0280.11
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VoxCelebGermanPublicTV
FOMM0.323 0.456
MRAA 0.3080.454
Ours0.161 0.406
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VoxCelebTaichiHDTED-talks
MethodL1 LPIPSL1LPIPSL1LPIPS
wloDm0.049 0.1650.0620.1860.0310.12
Full0.041 0.1230.0570.1800.0280.11
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VoxCeleb(%)TaichiHD(%)TED-talk(%)
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MethodVoxCelebTaichiHDTED-talks
L1AKDAEDLPIPSL1(AKD,MKR)AEDLPIPSL1(AKD,MKR)AEDLPIPS
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Ours0.0411.3530.1380.1230.057(4.823,0.020)0.4310.1800.027(3.141,0.0095)0.3990.11
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It consists of a trunk network +3 (TNet), namely a unified encoder, and multiple branch networks (BNets), namely +4 resolution adapters. As per the input, a resolution-specific BNet is used and the +5 output are implanted as feature maps in the feature pyramid of TNet, at a layer with +6 the same resolution. The discriminability of tiny faces is significantly improved, as +7 the interpolation error introduced by rescaling, especially up-sampling, is mitigated +8 on the inputs. With branch distillation and backward-compatible training, BTNet +9 transfers discriminative high-resolution information to multiple branches while +10 guaranteeing representation compatibility. Our experiments demonstrate strong +11 performance on face recognition benchmarks, both for multi-resolution identity +12 matching and feature aggregation, with much less computation amount and param +13 eter storage. We establish new state-of-the-art on the challenging QMUL-SurvFace +14 1: N face identification task. + +# 15 1 Introduction + +16 Machine learning has advanced tremendously driven by deep learning methods, but is still severely +17 challenged by various data specifications, such as data type, structure, scale and size, etc. For +18 instance, face recognition (FR) is a well-established deep learning task, while the performance +19 degrades dramatically in the testing domain that differs from the training one, influenced by factors +20 of variance like resolution, illumination, occlusion, etc. +21 Most face recognition methods map each image to a point embedding in the common metric space +22 by deep neural networks (DNNs). The dissimilarity of images can be then calculated using various +23 distance metrics (e.g., cosine similarity, Euclidean distance, etc.) for face recognition tasks. +24 Recent advancements in margin-based loss (e.g., ArcFace [1], MV-Arc-Softmax [2], CurricularFace +25 [3], etc) enhanced discriminability of the metric space, with small intra-identity distance and large +26 inter-identity distance. However, lack of variation in training data still leads to poor generalizability. +27 Various useful methods are utilized to mitigate this issue. The model adapts to factors of variance +28 by augmenting datasets, whereas the large discrepancy in data distribution could potentially weaken +29 the model’s ability to extract discriminative features with the same data scale and model structure +30 (see Section 4.3). Fine-tuning is widely used to transfer large pretrained models to new domains with +31 different data specifications. However, this strategy requires one to store and deploy a separate copy +32 of the backbone parameters for every single new domain, which is expensive and often infeasible. +33 As known, the resolutions of face images in reality may be far beyond the scope covered by the +34 model. As the small feature maps with a fixed spatial extent (e.g., $7 \times 7 )$ are mapped to an embedding +35 with a predefined dimension (e.g., $1 2 8 - d$ , $5 1 2 - d$ , etc.) by a fully connected (fc) layer, input +36 images need to be rescaled to a canonical spatial size (e.g., $1 1 2 \times 1 1 2 )$ before fed into the network. +37 However, up-sampling low-resolution (LR) images introduces the interpolation error (see Section 3.1), +38 deteriorating the recognizable ones which contain enough clues to identify the subject. Even though +39 super-resolution methods [4–10] are widely used to build faces with good visualization, they inevitably +40 introduce feature information of other identities when reconstructing high-resolution (HR) faces. +41 This may lead to erroneous identity-specific features, which are detrimental to risk-controlled face +42 recognition. +43 Empirically, we can divide inputs by resolution distribution and learn to operate on them via multiple +44 models to achieve high accuracy and efficiency. However, multi-model fashion cannot be applied +45 directly for cross-resolution recognition as representation compatibility among models need to be +46 guaranteed [11–15]. +47 To improve discriminability while ensure the compatibility of the metric space for multi-resolution +48 face representation, we learn the “unified” representation by a partially-coupled Branch-to-Trunk +49 Network (BTNet). It is composed of multiple independent branch networks (BNets) and a shared +50 trunk network (TNet). A resolution-specific BNet is used for a given image, and the output are +51 implanted as feature maps in the feature pyramid of TNet, at a layer with the same resolution. +52 Furthermore, we find that multi-resolution training can be beneficial to building a strong and robust +53 TNet, and backward-compatible training (BCT) [11] can improve the representation compatibility +54 during the training process of BTNet. To ameliorate the discriminability of tiny faces, we propose +55 branch distillation in intermediate layers, utilizing information extracted from HR images to help the +56 extraction of discriminative features for resolution-specific branches. +57 Our method is simple and efficient, which breaks the convention of up-sampling the inputs and +58 serves as a general framework that can be easily implemented by several existing methods due to +59 conceptual simplicity. Meanwhile, BTNet is able to reduce the number of FLOPS by operating the +60 inputs without up-sampling, and per-resolution storage cost by only storing the learned branches and +61 resolution-aware BNs [16], while re-using the copy of the trunk model. +62 We demonstrate that our method performs comparably in various open-set face recognition tasks (1:1 +63 face verification and 1: N face identification), in both settings of multi-resolution identity matching +64 and feature aggregation, while meaningfully reduces the redundant computation cost and parameter +65 storage. In the challenging QMUL-SurvFace 1: N face identification task [17], we establish new +66 state-of-the-art by outperforming prior models. Furthermore, by avoiding the ill-posed problem (i.e., +67 image up-sampling), our approach also effectively reduces the additional noise and uncertainty of the +68 representation, which plays a key role in reliable risk-controlled face recognition. + +# 69 2 Related Work + +70 Compatible Representation Learning: The task of compatible representation learning aims at +71 encoding features that are interoperable with the features extracted from other models. Shen et. al. +72 [11] first formulated the problem of backward-compatible learning (BCT) and proposed to utilize the +73 old classifier for compatible feature learning. Since the multi-model fashion benefits representation +74 learning with lower computation, our idea of cross-resolution representation learning can be modeled +75 similar to cross-model compatibility [11–15], as metric space alignment for different resolutions. Our +76 goal is achieved by both compatibility-aware network architecture and training strategy. +77 Knowledge Distillation and Transfer: The concept of knowledge distillation (KD) was first +78 proposed by Hinton et. al. in [18], which can be summarized as employing a large parameter +79 model (teacher) to supervise the learning of a small parameter model (student). Distillation from +80 intermediate features [19–29] is widely adopted to enhance the effectiveness of knowledge transfer. +81 However, due to the “dark knowledge” hidden in the intermediate layers, additional subtle design is +82 often required to match and rescale intermediate features. Instead, our approach can easily locate the +83 distillation features without rescaling and effectively transfer knowledge from the HR domain to LR +84 branches. +85 Low Resolution Face Recognition: Its task includes low resolution-to-low resolution (LR-to-LR) +86 matching and low resolution-to-high resolution (LR-to-HR) matching [30]. The work can be divided +87 into two categories [31]: (1) Super-resolution (SR) based methods aim to upscale LR images to +88 construct HR images and use them for feature extraction [4–10]. (2) Projection-based methods aim to +89 extract adequate representations in different domains and project them into a common feature space +90 [32–34]. SR approaches are able to build faces with good visualization, but inevitably introduce +91 feature information of other identities when reconstructing corresponding HR faces, thus introducing +92 noise for identity-specific features. Compared to previous projection methods, our approach directly +93 learns discriminative representations in a common feature space for HR and LR inputs, without +94 additional projection heads for feature transformation. + +Pseudo-Siamese Networks: Siamese networks are a coupling architecture based on DNNs, which are widely used for signature verification [35], face verification [36, 37], tracking [38], etc. PseudoSiamese networks [39] are decoupled Siamese networks, as the weights of the two branches are not shared, resulting in a more flexible representation way for the two entities. Hughes et. al. in [40] proposed a pseudo-Siamese CNN for identifying corresponding patches in SAR and optical images. Inspired by pseudo-Siamese networks, we propose a resolution-adaptive partially coupled Siamese network architecture, extracting specific-shared features for images with different resolutions. + +# 3 Learning Specific-Shared Feature Transfer + +Instead of rescaling the inputs to a canonical size, we build multiple resolution-specific branches (BNets) that are used to map inputs to intermediate features with the same resolution and a resolutionshared trunk (TNet) to map feature maps with different resolutions to a high-dimension embedding. We gain several important properties by doing so: (1) Processing inputs on its original resolution can diminish the inevitably introduced error via up-sampling or information loss via down-sampling, thus preserving the discriminability of visual information with different resolutions. (2) Information streams of different resolutions are encoded uniformly, thus enabling the representation compatibility, which is particularly beneficial to open-set face recognition considering that a compatible metric space is the prerequisite for computing similarity. (3) This also effectively reduce the computation for LR images by supplying computational resources conditioned on the input resolution. + +# 113 3.1 Up-Sampling Error Analysis + +![](images/bcaf8fb368f804daba922e50885b5d890a9bd70ea33f0a8743d4e42864a72421.jpg) +Figure 1: Estimated Error Upperbound. (bilinear interpolation, average value for over 100 images) with the change of image resolution relative to resolution 112. + +![](images/49c6de557e00d2a1971a3a5c302649e9eb1af1a26b38f484c1a1576206274175.jpg) +Figure 2: Basic ideas of the proposed BTNet. Images of a certain identity are first projected to the feature maps with the same resolution respectively (Adapt) and then projected to a unified feature representation (Encode). In this figure, feature maps with the same resolution are indicated by outlines in the same color. + +114 Figure 1 illustrates the experimental estimation of interpolation error, whose upper bound increases +115 with the decline of the image resolution (see detailed theoretical derivation in Appendix A.1). Note +116 that the error soars up when the resolution drops below 32 approximately which can be viewed as LR +117 face images, consistent with the tiny-object criterion [41]. +118 The results show that: (1) inputs with a resolution higher than around 32 can be considered in the +119 same HR domain, since the error information introduced by up-sampling via interpolation can be +120 ignored to a certain extent; (2) inputs with a resolution lower than around 32 should be treated as in +121 various LR domains due to the high sensitivity of the resolution to errors. + +# 3.2 Branch-to-Trunk Network + +Let $X$ be an input RGB image with a space shape: $X \in \mathbb { R } ^ { H \times W \times 3 }$ where $H \times W$ corresponds to the spatial dimension of the input. For efficient batch training and inference, we predefine a canonical size $S \times S$ (e.g., $1 1 2 \times 1 1 2$ for typical face recognition models like ArcFace [1]). + +We build a trunk network $\boldsymbol { T } : \mathbb { R } ^ { H \times W \times 3 } \mathbb { R } ^ { C _ { e m b } }$ capable of extracting discriminative information with different resolutions, where $C _ { e m b }$ is the number of embedding channels. For every resolution $r$ in the candidate set, we formulate a resolution-specific branch, $z _ { r } = \textit { B } _ { r } ( X _ { r } )$ , which maps the input image $X _ { r }$ to feature maps with the same resolution and expanded channels $\boldsymbol { z } _ { r } : \mathbb { R } ^ { r \times r \times 3 } \mathbb { R } ^ { r \times r \times C _ { r } }$ . The idea is to learn our branches $B$ to focus on resolution-specific feature transfer independently. Feature maps will then be coupled to the trunk network $T$ in the feature pyramid with the same spatial resolution $r \times r$ , allowing for further mapping to the unified presentation space by $T _ { r } : \mathbb { R } ^ { r \times r \times C _ { r } } $ $\mathbb { R } ^ { C _ { e m b } }$ . + +Here, we follow the idea of “avoiding redundant up-sampling”. Our branches $B$ are implemented with same-resolution mapping: i.e., the model preserves the network architecture of $T$ from input to the layer with resolution $r$ and abandons down-sampling operations (e.g., replacing the convolution of stride 2 with stride 1, abandoning the pooling layers, etc.) to keep the same-resolution flow. + +We specifically name our specific-shared feature transfer network as Branch-to-Trunk Network, abbreviated as "BTNet". Figure 2 visually summarizes the main ideas of BTNet. + +# 3.3 Training Objectives + +We now describe the training objectives. The training of BTNet includes training the trunk network $T$ such that it can produce discriminative and compatible representations for multi-resolution information, and fine-tuning the branch networks $B$ to encourage them to learn resolution-specific feature transfer, so as to improve accuracy without compromising compatibility. + +Influence Loss. It is a compatibility-aware classification loss which is implemented by feeding the embeddings of the new model to the classifier of the old model [11]. Since the difficulties of samples vary due to image resolution, we compute CurricularFace [3] as our classification loss, in the form of: + +$$ +L _ { c u r } = - \log ( \frac { e ^ { s \cos ( \theta _ { y _ { i } } + m ) } } { e ^ { s \cos ( \theta _ { y _ { i } } + m ) } + \sum _ { j = 1 , j \neq y _ { i } } ^ { n } e ^ { s N ( t ^ { ( k ) } , \cos ( \theta _ { j } ) ) } } ) +$$ + +$$ +\begin{array} { c } { { N ( t , \cos \theta _ { j } ) = \displaystyle \left\{ \begin{array} { l c } { { \cos ( \theta _ { j } ) , } } & { { \cos ( \theta _ { y _ { i } } + m ) - \cos ( \theta _ { j } ) \ge 0 } } \\ { { \cos ( \theta _ { j } ) ( t + \cos ( \theta _ { j } ) ) , } } & { { e l s e } } \end{array} \right. } } \\ { { t ^ { ( k ) } = \displaystyle \alpha \sum _ { i } \cos \theta _ { y _ { i } } + ( 1 - \alpha ) t ^ { ( k - 1 ) } } } \end{array} +$$ + +150 which distinguishes both the difficultness of different samples in each stage and relative importance +151 of easy and hard samples during different training stages. Thus, we refine CurricularFace loss as our +152 influence loss: + +$$ +L _ { i n f l u e n c e } = L _ { c u r } ( \varphi _ { b t } , \kappa ^ { * } ) +$$ + +where 153 $\varphi _ { b t }$ is BTNet backbone (both $B _ { r }$ and $T _ { r }$ ), and $\kappa ^ { * }$ is the classifier of the pretrained trunk $T$ . + +154 Branch Distillation Loss. Due to the +155 continuity of the scale change of both the +156 image pyramid and the feature pyramid +157 [42], we can get a qualitative sense of +158 the similarity between images and feature +159 maps with the same resolution (see Figure +160 3). Furthermore, features extracted from +161 HR images have richer and clearer infor +162 mation than those from LR images [43]. +163 Motivated by these analyses, we utilize an +164 MSE loss to encourage the branch output +165 $z _ { r }$ to be similar to the corresponding fea +166 ture maps of the pretrained trunk network +167 $z _ { s }$ : + +![](images/e434a9a21fd643648ac4896d6cdd3ccf6e9342a751aac825373efc28d542ddee.jpg) +Figure 3: Visual comparison of face image-feature map pairs with different resolutions (resized to a common size here for illustration). + +$$ +L _ { b r a n c h } = \frac { 1 } { V } \sum _ { v = 1 } ^ { V } { ( z _ { r _ { v } } - z _ { s _ { v } } ) } ^ { 2 } +$$ + +168 where $V$ denotes the batch size. + +169 The whole training objective is a combination of the above objectives: + +$$ +L = L _ { i n f l u e n c e } + \lambda _ { b r a n c h } L _ { b r a n c h } +$$ + +170 where $\lambda _ { b r a n c h }$ is a hyper-parameter to weigh the losses and we set $\lambda _ { b r a n c h } = 0 . 5$ in all our experi +171 ments. + +![](images/22946257e92b509279813a29ef0c3ac19fcb89eb7b3c4b6d171707c4f9ae79f6.jpg) +Figure 4: Comparison of $\#$ Params $\mathbf { \Psi } ^ { ( \mathbf { M } ) }$ between fully finetuning and $\varphi _ { b t }$ . + +![](images/1df08d5fa7d736b80d3f65aa605b4972425929210f8a903a7b8e250ca889addb.jpg) +Figure 5: Comparison of FLOPs (G) between baselines and $\varphi _ { b t }$ . + +# 172 3.4 Storing Branch Networks + +An obvious adaptation strategy is fully finetuning of the model on each resolution. However, this strategy requires one to store and deploy a separate copy of the backbone parameters for every resolution, which is an expensive proposition and difficult to expand into more segmented resolution branches. Our BTNet is beneficial in the scenario of multi-resolution face recognition which achieves better parameter/accuracy trade-offs. Since activation statistics including means and variances under different resolutions are incompatible [44], we update and store Batch Normalization (BN) [45] parameters in all layers of $B _ { r }$ and $T _ { r }$ for each resolution, whose amount is negligible. Apart from this, we only need to store the learned branches and re-use the original copy of the pretrained trunk model, significantly reducing the storage cost. Figure 4 shows that BTNet requires only $1 . 1 \% \sim 4 8 . 9 \%$ of all the parameters compared to fully updating all the parameters of TNet. + +# 4 Experiments + +184 To validate BTNet on face recognition tasks in open universe, we perform 1:1 verification and $1 : N$ +185 identification tasks in two different settings, including (a) multi-resolution identity matching, and + +(b) multi-resolution feature aggregation. For 1:1 verification, a pair of templates are provided and the model is to decide whether they belong to the same identity or not. For 1:N identification, a set of gallery images are first mapped onto their embedding vectors (indexing) and the embeddings of query images are extracted to perform search against indexed gallery. + +# 4.1 Implementation Details + +Datasets. We use MS1Mv3 [46] for training face embedding models. The MS1Mv3 dataset contains 5,179,510 images of 93,431 celebrities. According to the test setting, different test datasets are used. + +·Multi-Resolution Identity Matching. We try on six widely adopted face verification benchmarks: LFW [47], CFP-FF [48], CFP-FP [48], AgeDB-30 [49], CALFW [50], and CPLFW [51], while the large-scale surveillance face dataset QMUL-SurvFace [17] is used for 1:N face identification, which contains native LR surveillance faces across wide space and time. The spatial resolution for QMUL-SurvFace ranges from 6/5 to 124/106 in height/width with an average of 24/20. + +·Multi-Resolution Feature Aggregation. We adopt a top challenging benchmark IJB-C [52], which has around 130k images from 3,531 identities, for two standard testing protocols: $1 : 1$ verification and 1:N identification. + +Training. All the models are trained on four RTX 2080 Tis with batch size 128 by stochastic gradient descent. For TNet, we train for 25 epochs, with learning rate initialized at 0.2 with 2 warmup epochs and decaying as a quadratic polynomial. We augment training samples by random horizonal flipping and multi-resolution training. For BNets, we initialize the learning rate by 0.02 without warm-up epochs. The training all stops at the $1 0 t h$ epoch for a fair comparison. The recommended hyper-parameters are used for classification loss from the original paper (e.g., $m = 0 . 5 , s = 6 4$ for ArcFace [1], and $\alpha = 0 . 9 9 , t ^ { 0 } = 0$ for CurricularFace [3]). Only horizonal flipping is used as augmentation when training BNets. + +Baselines. In our experiment, several baselines are used to validate BTNet in learning discriminative and compatible representations for multi-resolution face recognition. + +·High-Resolution Trained $\varphi _ { h r }$ . Naive baseline trained with HR data. + +·Independently Trained $\varphi _ { m m }$ . Multi-model fashion: is it possible to achieve better results if we train a specific model for each resolution independently? Specifically, we train $\varphi _ { r }$ for data with resolution $r$ and denote the multi-model collections as $\varphi _ { m m }$ . + +·Multi-Resolution Trained $\varphi _ { m r }$ . Trained with multi-resolution data which adapts to resolutionvariance. Specifically, each image is randomly down-sampled to a size in the candidate set $\textstyle \{ { \frac { 1 1 2 } { 2 ^ { i } } } \ x$ $\textstyle { \frac { 1 1 2 } { 2 ^ { i } } } | i = 0 , 1 , 2 , 3 , 4 \}$ with equal probability of being chosen, and then up-sampled back to $1 1 2 \times 1 1 2$ + +Instantiation of Network Architecture. The BTNet and baselines are implemented with ResNet50 [53], and they could be extended easily with other implementations. Dubbed as $\varphi _ { b t }$ , the detailed instantiation of BTNet based on ResNet50 is illustrated in Appendix A.2. + +# 4.2 Evaluation Metrics + +On the benchmarks for face verification, we use 1:1 verification accuracy as the basic metrics. The rank-20 true positive identification rates (TPIR20) at varying false positive identification rates (FPIR) and AUC are used to report the identification results on QMUL-SurvFace. The evaluation metrics for IJB-C 1:1 verification protocol are true acceptance rates (TAR) at different false acceptance rate (FAR). For 1:N identification, the basic evaluation metrics are the true positive identification rates (TPIR) at different false positive identification rates (FPIR). + +229 For better evaluation, we define another two metrics to assess the relative performance gain similar to +230 [11, 14]. + +(a) Cross-resolution identity matching. + +Table 1: Comparison of different methods on six face verification benchmarks. “Acc.” denotes average 1:1 verification accuracy. + +
112&7112&14112&28
Acc.GainAcc.GainAcc.Gain
hr57.75-81.02=95.90
mm50.58-0.8949.90-4.8250.03-305.80
mr65.85+1.0087.47+1.0096.05+1.00
bt(Ours)86.10+3.5094.08+2.0296.65+5.00
+ +(b) Same-resolution identity matching. + +
7&714&1428&28112&112
Acc.GainAcc.GainAcc.GainAcc.Gain
60.7073.8893.58=97.68-
62.57+1.0078.00+1.0094.68+1.0097.68-
61.02+0.1780.32+1.5695.12+1.4097.25-
77.78+9.1390.90+4.1396.27+2.4597.25=
+ +231 Cross-Resolution Gain. With the purpose towards the cross-resolution compatible representations, +232 we define the performance gain as follows: + +$$ +G a i n _ { r _ { 1 } \& r _ { 2 } } ( \varphi ) = \frac { M _ { r _ { 1 } \& r _ { 2 } } ( \varphi ) - M _ { r _ { 1 } \& r _ { 2 } } ( \varphi _ { h r } ) } { | M _ { r _ { 1 } \& r _ { 2 } } ( \varphi _ { m r } ) - M _ { r _ { 1 } \& r _ { 2 } } ( \varphi _ { h r } ) | } +$$ + +Here 33 $M _ { r _ { 1 } \& r _ { 2 } } ( \cdot )$ are metrics when the resolutions of the image/template pair are $r _ { 1 } \times r _ { 1 }$ and $r _ { 2 } \times r _ { 2 }$ 4 $( r _ { 1 } \neq r _ { 2 } )$ , respectively. $\varphi _ { m r }$ shares the same architecture with $\varphi _ { h r }$ while is trained on multi-resolution 5 images and thus serves as the baseline of cross-resolution gain. + +36 Same-Resolution Gain. For the scenario of multi-resolution face recognition, the performance of +37 same-resolution verification/identification is also vital besides cross-resolution one. Therefore, we +38 report the relative performance improvement from base model $\varphi _ { h r }$ in the scenario of same-resolution. + +$$ +G a i n _ { r \& r } ( \varphi ) = \frac { M _ { r \& r } ( \varphi ) - M _ { r \& r } ( \varphi _ { h r } ) } { | M _ { r \& r } ( \varphi _ { r } ) - M _ { r \& r } ( \varphi _ { h r } ) | } +$$ + +39 Here $M _ { r \& r } \left( \cdot \right)$ are metrics when the resolutions of the image/template pair are both $r \times r$ . $\varphi _ { r }$ is +40 a model of the set $\{ \varphi _ { m m } = \varphi _ { r } | r = 7 , 1 4 , 2 8 \}$ trained on images with resolution $r \times r$ without +41 considering cross-resolution representation compatibility, which serves as the baseline of same +42 resolution gain on resolution $r$ . Note that for both metrics we add the absolute symbol to the +43 denominator as they can be negative in some test settings (detailed in Section 4.3 and 4.4). + +# 4.3 Multi-Resolution Identity Matching + +We now conduct experiments on the proposed BTNet framework for multi-resolution identity matching. Two different settings are included : (1) same-resolution matching, and (2) cross-resolution matching. Table 1 compares the average performance on popular benchmarks for $\varphi _ { h r } , \varphi _ { m m } , \varphi _ { m r } ,$ $\varphi _ { b t }$ . The experimental results on each dataset are detailed in Appendix A.5. + +249 When directly applied to test data with the resolution lower than training data, $\varphi _ { h r }$ suffers a severe +250 performance degradation. Up-sampling images via interpolation can increase the amount of data +251 but not the amount of information, only to improve the detailed part of the image and the spatial +252 resolution (size) [64]. Moreover, it also brings various noise and artificial processing traces [65]. +253 Up-sampling images via interpolation-typically bilinear interpolation or bicubic interpolation of +254 4x4 pixel neighborhoods, essentially a function approximation method, is bound to introduce error +255 information (detailed in Appendix A.1), thus potentially confusing identity information, which is +256 especially crucial for LR images with limited details. We are able to observe improvement of $\varphi _ { m m }$ in +257 same-resolution matching but its cross-resolution gain is negative with approximately $50 \%$ accuracy. +258 Unsurprisingly, independently trained $\varphi _ { r }$ is unaware of representation compatibility, and thus does +259 not naturally suitable for cross-resolution recognition. The results show that $\varphi _ { m r }$ improved both +260 cross-resolution and same-resolution accuracy by a large margin, as it learns to adapt to resolution +261 variance and maintain discriminability of multi-resolution inputs. Note that the model size and +262 training data scale stay the same, while only the resolution distribution of the data changes for +263 $\varphi _ { m r }$ , and thus there is a marginal accuracy drop in the setting of 112&112 matching. Comparably, +264 $\varphi _ { b t }$ substantially outperforms all baselines with $2 . 0 2 \sim 5 . 0 0$ cross-resolution gain and 2.45\~9.13 +265 same-resolution gain. Importantly, due to the multi-resolution branches, our approach has a cost same +266 with $\varphi _ { m m }$ , significantly lower than $\varphi _ { h r }$ and $\varphi _ { m r }$ (see Figure 5). +267 For inference on inputs with resolutions not strictly matched to the branch, we validate three selection +268 strategies based on three resolution indicators (see Figure 6). Table 2 compares BTNet against the +269 state-of-the-arts models on QMUL-SurvFace 1:N identification benchmark. We are able to observe +270 that our proposed approach extends the state-of-the-arts while being more computationally efficient. +271 We believe the performance of BTNet (max $^ +$ ceil) is the highest that have been reported so far, and +272 we believe it is meaningful with the increased focus on unconstrained surveillance applications. + +Table 2: Performance of face identification on QMUL-SurvFace. Most compared results are cited from [17, 54] except BTNet. + +
TPIR20(%)@FPIR
AUC0.30.20.10.01
VGG-Face [55]14.05.12.60.80.1
DeepID2 [56]20.812.88.13.40.8
FaceNet [57]19.812.78.14.31.0
SphereFace [58]28.121.315.78.31.0
SRCNN [59]27.020.014.96.20.6
FSRCNN [60]27.320.014.46.10.7
VDSR [61]27.320.114.56.10.8
DRRN [62]27.520.314.96.30.6
LapSRN [63]27.420.214.76.30.7
ArcFace [1]25.318.715.110.12.0
RAN [54]32.326.521.614.93.8
BTNet (avg.+floor)32.627.923.416.51.4
BTNet (avg.+near)34.630.325.718.91.5
BTNet (avg.+ceil)35.431.126.820.32.2
BTNet (min+floor)32.327.623.216.11.4
BTNet (min+near)34.029.625.018.01.4
BTNet (min+ceil)35.331.026.619.92.0
BTNet (max+floor)33.629.124.517.61.3
BTNet (max+near)35.231.026.419.61.7
BTNet (max+ceil)35.431.226.920.62.5
+ +# 4.4 Multi-Resolution Feature Aggregation + +Multi-resolution feature aggregation is common in set-based recognition tasks where the model needs to determine the similarity of sets (templates), instead of images. Each set could contain images of the same identity with different resolutions. In our experiment, we rescale the original and flipped images in each set to different resolutions and aggregate their features into a representation of the template. Detailed experimental results can be seen in Appendix A.5. + +Table 3 (a) compares the cross-resolution results of $\mathrm { T A R } @ \mathrm { F A R } = 1 0 ^ { - 4 }$ for 1:1 verification. The cross-resolution features are ensured to be mapped to the same vector space where the aggregation is conducted for $\varphi _ { h r }$ and $\varphi _ { m r }$ , but we can observe that $\varphi _ { h r }$ performs much better than $\varphi _ { m r }$ . One possible reason is that $\varphi _ { h r }$ has outstanding discriminability to extract HR features, while LR features may not overly deteriorate the HR information. This phenomenon also suggests that $\varphi _ { m r }$ sacrifices its discriminability in exchange for the adaptability for resolution-variance. We can see $\varphi _ { b t }$ is comparable with $\varphi _ { h r }$ , demonstrating the discriminative power of BTNet for aggregating multi-resolution features. + +Table 3 (b) compares the same-resolution results of $\mathrm { T A R } @ \mathrm { F A R } = 1 0 ^ { - 4 }$ for 1:1 verification. When HR information is removed from the template representation (i.e., test settings 7&7, 14&14, 28&28), $\varphi _ { h r }$ suffers from performance degradation as well, as the informative embedding cannot catch the lost details of the LR images [54]. Both $\varphi _ { m m }$ and $\varphi _ { m r }$ improve with a limited same-resolution gain, while $\varphi _ { b t }$ surpasses the baselines by a large margin while also reducing the compute. + +In Table 4 we show the results of291 $\mathrm { T P I R } @ \mathrm { F P I R } { = } 1 0 ^ { - 1 }$ for 1:N identification protocol. Similar to our 292 results for 1:1 verification, we are able to observe that $\varphi _ { b t }$ is comparable or even better than $\varphi _ { h r }$ with + +Table 3: Comparison of different methods on the IJB-C dataset 1:1 face verification task. +“TAR” denotes TAR $( \% @ \mathsf { F A R = } 1 \mathsf { e } { \mathrm { - } } 4 )$ . + +(a) Cross-resolution feature aggregation. + +(b) Same-resolution feature aggregation. + +
7&714&1428&28112&112
TARGainTARGainTARGainTARGain
4.83=33.7489.6596.40
4.83+0.0029.26-1.0092.58+1.0096.40=
4.48-40.51+1.5192.81+1.0896.06
35.47-82.08+10.7994.50+1.6696.06=
+ +
112&7112&14112&28
TARGainTARGainTARGain
hr88.8992.40=95.62
mm74.54-0.5693.52+1.3395.42-0.69
mr63.11-1.0091.56-1.0095.33-1.00
bt(Ours)88.17-0.0393.97+1.8795.62+0.00
+ +Table 4: Comparison of different methods on the IJB-C dataset 1: N face identification task. +“TPIR” denotes TPIR $( \% @ \mathsf { G P I R { = } } 0 . 1 $ ). + +(a) Cross-resolution feature aggregation. + +
112&7112&14112&28
TPIRGainTPIRGainTPIRGain
hr85.60-90.11-94.27-
mm69.70-0.5591.73+1.5394.13-0.33
mr56.64-1.0089.05-1.0093.84-1.00
bt(Ours)83.93-0.0691.87+1.6694.33+0.14
+ +(b) Same-resolution feature aggregation. + +
7&714&1428&28112&112
TPIRGainTPIRGainTPIRGainTPIRGain
3.12-26.37-86.06=95.57-
3.24+1.0021.84-1.0089.76+1.0095.57=
3.25+1.0837.58+2.4791.02+1.3494.85-
27.70+204.8376.65+11.1092.89+1.8594.85-
+ +293 HR information involved and can preserve superior discriminability with limited LR information, +294 while also being more computationally efficient. + +![](images/a8c4e8888c6239a2f78ab3386fd66c4578ddb9b6f3764bbdd536c5e5c274bf6e.jpg) +Figure 6: Branch selection process. Max/min/average is used on (W, H) to obtain a resolution indicator for further allocation (floor/near/ceil) to a certain branch. + +# 295 5 Discussion and Conclusion + +This paper works on the problem of multi-resolution face recognition, and provides a new scheme to operate images conditioned on its input resolution without large span rescaling. The error introduced by up-sampling via interpolation is investigated and analyzed. Decoupled as branches for discriminative representation learning and coupled as the trunk for compatible representation learning, our Branch-to-Trunk Network (BTNet) achieves significant improvements on multi-resolution face verification and identification tasks. Besides, the superiority of BTNet in reducing computational cost and parameter storage cost is also demonstrated. It is worth noting that our approach is easy to expand to recognition tasks for other classes of objects and has the potential to serve as a general network architecture for multi-resolution visual recognition. + +Limitations and Future Work. The dislocation between the underlying optical resolution of native face images and that of a certain branch may limit the power of the model, which may be improved by selecting the optimal processing branch for the input in combination with the image quality, rather than by image size alone. The optimal branch selection strategy is not fully investigated though we have provided an intuitive way to select the branch for inputs (see Figure 6). 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High-frequency component helps explain the generalization of convolutional neural networks. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition, CVPR 2020, Seattle, WA, USA, June 13-19, 2020, pages 8681–8691. Computer Vision Foundation / IEEE, 2020. + +# Checklist + +1. For all authors... + +(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] +(b) Did you describe the limitations of your work? [Yes] See Section 5.The dislocation between the underlying optical resolution of native face images and that of a certain branch may limit the power of the model, which may be improved by selecting the optimal processing branch for the input in combination with the image quality, rather than by image size alone. The optimal branch selection strategy is not fully investigated though we have provided an intuitive way to select the branch for inputs. +(c) Did you discuss any potential negative societal impacts of your work? [N/A] We study a general framework for multi-resolution face recognition. Our method is not for specific applications, which does not directly involve societal issues. +(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] + +2. If you are including theoretical results... + +(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Appendix A.1 +(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix A.1 + +3. If you ran experiments... + +(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We follow the common practice in previous works, where they didn’t report the error bars. +(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See our implementation details in Section 4. + +4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... + +(a) If your work uses existing assets, did you cite the creators? [Yes] +(b) Did you mention the license of the assets? [N/A] +(c) Did you include any new assets either in the supplemental material or as a URL? [No] +(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] +(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] + +5. If you used crowdsourcing or conducted research with human subjects... + +591 +592 +593 +594 +595 +596 +597 + +(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] +(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] +(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] \ No newline at end of file diff --git a/parse/dev/AIqC7F7xV-d/AIqC7F7xV-d_content_list.json b/parse/dev/AIqC7F7xV-d/AIqC7F7xV-d_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..b8be2e2b4fc235ef93eba0a35253c1c305d791fc --- /dev/null +++ b/parse/dev/AIqC7F7xV-d/AIqC7F7xV-d_content_list.json @@ -0,0 +1,1495 @@ +[ + { + "type": "text", + "text": "Learning Unified Representations for Multi-Resolution Face Recognition ", + "text_level": 1, + "bbox": [ + 174, + 122, + 825, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous Author(s) \nAffiliation \nAddress \nemail ", + "bbox": [ + 423, + 226, + 580, + 281 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 306, + 535, + 323 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 In this work, we propose Branch-to-Trunk network (BTNet), a novel representation \n2 learning method for multi-resolution face recognition. It consists of a trunk network \n3 (TNet), namely a unified encoder, and multiple branch networks (BNets), namely \n4 resolution adapters. As per the input, a resolution-specific BNet is used and the \n5 output are implanted as feature maps in the feature pyramid of TNet, at a layer with \n6 the same resolution. The discriminability of tiny faces is significantly improved, as \n7 the interpolation error introduced by rescaling, especially up-sampling, is mitigated \n8 on the inputs. With branch distillation and backward-compatible training, BTNet \n9 transfers discriminative high-resolution information to multiple branches while \n10 guaranteeing representation compatibility. Our experiments demonstrate strong \n11 performance on face recognition benchmarks, both for multi-resolution identity \n12 matching and feature aggregation, with much less computation amount and param \n13 eter storage. We establish new state-of-the-art on the challenging QMUL-SurvFace \n14 1: N face identification task. ", + "bbox": [ + 148, + 339, + 766, + 550 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "15 1 Introduction ", + "text_level": 1, + "bbox": [ + 148, + 569, + 312, + 587 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "16 Machine learning has advanced tremendously driven by deep learning methods, but is still severely \n17 challenged by various data specifications, such as data type, structure, scale and size, etc. For \n18 instance, face recognition (FR) is a well-established deep learning task, while the performance \n19 degrades dramatically in the testing domain that differs from the training one, influenced by factors \n20 of variance like resolution, illumination, occlusion, etc. \n21 Most face recognition methods map each image to a point embedding in the common metric space \n22 by deep neural networks (DNNs). The dissimilarity of images can be then calculated using various \n23 distance metrics (e.g., cosine similarity, Euclidean distance, etc.) for face recognition tasks. \n24 Recent advancements in margin-based loss (e.g., ArcFace [1], MV-Arc-Softmax [2], CurricularFace \n25 [3], etc) enhanced discriminability of the metric space, with small intra-identity distance and large \n26 inter-identity distance. However, lack of variation in training data still leads to poor generalizability. \n27 Various useful methods are utilized to mitigate this issue. The model adapts to factors of variance \n28 by augmenting datasets, whereas the large discrepancy in data distribution could potentially weaken \n29 the model’s ability to extract discriminative features with the same data scale and model structure \n30 (see Section 4.3). Fine-tuning is widely used to transfer large pretrained models to new domains with \n31 different data specifications. However, this strategy requires one to store and deploy a separate copy \n32 of the backbone parameters for every single new domain, which is expensive and often infeasible. \n33 As known, the resolutions of face images in reality may be far beyond the scope covered by the \n34 model. As the small feature maps with a fixed spatial extent (e.g., $7 \\times 7 )$ are mapped to an embedding \n35 with a predefined dimension (e.g., $1 2 8 - d$ , $5 1 2 - d$ , etc.) by a fully connected (fc) layer, input \n36 images need to be rescaled to a canonical spatial size (e.g., $1 1 2 \\times 1 1 2 )$ before fed into the network. \n37 However, up-sampling low-resolution (LR) images introduces the interpolation error (see Section 3.1), \n38 deteriorating the recognizable ones which contain enough clues to identify the subject. Even though \n39 super-resolution methods [4–10] are widely used to build faces with good visualization, they inevitably \n40 introduce feature information of other identities when reconstructing high-resolution (HR) faces. \n41 This may lead to erroneous identity-specific features, which are detrimental to risk-controlled face \n42 recognition. \n43 Empirically, we can divide inputs by resolution distribution and learn to operate on them via multiple \n44 models to achieve high accuracy and efficiency. However, multi-model fashion cannot be applied \n45 directly for cross-resolution recognition as representation compatibility among models need to be \n46 guaranteed [11–15]. \n47 To improve discriminability while ensure the compatibility of the metric space for multi-resolution \n48 face representation, we learn the “unified” representation by a partially-coupled Branch-to-Trunk \n49 Network (BTNet). It is composed of multiple independent branch networks (BNets) and a shared \n50 trunk network (TNet). A resolution-specific BNet is used for a given image, and the output are \n51 implanted as feature maps in the feature pyramid of TNet, at a layer with the same resolution. \n52 Furthermore, we find that multi-resolution training can be beneficial to building a strong and robust \n53 TNet, and backward-compatible training (BCT) [11] can improve the representation compatibility \n54 during the training process of BTNet. To ameliorate the discriminability of tiny faces, we propose \n55 branch distillation in intermediate layers, utilizing information extracted from HR images to help the \n56 extraction of discriminative features for resolution-specific branches. \n57 Our method is simple and efficient, which breaks the convention of up-sampling the inputs and \n58 serves as a general framework that can be easily implemented by several existing methods due to \n59 conceptual simplicity. Meanwhile, BTNet is able to reduce the number of FLOPS by operating the \n60 inputs without up-sampling, and per-resolution storage cost by only storing the learned branches and \n61 resolution-aware BNs [16], while re-using the copy of the trunk model. \n62 We demonstrate that our method performs comparably in various open-set face recognition tasks (1:1 \n63 face verification and 1: N face identification), in both settings of multi-resolution identity matching \n64 and feature aggregation, while meaningfully reduces the redundant computation cost and parameter \n65 storage. In the challenging QMUL-SurvFace 1: N face identification task [17], we establish new \n66 state-of-the-art by outperforming prior models. Furthermore, by avoiding the ill-posed problem (i.e., \n67 image up-sampling), our approach also effectively reduces the additional noise and uncertainty of the \n68 representation, which plays a key role in reliable risk-controlled face recognition. ", + "bbox": [ + 147, + 595, + 825, + 670 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 679, + 825, + 723 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 731, + 825, + 866 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 873, + 826, + 904 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 90, + 826, + 212 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 218, + 825, + 279 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 286, + 825, + 361 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 368, + 825, + 444 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 450, + 825, + 526 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 532, + 825, + 638 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "69 2 Related Work ", + "text_level": 1, + "bbox": [ + 148, + 647, + 321, + 665 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "70 Compatible Representation Learning: The task of compatible representation learning aims at \n71 encoding features that are interoperable with the features extracted from other models. Shen et. al. \n72 [11] first formulated the problem of backward-compatible learning (BCT) and proposed to utilize the \n73 old classifier for compatible feature learning. Since the multi-model fashion benefits representation \n74 learning with lower computation, our idea of cross-resolution representation learning can be modeled \n75 similar to cross-model compatibility [11–15], as metric space alignment for different resolutions. Our \n76 goal is achieved by both compatibility-aware network architecture and training strategy. \n77 Knowledge Distillation and Transfer: The concept of knowledge distillation (KD) was first \n78 proposed by Hinton et. al. in [18], which can be summarized as employing a large parameter \n79 model (teacher) to supervise the learning of a small parameter model (student). Distillation from \n80 intermediate features [19–29] is widely adopted to enhance the effectiveness of knowledge transfer. \n81 However, due to the “dark knowledge” hidden in the intermediate layers, additional subtle design is \n82 often required to match and rescale intermediate features. Instead, our approach can easily locate the \n83 distillation features without rescaling and effectively transfer knowledge from the HR domain to LR \n84 branches. \n85 Low Resolution Face Recognition: Its task includes low resolution-to-low resolution (LR-to-LR) \n86 matching and low resolution-to-high resolution (LR-to-HR) matching [30]. The work can be divided \n87 into two categories [31]: (1) Super-resolution (SR) based methods aim to upscale LR images to \n88 construct HR images and use them for feature extraction [4–10]. (2) Projection-based methods aim to \n89 extract adequate representations in different domains and project them into a common feature space \n90 [32–34]. SR approaches are able to build faces with good visualization, but inevitably introduce \n91 feature information of other identities when reconstructing corresponding HR faces, thus introducing \n92 noise for identity-specific features. Compared to previous projection methods, our approach directly \n93 learns discriminative representations in a common feature space for HR and LR inputs, without \n94 additional projection heads for feature transformation. ", + "bbox": [ + 147, + 679, + 825, + 785 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 796, + 825, + 917 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 90, + 825, + 242 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Pseudo-Siamese Networks: Siamese networks are a coupling architecture based on DNNs, which are widely used for signature verification [35], face verification [36, 37], tracking [38], etc. PseudoSiamese networks [39] are decoupled Siamese networks, as the weights of the two branches are not shared, resulting in a more flexible representation way for the two entities. Hughes et. al. in [40] proposed a pseudo-Siamese CNN for identifying corresponding patches in SAR and optical images. Inspired by pseudo-Siamese networks, we propose a resolution-adaptive partially coupled Siamese network architecture, extracting specific-shared features for images with different resolutions. ", + "bbox": [ + 160, + 252, + 826, + 358 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Learning Specific-Shared Feature Transfer ", + "text_level": 1, + "bbox": [ + 160, + 372, + 563, + 390 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Instead of rescaling the inputs to a canonical size, we build multiple resolution-specific branches (BNets) that are used to map inputs to intermediate features with the same resolution and a resolutionshared trunk (TNet) to map feature maps with different resolutions to a high-dimension embedding. We gain several important properties by doing so: (1) Processing inputs on its original resolution can diminish the inevitably introduced error via up-sampling or information loss via down-sampling, thus preserving the discriminability of visual information with different resolutions. (2) Information streams of different resolutions are encoded uniformly, thus enabling the representation compatibility, which is particularly beneficial to open-set face recognition considering that a compatible metric space is the prerequisite for computing similarity. (3) This also effectively reduce the computation for LR images by supplying computational resources conditioned on the input resolution. ", + "bbox": [ + 163, + 400, + 826, + 550 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "113 3.1 Up-Sampling Error Analysis ", + "text_level": 1, + "bbox": [ + 143, + 561, + 411, + 577 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/bcaf8fb368f804daba922e50885b5d890a9bd70ea33f0a8743d4e42864a72421.jpg", + "image_caption": [ + "Figure 1: Estimated Error Upperbound. (bilinear interpolation, average value for over 100 images) with the change of image resolution relative to resolution 112. " + ], + "image_footnote": [], + "bbox": [ + 178, + 597, + 457, + 742 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/49c6de557e00d2a1971a3a5c302649e9eb1af1a26b38f484c1a1576206274175.jpg", + "image_caption": [ + "Figure 2: Basic ideas of the proposed BTNet. Images of a certain identity are first projected to the feature maps with the same resolution respectively (Adapt) and then projected to a unified feature representation (Encode). In this figure, feature maps with the same resolution are indicated by outlines in the same color. " + ], + "image_footnote": [], + "bbox": [ + 483, + 563, + 813, + 724 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "114 Figure 1 illustrates the experimental estimation of interpolation error, whose upper bound increases \n115 with the decline of the image resolution (see detailed theoretical derivation in Appendix A.1). Note \n116 that the error soars up when the resolution drops below 32 approximately which can be viewed as LR \n117 face images, consistent with the tiny-object criterion [41]. \n118 The results show that: (1) inputs with a resolution higher than around 32 can be considered in the \n119 same HR domain, since the error information introduced by up-sampling via interpolation can be \n120 ignored to a certain extent; (2) inputs with a resolution lower than around 32 should be treated as in \n121 various LR domains due to the high sensitivity of the resolution to errors. ", + "bbox": [ + 140, + 851, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 90, + 825, + 151 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 Branch-to-Trunk Network ", + "text_level": 1, + "bbox": [ + 174, + 162, + 397, + 178 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Let $X$ be an input RGB image with a space shape: $X \\in \\mathbb { R } ^ { H \\times W \\times 3 }$ where $H \\times W$ corresponds to the spatial dimension of the input. For efficient batch training and inference, we predefine a canonical size $S \\times S$ (e.g., $1 1 2 \\times 1 1 2$ for typical face recognition models like ArcFace [1]). ", + "bbox": [ + 171, + 184, + 825, + 229 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We build a trunk network $\\boldsymbol { T } : \\mathbb { R } ^ { H \\times W \\times 3 } \\mathbb { R } ^ { C _ { e m b } }$ capable of extracting discriminative information with different resolutions, where $C _ { e m b }$ is the number of embedding channels. For every resolution $r$ in the candidate set, we formulate a resolution-specific branch, $z _ { r } = \\textit { B } _ { r } ( X _ { r } )$ , which maps the input image $X _ { r }$ to feature maps with the same resolution and expanded channels $\\boldsymbol { z } _ { r } : \\mathbb { R } ^ { r \\times r \\times 3 } \\mathbb { R } ^ { r \\times r \\times C _ { r } }$ . The idea is to learn our branches $B$ to focus on resolution-specific feature transfer independently. Feature maps will then be coupled to the trunk network $T$ in the feature pyramid with the same spatial resolution $r \\times r$ , allowing for further mapping to the unified presentation space by $T _ { r } : \\mathbb { R } ^ { r \\times r \\times C _ { r } } $ $\\mathbb { R } ^ { C _ { e m b } }$ . ", + "bbox": [ + 168, + 234, + 825, + 357 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here, we follow the idea of “avoiding redundant up-sampling”. Our branches $B$ are implemented with same-resolution mapping: i.e., the model preserves the network architecture of $T$ from input to the layer with resolution $r$ and abandons down-sampling operations (e.g., replacing the convolution of stride 2 with stride 1, abandoning the pooling layers, etc.) to keep the same-resolution flow. ", + "bbox": [ + 158, + 364, + 825, + 425 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We specifically name our specific-shared feature transfer network as Branch-to-Trunk Network, abbreviated as \"BTNet\". Figure 2 visually summarizes the main ideas of BTNet. ", + "bbox": [ + 165, + 431, + 823, + 462 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 Training Objectives ", + "text_level": 1, + "bbox": [ + 174, + 473, + 351, + 489 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We now describe the training objectives. The training of BTNet includes training the trunk network $T$ such that it can produce discriminative and compatible representations for multi-resolution information, and fine-tuning the branch networks $B$ to encourage them to learn resolution-specific feature transfer, so as to improve accuracy without compromising compatibility. ", + "bbox": [ + 173, + 494, + 825, + 555 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Influence Loss. It is a compatibility-aware classification loss which is implemented by feeding the embeddings of the new model to the classifier of the old model [11]. Since the difficulties of samples vary due to image resolution, we compute CurricularFace [3] as our classification loss, in the form of: ", + "bbox": [ + 171, + 566, + 825, + 627 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d6b9123892a8d3b50dd58976ab5588ce77e9987f40e0e996073d9ea59c4e0169.jpg", + "text": "$$\nL _ { c u r } = - \\log ( \\frac { e ^ { s \\cos ( \\theta _ { y _ { i } } + m ) } } { e ^ { s \\cos ( \\theta _ { y _ { i } } + m ) } + \\sum _ { j = 1 , j \\neq y _ { i } } ^ { n } e ^ { s N ( t ^ { ( k ) } , \\cos ( \\theta _ { j } ) ) } } )\n$$", + "text_format": "latex", + "bbox": [ + 303, + 645, + 694, + 685 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8f657c9f655cdab89de25260c177752739bba8583601187e6f0314079daf299b.jpg", + "text": "$$\n\\begin{array} { c } { { N ( t , \\cos \\theta _ { j } ) = \\displaystyle \\left\\{ \\begin{array} { l c } { { \\cos ( \\theta _ { j } ) , } } & { { \\cos ( \\theta _ { y _ { i } } + m ) - \\cos ( \\theta _ { j } ) \\ge 0 } } \\\\ { { \\cos ( \\theta _ { j } ) ( t + \\cos ( \\theta _ { j } ) ) , } } & { { e l s e } } \\end{array} \\right. } } \\\\ { { t ^ { ( k ) } = \\displaystyle \\alpha \\sum _ { i } \\cos \\theta _ { y _ { i } } + ( 1 - \\alpha ) t ^ { ( k - 1 ) } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 256, + 710, + 730, + 792 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "150 which distinguishes both the difficultness of different samples in each stage and relative importance \n151 of easy and hard samples during different training stages. Thus, we refine CurricularFace loss as our \n152 influence loss: ", + "bbox": [ + 140, + 803, + 825, + 848 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2f5169b65aef79ee9ba9f8724552804ab1c162a4b97e4bdecf71cc03dc3589ad.jpg", + "text": "$$\nL _ { i n f l u e n c e } = L _ { c u r } ( \\varphi _ { b t } , \\kappa ^ { * } )\n$$", + "text_format": "latex", + "bbox": [ + 403, + 867, + 594, + 885 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where 153 $\\varphi _ { b t }$ is BTNet backbone (both $B _ { r }$ and $T _ { r }$ ), and $\\kappa ^ { * }$ is the classifier of the pretrained trunk $T$ . ", + "bbox": [ + 135, + 897, + 810, + 915 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "154 Branch Distillation Loss. Due to the \n155 continuity of the scale change of both the \n156 image pyramid and the feature pyramid \n157 [42], we can get a qualitative sense of \n158 the similarity between images and feature \n159 maps with the same resolution (see Figure \n160 3). Furthermore, features extracted from \n161 HR images have richer and clearer infor \n162 mation than those from LR images [43]. \n163 Motivated by these analyses, we utilize an \n164 MSE loss to encourage the branch output \n165 $z _ { r }$ to be similar to the corresponding fea \n166 ture maps of the pretrained trunk network \n167 $z _ { s }$ : ", + "bbox": [ + 140, + 90, + 452, + 301 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/e434a9a21fd643648ac4896d6cdd3ccf6e9342a751aac825373efc28d542ddee.jpg", + "image_caption": [ + "Figure 3: Visual comparison of face image-feature map pairs with different resolutions (resized to a common size here for illustration). " + ], + "image_footnote": [], + "bbox": [ + 467, + 82, + 820, + 231 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/f2246fd21423a49aa7cc30c51c18bb60978ccaa1ffe7a96330ec0d7e5b0584e2.jpg", + "text": "$$\nL _ { b r a n c h } = \\frac { 1 } { V } \\sum _ { v = 1 } ^ { V } { ( z _ { r _ { v } } - z _ { s _ { v } } ) } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 395, + 313, + 602, + 356 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "168 where $V$ denotes the batch size. ", + "bbox": [ + 147, + 358, + 383, + 373 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "169 The whole training objective is a combination of the above objectives: ", + "bbox": [ + 145, + 380, + 632, + 395 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/cbd4434da50cb2dcda48da9a71a6dcc4aa4054c36bb94eaea39636b77d6c7fe8.jpg", + "text": "$$\nL = L _ { i n f l u e n c e } + \\lambda _ { b r a n c h } L _ { b r a n c h }\n$$", + "text_format": "latex", + "bbox": [ + 382, + 397, + 614, + 415 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "170 where $\\lambda _ { b r a n c h }$ is a hyper-parameter to weigh the losses and we set $\\lambda _ { b r a n c h } = 0 . 5$ in all our experi \n171 ments. ", + "bbox": [ + 140, + 421, + 833, + 452 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/22946257e92b509279813a29ef0c3ac19fcb89eb7b3c4b6d171707c4f9ae79f6.jpg", + "image_caption": [ + "Figure 4: Comparison of $\\#$ Params $\\mathbf { \\Psi } ^ { ( \\mathbf { M } ) }$ between fully finetuning and $\\varphi _ { b t }$ . " + ], + "image_footnote": [], + "bbox": [ + 209, + 468, + 460, + 617 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/1df08d5fa7d736b80d3f65aa605b4972425929210f8a903a7b8e250ca889addb.jpg", + "image_caption": [ + "Figure 5: Comparison of FLOPs (G) between baselines and $\\varphi _ { b t }$ . " + ], + "image_footnote": [], + "bbox": [ + 526, + 468, + 777, + 617 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "172 3.4 Storing Branch Networks ", + "text_level": 1, + "bbox": [ + 147, + 678, + 392, + 693 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "An obvious adaptation strategy is fully finetuning of the model on each resolution. However, this strategy requires one to store and deploy a separate copy of the backbone parameters for every resolution, which is an expensive proposition and difficult to expand into more segmented resolution branches. Our BTNet is beneficial in the scenario of multi-resolution face recognition which achieves better parameter/accuracy trade-offs. Since activation statistics including means and variances under different resolutions are incompatible [44], we update and store Batch Normalization (BN) [45] parameters in all layers of $B _ { r }$ and $T _ { r }$ for each resolution, whose amount is negligible. Apart from this, we only need to store the learned branches and re-use the original copy of the pretrained trunk model, significantly reducing the storage cost. Figure 4 shows that BTNet requires only $1 . 1 \\% \\sim 4 8 . 9 \\%$ of all the parameters compared to fully updating all the parameters of TNet. ", + "bbox": [ + 173, + 698, + 826, + 849 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 169, + 856, + 313, + 872 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "184 To validate BTNet on face recognition tasks in open universe, we perform 1:1 verification and $1 : N$ \n185 identification tasks in two different settings, including (a) multi-resolution identity matching, and ", + "bbox": [ + 148, + 882, + 825, + 912 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "(b) multi-resolution feature aggregation. For 1:1 verification, a pair of templates are provided and the model is to decide whether they belong to the same identity or not. For 1:N identification, a set of gallery images are first mapped onto their embedding vectors (indexing) and the embeddings of query images are extracted to perform search against indexed gallery. ", + "bbox": [ + 165, + 90, + 825, + 151 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 Implementation Details ", + "text_level": 1, + "bbox": [ + 166, + 164, + 377, + 179 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Datasets. We use MS1Mv3 [46] for training face embedding models. The MS1Mv3 dataset contains 5,179,510 images of 93,431 celebrities. According to the test setting, different test datasets are used. ", + "bbox": [ + 174, + 189, + 825, + 233 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "·Multi-Resolution Identity Matching. We try on six widely adopted face verification benchmarks: LFW [47], CFP-FF [48], CFP-FP [48], AgeDB-30 [49], CALFW [50], and CPLFW [51], while the large-scale surveillance face dataset QMUL-SurvFace [17] is used for 1:N face identification, which contains native LR surveillance faces across wide space and time. The spatial resolution for QMUL-SurvFace ranges from 6/5 to 124/106 in height/width with an average of 24/20. ", + "bbox": [ + 173, + 246, + 825, + 321 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "·Multi-Resolution Feature Aggregation. We adopt a top challenging benchmark IJB-C [52], which has around 130k images from 3,531 identities, for two standard testing protocols: $1 : 1$ verification and 1:N identification. ", + "bbox": [ + 173, + 334, + 825, + 378 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Training. All the models are trained on four RTX 2080 Tis with batch size 128 by stochastic gradient descent. For TNet, we train for 25 epochs, with learning rate initialized at 0.2 with 2 warmup epochs and decaying as a quadratic polynomial. We augment training samples by random horizonal flipping and multi-resolution training. For BNets, we initialize the learning rate by 0.02 without warm-up epochs. The training all stops at the $1 0 t h$ epoch for a fair comparison. The recommended hyper-parameters are used for classification loss from the original paper (e.g., $m = 0 . 5 , s = 6 4$ for ArcFace [1], and $\\alpha = 0 . 9 9 , t ^ { 0 } = 0$ for CurricularFace [3]). Only horizonal flipping is used as augmentation when training BNets. ", + "bbox": [ + 173, + 391, + 826, + 512 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Baselines. In our experiment, several baselines are used to validate BTNet in learning discriminative and compatible representations for multi-resolution face recognition. ", + "bbox": [ + 176, + 523, + 823, + 554 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "·High-Resolution Trained $\\varphi _ { h r }$ . Naive baseline trained with HR data. ", + "bbox": [ + 171, + 566, + 643, + 582 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "·Independently Trained $\\varphi _ { m m }$ . Multi-model fashion: is it possible to achieve better results if we train a specific model for each resolution independently? Specifically, we train $\\varphi _ { r }$ for data with resolution $r$ and denote the multi-model collections as $\\varphi _ { m m }$ . ", + "bbox": [ + 174, + 594, + 825, + 640 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "·Multi-Resolution Trained $\\varphi _ { m r }$ . Trained with multi-resolution data which adapts to resolutionvariance. Specifically, each image is randomly down-sampled to a size in the candidate set $\\textstyle \\{ { \\frac { 1 1 2 } { 2 ^ { i } } } \\ x$ $\\textstyle { \\frac { 1 1 2 } { 2 ^ { i } } } | i = 0 , 1 , 2 , 3 , 4 \\}$ with equal probability of being chosen, and then up-sampled back to $1 1 2 \\times 1 1 2$ ", + "bbox": [ + 174, + 651, + 826, + 696 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Instantiation of Network Architecture. The BTNet and baselines are implemented with ResNet50 [53], and they could be extended easily with other implementations. Dubbed as $\\varphi _ { b t }$ , the detailed instantiation of BTNet based on ResNet50 is illustrated in Appendix A.2. ", + "bbox": [ + 174, + 708, + 825, + 753 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 Evaluation Metrics ", + "text_level": 1, + "bbox": [ + 174, + 766, + 346, + 781 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "On the benchmarks for face verification, we use 1:1 verification accuracy as the basic metrics. The rank-20 true positive identification rates (TPIR20) at varying false positive identification rates (FPIR) and AUC are used to report the identification results on QMUL-SurvFace. The evaluation metrics for IJB-C 1:1 verification protocol are true acceptance rates (TAR) at different false acceptance rate (FAR). For 1:N identification, the basic evaluation metrics are the true positive identification rates (TPIR) at different false positive identification rates (FPIR). ", + "bbox": [ + 173, + 789, + 825, + 878 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "229 For better evaluation, we define another two metrics to assess the relative performance gain similar to \n230 [11, 14]. ", + "bbox": [ + 150, + 886, + 825, + 916 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "(a) Cross-resolution identity matching. ", + "bbox": [ + 220, + 102, + 449, + 116 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/5bc6cd3f78ae7b22f0215c5956576a0f985cb5a161b0a0711b0085a5814c9685.jpg", + "table_caption": [ + "Table 1: Comparison of different methods on six face verification benchmarks. “Acc.” denotes average 1:1 verification accuracy. " + ], + "table_footnote": [], + "table_body": "
112&7112&14112&28
Acc.GainAcc.GainAcc.Gain
hr57.75-81.02=95.90
mm50.58-0.8949.90-4.8250.03-305.80
mr65.85+1.0087.47+1.0096.05+1.00
bt(Ours)86.10+3.5094.08+2.0296.65+5.00
", + "bbox": [ + 173, + 122, + 480, + 218 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/e4b339b356e5710a821d674e46406b978e735e5547439a4c5a002b9ecd815938.jpg", + "table_caption": [ + "(b) Same-resolution identity matching. " + ], + "table_footnote": [], + "table_body": "
7&714&1428&28112&112
Acc.GainAcc.GainAcc.GainAcc.Gain
60.7073.8893.58=97.68-
62.57+1.0078.00+1.0094.68+1.0097.68-
61.02+0.1780.32+1.5695.12+1.4097.25-
77.78+9.1390.90+4.1396.27+2.4597.25=
", + "bbox": [ + 495, + 122, + 812, + 218 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "231 Cross-Resolution Gain. With the purpose towards the cross-resolution compatible representations, \n232 we define the performance gain as follows: ", + "bbox": [ + 138, + 232, + 826, + 262 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/4b139041d6657e17e2e1fa3469d8912dd4a8543ae9b6b2451d4ccd7624681030.jpg", + "text": "$$\nG a i n _ { r _ { 1 } \\& r _ { 2 } } ( \\varphi ) = \\frac { M _ { r _ { 1 } \\& r _ { 2 } } ( \\varphi ) - M _ { r _ { 1 } \\& r _ { 2 } } ( \\varphi _ { h r } ) } { | M _ { r _ { 1 } \\& r _ { 2 } } ( \\varphi _ { m r } ) - M _ { r _ { 1 } \\& r _ { 2 } } ( \\varphi _ { h r } ) | }\n$$", + "text_format": "latex", + "bbox": [ + 330, + 273, + 668, + 308 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Here 33 $M _ { r _ { 1 } \\& r _ { 2 } } ( \\cdot )$ are metrics when the resolutions of the image/template pair are $r _ { 1 } \\times r _ { 1 }$ and $r _ { 2 } \\times r _ { 2 }$ 4 $( r _ { 1 } \\neq r _ { 2 } )$ , respectively. $\\varphi _ { m r }$ shares the same architecture with $\\varphi _ { h r }$ while is trained on multi-resolution 5 images and thus serves as the baseline of cross-resolution gain. ", + "bbox": [ + 156, + 318, + 825, + 363 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "36 Same-Resolution Gain. For the scenario of multi-resolution face recognition, the performance of \n37 same-resolution verification/identification is also vital besides cross-resolution one. Therefore, we \n38 report the relative performance improvement from base model $\\varphi _ { h r }$ in the scenario of same-resolution. ", + "bbox": [ + 151, + 372, + 826, + 417 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/afc66ac1bfdfe86e199dea6044bcececcbb9f300a5752f45c8950c00da6e810e.jpg", + "text": "$$\nG a i n _ { r \\& r } ( \\varphi ) = \\frac { M _ { r \\& r } ( \\varphi ) - M _ { r \\& r } ( \\varphi _ { h r } ) } { | M _ { r \\& r } ( \\varphi _ { r } ) - M _ { r \\& r } ( \\varphi _ { h r } ) | }\n$$", + "text_format": "latex", + "bbox": [ + 354, + 422, + 643, + 457 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "39 Here $M _ { r \\& r } \\left( \\cdot \\right)$ are metrics when the resolutions of the image/template pair are both $r \\times r$ . $\\varphi _ { r }$ is \n40 a model of the set $\\{ \\varphi _ { m m } = \\varphi _ { r } | r = 7 , 1 4 , 2 8 \\}$ trained on images with resolution $r \\times r$ without \n41 considering cross-resolution representation compatibility, which serves as the baseline of same \n42 resolution gain on resolution $r$ . Note that for both metrics we add the absolute symbol to the \n43 denominator as they can be negative in some test settings (detailed in Section 4.3 and 4.4). ", + "bbox": [ + 150, + 467, + 825, + 542 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 Multi-Resolution Identity Matching ", + "text_level": 1, + "bbox": [ + 168, + 551, + 462, + 568 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We now conduct experiments on the proposed BTNet framework for multi-resolution identity matching. Two different settings are included : (1) same-resolution matching, and (2) cross-resolution matching. Table 1 compares the average performance on popular benchmarks for $\\varphi _ { h r } , \\varphi _ { m m } , \\varphi _ { m r } ,$ $\\varphi _ { b t }$ . The experimental results on each dataset are detailed in Appendix A.5. ", + "bbox": [ + 173, + 573, + 826, + 633 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "249 When directly applied to test data with the resolution lower than training data, $\\varphi _ { h r }$ suffers a severe \n250 performance degradation. Up-sampling images via interpolation can increase the amount of data \n251 but not the amount of information, only to improve the detailed part of the image and the spatial \n252 resolution (size) [64]. Moreover, it also brings various noise and artificial processing traces [65]. \n253 Up-sampling images via interpolation-typically bilinear interpolation or bicubic interpolation of \n254 4x4 pixel neighborhoods, essentially a function approximation method, is bound to introduce error \n255 information (detailed in Appendix A.1), thus potentially confusing identity information, which is \n256 especially crucial for LR images with limited details. We are able to observe improvement of $\\varphi _ { m m }$ in \n257 same-resolution matching but its cross-resolution gain is negative with approximately $50 \\%$ accuracy. \n258 Unsurprisingly, independently trained $\\varphi _ { r }$ is unaware of representation compatibility, and thus does \n259 not naturally suitable for cross-resolution recognition. The results show that $\\varphi _ { m r }$ improved both \n260 cross-resolution and same-resolution accuracy by a large margin, as it learns to adapt to resolution \n261 variance and maintain discriminability of multi-resolution inputs. Note that the model size and \n262 training data scale stay the same, while only the resolution distribution of the data changes for \n263 $\\varphi _ { m r }$ , and thus there is a marginal accuracy drop in the setting of 112&112 matching. Comparably, \n264 $\\varphi _ { b t }$ substantially outperforms all baselines with $2 . 0 2 \\sim 5 . 0 0$ cross-resolution gain and 2.45\\~9.13 \n265 same-resolution gain. Importantly, due to the multi-resolution branches, our approach has a cost same \n266 with $\\varphi _ { m m }$ , significantly lower than $\\varphi _ { h r }$ and $\\varphi _ { m r }$ (see Figure 5). \n267 For inference on inputs with resolutions not strictly matched to the branch, we validate three selection \n268 strategies based on three resolution indicators (see Figure 6). Table 2 compares BTNet against the \n269 state-of-the-arts models on QMUL-SurvFace 1:N identification benchmark. We are able to observe \n270 that our proposed approach extends the state-of-the-arts while being more computationally efficient. \n271 We believe the performance of BTNet (max $^ +$ ceil) is the highest that have been reported so far, and \n272 we believe it is meaningful with the increased focus on unconstrained surveillance applications. ", + "bbox": [ + 142, + 638, + 826, + 912 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/a917f84eb65923fef0ec3ac3270ad45b1e02c6a459f4c0c4aa26583a3e31ddf9.jpg", + "table_caption": [ + "Table 2: Performance of face identification on QMUL-SurvFace. Most compared results are cited from [17, 54] except BTNet. " + ], + "table_footnote": [], + "table_body": "
TPIR20(%)@FPIR
AUC0.30.20.10.01
VGG-Face [55]14.05.12.60.80.1
DeepID2 [56]20.812.88.13.40.8
FaceNet [57]19.812.78.14.31.0
SphereFace [58]28.121.315.78.31.0
SRCNN [59]27.020.014.96.20.6
FSRCNN [60]27.320.014.46.10.7
VDSR [61]27.320.114.56.10.8
DRRN [62]27.520.314.96.30.6
LapSRN [63]27.420.214.76.30.7
ArcFace [1]25.318.715.110.12.0
RAN [54]32.326.521.614.93.8
BTNet (avg.+floor)32.627.923.416.51.4
BTNet (avg.+near)34.630.325.718.91.5
BTNet (avg.+ceil)35.431.126.820.32.2
BTNet (min+floor)32.327.623.216.11.4
BTNet (min+near)34.029.625.018.01.4
BTNet (min+ceil)35.331.026.619.92.0
BTNet (max+floor)33.629.124.517.61.3
BTNet (max+near)35.231.026.419.61.7
BTNet (max+ceil)35.431.226.920.62.5
", + "bbox": [ + 318, + 121, + 676, + 460 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 481, + 825, + 571 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.4 Multi-Resolution Feature Aggregation ", + "text_level": 1, + "bbox": [ + 166, + 583, + 480, + 599 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Multi-resolution feature aggregation is common in set-based recognition tasks where the model needs to determine the similarity of sets (templates), instead of images. Each set could contain images of the same identity with different resolutions. In our experiment, we rescale the original and flipped images in each set to different resolutions and aggregate their features into a representation of the template. Detailed experimental results can be seen in Appendix A.5. ", + "bbox": [ + 173, + 604, + 825, + 679 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 3 (a) compares the cross-resolution results of $\\mathrm { T A R } @ \\mathrm { F A R } = 1 0 ^ { - 4 }$ for 1:1 verification. The cross-resolution features are ensured to be mapped to the same vector space where the aggregation is conducted for $\\varphi _ { h r }$ and $\\varphi _ { m r }$ , but we can observe that $\\varphi _ { h r }$ performs much better than $\\varphi _ { m r }$ . One possible reason is that $\\varphi _ { h r }$ has outstanding discriminability to extract HR features, while LR features may not overly deteriorate the HR information. This phenomenon also suggests that $\\varphi _ { m r }$ sacrifices its discriminability in exchange for the adaptability for resolution-variance. We can see $\\varphi _ { b t }$ is comparable with $\\varphi _ { h r }$ , demonstrating the discriminative power of BTNet for aggregating multi-resolution features. ", + "bbox": [ + 173, + 685, + 825, + 792 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 3 (b) compares the same-resolution results of $\\mathrm { T A R } @ \\mathrm { F A R } = 1 0 ^ { - 4 }$ for 1:1 verification. When HR information is removed from the template representation (i.e., test settings 7&7, 14&14, 28&28), $\\varphi _ { h r }$ suffers from performance degradation as well, as the informative embedding cannot catch the lost details of the LR images [54]. Both $\\varphi _ { m m }$ and $\\varphi _ { m r }$ improve with a limited same-resolution gain, while $\\varphi _ { b t }$ surpasses the baselines by a large margin while also reducing the compute. ", + "bbox": [ + 171, + 799, + 825, + 875 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Table 4 we show the results of291 $\\mathrm { T P I R } @ \\mathrm { F P I R } { = } 1 0 ^ { - 1 }$ for 1:N identification protocol. Similar to our 292 results for 1:1 verification, we are able to observe that $\\varphi _ { b t }$ is comparable or even better than $\\varphi _ { h r }$ with ", + "bbox": [ + 145, + 881, + 823, + 912 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 3: Comparison of different methods on the IJB-C dataset 1:1 face verification task. \n“TAR” denotes TAR $( \\% @ \\mathsf { F A R = } 1 \\mathsf { e } { \\mathrm { - } } 4 )$ . ", + "bbox": [ + 168, + 65, + 753, + 93 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(a) Cross-resolution feature aggregation. ", + "bbox": [ + 215, + 103, + 454, + 116 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/5b1f1bff4f8c3abafbab07138d8023950fef2ba90c0c9e80acbd2e7a3f3c7011.jpg", + "table_caption": [ + "(b) Same-resolution feature aggregation. " + ], + "table_footnote": [], + "table_body": "
7&714&1428&28112&112
TARGainTARGainTARGainTARGain
4.83=33.7489.6596.40
4.83+0.0029.26-1.0092.58+1.0096.40=
4.48-40.51+1.5192.81+1.0896.06
35.47-82.08+10.7994.50+1.6696.06=
", + "bbox": [ + 496, + 122, + 818, + 218 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/2443428c5deb0919a8302f752c3c6d53991627ec9633ca5d82d7fb69c4bdb157.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
112&7112&14112&28
TARGainTARGainTARGain
hr88.8992.40=95.62
mm74.54-0.5693.52+1.3395.42-0.69
mr63.11-1.0091.56-1.0095.33-1.00
bt(Ours)88.17-0.0393.97+1.8795.62+0.00
", + "bbox": [ + 173, + 122, + 470, + 217 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 4: Comparison of different methods on the IJB-C dataset 1: N face identification task. \n“TPIR” denotes TPIR $( \\% @ \\mathsf { G P I R { = } } 0 . 1 $ ). ", + "bbox": [ + 169, + 234, + 774, + 262 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(a) Cross-resolution feature aggregation. ", + "bbox": [ + 215, + 271, + 455, + 285 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/6065fcbea5b19bea017c03a6374b0469b7a188b7d66b4092782fe3b712c6e307.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
112&7112&14112&28
TPIRGainTPIRGainTPIRGain
hr85.60-90.11-94.27-
mm69.70-0.5591.73+1.5394.13-0.33
mr56.64-1.0089.05-1.0093.84-1.00
bt(Ours)83.93-0.0691.87+1.6694.33+0.14
", + "bbox": [ + 173, + 291, + 470, + 387 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/7f48a4a8376d99bea53855f456f39b4b1f9bc918498a85209e6e981ad9516181.jpg", + "table_caption": [ + "(b) Same-resolution feature aggregation. " + ], + "table_footnote": [], + "table_body": "
7&714&1428&28112&112
TPIRGainTPIRGainTPIRGainTPIRGain
3.12-26.37-86.06=95.57-
3.24+1.0021.84-1.0089.76+1.0095.57=
3.25+1.0837.58+2.4791.02+1.3494.85-
27.70+204.8376.65+11.1092.89+1.8594.85-
", + "bbox": [ + 496, + 291, + 834, + 387 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "293 HR information involved and can preserve superior discriminability with limited LR information, \n294 while also being more computationally efficient. ", + "bbox": [ + 142, + 414, + 826, + 444 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/a8c4e8888c6239a2f78ab3386fd66c4578ddb9b6f3764bbdd536c5e5c274bf6e.jpg", + "image_caption": [ + "Figure 6: Branch selection process. Max/min/average is used on (W, H) to obtain a resolution indicator for further allocation (floor/near/ceil) to a certain branch. " + ], + "image_footnote": [], + "bbox": [ + 186, + 459, + 799, + 549 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "295 5 Discussion and Conclusion ", + "text_level": 1, + "bbox": [ + 151, + 606, + 428, + 623 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This paper works on the problem of multi-resolution face recognition, and provides a new scheme to operate images conditioned on its input resolution without large span rescaling. The error introduced by up-sampling via interpolation is investigated and analyzed. Decoupled as branches for discriminative representation learning and coupled as the trunk for compatible representation learning, our Branch-to-Trunk Network (BTNet) achieves significant improvements on multi-resolution face verification and identification tasks. Besides, the superiority of BTNet in reducing computational cost and parameter storage cost is also demonstrated. It is worth noting that our approach is easy to expand to recognition tasks for other classes of objects and has the potential to serve as a general network architecture for multi-resolution visual recognition. ", + "bbox": [ + 173, + 633, + 825, + 768 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Limitations and Future Work. The dislocation between the underlying optical resolution of native face images and that of a certain branch may limit the power of the model, which may be improved by selecting the optimal processing branch for the input in combination with the image quality, rather than by image size alone. The optimal branch selection strategy is not fully investigated though we have provided an intuitive way to select the branch for inputs (see Figure 6). Importantly, based on the unified multi-resolution metric space, the underlying resolution of the inputs (integrated spatial resolution with quality assessment) can be utilized to provide the reliability of the representation and contribute to risk-controlled face recognition. They will be our future research directions. ", + "bbox": [ + 168, + 779, + 825, + 900 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "References \n[1] Jiankang Deng, Jia Guo, Niannan Xue, and Stefanos Zafeiriou. Arcface: Additive angular margin loss for deep face recognition. In IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2019, Long Beach, CA, USA, June 16-20, 2019, pages 4690–4699. Computer Vision Foundation / IEEE, 2019. \n[2] Xiaobo Wang, Shifeng Zhang, Shuo Wang, Tianyu Fu, Hailin Shi, and Tao Mei. Mis-classified vector guided softmax loss for face recognition. 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", + "bbox": [ + 163, + 101, + 826, + 922 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "", + "bbox": [ + 163, + 89, + 826, + 372 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Checklist ", + "text_level": 1, + "bbox": [ + 142, + 396, + 254, + 412 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "1. For all authors... ", + "bbox": [ + 214, + 421, + 330, + 434 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] \n(b) Did you describe the limitations of your work? [Yes] See Section 5.The dislocation between the underlying optical resolution of native face images and that of a certain branch may limit the power of the model, which may be improved by selecting the optimal processing branch for the input in combination with the image quality, rather than by image size alone. The optimal branch selection strategy is not fully investigated though we have provided an intuitive way to select the branch for inputs. \n(c) Did you discuss any potential negative societal impacts of your work? [N/A] We study a general framework for multi-resolution face recognition. Our method is not for specific applications, which does not directly involve societal issues. \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ", + "bbox": [ + 240, + 439, + 825, + 598 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 215, + 603, + 467, + 616 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Appendix A.1 \n(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix A.1 ", + "bbox": [ + 240, + 619, + 799, + 648 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "3. If you ran experiments... ", + "bbox": [ + 215, + 654, + 377, + 666 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We follow the common practice in previous works, where they didn’t report the error bars. \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See our implementation details in Section 4. 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", + "bbox": [ + 220, + 92, + 656, + 106 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "591 \n592 \n593 \n594 \n595 \n596 \n597 ", + "bbox": [ + 138, + 101, + 163, + 194 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? 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As the small feature maps with a fixed spatial extent (e.g.,", + "type": "text" + }, + { + "bbox": [ + 363, + 704, + 387, + 715 + ], + "score": 0.87, + "content": "7 \\times 7 )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 703, + 505, + 718 + ], + "score": 1.0, + "content": "are mapped to an embedding", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 89, + 75, + 99, + 84 + ], + "score": 1.0, + "content": "35", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 73, + 250, + 84 + ], + "score": 1.0, + "content": "with a predefined dimension (e.g.,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 250, + 73, + 285, + 84 + ], + "score": 0.38, + "content": "1 2 8 - d", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 285, + 73, + 288, + 84 + ], + "score": 1.0, + "content": ",", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 289, + 73, + 324, + 84 + ], + "score": 0.47, + "content": "5 1 2 - d", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 324, + 73, + 505, + 84 + ], + "score": 1.0, + "content": ", etc.) by a fully connected (fc) layer, input", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 84, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 89, + 87, + 100, + 96 + ], + "score": 1.0, + "content": "36", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 84, + 344, + 97 + ], + "score": 1.0, + "content": "images need to be rescaled to a canonical spatial size (e.g.,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 345, + 85, + 389, + 96 + ], + "score": 0.88, + "content": "1 1 2 \\times 1 1 2 )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 390, + 84, + 506, + 97 + ], + "score": 1.0, + "content": "before fed into the network.", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 96, + 506, + 109 + ], + "spans": [ + { + "bbox": [ + 89, + 99, + 100, + 108 + ], + "score": 1.0, + "content": "37", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 96, + 506, + 109 + ], + "score": 1.0, + "content": "However, up-sampling low-resolution (LR) images introduces the interpolation error (see Section 3.1),", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 109, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 89, + 110, + 100, + 120 + ], + "score": 1.0, + "content": "38", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 109, + 505, + 120 + ], + "score": 1.0, + "content": "deteriorating the recognizable ones which contain enough clues to identify the subject. Even though", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 120, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 89, + 123, + 100, + 132 + ], + "score": 1.0, + "content": "39", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 120, + 505, + 133 + ], + "score": 1.0, + "content": "super-resolution methods [4–10] are widely used to build faces with good visualization, they inevitably", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 89, + 135, + 100, + 144 + ], + "score": 1.0, + "content": "40", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "introduce feature information of other identities when reconstructing high-resolution (HR) faces.", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 144, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 89, + 146, + 99, + 156 + ], + "score": 1.0, + "content": "41", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 144, + 506, + 157 + ], + "score": 1.0, + "content": "This may lead to erroneous identity-specific features, which are detrimental to risk-controlled face", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 156, + 157, + 169 + ], + "spans": [ + { + "bbox": [ + 88, + 158, + 100, + 169 + ], + "score": 1.0, + "content": "42", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 156, + 157, + 169 + ], + "score": 1.0, + "content": "recognition.", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 174, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 90, + 176, + 99, + 185 + ], + "score": 1.0, + "content": "43", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 174, + 506, + 186 + ], + "score": 1.0, + "content": "Empirically, we can divide inputs by resolution distribution and learn to operate on them via multiple", + "type": "text", + "cross_page": true + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 186, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 89, + 188, + 99, + 197 + ], + "score": 1.0, + "content": "44", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 186, + 505, + 198 + ], + "score": 1.0, + "content": "models to achieve high accuracy and efficiency. However, multi-model fashion cannot be applied", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 89, + 200, + 99, + 209 + ], + "score": 1.0, + "content": "45", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "directly for cross-resolution recognition as representation compatibility among models need to be", + "type": "text", + "cross_page": true + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 209, + 190, + 222 + ], + "spans": [ + { + "bbox": [ + 89, + 212, + 100, + 221 + ], + "score": 1.0, + "content": "46", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 209, + 190, + 222 + ], + "score": 1.0, + "content": "guaranteed [11–15].", + "type": "text", + "cross_page": true + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 227, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 90, + 229, + 100, + 239 + ], + "score": 1.0, + "content": "47", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 227, + 506, + 241 + ], + "score": 1.0, + "content": "To improve discriminability while ensure the compatibility of the metric space for multi-resolution", + "type": "text", + "cross_page": true + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 89, + 241, + 100, + 251 + ], + "score": 1.0, + "content": "48", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "face representation, we learn the “unified” representation by a partially-coupled Branch-to-Trunk", + "type": "text", + "cross_page": true + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 89, + 253, + 100, + 263 + ], + "score": 1.0, + "content": "49", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "Network (BTNet). It is composed of multiple independent branch networks (BNets) and a shared", + "type": "text", + "cross_page": true + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 262, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 89, + 265, + 100, + 275 + ], + "score": 1.0, + "content": "50", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 262, + 506, + 276 + ], + "score": 1.0, + "content": "trunk network (TNet). A resolution-specific BNet is used for a given image, and the output are", + "type": "text", + "cross_page": true + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 275, + 481, + 288 + ], + "spans": [ + { + "bbox": [ + 89, + 276, + 100, + 286 + ], + "score": 1.0, + "content": "51", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 275, + 481, + 288 + ], + "score": 1.0, + "content": "implanted as feature maps in the feature pyramid of TNet, at a layer with the same resolution.", + "type": "text", + "cross_page": true + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 89, + 294, + 100, + 304 + ], + "score": 1.0, + "content": "52", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 292, + 505, + 305 + ], + "score": 1.0, + "content": "Furthermore, we find that multi-resolution training can be beneficial to building a strong and robust", + "type": "text", + "cross_page": true + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 89, + 306, + 100, + 316 + ], + "score": 1.0, + "content": "53", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "TNet, and backward-compatible training (BCT) [11] can improve the representation compatibility", + "type": "text", + "cross_page": true + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 315, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 89, + 318, + 100, + 328 + ], + "score": 1.0, + "content": "54", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 315, + 506, + 330 + ], + "score": 1.0, + "content": "during the training process of BTNet. 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For efficient batch training and inference, we predefine a canonical", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 171, + 434, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 124, + 183 + ], + "score": 1.0, + "content": "size", + "type": "text" + }, + { + "bbox": [ + 124, + 171, + 151, + 181 + ], + "score": 0.88, + "content": "S \\times S", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 171, + 175, + 183 + ], + "score": 1.0, + "content": "(e.g.,", + "type": "text" + }, + { + "bbox": [ + 176, + 171, + 219, + 181 + ], + "score": 0.89, + "content": "1 1 2 \\times 1 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 171, + 434, + 183 + ], + "score": 1.0, + "content": "for typical face recognition models like ArcFace [1]).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 143, + 506, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 103, + 186, + 505, + 283 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 210, + 201 + ], + "score": 1.0, + "content": "We build a trunk network", + "type": "text" + }, + { + "bbox": [ + 210, + 186, + 308, + 198 + ], + "score": 0.93, + "content": "\\boldsymbol { T } : \\mathbb { R } ^ { H \\times W \\times 3 } \\mathbb { R } ^ { C _ { e m b } }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 185, + 506, + 201 + ], + "score": 1.0, + "content": "capable of extracting discriminative information", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 199, + 504, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 237, + 213 + ], + "score": 1.0, + "content": "with different resolutions, where", + "type": "text" + }, + { + "bbox": [ + 237, + 200, + 261, + 212 + ], + "score": 0.92, + "content": "C _ { e m b }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 199, + 497, + 213 + ], + "score": 1.0, + "content": "is the number of embedding channels. 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All the models are trained on four RTX 2080 Tis with batch size 128 by stochastic", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 323, + 507, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 507, + 335 + ], + "score": 1.0, + "content": "gradient descent. For TNet, we train for 25 epochs, with learning rate initialized at 0.2 with 2 warm-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "up epochs and decaying as a quadratic polynomial. We augment training samples by random horizonal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "flipping and multi-resolution training. For BNets, we initialize the learning rate by 0.02 without", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 289, + 370 + ], + "score": 1.0, + "content": "warm-up epochs. The training all stops at the", + "type": "text" + }, + { + "bbox": [ + 290, + 358, + 311, + 369 + ], + "score": 0.78, + "content": "1 0 t h", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 359, + 505, + 370 + ], + "score": 1.0, + "content": "epoch for a fair comparison. The recommended", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 431, + 383 + ], + "score": 1.0, + "content": "hyper-parameters are used for classification loss from the original paper (e.g.,", + "type": "text" + }, + { + "bbox": [ + 431, + 370, + 505, + 382 + ], + "score": 0.91, + "content": "m = 0 . 5 , s = 6 4", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 381, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 192, + 396 + ], + "score": 1.0, + "content": "for ArcFace [1], and", + "type": "text" + }, + { + "bbox": [ + 192, + 382, + 264, + 394 + ], + "score": 0.9, + "content": "\\alpha = 0 . 9 9 , t ^ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 381, + 506, + 396 + ], + "score": 1.0, + "content": "for CurricularFace [3]). Only horizonal flipping is used as", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 395, + 250, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 250, + 407 + ], + "score": 1.0, + "content": "augmentation when training BNets.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 108, + 415, + 504, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "Baselines. In our experiment, several baselines are used to validate BTNet in learning discriminative", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 428, + 383, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 383, + 441 + ], + "score": 1.0, + "content": "and compatible representations for multi-resolution face recognition.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 105, + 449, + 394, + 461 + ], + "lines": [ + { + "bbox": [ + 106, + 448, + 395, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 217, + 463 + ], + "score": 1.0, + "content": "·High-Resolution Trained", + "type": "text" + }, + { + "bbox": [ + 217, + 451, + 234, + 461 + ], + "score": 0.87, + "content": "\\varphi _ { h r }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 448, + 395, + 463 + ], + "score": 1.0, + "content": ". Naive baseline trained with HR data.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 210, + 486 + ], + "score": 1.0, + "content": "·Independently Trained", + "type": "text" + }, + { + "bbox": [ + 210, + 473, + 231, + 483 + ], + "score": 0.89, + "content": "\\varphi _ { m m }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 469, + 506, + 486 + ], + "score": 1.0, + "content": ". Multi-model fashion: is it possible to achieve better results if we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 484, + 504, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 435, + 495 + ], + "score": 1.0, + "content": "train a specific model for each resolution independently? 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Specifically, each image is randomly down-sampled to a size in the candidate set", + "type": "text" + }, + { + "bbox": [ + 474, + 527, + 506, + 541 + ], + "score": 0.9, + "content": "\\textstyle \\{ { \\frac { 1 1 2 } { 2 ^ { i } } } \\ x", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 535, + 503, + 559 + ], + "spans": [ + { + "bbox": [ + 107, + 540, + 189, + 554 + ], + "score": 0.92, + "content": "\\textstyle { \\frac { 1 1 2 } { 2 ^ { i } } } | i = 0 , 1 , 2 , 3 , 4 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 535, + 460, + 559 + ], + "score": 1.0, + "content": "with equal probability of being chosen, and then up-sampled back to", + "type": "text" + }, + { + "bbox": [ + 461, + 541, + 503, + 551 + ], + "score": 0.88, + "content": "1 1 2 \\times 1 1 2", + "type": "inline_equation" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "Instantiation of Network Architecture. 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Dubbed as", + "type": "text" + }, + { + "bbox": [ + 436, + 576, + 451, + 586 + ], + "score": 0.87, + "content": "\\varphi _ { b t }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 574, + 505, + 586 + ], + "score": 1.0, + "content": ", the detailed", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 586, + 401, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 401, + 598 + ], + "score": 1.0, + "content": "instantiation of BTNet based on ResNet50 is illustrated in Appendix A.2.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 107, + 607, + 212, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 213, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 213, + 621 + ], + "score": 1.0, + "content": "4.2 Evaluation Metrics", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 625, + 505, + 696 + ], + "lines": [ + { + "bbox": [ + 106, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "On the benchmarks for face verification, we use 1:1 verification accuracy as the basic metrics. 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For 1:N identification, a set", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 96, + 506, + 110 + ], + "spans": [ + { + "bbox": [ + 105, + 96, + 506, + 110 + ], + "score": 1.0, + "content": "of gallery images are first mapped onto their embedding vectors (indexing) and the embeddings of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 108, + 385, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 385, + 122 + ], + "score": 1.0, + "content": "query images are extracted to perform search against indexed gallery.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 73, + 506, + 122 + ] + }, + { + "type": "title", + "bbox": [ + 102, + 130, + 231, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 231, + 143 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 231, + 143 + ], + "score": 1.0, + "content": "4.1 Implementation Details", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 150, + 505, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "Datasets. We use MS1Mv3 [46] for training face embedding models. The MS1Mv3 dataset", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 175 + ], + "score": 1.0, + "content": "contains 5,179,510 images of 93,431 celebrities. According to the test setting, different test datasets", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 174, + 144, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 144, + 185 + ], + "score": 1.0, + "content": "are used.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 150, + 506, + 185 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 195, + 505, + 255 + ], + "lines": [ + { + "bbox": [ + 105, + 195, + 507, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 507, + 209 + ], + "score": 1.0, + "content": "·Multi-Resolution Identity Matching. We try on six widely adopted face verification benchmarks:", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "LFW [47], CFP-FF [48], CFP-FP [48], AgeDB-30 [49], CALFW [50], and CPLFW [51], while", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 220, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 232 + ], + "score": 1.0, + "content": "the large-scale surveillance face dataset QMUL-SurvFace [17] is used for 1:N face identification,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 231, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 245 + ], + "score": 1.0, + "content": "which contains native LR surveillance faces across wide space and time. The spatial resolution for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 244, + 457, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 457, + 256 + ], + "score": 1.0, + "content": "QMUL-SurvFace ranges from 6/5 to 124/106 in height/width with an average of 24/20.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 195, + 507, + 256 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 265, + 505, + 300 + ], + "lines": [ + { + "bbox": [ + 106, + 264, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "·Multi-Resolution Feature Aggregation. We adopt a top challenging benchmark IJB-C [52], which", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 435, + 290 + ], + "score": 1.0, + "content": "has around 130k images from 3,531 identities, for two standard testing protocols:", + "type": "text" + }, + { + "bbox": [ + 436, + 277, + 456, + 288 + ], + "score": 0.58, + "content": "1 : 1", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "verification", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 289, + 198, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 198, + 301 + ], + "score": 1.0, + "content": "and 1:N identification.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 264, + 506, + 301 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 310, + 506, + 406 + ], + "lines": [ + { + "bbox": [ + 106, + 310, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 323 + ], + "score": 1.0, + "content": "Training. All the models are trained on four RTX 2080 Tis with batch size 128 by stochastic", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 323, + 507, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 507, + 335 + ], + "score": 1.0, + "content": "gradient descent. For TNet, we train for 25 epochs, with learning rate initialized at 0.2 with 2 warm-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "up epochs and decaying as a quadratic polynomial. We augment training samples by random horizonal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "flipping and multi-resolution training. 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The recommended", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 431, + 383 + ], + "score": 1.0, + "content": "hyper-parameters are used for classification loss from the original paper (e.g.,", + "type": "text" + }, + { + "bbox": [ + 431, + 370, + 505, + 382 + ], + "score": 0.91, + "content": "m = 0 . 5 , s = 6 4", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 381, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 192, + 396 + ], + "score": 1.0, + "content": "for ArcFace [1], and", + "type": "text" + }, + { + "bbox": [ + 192, + 382, + 264, + 394 + ], + "score": 0.9, + "content": "\\alpha = 0 . 9 9 , t ^ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 381, + 506, + 396 + ], + "score": 1.0, + "content": "for CurricularFace [3]). Only horizonal flipping is used as", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 395, + 250, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 250, + 407 + ], + "score": 1.0, + "content": "augmentation when training BNets.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 310, + 507, + 407 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 415, + 504, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "Baselines. In our experiment, several baselines are used to validate BTNet in learning discriminative", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 428, + 383, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 383, + 441 + ], + "score": 1.0, + "content": "and compatible representations for multi-resolution face recognition.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 106, + 415, + 505, + 441 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 449, + 394, + 461 + ], + "lines": [ + { + "bbox": [ + 106, + 448, + 395, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 217, + 463 + ], + "score": 1.0, + "content": "·High-Resolution Trained", + "type": "text" + }, + { + "bbox": [ + 217, + 451, + 234, + 461 + ], + "score": 0.87, + "content": "\\varphi _ { h r }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 448, + 395, + 463 + ], + "score": 1.0, + "content": ". Naive baseline trained with HR data.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 448, + 395, + 463 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 210, + 486 + ], + "score": 1.0, + "content": "·Independently Trained", + "type": "text" + }, + { + "bbox": [ + 210, + 473, + 231, + 483 + ], + "score": 0.89, + "content": "\\varphi _ { m m }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 469, + 506, + 486 + ], + "score": 1.0, + "content": ". Multi-model fashion: is it possible to achieve better results if we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 484, + 504, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 435, + 495 + ], + "score": 1.0, + "content": "train a specific model for each resolution independently? Specifically, we train", + "type": "text" + }, + { + "bbox": [ + 435, + 485, + 448, + 495 + ], + "score": 0.85, + "content": "\\varphi _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 484, + 504, + 495 + ], + "score": 1.0, + "content": "for data with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 493, + 351, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 149, + 510 + ], + "score": 1.0, + "content": "resolution", + "type": "text" + }, + { + "bbox": [ + 149, + 497, + 155, + 505 + ], + "score": 0.7, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 493, + 325, + 510 + ], + "score": 1.0, + "content": "and denote the multi-model collections as", + "type": "text" + }, + { + "bbox": [ + 325, + 497, + 347, + 507 + ], + "score": 0.89, + "content": "\\varphi _ { m m }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 493, + 351, + 510 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 469, + 506, + 510 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 516, + 506, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 507, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 222, + 530 + ], + "score": 1.0, + "content": "·Multi-Resolution Trained", + "type": "text" + }, + { + "bbox": [ + 222, + 518, + 241, + 528 + ], + "score": 0.87, + "content": "\\varphi _ { m r }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 516, + 507, + 530 + ], + "score": 1.0, + "content": ". Trained with multi-resolution data which adapts to resolution-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 102, + 523, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 102, + 523, + 473, + 548 + ], + "score": 1.0, + "content": "variance. Specifically, each image is randomly down-sampled to a size in the candidate set", + "type": "text" + }, + { + "bbox": [ + 474, + 527, + 506, + 541 + ], + "score": 0.9, + "content": "\\textstyle \\{ { \\frac { 1 1 2 } { 2 ^ { i } } } \\ x", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 535, + 503, + 559 + ], + "spans": [ + { + "bbox": [ + 107, + 540, + 189, + 554 + ], + "score": 0.92, + "content": "\\textstyle { \\frac { 1 1 2 } { 2 ^ { i } } } | i = 0 , 1 , 2 , 3 , 4 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 535, + 460, + 559 + ], + "score": 1.0, + "content": "with equal probability of being chosen, and then up-sampled back to", + "type": "text" + }, + { + "bbox": [ + 461, + 541, + 503, + 551 + ], + "score": 0.88, + "content": "1 1 2 \\times 1 1 2", + "type": "inline_equation" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 102, + 516, + 507, + 559 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "Instantiation of Network Architecture. The BTNet and baselines are implemented with ResNet50", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 435, + 586 + ], + "score": 1.0, + "content": "[53], and they could be extended easily with other implementations. Dubbed as", + "type": "text" + }, + { + "bbox": [ + 436, + 576, + 451, + 586 + ], + "score": 0.87, + "content": "\\varphi _ { b t }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 574, + 505, + 586 + ], + "score": 1.0, + "content": ", the detailed", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 586, + 401, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 401, + 598 + ], + "score": 1.0, + "content": "instantiation of BTNet based on ResNet50 is illustrated in Appendix A.2.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 106, + 561, + 505, + 598 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 607, + 212, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 213, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 213, + 621 + ], + "score": 1.0, + "content": "4.2 Evaluation Metrics", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 625, + 505, + 696 + ], + "lines": [ + { + "bbox": [ + 106, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "On the benchmarks for face verification, we use 1:1 verification accuracy as the basic metrics. 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Moreover, it also brings various noise and artificial processing traces [65].", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 85, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 85, + 557, + 100, + 567 + ], + "score": 1.0, + "content": "253", + "type": "text" + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "Up-sampling images via interpolation-typically bilinear interpolation or bicubic interpolation of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 85, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 85, + 569, + 100, + 579 + ], + "score": 1.0, + "content": "254", + "type": "text" + }, + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "4x4 pixel neighborhoods, essentially a function approximation method, is bound to introduce error", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 86, + 580, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 86, + 582, + 100, + 591 + ], + "score": 1.0, + "content": "255", + "type": "text" + }, + { + "bbox": [ + 106, + 580, + 506, + 592 + ], + "score": 1.0, + "content": "information (detailed in Appendix A.1), thus potentially confusing identity information, which is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 85, + 591, + 507, + 605 + ], + "spans": [ + { + "bbox": [ + 85, + 593, + 100, + 603 + ], + "score": 1.0, + "content": "256", + "type": "text" + }, + { + "bbox": [ + 105, + 591, + 471, + 605 + ], + "score": 1.0, + "content": "especially crucial for LR images with limited details. 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112&7112&14112&28
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7&714&1428&28112&112
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Up-sampling images via interpolation can increase the amount of data", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 532, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 85, + 533, + 99, + 543 + ], + "score": 1.0, + "content": "251", + "type": "text" + }, + { + "bbox": [ + 106, + 532, + 505, + 543 + ], + "score": 1.0, + "content": "but not the amount of information, only to improve the detailed part of the image and the spatial", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 85, + 545, + 100, + 556 + ], + "score": 1.0, + "content": "252", + "type": "text" + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "resolution (size) [64]. 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TPIR20(%)@FPIR
AUC0.30.20.10.01
VGG-Face [55]14.05.12.60.80.1
DeepID2 [56]20.812.88.13.40.8
FaceNet [57]19.812.78.14.31.0
SphereFace [58]28.121.315.78.31.0
SRCNN [59]27.020.014.96.20.6
FSRCNN [60]27.320.014.46.10.7
VDSR [61]27.320.114.56.10.8
DRRN [62]27.520.314.96.30.6
LapSRN [63]27.420.214.76.30.7
ArcFace [1]25.318.715.110.12.0
RAN [54]32.326.521.614.93.8
BTNet (avg.+floor)32.627.923.416.51.4
BTNet (avg.+near)34.630.325.718.91.5
BTNet (avg.+ceil)35.431.126.820.32.2
BTNet (min+floor)32.327.623.216.11.4
BTNet (min+near)34.029.625.018.01.4
BTNet (min+ceil)35.331.026.619.92.0
BTNet (max+floor)33.629.124.517.61.3
BTNet (max+near)35.231.026.419.61.7
BTNet (max+ceil)35.431.226.920.62.5
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TPIR20(%)@FPIR
AUC0.30.20.10.01
VGG-Face [55]14.05.12.60.80.1
DeepID2 [56]20.812.88.13.40.8
FaceNet [57]19.812.78.14.31.0
SphereFace [58]28.121.315.78.31.0
SRCNN [59]27.020.014.96.20.6
FSRCNN [60]27.320.014.46.10.7
VDSR [61]27.320.114.56.10.8
DRRN [62]27.520.314.96.30.6
LapSRN [63]27.420.214.76.30.7
ArcFace [1]25.318.715.110.12.0
RAN [54]32.326.521.614.93.8
BTNet (avg.+floor)32.627.923.416.51.4
BTNet (avg.+near)34.630.325.718.91.5
BTNet (avg.+ceil)35.431.126.820.32.2
BTNet (min+floor)32.327.623.216.11.4
BTNet (min+near)34.029.625.018.01.4
BTNet (min+ceil)35.331.026.619.92.0
BTNet (max+floor)33.629.124.517.61.3
BTNet (max+near)35.231.026.419.61.7
BTNet (max+ceil)35.431.226.920.62.5
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Max/min/average is used on (W, H) to obtain a resolution", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 454, + 373, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 373, + 466 + ], + "score": 1.0, + "content": "indicator for further allocation (floor/near/ceil) to a certain branch.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + } + ], + "index": 32.25 + }, + { + "type": "title", + "bbox": [ + 93, + 480, + 262, + 494 + ], + "lines": [ + { + "bbox": [ + 88, + 479, + 263, + 496 + ], + "spans": [ + { + "bbox": [ + 88, + 479, + 263, + 496 + ], + "score": 1.0, + "content": "295 5 Discussion and Conclusion", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "score": 1.0, + "content": "This paper works on the problem of multi-resolution face recognition, and provides a new scheme", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 515, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 506, + 526 + ], + "score": 1.0, + "content": "to operate images conditioned on its input resolution without large span rescaling. The error intro-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "duced by up-sampling via interpolation is investigated and analyzed. Decoupled as branches for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 536, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 552 + ], + "score": 1.0, + "content": "discriminative representation learning and coupled as the trunk for compatible representation learning,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "our Branch-to-Trunk Network (BTNet) achieves significant improvements on multi-resolution face", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "verification and identification tasks. Besides, the superiority of BTNet in reducing computational", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "cost and parameter storage cost is also demonstrated. It is worth noting that our approach is easy to", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "expand to recognition tasks for other classes of objects and has the potential to serve as a general", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 598, + 349, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 349, + 610 + ], + "score": 1.0, + "content": "network architecture for multi-resolution visual recognition.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 103, + 617, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "Limitations and Future Work. The dislocation between the underlying optical resolution of native", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 630, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 643 + ], + "score": 1.0, + "content": "face images and that of a certain branch may limit the power of the model, which may be improved", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "by selecting the optimal processing branch for the input in combination with the image quality, rather", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 653, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 506, + 668 + ], + "score": 1.0, + "content": "than by image size alone. 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The error intro-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "duced by up-sampling via interpolation is investigated and analyzed. Decoupled as branches for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 536, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 552 + ], + "score": 1.0, + "content": "discriminative representation learning and coupled as the trunk for compatible representation learning,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "our Branch-to-Trunk Network (BTNet) achieves significant improvements on multi-resolution face", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "verification and identification tasks. 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[N/A] We study a general", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 161, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 161, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "framework for multi-resolution face recognition. Our method is not for specific applications,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 162, + 452, + 332, + 463 + ], + "spans": [ + { + "bbox": [ + 162, + 452, + 332, + 463 + ], + "score": 1.0, + "content": "which does not directly involve societal issues.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 146, + 463, + 503, + 475 + ], + "spans": [ + { + "bbox": [ + 146, + 463, + 503, + 475 + ], + "score": 1.0, + "content": "(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 132, + 478, + 286, + 488 + ], + "lines": [ + { + "bbox": [ + 130, + 477, + 288, + 489 + ], + "spans": [ + { + "bbox": [ + 130, + 477, + 288, + 489 + ], + "score": 1.0, + "content": "2. If you are including theoretical results...", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 147, + 491, + 489, + 514 + ], + "lines": [ + { + "bbox": [ + 146, + 491, + 489, + 502 + ], + "spans": [ + { + "bbox": [ + 146, + 491, + 489, + 502 + ], + "score": 1.0, + "content": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Appendix A.1", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 146, + 502, + 464, + 514 + ], + "spans": [ + { + "bbox": [ + 146, + 502, + 464, + 514 + ], + "score": 1.0, + "content": "(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix A.1", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 132, + 518, + 231, + 528 + ], + "lines": [ + { + "bbox": [ + 129, + 516, + 232, + 529 + ], + "spans": [ + { + "bbox": [ + 129, + 516, + 232, + 529 + ], + "score": 1.0, + "content": "3. If you ran experiments...", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 147, + 531, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 146, + 530, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 146, + 530, + 505, + 542 + ], + "score": 1.0, + "content": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 162, + 541, + 392, + 552 + ], + "spans": [ + { + "bbox": [ + 162, + 541, + 392, + 552 + ], + "score": 1.0, + "content": "results (either in the supplemental material or as a URL)? [Yes]", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 146, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 146, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 161, + 562, + 185, + 575 + ], + "spans": [ + { + "bbox": [ + 161, + 562, + 185, + 575 + ], + "score": 1.0, + "content": "[Yes]", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 148, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 148, + 576, + 160, + 585 + ], + "score": 1.0, + "content": "(c)", + "type": "text" + }, + { + "bbox": [ + 161, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "Did you report error bars (e.g., with respect to the random seed after running experiments", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 162, + 585, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 162, + 585, + 505, + 595 + ], + "score": 1.0, + "content": "multiple times)? [No] We follow the common practice in previous works, where they didn’t", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 162, + 595, + 238, + 606 + ], + "spans": [ + { + "bbox": [ + 162, + 595, + 238, + 606 + ], + "score": 1.0, + "content": "report the error bars.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 146, + 606, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 146, + 606, + 506, + 618 + ], + "score": 1.0, + "content": "(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 162, + 617, + 474, + 627 + ], + "spans": [ + { + "bbox": [ + 162, + 617, + 474, + 627 + ], + "score": 1.0, + "content": "internal cluster, or cloud provider)? 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[Yes]", + "type": "text" + } + ], + "index": 19, + "is_list_end_line": true + }, + { + "bbox": [ + 147, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 147, + 371, + 160, + 380 + ], + "score": 1.0, + "content": "(b)", + "type": "text" + }, + { + "bbox": [ + 161, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "Did you describe the limitations of your work? 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The optimal", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 162, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 162, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "branch selection strategy is not fully investigated though we have provided an intuitive way to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 162, + 420, + 264, + 431 + ], + "spans": [ + { + "bbox": [ + 162, + 420, + 264, + 431 + ], + "score": 1.0, + "content": "select the branch for inputs.", + "type": "text" + } + ], + "index": 25, + "is_list_end_line": true + }, + { + "bbox": [ + 147, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 147, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "(c) Did you discuss any potential negative societal impacts of your work? [N/A] We study a general", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 161, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "framework for multi-resolution face recognition. Our method is not for specific applications,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 162, + 452, + 332, + 463 + ], + "spans": [ + { + "bbox": [ + 162, + 452, + 332, + 463 + ], + "score": 1.0, + "content": "which does not directly involve societal issues.", + "type": "text" + } + ], + "index": 28, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 463, + 503, + 475 + ], + "spans": [ + { + "bbox": [ + 146, + 463, + 503, + 475 + ], + "score": 1.0, + "content": "(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + } + ], + "index": 23.5, + "bbox_fs": [ + 146, + 348, + 506, + 475 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 478, + 286, + 488 + ], + "lines": [ + { + "bbox": [ + 130, + 477, + 288, + 489 + ], + "spans": [ + { + "bbox": [ + 130, + 477, + 288, + 489 + ], + "score": 1.0, + "content": "2. If you are including theoretical results...", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 130, + 477, + 288, + 489 + ] + }, + { + "type": "index", + "bbox": [ + 147, + 491, + 489, + 514 + ], + "lines": [ + { + "bbox": [ + 146, + 491, + 489, + 502 + ], + "spans": [ + { + "bbox": [ + 146, + 491, + 489, + 502 + ], + "score": 1.0, + "content": "(a) Did you state the full set of assumptions of all theoretical results? 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[Yes] See our implementation details in Section 4.", + "type": "text" + } + ], + "index": 42, + "is_list_end_line": true + } + ], + "index": 38, + "bbox_fs": [ + 146, + 530, + 506, + 627 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 630, + 470, + 641 + ], + "lines": [ + { + "bbox": [ + 129, + 629, + 471, + 642 + ], + "spans": [ + { + "bbox": [ + 129, + 629, + 471, + 642 + ], + "score": 1.0, + "content": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43, + "bbox_fs": [ + 129, + 629, + 471, + 642 + ] + }, + { + "type": "list", + "bbox": [ + 147, + 644, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 146, + 644, + 398, + 656 + ], + "spans": [ + { + "bbox": [ + 146, + 644, + 398, + 656 + ], + "score": 1.0, + "content": "(a) If your work uses existing assets, did you cite the creators? 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TPIR20(%)@FPIR
AUC0.30.20.10.01
VGG-Face [55]14.05.12.60.80.1
DeepID2 [56]20.812.88.13.40.8
FaceNet [57]19.812.78.14.31.0
SphereFace [58]28.121.315.78.31.0
SRCNN [59]27.020.014.96.20.6
FSRCNN [60]27.320.014.46.10.7
VDSR [61]27.320.114.56.10.8
DRRN [62]27.520.314.96.30.6
LapSRN [63]27.420.214.76.30.7
ArcFace [1]25.318.715.110.12.0
RAN [54]32.326.521.614.93.8
BTNet (avg.+floor)32.627.923.416.51.4
BTNet (avg.+near)34.630.325.718.91.5
BTNet (avg.+ceil)35.431.126.820.32.2
BTNet (min+floor)32.327.623.216.11.4
BTNet (min+near)34.029.625.018.01.4
BTNet (min+ceil)35.331.026.619.92.0
BTNet (max+floor)33.629.124.517.61.3
BTNet (max+near)35.231.026.419.61.7
BTNet (max+ceil)35.431.226.920.62.5
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AlgorithmDomainArchEncoderInputsLenRL
Duan et al. (2016)Meta RLseparateGRUoard1000TRPO,PPO
Wang et al. (2017)Meta RLsharedLSTMoart5-150A2C
Baseline from Rakelly et al. (2019)Meta RLseparateGRUoard100PPO
Baseline from Zintgraf et al. (2020)Meta RLseparateGRUoardMaxA2C,PPO
Baseline from Fakoor et al. (2020)MetaRLseparateGRUoar10-25TD3
Baseline from Yu et al. (2019)Meta RLseparateGRUoard500PPO
Kostrikov (2018)POMDPsharedGRU05-2048PPO,A2C
Wang et al. (2020)POMDPseparateLSTMoa150TD3, SAC
Meng et al. (2021)POMDPseparateLSTMoa1-5TD3
Yang & Nguyen (2021)POMDPseparatebothoaMaxTD3, SAC
Baseline from Igl et al. (2018)POMDPsharedGRUoa25A2C
Baseline from Han et al. (2020)POMDPsharedLSTM064SAC
Baseline from Zhang et al. (2021)Robust RLseparateLSTM0100PPO
Baseline1 from Packer et al. (2018)GeneralizationsharedLSTM0128-512PPO,A2C
Baseline2 from Packer et al. (2018)GeneralizationseparateLSTMoard128-512PPO,A2C
Our methodMeta RLseparateLSTMoard64TD3
Our methodPOMDPseparateGRUoa64TD3
Our methodRobust RLseparateLSTM064TD3
Our methodGeneralizationseparateLSTM064TD3
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Change in one decision factorSubareaEnvPerformance comparison
Inputs:oa→oar“Standard”POMDPWalker-P981.6 → 1345.0 (1.3×)
RL: TD3 →SAC“Standard”POMDPAnt-P310.7 → 2123.5 (6.8×)
Encoder: LSTM→GRURobust RLWalker-Robust765.9 → 931.3 (1.2×)
Len: 64 →400Meta RLCheetah-Vel-85.2 -→ -74.6 (+14%)
Len: 64→5GeneralizationHopper-Generalize0.292 -→ 0.415 (1.4×)
Len: 64→5“Standard"POMDPWalker-V121.4 -→ 264.3 (2.2×)
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MethodEnvironmentTime costRAMGPU memory
Ours (GRU)Hopper-V22.5 h0(1)1.2 GB
VRM (Han et al., 2020)Hopper-V102 h0(200)N/A
OursSemi-Circle12h0(1)1 GB
VariBAD (Dorfman et al., 2020)Semi-Circle2.3 h0(1)*9.5 GB
OursCheetah-Robust7h0(1)1.1 GB
MRPO (Jiang et al., 2021)Cheetah-Robust0.4 hN/AN/A
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Meta-RL* POMDP Robust RL Gen. in RL
HiddenlayersizeObserv.encoderPrev. Action encoder*Reward encoder*RNNMLP[32][16][16][128][128,128,128] [256,256]
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TraininghparamsEnvironment stepsGradient steps5M 1.5M 3M0.1M 1.5M 0.6M
PolicyinputsLargest input spaceBest input spaceoard oar oa oaroard oa 0 0
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Meta-RL*POMDPRobustRLGen. in RL
Approach(off-policy) VariBADVRMMRPOEPOPT
Memory-based?√√XX
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292.0, + 669.0 + ], + "score": 1.0, + "text": "" + } + ], + "page_info": { + "page_no": 33, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/dev/KLaDXLAzzFT/KLaDXLAzzFT.md b/parse/dev/KLaDXLAzzFT/KLaDXLAzzFT.md new file mode 100644 index 0000000000000000000000000000000000000000..0144064e79e11a737587a535c91347e9b879576d --- /dev/null +++ b/parse/dev/KLaDXLAzzFT/KLaDXLAzzFT.md @@ -0,0 +1,1388 @@ +# NEAR-OPTIMAL OFFLINE REINFORCEMENT LEARNING WITH LINEAR REPRESENTATION: LEVERAGING VARIANCE INFORMATION WITH PESSIMISM + +Ming Yin∗ Yaqi Duan† ∗Department of Computer Science University of California, Santa Barbara {ming yin,yuxiangw}@cs.ucsb.edu + +Mengdi Wang† Yu-Xiang Wang∗ †Department of Electrical and Computer Engineering Princeton University {yaqid,mengdiw}@princeton.edu + +# ABSTRACT + +Offline reinforcement learning, which seeks to utilize offline/historical data to optimize sequential decision-making strategies, has gained surging prominence in recent studies. Due to the advantage that appropriate function approximators can help mitigate the sample complexity burden in modern reinforcement learning problems, existing endeavors usually enforce powerful function representation models (e.g. neural networks) to learn the optimal policies. However, a precise understanding of the statistical limits with function representations, remains elusive, even when such a representation is linear. + +Towards this goal, we study the statistical limits of offline reinforcement learning with linear model representations. To derive the tight offline learning bound, we design the variance-aware pessimistic value iteration (VAPVI), which adopts the conditional variance information of the value function for time-inhomogeneous episodic linear Markov decision processes (MDPs). VAPVI leverages estimated variances of the value functions to reweight the Bellman residuals in the leastsquare pessimistic value iteration and provides improved offline learning bounds over the best-known existing results (whereas the Bellman residuals are equally weighted by design). More importantly, our learning bounds are expressed in terms of system quantities, which provide natural instance-dependent characterizations that previous results are short of. We hope our results draw a clearer picture of what offline learning should look like when linear representations are provided. + +# 1 INTRODUCTION + +Offline reinforcement learning (offline RL or batch RL Lange et al. (2012); Levine et al. (2020)) is the framework for learning a reward-maximizing policy in an unknown environment (Markov Decision Process or MDP)1 using the logged data coming from some behavior policy $\mu$ . Function approximations, on the other hand, are well-known for generalization in the standard supervised learning. Offline RL with function representation/approximation, as a result, provides generalization across large state-action spaces for the challenging sequential decision-making problems when no iteration is allowed (as opposed to online learning). This paradigm is crucial to the success of modern RL problems as many deep RL algorithms find their prototypes in the literature of offline RL. For example, Xie and Jiang (2020) provides a view that Fitted $Q$ -Iteration (Gordon, 1999; Ernst et al., 2005) can be considered as the theoretical prototype of the deep $Q$ -networks algorithm (DQN) Mnih et al. (2015) with neural networks being the function representors. On the empirical side, there are a huge body of deep RL-based algorithms (Mnih et al., 2015; Silver et al., 2017; Fujimoto et al., 2019; Kumar et al., 2019; Wu et al., 2019; Kidambi et al., 2020; Yu et al., 2020; Kumar et al., 2020; Janner et al., 2021; Chen et al., 2021a; Kostrikov et al., 2022) that utilize function approximations to achieve respective successes in the offline regime. However, it is also realized that practical function approximation schemes can be quite sample inefficient (e.g. millions of samples are needed for deep $Q$ -network to solve certain Atari games Mnih et al. (2015)). + +To understand this phenomenon, there are numerous studies consider how to achieve sample efficiency with function approximation from the theoretical side, as researchers find sample efficient algorithms are possible with particular model representations, in either online RL (e.g. Yang and Wang (2019; 2020); Modi et al. (2020); Jin et al. (2020); Ayoub et al. (2020); Jiang et al. (2017); Du et al. (2019); Sun et al. (2019); Zanette et al. (2020); Zhou et al. (2021a); Jin et al. (2021a); Du et al. (2021)) or offline RL (e.g. Munos (2003); Chen and Jiang (2019); Xie and Jiang (2020); Jin et al. (2021b); Xie et al. (2021a); Min et al. (2021); Duan et al. (2021); Nguyen-Tang et al. (2021); Zanette et al. (2021)). + +Among them, the linear MDP model (Yang and Wang, 2020; Jin et al., 2020), where the transition is represented as a linear combinations of the given $d$ -dimensional feature, is (arguably) the most studied setting in function approximation and there are plenty of extensions based upon it (e.g. generalized linear model (Wang et al., 2021b), reward-free RL (Wang et al., 2020), gap-dependent analysis (He et al., 2021) or generative adversarial learning (Liu et al., 2021)). Given its prosperity, however, there are still unknowns for understanding function representations in RL, especially in the offline case. + +• While there are surging researches in showing provable sample efficiency (polynomial sample complexity is possible) under a variety of function approximation schemes, how to improve the sample efficiency for a given class of function representations remains understudied. For instance, given a neural network approximation class, an algorithm that learns the optimal policy with complexity ${ \cal O } ( H ^ { 1 0 } )$ is far worse than the one that can learn in $O ( H ^ { 3 } )$ sample complexity, despite that both algorithms are considered sample efficient. Therefore, how to achieve the optimal/tight sample complexity when function approximation is provided is a valuable question to consider. On the other hand, it is known that tight sample complexity, due to the limit of the existing statistical analysis tools, can be very tough to establish when function representation has a very complicated form. However, does this mean tight analysis is not hopeful even when the representation is linear? + +• Second, in the existing analysis of offline RL (with function approximation or simply the tabular MDPs), the learning bounds depend either explicitly on the data-coverage quantities (e.g. uniform concentrability coefficients Chen and Jiang (2019); Xie and Jiang (2020), uniform visitation measure Yin et al. (2021); Yin and Wang (2021a) and single concentrability Rashidinejad et al. (2021); Xie et al. (2021b)) or the horizon length $H$ (Jin et al., 2021b; Uehara and Sun, 2021). While those results are valuable as they do not depend on the structure of the particular problem (therefore, remain valid even for pathological MDPs), in practice, the empirical performances of offline reinforcement learning are often far better than those non-adaptive bounds would indicate. Can the learning bounds reflect the nature of individual MDP instances when the MDP model has a certain function representation? + +In this work, we think about offline RL from the above two aspects. In particular, we consider the fundamental linear model representations and ask the following question of interest: + +Can we achieve the statistical limits for offline RL when models have linear representations? + +# 1.1 RELATED WORKS + +Offline RL with general function representations. The finite sample analysis of offline RL with function approximation is initially conducted by Fitted $Q$ -Iteration (FQI) type algorithms and can be dated back to (Munos, 2003; Szepesvari and Munos ´ , 2005; Antos et al., 2008a;b). Later, Chen and Jiang (2019); Le et al. (2019); Xie and Jiang (2020) follow this line of research and derive the improved learning results. However, owing to the aim for tackling general function approximation, those learning bounds are expressed in terms of the stringent concentrability coefficients (therefore, are less adaptive to individual instances) and are usually only information-theoretical, due to the computational intractability of the optimization procedure over the general function classes. Other works impose weaker assumptions (e.g. partial coverage (Liu et al., 2020; Kidambi et al., 2020; Uehara and Sun, 2021)), and their finite sample analysis are generally suboptimal in terms of $H$ or the effective horizon $( 1 - \gamma ) ^ { - 1 }$ . + +Offline RL with tabular models. For tabular MDPs, tight learning bounds can be achieved under several data-coverage assumptions. For the class of problems with uniform data-visitation measure $d _ { m }$ , the near-optimal sample complexity bound has the rate $O ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } )$ for time-inhomogeneous MDPs (Yin et al., 2021) and $O ( H ^ { 2 } / d _ { m } \dot { \epsilon } )$ for time-homogeneous MDPs (Yin and Wang, 2021a; Ren et al., 2021). Under the single concentrability assumption, the tight rate ${ \cal O } ( H ^ { 3 } S C ^ { \star } / \epsilon ^ { 2 } )$ is obtained by Xie et al. (2021b). In particular, the recent study Yin and Wang (2021b) introduces the intrinsic offline learning bound that is not only instance-dependent but also subsumes previous optimal results. + +Offline RL with linear model representations. Recently, there is more focus on studying the provable efficient offline RL under the linear model representations. Jin et al. (2021b) first shows offline RL with linear MDP is provably efficient by the pessimistic value iteration. Their analysis deviates from their lower bound by a factor of $d \cdot H$ (check their Theorem 4.4 and 4.6). Later, Xie et al. (2021a) considers function approximation under the Bellman-consistent assumptions, and, when realized to linear MDP setting, improves the sample complexity guarantee of Jin et al. (2021b) by an order $O ( d )$ (Theorem 3.2).2 However, their improvement only holds for finite action space (due to the dependence $\log \left| { \mathcal { A } } \right| )$ and by the direct reduction (from Theorem 3.1) their result does not imply a computationally tractable algorithm with the same guarantee. Concurrently, Zanette et al. (2021) considers the Linear Bellman Complete model and designs the actor-critic style algorithm that achieves tight result under the assumption that the value function is bounded by 1. While their algorithm is efficient (which is based on solving a sequence of second-order cone programs), the resulting learning bound requires the action space to be finite due to the mirror descent updates in the Actor procedure (Agarwal et al., 2021). Besides, assuming the value function to be less than 1 simplifies the challenges in dealing with horizon $H$ since when rescaling their result to $[ 0 , H ]$ , there is a $H$ factor blow-up, which makes no horizon improvement comparing to Jin et al. (2021b). As a result, none of the existing algorithms can achieve the statistical limit for the well-structured linear MDP model with the general (infinite or continuous) state-action spaces. On the other hand, Wang et al. (2021a); Zanette (2021) study the statistical hardness of offline RL with linear representations by proving the exponential lower bounds. Recently, Foster et al. (2021) shows realizability and concentrability are not sufficient for offline learning when state space is arbitrary large. + +Variance-aware studies. Talebi and Maillard (2018) first incorporates the variance structure in online tabular MDPs and Zanette and Brunskill (2019) tightens the result. For linear mixture MDPs, Zhou et al. (2021a) first uses variance structure to achieve near-optimal result and the Weighted OFUL incorporates the variance structure explicitly in the regret bound. Recently, Variance-awareness is also considered in Zhang et al. (2021) for horizon-free setting and for OPE problem (Min et al., 2021). In particular, We point out that Min et al. (2021) is the first work that uses variance reweighting for policy evaluation in offline RL, which inspires our study for policy optimization problem. The guarantee of Min et al. (2021) strictly improves over Duan et al. (2020) for OPE problem. + +# 1.2 OUR CONTRIBUTION + +In this work, we study offline RL for time-inhomogeneous episodic linear Markov decision processes. Linear MDPs serve as one critical step towards understanding function approximation in RL since: 1. unlike general function representation, linear MDP representation has the well-structured form by the given feature representors, which makes delicate statistical analysis hopeful; 2. unlike tabular representation, which only works for finite models, linear MDP provides generalization as it adapts to infinite or continuous state-action spaces. Especially, we design the variance-aware pessimistic value iteration (VAPVI, Algorithm 1) which incorporates the conditional variance information of the value function and, by the variance structure, Theorem 3.2 is able to improve over the aforementioned stateof-the-art guarantees. In addition, we further improve the state-action guarantee by designing an even tighter bonus (4). VAPVI-Improved (Theorem 3.3) is near-minimax optimal as indicated by our lower bound (Theorem 3.5). Importantly, the resulting learning bounds from VAPVI/VAPVI-Improved are able to characterize the adaptive nature of individual instances and yield different convergence rates for different problems. Algorithmically, our algorithm builds upon the nice Min et al. (2021) with pessimism as we use the estimated variances to reweight the Bellman residual learning objective so that the (training) samples with high uncertainty get less attention (Section 3). This is the key to obtaining instance-adaptive guarantees. + +# 2 PRELIMINARIES + +# 2.1 PROBLEM SETTINGS + +Episodic time-inhomogeneous linear Markov decision process. A finite-horizon Markov Decision Process (MDP) is denoted as $M = \left( S , A , P , r , H , d _ { 1 } \right)$ (Sutton and Barto, 2018), where $s$ is the arbitrary state space and $\mathcal { A }$ is the arbitrary action space which can be infinite or even continuous. A time-inhomogeneous transition kernel $\begin{array} { r } { \dot { P } _ { h } : S \times \bar { \mathcal { A } } \mapsto \Delta ^ { \mathcal { S } } } \end{array}$ ( $\Delta ^ { s }$ represents a probability simplex) maps each state action $\left( s _ { h } , a _ { h } \right)$ to a probability distribution $P _ { h } ( \cdot | s _ { h } , a _ { h } )$ and $P _ { h }$ can be different across time. In addition, $r : \mathcal { S } \times A \mapsto \mathbb { R }$ is the mean reward function satisfying $0 \leq r \leq 1$ . $d _ { 1 }$ is the initial state distribution. $H$ is the horizon. A policy $\pi = ( \pi _ { 1 } , \ldots , \pi _ { H } ) $ assigns each state $s _ { h } \in S$ a probability distribution over actions according to the map $s _ { h } \mapsto \pi _ { h } \bigl ( \cdot | s _ { h } \bigr ) \bar { \forall } h \in [ H ]$ and induces a random trajectory $s _ { 1 } , a _ { 1 } , r _ { 1 } , \ldots , s _ { H } , a _ { H } , r _ { H } , s _ { H + 1 }$ with $s _ { 1 } \sim d _ { 1 } , a _ { h } \sim \pi ( \cdot | s _ { h } ) , s _ { h + 1 } \sim$ $P _ { h } ( \cdot | s _ { h } , a _ { h } ) , \forall h \in [ \bar { H } ]$ . In particular, we adopts the linear MDP protocol from Jin et al. (2020; 2021b), meaning that the transition kernel and the mean reward function admit linear structures in the feature map. + +Definition 2.1 (Linear MDPs). 3 An episodic MDP $( S , { \mathcal { A } } , H , P , r )$ is called a linear MDP with a known (unsigned) feature map $\phi : \mathcal { S } \times \mathcal { A } \mathbb { R } ^ { d }$ if there exist $d$ unknown (unsigned) measures $\nu _ { h } = ( \nu _ { h } ^ { ( 1 ) } , \dots , \nu _ { h } ^ { ( d ) } )$ over $s$ and an unknown vector $\theta _ { h } \in \mathbb { R } ^ { d }$ such that + +$P _ { h } \left( \boldsymbol { s } ^ { \prime } \mid \boldsymbol { s } , \boldsymbol { a } \right) = \left. \phi ( \boldsymbol { s } , \boldsymbol { a } ) , \nu _ { h } \left( \boldsymbol { s } ^ { \prime } \right) \right. , \quad r _ { h } \left( \boldsymbol { s } , \boldsymbol { a } \right) = \left. \phi ( \boldsymbol { x } , \boldsymbol { a } ) , \theta _ { h } \right. , \quad \forall \boldsymbol { s } ^ { \prime } , \boldsymbol { s } \in \mathcal { S } , \boldsymbol { a } \in \mathcal { A } , \ : h \in [ H ] .$ . where $\| \nu _ { h } ( S ) \| _ { 2 } \leq \sqrt { d }$ and $\operatorname* { m a x } ( \left\| \phi ( s , a ) \right\| _ { 2 } , \left\| \theta _ { h } \right\| _ { 2 } ) \leq 1$ for all $h \in [ H ]$ and $\forall s , a \in S \times A$ . $\begin{array} { r } { \| \mu _ { h } ( S ) \| = \int _ { S } \| \mu _ { h } ( s ) \| d s } \end{array}$ . + +$V$ $Q _ { h } ^ { \pi } ( \cdot , \cdot ) \in \mathbb { R } ^ { S \times A }$ -values and $Q$ -values. For any policy fined as: . The perf $\begin{array} { r } { V _ { h } ^ { \pi } ( s ) = \mathbb { E } _ { \pi } [ \sum _ { t = h } ^ { H } r _ { t } | s _ { h } = s ] , \quad Q _ { h } ^ { \pi } ( s , a ) = \mathbb { E } _ { \pi } [ \sum _ { t = h } ^ { H } r _ { \frac { t } { s } } | s _ { h } , a _ { h } = \frac { 1 } { s } ] } \end{array}$ $\pi$ , the $V$ -value functions $V _ { h } ^ { \pi } ( \cdot ) \in \mathbb { R } ^ { S }$ and Q-value functions $s , a ] , \forall s , a , h \in S , { \mathcal { A } } , [ H ]$ $\begin{array} { r } { \boldsymbol { v } ^ { \pi } : = \mathbb { E } _ { d _ { 1 } } \left[ V _ { 1 } ^ { \pi } \right] = \mathbb { E } _ { \pi , d _ { 1 } } \left[ \sum _ { t = 1 } ^ { H } \boldsymbol { r } _ { t } \right| } \end{array}$ The Bellman (optimality) equations follow $\iota \in [ H ] \colon Q _ { h } ^ { \pi } = r _ { h } + P _ { h } V _ { h + 1 } ^ { \pi } , ~ V _ { h } ^ { \pi } = \mathbb { E } _ { a \sim \pi _ { h } } [ Q _ { h } ^ { \bar { \pi } } ] , ~ Q _ { h } ^ { \star } \stackrel { = } { = }$ $r _ { h } + P _ { h } V _ { h + 1 } ^ { \star }$ , $V _ { h } ^ { \star } = \operatorname* { m a x } _ { a } Q _ { h } ^ { \star } ( \cdot , a )$ (where $Q _ { h } , V _ { h } , P _ { h }$ are vectors). By Definition 2.1, the $Q$ -values also admit linear structures, i.e. $Q _ { h } ^ { \pi } = \langle \phi , w _ { h } ^ { \pi } \rangle$ for some $w _ { h } ^ { \pi } \in \mathbb { R } ^ { d }$ (Lemma H.9). Lastly, for a policy $\pi$ , we denote the induced occupancy measure over the state-action space at any time $h \in [ H ]$ to be: for any $E \subseteq S \times A$ , $d _ { h } ^ { \pi } ( E ) : = \mathbb { E } [ ( s _ { h } , a _ { h } ) \in E | s _ { 1 } \sim d _ { 1 } , a _ { i } \sim \pi ( \cdot | s _ { i } ) , s _ { i } \sim P _ { i - 1 } ( \cdot | s _ { i - 1 } , \bar { a _ { i - 1 } } ) , 1 \le h | s _ { i - 1 } \sim \pi ( \cdot | s _ { i - 1 } , \bar { a _ { i - 1 } } ) ]$ $i \leq h \}$ and $\begin{array} { r } { \mathbb { E } _ { \pi , h } [ f ( s , a ) ] : = \int _ { \mathcal { S } \times \mathcal { A } } f ( s , a ) d _ { h } ^ { \pi } ( s , a ) d s d a } \end{array}$ . Here for notation simplicity we abuse $d _ { h } ^ { \pi } ( \cdot )$ to denote either probability measure or density function. + +Offline learning setting. Offlprovided with the historical data learn the policy rolled out from s $\pi$ that maximizes me behavior polic $v ^ { \pi }$ ,. $\mathcal { D } = \{ ( s _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } , r _ { h } ^ { \tau } , s _ { h + 1 } ^ { \tau } ) \} _ { \tau \in [ K ] } ^ { h \in [ H ] }$ $\mu$ The offline nature requires we cannot change $\mu$ and in particular we do not know the data generating distribution of $\mu$ . To sum up, the agent seeks to find a policy $\pi _ { \mathrm { a l g } }$ such that $v ^ { \star } - v ^ { \pi _ { \mathrm { a l g } } } \leq \epsilon$ for the given batch data $\mathcal { D }$ and a given targeted accuracy $\epsilon > 0$ . + +# 2.2 ASSUMPTIONS + +It is known that learning a near-optimal policy from the offline data $\mathcal { D }$ cannot be sample efficient without certain data-coverage assumptions (Wang et al., 2021a; Yin and Wang, 2021b). To begin with, we define the population covariance matrix under the behavior policy $\mu$ for all $h \in [ H ]$ : + +$$ +\Sigma _ { h } ^ { p } : = \mathbb { E } _ { \mu , h } \left[ \phi ( s , a ) \phi ( s , a ) ^ { \top } \right] , +$$ + +since $\Sigma _ { h } ^ { p }$ measure the coverage of state-action space for data $\mathcal { D }$ , we make the following assumption. + +Assumption 2.2 (Feature Coverage). The data distributions $\mu$ satisfy the minimum eigenvalue condition: $\forall h \in [ H ]$ , $\kappa _ { h } : = \lambda _ { \operatorname* { m i n } } ( \bar { \Sigma } _ { h } ^ { p } ) > 0$ and denote $\kappa = \operatorname* { m i n } _ { h } \kappa _ { h }$ . Note $\kappa$ is a system-dependent (non-universal) quantity as it is upper bounded by $1 / d$ (Assumption 2 in Wang et al. (2021a)). + +We make this assumption for the following reasons. First of all, our offline learning guarantee (Theorem 3.2) provides simultaneously comparison to all the policies, which is stronger than only competing with the optimal policy (whereas relaxed assumption suffices, e.g. $\begin{array} { r } { \operatorname* { s u p } _ { x \in \mathbb { R } ^ { d } } \frac { \bar { x } \Sigma _ { \pi ^ { \star } } x ^ { \top } } { x \Sigma _ { \mu } x ^ { \top } } < \infty } \end{array}$ (Uehara and Sun, 2021)). As a consequence, the behavior distribution $\mu$ must be able to explore each feature dimension for the result to be valid. Second, even if Assumption 2.2 does not hold, we can always restrict our algorithmic design to the effective subspan of $\bar { \Sigma _ { h } ^ { p } }$ , which causes the alternative notion of $\begin{array} { r } { \kappa : = \operatorname* { m i n } _ { h \in [ H ] } \left\{ \kappa _ { h } : s . t . \right. } \end{array}$ $\kappa _ { h } =$ smallest positive eigenvalue at time $h \}$ (see Appendix G.1 for detailed discussions). In this scenario, learning the optimal policy cannot be guaranteed as a constant suboptimality gap needs to be suffered due to the lack of coverage and this is formed as assumption-free $R L$ in Yin and Wang (2021b). Lastly, previous works analyzing the linear MDPs impose very similar assumptions, e.g. Xie et al. (2021a) Theorem 3.2 where $\dot { \Sigma _ { \mathcal { D } } } ^ { - 1 }$ exists and Min et al. (2021) for the OPE problem. + +Next, for any function $V _ { h + 1 } ( \cdot ) \in [ 0 , H - h ]$ , we define the conditional variance $\sigma _ { V _ { h + 1 } } : S \times \mathcal { A } \to \mathbb { R } _ { + }$ as $\sigma _ { V _ { h + 1 } } ( s , a ) ^ { 2 } : = \operatorname* { m a x } \{ 1 , \operatorname { V a r } _ { P _ { h } } ( V _ { h + 1 } ) ( s , a ) \}$ .4 Based on this definition, we can define the variance-involved population covariance matrices as: $\mathbf { \Sigma } _ { h } ^ { p } : = \mathbb { E } _ { \mu , h } \left[ \sigma _ { V _ { h + 1 } } ( s , a ) ^ { - 2 } \phi ( s , a ) \phi ( s , a ) ^ { \top } \right]$ . In particular, when ${ \cal V } _ { h } = { \cal V } _ { h } ^ { \star }$ , we use the notation $\Lambda _ { h } ^ { \star p }$ instead. + +# 3 ALGORITHM + +Least square regression is usually considered as one of the “default” tools for handling problems with linear structures (e.g. LinUCB algorithm for linear Bandits) and finds its popularity in RL as well since Least-Square Value Iteration (LSVI, Jin et al. (2020)) is shown to be provably efficient for linear MDPs, due to that $V _ { h + 1 } ( s ^ { \prime } )$ is an unbiased estimator of $[ P _ { h } V _ { h + 1 } ] ( s , a )$ . Concretely, it solves the ridge regression problems at each time steps (with $\lambda > 0$ being the regularization parameter): + +$$ +\widehat { w } _ { h } : = \underset { w \in \mathbb { R } ^ { d } } { \mathrm { a r g m i n } } \ \lambda \| w \| _ { 2 } ^ { 2 } + \sum _ { k = 1 } ^ { K } \left[ \langle \phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) , w \rangle - r _ { h } ^ { k } - V _ { h + 1 } \big ( s _ { h + 1 } ^ { \prime k } \big ) \right] ^ { 2 } +$$ + +and has the closed-form solution $\begin{array} { r } { \widehat { w } _ { h } \ = \ \Sigma _ { h } ^ { - 1 } \sum _ { k = 1 } ^ { K } \phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) [ r _ { k , h } + V _ { h + 1 } ( s _ { h } ^ { \prime k } ) ] } \end{array}$ with $\Sigma _ { h } ^ { - 1 } ~ =$ $\begin{array} { r } { \sum _ { k = 1 } ^ { K } \phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) \phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) ^ { \top } + \lambda I } \end{array}$ b. In offline RL, this has also been leveraged in pessimistic value iteration (Jin et al., 2021b) and fitted $Q$ -evaluation (Duan et al., 2020). Nevertheless, LSVI could only yield suboptimal guarantees, as illustrated by the following example. + +Example 3.1. Instantiate PEVI (Theorem 4.4 in Jin et al. (2021b)) with $\phi ( s , a ) = \mathbf { 1 } _ { s , a }$ (i.e. tabular $M D P s ) ^ { 5 }$ , by direct calculation the learning bound has the form $\begin{array} { r l } & { O ( d H \cdot \sum _ { \boldsymbol { h } , \boldsymbol { s } , \boldsymbol { a } } { d _ { \boldsymbol { h } } ^ { \pi } } ^ { \star } ( \boldsymbol { s } , \boldsymbol { a } ) \sqrt { \frac { 1 } { K \cdot d _ { \boldsymbol { h } } ^ { \mu } ( \boldsymbol { s } , \boldsymbol { a } ) } } ) } \end{array}$ q 1K·dµh(s,a) ) and the optimal result (Yin and Wang (2021b) Theorem 4.1) gives $\begin{array} { r l } & { O ( \sum _ { h , s , a } { d _ { h } ^ { \pi ^ { \star } } ( s , a ) \sqrt { \frac { \operatorname { V a r } _ { P _ { s , a } } ( r + V _ { h + 1 } ^ { \star } ) } { K \cdot d _ { h } ^ { \mu } ( s , a ) } } } ) } \end{array}$ The former has the horizon dependence $H ^ { 2 }$ and the latter is $H ^ { 3 / 2 }$ by law of total variance. + +Motivation. By comparing the above two expressions, it can be seen that PEVI cannot get rid of the explicit $H$ factor due to missing the variance information $( w . r . t ~ V ^ { \star } )$ . If we go deeper, one could find that it might not be all that ideal to put equal weights on all the training samples in the least square objective (2), since, unlike linear regression where the randomness coming from one source distribution, we are regressing over a sequence of distributions in RL (i.e. each $s _ { h } , a _ { h }$ corresponds to a different distribution $P ( \cdot | s _ { h } , a _ { h } )$ and there are possibly infinite many of them). Therefore, conceptually, the sample piece $( s _ { h } , a _ { h } , s _ { h + 1 } )$ that has higher variance distribution $P ( \cdot | s _ { h } , a _ { h } )$ tends to be less “reliable” than the one $( s _ { h } ^ { \prime } , a _ { h } ^ { \prime } , s _ { h + 1 } ^ { \prime } )$ with lower variance (hence should not have equal weight in (2)). This suggests reweighting scheme might help improve the learning guarantee and reweighting over the variance of the value function stands as a natural choice. + +# 3.1 VARIANCE-AWARE PESSIMISTIC VALUE ITERATION + +Now we explain our framework that incorporates the variance information. Our design is motivated by previous Zhou et al. (2021a) (for online learning) and MinBy the offline nature, we can use the independent episodic data $\mathcal { D } ^ { \prime } = \{ ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } , \bar { r } _ { h } ^ { \tau } , \bar { s } _ { h } ^ { \tau \prime } ) \} _ { \tau \in [ K ] } ^ { h \in [ H ] }$ ation).(from $\mu _ { . }$ ) to estimate the conditional variance of any -values $V _ { h + 1 }$ via the definition $[ \mathrm { V a r } _ { h } V _ { h + 1 } ] ( s , a ) =$ + +$[ P _ { h } ( V _ { h + 1 } ) ^ { 2 } ] ( s , a ) - ( [ P _ { h } V _ { h + 1 } ] ( s , a ) ) ^ { 2 } .$ . For the second order moment, by Definition 2.1, it holds + +$$ +\left[ P _ { h } V _ { h + 1 } ^ { 2 } \right] ( s , a ) = \int _ { S } V _ { h + 1 } ^ { 2 } \left( s ^ { \prime } \right) \mathrm { d } { \cal P } _ { h } \left( s ^ { \prime } \mid s , a \right) = \phi ( s , a ) ^ { \top } \int _ { S } V _ { h + 1 } ^ { 2 } \left( s ^ { \prime } \right) \mathrm { d } \nu _ { h } \left( s ^ { \prime } \right) . +$$ + +Denote $\begin{array} { r } { \beta _ { h } : = \int _ { S } V _ { h + 1 } ^ { 2 } \left( s ^ { \prime } \right) \mathrm { d } \nu _ { h } \left( s ^ { \prime } \right) } \end{array}$ , then $P _ { h } V _ { h + 1 } ^ { 2 } = \left. \phi , \beta _ { h } \right.$ and we can estimator it via: + +$$ +\bar { \beta } _ { h } = \underset { \beta \in \mathbb { R } ^ { d } } { \mathrm { a r g m i n } } \sum _ { k = 1 } ^ { K } \left[ \left. \phi ( \bar { s } _ { h } ^ { k } , \bar { a } _ { h } ^ { k } ) , \beta \right. - V _ { h + 1 } ^ { 2 } \left( \bar { s } _ { h + 1 } ^ { k } \right) \right] ^ { 2 } + \lambda \| \beta \| _ { 2 } ^ { 2 } = \bar { \Sigma } _ { h } ^ { - 1 } \sum _ { k = 1 } ^ { K } \phi ( \bar { s } _ { h } ^ { k } , \bar { a } _ { h } ^ { k } ) V _ { h + 1 } ^ { 2 } \left( \bar { s } _ { h + 1 } ^ { k } \right) +$$ + +and, similarly, the first order moment $P _ { h } V _ { h + 1 } : = \left. \phi , \theta _ { h } \right.$ can be estimated via: + +$$ +\bar { \theta } _ { h } = \underset { \theta \in \mathbb { R } ^ { d } } { \operatorname { a r g m i n } } \sum _ { k = 1 } ^ { K } \left[ \left. \phi ( \bar { s } _ { h } ^ { k } , \bar { a } _ { h } ^ { k } ) , \theta \right. - V _ { h + 1 } \left( \bar { s } _ { h + 1 } ^ { k } \right) \right] ^ { 2 } + \lambda \| \theta \| _ { 2 } ^ { 2 } = \bar { \Sigma } _ { h } ^ { - 1 } \sum _ { k = 1 } ^ { K } \phi ( \bar { s } _ { h } ^ { k } , \bar { a } _ { h } ^ { k } ) V _ { h + 1 } \left( \bar { s } _ { h + 1 } ^ { k } \right) +$$ + +The final estimator is defined as $\widehat { \sigma } _ { V _ { h } } ^ { 2 } ( \cdot , \cdot ) \ : = \ \operatorname* { m a x } \{ 1 , \widehat { \mathrm { V a r } } _ { h } V _ { h + 1 } ( \cdot , \cdot ) \}$ with $\widehat { \mathrm { V a r } _ { h } } V _ { h + 1 } ( \cdot , \cdot ) \ =$ $\langle \phi ( \cdot , \cdot ) , \bar { \beta } _ { h } \rangle _ { [ 0 , ( H - h + 1 ) ^ { 2 } ] } - \left[ \langle \phi ( \cdot , \cdot ) , \bar { \theta } _ { h } \rangle _ { [ 0 , H - h + 1 ] } \right] ^ { 2 }$ .6 In particular, when setting $V _ { h + 1 } = \widehat { V } _ { h + 1 }$ , it recovers $\widehat { \sigma } _ { h }$ in Algorithm 1 line 8. Here $\begin{array} { r } { \bar { \Sigma } _ { h } = \sum _ { \tau = 1 } ^ { K } \phi ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } ) \phi ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } ) ^ { \top } + \lambda I _ { d } . } \end{array}$ . + +Variance-weighted LSVI. The idea of LSVI (2) is based on approximate the Bellman updates: ${ \mathcal { T } } _ { h } ( V ) ( s , a ) { \stackrel { \smile } { = } } r _ { h } ( s , a ) + ( P _ { h } V ) ( s , a )$ . With variance estimator $\bar { \widehat { \sigma } } _ { h }$ at hand, we can modify (2) to bsolve the variance-weighted LSVI instead (Line 10 of Algorithm 1) + +$$ +\widehat { w } _ { h } : = \operatorname { a r g m i n } _ { w \in \mathbb { R } ^ { d } } \lambda \| w \| _ { 2 } ^ { 2 } + \sum _ { k = 1 } ^ { K } \frac { \left[ \langle \phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) , w \rangle - r _ { h } ^ { k } - \widehat { V } _ { h + 1 } ( s _ { h + 1 } ^ { \prime k } ) \right] ^ { 2 } } { \widehat { \sigma } _ { h } ^ { 2 } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) } = \widehat { \Lambda } _ { h } ^ { - 1 } \sum _ { k = 1 } ^ { K } \frac { \phi \left( s _ { h } ^ { k } , a _ { h } ^ { k } \right) \cdot \left[ r _ { h } ^ { k } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { k } \right) \right] } { \widehat { \sigma } ^ { 2 } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) } +$$ + +where $\begin{array} { r } { \widehat { \Lambda } _ { h } = \sum _ { k = 1 } ^ { K } \phi \big ( s _ { h } ^ { k } , a _ { h } ^ { k } \big ) \phi \big ( s _ { h } ^ { k } , a _ { h } ^ { k } \big ) ^ { \top } \big / \widehat \sigma _ { h } ^ { 2 } \big ( s _ { h } ^ { k } , a _ { h } ^ { k } \big ) + \lambda I _ { d } } \end{array}$ . The estimated Bellman update $\widehat { \mathcal { T } } _ { h }$ (acts on $\widehat { V } _ { h + 1 } )$ is defined as: $( \widehat { T } _ { h } \widehat { V } _ { h + 1 } ) ( \cdot , \cdot ) = \phi ( \cdot , \cdot ) ^ { \top } \widehat { w } _ { h }$ and the pessimism $\Gamma _ { h }$ is assigned to update $\widehat { Q } _ { h } \approx \widehat { \mathcal { T } } _ { h } \widehat { V } _ { h + 1 } - \Gamma _ { h }$ , i.e. Bellman update $^ +$ Pessimism (Line 10-12 in Algorithm 1). + +Tighter Pessimistic Design. To improve the learning guarantee, we create a tighter penalty design that includes $\widehat { \Lambda } _ { h } ^ { - 1 }$ rather than $\bar { \Sigma } _ { h } ^ { - 1 }$ and an extra higher order $O ( \textstyle { \frac { 1 } { K } } )$ term: + +$$ +\Gamma _ { h } O ( \sqrt { d } \cdot ( \phi ( \cdot , \cdot ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \phi ( \cdot , \cdot ) ) ^ { 1 / 2 } ) + \frac { 2 H ^ { 3 } \sqrt { d } } { K } +$$ + +Note such a design admits no explicit factor in $H$ in the main term (as opposed to Jin et al. (2021b)) therefore is the key for achieving adaptive/problem-dependent results (as we shall discuss later). The full algorithm VAindependent parts with $\mathcal { D } = \{ ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } , r _ { h } ^ { \tau } , s _ { h } ^ { \tau \prime } ) \} _ { \tau \in [ K ] } ^ { h \in [ H ] }$ In pand $\mathcal { D } ^ { \prime } = \{ ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } , \bar { r } _ { h } ^ { \tau } , \bar { s } _ { h } ^ { \tau \prime } ) \} _ { \tau \in [ K ] } ^ { h \in [ H ] }$ data into twofor different purposes (estimating variance and updating $Q$ -values). + +# 3.2 MAIN RESULT + +We denote quantities $\mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } , \mathcal { M } _ { 3 } , \mathcal { M } _ { 4 }$ as in the notation list A. Then VAPVI provides the following result. The complete proof is provided in Appendix C. + +Theorem 3.2. Let $K$ be the number of episodes. If $K > \operatorname* { m a x } \{ \mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } , \mathcal { M } _ { 3 } , \mathcal { M } _ { 4 } \}$ and $\sqrt { d } > \xi$ , $\begin{array} { r } { \xi : = \operatorname* { s u p } _ { V \in [ 0 , H ] , \ : s ^ { \prime } \sim P _ { h } ( s , a ) , \ : h \in [ H ] } \left| \frac { r _ { h } + V \left( s ^ { \prime } \right) - ( \mathcal T _ { h } V ) ( s , a ) } { \sigma _ { V } ( s , a ) } \right| . } \end{array}$ Then for any $0 < \lambda < \kappa$ , with probability $1 - \delta ,$ , for all policy $\pi$ simultaneously, the output $\widehat { \pi }$ of Algorithm 1 satisfies + +$$ +v ^ { \pi } - v ^ { \widehat { \pi } } \leq \widetilde { O } \big ( \sqrt { d } \cdot \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \pi } \bigg [ \sqrt { \phi ( \cdot , \cdot ) ^ { \top } \Lambda _ { h } ^ { - 1 } \phi ( \cdot , \cdot ) } \bigg ] \big ) + \frac { 2 H ^ { 4 } \sqrt { d } } { K } +$$ + +$\begin{array} { r } { \Lambda _ { h } = \sum _ { k = 1 } ^ { K } \frac { \phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) \cdot \phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) ^ { \top } } { \sigma _ { \hat { V } _ { h + 1 } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) } ^ { 2 } } + \lambda I _ { d } } \end{array}$ . In particular, we have with probability $1 - \delta$ , + +$$ +\boldsymbol { v } ^ { \star } - \boldsymbol { v } ^ { \widehat { \pi } } \leq \widetilde { O } \big ( \sqrt { d } \cdot \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \pi ^ { \star } } \bigg [ \sqrt { \phi ( \cdot , \cdot ) ^ { \top } \Lambda _ { h } ^ { \star - 1 } \phi ( \cdot , \cdot ) } \bigg ] \big ) + \frac { 2 H ^ { 4 } \sqrt { d } } { K } +$$ + +$\begin{array} { r } { \Lambda _ { h } ^ { \star } = \sum _ { k = 1 } ^ { K } \frac { \phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) \cdot \phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) ^ { \top } } { \sigma _ { V _ { h + 1 } ^ { \star } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) } ^ { 2 } } + \lambda I _ { d } } \end{array}$ and $\widetilde O$ hides universal constants and the Polylog terms. + +# Algorithm 1 Variance-Aware Pessimistic Value Iteration (VAPVI) + +1: Input: Dataset $\begin{array} { r } { \mathcal { D } = \{ \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } , r _ { h } ^ { \tau } \right) \} _ { \tau , h = 1 } ^ { K , H } \mathcal { D } ^ { \prime } = \{ \left( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } , \bar { r } _ { h } ^ { \tau } \right) \} _ { \tau , h = 1 } ^ { K , H } } \end{array}$ . Universal constant $C$ . +2: Initialization: Set $\widehat { V } _ { H + 1 } ( \cdot ) 0$ . +3: for $h = H , H - 1 , \ldots , 1$ do +4: $\diamond$ Phase1: Regular Least-square Value Iteration for conditional variances +5: Set $\begin{array} { r } { \bar { \Sigma } _ { h } \sum _ { \tau = 1 } ^ { K } \phi ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } ) \phi ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } ) ^ { \top } + \lambda I } \end{array}$ +6: Set $\begin{array} { r } { \bar { \beta } _ { h } \gets \bar { \Sigma } _ { h } ^ { - 1 } \sum _ { \tau = 1 } ^ { K } \phi ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } ) \cdot \widehat { V } _ { h + 1 } ( \bar { s } _ { h + 1 } ^ { \tau } ) ^ { 2 } } \end{array}$ +7: Set $\begin{array} { r } { \bar { \theta } _ { h } \gets \bar { \Sigma } _ { h } ^ { - 1 } \sum _ { \tau = 1 } ^ { K } \phi ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } ) \cdot \widehat { V } _ { h + 1 } \big ( \bar { s } _ { h + 1 } ^ { \tau } \big ) } \end{array}$ +8: Set $\begin{array} { r } { \left[ \widehat { \mathrm { V a r } _ { h } } \widehat { V } _ { h + 1 } \right] ( \cdot , \cdot ) = \left. \phi ( \cdot , \cdot ) , \bar { \beta } _ { h } \right. _ { \left[ 0 , ( H - h + 1 ) ^ { 2 } \right] } - \left[ \left. \phi ( \cdot , \cdot ) , \bar { \theta } _ { h } \right. _ { \left[ 0 , H - h + 1 \right] } \right] ^ { 2 } } \end{array}$ +9: Set $\widehat { \sigma } _ { h } ( \cdot , \cdot ) ^ { 2 } \gets \operatorname* { m a x } \{ 1 , \widehat { \mathrm { V a r } } _ { P _ { h } } \widehat { V } _ { h + 1 } ( \cdot , \cdot ) \}$ +10: $\diamond$ bPhase2: Weighted Least-square Value Iteration for pessimistic updates +11: $\begin{array} { r l } & { \mathrm { S e t } \widehat { \Lambda } _ { h } \gets \sum _ { \tau = 1 } ^ { \overline { { K } } } \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) ^ { \widehat { \circ } } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) ^ { \top } / \widehat { \sigma } ^ { 2 } \big ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \big ) ^ { \widehat { \mathbf { \phi } } } + \lambda \cdot I , } \\ & { \mathrm { S e t } \widehat { w } _ { h } \gets \widehat { \Lambda } _ { h } ^ { - 1 } ( \sum _ { \tau = 1 } ^ { K } \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \cdot ( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } ( s _ { h + 1 } ^ { \tau } ) ) / \widehat { \sigma } ^ { 2 } \big ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \big ) ) } \\ & { \mathrm { S e t } \Gamma _ { h } ( \cdot , \cdot ) \gets C \sqrt { d } \cdot ( \phi ( \cdot , \cdot ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \phi ( \cdot , \cdot ) ) ^ { 1 / 2 } + \frac { 2 H ^ { 3 } \sqrt { d } } { K } \qquad \mathrm { ( U s e ~ \Gamma } _ { h } ^ { I } \mathrm { ~ f o r ~ t h e ~ i m p r o v e d ~ v e r s i o n ) } } \\ & { \mathrm { S e t } \widehat { \bar { Q } } _ { h } ( \cdot , \cdot ) \gets \phi \big ( \cdot , \cdot \big ) ^ { \top } \widehat { w } _ { h } - \Gamma _ { h } ( \cdot , \cdot ) } \\ & { \mathrm { S e t } \widehat { \widehat { \hat { Q } } } _ { h } ( \cdot , \cdot ) \gets \operatorname* { m i n } \{ \widehat { \bar { Q } } _ { h } ( \cdot , \cdot ) , H - h + 1 \} ^ { + } } \\ & \mathrm { S e t } \widehat { \hat { \mathcal { R } } _ { h } } ( \cdot , \cdot ) \gets \operatorname* { a r g m a x } _ { \pi _ { h } } \widehat { Q } _ { h } ( \cdot , \cdot ) , \pi _ { h } ( \cdot \cdot ) _ { A } , \widehat { V } _ { h } ( \cdot ) \gets \operatorname* { m a x } _ { \pi _ { h } } \widehat { Q } _ { h } ( \cdot , \cdot ) , \pi _ { h } \end{array}$ +12: +13: +14: +15: +16: +17: end for18: Output: $\left\{ \widehat { \pi } _ { h } \right\} _ { h = 1 } ^ { H }$ . + +Theorem 3.2 provides improvements over the existing best-known results and we now explain it. +However, before that, we first discuss about our theorem condition. + +Comparing to Zhou et al. (2021a). In the online regime, Zhou et al. (2021a) is the first result that achieves optimal regret rate with $O ( d H \sqrt { T } )$ in the linear (mixture) MDPs. However, this result requires the condition $d \geq H$ (their Theorem 6 and Remark 7). In offline RL, VAPVI only requires a milder condition $\sqrt { d } > \xi$ comparing to $d \ge H$ (since for any fixed $V \in [ 0 , H ]$ , the standardized quantity $\frac { r + V ( s ^ { \prime } ) - ( \mathcal { T } _ { h } V ) ( s , a ) } { \sigma _ { V } ( s , a ) }$ is bounded by constant with high probability, e.g. by chebyshev inequality), which makes our result apply to a wider range of linear MDPs. + +Comparing to Jin et al. (2021b). Jin et al. (2021b) first shows pessimistic value iteration (PEVI) is provably efficient for Linear MDPs in offline RL. VAPVI improves PEVI over $O ( { \sqrt { d } } )$ on the feature dimension, and improves the horizon dependence as $\begin{array} { r } { \Lambda _ { h } \ \succcurlyeq \ \frac { 1 } { H ^ { 2 } } \sum _ { h } } \end{array}$ implies $\Lambda _ { h } ^ { - 1 } \preccurlyeq$ $H ^ { 2 } \Sigma _ { h } ^ { - 1 }$ . In addition, when instantiate to the tabular case, i.e. $\phi ( s , a ) \ : = \ : \mathbf { 1 } _ { s , a }$ , VAPVI gives $\begin{array} { r l } & { O ( \sqrt { d } \sum _ { h , s , a } { d _ { h } ^ { \pi } } ^ { \star } ( s , a ) \sqrt { \frac { \operatorname { V a r } _ { P _ { s , a } } ( r + V _ { h + 1 } ^ { \star } ) } { K \cdot d _ { h } ^ { \mu } ( s , a ) } } ) } \end{array}$ r VarPs,a (r+V ?h+1)K·dµ(s,a) ), which enjoys O(√H) improvement over PEVI (recall Example 3.1) and the order ${ \cal O } ( H ^ { 3 / 2 } )$ is tight (check Section G for the detailed derivation). + +Comparing to Xie et al. (2021a). Their linear MDP guarantee in Theorem 3.2. enjoys the same rate as VAPVI in feature dimension but the horizon dependence is essentially the same as Jin et al. (2021b) (by translating $\begin{array} { r } { H \approx O \bigl ( \frac { 1 } { 1 - \gamma } \bigr ) , } \end{array}$ ) therefore is not optimal. The general function approximation scheme in Xie et al. (2021a) provides elegant characterizations for on-support error and off-support error, but the algorithmic framework is information-theoretical only (and the practical version PSPI will not yield the same learning guarantee). Also, due to the use finite function class and policy class, the reduction to linear MDP only works with finite action space. As a comparison, VAPVI has no constraints on any of these. + +Comparing to Zanette et al. (2021). Concurrently, Zanette et al. (2021) considers offline RL with the linear Bellman complete model, which is more general than linear MDPs and, with the assumption $Q ^ { \pi } \leq 1$ , their PACLE algorithm provides near-minimax optimal guarantee in this setting. However, when recovering to the standard setting $Q ^ { \pi } \in [ 0 , H ]$ , their bound will rescale by an $H$ factor,7 which could be suboptimal due to the variance-unawareness. The reason behind this is: when $Q ^ { \pi } \leq 1$ , lack of variance information encoding will not matter, since in this case $\mathrm { V a r } _ { P } ( V ^ { \pi } ) \leq 1$ has constant order (therefore will not affect the optimal rate); when $Q ^ { \pi } \in [ 0 , H ]$ , $\operatorname { V a r } _ { P } ( V ^ { \pi } )$ can be as large as $H ^ { 2 }$ , effectively leveraging the variance information can help improve the sample efficiency, e.g. via law of total variances, just like VAPVI does. On the other hand, their guarantee also requires finite action space, due to the mirror descent style analysis. Nevertheless, we do point out Zanette et al. (2021) has improved state-action measure than VAPVI, as $\| E _ { \pi } [ \phi ( \cdot , \cdot ) ] \| _ { M ^ { - 1 } } \leq \mathbb { E } _ { \pi } [ \| \phi ( \cdot , \cdot ) \| _ { M ^ { - 1 } } ]$ by Jensen’s inequality and that norm $\lVert \cdot \rVert _ { M ^ { - 1 } }$ is convex for some positive-definite matrix $M$ . + +Adaptive characterization and faster convergence. Comparing to existing works, one major improvement is that the main term for VAPVI $\begin{array} { r l } { { \sqrt { d } \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \pi ^ { \star } } \bigl [ \sqrt { \phi ( \cdot , \cdot ) ^ { \top } \Lambda _ { h } ^ { \star - 1 } \phi ( \cdot , \cdot ) } \bigr ] } \quad } & { { } } \end{array}$ admits no explicit dependence on $H$ , which provides a more adaptive/instance-dependent characterization. For instance, if we ignore the technical treatment by taking $\lambda = 0$ and $\sigma _ { h } ^ { \star } \widetilde { \approx } \mathrm { V a r } _ { P } ( V _ { h + 1 } ^ { \star } )$ , then for the partially deterministic systems (where there are $t$ stochastic $P _ { h }$ ’s and $H - t$ deterministic $P _ { h }$ ’s), the main term diminishes to $\begin{array} { r l } { \sqrt { d } \sum _ { i = 1 } ^ { t } \mathbb { E } _ { \pi ^ { \star } } \left[ \sqrt { \phi ( \cdot , \cdot ) ^ { \top } \Lambda _ { h _ { i } } ^ { \star - 1 } \phi ( \cdot , \cdot ) } \right] } & { { } } \end{array}$ with $h _ { i } \in \{ h : ~ s . t . ~ P _ { h }$ is stochastic $\}$ and can be a much smaller quantity when $t \ll H$ . Furthermore, for the fully deterministic system, VAPVI automatically provides faster convergence rate $O ( \textstyle { \frac { 1 } { K } } )$ from the higher order term, given that the main term degenerates to 0. Those adaptive/instance-dependent features are not enjoyed by (Xie et al., 2021a; Zanette et al., 2021), as they always provide the standard statistical rate $\scriptstyle { \dot { O ( } } { \frac { 1 } { \sqrt { K } } } )$ (also check Remark C.9 for a related discussion). + +# 3.3 VAPVI-IMPROVED: FURTHER IMPROVEMENT IN STATE-ACTION DIMENSION + +Can we further improve the VAPVI? Indeed, by deploying a carefully tuned tighter penalty, we are able to further improve the state-action dependence if the feature is non-negative $( \phi \ge 0 )$ ). Concretely, we replace the following $\Gamma _ { h } ^ { I }$ in Algorithm 1 instead, and call the algorithm VAPVI-Improved (or VAPVI-I for short). The proof can be found in Appendix D. + +$$ +\Gamma _ { h } ^ { I } ( s , a ) \gets \phi ( s , a ) ^ { \top } \left| \widehat { \Lambda } _ { h } ^ { - 1 } \sum _ { \tau = 1 } ^ { K } \frac { \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \widehat { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) } { \widehat { \sigma } _ { h } ^ { 2 } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) } \right| + \widetilde { O } ( \frac { H ^ { 3 } d / \kappa } { K } ) +$$ + +Theorem 3.3. Suppose the feature is non-negative √ $\left( \phi \geq 0 \right)$ ). Let $K$ be the number of episodes. If $K > \operatorname* { m a x } \{ \mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } , \mathcal { M } _ { 3 } , \mathcal { M } _ { 4 } \}$ and $\sqrt { d } > \xi$ . Deploying $\Gamma _ { h } ^ { I }$ (4) in Algorithm $^ { l }$ . Then for any $0 < \lambda < \kappa$ , with probability $1 - \delta$ , for all policy $\pi$ simultaneously, the output $\widehat { \pi }$ of Algorithm $\boldsymbol { l }$ (VAPVI-I) satisfies + +$$ +v ^ { \pi } - v ^ { \widehat { \pi } } \leq \widetilde { O } \big ( \sqrt { d } \cdot \sum _ { h = 1 } ^ { H } \sqrt { \mathbb { E } _ { \pi } [ \phi ( \cdot , \cdot ) ] ^ { \top } \Lambda _ { h } ^ { - 1 } \mathbb { E } _ { \pi } [ \phi ( \cdot , \cdot ) ] } \big ) + \widetilde { O } ( \frac { H ^ { 4 } d / \kappa } { K } ) +$$ + +In particular, when choosing $\pi = \pi ^ { \star }$ , the above guarantee holds true with $\Lambda _ { h } ^ { - 1 }$ replaced by $\Lambda _ { h } ^ { \star - 1 }$ Here $\Lambda _ { h } ^ { - 1 }$ , $\Lambda _ { h } ^ { \star - 1 }$ , $\xi$ are defined the same as Theorem 3.2. + +Theorem 3.3 maintains nearly all the features of Theorem 3.2 (except higher order term is slightly worse) and the dominate term evolves from $\mathbb { E } _ { \pi } \left. \phi \right. _ { \Lambda _ { h } ^ { - 1 } }$ to $\| \mathbb { E } _ { \pi } [ \phi ] \| _ { \Lambda _ { h } ^ { - 1 } }$ . Clearly, the two bounds differ by the magnitude of Jensen’s inequality. To provide a concrete view of how much improvement is made, we check the parameter dependence in the context of tabular MDPs (where we ignore the higher order term for conciseness). In particular, we compare the results under the single-policy concentrability. + +Assumption 3.4 (Rashidinejad et al. (2021); Xie et al. (2021b)). There exists a optimal policy $\pi ^ { \star }$ , s.t. +$\begin{array} { r } { \operatorname* { s u p } _ { h , s , a } \bar { d } _ { h } ^ { \pi ^ { \star } } ( s , a ) / d _ { h } ^ { \mu } ( s , a ) : = C ^ { \star } < \infty } \end{array}$ , where $d ^ { \pi }$ is the marginal state-action probability under $\pi$ . + +In tabular RL, $\phi ( s , a ) = \mathbf { 1 } _ { s , a }$ and $d = S \cdot A \left( S , A \right.$ be the finite state, action cardinality), then + +$$ +\begin{array} { r l } & { \mathrm { T h e o r e m ~ 3 . 2 } \sqrt { S A } \displaystyle \sum _ { h } ^ { H } \sum _ { s , a } d _ { h } ^ { \pi ^ { \star } } ( s , a ) \sqrt { \frac { \mathrm { V a r } _ { P s , a } ( r + V _ { h + 1 } ^ { \star } ) } { K \cdot d _ { h } ^ { \mu } ( s , a ) } } \leq \sqrt { \frac { H ^ { 3 } C ^ { \star } S ^ { 2 } A } { K } } ; } \\ & { \mathrm { T h e o r e m ~ 3 . 3 } \sqrt { S A } \displaystyle \sum _ { h } ^ { H } \sqrt { \displaystyle \sum _ { s , a } d _ { h } ^ { \pi ^ { \star } } ( s , a ) ^ { 2 } \frac { \mathrm { V a r } _ { P s , a } ( r + V _ { h + 1 } ^ { \star } ) } { K \cdot d _ { h } ^ { \mu } ( s , a ) } } \leq \sqrt { \frac { H ^ { 3 } C ^ { \star } S A } { K } } . } \end{array} +$$ + +Theorem 3.3 enjoys a $S$ state improvement over Theorem 3.2 and nearly recovers the minimax rate $\sqrt { \frac { H ^ { 3 } C ^ { \star } { \cal S } } { K } }$ (Xie et al., 2021b). The detailed derivation can be found in Appendix G. Also, to show our result is near-optimal, we provide the corresponding lower bound. The proof is in Appendix E. + +Theorem 3.5 (Minimax lower bound). There exist a pair of universal constants $c , c ^ { \prime } > 0$ such that given dimension $d$ , horizon $H$ and sample size $K > c ^ { \prime } d ^ { 3 }$ , one can always find a family of linear MDP instances M such that (where Λ?h = PKk=1 φ(s h,a h)·φ(s h,a h) Varh(V ? )(sk ,ak ) satisfies $( \Lambda _ { h } ^ { \star } ) ^ { - 1 }$ exists and + +$$ +\begin{array} { r l } & { \mathrm { V a r } _ { h } ( V _ { h + 1 } ^ { \star } ) ( s _ { h } ^ { k } , a _ { h } ^ { k } ) > 0 \forall M \in \mathcal { M } ) } \\ & { \qquad \quad \frac { \operatorname* { i n f } } { \pi } \underset { M \in \mathcal { M } } { \operatorname* { s u p } } \mathbb { E } _ { M } \left[ v ^ { \star } - v ^ { \hat { \pi } } \right] / \left( \sqrt { d } \cdot \displaystyle \sum _ { h = 1 } ^ { H } \sqrt { \mathbb { E } _ { \pi ^ { \star } } \left[ \phi \right] ^ { \top } ( \Lambda _ { h } ^ { \star } ) ^ { - 1 } \mathbb { E } _ { \pi ^ { \star } } \left[ \phi \right] } \right) \geq c . } \end{array} +$$ + +Theorem 3.5 nearly matches the main term in VAPVI-I (Theorem 3.3) and certifies it is near-optimal. + +# 4 PROOF OVERVIEW + +Due to the space constraint, we could only provide a brief overview of the key proving ideas of the theorems. We begin with Theorem 3.2. First, by the extended value difference lemma (Lemma H.7), we can convert bounding the suboptimality gap of $\boldsymbol { v } ^ { \star } - \boldsymbol { v } ^ { \widehat { \pi } }$ to bounding ${ \begin{array} { r } { \sum _ { h = 1 } ^ { H } 2 \cdot \mathbb { E } _ { \pi } \left[ \Gamma _ { h } { \left( s _ { h } , a _ { h } \right) } \right] } \end{array} }$ , given that $| ( \mathcal T _ { h } \widehat { V } _ { h + 1 } - \widehat { \mathcal T } _ { h } \widehat { V } _ { h + 1 } ) ( s , a ) | \leq \Gamma _ { h } ( s , a )$ for all $s , a , h$ . To bound $\mathcal { T } _ { h } \widehat { V } _ { h + 1 } - \widehat { \mathcal { T } } _ { h } \widehat { V } _ { h + 1 }$ , by decomposing it reduces to bounding the key quantity + +$$ +\phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \left[ \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) / \widehat { \sigma } _ { h } ^ { 2 } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right] +$$ + +The term is treated in two steps. First, we bound the gap of $\left\| \sigma _ { \widehat { V } _ { h + 1 } } ^ { 2 } - \widehat { \sigma } _ { h } ^ { 2 } \right\|$ so we can convert $\widehat { \sigma } _ { h } ^ { 2 }$ to $\sigma _ { \widehat { V } _ { h + 1 } } ^ { 2 }$ . Next, since $\begin{array} { r } { \mathrm { V a r } \left[ r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \mid s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right] \approx \sigma _ { \widehat { V } _ { h + 1 } } ^ { 2 } } \end{array}$ , therefore by the variance-weighted scheme in (equation 7), we can leverage the recent technical development Bernstein inequality for self-normalized martingale (Lemma H.3) for acquiring the tight result, in contrast to the previous treatment of Hoeffding inequality for self-normalized martingale $^ +$ Covering.8 For the second part, one needs to further convert $\sigma _ { \widehat { V } _ { h + 1 } } ^ { 2 ^ { - } }$ to $\sigma _ { h } ^ { \star 2 }$ $\cdot \Lambda _ { h } ^ { - 1 }$ to $\Lambda _ { h } ^ { \star - 1 }$ ) with appropriate concentrations. The proof of Theorem 3.3 is similar but with more complicated computations and relies on using the linear representation of $\phi$ in $\Gamma _ { h } ^ { I }$ (4), so that the expectation over $\pi$ is inside the square root by taking expectation over the linear representation at the beginning. The lower bound proof uses a simple modification of Zanette et al. (2021) which consists of the reduction from learning to testing with Assouad’s method, and the use of standard information inequalities (e.g. from total variation to KL divergence). For completeness, we provide the full proof in Appendix E. + +# 5 DISCUSSION AND CONCLUSION + +This work studies offline RL with linear MDP representation and contributes Variance Aware Pessimistic Value Iteration (VAPVI) which adopts the conditional variance information of the value function. VAPVI uses the estimated variances to reweight the Bellman residuals in the least-square pessimistic value iteration and provides improved offline learning bounds over the existing best-known results. VAPVI-I further improves over VAPVI in the state-action dimension and is near-minimax optimal. One highlight of the theorems is that our learning bounds are expressed in terms of system quantities, which automatically provide natural instance-dependent characterizations that previous results are short of. + +On the other hand, while VAPVI/VAPVI-I close the existing gap from previous literature (Jin et al., 2021b; Xie et al., 2021a), the optimal guarantee is in the minimax sense. Although our upper bounds possess instance-dependent characterizations, the lower bound only holds true for a class of hard instances. In this sense, whether “instance-dependent optimality” can be achieved remains elusive in the current linear MDP setting (such a discussion is recently initiated in MAB problems (Xiao et al., 2021)). We leave this as future work. + +# ACKNOWLEDGMENTS + +The authors would like to thank Quanquan Gu for explaining Min et al. (2021) and introducing a couple of related literatures. Ming Yin would like to thank Zhuoran Yang for the helpful suggestions and Dan Qiao for a careful proofreading. Mengdi Wang gratefully acknowledges funding from Office of Naval Research (ONR) N00014-21-1-2288, Air Force Office of Scientific Research (AFOSR) FA9550-19-1-0203, and NSF 19-589, CMMI-1653435. Yu-Xiang Wang gratefully acknowledges funding from National Science Foundation (NSF) #2007117 and #2003257. + +# REFERENCES + +Yasin Abbasi-Yadkori, David P ´ al, and Csaba Szepesv ´ ari. Improved algorithms for linear stochastic ´ bandits. 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PMLR, 2021b. + +# Appendix + +# A NOTATION LIST + +$\Sigma _ { h } ^ { p }$ $\begin{array} { r l } & { \mathbb { E } _ { \mu , h } \left[ \phi ( s , a ) \phi ( s , a ) ^ { \top } \right] } \\ & { \mathbb { E } _ { \mu , h } \left[ \sigma _ { V _ { h + 1 } } ( s , a ) ^ { - 2 } \phi ( s , a ) \phi ( s , a ) ^ { \top } \right] } \\ & { \operatorname* { m i n } _ { h } \lambda _ { \operatorname* { m i n } } ( \Sigma _ { h } ^ { p } ) } \end{array}$ +$\Lambda _ { h } ^ { p }$ +$\kappa$ +ι $\operatorname* { m i n } _ { h } \lambda _ { \operatorname* { m i n } } ( \Lambda _ { h } ^ { p } ) \ge \kappa / H ^ { 2 }$ for any $V _ { h }$ +$\sigma _ { V } ^ { 2 } ( s , a )$ $\operatorname* { m a x } \{ 1 , \operatorname { V a r } _ { P _ { h } } ( V ) ( s , a ) \}$ for any $V$ +σ2Vh+1 (s, a) σ2Vh (s, a) $\begin{array} { r l } & { \operatorname* { m a x } \{ 1 , \nabla \mathrm { a r } _ { h } ( V _ { h + 1 } ) ( s , a ) \} } \\ & { \operatorname* { m a x } \{ 1 , \widehat { \mathrm { V a r } } _ { h } h _ { h + 1 } ( s , a ) \} } \\ & { \sum _ { \tau = 1 } ^ { K } \phi ( \tilde { s } _ { h } ^ { \tau } , \tilde { a } _ { h } ^ { \tau } ) \phi ( \tilde { s } _ { h } ^ { \tau } , \tilde { a } _ { h } ^ { \tau } ) ^ { \top } + \lambda I _ { d } } \\ & { \sum _ { k = 1 } ^ { K } \phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) \phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) ^ { \top } / \widehat { \sigma } _ { h } ^ { 2 } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) + \lambda I _ { d } } \\ & { \operatorname* { m a x } \{ 2 \lambda , 1 2 8 \log ( 2 d / \delta ) , 1 2 8 H ^ { 4 } \log ( 2 d / \delta ) / \kappa ^ { 2 } \} } \\ & { \operatorname* { m a x } \{ \frac { \lambda ^ { 2 } } { \kappa \log ( ( \lambda + K ) H / \lambda \delta ) } , 9 6 ^ { 2 } H ^ { 1 2 } d \log ( ( \lambda + K ) H / \lambda \delta ) / \kappa ^ { 5 } \} } \\ & { \operatorname* { m a x } \{ 5 1 2 H ^ { 4 } / \kappa ^ { 2 } \log \left( \frac { 2 d } { \delta } \right) , 4 \lambda H ^ { 2 } / \kappa \} } \\ & { 1 2 \sqrt { H ^ { 4 } d \log ( ( \lambda + K ) H / \lambda \delta ) / \kappa } } \end{array}$ +Σ¯ h +$\widehat { \Lambda } _ { h }$ +$\mathcal { M } _ { 1 }$ +$\mathcal { M } _ { 2 }$ +M3 +$\mathcal { M } _ { 4 }$ +δ Failure probability +$\xi$ $\begin{array} { r l } & { \operatorname* { s u p } _ { V \in [ 0 , H ] , \ s ^ { \prime } \sim P _ { h } ( s , a ) , \ h \in [ H ] } \left| \frac { r _ { h } + V \left( s ^ { \prime } \right) - ( \mathcal { T } _ { h } V ) ( s , a ) } { \sigma _ { V } ( s , a ) } \right| } \\ & { 3 6 \sqrt { \frac { H ^ { 4 } d ^ { 3 } } { \kappa } \log \left( \frac { ( \lambda + K ) 2 K d H ^ { 2 } } { \lambda \delta } \right) } + 1 2 \lambda \frac { H ^ { 2 } \sqrt { d } } { \kappa } } \end{array}$ +$C _ { H , d , \kappa , K }$ + +# B EXTENDED LITERATURE REVIEW + +# B.1 LINEAR MODEL REPRESENTATION AND ITS EXTENSION IN ONLINE RL + +There are numerous works in online RL that study linear model representations. Yang and Wang (2019; 2020); Jin et al. (2020) propose Linear MDP, which assumes the transition kernel and the reward are linear in given features. Cai et al. (2020); Ayoub et al. (2020); Modi et al. (2020); Zhou et al. (2021b) propose Linear mixture MDP, which assumes the transition probability is a linear combination of some base kernels. Linear Bellman Complete model (Zanette et al., 2020) generalizes linear MDP model by allowing linear functions to approximate the $Q$ -function and the function class is closed under the Bellman update. The notion of low Bellman rank (Jiang et al., 2017) subsumes not only linear MDPs but also models including linear quadratic regulator (LQR), Reactive POMDP (Krishnamurthy et al., 2016) and Block MDP (Du et al., 2019). There are also other models, e.g. factored MDP (Sun et al., 2019), Bellman Eluder dimension (Jin et al., 2021a) and Bilinear class (Du et al., 2021). With the linear MDP model itself, there are also fruitful extensions, e.g. gap-dependent analysis with logarithmic regret (He et al., 2021), low-switching cost RL (Gao et al., 2021), safe RL (Amani et al., 2021), reward-free RL (Wang et al., 2020), generalized linear model (GLM) (Wang et al., 2021b), two-player markov game (Chen et al., 2021b) and generative adversarial learning (Liu et al., 2021). In particular, Zhou et al. (2021a) shows UCRL- $\mathrm { { V T R + } }$ is near-minimax optimal when feature dimension $d \geq H$ . + +# B.2 EXISTING RESULTS IN OFFLINE RL WITH MODEL REPRESENTATIONS + +Offline RL with general function representations. The finite sample analysis of offline RL with function approximation is initially conducted by Fitted $Q$ -Iteration (FQI) type algorithms and can be dated back to (Munos, 2003; Szepesvari and Munos ´ , 2005; Antos et al., 2008a;b). Later, Chen and Jiang (2019); Le et al. (2019); Xie and Jiang (2020) follow this line of research and derive the improved learning results. However, owing to the aim for tackling general function approximation, those learning bounds are expressed in terms of the stringent concentrability coefficients (therefore, are less adaptive to individual instances) and are usually only information-theoretical, due to the computational intractability of the optimization procedure over the general function classes. Other works impose weaker assumptions (e.g. partial coverage (Liu et al., 2020; Kidambi et al., 2020; Uehara and Sun, 2021)), and their finite sample analysis are generally suboptimal in terms of $H$ or the effective horizon $( 1 - \gamma ) ^ { - 1 }$ . + +Offline RL with tabular models. For tabular MDPs, tight learning bounds can be achieved under several data-coverage assumptions. For the class of problems with uniform data-visitation measure $d _ { m }$ , the near-optimal sample complexity bound has the rate $O ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } )$ for time-inhomogeneous MDPs (Yin et al., 2021) and $O ( H ^ { 2 } / d _ { m } \dot { \epsilon } )$ for time-homogeneous MDPs (Yin and Wang, 2021a; Ren et al., 2021). Under the single concentrability assumption, the tight rate ${ \cal O } ( H ^ { 3 } S C ^ { \star } / \epsilon ^ { \top } )$ is obtained by Xie et al. (2021b). In particular, the recent study Yin and Wang (2021b) introduces the intrinsic offline learning bound that is not only instance-dependent but also subsumes previous optimal results. More recently, Shi et al. (2022) uses model-free approach to achieve minimax rate with a larger $\epsilon$ -range. + +Offline RL with linear model representations. Recently, there are more focus on studying the provable efficient offline RL under the linear model representations. Jin et al. (2021b) first shows offline RL with linear MDP is provably efficient by the pessimistic value iteration (PEVI), which is an offline counterpart of LSVI-UCB in Jin et al. (2020). Their analysis deviates from their lower bound by a factor of $d \cdot H$ (check their Theorem 4.4 and 4.6). Later, Xie et al. (2021a) considers function approximation under the Bellman-consistent assumptions, and, when realized to linear MDP setting, improve the sample complexity guarantee of Jin et al. (2021b) by a order $O ( d )$ (Theorem 3.2). However, their improvement only holds for finite action space (due to the dependence $\log \left| { \mathcal { A } } \right| )$ and by the direct reduction (from Theorem 3.1) their result does not imply a computationally tractable algorithm. In addition, there is no improvement on the horizon dependence. Concurrently, Zanette et al. (2021) considers the Linear Bellman Complete model (which originates from its online version Zanette et al. (2020)) and designs the actor-critic style algorithm that achieves tight result under the assumption that the value function is bounded by 1. While their algorithm is efficient (which is based on solving a sequence of second-order cone programs), the resulting learning bound requires the action space to be finite due to the mirror descent/natural policy gradient updates in the Actor procedure (Agarwal et al., 2021). Besides, assuming the value function to be less than 1 simplifies the challenges in dealing with horizon $H$ since when rescale their result to $[ 0 , H ]$ , there is a $H$ factor blow-up, which makes no improvement in the horizon dependence comparing to Jin et al. (2021b). On the other hand, Wang et al. (2021a); Zanette (2021) study the statistical hardness of offline RL with linear representations by proofing the exponential lower bounds. As a result, none of the existing algorithms can achieve the statistical limit for the well-structured linear MDP model with the general (infinite or continuous) state-action spaces in the offline regime. + +# C PROOFS IN SECTION 3.2 + +Instead of proofing the result for $v ^ { \star } - v ^ { \widehat { \pi } }$ , in most parts of the proof we deal with $V _ { 1 } ^ { \star } - V _ { 1 } ^ { \widehat \pi }$ , which is more general. + +# C.1 SOME PREPARATIONS + +Define the Bellman update error $\zeta _ { h } ( s , a ) \ : = \ ( { \mathcal T } _ { h } \widehat V _ { h + 1 } ) ( s , a ) \ - \ - \ - \widetilde Q _ { h } ( s , a )$ and recall $\widehat { \pi } _ { h } ( s ) \ =$ arg $\operatorname* { m a x } _ { \pi _ { h } } \langle \widehat { Q } _ { h } ( s , \cdot ) , \pi _ { h } ( \cdot \mid s ) \rangle _ { \mathcal { A } }$ , then by the direct application of Lemma H.8 + +$$ +V _ { 1 } ^ { \pi } ( s ) - V _ { 1 } ^ { \widehat { \pi } } ( s ) \leq \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \pi } \left[ \zeta _ { h } ( s _ { h } , a _ { h } ) \ | \ s _ { 1 } = s \right] - \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \widehat { \pi } } \left[ \zeta _ { h } ( s _ { h } , a _ { h } ) \ | \ s _ { 1 } = s \right] . +$$ + +The next lemma shows it is sufficient to bound the pessimistic penalty, which is the key in the proof. + +Lemma C.1. Suppose with probability $1 - \delta$ , it holds for all $h , s , a \in [ H ] \times S \times A$ that $| ( \widehat { T _ { h } } \widehat { V _ { h + 1 } } -$ $\widehat { \mathcal { T } } _ { h } \widehat { V } _ { h + 1 } ) ( s , a ) | \ \leq \ \Gamma _ { h } ( s , a )$ , then it implies $\forall s , a , h \in \mathcal { S } \times \mathcal { A } \times [ H ]$ , $0 \leq \zeta _ { h } ( s , a ) \leq 2 \Gamma _ { h } ( s , a )$ . + +Furthermore, it holds for any policy $\pi$ simultaneously, with probability $1 - \delta$ , + +$$ +V _ { 1 } ^ { \pi } ( s ) - V _ { 1 } ^ { \widehat { \pi } } ( s ) \leq \sum _ { h = 1 } ^ { H } 2 \cdot \mathbb { E } _ { \pi } \left[ \Gamma _ { h } ( s _ { h } , a _ { h } ) \mid s _ { 1 } = s \right] . +$$ + +Proof of Lemma C.1. We first show given $| ( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } - \widehat { \mathcal { T } } _ { h } \widehat { V } _ { h + 1 } ) ( s , a ) | \ \leq \ \Gamma _ { h } ( s , a )$ , then $0 ~ \leq$ $\zeta _ { h } ( s , a ) \le 2 \Gamma _ { h } ( s , a ) , \forall s , a , h \in \mathcal { S } \times \bar { \mathcal { A } } \times [ H ]$ . + +Step1: we first show $0 \leq \zeta _ { h } ( s , a ) , \forall s , a , h \in S \times \mathcal { A } \times [ H ] .$ . + +Indeed, if $\bar { Q } _ { h } ( s , a ) ~ \leq ~ 0$ , then by definition $\widehat { Q } _ { h } ( s , a ) ~ = ~ 0$ and in this case $\zeta _ { h } ( s , a ) : =$ $( { \mathcal { T } } _ { h } { \widehat { V } } _ { h + 1 } ) ( s , a ) - { \widehat { Q } } _ { h } ( s , a ) = ( { \mathcal { T } } _ { h } { \widehat { V } } _ { h + 1 } ) ( s , a ) \geq 0$ ; if $\bar { Q } _ { h } ( s , a ) > 0$ , then $\widehat { Q } _ { h } ( s , a ) \ \leq \ \bar { Q } _ { h } ( s , a )$ and + +$$ +\begin{array} { r l } & { \zeta _ { h } ( s , a ) : = ( { \mathcal { T } } _ { h } \widehat { V } _ { h + 1 } ) ( s , a ) - \widehat { Q } _ { h } ( s , a ) \geq ( { \mathcal { T } } _ { h } \widehat { V } _ { h + 1 } ) ( s , a ) - \bar { Q } _ { h } ( s , a ) } \\ & { \qquad = ( { \mathcal { T } } _ { h } \widehat { V } _ { h + 1 } ) ( s , a ) - ( \widehat { { \mathcal { T } } _ { h } } \widehat { V } _ { h + 1 } ) ( s , a ) + \Gamma _ { h } ( s , a ) \geq 0 . } \end{array} +$$ + +Step2: next we show $\zeta _ { h } ( s , a ) \leq 2 \Gamma _ { h } ( s , a ) , \forall s , a , h \in \mathcal { S } \times \mathcal { A } \times [ H ] .$ + +Indeed, we have $\widehat { Q } _ { h } ( s , a ) = \operatorname* { m a x } ( \bar { Q } _ { h } ( s , a ) , 0 )$ and this is because: $\bar { Q } _ { h } ( x , a ) = ( \widehat { T } _ { h } \widehat { V } _ { h + 1 } ) ( x , a ) -$ $\Gamma _ { h } ( x , a ) \leq ( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } ) ( x , a ) \leq H - h + 1$ . Therefore, in this case we have: + +$$ +\begin{array} { r l } & { \zeta _ { h } ( s , a ) : = ( { \mathcal { T } } _ { h } { \widehat { V } } _ { h + 1 } ) ( s , a ) - { \widehat { Q } } _ { h } ( s , a ) \leq ( { \mathcal { T } } _ { h } { \widehat { V } } _ { h + 1 } ) ( s , a ) - { \bar { Q } } _ { h } ( s , a ) } \\ & { \qquad = ( { \mathcal { T } } _ { h } { \widehat { V } } _ { h + 1 } ) ( s , a ) - ( { \widehat { \mathcal { T } } } _ { h } { \widehat { V } } _ { h + 1 } ) ( s , a ) + \Gamma _ { h } ( s , a ) \leq 2 \cdot \Gamma _ { h } ( s , a ) . } \end{array} +$$ + +For the last statement, denote $\mathfrak { F } : = \{ 0 \le \zeta _ { h } ( s , a ) \le 2 \Gamma _ { h } ( s , a ) , \forall s , a , h \in \mathcal { S } \times \mathcal { A } \times [ H ]$ . Note conditional on $\mathfrak { F }$ , then by equation 8, $\begin{array} { r } { , V _ { 1 } ^ { \pi } ( s ) - V _ { 1 } ^ { \widehat { \pi } } ( s ) \leq \sum _ { h = 1 } ^ { H } 2 \cdot \mathbb { E } _ { \pi } [ \Gamma _ { h } ( s _ { h } , a _ { h } ) \mid s _ { 1 } = s ] } \end{array}$ holds for any policy $\pi$ almost surely. Therefore, + +$$ +\begin{array} { r l } & { \mathbb { P } \left[ \forall \pi , \ V _ { 1 } ^ { \pi } ( s ) - V _ { 1 } ^ { \widehat { \pi } } ( s ) \leq \displaystyle \sum _ { h = 1 } ^ { H } 2 \cdot \mathbb { E } _ { \pi } \big [ \Gamma _ { h } ( s _ { h } , a _ { h } ) \mid s _ { 1 } = s \big ] . \right] } \\ & { = \mathbb { P } \Bigg [ \forall \pi , \ V _ { 1 } ^ { \pi } ( s ) - V _ { 1 } ^ { \widehat { \pi } } ( s ) \leq \displaystyle \sum _ { h = 1 } ^ { H } 2 \cdot \mathbb { E } _ { \pi } \big [ \Gamma _ { h } ( s _ { h } , a _ { h } ) \mid s _ { 1 } = s \bigg ] \Bigg | \mathfrak { F } \Bigg ] \cdot \mathbb { P } [ \mathfrak { F } ] } \\ & { + \mathbb { P } \left[ \forall \pi , \ V _ { 1 } ^ { \pi } ( s ) - V _ { 1 } ^ { \widehat { \pi } } ( s ) \leq \displaystyle \sum _ { h = 1 } ^ { H } 2 \cdot \mathbb { R } _ { \pi } \big [ \Gamma _ { h } ( s _ { h } , a _ { h } ) \mid s _ { 1 } = s \big ] \Bigg | \mathfrak { F } ^ { \overline { { \kappa } } } \right] \cdot \mathbb { P } [ \mathfrak { F } ^ { c } ] } \\ & { \geq \mathbb { P } \Bigg [ \forall \pi , \ V _ { 1 } ^ { \pi } ( s ) - V _ { 1 } ^ { \widehat { \pi } } ( s ) \leq \displaystyle \sum _ { h = 1 } ^ { H } 2 \cdot \mathbb { E } _ { \pi } \big [ \Gamma _ { h } ( s _ { h } , a _ { h } ) \mid s _ { 1 } = s \big ] \Bigg | \mathfrak { F } \Bigg ] \cdot \mathbb { P } [ \mathfrak { F } ] \geq 1 \cdot \mathbb { P } [ \mathfrak { F } ] \geq 1 - \delta , } \end{array} +$$ + +which finishes the proof. + +# C.2 BOUNDING $\Big | ( { \mathcal T } _ { h } \widehat { V } _ { h + 1 } ) ( s , a ) - ( \widehat { { \mathcal T } } _ { h } \widehat { V } _ { h + 1 } ) ( s , a ) \Big | .$ + +By Lemma C.1, it remains to bound $| ( { \mathcal T } _ { h } \widehat { V } _ { h + 1 } ) ( s , a ) - ( \widehat { \mathcal T } _ { h } \widehat { V } _ { h + 1 } ) ( s , a ) |$ . Suppose $w _ { h }$ is the coefficient corresponding to the $\mathcal { T } _ { h } \widehat { V } _ { h + 1 }$ (such $w _ { h }$ exists by Lemma H.9), i.e. $\mathcal { T } _ { h } \widehat { V } _ { h + 1 } = \phi ^ { \top } w _ { h }$ , and recall + +$( \widehat { T } _ { h } \widehat { V } _ { h + 1 } ) ( s , a ) = \phi ( s , a ) ^ { \top } \widehat { w } _ { h }$ , then: + +$$ +\begin{array} { r l } & { \quad \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) ( s , a ) - \left( \widehat { \mathcal { T } } _ { h } \widehat { V } _ { h + 1 } \right) ( s , a ) = \phi ( s , a ) ^ { \top } ( w _ { h } - \widehat { w } _ { h } ) } \\ & { = \phi ( s , a ) ^ { \top } w _ { h } - \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \left( \displaystyle \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) \right) / \widehat { \sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \right) } \\ & { = \phi ( s , a ) ^ { \top } w _ { h } - \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \left( \displaystyle \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) / \widehat { \sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \right) } \\ & { { \overset { { , } } { , } } } \end{array} +$$ + +$$ ++ \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \left( \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \tau _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) / \widehat { \sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \right) . +$$ + +The term (i) is dealt by the following lemma. + +Lemma C.2. Recall $\kappa$ in Assumption 2.2. Suppose $\begin{array} { r } { K \ge \operatorname* { m a x } \left\{ 5 1 2 H ^ { 4 } / \kappa ^ { 2 } \log \left( \frac { 2 d } { \delta } \right) , 4 \lambda H ^ { 2 } / \kappa \right\} } \end{array}$ , then with probability $1 - \delta _ { : }$ , for all $s , a , h \in \mathcal S \times \mathcal A \times [ H ]$ + +$$ +\phi ( s , a ) ^ { \top } w _ { h } - \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \left( \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) / \widehat { \sigma } ^ { 2 } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) \Bigg | \leq \frac { 2 \lambda H ^ { 3 } \sqrt { d } / \kappa } { K } . +$$ + +Proof. Recall $\mathcal { T } _ { h } \widehat { V } _ { h + 1 } = \phi ^ { \top } w _ { h }$ and apply Lemma H.6, we obtain with probability $1 - \delta$ , for all $s , a , h \in \mathcal S \times \mathcal A \times [ H ]$ , + +$$ +\begin{array} { r l } & { \phi ( s , a ) ^ { \top } w _ { h } - \phi ( s , a ) ^ { \top } \widehat \Lambda _ { h } ^ { - 1 } \left( \displaystyle \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) / \widehat \sigma ^ { 2 } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) } \\ & { = \phi ( s , a ) ^ { \top } w _ { h } - \phi ( s , a ) ^ { \top } \widehat \Lambda _ { h } ^ { - 1 } \left( \displaystyle \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) ^ { \top } w _ { h } / \widehat \sigma ^ { 2 } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) } \\ & { = \phi ( s , a ) ^ { \top } w _ { h } - \phi ( s , a ) ^ { \top } \widehat \Lambda _ { h } ^ { - 1 } \left( \widehat { \Lambda } _ { h } - \lambda I \right) w _ { h } = \lambda \cdot \phi ( s , a ) ^ { \top } \widehat \Lambda _ { h } ^ { - 1 } w _ { h } } \\ & { \leq \lambda \| \phi ( s , a ) \| _ { \widehat \Lambda _ { h } ^ { - 1 } } \cdot \| w _ { h } \| _ { \widehat \Lambda _ { h } ^ { - 1 } } \leq \displaystyle \frac { \lambda } { K } \left\| \phi ( s , a ) \right\| _ { \{ \widehat \Lambda _ { h } ^ { p } } - 1 } \cdot \left\| w _ { h } \right\| _ { ( \widehat { \Lambda } _ { h } ^ { p } ) ^ { - 1 } } \\ & { \leq \displaystyle \frac { \lambda } { K } 1 \cdot \sqrt { \left\| ( \widehat { \Lambda } _ { h } ^ { p } ) ^ { - 1 } \right\| } \cdot 2 H \sqrt { d } \cdot \sqrt { \left\| ( \widehat { \Lambda } _ { h } ^ { p } ) ^ { - 1 } \right\| } } \end{array} +$$ + +where $\tilde { \Lambda } _ { h } ^ { p } : = \mathbb { E } _ { \mu , h } \left[ \widehat { \sigma } _ { h } ( s , a ) ^ { - 2 } \phi ( s , a ) \phi ( s , a ) ^ { \top } \right]$ and the second inequality is by Lemma H.6 (with $\phi ^ { \prime } ~ = ~ \phi / \widehat { \sigma } _ { h }$ and $\| \phi / { \widehat { \sigma } } _ { h } \| \ \leq \ \| \phi \| \ \leq \ 1 \ : = \ C )$ and the third inequality uses ${ \sqrt { a ^ { \top } \cdot A \cdot a } } \leq$ $\sqrt { \| a \| _ { 2 } \| A \| _ { 2 } \| a \| _ { 2 } } = \| a \| _ { 2 } \sqrt { \| A \| _ { 2 } }$ with $a$ to be either $\phi$ or $w _ { h }$ . Moreover, $\lambda _ { \operatorname* { m i n } } ( \tilde { \Lambda } _ { h } ^ { p } ) \geq$ $\kappa / \operatorname* { m a x } _ { h , s , a } \widehat { \sigma } _ { h } ( s , a ) ^ { 2 } \geq \kappa / H ^ { 2 }$ implies $\left\| ( \tilde { \Lambda } _ { h } ^ { p } ) ^ { - 1 } \right\| \leq H ^ { 2 } / \kappa$ , therefore for all $s , a , h \in \mathcal S \times \mathcal A \times [ H ]$ , with probability $1 - \delta$ + +$$ +\phi ( s , a ) ^ { \top } w _ { h } - \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \left( \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) / \widehat { \sigma } ^ { 2 } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) \Bigg | \leq \frac { 2 \lambda H ^ { 3 } \sqrt { d } / \kappa } { K } . +$$ + +For term (ii), denote: xτ = $\begin{array} { r } { x _ { \tau } = \frac { \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) } { \widehat { \sigma } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) } , \quad \eta _ { \tau } = \left( r _ { h } ^ { \tau } + { \widehat V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( { \mathcal T } _ { h } { \widehat V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) / { \widehat { \sigma } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) } , } \end{array}$ , bthen by Cauchy inequality it follows + +$$ +\begin{array} { l } { \displaystyle \left| \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \left( \displaystyle \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) / \widehat { \sigma } _ { h } ^ { 2 } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) \right| } \\ { \leq \sqrt { \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \phi ( s , a ) } \cdot \displaystyle \| \displaystyle \sum _ { \tau = 1 } ^ { K } x _ { \tau } \eta _ { \tau } \big \| _ { \widehat { \Lambda } _ { h } ^ { - 1 } } } \\ { \leq \sqrt { \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \phi ( s , a ) } \cdot \displaystyle \| \displaystyle \sum _ { \tau = 1 } ^ { K } x _ { \tau } \eta _ { \tau } \big \| _ { \widehat { \Lambda } _ { h } ^ { - 1 } } } \end{array} +$$ + +C.2.1 ANALYZING THE TERM $\sqrt { \phi ( s , a ) \widehat { \Lambda } _ { h } ^ { - 1 } \phi ( s , a ) }$ + +Recall (in Theorem 3.2) the estimated $\begin{array} { r } { \widehat { \Lambda } _ { h } = \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) ^ { \top } / \widehat { \sigma } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) + \lambda \cdot I } \end{array}$ and $\begin{array} { r } { \Lambda _ { h } = \sum _ { \tau = 1 } ^ { K } \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) ^ { \top } \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) / \sigma _ { \widehat { V } _ { h + 1 } } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) + \lambda I } \end{array}$ h h h h b h h . Then we have the following lemma to control the term $\sqrt { \phi ( s , a ) \widehat { \Lambda } _ { h } ^ { - 1 } \phi ( s , a ) }$ . + +Lemma C.3. Denote the quantities $C _ { 1 } \ = \ \operatorname* { m a x } \{ 2 \lambda , 1 2 8 \log ( 2 d / \delta ) , 1 2 8 H ^ { 4 } \log ( 2 d / \delta ) / \kappa ^ { 2 } \}$ and $\begin{array} { r } { C _ { 2 } = \operatorname* { m a x } \{ \frac { \lambda ^ { 2 } } { \kappa \log ( ( \lambda + K ) H / \lambda \delta ) } , 9 6 ^ { 2 } H ^ { 1 2 } d \log ( ( \lambda + K ) H / \lambda \delta ) / \kappa ^ { 5 } \} } \end{array}$ . Suppose the number of episode $K$ satisfies $K > \operatorname* { m a x } \{ C _ { 1 } , C _ { 2 } \}$ , then with probability $1 - \delta$ , + +$$ +\sqrt { \phi ( s , a ) \widehat \Lambda _ { h } ^ { - 1 } \phi ( s , a ) } \le 2 \sqrt { \phi ( s , a ) \Lambda _ { h } ^ { - 1 } \phi ( s , a ) } , \quad \forall s , a \in \mathcal { S } \times \mathcal { A } . +$$ + +Proof of Lemma C.3. By definition $\sqrt { \phi ( s , a ) \widehat { \Lambda } _ { h } ^ { - 1 } \phi ( s , a ) } = \| \phi ( s , a ) \| _ { \widehat { \Lambda } _ { h } ^ { - 1 } }$ . Then denote + +$$ +\widehat { \Lambda } _ { h } ^ { \prime } = \frac { 1 } { K } \widehat { \Lambda } _ { h } , \quad \Lambda _ { h } ^ { \prime } = \frac { 1 } { K } \Lambda _ { h } , +$$ + +where $\begin{array} { r } { \Lambda _ { h } \ = \ \sum _ { \tau = 1 } ^ { K } \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) ^ { \top } \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) / \sigma _ { \widehat { V } _ { h + 1 } } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) + \lambda I . } \end{array}$ . Under the condition of $K$ , by Lemma C.7, with probability $1 - \delta$ + +$$ +\begin{array} { r l } & { \left\| \widehat { \Lambda } _ { h } ^ { \prime } - \Lambda _ { h } ^ { \prime } \right\| \leq \underset { s , a } { \operatorname* { s u p } } \left\| \frac { \phi ( s , a ) \phi ( s , a ) ^ { \top } } { \widehat { \sigma } _ { h } ^ { 2 } ( s , a ) } - \frac { \phi ( s , a ) \phi ( s , a ) ^ { \top } } { \widehat { \sigma } _ { \widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \right\| } \\ & { \leq \underset { s , a } { \operatorname* { s u p } } \left| \frac { \widehat { \sigma } _ { h } ^ { 2 } ( s , a ) - \sigma _ { \widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { \widehat { \sigma } _ { h } ^ { 2 } ( s , a ) \sigma _ { \widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \right| \cdot \left\| \phi ( s , a ) \right\| ^ { 2 } \leq \underset { s , a } { \operatorname* { s u p } } \left| \frac { \widehat { \sigma } _ { h } ^ { 2 } ( s , a ) - \sigma _ { \widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { 1 } \right| \cdot 1 } \\ & { \leq 1 2 \sqrt { \frac { H ^ { 4 } d } { \kappa K } \log \left( \frac { \left( \lambda + K \right) H } { \lambda \delta } \right) } + 1 2 \lambda \frac { H ^ { 2 } \sqrt { d } } { \kappa K } . } \end{array} +$$ + +Next by Lemma H.5 (with $\phi$ to be $\phi / \sigma _ { \widehat { V } _ { h + 1 } }$ and $C = 1$ ), it holds with probability $1 - \delta$ , + +$$ +\left\| \Lambda _ { h } ^ { \prime } - \left( \mathbb { E } _ { \mu , h } [ \phi ( s , a ) \phi ( s , a ) ^ { \top } / \sigma _ { \hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) ] + \frac { \lambda } { K } I _ { d } \right) \right\| \leq \frac { 4 \sqrt { 2 } } { \sqrt { K } } \left( \log \frac { 2 d } { \delta } \right) ^ { 1 / 2 } . +$$ + +Therefore by Weyl’s spectrum theorem and the condition $K$ > $\operatorname* { m a x } \{ 2 \lambda , 1 2 8 \log ( 2 d / \delta ) , 1 2 8 H ^ { 4 } \log ( 2 d / \delta ) / \kappa ^ { 2 } \} .$ , the above implies + +$$ +\begin{array} { r l } & { \| \Lambda _ { t } ^ { \prime } \| = \lambda _ { \operatorname* { m a x } } ( N _ { h } ) \leq \lambda _ { \operatorname* { m a x } } \Big ( \mathrm { E } _ { \mu h } \big [ \phi ( s , a ) \phi ( s , a ) \big ] \Big ) + \frac { \lambda } { K } + \frac { 4 \sqrt { 2 } } { \sqrt { K } } \Big ( \log \frac { 2 d } { \delta } \Big ) ^ { 1 / 2 } } \\ & { \qquad - \Big \| \mathrm { E } _ { \mu h } [ \delta ( s , a ) \phi ( s , a ) ^ { \top } / \sigma _ { b \mathrm { t } \mathrm { \scriptsize \ ' \ ' } } ^ { 2 } ( s , a ) \mathbb { S } , a ] \Big \| _ { 2 } + \frac { \lambda } { K } + \frac { 4 \sqrt { 2 } } { \sqrt { K } } \Big ( \log \frac { 2 d } { \delta } \Big ) ^ { 1 / 2 } } \\ & { \qquad \leq \| \phi ( s , a ) \| ^ { 2 } + \frac { \lambda } { K } + \frac { 4 \sqrt { 2 } } { \sqrt { K } } \Big ( \log \frac { 2 d } { \delta } \Big ) ^ { 1 / 2 } \leq 1 + \frac { \lambda } { K } + \frac { 4 \sqrt { 2 } } { \sqrt { K } } \Big ( \log \frac { 2 d } { \delta } \Big ) ^ { 1 / 2 } \leq 2 , } \\ & { \qquad \quad \times _ { \operatorname* { m i n } } \big ( \Delta _ { h } ^ { \prime } \big ) \leq \lambda _ { \operatorname* { m i n } } \Big ( \mathrm { E } _ { \mu h } \big ) \big ( \tilde { \phi } ( s , a ) \phi ( s , a ) ^ { \top } / \sigma _ { b \mathrm { t } \mathrm { \scriptsize \ ' } + 1 } ^ { 2 } ( s , a ) \big ] \Big ) + \frac { \lambda } { K } - \frac { 4 \sqrt { 2 } } { \sqrt { K } } \Big ( \log \frac { 2 d } { \delta } \Big ) ^ { 1 / 2 } } \\ & { \qquad \quad \geq \lambda _ { \operatorname* { m i n } } \Big ( \mathrm { E } _ { \mu h } \big [ \phi ( s , a ) \phi ( s , a ) ^ { \top } / \sigma _ { b \mathrm { t } \mathrm { \scriptsize \ ' } + 1 } ^ { 2 } ( s , a ) \big ] \Big ) - \frac { 4 \sqrt { 2 } } { \sqrt { K } } \Big ( \log \frac { 2 d } { \delta } \Big ) ^ { 1 / 2 } } \\ & \qquad \quad \geq \frac { \kappa } { H ^ { 2 } } - \frac { 4 \sqrt { 2 } } { \sqrt { K } } \Big ( \log \frac \end{array} +$$ + +Hence with probability $1 - \delta , \| \Lambda _ { h } ^ { \prime } \| \leq 2$ and $\left\| \Lambda _ { h } ^ { \prime - 1 } \right\| = 1 / \lambda _ { \operatorname* { m i n } } ( \Lambda _ { h } ^ { \prime } ) \le 2 H ^ { 2 } / \kappa$ . Similarly, one can show $\left\| \widehat { \Lambda } _ { h } ^ { \prime - 1 } \right\| \leq 2 H ^ { 2 } / \kappa$ with high probability. + +Now apply Lemma H.4 to $\widehat { \Lambda } _ { h } ^ { \prime }$ and $\Lambda _ { h } ^ { \prime }$ and a union bound, we obtain with probability $1 - \delta$ , for all $s , a$ + +$$ +\begin{array} { r l } { \left\| \phi ( s , a ) \right\| _ { \widetilde { \Lambda } _ { h } ^ { - 1 } } \leq \left[ 1 + \sqrt { \left\| \Lambda _ { h } ^ { \prime - 1 } \right\| \left\| \Lambda _ { h } ^ { \prime } \right\| } \cdot \left\| \hat { \Lambda } _ { h } ^ { \prime - 1 } \right\| \cdot \left\| \hat { \Lambda } _ { h } ^ { \prime } - \Lambda _ { h } ^ { \prime } \right\| \right] \cdot \left\| \phi ( s , a ) \right\| _ { \Lambda _ { h } ^ { \prime - 1 } } } & { } \\ { \leq \left[ 1 + \sqrt { \frac { 2 H ^ { 2 } } { \kappa } } \cdot 1 \cdot \frac { 2 H ^ { 2 } } { \kappa } \cdot \left\| \hat { \Lambda } _ { h } ^ { \prime } - \Lambda _ { h } ^ { \prime } \right\| \right] \cdot \left\| \phi ( s , a ) \right\| _ { \Lambda _ { h } ^ { \prime - 1 } } } & { } \\ { \leq \left[ 1 + \sqrt { \frac { 4 \Delta H ^ { 4 } } { \kappa ^ { 2 } } \left( \sqrt { \frac { H ^ { 4 } d } { \kappa K } } \log \left( \frac { ( \lambda + K ) H } { \lambda \delta } \right) + \lambda \frac { H ^ { 2 } \sqrt { d } } { \kappa K } \right) } \right] \cdot \left\| \phi ( s , a ) \right\| _ { \Lambda _ { h } ^ { \prime - 1 } } } & { } \\ { \leq \left[ 1 + \sqrt { \frac { 9 6 H ^ { 4 } } { \kappa ^ { 2 } } \sqrt { \frac { H ^ { 4 } d } { \kappa K } } \log \left( \frac { ( \lambda + K ) H } { \lambda \delta } \right) } \right] \cdot \left\| \phi ( s , a ) \right\| _ { \Lambda _ { h } ^ { \prime - 1 } } \leq 2 \left\| \phi ( s , a ) \right\| _ { \Lambda _ { h } ^ { \prime - 1 } } } & { } \end{array} +$$ + +where the third inequality uses equation 11 and the last and the second last inequality use $\begin{array} { r } { K > \operatorname* { m a x } \lbrace \frac { \lambda ^ { 2 } } { \kappa \log ( ( \lambda + K ) H / \lambda \delta ) } , 9 6 ^ { 2 } H ^ { 1 2 } d \log ( ( \lambda + K ) H / \lambda \delta ) / \kappa ^ { 5 } \rbrace . } \end{array}$ . Note the above is equivalent to $\sqrt { \phi ( s , a ) \widehat \Lambda _ { h } ^ { - 1 } \phi ( s , a ) } \le 2 \sqrt { \phi ( s , a ) \Lambda _ { h } ^ { - 1 } \phi ( s , a ) }$ by multiplying $1 / \sqrt { K }$ on both sides. + +C.2.2 ANALYZING THE TERM $\begin{array} { r } { | | \sum _ { \tau = 1 } ^ { K } x _ { \tau } \eta _ { \tau } | | _ { \widehat { \Lambda } ^ { - 1 } } } \end{array}$ + +Lemma C.4. Recall $\begin{array} { r } { x _ { \tau } = \frac { \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) } { \widehat { \sigma } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) } } \end{array}$ $\begin{array} { r } { \frac { a _ { h } ^ { \tau } } { a _ { h } ^ { \tau } ) } a n d \eta _ { \tau } = \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \right) / \widehat { \sigma } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) . } \end{array}$ . Let $\begin{array} { r } { C _ { H , d , \kappa , K } : = 3 6 \sqrt { \frac { H ^ { 4 } d ^ { 3 } } { \kappa } \log \left( \frac { ( \lambda + K ) 2 K d H ^ { 2 } } { \lambda \delta } \right) } + 1 2 \lambda \frac { H ^ { 2 } \sqrt { d } } { \kappa } } \end{array}$ and denote + +$$ +\xi : = \operatorname* { s u p } _ { V \in \left[ 0 , H \right] , s ^ { \prime } \sim P _ { h } \left( s , a \right) , h \in \left[ H \right] } \left| \frac { r _ { h } + V \left( s ^ { \prime } \right) - \left( \mathcal T _ { h } V \right) \left( s , a \right) } { \sigma _ { V } \left( s , a \right) } \right| . +$$ + +If $K \ge 4 C _ { H , d , \kappa , K } ^ { 2 }$ and $K \ge \widetilde { \cal O } ( H ^ { 6 } d / \kappa ) .$ , then with probability $1 - \delta$ , + +$$ +\left\| \sum _ { \tau = 1 } ^ { K } x _ { \tau } \eta _ { \tau } \right\| _ { \widehat { \Lambda } ^ { - 1 } } \leq 1 6 \sqrt { d \log \left( 1 + \frac { K } { \lambda d } \right) } \cdot \log \left( \frac { 4 K ^ { 2 } } { \delta } \right) + 4 \xi \log \left( \frac { 4 K ^ { 2 } } { \delta } \right) \leq \widetilde { O } \operatorname* { m a x } \left\{ \sqrt { d } , \xi \right\} , +$$ + +where $\widetilde O$ absorbs the constants and Polylog terms. + +Proof of Lemma C.4. By construction, we have $\| x _ { \tau } \| \le \| \phi / \widehat { \sigma } \| \le 1$ and by Lemma C.7, with probability $1 - \delta / 3$ , + +$$ +\left\| \sigma _ { \widehat { V } _ { h + 1 } } - \widehat { \sigma } _ { h } \right\| _ { \infty } = \operatorname* { s u p } _ { s , a } \left| \frac { \sigma _ { \widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) - \widehat { \sigma } _ { h } ^ { 2 } ( s , a ) } { \sigma _ { \widehat { V } _ { h + 1 } } ( s , a ) + \widehat { \sigma } _ { h } ( s , a ) } \right| \leq \frac { 1 } { 2 } \left\| \sigma _ { \widehat { V } _ { h + 1 } } ^ { 2 } - \widehat { \sigma } _ { h } ^ { 2 } \right\| _ { \infty } \leq C _ { H , d , \kappa , K } \sqrt { \frac { 1 } { K } } +$$ + +Therefore, when $K \geq 4 C _ { H , d , \kappa , K } ^ { 2 } , C _ { H , d , \kappa , K } \sqrt { \frac { 1 } { K } } \leq 1 / 2 \leq \sigma _ { \widehat { V } _ { h + 1 } } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) / 2$ and hence + +$$ +\begin{array} { r l } & { \eta _ { \tau } | \leq \left| \frac { r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) } { \sigma _ { \widehat { V } _ { h + 1 } } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) - \frac { C _ { H , d , \kappa , K } } { K ^ { 1 / 2 } } } \right| \leq 2 \left| \frac { r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) } { \sigma _ { \widehat { V } _ { h + 1 } } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) } \right| } \\ & { \quad \leq 2 \underset { V \in [ 0 , H ] , s ^ { \prime } \sim P _ { h } ( s , a ) } { \operatorname* { s u p } } \left| \frac { r + V \left( s ^ { \prime } \right) - \left( \mathcal { T } _ { h } V \right) \left( s , a \right) } { \sigma _ { V } \left( s , a \right) } \right| : = \xi . } \end{array} +$$ + +Next, for a fixed function $V$ , we define the Bellman error as $B _ { h } ( V ) ( s , a ) = r _ { h } + V ( s ^ { \prime } ) - ( T _ { h } V ) ( s , a )$ , then + +$$ +\begin{array} { r l } & { - \frac { \nabla \cdot \mathbf { S } _ { \perp + } [ \hat { \mathbf { x } } _ { \perp } + \hat { \mathbf { x } } _ { \perp } - ( \hat { \mathbf { x } } _ { \perp } - [ ( \hat { \mathbf { x } } _ { \perp } \hat { \mathbf { e } } _ { \perp + } ^ { - \mathbf { A } } ) ] ) \cdot \mathbf { \bar { x } } _ { \perp } \hat { \mathbf { e } } _ { \perp } ^ { - \mathbf { A } } ] \cdot [ \hat { \mathbf { x } } _ { \perp - } ] } { \hat { \mathcal { Q } } _ { \perp + } [ \hat { \mathbf { x } } _ { \perp } + \hat { \mathbf { x } } _ { \perp } ] } } \\ & { - \frac { \nabla \cdot \mathbf { S } _ { \perp + } [ \hat { \mathbf { x } } _ { \perp } + \hat { \mathbf { x } } _ { \perp } - \hat { \mathbf { x } } _ { \perp } ] } { 2 } \cdot \frac { \partial \hat { \mathbf { x } } _ { \perp + } [ \hat { \mathbf { x } } _ { \perp } + \hat { \mathbf { x } } _ { \perp } ] \cdot ( \hat { \mathbf { x } } _ { \perp } - \hat { \mathbf { x } } _ { \perp } ) } { \partial \hat { \mathbf { x } } _ { \perp } [ \hat { \mathbf { x } } _ { \perp } + \hat { \mathbf { x } } _ { \perp } ] } [ \hat { \mathbf { x } } _ { \perp } \hat { \mathbf { e } } _ { \perp + } ^ { \mathbf { A } } ] \hat { \mathcal { Q } } _ { \perp - } \Bigg ] } \\ & \phantom { = } \frac { \nabla \cdot \mathbf { S } _ { \perp + } [ \hat { \mathbf { x } } _ { \perp + } ^ { \mathbf { B } } \cdot \mathbf { \bar { x } } _ { \perp } \hat { \mathbf { e } } _ { \perp + } ^ { \mathbf { B } } ] \cdot [ \hat { \mathbf { x } } _ { \perp } \hat { \mathbf { e } } _ { \perp + } ^ { \mathbf { B } } ] \cdot [ \hat { \mathbf { x } } _ { \perp - } \mathbf { \bar { x } } _ { \perp } \hat { \mathbf { x } } _ { \perp } \hat { \mathbf { x } } _ { \perp } ] \hat { \mathcal { Q } } _ { \perp - } } \\ & \phantom { = } \frac \nabla \cdot \mathbf { S } _ { \perp + } [ \hat { \mathbf { x } } _ \ \end{array} +$$ + +where the first inequality is by Lemma H.11, the second inequality is by $\mathcal { T } _ { h }$ is non-expansive, the third inequality is by Lemma C.8, the next equality is by Markovian property, and the fourth inequality is by Lemma C.7 and Lemma C.10. The fifth inequality uses definition $\sigma _ { h , V } ( s , a ) ^ { 2 } : =$ $\operatorname* { m a x } \{ 1 , \operatorname { V a r } _ { P _ { h } } ( V ) ( s , a ) \}$ and the last one is by condition $K \ge \widetilde { O } ( H ^ { 6 } d / \kappa )$ and $\sigma _ { h , V ^ { \star } } ( s , a ) ^ { 2 } : =$ $\operatorname* { n a x } \{ 1 , \operatorname { V a r } _ { P _ { h } } ( V ^ { \star } ) ( s , a ) \} \ \geq \ 1$ . Thus, by Bernstein inequality for self-normalized martingale + +(Lemma H.3),9 with probability $1 - \delta$ , + +$$ +\left\| \sum _ { \tau = 1 } ^ { K } x _ { \tau } \eta _ { \tau } \right\| _ { \widehat { \Lambda } ^ { - 1 } } \leq \widetilde { \mathcal { O } } \left( \sqrt { d \log \left( 1 + \frac { K } { \lambda d } \right) } \cdot \log \left( \frac { 4 K ^ { 2 } } { \delta } \right) \right) + 4 \xi \log \left( \frac { 4 K ^ { 2 } } { \delta } \right) \leq \widetilde { \mathcal { O } } \operatorname* { m a x } \left\{ \sqrt { d } , \xi \right\} +$$ + +where $\widetilde O$ absorbs the constants and Polylog terms. + +Recall $\mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } , \mathcal { M } _ { 3 } , \mathcal { M } _ { 4 }$ in List A. Based on the above results, we have the following key lemma: + +Lemma C.5. Assume $K > \operatorname* { m a x } \{ \mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } , \mathcal { M } _ { 3 } , \mathcal { M } _ { 4 } \}$ , for any $0 < \lambda < \kappa ,$ , suppose $\sqrt { d } > \xi$ , $\xi : = \operatorname* { s u p } _ { V \in [ 0 , H ] , \ s ^ { \prime } \sim P _ { h } ( s , a ) , \ h \in [ H ] } \left| \frac { r _ { h } + V \big ( s ^ { \prime } \big ) - ( T _ { h } V ) ( s , a ) } { \sigma _ { V } ( s , a ) } \right| .$ Then with probability $1 - \delta _ { : }$ , for all $h , s , a \in [ H ] \times \mathcal { S } \times \mathcal { A } ,$ , + +$$ +\left| ( { \cal T } _ { h } \widehat V _ { h + 1 } - \widehat T _ { h } \widehat V _ { h + 1 } ) ( s , a ) \right| \leq \widetilde O \left( \sqrt { d } \sqrt { \phi ( s , a ) \Lambda _ { h } ^ { - 1 } \phi ( s , a ) } \right) + \frac { 2 H ^ { 3 } \sqrt { d } } { K } , +$$ + +where $\begin{array} { r } { \Lambda _ { h } = \sum _ { \tau = 1 } ^ { K } \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) ^ { \top } \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) / \sigma _ { \widehat { V } _ { h + 1 } } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) + \lambda I } \end{array}$ and $\widetilde O$ absorbs the universal constants and Polylog terms. + +Proof of Lemma C.5. Combing equation 9, Lemma C.2, equation 10, Lemma C.3 and C.4 and a union bound to finish the proof. □ + +# C.3 PROOF OF THE FIRST PART OF THEOREM 3.2 + +Theorem C.6 (First part of Theorem 3.2). Let $K$ be the number of episodes. Suppose $\sqrt { d } > \xi$ , where $\begin{array} { r } { \xi : = \operatorname* { s u p } _ { V \in [ 0 , H ] , \ s ^ { \prime } \sim P _ { h } ( s , a ) , \ h \in [ H ] } \left| \frac { r _ { h } + V \left( s ^ { \prime } \right) - ( \mathcal T _ { h } V ) ( s , a ) } { \sigma _ { V } ( s , a ) } \right| } \end{array}$ and $K > \operatorname* { m a x } \{ \mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } , \mathcal { M } _ { 3 } , \mathcal { M } _ { 4 } \} ^ { 1 0 }$ . Then for any $0 < \lambda < \kappa$ , with probability $1 - \delta$ , for all policy $\pi$ simultaneously, the output $\widehat { \pi }$ of Algorithm $^ { l }$ satisfies + +$$ +v ^ { \pi } - v ^ { \widehat { \pi } } \leq \widetilde { O } \left( \sqrt { d } \cdot \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \pi } \left[ \left( \phi ( \cdot , \cdot ) ^ { \top } \Lambda _ { h } ^ { - 1 } \phi ( \cdot , \cdot ) \right) ^ { 1 / 2 } \right] \right) + \frac { 2 H ^ { 4 } \sqrt { d } } { K } +$$ + +where $\begin{array} { r } { \Lambda _ { h } = \sum _ { \tau = 1 } ^ { K } \frac { \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \cdot \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) ^ { \top } } { \sigma _ { \hat { V } _ { h + 1 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) } ^ { 2 } } + \lambda I _ { d } } \end{array}$ 1 φ(sτh,aτh)·φ(sτh,aτh)>2 + λId and O absorbs the universal constants and the Polylog terms. + +Proof of Theorem C.6. Combing Lemma C.1 and Lemma C.5, we directly have with probability $1 - \delta$ , for all policy $\pi$ simultaneously, + +$$ +V _ { 1 } ^ { \pi } ( s ) - V _ { 1 } ^ { \widehat { \pi } } ( s ) \leq \widetilde { O } \left( \sqrt { d } \cdot \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \pi } \left[ \left( \phi ( \cdot , \cdot ) ^ { \top } \Lambda _ { h } ^ { - 1 } \phi ( \cdot , \cdot ) \right) ^ { 1 / 2 } \middle | s _ { 1 } = s \right] \right) + \frac { 2 H ^ { 4 } \sqrt { d } } { K } , +$$ + +now take the initial distribution $d _ { 1 }$ on both sides to get the stated result. + +# C.4 TWO INTERMEDIATE RESULTS + +The next two lemmas provide intermediate results in finishing the whole proofs. + +# C.4.1 BOUNDING THE VARIANCE + +Lemma C.7. Recall the definition $\widehat { \sigma } _ { h } ( \cdot , \cdot ) ^ { 2 } = \operatorname* { m a x } \{ 1 , \widehat { \mathrm { V a r } } _ { P _ { h } } \widehat { V } _ { h + 1 } ( \cdot , \cdot ) \} + 1$ and $\sigma _ { \widehat { V } _ { h + 1 } } ( \cdot , \cdot ) ^ { 2 } : =$ $\operatorname * { m a x } \{ 1 , \mathrm { V a r } _ { P _ { h } } \widehat { V } _ { h + 1 } ( \cdot , \cdot ) \} \ + \ 1 .$ Moreover, $\begin{array} { r l r } { \left[ \widehat { \mathrm { V a r } } _ { h } \widehat { V } _ { h + 1 } \right] ( \cdot , \cdot ) } & { { } = } & { \big \langle \phi ( \cdot , \cdot ) , \bar { \beta } _ { h } \big \rangle _ { \left[ 0 , ( H - h + 1 ) ^ { 2 } \right] } - } \end{array}$ $\left[ \left. \phi ( \cdot , \cdot ) , \bar { \theta } _ { h } \right. _ { [ 0 , H - h + 1 ] } \right] ^ { 2 }$ (where $\bar { \beta } _ { h }$ and ${ \bar { \theta } } _ { h }$ are defined in Algorithm 1). Let $K \_ { \mathbf { \alpha } } \geq$ max $\textstyle \left\{ 5 1 2 ( 1 / \kappa ) ^ { 2 } \log \left( { \frac { 4 H d } { \delta } } \right) , 4 \lambda / \kappa \right\}$ , then with probability $1 - \delta$ , + +$$ +\operatorname* { s u p } _ { h } \lvert | \widehat { \sigma } _ { h } ^ { 2 } - \sigma _ { \widehat { V } _ { h + 1 } } ^ { 2 } | | _ { \infty } \leq 3 6 \sqrt { \frac { H ^ { 4 } d ^ { 3 } } { \kappa K } \log \left( \frac { ( \lambda + K ) 2 K d H ^ { 2 } } { \lambda \delta } \right) } + 1 2 \lambda \frac { H ^ { 2 } \sqrt { d } } { \kappa K } . +$$ + +Proof. Step1: we first show for all $h , s , a \in [ H ] \times \mathcal { S } \times \mathcal { A }$ , with probability $1 - \delta$ + +$$ +\langle \phi ( s , a ) , \bar { \beta } _ { h } \rangle _ { [ 0 , ( H - h + 1 ) ^ { 2 } ] } - \mathbb { P } _ { h } ( \widehat { V } _ { h + 1 } ) ^ { 2 } ( s , a ) \bigg | \leq 1 2 \sqrt { \frac { H ^ { 4 } d ^ { 3 } } { \kappa K } \log \left( \frac { ( \lambda + K ) 2 K d H ^ { 2 } } { \lambda \delta } \right) } + 4 \lambda \frac { H ^ { 2 } \sqrt { d } } { \kappa K } . +$$ + +# Proof of Step1. Note + +$$ +\begin{array} { r l } & { | \phi ( s , a ) , \widetilde { \beta } _ { h } [ 0 , ( J - h + 1 ) ^ { 2 } ] - \widetilde { \Gamma } _ { h } ( \widehat { V } _ { h + 1 } ) ^ { 2 } ( s , a ) | \leq | \phi ( s , a ) , \widetilde { \beta } _ { h } - \widetilde { \Gamma } _ { h } ( \widehat { V } _ { h + 1 } ) ^ { 2 } ( s , a ) | } \\ & { = | \phi ( s , a ) ^ { \top } \widetilde { \Sigma } _ { h } ^ { - 1 } \displaystyle \sum _ { \tau = 1 } ^ { K } \phi ( \widehat { s } _ { h } ^ { \top } , \widetilde { a } _ { h } ^ { \top } ) \cdot \widehat { V } _ { h + 1 } ( \widehat { s } _ { h + 1 } ^ { \top } ) ^ { 2 } - \widetilde { \Gamma } _ { h } ( \widehat { V } _ { h + 1 } ) ^ { 2 } ( s , a ) | } \\ & { = | \phi ( s , a ) ^ { \top } \widetilde { \Sigma } _ { h } ^ { - 1 } \displaystyle \sum _ { \tau = 1 } ^ { K } \phi ( \widehat { s } _ { h } ^ { \top } , \widetilde { a } _ { h } ^ { \top } ) \cdot \widehat { V } _ { h + 1 } ( \widehat { s } _ { h + 1 } ^ { \top } ) ^ { 2 } - \phi ( s , a ) ^ { \top } \int _ { \gamma } ( \widehat { V } _ { h + 1 } ) ^ { 2 } ( s ^ { \top } ) d r _ { h } ( s ^ { \top } ) | } \\ & { = | \phi ( s , a ) ^ { \top } \widetilde { \Sigma } _ { h } ^ { - 1 } \displaystyle \sum _ { \tau = 1 } ^ { K } \phi ( \widehat { s } _ { h } ^ { \top } , a ) ^ { \top } \cdot \widehat { V } _ { h + 1 } ( \widehat { s } _ { h + 1 } ^ { \top } ) ^ { 2 } - \phi ( s , a ) ^ { \top } \displaystyle \sum _ { \tau = 1 } ^ { K } \phi ( \widehat { s } _ { h } ^ { \top } , a _ { h } ^ { \top } ) ( \phi ( \widehat { s } _ { h } ^ { \top } , a ) ^ { \top } ) ^ { \top } + \lambda I ) \int _ { \mathcal { O } } ( \widehat { V } _ { h + 1 } ) ^ { 2 } ( s ^ { \top } ) d r _ { h } } \\ & \ : \sum _ { \tau = 1 } ^ { K } | \phi ( s , a ) ^ { \top } \widetilde { \Sigma } _ h \end{array} +$$ + +For $\textcircled{2}$ , since $\begin{array} { r } { K \ge \operatorname* { m a x } \left\{ 5 1 2 ( 1 / \kappa ) ^ { 2 } \log \left( \frac { 4 H d } { \delta } \right) , 4 \lambda / \kappa \right\} } \end{array}$ , by Lemma H.6 and a union bound over $h \in [ H ]$ , with probability $1 - \delta$ for all $h , s , a \in [ H ] \times \mathcal { S } \times \mathcal { A }$ , + +$$ +\begin{array} { l } { \displaystyle \mathfrak { D } \le \lambda \| \phi ( s , a ) \| _ { \bar { \Sigma } _ { h } ^ { - 1 } } \left\| \int _ { S } ( \widehat { V } _ { h + 1 } ) ^ { 2 } ( s ^ { \prime } ) d \nu _ { h } ( s ^ { \prime } ) \right\| _ { \bar { \Sigma } _ { h } ^ { - 1 } } } \\ { \displaystyle \quad \le \lambda \frac { 2 } { \sqrt { K } } \| \phi ( s , a ) \| _ { ( \Sigma _ { h } ^ { p } ) ^ { - 1 } } \frac { 2 } { \sqrt { K } } \left\| \int _ { S } ( \widehat { V } _ { h + 1 } ) ^ { 2 } ( s ^ { \prime } ) d \nu _ { h } ( s ^ { \prime } ) \right\| _ { ( \Sigma _ { h } ^ { p } ) ^ { - 1 } } \le 4 \lambda \left\| ( \Sigma _ { h } ^ { p } ) ^ { - 1 } \right\| \frac { H ^ { 2 } \sqrt { d } } { K } \le 4 \lambda \frac { H ^ { 2 } \vee \lambda } { \kappa K } } \end{array} +$$ + +For $\textcircled{1}$ , we have + +$$ +\mathbb { O } \leq \| \phi ( s , a ) \| _ { \bar { \Sigma } _ { h } ^ { - 1 } } \left\| \sum _ { \tau = 1 } ^ { K } \phi ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } ) \cdot \Big ( \widehat { V } _ { h + 1 } ( \bar { s } _ { h + 1 } ^ { \tau } ) ^ { 2 } - \mathbb { P } _ { h } ( \widehat { V } _ { h + 1 } ) ^ { 2 } ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } ) \Big ) \right\| _ { \bar { \Sigma } _ { h } ^ { - 1 } } +$$ + +Bounding using covering. Note for any fix $V _ { h + 1 }$ , we can define $x _ { \tau } = \phi ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } )$ $( \| \phi \| _ { 2 } \leq 1 )$ and $\eta _ { \tau } = V _ { h + 1 } ( \bar { s } _ { h + 1 } ^ { \tau } ) ^ { 2 } - \mathbb { P } _ { h } ( V _ { h + 1 } ) ^ { 2 } ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } )$ is $H ^ { 2 }$ -subgaussian, by Lemma H.2 (where $t = K$ and $L = 1$ ) with probability $1 - \delta$ , + +$$ +\left\| \sum _ { \tau = 1 } ^ { K } \phi ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } ) \cdot \left( V _ { h + 1 } \big ( \bar { s } _ { h + 1 } ^ { \tau } \big ) ^ { 2 } - \mathbb { P } _ { h } \big ( V _ { h + 1 } \big ) ^ { 2 } \big ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } \big ) \right) \right\| _ { \bar { \Sigma } _ { h } ^ { - 1 } } \leq \sqrt { 8 H ^ { 4 } \cdot \frac d 2 \log \left( \frac { \lambda + K } { \lambda \delta } \right) } +$$ + +let $\mathcal { N } _ { h } ( \epsilon )$ be the minimal $\epsilon$ -cover (with respect the supremum norm) of $\mathcal { V } _ { h } : = \{ V _ { h } : V _ { h } ( \cdot ) =$ $\begin{array} { r } { \operatorname* { m a x } _ { a \in { \cal A } } \{ \operatorname* { m i n } \{ \phi ( s , a ) ^ { \top } \theta - C _ { 1 } \sqrt { d \cdot \phi ( \cdot , \cdot ) ^ { \top } \widehat \Lambda _ { h } ^ { - 1 } \phi ( \cdot , \cdot ) } - C _ { 2 } , H - h + 1 \} ^ { + } \} \} . } \end{array}$ . That is, for any + +$V \in \mathcal { V } _ { h }$ , there exists a value function $V ^ { \prime } \in \mathcal { N } _ { h } ( \epsilon )$ such that $\begin{array} { r } { \operatorname* { s u p } _ { s \in \mathcal { S } } | V ( s ) - V ^ { \prime } ( s ) | < \epsilon . } \end{array}$ . Now by a union bound, we obtain with probability $1 - \delta$ + +$$ +\operatorname* { s u p } _ { \tilde { h } + 1 \in \mathcal { N } _ { h + 1 } ( \epsilon ) } \left\| \sum _ { \tau = 1 } ^ { K } \phi ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } ) \cdot \left( V _ { h + 1 } ( \bar { s } _ { h + 1 } ^ { \tau } ) ^ { 2 } - \mathbb { P } _ { h } ( V _ { h + 1 } ) ^ { 2 } ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } ) \right) \right\| _ { \bar { \Sigma } _ { h } ^ { - 1 } } \leq \sqrt { 8 H ^ { 4 } \cdot \frac { d } { 2 } \log \left( \frac { \lambda + K } { \lambda \delta } | \mathcal { N } _ { h + 1 } ( 6 , 7 ) | \bar { \mu } _ { h } ( 7 ) | \right) } +$$ + +which implies + +$$ +\begin{array} { r l } & { \displaystyle \left\| \displaystyle \sum _ { \tau = 1 } ^ { K } \phi ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } ) \cdot \Big ( \widehat { V } _ { h + 1 } ( \bar { s } _ { h + 1 } ^ { \tau } ) ^ { 2 } - \mathbb { P } _ { h } ( \widehat { V } _ { h + 1 } ) ^ { 2 } ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } ) \Big ) \right\| _ { \bar { \Sigma } _ { h } ^ { - 1 } } } \\ & { \displaystyle \leq \sqrt { 8 H ^ { 4 } \cdot \frac { d } { 2 } \log \Big ( \frac { \lambda + K } { \lambda \delta } | \mathcal { N } _ { h + 1 } ( \epsilon ) | \Big ) } + 4 H ^ { 2 } \sqrt { \epsilon ^ { 2 } K ^ { 2 } / \lambda } } \end{array} +$$ + +choosing $\epsilon = d \sqrt { \lambda } / K$ , applying Lemma B.3 of Jin et al. (2021b)11 to the covering number $\mathcal { N } _ { h + 1 } ( \epsilon )$ w.r.t. $\nu _ { h + 1 }$ , we can further bound above by + +$$ +\leq \sqrt { 8 H ^ { 4 } \cdot \frac { d ^ { 3 } } { 2 } \log \left( \frac { \lambda + K } { \lambda \delta } 2 d H K \right) } + 4 H ^ { 2 } \sqrt { d ^ { 2 } } \leq 6 \sqrt { H ^ { 4 } \cdot d ^ { 3 } \log \left( \frac { \lambda + K } { \lambda \delta } 2 d H K \right) } +$$ + +Apply a union bound for $h \in [ H ]$ , we have with probability $1 - \delta$ , for all $h \in [ H ]$ , + +$$ +\left| \sum _ { \tau = 1 } ^ { K } \phi ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } ) \cdot \left( \widehat { V } _ { h + 1 } ( \bar { s } _ { h + 1 } ^ { \tau } ) ^ { 2 } - \mathbb { P } _ { h } ( \widehat { V } _ { h + 1 } ) ^ { 2 } ( \bar { s } _ { h } ^ { \tau } , \bar { a } _ { h } ^ { \tau } ) \right) \right| _ { \bar { \Sigma } _ { h } ^ { - 1 } } \leq 6 \sqrt { H ^ { 4 } d ^ { 3 } \log \left( \frac { ( \lambda + K ) 2 K d H ^ { 2 } } { \lambda \delta } \right) } +$$ + +and similar to $\textcircled{2}$ , with probability $1 - \delta$ for all $h , s , a \in [ H ] \times \mathcal { S } \times \mathcal { A }$ , + +$$ +\left. \phi ( s , a ) \right. _ { \bar { \Sigma } _ { h } ^ { - 1 } } \leq \frac { 2 \left. ( \Sigma _ { h } ^ { p } ) ^ { - 1 } \right. ^ { 1 / 2 } } { \sqrt { K } } \leq \frac { 2 } { \sqrt { \kappa K } } . +$$ + +Combing equation 13, equation 14, equation 15 and equation 16 we obtain with probability $1 - \delta$ for all $h , s , a \in [ H ] \times \mathcal { S } \times \mathcal { A }$ , + +$$ +\langle \phi ( s , a ) , \bar { \beta } _ { h } \rangle _ { [ 0 , ( H - h + 1 ) ^ { 2 } ] } - \mathbb { P } _ { h } ( \widehat { V } _ { h + 1 } ) ^ { 2 } ( s , a ) \bigg | \leq 1 2 \sqrt { \frac { H ^ { 4 } d ^ { 3 } } { \kappa K } \log \left( \frac { ( \lambda + K ) 2 K d H ^ { 2 } } { \lambda \delta } \right) } + 4 \lambda \frac { H ^ { 2 } \sqrt { d } } { \kappa K } . +$$ + +Step2: we show for all $h , s , a \in [ H ] \times \mathcal { S } \times \mathcal { A }$ , with probability $1 - \delta$ + +$$ +\left| \langle \phi ( s , a ) , \bar { \theta } _ { h } \rangle _ { [ 0 , H - h + 1 ] } - \mathbb { P } _ { h } ( \widehat { V } _ { h + 1 } ) ( s , a ) \right| \leq 1 2 \sqrt { \frac { H ^ { 2 } d ^ { 3 } } { \kappa K } \log \left( \frac { ( \lambda + K ) 2 K d H ^ { 2 } } { \lambda \delta } \right) } + 4 \lambda \frac { H \sqrt { d } } { \kappa K } . +$$ + +The proof of Step2 follows nearly the identical way as Step1 except $\widehat { V } _ { h } ^ { 2 }$ is replaced by $\widehat { V } _ { h }$ . + +$$ +\begin{array} { r } { \mathbf { S t e p 3 : W e ~ p r o v e ~ s u p } _ { h } \vert \vert \widehat \sigma _ { h } ^ { 2 } - \sigma _ { \widehat { V } _ { h } } ^ { 2 } \vert \vert _ { \infty } \leq 3 6 \sqrt { \frac { H ^ { 4 } d ^ { 3 } } { \kappa K } \log \left( \frac { ( \lambda + K ) 2 K d H ^ { 2 } } { \lambda \delta } \right) } + 1 2 \lambda \frac { H ^ { 2 } \sqrt { d } } { \kappa K } . } \end{array} +$$ + +Proof of Step3. By equation 17, + +$$ +\begin{array} { r l } & { \quad \left| \left[ \big < \phi ( \cdot , \cdot ) , \bar { \theta } _ { h } \big > _ { [ 0 , H - h + 1 ] } \right] ^ { 2 } - \left[ \mathbb { P } _ { h } ( \widehat { V } _ { h + 1 } ) ( s , a ) \right] ^ { 2 } \right| } \\ & { = \left| \big < \phi ( s , a ) , \bar { \theta } _ { h } \big > _ { [ 0 , H - h + 1 ] } + \mathbb { P } _ { h } ( \widehat { V } _ { h + 1 } ) ( s , a ) \right| \cdot \left| \big < \phi ( s , a ) , \bar { \theta } _ { h } \big > _ { [ 0 , H - h + 1 ] } - \mathbb { P } _ { h } ( \widehat { V } _ { h + 1 } ) ( s , a ) \right| } \\ & { \leq 2 H \cdot \left| \big < \phi ( s , a ) , \bar { \theta } _ { h } \big > _ { [ 0 , H - h + 1 ] } - \mathbb { P } _ { h } ( \widehat { V } _ { h + 1 } ) ( s , a ) \right| \leq 2 4 \sqrt { \frac { H ^ { 4 } d ^ { 3 } } { \kappa K } \log \left( \frac { \big ( \lambda + K \big ) 2 K d H ^ { 2 } } { \lambda \delta } \right) } + 8 \lambda \frac { H ^ { 2 } \sqrt { \lambda + 1 } } { \kappa K } } \end{array} +$$ + +Combining this with Step1 we receive $\forall h , s , a \in [ H ] \times S \times A$ , with probability $1 - \delta$ + +$$ +\left| \widehat { \mathrm { V a r } } _ { h } \widehat { V } _ { h + 1 } ( s , a ) - \mathrm { V a r } _ { P _ { h } } \widehat { V } _ { h + 1 } ( s , a ) \right| \leq 3 6 \sqrt { \frac { H ^ { 4 } d ^ { 3 } } { \kappa K } \log \left( \frac { ( \lambda + K ) 2 K d H ^ { 2 } } { \lambda \delta } \right) } + 1 2 \lambda \frac { H ^ { 2 } \sqrt { d } } { \kappa K } . +$$ + +Finally, by the non-expansiveness of operator $\operatorname* { m a x } \{ 1 , \cdot \}$ , we have the stated result. + +C.4.2 A CRUDE BOUND ON $\operatorname* { s u p } _ { h } \lvert | V _ { h } ^ { \star } - \widehat { V } _ { h } \rvert | _ { \infty }$ . + +Lemma C.8. Define $\begin{array} { r l r } { \widehat { \sigma } _ { h } ( s , a ) } & { { } = } & { \sqrt { \operatorname* { m a x } \left\{ 1 , \widehat { \mathrm { V a r } } _ { P _ { h } } \widehat { V } _ { h + 1 } ( s , a ) \right\} + 1 } , \quad i f \quad K \quad \geq } \end{array}$ $\operatorname* { m a x } \{ \mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } , \mathcal { M } _ { 3 } , \mathcal { M } _ { 4 } \}$ and $K > C \cdot H ^ { 4 } \kappa ^ { 2 }$ , then with probability at least $1 - \delta$ , + +$$ +\operatorname* { s u p } _ { h } \left\| V _ { h } ^ { \star } - \widehat { V } _ { h } \right\| _ { \infty } \leq \widetilde { O } \left( \frac { H ^ { 2 } \sqrt { d } } { \sqrt { \kappa K } } \right) . +$$ + +Proof. Step1: We show with probability at least $1 - \delta$ $\begin{array} { r } { \delta , \operatorname* { s u p } _ { h } \left\| V _ { h } ^ { \star } - V _ { h } ^ { \widehat { \pi } } \right\| _ { \infty } \leq \widetilde { O } \left( \frac { H ^ { 2 } \sqrt { d } } { \sqrt { \kappa K } } \right) . } \end{array}$ + +Indeed, combing Lemma C.1 and Lemma C.5, similar to the proof of Theorem C.6, we directly have with probability $1 - \delta$ , for all policy $\pi$ simultaneously, and for all $s \in S$ , $h \in [ H ]$ + +$$ +V _ { h } ^ { \pi } ( s ) - V _ { h } ^ { \widehat { \pi } } ( s ) \leq \widetilde { O } \left( \sqrt { d } \cdot \sum _ { t = h } ^ { H } \mathbb { E } _ { \pi } \left[ \left( \phi ( \cdot , \cdot ) ^ { \top } \Lambda _ { t } ^ { - 1 } \phi ( \cdot , \cdot ) \right) ^ { 1 / 2 } \middle | s _ { h } = s \right] \right) + \frac { 2 H ^ { 4 } \sqrt { d } } { K } , +$$ + +Next, since $\begin{array} { r } { K \ge \operatorname* { m a x } \left\{ 5 1 2 ( 1 / \kappa ) ^ { 2 } \log \left( \frac { 4 H d } { \delta } \right) , 4 \lambda / \kappa \right\} } \end{array}$ , by Lemma H.6 and a union bound over $h \in [ H ]$ , with probability $1 - \delta$ + +$$ +\operatorname* { s u p } _ { s , a } \| \phi ( s , a ) \| _ { \hat { \Lambda } _ { h } ^ { - 1 } } \leq \frac { 2 } { \sqrt { K } } \operatorname* { s u p } _ { s , a } \| \phi ( s , a ) \| _ { \Lambda _ { h } ^ { p - 1 } } \leq \frac { 2 H } { \sqrt { \kappa K } } , \forall h \in [ H ] . +$$ + +Lastly, taking $\pi = \pi ^ { \star }$ in equation 18 to obtain + +$$ +\begin{array} { c l } { \displaystyle 0 \leq { V _ { h } ^ { \pi ^ { * } } ( s ) - V _ { h } ^ { \widehat { \pi } } ( s ) \leq \widetilde O \left( \sqrt { d } \cdot \displaystyle \sum _ { t = h } ^ { H } \mathbb { E } _ { \pi ^ { * } } \left[ \left( \phi ( \cdot , \cdot ) ^ { \top } \Lambda _ { t } ^ { - 1 } \phi ( \cdot , \cdot ) \right) ^ { 1 / 2 } \Big | s _ { h } = s \right] \right) + \frac { 2 H ^ { 4 } \sqrt { d } } { K } } } \\ { \displaystyle \leq \widetilde O \left( \frac { H ^ { 2 } \sqrt { d } } { \sqrt { \kappa K } } \right) + \frac { 2 H ^ { 4 } \sqrt { d } } { K } . } \end{array} +$$ + +This implies by using the condition $K > C \cdot H ^ { 4 } \kappa ^ { 2 }$ , we finish the proof of Step1. + +Step2: We show with probability $\begin{array} { r } { 1 - \delta , \operatorname* { s u p } _ { h } { \left\| { \widehat { V } } _ { h } - V _ { h } ^ { \widehat { \pi } } \right\| _ { \infty } } \leq \widetilde { O } \left( \frac { H ^ { 2 } \sqrt { d } } { \sqrt { \kappa K } } \right) . } \end{array}$ + +Indeed, applying Extended Value Difference Lemma H.7 for $\pi = \pi ^ { \prime } = \widehat { \pi }$ , then with probability $1 - \delta$ , for all $s , h$ + +$$ +\begin{array} { r l } & { ~ \displaystyle | \widehat { V } _ { h } ( s ) - V _ { h } ^ { \widehat { \pi } } ( s ) | = \displaystyle | \sum _ { t = h } ^ { H } \mathbb { E } _ { \widehat { \pi } } [ \widehat { Q } _ { h } ( s _ { h } , a _ { h } ) - ( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } ) ( s _ { h } , a _ { h } ) | s _ { h } = s ] | } \\ & { \leq \displaystyle \sum _ { t = h } ^ { H } \| ( \widehat { \mathcal { T } } _ { h } \widehat { V } _ { h + 1 } - \mathcal { T } _ { h } \widehat { V } _ { h + 1 } ) ( s , a ) \| + \| \Gamma _ { h } ( s , a ) \| } \\ & { { \leq } \widetilde { O } ( H \sqrt { d } \| \sqrt { \phi ( s , a ) \Lambda _ { h } ^ { - 1 } \phi ( s , a ) } \| ) + \frac { 4 H ^ { 4 } \sqrt { d } } { K } \leq \widetilde { O } ( \frac { H ^ { 2 } \sqrt { d } } { \sqrt { \kappa K } } ) } \end{array} +$$ + +where the second inequality uses Lemma ${ \mathrm { C } } . 5 ^ { 1 2 }$ and the last inequality follows the same procedure as Step1. + +Step3: Combine Step1 and Step2, by triangular inequality and a union bound we finish the proof of the lemma. + +Remark C.9. Note as an intermediate calculation, equation $^ { 1 9 }$ ensures a learning bound with order $\widetilde { \cal O } \big ( \frac { H ^ { 2 } \sqrt { d } } { \sqrt { \kappa K } } \big )$ . Here, the convergence rate is the standard statistical rate $\textstyle { \frac { 1 } { \sqrt { K } } }$ and the $H ^ { 2 }$ dependence is loose. However, the feature dependence $\sqrt { d / \kappa }$ is roughly tight, since, in the well-explored case (Assumption 2 of Wang et al. (2021a)), √ $\kappa = 1 / d$ and the $\sqrt { d / \kappa } = \sqrt { d ^ { 2 } }$ recovers the optimal feature dependence $d H \sqrt { T }$ in the online setting (Zhou et al., 2021a). If $\kappa \ll 1 / d ,$ , then doing offline learning requires sample size proportional to $d / \kappa ,$ , which reveals offline $R L$ is harder when the exploration of behavior policy is insufficient. When $\kappa = 0$ , learning the optimal policy accurately cannot be guaranteed even if the sample/episode size $K \infty$ . + +C.5 PROOF OF THE SECOND PART OF THEOREM 3.2 + +Lemma C.10. Recall $\widehat { \sigma } _ { h } = \sqrt { \operatorname* { m a x } \left\{ 1 , \widehat { \mathrm { V a r } } _ { P _ { h } } \widehat { V } _ { h + 1 } \right\} + 1 }$ and $\sigma _ { h } ^ { \star } = \sqrt { \operatorname* { m a x } \left\{ 1 , \mathrm { V a r } _ { P _ { h } } V _ { h + 1 } ^ { \star } \right\} + 1 } .$ . Let $\begin{array} { r } { K \ge \operatorname* { m a x } \left\{ 5 1 2 ( 1 / \kappa ) ^ { 2 } \log \left( \frac { 4 H d } { \delta } \right) , 4 \lambda / \kappa \right\} } \end{array}$ and $K \ge \operatorname* { m a x } \{ \mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } , \mathcal { M } _ { 3 } , \mathcal { M } _ { 4 } \}$ , then with probability $1 - \delta$ , + +$$ +\operatorname* { s u p } _ { h } \vert \vert \widehat { \sigma } _ { h } ^ { 2 } - \sigma _ { h } ^ { \star 2 } \vert \vert _ { \infty } \leq \widetilde { O } \left( \frac { H ^ { 3 } \sqrt { d } } { \sqrt { \kappa K } } \right) . +$$ + +Proof. By definition and the non-expansiveness of $\operatorname* { m a x } \{ 1 , \cdot \} + 1$ , we have + +$$ +\begin{array} { r l } & { \quad \left\| \sigma _ { \widehat { V } _ { h + 1 } } ^ { 2 } - \sigma _ { h } ^ { \star 2 } \right\| _ { \infty } \leq \left\| \operatorname { V a r } \widehat { V } _ { h + 1 } - \operatorname { V a r } V _ { h + 1 } ^ { \star } \right\| _ { \infty } } \\ & { \leq \left\| \mathbb { P } _ { h } \left( \widehat { V } _ { h + 1 } ^ { 2 } - V _ { h + 1 } ^ { \star 2 } \right) \right\| _ { \infty } + \left\| ( \mathbb { P } _ { h } \widehat { V } _ { h + 1 } ) ^ { 2 } - ( \mathbb { P } _ { h } V _ { h + 1 } ^ { \star } ) ^ { 2 } \right\| _ { \infty } } \\ & { \leq \left\| \widehat { V } _ { h + 1 } ^ { 2 } - V _ { h + 1 } ^ { \star 2 } \right\| _ { \infty } + \left\| ( \mathbb { P } _ { h } \widehat { V } _ { h + 1 } + \mathbb { P } _ { h } V _ { h + 1 } ^ { \star } ) ( \mathbb { P } _ { h } \widehat { V } _ { h + 1 } - \mathbb { P } _ { h } V _ { h + 1 } ^ { \star } ) \right\| _ { \infty } } \\ & { \leq 2 H \left\| \widehat { V } _ { h + 1 } - V _ { h + 1 } ^ { \star } \right\| _ { \infty } + 2 H \left\| \mathbb { P } _ { h } \widehat { V } _ { h + 1 } - \mathbb { P } _ { h } V _ { h + 1 } ^ { \star } \right\| _ { \infty } \leq \widetilde { \mathcal { O } } \left( \frac { H ^ { 3 } \sqrt { d } } { \sqrt { \kappa K } } \right) . } \end{array} +$$ + +with probability $1 - \delta$ for all $h \in [ H ]$ , where the last inequality comes from Lemma C.8. Combining this with Lemma C.7, we have the stated result. + +Lemma C.11. Denote the quantities $C _ { 1 } = \operatorname* { m a x } \{ 2 \lambda , 1 2 8 \log ( 2 d / \delta ) , 1 2 8 H ^ { 4 } \log ( 2 d / \delta ) / \kappa ^ { 2 } \}$ and $\begin{array} { r } { C _ { 2 } = \operatorname* { m a x } \{ \frac { \lambda ^ { 2 } } { \kappa \log ( ( \lambda + K ) H / \lambda \delta ) } , 9 6 ^ { 2 } H ^ { 1 2 } d \log ( ( \lambda + K ) H / \lambda \delta ) / \kappa ^ { 5 } \} } \end{array}$ . Suppose the number of episode $K$ satisfies $K > \operatorname* { m a x } \{ C _ { 1 } , C _ { 2 } \}$ , then with probability $1 - \delta$ , + +$$ +\sqrt { \phi ( s , a ) \Lambda _ { h } ^ { - 1 } \phi ( s , a ) } \le 2 \sqrt { \phi ( s , a ) \Lambda _ { h } ^ { \star - 1 } \phi ( s , a ) } , \quad \forall s , a \in \mathcal { S } \times \mathcal { A } , +$$ + +Proof of Lemma C.11. By definition $\sqrt { \phi ( s , a ) \Lambda _ { h } ^ { - 1 } \phi ( s , a ) } = \| \phi ( s , a ) \| _ { \Lambda _ { h } ^ { - 1 } } .$ . Then denote + +$$ +\Lambda _ { h } ^ { \prime } = \frac { 1 } { K } \Lambda _ { h } , \quad { \Lambda _ { h } ^ { \star } } ^ { \prime } = \frac { 1 } { K } \Lambda _ { h } ^ { \star } , +$$ + +where $\begin{array} { r } { \Lambda _ { h } \ = \ \sum _ { \tau = 1 } ^ { K } \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) ^ { \top } \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) / \sigma _ { V _ { h + 1 } ^ { \star } } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) + \lambda I . } \end{array}$ . Under the condition of $K$ , by Lemma C.10, with probability $1 - \delta$ + +$$ +\begin{array} { r l } & { \left. \Lambda _ { h } ^ { \star ^ { \prime } } - \Lambda _ { h } ^ { \prime } \right. \leq \underset { s , a } { \operatorname* { s u p } } \left. \frac { \phi ( s , a ) \phi ( s , a ) ^ { \top } } { \sigma _ { h } ^ { \star 2 } ( s , a ) } - \frac { \phi ( s , a ) \phi ( s , a ) ^ { \top } } { \sigma _ { \hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \right. } \\ & { \leq \underset { s , a } { \operatorname* { s u p } } \left. \frac { \sigma _ { h } ^ { \star 2 } ( s , a ) - \sigma _ { \hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { \sigma _ { h } ^ { \star 2 } ( s , a ) \sigma _ { \hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \right. \cdot \left. \phi ( s , a ) \right. ^ { 2 } \leq \underset { s , a } { \operatorname* { s u p } } \left. \frac { \sigma _ { h } ^ { \star 2 } ( s , a ) - \sigma _ { \hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { 1 } \right. \cdot 1 } \\ & { \leq \tilde { O } \left( \frac { H ^ { 3 } \sqrt { d } } { \sqrt { \kappa K } } \right) . } \end{array} +$$ + +Next by Lemma H.5 (with $\phi$ to be $\phi / \sigma _ { V _ { h + 1 } ^ { \star } }$ and $C = 1$ ), it holds with probability $1 - \delta$ , + +$$ +\left\| \Lambda _ { h } ^ { \star ^ { \prime } } - \left( \mathbb { E } _ { \mu , h } [ \phi ( s , a ) \phi ( s , a ) ^ { \top } / \sigma _ { V _ { h + 1 } ^ { \star } } ^ { 2 } ( s , a ) ] + \frac { \lambda } { K } I _ { d } \right) \right\| \leq \frac { 4 \sqrt { 2 } } { \sqrt { K } } \left( \log \frac { 2 d } { \delta } \right) ^ { 1 / 2 } . +$$ + +Therefore by Weyl’s spectrum theorem and the condition $K$ > $\operatorname* { m a x } \{ 2 \lambda , 1 2 8 \log ( 2 d / \delta ) , 1 2 8 H ^ { 4 } \log ( 2 d / \delta ) / \kappa ^ { 2 } \}$ , the above implies + +$$ +\begin{array} { r l } { \left\| { { { \Lambda } _ { k } ^ { \prime } } } \right\| = \lambda { \operatorname* { m s u p } _ { i } } ( \Lambda _ { k } ^ { \prime } ) \le \lambda { \operatorname* { m s u p } _ { i } } \left( { \operatorname* { \sum } _ { n , h } } [ \phi ( s , a ) \phi ( s , a ) ] ^ { \top } \rho _ { \mathcal { N } _ { k + 1 } ^ { \prime } } ^ { \sigma } ( s , a ) \right) + \frac { \lambda } { K } + \frac { 4 \sqrt { 2 } } { \sqrt { K } } \left( \log \frac { 2 d } { \delta } \right) ^ { 1 / 2 } } & \\ { \le \left\| { { \mathbb { E } _ { \rho , h } } [ \phi ( s , a ) \phi ( s , a ) ] ^ { \top } } \rho _ { \mathcal { N } _ { k + 1 } ^ { \prime } } ^ { \sigma } ( s , a ) \right\| + \frac { \lambda } { K } + \frac { 4 \sqrt { 2 } } { \sqrt { K } } \left( \log \frac { 2 d } { \delta } \right) ^ { 1 / 2 } } & \\ { \le \| \phi ( s , a ) \| ^ { 2 } + \frac { \lambda } { K } + \frac { 4 \sqrt { 2 } } { \sqrt { K } } \left( \log \frac { 2 d } { \delta } \right) ^ { 1 / 2 } \le 1 + \frac { \lambda } { K } + \frac { 4 \sqrt { 2 } } { \sqrt { K } } \left( \log \frac { 2 d } { \delta } \right) ^ { 1 / 2 } \le 2 , } & \\ { \operatorname* { m i n } ( \Lambda _ { k } ^ { \prime } ) \le \lambda { \operatorname* { m i n } } \left( { \mathbb { E } _ { \rho , h } } [ \phi ( s , a ) \phi ( s , a ) ] ^ { \top } \rho _ { \mathcal { N } _ { k + 1 } ^ { \prime } ( s , a ) } ^ { \sigma } | \right) + \frac { \lambda } { K } - \frac { 4 \sqrt { 2 } } { \sqrt { K } } \left( \log \frac { 2 d } { \delta } \right) ^ { 1 / 2 } } & \\ { \le \lambda { \operatorname* { m i n } } \left( { \mathbb { E } _ { \rho , h } } [ \phi ( s , a ) \phi ( s , a ) ] ^ { \top } \rho _ { \mathcal { N } _ { k + 1 } ^ { \prime } ( s , a ) } ^ { \sigma } | \right) - \frac { 4 \sqrt { 2 } } { \sqrt { K } } \left( \log \frac { 2 d } { \delta } \right) ^ { 1 / 2 } } & \\ \ge \frac { \kappa } { L } - \frac { 4 \sqrt { 2 } } \end{array} +$$ + +Hence with probability $1 - \delta$ , $\left\| \Lambda _ { h } ^ { \star ^ { \prime } } \right\| \leq 2$ and $\left\| { \boldsymbol { \Lambda } } _ { h } ^ { \star ^ { \prime } - 1 } \right\| = 1 / { \lambda _ { \operatorname* { m i n } } ( \boldsymbol { \Lambda } _ { h } ^ { \star ^ { \prime } } ) } \le 2 H ^ { 2 } / \kappa$ . Similarly, $\left\| \Lambda _ { h } ^ { ' - 1 } \right\| \le 2 H ^ { 2 } / \kappa$ with high probability. + +Now apply Lemma H.4 to $\Lambda _ { h } ^ { \star ^ { \prime } }$ and $\Lambda _ { h } ^ { \prime }$ and a union bound, we obtain with probability $1 - \delta$ , for all $s , a$ + +$$ +\begin{array} { r l } { \| \phi ( s , a ) \| _ { \Lambda _ { h } ^ { \prime - 1 } } \leq \left[ 1 + \sqrt { \left\| \Lambda _ { h } ^ { \star ^ { \prime } - 1 } \right\| \left\| \Lambda _ { h } ^ { \star ^ { \prime } } \right\| \cdot \left\| \Lambda _ { h } ^ { \prime - 1 } \right\| \cdot \left\| \Lambda _ { h } ^ { \star ^ { \prime } } - \Lambda _ { h } ^ { \prime } \right\| } \right] \cdot \| \phi ( s , a ) \| _ { \Lambda _ { h } ^ { \star ^ { \prime } - 1 } } } & { } \\ { \leq \left[ 1 + \sqrt { \frac { 2 H ^ { 2 } } { \kappa } \cdot 1 \cdot \frac { 2 H ^ { 2 } } { \kappa } \cdot \left\| \Lambda _ { h } ^ { \star ^ { \prime } } - \Lambda _ { h } ^ { \prime } \right\| } \right] \cdot \| \phi ( s , a ) \| _ { \Lambda _ { h } ^ { \star ^ { \prime } - 1 } } } & { } \\ { \leq \left[ 1 + \sqrt { \frac { H ^ { 4 } } { \kappa ^ { 2 } } \left[ \tilde { \mathcal { O } } \left( \frac { H ^ { 3 } \sqrt { d } } { \sqrt { \kappa K } } \right) \right] } \right] \cdot \| \phi ( s , a ) \| _ { \Lambda _ { h } ^ { \star ^ { \prime } - 1 } } \leq 2 \left\| \phi ( s , a ) \right\| _ { \Lambda _ { h } ^ { \star ^ { \prime } - 1 } } } & { } \end{array} +$$ + +where the third inequality uses equation 20 and the last inequality uses $K \_ { > }$ $\begin{array} { r } { \operatorname* { m a x } \{ \frac { \lambda ^ { 2 } } { \kappa \log ( ( \lambda + K ) H / \lambda \delta ) } , \overset { } { 9 } 6 ^ { 2 } \dot { H } ^ { 1 2 } d \log ( ( \lambda + K ) H / \lambda \delta ) / \kappa ^ { 5 } \} } \end{array}$ . The claimed result follows straightforwardly by multiplying $1 / \sqrt { K }$ on both sides of the above. + +Proof of Theorem 3.2. The first part of the theorem has been shown in Theorem C.6. For the second part, apply Theorem C.6 with $\pi = \pi ^ { \star }$ , then with probability $1 - \delta$ , + +$$ +v ^ { \pi ^ { \star } } - v ^ { \widehat { \pi } } \leq \widetilde O \left( \sqrt { d } \cdot \sum _ { h = 1 } ^ { H } \mathbb E _ { \pi ^ { \star } } \left[ \left( \phi ( \cdot , \cdot ) ^ { \top } \Lambda _ { h } ^ { - 1 } \phi ( \cdot , \cdot ) \right) ^ { 1 / 2 } \right] \right) + \frac { 2 H ^ { 4 } \sqrt { d } } { K } , +$$ + +Now apply Lemma C.11 and a union bound, with probability $1 - \delta$ , + +$$ +0 \leq v ^ { \star } - v ^ { \widehat { \pi } } \leq \widetilde { O } \left( \sqrt { d } \cdot \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \pi ^ { \star } } \left[ \left( \phi ( \cdot , \cdot ) ^ { \top } \Lambda _ { h } ^ { \star - 1 } \phi ( \cdot , \cdot ) \right) ^ { 1 / 2 } \right] \right) + \frac { 2 H ^ { 4 } \sqrt { d } } { K } . +$$ + +# D PROOF OF THEOREM 3.3 + +First of all, we show the following lemma. + +Lemma D.1. Suppose $K > \operatorname* { m a x } \{ \mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } , \mathcal { M } _ { 3 } , \mathcal { M } _ { 4 } \}$ . Plug + +$$ +\mathbf { \Sigma } _ { h } ^ { \scriptscriptstyle { T } } ( s , a ) \gets \phi ( s , a ) ^ { \scriptscriptstyle { T } } \left| \widehat { \Lambda } _ { h } ^ { - 1 } \sum _ { \tau = 1 } ^ { K } \frac { \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \widehat { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) } { \widehat { \sigma } _ { h } ^ { 2 } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) } \right| + \widetilde { O } ( \frac { H ^ { 3 } d / \kappa } { K } ) +$$ + +in Algorithm $I$ and let $\mathcal { T } _ { h }$ be the Bellman operator and $\widehat { \mathcal { T } } _ { h }$ be the approximated Bellman operator. Then we have with probability $1 - \delta$ : + +$$ +\vert ( \mathcal T _ { h } \widehat V _ { h + 1 } - \widehat T _ { h } \widehat V _ { h + 1 } ) ( s , a ) \vert \le \Gamma _ { h } ^ { I } ( s , a ) , \quad \forall s , a \in \mathcal S \times \mathcal A . +$$ + +Proof of Lemma $D . I$ . Suppose $w _ { h }$ is the coefficient corresponding to the $\mathcal { T } _ { h } \widehat { V } _ { h + 1 }$ (such $w _ { h }$ exists by Lemma H.9), i.e. $\mathcal { T } _ { h } \widehat { V } _ { h + 1 } = \phi ^ { \top } w _ { h }$ , and recall $( \widehat { T } _ { h } \widehat { V } _ { h + 1 } ) ( s , a ) = \phi ( s , a ) ^ { \top } \widehat { w } _ { h }$ , then: + +$$ +\begin{array} { r l } & { \quad \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) ( s , a ) - \left( \widehat { \mathcal { T } } _ { h } \widehat { V } _ { h + 1 } \right) ( s , a ) = \phi ( s , a ) ^ { \top } \left( w _ { h } - \widehat { w } _ { h } \right) } \\ & { = \phi ( s , a ) ^ { \top } w _ { h } - \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \left( \displaystyle \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) \right) / \widehat { \sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \right) } \\ & { = \phi ( s , a ) ^ { \top } w _ { h } - \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \left( \displaystyle \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) / \widehat { \sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \right) } \end{array} +$$ + +$$ ++ \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } ( \sum _ { \tau = 1 } ^ { K } \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \cdot ( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } ( s _ { h + 1 } ^ { \tau } ) - ( \widehat { T } _ { h } \widehat { V } _ { h + 1 } ) ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) ) / \widehat { \sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) ) ) +$$ + +$$ ++ \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \left( \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( \left( \widehat { \mathcal { T } } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) - \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) / \widehat { \sigma } _ { h } ^ { 2 } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) +$$ + +For term (i), by Lemma C.2 it is bounded by $\frac { 2 \lambda H ^ { 3 } \sqrt { d } / \kappa } { K }$ with probability $1 - \delta / 2$ . 13 + +For term (ii), it is bounded by + +$$ +\phi ( s , a ) ^ { \top } \left| \widehat { \Lambda } _ { h } ^ { - 1 } \sum _ { \tau = 1 } ^ { K } \frac { \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \widehat { \mathcal { T } } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) } { \widehat { \sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) } \right| . +$$ + +For term (iii), by Cauchy inequality + +$$ +\begin{array} { l } { \displaystyle \phi ( s , a ) ^ { \top } \widehat \Lambda _ { h } ^ { - 1 } \left( \displaystyle \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( \left( \widehat { \gamma } _ { h } \widehat V _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) - \left( \widehat { \gamma } _ { h } \widehat V _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) / \widehat \sigma _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \right) } \\ { \displaystyle \leq \left\| \phi ( s , a ) \right\| _ { \widehat \Lambda _ { h } ^ { - 1 } } \cdot \displaystyle \left\| \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( \left( \widehat { \gamma } _ { h } \widehat V _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) - \left( \widehat { \gamma } _ { h } \widehat V _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) / \widehat \sigma _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \right\| _ { \widehat \Lambda _ { h } ^ { - 1 } } } \\ { \displaystyle \leq \displaystyle \frac { 2 H } { \sqrt { \kappa K } } \cdot \displaystyle \left\| \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( \left( \widehat { \gamma } _ { h } \widehat V _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) - \left( \widehat { \gamma } _ { h } \widehat V _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) / \widehat \sigma _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \right\| _ { \widehat \Lambda _ { h } ^ { - 1 } } } \\ \displaystyle \leq \frac { 2 H } { \sqrt { \kappa K } } \cdot \displaystyle \tilde { O } ( \frac { H ^ { 2 } \sqrt { d / \kappa } } { \sqrt { K } } ) \cdot \sqrt { d } = \tilde { O } ( \frac H ^ { 3 } d / \kappa \end{array} +$$ + +where the first inequality is by Lemma H.6 (with $\phi ^ { \prime } = \phi / \widehat { \sigma } _ { h }$ and $\| \phi / { \widehat { \sigma } } _ { h } \| \leq \| \phi \| \leq 1 : = C )$ and the third inequality uses ${ \sqrt { a ^ { \top } \cdot A \cdot a } } \leq { \sqrt { \left\| a \right\| _ { 2 } \left\| A \right\| _ { 2 } \left\| a \right\| _ { 2 } } } = \left\| a \right\| _ { 2 } { \sqrt { \left\| A \right\| _ { 2 } } }$ with $a$ to be either $\phi$ or $w _ { h }$ . Moreover, $\begin{array} { r } { \lambda _ { \operatorname* { m i n } } ( \widetilde { \Lambda } _ { h } ^ { p } ) \ge \kappa / \operatorname* { m a x } _ { h , s , a } \widehat { \sigma } _ { h } ( s , a ) ^ { 2 } \ge \kappa / H ^ { 2 } } \end{array}$ implies $\left\| ( \tilde { \Lambda } _ { h } ^ { p } ) ^ { - 1 } \right\| \leq H ^ { 2 } / \kappa$ . + +The second inequality is true by denoting $x _ { \tau } = \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) / \widehat { \sigma } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } )$ and + +$$ +\eta _ { \tau } = \left( \left( \widehat { T } _ { h } \widehat { V } _ { h + 1 } \right) ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) - \left( T _ { h } \widehat { V } _ { h + 1 } \right) ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \right) / \widehat { \sigma } _ { h } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) +$$ + +and use Lemma H.10 as the condition for applying Lemma H.2. By collecting those three terms together we have the result. + +# D.1 PROOF OF THEOREM 3.3 + +Proof. Use Lemma D.1 as the condition for Lemma C.1 and average over initial distribution $d _ { 1 }$ , we obtain with probability $1 - \delta$ , + +$$ +\begin{array} { l } { v ^ { \pi } - v ^ { \pi } \leq } \\ { \displaystyle \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \pi _ { h } } \left[ \phi ( s , a ) \right] ^ { \top } \left| \widehat { \Lambda } _ { h } ^ { - 1 } \displaystyle \sum _ { \tau = 1 } ^ { K } \frac { \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \widehat { \mathcal { T } } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) } { \widehat { \sigma } _ { h } ^ { 2 } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) } \right| + \widetilde { O } ( \frac { H ^ { 4 } d / } { K } ) } \end{array} +$$ + +Denote $\begin{array} { r } { A _ { h } : = \sum _ { \tau = 1 } ^ { K } \frac { \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \cdot \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \tau _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) } { \widehat { \sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) } } \end{array}$ + +$$ +\begin{array} { r l } & { \mathbb { E } _ { \pi _ { h } } \left[ \phi ( s , a ) \right] ^ { \top } \left| \widehat { \Lambda } _ { h } ^ { - 1 } \displaystyle \sum _ { \tau = 1 } ^ { K } \frac { \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \widehat { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) } { \widehat { \sigma } _ { h } ^ { 2 } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) } \right| } \\ & { \leq \mathbb { E } _ { \pi _ { h } } \left[ \phi \right] ^ { \top } \cdot \left| \widehat { \Lambda } _ { h } ^ { - 1 } A _ { h } \right| + \mathbb { E } _ { \pi _ { h } } \left[ \phi \right] ^ { \top } \left| \widehat { \Lambda } _ { h } ^ { - 1 } \displaystyle \sum _ { \tau = 1 } ^ { K } \frac { \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( 7 _ { h } \widehat { V } _ { h + 1 } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) - \widehat { T } _ { h } \widehat { V } _ { h + 1 } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) } { \widehat { \sigma } _ { h } ^ { 2 } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) } \right| } \end{array} +$$ + +For the second term, it can be bounded similar to term (iii) in Lemma D.1 and for the first term we have the following: + +$$ +\begin{array} { r l } & { \quad \mathbb E _ { \pi _ { h } } \left[ \phi \right] ^ { \top } \cdot \left| \widehat \Lambda _ { h } ^ { - 1 } A _ { h } \right| = \mathbb E _ { \pi _ { h } } \left[ \phi \right] ^ { \top } \cdot \widehat \Lambda _ { h } ^ { - 1 } \cdot \widehat \Lambda _ { h } \left| \widehat \Lambda _ { h } ^ { - 1 } A _ { h } \right| \leq \| \mathbb E _ { \pi _ { h } } \left[ \phi \right] \| _ { \widehat \Lambda _ { h } ^ { - 1 } } \cdot \left\| \widehat \Lambda _ { h } | \widehat \Lambda _ { h } ^ { - 1 } A _ { h } \right| \Big \| _ { \widehat \Lambda _ { h } ^ { - 1 } } } \\ & { \leq \| \mathbb E _ { \pi _ { h } } [ \phi ] \| _ { \widehat \Lambda _ { h } ^ { - 1 } } \cdot \| | A _ { h } | \| _ { \widehat \Lambda _ { h } ^ { - 1 } } \leq \widetilde O ( \sqrt d ) \| \mathbb E _ { \pi _ { h } } [ \phi ] \| _ { \widehat \Lambda _ { h } ^ { - 1 } } \leq \widetilde O ( \sqrt d \| \mathbb E _ { \pi _ { h } } [ \phi ] \| _ { \Lambda _ { h } ^ { - 1 } } ) , } \end{array} +$$ + +where the first inequality uses Cauchy’s inequality, the second inequality uses $\widehat { \Lambda } _ { h }$ is coordinate-wise positive (since we assume here $\phi \geq 0$ ), the third inequality is identical to the analysis in Section C.2.2 and the fourth inequality is identical to the analysis in Section C.2.1 with $\phi$ replaced by $\mathbb { E } [ \phi ]$ . Plug this back to equation 22 we finish the proof for the first part. For the second part, converting $\Lambda _ { h } ^ { - 1 }$ to $\Lambda _ { h } ^ { \star - 1 }$ is identical to Section C.5. This finishes the proof. □ + +# E PROOF OF MINIMAX LOWER BOUND THEOREM 3.5 + +The proof follows the lower bound proof of Zanette et al. (2021). For completeness, we provide all the details in below. + +# E.1 CONSTRUCTION + +Similar to the proof of [Zanette et al. (2021), Theorem 2], we construct a family of MDPs, each parameterized by a Boolean vector $u = ( u _ { 1 } , \dotsc , u _ { H } )$ with each $u _ { h } \in \{ - 1 , + 1 \} ^ { d - 2 }$ for $h \in [ H ]$ . The MDPs share the same transition kernel and are only different in the reward observations. + +State space: At each time step $h$ , there are two states $S = \{ + 1 , - 1 \}$ . + +Action space: The action space $\mathcal { A } = \{ - 1 , 0 , + 1 \} ^ { d - 2 }$ + +Feature map: The feature map $\phi : \mathcal { S } \times \mathcal { A } \mapsto \mathbb { R } ^ { d }$ is given by + +$$ +\phi ( + 1 , a ) = \left( \frac { \frac { a } { \sqrt { 2 d } } } { \frac { 1 } { \sqrt { 2 } } } \right) \in \mathbb R ^ { d } , \qquad \phi ( - 1 , a ) = \left( \begin{array} { c } { \frac { a } { \sqrt { 2 d } } } \\ { 0 } \\ { \frac { 1 } { \sqrt { 2 } } } \end{array} \right) \in \mathbb R ^ { d } . +$$ + +The construction ensures the condition $\| \phi ( s , a ) \| _ { 2 } \leq 1$ for any $( s , a ) \in S \times A$ . + +Transition kernel: The transition probability $P _ { h } ( s ^ { \prime } \mid s , a )$ is independent of action $a$ . In other words, the Markov decision process reduces to a homogeneous Markov chain with transition matrix + +$$ +\mathbf { P } = \left( \begin{array} { l l } { \frac { 1 } { 2 } } & { \frac { 1 } { 2 } } \\ { \frac { 1 } { 2 } } & { \frac { 1 } { 2 } } \end{array} \right) \in \mathbb { R } ^ { 2 } . +$$ + +By letting + +$$ +\nu _ { h } ( + 1 ) = \nu _ { h } ( - 1 ) = \left( \begin{array} { l } { \mathbf { 0 } _ { d - 2 } } \\ { \frac { 1 } { \sqrt { 2 } } } \\ { \frac { 1 } { \sqrt { 2 } } } \end{array} \right) \in \mathbb { R } ^ { d } , +$$ + +we have $P _ { h } ( s ^ { \prime } \mid s , a ) = \langle \phi ( s , a ) , \nu _ { h } ( s ^ { \prime } ) \rangle$ to be a valid probability transition. + +Reward observations: For any MDP $M _ { u }$ , at each times step $h$ , the reward follows a Gaussian distribution with + +$$ +R _ { u , h } ( s , a ) \sim \mathcal { N } \bigg ( \frac { s } { \sqrt { 6 } } + \frac { \delta } { \sqrt { 2 d } } \langle a , u _ { h } \rangle , 1 \bigg ) , +$$ + +where $\delta \in \left[ 0 , \frac { 1 } { \sqrt { 3 d } } \right]$ determines to what extent the MDP models are different from each other. The mean reward function satisfies $r _ { u , h } ( s , a ) = \langle \phi ( s , a ) , \theta _ { u , h } \rangle$ with + +$$ +\theta _ { u , h } = \left( \begin{array} { c } { \delta u _ { h } } \\ { \frac { 1 } { \sqrt { 3 } } } \\ { - \frac { 1 } { \sqrt { 3 } } } \end{array} \right) \in \mathbb { R } ^ { d } . +$$ + +Offline data collection Scheme: The dataset D = {(sτh, aτh, rτh, sτh+1)}h∈[H]τ∈[K] consist of $K$ i.i.d. trajectories. All the trajectories initiate from uniform distribution. We take a behavior policy $\textstyle \mu ( \cdot \mid s )$ that is independent of state $s$ . Let $\{ e _ { 1 } , e _ { 2 } , \ldots , e _ { d - 2 } \}$ be the canonical bases of $\mathbb { R } ^ { d - 2 }$ and ${ \bf 0 } _ { d - 2 } \in \mathbb { R } ^ { \hat { d } - 2 }$ be the zero vector. The behavior policy $\mu$ is set as + +$$ +\begin{array} { r } { \mu ( e _ { j } \mid s ) = \frac { 1 } { d } \quad \mathrm { f o r ~ a n y ~ } j \in [ d - 2 ] \qquad \mathrm { a n d } \qquad \mu ( \mathbf { 0 } _ { d - 2 } \mid s ) = \frac { 2 } { d } . } \end{array} +$$ + +# E.2 OVERVIEW OF PROOF + +The proof of the theorem is based on Assouad’s method, where we first reduce the problem to binary hypothesis tests (Lemmas E.1 and E.2) and then connect the testing error to the uncertainty quantity in the upper bound (Lemma E.3). + +Lemma E.1 (Reduction to testing). There exists a universal constant $c _ { 1 } > 0$ such thata + +$$ +\operatorname* { i n f } _ { \hat { \pi } } \operatorname* { m a x } _ { u \in \mathcal { U } } \mathbb { E } _ { u } \left[ V _ { u } ^ { \star } - V _ { u } ^ { \widehat { \pi } } \right] \geq c _ { 1 } \delta \sqrt { d } H \operatorname* { m i n } _ { u , u ^ { \prime } \in \mathcal { U } : D _ { H } ( u ^ { \prime } ; u ) = 1 } \operatorname* { i n f } _ { \psi } \big [ \mathbb { P } _ { u } ( \psi \neq u ) + \mathbb { P } _ { u ^ { \prime } } ( \psi \neq u ^ { \prime } ) \big ] , +$$ + +where $\widehat { \pi }$ denotes the output of any algorithm that maps from observations to an estimated policy. $\psi$ is bany test function for parameter $u$ and $D _ { H }$ is the hamming distance. + +Lemma E.2. There exists a universal constant $c _ { 2 } > 0$ such that when taking $\begin{array} { r } { \delta : = \frac { c _ { 2 } d } { \sqrt { K } } } \end{array}$ + +$$ +\operatorname* { m i n } _ { u , u ^ { \prime } \in \mathcal { U } : D _ { H } ( u ^ { \prime } ; u ) = 1 } \operatorname* { i n f } _ { \psi } \left[ \mathbb { P } _ { u } ( \psi \neq u ) + \mathbb { P } _ { u ^ { \prime } } ( \psi \neq u ^ { \prime } ) \right] \geq \frac { 1 } { 2 } . +$$ + +When $\begin{array} { r } { K \gtrsim d ^ { 3 } , \delta : = \frac { c _ { 2 } d } { \sqrt { K } } } \end{array}$ ensures that $\delta \leq 1 / \sqrt { 3 d }$ . Combining Lemmas E.1 and E.2 yields a lower bound + +$$ +\operatorname* { i n f } _ { \widehat { \pi } } \operatorname* { m a x } _ { u \in \mathcal { U } } \mathbb { E } _ { u } \left[ V _ { u } ^ { \star } - V _ { u } ^ { \widehat { \pi } } \right] \geq c \ \frac { d \sqrt { d } H } { \sqrt { K } } , +$$ + +where lower $c > 0$ isto universal constan uncertainty term E.3 to connect the abovefor the chosen linear MDP $\begin{array} { r } { \sqrt { d } \cdot \sum _ { h = 1 } ^ { H } \sqrt { \mathbb { E } _ { \pi ^ { \star } } [ \phi ] ^ { \top } ( \Lambda _ { h } ^ { \star } ) ^ { - 1 } \mathbb { E } _ { \pi ^ { \star } } [ \phi ] } } \end{array}$ $\mathcal { M }$ + +Lemma E.3. There exists a universal constant $c _ { 3 } > 0$ such that for all $M \in \mathcal { M }$ , + +$$ +\sum _ { h = 1 } ^ { H } \sqrt { \mathbb { E } _ { \pi ^ { \star } } [ \phi ] ^ { \top } ( \Lambda _ { h } ^ { \star } ) ^ { - 1 } \mathbb { E } _ { \pi ^ { \star } } [ \phi ] } \le c _ { 3 } \frac { d H } { \sqrt { K } } . +$$ + +Plugging inequality equation E.3 into the bound equation 25, we obtain the minimax lower bound equation 6 in the statement of theorem. + +# E.3 REDUCTION TO TESTING VIA ASSOUAD’S METHOD + +Proof of Lemma E.1. For any index vector $u = ( u _ { 1 } , \ldots , u _ { H } ) \in \mathcal { U } = \{ - 1 , + 1 \} ^ { ( d - 2 ) \times H }$ , the optimal policy for MDP instance $M _ { u }$ is simply + +$$ +\pi _ { h } ^ { \star } ( \cdot ) = u _ { h } \qquad { \mathrm { f o r ~ } } h \in [ H ] . +$$ + +Similar to the proof of Lemma 9 in Zanette et al. (2021), we can show that the value suboptimality of policy $\pi$ on MDP $M _ { u }$ is given by + +$$ +V _ { u } ^ { \star } - V _ { u } ^ { \pi } = \frac { \delta } { \sqrt { 2 d } } \sum _ { h = 1 } ^ { H } \left\| u _ { h } - \mathbb { E } _ { \pi } [ a _ { h } ] \right\| _ { 1 } . +$$ + +Define $u ^ { \pi } = ( u _ { 1 } ^ { \pi } , \ldots , u _ { H } ^ { \pi } )$ with $u _ { h } ^ { \pi } : = \mathrm { s i g n } \left( \mathbb { E } _ { \pi } [ a _ { h } ] \right)$ , then the $\ell _ { 1 }$ -norm is lower bounded as + +$$ +\begin{array} { r } { \left\| u _ { h } - \mathbb { E } _ { \pi } [ a _ { h } ] \right\| _ { 1 } \ge D _ { H } ( u _ { h } ^ { \pi } ; u _ { h } ) , } \end{array} +$$ + +where $D _ { H } ( \cdot ; \cdot )$ denotes the Hamming distance. It follows that + +$$ +V _ { u } ^ { \star } - V _ { u } ^ { \pi } \geq \frac { \delta } { \sqrt { 2 d } } D _ { H } ( u ^ { \pi } ; u ) . +$$ + +We then apply Assouad’s method (Lemma 2.12 in Sampson and Guttorp (1992)) and obtain that + +$$ +\operatorname* { i n f } _ { \substack { \hat { u } \in \mathcal { U } } } \operatorname* { m a x } _ { \boldsymbol { u } \in \mathcal { U } } \mathbb { E } _ { \boldsymbol { u } } \big [ D _ { \boldsymbol { H } } ( \hat { u } ; \boldsymbol { u } ) \big ] \geq \frac { ( d - 2 ) H } { 2 } \operatorname* { m i n } _ { \substack { \boldsymbol { u } , \boldsymbol { u } ^ { \prime } \in \mathcal { U } : D _ { \boldsymbol { H } } ( \boldsymbol { u } ^ { \prime } ; \boldsymbol { u } ) = 1 } } \operatorname* { i n f } _ { \boldsymbol { \psi } } \big [ \mathbb { P } _ { \boldsymbol { u } } ( \boldsymbol { \psi } \neq \boldsymbol { u } ) + \mathbb { P } _ { \boldsymbol { u } ^ { \prime } } ( \boldsymbol { \psi } \neq \boldsymbol { u } ^ { \prime } ) \big ] , +$$ + +where $\psi$ is any test functions mapping from observations to $\{ u , u ^ { \prime } \}$ . Combining inequalities equation 27 and equation 28, we finish the proof. □ + +# E.4 LOWER BOUND ON THE TESTING ERROR + +Proof of Lemma $E . 2$ . The proof of Lemma E.2 is similar to that of Lemma 10 in Zanette et al. (2021). We first apply Theorem 2.12 in Sampson and Guttorp (1992) to lower bound the testing error using Kullback–Leibler divergence and obtain + +$$ +\operatorname* { m i n } _ { \substack { \tau , u ^ { \prime } \in \mathcal { U } : D _ { H } ( u ^ { \prime } ; u ) = 1 } } \operatorname* { i n f } _ { \psi } \left[ \mathbb { P } _ { u } ( \psi \neq u ) + \mathbb { P } _ { u ^ { \prime } } ( \psi \neq u ^ { \prime } ) \right] \geq 1 - \left( \frac { 1 } { 2 } \operatorname* { m a x } _ { \substack { u , u ^ { \prime } \in \mathcal { U } : D _ { H } ( u ^ { \prime } ; u ) = 1 } } D _ { \mathrm { K L } } ( Q _ { u } \| Q _ { u ^ { \prime } } ) \right) ^ { 1 / 2 } . +$$ + +It only remains to estimate $D _ { \mathrm { K L } } ( \mathcal { Q } _ { u } \Vert \mathcal { Q } _ { u ^ { \prime } } )$ . + +The probability density $\mathcal { Q } _ { u }$ takes the form + +$$ +{ \mathcal { Q } } _ { u } ( { \mathcal { D } } ) = \prod _ { k = 1 } ^ { K } \xi _ { 1 } ( s _ { 1 } ^ { k } ) \prod _ { h = 1 } ^ { H } \mu ( a _ { h } ^ { k } \mid s _ { h } ^ { k } ) \left[ R _ { u , h } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) \right] ( r _ { h } ^ { k } ) \mathbb { P } _ { h } ( s _ { h + 1 } ^ { k } \mid s _ { h } ^ { k } , a _ { h } ^ { k } ) +$$ + +where $\xi _ { 1 } = \left[ \textstyle { \frac { 1 } { 2 } } , \frac { 1 } { 2 } \right]$ is the initial distribution. It follows that + +$$ +\begin{array} { l } { { \displaystyle D _ { K L } \big ( \mathcal { Q } _ { u } \big \| \mathcal { Q } _ { u ^ { \prime } } \big ) = \mathbb { E } _ { u } \big [ \log \big ( \mathcal { Q } _ { u } / \mathcal { Q } _ { u ^ { \prime } } \big ) \big ] } \ ~ } \\ { { \displaystyle = K \cdot \sum _ { h = 1 } ^ { H } \mathbb { E } _ { u } \Big [ \log \big ( \big [ R _ { u , h } ( s _ { h } ^ { 1 } , a _ { h } ^ { 1 } ) \big ] ( r _ { h } ^ { 1 } ) \big / \big [ R _ { u ^ { \prime } , h } ( s _ { h } ^ { 1 } , a _ { h } ^ { 1 } ) \big ] ( r _ { h } ^ { 1 } ) \big ) \Big ] } \ ~ } \\ { { \displaystyle = \frac { K } { d } \sum _ { j = 1 } ^ { d - 2 } D _ { \mathrm { K L } } \Big ( \mathcal { N } \big ( \frac { \delta } { \sqrt { 2 d } } \langle e _ { j } , u _ { h } \rangle , 1 \big ) \ \Big \| \mathcal { N } \big ( \frac { \delta } { \sqrt { 2 d } } \langle e _ { j } , u _ { h } ^ { \prime } \rangle , 1 \big ) \Big ) . } } \end{array} +$$ + +If we take $\begin{array} { r } { \delta = \frac { c _ { 2 } d } { \sqrt { K } } } \end{array}$ , then inequality equation 29 ensures inequality equation 24, as claimed in the statement of the lemma. + +# E.5 CONNECTION TO THE UNCERTAINTY TERM + +Proof of Lemma E.3. We first calculate the explicit form of the inverse of variance-rescaled covariance matrix $\Lambda _ { h } ^ { \star , p }$ . For each time step $h \in [ H ]$ , the value function $V _ { u , h + 1 } ^ { \star }$ takes the form + +$$ +\begin{array} { r } { V _ { u , h + 1 } ^ { \star } = \mathbb { E } _ { \pi ^ { \star } } r _ { u , h + 1 } + \bigl ( \mathbb { P } _ { h + 1 } ^ { \pi ^ { \star } } V _ { u , h + 2 } ^ { \star } \bigr ) . } \end{array} +$$ + +Sinc $\begin{array} { r } { \colon \bigl ( \mathbb { P } _ { h + 1 } V _ { u , h + 2 } ^ { \star } \bigr ) ( + 1 ) = \bigl ( \mathbb { P } _ { h + 1 } V _ { u , h + 2 } ^ { \star } \bigr ) ( - 1 ) \mathrm { ~ a n d ~ } r _ { u , h + 1 } ( + 1 , a ) - r _ { u , h + 1 } ( - 1 , a ) = 2 / \sqrt { 6 } } \end{array}$ , we have + +$$ +\operatorname { V a r } _ { P _ { h } } ( V _ { u , h + 1 } ^ { \star } ) ( + 1 , a ) = \operatorname { V a r } _ { P _ { h } } ( \mathbb { E } _ { \pi ^ { \star } } r _ { u , h + 1 } ) ( + 1 , a ) = \frac 1 6 . +$$ + +Similarly, + +$$ +\operatorname { V a r } _ { P _ { h } } ( V _ { u , h + 1 } ^ { \star } ) ( - 1 , a ) = \operatorname { V a r } _ { P _ { h } } ( V _ { u , h + 1 } ^ { \star } ) ( + 1 , a ) = \frac { 1 } { 6 } . +$$ + +By routine calculation, we find that the population-level rescaled covariance matrix takes the form + +$$ +\Lambda _ { h } ^ { \star , p } = \frac { 3 K } { 2 } \left( \begin{array} { c c } { \frac { 2 } { d ^ { 2 } } \mathbf { I } _ { d - 2 } } & { \frac { 1 } { d \sqrt { d } } \mathbf { 1 } _ { ( d - 2 ) \times 2 } } \\ { \frac { 1 } { d \sqrt { d } } \mathbf { 1 } _ { 2 \times ( d - 2 ) } } & { \mathbf { I } _ { 2 } } \end{array} \right) \in \mathbb { R } ^ { d \times d } +$$ + +for any $h \in [ H ]$ . Applying Gaussian elimination on $\Lambda _ { h } ^ { \star , p }$ , we have + +$$ +( \Lambda _ { h } ^ { \star , p } ) ^ { - 1 } = \frac { 2 } { 3 K } \left( \begin{array} { c c } { { \frac { d ^ { 2 } } { 2 } \bigl \{ { \bf I } _ { d - 2 } + \frac { 1 } { d - 2 } { \bf 1 } _ { ( d - 2 ) \times ( d - 2 ) } \bigr \} } } & { { - \frac { d \sqrt { d } } { 2 ( d - 2 ) } { \bf 1 } _ { ( d - 2 ) \times 2 } } } \\ { { - \frac { d \sqrt { d } } { 2 ( d - 2 ) } { \bf 1 } _ { 2 \times ( d - 2 ) } } } & \frac { 1 } { d - 2 } \left( \begin{array} { c c } { { d - 1 } } & { { 1 } } \\ { { 1 } } & { { d - 1 } } \end{array} \right) \right) . \end{array} +$$ + +For each time step $h \in [ H ]$ , we have (by Jensen’s inequality) + +$$ +\sqrt { \mathbb { E } _ { \pi ^ { \star } } [ \phi ] ^ { \top } ( \Lambda _ { h } ^ { \star } ) ^ { - 1 } \mathbb { E } _ { \pi ^ { \star } } [ \phi ] } \le \frac { 1 } { 2 } \bigl \| \phi ( + 1 , u _ { h } ) \bigr \| _ { ( \Lambda _ { h } ^ { \star , p } ) ^ { - 1 } } + \frac { 1 } { 2 } \bigl \| \phi ( - 1 , u _ { h } ) \bigr \| _ { ( \Lambda _ { h } ^ { \star , p } ) ^ { - 1 } } . +$$ + +Recall that by our construction, + +$$ +\phi ( + 1 , u _ { h } ) = \left( \begin{array} { c c } { \frac { u _ { h } } { \sqrt { 2 d } } } \\ { \frac { 1 } { \sqrt { 2 } } } \\ { 0 } \end{array} \right) \in \mathbb R ^ { d } , \qquad \phi ( - 1 , u _ { h } ) = \left( \begin{array} { c c } { \frac { u _ { h } } { \sqrt { 2 d } } } \\ { 0 } \\ { \frac { 1 } { \sqrt { 2 } } } \end{array} \right) \in \mathbb R ^ { d } . +$$ + +It follows that + +$$ +\begin{array} { l } { \displaystyle \left\| \phi ( + 1 , u _ { h } ) \right\| _ { ( \Lambda _ { h } ^ { \star , p } ) ^ { - 1 } } ^ { 2 } = \left\| \phi ( - 1 , u _ { h } ) \right\| _ { ( \Lambda _ { h } ^ { \star , p } ) ^ { - 1 } } ^ { 2 } } \\ { \displaystyle = \frac 2 { 3 K } \biggl \{ \frac d 4 u _ { h } ^ { \top } \{ \mathbf I _ { d - 2 } + \frac 1 { d - 2 } \mathbf 1 _ { ( d - 2 ) \times ( d - 2 ) } \} u _ { h } - \frac d { 2 ( d - 2 ) } \mathbf 1 _ { d - 2 } ^ { \top } u _ { h } + \frac d { 2 ( d - 2 ) } \biggr \} } \\ { \displaystyle = \frac 2 { 3 K } \biggl \{ \frac { d ^ { 2 } } + \frac { d } { 4 ( d - 2 ) } \bigl ( 1 - \mathbf 1 _ { d - 2 } ^ { \top } u _ { h } \bigr ) ^ { 2 } + \frac 1 4 \biggr \} } \\ { \displaystyle \le \frac 2 { 3 K } \biggl \{ \frac { d ^ { 2 } } 4 + \frac { d ( d - 1 ) ^ { 2 } } { 4 ( d - 2 ) } + \frac 1 4 \biggr \} = \frac 2 { 3 K } \biggl \{ \frac { d ^ { 2 } } 2 + \frac { d - 1 } { 2 ( d - 2 ) } \biggr \} \lesssim d ^ { 2 } / K . } \end{array} +$$ + +Therefore, + +$$ +\sqrt { \mathbb { E } _ { \pi ^ { \star } } [ \phi ] ^ { \top } ( \Lambda _ { h } ^ { \star } ) ^ { - 1 } \mathbb { E } _ { \pi ^ { \star } } [ \phi ] } \lesssim d / \sqrt { K } . +$$ + +Taking the summation over $h \in [ H ]$ , we obtain the bound equation 26 as claimed in the lemma statement. + +# F A NUMERICAL SIMULATION + +# F.1 A LINEAR MDP CONSTRUCTION + +We consider a synthetic linear MDP example that is similar to Min et al. (2021) but with some modifications for the offline learning task. The MDP instance we use consists of $| S | = 2$ states and $| { \mathcal { A } } | = 1 0 0$ actions, and feature dimension $d = 1 0 0$ . We set ${ \mathcal { S } } = \{ 0 , 1 \}$ and $\mathcal { A } = \{ 0 , 1 , \ldots , 9 9 \}$ respectively. For each action $a \ \in \ \{ 0 , 1 , \ldots , 9 9 \}$ , we use binary encoding to obtain a vector $\textbf { a } \in \ \bar { \mathbb { R } } ^ { 8 }$ using its binary representation (i.e. each coordinate is either 0 or 1). we interchangebly use $a$ and and its vector representation a for the ease of explanation. We first define $\delta ( s , a ) = { \left\{ \begin{array} { l l } { 1 } & { { \mathrm { ~ i f ~ } } \mathbf { 1 } \{ s = 0 \} = \mathbf { 1 } \{ a = 0 \} } \\ { 0 } & { { \mathrm { ~ o t h e r w i s e } } } \end{array} \right. }$ , then the non-stationary linear MDP is specified by the following configuration + +• Feature mapping: + +$$ +\phi ( s , a ) = \left( \mathbf { a } ^ { \top } , \delta ( s , a ) , 1 - \delta ( s , a ) \right) ^ { \top } \in \mathbb { R } ^ { 1 0 } +$$ + +• The true measure $\nu _ { h }$ + +$$ +\pmb { \nu } _ { h } ( s ) = ( 0 , \dots , 0 , ( 1 - s ) \oplus \alpha _ { h } , s \oplus \alpha _ { h } ) , +$$ + +where $\{ \alpha _ { h } \} _ { h \in [ H ] }$ is a sequence of integers taking values 0 or 1 and $\oplus$ is the standard XOR operator. We define + +$$ +\theta _ { h } \equiv ( 0 , \ldots , 0 , r , 1 - r ) \in \mathbb { R } ^ { 1 0 } +$$ + +with the choice of $r = 0 . 9$ . The transition follows $P _ { h } ( s ^ { \prime } | s , a ) = \langle \phi ( s , a ) , \nu _ { h } ( s ^ { \prime } ) \rangle$ and the mean reward function $r _ { h } ( s , a ) = \langle \phi ( s , a ) , \theta _ { h } \rangle$ . + +• Behavior policy: always choose action $a = 0$ with probability $p$ , and other actions uniformly with probability $( 1 - p ) / 9 9$ . The initial distribution chooses $s = 0$ and $s = 1$ with equal probability $1 / 2$ . We use $p = 0 . 6$ . + +![](images/34259f6dd9b65de1f148aa38cd9fca4ea9b5c62a39196fb77cf679acac11f205.jpg) +Figure 1: Comparison between PEVI and VAPVI in the non-stationary linear MDP instance described above. In each figure, y-axis denotes suboptimality gap $v ^ { \star } - v ^ { \widehat { \pi } }$ , $\mathbf { X }$ -axis denotes number of episodes $K$ . The problem horizons are fixed to be $H = 2 0 , 3 0 , 5 0$ . The solid line denotes the average suboptimality gap over 50 trials and the error bar area is the corresponding standard deviation. The range of $K$ is from 5 to 1000. + +# F.2 EMPIRICAL COMPARISON BETWEEN PEVI AND VAPVI ON THE CONSTRUCTED LINEAR MDP + +We compare Pessimistic Value Iteration (PEVI) in Jin et al. (2021b) and our VAPVI Algorithm 1 in Figure 1, with horizon to be $H = 2 0$ , 30, 50. In addition, we add the non-pessimistic version for both algorithms, i.e. least-square value iteration (LSVI) and variance-aware value iteration (VAVI). The true optimal value $v ^ { \star }$ is computed via value iteration using the underlying transition kernels. For the empirical validation of VAPVI, we do not split the data and, in particular, in all the methods we choose $\lambda = 0 . 0 1$ (instead of $\lambda = 1$ used in theory (Jin et al., 2021b) which causes over-regularization in the simulation). + +We can observe VAPVI outperforms PEVI and the gap becomes larger when horizon $H$ increases. One main reason for this to happen is due to the bonus used in PEVI (Jin et al., 2021b) + +$$ +O \left[ d H \cdot \left( \phi ( \cdot , \cdot ) ^ { \top } \Sigma _ { h } ^ { - 1 } \phi ( \cdot , \cdot ) \right) ^ { 1 / 2 } \right] +$$ + +is overly pessimistic comparing to our + +$$ +O \left[ \sqrt { d } \cdot \left( \phi ( \cdot , \cdot ) ^ { \top } \Lambda _ { h } ^ { - 1 } \phi ( \cdot , \cdot ) \right) ^ { 1 / 2 } \right] +$$ + +when $H$ becomes larger and this could potentially make the learning less accurate. In addition, both non-pessimistic algorithms exhibit similar accuracy, and this is partially owing to our truncation scheme $\widehat { \sigma } _ { h } ( \cdot , \cdot ) ^ { 2 } \gets \operatorname* { m a x } \{ 1 , \widehat { \mathrm { V a r } } _ { P _ { h } } \widehat { V } _ { h + 1 } ( \cdot , \cdot ) \}$ so $\widehat { \sigma } _ { h } ( \cdot , \cdot ) ^ { 2 }$ will just be 1 when the estimated variance b bis small. Lastly, variance-aware pessimism eventually outperforms non-pessimism algorithms when sample size is large and this might come from that the pessimistic bonus is estimated more accurately when more samples are collected. + +# G SOME MISSING DERIVATIONS AND DISCUSSIONS + +# G.1 REGARDING COVERAGE ASSUMPTION + +Now we discuss the feature coverage assumption. Indeed, even if Assumption 2.2 is not satisfied, we can still learn in the effective subspan of $\dot { \Sigma } _ { h } ^ { p } : = \mathbb { E } _ { \boldsymbol { \mu } , h } \left[ \boldsymbol { \phi } ( \boldsymbol { s } , \boldsymbol { a } ) \boldsymbol { \phi } ( \boldsymbol { s } , \boldsymbol { a } ) ^ { \top } \right] ^ { - }$ . Concretely, since $\Sigma _ { h } ^ { p }$ is symmetric, by orthogonal decomposition we have $\Sigma _ { h } ^ { p } = Z _ { h } \Lambda Z _ { h } ^ { \top }$ , where $Z _ { h }$ (can be estimated using the samples for practical purpose) consists of orthogonal basis and $\Lambda$ consists of eigenvalues of $\Sigma _ { h } ^ { p }$ in the diagonal. Suppose we do not have a full coverage, i.e. + +$$ +\Lambda = \mathrm { d i a g } [ \lambda _ { 1 } , \lambda _ { 2 } , . . . , \lambda _ { d ^ { \prime } } , 0 , . . . , 0 ] \quad \mathrm { w i t h } \quad d ^ { \prime } < d +$$ + +then we can create transformed features $\phi _ { h } ^ { \prime } ( s , a ) = Z _ { h } \cdot \phi _ { h } ( s , a )$ , and then + +$$ +\mathbb { E } _ { \mu , h } \left[ \phi _ { h } ^ { \prime } ( s , a ) \phi _ { h } ^ { \prime } ( s , a ) ^ { \top } \right] = \Lambda = \mathrm { d i a g } [ \lambda _ { 1 } , \lambda _ { 2 } , . . . , \lambda _ { d ^ { \prime } } , 0 , . . . , 0 ] . +$$ + +Then we can do learning w.r.t. the truncated features $\phi _ { h } ^ { \prime } | _ { 1 : d ^ { \prime } }$ ’s instead of the original $\phi$ . It reduces to the weaker notion of $\begin{array} { r } { \kappa : = \operatorname* { m i n } _ { h \in [ H ] } \{ \kappa _ { h } : s . t } \end{array}$ . $\kappa _ { h } =$ smallest positive eigenvalue at time $h \}$ . + +When reducing Theorem 3.2,3.3 to the tabular case, set $\phi ( s , a ) = \mathbf { 1 } _ { s , a }$ , $d = S A$ , $\lambda = 0$ , and recall by Assumption 3.4 (let’s assume $\pi ^ { \star }$ is a deterministic policy as it always exists in tabular MDP) $\begin{array} { r } { C ^ { \star } : = \operatorname* { s u p } _ { h , s , a } { d _ { h } ^ { \pi ^ { \star } } ( s , a ) / d _ { h } ^ { \mu } ( s , a ) } } \end{array}$ , then for Theorem 3.2 + +$$ +\begin{array} { r l } & { \mathcal { U } _ { \ge } \displaystyle \sum _ { u = 1 } ^ { N } \frac { \mu } { \mu } \left. \nabla ^ { \mu } u ^ { \mu } ( x ) \frac { \mu } { \mu } \nabla ^ { \nu } \cdot \nabla ^ { \mu } u ^ { \nu } \right. _ { \perp } - \mathcal { U } _ { \ge } \displaystyle \sum _ { u = 1 } ^ { N } \frac { \mu } { \mu } \sum _ { u = 1 } ^ { N } \mathcal { U } _ { \ge } \nu ( u ) \hat { \nu } ^ { \mu } \mathbb { I } _ { \ge } \mathbb { I } _ { \ge u } ^ { \nu } } \\ & { - \mathcal { U } _ { \ge } \displaystyle \sum _ { u = 1 } ^ { N } \frac { \mu } { \mu } \sum _ { u = 1 } ^ { N } \mathcal { U } _ { \ge } \mathbb { I } _ { \ge u } ^ { \nu } \mathbb { I } _ { \ge u } ^ { \nu } \mathbb { I } _ { \ge u } ^ { \nu } \mathbb { I } _ { \ge u } ^ { \nu } } \\ & { = - \mathcal { U } _ { \ge } \displaystyle \sum _ { u = 1 } ^ { N } \mathbb { I } _ { \ge u \ne 1 } ^ { N } \mathbb { I } _ { \ge u } ^ { \nu } \mathbb { I } _ { \ge u } ^ { \nu } \mathbb { I } _ { \ge u } ^ { \nu } \mathbb { I } _ { \ge u } ^ { \nu } } \\ & { \le \mathcal { U } _ { \ge } \displaystyle \sum _ { u = 1 } ^ { N } \mathbb { I } _ { \ge u \ne 1 } ^ { N } \mathbb { I } _ { \ge u \ne 1 } ^ { N } \mathbb { I } _ { \ge u \ne 1 } ^ { N } \mathbb { I } _ { \ge u \ne 1 } ^ { N } \mathbb { I } _ { \ge u \ne 1 } ^ { N } - \mathbb { I } _ { \ge } ^ { N } \mathbb { I } _ { \ge u \ne 1 } ^ { N } \mathbb { I } _ { \ge u \ne 1 } ^ { N } } \\ & { \le \mathcal { U } _ { \ge } \displaystyle \sum _ { u = 1 } ^ { N } \mathbb { I } _ { \ge u \ne 1 } ^ { N } \mathbb { I } _ { \ge u \ne 1 } ^ { N } \mathbb { I } _ { \ge u \ne 1 } ^ { N } \mathbb { I } _ { \ge u \ne 1 } ^ { N } \mathbb { I } _ { \ge u \ne 1 } ^ { N } \mathbb { I } _ { \ge u \ne 1 } ^ { N } } \\ & - \mathcal { U } _ { \ge } \displaystyle \sum _ { u = 1 } ^ { N } \mathbb { I } _ { \ge u \ne 1 } ^ \end{array} +$$ + +where the first inequality is by Chernoff bound and the last one is by Lemma 3.4. of Yin and Wang (2020) (Law of total variances). The rest of them are from Cauchy’s inequality. Similarly, for Theorem 3.3, we also have + +$$ +\begin{array} { r l } & { \forall \widehat { u } _ { \xi } ^ { \mu } = \sqrt { \frac { \mu } { \mu } \sqrt { \mu } \frac { \sqrt { \mu } } { \mu } \left\{ \Gamma ^ { 2 } + \widehat { u } _ { \xi } ^ { \mu } \left[ \Gamma ^ { 2 } + \widehat { u } _ { \xi } ^ { \mu } - \widehat { u } _ { \xi } ^ { \mu } \right] \right\} } } \\ & = \sqrt { \widehat { u } _ { \xi } ^ { \mu } - \widehat { u } _ { \xi } ^ { \mu } } \sqrt { \Gamma ^ { 2 } + \widehat { u } _ { \xi } ^ { \mu } \left[ \Gamma ^ { 2 } + \widehat { u } _ { \xi } ^ { \mu } - \widehat { u } _ { \xi } ^ { \mu } \right] \right\} \sqrt { \textnormal { w a r } \Gamma ^ { 2 } + \widehat { u } _ { \xi } ^ { \mu } \left[ \Gamma ^ { 2 } + \widehat { u } _ { \xi } ^ { \mu } - \widehat { u } _ { \xi } ^ { \mu } \right] } } \\ & { \quad - \sqrt { \widehat { u } _ { \xi } ^ { \mu } - \widehat { u } _ { \xi } ^ { \mu } } \sqrt { \Gamma ^ { 2 } + \widehat { u } _ { \xi } ^ { \mu } \left[ \Gamma ^ { 2 } + \widehat { u } _ { \xi } ^ { \mu } - \widehat { u } _ { \xi } ^ { \mu } \right] } } \\ & = - \sqrt { \widehat { u } _ { \xi } ^ { \mu } - \widehat { u } _ { \xi } ^ { \mu } } \sqrt { \frac { \sqrt { \mu } } { \mu } \sqrt { \mu } \frac { \sqrt { \mu } \left\{ \Gamma ^ { 2 } + \widehat { u } _ { \xi } ^ { \mu } \left[ \Gamma ^ { 2 } + \widehat { u } _ { \xi } ^ { \mu } - \widehat { u } _ { \xi } ^ { \mu } \right] \right\} } } \\ & \quad - \sqrt { \widehat { u } _ { \xi } ^ { \mu } - \widehat { u } _ { \xi } ^ { \mu } } \sqrt { \frac { \sqrt { \mu } } { \mu } \sqrt { \mu } \frac { \sqrt { \mu } \left\{ \Gamma ^ { 2 } + \widehat { u } _ { \xi } ^ { \mu } \left[ \Gamma ^ { 2 } + \widehat { u } _ { \xi } ^ { \mu } - \widehat { u } _ { \xi } ^ { \mu } \right] } { \Delta \xi \sqrt { \mu } } } \\ & \quad \leq \sqrt { \widehat { u } _ { \xi } \sqrt { \mu } \frac { \mu } { \mu } } \sqrt \ \end{array} +$$ + +# H AUXILIARY LEMMAS + +Lemma H.1 (Matrix McDiarmid inequality / Matrix Chernoff bound (Tropp, 2012)). Let $z _ { k }$ , $k =$ $1 , \ldots , K$ be independent random vectors in $\mathbb { R } ^ { d }$ , and let $H$ be a mapping that maps $K$ vectors to $a$ $d \times d$ symmetric matrix. Assume there exists a sequence of fixed symmetric matrices $\{ A _ { k } \} _ { k \in [ K ] }$ such that for $z _ { k } , z _ { k } ^ { \prime }$ ranges over all possible values for each $k \in [ K ]$ , it holds + +$$ +( H ( z _ { 1 } , \ldots , z _ { k } , \ldots , z _ { K } ) - H ( z _ { 1 } , \ldots , z _ { k } ^ { \prime } , \ldots , z _ { K } ) ) ^ { 2 } \preceq A _ { k } ^ { 2 } . +$$ + +Define $\textstyle \sigma ^ { 2 } : = \left\| \sum _ { k } A _ { k } ^ { 2 } \right\|$ . Then for any $t > 0$ , + +$$ +\begin{array} { r } { \mathbb { P } \left\{ \| H ( z _ { 1 } , \ldots , z _ { K } ) - \mathbb { E } H ( z _ { 1 } , \ldots , z _ { K } ) \| \ge t \right\} \le d \cdot \exp \left( \frac { - t ^ { 2 } } { 8 \sigma ^ { 2 } } \right) } \end{array} +$$ + +Lemma H.2 (Hoeffding inequality for self-normalized martingales (Abbasi-Yadkori et al., 2011)). Let $\{ \eta _ { t } \} _ { t = 1 } ^ { \infty }$ be a real-valued stochastic process. Let $\{ \mathcal { F } _ { t } \} _ { t = 0 } ^ { \infty }$ be a filtration, such that $\eta _ { t }$ is $\mathcal { F } _ { t }$ - measurable. Assume $\eta _ { t }$ also satisfies $\eta _ { t }$ given $\mathcal { F } _ { t - 1 }$ is zero-mean and $R$ -subgaussian, i.e. + +$$ +\forall \lambda \in \mathbb { R } , \quad \mathbb { E } \left[ e ^ { \lambda \eta _ { t } } \mid \mathcal { F } _ { t - 1 } \right] \le e ^ { \lambda ^ { 2 } R ^ { 2 } / 2 } +$$ + +Let $\begin{array} { r } { \Lambda _ { t } = \lambda I _ { d } + \sum _ { s = 1 } ^ { t } x _ { s } x _ { s } ^ { \top } } \end{array}$ $\{ x _ { t } \} _ { t = 1 } ^ { \infty }$ be an $\mathbb { R } ^ { d }$ -valued stochastic process where . Then for any $\delta > 0$ t , with probability is $\mathcal { F } _ { t - 1 }$ $1 - \delta _ { : }$ measurable and , for all $t > 0$ , $\| x _ { t } \| \leq L$ . Let + +$$ +\left\| \sum _ { s = 1 } ^ { t } x _ { s } \eta _ { s } \right\| _ { \Lambda _ { t } ^ { - 1 } } ^ { 2 } \leq 8 R ^ { 2 } \cdot \frac { d } { 2 } \log \left( \frac { \lambda + t L } { \lambda \delta } \right) . +$$ + +Lemma H.3 (Bernstein inequality for self-normalized martingales (Zhou et al., 2021a)). Let $\{ \eta _ { t } \} _ { t = 1 } ^ { \infty }$ be a real-valued stochastic process. Let $\{ \mathcal { F } _ { t } \} _ { t = 0 } ^ { \infty }$ be a filtration, such that $\eta _ { t }$ is $\mathcal { F } _ { t }$ -measurable. Assume $\eta _ { t }$ also satisfies + +$$ +| \eta _ { t } | \leq R , \mathbb { E } \left[ \eta _ { t } \mid \mathcal { F } _ { t - 1 } \right] = 0 , \mathbb { E } \left[ \eta _ { t } ^ { 2 } \mid \mathcal { F } _ { t - 1 } \right] \leq \sigma ^ { 2 } . +$$ + +Let $\{ x _ { t } \} _ { t = 1 } ^ { \infty }$ be an $\mathbb { R } ^ { d }$ -valued stochastic process where $x _ { t }$ is $\mathcal { F } _ { t - 1 }$ measurable and $\| x _ { t } \| \leq L$ . Let $\begin{array} { r } { \Lambda _ { t } = \lambda I _ { d } + \sum _ { s = 1 } ^ { t } x _ { s } x _ { s } ^ { \top } } \end{array}$ . Then for any $\delta > 0$ , with probability $1 - \delta _ { : }$ , for all $t > 0$ , + +$$ +\left\| \sum _ { s = 1 } ^ { t } \mathbf { x } _ { s } \eta _ { s } \right\| _ { \Lambda _ { t } ^ { - 1 } } \leq 8 \sigma \sqrt { d \log \left( 1 + \frac { t L ^ { 2 } } { \lambda d } \right) \cdot \log \left( \frac { 4 t ^ { 2 } } { \delta } \right) } + 4 R \log \left( \frac { 4 t ^ { 2 } } { \delta } \right) +$$ + +Lemma H.4 (Converting the variance under the matrix norm). Let $\Lambda _ { 1 }$ and $\Lambda _ { 2 } \in \mathbb { R } ^ { d \times d }$ are two positive semi-definite matrices. Then: + +$$ +\left\| \Lambda _ { 1 } ^ { - 1 } \right\| \leq \left\| \Lambda _ { 2 } ^ { - 1 } \right\| + \left\| \Lambda _ { 1 } ^ { - 1 } \right\| \cdot \left\| \Lambda _ { 2 } ^ { - 1 } \right\| \cdot \left\| \Lambda _ { 1 } - \Lambda _ { 2 } \right\| +$$ + +and + +$$ +\left\| \phi \right\| _ { \Lambda _ { 1 } ^ { - 1 } } \leq \left[ 1 + \sqrt { \left\| \Lambda _ { 2 } ^ { - 1 } \right\| \left\| \Lambda _ { 2 } \right\| \cdot \left\| \Lambda _ { 1 } ^ { - 1 } \right\| \cdot \left\| \Lambda _ { 1 } - \Lambda _ { 2 } \right\| } \right] \cdot \left\| \phi \right\| _ { \Lambda _ { 2 } ^ { - 1 } } . +$$ + +for all $\phi \in \mathbb { R } ^ { d }$ + +Proof. For the first part, note + +$$ +\left\| \Lambda _ { 1 } ^ { - 1 } \right\| \leq \left\| \Lambda _ { 2 } ^ { - 1 } \right\| + \left\| \Lambda _ { 1 } ^ { - 1 } - \Lambda _ { 2 } ^ { - 1 } \right\| \leq \left\| \Lambda _ { 2 } ^ { - 1 } \right\| + \left\| \Lambda _ { 2 } ^ { - 1 } \right\| \left\| \Lambda _ { 1 } - \Lambda _ { 2 } \right\| \left\| \Lambda _ { 1 } ^ { - 1 } \right\| +$$ + +For the second one, + +$$ +\begin{array} { r l } & { \left\| \phi \right\| _ { \Lambda _ { 1 } ^ { - 1 } } = \sqrt { \phi ^ { \top } \Lambda _ { 1 } ^ { - 1 } \phi } = \sqrt { \phi ^ { \top } \left( \Lambda _ { 1 } ^ { - 1 } - \Lambda _ { 2 } ^ { - 1 } \right) \phi + \phi ^ { \top } \Lambda _ { 2 } ^ { - 1 } \phi } } \\ & { = \sqrt { \phi ^ { \top } \Lambda _ { 2 } ^ { - 1 / 2 } \left( \Lambda _ { 2 } ^ { 1 / 2 } \Lambda _ { 1 } ^ { - 1 } \Lambda _ { 2 } ^ { 1 / 2 } - I + I \right) \Lambda _ { 2 } ^ { - 1 / 2 } \phi } \leq \sqrt { \| \phi \| _ { \Lambda _ { 2 } ^ { - 1 } } \cdot \left( 1 + \left\| \Lambda _ { 2 } ^ { 1 / 2 } \Lambda _ { 1 } ^ { - 1 } \Lambda _ { 2 } ^ { 1 / 2 } - I \right\| \right) \| \phi \| _ { \Lambda _ { 2 } ^ { - 1 } } } } \\ & { \leq \left( 1 + \left\| \Lambda _ { 2 } ^ { 1 / 2 } \Lambda _ { 1 } ^ { - 1 } \Lambda _ { 2 } ^ { 1 / 2 } - I \right\| ^ { 1 / 2 } \right) \cdot \left\| \phi \right\| _ { \Lambda _ { 2 } ^ { - 1 } } = \left( 1 + \left\| \Lambda _ { 2 } ^ { 1 / 2 } \Lambda _ { 1 } ^ { - 1 } \left( \Lambda _ { 2 } - \Lambda _ { 1 } \right) \Lambda _ { 2 } ^ { - 1 } \Lambda _ { 2 } ^ { 1 / 2 } \right\| ^ { 1 / 2 } \right) \cdot \left\| \phi \right\| _ { \Lambda _ { 2 } ^ { - 1 } } } \\ & { \leq \left( 1 + \sqrt { \| \Lambda _ { 2 } \| \left\| \Lambda _ { 1 } ^ { - 1 } \right\| \left\| \Lambda _ { 2 } ^ { - 1 } \right\| \left\| \Lambda _ { 1 } ^ { - 1 } - \Lambda _ { 2 } \right\| } \right) \cdot \left\| \phi \right\| _ { \Lambda _ { 2 } ^ { - 1 } } } \end{array} +$$ + +Lemma H.5 (Lemma H.4 of Min et al. (2021)). let $\phi : \mathcal { S } \times \mathcal { A } \mathbb { R } ^ { d }$ satisfies $\| \phi ( s , a ) \| \leq C$ for $s , a \in \mathcal S \times \mathcal A$ . For any $K > 0 , \lambda > 0 ,$ , define $\begin{array} { r } { \bar { G } _ { K } = \sum _ { k = 1 } ^ { K } \phi ( s _ { k } , a _ { k } ) \phi ( s _ { k } , \underline { { a } } _ { k } ) ^ { \top } + \lambda I _ { d } } \end{array}$ where $( s _ { k } , a _ { k } )$ $\nu$ $1 - \delta$ + +$$ +\left\| \frac { { { { \bar { G } } _ { K } } } } { K } - \mathbb { E } _ { \nu } \left[ \frac { { { { \bar { G } } _ { K } } } } { K } \right] \right\| \le \frac { 4 \sqrt 2 C ^ { 2 } } { \sqrt K } \left( \log \frac { { 2 d } } { \delta } \right) ^ { 1 / 2 } . +$$ + +Proof of Lemma $H . 5$ . For completeness, we provide the proof of Lemma H.5. Let $x _ { k } = \phi ( s _ { k } , a _ { k } )$ . Denote $\widetilde { \Sigma } _ { h }$ as the matrix obtained by replacing the $\mathrm { k }$ -th vector $x _ { k }$ in $\widehat { \Sigma } _ { h }$ by $\widetilde { x } _ { k }$ and leaving the rest $K - 1$ vectors unchanged. Then + +$$ +\left( \frac { \widehat \Sigma _ { h } } { K } - \frac { \widetilde \Sigma _ { h } } { K } \right) ^ { 2 } = \left( \frac { x _ { k } x _ { k } ^ { \top } - \widetilde x _ { k } \widetilde x _ { k } ^ { \top } } { K } \right) \preceq \frac { 1 } { K ^ { 2 } } \left( 2 x _ { k } x _ { k } ^ { \top } x _ { k } x _ { k } ^ { \top } + 2 \widetilde x _ { k } \widetilde x _ { k } ^ { \top } \widetilde x _ { k } \widetilde x _ { k } ^ { \top } \right) \preceq \frac { 4 C ^ { 4 } } { K ^ { 2 } } I _ { d } : = A _ { k } ^ { 2 } . +$$ + +Notice that $\begin{array} { r } { \left\| \sum _ { k } ^ { K } A _ { k } ^ { 2 } \right\| = \frac { 4 C ^ { 4 } } { K } } \end{array}$ , by Lemma H.1 we have the result. + +Lemma H.6 (Lemma H.5. of Min et al. (2021)). Let $\| \phi \| _ { 2 } \leq C$ . Define $\begin{array} { r } { \bar { G } _ { K } = \sum _ { k = 1 } ^ { K } \phi ( s _ { k } , a _ { k } ) \phi ( s _ { k } , a _ { k } ) ^ { \top } + \lambda I _ { d } } \end{array}$ $\phi : \mathcal { S } \times \mathcal { A } \mathbb { R } ^ { d }$ where $( s _ { k } , a _ { k } )$ be a bounded function s.t. ’s are i.i.d samples from some distribution $\nu$ . Let $G = \mathbb { E } _ { \nu } [ \phi ( s , a ) \phi ( s , a ) ^ { \top } ]$ . Then for any $\delta \in ( 0 , 1 )$ , if $K$ satisfies + +$$ +K \geq \operatorname* { m a x } \left\{ 5 1 2 C ^ { 4 } \left\| \mathbf { G } ^ { - 1 } \right\| ^ { 2 } \log \left( \frac { 2 d } { \delta } \right) , 4 \lambda \left\| \mathbf { G } ^ { - 1 } \right\| \right\} . +$$ + +Then with probability at least $1 - \delta$ , it holds simultaneously for all $u \in \mathbb { R } ^ { d }$ that + +$$ +\| u \| _ { \bar { G } _ { K } ^ { - 1 } } \leq \frac { 2 } { \sqrt { K } } \| u \| _ { G ^ { - 1 } } . +$$ + +Lemma H.7 (Extended Value Difference (Section B.1 in Cai et al. (2020))). Let $\pi = \{ \pi _ { h } \} _ { h = 1 } ^ { H }$ and $\pi ^ { \prime } = \{ \pi _ { h } ^ { \prime } \} _ { h = 1 } ^ { H }$ be two arbitrary policies and let $\{ \widehat { Q } _ { h } \} _ { h = 1 } ^ { H }$ be any given $Q$ -functions. Then define $\widehat { V } _ { h } ( s ) : = \langle \widehat { Q } _ { h } ( s , \cdot ) , \pi _ { h } ( \cdot \mid s ) \rangle$ for all $s \in S$ . Then for all $s \in S$ , + +$$ +\begin{array} { r } { \widehat { V } _ { 1 } ( s ) - V _ { 1 } ^ { \pi ^ { \prime } } ( s ) = \displaystyle \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \pi ^ { \prime } } \left[ \langle \widehat { Q } _ { h } \left( s _ { h } , \cdot \right) , \pi _ { h } \left( \cdot \mid s _ { h } \right) - \pi _ { h } ^ { \prime } \left( \cdot \mid s _ { h } \right) \rangle \mid s _ { 1 } = s \right] } \\ { + \displaystyle \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \pi ^ { \prime } } \left[ \widehat { Q } _ { h } \left( s _ { h } , a _ { h } \right) - \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } , a _ { h } \right) \mid s _ { 1 } = s \right] } \end{array} +$$ + +where $( T _ { h } V ) ( \cdot , \cdot ) : = r _ { h } ( \cdot , \cdot ) + ( P _ { h } V ) ( \cdot , \cdot ) .$ for any $V \in \mathbb { R } ^ { S }$ . + +Proof. Denote $\xi _ { h } = \widehat { Q } _ { h } - \mathcal { T } _ { h } \widehat { V } _ { h + 1 }$ . For any $h \in [ H ]$ , we have + +$$ +\begin{array} { r l } & { \widehat { V } _ { h } - V _ { h } ^ { \pi ^ { \prime } } = \langle \widehat { Q } _ { h } , \pi _ { h } \rangle - \langle Q _ { h } ^ { \pi ^ { \prime } } , \pi _ { h } ^ { \prime } \rangle } \\ & { \qquad = \langle \widehat { Q } _ { h } , \pi _ { h } - \pi _ { h } ^ { \prime } \rangle + \langle \widehat { Q } _ { h } - Q _ { h } ^ { \pi ^ { \prime } } , \pi _ { h } ^ { \prime } \rangle } \\ & { \qquad = \langle \widehat { Q } _ { h } , \pi _ { h } - \pi _ { h } ^ { \prime } \rangle + \langle P _ { h } ( \widehat { V } _ { h + 1 } - V _ { h + 1 } ^ { \pi ^ { \prime } } ) + \xi _ { h } , \pi _ { h } ^ { \prime } \rangle } \\ & { \qquad = \langle \widehat { Q } _ { h } , \pi _ { h } - \pi _ { h } ^ { \prime } \rangle + \langle P _ { h } ( \widehat { V } _ { h + 1 } - V _ { h + 1 } ^ { \pi ^ { \prime } } ) , \pi _ { h } ^ { \prime } \rangle + \langle \xi _ { h } , \pi _ { h } ^ { \prime } \rangle } \end{array} +$$ + +reof y apply the above for we can finish the pro $\widehat { V } _ { h + 1 } - V _ { h + 1 } ^ { \pi ^ { \prime } }$ and use the ma. $\mathbb { E } _ { \pi ^ { \prime } }$ notation (instead of the inner product $P _ { h } , \pi _ { h } ^ { \prime } )$ + +Lemma H.8. Let $\widehat { \pi } = \left\{ \widehat { \pi } _ { h } \right\} _ { h = 1 } ^ { H }$ and $\widehat { Q } _ { h } ( \cdot , \cdot )$ be the arbitrary policy and $Q$ -function and also $\widehat { V } _ { h } ( s ) = \langle \widehat { Q } _ { h } ( s , \cdot ) , \widehat { \pi } _ { h } ( \cdot | s ) \rangle \vee _ { s } ^ { \cdot } \in \mathcal { S }$ . and $\zeta _ { h } ( s , a ) : = ( \mathscr { T } _ { h } \widehat { V } _ { h + 1 } ) ( s , a ) - \widehat { Q } _ { h } ( s , a )$ (element-wisely) + +to be the Bellman update error. Then for any arbitrary $\pi$ , we have + +$$ +\begin{array} { c } { { { \displaystyle V _ { 1 } ^ { \pi } ( s ) - V _ { 1 } ^ { \widehat { \pi } } ( s ) = \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \pi } \left[ \zeta _ { h } ( s _ { h } , a _ { h } ) \mid s _ { 1 } = s \right] - \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \widehat { \pi } } \left[ \zeta _ { h } ( s _ { h } , a _ { h } ) \mid s _ { 1 } = s \right] } } } \\ { { { + \displaystyle \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \pi } \left[ \left. \widehat { Q } _ { h } \left( s _ { h } , \cdot \right) , \pi _ { h } \left( \cdot | s _ { h } \right) - \widehat { \pi } _ { h } \left( \cdot | s _ { h } \right) \right. \mid s _ { 1 } = x \right] } } } \end{array} +$$ + +where the expectation are taken over $s _ { h } , a _ { h }$ . + +Proof. Note the gap can be rewritten as + +$$ +\begin{array} { r } { V _ { 1 } ^ { \pi } ( s ) - V _ { 1 } ^ { \widehat { \pi } } ( s ) = V _ { 1 } ^ { \pi } ( s ) - \widehat { V } _ { 1 } ( s ) + \widehat { V } _ { 1 } ( s ) - V _ { 1 } ^ { \widehat { \pi } } ( s ) . } \end{array} +$$ + +By Lemma H.7 with $\pi = { \widehat { \pi } }$ , $\pi ^ { \prime } = \pi$ , we directly have + +$$ +\gamma _ { 1 } ^ { \pi } ( s ) - \widehat { V } _ { 1 } ( s ) = \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \pi } \left[ \zeta _ { h } ( s _ { h } , a _ { h } ) \mid s _ { 1 } = s \right] + \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \pi } \left[ \langle \widehat { Q } _ { h } \left( s _ { h } , \cdot \right) , \pi _ { h } \left( \cdot | s _ { h } \right) - \widehat { \pi } _ { h } \left( \cdot | s _ { h } \right) \rangle \mid s _ { 1 } = s \right] +$$ + +Next apply Lemma H.7 again with $\pi = \pi ^ { \prime } = \widehat { \pi }$ , we directly have + +$$ +\widehat { V } _ { 1 } ( s ) - V _ { 1 } ^ { \widehat { \pi } } ( s ) = - \sum _ { h = 1 } ^ { H } \mathbb { E } _ { \widehat { \pi } } \left[ \zeta _ { h } ( s _ { h } , a _ { h } ) \mid s _ { 1 } = s \right] . +$$ + +Combine the above two results we prove the stated result. + +Lemma H.9. For a linear MDP, for any $0 \le V ( \cdot ) \le H$ , then there exists a $w _ { h } \in \mathbb { R } ^ { d }$ s.t. $\mathcal { T } _ { h } V =$ +$\langle \phi , w _ { h } \rangle$ and $\| w _ { h } \| _ { 2 } ~ \le ~ 2 H \sqrt { d }$ for all $h \in [ H ]$ . Here ${ \mathcal { T } } _ { h } ( V ) ( s , a ) = r _ { h } ( x , a ) + ( P _ { h } V ) ( s , a )$ . +Similarly, for any $\pi$ , there exists $w _ { h } ^ { \pi } \in \mathbb { R } ^ { d }$ , such that $Q _ { h } ^ { \pi } = \langle \phi , w _ { h } ^ { \pi } \rangle$ with $\| w _ { h } ^ { \pi } \| _ { 2 } \leq 2 ( H - h + 1 ) \sqrt { d }$ . + +Proof. By definition, + +$$ +\begin{array} { c } { { \displaystyle { \mathcal { T } } _ { h } V = r _ { h } + ( P _ { h } V ) = \langle \phi , \theta _ { h } \rangle + \langle \phi , \displaystyle { \int _ { S } V ( s ) d \nu _ { h } ( s ) \rangle } } } \\ { { \displaystyle { \Rightarrow } w _ { h } = \theta _ { h } + \int _ { S } V ( s ) d \nu _ { h } ( s ) , } } \end{array} +$$ + +therefore $\| w _ { h } \| _ { 2 } \leq \| \theta _ { h } \| _ { 2 } + H \cdot \| \nu _ { h } ( S ) \| \leq 1 + H { \sqrt { d } } \leq 2 H { \sqrt { d } }$ . The proof of the second part is similar by backward induction and the fact $V _ { h } ^ { \pi } \leq H - h + 1$ for any $\pi$ . □ + +Lemma H.10. For any pessimistic bonus design $\Gamma _ { h }$ , suppose $K > \operatorname* { m a x } \{ \mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } , \mathcal { M } _ { 3 } , \mathcal { M } _ { 4 } \}$ , then with probability $1 - \delta$ , Algorithm $^ { l }$ yields + +$$ +\left\| \mathcal { T } _ { h } \widehat { V } _ { h + 1 } - \widehat { \mathcal { T } } _ { h } \widehat { V } _ { h + 1 } \right\| _ { \infty } \leq \widetilde { O } ( \frac { H ^ { 2 } \sqrt { d / \kappa } } { \sqrt { K } } ) +$$ + +Proof of Lemma H.10. Suppose $w _ { h }$ is the coefficient corresponding to the $\mathcal { T } _ { h } \widehat { V } _ { h + 1 }$ (such $w _ { h }$ exists by Lemma H.9), i.e. $\mathcal { T } _ { h } \widehat { V } _ { h + 1 } = \phi ^ { \top } w _ { h }$ , and recall $( \widehat { T } _ { h } \widehat { V } _ { h + 1 } ) ( s , a ) = \phi ( s , a ) ^ { \top } \widehat { w } _ { h }$ , then: + +$$ +\begin{array} { r l } & { \quad \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) ( s , a ) - \left( \widehat { \mathcal { T } } _ { h } \widehat { V } _ { h + 1 } \right) ( s , a ) = \phi ( s , a ) ^ { \top } ( w _ { h } - \widehat { w } _ { h } ) } \\ & { = \phi ( s , a ) ^ { \top } w _ { h } - \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \left( \displaystyle \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) \right) / \widehat { \sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \right) } \\ & { = \phi ( s , a ) ^ { \top } w _ { h } - \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \left( \displaystyle \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) / \widehat { \sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \right) } \\ & { \enspace \enspace \overset { \mathrm { ( ) } } { \underset { \mathrm { ( ) } } { = } } \phi ( s , a ) ^ { \top } w _ { h } - \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \left( \displaystyle \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) / \widehat { \sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \right) } \end{array} +$$ + +$$ ++ \phi ( s , a ) ^ { \top } \widehat { \Lambda } _ { h } ^ { - 1 } \left( \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) / \widehat { \sigma } _ { h } ^ { 2 } \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) . +$$ + +For term (i), it is bounded by 2λH3 d/κ with probability 1 − δ by Lemma C.2. + +For term (ii), by Cauchy inequality it is bounded by + +$$ +\begin{array} { r l } & { \quad \left. \phi ( s , a ) \right. _ { \widehat { \Lambda } _ { h } ^ { - 1 } } \cdot \left. \displaystyle \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) / \widehat { \sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \right. _ { \widehat { \Lambda } _ { h } ^ { - 1 } } } \\ & { \leq \frac { 2 H } { \sqrt { K K } } \left. \displaystyle \sum _ { \tau = 1 } ^ { K } \phi \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \cdot \left( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) - \left( \mathcal { T } _ { h } \widehat { V } _ { h + 1 } \right) \left( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } \right) \right) / \widehat { \sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) \right. _ { \widehat { \Lambda } _ { h } ^ { - 1 } } } \\ & { \leq \frac { 2 H } { \sqrt { K K } } \cdot \tilde { O } ( H \sqrt { d } ) = \widetilde { O } ( \frac { H ^ { 2 } \sqrt { d / \kappa } } { \sqrt { K } } ) , } \end{array} +$$ + +where the first inequality is by Lemma H.6 (with $\phi ^ { \prime } = \phi / \widehat { \sigma } _ { h }$ and $\| \phi / { \widehat { \sigma } } _ { h } \| \leq \| \phi \| \leq 1 : = C )$ and the third inequality uses ${ \sqrt { a ^ { \top } \cdot A \cdot a } } \leq { \sqrt { \| a \| _ { 2 } \| A \| _ { 2 } \| a \| _ { 2 } } } = \| a \| _ { 2 } { \sqrt { \| A \| _ { 2 } } }$ with $a$ to be either $\phi$ or $w _ { h }$ . Moreover, $\begin{array} { r } { \lambda _ { \operatorname* { m i n } } ( \widetilde { \Lambda } _ { h } ^ { p } ) \ge \kappa / \operatorname* { m a x } _ { h , s , a } \widehat { \sigma } _ { h } ( s , a ) ^ { 2 } \ge \kappa / H ^ { 2 } } \end{array}$ implies $\left\| ( \tilde { \Lambda } _ { h } ^ { p } ) ^ { - 1 } \right\| \leq H ^ { 2 } / \kappa$ . The second inequality comes from Lemma H.2 with $R = H$ since $| \eta _ { \tau } | = | ( r _ { h } ^ { \tau } + \widehat { V } _ { h + 1 } \left( s _ { h + 1 } ^ { \tau } \right) -$ $( \mathcal T _ { h } \widehat V _ { h + 1 } ) ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) ) / \widehat \sigma _ { h } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) | \leq H$ and $| x _ { \tau } | = | \phi ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) / \widehat { \sigma } _ { h } ( s _ { h } ^ { \tau } , a _ { h } ^ { \tau } ) | \leq 1$ . + +The final result is obtained by absorbing the term (i) via the condition $K \_ { > }$ $\operatorname* { m a x } \{ \mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } , \mathcal { M } _ { 3 } , \mathcal { M } _ { 4 } \}$ . + +Lemma H.11. Suppose random variables $\| X \| _ { \infty } \leq 2 H$ , $\| Y \| _ { \infty } \leq 2 H$ , then + +$$ +\left| \operatorname { V a r } ( X ) - \operatorname { V a r } ( Y ) \right| \leq 8 H \cdot \left\| X - Y \right\| _ { \infty } . +$$ + +Proof of Lemma H.11. + +$$ +\begin{array} { r l } & { \mathrm { ) - V a r } ( Y ) \vert = \vert \mathbb { E } [ X ^ { 2 } ] - \mathbb { E } [ Y ^ { 2 } ] - ( \mathbb { E } [ X ] ^ { 2 } - \mathbb { E } [ Y ] ^ { 2 } ) \vert = \vert \mathbb { E } [ ( X + Y ) ( X - Y ) ] - ( \mathbb { E } [ X + Y ] ) ( \mathbb { E } [ X - Y ] - \mathbb { E } [ X - Y ] ) \vert } \\ & { \qquad \le \mathbb { E } [ \vert X + Y \vert \cdot \vert X - Y \vert ] + 4 H \cdot \Vert X - Y \Vert _ { \infty } } \\ & { \qquad \le 4 H \mathbb { E } [ \vert X - Y \vert ] + 4 H \cdot \Vert X - Y \Vert _ { \infty } = 8 H \cdot \Vert X - Y \Vert _ { \infty } . } \end{array} +$$ \ No newline at end of file diff --git a/parse/dev/NbkjMn7X8H/NbkjMn7X8H_model.json 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Q. Chen1 Heli Ben-Hamu2 Maximilian Nickel1 Matt Le1 1Meta AI (FAIR) 2Weizmann Institute of Science ", + "bbox": [ + 184, + 143, + 795, + 174 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 210, + 544, + 226 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We introduce a new paradigm for generative modeling built on Continuous Normalizing Flows (CNFs), allowing us to train CNFs at unprecedented scale. Specifically, we present the notion of Flow Matching (FM), a simulation-free approach for training CNFs based on regressing vector fields of fixed conditional probability paths. Flow Matching is compatible with a general family of Gaussian probability paths for transforming between noise and data samples—which subsumes existing diffusion paths as specific instances. Interestingly, we find that employing FM with diffusion paths results in a more robust and stable alternative for training diffusion models. Furthermore, Flow Matching opens the door to training CNFs with other, non-diffusion probability paths. An instance of particular interest is using Optimal Transport (OT) displacement interpolation to define the conditional probability paths. These paths are more efficient than diffusion paths, provide faster training and sampling, and result in better generalization. Training CNFs using Flow Matching on ImageNet leads to consistently better performance than alternative diffusion-based methods in terms of both likelihood and sample quality, and allows fast and reliable sample generation using off-the-shelf numerical ODE solvers. ", + "bbox": [ + 233, + 244, + 764, + 479 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 505, + 334, + 521 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep generative models are a class of deep learning algorithms aimed at estimating and sampling from an unknown data distribution. The recent influx of amazing advances in generative modeling, e.g., for image generation Ramesh et al. (2022); Rombach et al. (2022), is mostly facilitated by the scalable and relatively stable training of diffusion-based models Ho et al. (2020); Song et al. (2020b). However, the restriction to simple diffusion processes leads to a rather confined space of sampling probability paths, resulting in very long training times and the need to adopt specialized methods (e.g., Song et al. (2020a); Zhang & Chen (2022)) for efficient sampling. ", + "bbox": [ + 174, + 535, + 825, + 632 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work we consider the general and deterministic framework of Continuous Normalizing Flows (CNFs; Chen et al. (2018)). CNFs are capable of modeling arbitrary probability path and are in particular known to encompass the probability paths modeled by diffusion processes (Song et al., 2021). ", + "bbox": [ + 174, + 638, + 820, + 681 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, aside from diffusion that can be trained efficiently via, e.g., denoising score matching (Vincent, 2011), no scalable CNF training algorithms are known. Indeed, maximum likelihood training (e.g., Grathwohl et al. (2018)) require expensive numerical ODE simulations, while existing simulation-free methods either involve intractable integrals (Rozen et al., 2021) or biased gradients (Ben-Hamu et al., 2022). ", + "bbox": [ + 174, + 681, + 509, + 805 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The goal of this work is to propose Flow Matching (FM), an efficient simulation-free approach to training CNF models, allowing the adoption of general probability paths to supervise CNF training. Importantly, FM breaks the barriers for scalable CNF training beyond diffusion, and sidesteps the need to reason about diffusion processes to directly work with probability paths. ", + "bbox": [ + 174, + 813, + 509, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/89a2d7265bf5515ae6bdd96c42b9296f4399a3c4ea7884491774afd2867ed3d1.jpg", + "image_caption": [ + "Figure 1: Unconditional ImageNet-128 samples of a CNF trained using Flow Matching with Optimal Transport probability paths. " + ], + "image_footnote": [], + "bbox": [ + 527, + 685, + 818, + 872 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In particular, we propose the Flow Matching objective (Section 3), a simple and intuitive training objective to regress onto a target vector field that generates a desired probability path. We first show that we can construct such target vector fields through per-example (i.e., conditional) formulations. Then, inspired by denoising score matching, we show that a per-example training objective, termed Conditional Flow Matching (CFM), provides equivalent gradients and does not require explicit knowledge of the intractable target vector field. Furthermore, we discuss a general family of per-example probability paths (Section 4) that can be used for Flow Matching, which subsumes existing diffusion paths as special instances. Even on diffusion paths, we find that using FM provides more robust and stable training, and achieves superior performance compared to score matching. Furthermore, this family of probability paths also includes a particularly interesting case: the vector field that corresponds to an Optimal Transport (OT) displacement interpolant (McCann, 1997). We find that conditional OT paths are simpler than diffusion paths, forming straight line trajectories whereas diffusion paths result in curved paths. These properties seem to empirically translate to faster training, faster generation, and better performance. ", + "bbox": [ + 174, + 103, + 825, + 299 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We empirically validate Flow Matching and the construction via Optimal Transport paths on ImageNet, a large and highly diverse image dataset. We find that we can easily train models to achieve favorable performance in both likelihood estimation and sample quality amongst competing diffusion-based methods. Furthermore, we find that our models produce better trade-offs between computational cost and sample quality compared to prior methods. Figure 1 depicts selected unconditional ImageNet $1 2 8 \\times 1 2 8$ samples from our model. ", + "bbox": [ + 174, + 304, + 825, + 388 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PRELIMINARIES: CONTINUOUS NORMALIZING FLOWS ", + "text_level": 1, + "bbox": [ + 176, + 409, + 658, + 425 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Let $\\mathbb { R } ^ { d }$ denote the data space with data points $x = ( x ^ { 1 } , \\ldots , x ^ { d } ) \\in \\mathbb { R } ^ { d }$ . Two important objects we use in this paper are: the probability density path $p : [ 0 , 1 ] \\times \\mathbb { R } ^ { d } \\mathbb { R } _ { > 0 }$ , which is a time dependent1 probability density function, i.e., $\\textstyle \\int p _ { t } ( x ) d x \\ = \\ 1$ , and a time-dependent vector field, $v : [ 0 , 1 ] \\times \\mathbb { R } ^ { d } \\to \\mathbb { R } ^ { d }$ . A vector field $v _ { t }$ can be used to construct a time-dependent diffeomorphic map, called a flow, $\\phi : [ 0 , 1 ] \\times \\mathbb { R } ^ { d } \\mathbb { R } ^ { d }$ , defined via the ordinary differential equation (ODE): ", + "bbox": [ + 173, + 438, + 825, + 512 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/1de48455949d4c09900a87474c24e3c99c24ea9a1198b79022133a427b3117da.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\frac { d } { d t } \\phi _ { t } ( x ) = v _ { t } ( \\phi _ { t } ( x ) ) } } \\\\ { { \\phi _ { 0 } ( x ) = x } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 426, + 516, + 571, + 566 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Previously, Chen et al. (2018) suggested modeling the vector field $v _ { t }$ with a neural network, $v _ { t } ( x ; \\theta )$ , where $\\theta \\in \\mathbb { R } ^ { p }$ are its learnable parameters, which in turn leads to a deep parametric model of the flow $\\phi _ { t }$ , called a Continuous Normalizing Flow (CNF). A CNF is used to reshape a simple prior density $p _ { 0 }$ (e.g., pure noise) to a more complicated one, $p _ { 1 }$ , via the push-forward equation ", + "bbox": [ + 174, + 569, + 823, + 626 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/2bcc40f70aee3d812aabbeab9278a919ff470f3e73c36843e6a77615cd2fbced.jpg", + "text": "$$\np _ { t } = [ \\phi _ { t } ] _ { * } p _ { 0 }\n$$", + "text_format": "latex", + "bbox": [ + 455, + 630, + 542, + 647 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where the push-forward (or change of variables) operator $^ *$ is defined by ", + "bbox": [ + 173, + 651, + 648, + 666 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/2d1f6666ce707fab565ac3daeaabb73d42f1916aaaeaa69713a6786b363b67c5.jpg", + "text": "$$\n[ \\phi _ { t } ] _ { * } p _ { 0 } ( x ) = p _ { 0 } ( \\phi _ { t } ^ { - 1 } ( x ) ) \\operatorname* { d e t } \\left[ \\frac { \\partial \\phi _ { t } ^ { - 1 } } { \\partial x } ( x ) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 354, + 670, + 643, + 707 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A vector field $v _ { t }$ is said to generate a probability density path $p _ { t }$ if its flow $\\phi _ { t }$ satisfies equation 3. One practical way to test if a vector field generates a probability path is using the continuity equation, which is a key component in our proofs, see Appendix A. We recap more information on CNFs, in particular how to compute the probability $p _ { 1 } ( x )$ at an arbitrary point $x \\in \\mathbb { R } ^ { d }$ in Appendix C. ", + "bbox": [ + 174, + 710, + 825, + 767 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 FLOW MATCHING ", + "text_level": 1, + "bbox": [ + 176, + 785, + 354, + 803 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Let $x _ { 1 }$ denote a random variable distributed according to some unknown data distribution $q ( x _ { 1 } )$ . We assume we only have access to data samples from $q ( x _ { 1 } )$ but have no access to the density function itself. Furthermore, we let $p _ { t }$ be a probability path such that $p _ { 0 } = p$ is a simple distribution, e.g., the standard normal distribution $p ( x ) = \\mathcal { N } ( x | 0 , I )$ , and let $p _ { 1 }$ be approximately equal in distribution to $q$ . We will later discuss how to construct such a path. The Flow Matching objective is then designed to match this target probability path, which will allow us to flow from $p _ { 0 }$ to $p _ { 1 }$ . ", + "bbox": [ + 173, + 816, + 825, + 901 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Given a target probability density path $p _ { t } ( x )$ and a corresponding vector field $u _ { t } ( x )$ , which generates $p _ { t } ( x )$ , we define the Flow Matching (FM) objective as ", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/1451c47685e9898cb6f6de1ffd4100aa7162dfa86fcc119a25f31c108e1f3b2c.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\mathtt { F M } } ( \\theta ) = \\mathbb { E } _ { t , p _ { t } ( x ) } \\| v _ { t } ( x ) - u _ { t } ( x ) \\| ^ { 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 377, + 141, + 619, + 160 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\theta$ denotes the learnable parameters of the CNF vector field $v _ { t }$ (as defined in Section 2), $t \\sim$ $\\boldsymbol { \\mathcal { U } } [ 0 , 1 ]$ (uniform distribution), and $x \\sim p _ { t } ( x )$ . Simply put, the FM loss regresses the vector field $u _ { t }$ with a neural network $v _ { t }$ . Upon reaching zero loss, the learned CNF model will generate $p _ { t } ( x )$ . ", + "bbox": [ + 174, + 167, + 825, + 210 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Flow Matching is a simple and attractive objective, but na¨ıvely on its own, it is intractable to use in practice since we have no prior knowledge for what an appropriate $p _ { t }$ and $u _ { t }$ are. There are many choices of probability paths that can satisfy $p _ { 1 } ( x ) \\approx q \\bar { ( x ) }$ , and more importantly, we generally don’t have access to a closed form $u _ { t }$ that generates the desired $p _ { t }$ . In this section, we show that we can construct both $p _ { t }$ and $u _ { t }$ using probability paths and vector fields that are only defined per sample, and an appropriate method of aggregation provides the desired $p _ { t }$ and $u _ { t }$ . Furthermore, this construction allows us to create a much more tractable objective for Flow Matching. ", + "bbox": [ + 173, + 217, + 825, + 314 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 CONSTRUCTING $p _ { t } , u _ { t }$ FROM CONDITIONAL PROBABILITY PATHS AND VECTOR FIELDS ", + "text_level": 1, + "bbox": [ + 173, + 333, + 803, + 348 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A simple way to construct a target probability path is via a mixture of simpler probability paths: Given a particular data sample $x _ { 1 }$ we denote by $p _ { t } ( x | x _ { 1 } )$ a conditional probability path such that it satisfies $p _ { 0 } ( x | x _ { 1 } ) = p ( x )$ at time $t = 0$ , and we design $p _ { 1 } ( x | x _ { 1 } )$ at $t = 1$ to be a distribution concentrated around $x = x _ { 1 }$ , e.g., $p _ { 1 } ( x | x _ { 1 } ) = \\mathcal { N } ( x | x _ { 1 } , \\sigma ^ { 2 } I )$ , a normal distribution with $x _ { 1 }$ mean and a sufficiently small standard deviation $\\sigma > 0$ . Marginalizing the conditional probability paths over $q ( x _ { 1 } )$ give rise to the marginal probability path ", + "bbox": [ + 173, + 358, + 825, + 444 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/501b9652caf23060d2ba47eaf0cf99a428141881e184e301556c768ef24ae160.jpg", + "text": "$$\np _ { t } ( x ) = \\int p _ { t } ( x | x _ { 1 } ) q ( x _ { 1 } ) d x _ { 1 } ,\n$$", + "text_format": "latex", + "bbox": [ + 397, + 452, + 599, + 484 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where in particular at time $t = 1$ , the marginal probability $p _ { 1 }$ is a mixture distribution that closely approximates the data distribution $q$ , ", + "bbox": [ + 174, + 492, + 823, + 521 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/64bed25ed01770fd8600a10cdbd3987963c5847e2ec65f194cce73f02954eb41.jpg", + "text": "$$\np _ { 1 } ( x ) = \\int p _ { 1 } ( x | x _ { 1 } ) q ( x _ { 1 } ) d x _ { 1 } \\approx q ( x ) .\n$$", + "text_format": "latex", + "bbox": [ + 369, + 529, + 629, + 561 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Interestingly, we can also define a marginal vector field, by “marginalizing” over the conditional vector fields in the following sense (we assume $p _ { t } ( x ) > 0$ for all $t$ and $x$ ): ", + "bbox": [ + 173, + 568, + 823, + 597 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d7382d0548f52b9bc962a349c1ddd4e29ebc15e859cdd6d34aaa6a19dd90f5cf.jpg", + "text": "$$\nu _ { t } ( x ) = \\int u _ { t } ( x | x _ { 1 } ) \\frac { p _ { t } ( x | x _ { 1 } ) q ( x _ { 1 } ) } { p _ { t } ( x ) } d x _ { 1 } ,\n$$", + "text_format": "latex", + "bbox": [ + 364, + 606, + 632, + 640 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $u _ { t } ( \\cdot | x _ { 1 } ) : \\mathbb { R } ^ { d } \\to \\mathbb { R } ^ { d }$ is a conditional vector field that generates $p _ { t } ( \\cdot | x _ { 1 } )$ . It may not seem apparent, but this way of aggregating the conditional vector fields actually results in the correct vector field for modeling the marginal probability path. ", + "bbox": [ + 174, + 648, + 825, + 693 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our first key observation is this: ", + "bbox": [ + 174, + 699, + 383, + 713 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The marginal vector field (equation 8) generates the marginal probability path (equation 6). ", + "bbox": [ + 191, + 722, + 799, + 737 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This provides a surprising connection between the conditional VFs (those that generate conditional probability paths) and the marginal VF (those that generate the marginal probability path). This connection allows us to break down the unknown and intractable marginal VF into simpler conditional VFs, which are much simpler to define as these only depend on a single data sample. We formalize this in the following theorem. ", + "bbox": [ + 174, + 751, + 825, + 821 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Theorem 1. Given vector fields $u _ { t } ( x | x _ { 1 } )$ that generate conditional probability paths $p _ { t } ( x | x _ { 1 } )$ , for any distribution $q ( x _ { 1 } )$ , the marginal vector field $u _ { t }$ in equation 8 generates the marginal probability path $p _ { t }$ in equation $6$ , i.e., $u _ { t }$ and $p _ { t }$ satisfy the continuity equation (equation 25). ", + "bbox": [ + 173, + 827, + 821, + 869 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The full proofs for our theorems are all provided in Appendix B. Theorem 1 can also be derived from the Diffusion Mixture Representation Theorem in Peluchetti (2021) that provides a formula for the marginal drift and diffusion coefficients in diffusion SDEs. ", + "bbox": [ + 174, + 881, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 CONDITIONAL FLOW MATCHING ", + "text_level": 1, + "bbox": [ + 176, + 103, + 442, + 118 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Unfortunately, due to the intractable integrals in the definitions of the marginal probability path and VF (equations 6 and 8), it is still intractable to compute $u _ { t }$ , and consequently, intractable to na¨ıvely compute an unbiased estimator of the original Flow Matching objective. Instead, we propose a simpler objective, which surprisingly will result in the same optima as the original objective. Specifically, we consider the Conditional Flow Matching (CFM) objective, ", + "bbox": [ + 173, + 128, + 825, + 199 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/a37854b6758e53e2db731ad85cfc4fef4a04dbaca66ac982f8c3c52d72d0825e.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\mathtt { C F M } } ( \\theta ) = \\mathbb { E } _ { t , q ( x _ { 1 } ) , p _ { t } ( x | x _ { 1 } ) } \\big | \\big | v _ { t } ( x ) - u _ { t } ( x | x _ { 1 } ) \\big | \\big | ^ { 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 338, + 202, + 658, + 223 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $t \\sim \\mathcal { U } [ 0 , 1 ]$ , $x _ { 1 } \\sim q ( x _ { 1 } )$ , and now $x \\sim p _ { t } ( x | x _ { 1 } )$ . Unlike the FM objective, the CFM objective allows us to easily sample unbiased estimates as long as we can efficiently sample from $p _ { t } ( x | x _ { 1 } )$ and compute $u _ { t } ( x | x _ { 1 } )$ , both of which can be easily done as they are defined on a per-sample basis. Our second key observation is therefore: ", + "bbox": [ + 173, + 224, + 825, + 281 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The FM (equation 5) and CFM (equation 9) objectives have identical gradients w.r.t. $\\theta$ . ", + "bbox": [ + 197, + 286, + 785, + 303 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "That is, optimizing the CFM objective is equivalent (in expectation) to optimizing the FM objective. Consequently, this allows us to train a CNF to generate the marginal probability path $p _ { t }$ —which in particular, approximates the unknown data distribution $q$ at $t { = } 1 { - }$ without ever needing access to either the marginal probability path or the marginal vector field. We simply need to design suitable conditional probability paths and vector fields. We formalize this property in the following theorem. ", + "bbox": [ + 173, + 313, + 825, + 385 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 2. Assuming that $p _ { t } ( x ) > 0$ for all $x ~ \\in ~ \\mathbb { R } ^ { d }$ and $t \\in [ 0 , 1 ]$ , then, up to a constant independent of $\\theta$ , $\\mathcal { L } _ { c F M }$ and $\\mathcal { L } _ { \\boldsymbol { F M } }$ are equal. Hence, $\\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\boldsymbol { F M } } ( \\boldsymbol { \\theta } ) = \\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\boldsymbol { c F M } } ( \\boldsymbol { \\theta } )$ . ", + "bbox": [ + 173, + 398, + 823, + 429 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 CONDITIONAL PROBABILITY PATHS AND VECTOR FIELDS ", + "text_level": 1, + "bbox": [ + 174, + 446, + 687, + 464 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The Conditional Flow Matching objective works with any choice of conditional probability path and conditional vector fields. In this section, we discuss the construction of $p _ { t } ( x | x _ { 1 } )$ and $u _ { t } ( x | x _ { 1 } )$ for a general family of Gaussian conditional probability paths. Namely, we consider conditional probability paths of the form ", + "bbox": [ + 173, + 478, + 825, + 534 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9abc3007f9552a0ae6ac16cf7afdeb304b3c57e8d53de54ffa5f9fb0d145d9d5.jpg", + "text": "$$\np _ { t } ( x | x _ { 1 } ) = \\mathcal { N } ( x \\mid \\mu _ { t } ( x _ { 1 } ) , \\sigma _ { t } ( x _ { 1 } ) ^ { 2 } I ) ,\n$$", + "text_format": "latex", + "bbox": [ + 375, + 535, + 620, + 554 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mu : [ 0 , 1 ] \\times \\mathbb { R } ^ { d } \\mathbb { R } ^ { d }$ is the time-dependent mean of the Gaussian distribution, while $\\sigma :$ $[ 0 , 1 ] \\times \\mathbb { R } \\to \\mathbb { R } _ { > 0 }$ describes a time-dependent scalar standard deviation (std). We set $\\mu _ { 0 } ( x _ { 1 } ) = 0$ and $\\sigma _ { 0 } ( x _ { 1 } ) = 1$ , so that all conditional probability paths converge to the same standard Gaussian noise distribution at $t = 0$ , $p ( x ) = \\mathcal { N } ( x | 0 , I )$ . We then set $\\mu _ { 1 } ( x _ { 1 } ) = x _ { 1 }$ and $\\sigma _ { 1 } ( x _ { 1 } ) = \\sigma _ { \\mathrm { { m i n } } }$ , which is set sufficiently small so that $p _ { 1 } ( x | x _ { 1 } )$ is a concentrated Gaussian distribution centered at $x _ { 1 }$ . ", + "bbox": [ + 173, + 555, + 825, + 626 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "There is an infinite number of vector fields that generate any particular probability path (e.g., by adding a divergence free component to the continuity equation, see equation 25), but the vast majority of these is due to the presence of components that leave the underlying distribution invariant—for instance, rotational components when the distribution is rotation-invariant—leading to unnecessary extra compute. We decide to use the simplest vector field corresponding to a canonical transformation for Gaussian distributions. Specifically, consider the flow (conditioned on $x _ { 1 }$ ) ", + "bbox": [ + 173, + 631, + 825, + 715 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/4c355702a1a0bdba997be1404617a62bc24294bad2220c50205646683054859f.jpg", + "text": "$$\n\\psi _ { t } ( x ) = \\sigma _ { t } ( x _ { 1 } ) x + \\mu _ { t } ( x _ { 1 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 405, + 717, + 591, + 734 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "When $x$ is distributed as a standard Gaussian, $\\psi _ { t } ( x )$ is the affine transformation that maps to a normally-distributed random variable with mean $\\mu _ { t } ( x _ { 1 } )$ and std $\\sigma _ { t } ( x _ { 1 } )$ . That is to say, according to equation 4, $\\psi _ { t }$ pushes the noise distribution $p _ { 0 } ( x | x _ { 1 } ) = p ( x )$ to $p _ { t } ( x | x _ { 1 } )$ , i.e., ", + "bbox": [ + 174, + 736, + 825, + 780 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/224b5d4c34a64ce60fe55134dabcdcecaf991d338ee38dcb959d8fe7c22d6919.jpg", + "text": "$$\n[ \\psi _ { t } ] _ { * } p ( x ) = p _ { t } ( x | x _ { 1 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 423, + 781, + 575, + 797 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This flow then provides a vector field that generates the conditional probability path: ", + "bbox": [ + 176, + 799, + 727, + 814 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5a8171fff0f5cfd141dc3af0f5e475c4c1c609a78926375c51bc5759813fe592.jpg", + "text": "$$\n\\frac { d } { d t } \\psi _ { t } ( x ) = u _ { t } ( \\psi _ { t } ( x ) | x _ { 1 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 413, + 815, + 586, + 845 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Reparameterizing $p _ { t } ( x | x _ { 1 } )$ in terms of just $x _ { 0 }$ and plugging equation 13 in the CFM loss we get ", + "bbox": [ + 174, + 848, + 802, + 863 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/6e335c46f4ce4131b90bc44eff03b10c63a2e2437664974a728ab420e00d8bbf.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { c F M } } ( \\theta ) = \\mathbb { E } _ { t , q ( x _ { 1 } ) , p ( x _ { 0 } ) } \\Big | \\Big | v _ { t } \\big ( \\psi _ { t } ( x _ { 0 } ) \\big ) - \\frac { d } { d t } \\psi _ { t } ( x _ { 0 } ) \\Big | \\Big | ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 864, + 673, + 893 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Since $\\psi _ { t }$ is a simple (invertible) affine map we can use equation 13 to solve for $u _ { t }$ in a closed form. \nLet $f ^ { \\prime }$ denote the derivative with respect to time, i.e., $\\begin{array} { r } { f ^ { \\prime } \\overset { \\cdot } { = } \\frac { d } { d t } f } \\end{array}$ , for a time-dependent function $f$ . ", + "bbox": [ + 173, + 896, + 825, + 925 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 3. Let $p _ { t } ( x | x _ { 1 } )$ be a Gaussian probability path as in equation $I O$ , and $\\psi _ { t }$ its corresponding flow map as in equation $I I$ . Then, the unique vector field that defines $\\psi _ { t }$ has the form: ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/8357e81d453a16d874d0e6ff1ddff837d0722203e0d9703e5a2b0f3a22253f9e.jpg", + "text": "$$\nu _ { t } ( x | x _ { 1 } ) = \\frac { \\sigma _ { t } ^ { \\prime } ( x _ { 1 } ) } { \\sigma _ { t } ( x _ { 1 } ) } \\left( x - \\mu _ { t } ( x _ { 1 } ) \\right) + \\mu _ { t } ^ { \\prime } ( x _ { 1 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 354, + 136, + 643, + 170 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Consequently, $u _ { t } ( x | x _ { 1 } )$ generates the Gaussian path $p _ { t } ( x | x _ { 1 } )$ . ", + "bbox": [ + 173, + 174, + 584, + 190 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 SPECIAL INSTANCES OF GAUSSIAN CONDITIONAL PROBABILITY PATHS ", + "text_level": 1, + "bbox": [ + 174, + 205, + 705, + 220 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our formulation is fully general for arbitrary functions $\\mu _ { t } ( x _ { 1 } )$ and $\\sigma _ { t } ( x _ { 1 } )$ , and we can set them to any differentiable function satisfying the desired boundary conditions. We first discuss the special cases that recover probability paths corresponding to previously-used diffusion processes. Since we directly work with probability paths, we can simply depart from reasoning about diffusion processes altogether. Therefore, in the second example below, we directly formulate a probability path based on the Wasserstein-2 optimal transport solution as an interesting instance. ", + "bbox": [ + 173, + 231, + 825, + 315 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Example I: Diffusion conditional VFs. Diffusion models start with data points and gradually add noise until it approximates pure noise. These can be formulated as stochastic processes, which have strict requirements in order to obtain closed form representation at arbitrary times $t$ , resulting in Gaussian conditional probability paths $p _ { t } ( x | x _ { 1 } )$ with specific choices of mean $\\mu _ { t } ( x _ { 1 } )$ and std $\\sigma _ { t } ( x _ { 1 } )$ (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2020b). For example, the reversed (noise data) Variance Exploding (VE) path has the form ", + "bbox": [ + 173, + 329, + 825, + 415 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/62e0303b23085cdff20ed11f2ec07ee225e3f057172abb9148ea753ddc961c0f.jpg", + "text": "$$\np _ { t } ( x | x _ { 1 } ) = \\mathcal { N } ( x | x _ { 1 } , \\sigma _ { 1 - t } ^ { 2 } I ) ,\n$$", + "text_format": "latex", + "bbox": [ + 401, + 417, + 594, + 436 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\sigma _ { t }$ is an increasing function, $\\sigma _ { 0 } = 0$ , and $\\sigma _ { 1 } \\gg 1$ . Next, equation 16 provides the choices of $\\mu _ { t } ( x _ { 1 } ) = x _ { 1 }$ and $\\sigma _ { t } ( x _ { 1 } ) = \\sigma _ { 1 - t }$ . Plugging these into equation 15 of Theorem 3 we get ", + "bbox": [ + 174, + 440, + 825, + 469 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/f650442ee3a44d871a6aefe71dda418d70fe0f8b84c6898bc624c781e52ba98f.jpg", + "text": "$$\nu _ { t } ( x | x _ { 1 } ) = - \\frac { \\sigma _ { 1 - t } ^ { \\prime } } { \\sigma _ { 1 - t } } ( x - x _ { 1 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 400, + 472, + 596, + 507 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The reversed (noise data) Variance Preserving (VP) diffusion path has the form ", + "bbox": [ + 174, + 510, + 704, + 526 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/f2fc52ef289fe2c87792302a917fe84b1bf939f8751c73cceef094f832578acd.jpg", + "text": "$$\np _ { t } ( x | x _ { 1 } ) = \\mathcal { N } ( x \\mid \\alpha _ { 1 - t } x _ { 1 } , \\left( 1 - \\alpha _ { 1 - t } ^ { 2 } \\right) I ) , \\mathrm { w h e r e } \\alpha _ { t } = e ^ { - \\frac { 1 } { 2 } T ( t ) } , T ( t ) = \\int _ { 0 } ^ { t } \\beta ( s ) d s ,\n$$", + "text_format": "latex", + "bbox": [ + 210, + 529, + 761, + 565 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "and $\\beta$ is the noise scale function. Equation 18 provides the choices of $\\mu _ { t } ( x _ { 1 } ) \\ = \\ \\alpha _ { 1 - t } x _ { 1 }$ and $\\sigma _ { t } ( x _ { 1 } ) = \\sqrt { 1 - \\alpha _ { 1 - t } ^ { 2 } }$ . Plugging these into equation 15 of Theorem 3 we get ", + "bbox": [ + 173, + 569, + 831, + 606 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/54dadd827f7f295eb9a05701db88e37439fac63abce27382d105ae79ce7911f8.jpg", + "text": "$$\nu _ { t } ( x | x _ { 1 } ) = \\frac { \\alpha _ { 1 - t } ^ { \\prime } } { 1 - \\alpha _ { 1 - t } ^ { 2 } } \\left( \\alpha _ { 1 - t } x - x _ { 1 } \\right) = - \\frac { T ^ { \\prime } ( 1 - t ) } { 2 } \\left[ \\frac { e ^ { - T ( 1 - t ) } x - e ^ { - \\frac { 1 } { 2 } T ( 1 - t ) } x _ { 1 } } { 1 - e ^ { - T ( 1 - t ) } } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 209, + 612, + 761, + 655 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our construction of the conditional VF $\\ i \\ u _ { t } \\big ( \\boldsymbol { x } | \\boldsymbol { x } _ { 1 } \\big )$ does in fact coincide with the vector field previously used in the deterministic probability flow (Song et al. (2020b), equation 13) when restricted to these conditional diffusion processes; see details in Appendix D. Nevertheless, combining the diffusion conditional VF with the Flow Matching objective offers an attractive training alternative—which we find to be more stable and robust in our experiments—to existing score matching approaches. ", + "bbox": [ + 173, + 659, + 825, + 729 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Another important observation is that, as these probability paths were previously derived as solutions of diffusion processes, they do not actually reach a true noise distribution in finite time. In practice, $p _ { 0 } ( x )$ is simply approximated by a suitable Gaussian distribution for sampling and likelihood evaluation. Instead, our construction provides full control over the probability path, and we can just directly set $\\mu _ { t }$ and $\\sigma _ { t }$ , as we will do next. ", + "bbox": [ + 173, + 736, + 825, + 808 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Example II: Optimal Transport conditional VFs. An arguably more natural choice for conditional probability paths is to define the mean and the std to simply change linearly in time, i.e., ", + "bbox": [ + 174, + 820, + 823, + 849 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/ea66abe1bb4161c5481165a716a9be380d3bfefc656307e644089ae39d5d52ab.jpg", + "text": "$$\n\\mu _ { t } ( x ) = t x _ { 1 } , \\mathrm { a n d } \\sigma _ { t } ( x ) = 1 - ( 1 - \\sigma _ { \\operatorname* { m i n } } ) t .\n$$", + "text_format": "latex", + "bbox": [ + 354, + 853, + 642, + 871 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "According to Theorem 3 this path is generated by the VF ", + "bbox": [ + 173, + 875, + 549, + 890 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/eae9e16f6eb8bcf8112a04b3739845b4f56275c18dc4563372016d134b2e0407.jpg", + "text": "$$\nu _ { t } ( x | x _ { 1 } ) = \\frac { x _ { 1 } - ( 1 - \\sigma _ { \\operatorname* { m i n } } ) x } { 1 - ( 1 - \\sigma _ { \\operatorname* { m i n } } ) t } ,\n$$", + "text_format": "latex", + "bbox": [ + 397, + 895, + 601, + 928 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/1335c7404c50a3bda501acfed97e3870692f455cdf0e5ff569420338692aef09.jpg", + "image_caption": [ + "Figure 2: Compared to the diffusion path’s conditional score function, the OT path’s conditional vector field has constant direction in time and is arguably simpler to fit with a parametric model. Note the blue color denotes larger magnitude while red color denotes smaller magnitude. " + ], + "image_footnote": [], + "bbox": [ + 173, + 99, + 821, + 181 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "which, in contrast to the diffusion conditional VF (equation 19), is defined for all $t \\in [ 0 , 1 ]$ . The conditional flow that corresponds to $u _ { t } ( x | x _ { 1 } )$ is ", + "bbox": [ + 174, + 247, + 823, + 276 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/6797a74401e49b8f9049600d557bda8d9b5e32b53d90973db930cb733583e096.jpg", + "text": "$$\n\\psi _ { t } ( x ) = ( 1 - ( 1 - \\sigma _ { \\operatorname* { m i n } } ) t ) x + t x _ { 1 } ,\n$$", + "text_format": "latex", + "bbox": [ + 380, + 281, + 616, + 297 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "and in this case, the CFM loss (see equations 9, 14) takes the form: ", + "bbox": [ + 173, + 303, + 612, + 318 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/0cf49f17eae5a62b17ef80ec9020a472664e4fb93807fa9b6c8185b06c5f1dbb.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { c F M } } ( \\theta ) = \\mathbb { E } _ { t , q ( x _ { 1 } ) , p ( x _ { 0 } ) } \\Big \\| v _ { t } \\big ( \\psi _ { t } ( x _ { 0 } ) \\big ) - \\Big ( x _ { 1 } - ( 1 - \\sigma _ { \\operatorname* { m i n } } ) x _ { 0 } \\Big ) \\Big \\| ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 284, + 324, + 714, + 353 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Allowing the mean and std to change linearly not only leads to simple and intuitive paths, but it is actually also optimal in the following sense. The conditional flow $\\psi _ { t } ( x )$ is in fact the Optimal Transport (OT) displacement map between the two Gaussians $p _ { 0 } ( x | x _ { 1 } )$ and $p _ { 1 } ( x | x _ { 1 } )$ . The OT interpolant, which is a probability path, is defined to be (see Definition 1.1 in McCann (1997)): ", + "bbox": [ + 174, + 357, + 825, + 412 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/cb6633766a9f0494efce02178377a69560d001a11d13b2c368bd59c3d703226a.jpg", + "text": "$$\np _ { t } = [ ( 1 - t ) \\mathrm { i d } + t \\psi ] _ { \\star } p _ { 0 }\n$$", + "text_format": "latex", + "bbox": [ + 413, + 419, + 583, + 435 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\psi : \\mathbb { R } ^ { d } \\mathbb { R } ^ { d }$ is the OT map pushing $p _ { 0 }$ to $p _ { 1 }$ , id denotes the identity map, i.e., $\\operatorname { i d } ( x ) = x$ and $( 1 - t ) \\mathrm { i d } + t \\psi$ is called the OT displacement map. Example 1.7 in McCann (1997) shows, that in our case of two Gaussians where the first is a standard one, the OT displacement map takes the form of equation 22. ", + "bbox": [ + 173, + 440, + 821, + 496 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Intuitively, particles under the OT displacement map always move in straight line trajectories and with constant speed. Figure 3 depicts sampling paths for the diffusion and OT conditional VFs. Interestingly, we find that sampling trajectory from diffusion paths can “overshoot” the final sample, resulting in unnecessary backtracking, whilst the OT paths are guaranteed to stay straight. ", + "bbox": [ + 174, + 503, + 614, + 588 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/45f525ed726462dc8a17baf4798cdebe774a867bd01f7ef97f23a6c58a2da77f.jpg", + "image_caption": [ + "Figure 3: Diffusion and OT conditional trajectories. " + ], + "image_footnote": [], + "bbox": [ + 638, + 492, + 823, + 568 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figure 2 compares the diffusion conditional score function (the re \ngression target in a typical diffusion methods), i.e., $\\nabla \\log p _ { t } ( x | x _ { 1 } )$ with $p _ { t }$ defined as in equation 18, with the OT conditional VF (equation 21). The start $\\left( p _ { 0 } \\right)$ and end $( p _ { 1 } )$ Gaussians are identical in both examples. An interesting observation is that the OT VF has a constant direction in time, which arguably leads to a simpler regression task. This property can also be verified directly from equation 21 as the VF can be written in the form $u _ { t } ( x | x _ { 1 } ) = g ( t ) h ( x | x _ { 1 } )$ . Figure 8 in the Appendix shows a visualization of the Diffusion VF. Lastly, we note that although the conditional flow is optimal, this by no means imply that the marginal VF is an optimal transport solution. Nevertheless, we expect the marginal vector field to remain relatively simple. ", + "bbox": [ + 174, + 595, + 825, + 719 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 739, + 344, + 756 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Continuous Normalizing Flows were introduced in (Chen et al., 2018) as a continuous-time version of Normalizing Flows (see e.g., Kobyzev et al. (2020); Papamakarios et al. (2021) for an overview). Originally, CNFs are trained with the maximum likelihood objective, but this involves expensive ODE simulations for the forward and backward propagation, resulting in high time complexity due to the sequential nature of ODE simulations. Although some works demonstrated the capability of CNF generative models for image synthesis (Grathwohl et al., 2018), scaling up to very high dimensional images is inherently difficult. A number of works attempted to regularize the ODE to be easier to solve, e.g., using augmentation (Dupont et al., 2019), adding regularization terms (Yang & Karniadakis, 2019; Finlay et al., 2020; Onken et al., 2021; Tong et al., 2020; Kelly et al., 2020), or stochastically sampling the integration interval (Du et al., 2022). These works merely aim to regularize the ODE but do not change the fundamental training algorithm. ", + "bbox": [ + 173, + 770, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/730cd8e2f3d4e2c634c2621f1890aaa304191d480cfd3f7f6fb9e7ba0d6f1d75.jpg", + "image_caption": [ + "Figure 4: (left) Trajectories of CNFs trained with different objectives on 2D checkerboard data. The OT path introduces the checkerboard pattern much earlier, while FM results in more stable training. (right) FM with OT results in more efficient sampling, solved using the midpoint scheme. " + ], + "image_footnote": [], + "bbox": [ + 176, + 99, + 820, + 241 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In order to speed up CNF training, some works have developed simulation-free CNF training frameworks by explicitly designing the target probability path and the dynamics. For instance, Rozen et al. (2021) consider a linear interpolation between the prior and the target density but involves integrals that were difficult to estimate in high dimensions, while Ben-Hamu et al. (2022) consider general probability paths similar to this work but suffers from biased gradients in the stochastic minibatch regime. In contrast, the Flow Matching framework allows simulation-free training with unbiased gradients and readily scales to very high dimensions. ", + "bbox": [ + 174, + 319, + 825, + 417 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Another approach to simulation-free training relies on the construction of a diffusion process to indirectly define the target probability path (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song & Ermon, 2019). Song et al. (2020b) shows that diffusion models are trained using denoising score matching (Vincent, 2011), a conditional objective that provides unbiased gradients with respect to the score matching objective. Conditional Flow Matching draws inspiration from this result, but generalizes to matching vector fields directly. Due to the ease of scalability, diffusion models have received increased attention, producing a variety of improvements such as loss-rescaling (Song et al., 2021), adding classifier guidance along with architectural improvements (Dhariwal & Nichol, 2021), and learning the noise schedule (Nichol & Dhariwal, 2021; Kingma et al., 2021). However, (Nichol & Dhariwal, 2021) and (Kingma et al., 2021) only consider a restricted setting of Gaussian conditional paths defined by simple diffusion processes with a single parameter—in particular, it does not include our conditional OT path. In an another line of works, (De Bortoli et al., 2021; Wang et al., 2021; Peluchetti, 2021) proposed finite time diffusion constructions via diffusion bridges theory resolving the approximation error incurred by infinite time denoising constructions. While existing works make use of a connection between diffusion processes and continuous normalizing flows with the same probability path (Maoutsa et al., 2020b; Song et al., 2020b; 2021), our work allows us to generalize beyond the class of probability paths modeled by simple diffusion. With our work, it is possible to completely sidestep the diffusion process construction and reason directly with probability paths, while still retaining efficient training and log-likelihood evaluations. Lastly, concurrently to our work (Liu et al., 2022; Albergo & Vanden-Eijnden, 2022) arrived at similar conditional objectives for simulation-free training of CNFs, while Neklyudov et al. (2023) derived an implicit objective when $u _ { t }$ is assumed to be a gradient field. ", + "bbox": [ + 173, + 424, + 825, + 729 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 762, + 326, + 779 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We explore the empirical benefits of using Flow Matching on the image datasets of CIFAR10 (Krizhevsky et al., 2009) and ImageNet at resolutions 32, 64, and 128 (Chrabaszcz et al., 2017; Deng et al., 2009). We also ablate the choice of diffusion path in Flow Matching, particularly between the standard variance preserving diffusion path and the optimal transport path. We discuss how sample generation is improved by directly parameterizing the generating vector field and using the Flow Matching objective. Lastly we show Flow Matching can also be used in the conditional generation setting. Unless otherwise specified, we evaluate likelihood and samples from the model using dopri5 (Dormand & Prince, 1980) at absolute and relative tolerances of 1e-5. Generated samples can be found in the Appendix, and all implementation details are in Appendix E. ", + "bbox": [ + 174, + 801, + 825, + 928 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/349dab26d852c74fe0448b9ce57f11b4af9f5916b2955147dbd344e2ce06be7a.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
ModelCIFAR-10ImageNet 32×32ImageNet 64×64ImageNet 128×128
NLL↓FID↓NFE↓NLL↓FID↓NFE↓NLL↓FID↓NFE↓ModelNLL↓FID↓
AblationsMGAN (Hoang etal.,2018)158.9
DDPM3.127.482743.546.992623.3217.36264PacGAN2 (Lin et a.,2018)=57.5
Score Matching3.1619.942423.565.681783.4019.74441Logo-GAN-AE(Sag etal,018)一50.9
ScoreFlow3.0920.784283.5514.141953.3624.95601Self-cond. GAN (Lucic e al.,2019)一41.7
OursUncond.BigGAN (Lueic etal,19)25.3
FM W/ Diffusion3.108.061833.546.371933.3316.88187PGMGAN (Armandpour et al.,2021)21.7
FM W/OT2.996.351423.535.021223.3114.45138FM W/OT2.9020.9
", + "bbox": [ + 173, + 101, + 820, + 208 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 1: Likelihood (BPD), quality of generated samples (FID), and evaluation time (NFE) for the same model trained with different methods. ", + "bbox": [ + 173, + 218, + 823, + 246 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/3348602332065aba7df6510ef6c9dcf1f31d1e1f70730d577577f5cf6e06f403.jpg", + "image_caption": [ + "Figure 6: Sample paths from the same initial noise with models trained on ImageNet $6 4 \\times 6 4$ . The OT path reduces noise roughly linearly, while diffusion paths visibly remove noise only towards the end of the path. Note also the differences between the generated images. " + ], + "image_footnote": [], + "bbox": [ + 181, + 256, + 816, + 332 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.1 DENSITY MODELING AND SAMPLE QUALITY ON IMAGENET", + "text_level": 1, + "bbox": [ + 176, + 396, + 633, + 410 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We start by comparing the same model architecture, i.e., the U-Net architecture from Dhariwal & Nichol (2021) with minimal changes, trained on CIFAR-10, and ImageNet 32/64 with different popular diffusion-based losses: DDPM from (Ho et al., 2020), Score Matching (SM) (Song et al., 2020b), and Score Flow (SF) (Song et al., 2021); see Appendix E.1 for exact details. Table 1 (left) summarizes our results alongside these baselines reporting negative log-likelihood (NLL) in units of bits per dimension (BPD), sample quality as measured by the Frechet Inception Distance (FID; Heusel et al. (2017)), and averaged number of function evaluations (NFE) required for the adaptive solver to reach its a prespecified numerical tolerance, averaged over $5 0 \\mathrm { k }$ samples. All models are trained using the same architecture, hyperparameter values and number of training iterations, where baselines are allowed more iterations for better convergence. Note that these are unconditional models. On both CIFAR-10 and ImageNet, FM-OT consistently obtains best results across all our quantitative measures compared to competing methods. We are noticing a higher that usual FID performance in CIFAR-10 compared to previous works (Ho et al., 2020; Song et al., 2020b; 2021) that can possibly be explained by the fact that our used architecture was not optimized for CIFAR-10. ", + "bbox": [ + 173, + 416, + 825, + 611 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Secondly, Table 1 (right) compares a model trained using Flow Matching with the OT path on ImageNet at resolution $1 2 8 \\times 1 2 8$ . Our FID is state-of-the-art with the exception of IC-GAN (Casanova et al., 2021) which uses conditioning with a self-supervised ResNet50 model, and therefore is left out of this table. Figures 11, 12, 13 in the Appendix show non-curated samples from these models. ", + "bbox": [ + 173, + 617, + 823, + 674 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Faster training. While existing works train diffusion models with a very high number of iterations (e.g., $1 . 3 \\mathrm { m }$ and $1 0 \\mathrm { m }$ iterations are reported by Score Flow and VDM, respectively), we find that Flow Matching generally converges much faster. Figure 5 shows FID curves during training of Flow Matching and all baselines for ImageNet $6 4 \\times 6 4$ ; FM-OT is able to lower the FID faster and to a greater extent than the alternatives. For ImageNet-128 Dhariwal & Nichol (2021) train for $4 . 3 6 \\mathrm { m }$ iterations with batch size 256, while FM (with $2 5 \\%$ larger model) used $5 0 0 \\mathrm { k }$ iterations with batch size $1 . 5 \\mathrm { k } .$ , i.e., $33 \\%$ less image throughput; see Table 3 for exact details. Furthermore, the cost of sampling from a model can drastically change during training for score matching, whereas the sampling cost stays constant when training with Flow Matching (Figure 10 in Appendix). ", + "bbox": [ + 174, + 680, + 581, + 819 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/efb03c36b63c672277f0cbd4eeb7bb035945f2981aaf08e10a378e6b6bb08518.jpg", + "image_caption": [ + "Figure 5: Image quality during training, ImageNet $6 4 \\times 6 4$ . " + ], + "image_footnote": [], + "bbox": [ + 596, + 683, + 820, + 784 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 820, + 820, + 861 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.2 SAMPLING EFFICIENCY ", + "text_level": 1, + "bbox": [ + 174, + 875, + 379, + 888 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For sampling, we first draw a random noise sample $x _ { 0 } \\sim \\mathcal { N } ( 0 , I )$ then compute $\\phi _ { 1 } ( x _ { 0 } )$ by solving equation 1 with the trained VF, $v _ { t }$ , on the interval $t \\in [ 0 , 1 ]$ using an ODE solver. While diffusion models can also be sampled through an SDE formulation, this can be highly inefficient and many methods that propose fast samplers (e.g., Song et al. (2020a); Zhang & Chen (2022)) directly make use of the ODE perspective (see Appendix D). In part, this is due to ODE solvers being much more efficient—yielding lower error at similar computational costs (Kloeden et al., 2012)—and the multitude of available ODE solver schemes. When compared to our ablation models, we find that models trained using Flow Matching with the OT path always result in the most efficient sampler, regardless of ODE solver, as demonstrated next. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/d1afc5e0388f7cf7d6642a4f4574fe08ca41414fb88153f5ff7860879984dd6f.jpg", + "image_caption": [ + "Figure 7: Flow Matching, especially when using OT paths, allows us to use fewer evaluations for sampling while retaining similar numerical error (left) and sample quality (right). Results are shown for models trained on ImageNet $3 2 \\times 3 2$ , and numerical errors are for the midpoint scheme. " + ], + "image_footnote": [], + "bbox": [ + 191, + 102, + 808, + 186 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 250, + 825, + 347 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Sample paths. We first qualitatively visualize the difference in sampling paths between diffusion and OT. Figure 6 shows samples from ImageNet-64 models using identical random seeds, where we find that the OT path model starts generating images sooner than the diffusion path models, where noise dominates the image until the very last time point. We additionally depict the probability density paths in 2D generation of a checkerboard pattern, Figure 4 (left), noticing a similar trend. ", + "bbox": [ + 174, + 354, + 825, + 424 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Low-cost samples. We next switch to fixed-step solvers and compare low $( \\leq 1 0 0 )$ NFE samples computed with the ImageNet-32 models from Table 1. In Figure 7 (left), we compare the per-pixel MSE of low NFE solutions compared with $1 0 0 0 ~ \\mathrm { N F E }$ solutions (we use 256 random noise seeds), and notice that the FM with OT model produces the best numerical error, in terms of computational cost, requiring roughly only $60 \\%$ of the NFEs to reach the same error threshold as diffusion models. Secondly, Figure 7 (right) shows how FID changes as a result of the computational cost, where we find FM with OT is able to achieve decent FID even at very low NFE values, producing better tradeoff between sample quality and cost compared to ablated models. Figure 4 (right) shows low-cost sampling effects for the 2D checkerboard experiment. ", + "bbox": [ + 173, + 431, + 825, + 556 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6.3 CONDITIONAL SAMPLING FROM LOW-RESOLUTION IMAGES ", + "text_level": 1, + "bbox": [ + 176, + 570, + 625, + 583 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Lastly, we experimented with Flow Matching for condi \ntional image generation. In particular, upsampling images \nfrom $6 4 \\times 6 4$ to $2 5 6 \\times 2 5 6$ . We follow the evaluation proce \ndure in (Saharia et al., 2022) and compute the FID of the \nupsampled validation images; baselines include reference \n(FID of original validation set), and regression. Results \nare in Table 2. Upsampled image samples are shown in \nFigures 14, 15 in the Appendix. FM-OT achieves simi \nlar PSNR and SSIM values to (Saharia et al., 2022) while \nconsiderably improving on FID and IS, which as argued by (Saharia et al., 2022) is a better indication ", + "bbox": [ + 174, + 595, + 552, + 739 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "of generation quality. ", + "bbox": [ + 173, + 720, + 823, + 748 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/5b254a42af1abf9071aa58ce1d1b8ccacd4150a50a976cf2bb2111ed9748e69a.jpg", + "table_caption": [ + "Table 2: Image super-resolution on the ImageNet validation set. " + ], + "table_footnote": [], + "table_body": "
ModelFID↓IS↑PSNR↑SSIM↑
Reference1.9240.81一
Regression15.2121.127.90.801
SR3 (Saharia et al,22)5.2180.126.40.762
FM W/OT3.4200.824.70.747
", + "bbox": [ + 568, + 599, + 820, + 672 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 768, + 318, + 785 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We introduced Flow Matching, a new simulation-free framework for training Continuous Normalizing Flow models, relying on conditional constructions to effortlessly scale to very high dimensions. Furthermore, the FM framework provides an alternative view on diffusion models, and suggests forsaking the stochastic/diffusion construction in favor of more directly specifying the probability path, allowing us to, e.g., construct paths that allow faster sampling and/or improve generation. We experimentally showed the ease of training and sampling when using the Flow Matching framework, and in the future, we expect FM to open the door to allowing a multitude of probability paths (e.g., non-isotropic Gaussians or more general kernels altogether). ", + "bbox": [ + 174, + 796, + 825, + 909 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "SOCIAL RESPONSIBILITY ", + "text_level": 1, + "bbox": [ + 176, + 104, + 348, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Along side its many positive applications, image generation can also be used for harmful proposes. Using content-controlled training sets and image validation/classification can help reduce these uses. Furthermore, the energy demand for training large deep learning models is increasing at a rapid pace (Amodei et al., 2018; Thompson et al., 2020), focusing on methods that are able to train using less gradient updates / image throughput can lead to significant time and energy savings. ", + "bbox": [ + 174, + 130, + 825, + 199 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 218, + 334, + 229 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We thank Qinqing Zheng for her feedback. HB is supported by a grant from Israel CHE Program for Data Science Research Centers. Additionally, we acknowledge the Python community (Van Rossum & Drake Jr, 1995; Oliphant, 2007) for developing the core set of tools that enabled this work, including PyTorch (Paszke et al., 2019), PyTorch Lightning (Falcon & team, 2019), Hydra (Yadan, 2019), Jupyter (Kluyver et al., 2016), Matplotlib (Hunter, 2007), seaborn (Waskom et al., 2018), numpy (Oliphant, 2006; Van Der Walt et al., 2011), SciPy (Jones et al., 2014), and torchdiffeq (Chen, 2018). 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", + "bbox": [ + 173, + 797, + 825, + 840 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Qinsheng Zhang and Yongxin Chen. Fast sampling of diffusion models with exponential integrator. arXiv preprint arXiv:2204.13902, 2022. ", + "bbox": [ + 173, + 849, + 823, + 878 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A THE CONTINUITY EQUATION ", + "text_level": 1, + "bbox": [ + 176, + 102, + 447, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "One method of testing if a vector field $v _ { t }$ generates a probability path $p _ { t }$ is the continuity equation (Villani, 2009). It is a Partial Differential Equation (PDE) providing a necessary and sufficient condition to ensuring that a vector field $v _ { t }$ generates $p _ { t }$ , ", + "bbox": [ + 173, + 132, + 825, + 175 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/fe5b04412511cc63e0f66d8743d56a4f7bf84f07cd6cebc6e455547a8200de05.jpg", + "text": "$$\n\\frac { d } { d t } p _ { t } ( x ) + \\mathrm { d i v } ( p _ { t } ( x ) v _ { t } ( x ) ) = 0 ,\n$$", + "text_format": "latex", + "bbox": [ + 390, + 179, + 607, + 209 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where the divergence operator, div, is defined with respect to the spatial variable $\\boldsymbol { x } = ( x ^ { 1 } , \\dots , x ^ { d } )$ , i.e., div = Pdi=1 ∂∂xi . ", + "bbox": [ + 171, + 213, + 825, + 248 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B THEOREM PROOFS ", + "text_level": 1, + "bbox": [ + 176, + 263, + 366, + 281 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Theorem 1. Given vector fields $u _ { t } ( x | x _ { 1 } )$ that generate conditional probability paths $p _ { t } ( x | x _ { 1 } )$ , for any distribution $q ( x _ { 1 } )$ , the marginal vector field $u _ { t }$ in equation 8 generates the marginal probability path $p _ { t }$ in equation $6$ , i.e., $u _ { t }$ and $p _ { t }$ satisfy the continuity equation (equation 25). ", + "bbox": [ + 173, + 295, + 825, + 339 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof. To verify this, we check that $p _ { t }$ and $u _ { t }$ satisfy the continuity equation (equation 25): ", + "bbox": [ + 169, + 353, + 769, + 369 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/1798bfb32ea534f03ba1110ebe18649243e5914a188a84921fd57f51ef08d354.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\frac { d } { d t } p _ { t } ( x ) = \\int \\Big ( \\frac { d } { d t } p _ { t } ( x | x _ { 1 } ) \\Big ) q ( x _ { 1 } ) d x _ { 1 } = - \\int \\mathrm { d i v } \\Big ( u _ { t } ( x | x _ { 1 } ) p _ { t } ( x | x _ { 1 } ) \\Big ) q ( x _ { 1 } ) d x _ { 1 } } \\\\ { \\displaystyle \\qquad = - \\mathrm { d i v } \\Big ( \\int u _ { t } ( x | x _ { 1 } ) p _ { t } ( x | x _ { 1 } ) q ( x _ { 1 } ) d x _ { 1 } \\Big ) = - \\mathrm { d i v } \\Big ( u _ { t } ( x ) p _ { t } ( x ) \\Big ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 233, + 390, + 764, + 458 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where in the second equality we used the fact that $u _ { t } ( \\cdot | x _ { 1 } )$ generates $p _ { t } ( \\cdot | x _ { 1 } )$ , in the last equality we used equation 8. Furthermore, the first and third equalities are justified by assuming the integrands satisfy the regularity conditions of the Leibniz Rule (for exchanging integration and differentiation). ", + "bbox": [ + 178, + 459, + 821, + 505 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Theorem 2. Assuming that $p _ { t } ( x ) > 0$ for all $x ~ \\in ~ \\mathbb { R } ^ { d }$ and $t \\in [ 0 , 1 ]$ , then, up to a constant independent of $\\theta$ , $\\mathcal { L } _ { \\scriptscriptstyle C F M }$ and $\\mathcal { L } _ { \\boldsymbol { F M } }$ are equal. Hence, $\\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\boldsymbol { F M } } ( \\boldsymbol { \\theta } ) = \\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\boldsymbol { c F M } } ( \\boldsymbol { \\theta } )$ . ", + "bbox": [ + 173, + 522, + 825, + 553 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof. To ensure existence of all integrals and to allow the changing of integration order (by Fubini’s Theorem) in the following we assume that $q ( x )$ and $p _ { t } ( x | x _ { 1 } )$ are decreasing to zero at a sufficient speed as $\\| x \\| \\to \\infty$ , and that $u _ { t } , v _ { t } , \\nabla _ { \\theta } v _ { t }$ are bounded. ", + "bbox": [ + 173, + 566, + 826, + 609 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "First, using the standard bilinearity of the 2-norm we have that ", + "bbox": [ + 173, + 616, + 584, + 631 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/3e6dbd0d3876856c78f31592f78e6a1fe03df228e337ebccac85c32bddb91803.jpg", + "text": "$$\n\\begin{array} { c } { { \\left\\| v _ { t } ( x ) - u _ { t } ( x ) \\right\\| ^ { 2 } = \\left\\| v _ { t } ( x ) \\right\\| ^ { 2 } - 2 \\left. v _ { t } ( x ) , u _ { t } ( x ) \\right. + \\left\\| u _ { t } ( x ) \\right\\| ^ { 2 } } } \\\\ { { \\left\\| v _ { t } ( x ) - u _ { t } ( x | x _ { 1 } ) \\right\\| ^ { 2 } = \\left\\| v _ { t } ( x ) \\right\\| ^ { 2 } - 2 \\left. v _ { t } ( x ) , u _ { t } ( x | x _ { 1 } ) \\right. + \\left\\| u _ { t } ( x | x _ { 1 } ) \\right\\| ^ { 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 633, + 735, + 678 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Next, remember that $u _ { t }$ is independent of $\\theta$ and note that ", + "bbox": [ + 174, + 679, + 545, + 694 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/c338a43f682aac58436403bd1b876823bbce5f62e42c729b8080b138bbf3fa86.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathbb { E } _ { p _ { t } ( x ) } \\| v _ { t } ( x ) \\| ^ { 2 } = \\int \\| v _ { t } ( x ) \\| ^ { 2 } p _ { t } ( x ) d x = \\int \\| v _ { t } ( x ) \\| ^ { 2 } p _ { t } ( x | x _ { 1 } ) q ( x _ { 1 } ) d x _ { 1 } d x } \\\\ { \\displaystyle \\qquad = \\mathbb { E } _ { q ( x _ { 1 } ) , p _ { t } ( x | x _ { 1 } ) } \\| v _ { t } ( x ) \\| ^ { 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 251, + 696, + 746, + 751 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where in the second equality we use equation 6, and in the third equality we change the order of integration. Next, ", + "bbox": [ + 171, + 752, + 823, + 780 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/36081ff2181235c781967509e003805b29f5869e0ed73e57be90102cd2900914.jpg", + "text": "$$\n\\begin{array} { l } { \\mathbb { E } _ { p _ { t } ( x ) } \\left. v _ { t } ( x ) , u _ { t } ( x ) \\right. = \\displaystyle \\int \\left. v _ { t } ( x ) , \\frac { \\int u _ { t } ( x | x _ { 1 } ) p _ { t } ( x | x _ { 1 } ) q ( x _ { 1 } ) d x _ { 1 } } { p _ { t } ( x ) } \\right. p _ { t } ( x ) d x } \\\\ { \\displaystyle \\qquad = \\int \\left. v _ { t } ( x ) , \\int u _ { t } ( x | x _ { 1 } ) p _ { t } ( x | x _ { 1 } ) q ( x _ { 1 } ) d x _ { 1 } \\right. d x } \\\\ { \\displaystyle \\qquad = \\int \\left. v _ { t } ( x ) , u _ { t } ( x | x _ { 1 } ) \\right. p _ { t } ( x | x _ { 1 } ) q ( x _ { 1 } ) d x _ { 1 } d x } \\\\ { \\displaystyle \\qquad = \\mathbb { E } _ { q ( x _ { 1 } ) , p _ { t } ( x | x _ { 1 } ) } \\left. v _ { t } ( x ) , u _ { t } ( x | x _ { 1 } ) \\right. , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 246, + 784, + 751, + 907 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where in the last equality we change again the order of integration. ", + "bbox": [ + 173, + 909, + 611, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Theorem 3. Let $p _ { t } ( x | x _ { 1 } )$ be a Gaussian probability path as in equation $I O$ , and $\\psi _ { t }$ its corresponding flow map as in equation $I I$ . Then, the unique vector field that defines $\\psi _ { t }$ has the form: ", + "bbox": [ + 169, + 103, + 823, + 133 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/febfed21402c426ec001e37526a7ccb3d032e50c7dd37dbaa621376e31bb6bf4.jpg", + "text": "$$\nu _ { t } ( x | x _ { 1 } ) = \\frac { \\sigma _ { t } ^ { \\prime } ( x _ { 1 } ) } { \\sigma _ { t } ( x _ { 1 } ) } \\left( x - \\mu _ { t } ( x _ { 1 } ) \\right) + \\mu _ { t } ^ { \\prime } ( x _ { 1 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 354, + 137, + 643, + 172 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Consequently, $u _ { t } ( x | x _ { 1 } )$ generates the Gaussian path $p _ { t } ( x | x _ { 1 } )$ . ", + "bbox": [ + 174, + 178, + 584, + 194 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. For notational simplicity let $w _ { t } ( x ) = u _ { t } ( x | x _ { 1 } )$ . Now consider equation 1: ", + "bbox": [ + 173, + 207, + 705, + 224 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/ff7aea21c37034340b677056bb80d93ce868e087e755c5e023d243c2fa31ec62.jpg", + "text": "$$\n\\frac { d } { d t } \\psi _ { t } ( x ) = w _ { t } ( \\psi _ { t } ( x ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 421, + 229, + 576, + 261 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Since $\\psi _ { t }$ is invertible (as $\\sigma _ { t } ( x _ { 1 } ) > 0 ,$ ) we let $x = \\psi ^ { - 1 } ( y )$ and get ", + "bbox": [ + 173, + 266, + 602, + 282 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/9c285dda1582bfcda493913a5d4bee5cf66a83a39e3543f58f2881e6f3184fb0.jpg", + "text": "$$\n\\psi _ { t } ^ { \\prime } ( \\psi ^ { - 1 } ( y ) ) = w _ { t } ( y ) ,\n$$", + "text_format": "latex", + "bbox": [ + 424, + 287, + 571, + 306 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where we used the apostrophe notation for the derivative to emphasis that $\\psi _ { t } ^ { \\prime }$ is evaluated at $\\psi ^ { - 1 } ( y )$ . Now, inverting $\\psi _ { t } ( x )$ provides ", + "bbox": [ + 173, + 313, + 826, + 340 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/8f33f39f637677a7d6f426360164dcf778fb973c616f67efd5386eb61ffd4677.jpg", + "text": "$$\n\\psi _ { t } ^ { - 1 } ( y ) = \\frac { y - \\mu _ { t } ( x _ { 1 } ) } { \\sigma _ { t } ( x _ { 1 } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 419, + 338, + 578, + 373 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Differentiating $\\psi _ { t }$ with respect to $t$ gives ", + "bbox": [ + 174, + 376, + 441, + 390 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/12323b8cdd016082c6986994e4b03ca436ed52b715384a3d63433c72bc6a3efd.jpg", + "text": "$$\n\\psi _ { t } ^ { \\prime } ( x ) = \\sigma _ { t } ^ { \\prime } ( x _ { 1 } ) x + \\mu _ { t } ^ { \\prime } ( x _ { 1 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 405, + 395, + 593, + 414 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Plugging these last two equations in equation 26 we get ", + "bbox": [ + 173, + 420, + 540, + 435 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/4cac509816da004feffa199680fff84c267345a41c35b5d3283af3ed5d50b509.jpg", + "text": "$$\nw _ { t } ( y ) = \\frac { \\sigma _ { t } ^ { \\prime } ( x _ { 1 } ) } { \\sigma _ { t } ( x _ { 1 } ) } \\left( y - \\mu _ { t } ( x _ { 1 } ) \\right) + \\mu _ { t } ^ { \\prime } ( x _ { 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 366, + 440, + 632, + 474 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "as required. ", + "bbox": [ + 173, + 481, + 251, + 494 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C COMPUTING PROBABILITIES OF THE CNF MODEL ", + "text_level": 1, + "bbox": [ + 173, + 513, + 620, + 531 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We are given an arbitrary data point $x _ { 1 } \\in \\mathbb { R } ^ { d }$ and need to compute the model probability at that point, i.e., $p _ { 1 } ( x _ { 1 } )$ . Below we recap how this can be done covering the basic relevant ODEs, the scaling of the divergence computation, taking into account data transformations (e.g., centering of data), and Bits-Per-Dimension computation. ", + "bbox": [ + 173, + 545, + 825, + 603 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "ODE for computing $p _ { 1 } ( x _ { 1 } )$ . The continuity equation with equation 1 lead to the instantaneous change of variable (Chen et al., 2018; Ben-Hamu et al., 2022): ", + "bbox": [ + 171, + 617, + 823, + 646 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/38e9cca916579243d33e88e33265d929c1034a249edef99003745828fe8df29d.jpg", + "text": "$$\n\\frac { d } { d t } \\log p _ { t } ( \\phi _ { t } ( x ) ) + \\mathrm { d i v } \\big ( v _ { t } ( \\phi _ { t } ( x ) ) = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 369, + 651, + 629, + 683 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Integrating $t \\in [ 0 , 1 ]$ gives: ", + "bbox": [ + 173, + 688, + 352, + 704 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/794a24fca0231f7511e7df773c9487df090509a8b6349aee695025ecd9c9fc1e.jpg", + "text": "$$\n\\log p _ { 1 } ( \\phi _ { 1 } ( x ) ) - \\log p _ { 0 } ( \\phi _ { 0 } ( x ) ) = - \\int _ { 0 } ^ { 1 } \\mathrm { d i v } ( v _ { t } ( \\phi _ { t } ( x ) ) ) d t\n$$", + "text_format": "latex", + "bbox": [ + 307, + 709, + 691, + 746 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Therefore, the log probability can be computed together with the flow trajectory by solving the ODE: ", + "bbox": [ + 176, + 751, + 820, + 766 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/e12581f3f3ffcc91e8a45113f6f01a6de7eb3296e50ab9fe45445e521928443d.jpg", + "text": "$$\n\\frac { d } { d t } \\left[ \\phi _ { t } ( x ) \\right] = \\left[ \\begin{array} { c } { v _ { t } ( \\phi _ { t } ( x ) ) } \\\\ { - \\mathrm { d i v } \\big ( v _ { t } ( \\phi _ { t } ( x ) ) \\big ) } \\end{array} \\right]\n$$", + "text_format": "latex", + "bbox": [ + 383, + 771, + 614, + 806 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Given initial conditions ", + "bbox": [ + 174, + 811, + 330, + 825 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/51afdf1e8e6d35a94b292be5c1a0372adff62cde2cb3379094e63bd467f1d78e.jpg", + "text": "$$\n\\left[ \\begin{array} { c } { { \\phi _ { 0 } ( x ) } } \\\\ { { f ( 0 ) } } \\end{array} \\right] = \\left[ \\begin{array} { c } { { x _ { 0 } } } \\\\ { { c } } \\end{array} \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 436, + 829, + 560, + 866 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "the solution $\\left[ \\phi _ { t } ( x ) , f ( t ) \\right] ^ { T }$ is uniquely defined (up to some mild conditions on the $\\mathrm { V F } v _ { t } .$ ). Denote $x _ { 1 } = \\phi _ { 1 } ( x )$ , and according to equation 27, ", + "bbox": [ + 173, + 871, + 826, + 902 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/ee586360fd4d5d7f0bc70adf3c90696e1bb01e1542631737784c2bf7c42ddf3e.jpg", + "text": "$$\nf ( 1 ) = c + \\log p _ { 1 } ( x _ { 1 } ) - \\log p _ { 0 } ( x _ { 0 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 907, + 620, + 925 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Now, we are given an arbitrary $x _ { 1 }$ and want to compute $p _ { 1 } ( x _ { 1 } )$ . For this end, we will need to solve equation 28 in reverse. That is, ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/bf7b00b4701ea3ef4269ebe8eb251e0f823a73d5102a8d0a24e7bb5750937859.jpg", + "text": "$$\n{ \\frac { d } { d s } } \\left[ { \\phi } _ { 1 - s } ( x ) \\right] = \\left[ \\mathrm { - } v _ { 1 - s } ( { \\phi } _ { 1 - s } ( x ) ) \\right]\n$$", + "text_format": "latex", + "bbox": [ + 361, + 137, + 635, + 172 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "and we solve this equation for $s \\in [ 0 , 1 ]$ with the initial conditions at $s = 0$ ", + "bbox": [ + 176, + 178, + 668, + 194 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/5ec7749a3b17d695b104ff6922ce190292db97c755334ef08a6229291302dcad.jpg", + "text": "$$\n\\left[ \\begin{array} { c } { { \\phi _ { 1 } ( x ) } } \\\\ { { f ( 1 ) } } \\end{array} \\right] = \\left[ \\begin{array} { c } { { x _ { 1 } } } \\\\ { { 0 } } \\end{array} \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 437, + 199, + 560, + 234 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "From uniqueness of ODEs, the solution will be identical to the solution of equation 28 with initial conditions in equation 29 where $c = \\log p _ { 0 } ( x _ { 0 } ) - \\log p _ { 1 } ( x _ { 1 } )$ . This can be seen from equation 30 and setting $f ( 1 ) = 0$ . Therefore we get that ", + "bbox": [ + 173, + 238, + 825, + 281 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/492d44bdc9de504ecd1d093679f2b4e24fd9befc4ec2bc5ecf04c011fdc26138.jpg", + "text": "$$\nf ( 0 ) = \\log p _ { 0 } ( x _ { 0 } ) - \\log p _ { 1 } ( x _ { 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 392, + 286, + 606, + 305 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "and consequently ", + "bbox": [ + 173, + 310, + 289, + 324 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/239311dc78f3fe46fa589c3516c2c74da36ac8537af44f6899d4531d8d0522cc.jpg", + "text": "$$\n\\log p _ { 1 } ( x _ { 1 } ) = \\log p _ { 0 } ( x _ { 0 } ) - f ( 0 ) .\n$$", + "text_format": "latex", + "bbox": [ + 388, + 330, + 607, + 348 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "To summarize, to compute $p _ { 1 } ( x _ { 1 } )$ we first solve the ODE in equation 31 with initial conditions in equation 32, and the compute equation 33. ", + "bbox": [ + 174, + 353, + 825, + 383 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Unbiased estimator to $p _ { 1 } ( x _ { 1 } )$ . Solving equation 31 requires computation of div of VFs in $\\mathbb { R } ^ { d }$ which is costly. Grathwohl et al. (2018) suggest to replace the divergence by the (unbiased) Hutchinson trace estimator, ", + "bbox": [ + 173, + 396, + 825, + 439 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/055301d47bf988de36311256d3b6992a7bd09623e19c8ecd246d843237e4cf7c.jpg", + "text": "$$\n\\frac { d } { d s } \\left[ \\stackrel { \\phi _ { 1 - s } ( x ) } { \\tilde { f } ( 1 - s ) } \\right] = \\left[ z ^ { T } D v _ { 1 - s } ( \\phi _ { 1 - s } ( x ) ) z \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 354, + 444, + 642, + 479 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $z \\in \\mathbb { R } ^ { d }$ is a sample from a random variable such that $\\mathbb { E } z z ^ { T } = I$ . Solving the ODE in equation 34 exactly (in practice, with a small controlled error) with initial conditions in equation 32 leads to ", + "bbox": [ + 174, + 486, + 825, + 529 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/85f157c4e766b262cde9e0765dcb2c77f194a6c15e21b40311e6a5db93147f6d.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathbb { E } _ { z } \\left[ \\log p _ { 0 } ( x _ { 0 } ) - \\tilde { f } ( 0 ) \\right] = \\log p _ { 0 } ( x _ { 0 } ) - \\mathbb { E } _ { z } \\left[ \\tilde { f } ( 0 ) - \\tilde { f } ( 1 ) \\right] } & { } \\\\ & { = \\log p _ { 0 } ( x _ { 0 } ) - \\mathbb { E } _ { z } \\left[ \\int _ { 0 } ^ { 1 } z ^ { T } D v _ { 1 - s } ( \\phi _ { 1 - s } ( x ) ) z d s \\right] } \\\\ & { = \\log p _ { 0 } ( x _ { 0 } ) - \\int _ { 0 } ^ { 1 } \\mathbb { E } _ { z } \\left[ z ^ { T } D v _ { 1 - s } ( \\phi _ { 1 - s } ( x ) ) z \\right] d s } \\\\ & { = \\log p _ { 0 } ( x _ { 0 } ) - \\int _ { 0 } ^ { 1 } \\mathrm { d i v } ( v _ { 1 - s } ( \\phi _ { 1 - s } ( x ) ) ) d s } \\\\ & { = \\log p _ { 0 } ( x _ { 0 } ) - ( f ( 0 ) - f ( 1 ) ) } \\\\ & { = \\log p _ { 0 } ( x _ { 0 } ) - ( \\log p _ { 0 } ( x _ { 0 } ) - \\log p _ { 1 } ( x _ { 1 } ) ) } \\\\ & { = \\log p _ { 1 } ( x _ { 1 } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 531, + 746, + 723 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where in the third equality we switched order of integration assuming the sufficient condition of Fubini’s theorem hold, and in the previous to last equality we used equation 30. Therefore the random variable ", + "bbox": [ + 174, + 727, + 825, + 768 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/4ec6d502f5641697e10299a904bc0735b356fa81b3a22c4cba932eb87306f84d.jpg", + "text": "$$\n\\log p _ { 0 } ( x _ { 0 } ) - \\tilde { f } ( 0 )\n$$", + "text_format": "latex", + "bbox": [ + 436, + 772, + 562, + 792 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "is an unbiased estimator for $\\log p _ { 1 } ( x _ { 1 } )$ . To summarize, for a scalable unbiased estimation of $p _ { 1 } ( x _ { 1 } )$ we first solve the ODE in equation 34 with initial conditions in equation 32, and then output equation 35. ", + "bbox": [ + 173, + 797, + 825, + 840 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Transformed data. Often, before training our generative model we transform the data, e.g., we scale and/or translate the data. Such a transformation is denoted by $\\varphi ^ { - 1 } : \\mathbb { R } ^ { d } \\to \\mathbb { R } ^ { d }$ and our generative model becomes a composition ", + "bbox": [ + 174, + 854, + 825, + 898 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/ffabfa3922b884205e3f7c00ba68e7a84dc8fb3773c6d6a68ab60eda5d02f23c.jpg", + "text": "$$\n\\psi ( x ) = \\varphi \\circ \\phi ( x )\n$$", + "text_format": "latex", + "bbox": [ + 441, + 904, + 557, + 921 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $\\phi : \\mathbb { R } ^ { d } \\mathbb { R } ^ { d }$ is the model we train. Given a prior probability $p _ { 0 }$ we have that the push forward of this probability under $\\psi$ (equation 3 and equation 4) takes the form ", + "bbox": [ + 169, + 102, + 825, + 132 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/c711a083465757af9f39beacaacf0e892e2cc3d3ef6a42f168e026de550fd736.jpg", + "text": "$$\n\\begin{array} { c } { { p _ { 1 } ( x ) = \\psi _ { * } p _ { 0 } ( x ) = p _ { 0 } ( \\phi ^ { - 1 } ( \\varphi ^ { - 1 } ( x ) ) ) \\operatorname* { d e t } \\left[ D \\phi ^ { - 1 } ( \\varphi ^ { - 1 } ( x ) ) \\right] \\operatorname* { d e t } \\left[ D \\varphi ^ { - 1 } ( x ) \\right] } } \\\\ { { = \\left( \\phi _ { * } p _ { 0 } ( \\varphi ^ { - 1 } ( x ) ) \\right) \\operatorname* { d e t } \\left[ D \\varphi ^ { - 1 } ( x ) \\right] } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 138, + 754, + 181 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "and therefore ", + "bbox": [ + 173, + 188, + 263, + 202 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/5be21aea02ad7693d4cf8f2e82e691a51afbed430dc7ef97c09beaf60753c7aa.jpg", + "text": "$$\n\\begin{array} { r } { \\log p _ { 1 } ( x ) = \\log \\phi _ { * } p _ { 0 } ( \\varphi ^ { - 1 } ( x ) ) + \\log \\operatorname* { d e t } \\left[ D \\varphi ^ { - 1 } ( x ) \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 318, + 209, + 678, + 229 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For images $d = H \\times W \\times 3$ we consider a transform $\\phi$ that maps each pixel value from $[ - 1 , 1 ]$ to $[ 0 , 2 5 6 ]$ . Therefore, ", + "bbox": [ + 173, + 238, + 823, + 266 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/ac5cdb9d1eb7d43ea36e6a21f0df2fd07ddccf92640fc9dba09b7d511d69a731.jpg", + "text": "$$\n\\varphi ( y ) = 2 ^ { 7 } ( y + 1 ) ,\n$$", + "text_format": "latex", + "bbox": [ + 434, + 268, + 560, + 287 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 294, + 200, + 308 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/81107ca33fc70d84cd6f276c4782ee542ce4c1a77539baf06773f9afca3a1b8b.jpg", + "text": "$$\n\\varphi ^ { - 1 } ( x ) = 2 ^ { - 7 } x - 1 .\n$$", + "text_format": "latex", + "bbox": [ + 426, + 308, + 570, + 327 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For this case we have ", + "bbox": [ + 173, + 333, + 316, + 347 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/9613f04d34a658dacd796567c047deff292ce4d56e18b56f4763a35c45cf69b1.jpg", + "text": "$$\n\\log p _ { 1 } ( x ) = \\log \\phi _ { * } p _ { 0 } ( \\varphi ^ { - 1 } ( x ) ) - 7 d \\log 2 .\n$$", + "text_format": "latex", + "bbox": [ + 357, + 347, + 642, + 367 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Bits-Per-Dimension (BPD) computation. BPD is defined by ", + "text_level": 1, + "bbox": [ + 173, + 381, + 591, + 397 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/a793be7f55133886984e5a339286d130cac006a1bb762b9689eb884a782f6e7d.jpg", + "text": "$$\n\\mathrm { B P D } = \\mathbb { E } _ { x _ { 1 } } \\left[ - \\frac { \\log _ { 2 } p _ { 1 } ( x _ { 1 } ) } { d } \\right] = \\mathbb { E } _ { x _ { 1 } } \\left[ - \\frac { \\log p _ { 1 } ( x _ { 1 } ) } { d \\log 2 } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 323, + 405, + 674, + 440 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Following equation 36 we get ", + "bbox": [ + 174, + 449, + 372, + 463 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/2d92bb868561a5e9a025bb78fb133db33e2792a16fd033e3e5cd9fa895e6de81.jpg", + "text": "$$\n\\mathrm { B P D } = - \\frac { \\log \\phi _ { * } p _ { 0 } ( \\varphi ^ { - 1 } ( x ) ) } { d \\log 2 } + 7 .\n$$", + "text_format": "latex", + "bbox": [ + 383, + 470, + 614, + 506 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "and $\\log { \\phi _ { * } p _ { 0 } ( \\varphi ^ { - 1 } ( x ) ) }$ is approximated using the unbiased estimator in equation 35 over the transformed data $\\varphi ^ { - 1 } ( x _ { 1 } )$ . Averaging the unbiased estimator on a large test test $x _ { 1 }$ provides a good approximation to the test set BPD. ", + "bbox": [ + 174, + 515, + 825, + 558 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "D DIFFUSION CONDITIONAL VECTOR FIELDS ", + "text_level": 1, + "bbox": [ + 173, + 582, + 562, + 598 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We derive the vector field governing the Probability Flow ODE (equation 13 in Song et al. (2020b)) for the VE and VP diffusion paths (equation 18) and note that it coincides with the conditional vector fields we derive using Theorem 3, namely the vector fields defined in equations 16 and 19. ", + "bbox": [ + 173, + 613, + 825, + 656 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We start with a short primer on how to find a conditional vector field for the probability path described by the Fokker-Planck equation, then instantiate it for the VE and VP probability paths. ", + "bbox": [ + 174, + 662, + 823, + 691 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Since in the diffusion literature the diffusion process runs from data at time $t = 0$ to noise at time $t = 1$ , we will need the following lemma to translate the diffusion VFs to our convention of $t = 0$ corresponds to noise and $t = 1$ corresponds to data: ", + "bbox": [ + 174, + 698, + 825, + 741 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Lemma 1. Consider a flow defined by a vector field $u _ { t } ( x )$ generating probability density path $p _ { t } ( x )$ . Then, the vector field $\\tilde { u } _ { t } ( x ) = - u _ { 1 - t } ( x )$ generates the path $\\tilde { p } _ { t } ( x ) = p _ { 1 - t } ( x )$ when initiated from $\\tilde { p } _ { 0 } ( x ) = p _ { 1 } ( x )$ . ", + "bbox": [ + 173, + 746, + 825, + 790 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. We use the continuity equation (equation 25): ", + "bbox": [ + 173, + 810, + 522, + 827 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/a0aa6278e747bb1b91e56135d9f15d4dac7159aa9c4691ed3a635b7ba2930002.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\frac { d } { d t } \\tilde { p } _ { t } ( x ) = \\frac { d } { d t } p _ { 1 - t } ( x ) = - p _ { 1 - t } ^ { \\prime } ( x ) } \\\\ { \\displaystyle = \\mathrm { d i v } \\big ( p _ { 1 - t } ( x ) u _ { 1 - t } ( x ) \\big ) } \\\\ { \\displaystyle = - \\mathrm { d i v } \\big ( \\tilde { p } _ { t } ( x ) \\big ( - u _ { 1 - t } ( x ) \\big ) \\big ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 833, + 666, + 902 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "and therefore $\\tilde { u } _ { t } ( x ) = - u _ { 1 - t } ( x )$ generates $\\tilde { p } _ { t } ( x )$ . ", + "bbox": [ + 173, + 909, + 501, + 925 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Conditional VFs for Fokker-Planck probability paths Consider a Stochastic Differential Equation (SDE) of the standard form ", + "bbox": [ + 173, + 102, + 823, + 131 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/919d753c1a7b23e1a007d6584703fe8397c46395567fa599af96c3b5e2ddac03.jpg", + "text": "$$\nd y = f _ { t } d t + g _ { t } d w\n$$", + "text_format": "latex", + "bbox": [ + 436, + 131, + 562, + 147 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "with time parameter $t$ , drift $f _ { t }$ , diffusion coefficient $g _ { t }$ , and $d w$ is the Wiener process. The solution $y _ { t }$ to the SDE is a stochastic process, i.e., a continuous time-dependent random variable, the probability density of which, $p _ { t } ( y _ { t } )$ , is characterized by the Fokker-Planck equation: ", + "bbox": [ + 173, + 147, + 825, + 189 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/b3692e8b2b3ba9c00f8c55dd57fd29904ee9fbbb2d9a296481deb43b6b633eaf.jpg", + "text": "$$\n\\frac { d p _ { t } } { d t } = - \\mathrm { d i v } ( f _ { t } p _ { t } ) + \\frac { g _ { t } ^ { 2 } } { 2 } \\Delta p _ { t }\n$$", + "text_format": "latex", + "bbox": [ + 401, + 189, + 596, + 222 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $\\Delta$ represents the Laplace operator (in $y$ ), namely $\\operatorname { d i v } \\nabla$ , where $\\nabla$ is the gradient operator (also in $y$ ). Rewriting this equation in the form of the continuity equation can be done as follows (Maoutsa et al., 2020a): ", + "bbox": [ + 174, + 222, + 825, + 263 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/f1e7cd40b2808a9ad1907de052e44511326b9ba8a23a2631ee1871a32bf479c8.jpg", + "text": "$$\n\\frac { d p _ { t } } { d t } = - \\mathrm { d i v } \\Big ( f _ { t } p _ { t } - \\frac { g ^ { 2 } } { 2 } \\frac { \\nabla p _ { t } } { p _ { t } } p _ { t } \\Big ) = - \\mathrm { d i v } \\Big ( \\big ( f _ { t } - \\frac { g _ { t } ^ { 2 } } { 2 } \\nabla \\log p _ { t } \\big ) p _ { t } \\Big ) = - \\mathrm { d i v } \\Big ( w _ { t } p _ { t } \\Big )\n$$", + "text_format": "latex", + "bbox": [ + 230, + 263, + 767, + 297 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where the vector field ", + "bbox": [ + 174, + 297, + 318, + 313 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/036dd1ceb5d969588ec50be22d3fa4fa8bf621da3f5856541ce9fb781d80736b.jpg", + "text": "$$\nw _ { t } = f _ { t } - \\frac { g _ { t } ^ { 2 } } { 2 } \\nabla \\log p _ { t }\n$$", + "text_format": "latex", + "bbox": [ + 423, + 306, + 575, + 339 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "satisfies the continuity equation with the probability path $p _ { t }$ , and therefore generates $p _ { t }$ ", + "bbox": [ + 176, + 338, + 745, + 353 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Variance Exploding (VE) path The SDE for the VE path is ", + "bbox": [ + 173, + 364, + 584, + 381 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/0420adccd41dc71eb52284eeea10558b56454f63a964df7b6bfdde39ec81675b.jpg", + "text": "$$\nd y = { \\sqrt { { \\frac { d } { d t } } \\sigma _ { t } ^ { 2 } } } d w ,\n$$", + "text_format": "latex", + "bbox": [ + 439, + 382, + 557, + 416 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $\\sigma _ { 0 } = 0$ and increasing to infinity as $t 1$ . The SDE is moving from data, $y _ { 0 }$ , at $t = 0$ to noise, $y _ { 1 }$ , at $t = 1$ with the probability path ", + "bbox": [ + 174, + 416, + 823, + 444 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/4c5b959ba481298385e6e62ac8b84351a4f89cce1fb569ef239a932539ac795e.jpg", + "text": "$$\np _ { t } ( y | y _ { 0 } ) = \\mathcal { N } ( y | y _ { 0 } , \\sigma _ { t } ^ { 2 } I ) .\n$$", + "text_format": "latex", + "bbox": [ + 411, + 444, + 584, + 463 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The conditional VF according to equation 40 is: ", + "bbox": [ + 176, + 463, + 490, + 478 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/4161e550cae70a6e74c66f1d1d2a05dff98dda2c93704ffa9bdfc6f01121c788.jpg", + "text": "$$\nw _ { t } ( y | y _ { 0 } ) = \\frac { \\sigma _ { t } ^ { \\prime } } { \\sigma _ { t } } ( y - y _ { 0 } )\n$$", + "text_format": "latex", + "bbox": [ + 419, + 478, + 578, + 511 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Using Lemma 1 we get that the probability path ", + "bbox": [ + 176, + 511, + 488, + 525 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/48acef740825cbc0dae4d083ba02b10ae7712609a0c5909342b6604a10f40e48.jpg", + "text": "$$\n\\tilde { p } _ { t } ( y | y _ { 0 } ) = \\mathcal { N } ( y | y _ { 0 } , \\sigma _ { 1 - t } ^ { 2 } I )\n$$", + "text_format": "latex", + "bbox": [ + 405, + 525, + 591, + 544 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "is generated by ", + "bbox": [ + 173, + 545, + 274, + 559 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/8a8732a9f4f7eccaba0bbdae42c6eee0c1d4bf506ab40b34b707ae5107af9780.jpg", + "text": "$$\n\\tilde { w } _ { t } ( y | y _ { 0 } ) = - \\frac { \\sigma _ { 1 - t } ^ { \\prime } } { \\sigma _ { 1 - t } } ( y - y _ { 0 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 401, + 559, + 594, + 593 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "which coincides with equation 17. ", + "bbox": [ + 174, + 593, + 398, + 608 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Variance Preserving (VP) path The SDE for the VP path is ", + "bbox": [ + 173, + 621, + 586, + 637 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/d994865afc5547bda7055605c73fcae48c88f976ea142abd9ec59ba179d6bb14.jpg", + "text": "$$\nd y = - \\frac { T ^ { \\prime } ( t ) } { 2 } y + \\sqrt { T ^ { \\prime } ( t ) } d w ,\n$$", + "text_format": "latex", + "bbox": [ + 398, + 637, + 598, + 669 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { T ( t ) = \\int _ { 0 } ^ { t } \\beta ( s ) d s , t \\in [ 0 , 1 ] } \\end{array}$ . The SDE coefficients are therefore ", + "bbox": [ + 174, + 670, + 640, + 688 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/6aa767b290936be20a14d218eaf6b1d1f25989853538b6d560fe524f0b280d3e.jpg", + "text": "$$\nf _ { s } ( y ) = - \\frac { T ^ { \\prime } ( s ) } { 2 } y , \\quad g _ { s } = \\sqrt { T ^ { \\prime } ( s ) }\n$$", + "text_format": "latex", + "bbox": [ + 380, + 688, + 617, + 719 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 719, + 200, + 733 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/72ed9fa345eb3966144f1db93a8617af4bfd50a098a2d4b77c69b2389ca3136a.jpg", + "text": "$$\np _ { t } ( y | y _ { 0 } ) = \\mathcal { N } ( y | e ^ { - \\frac { 1 } { 2 } T ( t ) } y _ { 0 } , ( 1 - e ^ { - T ( t ) } ) I ) .\n$$", + "text_format": "latex", + "bbox": [ + 351, + 731, + 645, + 752 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Plugging these choices in equation 40 we get the conditional VF ", + "bbox": [ + 181, + 753, + 594, + 767 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/dc8e57f2caa10d1313053c019f2fc8de599f48865eedadab569d309880c0f3d5.jpg", + "text": "$$\nw _ { t } ( y | y _ { 0 } ) = { \\frac { T ^ { \\prime } ( t ) } { 2 } } \\left( { \\frac { y - e ^ { - { \\frac { 1 } { 2 } } T ( t ) } y _ { 0 } } { 1 - e ^ { - T ( t ) } } } - y \\right)\n$$", + "text_format": "latex", + "bbox": [ + 357, + 767, + 638, + 809 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Using Lemma 1 to reverse the time we get the conditional VF for the reverse probability path: ", + "bbox": [ + 176, + 809, + 785, + 824 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/ec62117d11f75b6ad8e3be0cd98c635f02e7e69ef04424b7174720dd9ef03630.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\tilde { w } _ { t } ( y | y _ { 0 } ) = - \\frac { T ^ { \\prime } ( 1 - t ) } { 2 } \\left( \\frac { y - e ^ { - \\frac 1 2 T ( 1 - t ) } y _ { 0 } } { 1 - e ^ { - T ( 1 - t ) } } - y \\right) } \\\\ { = - \\frac { T ^ { \\prime } ( 1 - t ) } { 2 } \\left[ \\frac { e ^ { - T ( 1 - t ) } y - e ^ { - \\frac 1 2 T ( 1 - t ) } y _ { 0 } } { 1 - e ^ { - T ( 1 - t ) } } \\right] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 313, + 825, + 683, + 910 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "which coincides with equation 19. ", + "bbox": [ + 174, + 909, + 398, + 924 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/fab5ea8c033deb403362d41d1fb6d0a7c5553b41abd30ba80e8f0ec3e0e2e6d7.jpg", + "image_caption": [ + "Figure 8: VP Diffusion path’s conditional vector field. Compare to Figure 2. " + ], + "image_footnote": [], + "bbox": [ + 323, + 99, + 669, + 188 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/10154dd1f03b66c681a67154d5dff0682668218e3c2e6793db94b67cd56d689e.jpg", + "image_caption": [ + "Figure 9: Trajectories of CNFs trained with ScoreFlow (Song et al., 2021) and DDPM (Ho et al., 2020) losses on 2D checkerboard data, using the same learning rate and other hyperparameters as Figure 4. " + ], + "image_footnote": [], + "bbox": [ + 276, + 226, + 720, + 325 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E IMPLEMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 404, + 434, + 420 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "For the 2D example we used an MLP with 5-layers of 512 neurons each, while for images we used the UNet architecture from Dhariwal & Nichol (2021). For images, we center crop images and resize to the appropriate dimension, whereas for the $3 2 \\times 3 2$ and $6 4 \\times 6 4$ resolutions we use the same pre-processing as (Chrabaszcz et al., 2017). The three methods (FM-OT, FM-Diffusion, and SMDiffusion) are always trained on the same architecture, same hyper-parameters, and for the same number of epochs. ", + "bbox": [ + 173, + 433, + 825, + 518 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E.1 DIFFUSION BASELINES", + "text_level": 1, + "bbox": [ + 174, + 534, + 375, + 547 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Losses. We consider three options as diffusion baselines that correspond to the most popular diffusion loss parametrizations (Song & Ermon, 2019; Song et al., 2021; Ho et al., 2020; Kingma et al., 2021). We will assume general Gaussian path form of equation 10, i.e., ", + "bbox": [ + 173, + 559, + 825, + 602 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/65bbc5204d2a2aa85b211e6da2dab8c6dfa941cf28880d553c357a2363194884.jpg", + "text": "$$\np _ { t } ( x | x _ { 1 } ) = \\mathcal { N } ( x | \\mu _ { t } ( x _ { 1 } ) , \\sigma _ { t } ^ { 2 } ( x _ { 1 } ) I ) .\n$$", + "text_format": "latex", + "bbox": [ + 380, + 604, + 616, + 623 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Score Matching loss is ", + "bbox": [ + 173, + 635, + 325, + 648 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/a5505a2c9c71dfb4c519371157a8e27a617ccc743d7e0e9e706d3b1f10fd0fa4.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { s x } } ( \\theta ) = \\mathbb { E } _ { t , q ( x _ { 1 } ) , p _ { t } ( x | x _ { 1 } ) } \\lambda ( t ) \\left\\| s _ { t } ( x ) - \\nabla \\log p _ { t } ( x | x _ { 1 } ) \\right\\| ^ { 2 } } \\\\ { = \\mathbb { E } _ { t , q ( x _ { 1 } ) , p _ { t } ( x | x _ { 1 } ) } \\lambda ( t ) \\left\\| s _ { t } ( x ) - \\frac { x - \\mu _ { t } ( x _ { 1 } ) } { \\sigma _ { t } ^ { 2 } ( x _ { 1 } ) } \\right\\| ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 651, + 687, + 713 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Taking $\\lambda ( t ) = \\sigma _ { t } ^ { 2 } ( x _ { 1 } )$ corresponds to the original Score Matching (SM) loss from Song & Ermon (2019), while considering $\\lambda ( t ) = \\beta ( 1 - t )$ ( $\\beta$ is defined below) corresponds to the Score Flow (SF) loss motivated by an NLL upper bound (Song et al., 2021); $s _ { t }$ is the learnable score function. DDPM (Noise Matching) loss from Ho et al. (2020) (equation 14) is ", + "bbox": [ + 176, + 715, + 825, + 771 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/8f2d887d11da49fccd5cad6658b6eff2b2a02e8a8041c7801ec71696a3f66136.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\mathcal { L } _ { \\mathrm { \\tiny { N M } } } ( \\theta ) = \\mathbb { E } _ { t , q ( x _ { 1 } ) , p _ { t } ( x | x _ { 1 } ) } \\left\\| \\epsilon _ { t } ( x ) - \\frac { x - \\mu _ { t } ( x _ { 1 } ) } { \\sigma _ { t } ( x _ { 1 } ) } \\right\\| ^ { 2 } } \\\\ & { \\quad \\quad \\quad = \\mathbb { E } _ { t , q ( x _ { 1 } ) , p _ { 0 } ( x _ { 0 } ) } \\Big \\| \\epsilon _ { t } \\big ( \\sigma _ { t } ( x _ { 1 } ) x _ { 0 } + \\mu _ { t } ( x _ { 1 } ) \\big ) - x _ { 0 } \\Big \\| ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 310, + 775, + 687, + 843 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $p _ { 0 } ( x ) = \\mathcal { N } ( x | 0 , I )$ is the standard Gaussian, and $\\epsilon _ { t }$ is the learnable noise function. ", + "bbox": [ + 171, + 845, + 758, + 861 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Diffusion path. For the diffusion path we use the standard VP diffusion (equation 19), namely, ", + "bbox": [ + 173, + 873, + 805, + 890 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/f39795b3a765702359bd1d39a59e9f4c5ea4cf002327c0fd84fbb0ec651bee88.jpg", + "text": "$$\n\\mu _ { t } ( x _ { 1 } ) = \\alpha _ { 1 - t } x _ { 1 } , \\quad \\sigma _ { t } ( x _ { 1 } ) = \\sqrt { 1 - \\alpha _ { 1 - t } ^ { 2 } } , \\quad \\mathrm { w h e r e ~ } \\alpha _ { t } = e ^ { - \\frac 1 2 T ( t ) } , \\quad T ( t ) = \\int _ { 0 } ^ { t } \\beta ( s ) d s ,\n$$", + "text_format": "latex", + "bbox": [ + 199, + 893, + 795, + 926 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/c2fe35d6fadc8127e35ced82d45a5c87471d24e16d0b69f3b4c7b971c4b2a6ee.jpg", + "table_caption": [ + "Table 3: Hyper-parameters used for training each model " + ], + "table_footnote": [], + "table_body": "
CIFAR10ImageNet-32ImageNet-64ImageNet-128
Channels256256192256
Depth2333
Channels multiple1,2,2,21,2,2,21,2,3,41,1,2,3,4
Heads4444
Heads Channels64646464
Attention resolution1616,832,16.832,16.8
Dropout0.00.00.00.0
Effective Batch size256102420481536
GPUs241632
Epochs1000200250571
Iterations391k250k157k500k
Learning Rate5e-41e-41e-41e-4
Learning Rate SchedulerPolynomial DecayPolynomial DecayConstantPolynomial Decay
Warmup Steps45k20k120k
", + "bbox": [ + 173, + 101, + 820, + 304 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "with, as suggested in Song et al. (2020b), $\\beta ( s ) = \\beta _ { \\mathrm { m i n } } + s ( \\beta _ { \\mathrm { m a x } } - \\beta _ { \\mathrm { m i n } } )$ and consequently ", + "bbox": [ + 173, + 359, + 776, + 377 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/a7a5f359f500d3836406bf3543b5010f35238b6d9d21f013790f071403e1cc4d.jpg", + "text": "$$\nT ( s ) = \\int _ { 0 } ^ { s } \\beta ( r ) d r = s \\beta _ { \\operatorname* { m i n } } + \\frac { 1 } { 2 } { s } ^ { 2 } ( \\beta _ { \\operatorname* { m a x } } - \\beta _ { \\operatorname* { m i n } } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 326, + 386, + 668, + 421 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $\\beta _ { \\mathrm { m i n } } = 0 . 1$ , $\\beta _ { \\mathrm { m a x } } = 2 0$ and time is sampled in $[ 0 , 1 - \\epsilon ] , \\epsilon = 1 0 ^ { - 5 }$ for training and likelihood and $\\epsilon = 1 0 ^ { - 5 }$ for sampling. ", + "bbox": [ + 174, + 433, + 823, + 462 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Sampling. Score matching samples are produced by solving the ODE (equation 1) with the vector field ", + "bbox": [ + 171, + 481, + 823, + 510 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/cfcf75b1648b066fac2214e199a616b4c1e047e3494c9771b11646871e59d8d4.jpg", + "text": "$$\nu _ { t } ( x ) = - \\frac { T ^ { \\prime } ( 1 - t ) } { 2 } \\left[ s _ { t } ( x ) - x \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 385, + 512, + 612, + 544 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "DDPM samples are computed with equation 46 after setting $s _ { t } ( x ) \\ = \\ \\epsilon _ { t } ( x ) / \\sigma _ { t }$ , where $\\sigma _ { t } =$ $\\sqrt { 1 - \\alpha _ { 1 - t } ^ { 2 } }$ . ", + "bbox": [ + 174, + 553, + 823, + 590 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E.2 TRAINING & EVALUATION DETAILS ", + "text_level": 1, + "bbox": [ + 178, + 613, + 462, + 627 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We report the hyper-parameters used in Table 3. We use full 32 bit-precision for training CIFAR10 and ImageNet-32 and 16-bit mixed precision for training ImageNet-64/128/256. All models are trained using the Adam optimizer with the following parameters: $\\beta _ { 1 } = 0 . 9$ , $\\beta _ { 2 } = 0 . 9 9 9$ , weight decay $= 0 . 0$ , and $\\epsilon = 1 e { - 8 }$ . All methods we trained (i.e., FM-OT, FM-Diffusion, SM-Diffusion) using identical architectures, with the same parameters for the the same number of Epochs (see Table 3 for details). We use either a constant learning rate schedule or a polynomial decay schedule (see Table 3). The polynomial decay learning rate schedule includes a warm-up phase for a specified number of training steps. In the warm-up phase, the learning rate is linearly increased from $1 e { - } 8$ to the peak learning rate (specified in Table 3). Once the peak learning rate is achieved, it linearly decays the learning rate down to $1 e { - } 8$ until the final training step. ", + "bbox": [ + 173, + 640, + 826, + 781 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "When reporting negative log-likelihood, we dequantize using the standard uniform dequantization. We report an importance-weighted estimate using ", + "bbox": [ + 173, + 786, + 823, + 816 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/ad3f34a875a458edcd71a8261a4372ccafb193124e3c0559737e4d22e479813e.jpg", + "text": "$$\n\\log \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } p _ { t } ( x + u _ { k } ) , \\mathrm { ~ w h e r e ~ } u _ { k } \\sim \\mathcal { U } ( 0 , 1 ) ,\n$$", + "text_format": "latex", + "bbox": [ + 349, + 827, + 647, + 871 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "with $x$ is in $\\{ 0 , . . . , 2 5 5 \\}$ and solved at $t \\ = \\ 1$ with an adaptive step size solver dopri5 with $\\mathtt { a t o l } { = } \\mathtt { r t o l } { = } 1 { \\mathtt { e } } { - } 5$ using the torchdiffeq (Chen, 2018) library. Estimated values for different values of $K$ are in Table 4. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/9a6a57f1ef2c528587b1fdcef26ca44535033f13752aabfd00f67b2c6bca63da.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
CIFAR-10ImageNet 32×32ImageNet 64×64
ModelK=1K=20K=50K=1K=5K=15K=1K=5K=10
Ablation
DDPM3.243.143.123.623.573.543.363.333.32
Score Matching3.283.183.163.653.593.573.433.413.40
ScoreFlow3.213.113.093.633.573.553.393.373.36
Ours
FM W/ Diffusion3.233.133.103.643.583.563.373.343.33
FM W/OT3.113.012.993.623.563.533.353.333.31
", + "bbox": [ + 171, + 101, + 820, + 258 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Table 4: Negative log-likelihood (in bits per dimension) on the test set with different values of $K$ using uniform dequantization. ", + "bbox": [ + 169, + 268, + 821, + 296 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/eb72d06ff7f414249525b415f5268f964d09ba927ba3832e5105240249f648a8.jpg", + "image_caption": [ + "Figure 10: Function evaluations for sampling during training, for models trained on CIFAR-10 using dopri5 solver with tolerance $1 e ^ { - 5 }$ . " + ], + "image_footnote": [], + "bbox": [ + 333, + 310, + 663, + 462 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "When computing FID/Inception scores for CIFAR10, ImageNet-32/64 we use the TensorFlow GAN library 2. To remain comparable to Dhariwal & Nichol (2021) for ImageNet-128 we use the evaluation script they include in their publicly available code repository 3 ", + "bbox": [ + 174, + 535, + 825, + 578 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "F ADDITIONAL TABLES AND FIGURES ", + "text_level": 1, + "bbox": [ + 173, + 598, + 503, + 613 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/bbd216bda2a45c32f64cbff5069528eba027c304db27b89b8278e2d54469e621.jpg", + "image_caption": [ + "Figure 11: Non-curated unconditional ImageNet-32 generated images of a CNF trained with FMOT. " + ], + "image_footnote": [], + "bbox": [ + 174, + 117, + 823, + 866 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/21c9532a95b63a45824c40001fb72d928ad3257051ae8b573500062a2340aacd.jpg", + "image_caption": [ + "Figure 12: Non-curated unconditional ImageNet-64 generated images of a CNF trained with FMOT. " + ], + "image_footnote": [], + "bbox": [ + 174, + 140, + 823, + 844 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/ff9bb9ac0a9e7091b4805c1cb3148f3a1c8ab55bc81f7c9a3cd466445096728d.jpg", + "image_caption": [ + "Figure 13: Non-curated unconditional ImageNet-128 generated images of a CNF trained with FMOT. " + ], + "image_footnote": [], + "bbox": [ + 174, + 165, + 823, + 821 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/a728a372aa5ccdaa014733a34baad675c706ed6d306c319c88ebf732f3195157.jpg", + "image_caption": [ + "Figure 14: Conditional generation $6 4 \\times 6 4 \\to 2 5 6 \\times 2 5 6 .$ Flow Matching OT upsampled images from validation set. " + ], + "image_footnote": [], + "bbox": [ + 174, + 87, + 841, + 878 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/249942f92d435160a5e3bf6685ea8bb6e47279c4b60525de85263b1f6ee36127.jpg", + "image_caption": [ + "Figure 15: Conditional generation $6 4 \\times 6 4 \\to 2 5 6 \\times 2 5 6$ . Flow Matching OT upsampled images from validation set. " + ], + "image_footnote": [], + "bbox": [ + 174, + 94, + 841, + 877 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/a4c5748ca5b3132645a197cee50cf33f4995449735f7a790c302e4fb4f957059.jpg", + "image_caption": [ + "Figure 16: Generated samples from the same initial noise, but with varying number of function evaluations (NFE). Flow matching with OT path trained on ImageNet-128. " + ], + "image_footnote": [], + "bbox": [ + 174, + 198, + 825, + 781 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/bd5aa70b81b2a7350e855dc0afed06b965323279c8a07921010964de77cc0e34.jpg", + "image_caption": [ + "Figure 17: Generated samples from the same initial noise, but with varying number of function evaluations (NFE). Flow matching with OT path trained on ImageNet $2 5 6 \\times 2 5 6$ . 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ModelCIFAR-10ImageNet 32×32ImageNet 64×64ImageNet 128×128
NLL↓FID↓NFE↓NLL↓FID↓NFE↓NLL↓FID↓NFE↓ModelNLL↓FID↓
AblationsMGAN (Hoang etal.,2018)158.9
DDPM3.127.482743.546.992623.3217.36264PacGAN2 (Lin et a.,2018)=57.5
Score Matching3.1619.942423.565.681783.4019.74441Logo-GAN-AE(Sag etal,018)一50.9
ScoreFlow3.0920.784283.5514.141953.3624.95601Self-cond. GAN (Lucic e al.,2019)一41.7
OursUncond.BigGAN (Lueic etal,19)25.3
FM W/ Diffusion3.108.061833.546.371933.3316.88187PGMGAN (Armandpour et al.,2021)21.7
FM W/OT2.996.351423.535.021223.3114.45138FM W/OT2.9020.9
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Reference1.9240.81一
Regression15.2121.127.90.801
SR3 (Saharia et al,22)5.2180.126.40.762
FM W/OT3.4200.824.70.747
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SuiteEnvironmentBCFilt. BCTD3+BCOnestepCQL-rCQL-pDTRvS-RRvS-GGCSL
D4RL AntMazeumaze-v254.6 45.660.0 46.578.6 71.464.3 60.744.8 23.474.065.664.4 70.165.4 60.9
umaze-diverse-v284.051.2
medium-play-v20.042.110.60.30.061.21.04.558.1
medium-diverse-v20.037.23.00.00.053.70.67.767.3
large-play-v20.028.00.20.00.015.80.03.532.4
large-diverse-v20.034.30.00.00.014.90.23.736.9
antmze-v2 average16.741.427.3*20.9*11.450.6*19.825.653.5
halfcheetah-random-v2 hopper-random-v22.32.011.06.918.62.23.9 7.7
4.84.18.57.89.37.5 2.0
walker2d-random-v21.71.71.66.12.5-0.2
random-v2 average2.92.66.93.93.8
halfcheetah-medium-replay-v236.640.67.0 44.642.410.1 47.336.638.0
hopper-medium-replay-v218.175.960.971.097.882.773.5
walker2d-medium-replay-v226.062.581.871.686.166.660.6
medium-replay-v2 average26.959.762.461.777.162.0
halfcheetah-medium-v242.642.542.657.4 41.6
hopper-medium-v252.956.948.3 59.355.6 83.349.1 64.667.660.2
walker2d-medium-v275.375.083.785.682.974.071.7
medium-v2 average56.958.174.865.561.457.8
halfcheetah-medium-expert-v255.292.963.8 90.793.585.886.892.2
hopper-medium-expert-v252.5110.998.0102.1102.0107.6101.7
walker2d-medium-expert-v2107.5109.0110.1110.9109.5108.1106.0
medium-expert-v2average71.7104.399.6102.299.1100.8100.0
gym-v2 average39.656.258.261.463.057.054.7
D4RL Kitchenkitchen-complete-v065.04.011.843.81.5
kitchen-mixed-v051.540.024.251.01.150.2 60.3
kitchen-partial-v038.066.020.849.80.551.4
kitchen-vO average54.0
51.536.718.948.21.014.828.0
GCSLclaw door28.0
four rooms95.8 63.081.0
lunar54.770.0
pusher81.883.0
gcsl average62.058.0
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HyperparameterValueEnvironment
Hidden layers2All
Layer width1024All
NonlinearityReLUAll
Learning rate1e-3All
Epochs10GCSL
50Kitchen
2000Gym
Gradient steps20000AntMaze
Batch size256GCSL,Kitchen
16384Gym,AntMaze
Dropout0.1GCSL,Kitchen
0Gym,AntMaze
Goal stateGivenGCSL
All subtasks completedKitchen
Reward target(x,y) locationAntMaze
110Gym medium-expert,AntMaze,Kitchen
90Gym {hopper,walker2d}-medium-replay,walker2d-medium
60Gym hopper-medium
Policy output40Gym random,halfcheetah-{medium,medium-replay}
Discrete categoricalGCSL
Unimodal GaussianKitchen, Gym,AntMaze
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"text", + "text": "MASTERING VISUAL CONTINUOUS CONTROL: IMPROVED DATA-AUGMENTED REINFORCEMENT LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 98, + 823, + 170 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 195, + 400, + 223 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 261, + 544, + 275 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We present DrQ-v2, a model-free reinforcement learning (RL) algorithm for visual continuous control. DrQ-v2 builds on DrQ, an off-policy actor-critic approach that uses data augmentation to learn directly from pixels. We introduce several improvements that yield state-of-the-art results on the DeepMind Control Suite. Notably, DrQ-v2 is able to solve complex humanoid locomotion tasks directly from pixel observations, previously unattained by model-free RL. DrQ-v2 is conceptually simple, easy to implement, and provides significantly better computational footprint compared to prior work, with the majority of tasks taking just 8 hours to train on a single GPU. Finally, DrQ-v2’s implementation is publicly released to provide RL practitioners with a strong and computationally efficient baseline. ", + "bbox": [ + 233, + 295, + 766, + 434 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 476, + 336, + 491 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Creating sample-efficient continuous control methods that observe high-dimensional images has been a long standing challenge in reinforcement learning (RL) . Over the last three years, the RL community has made significant headway on this problem, improving sample-efficiency significantly. The key insight to solving visual control is the learning of better low-dimensional representations, either through autoencoders (Yarats et al., 2019; Finn et al., 2015), variational inference (Hafner et al., 2018; 2019; Lee et al., 2019), contrastive learning (Srinivas et al., 2020; Yarats et al., 2021a), self-prediction (Schwarzer et al., 2020b), or data augmentations (Yarats et al., 2021b; Laskin et al., 2020). However, current state-of-the-art model-free methods are still limited in three ways. First, they are unable to solve the more challenging visual control problems such as quadruped and humanoid locomotion. Second, they often require significant computational resources, i.e. lengthy training times using distributed multi-GPU infrastructure. Lastly, it is often unclear how different design choices affect overall system performance. ", + "bbox": [ + 174, + 510, + 826, + 678 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/e84129af41009f6a5914fde2b150e79150a0bba702766235dd57287b087195b6.jpg", + "image_caption": [ + "Figure $1 : \\mathrm { D r Q - v } 2$ demonstrates significantly better sample efficiency and computational footprint compared to state-of-the-art model-free methods for visual continuous control while being conceptually simple and easy to implement. (Left two) Average performance results across 12 challenging tasks from the DeepMind Control Suite (the set of tasks can be seen in Figure 8). (Right two) Performance on the Humanoid Walk task from visual input, previously unsolved by model-free methods. In both cases we report sample complexity and wall-clock time axes for evaluation, with time being measured on a single GPU machine and using official implementations for each method. " + ], + "image_footnote": [], + "bbox": [ + 176, + 705, + 816, + 813 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper we present DrQ-v2, a simple model-free algorithm that builds on the idea of using data augmentations (Yarats et al., 2021b; Laskin et al., 2020) to solve hard visual control problems. Most notably, it is the first model-free method that solves complex humanoid tasks directly from pixels. Compared to previous state-of-the-art model-free methods, DrQ-v2 provides significant improvements in sample efficiency across tasks from the DeepMind Control Suite (Tassa et al., 2018). Conceptually simple, DrQ-v2 is also computationally efficient, which allows solving most tasks in DeepMind Control Suite in just 8 hours on a single GPU (see Figure 1). Recently, a model-based method, DreamerV2 (Hafner et al., 2020) was also shown to solve visual continuous control problems and it was first to solve the humanoid locomotion problem from pixels. While our model-free $\\mathrm { D r Q - v } 2$ matches DreamerV2 in terms sample efficiency and performance, it does so $4 \\times$ faster in terms of wall-clock time to train. We believe this makes DrQ-v2 a more accessible approach to support research in visual continuous control and it reinforces the question on whether model-free or model-based is the more suitable approach to solve this type of tasks. ", + "bbox": [ + 174, + 103, + 825, + 284 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "$\\mathrm { D r Q - v } 2$ , which is detailed in Section 3, improves upon DrQ (Yarats et al., 2021b) by making several algorithmic changes: (i) switching the base RL algorithm from SAC (Haarnoja et al., 2018a) to DDPG (Lillicrap et al., 2015a) with clipped double Q-learning from TD3 (Fujimoto et al., 2018), (ii) this allows us straightforwardly incorporating multi-step return, (iii) adding bilinear interpolation to the random shift image augmentation, (iv) introducing an exploration schedule, (v) selecting better hyper-parameters including a larger capacity of the replay buffer. A careful ablation study of these design choices is presented in Section 4.4. Furthermore, we re-examine the original implementation of DrQ and identify several computational bottlenecks such as replay buffer management, data augmentation processing, batch size, and frequency of learning updates (see Section 3.2). To remedy these, we have developed a new implementation that both achieves better performance and trains around 3.5 times faster with respect to wall-clock time than the previous implementation on the same hardware with an increase in environment frame throughput (FPS) from 28 to 96 (i.e., it takes $1 0 ^ { 6 } / 9 6 / 3 6 0 0 \\approx 2 . 9$ hours to train for 1M environment steps). DrQ-v2’s implementation is available at https://anonymous.4open.science/r/drqv2. ", + "bbox": [ + 174, + 291, + 825, + 484 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 176, + 507, + 326, + 525 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 REINFORCEMENT LEARNING FROM IMAGES ", + "text_level": 1, + "bbox": [ + 179, + 541, + 516, + 556 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We formulate image-based control as an infinite-horizon Markov Decision Process (MDP) (Bellman, 1957). Generally, in such a setting, an image rendering of the system is not sufficient to perfectly describe the system’s underlying state. To this end and per common practice (Mnih et al., 2013), we approximate the current state of the system by stacking three consecutive prior observations. With this in mind, such MDP can be described as a tuple $( \\mathcal { X } , \\mathcal { A } , P , R , \\gamma , d _ { 0 } )$ , where $\\mathcal { X }$ is the state space (a three-stack of image observations), $\\mathcal { A }$ is the action space, $P : \\mathcal { X } \\times \\mathcal { A } \\Delta ( \\mathcal { X } )$ is the transition function1 that defines a probability distribution over the next state given the current state and action, $R : \\mathcal { X } \\times \\mathcal { A } [ 0 , 1 ]$ is the reward function, $\\gamma \\in [ 0 , 1 )$ is a discount factor, and $d _ { 0 } \\in \\Delta ( { \\mathcal { X } } )$ is the distribution of the initial state $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ . The goal is to find a policy $\\pi : \\mathcal { X } \\Delta ( \\mathcal { A } )$ that maximizes the expected discounted sum of rewards $\\mathbb { E } _ { \\pi } \\big [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r _ { t } \\big ]$ , where $\\mathbf { x } _ { 0 } \\sim d _ { 0 }$ , and $\\forall t$ we have $\\mathbf { \\boldsymbol { a } } _ { t } \\sim \\pi ( \\cdot | \\mathbf { \\boldsymbol { x } } _ { t } )$ $\\mathbf { \\boldsymbol { x } } _ { t + 1 } \\sim P ( \\cdot | \\mathbf { \\boldsymbol { x } } _ { t } , \\mathbf { \\boldsymbol { a } } _ { t } )$ , and $r _ { t } = R ( \\pmb { x } _ { t } , \\pmb { a } _ { t } ) \\overline { { \\mathbf { \\phi } } }$ . ", + "bbox": [ + 173, + 568, + 825, + 722 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 DEEP DETERMINISTIC POLICY GRADIENT ", + "text_level": 1, + "bbox": [ + 176, + 741, + 506, + 756 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Deep Deterministic Policy Gradient (DDPG) (Lillicrap et al., 2015a) is an actor-critic algorithm for continuous control that concurrently learns a Q-function $Q _ { \\theta }$ and a deterministic policy $\\pi _ { \\phi }$ . For this, DDPG uses Q-learning (Watkins and Dayan, 1992) to learn $Q _ { \\theta }$ by minimizing the one-step Bellman residual $J _ { \\theta } ( \\mathcal { D } ) = \\mathbb { E } _ { ( \\pmb { x } _ { t } , \\underline { { a } } _ { t } , r _ { t } , \\pmb { x } _ { t + 1 } ) \\sim \\mathcal { D } } [ \\big ( Q _ { \\theta } ( \\pmb { x } _ { t } , \\underline { { a } } _ { t } ) _ { } r _ { t } - \\underline { { \\gamma Q _ { \\theta } } } ( \\pmb { x } _ { t + 1 } , \\pi _ { \\phi } ( \\pmb { x } _ { t + 1 } ) ) ^ { 2 } \\big ]$ . The policy $\\pi _ { \\phi }$ is learned by employing Deterministic Policy Gradient (DPG) (Silver et al., 2014) and maximizing $J _ { \\phi } ( \\mathcal { D } ) = \\mathbb { E } _ { \\pmb { x } _ { t } \\sim \\mathcal { D } } [ Q _ { \\theta } ( \\pmb { x } _ { t } , \\pi _ { \\phi } ( \\pmb { x } _ { t } ) ) ]$ , so $\\pi _ { \\phi } ( \\pmb { x } _ { t } )$ approximates argmax $\\mathbf { \\Sigma } _ { \\alpha } Q _ { \\theta } ( \\pmb { x } _ { t } , \\pmb { a } )$ . Here, $\\mathcal { D }$ is a replay buffer of environment transitions and $\\bar { \\theta }$ is an exponential moving average of the weights. DDPG is amenable to incorporate $n$ -step returns (Watkins, 1989; eng and Williams, 1996) when estimating TD error beyond a single step (Barth-Maron et al., 2018). In practice, $n$ -step returns allow for faster reward propagation and has been previously used in policy gradient and Q-learning methods (Mnih et al., 2016b; Barth-Maron et al., 2018; Hessel et al., 2017). ", + "bbox": [ + 174, + 767, + 825, + 895 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/549fe695324b96cc93df2881f3296d0b6333a6e41e473a27a33b6bfc84cf2f08.jpg", + "image_caption": [ + "Figure 2: (Left): $\\mathrm { D r Q - v } 2$ is an off-policy actor-critic algorithm for image-based RL. It alleviates encoder overfitting by applying random shift augmentation to pixel observations sampled from the replay buffer. (Right): Examples of walking and standing behaviors learned by $\\mathrm { D r Q - v } 2$ for a complex humanoid agent from DMC (Tassa et al., 2018) with 21 and 54 dimensional action and state spaces, respectively. DrQ-v2 does not have access to the internal state of the environment, only observing three consecutive pixel frames at a time. Despite this imperfect observational channel, our agent still manages to solve the tasks. To the best of our knowledge, this is the first successful demonstration by a model-free method, using pixel-based inputs of these tasks. " + ], + "image_footnote": [], + "bbox": [ + 209, + 68, + 789, + 227 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 378, + 823, + 406 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.3 DATA AUGMENTATION IN REINFORCEMENT LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 422, + 596, + 438 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Recently, it has been shown that data augmentation techniques, commonplace in Computer Vision, are also important for achieving the state-of-the-art performance in image-based RL (Yarats et al., 2021b; Laskin et al., 2020). For example, the state-of-the-art algorithm for visual RL, DrQ (Yarats et al., 2021b) builds on top of Soft Actor-Critic (Haarnoja et al., 2018a), a model-free actor-critic algorithm, by adding a convolutional encoder and data augmentation in the form of random shifts. The use of such data augmentations now forms an essential component of several recent visual RL algorithms (Srinivas et al., 2020; Raileanu et al., 2020; Yarats et al., 2021a; Stooke et al., 2020; Hansen and Wang, 2021; Schwarzer et al., 2020b). ", + "bbox": [ + 174, + 449, + 826, + 560 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 DRQ-V2: IMPROVED DATA-AUGMENTED REINFORCEMENT LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 580, + 794, + 597 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we describe DrQ-v2, a simple model-free actor-critic RL algorithm for image-based continuous control, that builds upon DrQ. ", + "bbox": [ + 174, + 612, + 823, + 641 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 ALGORITHMIC DETAILS ", + "text_level": 1, + "bbox": [ + 174, + 657, + 383, + 671 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Image Augmentation As in DrQ we apply random shifts image augmentation to pixel observations of the environment. In the settings of visual continuous control by DMC, this augmentation can be instantiated by first padding each side of $8 4 \\times 8 4$ observation rendering by 4 pixels (by repeating boundary pixels), and then selecting a random $8 4 \\times 8 4$ crop, yielding the original image shifted by $\\pm 4$ pixels. We also find it useful to apply bilinear interpolation on top of the shifted image (i.e, we replace each pixel value with the average of the four nearest pixel values). In our experiments, this modification provides an additional performance boost across the board. ", + "bbox": [ + 174, + 684, + 825, + 781 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Image Encoder The augmented image observation is then embedded into a low-dimensional latent vector by applying a convolutional encoder. We use the same encoder architecture as in DrQ, which first was introduced introduced in SAC-AE (Yarats et al., 2019). This process can be succinctly summarized as $\\pmb { h } = f _ { \\xi } ( \\mathrm { a u g } ( \\pmb { x } ) )$ , where $f _ { \\xi }$ is the encoder, aug is the random shifts augmentation, and $_ { \\textbf { \\em x } }$ is the original image observation. ", + "bbox": [ + 174, + 796, + 825, + 866 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Actor-Critic Algorithm We use DDPG (Lillicrap et al., 2015a) as a backbone actor-critic RL algorithm and, similarly to Barth-Maron et al. (2018), augment it with $n$ -step returns to estimate TD error. This results into faster reward propagation and overall learning progress (Mnih et al., 2016a). ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "While some methods (Hafner et al., 2020) employ more sophisticated techniques such as $\\mathrm { T D } ( \\lambda )$ or Retrace $( \\lambda )$ (Munos et al., 2016), they are often computationally demanding when $n$ is large. We find that using simple $n$ -step returns, without an importance sampling correction, strikes a good balance between performance and efficiency. We also employ clipped double Q-learning (Fujimoto et al., 2018) to reduce overestimation bias in the target value. Practically, this requires training two Qfunctions $Q _ { \\theta _ { 1 } }$ and $Q _ { \\theta _ { 2 } }$ . For this, we sample a mini-batch of transitions $\\tau = ( \\mathbf { x } _ { t } , \\mathbf { a } _ { t } , r _ { t : t + n - 1 } , \\mathbf { x } _ { t + n } )$ from the replay buffer $\\mathcal { D }$ and compute the following two losses: ", + "bbox": [ + 173, + 103, + 826, + 202 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/21f5b49e1058571b066a400cb4942e8b0691c5876a0567d3f54d57c99b7289be.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\boldsymbol { \\theta } _ { k } , \\boldsymbol { \\xi } } ( \\mathcal { D } ) = \\mathbb { E } _ { \\tau \\sim \\mathcal { D } } \\big [ ( Q _ { \\boldsymbol { \\theta } _ { k } } ( h _ { t } , \\boldsymbol { a } _ { t } ) - y ) ^ { 2 } \\big ] \\quad \\forall k \\in \\{ 1 , 2 \\} , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 315, + 210, + 681, + 231 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "with the TD target $y$ defined as: ", + "bbox": [ + 173, + 238, + 383, + 253 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/bb4d51d5aefb63d565fb1da6ccfcdc2b6e2cfc9e812e1c7364b45c1df85f8d92.jpg", + "text": "$$\ny = \\sum _ { i = 0 } ^ { n - 1 } \\gamma ^ { i } r _ { t + i } + \\gamma ^ { n } \\operatorname* { m i n } _ { k = 1 , 2 } Q _ { \\bar { \\theta } _ { k } } ( h _ { t + n } , \\mathbf { a } _ { t + n } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 346, + 263, + 650, + 308 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\pmb { h } _ { t } = f _ { \\xi } ( \\mathrm { a u g } ( \\pmb { x } _ { t } ) )$ , ${ h _ { t + n } } = f _ { \\xi } ( \\mathrm { a u g } ( { { \\bf x } _ { t + n } } ) )$ , $a _ { t + n } = \\pi _ { \\phi } ( h _ { t + n } ) + \\epsilon , \\bar { \\theta } _ { 1 }$ and ${ \\bar { \\theta } _ { 2 } }$ are the slowmoving weights for the Q target networks. We note, that in contrast to DrQ, we do not employ a target network for the encoder $f _ { \\xi }$ and always use the most recent weights $\\xi$ to embed $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ and $\\mathbf { \\Delta } _ { \\mathbf { x } _ { t + n } }$ . The exploration noise $\\epsilon$ is sampled from $\\mathrm { c l i p } ( \\mathcal { N } ( 0 , \\sigma ^ { 2 } ) , - c , c )$ similar to TD3 (Fujimoto et al., 2018), with the exception of decaying $\\sigma$ , which we describe below. Finally, we train the deterministic actor $\\pi _ { \\phi }$ using DPG with the following loss: ", + "bbox": [ + 173, + 318, + 826, + 404 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/19949ea5e4ecf0cc87814f68e6506792333b535c3a3583c7335970adf32fefdc.jpg", + "text": "$$\n\\mathcal { L } _ { \\phi } ( \\mathcal { D } ) = - \\mathbb { E } _ { x _ { t } \\sim \\mathcal { D } } \\big [ \\operatorname* { m i n } _ { k = 1 , 2 } Q _ { \\theta _ { k } } ( h _ { t } , \\pmb { a } _ { t } ) \\big ] ,\n$$", + "text_format": "latex", + "bbox": [ + 364, + 414, + 632, + 440 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\pmb { h } _ { t } = f _ { \\xi } ( \\mathrm { a u g } ( \\pmb { x } _ { t } ) )$ , ${ \\pmb a } _ { t } = \\pi _ { \\phi } ( { \\pmb h } _ { t } ) + \\epsilon$ , and $\\epsilon \\sim \\mathrm { c l i p } ( \\mathcal { N } ( 0 , \\sigma ^ { 2 } ) , - c , c )$ . Similar to $_ \\mathrm { D r Q }$ , we do not use actor’s gradients to update the encoder’s parameters $\\xi$ . ", + "bbox": [ + 173, + 450, + 825, + 481 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Scheduled Exploration Noise Empirically, we observe that it is helpful to have different levels of exploration at different stages of learning. At the beginning of training we want the agent to be more stochastic and explore the environment more effectively, while at the later stages of training, when the agent has already identified promising behaviors, it is better to be more deterministic and master those behaviors. Similar to Amos et al. (2020), we instantiate this idea by using linear decay $\\sigma ( t )$ for the variance $\\sigma ^ { 2 }$ of the exploration noise defined as: ", + "bbox": [ + 173, + 498, + 825, + 583 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/845554410f8da0899d23c194d909d3308487d8ae22f946d09cbaf11cb779a4d8.jpg", + "text": "$$\n\\sigma ( t ) = \\sigma _ { \\mathrm { i n i t } } + ( 1 - \\mathrm { m i n } ( \\frac { t } { T } , 1 ) ) ( \\sigma _ { \\mathrm { f i n a l } } - \\sigma _ { \\mathrm { i n i t } } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 339, + 593, + 656, + 623 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\sigma _ { \\mathrm { i n i t } }$ and $\\sigma _ { \\mathrm { f i n a l } }$ are the initial and final values for standard deviation, and $T$ is the decay horizon. ", + "bbox": [ + 174, + 632, + 823, + 647 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Key Hyper-Parameters We conduct an extensive hyper-parameter search and identify several hyper-parameter changes compared to DrQ. The three most important hyper-parameters are: (i) the size of the replay buffer, (ii) mini-batch size, and (iii) learning rate. Specifically, we use a 10 times larger replay buffer than DrQ. We also use a smaller mini-batch size of 256 without any noticeable performance degradation. This is in contrast to CURL (Srinivas et al., 2020) and $\\mathrm { D r Q }$ (Yarats et al., 2021b) that both use a larger batch size of 512 to attain more stable training in the expense of computational efficiency. Finally, we find that using smaller learning rate of $1 \\times 1 0 ^ { - 4 }$ , rather than DrQ’s learning rate of $\\mathrm { i \\times 1 0 ^ { - 3 } }$ , results into more stable training without any loss in learning speed. ", + "bbox": [ + 173, + 665, + 826, + 779 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 IMPLEMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 797, + 406, + 813 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Faster Image Augmentation We replace DrQ’s random shifts augmentation (i.e., kornia.augmentation.RandomCrop) by a custom implementation that uses flowfield image sampling provided in PyTorch (i.e., grid_sample). This is done for two reasons. First, we noticed that Kornia’s implementation does not fully utilize GPU pipelining since it has some intermediate CPU to GPU data transferring which breaks the computational flow. Second, using grid_sample allows straightforward addition of bilinear interpolation. Our custom random shifts augmentation improves training throughput by a factor of 2. ", + "bbox": [ + 173, + 825, + 826, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Faster Replay Buffer Another computational bottleneck of $\\mathrm { D r Q }$ was the replay buffer. The specific implementation had poor memory management which resulted in slow CPU to GPU data transfer, which also restricted the number of image-based transitions that could be stored. We reimplemented the replay buffer to address these issues which led to a ten-fold increase in storage capacity and faster data transfer. More details are available in our open-source release. We note that the improved training speed of DrQ-v2 was key to solving humanoid tasks as it enabled much faster experimentation. ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 209, + 326, + 226 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section we provide empirical evaluation of $\\mathrm { D r Q - v } 2$ on an extensive set of visual continuous control tasks from DMC (Tassa et al., 2018). We first present comparison to prior methods, both model-free and model-based, in terms of sample efficiency and wall-clock time. We then present a large scale ablation study that guided the final version of DrQ-v2. ", + "bbox": [ + 174, + 242, + 825, + 297 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 SETUP ", + "text_level": 1, + "bbox": [ + 174, + 316, + 261, + 330 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Environments We consider a set of MuJoCo tasks (Todorov et al., 2012) provided by DMC (Tassa et al., 2018), a widely used benchmark for continous control. DMC offers environments of various difficulty, ranging from the simple control problems such as the single degree of freedom (DOF) pendulum and cartpool, to the control of complex multi-joint bodies such as the humanoid (21 DOF). We consider learning from pixels. In this setting, environment observations are stacks of 3 consecutive RGB images of size $8 4 \\times 8 4$ , stacked along the channel dimension to enable inference of dynamic information like velocity and acceleration. In total, we consider 24 different tasks, which we group into three buckets, easy, medium, and hard, according to the sample complexity to reach near-optimal performance (see Appendix B). Our motivation for this is to encourage RL practitioners to focus on the medium and hard tasks and stop using the easy tasks for evaluation, as they are mostly solved at this point and may no longer provide any valuable signal in comparing different methods. ", + "bbox": [ + 174, + 343, + 825, + 496 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Training Details For all tasks in the suite an episode corresponds to 1000 steps, where a per-step reward is in the unit interval [0, 1]. This upper bounds the episode return to 1000 making it easier to compute aggregated performance measures across tasks. To facilitate fair wall-clock time comparison all algorithms are trained on the same hardware (i.e., a single NVIDIA V100 GPU machine) and evaluated with the same periodicity of 20000 environment steps. Each evaluation query averages episode returns over 10 episodes. Per common practice (Hafner et al., 2019), we employ action repeat of 2 and measure sample complexity in the environment steps, rather than the actor steps. In all the figures we plot the mean performance over 10 seeds together with the shaded regions which represent $9 \\hat { 5 } \\%$ confidence intervals. A full list of hyper-parameters can be found in Appendix E. ", + "bbox": [ + 174, + 513, + 825, + 638 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Comparison Axes In many real-world applications, taking a step in the environment incurs significant computational cost making sample efficiency a critical feature of an RL algorithm. It is hence important to compare RL algorithms in terms of their sample efficiency. We facilitate this comparison by computing an algorithm’s performance measured by episode return with respect to environment steps. On the other end, striving low sample complexity often comes at the cost of a poor computational efficiency. Unfortunately, recent deep RL literature has paid very little attention to this important axis, which has led to skyrocketing hardware requirements. Such a trend has made it virtually impossible for an RL practitioner with modest hardware capacity to contribute to advancements in image-based RL, leaving research in this area to a few well-equipped labs. To democratize research in visual RL, we additionally propose to compare the agents in terms of wall-clock training time given the same single GPU hardware. We note that it is possible to adapt DrQ-v2 to a distributed setup, as has been done for DDPG in prior work (Barth-Maron et al., 2018; Hoffman et al., 2020). ", + "bbox": [ + 174, + 655, + 825, + 821 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 COMPARISON TO MODEL-FREE METHODS ", + "text_level": 1, + "bbox": [ + 176, + 840, + 508, + 856 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Baselines We compare our method to several state-of-the-art model-free algorithms for visual RL including CURL (Srinivas et al., 2020), DrQ (Yarats et al., 2021b), and vanilla SAC (Haarnoja et al., 2018a) augmented with the convolutional encoder from SAC-AE (Yarats et al., 2019). Vanilla SAC is a weak baseline and only included as a ground point to showcase the recent progress in visual RL. ", + "bbox": [ + 174, + 868, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/ecbef5ca3f3d3542fd76d49b66dfb726dde336b060b81096054addb1a0048ea7.jpg", + "image_caption": [ + "Figure 3: We compare $\\mathrm { D r Q - v } 2$ on a subset of continuous control tasks that offer various challenges, including complex dynamics, sparse rewards, hard exploration, and more. (a) $\\mathrm { D r Q - v } 2$ demonstrates favorable sample efficiency and comfortably outperforms leading model-free baselines, as well as requiring less wall-clock training image (b). " + ], + "image_footnote": [], + "bbox": [ + 179, + 84, + 826, + 520 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Sample Efficiency Axis We present results on several medium and hard tasks in Figure 3a. Full results can be found in Appendix (Figure 6, Figure 8, and Figure 10). Our empirical study reveals that $\\mathrm { D r Q - v } 2$ outperforms prior model-free methods in terms of sample efficiency across the three benchmarks with different levels of difficulty. Importantly, DrQ-v2’s advantage is more pronounced on harder tasks (i.e., acrobot, quadruped, and humanoid), where exploration is especially challenging. Finally, DrQ-v2 solves the DMC humanoid locomotion tasks directly from pixels, which, to the best of our knowledge, is the first successful demonstration of such feat by a model-free method. ", + "bbox": [ + 174, + 613, + 825, + 712 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Compute Efficiency Axis To facilitate a fair comparison in terms of sheer wall-clock training time, besides employee the identical training protocol (see Section 4.1), we also use the same mini-batch size of 256 for each agent. In Figure 13, we evaluate $\\mathrm { D r Q - v } 2$ on a subset of DMC tasks for the sake of brevity only, and note that the demonstrated results can be easily extrapolated to the other tasks given the linear dependency between training time and sample complexity. In our benchmarks, $\\mathrm { D r Q - v } 2$ is able to achieve a throughput of 96 FPS, which favorably compares to DrQ’s 28 FPS (a $3 . 4 \\times$ increase), and CURL’s 16 FPS (a $6 \\times$ increase) throughputs. Practically, $\\mathrm { D r Q - v } 2$ solves easy, medium, and hard tasks within 2.9, 8.6, and 86 hours respectively. Full results can be found in Appendix (Figure 7, Figure 9, and Figure 11). ", + "bbox": [ + 174, + 727, + 825, + 852 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.3 COMPARISON TO MODEL-BASED METHODS ", + "text_level": 1, + "bbox": [ + 174, + 869, + 519, + 883 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Baseline To see how $\\mathrm { D r Q - v } 2$ stacks up against model-based methods, which tend to achieve better sample complexity in expense of a larger computational footprint, we also compare to recent and unpublished2 improvements to Dreamer-v2 (Hafner et al., 2020), a leading model-based approach for visual continuous control. The recent update shows that the model-based approach can solve the DMC humanoid tasks directly from pixel inputs. The open-source implementation of Dreamer-v2 (https://github.com/danijar/dreamerv2) only provides learning curves for Humanoid Walk. For this reason we run their code to obtain results on other DMC tasks. To limit hardware requirements of compute-expensive Dreamer-v2, we only run it on a subset of 12 out of 24 considered tasks. This subset, however, overlaps with all the three (i.e. easy, medium, and hard) benchmarks. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/545d212f4784001ae204bbbd2da7b19024aaeec84e241d0466ac5d3246eb06bc.jpg", + "image_caption": [ + "Figure 4: Model-based Dreamer-v2 needs to train a world model and thus performs more computations during training than model-free DrQ-v2. Still, (a) $\\mathrm { D r Q - v } 2$ is able to match Dreamer-v2’s sample efficiency, while $\\mathbf { ( b ) }$ requiring much less wall-clock training time. " + ], + "image_footnote": [], + "bbox": [ + 179, + 84, + 826, + 330 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 410, + 825, + 508 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Sample Efficiency Axis Our empirical study in Figure 4a reveals that in many cases, DrQ-v2, despite being a model-free method, can rival sample efficiency of state-of-the-art model-based Dreamer-v2. We note, however, that on several tasks (for example Acrobot Swingup) Dreamer-v2 outperforms DrQ-v2. We leave investigation of such discrepancy for future work. Full results are provided in Appendix D (Figure 12). ", + "bbox": [ + 174, + 525, + 825, + 594 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Compute Efficiency Axis A different picture emerges if comparison is done with respect to wallclock training time. Dreamer-v2, being a model-based method, performs significantly more floating point operations to reach its sample efficiency. In our benchmarks, Dreamer-v2 records a throughput of 24 FPS, which is $4 \\times$ less than DrQ-v2’s throughput of 96 FPS, measured on the same hardware. In Figure 4b we plot learning curves against wall-clock time and observe that $\\mathrm { D r Q - v } 2$ takes less time to solve the tasks. Full results can be found in Appendix (Figure 13). ", + "bbox": [ + 174, + 612, + 825, + 695 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.4 ABLATION STUDY ", + "text_level": 1, + "bbox": [ + 174, + 714, + 341, + 728 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section we present an extensive ablation study that guided us to the final version of $\\mathrm { D r Q - v } 2$ Here, for brevity we only discuss experiments that were most impactful and omit others that did not pan out. For computational reasons, we only ablate on 3 different control tasks of various difficulty levels. Our findings are summarized in Figure 5 and detailed below. ", + "bbox": [ + 174, + 741, + 825, + 796 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Switching from SAC to DDPG DrQ (Yarats et al., 2021b) leverages SAC (Haarnoja et al., 2018a) as the backbone RL algorithm. While it has been demonstrated by many works, including the original manuscripts (Haarnoja et al., 2018a;b) that SAC is superior to DDPG (Lillicrap et al., 2015b), our careful examination identifies two shortcomings that preclude SAC (within DrQ) to solve hard exploration-wise image-based tasks. First, the automatic entropy adjustment strategy, introduced in Haarnoja et al. (2018b), is inadequate and in some cases leads to a premature entropy collapse. ", + "bbox": [ + 174, + 813, + 825, + 897 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/ba1a4435887f6483f6e5539af32ee4c9abf61350f7e4d286d35d2b17585b6381.jpg", + "image_caption": [ + "(a) DrQ (dotted silver) relies on SAC as a base RL algorithm. Replacing SAC with DDPG results in a significant performance gain (blue). " + ], + "image_footnote": [], + "bbox": [ + 207, + 64, + 790, + 191 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/8cefa7b04095a5ad9cc5e618bf568a2e8cb599d743766c5d5f19dd90a7f61e8b.jpg", + "image_caption": [ + "(b) DDPG straightforwardly incorporates $n$ -step returns, a critical tool for exploration. We observe that the 3 (blue) and 5 (red) steps variants provide additional improvements to the previous version that uses single step TD-targets (silver). Going forward, we adopt 3-step returns (blue). " + ], + "image_footnote": [], + "bbox": [ + 207, + 236, + 790, + 361 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/7344d8d594db9770f16e401999ae3c88c9529605098182fbc253dd13972aed74.jpg", + "image_caption": [ + "(c) Increasing the size of the replay buffer (B) improves performance, over the original $1 0 ^ { 5 }$ used by DrQ (silver). Going forward, we use a buffer size of $1 \\dot { 0 } ^ { 6 }$ (red). " + ], + "image_footnote": [], + "bbox": [ + 202, + 417, + 789, + 544 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(d) Finally, a decaying schedule for the variance of the exploration noise (blue) helps on hard exploration tasks, versus the fixed variance variant (silver). ", + "bbox": [ + 178, + 718, + 816, + 744 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/9ac2122386f89d3dc6ae3ea32bbd49484f830769555b1e994b96ad2c05cfd549.jpg", + "image_caption": [ + "Figure 5: An ablation study that led us to the final version of $\\mathrm { D r Q - v } 2$ . We incrementally show each of the four key improvements to $\\mathrm { D r Q }$ that collectively form DrQ-v2. The silver dotted curves in the first row show the original DrQ. In subsequent rows they show progressive improvements, using the optimal choice from the previous rows (i.e., the silver curve in the third row shows DrQ with a DDPG base RL algorithm and 3-step returns). The red and blue curves show the effect of individual modifications. In the last row the blue curve corresponds to DrQ-v2. " + ], + "image_footnote": [], + "bbox": [ + 191, + 588, + 790, + 714 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "This prevents the agent from finding more optimal behaviors due to the insufficient exploration. In Figure 5a, we empirically verify our intuition and, indeed, observe that DDPG demonstrates better exploration properties than SAC. Here, DDPG uses constant $\\sigma = 0 . 2$ for the exploration noise. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "N-step Returns The second issue concerns the inability of soft Q-learning to incorporate $n$ -step returns to estimate TD error in a straightforward manner. The reason for this is that computing a target value for soft Q-function requires estimating per-step entropy of the policy, which is challenging to do for large $n$ in the off-policy regime. In contrast, DDPG does not require estimating per-step entropy to compute targets and is more amenable for $n$ -step returns. In Figure 5b we demonstrate that estimating TD error with $n$ -step returns improves sample efficiency over vanilla DDPG. We select 3-step returns as a sensible choice for our method. ", + "bbox": [ + 174, + 103, + 825, + 200 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Replay Buffer Size We hypothesize that a larger replay buffer plays an important role in circumventing the catastrophic forgetting problem (Fedus et al., 2020). This issue is especially prominent in tasks with more diverse initial state distributions (i.e., reacher or humanoid tasks), where the vast variety of possible behaviors requires significantly larger memory. We confirm this intuition by ablating the size of the replay buffer in Figure 5c, where we observe that a buffer size of 1M helps to improve performance on Reacher Hard considerably. ", + "bbox": [ + 174, + 217, + 825, + 300 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Scheduled Exploration Noise Finally, we demonstrate that it is useful to decay the variance of the exploration noise over the course of training according to Equation (3). In Figure 5d, we compare two versions of our algorithm, where the first variant uses a fixed standard deviation of $\\sigma = 0 . 2$ , while the second variant employes the decaying schedule $\\sigma ( t )$ , with parameters $\\sigma _ { \\mathrm { i n i t } } = 1 . 0$ , $\\sigma _ { \\mathrm { f i n a l } } = 0 . 1$ , and $T = 5 0 0 0 0 0$ . Having the exploration noise to decay linearly over time turns out to be helpful and provide an additional performance boost, which was especially useful for solving humanoid tasks. ", + "bbox": [ + 174, + 316, + 825, + 400 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 420, + 344, + 436 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Visual Reinforcement Learning Successes of visual representation learning in computer vision (Vincent et al., 2008; Doersch et al., 2015; Wang and Gupta, 2015; Noroozi and Favaro, 2016; Zhang et al., 2017; Gidaris et al., 2018) has inspired successes in visual RL, where coherent representations are learned alongside RL. Works such as SAC-AE (Yarats et al., 2019), PlaNet (Hafner et al., 2018), and SLAC (Lee et al., 2019), demonstrated how auto-encoders (Finn et al., 2015) could improve visual RL. Following this, other self-supervised objectives such as contrastive learning in CURL (Srinivas et al., 2020) and ATC (Stooke et al., 2020), self-prediction in SPR (Schwarzer et al., 2020a), contrastive cluster assignment in Proto-RL (Yarats et al., 2021a), and augmented data in DrQ (Yarats et al., 2021b) and RAD (Laskin et al., 2020), have significantly bridged the gap between state-based and image-based RL. Future prediction objectives (Hafner et al., 2018; 2019; Yan et al., 2020; Finn et al., 2015; Pinto et al., 2016; Agrawal et al., 2016) and other auxiliary objectives (Jaderberg et al., 2016; Zhan et al., 2020; Young et al., 2020; Chen et al., 2020) have shown improvements on a variety of problems ranging from gameplay, continuous control, and robotics. In the context of visual control settings, clever use of augmented data (Yarats et al., 2021b; Laskin et al., 2020) currently produces state-of-the-art results on visual tasks from DMC (Tassa et al., 2018). ", + "bbox": [ + 174, + 453, + 825, + 660 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Humanoid Control The humanoid control problem first presented in Tassa et al. (2012), has been studied as one of the hardest control problems due to its large state and action spaces. The earliest solutions to this problem use ideas in model-based optimal control to generate policies given an accurate model of the humanoid . Subsequent works in RL have shown that model-free policies can solve the humanoid control problem given access to proprioceptive state observations. However, solving such a problem from visual observations has been a challenging problem, with leading RL algorithms making little progress to solve the task (Tassa et al., 2018). Recently, Hafner et al. (2020) was able to solve this problem through a model-based technique in around 30M environment steps and 340 hours of training on a single GPU machine. DrQ-v2, presented in this paper, marks the first model-free RL method that can solve humanoid control from visual observations, taking also around 30M steps and 86 hours of training on the same hardware. ", + "bbox": [ + 174, + 676, + 825, + 829 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 849, + 318, + 866 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have introduced a conceptually simple model-free actor-critic RL agent for image-based continuous control – DrQ-v2. Our method provides significantly better computational footprint and masters tasks from DMC directly from pixels, most notably the humanoid locomotion tasks that ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "were previously unsolved by model-free approaches. 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", + "bbox": [ + 174, + 686, + 825, + 731 + ], + "page_idx": 12 + } +] \ No newline at end of file diff --git a/parse/dev/_VjQlMeSB_J/_VjQlMeSB_J.md b/parse/dev/_VjQlMeSB_J/_VjQlMeSB_J.md new file mode 100644 index 0000000000000000000000000000000000000000..9e605207af4e65aba4befec98807c8e5d2a4e9d0 --- /dev/null +++ b/parse/dev/_VjQlMeSB_J/_VjQlMeSB_J.md @@ -0,0 +1,295 @@ +# Chain-of-Thought Prompting Elicits Reasoning in Large Language Models + +Jason Wei Xuezhi Wang Dale Schuurmans Maarten Bosma + +Brian Ichter Fei Xia Ed H. Chi Quoc V. Le Denny Zhou + +Google Research, Brain Team {jasonwei,dennyzhou}@google.com + +# Abstract + +We explore how generating a chain of thought—a series of intermediate reasoning steps—significantly improves the ability of large language models to perform complex reasoning. In particular, we show how such reasoning abilities emerge naturally in sufficiently large language models via a simple method called chain-ofthought prompting, where a few chain of thought demonstrations are provided as exemplars in prompting. + +Experiments on three large language models show that chain-of-thought prompting improves performance on a range of arithmetic, commonsense, and symbolic reasoning tasks. The empirical gains can be striking. For instance, prompting a PaLM 540B with just eight chain-of-thought exemplars achieves state-of-the-art accuracy on the GSM8K benchmark of math word problems, surpassing even finetuned GPT-3 with a verifier. + +![](images/89ab3799d105e595086d7c7e3059c4c95016564718ada32028e60eef2d474c49.jpg) +Figure 1: Chain-of-thought prompting enables large language models to tackle complex arithmetic, commonsense, and symbolic reasoning tasks. Chain-of-thought reasoning processes are highlighted. + +# 1 Introduction + +The NLP landscape has recently been revolutionized by language models (Peters et al., 2018; Devlin et al., 2019; Brown et al., 2020, inter alia). Scaling up the size of language models has been shown to confer a range of benefits, such as improved performance and sample efficiency (Kaplan et al., 2020; Brown et al., 2020, inter alia). However, scaling up model size alone has not proved sufficient for achieving high performance on challenging tasks such as arithmetic, commonsense, and symbolic reasoning (Rae et al., 2021). + +This work explores how the reasoning ability of large language models can be unlocked by a simple method motivated by two ideas. First, techniques for arithmetic reasoning can benefit from generating natural language rationales that lead to the final answer. Prior work has given models the ability to generate natural language intermediate steps by training from scratch (Ling et al., 2017) or finetuning a pretrained model (Cobbe et al., 2021), in addition to neuro-symbolic methods that use formal languages instead of natural language (Roy and Roth, 2015; Chiang and Chen, 2019; Amini et al., 2019; Chen et al., 2019). Second, large language models offer the exciting + +Finetuned GPT-3 175B +Prior best PaLM 540B: standard prompting PaLM 540B: chain-of-thought prompting + +![](images/d7882b4649faefdd9a2b82d45815028e7e1d5a09149c74872bb38e2b8bb59569.jpg) +Figure 2: PaLM 540B uses chain-ofthought prompting to achieve new stateof-the-art performance on the GSM8K benchmark of math word problems. Finetuned GPT-3 and prior best are from Cobbe et al. (2021). + +prospect of in-context few-shot learning via prompting. That is, instead of finetuning a separate language model checkpoint for each new task, one can simply “prompt” the model with a few input–output exemplars demonstrating the task. Remarkably, this has been successful for a range of simple question-answering tasks (Brown et al., 2020). + +Both of the above ideas, however, have key limitations. For rationale-augmented training and finetuning methods, it is costly to create a large set of high quality rationales, which is much more complicated than simple input–output pairs used in normal machine learning. For the traditional fewshot prompting method used in Brown et al. (2020), it works poorly on tasks that require reasoning abilities, and often does not improve substantially with increasing language model scale (Rae et al., 2021). In this paper, we combine the strengths of these two ideas in a way that avoids their limitations. Specifically, we explore the ability of language models to perform few-shot prompting for reasoning tasks, given a prompt that consists of triples: hinput, chain of thought, outputi. A chain of thought is a series of intermediate natural language reasoning steps that lead to the final output, and we refer to this approach as chain-of-thought prompting. An example prompt is shown in Figure 1. + +We present empirical evaluations on arithmetic, commonsense, and symbolic reasoning benchmarks, showing that chain-of-thought prompting outperforms standard prompting, sometimes to a striking degree. Figure 2 illustrates one such result—on the GSM8K benchmark of math word problems (Cobbe et al., 2021), chain-of-thought prompting with PaLM 540B outperforms standard prompting by a large margin and achieves new state-of-the-art performance. A prompting only approach is important because it does not require a large training dataset and because a single model checkpoint can perform many tasks without loss of generality. This work underscores how large language models can learn via a few examples with natural language data about the task (c.f. automatically learning the patterns underlying inputs and outputs via a large training dataset). + +# 2 Chain-of-Thought Prompting + +Consider one’s own thought process when solving a complicated reasoning task such as a multi-step math word problem. It is typical to decompose the problem into intermediate steps and solve each before giving the final answer: “After Jane gives 2 flowers to her mom she has 10 . . . then after she gives 3 to her dad she will have 7 . . . so the answer is 7.” The goal of this paper is to endow language models with the ability to generate a similar chain of thought—a coherent series of intermediate reasoning steps that lead to the final answer for a problem. We will show that sufficiently large language models can generate chains of thought if demonstrations of chain-of-thought reasoning are provided in the exemplars for few-shot prompting. + +Figure 1 shows an example of a model producing a chain of thought to solve a math word problem that it would have otherwise gotten incorrect. The chain of thought in this case resembles a solution and can interpreted as one, but we still opt to call it a chain of thought to better capture the idea that it mimics a step-by-step thought process for arriving at the answer (and also, solutions/explanations typically come after the final answer (Narang et al., 2020; Wiegreffe et al., 2022; Lampinen et al., 2022, inter alia)). + +Chain-of-thought prompting has several attractive properties as an approach for facilitating reasoning in language models. + +1. First, chain of thought, in principle, allows models to decompose multi-step problems into intermediate steps, which means that additional computation can be allocated to problems that require more reasoning steps. +2. Second, a chain of thought provides an interpretable window into the behavior of the model, suggesting how it might have arrived at a particular answer and providing opportunities to debug where the reasoning path went wrong (although fully characterizing a model’s computations that support an answer remains an open question). +3. Third, chain-of-thought reasoning can be used for tasks such as math word problems, commonsense reasoning, and symbolic manipulation, and is potentially applicable (at least in principle) to any task that humans can solve via language. +4. Finally, chain-of-thought reasoning can be readily elicited in sufficiently large off-the-shelf language models simply by including examples of chain of thought sequences into the exemplars of few-shot prompting. + +In empirical experiments, we will observe the utility of chain-of-thought prompting for arithmetic reasoning (Section 3), commonsense reasoning (Section 4), and symbolic reasoning (Section 5). + +# 3 Arithmetic Reasoning + +We begin by considering math word problems of the form in Figure 1, which measure the arithmetic reasoning ability of language models. Though simple for humans, arithmetic reasoning is a task where language models often struggle (Hendrycks et al., 2021; Patel et al., 2021, inter alia). Strikingly, chainof-thought prompting when used with the 540B parameter language model performs comparably with task-specific finetuned models on several tasks, even achieving new state of the art on the challenging GSM8K benchmark (Cobbe et al., 2021). + +# 3.1 Experimental Setup + +We explore chain-of-thought prompting for various language models on multiple benchmarks. + +Benchmarks. We consider the following five math word problem benchmarks: (1) the GSM8K benchmark of math word problems (Cobbe et al., 2021), (2) the SVAMP dataset of math word problems with varying structures (Patel et al., 2021), (3) the ASDiv dataset of diverse math word problems (Miao et al., 2020), (4) the AQuA dataset of algebraic word problems, and (5) the MAWPS benchmark (Koncel-Kedziorski et al., 2016). Example problems are given in Appendix Table 12. + +Standard prompting. For the baseline, we consider standard few-shot prompting, popularized by Brown et al. (2020), in which a language model is given in-context exemplars of input–output pairs before outputting a prediction for a test-time example. Exemplars are formatted as questions and answers. The model gives the answer directly, as shown in Figure 1 (left). + +Chain-of-thought prompting. Our proposed approach is to augment each exemplar in few-shot prompting with a chain of thought for an associated answer, as illustrated in Figure 1 (right). As most of the datasets only have an evaluation split, we manually composed a set of eight few-shot exemplars with chains of thought for prompting—Figure 1 (right) shows one chain of thought exemplar, and the full set of exemplars is given in Appendix Table 20. (These particular exemplars did not undergo prompt engineering; robustness is studied in Section 3.4 and Appendix A.2.) To investigate whether chain-of-thought prompting in this form can successfully elicit successful reasoning across a range of math word problems, we used this single set of eight chain of thought exemplars for all benchmarks except AQuA, which is multiple choice instead of free response. For AQuA, we used four exemplars and solutions from the training set, as given in Appendix Table 21. + +![](images/89a5ee9245764678a62df985ae74ac1f355481b7fba3da340ff294060cdc69a1.jpg) +Figure 3: Examples of hinput, chain of thought, outputi triples for arithmetic, commonsense, and symbolic reasoning benchmarks. Chains of thought are highlighted. Full prompts in Appendix G. + +Language models. We evaluate five large language models. The first is GPT-3 (Brown et al., 2020), for which we use text-ada-001, text-babbage-001, text-curie-001, and text-davinci-002, which presumably correspond to InstructGPT models of 350M, 1.3B, 6.7B, and 175B parameters (Ouyang et al., 2022).The second is LaMDA (Thoppilan et al., 2022), which has models of 422M, 2B, 8B, 68B, and 137B parameters. The third is PaLM, which has models of 8B, 62B, and 540B parameters. The fourth is UL2 20B (Tay et al., 2022), and the fifth is Codex (Chen et al., 2021, code-davinci-002 in the OpenAI API). We sample from the models via greedy decoding (though follow-up work shows chain-of-thought prompting can be improved by taking the majority final answer over many sampled generations (Wang et al., 2022a)). For LaMDA, we report averaged results over five random seeds, where each seed had a different randomly shuffled order of exemplars. As LaMDA experiments did not show large variance among different seeds, to save compute we report results for a single exemplar order for all other models. + +# 3.2 Results + +The strongest results of chain-of-thought prompting are summarized in Figure 4, with all experimental outputs for each model collection, model size, and benchmark shown in Table 2 in the Appendix. There are three key takeaways. First, Figure 4 shows that chain-of-thought prompting is an emergent ability of model scale (Wei et al., 2022b). That is, chain-of-thought prompting does not positively impact performance for small models, and only yields performance gains when used with models of ${ \sim } 1 0 0 \mathrm { B }$ parameters. We qualitatively found that models of smaller scale produced fluent but illogical chains of thought, leading to lower performance than standard prompting. + +Second, chain-of-thought prompting has larger performance gains for more-complicated problems. For instance, for GSM8K (the dataset with the lowest baseline performance), performance more than doubled for the largest GPT and PaLM models. On the other hand, for SingleOp, the easiest subset of MAWPS which only requires a single step to solve, performance improvements were either negative or very small (see Appendix Table 3). + +Third, chain-of-thought prompting via GPT-3 175B and PaLM 540B compares favorably to prior state of the art, which typically finetunes a task-specific model on a labeled training dataset. Figure 4 shows how PaLM 540B uses chain-ofthought prompting to achieve new state of the art on GSM8K, SVAMP, and MAWPS (though note that standard prompting already passed the prior best for SVAMP). On the other two datasets, AQuA and ASDiv, PaLM with chain-of-thought prompting reaches within $2 \%$ of the state of the art (Appendix Table 2). + +To better understand why chain-of-thought prompting works, we manually examined modelgenerated chains of thought by LaMDA 137B for GSM8K. Of 50 random examples where the model returned the correct final answer, all of the generated chains of thought were also logically and mathematically correct except two that coincidentally arrived at the correct answer (see Appendix D.1, and Table 8 for examples of correct model-generated chains of thought). We also randomly examined 50 random samples for which the model gave the wrong answer. The summary of this analysis is that $46 \%$ of the chains of thought were almost correct, barring minor mistakes (calculator error, symbol map + +![](images/66c022898bff367d5860a3fc49e57a4fadc8dcb6199599bfe7c3100af93ef076.jpg) +Figure 4: Chain-of-thought prompting enables large language models to solve challenging math problems. Notably, chain-of-thought reasoning is an emergent ability of increasing model scale. Prior best numbers are from Cobbe et al. (2021) for GSM8K, Jie et al. (2022) for SVAMP, and Lan et al. (2021) for MAWPS. + +ping error, or one reasoning step missing), and that the other $54 \%$ of the chains of thought had major errors in semantic understanding or coherence (see Appendix D.2). To provide a small insight into why scaling improves chain-of-thought reasoning ability, we performed a similar analysis of errors made by PaLM 62B and whether those errors were fixed by scaling to PaLM 540B. The summary is that scaling PaLM to 540B fixes a large portion of one-step missing and semantic understanding errors in the 62B model (see Appendix A.1). + +# 3.3 Ablation Study + +The observed benefits of using chain-of-thought prompting raises the natural question of whether the same performance improvements can be conferred via other types of prompting. Figure 5 shows an ablation study with three variations of chain of thought described below. + +Equation only. One reason for why chain-of-thought prompting might help is that it produces the mathematical equation to be evaluated, and so we test a variation where the model is prompted to output only a mathematical equation before giving the answer. Figure 5 shows that equation only prompting does not help much for GSM8K, which implies that the semantics of the questions in GSM8K are too challenging to directly translate into an equation without the natural language reasoning steps in chain of thought. For datasets of one-step or two-step problems, however, we find that equation only prompting does improve performance, since the equation can be easily derived from the question (see Appendix Table 6). + +Variable compute only. Another intuition is that chain of thought allows the model to spend more computation (i.e., intermediate tokens) on harder problems. To isolate the effect of variable computation from chain-of-thought reasoning, we test a configuration where the model is prompted to output a only sequence of dots $( \ldots )$ equal to the number of characters in the equation needed to solve the problem. This variant performs about the same as the baseline, which suggests that variable computation by itself is not the reason for the success of chainof-thought prompting, and that there appears to be utility from expressing intermediate steps via natural language. + +Chain of thought after answer. Another potential benefit of chain-of-thought prompting could simply be that such prompts allow the model to better access relevant knowledge acquired during pretraining. Therefore, we test an alternative configuration where the chain of thought prompt is only given after the answer, isolating whether the model actually depends on the produced chain of thought to give the final answer. This variant performs about the same as the baseline, which suggests that the sequential reasoning embodied in the chain of thought is useful for reasons beyond just activating knowledge. + +Standard prompting Z Equation only Variable compute only 用 Reasoning after answer Chain-of-thought prompting + +![](images/e6ae1dc967e810730fda1618d18b35c7fc13ce62af4bf30f06c19dc149050fe2.jpg) +Figure 5: Ablation study for different variations of prompting using LaMDA 137B and PaLM 540B. Results for other datasets are given in Appendix Table 6 and Table 7. + +# 3.4 Robustness of Chain of Thought + +Sensitivity to exemplars is a key consideration of prompting approaches—for instance, varying the permutation of few-shot exemplars can cause the accuracy of GPT-3 on SST-2 to range from near chance $( 5 4 . 3 \% )$ to near state of the art $( 9 3 . 4 \% )$ (Zhao et al., 2021). In this final subsection, we evaluate robustness to chains of thought written by different annotators. In addition to the results above, which used chains of thought written by an Annotator A, two other co-authors of this paper (Annotators B and C) independently wrote chains of thought for the same few-shot exemplars (shown in Appendix H). Annotator A also wrote another chain of thought that was more concise than the original, following the style of solutions given in Cobbe et al. (2021).1 + +Figure 6 shows these results for LaMDA 137B on GSM8K and MAWPS (ablation results for other datasets are given in Appendix Table 6 / Table 7). Although there is variance among different chain of thought annotations, as would be expected when using exemplar-based prompting (Le Scao and Rush, 2021; Reynolds and McDonell, 2021; Zhao et al., 2021), all sets of chain of thought prompts outperform the standard baseline by a large margin. This result implies that successful use of chain of thought does not depend on a particular linguistic style. + +![](images/4e11e7bcb919c728e67a6b419ec469fa86d0f95c703a30a4d57759b25ff7f940.jpg) +Figure 6: Chain-of-thought prompting has variance for different prompt examples (as expected) but outperforms standard prompting for various annotators as well as for different exemplars. + +To confirm that successful chain-of-thought prompting works for other sets of exemplars, we also run experiments with three sets of eight exemplars randomly sampled from the GSM8K training set, an independent source (examples in this dataset already included reasoning steps like a chain of thought).2 Figure 6 shows that these prompts performed comparably with our manually written exemplars, also substantially outperforming standard prompting. + +In addition to robustness to annotators, independently-written chains of thought, different exemplars, and various language models, we also find that chain-of-thought prompting for arithmetic reasoning is robust to different exemplar orders and varying numbers of exemplars (see Appendix A.2). + +# 4 Commonsense Reasoning + +Although chain of thought is particularly suitable for math word problems, the language-based nature of chain of thought actually makes it applicable to a broad class of commonsense reasoning problems, which involve reasoning about physical and human interactions under the presumption of general background knowledge. Commonsense reasoning is key for interacting with the world and is still beyond the reach of current natural language understanding systems (Talmor et al., 2021). + +Benchmarks. We consider five datasets covering a diverse range of commonsense reasoning types. The popular CSQA (Talmor et al., 2019) asks commonsense questions about the world involving complex semantics that often require prior knowledge. StrategyQA (Geva et al., 2021) requires models to infer a multi-hop strategy to answer questions. We choose two specialized evaluation sets from the BIG-bench effort (BIG-bench collaboration, 2021): Date Understanding, which involves inferring a date from a given context, and Sports Understanding, which involves determining whether a sentence relating to sports is plausible or implausible. Finally, the SayCan dataset (Ahn et al., 2022) involves mapping a natural language instruction to a sequence of robot actions from a discrete set. Figure 3 shows examples with chain of thought annotations for all datasets. + +Prompts. We follow the same experimental setup as the prior section. For CSQA and StrategyQA, we randomly selected examples from the training set and manually composed chains of thought for them to use as few-shot exemplars. The two BIG-bench tasks do not have training sets, so we selected the first ten examples as exemplars in the evaluation set as few-shot exemplars and report numbers on the rest of the evaluation set. For SayCan, we use six examples from the training set used in Ahn et al. (2022) and also manually composed chains of thought. + +Results. Figure 7 highlights these results for PaLM (full results for LaMDA, GPT-3, and different model scales are shown in Table 4). For all tasks, scaling up model size improved the performance of standard prompting; chain-of-thought prompting led to further gains, with improvements appearing to be largest for PaLM 540B. With chain-of-thought prompting, PaLM 540B achieved strong performance relative to baselines, outperforming the prior state of the art on StrategyQA $7 5 . 6 \%$ vs $6 9 . 4 \%$ and outperforming an unaided sports enthusiast on sports understanding $9 5 . 4 \%$ vs $84 \%$ ). These results demonstrate that chain-of-thought prompting can also improve performance on tasks requiring a range of commonsense reasoning abilities (though note that gain was minimal on CSQA). + +![](images/314d6a9f471c937800bcc79eed382244c146f9aacc68828ea5e504cb8128e362.jpg) +Figure 7: Chain-of-thought prompting also improves the commonsense reasoning abilities of language models. The language model shown here is PaLM. Prior best numbers are from the leaderboards of CSQA (Talmor et al., 2019) and StrategyQA (Geva et al., 2021) (single-model only, as of May 5, 2022). Additional results using various sizes of LaMDA, GPT-3, and PaLM are shown in Table 4. + +# 5 Symbolic Reasoning + +Our final experimental evaluation considers symbolic reasoning, which is simple for humans but potentially challenging for language models. We show that chain-ofthought prompting not only enables language models to perform symbolic reasoning tasks that are challenging in the standard prompting setting, but also facilitates length generalization to inference-time inputs longer than those seen in the few-shot exemplars. + +Tasks. We use the following two toy tasks. + +• Last letter concatenation. This task asks the model to concatenate the last letters of words in a name (e.g., “Amy Brown” $ ^ { \ast } y n ^ { \prime \prime }$ ). It is a more challenging version of first letter concatenation, which language models can already perform without chain of thought.3 We generate full names by randomly concatenating names from the top one-thousand first and last names from name census data (https://namecensus.com/). + +![](images/e4f8153620820f36adc177bd4972c7b26a10341d1dec25b01e73852fd72bc7ae.jpg) +Figure 8: Using chain-of-thought prompting facilitates generalization to longer sequences in two symbolic reasoning tasks. + +• Coin flip. This task asks the model to answer whether a coin is still heads up after people either flip or don’t flip the coin (e.g., “A coin is heads up. Phoebe flips the coin. Osvaldo does not flip the coin. Is the coin still heads up? $^ { \prime \prime } \right. ^ { \left. } n o ^ { \prime \prime }$ ). + +As the construction of these symbolic reasoning tasks is well-defined, for each task we consider an in-domain test set for which examples had the same number of steps as the training/few-shot exemplars, as well as an out-of-domain (OOD) test set, for which evaluation examples had more steps than those in the exemplars. For last letter concatenation, the model only sees exemplars of names with two words, and then performs last letter concatenation on names with 3 and 4 words.4 We do the same for the number of potential flips in the coin flip task. Our experimental setup uses the same methods and models as in the prior two sections. We again manually compose chains of thought for the few-shot exemplars for each task, which are given in Figure 3. + +Results. The results of these in-domain and OOD evaluations are shown in Figure 8 for PaLM, with results for LaMDA shown in Appendix Table 5. With PaLM 540B, chain-of-thought prompting leads to almost $100 \%$ solve rates (note that standard prompting already solves coin flip with PaLM 540, though not for LaMDA 137B). Note that these in-domain evaluations are “toy tasks” in the sense that perfect solution structures are already provided by the chains of thought in the few-shot exemplars; all the model has to do is repeat the same steps with the new symbols in the test-time example. And yet, small models still fail—the ability to perform abstract manipulations on unseen symbols for these three tasks only arises at the scale of 100B model parameters. + +As for the OOD evaluations, standard prompting fails for both tasks. With chain-of-thought prompting, language models achieve upward scaling curves (though performance is lower than in the in-domain setting). Hence, chain-of-thought prompting facilitates length generalization beyond seen chains of thought for language models of sufficient scale. + +# 6 Discussion + +We have explored chain-of-thought prompting as a simple mechanism for eliciting multi-step reasoning behavior in large language models. We first saw that chain-of-thought prompting improves performance by a large margin on arithmetic reasoning, yielding improvements that are much stronger than ablations and robust to different annotators, exemplars, and language models (Section 3). Next, experiments on commonsense reasoning underscored how the linguistic nature of chain-of-thought reasoning makes it generally applicable (Section 4). Finally, we showed that for symbolic reasoning, chain-of-thought prompting facilitates OOD generalization to longer sequence lengths (Section 5). In all experiments, chain-of-thought reasoning is elicited simply by prompting an off-the-shelf language model. No language models were finetuned in the process of writing this paper. + +The emergence of chain-of-thought reasoning as a result of model scale has been a prevailing theme (Wei et al., 2022b). For many reasoning tasks where standard prompting has a flat scaling curve, chainof-thought prompting leads to dramatically increasing scaling curves. Chain-of-thought prompting appears to expand the set of tasks that large language models can perform successfully—in other words, our work underscores that standard prompting only provides a lower bound on the capabilities of large language models. This observation likely raises more questions than it answers—for instance, how much more can we expect reasoning ability to improve with a further increase in model scale? What other prompting methods might expand the range of tasks that language models can solve? + +As for limitations, we first qualify that although chain of thought emulates the thought processes of human reasoners, this does not answer whether the neural network is actually “reasoning,” which we leave as an open question. Second, although the cost of manually augmenting exemplars with chains of thought is minimal in the few-shot setting, such annotation costs could be prohibitive for finetuning (though this could potentially be surmounted with synthetic data generation, or zero-shot generalization). Third, there is no guarantee of correct reasoning paths, which can lead to both correct and incorrect answers; improving factual generations of language models is an open direction for future work (Rashkin et al., 2021; Ye and Durrett, 2022; Wiegreffe et al., 2022, inter alia). Finally, the emergence of chain-of-thought reasoning only at large model scales makes it costly to serve in real-world applications; further research could explore how to induce reasoning in smaller models. + +# 7 Related Work + +This work is inspired by many research areas, which we detail in an extended related work section (Appendix C). Here we describe two directions and associated papers that are perhaps most relevant. + +The first relevant direction is using intermediate steps to solve reasoning problems. Ling et al. (2017) pioneer the idea of using natural language rationales to solve math word problems through a series of intermediate steps. Their work is a remarkable contrast to the literature using formal languages to reason (Roy et al., 2015; Chiang and Chen, 2019; Amini et al., 2019; Chen et al., 2019). Cobbe et al. (2021) extend Ling et al. (2017) by creating a larger dataset and using it to finetune a pretrained language model rather than training a model from scratch. In the domain of program synthesis, Nye et al. (2021) leverage language models to predict the final outputs of Python programs via first line-to-line predicting the intermediate computational results, and show that their step-by-step prediction method performs better than directly predicting the final outputs. + +Naturally, this paper also relates closely to the large body of recent work on prompting. Since the popularization of few-shot prompting as given by Brown et al. (2020), several general approaches have improved the prompting ability of models, such as automatically learning prompts (Lester et al., 2021) or giving models instructions describing a task (Wei et al., 2022a; Sanh et al., 2022; Ouyang et al., 2022). Whereas these approaches improve or augment the input part of the prompt (e.g., instructions that are prepended to inputs), our work takes the orthogonal direction of augmenting the outputs of language models with a chain of thought. + +# 8 Conclusions + +We have explored chain-of-thought prompting as a simple and broadly applicable method for enhancing reasoning in language models. Through experiments on arithmetic, symbolic, and commonsense reasoning, we find that chain-of-thought reasoning is an emergent property of model scale that allows sufficiently large language models to perform reasoning tasks that otherwise have flat scaling curves. Broadening the range of reasoning tasks that language models can perform will hopefully inspire further work on language-based approaches to reasoning. + +# Acknowledgements + +We thank Jacob Devlin, Claire Cui, Andrew Dai, and Ellie Pavlick for providing feedback on the paper. 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NeurIPS. \ No newline at end of file diff --git a/parse/dev/_VjQlMeSB_J/_VjQlMeSB_J_content_list.json b/parse/dev/_VjQlMeSB_J/_VjQlMeSB_J_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..28ca2a848894a7c3fa92e532245f7c2469d32cb1 --- /dev/null +++ b/parse/dev/_VjQlMeSB_J/_VjQlMeSB_J_content_list.json @@ -0,0 +1,1424 @@ +[ + { + "type": "text", + "text": "Chain-of-Thought Prompting Elicits Reasoning in Large Language Models ", + "text_level": 1, + "bbox": [ + 214, + 122, + 785, + 174 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jason Wei Xuezhi Wang Dale Schuurmans Maarten Bosma ", + "bbox": [ + 243, + 224, + 754, + 241 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Brian Ichter Fei Xia Ed H. Chi Quoc V. Le Denny Zhou ", + "bbox": [ + 246, + 247, + 751, + 263 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Google Research, Brain Team {jasonwei,dennyzhou}@google.com ", + "bbox": [ + 366, + 272, + 632, + 301 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 348, + 535, + 364 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We explore how generating a chain of thought—a series of intermediate reasoning steps—significantly improves the ability of large language models to perform complex reasoning. In particular, we show how such reasoning abilities emerge naturally in sufficiently large language models via a simple method called chain-ofthought prompting, where a few chain of thought demonstrations are provided as exemplars in prompting. ", + "bbox": [ + 232, + 378, + 766, + 463 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Experiments on three large language models show that chain-of-thought prompting improves performance on a range of arithmetic, commonsense, and symbolic reasoning tasks. The empirical gains can be striking. For instance, prompting a PaLM 540B with just eight chain-of-thought exemplars achieves state-of-the-art accuracy on the GSM8K benchmark of math word problems, surpassing even finetuned GPT-3 with a verifier. ", + "bbox": [ + 232, + 465, + 766, + 547 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/89ab3799d105e595086d7c7e3059c4c95016564718ada32028e60eef2d474c49.jpg", + "image_caption": [ + "Figure 1: Chain-of-thought prompting enables large language models to tackle complex arithmetic, commonsense, and symbolic reasoning tasks. Chain-of-thought reasoning processes are highlighted. " + ], + "image_footnote": [], + "bbox": [ + 192, + 570, + 802, + 806 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 89, + 310, + 106 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The NLP landscape has recently been revolutionized by language models (Peters et al., 2018; Devlin et al., 2019; Brown et al., 2020, inter alia). Scaling up the size of language models has been shown to confer a range of benefits, such as improved performance and sample efficiency (Kaplan et al., 2020; Brown et al., 2020, inter alia). However, scaling up model size alone has not proved sufficient for achieving high performance on challenging tasks such as arithmetic, commonsense, and symbolic reasoning (Rae et al., 2021). ", + "bbox": [ + 174, + 125, + 550, + 262 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This work explores how the reasoning ability of large language models can be unlocked by a simple method motivated by two ideas. First, techniques for arithmetic reasoning can benefit from generating natural language rationales that lead to the final answer. Prior work has given models the ability to generate natural language intermediate steps by training from scratch (Ling et al., 2017) or finetuning a pretrained model (Cobbe et al., 2021), in addition to neuro-symbolic methods that use formal languages instead of natural language (Roy and Roth, 2015; Chiang and Chen, 2019; Amini et al., 2019; Chen et al., 2019). Second, large language models offer the exciting ", + "bbox": [ + 174, + 270, + 549, + 435 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Finetuned GPT-3 175B \nPrior best PaLM 540B: standard prompting PaLM 540B: chain-of-thought prompting ", + "bbox": [ + 565, + 126, + 836, + 185 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/d7882b4649faefdd9a2b82d45815028e7e1d5a09149c74872bb38e2b8bb59569.jpg", + "image_caption": [ + "Figure 2: PaLM 540B uses chain-ofthought prompting to achieve new stateof-the-art performance on the GSM8K benchmark of math word problems. Finetuned GPT-3 and prior best are from Cobbe et al. (2021). " + ], + "image_footnote": [], + "bbox": [ + 578, + 190, + 828, + 324 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "prospect of in-context few-shot learning via prompting. That is, instead of finetuning a separate language model checkpoint for each new task, one can simply “prompt” the model with a few input–output exemplars demonstrating the task. Remarkably, this has been successful for a range of simple question-answering tasks (Brown et al., 2020). ", + "bbox": [ + 174, + 435, + 825, + 489 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Both of the above ideas, however, have key limitations. For rationale-augmented training and finetuning methods, it is costly to create a large set of high quality rationales, which is much more complicated than simple input–output pairs used in normal machine learning. For the traditional fewshot prompting method used in Brown et al. (2020), it works poorly on tasks that require reasoning abilities, and often does not improve substantially with increasing language model scale (Rae et al., 2021). In this paper, we combine the strengths of these two ideas in a way that avoids their limitations. Specifically, we explore the ability of language models to perform few-shot prompting for reasoning tasks, given a prompt that consists of triples: hinput, chain of thought, outputi. A chain of thought is a series of intermediate natural language reasoning steps that lead to the final output, and we refer to this approach as chain-of-thought prompting. An example prompt is shown in Figure 1. ", + "bbox": [ + 174, + 497, + 825, + 636 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We present empirical evaluations on arithmetic, commonsense, and symbolic reasoning benchmarks, showing that chain-of-thought prompting outperforms standard prompting, sometimes to a striking degree. Figure 2 illustrates one such result—on the GSM8K benchmark of math word problems (Cobbe et al., 2021), chain-of-thought prompting with PaLM 540B outperforms standard prompting by a large margin and achieves new state-of-the-art performance. A prompting only approach is important because it does not require a large training dataset and because a single model checkpoint can perform many tasks without loss of generality. This work underscores how large language models can learn via a few examples with natural language data about the task (c.f. automatically learning the patterns underlying inputs and outputs via a large training dataset). ", + "bbox": [ + 173, + 641, + 825, + 766 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Chain-of-Thought Prompting ", + "text_level": 1, + "bbox": [ + 174, + 791, + 452, + 810 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Consider one’s own thought process when solving a complicated reasoning task such as a multi-step math word problem. It is typical to decompose the problem into intermediate steps and solve each before giving the final answer: “After Jane gives 2 flowers to her mom she has 10 . . . then after she gives 3 to her dad she will have 7 . . . so the answer is 7.” The goal of this paper is to endow language models with the ability to generate a similar chain of thought—a coherent series of intermediate reasoning steps that lead to the final answer for a problem. We will show that sufficiently large language models can generate chains of thought if demonstrations of chain-of-thought reasoning are provided in the exemplars for few-shot prompting. ", + "bbox": [ + 174, + 827, + 823, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 90, + 823, + 119 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Figure 1 shows an example of a model producing a chain of thought to solve a math word problem that it would have otherwise gotten incorrect. The chain of thought in this case resembles a solution and can interpreted as one, but we still opt to call it a chain of thought to better capture the idea that it mimics a step-by-step thought process for arriving at the answer (and also, solutions/explanations typically come after the final answer (Narang et al., 2020; Wiegreffe et al., 2022; Lampinen et al., 2022, inter alia)). ", + "bbox": [ + 173, + 126, + 825, + 209 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Chain-of-thought prompting has several attractive properties as an approach for facilitating reasoning in language models. ", + "bbox": [ + 174, + 215, + 823, + 244 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1. First, chain of thought, in principle, allows models to decompose multi-step problems into intermediate steps, which means that additional computation can be allocated to problems that require more reasoning steps. \n2. Second, a chain of thought provides an interpretable window into the behavior of the model, suggesting how it might have arrived at a particular answer and providing opportunities to debug where the reasoning path went wrong (although fully characterizing a model’s computations that support an answer remains an open question). \n3. Third, chain-of-thought reasoning can be used for tasks such as math word problems, commonsense reasoning, and symbolic manipulation, and is potentially applicable (at least in principle) to any task that humans can solve via language. \n4. Finally, chain-of-thought reasoning can be readily elicited in sufficiently large off-the-shelf language models simply by including examples of chain of thought sequences into the exemplars of few-shot prompting. ", + "bbox": [ + 210, + 251, + 825, + 440 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In empirical experiments, we will observe the utility of chain-of-thought prompting for arithmetic reasoning (Section 3), commonsense reasoning (Section 4), and symbolic reasoning (Section 5). ", + "bbox": [ + 173, + 448, + 823, + 476 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Arithmetic Reasoning ", + "text_level": 1, + "bbox": [ + 174, + 496, + 390, + 513 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We begin by considering math word problems of the form in Figure 1, which measure the arithmetic reasoning ability of language models. Though simple for humans, arithmetic reasoning is a task where language models often struggle (Hendrycks et al., 2021; Patel et al., 2021, inter alia). Strikingly, chainof-thought prompting when used with the 540B parameter language model performs comparably with task-specific finetuned models on several tasks, even achieving new state of the art on the challenging GSM8K benchmark (Cobbe et al., 2021). ", + "bbox": [ + 174, + 529, + 825, + 612 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 Experimental Setup ", + "text_level": 1, + "bbox": [ + 174, + 630, + 352, + 645 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We explore chain-of-thought prompting for various language models on multiple benchmarks. ", + "bbox": [ + 171, + 655, + 789, + 670 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Benchmarks. We consider the following five math word problem benchmarks: (1) the GSM8K benchmark of math word problems (Cobbe et al., 2021), (2) the SVAMP dataset of math word problems with varying structures (Patel et al., 2021), (3) the ASDiv dataset of diverse math word problems (Miao et al., 2020), (4) the AQuA dataset of algebraic word problems, and (5) the MAWPS benchmark (Koncel-Kedziorski et al., 2016). Example problems are given in Appendix Table 12. ", + "bbox": [ + 174, + 676, + 825, + 746 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Standard prompting. For the baseline, we consider standard few-shot prompting, popularized by Brown et al. (2020), in which a language model is given in-context exemplars of input–output pairs before outputting a prediction for a test-time example. Exemplars are formatted as questions and answers. The model gives the answer directly, as shown in Figure 1 (left). ", + "bbox": [ + 174, + 752, + 823, + 808 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Chain-of-thought prompting. Our proposed approach is to augment each exemplar in few-shot prompting with a chain of thought for an associated answer, as illustrated in Figure 1 (right). As most of the datasets only have an evaluation split, we manually composed a set of eight few-shot exemplars with chains of thought for prompting—Figure 1 (right) shows one chain of thought exemplar, and the full set of exemplars is given in Appendix Table 20. (These particular exemplars did not undergo prompt engineering; robustness is studied in Section 3.4 and Appendix A.2.) To investigate whether chain-of-thought prompting in this form can successfully elicit successful reasoning across a range of math word problems, we used this single set of eight chain of thought exemplars for all benchmarks except AQuA, which is multiple choice instead of free response. For AQuA, we used four exemplars and solutions from the training set, as given in Appendix Table 21. ", + "bbox": [ + 174, + 814, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/89a5ee9245764678a62df985ae74ac1f355481b7fba3da340ff294060cdc69a1.jpg", + "image_caption": [ + "Figure 3: Examples of hinput, chain of thought, outputi triples for arithmetic, commonsense, and symbolic reasoning benchmarks. Chains of thought are highlighted. Full prompts in Appendix G. " + ], + "image_footnote": [], + "bbox": [ + 176, + 85, + 825, + 486 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 549, + 826, + 590 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Language models. We evaluate five large language models. The first is GPT-3 (Brown et al., 2020), for which we use text-ada-001, text-babbage-001, text-curie-001, and text-davinci-002, which presumably correspond to InstructGPT models of 350M, 1.3B, 6.7B, and 175B parameters (Ouyang et al., 2022).The second is LaMDA (Thoppilan et al., 2022), which has models of 422M, 2B, 8B, 68B, and 137B parameters. The third is PaLM, which has models of 8B, 62B, and 540B parameters. The fourth is UL2 20B (Tay et al., 2022), and the fifth is Codex (Chen et al., 2021, code-davinci-002 in the OpenAI API). We sample from the models via greedy decoding (though follow-up work shows chain-of-thought prompting can be improved by taking the majority final answer over many sampled generations (Wang et al., 2022a)). For LaMDA, we report averaged results over five random seeds, where each seed had a different randomly shuffled order of exemplars. As LaMDA experiments did not show large variance among different seeds, to save compute we report results for a single exemplar order for all other models. ", + "bbox": [ + 173, + 598, + 826, + 763 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 Results ", + "text_level": 1, + "bbox": [ + 174, + 784, + 263, + 799 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The strongest results of chain-of-thought prompting are summarized in Figure 4, with all experimental outputs for each model collection, model size, and benchmark shown in Table 2 in the Appendix. There are three key takeaways. First, Figure 4 shows that chain-of-thought prompting is an emergent ability of model scale (Wei et al., 2022b). That is, chain-of-thought prompting does not positively impact performance for small models, and only yields performance gains when used with models of ${ \\sim } 1 0 0 \\mathrm { B }$ parameters. We qualitatively found that models of smaller scale produced fluent but illogical chains of thought, leading to lower performance than standard prompting. ", + "bbox": [ + 174, + 811, + 825, + 909 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Second, chain-of-thought prompting has larger performance gains for more-complicated problems. For instance, for GSM8K (the dataset with the lowest baseline performance), performance more than doubled for the largest GPT and PaLM models. On the other hand, for SingleOp, the easiest subset of MAWPS which only requires a single step to solve, performance improvements were either negative or very small (see Appendix Table 3). ", + "bbox": [ + 174, + 92, + 485, + 229 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Third, chain-of-thought prompting via GPT-3 175B and PaLM 540B compares favorably to prior state of the art, which typically finetunes a task-specific model on a labeled training dataset. Figure 4 shows how PaLM 540B uses chain-ofthought prompting to achieve new state of the art on GSM8K, SVAMP, and MAWPS (though note that standard prompting already passed the prior best for SVAMP). On the other two datasets, AQuA and ASDiv, PaLM with chain-of-thought prompting reaches within $2 \\%$ of the state of the art (Appendix Table 2). ", + "bbox": [ + 174, + 236, + 485, + 401 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To better understand why chain-of-thought prompting works, we manually examined modelgenerated chains of thought by LaMDA 137B for GSM8K. Of 50 random examples where the model returned the correct final answer, all of the generated chains of thought were also logically and mathematically correct except two that coincidentally arrived at the correct answer (see Appendix D.1, and Table 8 for examples of correct model-generated chains of thought). We also randomly examined 50 random samples for which the model gave the wrong answer. The summary of this analysis is that $46 \\%$ of the chains of thought were almost correct, barring minor mistakes (calculator error, symbol map", + "bbox": [ + 174, + 409, + 485, + 614 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/66c022898bff367d5860a3fc49e57a4fadc8dcb6199599bfe7c3100af93ef076.jpg", + "image_caption": [ + "Figure 4: Chain-of-thought prompting enables large language models to solve challenging math problems. Notably, chain-of-thought reasoning is an emergent ability of increasing model scale. Prior best numbers are from Cobbe et al. (2021) for GSM8K, Jie et al. (2022) for SVAMP, and Lan et al. (2021) for MAWPS. " + ], + "image_footnote": [], + "bbox": [ + 508, + 97, + 818, + 487 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "ping error, or one reasoning step missing), and that the other $54 \\%$ of the chains of thought had major errors in semantic understanding or coherence (see Appendix D.2). To provide a small insight into why scaling improves chain-of-thought reasoning ability, we performed a similar analysis of errors made by PaLM 62B and whether those errors were fixed by scaling to PaLM 540B. The summary is that scaling PaLM to 540B fixes a large portion of one-step missing and semantic understanding errors in the 62B model (see Appendix A.1). ", + "bbox": [ + 174, + 614, + 825, + 698 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 Ablation Study ", + "text_level": 1, + "bbox": [ + 174, + 720, + 316, + 736 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The observed benefits of using chain-of-thought prompting raises the natural question of whether the same performance improvements can be conferred via other types of prompting. Figure 5 shows an ablation study with three variations of chain of thought described below. ", + "bbox": [ + 174, + 750, + 825, + 791 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Equation only. One reason for why chain-of-thought prompting might help is that it produces the mathematical equation to be evaluated, and so we test a variation where the model is prompted to output only a mathematical equation before giving the answer. Figure 5 shows that equation only prompting does not help much for GSM8K, which implies that the semantics of the questions in GSM8K are too challenging to directly translate into an equation without the natural language reasoning steps in chain of thought. For datasets of one-step or two-step problems, however, we find that equation only prompting does improve performance, since the equation can be easily derived from the question (see Appendix Table 6). ", + "bbox": [ + 173, + 797, + 825, + 909 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Variable compute only. Another intuition is that chain of thought allows the model to spend more computation (i.e., intermediate tokens) on harder problems. To isolate the effect of variable computation from chain-of-thought reasoning, we test a configuration where the model is prompted to output a only sequence of dots $( \\ldots )$ equal to the number of characters in the equation needed to solve the problem. This variant performs about the same as the baseline, which suggests that variable computation by itself is not the reason for the success of chainof-thought prompting, and that there appears to be utility from expressing intermediate steps via natural language. ", + "bbox": [ + 174, + 92, + 583, + 243 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Chain of thought after answer. Another potential benefit of chain-of-thought prompting could simply be that such prompts allow the model to better access relevant knowledge acquired during pretraining. Therefore, we test an alternative configuration where the chain of thought prompt is only given after the answer, isolating whether the model actually depends on the produced chain of thought to give the final answer. This variant performs about the same as the baseline, which suggests that the sequential reasoning embodied in the chain of thought is useful for reasons beyond just activating knowledge. ", + "bbox": [ + 174, + 250, + 583, + 388 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Standard prompting Z Equation only Variable compute only 用 Reasoning after answer Chain-of-thought prompting ", + "bbox": [ + 617, + 95, + 812, + 169 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/e6ae1dc967e810730fda1618d18b35c7fc13ce62af4bf30f06c19dc149050fe2.jpg", + "image_caption": [ + "Figure 5: Ablation study for different variations of prompting using LaMDA 137B and PaLM 540B. Results for other datasets are given in Appendix Table 6 and Table 7. " + ], + "image_footnote": [], + "bbox": [ + 599, + 174, + 820, + 304 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.4 Robustness of Chain of Thought ", + "text_level": 1, + "bbox": [ + 176, + 415, + 436, + 429 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Sensitivity to exemplars is a key consideration of prompting approaches—for instance, varying the permutation of few-shot exemplars can cause the accuracy of GPT-3 on SST-2 to range from near chance $( 5 4 . 3 \\% )$ to near state of the art $( 9 3 . 4 \\% )$ (Zhao et al., 2021). In this final subsection, we evaluate robustness to chains of thought written by different annotators. In addition to the results above, which used chains of thought written by an Annotator A, two other co-authors of this paper (Annotators B and C) independently wrote chains of thought for the same few-shot exemplars (shown in Appendix H). Annotator A also wrote another chain of thought that was more concise than the original, following the style of solutions given in Cobbe et al. (2021).1 ", + "bbox": [ + 174, + 444, + 550, + 637 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figure 6 shows these results for LaMDA 137B on GSM8K and MAWPS (ablation results for other datasets are given in Appendix Table 6 / Table 7). Although there is variance among different chain of thought annotations, as would be expected when using exemplar-based prompting (Le Scao and Rush, 2021; Reynolds and McDonell, 2021; Zhao et al., 2021), all sets of chain of thought prompts outperform the standard baseline by a large margin. This result implies that successful use of chain of thought does not depend on a particular linguistic style. ", + "bbox": [ + 174, + 643, + 549, + 782 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/4e11e7bcb919c728e67a6b419ec469fa86d0f95c703a30a4d57759b25ff7f940.jpg", + "image_caption": [ + "Figure 6: Chain-of-thought prompting has variance for different prompt examples (as expected) but outperforms standard prompting for various annotators as well as for different exemplars. " + ], + "image_footnote": [], + "bbox": [ + 563, + 428, + 821, + 699 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To confirm that successful chain-of-thought prompting works for other sets of exemplars, we also run experiments with three sets of eight exemplars randomly sampled from the GSM8K training set, an independent source (examples in this dataset already included reasoning steps like a chain of thought).2 Figure 6 shows that these prompts performed comparably with our manually written exemplars, also substantially outperforming standard prompting. ", + "bbox": [ + 174, + 789, + 547, + 830 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 178, + 818, + 823, + 832 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 133 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In addition to robustness to annotators, independently-written chains of thought, different exemplars, and various language models, we also find that chain-of-thought prompting for arithmetic reasoning is robust to different exemplar orders and varying numbers of exemplars (see Appendix A.2). ", + "bbox": [ + 174, + 140, + 823, + 181 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 Commonsense Reasoning ", + "text_level": 1, + "bbox": [ + 176, + 202, + 418, + 219 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Although chain of thought is particularly suitable for math word problems, the language-based nature of chain of thought actually makes it applicable to a broad class of commonsense reasoning problems, which involve reasoning about physical and human interactions under the presumption of general background knowledge. Commonsense reasoning is key for interacting with the world and is still beyond the reach of current natural language understanding systems (Talmor et al., 2021). ", + "bbox": [ + 174, + 233, + 825, + 303 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Benchmarks. We consider five datasets covering a diverse range of commonsense reasoning types. The popular CSQA (Talmor et al., 2019) asks commonsense questions about the world involving complex semantics that often require prior knowledge. StrategyQA (Geva et al., 2021) requires models to infer a multi-hop strategy to answer questions. We choose two specialized evaluation sets from the BIG-bench effort (BIG-bench collaboration, 2021): Date Understanding, which involves inferring a date from a given context, and Sports Understanding, which involves determining whether a sentence relating to sports is plausible or implausible. Finally, the SayCan dataset (Ahn et al., 2022) involves mapping a natural language instruction to a sequence of robot actions from a discrete set. Figure 3 shows examples with chain of thought annotations for all datasets. ", + "bbox": [ + 174, + 309, + 825, + 434 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Prompts. We follow the same experimental setup as the prior section. For CSQA and StrategyQA, we randomly selected examples from the training set and manually composed chains of thought for them to use as few-shot exemplars. The two BIG-bench tasks do not have training sets, so we selected the first ten examples as exemplars in the evaluation set as few-shot exemplars and report numbers on the rest of the evaluation set. For SayCan, we use six examples from the training set used in Ahn et al. (2022) and also manually composed chains of thought. ", + "bbox": [ + 173, + 440, + 825, + 523 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Results. Figure 7 highlights these results for PaLM (full results for LaMDA, GPT-3, and different model scales are shown in Table 4). For all tasks, scaling up model size improved the performance of standard prompting; chain-of-thought prompting led to further gains, with improvements appearing to be largest for PaLM 540B. With chain-of-thought prompting, PaLM 540B achieved strong performance relative to baselines, outperforming the prior state of the art on StrategyQA $7 5 . 6 \\%$ vs $6 9 . 4 \\%$ and outperforming an unaided sports enthusiast on sports understanding $9 5 . 4 \\%$ vs $84 \\%$ ). These results demonstrate that chain-of-thought prompting can also improve performance on tasks requiring a range of commonsense reasoning abilities (though note that gain was minimal on CSQA). ", + "bbox": [ + 173, + 530, + 826, + 641 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/314d6a9f471c937800bcc79eed382244c146f9aacc68828ea5e504cb8128e362.jpg", + "image_caption": [ + "Figure 7: Chain-of-thought prompting also improves the commonsense reasoning abilities of language models. The language model shown here is PaLM. Prior best numbers are from the leaderboards of CSQA (Talmor et al., 2019) and StrategyQA (Geva et al., 2021) (single-model only, as of May 5, 2022). Additional results using various sizes of LaMDA, GPT-3, and PaLM are shown in Table 4. " + ], + "image_footnote": [], + "bbox": [ + 186, + 656, + 808, + 794 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 Symbolic Reasoning ", + "text_level": 1, + "bbox": [ + 174, + 89, + 374, + 107 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Our final experimental evaluation considers symbolic reasoning, which is simple for humans but potentially challenging for language models. We show that chain-ofthought prompting not only enables language models to perform symbolic reasoning tasks that are challenging in the standard prompting setting, but also facilitates length generalization to inference-time inputs longer than those seen in the few-shot exemplars. ", + "bbox": [ + 174, + 121, + 549, + 231 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Tasks. We use the following two toy tasks. ", + "bbox": [ + 173, + 246, + 467, + 261 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "• Last letter concatenation. This task asks the model to concatenate the last letters of words in a name (e.g., “Amy Brown” $ ^ { \\ast } y n ^ { \\prime \\prime }$ ). It is a more challenging version of first letter concatenation, which language models can already perform without chain of thought.3 We generate full names by randomly concatenating names from the top one-thousand first and last names from name census data (https://namecensus.com/). ", + "bbox": [ + 174, + 267, + 549, + 377 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/e4f8153620820f36adc177bd4972c7b26a10341d1dec25b01e73852fd72bc7ae.jpg", + "image_caption": [ + "Figure 8: Using chain-of-thought prompting facilitates generalization to longer sequences in two symbolic reasoning tasks. " + ], + "image_footnote": [], + "bbox": [ + 568, + 94, + 813, + 410 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "• Coin flip. This task asks the model to answer whether a coin is still heads up after people either flip or don’t flip the coin (e.g., “A coin is heads up. Phoebe flips the coin. Osvaldo does not flip the coin. Is the coin still heads up? $^ { \\prime \\prime } \\right. ^ { \\left. } n o ^ { \\prime \\prime }$ ). ", + "bbox": [ + 174, + 380, + 549, + 449 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As the construction of these symbolic reasoning tasks is well-defined, for each task we consider an in-domain test set for which examples had the same number of steps as the training/few-shot exemplars, as well as an out-of-domain (OOD) test set, for which evaluation examples had more steps than those in the exemplars. For last letter concatenation, the model only sees exemplars of names with two words, and then performs last letter concatenation on names with 3 and 4 words.4 We do the same for the number of potential flips in the coin flip task. Our experimental setup uses the same methods and models as in the prior two sections. We again manually compose chains of thought for the few-shot exemplars for each task, which are given in Figure 3. ", + "bbox": [ + 173, + 455, + 549, + 497 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 497, + 825, + 579 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Results. The results of these in-domain and OOD evaluations are shown in Figure 8 for PaLM, with results for LaMDA shown in Appendix Table 5. With PaLM 540B, chain-of-thought prompting leads to almost $100 \\%$ solve rates (note that standard prompting already solves coin flip with PaLM 540, though not for LaMDA 137B). Note that these in-domain evaluations are “toy tasks” in the sense that perfect solution structures are already provided by the chains of thought in the few-shot exemplars; all the model has to do is repeat the same steps with the new symbols in the test-time example. And yet, small models still fail—the ability to perform abstract manipulations on unseen symbols for these three tasks only arises at the scale of 100B model parameters. ", + "bbox": [ + 173, + 594, + 825, + 705 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As for the OOD evaluations, standard prompting fails for both tasks. With chain-of-thought prompting, language models achieve upward scaling curves (though performance is lower than in the in-domain setting). Hence, chain-of-thought prompting facilitates length generalization beyond seen chains of thought for language models of sufficient scale. ", + "bbox": [ + 174, + 712, + 825, + 767 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 Discussion ", + "text_level": 1, + "bbox": [ + 173, + 786, + 292, + 803 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We have explored chain-of-thought prompting as a simple mechanism for eliciting multi-step reasoning behavior in large language models. We first saw that chain-of-thought prompting improves performance by a large margin on arithmetic reasoning, yielding improvements that are much stronger than ablations and robust to different annotators, exemplars, and language models (Section 3). Next, experiments on commonsense reasoning underscored how the linguistic nature of chain-of-thought reasoning makes it generally applicable (Section 4). Finally, we showed that for symbolic reasoning, chain-of-thought prompting facilitates OOD generalization to longer sequence lengths (Section 5). In all experiments, chain-of-thought reasoning is elicited simply by prompting an off-the-shelf language model. No language models were finetuned in the process of writing this paper. ", + "bbox": [ + 176, + 818, + 825, + 873 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 161 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The emergence of chain-of-thought reasoning as a result of model scale has been a prevailing theme (Wei et al., 2022b). For many reasoning tasks where standard prompting has a flat scaling curve, chainof-thought prompting leads to dramatically increasing scaling curves. Chain-of-thought prompting appears to expand the set of tasks that large language models can perform successfully—in other words, our work underscores that standard prompting only provides a lower bound on the capabilities of large language models. This observation likely raises more questions than it answers—for instance, how much more can we expect reasoning ability to improve with a further increase in model scale? What other prompting methods might expand the range of tasks that language models can solve? ", + "bbox": [ + 173, + 166, + 825, + 279 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "As for limitations, we first qualify that although chain of thought emulates the thought processes of human reasoners, this does not answer whether the neural network is actually “reasoning,” which we leave as an open question. Second, although the cost of manually augmenting exemplars with chains of thought is minimal in the few-shot setting, such annotation costs could be prohibitive for finetuning (though this could potentially be surmounted with synthetic data generation, or zero-shot generalization). Third, there is no guarantee of correct reasoning paths, which can lead to both correct and incorrect answers; improving factual generations of language models is an open direction for future work (Rashkin et al., 2021; Ye and Durrett, 2022; Wiegreffe et al., 2022, inter alia). Finally, the emergence of chain-of-thought reasoning only at large model scales makes it costly to serve in real-world applications; further research could explore how to induce reasoning in smaller models. ", + "bbox": [ + 174, + 284, + 825, + 422 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 441, + 321, + 458 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This work is inspired by many research areas, which we detail in an extended related work section (Appendix C). Here we describe two directions and associated papers that are perhaps most relevant. ", + "bbox": [ + 176, + 473, + 825, + 501 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The first relevant direction is using intermediate steps to solve reasoning problems. Ling et al. (2017) pioneer the idea of using natural language rationales to solve math word problems through a series of intermediate steps. Their work is a remarkable contrast to the literature using formal languages to reason (Roy et al., 2015; Chiang and Chen, 2019; Amini et al., 2019; Chen et al., 2019). Cobbe et al. (2021) extend Ling et al. (2017) by creating a larger dataset and using it to finetune a pretrained language model rather than training a model from scratch. In the domain of program synthesis, Nye et al. (2021) leverage language models to predict the final outputs of Python programs via first line-to-line predicting the intermediate computational results, and show that their step-by-step prediction method performs better than directly predicting the final outputs. ", + "bbox": [ + 174, + 507, + 825, + 632 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Naturally, this paper also relates closely to the large body of recent work on prompting. Since the popularization of few-shot prompting as given by Brown et al. (2020), several general approaches have improved the prompting ability of models, such as automatically learning prompts (Lester et al., 2021) or giving models instructions describing a task (Wei et al., 2022a; Sanh et al., 2022; Ouyang et al., 2022). Whereas these approaches improve or augment the input part of the prompt (e.g., instructions that are prepended to inputs), our work takes the orthogonal direction of augmenting the outputs of language models with a chain of thought. ", + "bbox": [ + 174, + 638, + 825, + 736 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "8 Conclusions ", + "text_level": 1, + "bbox": [ + 174, + 755, + 307, + 772 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have explored chain-of-thought prompting as a simple and broadly applicable method for enhancing reasoning in language models. Through experiments on arithmetic, symbolic, and commonsense reasoning, we find that chain-of-thought reasoning is an emergent property of model scale that allows sufficiently large language models to perform reasoning tasks that otherwise have flat scaling curves. Broadening the range of reasoning tasks that language models can perform will hopefully inspire further work on language-based approaches to reasoning. ", + "bbox": [ + 174, + 786, + 825, + 869 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Acknowledgements ", + "text_level": 1, + "bbox": [ + 176, + 89, + 338, + 106 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We thank Jacob Devlin, Claire Cui, Andrew Dai, and Ellie Pavlick for providing feedback on the paper. We thank Jacob Austin, Yuhuai Wu, Henryk Michalewski, Aitor Lewkowycz, Charles Sutton, and Aakanksha Chowdhery for helpful discussions. We thank Sid Maxwell for notifying us about a mistake in the manual error analysis in the original manuscript. ", + "bbox": [ + 174, + 121, + 825, + 176 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References \nMichael Ahn, Anthony Brohan, Noah Brown, Yevgen Chebotar, Omar Cortes, Byron David, Chelsea Finn, Keerthana Gopalakrishnan, Karol Hausman, Alex Herzog, et al. 2022. 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However,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 337, + 175 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 337, + 175 + ], + "score": 1.0, + "content": "scaling up model size alone has not proved sufficient for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 175, + 336, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 336, + 187 + ], + "score": 1.0, + "content": "achieving high performance on challenging tasks such as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 186, + 336, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 336, + 198 + ], + "score": 1.0, + "content": "arithmetic, commonsense, and symbolic reasoning (Rae", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 196, + 159, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 159, + 209 + ], + "score": 1.0, + "content": "et al., 2021).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 336, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 337, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 337, + 227 + ], + "score": 1.0, + "content": "This work explores how the reasoning ability of large", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 225, + 336, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 336, + 237 + ], + "score": 1.0, + "content": "language models can be unlocked by a simple method", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 236, + 336, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 336, + 247 + ], + "score": 1.0, + "content": "motivated by two ideas. 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(2021).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 345, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "prospect of in-context few-shot learning via prompting. That is, instead of finetuning a separate", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 355, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 505, + 368 + ], + "score": 1.0, + "content": "language model checkpoint for each new task, one can simply “prompt” the model with a few", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "input–output exemplars demonstrating the task. Remarkably, this has been successful for a range of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 377, + 322, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 322, + 389 + ], + "score": 1.0, + "content": "simple question-answering tasks (Brown et al., 2020).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 394, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "Both of the above ideas, however, have key limitations. For rationale-augmented training and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "finetuning methods, it is costly to create a large set of high quality rationales, which is much more", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "complicated than simple input–output pairs used in normal machine learning. For the traditional few-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "shot prompting method used in Brown et al. 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In this paper, we combine the strengths of these two ideas in a way that avoids their limitations.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 458, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 473 + ], + "score": 1.0, + "content": "Specifically, we explore the ability of language models to perform few-shot prompting for reasoning", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "tasks, given a prompt that consists of triples: hinput, chain of thought, outputi. A chain of thought is", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "score": 1.0, + "content": "a series of intermediate natural language reasoning steps that lead to the final output, and we refer to", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 492, + 458, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 458, + 505 + ], + "score": 1.0, + "content": "this approach as chain-of-thought prompting. An example prompt is shown in Figure 1.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 106, + 508, + 505, + 607 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 506, + 520 + ], + "score": 1.0, + "content": "We present empirical evaluations on arithmetic, commonsense, and symbolic reasoning benchmarks,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 518, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 532 + ], + "score": 1.0, + "content": "showing that chain-of-thought prompting outperforms standard prompting, sometimes to a striking", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 529, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 543 + ], + "score": 1.0, + "content": "degree. Figure 2 illustrates one such result—on the GSM8K benchmark of math word problems", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 540, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 554 + ], + "score": 1.0, + "content": "(Cobbe et al., 2021), chain-of-thought prompting with PaLM 540B outperforms standard prompting", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "by a large margin and achieves new state-of-the-art performance. 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It is typical to decompose the problem into intermediate steps and solve each", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "before giving the final answer: “After Jane gives 2 flowers to her mom she has 10 . . . then after she", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 688, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 703 + ], + "score": 1.0, + "content": "gives 3 to her dad she will have 7 . . . so the answer is 7.” The goal of this paper is to endow language", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "models with the ability to generate a similar chain of thought—a coherent series of intermediate", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 710, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 725 + ], + "score": 1.0, + "content": "reasoning steps that lead to the final answer for a problem. 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Scaling up the size of lan-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 338, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 338, + 144 + ], + "score": 1.0, + "content": "guage models has been shown to confer a range of benefits,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 338, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 338, + 154 + ], + "score": 1.0, + "content": "such as improved performance and sample efficiency (Ka-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 338, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 338, + 166 + ], + "score": 1.0, + "content": "plan et al., 2020; Brown et al., 2020, inter alia). However,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 337, + 175 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 337, + 175 + ], + "score": 1.0, + "content": "scaling up model size alone has not proved sufficient for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 175, + 336, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 336, + 187 + ], + "score": 1.0, + "content": "achieving high performance on challenging tasks such as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 186, + 336, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 336, + 198 + ], + "score": 1.0, + "content": "arithmetic, commonsense, and symbolic reasoning (Rae", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 196, + 159, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 159, + 209 + ], + "score": 1.0, + "content": "et al., 2021).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 98, + 338, + 209 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 336, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 337, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 337, + 227 + ], + "score": 1.0, + "content": "This work explores how the reasoning ability of large", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 225, + 336, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 336, + 237 + ], + "score": 1.0, + "content": "language models can be unlocked by a simple method", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 236, + 336, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 336, + 247 + ], + "score": 1.0, + "content": "motivated by two ideas. 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Remarkably, this has been successful for a range of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 377, + 322, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 322, + 389 + ], + "score": 1.0, + "content": "simple question-answering tasks (Brown et al., 2020).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 344, + 505, + 389 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 394, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "Both of the above ideas, however, have key limitations. For rationale-augmented training and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "finetuning methods, it is costly to create a large set of high quality rationales, which is much more", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "complicated than simple input–output pairs used in normal machine learning. For the traditional few-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "shot prompting method used in Brown et al. 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In this paper, we combine the strengths of these two ideas in a way that avoids their limitations.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 458, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 473 + ], + "score": 1.0, + "content": "Specifically, we explore the ability of language models to perform few-shot prompting for reasoning", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "tasks, given a prompt that consists of triples: hinput, chain of thought, outputi. A chain of thought is", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "score": 1.0, + "content": "a series of intermediate natural language reasoning steps that lead to the final output, and we refer to", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 492, + 458, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 458, + 505 + ], + "score": 1.0, + "content": "this approach as chain-of-thought prompting. An example prompt is shown in Figure 1.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 393, + 506, + 505 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 508, + 505, + 607 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 506, + 520 + ], + "score": 1.0, + "content": "We present empirical evaluations on arithmetic, commonsense, and symbolic reasoning benchmarks,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 518, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 532 + ], + "score": 1.0, + "content": "showing that chain-of-thought prompting outperforms standard prompting, sometimes to a striking", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 529, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 543 + ], + "score": 1.0, + "content": "degree. Figure 2 illustrates one such result—on the GSM8K benchmark of math word problems", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 540, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 554 + ], + "score": 1.0, + "content": "(Cobbe et al., 2021), chain-of-thought prompting with PaLM 540B outperforms standard prompting", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "by a large margin and achieves new state-of-the-art performance. A prompting only approach is", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "score": 1.0, + "content": "important because it does not require a large training dataset and because a single model checkpoint", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "can perform many tasks without loss of generality. This work underscores how large language models", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 583, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 599 + ], + "score": 1.0, + "content": "can learn via a few examples with natural language data about the task (c.f. automatically learning", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 596, + 390, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 390, + 609 + ], + "score": 1.0, + "content": "the patterns underlying inputs and outputs via a large training dataset).", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 53, + "bbox_fs": [ + 105, + 509, + 506, + 609 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 627, + 277, + 642 + ], + "lines": [ + { + "bbox": [ + 104, + 625, + 279, + 646 + ], + "spans": [ + { + "bbox": [ + 104, + 625, + 279, + 646 + ], + "score": 1.0, + "content": "2 Chain-of-Thought Prompting", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 58 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 669 + ], + "score": 1.0, + "content": "Consider one’s own thought process when solving a complicated reasoning task such as a multi-step", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 668, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 679 + ], + "score": 1.0, + "content": "math word problem. It is typical to decompose the problem into intermediate steps and solve each", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "before giving the final answer: “After Jane gives 2 flowers to her mom she has 10 . . . then after she", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 688, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 703 + ], + "score": 1.0, + "content": "gives 3 to her dad she will have 7 . . . so the answer is 7.” The goal of this paper is to endow language", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "models with the ability to generate a similar chain of thought—a coherent series of intermediate", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 710, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 725 + ], + "score": 1.0, + "content": "reasoning steps that lead to the final answer for a problem. We will show that sufficiently large", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 71, + 505, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 505, + 87 + ], + "score": 1.0, + "content": "language models can generate chains of thought if demonstrations of chain-of-thought reasoning are", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 309, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 309, + 96 + ], + "score": 1.0, + "content": "provided in the exemplars for few-shot prompting.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 61.5, + "bbox_fs": [ + 105, + 656, + 506, + 725 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 505, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 505, + 87 + ], + "score": 1.0, + "content": "language models can generate chains of thought if demonstrations of chain-of-thought reasoning are", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 309, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 309, + 96 + ], + "score": 1.0, + "content": "provided in the exemplars for few-shot prompting.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 505, + 166 + ], + "lines": [ + { + "bbox": [ + 106, + 100, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 505, + 111 + ], + "score": 1.0, + "content": "Figure 1 shows an example of a model producing a chain of thought to solve a math word problem", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "that it would have otherwise gotten incorrect. The chain of thought in this case resembles a solution", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 122, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 506, + 133 + ], + "score": 1.0, + "content": "and can interpreted as one, but we still opt to call it a chain of thought to better capture the idea that it", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 133, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 505, + 144 + ], + "score": 1.0, + "content": "mimics a step-by-step thought process for arriving at the answer (and also, solutions/explanations", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "typically come after the final answer (Narang et al., 2020; Wiegreffe et al., 2022; Lampinen et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 153, + 180, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 180, + 167 + ], + "score": 1.0, + "content": "2022, inter alia)).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 171, + 504, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 169, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 185 + ], + "score": 1.0, + "content": "Chain-of-thought prompting has several attractive properties as an approach for facilitating reasoning", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 182, + 189, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 189, + 194 + ], + "score": 1.0, + "content": "in language models.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 129, + 199, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 130, + 199, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 130, + 199, + 505, + 211 + ], + "score": 1.0, + "content": "1. First, chain of thought, in principle, allows models to decompose multi-step problems into", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 141, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "intermediate steps, which means that additional computation can be allocated to problems", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 221, + 279, + 234 + ], + "spans": [ + { + "bbox": [ + 141, + 221, + 279, + 234 + ], + "score": 1.0, + "content": "that require more reasoning steps.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 128, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 128, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "2. Second, a chain of thought provides an interpretable window into the behavior of the model,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 141, + 245, + 506, + 258 + ], + "score": 1.0, + "content": "suggesting how it might have arrived at a particular answer and providing opportunities", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 257, + 506, + 270 + ], + "score": 1.0, + "content": "to debug where the reasoning path went wrong (although fully characterizing a model’s", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 268, + 401, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 401, + 279 + ], + "score": 1.0, + "content": "computations that support an answer remains an open question).", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 130, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 130, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "3. Third, chain-of-thought reasoning can be used for tasks such as math word problems,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 141, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "commonsense reasoning, and symbolic manipulation, and is potentially applicable (at least", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 302, + 386, + 316 + ], + "spans": [ + { + "bbox": [ + 141, + 302, + 386, + 316 + ], + "score": 1.0, + "content": "in principle) to any task that humans can solve via language.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 128, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 128, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "4. Finally, chain-of-thought reasoning can be readily elicited in sufficiently large off-the-shelf", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 141, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "language models simply by including examples of chain of thought sequences into the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 338, + 279, + 351 + ], + "spans": [ + { + "bbox": [ + 142, + 338, + 279, + 351 + ], + "score": 1.0, + "content": "exemplars of few-shot prompting.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 355, + 504, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 368 + ], + "score": 1.0, + "content": "In empirical experiments, we will observe the utility of chain-of-thought prompting for arithmetic", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 366, + 491, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 491, + 379 + ], + "score": 1.0, + "content": "reasoning (Section 3), commonsense reasoning (Section 4), and symbolic reasoning (Section 5).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 107, + 393, + 239, + 407 + ], + "lines": [ + { + "bbox": [ + 104, + 391, + 240, + 411 + ], + "spans": [ + { + "bbox": [ + 104, + 391, + 240, + 411 + ], + "score": 1.0, + "content": "3 Arithmetic Reasoning", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 505, + 485 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 504, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 504, + 431 + ], + "score": 1.0, + "content": "We begin by considering math word problems of the form in Figure 1, which measure the arithmetic", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "reasoning ability of language models. Though simple for humans, arithmetic reasoning is a task where", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "language models often struggle (Hendrycks et al., 2021; Patel et al., 2021, inter alia). Strikingly, chain-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "of-thought prompting when used with the 540B parameter language model performs comparably with", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 461, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 477 + ], + "score": 1.0, + "content": "task-specific finetuned models on several tasks, even achieving new state of the art on the challenging", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 474, + 274, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 274, + 485 + ], + "score": 1.0, + "content": "GSM8K benchmark (Cobbe et al., 2021).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 107, + 499, + 216, + 511 + ], + "lines": [ + { + "bbox": [ + 104, + 496, + 217, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 496, + 217, + 515 + ], + "score": 1.0, + "content": "3.1 Experimental Setup", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 105, + 519, + 483, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 484, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 484, + 533 + ], + "score": 1.0, + "content": "We explore chain-of-thought prompting for various language models on multiple benchmarks.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 547 + ], + "score": 1.0, + "content": "Benchmarks. We consider the following five math word problem benchmarks: (1) the GSM8K", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "benchmark of math word problems (Cobbe et al., 2021), (2) the SVAMP dataset of math word", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 558, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 569 + ], + "score": 1.0, + "content": "problems with varying structures (Patel et al., 2021), (3) the ASDiv dataset of diverse math word", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "problems (Miao et al., 2020), (4) the AQuA dataset of algebraic word problems, and (5) the MAWPS", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 578, + 495, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 495, + 592 + ], + "score": 1.0, + "content": "benchmark (Koncel-Kedziorski et al., 2016). Example problems are given in Appendix Table 12.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 596, + 504, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "Standard prompting. For the baseline, we consider standard few-shot prompting, popularized by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "Brown et al. (2020), in which a language model is given in-context exemplars of input–output pairs", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "before outputting a prediction for a test-time example. Exemplars are formatted as questions and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 630, + 402, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 402, + 641 + ], + "score": 1.0, + "content": "answers. The model gives the answer directly, as shown in Figure 1 (left).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "Chain-of-thought prompting. Our proposed approach is to augment each exemplar in few-shot", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "prompting with a chain of thought for an associated answer, as illustrated in Figure 1 (right). As most", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "of the datasets only have an evaluation split, we manually composed a set of eight few-shot exemplars", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "with chains of thought for prompting—Figure 1 (right) shows one chain of thought exemplar, and the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "full set of exemplars is given in Appendix Table 20. (These particular exemplars did not undergo", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "prompt engineering; robustness is studied in Section 3.4 and Appendix A.2.) To investigate whether", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "chain-of-thought prompting in this form can successfully elicit successful reasoning across a range of", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 504, + 95 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 71, + 505, + 96 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 505, + 166 + ], + "lines": [ + { + "bbox": [ + 106, + 100, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 505, + 111 + ], + "score": 1.0, + "content": "Figure 1 shows an example of a model producing a chain of thought to solve a math word problem", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "that it would have otherwise gotten incorrect. The chain of thought in this case resembles a solution", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 122, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 506, + 133 + ], + "score": 1.0, + "content": "and can interpreted as one, but we still opt to call it a chain of thought to better capture the idea that it", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 133, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 505, + 144 + ], + "score": 1.0, + "content": "mimics a step-by-step thought process for arriving at the answer (and also, solutions/explanations", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "typically come after the final answer (Narang et al., 2020; Wiegreffe et al., 2022; Lampinen et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 153, + 180, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 180, + 167 + ], + "score": 1.0, + "content": "2022, inter alia)).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 100, + 506, + 167 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 171, + 504, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 169, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 185 + ], + "score": 1.0, + "content": "Chain-of-thought prompting has several attractive properties as an approach for facilitating reasoning", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 182, + 189, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 189, + 194 + ], + "score": 1.0, + "content": "in language models.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 169, + 506, + 194 + ] + }, + { + "type": "list", + "bbox": [ + 129, + 199, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 130, + 199, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 130, + 199, + 505, + 211 + ], + "score": 1.0, + "content": "1. First, chain of thought, in principle, allows models to decompose multi-step problems into", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 141, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "intermediate steps, which means that additional computation can be allocated to problems", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 221, + 279, + 234 + ], + "spans": [ + { + "bbox": [ + 141, + 221, + 279, + 234 + ], + "score": 1.0, + "content": "that require more reasoning steps.", + "type": "text" + } + ], + "index": 12, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 128, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "2. Second, a chain of thought provides an interpretable window into the behavior of the model,", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 141, + 245, + 506, + 258 + ], + "score": 1.0, + "content": "suggesting how it might have arrived at a particular answer and providing opportunities", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 257, + 506, + 270 + ], + "score": 1.0, + "content": "to debug where the reasoning path went wrong (although fully characterizing a model’s", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 268, + 401, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 401, + 279 + ], + "score": 1.0, + "content": "computations that support an answer remains an open question).", + "type": "text" + } + ], + "index": 16, + "is_list_end_line": true + }, + { + "bbox": [ + 130, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 130, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "3. Third, chain-of-thought reasoning can be used for tasks such as math word problems,", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 141, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "commonsense reasoning, and symbolic manipulation, and is potentially applicable (at least", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 302, + 386, + 316 + ], + "spans": [ + { + "bbox": [ + 141, + 302, + 386, + 316 + ], + "score": 1.0, + "content": "in principle) to any task that humans can solve via language.", + "type": "text" + } + ], + "index": 19, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 128, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "4. Finally, chain-of-thought reasoning can be readily elicited in sufficiently large off-the-shelf", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 141, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "language models simply by including examples of chain of thought sequences into the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 338, + 279, + 351 + ], + "spans": [ + { + "bbox": [ + 142, + 338, + 279, + 351 + ], + "score": 1.0, + "content": "exemplars of few-shot prompting.", + "type": "text" + } + ], + "index": 22, + "is_list_end_line": true + } + ], + "index": 16, + "bbox_fs": [ + 128, + 199, + 506, + 351 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 355, + 504, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 368 + ], + "score": 1.0, + "content": "In empirical experiments, we will observe the utility of chain-of-thought prompting for arithmetic", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 366, + 491, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 491, + 379 + ], + "score": 1.0, + "content": "reasoning (Section 3), commonsense reasoning (Section 4), and symbolic reasoning (Section 5).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 354, + 506, + 379 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 393, + 239, + 407 + ], + "lines": [ + { + "bbox": [ + 104, + 391, + 240, + 411 + ], + "spans": [ + { + "bbox": [ + 104, + 391, + 240, + 411 + ], + "score": 1.0, + "content": "3 Arithmetic Reasoning", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 505, + 485 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 504, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 504, + 431 + ], + "score": 1.0, + "content": "We begin by considering math word problems of the form in Figure 1, which measure the arithmetic", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "reasoning ability of language models. Though simple for humans, arithmetic reasoning is a task where", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "language models often struggle (Hendrycks et al., 2021; Patel et al., 2021, inter alia). Strikingly, chain-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "of-thought prompting when used with the 540B parameter language model performs comparably with", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 461, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 477 + ], + "score": 1.0, + "content": "task-specific finetuned models on several tasks, even achieving new state of the art on the challenging", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 474, + 274, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 274, + 485 + ], + "score": 1.0, + "content": "GSM8K benchmark (Cobbe et al., 2021).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 419, + 506, + 485 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 499, + 216, + 511 + ], + "lines": [ + { + "bbox": [ + 104, + 496, + 217, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 496, + 217, + 515 + ], + "score": 1.0, + "content": "3.1 Experimental Setup", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 105, + 519, + 483, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 484, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 484, + 533 + ], + "score": 1.0, + "content": "We explore chain-of-thought prompting for various language models on multiple benchmarks.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 518, + 484, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 547 + ], + "score": 1.0, + "content": "Benchmarks. We consider the following five math word problem benchmarks: (1) the GSM8K", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "benchmark of math word problems (Cobbe et al., 2021), (2) the SVAMP dataset of math word", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 558, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 569 + ], + "score": 1.0, + "content": "problems with varying structures (Patel et al., 2021), (3) the ASDiv dataset of diverse math word", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "problems (Miao et al., 2020), (4) the AQuA dataset of algebraic word problems, and (5) the MAWPS", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 578, + 495, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 495, + 592 + ], + "score": 1.0, + "content": "benchmark (Koncel-Kedziorski et al., 2016). Example problems are given in Appendix Table 12.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 536, + 505, + 592 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 596, + 504, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "Standard prompting. For the baseline, we consider standard few-shot prompting, popularized by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "Brown et al. (2020), in which a language model is given in-context exemplars of input–output pairs", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "before outputting a prediction for a test-time example. Exemplars are formatted as questions and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 630, + 402, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 402, + 641 + ], + "score": 1.0, + "content": "answers. The model gives the answer directly, as shown in Figure 1 (left).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 595, + 506, + 641 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "Chain-of-thought prompting. Our proposed approach is to augment each exemplar in few-shot", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "prompting with a chain of thought for an associated answer, as illustrated in Figure 1 (right). As most", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "of the datasets only have an evaluation split, we manually composed a set of eight few-shot exemplars", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "with chains of thought for prompting—Figure 1 (right) shows one chain of thought exemplar, and the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "full set of exemplars is given in Appendix Table 20. (These particular exemplars did not undergo", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "prompt engineering; robustness is studied in Section 3.4 and Appendix A.2.) To investigate whether", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "chain-of-thought prompting in this form can successfully elicit successful reasoning across a range of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "math word problems, we used this single set of eight chain of thought exemplars for all benchmarks", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 459 + ], + "score": 1.0, + "content": "except AQuA, which is multiple choice instead of free response. For AQuA, we used four exemplars", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 457, + 374, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 374, + 469 + ], + "score": 1.0, + "content": "and solutions from the training set, as given in Appendix Table 21.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 645, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 68, + 505, + 385 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 68, + 505, + 385 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 68, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 108, + 68, + 505, + 385 + ], + "score": 0.971, + "type": "image", + "image_path": "89a5ee9245764678a62df985ae74ac1f355481b7fba3da340ff294060cdc69a1.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 68, + 505, + 173.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 173.66666666666669, + 505, + 279.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 279.33333333333337, + 505, + 385.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 394, + 505, + 418 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 394, + 504, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 504, + 407 + ], + "score": 1.0, + "content": "Figure 3: Examples of hinput, chain of thought, outputi triples for arithmetic, commonsense, and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 405, + 498, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 498, + 419 + ], + "score": 1.0, + "content": "symbolic reasoning benchmarks. Chains of thought are highlighted. Full prompts in Appendix G.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 506, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "math word problems, we used this single set of eight chain of thought exemplars for all benchmarks", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 459 + ], + "score": 1.0, + "content": "except AQuA, which is multiple choice instead of free response. For AQuA, we used four exemplars", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 457, + 374, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 374, + 469 + ], + "score": 1.0, + "content": "and solutions from the training set, as given in Appendix Table 21.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 474, + 506, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 474, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "Language models. We evaluate five large language models. The first is GPT-3 (Brown et al.,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "2020), for which we use text-ada-001, text-babbage-001, text-curie-001, and text-davinci-002, which", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "presumably correspond to InstructGPT models of 350M, 1.3B, 6.7B, and 175B parameters (Ouyang", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "et al., 2022).The second is LaMDA (Thoppilan et al., 2022), which has models of 422M, 2B, 8B,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "68B, and 137B parameters. The third is PaLM, which has models of 8B, 62B, and 540B parameters.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "The fourth is UL2 20B (Tay et al., 2022), and the fifth is Codex (Chen et al., 2021, code-davinci-002", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "in the OpenAI API). We sample from the models via greedy decoding (though follow-up work shows", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "score": 1.0, + "content": "chain-of-thought prompting can be improved by taking the majority final answer over many sampled", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "generations (Wang et al., 2022a)). For LaMDA, we report averaged results over five random seeds,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "where each seed had a different randomly shuffled order of exemplars. As LaMDA experiments", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "did not show large variance among different seeds, to save compute we report results for a single", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 595, + 252, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 252, + 605 + ], + "score": 1.0, + "content": "exemplar order for all other models.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 107, + 621, + 161, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 162, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 162, + 634 + ], + "score": 1.0, + "content": "3.2 Results", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 720 + ], + "lines": [ + { + "bbox": [ + 106, + 643, + 504, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 504, + 655 + ], + "score": 1.0, + "content": "The strongest results of chain-of-thought prompting are summarized in Figure 4, with all experimental", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 654, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 506, + 666 + ], + "score": 1.0, + "content": "outputs for each model collection, model size, and benchmark shown in Table 2 in the Appendix.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "score": 1.0, + "content": "There are three key takeaways. First, Figure 4 shows that chain-of-thought prompting is an emergent", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "ability of model scale (Wei et al., 2022b). That is, chain-of-thought prompting does not positively", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 686, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 699 + ], + "score": 1.0, + "content": "impact performance for small models, and only yields performance gains when used with models of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 696, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 137, + 708 + ], + "score": 0.84, + "content": "{ \\sim } 1 0 0 \\mathrm { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 696, + 506, + 712 + ], + "score": 1.0, + "content": "parameters. We qualitatively found that models of smaller scale produced fluent but illogical", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 707, + 402, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 402, + 722 + ], + "score": 1.0, + "content": "chains of thought, leading to lower performance than standard prompting.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 11, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 68, + 505, + 385 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 68, + 505, + 385 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 68, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 108, + 68, + 505, + 385 + ], + "score": 0.971, + "type": "image", + "image_path": "89a5ee9245764678a62df985ae74ac1f355481b7fba3da340ff294060cdc69a1.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 68, + 505, + 173.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 173.66666666666669, + 505, + 279.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 279.33333333333337, + 505, + 385.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 394, + 505, + 418 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 394, + 504, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 504, + 407 + ], + "score": 1.0, + "content": "Figure 3: Examples of hinput, chain of thought, outputi triples for arithmetic, commonsense, and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 405, + 498, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 498, + 419 + ], + "score": 1.0, + "content": "symbolic reasoning benchmarks. Chains of thought are highlighted. Full prompts in Appendix G.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 506, + 468 + ], + "lines": [], + "index": 6, + "bbox_fs": [ + 105, + 435, + 506, + 469 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 474, + 506, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 474, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "Language models. We evaluate five large language models. The first is GPT-3 (Brown et al.,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "2020), for which we use text-ada-001, text-babbage-001, text-curie-001, and text-davinci-002, which", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "presumably correspond to InstructGPT models of 350M, 1.3B, 6.7B, and 175B parameters (Ouyang", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "et al., 2022).The second is LaMDA (Thoppilan et al., 2022), which has models of 422M, 2B, 8B,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "68B, and 137B parameters. The third is PaLM, which has models of 8B, 62B, and 540B parameters.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "The fourth is UL2 20B (Tay et al., 2022), and the fifth is Codex (Chen et al., 2021, code-davinci-002", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "in the OpenAI API). We sample from the models via greedy decoding (though follow-up work shows", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "score": 1.0, + "content": "chain-of-thought prompting can be improved by taking the majority final answer over many sampled", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "generations (Wang et al., 2022a)). For LaMDA, we report averaged results over five random seeds,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "where each seed had a different randomly shuffled order of exemplars. As LaMDA experiments", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "did not show large variance among different seeds, to save compute we report results for a single", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 595, + 252, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 252, + 605 + ], + "score": 1.0, + "content": "exemplar order for all other models.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 474, + 506, + 605 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 621, + 161, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 162, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 162, + 634 + ], + "score": 1.0, + "content": "3.2 Results", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 720 + ], + "lines": [ + { + "bbox": [ + 106, + 643, + 504, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 504, + 655 + ], + "score": 1.0, + "content": "The strongest results of chain-of-thought prompting are summarized in Figure 4, with all experimental", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 654, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 506, + 666 + ], + "score": 1.0, + "content": "outputs for each model collection, model size, and benchmark shown in Table 2 in the Appendix.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "score": 1.0, + "content": "There are three key takeaways. First, Figure 4 shows that chain-of-thought prompting is an emergent", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "ability of model scale (Wei et al., 2022b). That is, chain-of-thought prompting does not positively", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 686, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 699 + ], + "score": 1.0, + "content": "impact performance for small models, and only yields performance gains when used with models of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 696, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 137, + 708 + ], + "score": 0.84, + "content": "{ \\sim } 1 0 0 \\mathrm { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 696, + 506, + 712 + ], + "score": 1.0, + "content": "parameters. We qualitatively found that models of smaller scale produced fluent but illogical", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 707, + 402, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 402, + 722 + ], + "score": 1.0, + "content": "chains of thought, leading to lower performance than standard prompting.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 643, + 506, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 297, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 298, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 298, + 85 + ], + "score": 1.0, + "content": "Second, chain-of-thought prompting has larger", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 298, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 298, + 95 + ], + "score": 1.0, + "content": "performance gains for more-complicated prob-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 298, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 298, + 106 + ], + "score": 1.0, + "content": "lems. For instance, for GSM8K (the dataset", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 105, + 298, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 298, + 117 + ], + "score": 1.0, + "content": "with the lowest baseline performance), perfor-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 298, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 298, + 127 + ], + "score": 1.0, + "content": "mance more than doubled for the largest GPT", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 298, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 298, + 138 + ], + "score": 1.0, + "content": "and PaLM models. On the other hand, for Sin-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 298, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 298, + 150 + ], + "score": 1.0, + "content": "gleOp, the easiest subset of MAWPS which only", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 298, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 298, + 161 + ], + "score": 1.0, + "content": "requires a single step to solve, performance im-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 297, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 297, + 172 + ], + "score": 1.0, + "content": "provements were either negative or very small", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 171, + 204, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 204, + 183 + ], + "score": 1.0, + "content": "(see Appendix Table 3).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 297, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 297, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 297, + 200 + ], + "score": 1.0, + "content": "Third, chain-of-thought prompting via GPT-3", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 297, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 297, + 211 + ], + "score": 1.0, + "content": "175B and PaLM 540B compares favorably to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 298, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 298, + 222 + ], + "score": 1.0, + "content": "prior state of the art, which typically finetunes a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 221, + 298, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 298, + 231 + ], + "score": 1.0, + "content": "task-specific model on a labeled training dataset.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 230, + 299, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 299, + 243 + ], + "score": 1.0, + "content": "Figure 4 shows how PaLM 540B uses chain-of-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 242, + 297, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 297, + 254 + ], + "score": 1.0, + "content": "thought prompting to achieve new state of the art", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 252, + 297, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 297, + 265 + ], + "score": 1.0, + "content": "on GSM8K, SVAMP, and MAWPS (though note", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 264, + 298, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 298, + 276 + ], + "score": 1.0, + "content": "that standard prompting already passed the prior", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 274, + 298, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 298, + 286 + ], + "score": 1.0, + "content": "best for SVAMP). On the other two datasets,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 285, + 297, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 297, + 297 + ], + "score": 1.0, + "content": "AQuA and ASDiv, PaLM with chain-of-thought", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 296, + 297, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 209, + 308 + ], + "score": 1.0, + "content": "prompting reaches within", + "type": "text" + }, + { + "bbox": [ + 210, + 297, + 225, + 307 + ], + "score": 0.85, + "content": "2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 296, + 297, + 308 + ], + "score": 1.0, + "content": "of the state of the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 307, + 202, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 202, + 319 + ], + "score": 1.0, + "content": "art (Appendix Table 2).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 324, + 297, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 323, + 297, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 297, + 336 + ], + "score": 1.0, + "content": "To better understand why chain-of-thought", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 335, + 298, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 298, + 347 + ], + "score": 1.0, + "content": "prompting works, we manually examined model-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 345, + 297, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 297, + 357 + ], + "score": 1.0, + "content": "generated chains of thought by LaMDA 137B", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 356, + 297, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 297, + 368 + ], + "score": 1.0, + "content": "for GSM8K. 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(2021) for MAWPS.", + "type": "text" + } + ], + "index": 68 + } + ], + "index": 65 + } + ], + "index": 57.0 + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 346, + 500 + ], + "score": 1.0, + "content": "ping error, or one reasoning step missing), and that the other", + "type": "text" + }, + { + "bbox": [ + 347, + 487, + 366, + 498 + ], + "score": 0.85, + "content": "54 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "of the chains of thought had major", + "type": "text" + } + ], + "index": 69 + }, + { + "bbox": [ + 106, + 499, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 505, + 510 + ], + "score": 1.0, + "content": "errors in semantic understanding or coherence (see Appendix D.2). 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On the other hand, for Sin-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 298, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 298, + 150 + ], + "score": 1.0, + "content": "gleOp, the easiest subset of MAWPS which only", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 298, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 298, + 161 + ], + "score": 1.0, + "content": "requires a single step to solve, performance im-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 297, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 297, + 172 + ], + "score": 1.0, + "content": "provements were either negative or very small", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 171, + 204, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 204, + 183 + ], + "score": 1.0, + "content": "(see Appendix Table 3).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 73, + 298, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 297, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 297, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 297, + 200 + ], + "score": 1.0, + "content": "Third, chain-of-thought prompting via GPT-3", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 297, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 297, + 211 + ], + "score": 1.0, + "content": "175B and PaLM 540B compares favorably to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 298, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 298, + 222 + ], + "score": 1.0, + "content": "prior state of the art, which typically finetunes a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 221, + 298, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 298, + 231 + ], + "score": 1.0, + "content": "task-specific model on a labeled training dataset.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 230, + 299, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 299, + 243 + ], + "score": 1.0, + "content": "Figure 4 shows how PaLM 540B uses chain-of-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 242, + 297, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 297, + 254 + ], + "score": 1.0, + "content": "thought prompting to achieve new state of the art", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 252, + 297, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 297, + 265 + ], + "score": 1.0, + "content": "on GSM8K, SVAMP, and MAWPS (though note", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 264, + 298, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 298, + 276 + ], + "score": 1.0, + "content": "that standard prompting already passed the prior", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 274, + 298, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 298, + 286 + ], + "score": 1.0, + "content": "best for SVAMP). 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Of 50 random examples where the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 367, + 298, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 298, + 379 + ], + "score": 1.0, + "content": "model returned the correct final answer, all of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 378, + 298, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 298, + 391 + ], + "score": 1.0, + "content": "the generated chains of thought were also log-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 389, + 298, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 298, + 402 + ], + "score": 1.0, + "content": "ically and mathematically correct except two", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 400, + 298, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 298, + 412 + ], + "score": 1.0, + "content": "that coincidentally arrived at the correct answer", + "type": "text" + } + ], + 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(2021) for MAWPS.", + "type": "text" + } + ], + "index": 68 + } + ], + "index": 65 + } + ], + "index": 57.0 + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 346, + 500 + ], + "score": 1.0, + "content": "ping error, or one reasoning step missing), and that the other", + "type": "text" + }, + { + "bbox": [ + 347, + 487, + 366, + 498 + ], + "score": 0.85, + "content": "54 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "of the chains of thought had major", + "type": "text" + } + ], + "index": 69 + }, + { + "bbox": [ + 106, + 499, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 505, + 510 + ], + "score": 1.0, + "content": "errors in semantic understanding or coherence (see Appendix D.2). To provide a small insight into", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 106, + 510, + 504, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 504, + 521 + ], + "score": 1.0, + "content": "why scaling improves chain-of-thought reasoning ability, we performed a similar analysis of errors", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "score": 1.0, + "content": "made by PaLM 62B and whether those errors were fixed by scaling to PaLM 540B. 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Figure 5 shows that equation", + "type": "text" + } + ], + "index": 81 + }, + { + "bbox": [ + 105, + 665, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 677 + ], + "score": 1.0, + "content": "only prompting does not help much for GSM8K, which implies that the semantics of the questions", + "type": "text" + } + ], + "index": 82 + }, + { + "bbox": [ + 104, + 675, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 104, + 675, + 505, + 689 + ], + "score": 1.0, + "content": "in GSM8K are too challenging to directly translate into an equation without the natural language", + "type": "text" + } + ], + "index": 83 + }, + { + "bbox": [ + 105, + 687, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 698 + ], + "score": 1.0, + "content": "reasoning steps in chain of thought. For datasets of one-step or two-step problems, however, we find", + "type": "text" + } + ], + "index": 84 + }, + { + "bbox": [ + 105, + 698, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 710 + ], + "score": 1.0, + "content": "that equation only prompting does improve performance, since the equation can be easily derived", + "type": "text" + } + ], + "index": 85 + }, + { + "bbox": [ + 105, + 708, + 278, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 278, + 721 + ], + "score": 1.0, + "content": "from the question (see Appendix Table 6).", + "type": "text" + } + ], + "index": 86 + } + ], + "index": 82.5, + "bbox_fs": [ + 104, + 631, + 505, + 721 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 357, + 193 + ], + "lines": [ + { + "bbox": [ + 107, + 73, + 357, + 84 + ], + "spans": [ + { + "bbox": [ + 107, + 73, + 357, + 84 + ], + "score": 1.0, + "content": "Variable compute only. Another intuition is that chain of", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 357, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 357, + 96 + ], + "score": 1.0, + "content": "thought allows the model to spend more computation (i.e.,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 94, + 357, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 357, + 106 + ], + "score": 1.0, + "content": "intermediate tokens) on harder problems. To isolate the effect", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 105, + 356, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 356, + 118 + ], + "score": 1.0, + "content": "of variable computation from chain-of-thought reasoning, we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 357, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 357, + 128 + ], + "score": 1.0, + "content": "test a configuration where the model is prompted to output a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 357, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 194, + 139 + ], + "score": 1.0, + "content": "only sequence of dots", + "type": "text" + }, + { + "bbox": [ + 194, + 128, + 213, + 138 + ], + "score": 0.25, + "content": "( \\ldots )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 127, + 357, + 139 + ], + "score": 1.0, + "content": "equal to the number of characters in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 356, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 356, + 150 + ], + "score": 1.0, + "content": "the equation needed to solve the problem. This variant performs", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 149, + 356, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 356, + 161 + ], + "score": 1.0, + "content": "about the same as the baseline, which suggests that variable", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 160, + 357, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 357, + 172 + ], + "score": 1.0, + "content": "computation by itself is not the reason for the success of chain-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 171, + 356, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 356, + 183 + ], + "score": 1.0, + "content": "of-thought prompting, and that there appears to be utility from", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 181, + 313, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 313, + 196 + ], + "score": 1.0, + "content": "expressing intermediate steps via natural language.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 357, + 308 + ], + "lines": [ + { + "bbox": [ + 107, + 198, + 357, + 210 + ], + "spans": [ + { + "bbox": [ + 107, + 198, + 357, + 210 + ], + "score": 1.0, + "content": "Chain of thought after answer. Another potential benefit of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 209, + 356, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 356, + 222 + ], + "score": 1.0, + "content": "chain-of-thought prompting could simply be that such prompts", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 219, + 357, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 357, + 232 + ], + "score": 1.0, + "content": "allow the model to better access relevant knowledge acquired", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 231, + 357, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 357, + 243 + ], + "score": 1.0, + "content": "during pretraining. Therefore, we test an alternative configura-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 241, + 357, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 357, + 254 + ], + "score": 1.0, + "content": "tion where the chain of thought prompt is only given after the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 253, + 356, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 356, + 264 + ], + "score": 1.0, + "content": "answer, isolating whether the model actually depends on the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 264, + 357, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 357, + 275 + ], + "score": 1.0, + "content": "produced chain of thought to give the final answer. 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In this final subsec-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 408, + 336, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 336, + 418 + ], + "score": 1.0, + "content": "tion, we evaluate robustness to chains of thought written", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 417, + 338, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 338, + 430 + ], + "score": 1.0, + "content": "by different annotators. 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exemplars (see Appendix A.2).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 108, + 160, + 256, + 174 + ], + "lines": [ + { + "bbox": [ + 104, + 157, + 258, + 177 + ], + "spans": [ + { + "bbox": [ + 104, + 157, + 258, + 177 + ], + "score": 1.0, + "content": "4 Commonsense Reasoning", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 185, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "score": 1.0, + "content": "Although chain of thought is particularly suitable for math word problems, the language-based nature", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 197, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 505, + 209 + ], + "score": 1.0, + "content": "of chain of thought actually makes it applicable to a broad class of commonsense reasoning problems,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "score": 1.0, + "content": "which involve reasoning about physical and human interactions under the presumption of general", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "background knowledge. Commonsense reasoning is key for interacting with the world and is still", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 466, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 466, + 242 + ], + "score": 1.0, + "content": "beyond the reach of current natural language understanding systems (Talmor et al., 2021).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 507, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 507, + 258 + ], + "score": 1.0, + "content": "Benchmarks. We consider five datasets covering a diverse range of commonsense reasoning types.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 255, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 505, + 270 + ], + "score": 1.0, + "content": "The popular CSQA (Talmor et al., 2019) asks commonsense questions about the world involving", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "complex semantics that often require prior knowledge. StrategyQA (Geva et al., 2021) requires", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "models to infer a multi-hop strategy to answer questions. We choose two specialized evaluation sets", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 505, + 302 + ], + "score": 1.0, + "content": "from the BIG-bench effort (BIG-bench collaboration, 2021): Date Understanding, which involves", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "inferring a date from a given context, and Sports Understanding, which involves determining whether", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "score": 1.0, + "content": "a sentence relating to sports is plausible or implausible. Finally, the SayCan dataset (Ahn et al.,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 335 + ], + "score": 1.0, + "content": "2022) involves mapping a natural language instruction to a sequence of robot actions from a discrete", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 333, + 425, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 425, + 345 + ], + "score": 1.0, + "content": "set. Figure 3 shows examples with chain of thought annotations for all datasets.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "Prompts. We follow the same experimental setup as the prior section. For CSQA and StrategyQA,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 360, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 371 + ], + "score": 1.0, + "content": "we randomly selected examples from the training set and manually composed chains of thought for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "them to use as few-shot exemplars. The two BIG-bench tasks do not have training sets, so we selected", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "the first ten examples as exemplars in the evaluation set as few-shot exemplars and report numbers on", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 393, + 507, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 507, + 405 + ], + "score": 1.0, + "content": "the rest of the evaluation set. For SayCan, we use six examples from the training set used in Ahn et al.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 403, + 327, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 327, + 416 + ], + "score": 1.0, + "content": "(2022) and also manually composed chains of thought.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 506, + 508 + ], + "lines": [ + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "Results. Figure 7 highlights these results for PaLM (full results for LaMDA, GPT-3, and different", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "model scales are shown in Table 4). For all tasks, scaling up model size improved the performance", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "of standard prompting; chain-of-thought prompting led to further gains, with improvements appear-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 104, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "ing to be largest for PaLM 540B. With chain-of-thought prompting, PaLM 540B achieved strong", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 464, + 465, + 477 + ], + "score": 1.0, + "content": "performance relative to baselines, outperforming the prior state of the art on StrategyQA", + "type": "text" + }, + { + "bbox": [ + 466, + 464, + 492, + 474 + ], + "score": 0.84, + "content": "7 5 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "vs", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 474, + 507, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 136, + 486 + ], + "score": 0.83, + "content": "6 9 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 474, + 438, + 488 + ], + "score": 1.0, + "content": "and outperforming an unaided sports enthusiast on sports understanding", + "type": "text" + }, + { + "bbox": [ + 438, + 475, + 466, + 486 + ], + "score": 0.83, + "content": "9 5 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 474, + 479, + 488 + ], + "score": 1.0, + "content": "vs", + "type": "text" + }, + { + "bbox": [ + 480, + 475, + 500, + 486 + ], + "score": 0.84, + "content": "84 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 474, + 507, + 488 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 485, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 506, + 498 + ], + "score": 1.0, + "content": "These results demonstrate that chain-of-thought prompting can also improve performance on tasks", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "requiring a range of commonsense reasoning abilities (though note that gain was minimal on CSQA).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "image", + "bbox": [ + 114, + 520, + 495, + 629 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 520, + 495, + 629 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 520, + 495, + 629 + ], + "spans": [ + { + "bbox": [ + 114, + 520, + 495, + 629 + ], + "score": 0.965, + "type": "image", + "image_path": "314d6a9f471c937800bcc79eed382244c146f9aacc68828ea5e504cb8128e362.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 114, + 520, + 495, + 556.3333333333334 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 114, + 556.3333333333334, + 495, + 592.6666666666667 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 114, + 592.6666666666667, + 495, + 629.0000000000001 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 639, + 505, + 695 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "Figure 7: Chain-of-thought prompting also improves the commonsense reasoning abilities of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 650, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 664 + ], + "score": 1.0, + "content": "language models. The language model shown here is PaLM. Prior best numbers are from the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 660, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 104, + 660, + 506, + 676 + ], + "score": 1.0, + "content": "leaderboards of CSQA (Talmor et al., 2019) and StrategyQA (Geva et al., 2021) (single-model only,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 672, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 506, + 685 + ], + "score": 1.0, + "content": "as of May 5, 2022). Additional results using various sizes of LaMDA, GPT-3, and PaLM are shown", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 683, + 151, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 151, + 695 + ], + "score": 1.0, + "content": "in Table 4.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + } + ], + "index": 38.0 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 105, + 701, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 699, + 504, + 714 + ], + "spans": [ + { + "bbox": [ + 118, + 699, + 200, + 714 + ], + "score": 1.0, + "content": "2We sample examples", + "type": "text" + }, + { + "bbox": [ + 200, + 702, + 221, + 712 + ], + "score": 0.84, + "content": "\\leq 6 0", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 699, + 488, + 714 + ], + "score": 1.0, + "content": "tokens to fit into our input context window, and also limit the examples to", + "type": "text" + }, + { + "bbox": [ + 488, + 702, + 504, + 712 + ], + "score": 0.82, + "content": "\\leq 2", + "type": "inline_equation" + } + ] + }, + { + "bbox": [ + 106, + 711, + 394, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 394, + 722 + ], + "score": 1.0, + "content": "steps to solve for a fair comparison with the eight exemplars that we composed.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 106 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 104, + 71, + 506, + 108 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 111, + 504, + 144 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "score": 1.0, + "content": "In addition to robustness to annotators, independently-written chains of thought, different exemplars,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 120, + 506, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 506, + 136 + ], + "score": 1.0, + "content": "and various language models, we also find that chain-of-thought prompting for arithmetic reasoning", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 133, + 479, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 479, + 145 + ], + "score": 1.0, + "content": "is robust to different exemplar orders and varying numbers of exemplars (see Appendix A.2).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 110, + 506, + 145 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 160, + 256, + 174 + ], + "lines": [ + { + "bbox": [ + 104, + 157, + 258, + 177 + ], + "spans": [ + { + "bbox": [ + 104, + 157, + 258, + 177 + ], + "score": 1.0, + "content": "4 Commonsense Reasoning", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 185, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "score": 1.0, + "content": "Although chain of thought is particularly suitable for math word problems, the language-based nature", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 197, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 505, + 209 + ], + "score": 1.0, + "content": "of chain of thought actually makes it applicable to a broad class of commonsense reasoning problems,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "score": 1.0, + "content": "which involve reasoning about physical and human interactions under the presumption of general", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "background knowledge. Commonsense reasoning is key for interacting with the world and is still", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 466, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 466, + 242 + ], + "score": 1.0, + "content": "beyond the reach of current natural language understanding systems (Talmor et al., 2021).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 185, + 505, + 242 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 507, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 507, + 258 + ], + "score": 1.0, + "content": "Benchmarks. We consider five datasets covering a diverse range of commonsense reasoning types.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 255, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 505, + 270 + ], + "score": 1.0, + "content": "The popular CSQA (Talmor et al., 2019) asks commonsense questions about the world involving", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "complex semantics that often require prior knowledge. StrategyQA (Geva et al., 2021) requires", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "models to infer a multi-hop strategy to answer questions. We choose two specialized evaluation sets", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 505, + 302 + ], + "score": 1.0, + "content": "from the BIG-bench effort (BIG-bench collaboration, 2021): Date Understanding, which involves", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "inferring a date from a given context, and Sports Understanding, which involves determining whether", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "score": 1.0, + "content": "a sentence relating to sports is plausible or implausible. Finally, the SayCan dataset (Ahn et al.,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 335 + ], + "score": 1.0, + "content": "2022) involves mapping a natural language instruction to a sequence of robot actions from a discrete", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 333, + 425, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 425, + 345 + ], + "score": 1.0, + "content": "set. Figure 3 shows examples with chain of thought annotations for all datasets.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 245, + 507, + 345 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "Prompts. We follow the same experimental setup as the prior section. For CSQA and StrategyQA,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 360, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 371 + ], + "score": 1.0, + "content": "we randomly selected examples from the training set and manually composed chains of thought for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "them to use as few-shot exemplars. The two BIG-bench tasks do not have training sets, so we selected", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "the first ten examples as exemplars in the evaluation set as few-shot exemplars and report numbers on", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 393, + 507, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 507, + 405 + ], + "score": 1.0, + "content": "the rest of the evaluation set. For SayCan, we use six examples from the training set used in Ahn et al.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 403, + 327, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 327, + 416 + ], + "score": 1.0, + "content": "(2022) and also manually composed chains of thought.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 349, + 507, + 416 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 506, + 508 + ], + "lines": [ + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "Results. Figure 7 highlights these results for PaLM (full results for LaMDA, GPT-3, and different", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "model scales are shown in Table 4). For all tasks, scaling up model size improved the performance", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "of standard prompting; chain-of-thought prompting led to further gains, with improvements appear-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 104, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "ing to be largest for PaLM 540B. With chain-of-thought prompting, PaLM 540B achieved strong", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 464, + 465, + 477 + ], + "score": 1.0, + "content": "performance relative to baselines, outperforming the prior state of the art on StrategyQA", + "type": "text" + }, + { + "bbox": [ + 466, + 464, + 492, + 474 + ], + "score": 0.84, + "content": "7 5 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "vs", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 474, + 507, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 136, + 486 + ], + "score": 0.83, + "content": "6 9 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 474, + 438, + 488 + ], + "score": 1.0, + "content": "and outperforming an unaided sports enthusiast on sports understanding", + "type": "text" + }, + { + "bbox": [ + 438, + 475, + 466, + 486 + ], + "score": 0.83, + "content": "9 5 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 474, + 479, + 488 + ], + "score": 1.0, + "content": "vs", + "type": "text" + }, + { + "bbox": [ + 480, + 475, + 500, + 486 + ], + "score": 0.84, + "content": "84 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 474, + 507, + 488 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 485, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 506, + 498 + ], + "score": 1.0, + "content": "These results demonstrate that chain-of-thought prompting can also improve performance on tasks", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "requiring a range of commonsense reasoning abilities (though note that gain was minimal on CSQA).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 420, + 507, + 509 + ] + }, + { + "type": "image", + "bbox": [ + 114, + 520, + 495, + 629 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 520, + 495, + 629 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 520, + 495, + 629 + ], + "spans": [ + { + "bbox": [ + 114, + 520, + 495, + 629 + ], + "score": 0.965, + "type": "image", + "image_path": "314d6a9f471c937800bcc79eed382244c146f9aacc68828ea5e504cb8128e362.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 114, + 520, + 495, + 556.3333333333334 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 114, + 556.3333333333334, + 495, + 592.6666666666667 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 114, + 592.6666666666667, + 495, + 629.0000000000001 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 639, + 505, + 695 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "Figure 7: Chain-of-thought prompting also improves the commonsense reasoning abilities of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 650, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 664 + ], + "score": 1.0, + "content": "language models. The language model shown here is PaLM. Prior best numbers are from the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 660, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 104, + 660, + 506, + 676 + ], + "score": 1.0, + "content": "leaderboards of CSQA (Talmor et al., 2019) and StrategyQA (Geva et al., 2021) (single-model only,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 672, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 506, + 685 + ], + "score": 1.0, + "content": "as of May 5, 2022). Additional results using various sizes of LaMDA, GPT-3, and PaLM are shown", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 683, + 151, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 151, + 695 + ], + "score": 1.0, + "content": "in Table 4.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + } + ], + "index": 38.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 229, + 85 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 231, + 88 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 231, + 88 + ], + "score": 1.0, + "content": "5 Symbolic Reasoning", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 336, + 183 + ], + "lines": [ + { + "bbox": [ + 106, + 95, + 338, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 338, + 108 + ], + "score": 1.0, + "content": "Our final experimental evaluation considers symbolic rea-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 338, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 338, + 118 + ], + "score": 1.0, + "content": "soning, which is simple for humans but potentially chal-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 339, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 339, + 129 + ], + "score": 1.0, + "content": "lenging for language models. We show that chain-of-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 128, + 337, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 337, + 140 + ], + "score": 1.0, + "content": "thought prompting not only enables language models to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 140, + 337, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 337, + 151 + ], + "score": 1.0, + "content": "perform symbolic reasoning tasks that are challenging in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 150, + 337, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 337, + 162 + ], + "score": 1.0, + "content": "the standard prompting setting, but also facilitates length", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 162, + 336, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 336, + 173 + ], + "score": 1.0, + "content": "generalization to inference-time inputs longer than those", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 171, + 235, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 235, + 185 + ], + "score": 1.0, + "content": "seen in the few-shot exemplars.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 195, + 286, + 207 + ], + "lines": [ + { + "bbox": [ + 106, + 193, + 287, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 287, + 209 + ], + "score": 1.0, + "content": "Tasks. We use the following two toy tasks.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 212, + 336, + 299 + ], + "lines": [ + { + "bbox": [ + 107, + 212, + 336, + 223 + ], + "spans": [ + { + "bbox": [ + 107, + 212, + 336, + 223 + ], + "score": 1.0, + "content": "• Last letter concatenation. This task asks the model", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 114, + 221, + 338, + 236 + ], + "spans": [ + { + "bbox": [ + 114, + 221, + 338, + 236 + ], + "score": 1.0, + "content": "to concatenate the last letters of words in a name (e.g.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 113, + 233, + 337, + 245 + ], + "spans": [ + { + "bbox": [ + 113, + 233, + 169, + 245 + ], + "score": 1.0, + "content": "“Amy Brown”", + "type": "text" + }, + { + "bbox": [ + 169, + 234, + 205, + 245 + ], + "score": 0.31, + "content": " ^ { \\ast } y n ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 233, + 337, + 245 + ], + "score": 1.0, + "content": "). 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This task asks the model to answer whether", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 115, + 312, + 337, + 324 + ], + "spans": [ + { + "bbox": [ + 115, + 312, + 337, + 324 + ], + "score": 1.0, + "content": "a coin is still heads up after people either flip or don’t", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 116, + 322, + 337, + 335 + ], + "spans": [ + { + "bbox": [ + 116, + 322, + 337, + 335 + ], + "score": 1.0, + "content": "flip the coin (e.g., “A coin is heads up. Phoebe flips the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 116, + 333, + 337, + 345 + ], + "spans": [ + { + "bbox": [ + 116, + 333, + 337, + 345 + ], + "score": 1.0, + "content": "coin. Osvaldo does not flip the coin. Is the coin still", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 116, + 344, + 208, + 357 + ], + "spans": [ + { + "bbox": [ + 116, + 344, + 155, + 357 + ], + "score": 1.0, + "content": "heads up?", + "type": "text" + }, + { + "bbox": [ + 155, + 345, + 200, + 356 + ], + "score": 0.72, + "content": "^ { \\prime \\prime } \\right. ^ { \\left. } n o ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 344, + 208, + 357 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 336, + 394 + ], + "lines": [ + { + "bbox": [ + 106, + 361, + 337, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 337, + 373 + ], + "score": 1.0, + "content": "As the construction of these symbolic reasoning tasks is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 371, + 337, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 337, + 383 + ], + "score": 1.0, + "content": "well-defined, for each task we consider an in-domain test", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 382, + 337, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 337, + 395 + ], + "score": 1.0, + "content": "set for which examples had the same number of steps as", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 394, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "score": 1.0, + "content": "the training/few-shot exemplars, as well as an out-of-domain (OOD) test set, for which evaluation", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "score": 1.0, + "content": "examples had more steps than those in the exemplars. For last letter concatenation, the model only", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "sees exemplars of names with two words, and then performs last letter concatenation on names with 3", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "score": 1.0, + "content": "and 4 words.4 We do the same for the number of potential flips in the coin flip task. Our experimental", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "setup uses the same methods and models as in the prior two sections. We again manually compose", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 448, + 458, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 458, + 461 + ], + "score": 1.0, + "content": "chains of thought for the few-shot exemplars for each task, which are given in Figure 3.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "Results. The results of these in-domain and OOD evaluations are shown in Figure 8 for PaLM,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 104, + 480, + 506, + 495 + ], + "score": 1.0, + "content": "with results for LaMDA shown in Appendix Table 5. With PaLM 540B, chain-of-thought prompting", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 169, + 505 + ], + "score": 1.0, + "content": "leads to almost", + "type": "text" + }, + { + "bbox": [ + 170, + 493, + 194, + 504 + ], + "score": 0.87, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "solve rates (note that standard prompting already solves coin flip with PaLM", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 505, + 504, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 504, + 515 + ], + "score": 1.0, + "content": "540, though not for LaMDA 137B). Note that these in-domain evaluations are “toy tasks” in the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "sense that perfect solution structures are already provided by the chains of thought in the few-shot", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 525, + 504, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 504, + 537 + ], + "score": 1.0, + "content": "exemplars; all the model has to do is repeat the same steps with the new symbols in the test-time", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "example. And yet, small models still fail—the ability to perform abstract manipulations on unseen", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 548, + 427, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 427, + 560 + ], + "score": 1.0, + "content": "symbols for these three tasks only arises at the scale of 100B model parameters.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 507, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 507, + 577 + ], + "score": 1.0, + "content": "As for the OOD evaluations, standard prompting fails for both tasks. With chain-of-thought prompting,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "language models achieve upward scaling curves (though performance is lower than in the in-domain", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 586, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 598 + ], + "score": 1.0, + "content": "setting). Hence, chain-of-thought prompting facilitates length generalization beyond seen chains of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 596, + 299, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 299, + 609 + ], + "score": 1.0, + "content": "thought for language models of sufficient scale.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5 + }, + { + "type": "title", + "bbox": [ + 106, + 623, + 179, + 636 + ], + "lines": [ + { + "bbox": [ + 104, + 621, + 181, + 639 + ], + "spans": [ + { + "bbox": [ + 104, + 621, + 181, + 639 + ], + "score": 1.0, + "content": "6 Discussion", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 108, + 648, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "We have explored chain-of-thought prompting as a simple mechanism for eliciting multi-step rea-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "soning behavior in large language models. We first saw that chain-of-thought prompting improves", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "score": 1.0, + "content": "performance by a large margin on arithmetic reasoning, yielding improvements that are much stronger", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 680, + 507, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 507, + 694 + ], + "score": 1.0, + "content": "than ablations and robust to different annotators, exemplars, and language models (Section 3). Next,", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 51.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 118, + 700, + 469, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 698, + 419, + 714 + ], + "spans": [ + { + "bbox": [ + 118, + 698, + 419, + 714 + ], + "score": 1.0, + "content": "3We tested 10 common names using GPT-3 davinci and it got all but one correct.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 709, + 467, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 467, + 724 + ], + "score": 1.0, + "content": "4For names of length longer than 2 words, we concatenate multiple first and last names together.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 229, + 85 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 231, + 88 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 231, + 88 + ], + "score": 1.0, + "content": "5 Symbolic Reasoning", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 336, + 183 + ], + "lines": [ + { + "bbox": [ + 106, + 95, + 338, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 338, + 108 + ], + "score": 1.0, + "content": "Our final experimental evaluation considers symbolic rea-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 338, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 338, + 118 + ], + "score": 1.0, + "content": "soning, which is simple for humans but potentially chal-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 339, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 339, + 129 + ], + "score": 1.0, + "content": "lenging for language models. We show that chain-of-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 128, + 337, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 337, + 140 + ], + "score": 1.0, + "content": "thought prompting not only enables language models to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 140, + 337, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 337, + 151 + ], + "score": 1.0, + "content": "perform symbolic reasoning tasks that are challenging in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 150, + 337, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 337, + 162 + ], + "score": 1.0, + "content": "the standard prompting setting, but also facilitates length", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 162, + 336, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 336, + 173 + ], + "score": 1.0, + "content": "generalization to inference-time inputs longer than those", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 171, + 235, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 235, + 185 + ], + "score": 1.0, + "content": "seen in the few-shot exemplars.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 95, + 339, + 185 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 195, + 286, + 207 + ], + "lines": [ + { + "bbox": [ + 106, + 193, + 287, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 287, + 209 + ], + "score": 1.0, + "content": "Tasks. We use the following two toy tasks.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 193, + 287, + 209 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 212, + 336, + 299 + ], + "lines": [ + { + "bbox": [ + 107, + 212, + 336, + 223 + ], + "spans": [ + { + "bbox": [ + 107, + 212, + 336, + 223 + ], + "score": 1.0, + "content": "• Last letter concatenation. This task asks the model", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 114, + 221, + 338, + 236 + ], + "spans": [ + { + "bbox": [ + 114, + 221, + 338, + 236 + ], + "score": 1.0, + "content": "to concatenate the last letters of words in a name (e.g.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 113, + 233, + 337, + 245 + ], + "spans": [ + { + "bbox": [ + 113, + 233, + 169, + 245 + ], + "score": 1.0, + "content": "“Amy Brown”", + "type": "text" + }, + { + "bbox": [ + 169, + 234, + 205, + 245 + ], + "score": 0.31, + "content": " ^ { \\ast } y n ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 233, + 337, + 245 + ], + "score": 1.0, + "content": "). It is a more challenging version", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 116, + 244, + 336, + 256 + ], + "spans": [ + { + "bbox": [ + 116, + 244, + 336, + 256 + ], + "score": 1.0, + "content": "of first letter concatenation, which language models can", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 115, + 255, + 337, + 267 + ], + "spans": [ + { + "bbox": [ + 115, + 255, + 337, + 267 + ], + "score": 1.0, + "content": "already perform without chain of thought.3 We generate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 115, + 265, + 337, + 278 + ], + "spans": [ + { + "bbox": [ + 115, + 265, + 337, + 278 + ], + "score": 1.0, + "content": "full names by randomly concatenating names from the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 115, + 276, + 338, + 289 + ], + "spans": [ + { + "bbox": [ + 115, + 276, + 338, + 289 + ], + "score": 1.0, + "content": "top one-thousand first and last names from name census", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 116, + 288, + 255, + 299 + ], + "spans": [ + { + "bbox": [ + 116, + 288, + 255, + 299 + ], + "score": 1.0, + "content": "data (https://namecensus.com/).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5, + "bbox_fs": [ + 107, + 212, + 338, + 299 + ] + }, + { + "type": "image", + "bbox": [ + 348, + 75, + 498, + 325 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 348, + 75, + 498, + 325 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 348, + 75, + 498, + 325 + ], + "spans": [ + { + "bbox": [ + 348, + 75, + 498, + 325 + ], + "score": 0.966, + "type": "image", + "image_path": "e4f8153620820f36adc177bd4972c7b26a10341d1dec25b01e73852fd72bc7ae.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 348, + 75, + 498, + 325 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 344, + 336, + 505, + 381 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 343, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 343, + 336, + 388, + 349 + ], + "score": 1.0, + "content": "Figure 8:", + "type": "text" + }, + { + "bbox": [ + 403, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "Using chain-of-thought", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 343, + 348, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 343, + 348, + 505, + 359 + ], + "score": 1.0, + "content": "prompting facilitates generalization to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 343, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 343, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "longer sequences in two symbolic rea-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 343, + 370, + 398, + 382 + ], + "spans": [ + { + "bbox": [ + 343, + 370, + 398, + 382 + ], + "score": 1.0, + "content": "soning tasks.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + } + ], + "index": 14.75 + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 336, + 356 + ], + "lines": [ + { + "bbox": [ + 107, + 300, + 338, + 313 + ], + "spans": [ + { + "bbox": [ + 107, + 300, + 338, + 313 + ], + "score": 1.0, + "content": "• Coin flip. This task asks the model to answer whether", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 115, + 312, + 337, + 324 + ], + "spans": [ + { + "bbox": [ + 115, + 312, + 337, + 324 + ], + "score": 1.0, + "content": "a coin is still heads up after people either flip or don’t", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 116, + 322, + 337, + 335 + ], + "spans": [ + { + "bbox": [ + 116, + 322, + 337, + 335 + ], + "score": 1.0, + "content": "flip the coin (e.g., “A coin is heads up. Phoebe flips the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 116, + 333, + 337, + 345 + ], + "spans": [ + { + "bbox": [ + 116, + 333, + 337, + 345 + ], + "score": 1.0, + "content": "coin. Osvaldo does not flip the coin. Is the coin still", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 116, + 344, + 208, + 357 + ], + "spans": [ + { + "bbox": [ + 116, + 344, + 155, + 357 + ], + "score": 1.0, + "content": "heads up?", + "type": "text" + }, + { + "bbox": [ + 155, + 345, + 200, + 356 + ], + "score": 0.72, + "content": "^ { \\prime \\prime } \\right. ^ { \\left. } n o ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 344, + 208, + 357 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 107, + 300, + 338, + 357 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 336, + 394 + ], + "lines": [ + { + "bbox": [ + 106, + 361, + 337, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 337, + 373 + ], + "score": 1.0, + "content": "As the construction of these symbolic reasoning tasks is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 371, + 337, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 337, + 383 + ], + "score": 1.0, + "content": "well-defined, for each task we consider an in-domain test", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 382, + 337, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 337, + 395 + ], + "score": 1.0, + "content": "set for which examples had the same number of steps as", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "score": 1.0, + "content": "the training/few-shot exemplars, as well as an out-of-domain (OOD) test set, for which evaluation", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "score": 1.0, + "content": "examples had more steps than those in the exemplars. For last letter concatenation, the model only", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "sees exemplars of names with two words, and then performs last letter concatenation on names with 3", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "score": 1.0, + "content": "and 4 words.4 We do the same for the number of potential flips in the coin flip task. Our experimental", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "setup uses the same methods and models as in the prior two sections. We again manually compose", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 448, + 458, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 458, + 461 + ], + "score": 1.0, + "content": "chains of thought for the few-shot exemplars for each task, which are given in Figure 3.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 361, + 337, + 395 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 394, + 505, + 459 + ], + "lines": [], + "index": 33.5, + "bbox_fs": [ + 105, + 394, + 506, + 461 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "Results. The results of these in-domain and OOD evaluations are shown in Figure 8 for PaLM,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 104, + 480, + 506, + 495 + ], + "score": 1.0, + "content": "with results for LaMDA shown in Appendix Table 5. With PaLM 540B, chain-of-thought prompting", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 169, + 505 + ], + "score": 1.0, + "content": "leads to almost", + "type": "text" + }, + { + "bbox": [ + 170, + 493, + 194, + 504 + ], + "score": 0.87, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "solve rates (note that standard prompting already solves coin flip with PaLM", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 505, + 504, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 504, + 515 + ], + "score": 1.0, + "content": "540, though not for LaMDA 137B). Note that these in-domain evaluations are “toy tasks” in the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "sense that perfect solution structures are already provided by the chains of thought in the few-shot", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 525, + 504, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 504, + 537 + ], + "score": 1.0, + "content": "exemplars; all the model has to do is repeat the same steps with the new symbols in the test-time", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "example. And yet, small models still fail—the ability to perform abstract manipulations on unseen", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 548, + 427, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 427, + 560 + ], + "score": 1.0, + "content": "symbols for these three tasks only arises at the scale of 100B model parameters.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 471, + 506, + 560 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 507, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 507, + 577 + ], + "score": 1.0, + "content": "As for the OOD evaluations, standard prompting fails for both tasks. With chain-of-thought prompting,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "language models achieve upward scaling curves (though performance is lower than in the in-domain", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 586, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 598 + ], + "score": 1.0, + "content": "setting). Hence, chain-of-thought prompting facilitates length generalization beyond seen chains of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 596, + 299, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 299, + 609 + ], + "score": 1.0, + "content": "thought for language models of sufficient scale.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 561, + 507, + 609 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 623, + 179, + 636 + ], + "lines": [ + { + "bbox": [ + 104, + 621, + 181, + 639 + ], + "spans": [ + { + "bbox": [ + 104, + 621, + 181, + 639 + ], + "score": 1.0, + "content": "6 Discussion", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 108, + 648, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "We have explored chain-of-thought prompting as a simple mechanism for eliciting multi-step rea-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "soning behavior in large language models. We first saw that chain-of-thought prompting improves", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "score": 1.0, + "content": "performance by a large margin on arithmetic reasoning, yielding improvements that are much stronger", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 680, + 507, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 507, + 694 + ], + "score": 1.0, + "content": "than ablations and robust to different annotators, exemplars, and language models (Section 3). Next,", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "experiments on commonsense reasoning underscored how the linguistic nature of chain-of-thought", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 82, + 507, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 507, + 98 + ], + "score": 1.0, + "content": "reasoning makes it generally applicable (Section 4). Finally, we showed that for symbolic reasoning,", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "chain-of-thought prompting facilitates OOD generalization to longer sequence lengths (Section 5). In", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 104, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 120 + ], + "score": 1.0, + "content": "all experiments, chain-of-thought reasoning is elicited simply by prompting an off-the-shelf language", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 425, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 425, + 129 + ], + "score": 1.0, + "content": "model. No language models were finetuned in the process of writing this paper.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 51.5, + "bbox_fs": [ + 105, + 648, + 507, + 694 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "experiments on commonsense reasoning underscored how the linguistic nature of chain-of-thought", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 82, + 507, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 507, + 98 + ], + "score": 1.0, + "content": "reasoning makes it generally applicable (Section 4). Finally, we showed that for symbolic reasoning,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "chain-of-thought prompting facilitates OOD generalization to longer sequence lengths (Section 5). In", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 104, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 120 + ], + "score": 1.0, + "content": "all experiments, chain-of-thought reasoning is elicited simply by prompting an off-the-shelf language", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 425, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 425, + 129 + ], + "score": 1.0, + "content": "model. No language models were finetuned in the process of writing this paper.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "The emergence of chain-of-thought reasoning as a result of model scale has been a prevailing theme", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "(Wei et al., 2022b). For many reasoning tasks where standard prompting has a flat scaling curve, chain-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "score": 1.0, + "content": "of-thought prompting leads to dramatically increasing scaling curves. Chain-of-thought prompting", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "appears to expand the set of tasks that large language models can perform successfully—in other", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "words, our work underscores that standard prompting only provides a lower bound on the capabilities", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "of large language models. This observation likely raises more questions than it answers—for instance,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "how much more can we expect reasoning ability to improve with a further increase in model scale?", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 208, + 494, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 494, + 222 + ], + "score": 1.0, + "content": "What other prompting methods might expand the range of tasks that language models can solve?", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 225, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 236 + ], + "score": 1.0, + "content": "As for limitations, we first qualify that although chain of thought emulates the thought processes of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "human reasoners, this does not answer whether the neural network is actually “reasoning,” which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "we leave as an open question. Second, although the cost of manually augmenting exemplars with", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "score": 1.0, + "content": "chains of thought is minimal in the few-shot setting, such annotation costs could be prohibitive for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "finetuning (though this could potentially be surmounted with synthetic data generation, or zero-shot", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "score": 1.0, + "content": "generalization). Third, there is no guarantee of correct reasoning paths, which can lead to both correct", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 291, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 506, + 303 + ], + "score": 1.0, + "content": "and incorrect answers; improving factual generations of language models is an open direction for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "score": 1.0, + "content": "future work (Rashkin et al., 2021; Ye and Durrett, 2022; Wiegreffe et al., 2022, inter alia). Finally,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "the emergence of chain-of-thought reasoning only at large model scales makes it costly to serve in", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 324, + 502, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 502, + 336 + ], + "score": 1.0, + "content": "real-world applications; further research could explore how to induce reasoning in smaller models.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 107, + 350, + 197, + 363 + ], + "lines": [ + { + "bbox": [ + 105, + 349, + 198, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 198, + 365 + ], + "score": 1.0, + "content": "7 Related Work", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 108, + 375, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "This work is inspired by many research areas, which we detail in an extended related work section", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 506, + 399 + ], + "score": 1.0, + "content": "(Appendix C). Here we describe two directions and associated papers that are perhaps most relevant.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "The first relevant direction is using intermediate steps to solve reasoning problems. Ling et al. (2017)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "pioneer the idea of using natural language rationales to solve math word problems through a series", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "of intermediate steps. Their work is a remarkable contrast to the literature using formal languages", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "to reason (Roy et al., 2015; Chiang and Chen, 2019; Amini et al., 2019; Chen et al., 2019). Cobbe", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "et al. (2021) extend Ling et al. (2017) by creating a larger dataset and using it to finetune a pretrained", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "language model rather than training a model from scratch. 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(2021) leverage language models to predict the final outputs of Python programs via", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 477, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 493 + ], + "score": 1.0, + "content": "first line-to-line predicting the intermediate computational results, and show that their step-by-step", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 489, + 411, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 411, + 503 + ], + "score": 1.0, + "content": "prediction method performs better than directly predicting the final outputs.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "Naturally, this paper also relates closely to the large body of recent work on prompting. Since the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "score": 1.0, + "content": "popularization of few-shot prompting as given by Brown et al. (2020), several general approaches", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 528, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 540 + ], + "score": 1.0, + "content": "have improved the prompting ability of models, such as automatically learning prompts (Lester et al.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 537, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 553 + ], + "score": 1.0, + "content": "2021) or giving models instructions describing a task (Wei et al., 2022a; Sanh et al., 2022; Ouyang", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 548, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 564 + ], + "score": 1.0, + "content": "et al., 2022). Whereas these approaches improve or augment the input part of the prompt (e.g.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 560, + 504, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 504, + 573 + ], + "score": 1.0, + "content": "instructions that are prepended to inputs), our work takes the orthogonal direction of augmenting the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 572, + 316, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 316, + 584 + ], + "score": 1.0, + "content": "outputs of language models with a chain of thought.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38 + }, + { + "type": "title", + "bbox": [ + 107, + 598, + 188, + 612 + ], + "lines": [ + { + "bbox": [ + 104, + 595, + 190, + 614 + ], + "spans": [ + { + "bbox": [ + 104, + 595, + 190, + 614 + ], + "score": 1.0, + "content": "8 Conclusions", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 623, + 505, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "We have explored chain-of-thought prompting as a simple and broadly applicable method for enhanc-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "ing reasoning in language models. Through experiments on arithmetic, symbolic, and commonsense", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "reasoning, we find that chain-of-thought reasoning is an emergent property of model scale that allows", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "sufficiently large language models to perform reasoning tasks that otherwise have flat scaling curves.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "score": 1.0, + "content": "Broadening the range of reasoning tasks that language models can perform will hopefully inspire", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 676, + 336, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 336, + 691 + ], + "score": 1.0, + "content": "further work on language-based approaches to reasoning.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 128 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 73, + 507, + 129 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "The emergence of chain-of-thought reasoning as a result of model scale has been a prevailing theme", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "(Wei et al., 2022b). For many reasoning tasks where standard prompting has a flat scaling curve, chain-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "score": 1.0, + "content": "of-thought prompting leads to dramatically increasing scaling curves. Chain-of-thought prompting", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "appears to expand the set of tasks that large language models can perform successfully—in other", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "words, our work underscores that standard prompting only provides a lower bound on the capabilities", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "of large language models. This observation likely raises more questions than it answers—for instance,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "how much more can we expect reasoning ability to improve with a further increase in model scale?", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 208, + 494, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 494, + 222 + ], + "score": 1.0, + "content": "What other prompting methods might expand the range of tasks that language models can solve?", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 132, + 506, + 222 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 225, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 236 + ], + "score": 1.0, + "content": "As for limitations, we first qualify that although chain of thought emulates the thought processes of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "human reasoners, this does not answer whether the neural network is actually “reasoning,” which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "we leave as an open question. Second, although the cost of manually augmenting exemplars with", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "score": 1.0, + "content": "chains of thought is minimal in the few-shot setting, such annotation costs could be prohibitive for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "finetuning (though this could potentially be surmounted with synthetic data generation, or zero-shot", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "score": 1.0, + "content": "generalization). Third, there is no guarantee of correct reasoning paths, which can lead to both correct", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 291, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 506, + 303 + ], + "score": 1.0, + "content": "and incorrect answers; improving factual generations of language models is an open direction for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "score": 1.0, + "content": "future work (Rashkin et al., 2021; Ye and Durrett, 2022; Wiegreffe et al., 2022, inter alia). Finally,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "the emergence of chain-of-thought reasoning only at large model scales makes it costly to serve in", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 324, + 502, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 502, + 336 + ], + "score": 1.0, + "content": "real-world applications; further research could explore how to induce reasoning in smaller models.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 225, + 506, + 336 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 350, + 197, + 363 + ], + "lines": [ + { + "bbox": [ + 105, + 349, + 198, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 198, + 365 + ], + "score": 1.0, + "content": "7 Related Work", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 108, + 375, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "This work is inspired by many research areas, which we detail in an extended related work section", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 506, + 399 + ], + "score": 1.0, + "content": "(Appendix C). Here we describe two directions and associated papers that are perhaps most relevant.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 106, + 375, + 506, + 399 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "The first relevant direction is using intermediate steps to solve reasoning problems. Ling et al. (2017)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "pioneer the idea of using natural language rationales to solve math word problems through a series", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "of intermediate steps. Their work is a remarkable contrast to the literature using formal languages", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "to reason (Roy et al., 2015; Chiang and Chen, 2019; Amini et al., 2019; Chen et al., 2019). 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(2021) leverage language models to predict the final outputs of Python programs via", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 477, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 493 + ], + "score": 1.0, + "content": "first line-to-line predicting the intermediate computational results, and show that their step-by-step", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 489, + 411, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 411, + 503 + ], + "score": 1.0, + "content": "prediction method performs better than directly predicting the final outputs.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 402, + 506, + 503 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "Naturally, this paper also relates closely to the large body of recent work on prompting. Since the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "score": 1.0, + "content": "popularization of few-shot prompting as given by Brown et al. (2020), several general approaches", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 528, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 540 + ], + "score": 1.0, + "content": "have improved the prompting ability of models, such as automatically learning prompts (Lester et al.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 537, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 553 + ], + "score": 1.0, + "content": "2021) or giving models instructions describing a task (Wei et al., 2022a; Sanh et al., 2022; Ouyang", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 548, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 564 + ], + "score": 1.0, + "content": "et al., 2022). Whereas these approaches improve or augment the input part of the prompt (e.g.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 560, + 504, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 504, + 573 + ], + "score": 1.0, + "content": "instructions that are prepended to inputs), our work takes the orthogonal direction of augmenting the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 572, + 316, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 316, + 584 + ], + "score": 1.0, + "content": "outputs of language models with a chain of thought.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 506, + 506, + 584 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 598, + 188, + 612 + ], + "lines": [ + { + "bbox": [ + 104, + 595, + 190, + 614 + ], + "spans": [ + { + "bbox": [ + 104, + 595, + 190, + 614 + ], + "score": 1.0, + "content": "8 Conclusions", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 623, + 505, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "We have explored chain-of-thought prompting as a simple and broadly applicable method for enhanc-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "ing reasoning in language models. Through experiments on arithmetic, symbolic, and commonsense", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "reasoning, we find that chain-of-thought reasoning is an emergent property of model scale that allows", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "sufficiently large language models to perform reasoning tasks that otherwise have flat scaling curves.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "score": 1.0, + "content": "Broadening the range of reasoning tasks that language models can perform will hopefully inspire", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 676, + 336, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 336, + 691 + ], + "score": 1.0, + "content": "further work on language-based approaches to reasoning.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 623, + 506, + 691 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 71, + 207, + 84 + ], + "lines": [ + { + "bbox": [ + 106, + 69, + 208, + 87 + ], + "spans": [ + { + "bbox": [ + 106, + 69, + 208, + 87 + ], + "score": 1.0, + "content": "Acknowledgements", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 505, + 140 + ], + "lines": [ + { + "bbox": [ + 105, + 95, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 505, + 108 + ], + "score": 1.0, + "content": "We thank Jacob Devlin, Claire Cui, Andrew Dai, and Ellie Pavlick for providing feedback on the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 106, + 507, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 507, + 120 + ], + "score": 1.0, + "content": "paper. 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We thank Jacob Austin, Yuhuai Wu, Henryk Michalewski, Aitor Lewkowycz, Charles Sutton,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 118, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 506, + 130 + ], + "score": 1.0, + "content": "and Aakanksha Chowdhery for helpful discussions. 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A common task of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "interest is causal structure learning (Pearl, 2009; Peters et al., 2017), which aims at learning a directed", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 478, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 489 + ], + "score": 1.0, + "content": "acyclic graph (DAG) in which edges represent causal relations between variables. While observational", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "data alone is in general not sufficient to identify the DAG (Yang et al., 2018; Hauser & Bühlmann,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "2012), interventional data can improve identifiability up to finding the exact graph (Eberhardt et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 509, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 523 + ], + "score": 1.0, + "content": "2005; Eberhardt, 2008). 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A promising", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 255, + 470, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 255, + 470, + 268 + ], + "score": 1.0, + "content": "direction is continuous optimization for score-based methods, which, however, re-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 267, + 471, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 471, + 279 + ], + "score": 1.0, + "content": "quire constrained optimization to enforce acyclicity or lack convergence guarantees.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 277, + 471, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 471, + 290 + ], + "score": 1.0, + "content": "In this paper, we present ENCO, an efficient structure learning method for directed,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 289, + 470, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 470, + 300 + ], + "score": 1.0, + "content": "acyclic causal graphs leveraging observational and interventional data. ENCO", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 300, + 470, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 470, + 311 + ], + "score": 1.0, + "content": "formulates the graph search as an optimization of independent edge likelihoods,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 310, + 470, + 323 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 470, + 323 + ], + "score": 1.0, + "content": "with the edge orientation being modeled as a separate parameter. Consequently, we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 322, + 470, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 470, + 333 + ], + "score": 1.0, + "content": "provide for ENCO convergence guarantees when interventions on all variables are", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 332, + 470, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 332, + 470, + 344 + ], + "score": 1.0, + "content": "available, without having to constrain the score function with respect to acyclicity.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 343, + 469, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 469, + 355 + ], + "score": 1.0, + "content": "In experiments, we show that ENCO can efficiently recover graphs with hundreds", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 354, + 469, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 469, + 366 + ], + "score": 1.0, + "content": "of nodes, an order of magnitude larger than what was previously possible, while", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 365, + 417, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 365, + 417, + 378 + ], + "score": 1.0, + "content": "handling deterministic variables and discovering latent confounders.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20, + "bbox_fs": [ + 140, + 234, + 471, + 378 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 398, + 206, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 208, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 208, + 413 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "Uncovering and understanding causal mechanisms is an important problem not only in machine", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 434, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 445 + ], + "score": 1.0, + "content": "learning (Schölkopf et al., 2021; Pearl, 2009) but also in various scientific disciplines such as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "score": 1.0, + "content": "computational biology (Friedman et al., 2000; Sachs et al., 2005), epidemiology (Robins et al., 2000;", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 456, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 466 + ], + "score": 1.0, + "content": "Vandenbroucke et al., 2016), and economics (Pearl, 2009; Hicks et al., 1980). A common task of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "interest is causal structure learning (Pearl, 2009; Peters et al., 2017), which aims at learning a directed", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 478, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 489 + ], + "score": 1.0, + "content": "acyclic graph (DAG) in which edges represent causal relations between variables. While observational", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "data alone is in general not sufficient to identify the DAG (Yang et al., 2018; Hauser & Bühlmann,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "2012), interventional data can improve identifiability up to finding the exact graph (Eberhardt et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 509, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 523 + ], + "score": 1.0, + "content": "2005; Eberhardt, 2008). Unfortunately, the solution space of DAGs grows super-exponentially with", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "the variable count, requiring efficient methods for large graphs. Current methods are typically applied", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 532, + 504, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 504, + 543 + ], + "score": 1.0, + "content": "to a few dozens of variables and cannot scale so well, which is imperative for modern applications", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "like learning causal relations with gene editing interventions (Dixit et al., 2016; Macosko et al., 2015).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 423, + 506, + 555 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "A promising new direction for scaling up DAG discovery methods are continuous-optimization", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "methods (Zheng et al., 2018; 2020; Zhu et al., 2020; Ke et al., 2019; Brouillard et al., 2020; Yu", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "et al., 2019). In contrast to score-based and constrained-based (Peters et al., 2017; Guo et al., 2020)", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "methods, continuous-optimization methods reinterpret the search over discrete graph topologies as", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "a continuous problem with neural networks as function approximators, for which efficient solvers", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 615, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 627 + ], + "score": 1.0, + "content": "are amenable. To restrict the search space to acyclic graphs, Zheng et al. (2018) first proposed to", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "score": 1.0, + "content": "view the search as a constrained optimization problem using an augmented Lagrangian procedure to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 636, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 649 + ], + "score": 1.0, + "content": "solve it. 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While simpler to optimize, methods relying on acyclicity regularizers commonly lack", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "score": 1.0, + "content": "guarantees for finding the correct causal graph, often converging to suboptimal solutions. Despite the", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 691, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 284, + 704 + ], + "score": 1.0, + "content": "advances, beyond linear, continuous settings", + "type": "text" + }, + { + "bbox": [ + 284, + 692, + 298, + 703 + ], + "score": 0.36, + "content": "\\mathrm { N g }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 691, + 505, + 704 + ], + "score": 1.0, + "content": "et al., 2020; Varando, 2020) continuous optimization", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 702, + 506, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 506, + 716 + ], + "score": 1.0, + "content": "methods still cannot scale to more than 100 variables, often due to difficulties in enforcing acyclicity.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 560, + 506, + 716 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "In this work, we address both problems. 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Yet,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "if we are able to intervene on all variables, the proposed optimization is guaranteed to converge to the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "correct, acyclic graph. Importantly, since such interventions might not always be available, we show", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "that our algorithm performs robustly even when intervening on fewer variables and having small", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 429, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 429, + 162 + ], + "score": 1.0, + "content": "sample sizes. We call our method ENCO for Efficient Neural Causal Discovery.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 164, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 505, + 179 + ], + "score": 1.0, + "content": "We make the following four contributions. 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While often", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 320, + 524 + ], + "score": 1.0, + "content": "performed from observational data, i.e. samples from", + "type": "text" + }, + { + "bbox": [ + 320, + 512, + 328, + 521 + ], + "score": 0.78, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "(see Glymour et al. (2019) for an overview),", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "we focus in this paper on algorithms that recover graphs from joint observational and interventional", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 534, + 487, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 487, + 546 + ], + "score": 1.0, + "content": "data. Commonly, such methods are grouped into constraint-based and score-based approaches.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "Constraint-based methods use conditional independence tests to identify causal relations (Monti", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "et al., 2019; Spirtes et al., 2000; Kocaoglu et al., 2019; Jaber et al., 2020; Sun et al., 2007; Hyttinen", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "et al., 2014). For instance, the Invariant Causal Prediction (ICP) algorithm (Peters et al., 2016;", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "Christina et al., 2018) exploits that causal mechanisms remain unchanged under an intervention", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "except the one intervened on (Pearl, 2009; Schölkopf et al., 2012). We rely on a similar idea by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "testing for mechanisms that generalize from the observational to the interventional setting. Another", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "line of work is to extend methods working on observations only to interventions by incorporating", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "those as additional constraints in the structure learning process (Mooij et al., 2020; Jaber et al., 2020).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "Score-based methods, on the other hand, search through the space of all possible causal structures", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "with the goal of optimizing a specified metric (Tsamardinos et al., 2006; Ke et al., 2019; Goudet et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 665, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 506, + 677 + ], + "score": 1.0, + "content": "2017; Zhu et al., 2020). This metric, also referred to as score function, is usually a combination of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "how well the structure fits the data, for instance in terms of log-likelihood, as well as regularizers for", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 688, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 699 + ], + "score": 1.0, + "content": "encouraging sparsity. Since the search space of DAGs is super-exponential in the number of nodes,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "many methods rely on a greedy search, yet returning graphs in the true equivalence class (Meek,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 709, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 721 + ], + "score": 1.0, + "content": "1997; Hauser & Bühlmann, 2012; Wang et al., 2017; Yang et al., 2018). For instance, GIES (Hauser", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "& Bühlmann, 2012) repeatedly adds, removes, and flips the directions of edges in a proposal graph", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "In this work, we address both problems. By modeling the orientation of an edge as a separate", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "parameter, we can define the score function without any acyclicity constraints or regularizers. This", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "allows for unbiased low-variance gradient estimators that scale learning to much larger graphs. Yet,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "if we are able to intervene on all variables, the proposed optimization is guaranteed to converge to the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "correct, acyclic graph. Importantly, since such interventions might not always be available, we show", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "that our algorithm performs robustly even when intervening on fewer variables and having small", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 429, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 429, + 162 + ], + "score": 1.0, + "content": "sample sizes. We call our method ENCO for Efficient Neural Causal Discovery.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 82, + 505, + 162 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 164, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 505, + 179 + ], + "score": 1.0, + "content": "We make the following four contributions. Firstly, we propose ENCO, a causal structure learning", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "method for observational and interventional data using continuous optimization. Different from", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "recent methods, ENCO models the edge orientation as a separate parameter. Secondly, we derive", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "unbiased, low-variance gradient estimators, which is crucial for scaling up the model to large numbers", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "of variables. Thirdly, we show that ENCO is guaranteed to converge to the correct causal graph if", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "interventions on all variables are available, despite not having any acyclicity constraints. Yet, we show", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "in practice that the algorithm works on partial intervention sets as well. Fourthly, we extend ENCO", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 507, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 507, + 255 + ], + "score": 1.0, + "content": "to detecting latent confounders. 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An important concept which distinguishes CGMs from", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 409, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 414, + 420 + ], + "score": 1.0, + "content": "standard Bayesian Networks are interventions. 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While often", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 320, + 524 + ], + "score": 1.0, + "content": "performed from observational data, i.e. samples from", + "type": "text" + }, + { + "bbox": [ + 320, + 512, + 328, + 521 + ], + "score": 0.78, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "(see Glymour et al. (2019) for an overview),", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "we focus in this paper on algorithms that recover graphs from joint observational and interventional", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 534, + 487, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 487, + 546 + ], + "score": 1.0, + "content": "data. Commonly, such methods are grouped into constraint-based and score-based approaches.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 489, + 506, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "Constraint-based methods use conditional independence tests to identify causal relations (Monti", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "et al., 2019; Spirtes et al., 2000; Kocaoglu et al., 2019; Jaber et al., 2020; Sun et al., 2007; Hyttinen", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "et al., 2014). For instance, the Invariant Causal Prediction (ICP) algorithm (Peters et al., 2016;", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "Christina et al., 2018) exploits that causal mechanisms remain unchanged under an intervention", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "except the one intervened on (Pearl, 2009; Schölkopf et al., 2012). We rely on a similar idea by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "testing for mechanisms that generalize from the observational to the interventional setting. Another", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "line of work is to extend methods working on observations only to interventions by incorporating", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "those as additional constraints in the structure learning process (Mooij et al., 2020; Jaber et al., 2020).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 550, + 506, + 639 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "Score-based methods, on the other hand, search through the space of all possible causal structures", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "with the goal of optimizing a specified metric (Tsamardinos et al., 2006; Ke et al., 2019; Goudet et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 665, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 506, + 677 + ], + "score": 1.0, + "content": "2017; Zhu et al., 2020). This metric, also referred to as score function, is usually a combination of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "how well the structure fits the data, for instance in terms of log-likelihood, as well as regularizers for", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 688, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 699 + ], + "score": 1.0, + "content": "encouraging sparsity. Since the search space of DAGs is super-exponential in the number of nodes,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "many methods rely on a greedy search, yet returning graphs in the true equivalence class (Meek,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 709, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 721 + ], + "score": 1.0, + "content": "1997; Hauser & Bühlmann, 2012; Wang et al., 2017; Yang et al., 2018). 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The graph on the right further shows how the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 154, + 201 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 154, + 190, + 163, + 200 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 189, + 182, + 201 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 182, + 189, + 190, + 198 + ], + "score": 0.65, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "correspond to edge probabilities. We learn those parameters by comparing", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 199, + 483, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 483, + 211 + ], + "score": 1.0, + "content": "multiple graph samples on how well they generalize from observational to interventional data.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 105, + 222, + 504, + 245 + ], + "lines": [ + { + "bbox": [ + 105, + 220, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 236 + ], + "score": 1.0, + "content": "until no higher-scoring graph can be found. The Interventional Greedy SP (IGSP) algorithm (Wang", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 234, + 465, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 465, + 245 + ], + "score": 1.0, + "content": "et al., 2017) is a hybrid method using conditional independence tests in its score function.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 250, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 504, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 504, + 262 + ], + "score": 1.0, + "content": "Continuous-optimization methods are score-based methods that avoid the combinatorial greedy", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 260, + 504, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 504, + 273 + ], + "score": 1.0, + "content": "search over DAGs by using gradient-based methods (Zheng et al., 2018; Ke et al., 2019; Lachapelle", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 272, + 504, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 504, + 284 + ], + "score": 1.0, + "content": "et al., 2020; Yu et al., 2019; Zheng et al., 2020; Zhu et al., 2020; Brouillard et al., 2020). Thereby, the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 282, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 296 + ], + "score": 1.0, + "content": "adjacency matrix is parameterized by weights that represent linear factors or probabilities of having", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "1an edge between a pair of nodes. The main challenge of such methods is how to limit the search space", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "score": 1.0, + "content": "1to acyclic graphs. One common approach is to view the search as a constrained optimization problem", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 316, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 328 + ], + "score": 1.0, + "content": "and deploy an augmented Lagrangian procedure to solve it (Zheng et al., 2018; 2020; Yu et al., 2019;", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 327, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 338 + ], + "score": 1.0, + "content": "Brouillard et al., 2020), including NOTEARS (Zheng et al., 2018) and DCDI (Brouillard et al., 2020).", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 336, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 227, + 350 + ], + "score": 1.0, + "content": "Alternatively, Ke et al. (2019);", + "type": "text" + }, + { + "bbox": [ + 228, + 338, + 242, + 349 + ], + "score": 0.37, + "content": "\\mathrm { N g }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 336, + 505, + 350 + ], + "score": 1.0, + "content": "et al. (2020) propose to use a regularization term penalizing cyclic", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 348, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 361 + ], + "score": 1.0, + "content": "graphs while allowing unconstrained optimization. However, the regularizer must be designed and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 360, + 491, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 491, + 372 + ], + "score": 1.0, + "content": "weighted such that the correct, acyclic causal graph is the global optimum of the score function.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 388, + 335, + 402 + ], + "lines": [ + { + "bbox": [ + 104, + 387, + 338, + 404 + ], + "spans": [ + { + "bbox": [ + 104, + 387, + 338, + 404 + ], + "score": 1.0, + "content": "3 EFFICIENT NEURAL CAUSAL DISCOVERY", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 108, + 415, + 246, + 426 + ], + "lines": [ + { + "bbox": [ + 106, + 415, + 247, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 247, + 427 + ], + "score": 1.0, + "content": "3.1 SCOPE AND ASSUMPTIONS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 325, + 450 + ], + "score": 1.0, + "content": "We consider the task of finding a directed acyclic graph", + "type": "text" + }, + { + "bbox": [ + 325, + 437, + 374, + 448 + ], + "score": 0.93, + "content": "G = ( V , E )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 435, + 395, + 450 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 395, + 437, + 406, + 446 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 435, + 506, + 450 + ], + "score": 1.0, + "content": "variables of an unknown", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "CGM given observational and interventional samples. Firstly, we assume that: (1) The CGM is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 458, + 504, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 493, + 470 + ], + "score": 1.0, + "content": "causally sufficient, i.e., all common causes of variables are included and observable; (2) We have", + "type": "text" + }, + { + "bbox": [ + 494, + 459, + 504, + 468 + ], + "score": 0.76, + "content": "N", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 470, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 481 + ], + "score": 1.0, + "content": "interventional datasets, each sparsely intervening on a different variable; (3) The interventions are", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 481, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 492 + ], + "score": 1.0, + "content": "“perfect” and “stochastic”, meaning the intervention does not set the variable necessarily to a single", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 489, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 104, + 489, + 506, + 505 + ], + "score": 1.0, + "content": "value. Thereby, we do not strictly require faithfulness, thus also recovering some graphs violating", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 516 + ], + "score": 1.0, + "content": "faithfulness. We emphasize that we place no constraints on the domains of the variables (they can", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 513, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 506, + 525 + ], + "score": 1.0, + "content": "be discrete, continuous, or mixed) or the distributions of the interventions. We discuss later how to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 524, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 536 + ], + "score": 1.0, + "content": "extend the algorithm to infer causal mechanisms in graphs with latent confounding causal variables.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 535, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 506, + 548 + ], + "score": 1.0, + "content": "Further, we discuss how to extend the algorithm to support interventions to subsets of variables only.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 107, + 561, + 178, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 180, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 180, + 574 + ], + "score": 1.0, + "content": "3.2 OVERVIEW", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "ENCO learns a causal graph from observational and interventional data by modelling a probability", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "for every possible directed edge between pairs of variables. The goal is that the probabilities", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "corresponding to the edges of the ground truth graph converge to one, while the probabilities of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "all other edges converge to zero. For this to happen, we exploit the idea of independent causal", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "mechanisms (Pearl, 2009; Peters et al., 2016), according to which the conditional distributions for all", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "variables in the ground-truth CGM stay invariant under an intervention, except for the intervened", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "ones. 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We learn those parameters by comparing", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 199, + 483, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 483, + 211 + ], + "score": 1.0, + "content": "multiple graph samples on how well they generalize from observational to interventional data.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 105, + 222, + 504, + 245 + ], + "lines": [], + "index": 7.5, + "bbox_fs": [ + 105, + 220, + 506, + 245 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 250, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 504, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 504, + 262 + ], + "score": 1.0, + "content": "Continuous-optimization methods are score-based methods that avoid the combinatorial greedy", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 260, + 504, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 504, + 273 + ], + "score": 1.0, + "content": "search over DAGs by using gradient-based methods (Zheng et al., 2018; Ke et al., 2019; Lachapelle", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 272, + 504, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 504, + 284 + ], + "score": 1.0, + "content": "et al., 2020; Yu et al., 2019; Zheng et al., 2020; Zhu et al., 2020; Brouillard et al., 2020). Thereby, the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 282, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 296 + ], + "score": 1.0, + "content": "adjacency matrix is parameterized by weights that represent linear factors or probabilities of having", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "1an edge between a pair of nodes. The main challenge of such methods is how to limit the search space", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "score": 1.0, + "content": "1to acyclic graphs. One common approach is to view the search as a constrained optimization problem", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 316, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 328 + ], + "score": 1.0, + "content": "and deploy an augmented Lagrangian procedure to solve it (Zheng et al., 2018; 2020; Yu et al., 2019;", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 327, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 338 + ], + "score": 1.0, + "content": "Brouillard et al., 2020), including NOTEARS (Zheng et al., 2018) and DCDI (Brouillard et al., 2020).", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 336, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 227, + 350 + ], + "score": 1.0, + "content": "Alternatively, Ke et al. 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However, the regularizer must be designed and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 360, + 491, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 491, + 372 + ], + "score": 1.0, + "content": "weighted such that the correct, acyclic causal graph is the global optimum of the score function.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 249, + 506, + 372 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 388, + 335, + 402 + ], + "lines": [ + { + "bbox": [ + 104, + 387, + 338, + 404 + ], + "spans": [ + { + "bbox": [ + 104, + 387, + 338, + 404 + ], + "score": 1.0, + "content": "3 EFFICIENT NEURAL CAUSAL DISCOVERY", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 108, + 415, + 246, + 426 + ], + "lines": [ + { + "bbox": [ + 106, + 415, + 247, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 247, + 427 + ], + "score": 1.0, + "content": "3.1 SCOPE AND ASSUMPTIONS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 325, + 450 + ], + "score": 1.0, + "content": "We consider the task of finding a directed acyclic graph", + "type": "text" + }, + { + "bbox": [ + 325, + 437, + 374, + 448 + ], + "score": 0.93, + "content": "G = ( V , E )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 435, + 395, + 450 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 395, + 437, + 406, + 446 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 435, + 506, + 450 + ], + "score": 1.0, + "content": "variables of an unknown", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "CGM given observational and interventional samples. 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For categorical random variables", + "type": "text" + }, + { + "bbox": [ + 378, + 168, + 390, + 178 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 165, + 505, + 178 + ], + "score": 1.0, + "content": ", we apply a softmax output", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 177, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 161, + 190 + ], + "score": 1.0, + "content": "activation for", + "type": "text" + }, + { + "bbox": [ + 161, + 179, + 175, + 191 + ], + "score": 0.88, + "content": "f _ { \\phi _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 177, + 506, + 190 + ], + "score": 1.0, + "content": ", and for continuous ones, we use Normalizing Flows (Rezende & Mohamed, 2015).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 195, + 506, + 360 + ], + "lines": [ + { + "bbox": [ + 106, + 195, + 507, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 507, + 208 + ], + "score": 1.0, + "content": "Graph fitting uses the learned networks to score and compare different graphs on interventional data.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 203, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 104, + 203, + 410, + 219 + ], + "score": 1.0, + "content": "For parameterizing the edge probabilities, we use two sets of parameters:", + "type": "text" + }, + { + "bbox": [ + 411, + 205, + 459, + 217 + ], + "score": 0.92, + "content": "\\gamma \\in \\mathbb { R } ^ { N \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 203, + 506, + 219 + ], + "score": 1.0, + "content": "represents", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 214, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 258, + 230 + ], + "score": 1.0, + "content": "the existence of edges in a graph, and", + "type": "text" + }, + { + "bbox": [ + 259, + 216, + 306, + 227 + ], + "score": 0.92, + "content": "\\pmb \\theta \\in \\mathbb R ^ { N \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 214, + 506, + 230 + ], + "score": 1.0, + "content": "the orientation of the edges. The likelihood of an", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 227, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 199, + 241 + ], + "score": 1.0, + "content": "edge is determined by", + "type": "text" + }, + { + "bbox": [ + 199, + 228, + 326, + 240 + ], + "score": 0.91, + "content": "p ( X _ { i } \\to X _ { j } ) = \\sigma ( \\gamma _ { i j } ) \\cdot \\sigma ( \\theta _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 227, + 351, + 241 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 351, + 228, + 374, + 240 + ], + "score": 0.92, + "content": "\\sigma ( . . . )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 227, + 506, + 241 + ], + "score": 1.0, + "content": "being the sigmoid function and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 237, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 107, + 239, + 153, + 251 + ], + "score": 0.91, + "content": "\\theta _ { i j } = - \\theta _ { j i }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 237, + 506, + 253 + ], + "score": 1.0, + "content": ". The probability of the two orientations always sum to one. The benefit of separating", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 327, + 263 + ], + "score": 1.0, + "content": "the edge probabilities into two independent parameters", + "type": "text" + }, + { + "bbox": [ + 328, + 252, + 336, + 261 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 250, + 354, + 263 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 354, + 250, + 361, + 260 + ], + "score": 0.78, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "is that it gives us more control over", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "score": 1.0, + "content": "the gradient updates. The existence of an (undirected) edge can usually be already learned from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 272, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 506, + 284 + ], + "score": 1.0, + "content": "observational or arbitrary interventional data alone, excluding deterministic variables (Pearl, 2009).", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "score": 1.0, + "content": "In contrast, the orientation can only be reliably detected from data for which an intervention is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 254, + 307 + ], + "score": 1.0, + "content": "performed on its adjacent nodes, i.e.,", + "type": "text" + }, + { + "bbox": [ + 254, + 294, + 267, + 305 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 293, + 279, + 307 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 279, + 294, + 292, + 306 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 293, + 342, + 307 + ], + "score": 1.0, + "content": "for learning", + "type": "text" + }, + { + "bbox": [ + 342, + 294, + 355, + 306 + ], + "score": 0.89, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 293, + 506, + 307 + ], + "score": 1.0, + "content": ". While other interventions eventually", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 303, + 504, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 395, + 317 + ], + "score": 1.0, + "content": "provide information on the edge direction, e.g., intervening on a node", + "type": "text" + }, + { + "bbox": [ + 396, + 305, + 410, + 316 + ], + "score": 0.89, + "content": "X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 303, + 491, + 317 + ], + "score": 1.0, + "content": "which is a child of", + "type": "text" + }, + { + "bbox": [ + 491, + 305, + 504, + 316 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 169, + 328 + ], + "score": 1.0, + "content": "and a parent of", + "type": "text" + }, + { + "bbox": [ + 170, + 316, + 183, + 328 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 315, + 314, + 328 + ], + "score": 1.0, + "content": ", we do not know the relation of", + "type": "text" + }, + { + "bbox": [ + 314, + 316, + 328, + 327 + ], + "score": 0.9, + "content": "X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 315, + 340, + 328 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 340, + 316, + 353, + 327 + ], + "score": 0.9, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 315, + 371, + 328 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 371, + 316, + 385, + 328 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 315, + 505, + 328 + ], + "score": 1.0, + "content": "at this stage, as we are in the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 325, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 444, + 340 + ], + "score": 1.0, + "content": "process of learning the structure. 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One reason is that on interventional data, an edge can improve the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 349, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 361 + ], + "score": 1.0, + "content": "log-likelihood estimate in one direction, but not necessarily the other, leading to conflicting gradients.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 340, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 341, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 244, + 380 + ], + "score": 1.0, + "content": "We optimize the graph parameters", + "type": "text" + }, + { + "bbox": [ + 245, + 367, + 253, + 377 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 362, + 271, + 380 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 271, + 366, + 278, + 375 + ], + "score": 0.8, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 362, + 341, + 380 + ], + "score": 1.0, + "content": "by minimizing", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 381, + 463, + 417 + ], + "lines": [ + { + "bbox": [ + 148, + 381, + 463, + 417 + ], + "spans": [ + { + "bbox": [ + 148, + 381, + 463, + 417 + ], + "score": 0.94, + "content": "\\tilde { \\mathcal { L } } = \\mathbb { E } _ { \\hat { I } \\sim p _ { I } ( I ) } \\mathbb { E } _ { \\tilde { p } _ { \\hat { I } } ( \\pmb { X } ) } \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C ) } \\left[ \\sum _ { i = 1 } ^ { N } \\mathcal { L } _ { C } ( X _ { i } ) \\right] + \\lambda _ { \\mathrm { s p a r s e } } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } \\sigma ( \\gamma _ { i j } ) \\cdot \\sigma ( \\theta _ { i j } )", + "type": "interline_equation", + "image_path": "4df70759b4f1ca4335a1089e18ef4f732a14254dc5fe76aad4bc4f4174b2801d.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 148, + 381, + 463, + 393.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 148, + 393.0, + 463, + 405.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 148, + 405.0, + 463, + 417.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 132, + 434 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 421, + 156, + 434 + ], + "score": 0.92, + "content": "p _ { I } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 421, + 460, + 434 + ], + "score": 1.0, + "content": "is the distribution over which variable to intervene on (usually uniform), and", + "type": "text" + }, + { + "bbox": [ + 461, + 421, + 489, + 434 + ], + "score": 0.93, + "content": "\\tilde { p } _ { \\hat { I } } ( \\boldsymbol { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 433, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 333, + 448 + ], + "score": 1.0, + "content": "joint distribution of all variables under the intervention", + "type": "text" + }, + { + "bbox": [ + 334, + 433, + 340, + 445 + ], + "score": 0.76, + "content": "\\hat { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 433, + 506, + 448 + ], + "score": 1.0, + "content": ". 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With", + "type": "text" + }, + { + "bbox": [ + 305, + 446, + 339, + 458 + ], + "score": 0.9, + "content": "p _ { \\gamma , \\theta } ( C )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 445, + 506, + 460 + ], + "score": 1.0, + "content": ", we denote the distribution over adjacency", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 104, + 456, + 144, + 470 + ], + "score": 1.0, + "content": "matrices", + "type": "text" + }, + { + "bbox": [ + 144, + 458, + 154, + 467 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 456, + 181, + 470 + ], + "score": 1.0, + "content": "under", + "type": "text" + }, + { + "bbox": [ + 182, + 458, + 200, + 469 + ], + "score": 0.88, + "content": "\\gamma , \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 456, + 233, + 470 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 234, + 457, + 338, + 470 + ], + "score": 0.88, + "content": "C _ { i j } \\sim \\mathrm { B e r } ( \\sigma ( \\gamma _ { i j } ) \\dot { \\sigma } ( \\theta _ { i j } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 456, + 345, + 470 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 345, + 457, + 380, + 469 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { C } ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "is the negative log-likelihood", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 467, + 504, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 187, + 482 + ], + "score": 1.0, + "content": "estimate of variable", + "type": "text" + }, + { + "bbox": [ + 187, + 469, + 200, + 479 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 467, + 360, + 482 + ], + "score": 1.0, + "content": "conditioned on the parents according to", + "type": "text" + }, + { + "bbox": [ + 361, + 469, + 369, + 479 + ], + "score": 0.65, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 468, + 504, + 481 + ], + "score": 0.88, + "content": ": { \\mathcal { L } } _ { C } ( X _ { i } ) = - \\log f _ { \\phi _ { i } } ( X _ { i } ; C . _ { , i } \\odot", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 107, + 479, + 128, + 491 + ], + "score": 0.83, + "content": "X _ { - i , }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 478, + 279, + 492 + ], + "score": 1.0, + "content": "). 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Although the orientation parameters do not guarantee the absence of loops with more", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 562, + 499, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 499, + 575 + ], + "score": 1.0, + "content": "variable, we show that under certain conditions ENCO yet converges to the correct, acyclic graph.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "title", + "bbox": [ + 108, + 587, + 408, + 598 + ], + "lines": [ + { + "bbox": [ + 105, + 586, + 409, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 409, + 599 + ], + "score": 1.0, + "content": "3.3 LOW-VARIANCE GRADIENT ESTIMATORS FOR EDGE PARAMETERS", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 607, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 104, + 606, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 147, + 621 + ], + "score": 1.0, + "content": "To update", + "type": "text" + }, + { + "bbox": [ + 147, + 609, + 155, + 619 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 606, + 173, + 621 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 173, + 608, + 180, + 617 + ], + "score": 0.78, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 606, + 506, + 621 + ], + "score": 1.0, + "content": "based on Equation 2, we need to determine their gradients through the expectation", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 617, + 507, + 632 + ], + "spans": [ + { + "bbox": [ + 107, + 619, + 142, + 632 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { p _ { \\Upsilon ; \\theta } ( C ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 617, + 174, + 632 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 174, + 619, + 183, + 628 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 617, + 507, + 632 + ], + "score": 1.0, + "content": "is a discrete variable. For this, we apply REINFORCE (Williams, 1992). For", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "clarity of exposition, we limit the discussion here to the final results and provide the detailed", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 640, + 428, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 275, + 653 + ], + "score": 1.0, + "content": "derivations in Appendix A. For parameter", + "type": "text" + }, + { + "bbox": [ + 276, + 642, + 289, + 653 + ], + "score": 0.87, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 640, + 428, + 653 + ], + "score": 1.0, + "content": ", we obtain the following gradient:", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5 + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 657, + 464, + 684 + ], + "lines": [ + { + "bbox": [ + 147, + 657, + 464, + 684 + ], + "spans": [ + { + "bbox": [ + 147, + 657, + 464, + 684 + ], + "score": 0.89, + "content": "\\frac { \\partial } { \\partial \\gamma _ { i j } } \\tilde { \\mathcal { L } } = \\sigma ^ { \\prime } ( \\gamma _ { i j } ) \\cdot \\sigma ( \\theta _ { i j } ) \\cdot \\mathbb { E } _ { X , C _ { - i j } } [ \\mathcal { L } _ { X _ { i } X _ { j } } ( X _ { j } ) - \\mathcal { L } _ { X _ { i } \\neq X _ { j } } ( X _ { j } ) + \\lambda _ { \\mathrm { s p a r s e } } ]", + "type": "interline_equation", + "image_path": "3b09b391bddad352d00a37996592c8bf2694447d2a318ef3e18b4a5dcb42faa1.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 147, + 657, + 464, + 684 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 686, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 104, + 686, + 134, + 701 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 688, + 170, + 700 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { X , C _ { - i j } }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 686, + 506, + 701 + ], + "score": 1.0, + "content": "summarizes for brevity the three expectations in Equation 2, excluding the edge", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 107, + 700, + 147, + 711 + ], + "score": 0.9, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 698, + 171, + 712 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 171, + 700, + 180, + 709 + ], + "score": 0.79, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 698, + 218, + 712 + ], + "score": 1.0, + "content": ". 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The likelihood of an", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 227, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 199, + 241 + ], + "score": 1.0, + "content": "edge is determined by", + "type": "text" + }, + { + "bbox": [ + 199, + 228, + 326, + 240 + ], + "score": 0.91, + "content": "p ( X _ { i } \\to X _ { j } ) = \\sigma ( \\gamma _ { i j } ) \\cdot \\sigma ( \\theta _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 227, + 351, + 241 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 351, + 228, + 374, + 240 + ], + "score": 0.92, + "content": "\\sigma ( . . . )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 227, + 506, + 241 + ], + "score": 1.0, + "content": "being the sigmoid function and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 237, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 107, + 239, + 153, + 251 + ], + "score": 0.91, + "content": "\\theta _ { i j } = - \\theta _ { j i }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 237, + 506, + 253 + ], + "score": 1.0, + "content": ". 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The benefit of separating", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 327, + 263 + ], + "score": 1.0, + "content": "the edge probabilities into two independent parameters", + "type": "text" + }, + { + "bbox": [ + 328, + 252, + 336, + 261 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 250, + 354, + 263 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 354, + 250, + 361, + 260 + ], + "score": 0.78, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "is that it gives us more control over", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "score": 1.0, + "content": "the gradient updates. The existence of an (undirected) edge can usually be already learned from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 272, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 506, + 284 + ], + "score": 1.0, + "content": "observational or arbitrary interventional data alone, excluding deterministic variables (Pearl, 2009).", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "score": 1.0, + "content": "In contrast, the orientation can only be reliably detected from data for which an intervention is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 254, + 307 + ], + "score": 1.0, + "content": "performed on its adjacent nodes, i.e.,", + "type": "text" + }, + { + "bbox": [ + 254, + 294, + 267, + 305 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 293, + 279, + 307 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 279, + 294, + 292, + 306 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 293, + 342, + 307 + ], + "score": 1.0, + "content": "for learning", + "type": "text" + }, + { + "bbox": [ + 342, + 294, + 355, + 306 + ], + "score": 0.89, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 293, + 506, + 307 + ], + "score": 1.0, + "content": ". While other interventions eventually", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 303, + 504, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 395, + 317 + ], + "score": 1.0, + "content": "provide information on the edge direction, e.g., intervening on a node", + "type": "text" + }, + { + "bbox": [ + 396, + 305, + 410, + 316 + ], + "score": 0.89, + "content": "X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 303, + 491, + 317 + ], + "score": 1.0, + "content": "which is a child of", + "type": "text" + }, + { + "bbox": [ + 491, + 305, + 504, + 316 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 169, + 328 + ], + "score": 1.0, + "content": "and a parent of", + "type": "text" + }, + { + "bbox": [ + 170, + 316, + 183, + 328 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 315, + 314, + 328 + ], + "score": 1.0, + "content": ", we do not know the relation of", + "type": "text" + }, + { + "bbox": [ + 314, + 316, + 328, + 327 + ], + "score": 0.9, + "content": "X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 315, + 340, + 328 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 340, + 316, + 353, + 327 + ], + "score": 0.9, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 315, + 371, + 328 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 371, + 316, + 385, + 328 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 315, + 505, + 328 + ], + "score": 1.0, + "content": "at this stage, as we are in the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 325, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 444, + 340 + ], + "score": 1.0, + "content": "process of learning the structure. 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One reason is that on interventional data, an edge can improve the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 349, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 361 + ], + "score": 1.0, + "content": "log-likelihood estimate in one direction, but not necessarily the other, leading to conflicting gradients.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 195, + 507, + 361 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 340, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 341, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 244, + 380 + ], + "score": 1.0, + "content": "We optimize the graph parameters", + "type": "text" + }, + { + "bbox": [ + 245, + 367, + 253, + 377 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 362, + 271, + 380 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 271, + 366, + 278, + 375 + ], + "score": 0.8, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 362, + 341, + 380 + ], + "score": 1.0, + "content": "by minimizing", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 362, + 341, + 380 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 381, + 463, + 417 + ], + "lines": [ + { + "bbox": [ + 148, + 381, + 463, + 417 + ], + "spans": [ + { + "bbox": [ + 148, + 381, + 463, + 417 + ], + "score": 0.94, + "content": "\\tilde { \\mathcal { L } } = \\mathbb { E } _ { \\hat { I } \\sim p _ { I } ( I ) } \\mathbb { E } _ { \\tilde { p } _ { \\hat { I } } ( \\pmb { X } ) } \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C ) } \\left[ \\sum _ { i = 1 } ^ { N } \\mathcal { L } _ { C } ( X _ { i } ) \\right] + \\lambda _ { \\mathrm { s p a r s e } } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } \\sigma ( \\gamma _ { i j } ) \\cdot \\sigma ( \\theta _ { i j } )", + "type": "interline_equation", + "image_path": "4df70759b4f1ca4335a1089e18ef4f732a14254dc5fe76aad4bc4f4174b2801d.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 148, + 381, + 463, + 393.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 148, + 393.0, + 463, + 405.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 148, + 405.0, + 463, + 417.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 132, + 434 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 421, + 156, + 434 + ], + "score": 0.92, + "content": "p _ { I } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 421, + 460, + 434 + ], + "score": 1.0, + "content": "is the distribution over which variable to intervene on (usually uniform), and", + "type": "text" + }, + { + "bbox": [ + 461, + 421, + 489, + 434 + ], + "score": 0.93, + "content": "\\tilde { p } _ { \\hat { I } } ( \\boldsymbol { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 433, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 333, + 448 + ], + "score": 1.0, + "content": "joint distribution of all variables under the intervention", + "type": "text" + }, + { + "bbox": [ + 334, + 433, + 340, + 445 + ], + "score": 0.76, + "content": "\\hat { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 433, + 506, + 448 + ], + "score": 1.0, + "content": ". 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With", + "type": "text" + }, + { + "bbox": [ + 305, + 446, + 339, + 458 + ], + "score": 0.9, + "content": "p _ { \\gamma , \\theta } ( C )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 445, + 506, + 460 + ], + "score": 1.0, + "content": ", we denote the distribution over adjacency", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 104, + 456, + 144, + 470 + ], + "score": 1.0, + "content": "matrices", + "type": "text" + }, + { + "bbox": [ + 144, + 458, + 154, + 467 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 456, + 181, + 470 + ], + "score": 1.0, + "content": "under", + "type": "text" + }, + { + "bbox": [ + 182, + 458, + 200, + 469 + ], + "score": 0.88, + "content": "\\gamma , \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 456, + 233, + 470 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 234, + 457, + 338, + 470 + ], + "score": 0.88, + "content": "C _ { i j } \\sim \\mathrm { B e r } ( \\sigma ( \\gamma _ { i j } ) \\dot { \\sigma } ( \\theta _ { i j } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 456, + 345, + 470 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 345, + 457, + 380, + 469 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { C } ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "is the negative log-likelihood", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 467, + 504, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 187, + 482 + ], + "score": 1.0, + "content": "estimate of variable", + "type": "text" + }, + { + "bbox": [ + 187, + 469, + 200, + 479 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 467, + 360, + 482 + ], + "score": 1.0, + "content": "conditioned on the parents according to", + "type": "text" + }, + { + "bbox": [ + 361, + 469, + 369, + 479 + ], + "score": 0.65, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 468, + 504, + 481 + ], + "score": 0.88, + "content": ": { \\mathcal { L } } _ { C } ( X _ { i } ) = - \\log f _ { \\phi _ { i } } ( X _ { i } ; C . _ { , i } \\odot", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 107, + 479, + 128, + 491 + ], + "score": 0.83, + "content": "X _ { - i , }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 478, + 279, + 492 + ], + "score": 1.0, + "content": "). The second term of Equation 2 is an", + "type": "text" + }, + { + "bbox": [ + 280, + 480, + 289, + 490 + ], + "score": 0.88, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "-regularizer on the edge probabilities. It acts as a prior,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "selecting the sparsest graph of those with similar likelihood estimates by removing redundant edges.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 421, + 506, + 504 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "Prediction. Alternating between the distribution and graph fitting stages allows us to fine-tune the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "neural networks to the most probable parent sets along the training. 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The orientation", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 349, + 552 + ], + "score": 1.0, + "content": "parameters prevent loops between any two variables, since", + "type": "text" + }, + { + "bbox": [ + 349, + 540, + 376, + 552 + ], + "score": 0.92, + "content": "\\sigma ( \\theta _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 541, + 505, + 552 + ], + "score": 1.0, + "content": "can only be greater than 0.5 in", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 551, + 504, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 504, + 563 + ], + "score": 1.0, + "content": "one direction. Although the orientation parameters do not guarantee the absence of loops with more", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 562, + 499, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 499, + 575 + ], + "score": 1.0, + "content": "variable, we show that under certain conditions ENCO yet converges to the correct, acyclic graph.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 506, + 506, + 575 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 587, + 408, + 598 + ], + "lines": [ + { + "bbox": [ + 105, + 586, + 409, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 409, + 599 + ], + "score": 1.0, + "content": "3.3 LOW-VARIANCE GRADIENT ESTIMATORS FOR EDGE PARAMETERS", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 607, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 104, + 606, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 147, + 621 + ], + "score": 1.0, + "content": "To update", + "type": "text" + }, + { + "bbox": [ + 147, + 609, + 155, + 619 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 606, + 173, + 621 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 173, + 608, + 180, + 617 + ], + "score": 0.78, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 606, + 506, + 621 + ], + "score": 1.0, + "content": "based on Equation 2, we need to determine their gradients through the expectation", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 617, + 507, + 632 + ], + "spans": [ + { + "bbox": [ + 107, + 619, + 142, + 632 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { p _ { \\Upsilon ; \\theta } ( C ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 617, + 174, + 632 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 174, + 619, + 183, + 628 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 617, + 507, + 632 + ], + "score": 1.0, + "content": "is a discrete variable. 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We then evaluate the log likelihoods of all variables for these graphs on the batch, and estimate", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 318, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 107, + 319, + 163, + 332 + ], + "score": 0.92, + "content": "\\mathscr { L } _ { X _ { i } \\to X _ { j } } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 318, + 181, + 333 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 181, + 319, + 237, + 331 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { X _ { i } \\not \\to X _ { j } } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 318, + 335, + 333 + ], + "score": 1.0, + "content": "for all pairs of variables", + "type": "text" + }, + { + "bbox": [ + 335, + 319, + 348, + 330 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 318, + 365, + 333 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 366, + 319, + 379, + 331 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 318, + 506, + 333 + ], + "score": 1.0, + "content": "by simply averaging the results", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 330, + 501, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 464, + 343 + ], + "score": 1.0, + "content": "for the two cases separately. Finally, the estimates are used to determine the gradients for", + "type": "text" + }, + { + "bbox": [ + 464, + 331, + 472, + 342 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 330, + 490, + 343 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 490, + 330, + 497, + 340 + ], + "score": 0.77, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 330, + 501, + 343 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 346, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "Low variance. Previous methods (Ke et al., 2019; Bengio et al., 2020) relied on a different", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "REINFORCE-like estimator proposed by Bengio et al. (2020). Adjusting their estimator to our", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 369, + 392, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 203, + 380 + ], + "score": 1.0, + "content": "setting of the parameter", + "type": "text" + }, + { + "bbox": [ + 203, + 370, + 216, + 381 + ], + "score": 0.86, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 369, + 392, + 380 + ], + "score": 1.0, + "content": ", for instance, the gradient looks as follows:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 383, + 428, + 410 + ], + "lines": [ + { + "bbox": [ + 182, + 383, + 428, + 410 + ], + "spans": [ + { + "bbox": [ + 182, + 383, + 428, + 410 + ], + "score": 0.95, + "content": "g _ { i j } = \\sigma ( \\theta _ { i j } ) \\cdot { \\mathbb { E } } _ { X } \\left[ \\frac { { \\mathbb { E } } _ { C } \\left[ \\left( \\sigma ( \\gamma _ { i j } ) - C _ { i j } \\right) \\cdot \\mathcal { L } _ { C } ( X _ { j } ) \\right] } { { \\mathbb { E } } _ { C } \\left[ \\mathcal { L } _ { C } ( X _ { j } ) \\right] } + \\lambda _ { \\mathrm { s p a r s e } } \\right]", + "type": "interline_equation", + "image_path": "5dab9c3c6883f0624fb48295791b6d777d5f0adca769c60e2802aa177c8fc7a6.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 182, + 383, + 428, + 392.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 182, + 392.0, + 428, + 401.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 182, + 401.0, + 428, + 410.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 348, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 411, + 349, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 133, + 423 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 414, + 146, + 424 + ], + "score": 0.87, + "content": "g _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 411, + 251, + 423 + ], + "score": 1.0, + "content": "represents the gradient of", + "type": "text" + }, + { + "bbox": [ + 252, + 413, + 264, + 424 + ], + "score": 0.88, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 411, + 349, + 423 + ], + "score": 1.0, + "content": ". Performing Monte-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 422, + 348, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 348, + 434 + ], + "score": 1.0, + "content": "Carlo sampling for estimating the gradient leads to a biased", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 433, + 349, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 349, + 445 + ], + "score": 1.0, + "content": "estimate which becomes asymptotically unbiased with in-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 444, + 349, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 349, + 456 + ], + "score": 1.0, + "content": "creasing number of samples (Bengio et al., 2020). The di-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 455, + 348, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 225, + 467 + ], + "score": 1.0, + "content": "vision by the expectation of", + "type": "text" + }, + { + "bbox": [ + 226, + 455, + 260, + 468 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { C } ( \\bar { X } _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 455, + 348, + 467 + ], + "score": 1.0, + "content": "is done for variance", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 466, + 348, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 348, + 478 + ], + "score": 1.0, + "content": "reduction (Mahmood et al., 2014). 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A major benefit of our gradient formulation in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 499, + 349, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 349, + 511 + ], + "score": 1.0, + "content": "Equation 3, instead, is that it removes this noise by con-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 510, + 348, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 348, + 522 + ], + "score": 1.0, + "content": "sidering the difference of the two independent Monte-Carlo", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 521, + 348, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 147, + 534 + ], + "score": 1.0, + "content": "estimates", + "type": "text" + }, + { + "bbox": [ + 147, + 521, + 204, + 534 + ], + "score": 0.93, + "content": "\\mathscr { L } _ { X _ { i } X _ { j } } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 521, + 223, + 534 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 223, + 521, + 279, + 534 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { X _ { i } \\not \\to X _ { j } } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 521, + 348, + 534 + ], + "score": 1.0, + "content": ". 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We", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 599, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 506, + 611 + ], + "score": 1.0, + "content": "show that not only does the global optimum of Equation 2 correspond to the true graph, but also that", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "there exist no other local minima ENCO can converge to. We outline the derivation and proof of these", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 621, + 502, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 502, + 633 + ], + "score": 1.0, + "content": "conditions in Appendix B, and limit our discussion here to the main assumptions and implications.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5 + }, + { + "type": "text", + "bbox": [ + 106, + 637, + 505, + 715 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "To construct a theoretical argument, we make the following assumptions. First, we assume that sparse", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 650, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 505, + 660 + ], + "score": 1.0, + "content": "interventions have been performed on all variables. 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A major benefit of our gradient formulation in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 499, + 349, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 349, + 511 + ], + "score": 1.0, + "content": "Equation 3, instead, is that it removes this noise by con-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 510, + 348, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 348, + 522 + ], + "score": 1.0, + "content": "sidering the difference of the two independent Monte-Carlo", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 521, + 348, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 147, + 534 + ], + "score": 1.0, + "content": "estimates", + "type": "text" + }, + { + "bbox": [ + 147, + 521, + 204, + 534 + ], + "score": 0.93, + "content": "\\mathscr { L } _ { X _ { i } X _ { j } } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 521, + 223, + 534 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 223, + 521, + 279, + 534 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { X _ { i } \\not \\to X _ { j } } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 521, + 348, + 534 + ], + "score": 1.0, + "content": ". Hence, we can", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 532, + 347, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 347, + 543 + ], + "score": 1.0, + "content": "use a smaller sample size than previous methods and attain", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 542, + 347, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 347, + 555 + ], + "score": 1.0, + "content": "10 times lower standard deviation, as visualized in Figure 2.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 411, + 349, + 555 + ] + }, + { + "type": "image", + "bbox": [ + 359, + 415, + 499, + 518 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 359, + 415, + 499, + 518 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 359, + 415, + 499, + 518 + ], + "spans": [ + { + "bbox": [ + 359, + 415, + 499, + 518 + ], + "score": 0.968, + "type": "image", + "image_path": "886c1d9e84f116c2eb1da0e2f11ee3aa375bf2ae8016ed31b7bd4860bf6d775d.jpg" + } + ] + } + ], + "index": 38.0, + "virtual_lines": [ + { + "bbox": [ + 359, + 415, + 499, + 466.5 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 359, + 466.5, + 499, + 518.0 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 355, + 526, + 505, + 559 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 354, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 354, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "Figure 2: ENCO estimates gradients", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 355, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 355, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "of significantly lower variance com-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 353, + 548, + 477, + 560 + ], + "spans": [ + { + "bbox": [ + 353, + 548, + 477, + 560 + ], + "score": 1.0, + "content": "pared to (Bengio et al., 2020).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43 + } + ], + "index": 40.5 + }, + { + "type": "title", + "bbox": [ + 108, + 567, + 257, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 567, + 259, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 259, + 580 + ], + "score": 1.0, + "content": "3.4 CONVERGENCE GUARANTEES", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "Next, we discuss the conditions under which ENCO convergences to the correct causal graph. We", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 599, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 506, + 611 + ], + "score": 1.0, + "content": "show that not only does the global optimum of Equation 2 correspond to the true graph, but also that", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "there exist no other local minima ENCO can converge to. We outline the derivation and proof of these", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 621, + 502, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 502, + 633 + ], + "score": 1.0, + "content": "conditions in Appendix B, and limit our discussion here to the main assumptions and implications.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 587, + 506, + 633 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 637, + 505, + 715 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "To construct a theoretical argument, we make the following assumptions. 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In other words, the parent set", + "type": "text" + }, + { + "bbox": [ + 401, + 671, + 431, + 682 + ], + "score": 0.9, + "content": "\\operatorname { p a } ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "reflects the inputs", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 681, + 503, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 266, + 695 + ], + "score": 1.0, + "content": "to the causal generation mechanism of", + "type": "text" + }, + { + "bbox": [ + 266, + 684, + 278, + 693 + ], + "score": 0.91, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 681, + 495, + 695 + ], + "score": 1.0, + "content": ". 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j } } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 255, + 506, + 269 + ], + "score": 1.0, + "content": "refers to the distribution", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 133, + 266, + 406, + 280 + ], + "spans": [ + { + "bbox": [ + 133, + 266, + 209, + 280 + ], + "score": 1.0, + "content": "over interventions", + "type": "text" + }, + { + "bbox": [ + 210, + 267, + 233, + 279 + ], + "score": 0.93, + "content": "p _ { I } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 266, + 389, + 280 + ], + "score": 1.0, + "content": "excluding the intervention on variable", + "type": "text" + }, + { + "bbox": [ + 389, + 268, + 402, + 279 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 266, + 406, + 280 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 504, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 214, + 296 + ], + "score": 1.0, + "content": "Further, for all other pairs", + "type": "text" + }, + { + "bbox": [ + 214, + 284, + 243, + 296 + ], + "score": 0.9, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 282, + 285, + 296 + ], + "score": 1.0, + "content": "for which", + "type": "text" + }, + { + "bbox": [ + 285, + 284, + 298, + 296 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 282, + 374, + 296 + ], + "score": 1.0, + "content": "is a descendant of", + "type": "text" + }, + { + "bbox": [ + 374, + 284, + 386, + 294 + ], + "score": 0.85, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 282, + 430, + 296 + ], + "score": 1.0, + "content": ", condition", + "type": "text" + }, + { + "bbox": [ + 431, + 284, + 437, + 293 + ], + "score": 0.32, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 282, + 506, + 296 + ], + "score": 1.0, + "content": "and 2 must hold.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "Condition 1 and 2 ensure that the orientations can be learned from interventions. Intuitively, ancestors", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "and descendants in the graph have to be dependent when intervening on the ancestors. This aligns", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 326, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 337 + ], + "score": 1.0, + "content": "with the technical interpretation in Theorem 3.1 that the likelihood estimate of the child variable must", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 335, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 350 + ], + "score": 1.0, + "content": "improve when intervening and conditioning on its ancestor variables. Condition 3 states intuitively", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "that the sparsity regularizer needs to be selected such that it chooses the sparsest graph among those", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 358, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 372 + ], + "score": 1.0, + "content": "graphs with equal joint distributions as the ground truth graph, without trading sparsity for worse", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "distribution estimates. The specific condition in Theorem 3.1 ensures thereby that the set can be", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "learned with a gradient-based algorithm. We emphasize that this condition only gives an upper bound", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 391, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 120, + 405 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 120, + 392, + 146, + 403 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 391, + 506, + 405 + ], + "score": 1.0, + "content": "when sufficiently large datasets are available. In practice, the graph can thus be recovered", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 402, + 507, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 507, + 415 + ], + "score": 1.0, + "content": "with a sufficiently small sparsity regularizer and dependencies among variables under interventions.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 414, + 431, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 431, + 426 + ], + "score": 1.0, + "content": "We provide more details for various settings and further intuition in Appendix B.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "Interventions on fewer variables. It is straightforward to extend ENCO to support interventions", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "on fewer variables. Normally, in the graph fitting stage, we sample one intervention at a time. We", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "can, thus, simply restrict the sampling only to the interventions that are possible (or provided in the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 339, + 476 + ], + "score": 1.0, + "content": "dataset). In this case, we update the orientation parameters", + "type": "text" + }, + { + "bbox": [ + 340, + 463, + 353, + 475 + ], + "score": 0.89, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 462, + 506, + 476 + ], + "score": 1.0, + "content": "of only those edges that connect to an", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 214, + 486 + ], + "score": 1.0, + "content": "intervened variable, either", + "type": "text" + }, + { + "bbox": [ + 214, + 474, + 227, + 485 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 474, + 239, + 486 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 239, + 474, + 253, + 486 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 474, + 505, + 486 + ], + "score": 1.0, + "content": ", as before. For all other orientation parameters, we extend the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 485, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 506, + 497 + ], + "score": 1.0, + "content": "gradient estimator to include interventions on all variables. Although this estimate is more noisy and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 496, + 478, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 478, + 508 + ], + "score": 1.0, + "content": "does not have convergence guarantees, it can still be informative about the edge orientations.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 526 + ], + "score": 1.0, + "content": "Enforcing acyclicity When the conditions are violated, e.g. by limited data, cycles can occur in the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "prediction. Since ENCO learns the orientations as a separate parameter, we can remove cycles by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 535, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 249, + 547 + ], + "score": 1.0, + "content": "finding the global order of variables", + "type": "text" + }, + { + "bbox": [ + 250, + 535, + 284, + 546 + ], + "score": 0.92, + "content": "O \\in S _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 535, + 308, + 547 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 308, + 535, + 323, + 546 + ], + "score": 0.88, + "content": "S _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 535, + 506, + 547 + ], + "score": 1.0, + "content": "being the set of permutations, that maximizes", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 546, + 507, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 302, + 564 + ], + "score": 1.0, + "content": "the pairwise orientation probabilities: arg maxO", + "type": "text" + }, + { + "bbox": [ + 302, + 546, + 403, + 561 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\prod _ { i = 1 } ^ { N } \\prod _ { j = i + 1 } ^ { N } \\sigma ( \\theta _ { O _ { i } , O _ { j } } ) } & { { } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 546, + 507, + 564 + ], + "score": 1.0, + "content": ". This utilizes the learned", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 560, + 494, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 494, + 572 + ], + "score": 1.0, + "content": "ancestor-descendant relations, making the algorithm more robust to noise in single interventions.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 109, + 584, + 282, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 584, + 283, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 283, + 597 + ], + "score": 1.0, + "content": "3.5 HANDLING LATENT CONFOUNDERS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "So far, we have assumed that all variables of the graph are observable and can be intervened on. A", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "common issue in causal discovery is the existence of latent confounders, i.e., an unobserved common", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "cause of two or more variables introducing dependencies between each other. In the presence of latent", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "confounders, structure learning methods may predict false positive edges. Interestingly, in the context", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 293, + 662 + ], + "score": 1.0, + "content": "of ENCO latent confounders for two variables", + "type": "text" + }, + { + "bbox": [ + 293, + 649, + 322, + 661 + ], + "score": 0.92, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "cause a unique pattern of learned parameters.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 195, + 673 + ], + "score": 1.0, + "content": "When intervening on", + "type": "text" + }, + { + "bbox": [ + 195, + 660, + 208, + 671 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 659, + 220, + 673 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 221, + 660, + 234, + 672 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 659, + 506, + 673 + ], + "score": 1.0, + "content": ", having an edge between the two variables is disadvantageous, as", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 203, + 683 + ], + "score": 1.0, + "content": "in the intervened graph", + "type": "text" + }, + { + "bbox": [ + 204, + 671, + 216, + 682 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 671, + 235, + 683 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 235, + 672, + 248, + 683 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "are (conditionally) independent. For interventions on all other", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 682, + 403, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 299, + 694 + ], + "score": 1.0, + "content": "variables, however, an edge can be beneficial as", + "type": "text" + }, + { + "bbox": [ + 299, + 684, + 311, + 693 + ], + "score": 0.9, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 682, + 330, + 694 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 330, + 684, + 343, + 694 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 682, + 403, + 694 + ], + "score": 1.0, + "content": "are correlated.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "Exploiting this, we extend ENCO to detect latent confounders. 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For all other edges besides between", + "type": "text" + }, + { + "bbox": [ + 491, + 721, + 504, + 732 + ], + "score": 0.86, + "content": "X _ { i }", + "type": "inline_equation" + } + ], + "index": 48 + } + ], + "index": 47 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 259, + 95 + ], + "score": 1.0, + "content": "Theorem 3.1. 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In practice, the graph can thus be recovered", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 402, + 507, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 507, + 415 + ], + "score": 1.0, + "content": "with a sufficiently small sparsity regularizer and dependencies among variables under interventions.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 414, + 431, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 431, + 426 + ], + "score": 1.0, + "content": "We provide more details for various settings and further intuition in Appendix B.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 303, + 507, + 426 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "Interventions on fewer variables. 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In this case, we update the orientation parameters", + "type": "text" + }, + { + "bbox": [ + 340, + 463, + 353, + 475 + ], + "score": 0.89, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 462, + 506, + 476 + ], + "score": 1.0, + "content": "of only those edges that connect to an", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 214, + 486 + ], + "score": 1.0, + "content": "intervened variable, either", + "type": "text" + }, + { + "bbox": [ + 214, + 474, + 227, + 485 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 474, + 239, + 486 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 239, + 474, + 253, + 486 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 474, + 505, + 486 + ], + "score": 1.0, + "content": ", as before. For all other orientation parameters, we extend the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 485, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 506, + 497 + ], + "score": 1.0, + "content": "gradient estimator to include interventions on all variables. Although this estimate is more noisy and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 496, + 478, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 478, + 508 + ], + "score": 1.0, + "content": "does not have convergence guarantees, it can still be informative about the edge orientations.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 430, + 506, + 508 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 526 + ], + "score": 1.0, + "content": "Enforcing acyclicity When the conditions are violated, e.g. by limited data, cycles can occur in the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "prediction. Since ENCO learns the orientations as a separate parameter, we can remove cycles by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 535, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 249, + 547 + ], + "score": 1.0, + "content": "finding the global order of variables", + "type": "text" + }, + { + "bbox": [ + 250, + 535, + 284, + 546 + ], + "score": 0.92, + "content": "O \\in S _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 535, + 308, + 547 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 308, + 535, + 323, + 546 + ], + "score": 0.88, + "content": "S _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 535, + 506, + 547 + ], + "score": 1.0, + "content": "being the set of permutations, that maximizes", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 546, + 507, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 302, + 564 + ], + "score": 1.0, + "content": "the pairwise orientation probabilities: arg maxO", + "type": "text" + }, + { + "bbox": [ + 302, + 546, + 403, + 561 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\prod _ { i = 1 } ^ { N } \\prod _ { j = i + 1 } ^ { N } \\sigma ( \\theta _ { O _ { i } , O _ { j } } ) } & { { } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 546, + 507, + 564 + ], + "score": 1.0, + "content": ". This utilizes the learned", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 560, + 494, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 494, + 572 + ], + "score": 1.0, + "content": "ancestor-descendant relations, making the algorithm more robust to noise in single interventions.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 511, + 507, + 572 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 584, + 282, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 584, + 283, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 283, + 597 + ], + "score": 1.0, + "content": "3.5 HANDLING LATENT CONFOUNDERS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "So far, we have assumed that all variables of the graph are observable and can be intervened on. A", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "common issue in causal discovery is the existence of latent confounders, i.e., an unobserved common", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "cause of two or more variables introducing dependencies between each other. In the presence of latent", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "confounders, structure learning methods may predict false positive edges. Interestingly, in the context", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 293, + 662 + ], + "score": 1.0, + "content": "of ENCO latent confounders for two variables", + "type": "text" + }, + { + "bbox": [ + 293, + 649, + 322, + 661 + ], + "score": 0.92, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "cause a unique pattern of learned parameters.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 195, + 673 + ], + "score": 1.0, + "content": "When intervening on", + "type": "text" + }, + { + "bbox": [ + 195, + 660, + 208, + 671 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 659, + 220, + 673 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 221, + 660, + 234, + 672 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 659, + 506, + 673 + ], + "score": 1.0, + "content": ", having an edge between the two variables is disadvantageous, as", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 203, + 683 + ], + "score": 1.0, + "content": "in the intervened graph", + "type": "text" + }, + { + "bbox": [ + 204, + 671, + 216, + 682 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 671, + 235, + 683 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 235, + 672, + 248, + 683 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "are (conditionally) independent. For interventions on all other", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 682, + 403, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 299, + 694 + ], + "score": 1.0, + "content": "variables, however, an edge can be beneficial as", + "type": "text" + }, + { + "bbox": [ + 299, + 684, + 311, + 693 + ], + "score": 0.9, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 682, + 330, + 694 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 330, + 684, + 343, + 694 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 682, + 403, + 694 + ], + "score": 1.0, + "content": "are correlated.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 604, + 506, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "Exploiting this, we extend ENCO to detect latent confounders. We focus on latent confounders", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "between two variables that do not have any direct edges with each other, and assume that the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 721, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 491, + 733 + ], + "score": 1.0, + "content": "confounder is not a child of any other observed variable. For all other edges besides between", + "type": "text" + }, + { + "bbox": [ + 491, + 721, + 504, + 732 + ], + "score": 0.86, + "content": "X _ { i }", + "type": "inline_equation" + } + ], + "index": 48 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 698, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 117, + 504, + 206 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 77, + 505, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 77, + 505, + 91 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 505, + 91 + ], + "score": 1.0, + "content": "Table 1: Comparing structure learning methods in terms of structural hamming distance (SHD) on", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 88, + 505, + 101 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 505, + 101 + ], + "score": 1.0, + "content": "common graph structures (lower is better), averaged over 25 graphs each. ENCO outperforms all", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 99, + 500, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 500, + 112 + ], + "score": 1.0, + "content": "baselines, and by enforcing acyclicity after training, can recover most graphs with minimal errors.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 106, + 117, + 504, + 206 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 117, + 504, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 504, + 206 + ], + "score": 0.983, + "html": "
Graph typebidiagchaincolliderfull junglerandom
GIES33.6 (±7.5)17.5 (±7.3)24.0 (±2.9)216.5 (±15.2)33.1(±2.9)57.5 (±14.2)
IGSP32.7(±5.1)14.6 (±2.3)23.7(±2.3)253.8(±12.6)35.9(±5.2)65.4(±8.0)
GES + Orientating14.8(±2.6)0.5 (±0.7)20.8 (±2.4)282.8(±4.2)14.7(±3.1)60.1(±8.9)
SDI9.0(±2.6)3.9 (±2.0)16.1 (±2.4)153.9 (±10.3)6.9 (±2.3)10.8(±3.9)
DCDI16.9 (±2.0)10.1(±1.1)10.9 (±3.6)21.0 (±4.8)17.9 (±4.1)7.7(±3.2)
ENCO (ours)2.2(±1.4)1.7 (±1.3)1.6 (±1.6)9.2 (±3.4)1.7 (±1.3)4.6 (±1.9)
ENCO-acyclic (ours)0.0 (±0.0)0.0 (±0.0)1.6 (±1.6)5.3 (±2.3)0.6 (±1.1)0.2 (±0.5)
", + "type": "table", + "image_path": "3394bfacd43eafa691aef5bff2cb82de2cfc98fc7a804dee2b0dcea1de4aaa52.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 106, + 117, + 504, + 146.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 106, + 146.66666666666666, + 504, + 176.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 106, + 176.33333333333331, + 504, + 205.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 223, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 124, + 235 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 223, + 138, + 235 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 223, + 505, + 235 + ], + "score": 1.0, + "content": ", we can still rely on the guarantees in Section 3.4 since Equation 7 already includes the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 234, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 246 + ], + "score": 1.0, + "content": "possibility of additional edges in such cases. After convergence, we score every pair of variables on", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 347, + 257 + ], + "score": 1.0, + "content": "how likely they share a latent confounder using a function", + "type": "text" + }, + { + "bbox": [ + 347, + 245, + 366, + 257 + ], + "score": 0.78, + "content": "\\operatorname { l c } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "that is maximized in the scenario", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 252, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 104, + 256, + 302, + 274 + ], + "score": 1.0, + "content": "mentioned above. For this, we define γij = γ(I)ij", + "type": "text" + }, + { + "bbox": [ + 259, + 256, + 333, + 273 + ], + "score": 0.93, + "content": "\\gamma _ { i j } = \\gamma _ { i j } ^ { ( I ) } + \\gamma _ { i j } ^ { ( O ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 252, + 386, + 279 + ], + "score": 1.0, + "content": ") where γ(I)ij i", + "type": "text" + }, + { + "bbox": [ + 380, + 258, + 505, + 272 + ], + "score": 1.0, + "content": "s only updated with gradients", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 100, + 266, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 100, + 266, + 268, + 295 + ], + "score": 1.0, + "content": "from Equation 3 under interventions on", + "type": "text" + }, + { + "bbox": [ + 268, + 275, + 280, + 285 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 266, + 301, + 295 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 302, + 272, + 321, + 288 + ], + "score": 0.92, + "content": "\\gamma _ { i j } ^ { ( O ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 266, + 505, + 295 + ], + "score": 1.0, + "content": "on all others. With this separation, we define", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 286, + 396, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 396, + 298 + ], + "score": 1.0, + "content": "the following score function which is maximized by latent confounders:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 161, + 301, + 449, + 323 + ], + "lines": [ + { + "bbox": [ + 161, + 301, + 449, + 323 + ], + "spans": [ + { + "bbox": [ + 161, + 301, + 449, + 323 + ], + "score": 0.92, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } ) = \\sigma \\left( \\gamma _ { i j } ^ { ( O ) } \\right) \\cdot \\sigma \\left( \\gamma _ { j i } ^ { ( O ) } \\right) \\cdot \\left( 1 - \\sigma \\left( \\gamma _ { i j } ^ { ( I ) } \\right) \\right) \\cdot \\left( 1 - \\sigma \\left( \\gamma _ { j i } ^ { ( I ) } \\right) \\right)", + "type": "interline_equation", + "image_path": "4723b19356180b7cca57524f7b7f04df8949604caadcb77b6d3f01d8fb6d93ec.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 161, + 301, + 449, + 323 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 101, + 325, + 511, + 347 + ], + "spans": [ + { + "bbox": [ + 101, + 325, + 305, + 347 + ], + "score": 1.0, + "content": "To converge to the mentioned values, especially of", + "type": "text" + }, + { + "bbox": [ + 305, + 327, + 325, + 344 + ], + "score": 0.92, + "content": "\\gamma _ { i j } ^ { ( O ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 325, + 511, + 347 + ], + "score": 1.0, + "content": ", we need a similar condition as in Equation 7:", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 341, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 372, + 354 + ], + "score": 1.0, + "content": "the improvement on the log-likelihood estimate gained by the edge", + "type": "text" + }, + { + "bbox": [ + 373, + 342, + 413, + 353 + ], + "score": 0.92, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 341, + 505, + 354 + ], + "score": 1.0, + "content": "and conditioned on all", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 352, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 170, + 367 + ], + "score": 1.0, + "content": "other parents of", + "type": "text" + }, + { + "bbox": [ + 171, + 353, + 184, + 365 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 352, + 276, + 367 + ], + "score": 1.0, + "content": "needs to be larger than", + "type": "text" + }, + { + "bbox": [ + 276, + 353, + 301, + 365 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 352, + 431, + 367 + ], + "score": 1.0, + "content": "on interventional data excluding", + "type": "text" + }, + { + "bbox": [ + 431, + 353, + 443, + 363 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 352, + 461, + 367 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 462, + 352, + 475, + 365 + ], + "score": 0.87, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 352, + 506, + 367 + ], + "score": 1.0, + "content": ". If this", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 362, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 414, + 377 + ], + "score": 1.0, + "content": "is not the case, the sparsity regularizer will instead remove the edge between", + "type": "text" + }, + { + "bbox": [ + 414, + 364, + 427, + 374 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 362, + 445, + 377 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 446, + 364, + 459, + 375 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 362, + 505, + 377 + ], + "score": 1.0, + "content": "preventing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "any false predictions among observed variables. For all other pairs of variables, at least one of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "terms in Equation 8 converges to zero. Thus, we can detect latent confounders by checking whether", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 180, + 408 + ], + "score": 1.0, + "content": "the score function", + "type": "text" + }, + { + "bbox": [ + 181, + 396, + 225, + 408 + ], + "score": 0.92, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 396, + 397, + 408 + ], + "score": 1.0, + "content": "is greater than a threshold hyperparameter", + "type": "text" + }, + { + "bbox": [ + 398, + 396, + 453, + 408 + ], + "score": 0.89, + "content": "\\tau \\in ( 0 . 0 , 1 . 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 396, + 505, + 408 + ], + "score": 1.0, + "content": ". We discuss", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 407, + 472, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 238, + 420 + ], + "score": 1.0, + "content": "possible guarantees in Appendix", + "type": "text" + }, + { + "bbox": [ + 239, + 408, + 246, + 417 + ], + "score": 0.28, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 407, + 472, + 420 + ], + "score": 1.0, + "content": ", and experimentally verify this approach in Section 4.5.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 108, + 435, + 200, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 201, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 201, + 449 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 507, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 507, + 472 + ], + "score": 1.0, + "content": "We evaluate ENCO on structure learning on synthetic datasets for systematic comparisons and real-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "world datasets for benchmarking against other methods in the literature. The experiments focus on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 482, + 507, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 507, + 494 + ], + "score": 1.0, + "content": "graphs with categorical variables, and experiments on continuous data are included in Appendix D.5.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 492, + 428, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 428, + 505 + ], + "score": 1.0, + "content": "Our code is publicly available at https://github.com/phlippe/ENCO.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 108, + 517, + 228, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 229, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 229, + 528 + ], + "score": 1.0, + "content": "4.1 EXPERIMENTAL SETUP", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 537, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "Graphs and datasets. Given a ground-truth causal graphical model, all methods are tasked to recover", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "the original DAG from a set of observational and interventional data points for each variable. In case", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "of synthetic graphs, we follow the setup of Ke et al. (2019) and create the conditional distributions", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 569, + 507, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 507, + 584 + ], + "score": 1.0, + "content": "from neural networks. These networks take as input the categorical values of its variable’s parents,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 582, + 373, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 373, + 594 + ], + "score": 1.0, + "content": "and are initialized orthogonally to output a non-trivial distribution.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 598, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "Baselines. We compare ENCO to GIES (Hauser & Bühlmann, 2012) and IGSP (Wang et al., 2017;", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "Yang et al., 2018) as greedy score-based approaches, and DCDI (Brouillard et al., 2020) and SDI (Ke", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "et al., 2019) as continuous optimization methods. Further, as a common observational baseline, we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "apply GES (Chickering, 2002) on the observational data to obtain a graph skeleton, and orient each", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "edge by learning the skeleton on the corresponding interventional distribution. We perform a separate", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "hyperparameter search for all baselines, and use the same neural network setup for SDI, DCDI, and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 664, + 481, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 481, + 677 + ], + "score": 1.0, + "content": "ENCO. Appendix C provides a detailed overview of the hyperparameters for all experiments.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35 + }, + { + "type": "title", + "bbox": [ + 108, + 690, + 415, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 416, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 416, + 702 + ], + "score": 1.0, + "content": "4.2 CAUSAL STRUCTURE LEARNING ON COMMON GRAPH STRUCTURES", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We first experiment on synthetic graphs. We pick six common graph structures and sample 5,000", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "observational data points and 200 per intervention. The graphs chain and full represent the", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 117, + 504, + 206 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 77, + 505, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 77, + 505, + 91 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 505, + 91 + ], + "score": 1.0, + "content": "Table 1: Comparing structure learning methods in terms of structural hamming distance (SHD) on", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 88, + 505, + 101 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 505, + 101 + ], + "score": 1.0, + "content": "common graph structures (lower is better), averaged over 25 graphs each. ENCO outperforms all", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 99, + 500, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 500, + 112 + ], + "score": 1.0, + "content": "baselines, and by enforcing acyclicity after training, can recover most graphs with minimal errors.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 106, + 117, + 504, + 206 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 117, + 504, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 504, + 206 + ], + "score": 0.983, + "html": "
Graph typebidiagchaincolliderfull junglerandom
GIES33.6 (±7.5)17.5 (±7.3)24.0 (±2.9)216.5 (±15.2)33.1(±2.9)57.5 (±14.2)
IGSP32.7(±5.1)14.6 (±2.3)23.7(±2.3)253.8(±12.6)35.9(±5.2)65.4(±8.0)
GES + Orientating14.8(±2.6)0.5 (±0.7)20.8 (±2.4)282.8(±4.2)14.7(±3.1)60.1(±8.9)
SDI9.0(±2.6)3.9 (±2.0)16.1 (±2.4)153.9 (±10.3)6.9 (±2.3)10.8(±3.9)
DCDI16.9 (±2.0)10.1(±1.1)10.9 (±3.6)21.0 (±4.8)17.9 (±4.1)7.7(±3.2)
ENCO (ours)2.2(±1.4)1.7 (±1.3)1.6 (±1.6)9.2 (±3.4)1.7 (±1.3)4.6 (±1.9)
ENCO-acyclic (ours)0.0 (±0.0)0.0 (±0.0)1.6 (±1.6)5.3 (±2.3)0.6 (±1.1)0.2 (±0.5)
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If this", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 362, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 414, + 377 + ], + "score": 1.0, + "content": "is not the case, the sparsity regularizer will instead remove the edge between", + "type": "text" + }, + { + "bbox": [ + 414, + 364, + 427, + 374 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 362, + 445, + 377 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 446, + 364, + 459, + 375 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 362, + 505, + 377 + ], + "score": 1.0, + "content": "preventing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "any false predictions among observed variables. For all other pairs of variables, at least one of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "terms in Equation 8 converges to zero. Thus, we can detect latent confounders by checking whether", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 180, + 408 + ], + "score": 1.0, + "content": "the score function", + "type": "text" + }, + { + "bbox": [ + 181, + 396, + 225, + 408 + ], + "score": 0.92, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 396, + 397, + 408 + ], + "score": 1.0, + "content": "is greater than a threshold hyperparameter", + "type": "text" + }, + { + "bbox": [ + 398, + 396, + 453, + 408 + ], + "score": 0.89, + "content": "\\tau \\in ( 0 . 0 , 1 . 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 396, + 505, + 408 + ], + "score": 1.0, + "content": ". We discuss", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 407, + 472, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 238, + 420 + ], + "score": 1.0, + "content": "possible guarantees in Appendix", + "type": "text" + }, + { + "bbox": [ + 239, + 408, + 246, + 417 + ], + "score": 0.28, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 407, + 472, + 420 + ], + "score": 1.0, + "content": ", and experimentally verify this approach in Section 4.5.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5, + "bbox_fs": [ + 101, + 325, + 511, + 420 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 435, + 200, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 201, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 201, + 449 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 507, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 507, + 472 + ], + "score": 1.0, + "content": "We evaluate ENCO on structure learning on synthetic datasets for systematic comparisons and real-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "world datasets for benchmarking against other methods in the literature. The experiments focus on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 482, + 507, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 507, + 494 + ], + "score": 1.0, + "content": "graphs with categorical variables, and experiments on continuous data are included in Appendix D.5.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 492, + 428, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 428, + 505 + ], + "score": 1.0, + "content": "Our code is publicly available at https://github.com/phlippe/ENCO.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 459, + 507, + 505 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 517, + 228, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 229, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 229, + 528 + ], + "score": 1.0, + "content": "4.1 EXPERIMENTAL SETUP", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 537, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "Graphs and datasets. Given a ground-truth causal graphical model, all methods are tasked to recover", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "the original DAG from a set of observational and interventional data points for each variable. In case", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "of synthetic graphs, we follow the setup of Ke et al. (2019) and create the conditional distributions", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 569, + 507, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 507, + 584 + ], + "score": 1.0, + "content": "from neural networks. These networks take as input the categorical values of its variable’s parents,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 582, + 373, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 373, + 594 + ], + "score": 1.0, + "content": "and are initialized orthogonally to output a non-trivial distribution.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 537, + 507, + 594 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 598, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "Baselines. We compare ENCO to GIES (Hauser & Bühlmann, 2012) and IGSP (Wang et al., 2017;", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "Yang et al., 2018) as greedy score-based approaches, and DCDI (Brouillard et al., 2020) and SDI (Ke", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "et al., 2019) as continuous optimization methods. Further, as a common observational baseline, we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "apply GES (Chickering, 2002) on the observational data to obtain a graph skeleton, and orient each", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "edge by learning the skeleton on the corresponding interventional distribution. We perform a separate", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "hyperparameter search for all baselines, and use the same neural network setup for SDI, DCDI, and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 664, + 481, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 481, + 677 + ], + "score": 1.0, + "content": "ENCO. Appendix C provides a detailed overview of the hyperparameters for all experiments.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 597, + 505, + 677 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 690, + 415, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 416, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 416, + 702 + ], + "score": 1.0, + "content": "4.2 CAUSAL STRUCTURE LEARNING ON COMMON GRAPH STRUCTURES", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We first experiment on synthetic graphs. We pick six common graph structures and sample 5,000", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "observational data points and 200 per intervention. The graphs chain and full represent the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 196, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 506, + 209 + ], + "score": 1.0, + "content": "minimally- and maximally-connected DAGs. The graph bidiag is a chain with 2-hop connections,", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 207, + 507, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 124, + 220 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 124, + 208, + 161, + 219 + ], + "score": 0.28, + "content": "{ \\dot { \\ ] } } \\ u \\mathrm { n g 1 e }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 161, + 207, + 507, + 220 + ], + "score": 1.0, + "content": "is a tree-like graph. In the collider graph, one node has all other nodes as parents.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 217, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 232 + ], + "score": 1.0, + "content": "Finally, random has a randomly sampled graph structure with a likelihood of 0.3 of two nodes being", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 229, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 505, + 242 + ], + "score": 1.0, + "content": "connected by a direct edge. For each graph structure, we generate 25 graphs with 25 nodes each, on", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "which we report the average performance and standard deviation. Following common practice, we", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "use structural hamming distance (SHD) as evaluation metric. SHD counts the number of edges that", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 262, + 425, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 425, + 275 + ], + "score": 1.0, + "content": "need to be removed, added, or flipped in order to obtain the ground truth graph.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 40.5, + "bbox_fs": [ + 106, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 76, + 504, + 151 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 76, + 504, + 151 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 76, + 504, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 76, + 504, + 151 + ], + "score": 0.955, + "type": "image", + "image_path": "6da330fa0e46687431ef43b89a89cdf23b9fe6c48c6b500c4216eb65af0da59c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 76, + 504, + 101.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 101.0, + 504, + 126.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 126.0, + 504, + 151.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 159, + 505, + 182 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 157, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 505, + 172 + ], + "score": 1.0, + "content": "Figure 4: Experiments on graphs with interventions on fewer variables. Additional graphs are shown", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 170, + 502, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 502, + 183 + ], + "score": 1.0, + "content": "in Appendix D.2. ENCO outperforms DCDI on bidiag and jungle, even for very few interventions.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 196, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 106, + 196, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 506, + 209 + ], + "score": 1.0, + "content": "minimally- and maximally-connected DAGs. The graph bidiag is a chain with 2-hop connections,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 207, + 507, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 124, + 220 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 208, + 161, + 219 + ], + "score": 0.28, + "content": "{ \\dot { \\ ] } } \\ u \\mathrm { n g 1 e }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 207, + 507, + 220 + ], + "score": 1.0, + "content": "is a tree-like graph. In the collider graph, one node has all other nodes as parents.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 217, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 232 + ], + "score": 1.0, + "content": "Finally, random has a randomly sampled graph structure with a likelihood of 0.3 of two nodes being", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 229, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 505, + 242 + ], + "score": 1.0, + "content": "connected by a direct edge. For each graph structure, we generate 25 graphs with 25 nodes each, on", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "which we report the average performance and standard deviation. Following common practice, we", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "use structural hamming distance (SHD) as evaluation metric. SHD counts the number of edges that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 262, + 425, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 425, + 275 + ], + "score": 1.0, + "content": "need to be removed, added, or flipped in order to obtain the ground truth graph.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 279, + 506, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "Table 1 shows that the continuous optimization methods outperform the greedy search approaches on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 290, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 506, + 302 + ], + "score": 1.0, + "content": "categorical variables. SDI works reasonably well on sparse graphs, but struggles with nodes that have", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 300, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 314 + ], + "score": 1.0, + "content": "many parents. DCDI can recover the collider and full graph to a better degree, yet degrades for sparse", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 311, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 104, + 311, + 506, + 325 + ], + "score": 1.0, + "content": "graphs. ENCO performs well on all graph structures, outperforming all baselines. For sparse graphs,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 323, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 335 + ], + "score": 1.0, + "content": "cycles can occur due to limited sample size. However, with enforcing acyclicity, ENCO-acyclic", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 334, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 346 + ], + "score": 1.0, + "content": "is able to recover four out of six graphs with less than one error on average. We further include", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 345, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 506, + 357 + ], + "score": 1.0, + "content": "experiments with various sample sizes in Appendix D.1. While other methods do not reliably recover", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 356, + 507, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 507, + 368 + ], + "score": 1.0, + "content": "the causal graph even for large sample sizes, ENCO attains low errors even with smaller sample sizes.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 383, + 190, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 192, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 192, + 395 + ], + "score": 1.0, + "content": "4.3 SCALABILITY", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 299, + 492 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 300, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 300, + 416 + ], + "score": 1.0, + "content": "Next, we test ENCO on graphs with large sets", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 414, + 300, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 300, + 427 + ], + "score": 1.0, + "content": "of variables. We create random graphs ranging", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 425, + 300, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 128, + 438 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 129, + 426, + 168, + 436 + ], + "score": 0.9, + "content": "N = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 425, + 179, + 438 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 180, + 425, + 226, + 437 + ], + "score": 0.9, + "content": "N = 1 { , } 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 425, + 300, + 438 + ], + "score": 1.0, + "content": "nodes with larger", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 437, + 300, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 300, + 448 + ], + "score": 1.0, + "content": "sample sizes. Every node has on average 8 edges", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 447, + 300, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 300, + 459 + ], + "score": 1.0, + "content": "and a maximum of 10 parents. The challenge of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 459, + 300, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 300, + 470 + ], + "score": 1.0, + "content": "large graphs is that the number of possible edges", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 470, + 299, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 299, + 480 + ], + "score": 1.0, + "content": "grows quadratically and the number of DAGs", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 480, + 300, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 300, + 492 + ], + "score": 1.0, + "content": "super-exponentially, requiring efficient methods.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 497, + 299, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 300, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 300, + 510 + ], + "score": 1.0, + "content": "We compare ENCO to the two best performing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 508, + 300, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 300, + 519 + ], + "score": 1.0, + "content": "baselines from Table 1, SDI and DCDI. All meth-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 519, + 299, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 299, + 531 + ], + "score": 1.0, + "content": "ods were given the same setup of neural networks", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 530, + 300, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 300, + 542 + ], + "score": 1.0, + "content": "and a maximum runtime which corresponds to 30", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 541, + 300, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 300, + 554 + ], + "score": 1.0, + "content": "epochs for ENCO. We plot the SHD over graph", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 552, + 300, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 300, + 564 + ], + "score": 1.0, + "content": "size and runtime in Figure 3. ENCO recovers the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 563, + 300, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 300, + 575 + ], + "score": 1.0, + "content": "causal graphs perfectly with no errors except for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 573, + 300, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 300, + 586 + ], + "score": 1.0, + "content": "the 1,000-node graph, for which it misses only", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5 + }, + { + "type": "image", + "bbox": [ + 309, + 405, + 498, + 530 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 309, + 405, + 498, + 530 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 309, + 405, + 498, + 530 + ], + "spans": [ + { + "bbox": [ + 309, + 405, + 498, + 530 + ], + "score": 0.967, + "type": "image", + "image_path": "a0a2b6c757105f0f64a7a68775247f2003eed08823a435e556cfb7218008d9dd.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 309, + 405, + 498, + 418.8888888888889 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 309, + 418.8888888888889, + 498, + 432.7777777777778 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 309, + 432.7777777777778, + 498, + 446.66666666666674 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 309, + 446.66666666666674, + 498, + 460.55555555555566 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 309, + 460.55555555555566, + 498, + 474.44444444444457 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 309, + 474.44444444444457, + 498, + 488.3333333333335 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 309, + 488.3333333333335, + 498, + 502.2222222222224 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 309, + 502.2222222222224, + 498, + 516.1111111111113 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 309, + 516.1111111111113, + 498, + 530.0000000000002 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 307, + 532, + 505, + 576 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 306, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 306, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "Figure 3: Evaluating SDI, DCDI, and ENCO on", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 306, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 306, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "large graphs in terms of SHD (lower is better).", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 306, + 553, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 306, + 553, + 506, + 565 + ], + "score": 1.0, + "content": "Dots represent single experiments, lines connect", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 306, + 564, + 495, + 576 + ], + "spans": [ + { + "bbox": [ + 306, + 564, + 495, + 576 + ], + "score": 1.0, + "content": "the averages. DCDI ran OOM for 1000 nodes.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5 + } + ], + "index": 44.25 + }, + { + "type": "text", + "bbox": [ + 108, + 585, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "one out of 1 million edges in 2 out of 10 experiments. SDI and DCDI achieve considerably worse", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "performance. This shows that ENCO can efficiently be applied to 1,000 variables while maintaining", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 607, + 468, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 468, + 619 + ], + "score": 1.0, + "content": "its convergence guarantees, underlining the benefit of its low-variance gradient estimators.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51 + }, + { + "type": "title", + "bbox": [ + 108, + 634, + 298, + 644 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 300, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 300, + 646 + ], + "score": 1.0, + "content": "4.4 INTERVENTIONS ON FEWER VARIABLES", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 53 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "We perform experiments on the same datasets as in Section 4.2, but provide interventional data only", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "for a randomly sampled subset of the 25 variables of each graph. We compare ENCO to DCDI,", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "which supports partial intervention sets, and plot the SHD over the number of intervened variables in", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Figure 4. Despite ENCO’s guarantees only holding for full interventions, it is still competitive and", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "outperforms DCDI in most settings. Importantly, enforcing acyclicity has an even greater impact on", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "fewer interventions as more orientations are trained on non-adjacent interventions (see Appendix B.4", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 720, + 507, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 507, + 734 + ], + "score": 1.0, + "content": "for detailed discussion). We conclude that ENCO works competitively with partial interventions too.", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 57 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 76, + 504, + 151 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 76, + 504, + 151 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 76, + 504, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 76, + 504, + 151 + ], + "score": 0.955, + "type": "image", + "image_path": "6da330fa0e46687431ef43b89a89cdf23b9fe6c48c6b500c4216eb65af0da59c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 76, + 504, + 101.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 101.0, + 504, + 126.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 126.0, + 504, + 151.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 159, + 505, + 182 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 157, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 505, + 172 + ], + "score": 1.0, + "content": "Figure 4: Experiments on graphs with interventions on fewer variables. Additional graphs are shown", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 170, + 502, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 502, + 183 + ], + "score": 1.0, + "content": "in Appendix D.2. ENCO outperforms DCDI on bidiag and jungle, even for very few interventions.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 196, + 505, + 273 + ], + "lines": [], + "index": 8, + "bbox_fs": [ + 105, + 196, + 507, + 275 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 279, + 506, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "Table 1 shows that the continuous optimization methods outperform the greedy search approaches on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 290, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 506, + 302 + ], + "score": 1.0, + "content": "categorical variables. SDI works reasonably well on sparse graphs, but struggles with nodes that have", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 300, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 314 + ], + "score": 1.0, + "content": "many parents. DCDI can recover the collider and full graph to a better degree, yet degrades for sparse", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 311, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 104, + 311, + 506, + 325 + ], + "score": 1.0, + "content": "graphs. ENCO performs well on all graph structures, outperforming all baselines. For sparse graphs,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 323, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 335 + ], + "score": 1.0, + "content": "cycles can occur due to limited sample size. However, with enforcing acyclicity, ENCO-acyclic", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 334, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 346 + ], + "score": 1.0, + "content": "is able to recover four out of six graphs with less than one error on average. We further include", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 345, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 506, + 357 + ], + "score": 1.0, + "content": "experiments with various sample sizes in Appendix D.1. While other methods do not reliably recover", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 356, + 507, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 507, + 368 + ], + "score": 1.0, + "content": "the causal graph even for large sample sizes, ENCO attains low errors even with smaller sample sizes.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 278, + 507, + 368 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 383, + 190, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 192, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 192, + 395 + ], + "score": 1.0, + "content": "4.3 SCALABILITY", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 299, + 492 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 300, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 300, + 416 + ], + "score": 1.0, + "content": "Next, we test ENCO on graphs with large sets", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 414, + 300, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 300, + 427 + ], + "score": 1.0, + "content": "of variables. 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Every node has on average 8 edges", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 447, + 300, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 300, + 459 + ], + "score": 1.0, + "content": "and a maximum of 10 parents. The challenge of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 459, + 300, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 300, + 470 + ], + "score": 1.0, + "content": "large graphs is that the number of possible edges", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 470, + 299, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 299, + 480 + ], + "score": 1.0, + "content": "grows quadratically and the number of DAGs", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 480, + 300, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 300, + 492 + ], + "score": 1.0, + "content": "super-exponentially, requiring efficient methods.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 403, + 300, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 497, + 299, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 300, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 300, + 510 + ], + "score": 1.0, + "content": "We compare ENCO to the two best performing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 508, + 300, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 300, + 519 + ], + "score": 1.0, + "content": "baselines from Table 1, SDI and DCDI. 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DCDI ran OOM for 1000 nodes.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5 + } + ], + "index": 44.25 + }, + { + "type": "text", + "bbox": [ + 108, + 585, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "one out of 1 million edges in 2 out of 10 experiments. SDI and DCDI achieve considerably worse", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "performance. This shows that ENCO can efficiently be applied to 1,000 variables while maintaining", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 607, + 468, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 468, + 619 + ], + "score": 1.0, + "content": "its convergence guarantees, underlining the benefit of its low-variance gradient estimators.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51, + "bbox_fs": [ + 106, + 585, + 505, + 619 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 634, + 298, + 644 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 300, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 300, + 646 + ], + "score": 1.0, + "content": "4.4 INTERVENTIONS ON FEWER VARIABLES", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 53 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "We perform experiments on the same datasets as in Section 4.2, but provide interventional data only", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "for a randomly sampled subset of the 25 variables of each graph. We compare ENCO to DCDI,", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "which supports partial intervention sets, and plot the SHD over the number of intervened variables in", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Figure 4. Despite ENCO’s guarantees only holding for full interventions, it is still competitive and", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "outperforms DCDI in most settings. Importantly, enforcing acyclicity has an even greater impact on", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "fewer interventions as more orientations are trained on non-adjacent interventions (see Appendix B.4", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 720, + 507, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 507, + 734 + ], + "score": 1.0, + "content": "for detailed discussion). We conclude that ENCO works competitively with partial interventions too.", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 57, + "bbox_fs": [ + 105, + 655, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 120, + 502, + 186 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 114 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "score": 1.0, + "content": "Table 3: Results on graphs from the BnLearn library measured in structural hamming distance (lower", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 505, + 104 + ], + "score": 1.0, + "content": "is better). Results are averaged over 5 seeds with standard deviations listed in Appendix C.5. Despite", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 102, + 474, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 474, + 114 + ], + "score": 1.0, + "content": "deterministic variables and rare events, ENCO can recover all graphs with almost no errors.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 107, + 120, + 502, + 186 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 120, + 502, + 186 + ], + "spans": [ + { + "bbox": [ + 107, + 120, + 502, + 186 + ], + "score": 0.977, + "html": "
Datasetcancer (5 nodes)asia (8 nodes)sachs (11 nodes)child (20 nodes)alarm (37 nodes)diabetes (413 nodes)pigs (44i nodes)
SDI3.04.07.011.224.4422.418.0
DCDI4.05.05.48.430.0--
ENCO (Ours)0.00.00.00.01.02.00.0
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The dataset is generated in the same way as", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 255, + 325, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 325, + 266 + ], + "score": 1.0, + "content": "before, except that we remove the latent variable from", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 266, + 326, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 326, + 277 + ], + "score": 1.0, + "content": "the input data and increase the observational and inter-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 277, + 325, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 325, + 289 + ], + "score": 1.0, + "content": "ventional sample size (see Appendix C.3 for ablation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 288, + 326, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 326, + 300 + ], + "score": 1.0, + "content": "studies). 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We choose", + "type": "text" + }, + { + "bbox": [ + 276, + 311, + 309, + 320 + ], + "score": 0.88, + "content": "\\tau = 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 310, + 325, + 322 + ], + "score": 1.0, + "content": "but", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11 + }, + { + "type": "table", + "bbox": [ + 333, + 259, + 502, + 312 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 332, + 224, + 505, + 258 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 331, + 223, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 331, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "Table 2: Results of ENCO on detecting la-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 331, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 331, + 235, + 505, + 248 + ], + "score": 1.0, + "content": "tent confounders. The missed confounders", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 332, + 246, + 478, + 258 + ], + "spans": [ + { + "bbox": [ + 332, + 246, + 478, + 258 + ], + "score": 1.0, + "content": "do not affect other edge predictions.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "table_body", + "bbox": [ + 333, + 259, + 502, + 312 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 333, + 259, + 502, + 312 + ], + "spans": [ + { + "bbox": [ + 333, + 259, + 502, + 312 + ], + "score": 0.971, + "html": "
MetricsENCO
SHD0.0 (±0.0)
Confounder recall96.8% (±9.5%)
Confounder precision100.0% (±0.0%)
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As shown in Table 2,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 330, + 507, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 210, + 345 + ], + "score": 1.0, + "content": "ENCO detects more than", + "type": "text" + }, + { + "bbox": [ + 210, + 332, + 230, + 343 + ], + "score": 0.88, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 330, + 507, + 345 + ], + "score": 1.0, + "content": "of the latent confounders without any false positives. What is more,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 342, + 478, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 478, + 356 + ], + "score": 1.0, + "content": "the few mistakes do not affect the detection of all other edges, which are recovered perfectly.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 369, + 258, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 259, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 259, + 381 + ], + "score": 1.0, + "content": "4.6 REAL-WORLD INSPIRED DATA", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "Finally, we evaluate ENCO on causal graphs from the Bayesian Network Repository (BnLearn)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "score": 1.0, + "content": "(Scutari, 2010). The repository contains graphs inspired by real-world applications that are used as", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "score": 1.0, + "content": "benchmarks in literature. In comparison to the synthetic graphs, the real-world graphs are sparser", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 422, + 507, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 507, + 436 + ], + "score": 1.0, + "content": "with a maximum of 6 parents per node and contain nodes with strongly peaked marginal distributions.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 433, + 479, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 479, + 448 + ], + "score": 1.0, + "content": "They also include deterministic variables, making the task challenging even for small graphs.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "We evaluate ENCO, SDI, and DCDI on 7 graphs with increasing sizes, see Table 3. We observe", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "that ENCO recovers almost all real-world causal graphs without errors, independent of their size. In", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 474, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "contrast, SDI suffers from more mistakes as the graphs become larger. An exception is pigs (Scutari,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 482, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 498 + ], + "score": 1.0, + "content": "2010), which has a maximum of 2 parents per node, and hence is easier to learn. The most challenging", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 495, + 507, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 507, + 507 + ], + "score": 1.0, + "content": "graph is diabetes (Andreassen et al., 1991) due to its large size and many deterministic variables.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "ENCO makes only two mistakes, showing that it can handle deterministic variables well. We discuss", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "results on small sample sizes in Appendix C.5, observing similar trends. We conclude that ENCO can", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 528, + 500, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 500, + 540 + ], + "score": 1.0, + "content": "reliably perform structure learning on a wide variety of settings, including deterministic variables.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 108, + 556, + 195, + 570 + ], + "lines": [ + { + "bbox": [ + 104, + 555, + 198, + 573 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 198, + 573 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "We propose ENCO, an efficient causal structure learning method leveraging observational and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "interventional data. Compared to previous work, ENCO models the edge orientations as separate", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "parameters and uses an objective unconstrained with respect to acyclicity. This allows for easier", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "optimization and low-variance gradient estimators while having convergence guarantees. As a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "consequence, the algorithm can efficiently scale to graphs that are at least one order of magnitude", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "larger graphs than what was possible. Experiments corroborate the capabilities of ENCO compared", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "to the state-of-the-art on an extensive array of settings on graph sizes, sizes of observational and", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 660, + 471, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 471, + 672 + ], + "score": 1.0, + "content": "interventional data, latent confounding, as well as on both partial and full intervention sets.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "Limitations. The convergence guarantees of ENCO require interventions on all variables, although", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "experiments on fewer interventions have shown promising results. Future work includes investigating", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "guarantee extensions of ENCO to this setting. A second limitation is that the orientations are", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 199, + 723 + ], + "score": 1.0, + "content": "missing transitivity: if", + "type": "text" + }, + { + "bbox": [ + 199, + 710, + 239, + 721 + ], + "score": 0.91, + "content": "X _ { 1 } \\succ X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 708, + 258, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 258, + 710, + 298, + 721 + ], + "score": 0.91, + "content": "X _ { 2 } \\succ X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 708, + 321, + 723 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 322, + 710, + 362, + 721 + ], + "score": 0.92, + "content": "X _ { 1 } \\succ X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "must hold. A potential direction is", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "incorporating transitive relations for improving convergence speed and results on fewer interventions.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 51 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 120, + 502, + 186 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 114 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "score": 1.0, + "content": "Table 3: Results on graphs from the BnLearn library measured in structural hamming distance (lower", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 505, + 104 + ], + "score": 1.0, + "content": "is better). Results are averaged over 5 seeds with standard deviations listed in Appendix C.5. Despite", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 102, + 474, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 474, + 114 + ], + "score": 1.0, + "content": "deterministic variables and rare events, ENCO can recover all graphs with almost no errors.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 107, + 120, + 502, + 186 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 120, + 502, + 186 + ], + "spans": [ + { + "bbox": [ + 107, + 120, + 502, + 186 + ], + "score": 0.977, + "html": "
Datasetcancer (5 nodes)asia (8 nodes)sachs (11 nodes)child (20 nodes)alarm (37 nodes)diabetes (413 nodes)pigs (44i nodes)
SDI3.04.07.011.224.4422.418.0
DCDI4.05.05.48.430.0--
ENCO (Ours)0.00.00.00.01.02.00.0
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We choose", + "type": "text" + }, + { + "bbox": [ + 276, + 311, + 309, + 320 + ], + "score": 0.88, + "content": "\\tau = 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 310, + 325, + 322 + ], + "score": 1.0, + "content": "but", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 321, + 507, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 411, + 333 + ], + "score": 1.0, + "content": "verify in Appendix C.3 that the method is not sensitive to the specific value of", + "type": "text" + }, + { + "bbox": [ + 411, + 324, + 418, + 331 + ], + "score": 0.68, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 321, + 507, + 333 + ], + "score": 1.0, + "content": ". As shown in Table 2,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 330, + 507, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 210, + 345 + ], + "score": 1.0, + "content": "ENCO detects more than", + "type": "text" + }, + { + "bbox": [ + 210, + 332, + 230, + 343 + ], + "score": 0.88, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 330, + 507, + 345 + ], + "score": 1.0, + "content": "of the latent confounders without any false positives. What is more,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 342, + 478, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 478, + 356 + ], + "score": 1.0, + "content": "the few mistakes do not affect the detection of all other edges, which are recovered perfectly.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 222, + 326, + 322 + ] + }, + { + "type": "table", + "bbox": [ + 333, + 259, + 502, + 312 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 332, + 224, + 505, + 258 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 331, + 223, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 331, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "Table 2: Results of ENCO on detecting la-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 331, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 331, + 235, + 505, + 248 + ], + "score": 1.0, + "content": "tent confounders. The missed confounders", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 332, + 246, + 478, + 258 + ], + "spans": [ + { + "bbox": [ + 332, + 246, + 478, + 258 + ], + "score": 1.0, + "content": "do not affect other edge predictions.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "table_body", + "bbox": [ + 333, + 259, + 502, + 312 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 333, + 259, + 502, + 312 + ], + "spans": [ + { + "bbox": [ + 333, + 259, + 502, + 312 + ], + "score": 0.971, + "html": "
MetricsENCO
SHD0.0 (±0.0)
Confounder recall96.8% (±9.5%)
Confounder precision100.0% (±0.0%)
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The repository contains graphs inspired by real-world applications that are used as", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "score": 1.0, + "content": "benchmarks in literature. In comparison to the synthetic graphs, the real-world graphs are sparser", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 422, + 507, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 507, + 436 + ], + "score": 1.0, + "content": "with a maximum of 6 parents per node and contain nodes with strongly peaked marginal distributions.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 433, + 479, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 479, + 448 + ], + "score": 1.0, + "content": "They also include deterministic variables, making the task challenging even for small graphs.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 389, + 507, + 448 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "We evaluate ENCO, SDI, and DCDI on 7 graphs with increasing sizes, see Table 3. We observe", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "that ENCO recovers almost all real-world causal graphs without errors, independent of their size. In", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 474, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "contrast, SDI suffers from more mistakes as the graphs become larger. An exception is pigs (Scutari,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 482, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 498 + ], + "score": 1.0, + "content": "2010), which has a maximum of 2 parents per node, and hence is easier to learn. The most challenging", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 495, + 507, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 507, + 507 + ], + "score": 1.0, + "content": "graph is diabetes (Andreassen et al., 1991) due to its large size and many deterministic variables.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "ENCO makes only two mistakes, showing that it can handle deterministic variables well. We discuss", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "results on small sample sizes in Appendix C.5, observing similar trends. We conclude that ENCO can", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 528, + 500, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 500, + 540 + ], + "score": 1.0, + "content": "reliably perform structure learning on a wide variety of settings, including deterministic variables.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 451, + 507, + 540 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 556, + 195, + 570 + ], + "lines": [ + { + "bbox": [ + 104, + 555, + 198, + 573 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 198, + 573 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "We propose ENCO, an efficient causal structure learning method leveraging observational and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "interventional data. Compared to previous work, ENCO models the edge orientations as separate", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "parameters and uses an objective unconstrained with respect to acyclicity. This allows for easier", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "optimization and low-variance gradient estimators while having convergence guarantees. As a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "consequence, the algorithm can efficiently scale to graphs that are at least one order of magnitude", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "larger graphs than what was possible. Experiments corroborate the capabilities of ENCO compared", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "to the state-of-the-art on an extensive array of settings on graph sizes, sizes of observational and", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 660, + 471, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 471, + 672 + ], + "score": 1.0, + "content": "interventional data, latent confounding, as well as on both partial and full intervention sets.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 583, + 506, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "Limitations. The convergence guarantees of ENCO require interventions on all variables, although", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "experiments on fewer interventions have shown promising results. Future work includes investigating", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "guarantee extensions of ENCO to this setting. A second limitation is that the orientations are", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 199, + 723 + ], + "score": 1.0, + "content": "missing transitivity: if", + "type": "text" + }, + { + "bbox": [ + 199, + 710, + 239, + 721 + ], + "score": 0.91, + "content": "X _ { 1 } \\succ X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 708, + 258, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 258, + 710, + 298, + 721 + ], + "score": 0.91, + "content": "X _ { 2 } \\succ X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 708, + 321, + 723 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 322, + 710, + 362, + 721 + ], + "score": 0.92, + "content": "X _ { 1 } \\succ X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "must hold. A potential direction is", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "incorporating transitive relations for improving convergence speed and results on fewer interventions.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 51, + "bbox_fs": [ + 105, + 676, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 210, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 213, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 213, + 95 + ], + "score": 1.0, + "content": "ETHICS STATEMENT", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "Causal structure learning algorithms such as the proposed method are mainly used to uncover and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "understand causal mechanisms from data. The knowledge of the underlying causal mechanisms can", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "then be applied to decide on specific actions that influence variables or factors in a desired way. For", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "score": 1.0, + "content": "instance, by knowing that the environmental pollution in a city has an impact on the risk of cancer", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "of its residents, one can try to reduce the pollution to decrease the risk of cancer. The applications", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "of causal structure learning are ranging across many scientific disciplines, including computational", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 170, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 184 + ], + "score": 1.0, + "content": "biology (Friedman et al., 2000; Sachs et al., 2005; Opgen-Rhein & Strimmer, 2007), epidemiology", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "(Robins et al., 2000; Vandenbroucke et al., 2016), and economics (Pearl, 2009; Hicks et al., 1980). We", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 193, + 504, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 504, + 205 + ], + "score": 1.0, + "content": "envision that our work can have positive impacts on those fields. One example we want to highlight", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "is the field of genomics. Recent advances have enabled to perform gene knockdown experiments", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "in a large scale, providing large amounts of interventional data (Dixit et al., 2016; Macosko et al.,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "2015). Gaining insights into how specific genes and diseases interact can lead to the development of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 248 + ], + "score": 1.0, + "content": "novel pharmaceutic methods for treating current diseases. Since the number of variables in those", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "experiments is tremendous, efficient causal structure learning algorithms are needed. The proposed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "score": 1.0, + "content": "method constitutes a first step towards this goal, and our work can foster future work for creating", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 285, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 285, + 282 + ], + "score": 1.0, + "content": "algorithms scaling beyond 10,000 variables.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 287, + 506, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 300 + ], + "score": 1.0, + "content": "Since the possible applications are fairly wide-ranging, there might be potential impacts we cannot", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 507, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 507, + 311 + ], + "score": 1.0, + "content": "forecast at the current time. This includes misuses of the method for unethical purposes. For instance,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "the method can be used to justify gender and race as causes for irrelevant variables if the output is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 320, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 506, + 331 + ], + "score": 1.0, + "content": "misinterpreted, initial assumptions of the model are ignored, or the input data has been manipulated.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 330, + 507, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 507, + 343 + ], + "score": 1.0, + "content": "Hence, the obligation to use this method in a correct way within ethical boundaries lies on the user.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 342, + 389, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 389, + 353 + ], + "score": 1.0, + "content": "We emphasize this responsibility of the user in the license of our code.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 108, + 369, + 267, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 368, + 270, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 270, + 382 + ], + "score": 1.0, + "content": "REPRODUCIBILITY STATEMENT", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 393, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 405 + ], + "score": 1.0, + "content": "To ensure reproducibility, we have published the source code of the proposed method ENCO at", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "https://github.com/phlippe/ENCO. The code includes instructions on how to download", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 415, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 427 + ], + "score": 1.0, + "content": "the datasets, and reproduce the experiments in Section 4 and additional experiments in Appendix D.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "Further, for all experiments of Section 4, we have included a detailed overview in Appendix C of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 436, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 450 + ], + "score": 1.0, + "content": "(a) the used data and its generation process, (b) all hyperparameters used for all methods, and (c)", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "additional details on the results. All experiments have been repeated with 5 to 25 seeds to obtain", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 457, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 472 + ], + "score": 1.0, + "content": "stable, reproducible results. Appendix C.1.2 outlines the packages that have been used for running", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 469, + 163, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 163, + 481 + ], + "score": 1.0, + "content": "the baselines.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 108, + 486, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "The computation resources deployed for all experiments are a 24-core CPU with a single NVIDIA", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "RTX3090 GPU. All experiments can be reproduced on a computer with a single GPU, and only the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 521 + ], + "score": 1.0, + "content": "experiments on graphs larger than 100 variables require a GPU memory of about 24GB. The other", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 520, + 333, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 333, + 531 + ], + "score": 1.0, + "content": "experiments can be performed on smaller GPUs as well.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 108, + 543, + 205, + 553 + ], + "lines": [ + { + "bbox": [ + 107, + 543, + 206, + 554 + ], + "spans": [ + { + "bbox": [ + 107, + 543, + 206, + 554 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "This work is financially supported by Qualcomm Technologies Inc., the University of Amsterdam", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "and the allowance Top consortia for Knowledge and Innovation (TKIs) from the Netherlands Ministry", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "score": 1.0, + "content": "of Economic Affairs and Climate Policy. We thank Pascal Mettes, Christina Winkler, and Sara", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "Magliacane for their valuable comments and feedback on an initial draft of this work. We thank", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "the anonymous reviewers for the reviews, suggestions, and engaging discussions which helped to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "improve this work further. Finally, we thank SURFsara for the support in using the Lisa Compute", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 626, + 140, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 140, + 640 + ], + "score": 1.0, + "content": "Cluster.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + }, + { + "type": "title", + "bbox": [ + 107, + 659, + 175, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 659, + 176, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 176, + 672 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 109, + 677, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 108, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 108, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "Steen Andreassen, Roman Hovorka, Jonathan Benn, Kristian G. Olesen, and Ewart R. Carson.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 115, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 115, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "A model-based approach to insulin adjustment. In M. Stefanelli, A. Hasman, M. Fieschi, and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 115, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 115, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "J. Talmon (eds.), Lecture Notes in Medical Informatics, volume 44 of Lecture Notes in Medical", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 115, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 115, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Informatics, pp. 239–249, Germany, 1991. Springer. 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The knowledge of the underlying causal mechanisms can", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "then be applied to decide on specific actions that influence variables or factors in a desired way. For", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "score": 1.0, + "content": "instance, by knowing that the environmental pollution in a city has an impact on the risk of cancer", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "of its residents, one can try to reduce the pollution to decrease the risk of cancer. 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We", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 193, + 504, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 504, + 205 + ], + "score": 1.0, + "content": "envision that our work can have positive impacts on those fields. One example we want to highlight", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "is the field of genomics. 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This includes misuses of the method for unethical purposes. 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Appendix C.1.2 outlines the packages that have been used for running", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 469, + 163, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 163, + 481 + ], + "score": 1.0, + "content": "the baselines.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 393, + 506, + 481 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 486, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "The computation resources deployed for all experiments are a 24-core CPU with a single NVIDIA", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "RTX3090 GPU. All experiments can be reproduced on a computer with a single GPU, and only the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 521 + ], + "score": 1.0, + "content": "experiments on graphs larger than 100 variables require a GPU memory of about 24GB. 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We thank Pascal Mettes, Christina Winkler, and Sara", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "Magliacane for their valuable comments and feedback on an initial draft of this work. We thank", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "the anonymous reviewers for the reviews, suggestions, and engaging discussions which helped to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "improve this work further. Finally, we thank SURFsara for the support in using the Lisa Compute", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 626, + 140, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 140, + 640 + ], + "score": 1.0, + "content": "Cluster.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 561, + 506, + 640 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 659, + 175, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 659, + 176, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 176, + 672 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 109, + 677, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 108, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 108, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "Steen Andreassen, Roman Hovorka, Jonathan Benn, Kristian G. Olesen, and Ewart R. 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This distribution can be set", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 141, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "as a hyperparameter to weight certain interventions higher than others. 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C . _ { , i } \\odot X _ { - i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 308, + 343, + 321 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 133, + 322, + 405, + 336 + ], + "spans": [ + { + "bbox": [ + 133, + 322, + 142, + 336 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 322, + 167, + 335 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 322, + 405, + 336 + ], + "score": 1.0, + "content": "is a hyperparameter representing the regularization weight.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 342, + 503, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "Based on this objective, we derive the gradient estimators for optimizing both edge existence and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 354, + 201, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 201, + 366 + ], + "score": 1.0, + "content": "orientation parameters.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 378, + 406, + 389 + ], + "lines": [ + { + "bbox": [ + 106, + 378, + 406, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 406, + 390 + ], + "score": 1.0, + "content": "A.1 LOW-VARIANCE GRADIENT ESTIMATOR FOR EDGE PARAMETERS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 506, + 449 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 504, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 442, + 411 + ], + "score": 1.0, + "content": "In order to optimize the edge parameters via SGD, we need to determine the gradient", + "type": "text" + }, + { + "bbox": [ + 442, + 397, + 467, + 414 + ], + "score": 0.93, + "content": "\\begin{array} { r l r } { { \\frac { \\partial } { \\partial \\gamma _ { i j } } \\tilde { \\mathcal { L } } } } & { { } } & { } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 396, + 495, + 411 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 496, + 398, + 504, + 408 + ], + "score": 0.77, + "content": "\\mathcal { L }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 412, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 424 + ], + "score": 1.0, + "content": "consists of a sum of two terms, i.e., the log-likelihood estimate and the regularization, we can look at", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 423, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 419, + 434 + ], + "score": 1.0, + "content": "both parts separately. To prevent any confusion of index variables, we will use", + "type": "text" + }, + { + "bbox": [ + 420, + 423, + 434, + 434 + ], + "score": 0.88, + "content": "k , l", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 423, + 506, + 434 + ], + "score": 1.0, + "content": "as indices for the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 103, + 429, + 477, + 454 + ], + "spans": [ + { + "bbox": [ + 103, + 429, + 149, + 454 + ], + "score": 1.0, + "content": "parameter", + "type": "text" + }, + { + "bbox": [ + 149, + 437, + 163, + 447 + ], + "score": 0.86, + "content": "\\gamma _ { k l }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 429, + 331, + 454 + ], + "score": 1.0, + "content": "for which we determine the gradient, i.e.,", + "type": "text" + }, + { + "bbox": [ + 331, + 433, + 357, + 450 + ], + "score": 0.92, + "content": "\\frac { \\partial } { \\partial \\gamma _ { k l } } \\tilde { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 429, + 377, + 454 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 378, + 435, + 391, + 447 + ], + "score": 0.88, + "content": "i , j", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 429, + 477, + 454 + ], + "score": 1.0, + "content": "as indices for sums.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 467 + ], + "score": 1.0, + "content": "As a first step, we determine the gradients for the regularization term. Those can be found by taking", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 464, + 225, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 225, + 476 + ], + "score": 1.0, + "content": "the derivative of the sigmoid:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 479, + 450, + 515 + ], + "lines": [ + { + "bbox": [ + 162, + 479, + 450, + 515 + ], + "spans": [ + { + "bbox": [ + 162, + 479, + 450, + 515 + ], + "score": 0.94, + "content": "\\frac { \\partial } { \\partial \\gamma _ { k l } } \\lambda _ { \\mathrm { s p a r s e } } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } \\sigma ( \\gamma _ { i j } ) \\sigma ( \\theta _ { i j } ) = \\sigma ( \\gamma _ { k l } ) \\cdot ( 1 - \\sigma ( \\gamma _ { k l } ) ) \\cdot \\sigma ( \\theta _ { k l } ) \\lambda _ { \\mathrm { s p a r s e } }", + "type": "interline_equation", + "image_path": "90301915a25fac37131703afc2c466de3beec8265fccfa244d9e189b4e3b02f8.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 162, + 479, + 450, + 491.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 162, + 491.0, + 450, + 503.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 162, + 503.0, + 450, + 515.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 517, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 467, + 531 + ], + "score": 1.0, + "content": "Thus, it is straight-forward to calculate for any edge parameter. In the following, we use", + "type": "text" + }, + { + "bbox": [ + 468, + 518, + 493, + 530 + ], + "score": 0.92, + "content": "\\sigma ^ { \\prime } ( \\ldots )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 528, + 388, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 262, + 543 + ], + "score": 1.0, + "content": "abbreviate the derivate of the sigmoid:", + "type": "text" + }, + { + "bbox": [ + 263, + 529, + 383, + 541 + ], + "score": 0.91, + "content": "\\sigma ^ { \\prime } ( \\gamma _ { k l } ) = \\sigma ( \\gamma _ { k l } ) ( 1 - \\sigma ( \\gamma _ { k l } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 528, + 388, + 543 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "score": 1.0, + "content": "For the log-likelihood term, we start by reorganizing the expectations to simplify the gradient", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 225, + 569 + ], + "score": 1.0, + "content": "expression. The derivate term", + "type": "text" + }, + { + "bbox": [ + 226, + 556, + 244, + 571 + ], + "score": 0.93, + "content": "\\frac { \\partial } { \\partial \\gamma _ { k l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "can be moved inside the two expectations over interventional data", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 569, + 400, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 400, + 582 + ], + "score": 1.0, + "content": "since those are independent of the graph parameters. Thus, we can write:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 583, + 420, + 618 + ], + "lines": [ + { + "bbox": [ + 191, + 583, + 420, + 618 + ], + "spans": [ + { + "bbox": [ + 191, + 583, + 420, + 618 + ], + "score": 0.93, + "content": "\\frac { \\partial } { \\partial \\gamma _ { k l } } \\tilde { \\mathcal { L } } ^ { \\prime } = \\mathbb { E } _ { \\hat { I } \\sim p _ { I } ( I ) } \\mathbb { E } _ { \\tilde { p } _ { \\hat { I } } ( \\pmb { X } ) } \\frac { \\partial } { \\partial \\gamma _ { k l } } \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C ) } \\left[ \\sum _ { i = 1 } ^ { N } \\mathcal { L } _ { C } ( X _ { i } ) \\right]", + "type": "interline_equation", + "image_path": "87e7763ce263534aeb6cee6da6876837b9378d6d9e48d47083b43b0dc3031f3e.jpg" + } + ] + } + ], + "index": 35.5, + "virtual_lines": [ + { + "bbox": [ + 191, + 583, + 420, + 600.5 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 191, + 600.5, + 420, + 618.0 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 451, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 452, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 213, + 636 + ], + "score": 1.0, + "content": "For readability, we denote", + "type": "text" + }, + { + "bbox": [ + 213, + 621, + 223, + 633 + ], + "score": 0.88, + "content": "\\tilde { \\mathcal { L } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 621, + 452, + 636 + ], + "score": 1.0, + "content": "to be the objective in Equation 9 without the regularizer.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 639, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 653 + ], + "score": 1.0, + "content": "Next, we take a closer look at the derivate of the expectation over adjacency matrices. Note that", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 103, + 650, + 507, + 669 + ], + "spans": [ + { + "bbox": [ + 103, + 650, + 317, + 669 + ], + "score": 1.0, + "content": "we have defined the adjacency matrix distribution as", + "type": "text" + }, + { + "bbox": [ + 317, + 650, + 481, + 667 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\bar { p } _ { \\gamma , \\theta } ( C ) = \\prod _ { i = 1 } ^ { N } \\mathbf { \\overbar { \\prod } } _ { j = 1 , j \\neq i } ^ { N } \\sigma ( \\gamma _ { i j } ) \\sigma ( \\theta _ { i j } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 650, + 507, + 669 + ], + "score": 1.0, + "content": ", with", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 107, + 665, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 107, + 665, + 140, + 677 + ], + "score": 0.9, + "content": "C _ { i j } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 665, + 228, + 677 + ], + "score": 1.0, + "content": "representing the edge", + "type": "text" + }, + { + "bbox": [ + 228, + 666, + 268, + 677 + ], + "score": 0.9, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 665, + 346, + 677 + ], + "score": 1.0, + "content": ". 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To prevent any confusion of index variables, we will use", + "type": "text" + }, + { + "bbox": [ + 420, + 423, + 434, + 434 + ], + "score": 0.88, + "content": "k , l", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 423, + 506, + 434 + ], + "score": 1.0, + "content": "as indices for the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 103, + 429, + 477, + 454 + ], + "spans": [ + { + "bbox": [ + 103, + 429, + 149, + 454 + ], + "score": 1.0, + "content": "parameter", + "type": "text" + }, + { + "bbox": [ + 149, + 437, + 163, + 447 + ], + "score": 0.86, + "content": "\\gamma _ { k l }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 429, + 331, + 454 + ], + "score": 1.0, + "content": "for which we determine the gradient, i.e.,", + "type": "text" + }, + { + "bbox": [ + 331, + 433, + 357, + 450 + ], + "score": 0.92, + "content": "\\frac { \\partial } { \\partial \\gamma _ { k l } } \\tilde { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 429, + 377, + 454 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 378, + 435, + 391, + 447 + ], + "score": 0.88, + "content": "i , j", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 429, + 477, + 454 + ], + "score": 1.0, + "content": "as indices for sums.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 103, + 396, + 506, + 454 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 467 + ], + "score": 1.0, + "content": "As a first step, we determine the gradients for the regularization term. Those can be found by taking", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 464, + 225, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 225, + 476 + ], + "score": 1.0, + "content": "the derivative of the sigmoid:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 451, + 505, + 476 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 479, + 450, + 515 + ], + "lines": [ + { + "bbox": [ + 162, + 479, + 450, + 515 + ], + "spans": [ + { + "bbox": [ + 162, + 479, + 450, + 515 + ], + "score": 0.94, + "content": "\\frac { \\partial } { \\partial \\gamma _ { k l } } \\lambda _ { \\mathrm { s p a r s e } } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } \\sigma ( \\gamma _ { i j } ) \\sigma ( \\theta _ { i j } ) = \\sigma ( \\gamma _ { k l } ) \\cdot ( 1 - \\sigma ( \\gamma _ { k l } ) ) \\cdot \\sigma ( \\theta _ { k l } ) \\lambda _ { \\mathrm { s p a r s e } }", + "type": "interline_equation", + "image_path": "90301915a25fac37131703afc2c466de3beec8265fccfa244d9e189b4e3b02f8.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 162, + 479, + 450, + 491.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 162, + 491.0, + 450, + 503.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 162, + 503.0, + 450, + 515.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 517, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 467, + 531 + ], + "score": 1.0, + "content": "Thus, it is straight-forward to calculate for any edge parameter. In the following, we use", + "type": "text" + }, + { + "bbox": [ + 468, + 518, + 493, + 530 + ], + "score": 0.92, + "content": "\\sigma ^ { \\prime } ( \\ldots )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 528, + 388, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 262, + 543 + ], + "score": 1.0, + "content": "abbreviate the derivate of the sigmoid:", + "type": "text" + }, + { + "bbox": [ + 263, + 529, + 383, + 541 + ], + "score": 0.91, + "content": "\\sigma ^ { \\prime } ( \\gamma _ { k l } ) = \\sigma ( \\gamma _ { k l } ) ( 1 - \\sigma ( \\gamma _ { k l } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 528, + 388, + 543 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 517, + 506, + 543 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "score": 1.0, + "content": "For the log-likelihood term, we start by reorganizing the expectations to simplify the gradient", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 225, + 569 + ], + "score": 1.0, + "content": "expression. The derivate term", + "type": "text" + }, + { + "bbox": [ + 226, + 556, + 244, + 571 + ], + "score": 0.93, + "content": "\\frac { \\partial } { \\partial \\gamma _ { k l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "can be moved inside the two expectations over interventional data", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 569, + 400, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 400, + 582 + ], + "score": 1.0, + "content": "since those are independent of the graph parameters. Thus, we can write:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 545, + 506, + 582 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 583, + 420, + 618 + ], + "lines": [ + { + "bbox": [ + 191, + 583, + 420, + 618 + ], + "spans": [ + { + "bbox": [ + 191, + 583, + 420, + 618 + ], + "score": 0.93, + "content": "\\frac { \\partial } { \\partial \\gamma _ { k l } } \\tilde { \\mathcal { L } } ^ { \\prime } = \\mathbb { E } _ { \\hat { I } \\sim p _ { I } ( I ) } \\mathbb { E } _ { \\tilde { p } _ { \\hat { I } } ( \\pmb { X } ) } \\frac { \\partial } { \\partial \\gamma _ { k l } } \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C ) } \\left[ \\sum _ { i = 1 } ^ { N } \\mathcal { L } _ { C } ( X _ { i } ) \\right]", + "type": "interline_equation", + "image_path": "87e7763ce263534aeb6cee6da6876837b9378d6d9e48d47083b43b0dc3031f3e.jpg" + } + ] + } + ], + "index": 35.5, + "virtual_lines": [ + { + "bbox": [ + 191, + 583, + 420, + 600.5 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 191, + 600.5, + 420, + 618.0 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 451, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 452, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 213, + 636 + ], + "score": 1.0, + "content": "For readability, we denote", + "type": "text" + }, + { + "bbox": [ + 213, + 621, + 223, + 633 + ], + "score": 0.88, + "content": "\\tilde { \\mathcal { L } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 621, + 452, + 636 + ], + "score": 1.0, + "content": "to be the objective in Equation 9 without the regularizer.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 621, + 452, + 636 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 639, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 653 + ], + "score": 1.0, + "content": "Next, we take a closer look at the derivate of the expectation over adjacency matrices. Note that", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 103, + 650, + 507, + 669 + ], + "spans": [ + { + "bbox": [ + 103, + 650, + 317, + 669 + ], + "score": 1.0, + "content": "we have defined the adjacency matrix distribution as", + "type": "text" + }, + { + "bbox": [ + 317, + 650, + 481, + 667 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\bar { p } _ { \\gamma , \\theta } ( C ) = \\prod _ { i = 1 } ^ { N } \\mathbf { \\overbar { \\prod } } _ { j = 1 , j \\neq i } ^ { N } \\sigma ( \\gamma _ { i j } ) \\sigma ( \\theta _ { i j } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 650, + 507, + 669 + ], + "score": 1.0, + "content": ", with", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 107, + 665, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 107, + 665, + 140, + 677 + ], + "score": 0.9, + "content": "C _ { i j } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 665, + 228, + 677 + ], + "score": 1.0, + "content": "representing the edge", + "type": "text" + }, + { + "bbox": [ + 228, + 666, + 268, + 677 + ], + "score": 0.9, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 665, + 346, + 677 + ], + "score": 1.0, + "content": ". 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The same process is performed for", + "type": "text" + }, + { + "bbox": [ + 454, + 276, + 467, + 286 + ], + "score": 0.88, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 273, + 502, + 288 + ], + "score": 1.0, + "content": "as well.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 106, + 306, + 504, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 306, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 133, + 321 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 307, + 247, + 320 + ], + "score": 0.92, + "content": "p _ { \\gamma , \\theta } ( C _ { k l } ) = \\sigma ( \\gamma _ { k l } ) \\cdot \\sigma ( \\theta _ { k l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 306, + 357, + 321 + ], + "score": 1.0, + "content": ". The first expectation over", + "type": "text" + }, + { + "bbox": [ + 358, + 308, + 405, + 320 + ], + "score": 0.93, + "content": "p _ { \\gamma , \\theta } ( C _ { - k l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 306, + 479, + 321 + ], + "score": 1.0, + "content": "is independent of", + "type": "text" + }, + { + "bbox": [ + 479, + 309, + 492, + 319 + ], + "score": 0.85, + "content": "\\gamma _ { k l }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 306, + 506, + 321 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 506, + 331 + ], + "score": 1.0, + "content": "we have defined the adjacency matrix distribution to be a product of independent edge probabilities.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 335, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 504, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 278, + 349 + ], + "score": 1.0, + "content": "The log-likelihood estimate of a variable,", + "type": "text" + }, + { + "bbox": [ + 279, + 335, + 312, + 348 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { C } ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 334, + 487, + 349 + ], + "score": 1.0, + "content": ", depends on the adjacency matrix column", + "type": "text" + }, + { + "bbox": [ + 488, + 337, + 504, + 348 + ], + "score": 0.86, + "content": "C _ { \\cdot , i }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 317, + 358 + ], + "score": 1.0, + "content": "which represents the input connections to the node", + "type": "text" + }, + { + "bbox": [ + 318, + 347, + 330, + 357 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 347, + 505, + 358 + ], + "score": 1.0, + "content": ". All other edges have no influence on the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 214, + 370 + ], + "score": 1.0, + "content": "log-likelihood estimate of", + "type": "text" + }, + { + "bbox": [ + 214, + 358, + 226, + 369 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 357, + 320, + 370 + ], + "score": 1.0, + "content": ". Hence, the parameter", + "type": "text" + }, + { + "bbox": [ + 320, + 360, + 334, + 369 + ], + "score": 0.88, + "content": "\\gamma _ { k l }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 357, + 399, + 370 + ], + "score": 1.0, + "content": "only influences", + "type": "text" + }, + { + "bbox": [ + 400, + 358, + 433, + 370 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { C } ( X _ { l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 357, + 505, + 370 + ], + "score": 1.0, + "content": ", and thus we can", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 368, + 271, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 271, + 380 + ], + "score": 1.0, + "content": "reduce the sum inside the expectation to:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 384, + 426, + 420 + ], + "lines": [ + { + "bbox": [ + 186, + 384, + 426, + 420 + ], + "spans": [ + { + "bbox": [ + 186, + 384, + 426, + 420 + ], + "score": 0.94, + "content": "\\frac { \\partial } { \\partial \\gamma _ { k l } } \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C _ { k l } ) } \\left[ \\sum _ { i = 1 } ^ { N } \\mathcal { L } _ { C } ( X _ { i } ) \\right] = \\frac { \\partial } { \\partial \\gamma _ { k l } } \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C _ { k l } ) } \\left[ \\mathcal { L } _ { C } ( X _ { l } ) \\right]", + "type": "interline_equation", + "image_path": "6e05bf211bc19209992ec97a6df18aedb927ed41bcfcce9f71a83e125594a9c1.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 384, + 426, + 402.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 186, + 402.0, + 426, + 420.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 429, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "The REINFORCE trick is a simple method to move the derivative of a discrete distribution inside the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 441, + 302, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 302, + 453 + ], + "score": 1.0, + "content": "expectation. Applied to our situation, we obtain:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 457, + 438, + 484 + ], + "lines": [ + { + "bbox": [ + 173, + 457, + 438, + 484 + ], + "spans": [ + { + "bbox": [ + 173, + 457, + 438, + 484 + ], + "score": 0.92, + "content": "\\frac { \\partial } { \\partial \\gamma _ { k l } } \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C _ { k l } ) } \\left[ \\mathcal { L } _ { C } ( X _ { l } ) \\right] = \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C _ { k l } ) } \\left[ \\mathcal { L } _ { C } ( X _ { l } ) \\frac { \\partial \\log p _ { \\gamma , \\theta } ( C _ { k l } ) } { \\partial \\gamma _ { k l } } \\right]", + "type": "interline_equation", + "image_path": "dcc6610ece1ac69abb4e8442dd55039b212687b799ee35c21c1809b5f82d4a46.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 173, + 457, + 438, + 466.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 173, + 466.0, + 438, + 475.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 173, + 475.0, + 438, + 484.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 506, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 304, + 501 + ], + "score": 1.0, + "content": "This leaves us with two cases in the expectation:", + "type": "text" + }, + { + "bbox": [ + 305, + 489, + 339, + 500 + ], + "score": 0.91, + "content": "C _ { k l } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 488, + 357, + 501 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 357, + 489, + 391, + 500 + ], + "score": 0.9, + "content": "C _ { k l } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 488, + 505, + 501 + ], + "score": 1.0, + "content": ". 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The first expectation over", + "type": "text" + }, + { + "bbox": [ + 358, + 308, + 405, + 320 + ], + "score": 0.93, + "content": "p _ { \\gamma , \\theta } ( C _ { - k l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 306, + 479, + 321 + ], + "score": 1.0, + "content": "is independent of", + "type": "text" + }, + { + "bbox": [ + 479, + 309, + 492, + 319 + ], + "score": 0.85, + "content": "\\gamma _ { k l }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 306, + 506, + 321 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 506, + 331 + ], + "score": 1.0, + "content": "we have defined the adjacency matrix distribution to be a product of independent edge probabilities.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 306, + 506, + 331 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 335, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 504, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 278, + 349 + ], + "score": 1.0, + "content": "The log-likelihood estimate of a variable,", + "type": "text" + }, + { + "bbox": [ + 279, + 335, + 312, + 348 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { C } ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 334, + 487, + 349 + ], + "score": 1.0, + "content": ", depends on the adjacency matrix column", + "type": "text" + }, + { + "bbox": [ + 488, + 337, + 504, + 348 + ], + "score": 0.86, + "content": "C _ { \\cdot , i }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 317, + 358 + ], + "score": 1.0, + "content": "which represents the input connections to the node", + "type": "text" + }, + { + "bbox": [ + 318, + 347, + 330, + 357 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 347, + 505, + 358 + ], + "score": 1.0, + "content": ". All other edges have no influence on the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 214, + 370 + ], + "score": 1.0, + "content": "log-likelihood estimate of", + "type": "text" + }, + { + "bbox": [ + 214, + 358, + 226, + 369 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 357, + 320, + 370 + ], + "score": 1.0, + "content": ". Hence, the parameter", + "type": "text" + }, + { + "bbox": [ + 320, + 360, + 334, + 369 + ], + "score": 0.88, + "content": "\\gamma _ { k l }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 357, + 399, + 370 + ], + "score": 1.0, + "content": "only influences", + "type": "text" + }, + { + "bbox": [ + 400, + 358, + 433, + 370 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { C } ( X _ { l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 357, + 505, + 370 + ], + "score": 1.0, + "content": ", and thus we can", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 368, + 271, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 271, + 380 + ], + "score": 1.0, + "content": "reduce the sum inside the expectation to:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 334, + 505, + 380 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 384, + 426, + 420 + ], + "lines": [ + { + "bbox": [ + 186, + 384, + 426, + 420 + ], + "spans": [ + { + "bbox": [ + 186, + 384, + 426, + 420 + ], + "score": 0.94, + "content": "\\frac { \\partial } { \\partial \\gamma _ { k l } } \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C _ { k l } ) } \\left[ \\sum _ { i = 1 } ^ { N } \\mathcal { L } _ { C } ( X _ { i } ) \\right] = \\frac { \\partial } { \\partial \\gamma _ { k l } } \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C _ { k l } ) } \\left[ \\mathcal { L } _ { C } ( X _ { l } ) \\right]", + "type": "interline_equation", + "image_path": "6e05bf211bc19209992ec97a6df18aedb927ed41bcfcce9f71a83e125594a9c1.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 384, + 426, + 402.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 186, + 402.0, + 426, + 420.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 429, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "The REINFORCE trick is a simple method to move the derivative of a discrete distribution inside the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 441, + 302, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 302, + 453 + ], + "score": 1.0, + "content": "expectation. Applied to our situation, we obtain:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 106, + 429, + 505, + 453 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 457, + 438, + 484 + ], + "lines": [ + { + "bbox": [ + 173, + 457, + 438, + 484 + ], + "spans": [ + { + "bbox": [ + 173, + 457, + 438, + 484 + ], + "score": 0.92, + "content": "\\frac { \\partial } { \\partial \\gamma _ { k l } } \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C _ { k l } ) } \\left[ \\mathcal { L } _ { C } ( X _ { l } ) \\right] = \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C _ { k l } ) } \\left[ \\mathcal { L } _ { C } ( X _ { l } ) \\frac { \\partial \\log p _ { \\gamma , \\theta } ( C _ { k l } ) } { \\partial \\gamma _ { k l } } \\right]", + "type": "interline_equation", + "image_path": "dcc6610ece1ac69abb4e8442dd55039b212687b799ee35c21c1809b5f82d4a46.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 173, + 457, + 438, + 466.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 173, + 466.0, + 438, + 475.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 173, + 475.0, + 438, + 484.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 506, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 304, + 501 + ], + "score": 1.0, + "content": "This leaves us with two cases in the expectation:", + "type": "text" + }, + { + "bbox": [ + 305, + 489, + 339, + 500 + ], + "score": 0.91, + "content": "C _ { k l } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 488, + 357, + 501 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 357, + 489, + 391, + 500 + ], + "score": 0.9, + "content": "C _ { k l } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 488, + 505, + 501 + ], + "score": 1.0, + "content": ". In other words, we need to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 231, + 512 + ], + "score": 1.0, + "content": "distinguish between samples of", + "type": "text" + }, + { + "bbox": [ + 231, + 500, + 240, + 510 + ], + "score": 0.84, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 500, + 336, + 512 + ], + "score": 1.0, + "content": "where we have the edge", + "type": "text" + }, + { + "bbox": [ + 336, + 500, + 377, + 511 + ], + "score": 0.92, + "content": "X _ { k } \\to X _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 500, + 505, + 512 + ], + "score": 1.0, + "content": ", and where we do not have such", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 511, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 141, + 523 + ], + "score": 1.0, + "content": "an edge", + "type": "text" + }, + { + "bbox": [ + 142, + 511, + 185, + 522 + ], + "score": 0.87, + "content": "( X _ { k } \\not \\to X _ { l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 511, + 506, + 523 + ], + "score": 1.0, + "content": "). Thus, we can also write the expectation as a weighted sum of those two cases:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 488, + 506, + 523 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 526, + 514, + 580 + ], + "lines": [ + { + "bbox": [ + 111, + 526, + 514, + 580 + ], + "spans": [ + { + "bbox": [ + 111, + 526, + 514, + 580 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { \\Xi _ { p _ { \\gamma , \\theta } ( C ) } \\left[ \\mathcal { L } _ { C } ( X _ { l } ) \\frac { \\partial \\log p _ { \\gamma , \\theta } ( C _ { k l } ) } { \\partial \\gamma _ { k l } } \\right] = \\sigma ( \\gamma _ { k l } ) \\cdot \\sigma ( \\theta _ { k l } ) \\cdot \\mathcal { L } _ { X _ { k } \\to X _ { l } } ( X _ { l } ) \\cdot \\frac { \\partial \\log \\sigma ( \\gamma _ { k l } ) \\cdot \\sigma ( \\theta _ { k l } ) } { \\partial \\gamma _ { k l } } + } & { { } } \\\\ { ( 1 - \\sigma ( \\gamma _ { k l } ) \\cdot \\sigma ( \\theta _ { k l } ) ) \\cdot \\mathcal { L } _ { X _ { l } \\ne y _ { X _ { k } } } ( X _ { k } ) \\cdot \\frac { \\partial \\log \\left( 1 - \\sigma ( \\gamma _ { k l } ) \\cdot \\sigma ( \\theta _ { k l } ) \\right) } { \\partial \\gamma _ { k l } } . } & { { } } \\end{array}", + "type": "interline_equation", + "image_path": "de0e5c154e555f86af0ed3b361eedca53f1803e4e5d4accbc72da747231354d9.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 111, + 526, + 514, + 544.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 111, + 544.0, + 514, + 562.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 111, + 562.0, + 514, + 580.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 598, + 504, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 136, + 612 + ], + "score": 1.0, + "content": "We use", + "type": "text" + }, + { + "bbox": [ + 136, + 599, + 191, + 610 + ], + "score": 0.92, + "content": "\\mathscr { L } _ { X _ { k } X _ { l } } ( X _ { l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 597, + 391, + 612 + ], + "score": 1.0, + "content": "to denote the (expected) negative log likelihood for", + "type": "text" + }, + { + "bbox": [ + 391, + 599, + 403, + 610 + ], + "score": 0.89, + "content": "X _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 597, + 505, + 612 + ], + "score": 1.0, + "content": "under adjacency matrices", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 609, + 266, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 223, + 622 + ], + "score": 1.0, + "content": "where we have an edge from", + "type": "text" + }, + { + "bbox": [ + 223, + 610, + 237, + 621 + ], + "score": 0.89, + "content": "X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 609, + 249, + 622 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 249, + 610, + 261, + 621 + ], + "score": 0.88, + "content": "X _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 609, + 266, + 622 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 597, + 505, + 622 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 625, + 397, + 658 + ], + "lines": [ + { + "bbox": [ + 214, + 625, + 397, + 658 + ], + "spans": [ + { + "bbox": [ + 214, + 625, + 397, + 658 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\begin{array} { r l } & { \\mathcal { L } _ { X _ { k } \\to X _ { l } } ( X _ { l } ) = \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C _ { - k l } ) , C _ { k l } = 1 } \\left[ \\mathcal { L } _ { C } ( X _ { l } ) \\right] } \\\\ & { \\mathcal { L } _ { X _ { k } \\not = X _ { l } } ( X _ { l } ) = \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C _ { - k l } ) , C _ { k l } = 0 } \\left[ \\mathcal { L } _ { C } ( X _ { l } ) \\right] } \\end{array} } \\end{array}", + "type": "interline_equation", + "image_path": "d418ae0b9a37214f6093ef6f151b765b625f34a7612793a1811532ff221a3f2c.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 214, + 625, + 397, + 641.5 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 214, + 641.5, + 397, + 658.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 660, + 464, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 465, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 465, + 673 + ], + "score": 1.0, + "content": "111The final step is to solve the two derivative terms in Equation 14. This is done as follows:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 659, + 465, + 673 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 676, + 449, + 733 + ], + "lines": [ + { + "bbox": [ + 163, + 676, + 449, + 733 + ], + "spans": [ + { + "bbox": [ + 163, + 676, + 449, + 733 + ], + "score": 0.94, + "content": "\\begin{array} { c } { { \\displaystyle \\frac { \\partial \\log \\sigma ( \\gamma _ { k l } ) \\cdot \\sigma ( \\theta _ { k l } ) } { \\partial \\gamma _ { k l } } = \\frac { \\partial \\log \\sigma ( \\gamma _ { k l } ) + \\log \\sigma ( \\theta _ { k l } ) } { \\partial \\gamma _ { k l } } = 1 - \\sigma ( \\gamma _ { k l } ) } } \\\\ { { \\displaystyle \\frac { \\partial \\log \\left( 1 - \\sigma ( \\gamma _ { k l } ) \\cdot \\sigma ( \\theta _ { k l } ) \\right) } { \\partial \\gamma _ { k l } } = - \\frac { \\sigma ( \\gamma _ { k l } ) \\cdot ( 1 - \\sigma ( \\gamma _ { k l } ) ) \\cdot \\sigma ( \\theta _ { k l } ) } { 1 - \\sigma ( \\gamma _ { k l } ) \\cdot \\sigma ( \\theta _ { k l } ) } } } \\end{array}", + "type": "interline_equation", + "image_path": "0115f70b3f63bab3a7676f202ef006f89c458d29834d78ae35073b80d19db9b8.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 163, + 676, + 449, + 695.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 163, + 695.0, + 449, + 714.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 163, + 714.0, + 449, + 733.0 + ], + "spans": [], + "index": 34 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 111, + 87, + 498, + 193 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 87, + 498, + 193 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 87, + 498, + 193 + ], + "spans": [ + { + "bbox": [ + 111, + 87, + 498, + 193 + ], + "score": 0.967, + "type": "image", + "image_path": "06ae892632176f196ccea6bcc78b02490098df4e234650971221b22d5c417447.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 111, + 87, + 498, + 122.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 111, + 122.33333333333334, + 498, + 157.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 157.66666666666669, + 498, + 193.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 208, + 506, + 253 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 442, + 220 + ], + "score": 1.0, + "content": "Figure 6: Plotting the gradient estimate variance of ENCO for three variables of", + "type": "text" + }, + { + "bbox": [ + 442, + 211, + 450, + 221 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "compared to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 219, + 507, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 464, + 232 + ], + "score": 1.0, + "content": "previous REINFORCE-like approach by Bengio et al. (2020) on an example graph with", + "type": "text" + }, + { + "bbox": [ + 464, + 220, + 503, + 230 + ], + "score": 0.9, + "content": "K = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 219, + 507, + 232 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "The gradients have been scaled to match in terms of averages. We can see a clear reduction in variance", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 242, + 411, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 411, + 254 + ], + "score": 1.0, + "content": "with the gradient estimator of ENCO allowing us to use lower sample sizes.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 105, + 275, + 478, + 287 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 480, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 480, + 289 + ], + "score": 1.0, + "content": "Putting these results back in the original equation and adding the sparsity regularizer, we get:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 292, + 483, + 320 + ], + "lines": [ + { + "bbox": [ + 111, + 292, + 483, + 320 + ], + "spans": [ + { + "bbox": [ + 111, + 292, + 483, + 320 + ], + "score": 0.86, + "content": "\\frac { \\partial } { \\partial \\gamma _ { i j } } \\tilde { \\mathcal { L } } = \\sigma ( \\gamma _ { i j } ) \\cdot ( 1 - \\sigma ( \\gamma _ { i j } ) ) \\cdot \\sigma ( \\theta _ { i j } ) \\cdot \\mathbb { E } _ { \\mathbf { X } , \\mathcal { C } _ { - i j } } \\left[ \\mathcal { L } _ { X _ { i } \\to X _ { j } } ( X _ { j } ) - \\mathcal { L } _ { X _ { i } \\neq X _ { j } } ( X _ { j } ) + \\lambda _ { \\mathrm { s p a r s } } \\right]", + "type": "interline_equation", + "image_path": "035301f075a0b71695d590d9ea10f7fcc5cd76cbd4a400d510bb3e15c403a63b.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 111, + 292, + 483, + 301.3333333333333 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 111, + 301.3333333333333, + 483, + 310.66666666666663 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 111, + 310.66666666666663, + 483, + 319.99999999999994 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 324, + 504, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 463, + 337 + ], + "score": 1.0, + "content": "To align the result with the gradient in Section 3.3, we switch the index notation from", + "type": "text" + }, + { + "bbox": [ + 464, + 325, + 477, + 335 + ], + "score": 0.86, + "content": "k , l", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 324, + 490, + 337 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 490, + 326, + 504, + 336 + ], + "score": 0.81, + "content": "i , j", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 331, + 503, + 352 + ], + "spans": [ + { + "bbox": [ + 104, + 331, + 204, + 352 + ], + "score": 1.0, + "content": "again. The expectation", + "type": "text" + }, + { + "bbox": [ + 205, + 335, + 240, + 348 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { X , C _ { - i j } }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 331, + 390, + 352 + ], + "score": 1.0, + "content": "is a short form for the expectations", + "type": "text" + }, + { + "bbox": [ + 391, + 335, + 503, + 349 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\hat { I } \\sim p _ { I } ( I ) } \\mathbb { E } _ { \\tilde { p } _ { \\hat { I } } ( \\pmb { X } ) } \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C _ { - i j } ) }", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 327, + 361 + ], + "score": 1.0, + "content": "From this expression, we can see that the gradients of", + "type": "text" + }, + { + "bbox": [ + 327, + 349, + 340, + 360 + ], + "score": 0.87, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "are proportional to the difference of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 358, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 248, + 372 + ], + "score": 1.0, + "content": "expected negative log-likelihood of", + "type": "text" + }, + { + "bbox": [ + 248, + 359, + 261, + 371 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 358, + 380, + 372 + ], + "score": 1.0, + "content": "with having an edge between", + "type": "text" + }, + { + "bbox": [ + 380, + 359, + 420, + 371 + ], + "score": 0.91, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 358, + 505, + 372 + ], + "score": 1.0, + "content": ", and the cases where", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 107, + 370, + 147, + 382 + ], + "score": 0.9, + "content": "X _ { i } \\not \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 369, + 505, + 382 + ], + "score": 1.0, + "content": ". 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Note that the adjacency matrix samples to estimate", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 419, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 107, + 419, + 163, + 432 + ], + "score": 0.92, + "content": "\\mathscr { L } _ { X _ { i } \\to X _ { j } } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 419, + 181, + 433 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 182, + 419, + 238, + 432 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { X _ { i } \\not \\to X _ { j } } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 419, + 506, + 433 + ], + "score": 1.0, + "content": "are not required to be the same. For efficiency, we instead sample", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 429, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 107, + 432, + 117, + 440 + ], + "score": 0.75, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 429, + 221, + 444 + ], + "score": 1.0, + "content": "adjacency matrices from", + "type": "text" + }, + { + "bbox": [ + 222, + 431, + 255, + 443 + ], + "score": 0.91, + "content": "p _ { \\gamma , \\theta } ( C )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 429, + 399, + 444 + ], + "score": 1.0, + "content": ", evaluate the likelihood of a batch", + "type": "text" + }, + { + "bbox": [ + 399, + 430, + 410, + 441 + ], + "score": 0.76, + "content": "\\boldsymbol { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 429, + 506, + 444 + ], + "score": 1.0, + "content": "under all these graphs.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 434, + 455 + ], + "score": 1.0, + "content": "Afterwards, we assign the evaluated samples to one of the two cases, depending on", + "type": "text" + }, + { + "bbox": [ + 434, + 442, + 450, + 453 + ], + "score": 0.9, + "content": "C _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "being zero or", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 422, + 465 + ], + "score": 1.0, + "content": "one. This way, we can reuse the same graph samples for all edge parameters", + "type": "text" + }, + { + "bbox": [ + 422, + 454, + 430, + 464 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 453, + 505, + 465 + ], + "score": 1.0, + "content": ". We visualize the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 445, + 476 + ], + "score": 1.0, + "content": "gradient calculation in Figure 5. In the cases where we perform an intervention on", + "type": "text" + }, + { + "bbox": [ + 446, + 464, + 458, + 474 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 462, + 506, + 476 + ], + "score": 1.0, + "content": ", we do not", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 473, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 144, + 488 + ], + "score": 1.0, + "content": "optimize", + "type": "text" + }, + { + "bbox": [ + 144, + 475, + 157, + 487 + ], + "score": 0.87, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 473, + 506, + 488 + ], + "score": 1.0, + "content": "for this step and set the gradients to zero. The same holds for gradient steps where we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 483, + 381, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 381, + 499 + ], + "score": 1.0, + "content": "do not have any samples for one of the two log-likelihood estimates.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 106, + 510, + 364, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 365, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 365, + 522 + ], + "score": 1.0, + "content": "A.1.1 COMPARISON TO PREVIOUS GRADIENT ESTIMATORS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "As discussed in Section 3, previous work on similar structure learning methods (Bengio et al., 2020;", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 539, + 504, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 504, + 550 + ], + "score": 1.0, + "content": "Ke et al., 2019) relied on a different estimator. In terms of derivation, the main difference is the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 222, + 563 + ], + "score": 1.0, + "content": "continuation from Equation", + "type": "text" + }, + { + "bbox": [ + 222, + 550, + 247, + 561 + ], + "score": 0.26, + "content": "1 4 ~ \\mathrm { o n }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 550, + 505, + 563 + ], + "score": 1.0, + "content": ". In our proposed method, we write the expectation as the sum", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "of two terms that can independently be approximated via Monte-Carlo sampling. In comparison,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 570, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 585 + ], + "score": 1.0, + "content": "Bengio et al. (2020) proposed to directly apply a Monte-Carlo sampler to Equation 14, and apply an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 583, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 594 + ], + "score": 1.0, + "content": "importance sampling weight to reduce the variance. This estimator is also used in the method SDI", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 593, + 400, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 400, + 606 + ], + "score": 1.0, + "content": "(Ke et al., 2019) to which we have experimentally compared our method.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "Figure 2 compared the gradient estimator in terms of standard deviation. The gradient estimator of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "ENCO achieves a 10 times lower standard deviation compared to Bengio et al. (2020) making it much", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 631, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 104, + 631, + 506, + 646 + ], + "score": 1.0, + "content": "more efficient. Since the estimator by Bengio et al. (2020) is biased and has a different mean, we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "have scaled both estimators to have the same mean. Specifically, we have applied ENCO to random", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "graphs from our experiments on synthetic graphs (see Section 4.2), and evaluated 64, 000 sampled", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "adjacency matrices in terms of log-likelihood estimates. These 64, 000 samples are grouped into sets", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 117, + 689 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 677, + 128, + 687 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 676, + 352, + 689 + ], + "score": 1.0, + "content": "samples which we have used to estimate the gradients of", + "type": "text" + }, + { + "bbox": [ + 352, + 678, + 360, + 688 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 676, + 492, + 689 + ], + "score": 1.0, + "content": ". We evaluated different values of", + "type": "text" + }, + { + "bbox": [ + 492, + 677, + 502, + 687 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 676, + 506, + 689 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 128, + 699 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 129, + 688, + 163, + 698 + ], + "score": 0.91, + "content": "K = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 687, + 174, + 699 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 175, + 688, + 219, + 698 + ], + "score": 0.9, + "content": "K = 4 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 687, + 505, + 699 + ], + "score": 1.0, + "content": ", and plotted the standard deviation of those estimates in Figure 2. We", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "have also visualized three examples as violin plots in Figure 6 that demonstrate that despite both", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 710, + 507, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 507, + 722 + ], + "score": 1.0, + "content": "estimators having a similar mean, the variance of gradient estimates is much higher for Bengio et al.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 720, + 137, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 137, + 732 + ], + "score": 1.0, + "content": "(2020).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 39 + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 111, + 87, + 498, + 193 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 87, + 498, + 193 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 87, + 498, + 193 + ], + "spans": [ + { + "bbox": [ + 111, + 87, + 498, + 193 + ], + "score": 0.967, + "type": "image", + "image_path": "06ae892632176f196ccea6bcc78b02490098df4e234650971221b22d5c417447.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 111, + 87, + 498, + 122.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 111, + 122.33333333333334, + 498, + 157.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 157.66666666666669, + 498, + 193.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 208, + 506, + 253 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 442, + 220 + ], + "score": 1.0, + "content": "Figure 6: Plotting the gradient estimate variance of ENCO for three variables of", + "type": "text" + }, + { + "bbox": [ + 442, + 211, + 450, + 221 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "compared to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 219, + 507, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 464, + 232 + ], + "score": 1.0, + "content": "previous REINFORCE-like approach by Bengio et al. (2020) on an example graph with", + "type": "text" + }, + { + "bbox": [ + 464, + 220, + 503, + 230 + ], + "score": 0.9, + "content": "K = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 219, + 507, + 232 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "The gradients have been scaled to match in terms of averages. We can see a clear reduction in variance", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 242, + 411, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 411, + 254 + ], + "score": 1.0, + "content": "with the gradient estimator of ENCO allowing us to use lower sample sizes.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 105, + 275, + 478, + 287 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 480, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 480, + 289 + ], + "score": 1.0, + "content": "Putting these results back in the original equation and adding the sparsity regularizer, we get:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 273, + 480, + 289 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 292, + 483, + 320 + ], + "lines": [ + { + "bbox": [ + 111, + 292, + 483, + 320 + ], + "spans": [ + { + "bbox": [ + 111, + 292, + 483, + 320 + ], + "score": 0.86, + "content": "\\frac { \\partial } { \\partial \\gamma _ { i j } } \\tilde { \\mathcal { L } } = \\sigma ( \\gamma _ { i j } ) \\cdot ( 1 - \\sigma ( \\gamma _ { i j } ) ) \\cdot \\sigma ( \\theta _ { i j } ) \\cdot \\mathbb { E } _ { \\mathbf { X } , \\mathcal { C } _ { - i j } } \\left[ \\mathcal { L } _ { X _ { i } \\to X _ { j } } ( X _ { j } ) - \\mathcal { L } _ { X _ { i } \\neq X _ { j } } ( X _ { j } ) + \\lambda _ { \\mathrm { s p a r s } } \\right]", + "type": "interline_equation", + "image_path": "035301f075a0b71695d590d9ea10f7fcc5cd76cbd4a400d510bb3e15c403a63b.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 111, + 292, + 483, + 301.3333333333333 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 111, + 301.3333333333333, + 483, + 310.66666666666663 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 111, + 310.66666666666663, + 483, + 319.99999999999994 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 324, + 504, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 463, + 337 + ], + "score": 1.0, + "content": "To align the result with the gradient in Section 3.3, we switch the index notation from", + "type": "text" + }, + { + "bbox": [ + 464, + 325, + 477, + 335 + ], + "score": 0.86, + "content": "k , l", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 324, + 490, + 337 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 490, + 326, + 504, + 336 + ], + "score": 0.81, + "content": "i , j", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 331, + 503, + 352 + ], + "spans": [ + { + "bbox": [ + 104, + 331, + 204, + 352 + ], + "score": 1.0, + "content": "again. The expectation", + "type": "text" + }, + { + "bbox": [ + 205, + 335, + 240, + 348 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { X , C _ { - i j } }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 331, + 390, + 352 + ], + "score": 1.0, + "content": "is a short form for the expectations", + "type": "text" + }, + { + "bbox": [ + 391, + 335, + 503, + 349 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\hat { I } \\sim p _ { I } ( I ) } \\mathbb { E } _ { \\tilde { p } _ { \\hat { I } } ( \\pmb { X } ) } \\mathbb { E } _ { p _ { \\gamma , \\theta } ( C _ { - i j } ) }", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 327, + 361 + ], + "score": 1.0, + "content": "From this expression, we can see that the gradients of", + "type": "text" + }, + { + "bbox": [ + 327, + 349, + 340, + 360 + ], + "score": 0.87, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "are proportional to the difference of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 358, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 248, + 372 + ], + "score": 1.0, + "content": "expected negative log-likelihood of", + "type": "text" + }, + { + "bbox": [ + 248, + 359, + 261, + 371 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 358, + 380, + 372 + ], + "score": 1.0, + "content": "with having an edge between", + "type": "text" + }, + { + "bbox": [ + 380, + 359, + 420, + 371 + ], + "score": 0.91, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 358, + 505, + 372 + ], + "score": 1.0, + "content": ", and the cases where", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 107, + 370, + 147, + 382 + ], + "score": 0.9, + "content": "X _ { i } \\not \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 369, + 505, + 382 + ], + "score": 1.0, + "content": ". 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This way, we can reuse the same graph samples for all edge parameters", + "type": "text" + }, + { + "bbox": [ + 422, + 454, + 430, + 464 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 453, + 505, + 465 + ], + "score": 1.0, + "content": ". We visualize the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 445, + 476 + ], + "score": 1.0, + "content": "gradient calculation in Figure 5. In the cases where we perform an intervention on", + "type": "text" + }, + { + "bbox": [ + 446, + 464, + 458, + 474 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 462, + 506, + 476 + ], + "score": 1.0, + "content": ", we do not", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 473, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 144, + 488 + ], + "score": 1.0, + "content": "optimize", + "type": "text" + }, + { + "bbox": [ + 144, + 475, + 157, + 487 + ], + "score": 0.87, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 473, + 506, + 488 + ], + "score": 1.0, + "content": "for this step and set the gradients to zero. The same holds for gradient steps where we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 483, + 381, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 381, + 499 + ], + "score": 1.0, + "content": "do not have any samples for one of the two log-likelihood estimates.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 396, + 506, + 499 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 510, + 364, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 365, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 365, + 522 + ], + "score": 1.0, + "content": "A.1.1 COMPARISON TO PREVIOUS GRADIENT ESTIMATORS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "As discussed in Section 3, previous work on similar structure learning methods (Bengio et al., 2020;", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 539, + 504, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 504, + 550 + ], + "score": 1.0, + "content": "Ke et al., 2019) relied on a different estimator. In terms of derivation, the main difference is the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 222, + 563 + ], + "score": 1.0, + "content": "continuation from Equation", + "type": "text" + }, + { + "bbox": [ + 222, + 550, + 247, + 561 + ], + "score": 0.26, + "content": "1 4 ~ \\mathrm { o n }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 550, + 505, + 563 + ], + "score": 1.0, + "content": ". In our proposed method, we write the expectation as the sum", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "of two terms that can independently be approximated via Monte-Carlo sampling. In comparison,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 570, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 585 + ], + "score": 1.0, + "content": "Bengio et al. (2020) proposed to directly apply a Monte-Carlo sampler to Equation 14, and apply an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 583, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 594 + ], + "score": 1.0, + "content": "importance sampling weight to reduce the variance. This estimator is also used in the method SDI", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 593, + 400, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 400, + 606 + ], + "score": 1.0, + "content": "(Ke et al., 2019) to which we have experimentally compared our method.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 527, + 506, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "Figure 2 compared the gradient estimator in terms of standard deviation. The gradient estimator of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "ENCO achieves a 10 times lower standard deviation compared to Bengio et al. (2020) making it much", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 631, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 104, + 631, + 506, + 646 + ], + "score": 1.0, + "content": "more efficient. Since the estimator by Bengio et al. (2020) is biased and has a different mean, we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "have scaled both estimators to have the same mean. Specifically, we have applied ENCO to random", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "graphs from our experiments on synthetic graphs (see Section 4.2), and evaluated 64, 000 sampled", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "adjacency matrices in terms of log-likelihood estimates. 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We", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "have also visualized three examples as violin plots in Figure 6 that demonstrate that despite both", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 710, + 507, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 507, + 722 + ], + "score": 1.0, + "content": "estimators having a similar mean, the variance of gradient estimates is much higher for Bengio et al.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 720, + 137, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 137, + 732 + ], + "score": 1.0, + "content": "(2020).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 39, + "bbox_fs": [ + 104, + 611, + 507, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "To verify that the improvement of ENCO is not just because of the gradient estimators, we have", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "performed an ablation study with ENCO deploying the gradient estimator of Bengio et al. 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This is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "done for two reasons. Firstly, the orientation parameter models only the direction of the edge, not", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 336, + 540 + ], + "score": 1.0, + "content": "whether it exists or not. The regularizer would increase", + "type": "text" + }, + { + "bbox": [ + 336, + 527, + 349, + 539 + ], + "score": 0.89, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "if the edge existence for the opposite", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 537, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 328, + 552 + ], + "score": 1.0, + "content": "direction would be greater than for the direction from", + "type": "text" + }, + { + "bbox": [ + 328, + 539, + 340, + 549 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 537, + 353, + 552 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 353, + 539, + 366, + 550 + ], + "score": 0.85, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 537, + 372, + 552 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 372, + 539, + 423, + 550 + ], + "score": 0.84, + "content": "i . e . \\gamma _ { i j } < \\gamma _ { j i }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 537, + 482, + 552 + ], + "score": 1.0, + "content": ", and decrease", + "type": "text" + }, + { + "bbox": [ + 482, + 538, + 495, + 550 + ], + "score": 0.89, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 537, + 506, + 552 + ], + "score": 1.0, + "content": "if", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 142, + 564 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 143, + 550, + 184, + 561 + ], + "score": 0.91, + "content": "\\gamma _ { i j } > \\gamma _ { j i }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 549, + 506, + 564 + ], + "score": 1.0, + "content": ". However, the orientation should only model the causal direction of an edge.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "Hence, we do not gain any value from a causal perspective when adding the regularizer to the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "score": 1.0, + "content": "gradient. Secondly, the regularizer would require us to take additional assumptions for guaranteeing", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "score": 1.0, + "content": "the discovery of the true graph upon convergence. In experiments with using a regularizer in the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 593, + 460, + 606 + ], + "spans": [ + { + "bbox": [ + 107, + 593, + 113, + 603 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 593, + 460, + 606 + ], + "score": 1.0, + "content": "-gradient, we did not observe any difference to the experiments without the regularizer.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 506, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "We note that the orientation parameters are considered to be pairwise independent. In other words,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 621, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 107, + 621, + 119, + 633 + ], + "score": 0.89, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 621, + 137, + 633 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 137, + 621, + 151, + 632 + ], + "score": 0.89, + "content": "\\theta _ { k l }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 621, + 316, + 633 + ], + "score": 1.0, + "content": "are considered independent parameters if", + "type": "text" + }, + { + "bbox": [ + 316, + 621, + 347, + 632 + ], + "score": 0.94, + "content": "i \\neq k , l", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 621, + 365, + 633 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 365, + 621, + 397, + 632 + ], + "score": 0.94, + "content": "j \\neq k , l", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 621, + 506, + 633 + ], + "score": 1.0, + "content": ". Global order distributions", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "such as Plackett-Luce (Plackett, 1975; Luce, 1959) can be used to also incorporate transitive relations.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "However, those require high variance gradient estimators and struggled with chains in early experi-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "ments. 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The distribution over", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 161, + 732 + ], + "score": 1.0, + "content": "interventions", + "type": "text" + }, + { + "bbox": [ + 161, + 721, + 180, + 732 + ], + "score": 0.92, + "content": "p ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "is set to a uniform distribution for all our experiments. 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The", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 423, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 399, + 435 + ], + "score": 1.0, + "content": "first line represents cases where we have an intervention on the variable", + "type": "text" + }, + { + "bbox": [ + 399, + 423, + 412, + 434 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 423, + 506, + 435 + ], + "score": 1.0, + "content": ", while we have it over", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 432, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 224, + 447 + ], + "score": 1.0, + "content": "interventions on the variable", + "type": "text" + }, + { + "bbox": [ + 224, + 434, + 237, + 445 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 432, + 505, + 447 + ], + "score": 1.0, + "content": "in the second line. 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This is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "done for two reasons. Firstly, the orientation parameter models only the direction of the edge, not", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 336, + 540 + ], + "score": 1.0, + "content": "whether it exists or not. 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Global order distributions", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "such as Plackett-Luce (Plackett, 1975; Luce, 1959) can be used to also incorporate transitive relations.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "However, those require high variance gradient estimators and struggled with chains in early experi-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "ments. 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Data: N variables {X1,.,XN}; observational dataset Dobs; interventional datasets Dint(I) for sparse,perfect interventions on all variables; distribution over interventions p(I) Result:A graph structure corresponding to the cuasal relations among variables nitializeγ=O,0=0
ornumberof epochsdo /* Distribution fitting */ for F iterations do X ~ Dobs
fori←1 to N do Mt~Ber(σ(ni)·σ(0li))
L =-logfρ(Xi;M_i⊙X_i) i ← Adam(Φi,VL)
end end
/* Graph fitting */
for G iterations do
/* Sample an intervention */ I ~ p(I) with intervention target Xt
X~Dint(I)
/* Evaluate multiple graph samples for gradient estimator */ for k = 1 to K do
C(k)~Ber(σ(γ)·σ(0))
for i= 1 to N do Lcw)(Xi)←-logfo(Xi;C） X-i)
end end
/* Update parameters */ for i= 1 to N do
for j=1 to Nwhere j≠iand j≠ t do
/* Considering edge X →Xj */
∑-1C）.Los） (xj） Lxi→xj(Xj）←
Ω1 ∑-1(-g）-（） (Xj）
LxiAxj(Xj）← Ωk1(1-0g)
Yij ←Yij-σ′(γij)·σ(0𝑖j)·[Cx𝑖→xj(Xj)-LxiAxj(Xj)+λsparse]
if i== t then
θij←0ij-σ'(0ij):σ(γij)·[Lxi→Xj(Xj)-LxiAx(Xj)]
θji←-ij end
end end end
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Data: N variables {X1,.,XN}; observational dataset Dobs; interventional datasets Dint(I) for sparse,perfect interventions on all variables; distribution over interventions p(I) Result:A graph structure corresponding to the cuasal relations among variables nitializeγ=O,0=0
ornumberof epochsdo /* Distribution fitting */ for F iterations do X ~ Dobs
fori←1 to N do Mt~Ber(σ(ni)·σ(0li))
L =-logfρ(Xi;M_i⊙X_i) i ← Adam(Φi,VL)
end end
/* Graph fitting */
for G iterations do
/* Sample an intervention */ I ~ p(I) with intervention target Xt
X~Dint(I)
/* Evaluate multiple graph samples for gradient estimator */ for k = 1 to K do
C(k)~Ber(σ(γ)·σ(0))
for i= 1 to N do Lcw)(Xi)←-logfo(Xi;C） X-i)
end end
/* Update parameters */ for i= 1 to N do
for j=1 to Nwhere j≠iand j≠ t do
/* Considering edge X →Xj */
∑-1C）.Los） (xj） Lxi→xj(Xj）←
Ω1 ∑-1(-g）-（） (Xj）
LxiAxj(Xj）← Ωk1(1-0g)
Yij ←Yij-σ′(γij)·σ(0𝑖j)·[Cx𝑖→xj(Xj)-LxiAxj(Xj)+λsparse]
if i== t then
θij←0ij-σ'(0ij):σ(γij)·[Lxi→Xj(Xj)-LxiAx(Xj)]
θji←-ij end
end end end
", + "type": "table", + "image_path": "bfba7038070d43af9c43e9493287af711fbcfab90a50053cfa7c6b414bee97f7.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 111, + 161, + 495, + 334.66666666666663 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 111, + 334.66666666666663, + 495, + 508.33333333333326 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 111, + 508.33333333333326, + 495, + 681.9999999999999 + ], + "spans": [], + "index": 8 + } + ] + } + ], + "index": 7 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 442, + 93 + ], + "lines": [ + { + "bbox": [ + 104, + 80, + 444, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 80, + 444, + 97 + ], + "score": 1.0, + "content": "B CONDITIONS FOR CONVERGING TO THE TRUE CAUSAL GRAPH", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 107, + 505, + 173 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "score": 1.0, + "content": "The following section gives an overview and proves the conditions under which ENCO converges to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 119, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 505, + 130 + ], + "score": 1.0, + "content": "the correct causal graph given sufficient time and data. We emphasize that we provide conditions here", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "for which no local optima exist, meaning that if ENCO converges, it returns the correct causal graph.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 506, + 153 + ], + "score": 1.0, + "content": "This is a stronger statement than showing that the global optimum corresponds to the true graph,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 506, + 163 + ], + "score": 1.0, + "content": "since a gradient-based algorithm can get stuck in a local optimum. We will discuss the conditions for", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 162, + 266, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 266, + 174 + ], + "score": 1.0, + "content": "the global optimum in Appendix B.2.5.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 179, + 505, + 223 + ], + "lines": [ + { + "bbox": [ + 106, + 180, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 505, + 190 + ], + "score": 1.0, + "content": "To make the proof more accessible, we will first discuss the assumptions that are needed for the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 190, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 506, + 202 + ], + "score": 1.0, + "content": "guarantee, and then give a sketch of the proof. The proof will first assume that we work in the data", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "limit, i.e. have given sufficient data, such that we can derive conditions that solely depend on the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 212, + 471, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 471, + 225 + ], + "score": 1.0, + "content": "causal graphical model. In Appendix B.2.3, we extend the proof to the limited data setting.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 107, + 238, + 196, + 249 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 197, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 197, + 251 + ], + "score": 1.0, + "content": "B.1 ASSUMPTIONS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 108, + 259, + 504, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 502, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 478, + 272 + ], + "score": 1.0, + "content": "Assumption 1 We are given a dataset of observational data from the joint distribution", + "type": "text" + }, + { + "bbox": [ + 479, + 259, + 502, + 272 + ], + "score": 0.92, + "content": "p ( X )", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 194, + 282 + ], + "score": 1.0, + "content": "Additionally, we have", + "type": "text" + }, + { + "bbox": [ + 194, + 271, + 204, + 280 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 270, + 308, + 282 + ], + "score": 1.0, + "content": "interventional datasets for", + "type": "text" + }, + { + "bbox": [ + 308, + 271, + 319, + 280 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "variables where in each intervention a different", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 282, + 407, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 407, + 294 + ], + "score": 1.0, + "content": "node is intervened on (the intervention size for each dataset is therefore 1).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 306, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "score": 1.0, + "content": "Assumption 2 A common assumption in causal structure learning is that the data distribution over", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 156, + 330 + ], + "score": 1.0, + "content": "all variables", + "type": "text" + }, + { + "bbox": [ + 157, + 318, + 181, + 330 + ], + "score": 0.92, + "content": "p ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "is Markovian and faithful with respect to the causal graph we are trying to model.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "This means that the graph represents the (conditional) independence relations between variables in the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 340, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 506, + 351 + ], + "score": 1.0, + "content": "data, and (conditional) independence relations in the data reflect the edges in the graph. For ENCO,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "score": 1.0, + "content": "faithfulness is not strictly required. This is because we work with interventional data. Instead, we rely", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 362, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 391, + 374 + ], + "score": 1.0, + "content": "on the Markov property and assume that for all variables, the parent set", + "type": "text" + }, + { + "bbox": [ + 391, + 362, + 421, + 373 + ], + "score": 0.82, + "content": "\\operatorname { p a } ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 362, + 506, + 374 + ], + "score": 1.0, + "content": "reflects the inputs to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 373, + 482, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 252, + 385 + ], + "score": 1.0, + "content": "the causal generation mechanism of", + "type": "text" + }, + { + "bbox": [ + 252, + 373, + 264, + 384 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 373, + 482, + 385 + ], + "score": 1.0, + "content": ". This allows us to also handle deterministic variables.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 397, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 506, + 410 + ], + "score": 1.0, + "content": "Assumption 3 For this proof, we assume that all variables of the graph are known and observable,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 409, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 420 + ], + "score": 1.0, + "content": "and no latent confounders exist. Latent confounders can introduce dependencies between variables", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "which are not reflected by the ground truth graph solely on the observed variables. We discuss the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 431, + 369, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 369, + 443 + ], + "score": 1.0, + "content": "extension of latent confounders in Section 3.5 and Appendix B.3.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 455, + 505, + 543 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "Assumption 4 ENCO relies on neural networks to determine the conditional data distributions", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 107, + 467, + 143, + 479 + ], + "score": 0.91, + "content": "p ( X _ { i } | \\ldots )", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 467, + 505, + 479 + ], + "score": 1.0, + "content": ". Hence, for providing a guarantee, we assume that in the graph learning step the neural", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "networks have been sufficiently trained such that they accurately model all possible conditional", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 154, + 502 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 154, + 489, + 191, + 501 + ], + "score": 0.92, + "content": "p ( X _ { i } | \\ldots )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 488, + 506, + 502 + ], + "score": 1.0, + "content": ". In practice, the neural networks might have a slight error. However, as long as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "enough data, network complexity, and training time is provided, it is fair to assume that the difference", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 510, + 502, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 497, + 523 + ], + "score": 1.0, + "content": "between the modeled distribution and the true conditional is smaller than an arbitrary constant", + "type": "text" + }, + { + "bbox": [ + 497, + 513, + 502, + 521 + ], + "score": 0.51, + "content": "\\epsilon", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "based on the universal approximation theorem (Hornik et al., 1989). For the limited data setting, see", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 533, + 175, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 175, + 545 + ], + "score": 1.0, + "content": "Appendix B.2.3.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "Assumption 5 We are given a sufficiently large interventional dataset such that sampling data", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "points from it models the exact interventional distribution under the true causal graph. This can", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "score": 1.0, + "content": "be achieved by, for example, sampling directly from the causal graph, or having an infinitely large", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 591, + 336, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 336, + 603 + ], + "score": 1.0, + "content": "dataset. For the limited data setting, see Appendix B.2.3.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 108, + 617, + 255, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 257, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 257, + 630 + ], + "score": 1.0, + "content": "B.2 CONVERGENCE CONDITIONS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 108, + 638, + 439, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 441, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 441, + 653 + ], + "score": 1.0, + "content": "The proof of the convergence conditions consists of the following three main steps:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 660, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 385, + 673 + ], + "score": 1.0, + "content": "Step 1 We show under which conditions the orientation parameters", + "type": "text" + }, + { + "bbox": [ + 386, + 661, + 398, + 672 + ], + "score": 0.9, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 659, + 468, + 673 + ], + "score": 1.0, + "content": "will converge to", + "type": "text" + }, + { + "bbox": [ + 469, + 662, + 488, + 671 + ], + "score": 0.86, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 659, + 506, + 673 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 142, + 671, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 142, + 671, + 190, + 684 + ], + "score": 0.9, + "content": "\\sigma ( \\theta _ { i j } ) 1", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 671, + 201, + 685 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 201, + 672, + 214, + 682 + ], + "score": 0.9, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 671, + 281, + 685 + ], + "score": 1.0, + "content": "is an ancestor of", + "type": "text" + }, + { + "bbox": [ + 281, + 672, + 294, + 683 + ], + "score": 0.84, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 671, + 347, + 685 + ], + "score": 1.0, + "content": ". Similarly, if", + "type": "text" + }, + { + "bbox": [ + 347, + 672, + 360, + 682 + ], + "score": 0.9, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 671, + 432, + 685 + ], + "score": 1.0, + "content": "is a descendant of", + "type": "text" + }, + { + "bbox": [ + 433, + 672, + 446, + 684 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 671, + 506, + 685 + ], + "score": 1.0, + "content": ", the parameter", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 142, + 682, + 313, + 695 + ], + "spans": [ + { + "bbox": [ + 142, + 683, + 155, + 695 + ], + "score": 0.87, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 682, + 223, + 695 + ], + "score": 1.0, + "content": "will converge to", + "type": "text" + }, + { + "bbox": [ + 223, + 684, + 242, + 693 + ], + "score": 0.87, + "content": "- \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 682, + 261, + 695 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 261, + 683, + 308, + 695 + ], + "score": 0.92, + "content": "\\sigma ( \\theta _ { i j } ) \\overset { \\cdot } { } 0", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 682, + 313, + 695 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "Step 2 Under the assumption that the orientation parameters have converged as in Step 1, we show", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 141, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 141, + 709, + 268, + 722 + ], + "score": 1.0, + "content": "that for edges in the true graph,", + "type": "text" + }, + { + "bbox": [ + 268, + 711, + 282, + 722 + ], + "score": 0.87, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "will converge to 1. Note that we need to take additional", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 141, + 720, + 349, + 735 + ], + "spans": [ + { + "bbox": [ + 141, + 720, + 299, + 735 + ], + "score": 1.0, + "content": "assumptions/conditions with respect to", + "type": "text" + }, + { + "bbox": [ + 300, + 721, + 324, + 733 + ], + "score": 0.92, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 720, + 349, + 735 + ], + "score": 1.0, + "content": "here.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44 + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 442, + 93 + ], + "lines": [ + { + "bbox": [ + 104, + 80, + 444, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 80, + 444, + 97 + ], + "score": 1.0, + "content": "B CONDITIONS FOR CONVERGING TO THE TRUE CAUSAL GRAPH", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 107, + 505, + 173 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "score": 1.0, + "content": "The following section gives an overview and proves the conditions under which ENCO converges to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 119, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 505, + 130 + ], + "score": 1.0, + "content": "the correct causal graph given sufficient time and data. We emphasize that we provide conditions here", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "for which no local optima exist, meaning that if ENCO converges, it returns the correct causal graph.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 506, + 153 + ], + "score": 1.0, + "content": "This is a stronger statement than showing that the global optimum corresponds to the true graph,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 506, + 163 + ], + "score": 1.0, + "content": "since a gradient-based algorithm can get stuck in a local optimum. We will discuss the conditions for", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 162, + 266, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 266, + 174 + ], + "score": 1.0, + "content": "the global optimum in Appendix B.2.5.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 106, + 506, + 174 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 179, + 505, + 223 + ], + "lines": [ + { + "bbox": [ + 106, + 180, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 505, + 190 + ], + "score": 1.0, + "content": "To make the proof more accessible, we will first discuss the assumptions that are needed for the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 190, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 506, + 202 + ], + "score": 1.0, + "content": "guarantee, and then give a sketch of the proof. The proof will first assume that we work in the data", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "limit, i.e. have given sufficient data, such that we can derive conditions that solely depend on the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 212, + 471, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 471, + 225 + ], + "score": 1.0, + "content": "causal graphical model. In Appendix B.2.3, we extend the proof to the limited data setting.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 180, + 506, + 225 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 238, + 196, + 249 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 197, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 197, + 251 + ], + "score": 1.0, + "content": "B.1 ASSUMPTIONS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 108, + 259, + 504, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 502, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 478, + 272 + ], + "score": 1.0, + "content": "Assumption 1 We are given a dataset of observational data from the joint distribution", + "type": "text" + }, + { + "bbox": [ + 479, + 259, + 502, + 272 + ], + "score": 0.92, + "content": "p ( X )", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 194, + 282 + ], + "score": 1.0, + "content": "Additionally, we have", + "type": "text" + }, + { + "bbox": [ + 194, + 271, + 204, + 280 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 270, + 308, + 282 + ], + "score": 1.0, + "content": "interventional datasets for", + "type": "text" + }, + { + "bbox": [ + 308, + 271, + 319, + 280 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "variables where in each intervention a different", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 282, + 407, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 407, + 294 + ], + "score": 1.0, + "content": "node is intervened on (the intervention size for each dataset is therefore 1).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 258, + 506, + 294 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 306, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "score": 1.0, + "content": "Assumption 2 A common assumption in causal structure learning is that the data distribution over", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 156, + 330 + ], + "score": 1.0, + "content": "all variables", + "type": "text" + }, + { + "bbox": [ + 157, + 318, + 181, + 330 + ], + "score": 0.92, + "content": "p ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "is Markovian and faithful with respect to the causal graph we are trying to model.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "This means that the graph represents the (conditional) independence relations between variables in the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 340, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 506, + 351 + ], + "score": 1.0, + "content": "data, and (conditional) independence relations in the data reflect the edges in the graph. For ENCO,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "score": 1.0, + "content": "faithfulness is not strictly required. This is because we work with interventional data. Instead, we rely", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 362, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 391, + 374 + ], + "score": 1.0, + "content": "on the Markov property and assume that for all variables, the parent set", + "type": "text" + }, + { + "bbox": [ + 391, + 362, + 421, + 373 + ], + "score": 0.82, + "content": "\\operatorname { p a } ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 362, + 506, + 374 + ], + "score": 1.0, + "content": "reflects the inputs to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 373, + 482, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 252, + 385 + ], + "score": 1.0, + "content": "the causal generation mechanism of", + "type": "text" + }, + { + "bbox": [ + 252, + 373, + 264, + 384 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 373, + 482, + 385 + ], + "score": 1.0, + "content": ". This allows us to also handle deterministic variables.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 307, + 506, + 385 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 397, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 506, + 410 + ], + "score": 1.0, + "content": "Assumption 3 For this proof, we assume that all variables of the graph are known and observable,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 409, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 420 + ], + "score": 1.0, + "content": "and no latent confounders exist. Latent confounders can introduce dependencies between variables", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "which are not reflected by the ground truth graph solely on the observed variables. We discuss the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 431, + 369, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 369, + 443 + ], + "score": 1.0, + "content": "extension of latent confounders in Section 3.5 and Appendix B.3.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 398, + 506, + 443 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 455, + 505, + 543 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "Assumption 4 ENCO relies on neural networks to determine the conditional data distributions", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 107, + 467, + 143, + 479 + ], + "score": 0.91, + "content": "p ( X _ { i } | \\ldots )", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 467, + 505, + 479 + ], + "score": 1.0, + "content": ". Hence, for providing a guarantee, we assume that in the graph learning step the neural", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "networks have been sufficiently trained such that they accurately model all possible conditional", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 154, + 502 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 154, + 489, + 191, + 501 + ], + "score": 0.92, + "content": "p ( X _ { i } | \\ldots )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 488, + 506, + 502 + ], + "score": 1.0, + "content": ". In practice, the neural networks might have a slight error. However, as long as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "enough data, network complexity, and training time is provided, it is fair to assume that the difference", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 510, + 502, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 497, + 523 + ], + "score": 1.0, + "content": "between the modeled distribution and the true conditional is smaller than an arbitrary constant", + "type": "text" + }, + { + "bbox": [ + 497, + 513, + 502, + 521 + ], + "score": 0.51, + "content": "\\epsilon", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "based on the universal approximation theorem (Hornik et al., 1989). For the limited data setting, see", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 533, + 175, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 175, + 545 + ], + "score": 1.0, + "content": "Appendix B.2.3.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 456, + 506, + 545 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "Assumption 5 We are given a sufficiently large interventional dataset such that sampling data", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "points from it models the exact interventional distribution under the true causal graph. This can", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "score": 1.0, + "content": "be achieved by, for example, sampling directly from the causal graph, or having an infinitely large", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 591, + 336, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 336, + 603 + ], + "score": 1.0, + "content": "dataset. For the limited data setting, see Appendix B.2.3.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 558, + 505, + 603 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 617, + 255, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 257, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 257, + 630 + ], + "score": 1.0, + "content": "B.2 CONVERGENCE CONDITIONS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 108, + 638, + 439, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 441, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 441, + 653 + ], + "score": 1.0, + "content": "The proof of the convergence conditions consists of the following three main steps:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 637, + 441, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 660, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 385, + 673 + ], + "score": 1.0, + "content": "Step 1 We show under which conditions the orientation parameters", + "type": "text" + }, + { + "bbox": [ + 386, + 661, + 398, + 672 + ], + "score": 0.9, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 659, + 468, + 673 + ], + "score": 1.0, + "content": "will converge to", + "type": "text" + }, + { + "bbox": [ + 469, + 662, + 488, + 671 + ], + "score": 0.86, + "content": "+ \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 659, + 506, + 673 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 142, + 671, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 142, + 671, + 190, + 684 + ], + "score": 0.9, + "content": "\\sigma ( \\theta _ { i j } ) 1", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 671, + 201, + 685 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 201, + 672, + 214, + 682 + ], + "score": 0.9, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 671, + 281, + 685 + ], + "score": 1.0, + "content": "is an ancestor of", + "type": "text" + }, + { + "bbox": [ + 281, + 672, + 294, + 683 + ], + "score": 0.84, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 671, + 347, + 685 + ], + "score": 1.0, + "content": ". 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Note that we need to take additional", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 141, + 720, + 349, + 735 + ], + "spans": [ + { + "bbox": [ + 141, + 720, + 299, + 735 + ], + "score": 1.0, + "content": "assumptions/conditions with respect to", + "type": "text" + }, + { + "bbox": [ + 300, + 721, + 324, + 733 + ], + "score": 0.92, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 720, + 349, + 735 + ], + "score": 1.0, + "content": "here.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 698, + 505, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 222, + 95 + ], + "score": 1.0, + "content": "Step 3 Once the parameters", + "type": "text" + }, + { + "bbox": [ + 222, + 84, + 236, + 95 + ], + "score": 0.87, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 81, + 253, + 95 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 83, + 266, + 95 + ], + "score": 0.89, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "have converged for the edges in the ground truth graph, we", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 142, + 93, + 418, + 106 + ], + "spans": [ + { + "bbox": [ + 142, + 93, + 418, + 106 + ], + "score": 1.0, + "content": "show that all other edges will be removed by the sparsity regularizer.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 114, + 505, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "score": 1.0, + "content": "The following paragraphs provide more details for each step. Note that causal graphs that do not", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 124, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 506, + 138 + ], + "score": 1.0, + "content": "fulfill all parts of the convergence guarantee can still eventually be recovered. The reason is that", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 460, + 149 + ], + "score": 1.0, + "content": "the conditions listed in the theorems below ensure that there exists no local minima for", + "type": "text" + }, + { + "bbox": [ + 460, + 137, + 468, + 146 + ], + "score": 0.76, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 135, + 485, + 149 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 485, + 137, + 494, + 147 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 135, + 506, + 149 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 146, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 505, + 160 + ], + "score": 1.0, + "content": "converge in. Although local minima exist, the optimization process might converge to the global", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 159, + 245, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 245, + 171 + ], + "score": 1.0, + "content": "minimum of the true causal graph.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 172, + 504, + 196 + ], + "lines": [ + { + "bbox": [ + 105, + 170, + 504, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 243, + 187 + ], + "score": 1.0, + "content": "Theorem B.1. Consider the edge", + "type": "text" + }, + { + "bbox": [ + 243, + 173, + 283, + 185 + ], + "score": 0.93, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 170, + 491, + 187 + ], + "score": 1.0, + "content": "in the true causal graph. The orientation parameter", + "type": "text" + }, + { + "bbox": [ + 491, + 173, + 504, + 185 + ], + "score": 0.92, + "content": "\\theta _ { i j }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 381, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 159, + 196 + ], + "score": 1.0, + "content": "converges to", + "type": "text" + }, + { + "bbox": [ + 159, + 183, + 204, + 196 + ], + "score": 0.93, + "content": "\\sigma ( \\theta _ { i j } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 183, + 381, + 196 + ], + "score": 1.0, + "content": "if the following two conditions are fulfilled:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 125, + 203, + 505, + 227 + ], + "lines": [ + { + "bbox": [ + 124, + 203, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 124, + 203, + 276, + 217 + ], + "score": 1.0, + "content": "(1) for all possible sets of parents of", + "type": "text" + }, + { + "bbox": [ + 276, + 205, + 289, + 217 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 203, + 332, + 217 + ], + "score": 1.0, + "content": "excluding", + "type": "text" + }, + { + "bbox": [ + 332, + 205, + 345, + 215 + ], + "score": 0.84, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 203, + 379, + 217 + ], + "score": 1.0, + "content": ", adding", + "type": "text" + }, + { + "bbox": [ + 379, + 205, + 391, + 215 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 203, + 506, + 217 + ], + "score": 1.0, + "content": "improves the log-likelihood", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 214, + 419, + 227 + ], + "spans": [ + { + "bbox": [ + 141, + 214, + 188, + 227 + ], + "score": 1.0, + "content": "estimate of", + "type": "text" + }, + { + "bbox": [ + 189, + 216, + 201, + 227 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 214, + 307, + 227 + ], + "score": 1.0, + "content": "under the intervention on", + "type": "text" + }, + { + "bbox": [ + 307, + 216, + 319, + 226 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 214, + 419, + 227 + ], + "score": 1.0, + "content": ", or leaves it unchanged:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 234, + 463, + 250 + ], + "lines": [ + { + "bbox": [ + 149, + 234, + 463, + 250 + ], + "spans": [ + { + "bbox": [ + 149, + 234, + 463, + 250 + ], + "score": 0.9, + "content": "\\forall \\widehat { p a } ( X _ { j } ) \\subseteq X _ { - i , j } : \\mathbb { E } _ { I _ { X _ { i } } , { \\pmb X } } \\left[ \\log p ( X _ { j } | \\widehat { p a } ( X _ { j } ) , X _ { i } ) - \\log p ( X _ { j } | \\widehat { p a } ( X _ { j } ) ) \\right] \\geq 0", + "type": "interline_equation", + "image_path": "81d9283e1ef81cc1d58dbc6f059293815551953765fa770963df73579466938f.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 149, + 234, + 463, + 250 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 124, + 257, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 124, + 255, + 503, + 272 + ], + "spans": [ + { + "bbox": [ + 124, + 255, + 244, + 272 + ], + "score": 1.0, + "content": "(2) there exists a set of nodes", + "type": "text" + }, + { + "bbox": [ + 245, + 258, + 276, + 270 + ], + "score": 0.92, + "content": "\\widehat { p a } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 255, + 490, + 272 + ], + "score": 1.0, + "content": ", for which the probability to be sampled as parents of", + "type": "text" + }, + { + "bbox": [ + 491, + 259, + 503, + 270 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 269, + 352, + 281 + ], + "spans": [ + { + "bbox": [ + 141, + 269, + 203, + 281 + ], + "score": 1.0, + "content": "is greater than", + "type": "text" + }, + { + "bbox": [ + 203, + 270, + 209, + 279 + ], + "score": 0.68, + "content": "O", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 269, + 352, + 281 + ], + "score": 1.0, + "content": ", and the following condition holds:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 288, + 462, + 303 + ], + "lines": [ + { + "bbox": [ + 149, + 288, + 462, + 303 + ], + "spans": [ + { + "bbox": [ + 149, + 288, + 462, + 303 + ], + "score": 0.91, + "content": "\\exists \\widehat { p a } ( X _ { j } ) \\subseteq X _ { - i , j } : \\mathbb { E } _ { I _ { X _ { i } } , { \\pmb X } } \\left[ \\log p ( X _ { j } | \\widehat { p a } ( X _ { j } ) , X _ { i } ) - \\log p ( X _ { j } | \\widehat { p a } ( X _ { j } ) ) \\right] > 0", + "type": "interline_equation", + "image_path": "c71b2f83052b488d5dde57e1944ab434c9038832d3ae9ead2e23bdfa40581649.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 149, + 288, + 462, + 303 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 314, + 505, + 348 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 504, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 491, + 328 + ], + "score": 1.0, + "content": "Proof. Based on the conditions in Equations 23 and 24, we need to show that the gradient of", + "type": "text" + }, + { + "bbox": [ + 491, + 315, + 504, + 327 + ], + "score": 0.88, + "content": "\\theta _ { i j }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 343, + 337 + ], + "score": 1.0, + "content": "is negative in expectation, independent of other values of", + "type": "text" + }, + { + "bbox": [ + 343, + 327, + 351, + 337 + ], + "score": 0.84, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 325, + 370, + 337 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 370, + 326, + 377, + 335 + ], + "score": 0.79, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 325, + 505, + 337 + ], + "score": 1.0, + "content": ". For readability, we define the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 336, + 186, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 186, + 349 + ], + "score": 1.0, + "content": "following function:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 352, + 426, + 367 + ], + "lines": [ + { + "bbox": [ + 185, + 352, + 426, + 367 + ], + "spans": [ + { + "bbox": [ + 185, + 352, + 426, + 367 + ], + "score": 0.9, + "content": "T ( X _ { k } , X _ { l } ) = \\mathbb { E } _ { I _ { X _ { k } } , X , C _ { - k l } } [ \\mathcal { L } _ { X _ { k } X _ { l } } ( X _ { l } ) - \\mathcal { L } _ { X _ { k } \\neq X _ { l } } ( X _ { l } ) ]", + "type": "interline_equation", + "image_path": "30bf8d25c065c9b2fdb63922bc57ba405007e63bc83790a7e6f6b6c7b8252d89.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 185, + 352, + 426, + 367 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 283, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 284, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 198, + 385 + ], + "score": 1.0, + "content": "Hence, the gradient of", + "type": "text" + }, + { + "bbox": [ + 198, + 372, + 210, + 384 + ], + "score": 0.9, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 370, + 284, + 385 + ], + "score": 1.0, + "content": "can be written as:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 388, + 467, + 415 + ], + "lines": [ + { + "bbox": [ + 143, + 388, + 467, + 415 + ], + "spans": [ + { + "bbox": [ + 143, + 388, + 467, + 415 + ], + "score": 0.9, + "content": "\\frac { \\partial } { \\partial \\theta _ { i j } } \\tilde { \\mathcal { L } } = \\sigma ^ { \\prime } ( \\theta _ { i j } ) \\cdot \\Big ( p ( I _ { X _ { i } } ) \\cdot \\sigma ( \\gamma _ { i j } ) \\cdot T ( X _ { i } , X _ { j } ) - p ( I _ { X _ { j } } ) \\cdot \\sigma ( \\gamma _ { j i } ) \\cdot T ( X _ { j } , X _ { i } ) \\Big )", + "type": "interline_equation", + "image_path": "9adcba2e4e1553c3ded944c991951a8d40345e04c9fd4de8415cf614740f3ef5.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 143, + 388, + 467, + 397.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 143, + 397.0, + 467, + 406.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 143, + 406.0, + 467, + 415.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 212, + 433 + ], + "score": 1.0, + "content": "Looking at the gradient of", + "type": "text" + }, + { + "bbox": [ + 212, + 420, + 225, + 432 + ], + "score": 0.9, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 418, + 403, + 433 + ], + "score": 1.0, + "content": "in Equation 26, the conditions correspond to", + "type": "text" + }, + { + "bbox": [ + 404, + 420, + 448, + 432 + ], + "score": 0.9, + "content": "T ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 418, + 506, + 433 + ], + "score": 1.0, + "content": "being smaller", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 327, + 443 + ], + "score": 1.0, + "content": "or equals to zero. Note that the sign is flipped because in", + "type": "text" + }, + { + "bbox": [ + 327, + 431, + 371, + 443 + ], + "score": 0.92, + "content": "T ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 430, + 505, + 443 + ], + "score": 1.0, + "content": ", we have negative log-likelihoods", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 169, + 454 + ], + "score": 1.0, + "content": "represented by", + "type": "text" + }, + { + "bbox": [ + 169, + 442, + 224, + 453 + ], + "score": 0.9, + "content": "\\mathscr { L } _ { X _ { k } X _ { l } } ( X _ { l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 441, + 505, + 454 + ], + "score": 1.0, + "content": ", while in Equations 23 and 24, we have log-likelihoods. 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i , j } : \\mathbb { E } _ { I _ { X _ { i } } , { \\pmb X } } \\left[ \\log p ( X _ { j } | \\widehat { p a } ( X _ { j } ) , X _ { i } ) - \\log p ( X _ { j } | \\widehat { p a } ( X _ { j } ) ) \\right] > 0", + "type": "interline_equation", + "image_path": "c71b2f83052b488d5dde57e1944ab434c9038832d3ae9ead2e23bdfa40581649.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 149, + 288, + 462, + 303 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 314, + 505, + 348 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 504, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 491, + 328 + ], + "score": 1.0, + "content": "Proof. 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For readability, we define the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 336, + 186, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 186, + 349 + ], + "score": 1.0, + "content": "following function:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 312, + 505, + 349 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 352, + 426, + 367 + ], + "lines": [ + { + "bbox": [ + 185, + 352, + 426, + 367 + ], + "spans": [ + { + "bbox": [ + 185, + 352, + 426, + 367 + ], + "score": 0.9, + "content": "T ( X _ { k } , X _ { l } ) = \\mathbb { E } _ { I _ { X _ { k } } , X , C _ { - k l } } [ \\mathcal { L } _ { X _ { k } X _ { l } } ( X _ { l } ) - \\mathcal { L } _ { X _ { k } \\neq X _ { l } } ( X _ { l } ) ]", + "type": "interline_equation", + "image_path": "30bf8d25c065c9b2fdb63922bc57ba405007e63bc83790a7e6f6b6c7b8252d89.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 185, + 352, + 426, + 367 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 283, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 284, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 198, + 385 + ], + "score": 1.0, + "content": "Hence, the gradient of", + "type": "text" + }, + { + "bbox": [ + 198, + 372, + 210, + 384 + ], + "score": 0.9, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 370, + 284, + 385 + ], + "score": 1.0, + "content": "can be written as:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 370, + 284, + 385 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 388, + 467, + 415 + ], + "lines": [ + { + "bbox": [ + 143, + 388, + 467, + 415 + ], + "spans": [ + { + "bbox": [ + 143, + 388, + 467, + 415 + ], + "score": 0.9, + "content": "\\frac { \\partial } { \\partial \\theta _ { i j } } \\tilde { \\mathcal { L } } = \\sigma ^ { \\prime } ( \\theta _ { i j } ) \\cdot \\Big ( p ( I _ { X _ { i } } ) \\cdot \\sigma ( \\gamma _ { i j } ) \\cdot T ( X _ { i } , X _ { j } ) - p ( I _ { X _ { j } } ) \\cdot \\sigma ( \\gamma _ { j i } ) \\cdot T ( X _ { j } , X _ { i } ) \\Big )", + "type": "interline_equation", + "image_path": "9adcba2e4e1553c3ded944c991951a8d40345e04c9fd4de8415cf614740f3ef5.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 143, + 388, + 467, + 397.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 143, + 397.0, + 467, + 406.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 143, + 406.0, + 467, + 415.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 212, + 433 + ], + "score": 1.0, + "content": "Looking at the gradient of", + "type": "text" + }, + { + "bbox": [ + 212, + 420, + 225, + 432 + ], + "score": 0.9, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 418, + 403, + 433 + ], + "score": 1.0, + "content": "in Equation 26, the conditions correspond to", + "type": "text" + }, + { + "bbox": [ + 404, + 420, + 448, + 432 + ], + "score": 0.9, + "content": "T ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 418, + 506, + 433 + ], + "score": 1.0, + "content": "being smaller", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 327, + 443 + ], + "score": 1.0, + "content": "or equals to zero. 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If", + "type": "text" + }, + { + "bbox": [ + 394, + 464, + 494, + 476 + ], + "score": 0.91, + "content": "T ( X _ { i } , X _ { j } ) - T ( X _ { j } , X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 473, + 394, + 489 + ], + "spans": [ + { + "bbox": [ + 104, + 473, + 260, + 489 + ], + "score": 1.0, + "content": "smaller than zero, then the gradient of", + "type": "text" + }, + { + "bbox": [ + 261, + 475, + 273, + 487 + ], + "score": 0.9, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 473, + 377, + 489 + ], + "score": 1.0, + "content": "is negative, i.e.increasing", + "type": "text" + }, + { + "bbox": [ + 378, + 475, + 390, + 487 + ], + "score": 0.88, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 473, + 394, + 489 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 418, + 506, + 489 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 491, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 490, + 504, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 199, + 505 + ], + "score": 1.0, + "content": "First, we look at when", + "type": "text" + }, + { + "bbox": [ + 199, + 491, + 262, + 504 + ], + "score": 0.94, + "content": "T ( X _ { i } , X _ { j } ) < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 490, + 490, + 505 + ], + "score": 1.0, + "content": ". 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If", + "type": "text" + }, + { + "bbox": [ + 490, + 723, + 504, + 732 + ], + "score": 0.89, + "content": "X _ { k }", + "type": "inline_equation" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 123, + 95 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 123, + 83, + 136, + 93 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 136, + 82, + 453, + 95 + ], + "score": 1.0, + "content": "again share a collider, we can apply this argumentation recursively until a node", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 453, + 83, + 468, + 93 + ], + "score": 0.88, + "content": "X _ { n }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 468, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "does not", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 190, + 106 + ], + "score": 1.0, + "content": "share a collider with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 191, + 94, + 203, + 105 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 203, + 93, + 506, + 106 + ], + "score": 1.0, + "content": ". 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The recursion will always come to an end as we have a finite set of nodes,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 291, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 291, + 117 + ], + "score": 1.0, + "content": "and the causal graph is assumed to be acyclic.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 108, + 121, + 504, + 155 + ], + "lines": [ + { + "bbox": [ + 106, + 119, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 366, + 134 + ], + "score": 1.0, + "content": "Thus, if the conditions in Equations 23 and 24 hold for an edge", + "type": "text" + }, + { + "bbox": [ + 366, + 122, + 407, + 133 + ], + "score": 0.92, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 119, + 505, + 134 + ], + "score": 1.0, + "content": "in the causal graph, we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 472, + 145 + ], + "score": 1.0, + "content": "can guarantee that with sufficient time and data, the corresponding orientation parameter", + "type": "text" + }, + { + "bbox": [ + 472, + 132, + 485, + 145 + ], + "score": 0.89, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "will", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 155, + 156 + ], + "score": 1.0, + "content": "converge to", + "type": "text" + }, + { + "bbox": [ + 155, + 143, + 200, + 156 + ], + "score": 0.96, + "content": "\\sigma ( \\theta _ { i j } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 144, + 204, + 156 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 144, + 506, + 155 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 503, + 195 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 281, + 174 + ], + "score": 1.0, + "content": "Theorem B.2. Consider a pair of variables", + "type": "text" + }, + { + "bbox": [ + 282, + 162, + 311, + 174 + ], + "score": 0.74, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 160, + 352, + 174 + ], + "score": 1.0, + "content": "for which", + "type": "text" + }, + { + "bbox": [ + 352, + 162, + 365, + 172 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 160, + 433, + 174 + ], + "score": 1.0, + "content": "is an ancestor of", + "type": "text" + }, + { + "bbox": [ + 434, + 162, + 447, + 173 + ], + "score": 0.86, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 160, + 505, + 174 + ], + "score": 1.0, + "content": "without direct", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 410, + 185 + ], + "score": 1.0, + "content": "edge in the true causal graph. Then, the orientation parameter of the edge", + "type": "text" + }, + { + "bbox": [ + 410, + 173, + 450, + 184 + ], + "score": 0.9, + "content": "X _ { i } \\to \\bar { X } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "converges to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 182, + 430, + 198 + ], + "spans": [ + { + "bbox": [ + 107, + 183, + 152, + 195 + ], + "score": 0.94, + "content": "\\sigma ( \\theta _ { i j } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 182, + 396, + 198 + ], + "score": 1.0, + "content": "if the same conditions as in Theorem B.1 hold for the pair of", + "type": "text" + }, + { + "bbox": [ + 396, + 184, + 425, + 195 + ], + "score": 0.91, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 182, + 430, + 198 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 207, + 505, + 251 + ], + "lines": [ + { + "bbox": [ + 105, + 206, + 504, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 441, + 219 + ], + "score": 1.0, + "content": "Proof. 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The variables", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 197, + 363 + ], + "score": 1.0, + "content": "can be independent if", + "type": "text" + }, + { + "bbox": [ + 197, + 351, + 210, + 362 + ], + "score": 0.9, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 351, + 386, + 363 + ], + "score": 1.0, + "content": "is conditioned on variables that d-separate", + "type": "text" + }, + { + "bbox": [ + 386, + 352, + 399, + 362 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 351, + 417, + 363 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 418, + 351, + 431, + 363 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "in the true causal", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 281, + 374 + ], + "score": 1.0, + "content": "graph. For instance, having the children of", + "type": "text" + }, + { + "bbox": [ + 281, + 362, + 294, + 372 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 361, + 348, + 374 + ], + "score": 1.0, + "content": "as parents of", + "type": "text" + }, + { + "bbox": [ + 349, + 362, + 361, + 372 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "creates this scenario. However, for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "this scenario to take place, one or more orientation parameters of parent-child or ancestor-descendant", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 371, + 396 + ], + "score": 1.0, + "content": "pairs must be incorrectly converged. In case of a parent-child pair", + "type": "text" + }, + { + "bbox": [ + 371, + 384, + 401, + 395 + ], + "score": 0.63, + "content": "X _ { i } , X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 383, + 506, + 396 + ], + "score": 1.0, + "content": ", Theorem B.1 shows that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 107, + 394, + 133, + 406 + ], + "score": 0.92, + "content": "\\sigma ( \\theta _ { i k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "will converge to one removing any possibility of a reversed edge to be sampled. In case of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 405, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 221, + 419 + ], + "score": 1.0, + "content": "an ancestor-descendant pair", + "type": "text" + }, + { + "bbox": [ + 222, + 406, + 250, + 417 + ], + "score": 0.91, + "content": "X _ { i } , X _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 405, + 411, + 419 + ], + "score": 1.0, + "content": ", we can apply a recursive argument: as", + "type": "text" + }, + { + "bbox": [ + 412, + 406, + 424, + 416 + ], + "score": 0.87, + "content": "X _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 405, + 474, + 419 + ], + "score": 1.0, + "content": "d-separates", + "type": "text" + }, + { + "bbox": [ + 474, + 406, + 487, + 416 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 405, + 506, + 419 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 415, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 107, + 417, + 120, + 429 + ], + "score": 0.82, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 415, + 124, + 430 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 124, + 416, + 136, + 427 + ], + "score": 0.84, + "content": "X _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 415, + 212, + 430 + ], + "score": 1.0, + "content": "must come before", + "type": "text" + }, + { + "bbox": [ + 213, + 417, + 226, + 429 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 415, + 380, + 430 + ], + "score": 1.0, + "content": "in the causal order. If for the gradient", + "type": "text" + }, + { + "bbox": [ + 380, + 416, + 392, + 427 + ], + "score": 0.89, + "content": "\\theta _ { i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 415, + 506, + 430 + ], + "score": 1.0, + "content": ", we have a similar scenario", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 126, + 440 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 428, + 139, + 438 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 427, + 282, + 440 + ], + "score": 1.0, + "content": "being conditionally independent of", + "type": "text" + }, + { + "bbox": [ + 282, + 428, + 295, + 439 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 427, + 505, + 440 + ], + "score": 1.0, + "content": ", the same argument applies. This can be recursively", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 333, + 451 + ], + "score": 1.0, + "content": "applied until no more variables except direct children of", + "type": "text" + }, + { + "bbox": [ + 333, + 438, + 345, + 449 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 438, + 363, + 451 + ], + "score": 1.0, + "content": "can", + "type": "text" + }, + { + "bbox": [ + 363, + 439, + 370, + 448 + ], + "score": 0.26, + "content": "\\mathrm { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 438, + 406, + 451 + ], + "score": 1.0, + "content": "-separate", + "type": "text" + }, + { + "bbox": [ + 407, + 438, + 419, + 449 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 438, + 437, + 451 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 438, + 438, + 451, + 450 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 438, + 506, + 451 + ], + "score": 1.0, + "content": ". In that case,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 107, + 449, + 134, + 461 + ], + "score": 0.92, + "content": "\\sigma ( \\theta _ { i k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "will converge to one, which leads to all other orientation parameters to converge to one as well.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 115, + 474 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 116, + 461, + 128, + 471 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 459, + 269, + 474 + ], + "score": 1.0, + "content": "is not conditionally independent of", + "type": "text" + }, + { + "bbox": [ + 269, + 461, + 282, + 472 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 459, + 506, + 474 + ], + "score": 1.0, + "content": ", we can rely back on the argumentation of Theorem B.1", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 469, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 197, + 485 + ], + "score": 1.0, + "content": "when we have an edge", + "type": "text" + }, + { + "bbox": [ + 198, + 471, + 238, + 483 + ], + "score": 0.92, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 469, + 375, + 485 + ], + "score": 1.0, + "content": ": as in the intervened causal graph,", + "type": "text" + }, + { + "bbox": [ + 376, + 471, + 388, + 482 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 469, + 406, + 485 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 407, + 471, + 420, + 483 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 469, + 506, + 485 + ], + "score": 1.0, + "content": "are independent, any", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "correlation learned from observational data can only lead to a worse log-likelihood estimate. In cases", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 473, + 505 + ], + "score": 1.0, + "content": "of colliders, we can rely on the recursive argument from before. Thus, under interventions on", + "type": "text" + }, + { + "bbox": [ + 474, + 493, + 487, + 505 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 493, + 505, + 505 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 503, + 430, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 152, + 519 + ], + "score": 1.0, + "content": "gradient of", + "type": "text" + }, + { + "bbox": [ + 153, + 505, + 165, + 516 + ], + "score": 0.89, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 503, + 412, + 519 + ], + "score": 1.0, + "content": "must be smaller or equals to zero in expectation, i.e.increases", + "type": "text" + }, + { + "bbox": [ + 413, + 505, + 425, + 516 + ], + "score": 0.9, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 503, + 430, + 519 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 503, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 504, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 239, + 534 + ], + "score": 1.0, + "content": "Therefore, we can conclude that", + "type": "text" + }, + { + "bbox": [ + 239, + 521, + 265, + 533 + ], + "score": 0.92, + "content": "\\sigma ( \\theta _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 519, + 474, + 534 + ], + "score": 1.0, + "content": "converges to one for any ancestor-descendant pairs", + "type": "text" + }, + { + "bbox": [ + 474, + 522, + 504, + 533 + ], + "score": 0.87, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 532, + 503, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 259, + 543 + ], + "score": 1.0, + "content": "under the conditions in Theorem B.1.", + "type": "text" + }, + { + "bbox": [ + 497, + 534, + 503, + 540 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 105, + 549, + 504, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 547, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 242, + 564 + ], + "score": 1.0, + "content": "Theorem B.3. Consider an edge", + "type": "text" + }, + { + "bbox": [ + 242, + 550, + 282, + 562 + ], + "score": 0.89, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 547, + 448, + 564 + ], + "score": 1.0, + "content": "in the true causal graph. The parameter", + "type": "text" + }, + { + "bbox": [ + 448, + 552, + 461, + 562 + ], + "score": 0.87, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 547, + 506, + 564 + ], + "score": 1.0, + "content": "converges", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 560, + 293, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 117, + 573 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 560, + 162, + 573 + ], + "score": 0.93, + "content": "\\sigma ( \\gamma _ { i j } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 560, + 293, + 573 + ], + "score": 1.0, + "content": "if the following condition holds:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 577, + 457, + 599 + ], + "lines": [ + { + "bbox": [ + 154, + 577, + 457, + 599 + ], + "spans": [ + { + "bbox": [ + 154, + 577, + 457, + 599 + ], + "score": 0.9, + "content": "\\operatorname* { m i n } _ { \\substack { \\hat { p } a \\subseteq g p a _ { i } ( X _ { j } ) } } \\mathbb { E } _ { \\hat { I } \\sim p _ { I _ { - j } } ( I ) } \\mathbb { E } _ { \\tilde { p } _ { \\hat { I } } ( { \\pmb X } ) } \\big [ \\log p ( X _ { j } | \\hat { p a } , X _ { i } ) - \\log p ( X _ { j } | \\hat { p a } ) \\big ] > \\lambda _ { s p a r s e }", + "type": "interline_equation", + "image_path": "b46900ea849423d1308a1dde70d9c78dcc2f8c989dbe9a59e0602b00214f125f.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 154, + 577, + 457, + 599 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 135, + 618 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 605, + 174, + 617 + ], + "score": 0.91, + "content": "g p a _ { i } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 603, + 302, + 618 + ], + "score": 1.0, + "content": "is the set of nodes excluding", + "type": "text" + }, + { + "bbox": [ + 302, + 605, + 315, + 616 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 603, + 506, + 618 + ], + "score": 1.0, + "content": "which, according to the ground truth graph,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 199, + 629 + ], + "score": 1.0, + "content": "could have an edge to", + "type": "text" + }, + { + "bbox": [ + 199, + 616, + 213, + 628 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 615, + 349, + 629 + ], + "score": 1.0, + "content": "without introducing a cycle, and", + "type": "text" + }, + { + "bbox": [ + 349, + 616, + 381, + 628 + ], + "score": 0.92, + "content": "p _ { I _ { - j } } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "refers to the distribution over", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 626, + 358, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 161, + 640 + ], + "score": 1.0, + "content": "interventions", + "type": "text" + }, + { + "bbox": [ + 161, + 627, + 185, + 639 + ], + "score": 0.92, + "content": "p _ { I } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 626, + 340, + 640 + ], + "score": 1.0, + "content": "excluding the intervention on variable", + "type": "text" + }, + { + "bbox": [ + 340, + 628, + 353, + 639 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 626, + 358, + 640 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 650, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "Proof. To show this convergence, we assume that the orientation parameters have converged cor-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 661, + 507, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 325, + 676 + ], + "score": 1.0, + "content": "responding to Theorem B.1 and B.2. The parameter", + "type": "text" + }, + { + "bbox": [ + 326, + 663, + 339, + 674 + ], + "score": 0.88, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 661, + 396, + 676 + ], + "score": 1.0, + "content": "converges to", + "type": "text" + }, + { + "bbox": [ + 396, + 661, + 444, + 674 + ], + "score": 0.93, + "content": "\\sigma ( \\gamma _ { i j } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 661, + 507, + 676 + ], + "score": 1.0, + "content": "if its gradient,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 107, + 668, + 508, + 693 + ], + "spans": [ + { + "bbox": [ + 107, + 673, + 132, + 689 + ], + "score": 0.92, + "content": "\\begin{array} { r l r } { { \\frac { \\partial } { \\partial \\gamma _ { i j } } \\tilde { \\mathcal { L } } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 668, + 299, + 693 + ], + "score": 1.0, + "content": ", is negative independent of other values of", + "type": "text" + }, + { + "bbox": [ + 300, + 677, + 308, + 686 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 668, + 415, + 693 + ], + "score": 1.0, + "content": "and orientation parameters", + "type": "text" + }, + { + "bbox": [ + 415, + 675, + 422, + 685 + ], + "score": 0.8, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 668, + 508, + 693 + ], + "score": 1.0, + "content": "that are not included", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 286, + 701 + ], + "score": 1.0, + "content": "in Theorem B.1 and B.2. The gradient of", + "type": "text" + }, + { + "bbox": [ + 286, + 689, + 299, + 700 + ], + "score": 0.88, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "includes an expectation over adjacency matrices", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 141, + 711 + ], + "score": 0.91, + "content": "p _ { \\gamma , \\theta } ( C )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 698, + 246, + 712 + ], + "score": 1.0, + "content": ". Based on the converged", + "type": "text" + }, + { + "bbox": [ + 247, + 700, + 253, + 708 + ], + "score": 0.82, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "-values, we only need to consider sets of nodes as parents for", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 107, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 107, + 711, + 120, + 722 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "that contain parents, ancestors, or (conditionally) independent nodes according to the ground", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 719, + 507, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 312, + 735 + ], + "score": 1.0, + "content": "truth graph. This sets of parents is represented by", + "type": "text" + }, + { + "bbox": [ + 312, + 721, + 351, + 733 + ], + "score": 0.92, + "content": "\\mathrm { g p a } _ { i } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 719, + 507, + 735 + ], + "score": 1.0, + "content": ". Among those remaining parent sets,", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47 + } + ], + "page_idx": 22, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 82, + 506, + 117 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 108, + 121, + 504, + 155 + ], + "lines": [ + { + "bbox": [ + 106, + 119, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 366, + 134 + ], + "score": 1.0, + "content": "Thus, if the conditions in Equations 23 and 24 hold for an edge", + "type": "text" + }, + { + "bbox": [ + 366, + 122, + 407, + 133 + ], + "score": 0.92, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 119, + 505, + 134 + ], + "score": 1.0, + "content": "in the causal graph, we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 472, + 145 + ], + "score": 1.0, + "content": "can guarantee that with sufficient time and data, the corresponding orientation parameter", + "type": "text" + }, + { + "bbox": [ + 472, + 132, + 485, + 145 + ], + "score": 0.89, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "will", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 155, + 156 + ], + "score": 1.0, + "content": "converge to", + "type": "text" + }, + { + "bbox": [ + 155, + 143, + 200, + 156 + ], + "score": 0.96, + "content": "\\sigma ( \\theta _ { i j } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 144, + 204, + 156 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 144, + 506, + 155 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 106, + 119, + 506, + 156 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 503, + 195 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 281, + 174 + ], + "score": 1.0, + "content": "Theorem B.2. Consider a pair of variables", + "type": "text" + }, + { + "bbox": [ + 282, + 162, + 311, + 174 + ], + "score": 0.74, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 160, + 352, + 174 + ], + "score": 1.0, + "content": "for which", + "type": "text" + }, + { + "bbox": [ + 352, + 162, + 365, + 172 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 160, + 433, + 174 + ], + "score": 1.0, + "content": "is an ancestor of", + "type": "text" + }, + { + "bbox": [ + 434, + 162, + 447, + 173 + ], + "score": 0.86, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 160, + 505, + 174 + ], + "score": 1.0, + "content": "without direct", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 410, + 185 + ], + "score": 1.0, + "content": "edge in the true causal graph. Then, the orientation parameter of the edge", + "type": "text" + }, + { + "bbox": [ + 410, + 173, + 450, + 184 + ], + "score": 0.9, + "content": "X _ { i } \\to \\bar { X } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "converges to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 182, + 430, + 198 + ], + "spans": [ + { + "bbox": [ + 107, + 183, + 152, + 195 + ], + "score": 0.94, + "content": "\\sigma ( \\theta _ { i j } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 182, + 396, + 198 + ], + "score": 1.0, + "content": "if the same conditions as in Theorem B.1 hold for the pair of", + "type": "text" + }, + { + "bbox": [ + 396, + 184, + 425, + 195 + ], + "score": 0.91, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 182, + 430, + 198 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 160, + 505, + 198 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 207, + 505, + 251 + ], + "lines": [ + { + "bbox": [ + 105, + 206, + 504, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 441, + 219 + ], + "score": 1.0, + "content": "Proof. 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The variables", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 197, + 363 + ], + "score": 1.0, + "content": "can be independent if", + "type": "text" + }, + { + "bbox": [ + 197, + 351, + 210, + 362 + ], + "score": 0.9, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 351, + 386, + 363 + ], + "score": 1.0, + "content": "is conditioned on variables that d-separate", + "type": "text" + }, + { + "bbox": [ + 386, + 352, + 399, + 362 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 351, + 417, + 363 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 418, + 351, + 431, + 363 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "in the true causal", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 281, + 374 + ], + "score": 1.0, + "content": "graph. 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If for the gradient", + "type": "text" + }, + { + "bbox": [ + 380, + 416, + 392, + 427 + ], + "score": 0.89, + "content": "\\theta _ { i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 415, + 506, + 430 + ], + "score": 1.0, + "content": ", we have a similar scenario", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 126, + 440 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 428, + 139, + 438 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 427, + 282, + 440 + ], + "score": 1.0, + "content": "being conditionally independent of", + "type": "text" + }, + { + "bbox": [ + 282, + 428, + 295, + 439 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 427, + 505, + 440 + ], + "score": 1.0, + "content": ", the same argument applies. This can be recursively", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 333, + 451 + ], + "score": 1.0, + "content": "applied until no more variables except direct children of", + "type": "text" + }, + { + "bbox": [ + 333, + 438, + 345, + 449 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 438, + 363, + 451 + ], + "score": 1.0, + "content": "can", + "type": "text" + }, + { + "bbox": [ + 363, + 439, + 370, + 448 + ], + "score": 0.26, + "content": "\\mathrm { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 438, + 406, + 451 + ], + "score": 1.0, + "content": "-separate", + "type": "text" + }, + { + "bbox": [ + 407, + 438, + 419, + 449 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 438, + 437, + 451 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 438, + 438, + 451, + 450 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 438, + 506, + 451 + ], + "score": 1.0, + "content": ". In that case,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 107, + 449, + 134, + 461 + ], + "score": 0.92, + "content": "\\sigma ( \\theta _ { i k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "will converge to one, which leads to all other orientation parameters to converge to one as well.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 115, + 474 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 116, + 461, + 128, + 471 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 459, + 269, + 474 + ], + "score": 1.0, + "content": "is not conditionally independent of", + "type": "text" + }, + { + "bbox": [ + 269, + 461, + 282, + 472 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 459, + 506, + 474 + ], + "score": 1.0, + "content": ", we can rely back on the argumentation of Theorem B.1", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 469, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 197, + 485 + ], + "score": 1.0, + "content": "when we have an edge", + "type": "text" + }, + { + "bbox": [ + 198, + 471, + 238, + 483 + ], + "score": 0.92, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 469, + 375, + 485 + ], + "score": 1.0, + "content": ": as in the intervened causal graph,", + "type": "text" + }, + { + "bbox": [ + 376, + 471, + 388, + 482 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 469, + 406, + 485 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 407, + 471, + 420, + 483 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 469, + 506, + 485 + ], + "score": 1.0, + "content": "are independent, any", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "correlation learned from observational data can only lead to a worse log-likelihood estimate. In cases", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 473, + 505 + ], + "score": 1.0, + "content": "of colliders, we can rely on the recursive argument from before. Thus, under interventions on", + "type": "text" + }, + { + "bbox": [ + 474, + 493, + 487, + 505 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 493, + 505, + 505 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 503, + 430, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 152, + 519 + ], + "score": 1.0, + "content": "gradient of", + "type": "text" + }, + { + "bbox": [ + 153, + 505, + 165, + 516 + ], + "score": 0.89, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 503, + 412, + 519 + ], + "score": 1.0, + "content": "must be smaller or equals to zero in expectation, i.e.increases", + "type": "text" + }, + { + "bbox": [ + 413, + 505, + 425, + 516 + ], + "score": 0.9, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 503, + 430, + 519 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 328, + 506, + 519 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 503, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 504, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 239, + 534 + ], + "score": 1.0, + "content": "Therefore, we can conclude that", + "type": "text" + }, + { + "bbox": [ + 239, + 521, + 265, + 533 + ], + "score": 0.92, + "content": "\\sigma ( \\theta _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 519, + 474, + 534 + ], + "score": 1.0, + "content": "converges to one for any ancestor-descendant pairs", + "type": "text" + }, + { + "bbox": [ + 474, + 522, + 504, + 533 + ], + "score": 0.87, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 532, + 503, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 259, + 543 + ], + "score": 1.0, + "content": "under the conditions in Theorem B.1.", + "type": "text" + }, + { + "bbox": [ + 497, + 534, + 503, + 540 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 519, + 504, + 543 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 549, + 504, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 547, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 242, + 564 + ], + "score": 1.0, + "content": "Theorem B.3. Consider an edge", + "type": "text" + }, + { + "bbox": [ + 242, + 550, + 282, + 562 + ], + "score": 0.89, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 547, + 448, + 564 + ], + "score": 1.0, + "content": "in the true causal graph. The parameter", + "type": "text" + }, + { + "bbox": [ + 448, + 552, + 461, + 562 + ], + "score": 0.87, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 547, + 506, + 564 + ], + "score": 1.0, + "content": "converges", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 560, + 293, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 117, + 573 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 560, + 162, + 573 + ], + "score": 0.93, + "content": "\\sigma ( \\gamma _ { i j } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 560, + 293, + 573 + ], + "score": 1.0, + "content": "if the following condition holds:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 547, + 506, + 573 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 577, + 457, + 599 + ], + "lines": [ + { + "bbox": [ + 154, + 577, + 457, + 599 + ], + "spans": [ + { + "bbox": [ + 154, + 577, + 457, + 599 + ], + "score": 0.9, + "content": "\\operatorname* { m i n } _ { \\substack { \\hat { p } a \\subseteq g p a _ { i } ( X _ { j } ) } } \\mathbb { E } _ { \\hat { I } \\sim p _ { I _ { - j } } ( I ) } \\mathbb { E } _ { \\tilde { p } _ { \\hat { I } } ( { \\pmb X } ) } \\big [ \\log p ( X _ { j } | \\hat { p a } , X _ { i } ) - \\log p ( X _ { j } | \\hat { p a } ) \\big ] > \\lambda _ { s p a r s e }", + "type": "interline_equation", + "image_path": "b46900ea849423d1308a1dde70d9c78dcc2f8c989dbe9a59e0602b00214f125f.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 154, + 577, + 457, + 599 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 135, + 618 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 605, + 174, + 617 + ], + "score": 0.91, + "content": "g p a _ { i } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 603, + 302, + 618 + ], + "score": 1.0, + "content": "is the set of nodes excluding", + "type": "text" + }, + { + "bbox": [ + 302, + 605, + 315, + 616 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 603, + 506, + 618 + ], + "score": 1.0, + "content": "which, according to the ground truth graph,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 199, + 629 + ], + "score": 1.0, + "content": "could have an edge to", + "type": "text" + }, + { + "bbox": [ + 199, + 616, + 213, + 628 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 615, + 349, + 629 + ], + "score": 1.0, + "content": "without introducing a cycle, and", + "type": "text" + }, + { + "bbox": [ + 349, + 616, + 381, + 628 + ], + "score": 0.92, + "content": "p _ { I _ { - j } } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "refers to the distribution over", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 626, + 358, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 161, + 640 + ], + "score": 1.0, + "content": "interventions", + "type": "text" + }, + { + "bbox": [ + 161, + 627, + 185, + 639 + ], + "score": 0.92, + "content": "p _ { I } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 626, + 340, + 640 + ], + "score": 1.0, + "content": "excluding the intervention on variable", + "type": "text" + }, + { + "bbox": [ + 340, + 628, + 353, + 639 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 626, + 358, + 640 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 603, + 506, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 650, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "Proof. To show this convergence, we assume that the orientation parameters have converged cor-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 661, + 507, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 325, + 676 + ], + "score": 1.0, + "content": "responding to Theorem B.1 and B.2. The parameter", + "type": "text" + }, + { + "bbox": [ + 326, + 663, + 339, + 674 + ], + "score": 0.88, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 661, + 396, + 676 + ], + "score": 1.0, + "content": "converges to", + "type": "text" + }, + { + "bbox": [ + 396, + 661, + 444, + 674 + ], + "score": 0.93, + "content": "\\sigma ( \\gamma _ { i j } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 661, + 507, + 676 + ], + "score": 1.0, + "content": "if its gradient,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 107, + 668, + 508, + 693 + ], + "spans": [ + { + "bbox": [ + 107, + 673, + 132, + 689 + ], + "score": 0.92, + "content": "\\begin{array} { r l r } { { \\frac { \\partial } { \\partial \\gamma _ { i j } } \\tilde { \\mathcal { L } } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 668, + 299, + 693 + ], + "score": 1.0, + "content": ", is negative independent of other values of", + "type": "text" + }, + { + "bbox": [ + 300, + 677, + 308, + 686 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 668, + 415, + 693 + ], + "score": 1.0, + "content": "and orientation parameters", + "type": "text" + }, + { + "bbox": [ + 415, + 675, + 422, + 685 + ], + "score": 0.8, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 668, + 508, + 693 + ], + "score": 1.0, + "content": "that are not included", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 286, + 701 + ], + "score": 1.0, + "content": "in Theorem B.1 and B.2. The gradient of", + "type": "text" + }, + { + "bbox": [ + 286, + 689, + 299, + 700 + ], + "score": 0.88, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "includes an expectation over adjacency matrices", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 141, + 711 + ], + "score": 0.91, + "content": "p _ { \\gamma , \\theta } ( C )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 698, + 246, + 712 + ], + "score": 1.0, + "content": ". Based on the converged", + "type": "text" + }, + { + "bbox": [ + 247, + 700, + 253, + 708 + ], + "score": 0.82, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "-values, we only need to consider sets of nodes as parents for", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 107, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 107, + 711, + 120, + 722 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "that contain parents, ancestors, or (conditionally) independent nodes according to the ground", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 719, + 507, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 312, + 735 + ], + "score": 1.0, + "content": "truth graph. This sets of parents is represented by", + "type": "text" + }, + { + "bbox": [ + 312, + 721, + 351, + 733 + ], + "score": 0.92, + "content": "\\mathrm { g p a } _ { i } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 719, + 507, + 735 + ], + "score": 1.0, + "content": ". Among those remaining parent sets,", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "we need to ensure that for any such set, the gradient is negative. The condition in Equation 28", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 507, + 111 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 230, + 109 + ], + "score": 1.0, + "content": "corresponds to the inequality", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 230, + 93, + 276, + 109 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { \\partial } { \\partial \\gamma _ { i j } } \\tilde { \\mathcal { L } } < 0 } \\end{array}", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 277, + 90, + 507, + 111 + ], + "score": 1.0, + "content": "since the term on the left represents the log-likelihood", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 106, + 507, + 124 + ], + "spans": [ + { + "bbox": [ + 104, + 106, + 150, + 124 + ], + "score": 1.0, + "content": "difference", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 150, + 109, + 274, + 122 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { X _ { i } X _ { j } } ( X _ { j } ) - \\mathcal { L } _ { X _ { i } \\not = \\dot { X } _ { j } } ( X _ { j } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 274, + 106, + 350, + 124 + ], + "score": 1.0, + "content": "in the gradients of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 351, + 110, + 364, + 121 + ], + "score": 0.88, + "content": "\\gamma _ { i j }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 364, + 106, + 507, + 124 + ], + "score": 1.0, + "content": "in Equation 20 with a flipped sign.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 118, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 104, + 118, + 264, + 134 + ], + "score": 1.0, + "content": "For readability and better interpretation,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 264, + 120, + 289, + 132 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 289, + 118, + 506, + 134 + ], + "score": 1.0, + "content": "has been moved on the right site of the inequality. This", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 506, + 143 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 160, + 143 + ], + "score": 1.0, + "content": "is possible as", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 161, + 131, + 186, + 142 + ], + "score": 0.89, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 186, + 131, + 506, + 143 + ], + "score": 1.0, + "content": "is independent of the two expectations in Equation 28. If the inequality holds for", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 141, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 162, + 155 + ], + "score": 1.0, + "content": "all parent sets", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 162, + 142, + 174, + 153 + ], + "score": 0.34, + "content": "\\hat { \\mathsf { p a } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 174, + 141, + 240, + 155 + ], + "score": 1.0, + "content": "), the gradient of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 240, + 143, + 253, + 154 + ], + "score": 0.87, + "content": "\\gamma _ { i j }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 254, + 141, + 506, + 155 + ], + "score": 1.0, + "content": "can be guaranteed to be negative in expectation, independent of", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 184, + 165 + ], + "score": 1.0, + "content": "the other values of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 184, + 154, + 192, + 164 + ], + "score": 0.8, + "content": "\\gamma", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 192, + 153, + 478, + 165 + ], + "score": 1.0, + "content": ". Since the distribution over parent sets paˆ ) depends on other values of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 478, + 154, + 486, + 164 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 487, + 153, + 505, + 165 + ], + "score": 1.0, + "content": ", the", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 176 + ], + "score": 1.0, + "content": "condition in Equation 28 ensures that even for the parent set with the lowest log-likelihood difference,", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 173, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 104, + 175, + 188, + 188 + ], + "score": 1.0, + "content": "it is still larger than", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 188, + 176, + 213, + 188 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 213, + 175, + 395, + 188 + ], + "score": 1.0, + "content": ". 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The condition in Equation 28", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 507, + 111 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 230, + 109 + ], + "score": 1.0, + "content": "corresponds to the inequality", + "type": "text" + }, + { + "bbox": [ + 230, + 93, + 276, + 109 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { \\partial } { \\partial \\gamma _ { i j } } \\tilde { \\mathcal { L } } < 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 90, + 507, + 111 + ], + "score": 1.0, + "content": "since the term on the left represents the log-likelihood", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 106, + 507, + 124 + ], + "spans": [ + { + "bbox": [ + 104, + 106, + 150, + 124 + ], + "score": 1.0, + "content": "difference", + "type": "text" + }, + { + "bbox": [ + 150, + 109, + 274, + 122 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { X _ { i } X _ { j } } ( X _ { j } ) - \\mathcal { L } _ { X _ { i } \\not = \\dot { X } _ { j } } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 106, + 350, + 124 + ], + "score": 1.0, + "content": "in the gradients of", + "type": "text" + }, + { + "bbox": [ + 351, + 110, + 364, + 121 + ], + "score": 0.88, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 106, + 507, + 124 + ], + "score": 1.0, + "content": "in Equation 20 with a flipped sign.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 118, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 104, + 118, + 264, + 134 + ], + "score": 1.0, + "content": "For readability and better interpretation,", + "type": "text" + }, + { + "bbox": [ + 264, + 120, + 289, + 132 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 118, + 506, + 134 + ], + "score": 1.0, + "content": "has been moved on the right site of the inequality. This", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 506, + 143 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 160, + 143 + ], + "score": 1.0, + "content": "is possible as", + "type": "text" + }, + { + "bbox": [ + 161, + 131, + 186, + 142 + ], + "score": 0.89, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 131, + 506, + 143 + ], + "score": 1.0, + "content": "is independent of the two expectations in Equation 28. If the inequality holds for", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 141, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 162, + 155 + ], + "score": 1.0, + "content": "all parent sets", + "type": "text" + }, + { + "bbox": [ + 162, + 142, + 174, + 153 + ], + "score": 0.34, + "content": "\\hat { \\mathsf { p a } }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 141, + 240, + 155 + ], + "score": 1.0, + "content": "), the gradient of", + "type": "text" + }, + { + "bbox": [ + 240, + 143, + 253, + 154 + ], + "score": 0.87, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 141, + 506, + 155 + ], + "score": 1.0, + "content": "can be guaranteed to be negative in expectation, independent of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 184, + 165 + ], + "score": 1.0, + "content": "the other values of", + "type": "text" + }, + { + "bbox": [ + 184, + 154, + 192, + 164 + ], + "score": 0.8, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 153, + 478, + 165 + ], + "score": 1.0, + "content": ". Since the distribution over parent sets paˆ ) depends on other values of", + "type": "text" + }, + { + "bbox": [ + 478, + 154, + 486, + 164 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 153, + 505, + 165 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 176 + ], + "score": 1.0, + "content": "condition in Equation 28 ensures that even for the parent set with the lowest log-likelihood difference,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 173, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 104, + 175, + 188, + 188 + ], + "score": 1.0, + "content": "it is still larger than", + "type": "text" + }, + { + "bbox": [ + 188, + 176, + 213, + 188 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 175, + 395, + 188 + ], + "score": 1.0, + "content": ". If this condition holds, then the gradient of", + "type": "text" + }, + { + "bbox": [ + 396, + 174, + 421, + 191 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { \\partial } { \\partial \\gamma _ { i j } } \\bar { \\mathcal { L } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 173, + 506, + 189 + ], + "score": 1.0, + "content": "will be smaller than", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 189, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 249, + 202 + ], + "score": 1.0, + "content": "zero independent of other values of", + "type": "text" + }, + { + "bbox": [ + 250, + 191, + 257, + 201 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 189, + 262, + 202 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 189, + 506, + 200 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 212, + 505, + 290 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 226 + ], + "score": 1.0, + "content": "The condition in Equation 28 introduces a dependency between convergence guarantees and the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 197, + 237 + ], + "score": 1.0, + "content": "regularizer parameter", + "type": "text" + }, + { + "bbox": [ + 197, + 224, + 222, + 236 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 223, + 410, + 237 + ], + "score": 1.0, + "content": ". The lower we set the regularization weight", + "type": "text" + }, + { + "bbox": [ + 410, + 224, + 435, + 236 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 223, + 505, + 237 + ], + "score": 1.0, + "content": ", the more edges", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 505, + 247 + ], + "score": 1.0, + "content": "we can guarantee to recover. If the regularization weight is set too high, we can eventually obtain", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "false negative edge predictions. If the regularization weight is set very low, we take a longer time", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "score": 1.0, + "content": "to converge as it requires lower gradient variance or more update steps, and is more sensitive in a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "limited data regime. Nonetheless, if sufficient computational resources and data is provided, any", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 278, + 237, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 141, + 292 + ], + "score": 1.0, + "content": "value of", + "type": "text" + }, + { + "bbox": [ + 141, + 279, + 184, + 291 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 278, + 237, + 292 + ], + "score": 1.0, + "content": "can be used.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 292, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 258, + 306 + ], + "score": 1.0, + "content": "Theorem B.4. Assume for all edges", + "type": "text" + }, + { + "bbox": [ + 258, + 293, + 300, + 305 + ], + "score": 0.92, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 290, + 407, + 306 + ], + "score": 1.0, + "content": "in the true causal graph,", + "type": "text" + }, + { + "bbox": [ + 408, + 293, + 434, + 305 + ], + "score": 0.92, + "content": "\\sigma ( \\theta _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 290, + 454, + 306 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 455, + 293, + 481, + 305 + ], + "score": 0.92, + "content": "\\sigma ( \\gamma _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 290, + 505, + 306 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 354, + 317 + ], + "score": 1.0, + "content": "converged to one. Then, the likelihood of all other edges, i.e.", + "type": "text" + }, + { + "bbox": [ + 354, + 304, + 414, + 316 + ], + "score": 0.92, + "content": "\\sigma ( \\theta _ { l k } ) \\cdot \\sigma ( \\theta _ { l k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 302, + 505, + 317 + ], + "score": 1.0, + "content": ", will converge to zero", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 314, + 253, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 205, + 328 + ], + "score": 1.0, + "content": "under the condition that", + "type": "text" + }, + { + "bbox": [ + 205, + 315, + 249, + 327 + ], + "score": 0.91, + "content": "\\lambda _ { s p a r s e } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 314, + 253, + 328 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 337, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 504, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 473, + 351 + ], + "score": 1.0, + "content": "Proof. If all edges in the ground truth graph have converged, all other pairs of variables", + "type": "text" + }, + { + "bbox": [ + 474, + 338, + 504, + 350 + ], + "score": 0.91, + "content": "X _ { l } , X _ { k }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "score": 1.0, + "content": "are (conditionally) independent in the graph. This statement follows from the Markov property of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 313, + 373 + ], + "score": 1.0, + "content": "the graph and excludes ancestor-descendant pairs", + "type": "text" + }, + { + "bbox": [ + 313, + 360, + 343, + 372 + ], + "score": 0.92, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 360, + 505, + 373 + ], + "score": 1.0, + "content": ". The possibility of having edges from", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 468, + 384 + ], + "score": 1.0, + "content": "descendants to ancestors has been removed by the fact that the orientation parameters", + "type": "text" + }, + { + "bbox": [ + 468, + 371, + 481, + 383 + ], + "score": 0.89, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 382, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 505, + 395 + ], + "score": 1.0, + "content": "converged according to Theorem B.2. Thus, for those cases, we already have the guarantee that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 392, + 242, + 407 + ], + "spans": [ + { + "bbox": [ + 107, + 393, + 166, + 406 + ], + "score": 0.93, + "content": "\\sigma ( \\theta _ { i j } ) \\cdot \\sigma ( \\theta _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 392, + 242, + 407 + ], + "score": 1.0, + "content": "converges to zero.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 409, + 504, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 409, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 259, + 423 + ], + "score": 1.0, + "content": "For a conditionally independent pair", + "type": "text" + }, + { + "bbox": [ + 260, + 410, + 289, + 421 + ], + "score": 0.89, + "content": "X _ { l } , X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 409, + 505, + 423 + ], + "score": 1.0, + "content": ", the difference of the log-likelihood estimate in the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 153, + 434 + ], + "score": 1.0, + "content": "gradient of", + "type": "text" + }, + { + "bbox": [ + 154, + 422, + 167, + 433 + ], + "score": 0.84, + "content": "\\gamma _ { l k }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 421, + 171, + 434 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 172, + 421, + 310, + 433 + ], + "score": 0.89, + "content": "i . e . \\mathcal { L } _ { X _ { l } X _ { k } } ( X _ { k } ) - \\mathcal { L } _ { X _ { l } \\not = \\mathcal { N } _ { k } } ( X _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 421, + 505, + 434 + ], + "score": 1.0, + "content": ", is zero in expectation since independent nodes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 432, + 357, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 357, + 445 + ], + "score": 1.0, + "content": "do not share any information. Thus, the gradient remaining is:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 447, + 373, + 473 + ], + "lines": [ + { + "bbox": [ + 237, + 447, + 373, + 473 + ], + "spans": [ + { + "bbox": [ + 237, + 447, + 373, + 473 + ], + "score": 0.95, + "content": "\\frac { \\partial } { \\partial \\gamma _ { l k } } \\tilde { \\mathcal { L } } = \\sigma ^ { \\prime } ( \\gamma _ { l k } ) \\cdot \\sigma ( \\theta _ { l k } ) \\cdot \\lambda _ { \\mathrm { s p a r s e } }", + "type": "interline_equation", + "image_path": "eed29822c5635f81159c76965bb2dc6d58f73e26d20e2b27b1c29a4fe9b291fd.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 237, + 447, + 373, + 473 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 506, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 350, + 488 + ], + "score": 1.0, + "content": "Since the gradient is positive independent of the values of", + "type": "text" + }, + { + "bbox": [ + 350, + 477, + 364, + 487 + ], + "score": 0.88, + "content": "\\gamma _ { l k }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 474, + 384, + 488 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 384, + 476, + 416, + 487 + ], + "score": 0.54, + "content": "\\theta _ { l k } , \\gamma _ { l k }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "will decrease until it", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 486, + 209, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 159, + 498 + ], + "score": 1.0, + "content": "converges to", + "type": "text" + }, + { + "bbox": [ + 159, + 486, + 205, + 498 + ], + "score": 0.93, + "content": "\\sigma ( \\gamma _ { l k } ) = \\bar { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 486, + 209, + 498 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 146, + 516 + ], + "score": 1.0, + "content": "Hence, if", + "type": "text" + }, + { + "bbox": [ + 146, + 505, + 160, + 515 + ], + "score": 0.86, + "content": "\\gamma _ { l k }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 502, + 417, + 516 + ], + "score": 1.0, + "content": "decreases for all pairs of (conditionally) independent variables", + "type": "text" + }, + { + "bbox": [ + 417, + 504, + 447, + 515 + ], + "score": 0.91, + "content": "X _ { l } , X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 502, + 505, + 516 + ], + "score": 1.0, + "content": "in the ground", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 514, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 173, + 528 + ], + "score": 1.0, + "content": "truth graph, and", + "type": "text" + }, + { + "bbox": [ + 173, + 514, + 200, + 527 + ], + "score": 0.92, + "content": "\\sigma ( \\theta _ { l k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 514, + 444, + 528 + ], + "score": 1.0, + "content": "converged to zero for children and descendants, the product", + "type": "text" + }, + { + "bbox": [ + 444, + 514, + 505, + 527 + ], + "score": 0.91, + "content": "\\sigma ( \\gamma _ { l k } ) \\cdot \\sigma ( \\theta _ { l k } )", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 525, + 504, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 397, + 538 + ], + "score": 1.0, + "content": "will converge to zero for all edges not existing in the ground truth graph.", + "type": "text" + }, + { + "bbox": [ + 495, + 527, + 504, + 535 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 549, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 548, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 563 + ], + "score": 1.0, + "content": "For graphs that fulfill all conditions in the Theorems B.1 and B.4, ENCO is guaranteed to converge", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "given sufficient data and time. 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The lower we set the regularization weight", + "type": "text" + }, + { + "bbox": [ + 410, + 224, + 435, + 236 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 223, + 505, + 237 + ], + "score": 1.0, + "content": ", the more edges", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 505, + 247 + ], + "score": 1.0, + "content": "we can guarantee to recover. If the regularization weight is set too high, we can eventually obtain", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "false negative edge predictions. If the regularization weight is set very low, we take a longer time", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "score": 1.0, + "content": "to converge as it requires lower gradient variance or more update steps, and is more sensitive in a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "limited data regime. Nonetheless, if sufficient computational resources and data is provided, any", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 278, + 237, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 141, + 292 + ], + "score": 1.0, + "content": "value of", + "type": "text" + }, + { + "bbox": [ + 141, + 279, + 184, + 291 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 278, + 237, + 292 + ], + "score": 1.0, + "content": "can be used.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 212, + 506, + 292 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 292, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 258, + 306 + ], + "score": 1.0, + "content": "Theorem B.4. Assume for all edges", + "type": "text" + }, + { + "bbox": [ + 258, + 293, + 300, + 305 + ], + "score": 0.92, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 290, + 407, + 306 + ], + "score": 1.0, + "content": "in the true causal graph,", + "type": "text" + }, + { + "bbox": [ + 408, + 293, + 434, + 305 + ], + "score": 0.92, + "content": "\\sigma ( \\theta _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 290, + 454, + 306 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 455, + 293, + 481, + 305 + ], + "score": 0.92, + "content": "\\sigma ( \\gamma _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 290, + 505, + 306 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 354, + 317 + ], + "score": 1.0, + "content": "converged to one. Then, the likelihood of all other edges, i.e.", + "type": "text" + }, + { + "bbox": [ + 354, + 304, + 414, + 316 + ], + "score": 0.92, + "content": "\\sigma ( \\theta _ { l k } ) \\cdot \\sigma ( \\theta _ { l k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 302, + 505, + 317 + ], + "score": 1.0, + "content": ", will converge to zero", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 314, + 253, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 205, + 328 + ], + "score": 1.0, + "content": "under the condition that", + "type": "text" + }, + { + "bbox": [ + 205, + 315, + 249, + 327 + ], + "score": 0.91, + "content": "\\lambda _ { s p a r s e } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 314, + 253, + 328 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 290, + 505, + 328 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 337, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 504, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 473, + 351 + ], + "score": 1.0, + "content": "Proof. If all edges in the ground truth graph have converged, all other pairs of variables", + "type": "text" + }, + { + "bbox": [ + 474, + 338, + 504, + 350 + ], + "score": 0.91, + "content": "X _ { l } , X _ { k }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "score": 1.0, + "content": "are (conditionally) independent in the graph. This statement follows from the Markov property of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 313, + 373 + ], + "score": 1.0, + "content": "the graph and excludes ancestor-descendant pairs", + "type": "text" + }, + { + "bbox": [ + 313, + 360, + 343, + 372 + ], + "score": 0.92, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 360, + 505, + 373 + ], + "score": 1.0, + "content": ". The possibility of having edges from", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 468, + 384 + ], + "score": 1.0, + "content": "descendants to ancestors has been removed by the fact that the orientation parameters", + "type": "text" + }, + { + "bbox": [ + 468, + 371, + 481, + 383 + ], + "score": 0.89, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 382, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 505, + 395 + ], + "score": 1.0, + "content": "converged according to Theorem B.2. Thus, for those cases, we already have the guarantee that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 392, + 242, + 407 + ], + "spans": [ + { + "bbox": [ + 107, + 393, + 166, + 406 + ], + "score": 0.93, + "content": "\\sigma ( \\theta _ { i j } ) \\cdot \\sigma ( \\theta _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 392, + 242, + 407 + ], + "score": 1.0, + "content": "converges to zero.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 338, + 505, + 407 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 409, + 504, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 409, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 259, + 423 + ], + "score": 1.0, + "content": "For a conditionally independent pair", + "type": "text" + }, + { + "bbox": [ + 260, + 410, + 289, + 421 + ], + "score": 0.89, + "content": "X _ { l } , X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 409, + 505, + 423 + ], + "score": 1.0, + "content": ", the difference of the log-likelihood estimate in the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 153, + 434 + ], + "score": 1.0, + "content": "gradient of", + "type": "text" + }, + { + "bbox": [ + 154, + 422, + 167, + 433 + ], + "score": 0.84, + "content": "\\gamma _ { l k }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 421, + 171, + 434 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 172, + 421, + 310, + 433 + ], + "score": 0.89, + "content": "i . e . \\mathcal { L } _ { X _ { l } X _ { k } } ( X _ { k } ) - \\mathcal { L } _ { X _ { l } \\not = \\mathcal { N } _ { k } } ( X _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 421, + 505, + 434 + ], + "score": 1.0, + "content": ", is zero in expectation since independent nodes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 432, + 357, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 357, + 445 + ], + "score": 1.0, + "content": "do not share any information. Thus, the gradient remaining is:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 409, + 505, + 445 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 447, + 373, + 473 + ], + "lines": [ + { + "bbox": [ + 237, + 447, + 373, + 473 + ], + "spans": [ + { + "bbox": [ + 237, + 447, + 373, + 473 + ], + "score": 0.95, + "content": "\\frac { \\partial } { \\partial \\gamma _ { l k } } \\tilde { \\mathcal { L } } = \\sigma ^ { \\prime } ( \\gamma _ { l k } ) \\cdot \\sigma ( \\theta _ { l k } ) \\cdot \\lambda _ { \\mathrm { s p a r s e } }", + "type": "interline_equation", + "image_path": "eed29822c5635f81159c76965bb2dc6d58f73e26d20e2b27b1c29a4fe9b291fd.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 237, + 447, + 373, + 473 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 506, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 350, + 488 + ], + "score": 1.0, + "content": "Since the gradient is positive independent of the values of", + "type": "text" + }, + { + "bbox": [ + 350, + 477, + 364, + 487 + ], + "score": 0.88, + "content": "\\gamma _ { l k }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 474, + 384, + 488 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 384, + 476, + 416, + 487 + ], + "score": 0.54, + "content": "\\theta _ { l k } , \\gamma _ { l k }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "will decrease until it", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 486, + 209, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 159, + 498 + ], + "score": 1.0, + "content": "converges to", + "type": "text" + }, + { + "bbox": [ + 159, + 486, + 205, + 498 + ], + "score": 0.93, + "content": "\\sigma ( \\gamma _ { l k } ) = \\bar { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 486, + 209, + 498 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 474, + 506, + 498 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 146, + 516 + ], + "score": 1.0, + "content": "Hence, if", + "type": "text" + }, + { + "bbox": [ + 146, + 505, + 160, + 515 + ], + "score": 0.86, + "content": "\\gamma _ { l k }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 502, + 417, + 516 + ], + "score": 1.0, + "content": "decreases for all pairs of (conditionally) independent variables", + "type": "text" + }, + { + "bbox": [ + 417, + 504, + 447, + 515 + ], + "score": 0.91, + "content": "X _ { l } , X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 502, + 505, + 516 + ], + "score": 1.0, + "content": "in the ground", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 514, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 173, + 528 + ], + "score": 1.0, + "content": "truth graph, and", + "type": "text" + }, + { + "bbox": [ + 173, + 514, + 200, + 527 + ], + "score": 0.92, + "content": "\\sigma ( \\theta _ { l k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 514, + 444, + 528 + ], + "score": 1.0, + "content": "converged to zero for children and descendants, the product", + "type": "text" + }, + { + "bbox": [ + 444, + 514, + 505, + 527 + ], + "score": 0.91, + "content": "\\sigma ( \\gamma _ { l k } ) \\cdot \\sigma ( \\theta _ { l k } )", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 525, + 504, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 397, + 538 + ], + "score": 1.0, + "content": "will converge to zero for all edges not existing in the ground truth graph.", + "type": "text" + }, + { + "bbox": [ + 495, + 527, + 504, + 535 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 106, + 502, + 505, + 538 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 549, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 548, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 563 + ], + "score": 1.0, + "content": "For graphs that fulfill all conditions in the Theorems B.1 and B.4, ENCO is guaranteed to converge", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "given sufficient data and time. The conditions in the theorems ensure that there exist no local minima", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 571, + 464, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 427, + 583 + ], + "score": 1.0, + "content": "or saddle points in the loss surface of the objective in Equation 2 with respect to", + "type": "text" + }, + { + "bbox": [ + 427, + 573, + 436, + 583 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 571, + 453, + 583 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 453, + 572, + 460, + 581 + ], + "score": 0.77, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 571, + 464, + 583 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 548, + 505, + 583 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 593, + 505, + 639 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 504, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 496, + 606 + ], + "score": 1.0, + "content": "Summary We can summarize the conditions discussed above as follows. 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For further details on the precise calculations, we provide a Jupyter Notebook", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 316, + 292, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 292, + 329 + ], + "score": 1.0, + "content": "that contains all calculations in this example1.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 294, + 506, + 329 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 333, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 332, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 309, + 347 + ], + "score": 1.0, + "content": "Suppose we have a graph with 3 binary variables,", + "type": "text" + }, + { + "bbox": [ + 311, + 334, + 359, + 345 + ], + "score": 0.26, + "content": "X _ { 1 } , X _ { 2 } , X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 332, + 477, + 347 + ], + "score": 1.0, + "content": ", with the causal graph being", + "type": "text" + }, + { + "bbox": [ + 477, + 334, + 505, + 345 + ], + "score": 0.88, + "content": "X _ { 1 } ", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 107, + 345, + 149, + 356 + ], + "score": 0.91, + "content": "X _ { 2 } X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 344, + 505, + 357 + ], + "score": 1.0, + "content": ", i.e., a small chain. For simplicity, let us assume that the true, conditional distributions are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 353, + 166, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 166, + 370 + ], + "score": 1.0, + "content": "the following:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 332, + 505, + 370 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 373, + 394, + 448 + ], + "lines": [ + { + "bbox": [ + 216, + 373, + 394, + 448 + ], + "spans": [ + { + "bbox": [ + 216, + 373, + 394, + 448 + ], + "score": 0.93, + "content": "\\begin{array} { r c l } { { p ( X _ { 1 } ) } } & { { = } } & { { { \\bf { B e r n } } ( 0 . 7 ) } } \\\\ { { } } & { { } } & { { } } \\\\ { { p ( X _ { 2 } | X _ { 1 } ) } } & { { = } } & { { \\left\\{ \\begin{array} { l l } { { X _ { 1 } } } & { { \\mathrm { w i t h ~ p r o b . ~ 0 . 6 } } } \\\\ { { X _ { 1 } \\oplus 1 } } & { { \\mathrm { w i t h ~ p r o b . ~ 0 . 4 } } } \\end{array} \\right. } } \\\\ { { } } & { { } } & { { } } \\\\ { { p ( X _ { 3 } | X _ { 2 } ) } } & { { = } } & { { \\left\\{ \\begin{array} { l l } { { X _ { 2 } } } & { { \\mathrm { w i t h ~ p r o b . ~ 0 . 2 } } } \\\\ { { X _ { 2 } \\oplus 1 } } & { { \\mathrm { w i t h ~ p r o b . ~ 0 . 8 } } } \\end{array} \\right. } } \\end{array}", + "type": "interline_equation", + "image_path": "77358055ccc780a69f12f294f93ec0c1a68c7857ad6a8f1270700cfa1dc8088c.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 216, + 373, + 394, + 388.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 216, + 388.0, + 394, + 403.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 216, + 403.0, + 394, + 418.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 216, + 418.0, + 394, + 433.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 216, + 433.0, + 394, + 448.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 454, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 454, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 171, + 469 + ], + "score": 1.0, + "content": "In other words,", + "type": "text" + }, + { + "bbox": [ + 172, + 456, + 185, + 466 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 454, + 286, + 469 + ], + "score": 1.0, + "content": "is equals to the value of", + "type": "text" + }, + { + "bbox": [ + 287, + 456, + 300, + 466 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 454, + 505, + 469 + ], + "score": 1.0, + "content": "with a probability of 0.6, and the opposite binary", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 213, + 479 + ], + "score": 1.0, + "content": "value otherwise. 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Therefore, the sample with the highest probability in this joint", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 487, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 196, + 501 + ], + "score": 1.0, + "content": "distribution would be", + "type": "text" + }, + { + "bbox": [ + 196, + 488, + 300, + 499 + ], + "score": 0.9, + "content": "X _ { 1 } = 1 , X _ { 2 } = 1 , X _ { 3 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 487, + 505, + 501 + ], + "score": 1.0, + "content": ". Further, we assume that all interventions replace", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 499, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 385, + 511 + ], + "score": 1.0, + "content": "the respective conditional distribution by a uniform distribution, i.e.,", + "type": "text" + }, + { + "bbox": [ + 386, + 499, + 477, + 511 + ], + "score": 0.85, + "content": "p _ { I _ { X _ { i } } } ( X _ { i } ) = \\mathbf { B e r n } ( 0 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 499, + 506, + 511 + ], + "score": 1.0, + "content": ". 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Thus, we need to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 364, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 364, + 430, + 378 + ], + "score": 1.0, + "content": "compare two parent sets for finding the minimum of the left-side term:", + "type": "text" + }, + { + "bbox": [ + 430, + 365, + 444, + 376 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 364, + 505, + 378 + ], + "score": 1.0, + "content": "and the empty", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 375, + 290, + 388 + ], + "spans": [ + { + "bbox": [ + 142, + 375, + 290, + 388 + ], + "score": 1.0, + "content": "set. First, we consider the empty set:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 391, + 518, + 423 + ], + "lines": [ + { + "bbox": [ + 142, + 391, + 518, + 423 + ], + "spans": [ + { + "bbox": [ + 142, + 391, + 518, + 423 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\hat { I } \\sim p _ { I - 3 } ( I ) } \\mathbb { E } _ { \\tilde { p } _ { \\hat { I } } ( \\mathbf { X } ) } \\left[ \\log p ( X _ { 3 } | X _ { 2 } ) - \\log p ( X _ { 3 } ) \\right] \\approx \\underbrace { 1 / 2 \\cdot 0 . 1 9 4 } _ { I _ { X _ { 1 } } } + \\underbrace { 1 / 2 \\cdot 0 . 1 9 4 } _ { I _ { X _ { 2 } } } = 0 . 1 9 4 > \\lambda _ { \\mathrm { s p a r e } } } \\end{array}", + "type": "interline_equation", + "image_path": "5134128b04532f3d49e47d547fe53cedfe4c2b9119fb699202c25f6b19ad091a.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 142, + 391, + 518, + 401.6666666666667 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 142, + 401.6666666666667, + 518, + 412.33333333333337 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 142, + 412.33333333333337, + 518, + 423.00000000000006 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 143, + 427, + 504, + 450 + ], + "lines": [ + { + "bbox": [ + 141, + 426, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 141, + 426, + 218, + 440 + ], + "score": 1.0, + "content": "Again, we exclude", + "type": "text" + }, + { + "bbox": [ + 219, + 428, + 235, + 439 + ], + "score": 0.91, + "content": "I _ { X _ { 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 426, + 330, + 440 + ], + "score": 1.0, + "content": "since we do not update", + "type": "text" + }, + { + "bbox": [ + 330, + 429, + 345, + 439 + ], + "score": 0.87, + "content": "\\gamma _ { 2 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 426, + 506, + 440 + ], + "score": 1.0, + "content": "in this case. The second case considers", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 437, + 266, + 451 + ], + "spans": [ + { + "bbox": [ + 142, + 439, + 156, + 450 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 437, + 266, + 451 + ], + "score": 1.0, + "content": "as additional parent set pa: ˆ", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 453, + 496, + 501 + ], + "lines": [ + { + "bbox": [ + 153, + 453, + 496, + 501 + ], + "spans": [ + { + "bbox": [ + 153, + 453, + 496, + 501 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { \\hat { I } \\sim p _ { I _ { - 3 } } ( I ) } \\mathbb { E } _ { \\hat { p } _ { \\hat { I } } ( \\mathbf { X } ) } \\big [ \\log p ( X _ { 3 } | X _ { 2 } , X _ { 1 } ) - \\log p ( X _ { 3 } | X _ { 1 } ) \\big ] \\approx \\underbrace { 1 / 2 \\cdot 0 . 1 8 6 } _ { I _ { X _ { 1 } } } + \\underbrace { 1 / 2 \\cdot 0 . 2 0 0 } _ { I _ { X _ { 2 } } } } \\\\ & { \\qquad = 0 . 1 9 3 > \\lambda _ { \\mathrm { s p a r s e } } } \\end{array}", + "type": "interline_equation", + "image_path": "0af0ec8cd54bfab172398d4de0a1390a688b7ba932f16d65079d5ec37509cf45.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 153, + 453, + 496, + 469.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 153, + 469.0, + 496, + 485.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 153, + 485.0, + 496, + 501.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 505, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 141, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 141, + 505, + 376, + 519 + ], + "score": 1.0, + "content": "The minimum of both values is 0.193. 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For", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 568, + 232, + 583 + ], + "spans": [ + { + "bbox": [ + 141, + 568, + 232, + 583 + ], + "score": 1.0, + "content": "the empty set, we get:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 586, + 425, + 601 + ], + "lines": [ + { + "bbox": [ + 222, + 586, + 425, + 601 + ], + "spans": [ + { + "bbox": [ + 222, + 586, + 425, + 601 + ], + "score": 0.87, + "content": "\\mathbb { E } _ { I _ { X _ { 1 } } , \\mathbf { X } } \\left[ \\log p ( X _ { 3 } | X _ { 1 } ) - \\log p ( X _ { 3 } ) \\right] \\approx 0 . 0 0 8 \\ge 0", + "type": "interline_equation", + "image_path": "1376d7ec89547d51acbca258da7de0145b5a331eb38f67a819062bba1975e8d9.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 222, + 586, + 425, + 601 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 604, + 285, + 616 + ], + "lines": [ + { + "bbox": [ + 141, + 603, + 286, + 618 + ], + "spans": [ + { + "bbox": [ + 141, + 603, + 224, + 618 + ], + "score": 1.0, + "content": "For conditioning on", + "type": "text" + }, + { + "bbox": [ + 224, + 605, + 237, + 615 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 603, + 286, + 618 + ], + "score": 1.0, + "content": ", we obtain:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 621, + 432, + 636 + ], + "lines": [ + { + "bbox": [ + 215, + 621, + 432, + 636 + ], + "spans": [ + { + "bbox": [ + 215, + 621, + 432, + 636 + ], + "score": 0.88, + "content": "\\mathbb { E } _ { I _ { X _ { 1 } } , { \\mathbf { X } } } \\left[ \\log p ( X _ { 3 } | X _ { 1 } , X _ { 2 } ) - \\log p ( X _ { 3 } | X _ { 2 } ) \\right] = 0 \\ge 0", + "type": "interline_equation", + "image_path": "6fb72492edc5d2247382d001a279f4d24cbbeaedf2b8664fdde1b97bc88d2e60.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 215, + 621, + 432, + 636 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 140, + 639, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 141, + 638, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 141, + 638, + 279, + 653 + ], + "score": 1.0, + "content": "The difference is zero because", + "type": "text" + }, + { + "bbox": [ + 279, + 640, + 293, + 651 + ], + "score": 0.91, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 638, + 376, + 653 + ], + "score": 1.0, + "content": "is independent of", + "type": "text" + }, + { + "bbox": [ + 376, + 640, + 390, + 651 + ], + "score": 0.9, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 638, + 488, + 653 + ], + "score": 1.0, + "content": "when conditioned on", + "type": "text" + }, + { + "bbox": [ + 488, + 640, + 502, + 651 + ], + "score": 0.88, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 638, + 506, + 653 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 143, + 650, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 143, + 651, + 260, + 663 + ], + "score": 0.91, + "content": "p ( X _ { 3 } | X _ { 1 } , X _ { 2 } ) \\ : = \\ : p ( X _ { 3 } | X _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 650, + 505, + 664 + ], + "score": 1.0, + "content": "Since both values are greater or equals to zero, condition", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 660, + 274, + 675 + ], + "spans": [ + { + "bbox": [ + 141, + 660, + 239, + 675 + ], + "score": 1.0, + "content": "1 is fulfilled for the pair", + "type": "text" + }, + { + "bbox": [ + 239, + 663, + 270, + 673 + ], + "score": 0.91, + "content": "X _ { 1 } , X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 660, + 274, + 675 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 135, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 133, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 133, + 676, + 404, + 690 + ], + "score": 1.0, + "content": "• Condition 2: from condition 1, we can see that the parent set of", + "type": "text" + }, + { + "bbox": [ + 405, + 677, + 435, + 689 + ], + "score": 0.92, + "content": "\\widehat { \\sf p a } ( X _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "being the empty", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 141, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "set is the only option that fulfills the condition being greater than zero. Since we start the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 141, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "optimization process with an initialization that assigns a non-zero probability to all possible", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 141, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 141, + 709, + 249, + 723 + ], + "score": 1.0, + "content": "parent sets, it follows that", + "type": "text" + }, + { + "bbox": [ + 250, + 710, + 280, + 722 + ], + "score": 0.92, + "content": "\\widehat { \\sf p a } ( X _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "being the empty set has a probability greater than zero", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 141, + 720, + 446, + 732 + ], + "spans": [ + { + "bbox": [ + 141, + 720, + 446, + 732 + ], + "score": 1.0, + "content": "throughout the optimization process. 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The second case considers", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 437, + 266, + 451 + ], + "spans": [ + { + "bbox": [ + 142, + 439, + 156, + 450 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 437, + 266, + 451 + ], + "score": 1.0, + "content": "as additional parent set pa: ˆ", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 141, + 426, + 506, + 451 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 453, + 496, + 501 + ], + "lines": [ + { + "bbox": [ + 153, + 453, + 496, + 501 + ], + "spans": [ + { + "bbox": [ + 153, + 453, + 496, + 501 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { \\hat { I } \\sim p _ { I _ { - 3 } } ( I ) } \\mathbb { E } _ { \\hat { p } _ { \\hat { I } } ( \\mathbf { X } ) } \\big [ \\log p ( X _ { 3 } | X _ { 2 } , X _ { 1 } ) - \\log p ( X _ { 3 } | X _ { 1 } ) \\big ] \\approx \\underbrace { 1 / 2 \\cdot 0 . 1 8 6 } _ { I _ { X _ { 1 } } } + \\underbrace { 1 / 2 \\cdot 0 . 2 0 0 } _ { I _ { X _ { 2 } } } } \\\\ & { \\qquad = 0 . 1 9 3 > \\lambda _ { \\mathrm { s p a r s e } } } \\end{array}", + "type": "interline_equation", + "image_path": "0af0ec8cd54bfab172398d4de0a1390a688b7ba932f16d65079d5ec37509cf45.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 153, + 453, + 496, + 469.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 153, + 469.0, + 496, + 485.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 153, + 485.0, + 496, + 501.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 505, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 141, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 141, + 505, + 376, + 519 + ], + "score": 1.0, + "content": "The minimum of both values is 0.193. 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For", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 568, + 232, + 583 + ], + "spans": [ + { + "bbox": [ + 141, + 568, + 232, + 583 + ], + "score": 1.0, + "content": "the empty set, we get:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 134, + 556, + 506, + 583 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 586, + 425, + 601 + ], + "lines": [ + { + "bbox": [ + 222, + 586, + 425, + 601 + ], + "spans": [ + { + "bbox": [ + 222, + 586, + 425, + 601 + ], + "score": 0.87, + "content": "\\mathbb { E } _ { I _ { X _ { 1 } } , \\mathbf { X } } \\left[ \\log p ( X _ { 3 } | X _ { 1 } ) - \\log p ( X _ { 3 } ) \\right] \\approx 0 . 0 0 8 \\ge 0", + "type": "interline_equation", + "image_path": "1376d7ec89547d51acbca258da7de0145b5a331eb38f67a819062bba1975e8d9.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 222, + 586, + 425, + 601 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 604, + 285, + 616 + ], + "lines": [ + { + "bbox": [ + 141, + 603, + 286, + 618 + ], + "spans": [ + { + "bbox": [ + 141, + 603, + 224, + 618 + ], + "score": 1.0, + "content": "For conditioning on", + "type": "text" + }, + { + "bbox": [ + 224, + 605, + 237, + 615 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 603, + 286, + 618 + ], + "score": 1.0, + "content": ", we obtain:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 141, + 603, + 286, + 618 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 621, + 432, + 636 + ], + "lines": [ + { + "bbox": [ + 215, + 621, + 432, + 636 + ], + "spans": [ + { + "bbox": [ + 215, + 621, + 432, + 636 + ], + "score": 0.88, + "content": "\\mathbb { E } _ { I _ { X _ { 1 } } , { \\mathbf { X } } } \\left[ \\log p ( X _ { 3 } | X _ { 1 } , X _ { 2 } ) - \\log p ( X _ { 3 } | X _ { 2 } ) \\right] = 0 \\ge 0", + "type": "interline_equation", + "image_path": "6fb72492edc5d2247382d001a279f4d24cbbeaedf2b8664fdde1b97bc88d2e60.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 215, + 621, + 432, + 636 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 140, + 639, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 141, + 638, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 141, + 638, + 279, + 653 + ], + "score": 1.0, + "content": "The difference is zero because", + "type": "text" + }, + { + "bbox": [ + 279, + 640, + 293, + 651 + ], + "score": 0.91, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 638, + 376, + 653 + ], + "score": 1.0, + "content": "is independent of", + "type": "text" + }, + { + "bbox": [ + 376, + 640, + 390, + 651 + ], + "score": 0.9, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 638, + 488, + 653 + ], + "score": 1.0, + "content": "when conditioned on", + "type": "text" + }, + { + "bbox": [ + 488, + 640, + 502, + 651 + ], + "score": 0.88, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 638, + 506, + 653 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 143, + 650, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 143, + 651, + 260, + 663 + ], + "score": 0.91, + "content": "p ( X _ { 3 } | X _ { 1 } , X _ { 2 } ) \\ : = \\ : p ( X _ { 3 } | X _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 650, + 505, + 664 + ], + "score": 1.0, + "content": "Since both values are greater or equals to zero, condition", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 660, + 274, + 675 + ], + "spans": [ + { + "bbox": [ + 141, + 660, + 239, + 675 + ], + "score": 1.0, + "content": "1 is fulfilled for the pair", + "type": "text" + }, + { + "bbox": [ + 239, + 663, + 270, + 673 + ], + "score": 0.91, + "content": "X _ { 1 } , X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 660, + 274, + 675 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 141, + 638, + 506, + 675 + ] + }, + { + "type": "text", + "bbox": [ + 135, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 133, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 133, + 676, + 404, + 690 + ], + "score": 1.0, + "content": "• Condition 2: from condition 1, we can see that the parent set of", + "type": "text" + }, + { + "bbox": [ + 405, + 677, + 435, + 689 + ], + "score": 0.92, + "content": "\\widehat { \\sf p a } ( X _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "being the empty", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 141, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "set is the only option that fulfills the condition being greater than zero. Since we start the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 141, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "optimization process with an initialization that assigns a non-zero probability to all possible", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 141, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 141, + 709, + 249, + 723 + ], + "score": 1.0, + "content": "parent sets, it follows that", + "type": "text" + }, + { + "bbox": [ + 250, + 710, + 280, + 722 + ], + "score": 0.92, + "content": "\\widehat { \\sf p a } ( X _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "being the empty set has a probability greater than zero", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 141, + 720, + 446, + 732 + ], + "spans": [ + { + "bbox": [ + 141, + 720, + 446, + 732 + ], + "score": 1.0, + "content": "throughout the optimization process. Hence, condition 2 is fulfilled as well.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43, + "bbox_fs": [ + 133, + 676, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "Summary: in conclusion, for the discussed example, we can guarantee that ENCO converges to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 215, + 107 + ], + "score": 1.0, + "content": "the correct causal graph if", + "type": "text" + }, + { + "bbox": [ + 215, + 94, + 271, + 106 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } < 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 93, + 506, + 107 + ], + "score": 1.0, + "content": ". To experimentally verify this results, we applied ENCO", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 422, + 118 + ], + "score": 1.0, + "content": "on this graph with two hyperparameter settings for the sparsity regularizer:", + "type": "text" + }, + { + "bbox": [ + 422, + 105, + 486, + 116 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 1 9", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 114, + 507, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 167, + 127 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 2 1", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 114, + 507, + 128 + ], + "score": 1.0, + "content": ". We considered a very large sample size, more specifically 10k per intervention and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 125, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 104, + 125, + 373, + 141 + ], + "score": 1.0, + "content": "100k observational samples, to simulate the data limit regime. For", + "type": "text" + }, + { + "bbox": [ + 373, + 127, + 434, + 137 + ], + "score": 0.88, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 1 9", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 125, + 506, + 141 + ], + "score": 1.0, + "content": ", ENCO was able", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 507, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 289, + 151 + ], + "score": 1.0, + "content": "to recover the graph without errors while for", + "type": "text" + }, + { + "bbox": [ + 289, + 137, + 349, + 149 + ], + "score": 0.93, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 2 1", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 137, + 390, + 151 + ], + "score": 1.0, + "content": ", the edge", + "type": "text" + }, + { + "bbox": [ + 390, + 138, + 432, + 148 + ], + "score": 0.91, + "content": "X _ { 1 } X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 137, + 507, + 151 + ], + "score": 1.0, + "content": "was, as expected,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 363, + 162 + ], + "score": 1.0, + "content": "missed. This verifies the theoretical result above with respect to", + "type": "text" + }, + { + "bbox": [ + 363, + 149, + 388, + 160 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 147, + 506, + 162 + ], + "score": 1.0, + "content": ". Note that if the condition is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "not fulfilled by selecting a too large sparsity regularizer, this does not necessarily mean that ENCO", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "will not be able to recover the graph. This is because we consider the ’worst-case’ parent set in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 507, + 195 + ], + "score": 1.0, + "content": "condition 3, while this case might not be in the true causal graph to which the other edges converge.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "title", + "bbox": [ + 107, + 207, + 313, + 219 + ], + "lines": [ + { + "bbox": [ + 106, + 207, + 314, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 314, + 220 + ], + "score": 1.0, + "content": "B.2.2 INTUITION BEHIND CONDITION 1 AND 2", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 228, + 505, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 229, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 239 + ], + "score": 1.0, + "content": "As mentioned in the text, condition 1 and 2 of Theorem 3.1 ensure that the orientation probabilities", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 251 + ], + "score": 1.0, + "content": "cannot converge to any local optima. Since the conditions explicitly involve the data distributions and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "implicitly the gradient estimators, we provide below an assumption from a data generation mechanism", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 404, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 404, + 272 + ], + "score": 1.0, + "content": "perspective as an alternative, that ensures condition 1 and 2 to be satisfied.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 278, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "Firstly, we assume that ancestors and descendants are not independent under interventions on the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 507, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 507, + 302 + ], + "score": 1.0, + "content": "ancestors. Note that there can exist graphs where the ancestors are independent of descendants,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 312 + ], + "score": 1.0, + "content": "for instance in a linear Gaussian setting when the ancestor has a weight of zero on the descendant.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 311, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 323 + ], + "score": 1.0, + "content": "However, those graphs, violating faithfulness, are impossible to find for any causal discovery method", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "since the variables are independent under any setting. In terms of condition 1 and 2, it would imply", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 333, + 242, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 242, + 345 + ], + "score": 1.0, + "content": "that the inequality is always zero.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 506, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "Next, we show that under the previous assumption, local optima of the orientation probabilities can", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 504, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 303, + 374 + ], + "score": 1.0, + "content": "only occur in the following structure: for an edge", + "type": "text" + }, + { + "bbox": [ + 304, + 361, + 344, + 373 + ], + "score": 0.91, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 360, + 490, + 374 + ], + "score": 1.0, + "content": ", there exist one or more parent(s) of", + "type": "text" + }, + { + "bbox": [ + 490, + 361, + 504, + 373 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 232, + 384 + ], + "score": 1.0, + "content": "sharing a common confounder", + "type": "text" + }, + { + "bbox": [ + 232, + 372, + 247, + 383 + ], + "score": 0.89, + "content": "X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 371, + 268, + 384 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 269, + 372, + 281, + 383 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 371, + 312, + 384 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 313, + 372, + 327, + 383 + ], + "score": 0.87, + "content": "X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 371, + 424, + 384 + ], + "score": 1.0, + "content": "is not a direct parent of", + "type": "text" + }, + { + "bbox": [ + 424, + 372, + 438, + 384 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 371, + 506, + 384 + ], + "score": 1.0, + "content": ". An example of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 232, + 396 + ], + "score": 1.0, + "content": "this structure is the following:", + "type": "text" + }, + { + "bbox": [ + 233, + 383, + 293, + 394 + ], + "score": 0.58, + "content": "X _ { 1 } X _ { 2 } , X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 383, + 298, + 396 + ], + "score": 1.0, + "content": ";", + "type": "text" + }, + { + "bbox": [ + 298, + 383, + 358, + 394 + ], + "score": 0.62, + "content": "X _ { 2 } , X _ { 3 } X _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "where the orientations of the edges", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 107, + 394, + 149, + 405 + ], + "score": 0.92, + "content": "X _ { 2 } X _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 393, + 167, + 406 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 167, + 394, + 209, + 405 + ], + "score": 0.93, + "content": "X _ { 3 } X _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "could have a local optimum. This statement can be proven as follows by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 404, + 504, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 455, + 418 + ], + "score": 1.0, + "content": "using the do-calculus (Pearl, 2009). Suppose a graph that includes the three variables", + "type": "text" + }, + { + "bbox": [ + 456, + 405, + 504, + 416 + ], + "score": 0.9, + "content": "X _ { 1 } , X _ { 2 } , X _ { 3 }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 127, + 429 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 416, + 171, + 426 + ], + "score": 0.87, + "content": "X _ { 1 } X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 415, + 175, + 429 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 175, + 416, + 219, + 426 + ], + "score": 0.88, + "content": "X _ { 3 } X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 415, + 240, + 429 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 241, + 416, + 255, + 426 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 415, + 365, + 429 + ], + "score": 1.0, + "content": "having no parents besides", + "type": "text" + }, + { + "bbox": [ + 365, + 416, + 379, + 426 + ], + "score": 0.87, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 415, + 398, + 429 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 398, + 416, + 412, + 426 + ], + "score": 0.87, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 415, + 427, + 429 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 428, + 416, + 441, + 426 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 415, + 461, + 429 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 461, + 416, + 475, + 426 + ], + "score": 0.88, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 415, + 506, + 429 + ], + "score": 1.0, + "content": "do not", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 343, + 439 + ], + "score": 1.0, + "content": "share a confounder, then, from do-calculus, we know that", + "type": "text" + }, + { + "bbox": [ + 343, + 427, + 505, + 438 + ], + "score": 0.89, + "content": "p ( X _ { 2 } | \\mathbf { d o } ( X _ { 1 } = x _ { 1 } ) ) = p ( X _ { 2 } | X _ { 1 } = x _ { 1 } )", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 124, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 437, + 297, + 450 + ], + "score": 0.9, + "content": "p ( X _ { 2 } | \\mathrm { d o } ( X _ { 3 } = x _ { 3 } ) ) = p ( X _ { 2 } | X _ { 3 } = x _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 437, + 506, + 450 + ], + "score": 1.0, + "content": ". Furthermore, since the conditional entropy of a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 340, + 461 + ], + "score": 1.0, + "content": "variable can only be smaller or equals to the marginal, i.e.", + "type": "text" + }, + { + "bbox": [ + 340, + 448, + 417, + 460 + ], + "score": 0.94, + "content": "H ( X ) \\geq H ( X | Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 448, + 465, + 461 + ], + "score": 1.0, + "content": ", estimating", + "type": "text" + }, + { + "bbox": [ + 465, + 449, + 479, + 459 + ], + "score": 0.88, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "under", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 458, + 507, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 176, + 473 + ], + "score": 1.0, + "content": "interventions on", + "type": "text" + }, + { + "bbox": [ + 176, + 460, + 190, + 470 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 458, + 368, + 473 + ], + "score": 1.0, + "content": "can only be improved by conditioning on", + "type": "text" + }, + { + "bbox": [ + 368, + 460, + 382, + 470 + ], + "score": 0.87, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 458, + 460, + 473 + ], + "score": 1.0, + "content": ", and similarly for", + "type": "text" + }, + { + "bbox": [ + 460, + 460, + 474, + 471 + ], + "score": 0.88, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 458, + 507, + 473 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "condition 1 is strictly fulfilled when parents do not share a confounder under the previous assumption", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 426, + 494 + ], + "score": 1.0, + "content": "of no independence in all possible settings. Now, consider the situation where", + "type": "text" + }, + { + "bbox": [ + 426, + 482, + 440, + 492 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 481, + 459, + 494 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 459, + 482, + 473, + 492 + ], + "score": 0.88, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "share a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "common confounder. Then, from do-calculus, we can state that there can exist a parameterization of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 501, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 264, + 517 + ], + "score": 1.0, + "content": "the conditional distributions for which", + "type": "text" + }, + { + "bbox": [ + 264, + 503, + 425, + 515 + ], + "score": 0.9, + "content": "p ( X _ { 2 } | \\mathbf { d o } ( X _ { 1 } = x _ { 1 } ) ) \\neq p ( X _ { 2 } | X _ { 1 } = x _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 501, + 506, + 517 + ], + "score": 1.0, + "content": ". Under this setting,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "we cannot guarantee that condition 1 is always fulfilled. However, whether this parent-confounder", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "structure above actually leads to a local optimum or not depends on the distributions, which condition", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "1 models. Intuitively, this requires the mutual information between the two or more parents to be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "score": 1.0, + "content": "very high, and the initial edge probabilities of those edges to be very low. Further, as the results show,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "this combination of events is not very common in practice, meaning that as long as the ancestor and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 568, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 582 + ], + "score": 1.0, + "content": "descendant are not independent under the interventions, we usually converge to a graph with the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 581, + 184, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 184, + 591 + ], + "score": 1.0, + "content": "correct orientation.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 596, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 209, + 609 + ], + "score": 1.0, + "content": "Besides, if the confounder", + "type": "text" + }, + { + "bbox": [ + 210, + 597, + 223, + 609 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 597, + 234, + 609 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 235, + 597, + 248, + 608 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 597, + 265, + 609 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 265, + 597, + 279, + 608 + ], + "score": 0.88, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 597, + 331, + 609 + ], + "score": 1.0, + "content": "is a parent of", + "type": "text" + }, + { + "bbox": [ + 331, + 597, + 345, + 608 + ], + "score": 0.87, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 597, + 505, + 609 + ], + "score": 1.0, + "content": ", then the local optimum would disappear", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 608, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 226, + 621 + ], + "score": 1.0, + "content": "with learning that edge since", + "type": "text" + }, + { + "bbox": [ + 226, + 608, + 425, + 620 + ], + "score": 0.85, + "content": "p ( X _ { 2 } | \\mathrm { d o } ( X _ { 1 } = x _ { 1 } ) , X _ { j } ) = p ( X _ { 2 } | X _ { 1 } = x _ { 1 } , X _ { j } ) ", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 608, + 506, + 621 + ], + "score": 1.0, + "content": ". In conclusion, for", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "many of the graph structures like chain, bidiag, collider, full and jungle, this shows that there does not", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "exist any local optima for the orientation probabilities. Only for the certain structures of confounded", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 640, + 484, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 484, + 654 + ], + "score": 1.0, + "content": "parents, there may exist local optima that depend on the specific distribution parameterization.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45 + }, + { + "type": "title", + "bbox": [ + 107, + 667, + 240, + 678 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 241, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 241, + 679 + ], + "score": 1.0, + "content": "B.2.3 LIMITED DATA REGIME", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "Assumption (3) and (4) are taken with respect to the data limit such that the conditions derived in the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "next section solely depend on the given causal graphical model. However, in practice, we often have", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "a limited data set. The proof presented for the data limit is straightforward to extend to this setting", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 721, + 238, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 238, + 732 + ], + "score": 1.0, + "content": "with the following modification:", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50.5 + } + ], + "page_idx": 26, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "Summary: in conclusion, for the discussed example, we can guarantee that ENCO converges to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 215, + 107 + ], + "score": 1.0, + "content": "the correct causal graph if", + "type": "text" + }, + { + "bbox": [ + 215, + 94, + 271, + 106 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } < 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 93, + 506, + 107 + ], + "score": 1.0, + "content": ". To experimentally verify this results, we applied ENCO", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 422, + 118 + ], + "score": 1.0, + "content": "on this graph with two hyperparameter settings for the sparsity regularizer:", + "type": "text" + }, + { + "bbox": [ + 422, + 105, + 486, + 116 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 1 9", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 114, + 507, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 167, + 127 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 2 1", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 114, + 507, + 128 + ], + "score": 1.0, + "content": ". We considered a very large sample size, more specifically 10k per intervention and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 125, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 104, + 125, + 373, + 141 + ], + "score": 1.0, + "content": "100k observational samples, to simulate the data limit regime. For", + "type": "text" + }, + { + "bbox": [ + 373, + 127, + 434, + 137 + ], + "score": 0.88, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 1 9", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 125, + 506, + 141 + ], + "score": 1.0, + "content": ", ENCO was able", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 507, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 289, + 151 + ], + "score": 1.0, + "content": "to recover the graph without errors while for", + "type": "text" + }, + { + "bbox": [ + 289, + 137, + 349, + 149 + ], + "score": 0.93, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 2 1", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 137, + 390, + 151 + ], + "score": 1.0, + "content": ", the edge", + "type": "text" + }, + { + "bbox": [ + 390, + 138, + 432, + 148 + ], + "score": 0.91, + "content": "X _ { 1 } X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 137, + 507, + 151 + ], + "score": 1.0, + "content": "was, as expected,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 363, + 162 + ], + "score": 1.0, + "content": "missed. This verifies the theoretical result above with respect to", + "type": "text" + }, + { + "bbox": [ + 363, + 149, + 388, + 160 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 147, + 506, + 162 + ], + "score": 1.0, + "content": ". Note that if the condition is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "not fulfilled by selecting a too large sparsity regularizer, this does not necessarily mean that ENCO", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "will not be able to recover the graph. This is because we consider the ’worst-case’ parent set in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 507, + 195 + ], + "score": 1.0, + "content": "condition 3, while this case might not be in the true causal graph to which the other edges converge.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 81, + 507, + 195 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 207, + 313, + 219 + ], + "lines": [ + { + "bbox": [ + 106, + 207, + 314, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 314, + 220 + ], + "score": 1.0, + "content": "B.2.2 INTUITION BEHIND CONDITION 1 AND 2", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 228, + 505, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 229, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 239 + ], + "score": 1.0, + "content": "As mentioned in the text, condition 1 and 2 of Theorem 3.1 ensure that the orientation probabilities", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 251 + ], + "score": 1.0, + "content": "cannot converge to any local optima. Since the conditions explicitly involve the data distributions and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "implicitly the gradient estimators, we provide below an assumption from a data generation mechanism", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 404, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 404, + 272 + ], + "score": 1.0, + "content": "perspective as an alternative, that ensures condition 1 and 2 to be satisfied.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 229, + 505, + 272 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 278, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "Firstly, we assume that ancestors and descendants are not independent under interventions on the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 507, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 507, + 302 + ], + "score": 1.0, + "content": "ancestors. Note that there can exist graphs where the ancestors are independent of descendants,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 312 + ], + "score": 1.0, + "content": "for instance in a linear Gaussian setting when the ancestor has a weight of zero on the descendant.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 311, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 323 + ], + "score": 1.0, + "content": "However, those graphs, violating faithfulness, are impossible to find for any causal discovery method", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "since the variables are independent under any setting. In terms of condition 1 and 2, it would imply", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 333, + 242, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 242, + 345 + ], + "score": 1.0, + "content": "that the inequality is always zero.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 277, + 507, + 345 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 506, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "Next, we show that under the previous assumption, local optima of the orientation probabilities can", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 504, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 303, + 374 + ], + "score": 1.0, + "content": "only occur in the following structure: for an edge", + "type": "text" + }, + { + "bbox": [ + 304, + 361, + 344, + 373 + ], + "score": 0.91, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 360, + 490, + 374 + ], + "score": 1.0, + "content": ", there exist one or more parent(s) of", + "type": "text" + }, + { + "bbox": [ + 490, + 361, + 504, + 373 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 232, + 384 + ], + "score": 1.0, + "content": "sharing a common confounder", + "type": "text" + }, + { + "bbox": [ + 232, + 372, + 247, + 383 + ], + "score": 0.89, + "content": "X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 371, + 268, + 384 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 269, + 372, + 281, + 383 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 371, + 312, + 384 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 313, + 372, + 327, + 383 + ], + "score": 0.87, + "content": "X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 371, + 424, + 384 + ], + "score": 1.0, + "content": "is not a direct parent of", + "type": "text" + }, + { + "bbox": [ + 424, + 372, + 438, + 384 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 371, + 506, + 384 + ], + "score": 1.0, + "content": ". An example of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 232, + 396 + ], + "score": 1.0, + "content": "this structure is the following:", + "type": "text" + }, + { + "bbox": [ + 233, + 383, + 293, + 394 + ], + "score": 0.58, + "content": "X _ { 1 } X _ { 2 } , X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 383, + 298, + 396 + ], + "score": 1.0, + "content": ";", + "type": "text" + }, + { + "bbox": [ + 298, + 383, + 358, + 394 + ], + "score": 0.62, + "content": "X _ { 2 } , X _ { 3 } X _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "where the orientations of the edges", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 107, + 394, + 149, + 405 + ], + "score": 0.92, + "content": "X _ { 2 } X _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 393, + 167, + 406 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 167, + 394, + 209, + 405 + ], + "score": 0.93, + "content": "X _ { 3 } X _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "could have a local optimum. This statement can be proven as follows by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 404, + 504, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 455, + 418 + ], + "score": 1.0, + "content": "using the do-calculus (Pearl, 2009). Suppose a graph that includes the three variables", + "type": "text" + }, + { + "bbox": [ + 456, + 405, + 504, + 416 + ], + "score": 0.9, + "content": "X _ { 1 } , X _ { 2 } , X _ { 3 }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 127, + 429 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 416, + 171, + 426 + ], + "score": 0.87, + "content": "X _ { 1 } X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 415, + 175, + 429 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 175, + 416, + 219, + 426 + ], + "score": 0.88, + "content": "X _ { 3 } X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 415, + 240, + 429 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 241, + 416, + 255, + 426 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 415, + 365, + 429 + ], + "score": 1.0, + "content": "having no parents besides", + "type": "text" + }, + { + "bbox": [ + 365, + 416, + 379, + 426 + ], + "score": 0.87, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 415, + 398, + 429 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 398, + 416, + 412, + 426 + ], + "score": 0.87, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 415, + 427, + 429 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 428, + 416, + 441, + 426 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 415, + 461, + 429 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 461, + 416, + 475, + 426 + ], + "score": 0.88, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 415, + 506, + 429 + ], + "score": 1.0, + "content": "do not", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 343, + 439 + ], + "score": 1.0, + "content": "share a confounder, then, from do-calculus, we know that", + "type": "text" + }, + { + "bbox": [ + 343, + 427, + 505, + 438 + ], + "score": 0.89, + "content": "p ( X _ { 2 } | \\mathbf { d o } ( X _ { 1 } = x _ { 1 } ) ) = p ( X _ { 2 } | X _ { 1 } = x _ { 1 } )", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 124, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 437, + 297, + 450 + ], + "score": 0.9, + "content": "p ( X _ { 2 } | \\mathrm { d o } ( X _ { 3 } = x _ { 3 } ) ) = p ( X _ { 2 } | X _ { 3 } = x _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 437, + 506, + 450 + ], + "score": 1.0, + "content": ". Furthermore, since the conditional entropy of a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 340, + 461 + ], + "score": 1.0, + "content": "variable can only be smaller or equals to the marginal, i.e.", + "type": "text" + }, + { + "bbox": [ + 340, + 448, + 417, + 460 + ], + "score": 0.94, + "content": "H ( X ) \\geq H ( X | Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 448, + 465, + 461 + ], + "score": 1.0, + "content": ", estimating", + "type": "text" + }, + { + "bbox": [ + 465, + 449, + 479, + 459 + ], + "score": 0.88, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "under", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 458, + 507, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 176, + 473 + ], + "score": 1.0, + "content": "interventions on", + "type": "text" + }, + { + "bbox": [ + 176, + 460, + 190, + 470 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 458, + 368, + 473 + ], + "score": 1.0, + "content": "can only be improved by conditioning on", + "type": "text" + }, + { + "bbox": [ + 368, + 460, + 382, + 470 + ], + "score": 0.87, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 458, + 460, + 473 + ], + "score": 1.0, + "content": ", and similarly for", + "type": "text" + }, + { + "bbox": [ + 460, + 460, + 474, + 471 + ], + "score": 0.88, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 458, + 507, + 473 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "condition 1 is strictly fulfilled when parents do not share a confounder under the previous assumption", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 426, + 494 + ], + "score": 1.0, + "content": "of no independence in all possible settings. Now, consider the situation where", + "type": "text" + }, + { + "bbox": [ + 426, + 482, + 440, + 492 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 481, + 459, + 494 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 459, + 482, + 473, + 492 + ], + "score": 0.88, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "share a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "common confounder. Then, from do-calculus, we can state that there can exist a parameterization of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 501, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 264, + 517 + ], + "score": 1.0, + "content": "the conditional distributions for which", + "type": "text" + }, + { + "bbox": [ + 264, + 503, + 425, + 515 + ], + "score": 0.9, + "content": "p ( X _ { 2 } | \\mathbf { d o } ( X _ { 1 } = x _ { 1 } ) ) \\neq p ( X _ { 2 } | X _ { 1 } = x _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 501, + 506, + 517 + ], + "score": 1.0, + "content": ". Under this setting,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "we cannot guarantee that condition 1 is always fulfilled. However, whether this parent-confounder", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "structure above actually leads to a local optimum or not depends on the distributions, which condition", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "1 models. Intuitively, this requires the mutual information between the two or more parents to be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "score": 1.0, + "content": "very high, and the initial edge probabilities of those edges to be very low. Further, as the results show,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "this combination of events is not very common in practice, meaning that as long as the ancestor and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 568, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 582 + ], + "score": 1.0, + "content": "descendant are not independent under the interventions, we usually converge to a graph with the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 581, + 184, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 184, + 591 + ], + "score": 1.0, + "content": "correct orientation.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 349, + 507, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 596, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 209, + 609 + ], + "score": 1.0, + "content": "Besides, if the confounder", + "type": "text" + }, + { + "bbox": [ + 210, + 597, + 223, + 609 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 597, + 234, + 609 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 235, + 597, + 248, + 608 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 597, + 265, + 609 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 265, + 597, + 279, + 608 + ], + "score": 0.88, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 597, + 331, + 609 + ], + "score": 1.0, + "content": "is a parent of", + "type": "text" + }, + { + "bbox": [ + 331, + 597, + 345, + 608 + ], + "score": 0.87, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 597, + 505, + 609 + ], + "score": 1.0, + "content": ", then the local optimum would disappear", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 608, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 226, + 621 + ], + "score": 1.0, + "content": "with learning that edge since", + "type": "text" + }, + { + "bbox": [ + 226, + 608, + 425, + 620 + ], + "score": 0.85, + "content": "p ( X _ { 2 } | \\mathrm { d o } ( X _ { 1 } = x _ { 1 } ) , X _ { j } ) = p ( X _ { 2 } | X _ { 1 } = x _ { 1 } , X _ { j } ) ", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 608, + 506, + 621 + ], + "score": 1.0, + "content": ". In conclusion, for", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "many of the graph structures like chain, bidiag, collider, full and jungle, this shows that there does not", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "exist any local optima for the orientation probabilities. Only for the certain structures of confounded", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 640, + 484, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 484, + 654 + ], + "score": 1.0, + "content": "parents, there may exist local optima that depend on the specific distribution parameterization.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 597, + 506, + 654 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 667, + 240, + 678 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 241, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 241, + 679 + ], + "score": 1.0, + "content": "B.2.3 LIMITED DATA REGIME", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "Assumption (3) and (4) are taken with respect to the data limit such that the conditions derived in the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "next section solely depend on the given causal graphical model. However, in practice, we often have", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "a limited data set. 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Still, we want to emphasize that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "despite graphs not fulfilling the conditions, ENCO might still converge to the correct DAG for those", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 671, + 463, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 362, + 684 + ], + "score": 1.0, + "content": "as the guarantee conditions assume the worst-case scenarios for", + "type": "text" + }, + { + "bbox": [ + 362, + 671, + 370, + 681 + ], + "score": 0.75, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 671, + 387, + 684 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 387, + 672, + 396, + 682 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 671, + 463, + 684 + ], + "score": 1.0, + "content": "in all situations.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "The first example we discuss is based on a fork structure where we have three binary variables,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 107, + 699, + 164, + 711 + ], + "score": 0.92, + "content": "\\{ X _ { 1 } , X _ { 2 } , X _ { 3 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 698, + 227, + 712 + ], + "score": 1.0, + "content": ", and the edges", + "type": "text" + }, + { + "bbox": [ + 227, + 699, + 270, + 710 + ], + "score": 0.91, + "content": "X _ { 1 } X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 698, + 289, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 289, + 699, + 332, + 710 + ], + "score": 0.92, + "content": "X _ { 2 } X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "(see Figure 8a). 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Still, there are situations where we cannot guarantee that ENCO convergences", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 356, + 651 + ], + "score": 1.0, + "content": "to the correct causal graph independent of the chosen value of", + "type": "text" + }, + { + "bbox": [ + 357, + 639, + 381, + 650 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 637, + 506, + 651 + ], + "score": 1.0, + "content": ". Here, we want to discuss two", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "scenarios visualized in Figure 8 under which the guarantees fail. Still, we want to emphasize that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "despite graphs not fulfilling the conditions, ENCO might still converge to the correct DAG for those", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 671, + 463, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 362, + 684 + ], + "score": 1.0, + "content": "as the guarantee conditions assume the worst-case scenarios for", + "type": "text" + }, + { + "bbox": [ + 362, + 671, + 370, + 681 + ], + "score": 0.75, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 671, + 387, + 684 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 387, + 672, + 396, + 682 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 671, + 463, + 684 + ], + "score": 1.0, + "content": "in all situations.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 616, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "The first example we discuss is based on a fork structure where we have three binary variables,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 107, + 699, + 164, + 711 + ], + "score": 0.92, + "content": "\\{ X _ { 1 } , X _ { 2 } , X _ { 3 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 698, + 227, + 712 + ], + "score": 1.0, + "content": ", and the edges", + "type": "text" + }, + { + "bbox": [ + 227, + 699, + 270, + 710 + ], + "score": 0.91, + "content": "X _ { 1 } X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 698, + 289, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 289, + 699, + 332, + 710 + ], + "score": 0.92, + "content": "X _ { 2 } X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "(see Figure 8a). The parameterization we", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 249, + 722 + ], + "score": 1.0, + "content": "look at is a (noisy) XOR-gate for", + "type": "text" + }, + { + "bbox": [ + 249, + 711, + 263, + 721 + ], + "score": 0.88, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 710, + 381, + 722 + ], + "score": 1.0, + "content": "with its two input variables", + "type": "text" + }, + { + "bbox": [ + 382, + 710, + 413, + 721 + ], + "score": 0.91, + "content": "X _ { 1 } , X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "being independent of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 416, + 733 + ], + "score": 1.0, + "content": "each other and uniformly distributed. The conditional probability distribution", + "type": "text" + }, + { + "bbox": [ + 417, + 720, + 475, + 732 + ], + "score": 0.93, + "content": "p ( X _ { 3 } | X _ { 1 } , \\mathbf { \\bar { X } } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 687, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 307, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 308, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 308, + 95 + ], + "score": 1.0, + "content": "summarized in the following probability function:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 100, + 410, + 154 + ], + "lines": [ + { + "bbox": [ + 200, + 100, + 410, + 154 + ], + "spans": [ + { + "bbox": [ + 200, + 100, + 410, + 154 + ], + "score": 0.95, + "content": "p ( X _ { 3 } = 1 | X _ { 1 } , X _ { 2 } ) = { \\left\\{ \\begin{array} { l l } { \\epsilon } & { { \\mathrm { i f ~ } } X _ { 1 } = 0 , X _ { 2 } = 0 } \\\\ { 1 - \\epsilon } & { { \\mathrm { i f ~ } } X _ { 1 } = 1 , X _ { 2 } = 0 } \\\\ { \\epsilon } & { { \\mathrm { i f ~ } } X _ { 1 } = 0 , X _ { 2 } = 1 } \\\\ { 1 - \\epsilon } & { { \\mathrm { i f ~ } } X _ { 1 } = 1 , X _ { 2 } = 1 } \\end{array} \\right. }", + "type": "interline_equation", + "image_path": "05f33398b7cba4bfebfd14afb02e4dd9b98cd0fe96ab2dce1b38858abc873cc8.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 200, + 100, + 410, + 113.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 200, + 113.5, + 410, + 127.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 200, + 127.0, + 410, + 140.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 200, + 140.5, + 410, + 154.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 164, + 506, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 179, + 178 + ], + "score": 1.0, + "content": "In other words, if", + "type": "text" + }, + { + "bbox": [ + 180, + 165, + 219, + 177 + ], + "score": 0.86, + "content": "X _ { 1 } \\neq X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 164, + 223, + 178 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 223, + 166, + 237, + 176 + ], + "score": 0.78, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 164, + 366, + 178 + ], + "score": 1.0, + "content": "is equals 1 with a likelihood of", + "type": "text" + }, + { + "bbox": [ + 366, + 166, + 389, + 176 + ], + "score": 0.79, + "content": "1 - \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 164, + 402, + 178 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 402, + 165, + 441, + 176 + ], + "score": 0.88, + "content": "X _ { 1 } = X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 164, + 445, + 178 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 446, + 166, + 460, + 176 + ], + "score": 0.83, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "is equals 1", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 185, + 189 + ], + "score": 1.0, + "content": "with a likelihood of", + "type": "text" + }, + { + "bbox": [ + 185, + 178, + 191, + 186 + ], + "score": 0.67, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 175, + 505, + 189 + ], + "score": 1.0, + "content": ". The issue that this probability table creates is the following. Knowing only one", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "out of the two variables does not improve the log likelihood estimate for the output. This is because", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 198, + 507, + 211 + ], + "spans": [ + { + "bbox": [ + 107, + 199, + 120, + 209 + ], + "score": 0.87, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 198, + 138, + 211 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 138, + 198, + 152, + 209 + ], + "score": 0.91, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 198, + 292, + 211 + ], + "score": 1.0, + "content": "are independent of each other, and", + "type": "text" + }, + { + "bbox": [ + 293, + 198, + 373, + 210 + ], + "score": 0.92, + "content": "p \\mathbf { \\bar { ( } } X _ { 3 } | X _ { 1 } ) = p ( X _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 198, + 507, + 211 + ], + "score": 1.0, + "content": "is a uniform distribution. Hence,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "score": 1.0, + "content": "the worst-case parent set in Equation 6 would be the empty set, and leads to an expected difference", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 219, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 255, + 234 + ], + "score": 1.0, + "content": "log-likelihood difference of zero. As", + "type": "text" + }, + { + "bbox": [ + 256, + 221, + 281, + 232 + ], + "score": 0.92, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 219, + 506, + 234 + ], + "score": 1.0, + "content": "is required to be greater than zero for Theorem B.4, we", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "cannot fulfill the condition for that graph. This means that an empty graph without any edges is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "score": 1.0, + "content": "a local minimum to which ENCO could converge. Yet, when the edge probabilities are non-zero,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 424, + 266 + ], + "score": 1.0, + "content": "we will sample adjacency matrices with both input variables being a parent of", + "type": "text" + }, + { + "bbox": [ + 424, + 253, + 438, + 264 + ], + "score": 0.9, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "with a non-zero", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 315, + 277 + ], + "score": 1.0, + "content": "probability. Hence, the log-likelihood difference for", + "type": "text" + }, + { + "bbox": [ + 316, + 264, + 329, + 275 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 264, + 348, + 277 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 348, + 264, + 362, + 275 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 264, + 373, + 277 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 373, + 264, + 387, + 275 + ], + "score": 0.9, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 264, + 506, + 277 + ], + "score": 1.0, + "content": "is unequal zero. Further, this", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "graph is still often correctly discovered despite ENCO not having a convergence guarantee for it.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 284, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 320, + 300 + ], + "score": 1.0, + "content": "We have conducted experiments on this graph with", + "type": "text" + }, + { + "bbox": [ + 320, + 286, + 437, + 298 + ], + "score": 0.92, + "content": "\\epsilon = \\{ 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 4 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 284, + 506, + 300 + ], + "score": 1.0, + "content": "using a sparsity", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 296, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 164, + 309 + ], + "score": 1.0, + "content": "regularizer of", + "type": "text" + }, + { + "bbox": [ + 164, + 297, + 220, + 308 + ], + "score": 0.89, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 1 \\mathrm { e } { \\mathrm { - } } 4", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 296, + 506, + 309 + ], + "score": 1.0, + "content": ", and in all cases, ENCO converged to the correct, acyclic graph. Note", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 210, + 320 + ], + "score": 1.0, + "content": "that values close to 0.5 for", + "type": "text" + }, + { + "bbox": [ + 211, + 310, + 217, + 318 + ], + "score": 0.64, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "are most challenging, because the difference between the true conditional", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 318, + 282, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 282, + 332 + ], + "score": 1.0, + "content": "and marginal distribution goes against zero.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 335, + 506, + 424 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "score": 1.0, + "content": "The second example we want to discuss aims at graphs that violate the condition in Theorem B.1,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 358 + ], + "score": 1.0, + "content": "more specifically Equation 23. The graph we consider is a fully connected graph with three variables", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 107, + 358, + 155, + 369 + ], + "score": 0.91, + "content": "X _ { 1 } , X _ { 2 } , X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 358, + 455, + 370 + ], + "score": 1.0, + "content": "(see Figure 8b). The scenario can be described as follows: if knowing", + "type": "text" + }, + { + "bbox": [ + 455, + 358, + 469, + 369 + ], + "score": 0.88, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "informs", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 369, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 229, + 381 + ], + "score": 1.0, + "content": "the log-likelihood estimate of", + "type": "text" + }, + { + "bbox": [ + 230, + 369, + 243, + 380 + ], + "score": 0.9, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 369, + 294, + 381 + ], + "score": 1.0, + "content": "more about", + "type": "text" + }, + { + "bbox": [ + 294, + 369, + 308, + 380 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 369, + 354, + 381 + ], + "score": 1.0, + "content": "than about", + "type": "text" + }, + { + "bbox": [ + 355, + 369, + 369, + 380 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 369, + 472, + 381 + ], + "score": 1.0, + "content": "itself, an intervention on", + "type": "text" + }, + { + "bbox": [ + 473, + 369, + 487, + 380 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 369, + 506, + 381 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 237, + 392 + ], + "score": 1.0, + "content": "a sampled graph with the edge", + "type": "text" + }, + { + "bbox": [ + 237, + 380, + 282, + 390 + ], + "score": 0.92, + "content": "X _ { 2 } X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 380, + 469, + 392 + ], + "score": 1.0, + "content": "could lead to a worse likelihood estimate of", + "type": "text" + }, + { + "bbox": [ + 469, + 380, + 483, + 390 + ], + "score": 0.88, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "than", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 287, + 403 + ], + "score": 1.0, + "content": "without the edge. For this scenario to happen,", + "type": "text" + }, + { + "bbox": [ + 288, + 390, + 329, + 402 + ], + "score": 0.92, + "content": "p ( X _ { 2 } | X _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "must be close to deterministic. Additionally,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 107, + 401, + 165, + 414 + ], + "score": 0.92, + "content": "p ( X _ { 3 } | X _ { 1 } , X _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 401, + 289, + 414 + ], + "score": 1.0, + "content": "must be much less reliant on", + "type": "text" + }, + { + "bbox": [ + 289, + 402, + 303, + 412 + ], + "score": 0.88, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 401, + 338, + 414 + ], + "score": 1.0, + "content": "than on", + "type": "text" + }, + { + "bbox": [ + 338, + 402, + 352, + 412 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 401, + 506, + 414 + ], + "score": 1.0, + "content": ", such as in the following probability", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 411, + 141, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 141, + 426 + ], + "score": 1.0, + "content": "density:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 423, + 412, + 477 + ], + "lines": [ + { + "bbox": [ + 198, + 423, + 412, + 477 + ], + "spans": [ + { + "bbox": [ + 198, + 423, + 412, + 477 + ], + "score": 0.94, + "content": "p ( X _ { 3 } = 1 | X _ { 1 } , X _ { 2 } ) = { \\left\\{ \\begin{array} { l l } { \\epsilon _ { 1 } } & { { \\mathrm { i f ~ } } X _ { 1 } = 0 , X _ { 2 } = 0 } \\\\ { 1 - \\epsilon _ { 1 } } & { { \\mathrm { i f ~ } } X _ { 1 } = 1 , X _ { 2 } = 0 } \\\\ { \\epsilon _ { 2 } } & { { \\mathrm { i f ~ } } X _ { 1 } = 0 , X _ { 2 } = 1 } \\\\ { 1 - \\epsilon _ { 2 } } & { { \\mathrm { i f ~ } } X _ { 1 } = 1 , X _ { 2 } = 1 } \\end{array} \\right. }", + "type": "interline_equation", + "image_path": "d305a9aa48d7e5a49a429ee7dd6b3728c17152f2b192fc04d771bc13283a3bb6.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 198, + 423, + 412, + 436.5 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 198, + 436.5, + 412, + 450.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 198, + 450.0, + 412, + 463.5 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 198, + 463.5, + 412, + 477.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 485, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 106, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 178, + 498 + ], + "score": 1.0, + "content": "The two variables", + "type": "text" + }, + { + "bbox": [ + 178, + 488, + 201, + 497 + ], + "score": 0.85, + "content": "\\epsilon _ { 1 } , \\epsilon _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "represent small constants close to zero. In this case, the graph can violate the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 291, + 509 + ], + "score": 1.0, + "content": "condition in Equation 23 since intervening on", + "type": "text" + }, + { + "bbox": [ + 292, + 497, + 306, + 507 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 497, + 436, + 509 + ], + "score": 1.0, + "content": "breaks the dependency between", + "type": "text" + }, + { + "bbox": [ + 437, + 497, + 450, + 507 + ], + "score": 0.9, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 497, + 468, + 509 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 469, + 497, + 482, + 507 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 497, + 505, + 509 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 200, + 520 + ], + "score": 1.0, + "content": "conditional distribution", + "type": "text" + }, + { + "bbox": [ + 201, + 507, + 242, + 519 + ], + "score": 0.92, + "content": "p ( X _ { 3 } | X _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "learned from observational data relies on the dependency between", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 517, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 107, + 519, + 120, + 529 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 517, + 139, + 532 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 140, + 519, + 153, + 529 + ], + "score": 0.88, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 517, + 333, + 532 + ], + "score": 1.0, + "content": "which can make it to a worse estimate than", + "type": "text" + }, + { + "bbox": [ + 333, + 518, + 360, + 531 + ], + "score": 0.92, + "content": "p ( X _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 517, + 451, + 532 + ], + "score": 1.0, + "content": ". Note that if the edge", + "type": "text" + }, + { + "bbox": [ + 451, + 519, + 494, + 529 + ], + "score": 0.92, + "content": "X _ { 1 } X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 517, + 506, + 532 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 528, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 104, + 528, + 488, + 543 + ], + "score": 1.0, + "content": "learned by ENCO though, this will not constitute a problem anymore since with conditioning on", + "type": "text" + }, + { + "bbox": [ + 489, + 530, + 502, + 540 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 528, + 506, + 543 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 539, + 504, + 554 + ], + "spans": [ + { + "bbox": [ + 104, + 539, + 122, + 554 + ], + "score": 1.0, + "content": "i.e.", + "type": "text" + }, + { + "bbox": [ + 122, + 541, + 181, + 552 + ], + "score": 0.92, + "content": "p ( X _ { 3 } | X _ { 2 } , X _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 539, + 220, + 554 + ], + "score": 1.0, + "content": ", the edge", + "type": "text" + }, + { + "bbox": [ + 221, + 541, + 262, + 552 + ], + "score": 0.92, + "content": "X _ { 2 } X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 539, + 496, + 554 + ], + "score": 1.0, + "content": "will gain a gradient towards the correct graph. Thus, when", + "type": "text" + }, + { + "bbox": [ + 496, + 542, + 504, + 552 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 551, + 504, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 123, + 564 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 552, + 130, + 561 + ], + "score": 0.74, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 551, + 390, + 564 + ], + "score": 1.0, + "content": "are not initialized with the worst-case values, the graph with both", + "type": "text" + }, + { + "bbox": [ + 391, + 552, + 405, + 562 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 551, + 423, + 564 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 423, + 552, + 437, + 562 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 551, + 490, + 564 + ], + "score": 1.0, + "content": "as parents of", + "type": "text" + }, + { + "bbox": [ + 490, + 552, + 504, + 563 + ], + "score": 0.89, + "content": "X _ { 3 }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 562, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 506, + 575 + ], + "score": 1.0, + "content": "can be sampled and provides gradients in the correct direction. Further, we did not observe any of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 409, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 409, + 586 + ], + "score": 1.0, + "content": "these situations in the synthetic and real-world graphs we experimented on.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 109, + 597, + 316, + 608 + ], + "lines": [ + { + "bbox": [ + 107, + 596, + 318, + 609 + ], + "spans": [ + { + "bbox": [ + 107, + 596, + 318, + 609 + ], + "score": 1.0, + "content": "B.2.5 CONDITIONS FOR THE GLOBAL OPTIMUM", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "So far, the discussion focused on proving that the optimization space does not contain any local", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 281, + 640 + ], + "score": 1.0, + "content": "optima with respect to the graph parameters", + "type": "text" + }, + { + "bbox": [ + 281, + 628, + 290, + 639 + ], + "score": 0.85, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 627, + 307, + 640 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 307, + 627, + 315, + 637 + ], + "score": 0.71, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "besides the global optimum. If these conditions", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "are violated, ENCO might still converge to the correct solution, since we are not guaranteed to find", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "and get stuck in one of these local optima. Thus, in this section, we provide conditions under which", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "the ground truth graph is the global optimum of the objective in Equation 2. Graphs that fulfill", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "these conditions are very likely to be correctly identified by ENCO, but with a suboptimal choice of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 682, + 441, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 441, + 694 + ], + "score": 1.0, + "content": "hyperparameters, initial starting conditions etc., we could return an incorrect graph.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "The conditions and proof follow a similar structure to those as before for the local optima. We first", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "discuss when we can guarantee that the global optima has the same orientation of edges, and then", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 721, + 376, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 376, + 733 + ], + "score": 1.0, + "content": "when we also find the correct parent set of the remaining variables.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50 + } + ], + "page_idx": 28, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 307, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 308, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 308, + 95 + ], + "score": 1.0, + "content": "summarized in the following probability function:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 82, + 308, + 95 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 100, + 410, + 154 + ], + "lines": [ + { + "bbox": [ + 200, + 100, + 410, + 154 + ], + "spans": [ + { + "bbox": [ + 200, + 100, + 410, + 154 + ], + "score": 0.95, + "content": "p ( X _ { 3 } = 1 | X _ { 1 } , X _ { 2 } ) = { \\left\\{ \\begin{array} { l l } { \\epsilon } & { { \\mathrm { i f ~ } } X _ { 1 } = 0 , X _ { 2 } = 0 } \\\\ { 1 - \\epsilon } & { { \\mathrm { i f ~ } } X _ { 1 } = 1 , X _ { 2 } = 0 } \\\\ { \\epsilon } & { { \\mathrm { i f ~ } } X _ { 1 } = 0 , X _ { 2 } = 1 } \\\\ { 1 - \\epsilon } & { { \\mathrm { i f ~ } } X _ { 1 } = 1 , X _ { 2 } = 1 } \\end{array} \\right. }", + "type": "interline_equation", + "image_path": "05f33398b7cba4bfebfd14afb02e4dd9b98cd0fe96ab2dce1b38858abc873cc8.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 200, + 100, + 410, + 113.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 200, + 113.5, + 410, + 127.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 200, + 127.0, + 410, + 140.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 200, + 140.5, + 410, + 154.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 164, + 506, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 179, + 178 + ], + "score": 1.0, + "content": "In other words, if", + "type": "text" + }, + { + "bbox": [ + 180, + 165, + 219, + 177 + ], + "score": 0.86, + "content": "X _ { 1 } \\neq X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 164, + 223, + 178 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 223, + 166, + 237, + 176 + ], + "score": 0.78, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 164, + 366, + 178 + ], + "score": 1.0, + "content": "is equals 1 with a likelihood of", + "type": "text" + }, + { + "bbox": [ + 366, + 166, + 389, + 176 + ], + "score": 0.79, + "content": "1 - \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 164, + 402, + 178 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 402, + 165, + 441, + 176 + ], + "score": 0.88, + "content": "X _ { 1 } = X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 164, + 445, + 178 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 446, + 166, + 460, + 176 + ], + "score": 0.83, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "is equals 1", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 185, + 189 + ], + "score": 1.0, + "content": "with a likelihood of", + "type": "text" + }, + { + "bbox": [ + 185, + 178, + 191, + 186 + ], + "score": 0.67, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 175, + 505, + 189 + ], + "score": 1.0, + "content": ". The issue that this probability table creates is the following. Knowing only one", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "out of the two variables does not improve the log likelihood estimate for the output. This is because", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 198, + 507, + 211 + ], + "spans": [ + { + "bbox": [ + 107, + 199, + 120, + 209 + ], + "score": 0.87, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 198, + 138, + 211 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 138, + 198, + 152, + 209 + ], + "score": 0.91, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 198, + 292, + 211 + ], + "score": 1.0, + "content": "are independent of each other, and", + "type": "text" + }, + { + "bbox": [ + 293, + 198, + 373, + 210 + ], + "score": 0.92, + "content": "p \\mathbf { \\bar { ( } } X _ { 3 } | X _ { 1 } ) = p ( X _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 198, + 507, + 211 + ], + "score": 1.0, + "content": "is a uniform distribution. Hence,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "score": 1.0, + "content": "the worst-case parent set in Equation 6 would be the empty set, and leads to an expected difference", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 219, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 255, + 234 + ], + "score": 1.0, + "content": "log-likelihood difference of zero. As", + "type": "text" + }, + { + "bbox": [ + 256, + 221, + 281, + 232 + ], + "score": 0.92, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 219, + 506, + 234 + ], + "score": 1.0, + "content": "is required to be greater than zero for Theorem B.4, we", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "cannot fulfill the condition for that graph. This means that an empty graph without any edges is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "score": 1.0, + "content": "a local minimum to which ENCO could converge. Yet, when the edge probabilities are non-zero,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 424, + 266 + ], + "score": 1.0, + "content": "we will sample adjacency matrices with both input variables being a parent of", + "type": "text" + }, + { + "bbox": [ + 424, + 253, + 438, + 264 + ], + "score": 0.9, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "with a non-zero", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 315, + 277 + ], + "score": 1.0, + "content": "probability. Hence, the log-likelihood difference for", + "type": "text" + }, + { + "bbox": [ + 316, + 264, + 329, + 275 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 264, + 348, + 277 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 348, + 264, + 362, + 275 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 264, + 373, + 277 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 373, + 264, + 387, + 275 + ], + "score": 0.9, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 264, + 506, + 277 + ], + "score": 1.0, + "content": "is unequal zero. Further, this", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "graph is still often correctly discovered despite ENCO not having a convergence guarantee for it.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 284, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 320, + 300 + ], + "score": 1.0, + "content": "We have conducted experiments on this graph with", + "type": "text" + }, + { + "bbox": [ + 320, + 286, + 437, + 298 + ], + "score": 0.92, + "content": "\\epsilon = \\{ 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 4 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 284, + 506, + 300 + ], + "score": 1.0, + "content": "using a sparsity", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 296, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 164, + 309 + ], + "score": 1.0, + "content": "regularizer of", + "type": "text" + }, + { + "bbox": [ + 164, + 297, + 220, + 308 + ], + "score": 0.89, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 1 \\mathrm { e } { \\mathrm { - } } 4", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 296, + 506, + 309 + ], + "score": 1.0, + "content": ", and in all cases, ENCO converged to the correct, acyclic graph. Note", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 210, + 320 + ], + "score": 1.0, + "content": "that values close to 0.5 for", + "type": "text" + }, + { + "bbox": [ + 211, + 310, + 217, + 318 + ], + "score": 0.64, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "are most challenging, because the difference between the true conditional", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 318, + 282, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 282, + 332 + ], + "score": 1.0, + "content": "and marginal distribution goes against zero.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 164, + 507, + 332 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 335, + 506, + 424 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "score": 1.0, + "content": "The second example we want to discuss aims at graphs that violate the condition in Theorem B.1,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 358 + ], + "score": 1.0, + "content": "more specifically Equation 23. The graph we consider is a fully connected graph with three variables", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 107, + 358, + 155, + 369 + ], + "score": 0.91, + "content": "X _ { 1 } , X _ { 2 } , X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 358, + 455, + 370 + ], + "score": 1.0, + "content": "(see Figure 8b). 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Further, we did not observe any of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 409, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 409, + 586 + ], + "score": 1.0, + "content": "these situations in the synthetic and real-world graphs we experimented on.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 485, + 506, + 586 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 597, + 316, + 608 + ], + "lines": [ + { + "bbox": [ + 107, + 596, + 318, + 609 + ], + "spans": [ + { + "bbox": [ + 107, + 596, + 318, + 609 + ], + "score": 1.0, + "content": "B.2.5 CONDITIONS FOR THE GLOBAL OPTIMUM", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "So far, the discussion focused on proving that the optimization space does not contain any local", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 281, + 640 + ], + "score": 1.0, + "content": "optima with respect to the graph parameters", + "type": "text" + }, + { + "bbox": [ + 281, + 628, + 290, + 639 + ], + "score": 0.85, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 627, + 307, + 640 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 307, + 627, + 315, + 637 + ], + "score": 0.71, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "besides the global optimum. If these conditions", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "are violated, ENCO might still converge to the correct solution, since we are not guaranteed to find", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "and get stuck in one of these local optima. Thus, in this section, we provide conditions under which", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "the ground truth graph is the global optimum of the objective in Equation 2. Graphs that fulfill", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "these conditions are very likely to be correctly identified by ENCO, but with a suboptimal choice of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 682, + 441, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 441, + 694 + ], + "score": 1.0, + "content": "hyperparameters, initial starting conditions etc., we could return an incorrect graph.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 616, + 506, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "The conditions and proof follow a similar structure to those as before for the local optima. We first", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "discuss when we can guarantee that the global optima has the same orientation of edges, and then", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 721, + 376, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 376, + 733 + ], + "score": 1.0, + "content": "when we also find the correct parent set of the remaining variables.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50, + "bbox_fs": [ + 106, + 698, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 505, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 283, + 96 + ], + "score": 1.0, + "content": "Theorem B.5. 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Under", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 174, + 268 + ], + "score": 1.0, + "content": "interventions on", + "type": "text" + }, + { + "bbox": [ + 174, + 256, + 187, + 266 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 255, + 282, + 268 + ], + "score": 1.0, + "content": ", the causal mechanism", + "type": "text" + }, + { + "bbox": [ + 282, + 255, + 340, + 268 + ], + "score": 0.92, + "content": "p ( X _ { j } | \\mathfrak { p a } ( X _ { j } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 255, + 365, + 268 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 366, + 255, + 396, + 267 + ], + "score": 0.9, + "content": "\\operatorname { p a } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "being the parent set of the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 225, + 280 + ], + "score": 1.0, + "content": "ground truth graph including", + "type": "text" + }, + { + "bbox": [ + 225, + 267, + 237, + 277 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 266, + 406, + 280 + ], + "score": 1.0, + "content": ", remains invariant under interventions on", + "type": "text" + }, + { + "bbox": [ + 407, + 267, + 419, + 277 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 266, + 506, + 280 + ], + "score": 1.0, + "content": ", and is strictly better", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 276, + 507, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 126, + 291 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 126, + 277, + 201, + 289 + ], + "score": 0.92, + "content": "p ( X _ { j } | \\mathsf { p a } ( X _ { j } ) \\backslash X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 276, + 258, + 291 + ], + "score": 1.0, + "content": "for estimating", + "type": "text" + }, + { + "bbox": [ + 258, + 278, + 272, + 289 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 276, + 489, + 291 + ], + "score": 1.0, + "content": "due to the direct causal relation. Under interventions on", + "type": "text" + }, + { + "bbox": [ + 489, + 278, + 502, + 289 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 276, + 507, + 291 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 198, + 301 + ], + "score": 1.0, + "content": "the causal mechanism", + "type": "text" + }, + { + "bbox": [ + 198, + 289, + 250, + 300 + ], + "score": 0.9, + "content": "p ( X _ { i } | X _ { j } , \\ldots )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 288, + 506, + 301 + ], + "score": 1.0, + "content": "leads to a strictly worse estimate as discussed in Theorem B.1,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 233, + 312 + ], + "score": 1.0, + "content": "since the dependency between", + "type": "text" + }, + { + "bbox": [ + 234, + 300, + 246, + 310 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 299, + 265, + 312 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 265, + 300, + 279, + 311 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "does not exist in the interventional regime. Hence, the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 310, + 504, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 230, + 324 + ], + "score": 1.0, + "content": "inverse orientation of the edge", + "type": "text" + }, + { + "bbox": [ + 230, + 311, + 270, + 323 + ], + "score": 0.92, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 310, + 289, + 324 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 289, + 311, + 330, + 322 + ], + "score": 0.92, + "content": "X _ { j } \\to X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 310, + 487, + 324 + ], + "score": 1.0, + "content": "cannot be part of the global optimum.", + "type": "text" + }, + { + "bbox": [ + 494, + 312, + 504, + 321 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 507, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 278, + 343 + ], + "score": 1.0, + "content": "Theorem B.6. For every pair of variables", + "type": "text" + }, + { + "bbox": [ + 278, + 330, + 307, + 342 + ], + "score": 0.91, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 329, + 335, + 343 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 335, + 330, + 348, + 341 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 329, + 507, + 343 + ], + "score": 1.0, + "content": "is an ancestor but not direct parents of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 120, + 356 + ], + "score": 0.85, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 342, + 166, + 356 + ], + "score": 1.0, + "content": ", the graph", + "type": "text" + }, + { + "bbox": [ + 166, + 342, + 175, + 353 + ], + "score": 0.83, + "content": "\\hat { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 342, + 439, + 356 + ], + "score": 1.0, + "content": "that optimizes objective in Equation 2 does not include the edge", + "type": "text" + }, + { + "bbox": [ + 439, + 343, + 480, + 355 + ], + "score": 0.92, + "content": "X _ { j } \\to X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "if the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 354, + 236, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 236, + 365 + ], + "score": 1.0, + "content": "conditions in Theorem B.5 hold.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 504, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 491, + 391 + ], + "score": 1.0, + "content": "Proof. To show this statement, we need to consider different independence relations between", + "type": "text" + }, + { + "bbox": [ + 491, + 380, + 504, + 391 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 391, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 123, + 403 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 391, + 137, + 403 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 391, + 174, + 403 + ], + "score": 1.0, + "content": ". First, if", + "type": "text" + }, + { + "bbox": [ + 174, + 391, + 187, + 402 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 391, + 205, + 403 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 205, + 391, + 219, + 403 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 391, + 506, + 403 + ], + "score": 1.0, + "content": "are independent in the observational dataset given any conditional set,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "the edge will be removed since any edge between two independent variables is removed for any", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 412, + 469, + 426 + ], + "spans": [ + { + "bbox": [ + 107, + 413, + 150, + 425 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 412, + 227, + 426 + ], + "score": 1.0, + "content": ". The same holds if", + "type": "text" + }, + { + "bbox": [ + 228, + 413, + 240, + 424 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 412, + 258, + 426 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 259, + 413, + 272, + 425 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 412, + 420, + 426 + ], + "score": 1.0, + "content": "are independent for interventions on", + "type": "text" + }, + { + "bbox": [ + 420, + 413, + 433, + 424 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 412, + 451, + 426 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 451, + 413, + 465, + 425 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 412, + 469, + 426 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 429, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "If they are dependent, we can follow a similar argument as in Theorem B.2. The causal mechanism", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 107, + 441, + 160, + 452 + ], + "score": 0.92, + "content": "p ( X _ { j } | X _ { i } , \\ldots )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 440, + 384, + 453 + ], + "score": 1.0, + "content": "transfers from observational to interventional data on", + "type": "text" + }, + { + "bbox": [ + 384, + 441, + 397, + 451 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "since on interventions on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 451, + 504, + 465 + ], + "spans": [ + { + "bbox": [ + 107, + 452, + 119, + 462 + ], + "score": 0.86, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 451, + 230, + 465 + ], + "score": 1.0, + "content": ", the causal mechanism of", + "type": "text" + }, + { + "bbox": [ + 230, + 452, + 244, + 464 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 451, + 438, + 465 + ], + "score": 1.0, + "content": "is not changed. Further, when intervening on", + "type": "text" + }, + { + "bbox": [ + 438, + 452, + 452, + 463 + ], + "score": 0.81, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 451, + 457, + 465 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 457, + 452, + 470, + 462 + ], + "score": 0.84, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 451, + 490, + 465 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 490, + 452, + 504, + 464 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 292, + 475 + ], + "score": 1.0, + "content": "become independent such that any mechanism", + "type": "text" + }, + { + "bbox": [ + 293, + 462, + 345, + 474 + ], + "score": 0.93, + "content": "p ( X _ { i } | X _ { j } , \\ldots )", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 462, + 444, + 475 + ], + "score": 1.0, + "content": "cannot transfer except if", + "type": "text" + }, + { + "bbox": [ + 445, + 463, + 457, + 473 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 462, + 475, + 475 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 476, + 463, + 489, + 474 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 472, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 250, + 487 + ], + "score": 1.0, + "content": "independent under interventions on", + "type": "text" + }, + { + "bbox": [ + 250, + 474, + 263, + 486 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 472, + 505, + 487 + ], + "score": 1.0, + "content": ". In this case, the edge will be again removed by the sparsity", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 396, + 497 + ], + "score": 1.0, + "content": "regularizer. This shows that for any setting, the orientation of the edge", + "type": "text" + }, + { + "bbox": [ + 396, + 484, + 437, + 496 + ], + "score": 0.91, + "content": "X _ { j } \\to X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "cannot lead to a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 187, + 509 + ], + "score": 1.0, + "content": "better estimate than", + "type": "text" + }, + { + "bbox": [ + 187, + 496, + 228, + 507 + ], + "score": 0.92, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 495, + 385, + 509 + ], + "score": 1.0, + "content": ", and in case of independence, the edge", + "type": "text" + }, + { + "bbox": [ + 385, + 496, + 426, + 507 + ], + "score": 0.92, + "content": "X _ { j } \\to X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "will be removed as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 129, + 518 + ], + "score": 1.0, + "content": "well.", + "type": "text" + }, + { + "bbox": [ + 495, + 507, + 505, + 516 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 504, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 212, + 539 + ], + "score": 1.0, + "content": "Theorem B.7. The graph", + "type": "text" + }, + { + "bbox": [ + 213, + 524, + 222, + 536 + ], + "score": 0.82, + "content": "\\hat { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 524, + 505, + 539 + ], + "score": 1.0, + "content": "that optimizes objective in Equation 2 models the same parent sets for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 536, + 291, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 291, + 550 + ], + "score": 1.0, + "content": "each variable under the following conditions:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 135, + 558, + 289, + 569 + ], + "lines": [ + { + "bbox": [ + 132, + 557, + 290, + 570 + ], + "spans": [ + { + "bbox": [ + 132, + 557, + 290, + 570 + ], + "score": 1.0, + "content": "• The conditions in Theorem B.5 hold.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 133, + 576, + 504, + 611 + ], + "lines": [ + { + "bbox": [ + 132, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 132, + 576, + 212, + 590 + ], + "score": 1.0, + "content": "• For any variable", + "type": "text" + }, + { + "bbox": [ + 213, + 578, + 226, + 588 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 576, + 320, + 590 + ], + "score": 1.0, + "content": "with its true parent set", + "type": "text" + }, + { + "bbox": [ + 320, + 577, + 351, + 589 + ], + "score": 0.92, + "content": "p a ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 576, + 505, + 590 + ], + "score": 1.0, + "content": ", there does not exist a smaller parent", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 140, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 140, + 587, + 162, + 601 + ], + "score": 1.0, + "content": "set p", + "type": "text" + }, + { + "bbox": [ + 163, + 588, + 218, + 599 + ], + "score": 0.85, + "content": "{ \\dot { \\boldsymbol { a } } } \\subset { \\boldsymbol { X } } \\setminus { \\boldsymbol { X } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 587, + 271, + 601 + ], + "score": 1.0, + "content": ", descendants", + "type": "text" + }, + { + "bbox": [ + 271, + 588, + 290, + 600 + ], + "score": 0.85, + "content": "( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 587, + 466, + 601 + ], + "score": 1.0, + "content": "which approximates the log-likelihood of", + "type": "text" + }, + { + "bbox": [ + 466, + 588, + 479, + 599 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "up to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 142, + 599, + 295, + 613 + ], + "spans": [ + { + "bbox": [ + 142, + 599, + 245, + 612 + ], + "score": 0.89, + "content": "\\lambda _ { s p a r s e } \\cdot ( | p a ( X _ { i } ) | - | \\hat { p a } | )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 599, + 295, + 613 + ], + "score": 1.0, + "content": "on average.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 133, + 618, + 370, + 630 + ], + "lines": [ + { + "bbox": [ + 132, + 616, + 372, + 632 + ], + "spans": [ + { + "bbox": [ + 132, + 616, + 262, + 632 + ], + "score": 1.0, + "content": "• The regularization parameter", + "type": "text" + }, + { + "bbox": [ + 263, + 619, + 288, + 631 + ], + "score": 0.83, + "content": "\\lambda _ { s p a r s e }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 616, + 372, + 632 + ], + "score": 1.0, + "content": "is greater than zero.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "Proof. If the orientations for all edges are according to the ground truth graph in the global optimum", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 351, + 668 + ], + "score": 1.0, + "content": "following Theorem B.5 and B.6, the parent set for a variable", + "type": "text" + }, + { + "bbox": [ + 351, + 655, + 364, + 666 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "is limited to those variables which", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 202, + 678 + ], + "score": 1.0, + "content": "are not descendants of", + "type": "text" + }, + { + "bbox": [ + 202, + 666, + 214, + 677 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 666, + 505, + 678 + ], + "score": 1.0, + "content": ". From the ground truth graph, we know that, conditioned on the true", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 147, + 690 + ], + "score": 1.0, + "content": "parent set", + "type": "text" + }, + { + "bbox": [ + 147, + 677, + 177, + 688 + ], + "score": 0.77, + "content": "\\operatorname { p a } ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 677, + 181, + 690 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 181, + 677, + 194, + 688 + ], + "score": 0.84, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "is independent of all other non-descendant variables. Thus, the log-likelihood", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 154, + 701 + ], + "score": 1.0, + "content": "estimate of", + "type": "text" + }, + { + "bbox": [ + 154, + 689, + 167, + 699 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 687, + 430, + 701 + ], + "score": 1.0, + "content": ", i.e. the left part of the objective in Equation 2, is optimized by", + "type": "text" + }, + { + "bbox": [ + 430, + 687, + 486, + 700 + ], + "score": 0.93, + "content": "p ( X _ { i } | \\bar { \\mathsf { p a } } ( X _ { i } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 687, + 505, + 701 + ], + "score": 1.0, + "content": ". To", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "show that this is also the global optimum when combining with the regularizer, we need to consider", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 184, + 722 + ], + "score": 1.0, + "content": "those parent sets of", + "type": "text" + }, + { + "bbox": [ + 185, + 710, + 197, + 721 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "which obtain a lower penalty, i.e. smaller parent sets. The difference between", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 170, + 734 + ], + "score": 1.0, + "content": "two parent sets", + "type": "text" + }, + { + "bbox": [ + 170, + 721, + 200, + 732 + ], + "score": 0.88, + "content": "\\operatorname { p a } ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 721, + 401, + 734 + ], + "score": 1.0, + "content": "and paˆ in terms of the regularizer corresponds to", + "type": "text" + }, + { + "bbox": [ + 402, + 720, + 502, + 733 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } \\cdot ( | \\mathrm { p a } ( X _ { i } ) | - | \\hat { \\mathrm { p a } } | )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 721, + 506, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43.5 + } + ], + "page_idx": 29, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "30", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 505, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 283, + 96 + ], + "score": 1.0, + "content": "Theorem B.5. For every pair of variables", + "type": "text" + }, + { + "bbox": [ + 284, + 83, + 313, + 95 + ], + "score": 0.91, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 81, + 342, + 96 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 343, + 83, + 355, + 93 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 81, + 415, + 96 + ], + "score": 1.0, + "content": "is a parent of", + "type": "text" + }, + { + "bbox": [ + 415, + 83, + 429, + 95 + ], + "score": 0.87, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 81, + 475, + 96 + ], + "score": 1.0, + "content": ", the graph", + "type": "text" + }, + { + "bbox": [ + 476, + 81, + 485, + 93 + ], + "score": 0.81, + "content": "\\hat { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 335, + 106 + ], + "score": 1.0, + "content": "optimizes objective in Equation 2 models the orientation", + "type": "text" + }, + { + "bbox": [ + 336, + 94, + 376, + 106 + ], + "score": 0.91, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 93, + 505, + 106 + ], + "score": 1.0, + "content": ", if there exists an edge between", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 105, + 304, + 120 + ], + "spans": [ + { + "bbox": [ + 107, + 106, + 119, + 118 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 105, + 138, + 120 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 138, + 106, + 151, + 119 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 105, + 163, + 120 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 163, + 105, + 172, + 117 + ], + "score": 0.79, + "content": "\\hat { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 105, + 304, + 120 + ], + "score": 1.0, + "content": ", under the following conditions:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 81, + 506, + 120 + ] + }, + { + "type": "list", + "bbox": [ + 131, + 127, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 133, + 127, + 376, + 141 + ], + "spans": [ + { + "bbox": [ + 133, + 127, + 142, + 141 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 128, + 155, + 139 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 127, + 174, + 141 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 174, + 128, + 187, + 140 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 127, + 376, + 141 + ], + "score": 1.0, + "content": "are not independent under observational data.", + "type": "text" + } + ], + "index": 3, + "is_list_end_line": true + }, + { + "bbox": [ + 131, + 145, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 131, + 145, + 236, + 162 + ], + "score": 1.0, + "content": "• Under interventions on", + "type": "text" + }, + { + "bbox": [ + 237, + 148, + 249, + 158 + ], + "score": 0.82, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 145, + 253, + 162 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 253, + 148, + 266, + 158 + ], + "score": 0.8, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 145, + 284, + 162 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 285, + 148, + 298, + 160 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 145, + 489, + 162 + ], + "score": 1.0, + "content": "are not independent given the true parent set of", + "type": "text" + }, + { + "bbox": [ + 490, + 148, + 502, + 160 + ], + "score": 0.86, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 145, + 506, + 162 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 3.5, + "bbox_fs": [ + 131, + 127, + 506, + 162 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 173, + 505, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 145, + 185 + ], + "score": 1.0, + "content": "Proof. 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Under interventions on", + "type": "text" + }, + { + "bbox": [ + 489, + 278, + 502, + 289 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 276, + 507, + 291 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 198, + 301 + ], + "score": 1.0, + "content": "the causal mechanism", + "type": "text" + }, + { + "bbox": [ + 198, + 289, + 250, + 300 + ], + "score": 0.9, + "content": "p ( X _ { i } | X _ { j } , \\ldots )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 288, + 506, + 301 + ], + "score": 1.0, + "content": "leads to a strictly worse estimate as discussed in Theorem B.1,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 233, + 312 + ], + "score": 1.0, + "content": "since the dependency between", + "type": "text" + }, + { + "bbox": [ + 234, + 300, + 246, + 310 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 299, + 265, + 312 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 265, + 300, + 279, + 311 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "does not exist in the interventional regime. Hence, the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 310, + 504, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 230, + 324 + ], + "score": 1.0, + "content": "inverse orientation of the edge", + "type": "text" + }, + { + "bbox": [ + 230, + 311, + 270, + 323 + ], + "score": 0.92, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 310, + 289, + 324 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 289, + 311, + 330, + 322 + ], + "score": 0.92, + "content": "X _ { j } \\to X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 310, + 487, + 324 + ], + "score": 1.0, + "content": "cannot be part of the global optimum.", + "type": "text" + }, + { + "bbox": [ + 494, + 312, + 504, + 321 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 221, + 507, + 324 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 507, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 278, + 343 + ], + "score": 1.0, + "content": "Theorem B.6. For every pair of variables", + "type": "text" + }, + { + "bbox": [ + 278, + 330, + 307, + 342 + ], + "score": 0.91, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 329, + 335, + 343 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 335, + 330, + 348, + 341 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 329, + 507, + 343 + ], + "score": 1.0, + "content": "is an ancestor but not direct parents of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 120, + 356 + ], + "score": 0.85, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 342, + 166, + 356 + ], + "score": 1.0, + "content": ", the graph", + "type": "text" + }, + { + "bbox": [ + 166, + 342, + 175, + 353 + ], + "score": 0.83, + "content": "\\hat { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 342, + 439, + 356 + ], + "score": 1.0, + "content": "that optimizes objective in Equation 2 does not include the edge", + "type": "text" + }, + { + "bbox": [ + 439, + 343, + 480, + 355 + ], + "score": 0.92, + "content": "X _ { j } \\to X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "if the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 354, + 236, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 236, + 365 + ], + "score": 1.0, + "content": "conditions in Theorem B.5 hold.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 329, + 507, + 365 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 504, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 491, + 391 + ], + "score": 1.0, + "content": "Proof. To show this statement, we need to consider different independence relations between", + "type": "text" + }, + { + "bbox": [ + 491, + 380, + 504, + 391 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 391, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 123, + 403 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 391, + 137, + 403 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 391, + 174, + 403 + ], + "score": 1.0, + "content": ". First, if", + "type": "text" + }, + { + "bbox": [ + 174, + 391, + 187, + 402 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 391, + 205, + 403 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 205, + 391, + 219, + 403 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 391, + 506, + 403 + ], + "score": 1.0, + "content": "are independent in the observational dataset given any conditional set,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "the edge will be removed since any edge between two independent variables is removed for any", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 412, + 469, + 426 + ], + "spans": [ + { + "bbox": [ + 107, + 413, + 150, + 425 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 412, + 227, + 426 + ], + "score": 1.0, + "content": ". The same holds if", + "type": "text" + }, + { + "bbox": [ + 228, + 413, + 240, + 424 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 412, + 258, + 426 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 259, + 413, + 272, + 425 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 412, + 420, + 426 + ], + "score": 1.0, + "content": "are independent for interventions on", + "type": "text" + }, + { + "bbox": [ + 420, + 413, + 433, + 424 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 412, + 451, + 426 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 451, + 413, + 465, + 425 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 412, + 469, + 426 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 380, + 506, + 426 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 429, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "If they are dependent, we can follow a similar argument as in Theorem B.2. The causal mechanism", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 107, + 441, + 160, + 452 + ], + "score": 0.92, + "content": "p ( X _ { j } | X _ { i } , \\ldots )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 440, + 384, + 453 + ], + "score": 1.0, + "content": "transfers from observational to interventional data on", + "type": "text" + }, + { + "bbox": [ + 384, + 441, + 397, + 451 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "since on interventions on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 451, + 504, + 465 + ], + "spans": [ + { + "bbox": [ + 107, + 452, + 119, + 462 + ], + "score": 0.86, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 451, + 230, + 465 + ], + "score": 1.0, + "content": ", the causal mechanism of", + "type": "text" + }, + { + "bbox": [ + 230, + 452, + 244, + 464 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 451, + 438, + 465 + ], + "score": 1.0, + "content": "is not changed. Further, when intervening on", + "type": "text" + }, + { + "bbox": [ + 438, + 452, + 452, + 463 + ], + "score": 0.81, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 451, + 457, + 465 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 457, + 452, + 470, + 462 + ], + "score": 0.84, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 451, + 490, + 465 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 490, + 452, + 504, + 464 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 292, + 475 + ], + "score": 1.0, + "content": "become independent such that any mechanism", + "type": "text" + }, + { + "bbox": [ + 293, + 462, + 345, + 474 + ], + "score": 0.93, + "content": "p ( X _ { i } | X _ { j } , \\ldots )", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 462, + 444, + 475 + ], + "score": 1.0, + "content": "cannot transfer except if", + "type": "text" + }, + { + "bbox": [ + 445, + 463, + 457, + 473 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 462, + 475, + 475 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 476, + 463, + 489, + 474 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 472, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 250, + 487 + ], + "score": 1.0, + "content": "independent under interventions on", + "type": "text" + }, + { + "bbox": [ + 250, + 474, + 263, + 486 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 472, + 505, + 487 + ], + "score": 1.0, + "content": ". In this case, the edge will be again removed by the sparsity", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 396, + 497 + ], + "score": 1.0, + "content": "regularizer. This shows that for any setting, the orientation of the edge", + "type": "text" + }, + { + "bbox": [ + 396, + 484, + 437, + 496 + ], + "score": 0.91, + "content": "X _ { j } \\to X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "cannot lead to a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 187, + 509 + ], + "score": 1.0, + "content": "better estimate than", + "type": "text" + }, + { + "bbox": [ + 187, + 496, + 228, + 507 + ], + "score": 0.92, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 495, + 385, + 509 + ], + "score": 1.0, + "content": ", and in case of independence, the edge", + "type": "text" + }, + { + "bbox": [ + 385, + 496, + 426, + 507 + ], + "score": 0.92, + "content": "X _ { j } \\to X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "will be removed as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 129, + 518 + ], + "score": 1.0, + "content": "well.", + "type": "text" + }, + { + "bbox": [ + 495, + 507, + 505, + 516 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 428, + 506, + 518 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 504, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 212, + 539 + ], + "score": 1.0, + "content": "Theorem B.7. The graph", + "type": "text" + }, + { + "bbox": [ + 213, + 524, + 222, + 536 + ], + "score": 0.82, + "content": "\\hat { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 524, + 505, + 539 + ], + "score": 1.0, + "content": "that optimizes objective in Equation 2 models the same parent sets for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 536, + 291, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 291, + 550 + ], + "score": 1.0, + "content": "each variable under the following conditions:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 524, + 505, + 550 + ] + }, + { + "type": "text", + "bbox": [ + 135, + 558, + 289, + 569 + ], + "lines": [ + { + "bbox": [ + 132, + 557, + 290, + 570 + ], + "spans": [ + { + "bbox": [ + 132, + 557, + 290, + 570 + ], + "score": 1.0, + "content": "• The conditions in Theorem B.5 hold.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 132, + 557, + 290, + 570 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 576, + 504, + 611 + ], + "lines": [ + { + "bbox": [ + 132, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 132, + 576, + 212, + 590 + ], + "score": 1.0, + "content": "• For any variable", + "type": "text" + }, + { + "bbox": [ + 213, + 578, + 226, + 588 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 576, + 320, + 590 + ], + "score": 1.0, + "content": "with its true parent set", + "type": "text" + }, + { + "bbox": [ + 320, + 577, + 351, + 589 + ], + "score": 0.92, + "content": "p a ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 576, + 505, + 590 + ], + "score": 1.0, + "content": ", there does not exist a smaller parent", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 140, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 140, + 587, + 162, + 601 + ], + "score": 1.0, + "content": "set p", + "type": "text" + }, + { + "bbox": [ + 163, + 588, + 218, + 599 + ], + "score": 0.85, + "content": "{ \\dot { \\boldsymbol { a } } } \\subset { \\boldsymbol { X } } \\setminus { \\boldsymbol { X } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 587, + 271, + 601 + ], + "score": 1.0, + "content": ", descendants", + "type": "text" + }, + { + "bbox": [ + 271, + 588, + 290, + 600 + ], + "score": 0.85, + "content": "( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 587, + 466, + 601 + ], + "score": 1.0, + "content": "which approximates the log-likelihood of", + "type": "text" + }, + { + "bbox": [ + 466, + 588, + 479, + 599 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "up to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 142, + 599, + 295, + 613 + ], + "spans": [ + { + "bbox": [ + 142, + 599, + 245, + 612 + ], + "score": 0.89, + "content": "\\lambda _ { s p a r s e } \\cdot ( | p a ( X _ { i } ) | - | \\hat { p a } | )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 599, + 295, + 613 + ], + "score": 1.0, + "content": "on average.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 132, + 576, + 505, + 613 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 618, + 370, + 630 + ], + "lines": [ + { + "bbox": [ + 132, + 616, + 372, + 632 + ], + "spans": [ + { + "bbox": [ + 132, + 616, + 262, + 632 + ], + "score": 1.0, + "content": "• The regularization parameter", + "type": "text" + }, + { + "bbox": [ + 263, + 619, + 288, + 631 + ], + "score": 0.83, + "content": "\\lambda _ { s p a r s e }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 616, + 372, + 632 + ], + "score": 1.0, + "content": "is greater than zero.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 132, + 616, + 372, + 632 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "Proof. If the orientations for all edges are according to the ground truth graph in the global optimum", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 351, + 668 + ], + "score": 1.0, + "content": "following Theorem B.5 and B.6, the parent set for a variable", + "type": "text" + }, + { + "bbox": [ + 351, + 655, + 364, + 666 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "is limited to those variables which", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 202, + 678 + ], + "score": 1.0, + "content": "are not descendants of", + "type": "text" + }, + { + "bbox": [ + 202, + 666, + 214, + 677 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 666, + 505, + 678 + ], + "score": 1.0, + "content": ". From the ground truth graph, we know that, conditioned on the true", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 147, + 690 + ], + "score": 1.0, + "content": "parent set", + "type": "text" + }, + { + "bbox": [ + 147, + 677, + 177, + 688 + ], + "score": 0.77, + "content": "\\operatorname { p a } ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 677, + 181, + 690 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 181, + 677, + 194, + 688 + ], + "score": 0.84, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "is independent of all other non-descendant variables. Thus, the log-likelihood", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 154, + 701 + ], + "score": 1.0, + "content": "estimate of", + "type": "text" + }, + { + "bbox": [ + 154, + 689, + 167, + 699 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 687, + 430, + 701 + ], + "score": 1.0, + "content": ", i.e. the left part of the objective in Equation 2, is optimized by", + "type": "text" + }, + { + "bbox": [ + 430, + 687, + 486, + 700 + ], + "score": 0.93, + "content": "p ( X _ { i } | \\bar { \\mathsf { p a } } ( X _ { i } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 687, + 505, + 701 + ], + "score": 1.0, + "content": ". To", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "show that this is also the global optimum when combining with the regularizer, we need to consider", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 184, + 722 + ], + "score": 1.0, + "content": "those parent sets of", + "type": "text" + }, + { + "bbox": [ + 185, + 710, + 197, + 721 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "which obtain a lower penalty, i.e. smaller parent sets. The difference between", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 170, + 734 + ], + "score": 1.0, + "content": "two parent sets", + "type": "text" + }, + { + "bbox": [ + 170, + 721, + 200, + 732 + ], + "score": 0.88, + "content": "\\operatorname { p a } ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 721, + 401, + 734 + ], + "score": 1.0, + "content": "and paˆ in terms of the regularizer corresponds to", + "type": "text" + }, + { + "bbox": [ + 402, + 720, + 502, + 733 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } \\cdot ( | \\mathrm { p a } ( X _ { i } ) | - | \\hat { \\mathrm { p a } } | )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 721, + 506, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 644, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 171, + 78, + 448, + 152 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 171, + 78, + 448, + 152 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 171, + 78, + 448, + 152 + ], + "spans": [ + { + "bbox": [ + 171, + 78, + 448, + 152 + ], + "score": 0.955, + "type": "image", + "image_path": "2c0836141ccd2da44a42014ac5247f5f6f58e0499a7b7730733c97a59efc4f30.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 171, + 78, + 448, + 102.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 171, + 102.66666666666667, + 448, + 127.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 171, + 127.33333333333334, + 448, + 152.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 161, + 505, + 205 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 160, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 174 + ], + "score": 1.0, + "content": "Figure 9: Visualization of the different graph structures we need to consider in the guarantee", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 172, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 339, + 184 + ], + "score": 1.0, + "content": "discussion of latent confounder detection. Latent variables", + "type": "text" + }, + { + "bbox": [ + 339, + 172, + 352, + 183 + ], + "score": 0.89, + "content": "X _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 172, + 505, + 184 + ], + "score": 1.0, + "content": "are shown in white, all other variables", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 183, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 261, + 195 + ], + "score": 1.0, + "content": "are observed. (a) The two children of", + "type": "text" + }, + { + "bbox": [ + 261, + 183, + 273, + 194 + ], + "score": 0.81, + "content": "X _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 183, + 277, + 195 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 277, + 183, + 291, + 194 + ], + "score": 0.79, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 183, + 310, + 195 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 310, + 183, + 324, + 194 + ], + "score": 0.9, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 183, + 468, + 195 + ], + "score": 1.0, + "content": ", are independent of each other. (b)", + "type": "text" + }, + { + "bbox": [ + 468, + 183, + 482, + 194 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 183, + 505, + 195 + ], + "score": 1.0, + "content": "is an", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 194, + 403, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 153, + 205 + ], + "score": 1.0, + "content": "ancestor of", + "type": "text" + }, + { + "bbox": [ + 153, + 194, + 167, + 205 + ], + "score": 0.89, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 194, + 386, + 205 + ], + "score": 1.0, + "content": ", and the two variables have a shared latent confounder", + "type": "text" + }, + { + "bbox": [ + 387, + 194, + 399, + 205 + ], + "score": 0.87, + "content": "X _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 194, + 403, + 205 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 232, + 505, + 256 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 246 + ], + "score": 1.0, + "content": "Thus, if there exists no parent set for which this difference is greater than the penalty for the worse", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 277, + 257 + ], + "score": 1.0, + "content": "log-likelihood estimate, the true parent set", + "type": "text" + }, + { + "bbox": [ + 277, + 244, + 307, + 256 + ], + "score": 0.85, + "content": "\\operatorname { p a } ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 244, + 437, + 257 + ], + "score": 1.0, + "content": "constitutes the global optimum.", + "type": "text" + }, + { + "bbox": [ + 495, + 245, + 505, + 255 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 106, + 276, + 424, + 288 + ], + "lines": [ + { + "bbox": [ + 106, + 276, + 425, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 425, + 289 + ], + "score": 1.0, + "content": "B.3 CONVERGENCE CONDITIONS FOR LATENT CONFOUNDER DETECTION", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 298, + 506, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 299, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 506, + 311 + ], + "score": 1.0, + "content": "In Section 3.5, we have discussed that ENCO can be extended to graph with latent confounders.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 293, + 324 + ], + "score": 1.0, + "content": "For this, we have to record the gradients of", + "type": "text" + }, + { + "bbox": [ + 293, + 312, + 307, + 322 + ], + "score": 0.85, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 309, + 434, + 324 + ], + "score": 1.0, + "content": "for the interventional data on", + "type": "text" + }, + { + "bbox": [ + 435, + 311, + 447, + 321 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 309, + 506, + 324 + ], + "score": 1.0, + "content": "and all other", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 322, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 267, + 339 + ], + "score": 1.0, + "content": "interventional data separately. We define", + "type": "text" + }, + { + "bbox": [ + 267, + 322, + 339, + 338 + ], + "score": 0.91, + "content": "\\gamma _ { i j } = \\gamma _ { i j } ^ { ( \\bar { I } ) } + \\gamma _ { i j } ^ { ( O ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 323, + 383, + 339 + ], + "score": 1.0, + "content": "where γ(I)ij", + "type": "text" + }, + { + "bbox": [ + 384, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "is only updated with gradients", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 338, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 276, + 353 + ], + "score": 1.0, + "content": "from Equation 3 under interventions on", + "type": "text" + }, + { + "bbox": [ + 276, + 340, + 289, + 351 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 339, + 312, + 353 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 312, + 338, + 332, + 354 + ], + "score": 0.92, + "content": "\\gamma _ { i j } ^ { ( O ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 340, + 505, + 351 + ], + "score": 1.0, + "content": "on all others. The score to detect latent", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 351, + 170, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 170, + 363 + ], + "score": 1.0, + "content": "confounders is:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 161, + 371, + 449, + 393 + ], + "lines": [ + { + "bbox": [ + 161, + 371, + 449, + 393 + ], + "spans": [ + { + "bbox": [ + 161, + 371, + 449, + 393 + ], + "score": 0.92, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } ) = \\sigma \\left( \\gamma _ { i j } ^ { ( O ) } \\right) \\cdot \\sigma \\left( \\gamma _ { j i } ^ { ( O ) } \\right) \\cdot \\left( 1 - \\sigma \\left( \\gamma _ { i j } ^ { ( I ) } \\right) \\right) \\cdot \\left( 1 - \\sigma \\left( \\gamma _ { j i } ^ { ( I ) } \\right) \\right)", + "type": "interline_equation", + "image_path": "12dc99b0df84e05af3803ba225d729a8fea714fd917dfb8cedd04378f3664ce2.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 161, + 371, + 449, + 393 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 504, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 336, + 418 + ], + "score": 1.0, + "content": "In this section, we show under which conditions the score", + "type": "text" + }, + { + "bbox": [ + 337, + 405, + 381, + 417 + ], + "score": 0.92, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 404, + 459, + 418 + ], + "score": 1.0, + "content": "converges to one if", + "type": "text" + }, + { + "bbox": [ + 460, + 405, + 472, + 416 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 404, + 490, + 418 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 490, + 405, + 504, + 417 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 416, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 506, + 427 + ], + "score": 1.0, + "content": "share a latent confounder. We restrict our discussion to latent confounders between two variables that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "do not have any direct edges with each other, and assume that the confounder is not a child of any", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "other observed variable. We assume that the causal graph based on the observed variable fulfills all", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 449, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 460 + ], + "score": 1.0, + "content": "conditions of Theorem B.1 to B.4 in Section B.2, meaning that without the latent confounders, the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 460, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 471 + ], + "score": 1.0, + "content": "graph could have been recovered without errors. Under those conditions, we can also show that the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "graph among observed variables with latent confounders is also correctly recovered. This is since the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 300, + 494 + ], + "score": 1.0, + "content": "latent confounders only affect Theorem B.4: if", + "type": "text" + }, + { + "bbox": [ + 300, + 482, + 313, + 492 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 481, + 331, + 494 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 332, + 482, + 345, + 494 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "share a latent confounder, they are not", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "conditionally independent given their observed parents. Thus, we can rely on the fact that all edges", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 504, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 506, + 515 + ], + "score": 1.0, + "content": "in the true causal graph will be found according to Theorem B.1 to B.4, and the edges with latent", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 515, + 268, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 268, + 526 + ], + "score": 1.0, + "content": "confounders do not fulfill Theorem B.4.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 505, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 363, + 545 + ], + "score": 1.0, + "content": "For all pairs of variables that do not share a latent confounder,", + "type": "text" + }, + { + "bbox": [ + 363, + 531, + 407, + 544 + ], + "score": 0.94, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 529, + 505, + 545 + ], + "score": 1.0, + "content": "converges to zero. The", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 102, + 543, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 102, + 543, + 313, + 560 + ], + "score": 1.0, + "content": "edges that are removed in Theorem B.4 converge to", + "type": "text" + }, + { + "bbox": [ + 313, + 543, + 365, + 559 + ], + "score": 0.93, + "content": "\\sigma ( \\gamma _ { i j } ^ { ( O ) } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 543, + 410, + 560 + ], + "score": 1.0, + "content": "which sets", + "type": "text" + }, + { + "bbox": [ + 410, + 545, + 455, + 558 + ], + "score": 0.92, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 543, + 506, + 560 + ], + "score": 1.0, + "content": "to zero. For", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "edges that have been recovered, we state in Equation 24 that the gradient for interventional data must", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 102, + 564, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 102, + 564, + 306, + 588 + ], + "score": 1.0, + "content": "be negative for interventions on the parent. Hence,", + "type": "text" + }, + { + "bbox": [ + 307, + 567, + 338, + 583 + ], + "score": 0.94, + "content": "\\sigma ( \\gamma _ { j i } ^ { ( I ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 564, + 460, + 588 + ], + "score": 1.0, + "content": "converges to one which brings", + "type": "text" + }, + { + "bbox": [ + 460, + 569, + 505, + 582 + ], + "score": 0.92, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } )", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 582, + 163, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 163, + 594 + ], + "score": 1.0, + "content": "to zero again.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 597, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 106, + 598, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 505, + 609 + ], + "score": 1.0, + "content": "For variables that share a latent confounder, we distinguish between two cases that are visualized", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 291, + 622 + ], + "score": 1.0, + "content": "in Figure 9. In the first case, we assume that", + "type": "text" + }, + { + "bbox": [ + 292, + 609, + 304, + 620 + ], + "score": 0.9, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 608, + 324, + 622 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 324, + 609, + 337, + 621 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 608, + 505, + 622 + ], + "score": 1.0, + "content": "are independent in the true causal graph", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 374, + 632 + ], + "score": 1.0, + "content": "excluding the latent confounder. This means that an intervention on", + "type": "text" + }, + { + "bbox": [ + 375, + 621, + 387, + 631 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 619, + 505, + 632 + ], + "score": 1.0, + "content": "does not cause any change in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 120, + 643 + ], + "score": 0.87, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 631, + 369, + 643 + ], + "score": 1.0, + "content": ", and vice versa. The second case describes the situation where", + "type": "text" + }, + { + "bbox": [ + 369, + 631, + 382, + 642 + ], + "score": 0.9, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 631, + 450, + 643 + ], + "score": 1.0, + "content": "is an ancestor of", + "type": "text" + }, + { + "bbox": [ + 450, + 631, + 463, + 643 + ], + "score": 0.91, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 631, + 505, + 643 + ], + "score": 1.0, + "content": ". The case", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 640, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 118, + 655 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 642, + 130, + 653 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 640, + 203, + 655 + ], + "score": 1.0, + "content": "being a parent of", + "type": "text" + }, + { + "bbox": [ + 204, + 642, + 217, + 654 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 640, + 506, + 655 + ], + "score": 1.0, + "content": "has been excluded in earlier assumptions as in those cases, we cannot", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 653, + 477, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 219, + 665 + ], + "score": 1.0, + "content": "separate the causal effect of", + "type": "text" + }, + { + "bbox": [ + 219, + 653, + 232, + 664 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 653, + 246, + 665 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 246, + 653, + 259, + 665 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 653, + 477, + 665 + ], + "score": 1.0, + "content": "based on its causal relation and the latent confounder.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 669, + 503, + 692 + ], + "lines": [ + { + "bbox": [ + 105, + 668, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 683 + ], + "score": 1.0, + "content": "In case that the two children of the latent confounder are not an ancestor-descendant pair, we can", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 680, + 315, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 315, + 693 + ], + "score": 1.0, + "content": "provide a guarantee under the following conditions.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 284, + 711 + ], + "score": 1.0, + "content": "Theorem B.8. Consider a pair of variables", + "type": "text" + }, + { + "bbox": [ + 284, + 701, + 313, + 711 + ], + "score": 0.92, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 698, + 437, + 711 + ], + "score": 1.0, + "content": "that share a latent confounder", + "type": "text" + }, + { + "bbox": [ + 438, + 699, + 450, + 710 + ], + "score": 0.82, + "content": "X _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 698, + 505, + 711 + ], + "score": 1.0, + "content": ". Assume that", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 107, + 710, + 119, + 721 + ], + "score": 0.86, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 709, + 138, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 139, + 710, + 152, + 722 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "are conditionally independent given the latent confounder and their observed parents.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 489, + 733 + ], + "score": 1.0, + "content": "Further, all other edges in the causal graph have been recovered under the conditions of Theorem", + "type": "text" + }, + { + "bbox": [ + 489, + 721, + 504, + 731 + ], + "score": 0.26, + "content": "B . 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Latent variables", + "type": "text" + }, + { + "bbox": [ + 339, + 172, + 352, + 183 + ], + "score": 0.89, + "content": "X _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 172, + 505, + 184 + ], + "score": 1.0, + "content": "are shown in white, all other variables", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 183, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 261, + 195 + ], + "score": 1.0, + "content": "are observed. (a) The two children of", + "type": "text" + }, + { + "bbox": [ + 261, + 183, + 273, + 194 + ], + "score": 0.81, + "content": "X _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 183, + 277, + 195 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 277, + 183, + 291, + 194 + ], + "score": 0.79, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 183, + 310, + 195 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 310, + 183, + 324, + 194 + ], + "score": 0.9, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 183, + 468, + 195 + ], + "score": 1.0, + "content": ", are independent of each other. 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The score to detect latent", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 351, + 170, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 170, + 363 + ], + "score": 1.0, + "content": "confounders is:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 299, + 506, + 363 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 161, + 371, + 449, + 393 + ], + "lines": [ + { + "bbox": [ + 161, + 371, + 449, + 393 + ], + "spans": [ + { + "bbox": [ + 161, + 371, + 449, + 393 + ], + "score": 0.92, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } ) = \\sigma \\left( \\gamma _ { i j } ^ { ( O ) } \\right) \\cdot \\sigma \\left( \\gamma _ { j i } ^ { ( O ) } \\right) \\cdot \\left( 1 - \\sigma \\left( \\gamma _ { i j } ^ { ( I ) } \\right) \\right) \\cdot \\left( 1 - \\sigma \\left( \\gamma _ { j i } ^ { ( I ) } \\right) \\right)", + "type": "interline_equation", + "image_path": "12dc99b0df84e05af3803ba225d729a8fea714fd917dfb8cedd04378f3664ce2.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 161, + 371, + 449, + 393 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 504, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 336, + 418 + ], + "score": 1.0, + "content": "In this section, we show under which conditions the score", + "type": "text" + }, + { + "bbox": [ + 337, + 405, + 381, + 417 + ], + "score": 0.92, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 404, + 459, + 418 + ], + "score": 1.0, + "content": "converges to one if", + "type": "text" + }, + { + "bbox": [ + 460, + 405, + 472, + 416 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 404, + 490, + 418 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 490, + 405, + 504, + 417 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 416, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 506, + 427 + ], + "score": 1.0, + "content": "share a latent confounder. We restrict our discussion to latent confounders between two variables that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "do not have any direct edges with each other, and assume that the confounder is not a child of any", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "other observed variable. 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This is since the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 300, + 494 + ], + "score": 1.0, + "content": "latent confounders only affect Theorem B.4: if", + "type": "text" + }, + { + "bbox": [ + 300, + 482, + 313, + 492 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 481, + 331, + 494 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 332, + 482, + 345, + 494 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "share a latent confounder, they are not", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "conditionally independent given their observed parents. Thus, we can rely on the fact that all edges", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 504, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 506, + 515 + ], + "score": 1.0, + "content": "in the true causal graph will be found according to Theorem B.1 to B.4, and the edges with latent", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 515, + 268, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 268, + 526 + ], + "score": 1.0, + "content": "confounders do not fulfill Theorem B.4.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 404, + 506, + 526 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 505, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 363, + 545 + ], + "score": 1.0, + "content": "For all pairs of variables that do not share a latent confounder,", + "type": "text" + }, + { + "bbox": [ + 363, + 531, + 407, + 544 + ], + "score": 0.94, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 529, + 505, + 545 + ], + "score": 1.0, + "content": "converges to zero. The", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 102, + 543, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 102, + 543, + 313, + 560 + ], + "score": 1.0, + "content": "edges that are removed in Theorem B.4 converge to", + "type": "text" + }, + { + "bbox": [ + 313, + 543, + 365, + 559 + ], + "score": 0.93, + "content": "\\sigma ( \\gamma _ { i j } ^ { ( O ) } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 543, + 410, + 560 + ], + "score": 1.0, + "content": "which sets", + "type": "text" + }, + { + "bbox": [ + 410, + 545, + 455, + 558 + ], + "score": 0.92, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 543, + 506, + 560 + ], + "score": 1.0, + "content": "to zero. For", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "edges that have been recovered, we state in Equation 24 that the gradient for interventional data must", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 102, + 564, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 102, + 564, + 306, + 588 + ], + "score": 1.0, + "content": "be negative for interventions on the parent. Hence,", + "type": "text" + }, + { + "bbox": [ + 307, + 567, + 338, + 583 + ], + "score": 0.94, + "content": "\\sigma ( \\gamma _ { j i } ^ { ( I ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 564, + 460, + 588 + ], + "score": 1.0, + "content": "converges to one which brings", + "type": "text" + }, + { + "bbox": [ + 460, + 569, + 505, + 582 + ], + "score": 0.92, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } )", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 582, + 163, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 163, + 594 + ], + "score": 1.0, + "content": "to zero again.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 102, + 529, + 506, + 594 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 597, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 106, + 598, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 505, + 609 + ], + "score": 1.0, + "content": "For variables that share a latent confounder, we distinguish between two cases that are visualized", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 291, + 622 + ], + "score": 1.0, + "content": "in Figure 9. In the first case, we assume that", + "type": "text" + }, + { + "bbox": [ + 292, + 609, + 304, + 620 + ], + "score": 0.9, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 608, + 324, + 622 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 324, + 609, + 337, + 621 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 608, + 505, + 622 + ], + "score": 1.0, + "content": "are independent in the true causal graph", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 374, + 632 + ], + "score": 1.0, + "content": "excluding the latent confounder. This means that an intervention on", + "type": "text" + }, + { + "bbox": [ + 375, + 621, + 387, + 631 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 619, + 505, + 632 + ], + "score": 1.0, + "content": "does not cause any change in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 120, + 643 + ], + "score": 0.87, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 631, + 369, + 643 + ], + "score": 1.0, + "content": ", and vice versa. The second case describes the situation where", + "type": "text" + }, + { + "bbox": [ + 369, + 631, + 382, + 642 + ], + "score": 0.9, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 631, + 450, + 643 + ], + "score": 1.0, + "content": "is an ancestor of", + "type": "text" + }, + { + "bbox": [ + 450, + 631, + 463, + 643 + ], + "score": 0.91, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 631, + 505, + 643 + ], + "score": 1.0, + "content": ". The case", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 640, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 118, + 655 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 642, + 130, + 653 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 640, + 203, + 655 + ], + "score": 1.0, + "content": "being a parent of", + "type": "text" + }, + { + "bbox": [ + 204, + 642, + 217, + 654 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 640, + 506, + 655 + ], + "score": 1.0, + "content": "has been excluded in earlier assumptions as in those cases, we cannot", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 653, + 477, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 219, + 665 + ], + "score": 1.0, + "content": "separate the causal effect of", + "type": "text" + }, + { + "bbox": [ + 219, + 653, + 232, + 664 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 653, + 246, + 665 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 246, + 653, + 259, + 665 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 653, + 477, + 665 + ], + "score": 1.0, + "content": "based on its causal relation and the latent confounder.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 598, + 506, + 665 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 669, + 503, + 692 + ], + "lines": [ + { + "bbox": [ + 105, + 668, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 683 + ], + "score": 1.0, + "content": "In case that the two children of the latent confounder are not an ancestor-descendant pair, we can", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 680, + 315, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 315, + 693 + ], + "score": 1.0, + "content": "provide a guarantee under the following conditions.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 668, + 505, + 693 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 284, + 711 + ], + "score": 1.0, + "content": "Theorem B.8. Consider a pair of variables", + "type": "text" + }, + { + "bbox": [ + 284, + 701, + 313, + 711 + ], + "score": 0.92, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 698, + 437, + 711 + ], + "score": 1.0, + "content": "that share a latent confounder", + "type": "text" + }, + { + "bbox": [ + 438, + 699, + 450, + 710 + ], + "score": 0.82, + "content": "X _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 698, + 505, + 711 + ], + "score": 1.0, + "content": ". Assume that", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 107, + 710, + 119, + 721 + ], + "score": 0.86, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 709, + 138, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 139, + 710, + 152, + 722 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "are conditionally independent given the latent confounder and their observed parents.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 489, + 733 + ], + "score": 1.0, + "content": "Further, all other edges in the causal graph have been recovered under the conditions of Theorem", + "type": "text" + }, + { + "bbox": [ + 489, + 721, + 504, + 731 + ], + "score": 0.26, + "content": "B . I", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 82, + 488, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 225, + 96 + ], + "score": 1.0, + "content": "to B.4. The confounder score", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 225, + 82, + 269, + 95 + ], + "score": 0.92, + "content": "l c ( X _ { i } , X _ { j } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 270, + 82, + 488, + 96 + ], + "score": 1.0, + "content": "converges to one if the following two conditions hold:", + "type": "text", + "cross_page": true + } + ], + "index": 0 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 698, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 487, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 488, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 225, + 96 + ], + "score": 1.0, + "content": "to B.4. The confounder score", + "type": "text" + }, + { + "bbox": [ + 225, + 82, + 269, + 95 + ], + "score": 0.92, + "content": "l c ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 82, + 488, + 96 + ], + "score": 1.0, + "content": "converges to one if the following two conditions hold:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 99, + 456, + 140 + ], + "lines": [ + { + "bbox": [ + 155, + 99, + 456, + 140 + ], + "spans": [ + { + "bbox": [ + 155, + 99, + 456, + 140 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\hat { I } \\sim p _ { I _ { - X _ { i } } } ( I ) } \\mathbb { E } _ { \\tilde { p } _ { \\hat { I } } ( \\mathbf { X } ) } \\big [ \\log p ( X _ { j } | p a ( X _ { j } ) , X _ { i } ) - \\log p ( X _ { j } | p a ( X _ { j } ) ) \\big ] > \\lambda _ { s p a r s e } } \\\\ { \\mathbb { E } _ { \\hat { I } \\sim p _ { I _ { - X _ { j } } } ( I ) } \\mathbb { E } _ { \\tilde { p } _ { \\hat { I } } ( \\mathbf { X } ) } \\big [ \\log p ( X _ { i } | p a ( X _ { i } ) , X _ { j } ) - \\log p ( X _ { i } | p a ( X _ { i } ) ) \\big ] > \\lambda _ { s p a r s e } } \\end{array}", + "type": "interline_equation", + "image_path": "4cb5fb3d160625a7e8ba288033703121bf0689adcbd900198e60febd6e38cdf2.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 155, + 99, + 456, + 112.66666666666667 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 155, + 112.66666666666667, + 456, + 126.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 155, + 126.33333333333334, + 456, + 140.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 150, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 104, + 148, + 507, + 169 + ], + "spans": [ + { + "bbox": [ + 104, + 148, + 349, + 169 + ], + "score": 1.0, + "content": "Proof. We need to show that under the two conditions above,", + "type": "text" + }, + { + "bbox": [ + 349, + 151, + 382, + 167 + ], + "score": 0.93, + "content": "\\sigma ( \\gamma _ { i j } ^ { ( O ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 148, + 400, + 169 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 401, + 151, + 434, + 167 + ], + "score": 0.94, + "content": "\\sigma ( \\gamma _ { j i } ^ { ( O ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 148, + 507, + 169 + ], + "score": 1.0, + "content": "are guaranteed to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 102, + 163, + 509, + 187 + ], + "spans": [ + { + "bbox": [ + 102, + 163, + 195, + 187 + ], + "score": 1.0, + "content": "converge to one while", + "type": "text" + }, + { + "bbox": [ + 195, + 166, + 226, + 182 + ], + "score": 0.93, + "content": "\\sigma ( \\gamma _ { i j } ^ { ( I ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 163, + 244, + 187 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 245, + 166, + 276, + 182 + ], + "score": 0.93, + "content": "\\sigma ( \\gamma _ { j i } ^ { ( I ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 163, + 412, + 187 + ], + "score": 1.0, + "content": "converge to zero. The distribution", + "type": "text" + }, + { + "bbox": [ + 412, + 169, + 447, + 182 + ], + "score": 0.91, + "content": "p _ { I _ { \\cdot X _ { k } } } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 163, + 509, + 187 + ], + "score": 1.0, + "content": "represents the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 408, + 194 + ], + "score": 1.0, + "content": "distribution over interventions excluding the ones performed on the variable", + "type": "text" + }, + { + "bbox": [ + 408, + 182, + 422, + 192 + ], + "score": 0.89, + "content": "X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 180, + 506, + 194 + ], + "score": 1.0, + "content": ". The two conditions", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "resemble Equation 28 with the difference that the intervention on the potential parent variable is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 203, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 506, + 215 + ], + "score": 1.0, + "content": "excluded, and the parent set is the true parent set of the correct causal graph. This is because all other", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 103, + 210, + 508, + 235 + ], + "spans": [ + { + "bbox": [ + 103, + 211, + 404, + 234 + ], + "score": 1.0, + "content": "edges have been correctly recovered, and the two conditions are concerning", + "type": "text" + }, + { + "bbox": [ + 415, + 210, + 508, + 235 + ], + "score": 1.0, + "content": "(γ(O)ij ). If the condition", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 102, + 226, + 507, + 250 + ], + "spans": [ + { + "bbox": [ + 102, + 226, + 375, + 250 + ], + "score": 1.0, + "content": "in Equation 41 holds, it corresponds to a negative gradient in γ(O)ij", + "type": "text" + }, + { + "bbox": [ + 375, + 228, + 507, + 247 + ], + "score": 1.0, + "content": "following the argumentation in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 102, + 238, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 102, + 238, + 242, + 267 + ], + "score": 1.0, + "content": "Theorem B.3. The same holds for", + "type": "text" + }, + { + "bbox": [ + 242, + 245, + 262, + 261 + ], + "score": 0.91, + "content": "\\gamma _ { j i } ^ { ( O ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 238, + 310, + 267 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 310, + 245, + 343, + 261 + ], + "score": 0.92, + "content": "\\sigma ( \\gamma _ { i j } ^ { ( O ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 238, + 362, + 267 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 362, + 245, + 395, + 261 + ], + "score": 0.92, + "content": "\\sigma ( \\gamma _ { j i } ^ { ( O ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 238, + 506, + 267 + ], + "score": 1.0, + "content": "are guaranteed to converge", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 257, + 311, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 311, + 271 + ], + "score": 1.0, + "content": "to one under the conditions given in Theorem B.8.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 275, + 506, + 370 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 509, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 263, + 292 + ], + "score": 1.0, + "content": "For the interventional parameters γ(I)ij", + "type": "text" + }, + { + "bbox": [ + 265, + 275, + 284, + 296 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 285, + 275, + 302, + 292 + ], + "score": 0.91, + "content": "\\gamma _ { j i } ^ { ( I ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 275, + 509, + 296 + ], + "score": 1.0, + "content": ", we show that the gradient can only be positive,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 288, + 508, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 182, + 308 + ], + "score": 1.0, + "content": "i.e.decreasing γ(I)ij", + "type": "text" + }, + { + "bbox": [ + 183, + 288, + 201, + 308 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 201, + 291, + 219, + 307 + ], + "score": 0.92, + "content": "\\gamma _ { j i } ^ { ( I ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 288, + 331, + 308 + ], + "score": 1.0, + "content": ". Under the intervention on", + "type": "text" + }, + { + "bbox": [ + 332, + 294, + 344, + 305 + ], + "score": 0.81, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 288, + 348, + 308 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 348, + 294, + 361, + 305 + ], + "score": 0.81, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 288, + 380, + 308 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 380, + 294, + 393, + 306 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 288, + 508, + 308 + ], + "score": 1.0, + "content": "become independent since", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 409, + 317 + ], + "score": 1.0, + "content": "we assume perfect interventions. In this case, the log-likelihood estimate of", + "type": "text" + }, + { + "bbox": [ + 409, + 305, + 423, + 317 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "cannot be improved", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 186, + 329 + ], + "score": 1.0, + "content": "by conditioning on", + "type": "text" + }, + { + "bbox": [ + 186, + 316, + 198, + 327 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 315, + 293, + 329 + ], + "score": 1.0, + "content": ". Hence, the difference", + "type": "text" + }, + { + "bbox": [ + 293, + 316, + 417, + 329 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { X _ { i } X _ { j } } ( X _ { j } ) - \\mathcal { L } _ { X _ { i } \\not X _ { j } } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "is greater or equal to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 326, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 324, + 341 + ], + "score": 1.0, + "content": "zero. When further considering the sparsity regularizer", + "type": "text" + }, + { + "bbox": [ + 324, + 327, + 349, + 339 + ], + "score": 0.87, + "content": "\\mathrm { \\dot { \\lambda } } _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 326, + 412, + 341 + ], + "score": 1.0, + "content": ", the gradient of", + "type": "text" + }, + { + "bbox": [ + 412, + 329, + 425, + 339 + ], + "score": 0.87, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 326, + 506, + 341 + ], + "score": 1.0, + "content": "under interventions", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 336, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 104, + 336, + 119, + 358 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 341, + 132, + 352 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 336, + 279, + 358 + ], + "score": 1.0, + "content": "can only be positive, i.e.decreasing", + "type": "text" + }, + { + "bbox": [ + 280, + 338, + 297, + 355 + ], + "score": 0.92, + "content": "\\gamma _ { i j } ^ { ( I ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 338, + 449, + 356 + ], + "score": 1.0, + "content": "). The same argument holds for", + "type": "text" + }, + { + "bbox": [ + 428, + 337, + 506, + 355 + ], + "score": 1.0, + "content": "γ(I)ji . Thus, we can", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 352, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 162, + 372 + ], + "score": 1.0, + "content": "conclude that", + "type": "text" + }, + { + "bbox": [ + 163, + 353, + 194, + 370 + ], + "score": 0.93, + "content": "\\sigma ( \\gamma _ { i j } ^ { ( I ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 352, + 212, + 372 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 212, + 353, + 243, + 370 + ], + "score": 0.94, + "content": "\\sigma ( \\gamma _ { j i } ^ { ( I ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 356, + 315, + 369 + ], + "score": 1.0, + "content": "converge to zero.", + "type": "text" + }, + { + "bbox": [ + 494, + 357, + 505, + 368 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 102, + 377, + 508, + 403 + ], + "spans": [ + { + "bbox": [ + 102, + 377, + 303, + 403 + ], + "score": 1.0, + "content": "If the conditions of Theorem B.8 are not fulfilled,", + "type": "text" + }, + { + "bbox": [ + 303, + 382, + 336, + 398 + ], + "score": 0.92, + "content": "\\sigma ( \\gamma _ { i j } ^ { ( O ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 377, + 354, + 403 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 355, + 382, + 388, + 398 + ], + "score": 0.93, + "content": "\\sigma ( \\gamma _ { j i } ^ { ( O ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 377, + 508, + 403 + ], + "score": 1.0, + "content": "might converge to zero. This", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 183, + 411 + ], + "score": 1.0, + "content": "results in the score", + "type": "text" + }, + { + "bbox": [ + 183, + 397, + 228, + 410 + ], + "score": 0.92, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 397, + 309, + 411 + ], + "score": 1.0, + "content": "being zero, but also", + "type": "text" + }, + { + "bbox": [ + 310, + 398, + 336, + 410 + ], + "score": 0.91, + "content": "\\sigma ( \\gamma _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "converging to zero. Hence, we do not get", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 408, + 449, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 449, + 421 + ], + "score": 1.0, + "content": "any false positive edge predictions as we have seen in the experiments of Section 4.5.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 506, + 525 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 222, + 438 + ], + "score": 1.0, + "content": "For the second case where", + "type": "text" + }, + { + "bbox": [ + 222, + 426, + 235, + 437 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 425, + 310, + 438 + ], + "score": 1.0, + "content": "is an ancestor of", + "type": "text" + }, + { + "bbox": [ + 311, + 426, + 324, + 438 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 425, + 506, + 438 + ], + "score": 1.0, + "content": ", we cannot give such a guarantee because", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 281, + 450 + ], + "score": 1.0, + "content": "of Theorem B.2. Theorem B.2 states that", + "type": "text" + }, + { + "bbox": [ + 281, + 436, + 307, + 449 + ], + "score": 0.93, + "content": "\\sigma ( \\theta _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 435, + 506, + 450 + ], + "score": 1.0, + "content": "converges to one for ancestor-descendant pairs.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 149, + 461 + ], + "score": 1.0, + "content": "However,", + "type": "text" + }, + { + "bbox": [ + 149, + 447, + 175, + 460 + ], + "score": 0.93, + "content": "\\sigma ( \\theta _ { j i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 447, + 302, + 461 + ], + "score": 1.0, + "content": "is a factor in the gradients of", + "type": "text" + }, + { + "bbox": [ + 302, + 450, + 315, + 460 + ], + "score": 0.88, + "content": "\\gamma _ { j i }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 447, + 401, + 461 + ], + "score": 1.0, + "content": ". This means that if", + "type": "text" + }, + { + "bbox": [ + 401, + 447, + 428, + 460 + ], + "score": 0.93, + "content": "\\sigma ( \\theta _ { j i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "converges to zero", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 458, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 333, + 471 + ], + "score": 1.0, + "content": "according to Theorem B.2, we cannot guarantee that", + "type": "text" + }, + { + "bbox": [ + 334, + 460, + 347, + 471 + ], + "score": 0.87, + "content": "\\gamma _ { j i }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 458, + 506, + 471 + ], + "score": 1.0, + "content": "converges to the desired value since", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 277, + 482 + ], + "score": 1.0, + "content": "its gradient becomes zero. Nevertheless,", + "type": "text" + }, + { + "bbox": [ + 278, + 469, + 305, + 480 + ], + "score": 0.88, + "content": "5 9 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "of the latent confounders in our experiments of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "Section 4.5 were on ancestor-descendant pairs. ENCO detects a majority of those confounders,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "showing that ENCO still works on such confounders despite not having guarantees. Further, we show", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 501, + 507, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 267, + 515 + ], + "score": 1.0, + "content": "in Section C.3 that the confounder scores", + "type": "text" + }, + { + "bbox": [ + 268, + 502, + 312, + 515 + ], + "score": 0.92, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 501, + 507, + 515 + ], + "score": 1.0, + "content": "indeed converge to one for detected confounders,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 512, + 221, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 221, + 526 + ], + "score": 1.0, + "content": "and zero for all other edges.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 105, + 538, + 345, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 348, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 348, + 550 + ], + "score": 1.0, + "content": "B.4 CONVERGENCE FOR PARTIAL INTERVENTION SETS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 108, + 558, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "In Section 4.4, we have experimentally shown that ENCO works on partial intervention sets as well.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "Here, we will discuss convergence guarantees in the situation when interventions are not provided on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 579, + 160, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 160, + 592 + ], + "score": 1.0, + "content": "all variables.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 347, + 609 + ], + "score": 1.0, + "content": "We start with discussing the case where, for a graph with", + "type": "text" + }, + { + "bbox": [ + 348, + 598, + 358, + 607 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "variables, we are given samples of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 607, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 173, + 622 + ], + "score": 1.0, + "content": "interventions on", + "type": "text" + }, + { + "bbox": [ + 174, + 609, + 201, + 619 + ], + "score": 0.88, + "content": "N - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 607, + 505, + 622 + ], + "score": 1.0, + "content": "variables. In this case, we can rely on the previous convergence guarantees", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 437, + 632 + ], + "score": 1.0, + "content": "discussed in Appendix B.2 with minor modifications. Specifically, for the variable", + "type": "text" + }, + { + "bbox": [ + 438, + 620, + 450, + 630 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 619, + 505, + 632 + ], + "score": 1.0, + "content": "on which we", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 631, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 320, + 643 + ], + "score": 1.0, + "content": "do not have interventions, the orientation parameters", + "type": "text" + }, + { + "bbox": [ + 320, + 631, + 329, + 641 + ], + "score": 0.87, + "content": "\\theta _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 631, + 506, + 643 + ], + "score": 1.0, + "content": "are only updated by interventions on other", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 641, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 241, + 653 + ], + "score": 1.0, + "content": "variables. 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X _ { i } } } ( I ) } \\mathbb { E } _ { \\tilde { p } _ { \\hat { I } } ( \\mathbf { X } ) } \\big [ \\log p ( X _ { j } | p a ( X _ { j } ) , X _ { i } ) - \\log p ( X _ { j } | p a ( X _ { j } ) ) \\big ] > \\lambda _ { s p a r s e } } \\\\ { \\mathbb { E } _ { \\hat { I } \\sim p _ { I _ { - X _ { j } } } ( I ) } \\mathbb { E } _ { \\tilde { p } _ { \\hat { I } } ( \\mathbf { X } ) } \\big [ \\log p ( X _ { i } | p a ( X _ { i } ) , X _ { j } ) - \\log p ( X _ { i } | p a ( X _ { i } ) ) \\big ] > \\lambda _ { s p a r s e } } \\end{array}", + "type": "interline_equation", + "image_path": "4cb5fb3d160625a7e8ba288033703121bf0689adcbd900198e60febd6e38cdf2.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 155, + 99, + 456, + 112.66666666666667 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 155, + 112.66666666666667, + 456, + 126.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 155, + 126.33333333333334, + 456, + 140.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 150, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 104, + 148, + 507, + 169 + ], + "spans": [ + { + "bbox": [ + 104, + 148, + 349, + 169 + ], + "score": 1.0, + "content": "Proof. 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The distribution", + "type": "text" + }, + { + "bbox": [ + 412, + 169, + 447, + 182 + ], + "score": 0.91, + "content": "p _ { I _ { \\cdot X _ { k } } } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 163, + 509, + 187 + ], + "score": 1.0, + "content": "represents the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 408, + 194 + ], + "score": 1.0, + "content": "distribution over interventions excluding the ones performed on the variable", + "type": "text" + }, + { + "bbox": [ + 408, + 182, + 422, + 192 + ], + "score": 0.89, + "content": "X _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 180, + 506, + 194 + ], + "score": 1.0, + "content": ". The two conditions", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "resemble Equation 28 with the difference that the intervention on the potential parent variable is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 203, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 506, + 215 + ], + "score": 1.0, + "content": "excluded, and the parent set is the true parent set of the correct causal graph. This is because all other", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 103, + 210, + 508, + 235 + ], + "spans": [ + { + "bbox": [ + 103, + 211, + 404, + 234 + ], + "score": 1.0, + "content": "edges have been correctly recovered, and the two conditions are concerning", + "type": "text" + }, + { + "bbox": [ + 415, + 210, + 508, + 235 + ], + "score": 1.0, + "content": "(γ(O)ij ). If the condition", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 102, + 226, + 507, + 250 + ], + "spans": [ + { + "bbox": [ + 102, + 226, + 375, + 250 + ], + "score": 1.0, + "content": "in Equation 41 holds, it corresponds to a negative gradient in γ(O)ij", + "type": "text" + }, + { + "bbox": [ + 375, + 228, + 507, + 247 + ], + "score": 1.0, + "content": "following the argumentation in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 102, + 238, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 102, + 238, + 242, + 267 + ], + "score": 1.0, + "content": "Theorem B.3. The same holds for", + "type": "text" + }, + { + "bbox": [ + 242, + 245, + 262, + 261 + ], + "score": 0.91, + "content": "\\gamma _ { j i } ^ { ( O ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 238, + 310, + 267 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 310, + 245, + 343, + 261 + ], + "score": 0.92, + "content": "\\sigma ( \\gamma _ { i j } ^ { ( O ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 238, + 362, + 267 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 362, + 245, + 395, + 261 + ], + "score": 0.92, + "content": "\\sigma ( \\gamma _ { j i } ^ { ( O ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 238, + 506, + 267 + ], + "score": 1.0, + "content": "are guaranteed to converge", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 257, + 311, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 311, + 271 + ], + "score": 1.0, + "content": "to one under the conditions given in Theorem B.8.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8, + "bbox_fs": [ + 102, + 148, + 509, + 271 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 275, + 506, + 370 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 509, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 263, + 292 + ], + "score": 1.0, + "content": "For the interventional parameters γ(I)ij", + "type": "text" + }, + { + "bbox": [ + 265, + 275, + 284, + 296 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 285, + 275, + 302, + 292 + ], + "score": 0.91, + "content": "\\gamma _ { j i } ^ { ( I ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 275, + 509, + 296 + ], + "score": 1.0, + "content": ", we show that the gradient can only be positive,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 288, + 508, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 182, + 308 + ], + "score": 1.0, + "content": "i.e.decreasing γ(I)ij", + "type": "text" + }, + { + "bbox": [ + 183, + 288, + 201, + 308 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 201, + 291, + 219, + 307 + ], + "score": 0.92, + "content": "\\gamma _ { j i } ^ { ( I ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 288, + 331, + 308 + ], + "score": 1.0, + "content": ". Under the intervention on", + "type": "text" + }, + { + "bbox": [ + 332, + 294, + 344, + 305 + ], + "score": 0.81, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 288, + 348, + 308 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 348, + 294, + 361, + 305 + ], + "score": 0.81, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 288, + 380, + 308 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 380, + 294, + 393, + 306 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 288, + 508, + 308 + ], + "score": 1.0, + "content": "become independent since", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 409, + 317 + ], + "score": 1.0, + "content": "we assume perfect interventions. In this case, the log-likelihood estimate of", + "type": "text" + }, + { + "bbox": [ + 409, + 305, + 423, + 317 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "cannot be improved", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 186, + 329 + ], + "score": 1.0, + "content": "by conditioning on", + "type": "text" + }, + { + "bbox": [ + 186, + 316, + 198, + 327 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 315, + 293, + 329 + ], + "score": 1.0, + "content": ". Hence, the difference", + "type": "text" + }, + { + "bbox": [ + 293, + 316, + 417, + 329 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { X _ { i } X _ { j } } ( X _ { j } ) - \\mathcal { L } _ { X _ { i } \\not X _ { j } } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "is greater or equal to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 326, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 324, + 341 + ], + "score": 1.0, + "content": "zero. When further considering the sparsity regularizer", + "type": "text" + }, + { + "bbox": [ + 324, + 327, + 349, + 339 + ], + "score": 0.87, + "content": "\\mathrm { \\dot { \\lambda } } _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 326, + 412, + 341 + ], + "score": 1.0, + "content": ", the gradient of", + "type": "text" + }, + { + "bbox": [ + 412, + 329, + 425, + 339 + ], + "score": 0.87, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 326, + 506, + 341 + ], + "score": 1.0, + "content": "under interventions", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 336, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 104, + 336, + 119, + 358 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 341, + 132, + 352 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 336, + 279, + 358 + ], + "score": 1.0, + "content": "can only be positive, i.e.decreasing", + "type": "text" + }, + { + "bbox": [ + 280, + 338, + 297, + 355 + ], + "score": 0.92, + "content": "\\gamma _ { i j } ^ { ( I ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 338, + 449, + 356 + ], + "score": 1.0, + "content": "). The same argument holds for", + "type": "text" + }, + { + "bbox": [ + 428, + 337, + 506, + 355 + ], + "score": 1.0, + "content": "γ(I)ji . Thus, we can", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 352, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 162, + 372 + ], + "score": 1.0, + "content": "conclude that", + "type": "text" + }, + { + "bbox": [ + 163, + 353, + 194, + 370 + ], + "score": 0.93, + "content": "\\sigma ( \\gamma _ { i j } ^ { ( I ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 352, + 212, + 372 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 212, + 353, + 243, + 370 + ], + "score": 0.94, + "content": "\\sigma ( \\gamma _ { j i } ^ { ( I ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 356, + 315, + 369 + ], + "score": 1.0, + "content": "converge to zero.", + "type": "text" + }, + { + "bbox": [ + 494, + 357, + 505, + 368 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 275, + 509, + 372 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 102, + 377, + 508, + 403 + ], + "spans": [ + { + "bbox": [ + 102, + 377, + 303, + 403 + ], + "score": 1.0, + "content": "If the conditions of Theorem B.8 are not fulfilled,", + "type": "text" + }, + { + "bbox": [ + 303, + 382, + 336, + 398 + ], + "score": 0.92, + "content": "\\sigma ( \\gamma _ { i j } ^ { ( O ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 377, + 354, + 403 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 355, + 382, + 388, + 398 + ], + "score": 0.93, + "content": "\\sigma ( \\gamma _ { j i } ^ { ( O ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 377, + 508, + 403 + ], + "score": 1.0, + "content": "might converge to zero. This", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 183, + 411 + ], + "score": 1.0, + "content": "results in the score", + "type": "text" + }, + { + "bbox": [ + 183, + 397, + 228, + 410 + ], + "score": 0.92, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 397, + 309, + 411 + ], + "score": 1.0, + "content": "being zero, but also", + "type": "text" + }, + { + "bbox": [ + 310, + 398, + 336, + 410 + ], + "score": 0.91, + "content": "\\sigma ( \\gamma _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "converging to zero. Hence, we do not get", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 408, + 449, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 449, + 421 + ], + "score": 1.0, + "content": "any false positive edge predictions as we have seen in the experiments of Section 4.5.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 102, + 377, + 508, + 421 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 506, + 525 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 222, + 438 + ], + "score": 1.0, + "content": "For the second case where", + "type": "text" + }, + { + "bbox": [ + 222, + 426, + 235, + 437 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 425, + 310, + 438 + ], + "score": 1.0, + "content": "is an ancestor of", + "type": "text" + }, + { + "bbox": [ + 311, + 426, + 324, + 438 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 425, + 506, + 438 + ], + "score": 1.0, + "content": ", we cannot give such a guarantee because", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 281, + 450 + ], + "score": 1.0, + "content": "of Theorem B.2. Theorem B.2 states that", + "type": "text" + }, + { + "bbox": [ + 281, + 436, + 307, + 449 + ], + "score": 0.93, + "content": "\\sigma ( \\theta _ { i j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 435, + 506, + 450 + ], + "score": 1.0, + "content": "converges to one for ancestor-descendant pairs.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 149, + 461 + ], + "score": 1.0, + "content": "However,", + "type": "text" + }, + { + "bbox": [ + 149, + 447, + 175, + 460 + ], + "score": 0.93, + "content": "\\sigma ( \\theta _ { j i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 447, + 302, + 461 + ], + "score": 1.0, + "content": "is a factor in the gradients of", + "type": "text" + }, + { + "bbox": [ + 302, + 450, + 315, + 460 + ], + "score": 0.88, + "content": "\\gamma _ { j i }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 447, + 401, + 461 + ], + "score": 1.0, + "content": ". This means that if", + "type": "text" + }, + { + "bbox": [ + 401, + 447, + 428, + 460 + ], + "score": 0.93, + "content": "\\sigma ( \\theta _ { j i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "converges to zero", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 458, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 333, + 471 + ], + "score": 1.0, + "content": "according to Theorem B.2, we cannot guarantee that", + "type": "text" + }, + { + "bbox": [ + 334, + 460, + 347, + 471 + ], + "score": 0.87, + "content": "\\gamma _ { j i }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 458, + 506, + 471 + ], + "score": 1.0, + "content": "converges to the desired value since", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 277, + 482 + ], + "score": 1.0, + "content": "its gradient becomes zero. Nevertheless,", + "type": "text" + }, + { + "bbox": [ + 278, + 469, + 305, + 480 + ], + "score": 0.88, + "content": "5 9 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "of the latent confounders in our experiments of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "Section 4.5 were on ancestor-descendant pairs. ENCO detects a majority of those confounders,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "showing that ENCO still works on such confounders despite not having guarantees. Further, we show", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 501, + 507, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 267, + 515 + ], + "score": 1.0, + "content": "in Section C.3 that the confounder scores", + "type": "text" + }, + { + "bbox": [ + 268, + 502, + 312, + 515 + ], + "score": 0.92, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 501, + 507, + 515 + ], + "score": 1.0, + "content": "indeed converge to one for detected confounders,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 512, + 221, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 221, + 526 + ], + "score": 1.0, + "content": "and zero for all other edges.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 425, + 507, + 526 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 538, + 345, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 348, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 348, + 550 + ], + "score": 1.0, + "content": "B.4 CONVERGENCE FOR PARTIAL INTERVENTION SETS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 108, + 558, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "In Section 4.4, we have experimentally shown that ENCO works on partial intervention sets as well.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "Here, we will discuss convergence guarantees in the situation when interventions are not provided on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 579, + 160, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 160, + 592 + ], + "score": 1.0, + "content": "all variables.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 558, + 506, + 592 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 347, + 609 + ], + "score": 1.0, + "content": "We start with discussing the case where, for a graph with", + "type": "text" + }, + { + "bbox": [ + 348, + 598, + 358, + 607 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "variables, we are given samples of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 607, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 173, + 622 + ], + "score": 1.0, + "content": "interventions on", + "type": "text" + }, + { + "bbox": [ + 174, + 609, + 201, + 619 + ], + "score": 0.88, + "content": "N - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 607, + 505, + 622 + ], + "score": 1.0, + "content": "variables. In this case, we can rely on the previous convergence guarantees", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 437, + 632 + ], + "score": 1.0, + "content": "discussed in Appendix B.2 with minor modifications. Specifically, for the variable", + "type": "text" + }, + { + "bbox": [ + 438, + 620, + 450, + 630 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 619, + 505, + 632 + ], + "score": 1.0, + "content": "on which we", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 631, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 320, + 643 + ], + "score": 1.0, + "content": "do not have interventions, the orientation parameters", + "type": "text" + }, + { + "bbox": [ + 320, + 631, + 329, + 641 + ], + "score": 0.87, + "content": "\\theta _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 631, + 506, + 643 + ], + "score": 1.0, + "content": "are only updated by interventions on other", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 641, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 241, + 653 + ], + "score": 1.0, + "content": "variables. 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i , j } : \\mathbb { E } _ { I _ { X _ { j } } , { \\pmb X } } \\left[ \\log p ( X _ { i } | \\widehat { \\mathbf { p a } } ( X _ { i } ) , X _ { j } ) - \\log p ( X _ { i } | \\widehat { \\mathbf { p a } } ( X _ { i } ) ) \\right] \\leq 0", + "type": "interline_equation", + "image_path": "198837553d2a64045a3ee91a866ac34957098f83097e0ebbbfb49f734a3ddd95.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 160, + 688, + 470, + 705 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 141, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 141, + 709, + 247, + 722 + ], + "score": 1.0, + "content": "For at least one parent set", + "type": "text" + }, + { + "bbox": [ + 248, + 711, + 277, + 722 + ], + "score": 0.92, + "content": "\\widehat { p a } ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 709, + 506, + 722 + ], + "score": 1.0, + "content": ", which has a probability greater than zero to be sampled,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 141, + 720, + 315, + 734 + ], + "spans": [ + { + "bbox": [ + 141, + 720, + 315, + 734 + ], + "score": 1.0, + "content": "this inequality is strictly smaller than zero.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5, + "bbox_fs": [ + 141, + 709, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 215, + 95 + ], + "score": 1.0, + "content": "This condition ensures that", + "type": "text" + }, + { + "bbox": [ + 216, + 83, + 228, + 95 + ], + "score": 0.9, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 82, + 383, + 95 + ], + "score": 1.0, + "content": "converges to the correct values, where", + "type": "text" + }, + { + "bbox": [ + 383, + 83, + 396, + 93 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "is the parent or ancestor of", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 94, + 120, + 106 + ], + "score": 0.86, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 93, + 371, + 106 + ], + "score": 1.0, + "content": ". 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With the conditions above, we", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 385, + 134 + ], + "score": 1.0, + "content": "can ensure that the all orientation parameters are learned, excluding", + "type": "text" + }, + { + "bbox": [ + 386, + 122, + 399, + 134 + ], + "score": 0.9, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 122, + 428, + 134 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 428, + 122, + 441, + 132 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 122, + 460, + 134 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 460, + 122, + 473, + 133 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 122, + 505, + 134 + ], + "score": 1.0, + "content": "are the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "variables for which we have not obtained interventions. 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Therefore, we cannot give guarantees of correctness for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 224, + 189 + ], + "score": 1.0, + "content": "incoming/outgoing edges of", + "type": "text" + }, + { + "bbox": [ + 225, + 177, + 237, + 187 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 176, + 256, + 189 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 257, + 177, + 270, + 189 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 176, + 506, + 189 + ], + "score": 1.0, + "content": ", and might make incorrect predictions of edges between", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 187, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 187, + 199 + ], + "score": 1.0, + "content": "these two variables.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 257, + 217 + ], + "score": 1.0, + "content": "When taking the next step to having", + "type": "text" + }, + { + "bbox": [ + 257, + 205, + 269, + 214 + ], + "score": 0.77, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "interventions provided, ENCO can create more incorrect", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "predictions. For the variables for which interventions are provided, we can use the same convergence", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 104, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "guarantees (Theorem B.1-B.4) since all conditions are independent across variables. For variables", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 491, + 249 + ], + "score": 1.0, + "content": "without interventions, we cannot rely on those. While we have observed that learning the missing", + "type": "text" + }, + { + "bbox": [ + 491, + 237, + 497, + 247 + ], + "score": 0.74, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "’s", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 246, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 506, + 261 + ], + "score": 1.0, + "content": "from other interventions give reasonable results, we see a degradation of performance the further the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "distance is between a node and the closest intervened variable. As an example, suppose we have a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 212, + 282 + ], + "score": 1.0, + "content": "chain with 5 variables, i.e.", + "type": "text" + }, + { + "bbox": [ + 212, + 270, + 339, + 281 + ], + "score": 0.91, + "content": "X _ { 1 } X _ { 2 } X _ { 3 } X _ { 4 } X _ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 270, + 506, + 282 + ], + "score": 1.0, + "content": ", and we are provided with an intervention", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 504, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 118, + 294 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 281, + 133, + 291 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 280, + 342, + 294 + ], + "score": 1.0, + "content": "only. This allows us to learn the orientation between", + "type": "text" + }, + { + "bbox": [ + 342, + 281, + 356, + 292 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 280, + 374, + 294 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 374, + 281, + 388, + 292 + ], + "score": 0.87, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 280, + 490, + 294 + ], + "score": 1.0, + "content": ". The orientation between", + "type": "text" + }, + { + "bbox": [ + 490, + 281, + 504, + 292 + ], + "score": 0.85, + "content": "X _ { 2 }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 504, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 123, + 304 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 292, + 138, + 303 + ], + "score": 0.89, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 291, + 375, + 304 + ], + "score": 1.0, + "content": "is often learned correctly as well because adding the edge", + "type": "text" + }, + { + "bbox": [ + 376, + 292, + 417, + 303 + ], + "score": 0.92, + "content": "X _ { 2 } X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 291, + 461, + 304 + ], + "score": 1.0, + "content": "instead of", + "type": "text" + }, + { + "bbox": [ + 462, + 292, + 504, + 303 + ], + "score": 0.91, + "content": "X _ { 3 } X _ { 2 }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 504, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 435, + 316 + ], + "score": 1.0, + "content": "gives a greater decrease in overall log-likelihood, since part of the information from", + "type": "text" + }, + { + "bbox": [ + 436, + 303, + 449, + 314 + ], + "score": 0.89, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 302, + 490, + 316 + ], + "score": 1.0, + "content": "to predict", + "type": "text" + }, + { + "bbox": [ + 490, + 304, + 504, + 314 + ], + "score": 0.88, + "content": "X _ { 2 }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 194, + 326 + ], + "score": 1.0, + "content": "is already included in", + "type": "text" + }, + { + "bbox": [ + 194, + 314, + 208, + 325 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 313, + 368, + 326 + ], + "score": 1.0, + "content": ". However, the further we go away from", + "type": "text" + }, + { + "bbox": [ + 369, + 314, + 382, + 325 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 313, + 506, + 326 + ], + "score": 1.0, + "content": ", the less information is shared", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 325, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 336 + ], + "score": 1.0, + "content": "between the child and the intervened variable. Moreover, the likelihood of a mistake occurring due to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 378, + 348 + ], + "score": 1.0, + "content": "limited data further increases. This is why the orientation of the edge", + "type": "text" + }, + { + "bbox": [ + 378, + 336, + 420, + 347 + ], + "score": 0.93, + "content": "X _ { 4 } X _ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "is not always learned", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 345, + 315, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 315, + 360 + ], + "score": 1.0, + "content": "correctly, which can also cause false positive edges.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 506, + 377 + ], + "score": 1.0, + "content": "Many scenarios of predicting false positive edges can, in theory, be solved by providing an undirected", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "skeleton of the graph, for example, obtained from observational data. Still, one of the cornerstones", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "of ENCO is that it does not assume faithfulness. Without faithfulness or any other assumption on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 409 + ], + "score": 1.0, + "content": "the functional form of the causal mechanisms, the correct undirected graph cannot be recovered", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "by any method. One of the future directions will be to include faithfulness in ENCO to solve the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 419, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 431 + ], + "score": 1.0, + "content": "scenarios mentioned above, although this would imply that we might not be able to recover edges of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 429, + 238, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 238, + 442 + ], + "score": 1.0, + "content": "deterministic variables anymore.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 109, + 455, + 298, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 454, + 300, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 300, + 467 + ], + "score": 1.0, + "content": "B.5 EXAMPLE FOR NON-FAITHFUL GRAPHS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 506, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "Below we give an example of a distribution which is not faithful with respect to its graph structure,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 414, + 498 + ], + "score": 1.0, + "content": "but can yet be found by ENCO. Suppose we have a chain of three variables,", + "type": "text" + }, + { + "bbox": [ + 414, + 486, + 484, + 497 + ], + "score": 0.93, + "content": "X _ { 1 } X _ { 2 } X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 486, + 506, + 498 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "simplicity, we assume here that all the variables are binary, but the argument can similarly hold for", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 508, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 265, + 521 + ], + "score": 1.0, + "content": "any categorical data. The distribution", + "type": "text" + }, + { + "bbox": [ + 266, + 508, + 292, + 520 + ], + "score": 0.92, + "content": "p ( X _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 508, + 413, + 521 + ], + "score": 1.0, + "content": "is an arbitrary function with", + "type": "text" + }, + { + "bbox": [ + 413, + 508, + 502, + 520 + ], + "score": 0.89, + "content": "0 < p ( X _ { 1 } \\stackrel { . } { = } 1 ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 508, + 506, + 521 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 343, + 532 + ], + "score": 1.0, + "content": "and the other two conditionals are deterministic functions:", + "type": "text" + }, + { + "bbox": [ + 343, + 519, + 447, + 531 + ], + "score": 0.81, + "content": "p ( X _ { 2 } | X _ { 1 } ) = \\delta [ X _ { 2 } = X _ { 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 519, + 451, + 532 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 451, + 519, + 506, + 531 + ], + "score": 0.86, + "content": "p ( X _ { 3 } | X _ { 2 } ) =", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 156, + 542 + ], + "score": 0.91, + "content": "\\delta [ X _ { 3 } = X _ { 2 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 529, + 380, + 542 + ], + "score": 1.0, + "content": ". This joint distribution is not faithful to the graph, since", + "type": "text" + }, + { + "bbox": [ + 380, + 531, + 394, + 541 + ], + "score": 0.88, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 529, + 466, + 542 + ], + "score": 1.0, + "content": "is independent of", + "type": "text" + }, + { + "bbox": [ + 466, + 531, + 480, + 541 + ], + "score": 0.88, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "given", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 540, + 504, + 554 + ], + "spans": [ + { + "bbox": [ + 107, + 542, + 120, + 552 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 540, + 460, + 554 + ], + "score": 1.0, + "content": ", which is not implied by the graph. We will now focus our discussion on the edge", + "type": "text" + }, + { + "bbox": [ + 461, + 542, + 504, + 552 + ], + "score": 0.91, + "content": "X _ { 1 } X _ { 3 }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "to show that, despite the independence, the proposed method ENCO can identify the true parent", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 563, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 132, + 575 + ], + "score": 1.0, + "content": "set of", + "type": "text" + }, + { + "bbox": [ + 132, + 563, + 146, + 574 + ], + "score": 0.89, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 563, + 506, + 575 + ], + "score": 1.0, + "content": ". 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With the conditions above, we", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 385, + 134 + ], + "score": 1.0, + "content": "can ensure that the all orientation parameters are learned, excluding", + "type": "text" + }, + { + "bbox": [ + 386, + 122, + 399, + 134 + ], + "score": 0.9, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 122, + 428, + 134 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 428, + 122, + 441, + 132 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 122, + 460, + 134 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 460, + 122, + 473, + 133 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 122, + 505, + 134 + ], + "score": 1.0, + "content": "are the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "variables for which we have not obtained interventions. 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Therefore, we cannot give guarantees of correctness for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 224, + 189 + ], + "score": 1.0, + "content": "incoming/outgoing edges of", + "type": "text" + }, + { + "bbox": [ + 225, + 177, + 237, + 187 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 176, + 256, + 189 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 257, + 177, + 270, + 189 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 176, + 506, + 189 + ], + "score": 1.0, + "content": ", and might make incorrect predictions of edges between", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 187, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 187, + 199 + ], + "score": 1.0, + "content": "these two variables.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 111, + 507, + 199 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 257, + 217 + ], + "score": 1.0, + "content": "When taking the next step to having", + "type": "text" + }, + { + "bbox": [ + 257, + 205, + 269, + 214 + ], + "score": 0.77, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "interventions provided, ENCO can create more incorrect", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "predictions. For the variables for which interventions are provided, we can use the same convergence", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 104, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "guarantees (Theorem B.1-B.4) since all conditions are independent across variables. For variables", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 491, + 249 + ], + "score": 1.0, + "content": "without interventions, we cannot rely on those. While we have observed that learning the missing", + "type": "text" + }, + { + "bbox": [ + 491, + 237, + 497, + 247 + ], + "score": 0.74, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "’s", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 246, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 506, + 261 + ], + "score": 1.0, + "content": "from other interventions give reasonable results, we see a degradation of performance the further the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "distance is between a node and the closest intervened variable. As an example, suppose we have a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 212, + 282 + ], + "score": 1.0, + "content": "chain with 5 variables, i.e.", + "type": "text" + }, + { + "bbox": [ + 212, + 270, + 339, + 281 + ], + "score": 0.91, + "content": "X _ { 1 } X _ { 2 } X _ { 3 } X _ { 4 } X _ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 270, + 506, + 282 + ], + "score": 1.0, + "content": ", and we are provided with an intervention", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 504, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 118, + 294 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 281, + 133, + 291 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 280, + 342, + 294 + ], + "score": 1.0, + "content": "only. This allows us to learn the orientation between", + "type": "text" + }, + { + "bbox": [ + 342, + 281, + 356, + 292 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 280, + 374, + 294 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 374, + 281, + 388, + 292 + ], + "score": 0.87, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 280, + 490, + 294 + ], + "score": 1.0, + "content": ". The orientation between", + "type": "text" + }, + { + "bbox": [ + 490, + 281, + 504, + 292 + ], + "score": 0.85, + "content": "X _ { 2 }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 504, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 123, + 304 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 292, + 138, + 303 + ], + "score": 0.89, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 291, + 375, + 304 + ], + "score": 1.0, + "content": "is often learned correctly as well because adding the edge", + "type": "text" + }, + { + "bbox": [ + 376, + 292, + 417, + 303 + ], + "score": 0.92, + "content": "X _ { 2 } X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 291, + 461, + 304 + ], + "score": 1.0, + "content": "instead of", + "type": "text" + }, + { + "bbox": [ + 462, + 292, + 504, + 303 + ], + "score": 0.91, + "content": "X _ { 3 } X _ { 2 }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 504, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 435, + 316 + ], + "score": 1.0, + "content": "gives a greater decrease in overall log-likelihood, since part of the information from", + "type": "text" + }, + { + "bbox": [ + 436, + 303, + 449, + 314 + ], + "score": 0.89, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 302, + 490, + 316 + ], + "score": 1.0, + "content": "to predict", + "type": "text" + }, + { + "bbox": [ + 490, + 304, + 504, + 314 + ], + "score": 0.88, + "content": "X _ { 2 }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 194, + 326 + ], + "score": 1.0, + "content": "is already included in", + "type": "text" + }, + { + "bbox": [ + 194, + 314, + 208, + 325 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 313, + 368, + 326 + ], + "score": 1.0, + "content": ". However, the further we go away from", + "type": "text" + }, + { + "bbox": [ + 369, + 314, + 382, + 325 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 313, + 506, + 326 + ], + "score": 1.0, + "content": ", the less information is shared", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 325, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 336 + ], + "score": 1.0, + "content": "between the child and the intervened variable. Moreover, the likelihood of a mistake occurring due to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 378, + 348 + ], + "score": 1.0, + "content": "limited data further increases. This is why the orientation of the edge", + "type": "text" + }, + { + "bbox": [ + 378, + 336, + 420, + 347 + ], + "score": 0.93, + "content": "X _ { 4 } X _ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "is not always learned", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 345, + 315, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 315, + 360 + ], + "score": 1.0, + "content": "correctly, which can also cause false positive edges.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 204, + 506, + 360 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 506, + 377 + ], + "score": 1.0, + "content": "Many scenarios of predicting false positive edges can, in theory, be solved by providing an undirected", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "skeleton of the graph, for example, obtained from observational data. Still, one of the cornerstones", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "of ENCO is that it does not assume faithfulness. Without faithfulness or any other assumption on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 409 + ], + "score": 1.0, + "content": "the functional form of the causal mechanisms, the correct undirected graph cannot be recovered", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "by any method. One of the future directions will be to include faithfulness in ENCO to solve the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 419, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 431 + ], + "score": 1.0, + "content": "scenarios mentioned above, although this would imply that we might not be able to recover edges of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 429, + 238, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 238, + 442 + ], + "score": 1.0, + "content": "deterministic variables anymore.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 363, + 506, + 442 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 455, + 298, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 454, + 300, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 300, + 467 + ], + "score": 1.0, + "content": "B.5 EXAMPLE FOR NON-FAITHFUL GRAPHS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 506, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "Below we give an example of a distribution which is not faithful with respect to its graph structure,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 414, + 498 + ], + "score": 1.0, + "content": "but can yet be found by ENCO. Suppose we have a chain of three variables,", + "type": "text" + }, + { + "bbox": [ + 414, + 486, + 484, + 497 + ], + "score": 0.93, + "content": "X _ { 1 } X _ { 2 } X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 486, + 506, + 498 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "simplicity, we assume here that all the variables are binary, but the argument can similarly hold for", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 508, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 265, + 521 + ], + "score": 1.0, + "content": "any categorical data. The distribution", + "type": "text" + }, + { + "bbox": [ + 266, + 508, + 292, + 520 + ], + "score": 0.92, + "content": "p ( X _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 508, + 413, + 521 + ], + "score": 1.0, + "content": "is an arbitrary function with", + "type": "text" + }, + { + "bbox": [ + 413, + 508, + 502, + 520 + ], + "score": 0.89, + "content": "0 < p ( X _ { 1 } \\stackrel { . } { = } 1 ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 508, + 506, + 521 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 343, + 532 + ], + "score": 1.0, + "content": "and the other two conditionals are deterministic functions:", + "type": "text" + }, + { + "bbox": [ + 343, + 519, + 447, + 531 + ], + "score": 0.81, + "content": "p ( X _ { 2 } | X _ { 1 } ) = \\delta [ X _ { 2 } = X _ { 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 519, + 451, + 532 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 451, + 519, + 506, + 531 + ], + "score": 0.86, + "content": "p ( X _ { 3 } | X _ { 2 } ) =", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 156, + 542 + ], + "score": 0.91, + "content": "\\delta [ X _ { 3 } = X _ { 2 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 529, + 380, + 542 + ], + "score": 1.0, + "content": ". This joint distribution is not faithful to the graph, since", + "type": "text" + }, + { + "bbox": [ + 380, + 531, + 394, + 541 + ], + "score": 0.88, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 529, + 466, + 542 + ], + "score": 1.0, + "content": "is independent of", + "type": "text" + }, + { + "bbox": [ + 466, + 531, + 480, + 541 + ], + "score": 0.88, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "given", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 540, + 504, + 554 + ], + "spans": [ + { + "bbox": [ + 107, + 542, + 120, + 552 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 540, + 460, + 554 + ], + "score": 1.0, + "content": ", which is not implied by the graph. We will now focus our discussion on the edge", + "type": "text" + }, + { + "bbox": [ + 461, + 542, + 504, + 552 + ], + "score": 0.91, + "content": "X _ { 1 } X _ { 3 }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "to show that, despite the independence, the proposed method ENCO can identify the true parent", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 563, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 132, + 575 + ], + "score": 1.0, + "content": "set of", + "type": "text" + }, + { + "bbox": [ + 132, + 563, + 146, + 574 + ], + "score": 0.89, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 563, + 506, + 575 + ], + "score": 1.0, + "content": ". 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Note that for", + "type": "text" + }, + { + "bbox": [ + 236, + 144, + 294, + 155 + ], + "score": 0.93, + "content": "p ( X _ { 3 } | X _ { 1 } , X _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 143, + 506, + 156 + ], + "score": 1.0, + "content": ", there are value combinations that have never been", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 162, + 167 + ], + "score": 1.0, + "content": "observed, i.e.", + "type": "text" + }, + { + "bbox": [ + 162, + 154, + 202, + 166 + ], + "score": 0.92, + "content": "X _ { 1 } \\neq X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 153, + 506, + 167 + ], + "score": 1.0, + "content": ", such that in practice we can’t guarantee a specific distribution to be learned", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "for such conditionals. 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In the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 260, + 588 + ], + "score": 1.0, + "content": "graph full, the parents of a variable", + "type": "text" + }, + { + "bbox": [ + 260, + 576, + 273, + 586 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 574, + 380, + 588 + ], + "score": 1.0, + "content": "are all previous variables:", + "type": "text" + }, + { + "bbox": [ + 381, + 575, + 503, + 587 + ], + "score": 0.92, + "content": "\\mathsf { p a } ( X _ { i } ) = \\{ X _ { 1 } , X _ { 2 } , . . . , X _ { i - 1 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 574, + 506, + 588 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 585, + 507, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 507, + 599 + ], + "score": 1.0, + "content": "Hence, it is the densest connected graph possible. The graph jungle represents a binary tree where a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "node is also connected to its parent’s parent. Finally, the graph random follows a randomly sampled", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 323, + 621 + ], + "score": 1.0, + "content": "adjacency matrix. For every possible pair of variables", + "type": "text" + }, + { + "bbox": [ + 323, + 608, + 353, + 620 + ], + "score": 0.92, + "content": "X _ { i } , X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 607, + 506, + 621 + ], + "score": 1.0, + "content": ", we sample an edge with a likelihood", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 442, + 632 + ], + "score": 1.0, + "content": "of 0.3. To determine the orientation of the edges, we assume the causal ordering of", + "type": "text" + }, + { + "bbox": [ + 442, + 619, + 489, + 630 + ], + "score": 0.91, + "content": "X _ { i } \\succ X _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 618, + 506, + 632 + ], + "score": 1.0, + "content": ". 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To model the ground-truth conditional distributions, we use randomly initialized neural", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "networks. Specifically, we use MLPs of two layers where the categorical inputs are represented", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 263, + 700 + ], + "score": 1.0, + "content": "by embedding vectors. For a variable", + "type": "text" + }, + { + "bbox": [ + 263, + 688, + 276, + 699 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 688, + 298, + 700 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 299, + 688, + 311, + 698 + ], + "score": 0.81, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 688, + 401, + 700 + ], + "score": 1.0, + "content": "parents, we stack the", + "type": "text" + }, + { + "bbox": [ + 401, + 688, + 413, + 698 + ], + "score": 0.76, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "embedding vectors to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "form the input to the following MLPs. Each embedding has a dimensionality of 4, hence the input", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 228, + 722 + ], + "score": 1.0, + "content": "size to the first linear layer is", + "type": "text" + }, + { + "bbox": [ + 228, + 710, + 245, + 720 + ], + "score": 0.73, + "content": "4 M", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ". The hidden size of the layers is 48, and we use a LeakyReLU", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "activation function in between the two linear layers. 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HyperparametersSDIENCO
Sparsity regularizer Xsparse{0.01, 0.02, 0.05, 0.1, 0.2}{0.002,0.004, 0.01}
DAG regularizer{0.2, 0.5, 1.0, 2.0, 5.0}
Distribution model2 layers, hidden size 64,LeakyReLU(α = 0.1)
Batch size128
Learning rate - model{2e-3,5e-3,2e-2,5e-2}
Weight decay - model{le-5, le-4, 1e-5}
Distribution fitting iterations F1000
Graph fiting iterations G100
Graph samples K100
Epochs5030
Learning rate - γ Learning rate - 0{5e-3,2e-2, 5e-2} ={5e-3,2e-2, 5e-2} {5e-3,2e-2, 5e-2,1e-1}
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The MLP and hyperparameters have been", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "chosen based on the design of networks used in ENCO, SDI (Ke et al., 2019) and DCDI (Brouillard", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "et al., 2020). For the initialization of the networks, we follow Ke et al. (2019) and use the orthogonal", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 427, + 352 + ], + "score": 1.0, + "content": "initialize with a gain of 2.5. The biases are thereby initialized uniformly between", + "type": "text" + }, + { + "bbox": [ + 427, + 341, + 449, + 352 + ], + "score": 0.77, + "content": "- 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "and 0.5. This", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "way, we obtain non-trivial, random distributions. Experiments with different synthetic distributions", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 363, + 231, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 231, + 375 + ], + "score": 1.0, + "content": "are provided in Appendix D.7.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 108, + 388, + 294, + 399 + ], + "lines": [ + { + "bbox": [ + 106, + 388, + 296, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 296, + 401 + ], + "score": 1.0, + "content": "C.1.2 METHODS AND HYPERPARAMETERS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 408, + 506, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "score": 1.0, + "content": "Baseline implementation We used existing implementations to run the baselines GIES (Hauser &", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "Bühlmann, 2012), IGSP (Wang et al., 2017), GES (Chickering, 2002) and DCDI (Brouillard et al.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 371, + 443 + ], + "score": 1.0, + "content": "2020). For GIES, we used the implementation from the R package", + "type": "text" + }, + { + "bbox": [ + 371, + 430, + 406, + 442 + ], + "score": 0.43, + "content": "\\mathtt { p c a l g } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 429, + 506, + 443 + ], + "score": 1.0, + "content": ". To run categorical data,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "we used the GaussL0penIntScore score function. For IGSP, we used the implementation of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "score": 1.0, + "content": "python package causaldag3. As IGSP uses conditional independence tests in its score function,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "score": 1.0, + "content": "we cast the categorical data into continuous space first and experiment with different kernel-based", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "independence tests. Due to its long runtime for large dataset sizes, we limit the interventional and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 485, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 225, + 497 + ], + "score": 1.0, + "content": "observational data set size to", + "type": "text" + }, + { + "bbox": [ + 225, + 486, + 242, + 496 + ], + "score": 0.49, + "content": "2 5 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 485, + 506, + 497 + ], + "score": 1.0, + "content": ". Larger dataset sizes did not show any significant improvements.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "For details on the observational GES experiments, see Section D.6. Finally, we have used the original", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "python code for DCDI published by the authors4. We have added the same neural networks used by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "ENCO into the framework to perform structure learning on categorical data. Bugs in the original", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "score": 1.0, + "content": "code were corrected to the best of our knowledge. 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HyperparametersSDIENCO
Sparsity regularizer Xsparse{0.01, 0.02, 0.05, 0.1, 0.2}{0.002,0.004, 0.01}
DAG regularizer{0.2, 0.5, 1.0, 2.0, 5.0}
Distribution model2 layers, hidden size 64,LeakyReLU(α = 0.1)
Batch size128
Learning rate - model{2e-3,5e-3,2e-2,5e-2}
Weight decay - model{le-5, le-4, 1e-5}
Distribution fitting iterations F1000
Graph fiting iterations G100
Graph samples K100
Epochs5030
Learning rate - γ Learning rate - 0{5e-3,2e-2, 5e-2} ={5e-3,2e-2, 5e-2} {5e-3,2e-2, 5e-2,1e-1}
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For GIES, we used the implementation from the R package", + "type": "text" + }, + { + "bbox": [ + 371, + 430, + 406, + 442 + ], + "score": 0.43, + "content": "\\mathtt { p c a l g } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 429, + 506, + 443 + ], + "score": 1.0, + "content": ". To run categorical data,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "we used the GaussL0penIntScore score function. For IGSP, we used the implementation of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "score": 1.0, + "content": "python package causaldag3. 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Due to its long runtime for large dataset sizes, we limit the interventional and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 485, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 225, + 497 + ], + "score": 1.0, + "content": "observational data set size to", + "type": "text" + }, + { + "bbox": [ + 225, + 486, + 242, + 496 + ], + "score": 0.49, + "content": "2 5 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 485, + 506, + 497 + ], + "score": 1.0, + "content": ". Larger dataset sizes did not show any significant improvements.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "For details on the observational GES experiments, see Section D.6. Finally, we have used the original", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "python code for DCDI published by the authors4. We have added the same neural networks used by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "ENCO into the framework to perform structure learning on categorical data. Bugs in the original", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "score": 1.0, + "content": "code were corrected to the best of our knowledge. 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The average run time of IGSP was 13mins.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 426, + 505, + 569 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 406, + 439 + ], + "score": 1.0, + "content": "SDI We focused the hyperparameter search for SDI on its two regularizers,", + "type": "text" + }, + { + "bbox": [ + 406, + 426, + 432, + 438 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 425, + 449, + 439 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 449, + 426, + 471, + 437 + ], + "score": 0.86, + "content": "\\lambda _ { \\mathrm { D A G } }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 425, + 505, + 439 + ], + "score": 1.0, + "content": ", as well", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 436, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 141, + 436, + 228, + 449 + ], + "score": 1.0, + "content": "as its learning rate for", + "type": "text" + }, + { + "bbox": [ + 228, + 439, + 235, + 448 + ], + "score": 0.8, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 436, + 506, + 449 + ], + "score": 1.0, + "content": ". The other hyperparameters with respect to the neural networks were", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 141, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "kept the same as ENCO for a fair comparison. We show all details of the hyperparameter", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 140, + 457, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 140, + 457, + 425, + 473 + ], + "score": 1.0, + "content": "search in Table 4. The best combination of regularizers found was", + "type": "text" + }, + { + "bbox": [ + 426, + 459, + 486, + 471 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 457, + 506, + 473 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 142, + 470, + 195, + 481 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { D A G } } ~ = ~ 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 469, + 273, + 482 + ], + "score": 1.0, + "content": ". Lower values of", + "type": "text" + }, + { + "bbox": [ + 273, + 470, + 298, + 481 + ], + "score": 0.88, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "lead to more false positives, especially in sparse", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 141, + 480, + 270, + 495 + ], + "score": 1.0, + "content": "graphs, while a lower value of", + "type": "text" + }, + { + "bbox": [ + 271, + 482, + 293, + 492 + ], + "score": 0.88, + "content": "\\lambda _ { \\mathrm { D A G } }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 480, + 506, + 495 + ], + "score": 1.0, + "content": "caused many two-variable loops. Compared to the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 140, + 490, + 507, + 506 + ], + "spans": [ + { + "bbox": [ + 140, + 490, + 322, + 506 + ], + "score": 1.0, + "content": "reported hyperparameter by Ke et al. (2019)", + "type": "text" + }, + { + "bbox": [ + 323, + 492, + 372, + 503 + ], + "score": 0.49, + "content": "( \\lambda _ { \\mathrm { D A G } } = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 490, + 375, + 506 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 375, + 492, + 427, + 504 + ], + "score": 0.62, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 1 ", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 490, + 507, + 506 + ], + "score": 1.0, + "content": "), we found a lower", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 140, + 502, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 140, + 502, + 506, + 516 + ], + "score": 1.0, + "content": "sparsity regularizer to work better. This is likely because of testing SDI on larger graphs.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 513, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 141, + 513, + 388, + 526 + ], + "score": 1.0, + "content": "In contrast to ENCO, SDI needed a lower learning rate for", + "type": "text" + }, + { + "bbox": [ + 388, + 515, + 396, + 525 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 513, + 506, + 526 + ], + "score": 1.0, + "content": "due to its higher variance", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 525, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 142, + 525, + 506, + 536 + ], + "score": 1.0, + "content": "gradient estimators. To compensate for it, we ran it for 50 instead of 30 epochs. In general,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 535, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 142, + 535, + 505, + 548 + ], + "score": 1.0, + "content": "SDI achieved lower scores than in the original experiments by Ke et al. (2019) which was", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 140, + 545, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 140, + 545, + 506, + 560 + ], + "score": 1.0, + "content": "because of the larger graph size and smaller dataset size. The average run time of SDI was", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 556, + 212, + 571 + ], + "spans": [ + { + "bbox": [ + 141, + 556, + 212, + 571 + ], + "score": 1.0, + "content": "4mins per graph.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 573, + 506, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 573, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 506, + 585 + ], + "score": 1.0, + "content": "DCDI The most crucial hyperparameter in DCDI was the initialization of the constraint factor", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 142, + 586, + 154, + 596 + ], + "score": 0.8, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 584, + 294, + 596 + ], + "score": 1.0, + "content": ". We experimented with a range of", + "type": "text" + }, + { + "bbox": [ + 294, + 584, + 345, + 596 + ], + "score": 0.82, + "content": "\\mu _ { 0 } \\in \\{ 1 \\mathrm { e } \\mathrm { - } 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 584, + 450, + 596 + ], + "score": 1.0, + "content": ", 1e-9, 1e-8, 1e-7, 1e-6, 1e-", + "type": "text" + }, + { + "bbox": [ + 450, + 585, + 461, + 596 + ], + "score": 0.67, + "content": "\\cdot 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "and found", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 595, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 142, + 596, + 185, + 606 + ], + "score": 0.89, + "content": "\\mu _ { 0 } = 1 \\mathrm { e } { - 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 595, + 506, + 607 + ], + "score": 1.0, + "content": "to work best. This is close to the reported value of 1e-8 by Brouillard et al. (2020).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 605, + 507, + 619 + ], + "spans": [ + { + "bbox": [ + 141, + 605, + 507, + 619 + ], + "score": 1.0, + "content": "Higher values lead to empty graphs, while lower values slowed down the optimization. Ad-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 616, + 504, + 630 + ], + "spans": [ + { + "bbox": [ + 141, + 616, + 373, + 630 + ], + "score": 1.0, + "content": "ditionally, we search over the regularizer hyperparameter", + "type": "text" + }, + { + "bbox": [ + 373, + 617, + 504, + 629 + ], + "score": 0.64, + "content": "\\lambda \\in \\{ 1 \\mathrm { e } \\mathrm { - } 3 , 1 \\mathrm { e } \\mathrm { - } 2 , 1 \\mathrm { e } \\mathrm { - } 1 , 1 . 0 , 1 0 . 0 \\}", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 141, + 627, + 208, + 640 + ], + "score": 1.0, + "content": "where we found", + "type": "text" + }, + { + "bbox": [ + 208, + 628, + 241, + 639 + ], + "score": 0.89, + "content": "\\lambda = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 627, + 506, + 640 + ], + "score": 1.0, + "content": ", which is the same value used by Brouillard et al. (2020). We stop", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 637, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 141, + 637, + 506, + 652 + ], + "score": 1.0, + "content": "the search after the Lagrangian constraint is below 1e-7, following Brouillard et al. (2020),", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 650, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 141, + 650, + 153, + 662 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 153, + 650, + 170, + 660 + ], + "score": 0.6, + "content": "5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 650, + 506, + 662 + ], + "score": 1.0, + "content": "iterations have been used which was sufficient to converge on all graphs. We have", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 142, + 661, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 142, + 661, + 505, + 673 + ], + "score": 1.0, + "content": "experimented with using weight decay to prevent overfitting, but did not find it to provide", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 672, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 141, + 672, + 506, + 684 + ], + "score": 1.0, + "content": "any gain in terms of structure learning performance. The average run time of DCDI was 16", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 684, + 179, + 694 + ], + "spans": [ + { + "bbox": [ + 141, + 684, + 179, + 694 + ], + "score": 1.0, + "content": "minutes.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "ENCO We outline the hyperparameters of ENCO in Table 4. As discussed in Section 3.4, the most", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 141, + 709, + 390, + 723 + ], + "score": 1.0, + "content": "crucial hyperparameter in ENCO is the sparsity regularizer", + "type": "text" + }, + { + "bbox": [ + 390, + 710, + 415, + 722 + ], + "score": 0.88, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 709, + 506, + 723 + ], + "score": 1.0, + "content": ". The larger it is, the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 141, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "faster ENCO converges, but at the same time might miss edges in the ground-truth graph.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + } + ], + "page_idx": 36, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 85, + 502, + 296 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 85, + 502, + 296 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 85, + 502, + 296 + ], + "spans": [ + { + "bbox": [ + 112, + 85, + 502, + 296 + ], + "score": 0.974, + "type": "image", + "image_path": "95507c42df0294de5201d6a7c23df27c78000536edbb1a4ce2aeeb61dbb828a2.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 85, + 502, + 155.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 155.33333333333331, + 502, + 225.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 225.66666666666663, + 502, + 295.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 310, + 505, + 365 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 310, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 323 + ], + "score": 1.0, + "content": "Figure 11: Learning curves of ENCO in terms of recall and precision on edge predictions for synthetic", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "graph structures. The orientations for the ground-truth edges are not plotted as they have usually been", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "score": 1.0, + "content": "correctly learned after the first epoch except for the graph full. Overall, we see that the edge recall", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 342, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 104, + 342, + 506, + 357 + ], + "score": 1.0, + "content": "starts very high for all graphs, and the precision catches up over the epochs. This is in line the steps", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 354, + 261, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 261, + 366 + ], + "score": 1.0, + "content": "in the convergence proof in Section B.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 106, + 388, + 504, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 388, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 399 + ], + "score": 1.0, + "content": "IGSP We experimented with two different conditional independence tests, kci and hsic, and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 235, + 411 + ], + "score": 1.0, + "content": "different cutoff values", + "type": "text" + }, + { + "bbox": [ + 235, + 399, + 349, + 411 + ], + "score": 0.71, + "content": "\\alpha = \\{ 1 \\mathrm { e } \\mathrm { - } 5 , 1 \\mathrm { e } \\mathrm { - } 4 , 1 \\mathrm { e } \\mathrm { - } 3 , 1 \\mathrm { e } \\mathrm { - } 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 398, + 505, + 411 + ], + "score": 1.0, + "content": ". The best hyperparameter setting was", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 409, + 400, + 423 + ], + "spans": [ + { + "bbox": [ + 141, + 409, + 183, + 423 + ], + "score": 1.0, + "content": "kci with", + "type": "text" + }, + { + "bbox": [ + 183, + 411, + 222, + 420 + ], + "score": 0.87, + "content": "\\alpha = 1 0 \\mathrm { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 409, + 400, + 423 + ], + "score": 1.0, + "content": ". The average run time of IGSP was 13mins.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 388, + 505, + 423 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 426, + 505, + 569 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 406, + 439 + ], + "score": 1.0, + "content": "SDI We focused the hyperparameter search for SDI on its two regularizers,", + "type": "text" + }, + { + "bbox": [ + 406, + 426, + 432, + 438 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 425, + 449, + 439 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 449, + 426, + 471, + 437 + ], + "score": 0.86, + "content": "\\lambda _ { \\mathrm { D A G } }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 425, + 505, + 439 + ], + "score": 1.0, + "content": ", as well", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 436, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 141, + 436, + 228, + 449 + ], + "score": 1.0, + "content": "as its learning rate for", + "type": "text" + }, + { + "bbox": [ + 228, + 439, + 235, + 448 + ], + "score": 0.8, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 436, + 506, + 449 + ], + "score": 1.0, + "content": ". The other hyperparameters with respect to the neural networks were", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 141, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "kept the same as ENCO for a fair comparison. We show all details of the hyperparameter", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 140, + 457, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 140, + 457, + 425, + 473 + ], + "score": 1.0, + "content": "search in Table 4. The best combination of regularizers found was", + "type": "text" + }, + { + "bbox": [ + 426, + 459, + 486, + 471 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 457, + 506, + 473 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 142, + 470, + 195, + 481 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { D A G } } ~ = ~ 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 469, + 273, + 482 + ], + "score": 1.0, + "content": ". Lower values of", + "type": "text" + }, + { + "bbox": [ + 273, + 470, + 298, + 481 + ], + "score": 0.88, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "lead to more false positives, especially in sparse", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 141, + 480, + 270, + 495 + ], + "score": 1.0, + "content": "graphs, while a lower value of", + "type": "text" + }, + { + "bbox": [ + 271, + 482, + 293, + 492 + ], + "score": 0.88, + "content": "\\lambda _ { \\mathrm { D A G } }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 480, + 506, + 495 + ], + "score": 1.0, + "content": "caused many two-variable loops. Compared to the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 140, + 490, + 507, + 506 + ], + "spans": [ + { + "bbox": [ + 140, + 490, + 322, + 506 + ], + "score": 1.0, + "content": "reported hyperparameter by Ke et al. (2019)", + "type": "text" + }, + { + "bbox": [ + 323, + 492, + 372, + 503 + ], + "score": 0.49, + "content": "( \\lambda _ { \\mathrm { D A G } } = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 490, + 375, + 506 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 375, + 492, + 427, + 504 + ], + "score": 0.62, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 1 ", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 490, + 507, + 506 + ], + "score": 1.0, + "content": "), we found a lower", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 140, + 502, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 140, + 502, + 506, + 516 + ], + "score": 1.0, + "content": "sparsity regularizer to work better. This is likely because of testing SDI on larger graphs.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 513, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 141, + 513, + 388, + 526 + ], + "score": 1.0, + "content": "In contrast to ENCO, SDI needed a lower learning rate for", + "type": "text" + }, + { + "bbox": [ + 388, + 515, + 396, + 525 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 513, + 506, + 526 + ], + "score": 1.0, + "content": "due to its higher variance", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 525, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 142, + 525, + 506, + 536 + ], + "score": 1.0, + "content": "gradient estimators. To compensate for it, we ran it for 50 instead of 30 epochs. In general,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 535, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 142, + 535, + 505, + 548 + ], + "score": 1.0, + "content": "SDI achieved lower scores than in the original experiments by Ke et al. (2019) which was", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 140, + 545, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 140, + 545, + 506, + 560 + ], + "score": 1.0, + "content": "because of the larger graph size and smaller dataset size. The average run time of SDI was", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 556, + 212, + 571 + ], + "spans": [ + { + "bbox": [ + 141, + 556, + 212, + 571 + ], + "score": 1.0, + "content": "4mins per graph.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 425, + 507, + 571 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 573, + 506, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 573, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 506, + 585 + ], + "score": 1.0, + "content": "DCDI The most crucial hyperparameter in DCDI was the initialization of the constraint factor", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 142, + 586, + 154, + 596 + ], + "score": 0.8, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 584, + 294, + 596 + ], + "score": 1.0, + "content": ". We experimented with a range of", + "type": "text" + }, + { + "bbox": [ + 294, + 584, + 345, + 596 + ], + "score": 0.82, + "content": "\\mu _ { 0 } \\in \\{ 1 \\mathrm { e } \\mathrm { - } 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 584, + 450, + 596 + ], + "score": 1.0, + "content": ", 1e-9, 1e-8, 1e-7, 1e-6, 1e-", + "type": "text" + }, + { + "bbox": [ + 450, + 585, + 461, + 596 + ], + "score": 0.67, + "content": "\\cdot 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "and found", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 595, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 142, + 596, + 185, + 606 + ], + "score": 0.89, + "content": "\\mu _ { 0 } = 1 \\mathrm { e } { - 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 595, + 506, + 607 + ], + "score": 1.0, + "content": "to work best. This is close to the reported value of 1e-8 by Brouillard et al. (2020).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 605, + 507, + 619 + ], + "spans": [ + { + "bbox": [ + 141, + 605, + 507, + 619 + ], + "score": 1.0, + "content": "Higher values lead to empty graphs, while lower values slowed down the optimization. Ad-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 616, + 504, + 630 + ], + "spans": [ + { + "bbox": [ + 141, + 616, + 373, + 630 + ], + "score": 1.0, + "content": "ditionally, we search over the regularizer hyperparameter", + "type": "text" + }, + { + "bbox": [ + 373, + 617, + 504, + 629 + ], + "score": 0.64, + "content": "\\lambda \\in \\{ 1 \\mathrm { e } \\mathrm { - } 3 , 1 \\mathrm { e } \\mathrm { - } 2 , 1 \\mathrm { e } \\mathrm { - } 1 , 1 . 0 , 1 0 . 0 \\}", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 141, + 627, + 208, + 640 + ], + "score": 1.0, + "content": "where we found", + "type": "text" + }, + { + "bbox": [ + 208, + 628, + 241, + 639 + ], + "score": 0.89, + "content": "\\lambda = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 627, + 506, + 640 + ], + "score": 1.0, + "content": ", which is the same value used by Brouillard et al. (2020). We stop", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 637, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 141, + 637, + 506, + 652 + ], + "score": 1.0, + "content": "the search after the Lagrangian constraint is below 1e-7, following Brouillard et al. (2020),", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 650, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 141, + 650, + 153, + 662 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 153, + 650, + 170, + 660 + ], + "score": 0.6, + "content": "5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 650, + 506, + 662 + ], + "score": 1.0, + "content": "iterations have been used which was sufficient to converge on all graphs. We have", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 142, + 661, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 142, + 661, + 505, + 673 + ], + "score": 1.0, + "content": "experimented with using weight decay to prevent overfitting, but did not find it to provide", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 672, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 141, + 672, + 506, + 684 + ], + "score": 1.0, + "content": "any gain in terms of structure learning performance. The average run time of DCDI was 16", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 684, + 179, + 694 + ], + "spans": [ + { + "bbox": [ + 141, + 684, + 179, + 694 + ], + "score": 1.0, + "content": "minutes.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 573, + 507, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "ENCO We outline the hyperparameters of ENCO in Table 4. As discussed in Section 3.4, the most", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 141, + 709, + 390, + 723 + ], + "score": 1.0, + "content": "crucial hyperparameter in ENCO is the sparsity regularizer", + "type": "text" + }, + { + "bbox": [ + 390, + 710, + 415, + 722 + ], + "score": 0.88, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 709, + 506, + 723 + ], + "score": 1.0, + "content": ". The larger it is, the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 141, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "faster ENCO converges, but at the same time might miss edges in the ground-truth graph.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 106, + 698, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 122, + 504, + 196 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 114 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "Table 5: Extension of Table 1 with the metric structural intervention distance (SID) (lower is", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 505, + 104 + ], + "score": 1.0, + "content": "better), averaged over 25 graphs each. The conclusion is the same as for SHD, namely that ENCO", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 103, + 429, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 429, + 115 + ], + "score": 1.0, + "content": "outperforms all baselines, while the acyclic heuristic has an even greater impact.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 108, + 122, + 504, + 196 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 122, + 504, + 196 + ], + "spans": [ + { + "bbox": [ + 108, + 122, + 504, + 196 + ], + "score": 0.981, + "html": "
Graph typebidiagchaincolliderfull junglerandom
GIES460.0 (±60.1)224.2(±87.3)83.6 (±143.5)527.9 (±35.7)441.4 (±26.1)471.9 (±33.7)
IGSP423.3(±48.2)240.1(±78.8)120.7(±51.4)554.8(±26.4)394.8(±73.5)524.0(±18.8)
SDI243.7(±55.1)70.2 (±46.8)24.0 (±0.0)537.1(±29.2)180.0(±56.4)317.9 (±62.7)
DCDI369.3 (±47.5)233.4(±24.8)10.9 (±3.6)183.6 (±54.5)339.4(±59.1)158.6 (±69.1)
ENCO (ours)28.2(±18.8)19.9 (±17.9)7.4 (±13.3)63.2 (±22.6)16.3 (±9.8)77.6 (±27.2)
ENCO-acyclic (ours)0.0(±0.0)0.0 (±0.0)7.4(±13.3)17.7(±8.9)4.6 (±8.1)5.3 (±11.8)
", + "type": "table", + "image_path": "cb36f83c6d4254f3321906edcf9b407115964ca711da30f079b4ad779ff5bb55.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 108, + 122, + 504, + 146.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 108, + 146.66666666666666, + 504, + 171.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 108, + 171.33333333333331, + 504, + 195.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 142, + 213, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 141, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 141, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "Lower values allow the detection of more edges for the price of longer training times. We", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 140, + 224, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 140, + 224, + 505, + 237 + ], + "score": 1.0, + "content": "have found that for the graph structures given, only the graph full was sensitive to the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 140, + 235, + 507, + 249 + ], + "spans": [ + { + "bbox": [ + 140, + 235, + 178, + 249 + ], + "score": 1.0, + "content": "value of", + "type": "text" + }, + { + "bbox": [ + 178, + 236, + 203, + 248 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 235, + 232, + 249 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 232, + 236, + 294, + 247 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 235, + 313, + 249 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 313, + 236, + 375, + 248 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 0 4", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 235, + 507, + 249 + ], + "score": 1.0, + "content": "performed almost equally well.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 142, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "In comparison to SDI, ENCO can make use of larger learning rates due to lower variance", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 258, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 258, + 280, + 270 + ], + "score": 1.0, + "content": "gradient estimators. Especially for", + "type": "text" + }, + { + "bbox": [ + 280, + 258, + 287, + 268 + ], + "score": 0.57, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 258, + 506, + 270 + ], + "score": 1.0, + "content": ", we have noticed that high learning rates are beneficial.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "This is in line with our theoretical guarantees which require the orientation parameters", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 280, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 280, + 351, + 291 + ], + "score": 1.0, + "content": "to converge first. We use the Adam optimizer for", + "type": "text" + }, + { + "bbox": [ + 351, + 281, + 360, + 291 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 280, + 379, + 291 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 379, + 280, + 387, + 290 + ], + "score": 0.72, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 280, + 425, + 291 + ], + "score": 1.0, + "content": "with the", + "type": "text" + }, + { + "bbox": [ + 425, + 280, + 433, + 291 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 280, + 505, + 291 + ], + "score": 1.0, + "content": "-hyperparameters", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 334, + 304 + ], + "score": 1.0, + "content": "(0.9, 0.9) and (0.9, 0.999) respectively. A lower", + "type": "text" + }, + { + "bbox": [ + 334, + 291, + 345, + 302 + ], + "score": 0.87, + "content": "\\beta _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 290, + 424, + 304 + ], + "score": 1.0, + "content": "hyperparameter for", + "type": "text" + }, + { + "bbox": [ + 424, + 292, + 432, + 302 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "allows the second", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "momentum to adapt faster to a change of gradient scale which happens for initial false", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 313, + 226, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 313, + 226, + 325 + ], + "score": 1.0, + "content": "positive predictions.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 143, + 325, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 141, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 141, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "The average run time of ENCO was 2mins per graph. The algorithm could be sped up even", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 141, + 336, + 337, + 349 + ], + "score": 1.0, + "content": "more by reducing the number of graph samples", + "type": "text" + }, + { + "bbox": [ + 338, + 336, + 348, + 346 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "and model fitting iterations. However,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 142, + 348, + 287, + 359 + ], + "score": 1.0, + "content": "for graphs of larger than 100 nodes,", + "type": "text" + }, + { + "bbox": [ + 288, + 347, + 326, + 358 + ], + "score": 0.9, + "content": "K = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 348, + 505, + 359 + ], + "score": 1.0, + "content": "and longer model fitting times showed to be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 358, + 491, + 371 + ], + "spans": [ + { + "bbox": [ + 141, + 358, + 491, + 371 + ], + "score": 1.0, + "content": "beneficial. The learning curves in terms of recall and precision are shown in Figure 10.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 394 + ], + "score": 1.0, + "content": "Enforcing acyclicity ENCO is guaranteed to converge to acyclic graphs in the data limit; arguably,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "an assumption that does not always hold. In the presence of cycles, which can occur especially", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "when low data is available, a simple heuristic is to keep the graph, which maximizes the orientation", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 320, + 427 + ], + "score": 1.0, + "content": "probabilities. Specifically, we aim to find the order", + "type": "text" + }, + { + "bbox": [ + 321, + 414, + 357, + 425 + ], + "score": 0.92, + "content": "O \\in S _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 414, + 390, + 427 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 390, + 414, + 405, + 425 + ], + "score": 0.9, + "content": "S _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 414, + 505, + 427 + ], + "score": 1.0, + "content": "represents the set of all", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 297, + 438 + ], + "score": 1.0, + "content": "permutations from 1 to the number of variables", + "type": "text" + }, + { + "bbox": [ + 297, + 425, + 307, + 435 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 425, + 505, + 438 + ], + "score": 1.0, + "content": ", for which we maximize the following objective:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 438, + 381, + 478 + ], + "lines": [ + { + "bbox": [ + 229, + 438, + 381, + 478 + ], + "spans": [ + { + "bbox": [ + 229, + 438, + 381, + 478 + ], + "score": 0.95, + "content": "\\hat { O } = \\arg \\operatorname* { m a x } _ { O } \\left[ \\prod _ { i = 1 } ^ { N } \\prod _ { j = i + 1 } ^ { N } \\sigma ( \\theta _ { O _ { i } , O _ { j } } ) \\right]", + "type": "interline_equation", + "image_path": "aa45076b07217f5bf8dbf708c25f22a1d49771f55716e82a4114e688f86d1e7b.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 229, + 438, + 381, + 458.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 229, + 458.0, + 381, + 478.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "For small cycles it is easy to do this exhaustively by checking all permutations. For larger cycles, we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 489, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 382, + 505 + ], + "score": 1.0, + "content": "apply a simple greedy search that works just as well. Once the order", + "type": "text" + }, + { + "bbox": [ + 383, + 489, + 392, + 502 + ], + "score": 0.85, + "content": "\\hat { O }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 490, + 505, + 505 + ], + "score": 1.0, + "content": "has been found, we remove", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 502, + 479, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 144, + 517 + ], + "score": 1.0, + "content": "all edges", + "type": "text" + }, + { + "bbox": [ + 144, + 504, + 185, + 516 + ], + "score": 0.92, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 503, + 213, + 517 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 213, + 505, + 218, + 514 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 503, + 267, + 517 + ], + "score": 1.0, + "content": "comes after", + "type": "text" + }, + { + "bbox": [ + 268, + 505, + 273, + 516 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 503, + 285, + 517 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 285, + 502, + 294, + 514 + ], + "score": 0.82, + "content": "\\hat { O }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 503, + 479, + 517 + ], + "score": 1.0, + "content": ". This guarantees to result in an acyclic graph.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 534 + ], + "score": 1.0, + "content": "The intuition behind this heuristic is the following. Cycles are often caused by a single orientation", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 447, + 545 + ], + "score": 1.0, + "content": "pair being incorrect due to noise in the interventional data. For example, in a chain", + "type": "text" + }, + { + "bbox": [ + 448, + 532, + 505, + 543 + ], + "score": 0.91, + "content": "X _ { 1 } X _ { 2 } ", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 107, + 543, + 177, + 554 + ], + "score": 0.92, + "content": "X _ { 3 } X _ { 4 } X _ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 542, + 350, + 555 + ], + "score": 1.0, + "content": ", it can happen that the orientation parameter", + "type": "text" + }, + { + "bbox": [ + 351, + 543, + 365, + 554 + ], + "score": 0.9, + "content": "\\theta _ { 1 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "is incorrectly learned as orientation", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 554, + 504, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 191, + 565 + ], + "score": 1.0, + "content": "of the edge between", + "type": "text" + }, + { + "bbox": [ + 192, + 554, + 206, + 565 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 554, + 225, + 565 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 225, + 554, + 239, + 565 + ], + "score": 0.89, + "content": "X _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 554, + 252, + 565 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 252, + 554, + 295, + 565 + ], + "score": 0.93, + "content": "X _ { 4 } X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 554, + 413, + 565 + ], + "score": 1.0, + "content": "if the interventional data on", + "type": "text" + }, + { + "bbox": [ + 413, + 555, + 427, + 565 + ], + "score": 0.89, + "content": "X _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 554, + 504, + 565 + ], + "score": 1.0, + "content": "does not show the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 173, + 578 + ], + "score": 1.0, + "content": "independence of", + "type": "text" + }, + { + "bbox": [ + 173, + 565, + 187, + 576 + ], + "score": 0.9, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 564, + 204, + 578 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 205, + 565, + 218, + 576 + ], + "score": 0.89, + "content": "X _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 564, + 415, + 578 + ], + "score": 1.0, + "content": ". 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Thus, it is easy to spot that", + "type": "text" + }, + { + "bbox": [ + 357, + 576, + 371, + 587 + ], + "score": 0.9, + "content": "\\theta _ { 1 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "is an outlier, and this is what the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 587, + 250, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 250, + 600 + ], + "score": 1.0, + "content": "simple heuristic above implements.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "Results Besides the structural hamming distance, a common alternative metric is structural inter-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "score": 1.0, + "content": "vention distance (SID) (Peters & Bühlmann, 2015). In contrast to SHD, SID quantifies the closeness", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "between two DAGs in terms of their corresponding causal inference statements. Hence, it is suited for", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "comparing causal discovery methods. The results of the experiments on the synthetic graphs in terms", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 651, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 104, + 651, + 506, + 668 + ], + "score": 1.0, + "content": "of SID are shown in Table 5, and show a similar trend as before, namely that ENCO is outperforming", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 663, + 160, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 160, + 677 + ], + "score": 1.0, + "content": "all baselines.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5 + }, + { + "type": "title", + "bbox": [ + 108, + 689, + 255, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 256, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 256, + 701 + ], + "score": 1.0, + "content": "C.2 SCALABILITY EXPERIMENTS", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Graph generation For generating the graphs, we use the same strategy as for the graph random", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 412, + 733 + ], + "score": 1.0, + "content": "in the previous experiments. The probability of sampling an edge is set to", + "type": "text" + }, + { + "bbox": [ + 412, + 720, + 432, + 732 + ], + "score": 0.87, + "content": "8 / N", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 720, + 505, + 733 + ], + "score": 1.0, + "content": ", meaning that on", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5 + } + ], + "page_idx": 37, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "38", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 122, + 504, + 196 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 114 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "Table 5: Extension of Table 1 with the metric structural intervention distance (SID) (lower is", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 505, + 104 + ], + "score": 1.0, + "content": "better), averaged over 25 graphs each. The conclusion is the same as for SHD, namely that ENCO", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 103, + 429, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 429, + 115 + ], + "score": 1.0, + "content": "outperforms all baselines, while the acyclic heuristic has an even greater impact.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 108, + 122, + 504, + 196 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 122, + 504, + 196 + ], + "spans": [ + { + "bbox": [ + 108, + 122, + 504, + 196 + ], + "score": 0.981, + "html": "
Graph typebidiagchaincolliderfull junglerandom
GIES460.0 (±60.1)224.2(±87.3)83.6 (±143.5)527.9 (±35.7)441.4 (±26.1)471.9 (±33.7)
IGSP423.3(±48.2)240.1(±78.8)120.7(±51.4)554.8(±26.4)394.8(±73.5)524.0(±18.8)
SDI243.7(±55.1)70.2 (±46.8)24.0 (±0.0)537.1(±29.2)180.0(±56.4)317.9 (±62.7)
DCDI369.3 (±47.5)233.4(±24.8)10.9 (±3.6)183.6 (±54.5)339.4(±59.1)158.6 (±69.1)
ENCO (ours)28.2(±18.8)19.9 (±17.9)7.4 (±13.3)63.2 (±22.6)16.3 (±9.8)77.6 (±27.2)
ENCO-acyclic (ours)0.0(±0.0)0.0 (±0.0)7.4(±13.3)17.7(±8.9)4.6 (±8.1)5.3 (±11.8)
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We", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 140, + 224, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 140, + 224, + 505, + 237 + ], + "score": 1.0, + "content": "have found that for the graph structures given, only the graph full was sensitive to the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 140, + 235, + 507, + 249 + ], + "spans": [ + { + "bbox": [ + 140, + 235, + 178, + 249 + ], + "score": 1.0, + "content": "value of", + "type": "text" + }, + { + "bbox": [ + 178, + 236, + 203, + 248 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 235, + 232, + 249 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 232, + 236, + 294, + 247 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 235, + 313, + 249 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 313, + 236, + 375, + 248 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 0 4", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 235, + 507, + 249 + ], + "score": 1.0, + "content": "performed almost equally well.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 142, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "In comparison to SDI, ENCO can make use of larger learning rates due to lower variance", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 258, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 258, + 280, + 270 + ], + "score": 1.0, + "content": "gradient estimators. Especially for", + "type": "text" + }, + { + "bbox": [ + 280, + 258, + 287, + 268 + ], + "score": 0.57, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 258, + 506, + 270 + ], + "score": 1.0, + "content": ", we have noticed that high learning rates are beneficial.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "This is in line with our theoretical guarantees which require the orientation parameters", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 280, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 280, + 351, + 291 + ], + "score": 1.0, + "content": "to converge first. We use the Adam optimizer for", + "type": "text" + }, + { + "bbox": [ + 351, + 281, + 360, + 291 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 280, + 379, + 291 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 379, + 280, + 387, + 290 + ], + "score": 0.72, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 280, + 425, + 291 + ], + "score": 1.0, + "content": "with the", + "type": "text" + }, + { + "bbox": [ + 425, + 280, + 433, + 291 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 280, + 505, + 291 + ], + "score": 1.0, + "content": "-hyperparameters", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 334, + 304 + ], + "score": 1.0, + "content": "(0.9, 0.9) and (0.9, 0.999) respectively. A lower", + "type": "text" + }, + { + "bbox": [ + 334, + 291, + 345, + 302 + ], + "score": 0.87, + "content": "\\beta _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 290, + 424, + 304 + ], + "score": 1.0, + "content": "hyperparameter for", + "type": "text" + }, + { + "bbox": [ + 424, + 292, + 432, + 302 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "allows the second", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "momentum to adapt faster to a change of gradient scale which happens for initial false", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 313, + 226, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 313, + 226, + 325 + ], + "score": 1.0, + "content": "positive predictions.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5, + "bbox_fs": [ + 140, + 214, + 507, + 325 + ] + }, + { + "type": "text", + "bbox": [ + 143, + 325, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 141, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 141, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "The average run time of ENCO was 2mins per graph. The algorithm could be sped up even", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 141, + 336, + 337, + 349 + ], + "score": 1.0, + "content": "more by reducing the number of graph samples", + "type": "text" + }, + { + "bbox": [ + 338, + 336, + 348, + 346 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "and model fitting iterations. However,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 142, + 348, + 287, + 359 + ], + "score": 1.0, + "content": "for graphs of larger than 100 nodes,", + "type": "text" + }, + { + "bbox": [ + 288, + 347, + 326, + 358 + ], + "score": 0.9, + "content": "K = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 348, + 505, + 359 + ], + "score": 1.0, + "content": "and longer model fitting times showed to be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 358, + 491, + 371 + ], + "spans": [ + { + "bbox": [ + 141, + 358, + 491, + 371 + ], + "score": 1.0, + "content": "beneficial. The learning curves in terms of recall and precision are shown in Figure 10.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 141, + 325, + 506, + 371 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 394 + ], + "score": 1.0, + "content": "Enforcing acyclicity ENCO is guaranteed to converge to acyclic graphs in the data limit; arguably,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "an assumption that does not always hold. In the presence of cycles, which can occur especially", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "when low data is available, a simple heuristic is to keep the graph, which maximizes the orientation", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 320, + 427 + ], + "score": 1.0, + "content": "probabilities. Specifically, we aim to find the order", + "type": "text" + }, + { + "bbox": [ + 321, + 414, + 357, + 425 + ], + "score": 0.92, + "content": "O \\in S _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 414, + 390, + 427 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 390, + 414, + 405, + 425 + ], + "score": 0.9, + "content": "S _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 414, + 505, + 427 + ], + "score": 1.0, + "content": "represents the set of all", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 297, + 438 + ], + "score": 1.0, + "content": "permutations from 1 to the number of variables", + "type": "text" + }, + { + "bbox": [ + 297, + 425, + 307, + 435 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 425, + 505, + 438 + ], + "score": 1.0, + "content": ", for which we maximize the following objective:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 380, + 506, + 438 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 438, + 381, + 478 + ], + "lines": [ + { + "bbox": [ + 229, + 438, + 381, + 478 + ], + "spans": [ + { + "bbox": [ + 229, + 438, + 381, + 478 + ], + "score": 0.95, + "content": "\\hat { O } = \\arg \\operatorname* { m a x } _ { O } \\left[ \\prod _ { i = 1 } ^ { N } \\prod _ { j = i + 1 } ^ { N } \\sigma ( \\theta _ { O _ { i } , O _ { j } } ) \\right]", + "type": "interline_equation", + "image_path": "aa45076b07217f5bf8dbf708c25f22a1d49771f55716e82a4114e688f86d1e7b.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 229, + 438, + 381, + 458.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 229, + 458.0, + 381, + 478.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "For small cycles it is easy to do this exhaustively by checking all permutations. For larger cycles, we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 489, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 382, + 505 + ], + "score": 1.0, + "content": "apply a simple greedy search that works just as well. Once the order", + "type": "text" + }, + { + "bbox": [ + 383, + 489, + 392, + 502 + ], + "score": 0.85, + "content": "\\hat { O }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 490, + 505, + 505 + ], + "score": 1.0, + "content": "has been found, we remove", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 502, + 479, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 144, + 517 + ], + "score": 1.0, + "content": "all edges", + "type": "text" + }, + { + "bbox": [ + 144, + 504, + 185, + 516 + ], + "score": 0.92, + "content": "X _ { i } \\to X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 503, + 213, + 517 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 213, + 505, + 218, + 514 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 503, + 267, + 517 + ], + "score": 1.0, + "content": "comes after", + "type": "text" + }, + { + "bbox": [ + 268, + 505, + 273, + 516 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 503, + 285, + 517 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 285, + 502, + 294, + 514 + ], + "score": 0.82, + "content": "\\hat { O }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 503, + 479, + 517 + ], + "score": 1.0, + "content": ". This guarantees to result in an acyclic graph.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 478, + 505, + 517 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 534 + ], + "score": 1.0, + "content": "The intuition behind this heuristic is the following. Cycles are often caused by a single orientation", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 447, + 545 + ], + "score": 1.0, + "content": "pair being incorrect due to noise in the interventional data. For example, in a chain", + "type": "text" + }, + { + "bbox": [ + 448, + 532, + 505, + 543 + ], + "score": 0.91, + "content": "X _ { 1 } X _ { 2 } ", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 107, + 543, + 177, + 554 + ], + "score": 0.92, + "content": "X _ { 3 } X _ { 4 } X _ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 542, + 350, + 555 + ], + "score": 1.0, + "content": ", it can happen that the orientation parameter", + "type": "text" + }, + { + "bbox": [ + 351, + 543, + 365, + 554 + ], + "score": 0.9, + "content": "\\theta _ { 1 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "is incorrectly learned as orientation", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 554, + 504, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 191, + 565 + ], + "score": 1.0, + "content": "of the edge between", + "type": "text" + }, + { + "bbox": [ + 192, + 554, + 206, + 565 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 554, + 225, + 565 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 225, + 554, + 239, + 565 + ], + "score": 0.89, + "content": "X _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 554, + 252, + 565 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 252, + 554, + 295, + 565 + ], + "score": 0.93, + "content": "X _ { 4 } X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 554, + 413, + 565 + ], + "score": 1.0, + "content": "if the interventional data on", + "type": "text" + }, + { + "bbox": [ + 413, + 555, + 427, + 565 + ], + "score": 0.89, + "content": "X _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 554, + 504, + 565 + ], + "score": 1.0, + "content": "does not show the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 173, + 578 + ], + "score": 1.0, + "content": "independence of", + "type": "text" + }, + { + "bbox": [ + 173, + 565, + 187, + 576 + ], + "score": 0.9, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 564, + 204, + 578 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 205, + 565, + 218, + 576 + ], + "score": 0.89, + "content": "X _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 564, + 415, + 578 + ], + "score": 1.0, + "content": ". However, most other orientation parameters, e.g.", + "type": "text" + }, + { + "bbox": [ + 416, + 565, + 430, + 576 + ], + "score": 0.76, + "content": "\\theta _ { 1 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 564, + 433, + 578 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 433, + 565, + 448, + 576 + ], + "score": 0.69, + "content": "\\theta _ { 1 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 564, + 451, + 578 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 452, + 565, + 466, + 576 + ], + "score": 0.62, + "content": "\\theta _ { 2 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 564, + 469, + 578 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 469, + 565, + 484, + 576 + ], + "score": 0.65, + "content": "\\theta _ { 3 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 564, + 506, + 578 + ], + "score": 1.0, + "content": ", etc.,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 357, + 588 + ], + "score": 1.0, + "content": "have been likely learned correctly. Thus, it is easy to spot that", + "type": "text" + }, + { + "bbox": [ + 357, + 576, + 371, + 587 + ], + "score": 0.9, + "content": "\\theta _ { 1 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "is an outlier, and this is what the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 587, + 250, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 250, + 600 + ], + "score": 1.0, + "content": "simple heuristic above implements.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 519, + 506, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "Results Besides the structural hamming distance, a common alternative metric is structural inter-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "score": 1.0, + "content": "vention distance (SID) (Peters & Bühlmann, 2015). In contrast to SHD, SID quantifies the closeness", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "between two DAGs in terms of their corresponding causal inference statements. Hence, it is suited for", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "comparing causal discovery methods. The results of the experiments on the synthetic graphs in terms", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 651, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 104, + 651, + 506, + 668 + ], + "score": 1.0, + "content": "of SID are shown in Table 5, and show a similar trend as before, namely that ENCO is outperforming", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 663, + 160, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 160, + 677 + ], + "score": 1.0, + "content": "all baselines.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5, + "bbox_fs": [ + 104, + 609, + 506, + 677 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 689, + 255, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 256, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 256, + 701 + ], + "score": 1.0, + "content": "C.2 SCALABILITY EXPERIMENTS", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Graph generation For generating the graphs, we use the same strategy as for the graph random", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 412, + 733 + ], + "score": 1.0, + "content": "in the previous experiments. The probability of sampling an edge is set to", + "type": "text" + }, + { + "bbox": [ + 412, + 720, + 432, + 732 + ], + "score": 0.87, + "content": "8 / N", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 720, + 505, + 733 + ], + "score": 1.0, + "content": ", meaning that on", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 300, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 314 + ], + "score": 1.0, + "content": "average, every node has 8 in- and outgoing edges. We limit the number of parents to 10 per node since,", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "otherwise, we cannot guarantee that the randomly sampled distributions take all parents faithfully into", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "account. This is also in line with the real-world inspired graphs of the BnLearn repository, which have", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "a maximum of 6 parents. To give an intuition on the complexity of such graphs, we show an example", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "graph of 100 nodes in Figure 12a. Accordingly to the number of variables, we have increased the", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 356, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 307, + 368 + ], + "score": 1.0, + "content": "data set size to 4096 samples per intervention, and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 307, + 356, + 328, + 366 + ], + "score": 0.56, + "content": "1 0 0 \\mathrm { k }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 328, + 356, + 505, + 368 + ], + "score": 1.0, + "content": "observational samples. We did not apply the", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "order heuristic on the predictions, since ENCO was able to recover acyclic graphs by itself with the", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 377, + 153, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 153, + 390 + ], + "score": 1.0, + "content": "given data.", + "type": "text", + "cross_page": true + } + ], + "index": 14 + } + ], + "index": 44.5, + "bbox_fs": [ + 106, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 132, + 79, + 471, + 223 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 132, + 79, + 471, + 223 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 132, + 79, + 471, + 223 + ], + "spans": [ + { + "bbox": [ + 132, + 79, + 471, + 223 + ], + "score": 0.958, + "type": "image", + "image_path": "6e785f3648cb305968d191d3b40631d09a17a3a4e0600ffbe41e934c38637247.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 132, + 79, + 471, + 127.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 132, + 127.0, + 471, + 175.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 132, + 175.0, + 471, + 223.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 233, + 505, + 277 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "score": 1.0, + "content": "Figure 12: (a) Example graph of 100 variables (best viewed electronically). Every node has on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "score": 1.0, + "content": "average 8 edges and a maximum of 10 parents. (b) Plotting recall and precision of the edge predictions", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "for the training on graphs with 1, 000 nodes. The small standard deviation across graphs shows that", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 266, + 271, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 271, + 279 + ], + "score": 1.0, + "content": "ENCO can reliably recover large graphs.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 300, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 300, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 314 + ], + "score": 1.0, + "content": "average, every node has 8 in- and outgoing edges. We limit the number of parents to 10 per node since,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "otherwise, we cannot guarantee that the randomly sampled distributions take all parents faithfully into", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "account. This is also in line with the real-world inspired graphs of the BnLearn repository, which have", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "a maximum of 6 parents. 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We did not apply the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "order heuristic on the predictions, since ENCO was able to recover acyclic graphs by itself with the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 377, + 153, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 153, + 390 + ], + "score": 1.0, + "content": "given data.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 113, + 415 + ], + "score": 0.78, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 403, + 482, + 417 + ], + "score": 1.0, + "content": "-freezing stage For ENCO, one challenge of large graphs is that the orientation parameters", + "type": "text" + }, + { + "bbox": [ + 482, + 404, + 489, + 414 + ], + "score": 0.77, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 269, + 428 + ], + "score": 1.0, + "content": "updated very sparsely. The gradients for", + "type": "text" + }, + { + "bbox": [ + 269, + 415, + 282, + 427 + ], + "score": 0.89, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "require data from an intervention on one of its adjacent", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 426, + 504, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 131, + 437 + ], + "score": 1.0, + "content": "nodes", + "type": "text" + }, + { + "bbox": [ + 132, + 426, + 144, + 437 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 426, + 156, + 437 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 156, + 426, + 169, + 438 + ], + "score": 0.88, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 426, + 370, + 437 + ], + "score": 1.0, + "content": ", which we evaluate less frequently with increasing", + "type": "text" + }, + { + "bbox": [ + 370, + 426, + 380, + 435 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 426, + 504, + 437 + ], + "score": 1.0, + "content": "as we iterate over interventions", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 436, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 131, + 449 + ], + "score": 1.0, + "content": "on all", + "type": "text" + }, + { + "bbox": [ + 131, + 437, + 141, + 446 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 436, + 505, + 449 + ], + "score": 1.0, + "content": "nodes. Hence, we require more iterations/epochs just for training the orientation parameters", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 446, + 504, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 496, + 461 + ], + "score": 1.0, + "content": "while wasting a lot of computational resources. To accelerate training of large graphs, we freeze", + "type": "text" + }, + { + "bbox": [ + 496, + 449, + 504, + 459 + ], + "score": 0.77, + "content": "\\gamma", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 456, + 504, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 456, + 312, + 474 + ], + "score": 1.0, + "content": "in every second graph fitting stage. Updating only", + "type": "text" + }, + { + "bbox": [ + 312, + 459, + 319, + 468 + ], + "score": 0.74, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 456, + 482, + 474 + ], + "score": 1.0, + "content": "allows us to use the same graph sample", + "type": "text" + }, + { + "bbox": [ + 482, + 459, + 504, + 471 + ], + "score": 0.9, + "content": "C _ { - i j }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 468, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 104, + 468, + 141, + 483 + ], + "score": 1.0, + "content": "for both", + "type": "text" + }, + { + "bbox": [ + 142, + 470, + 198, + 482 + ], + "score": 0.91, + "content": "\\mathscr { L } _ { X _ { i } X _ { j } } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 468, + 217, + 483 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 218, + 470, + 273, + 482 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { X _ { i } \\not \\to X _ { j } } ( X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 468, + 421, + 483 + ], + "score": 1.0, + "content": "since the log-likelihood estimate of", + "type": "text" + }, + { + "bbox": [ + 421, + 470, + 434, + 482 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 468, + 506, + 483 + ], + "score": 1.0, + "content": "only needs to be", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 479, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 162, + 494 + ], + "score": 1.0, + "content": "evaluated for", + "type": "text" + }, + { + "bbox": [ + 163, + 482, + 175, + 493 + ], + "score": 0.88, + "content": "\\theta _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 479, + 505, + 494 + ], + "score": 1.0, + "content": ". With this gradient estimator, we experience that as little as 4 graph samples are", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "sufficient to obtain a reasonable gradient variance. Hence, it is possible to perform more gradient", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 149, + 515 + ], + "score": 1.0, + "content": "updates of", + "type": "text" + }, + { + "bbox": [ + 150, + 503, + 156, + 512 + ], + "score": 0.79, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 502, + 485, + 515 + ], + "score": 1.0, + "content": "in the same computation time. Note that this is estimator not efficient when training", + "type": "text" + }, + { + "bbox": [ + 485, + 504, + 494, + 514 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 186, + 527 + ], + "score": 1.0, + "content": "we require different", + "type": "text" + }, + { + "bbox": [ + 187, + 514, + 208, + 525 + ], + "score": 0.89, + "content": "C _ { - i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 513, + 281, + 527 + ], + "score": 1.0, + "content": "samples for every", + "type": "text" + }, + { + "bbox": [ + 281, + 514, + 286, + 523 + ], + "score": 0.58, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 513, + 506, + 527 + ], + "score": 1.0, + "content": ". In experiments, we alternate the standard graph fitting", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 525, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 183, + 536 + ], + "score": 1.0, + "content": "step with this pure", + "type": "text" + }, + { + "bbox": [ + 184, + 525, + 190, + 534 + ], + "score": 0.82, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 525, + 506, + 536 + ], + "score": 1.0, + "content": "-training stage. We want to emphasize that this approach can also be used for", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "small graphs obtaining similar results as in Table 1. However, it is less needed because the orientation", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 547, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 558 + ], + "score": 1.0, + "content": "parameters are more frequently updated in the first place. Such an approach is not possible for the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 556, + 471, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 471, + 569 + ], + "score": 1.0, + "content": "baselines, SDI and DCDI, because they do not model the orientation as a separate variable.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "Hyperparameters To experiment with large graphs, we mostly keep to the same hyperparameters", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "as reported in Section C.1. However, all methods showed to gain by a small hyperparameter search.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "For SDI and ENCO, we increase the number of distribution fitting iterations as the neural networks", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 617, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 462, + 628 + ], + "score": 1.0, + "content": "need to model a larger set of possible parents. We also increase the learning rate of", + "type": "text" + }, + { + "bbox": [ + 462, + 618, + 471, + 628 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 617, + 506, + 628 + ], + "score": 1.0, + "content": "to 2e-2.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "However, while SDI reaches better performance with the increased learning rate at epoch 30, it", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "showed to perform worse when training for longer. This indicates that high learning rates can lead to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "local minima in SDI. Additionally, we noticed that a slightly higher sparsity regularizer improved", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "convergence speed for ENCO while SDI did not improve with a higher sparsity regularizer. Table 6", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "shows a hyperparameter overview of ENCO on large-scale graphs, and Figure 12b the learning curve", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 682, + 214, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 214, + 694 + ], + "score": 1.0, + "content": "on graphs of 1, 000 nodes.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "For DCDI, we noticed that the hyperparameters around the Lagrangian constraint needed to be", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "carefully fine-tuned. 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(2020), we", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + } + ], + "page_idx": 38, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 132, + 79, + 471, + 223 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 132, + 79, + 471, + 223 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 132, + 79, + 471, + 223 + ], + "spans": [ + { + "bbox": [ + 132, + 79, + 471, + 223 + ], + "score": 0.958, + "type": "image", + "image_path": "6e785f3648cb305968d191d3b40631d09a17a3a4e0600ffbe41e934c38637247.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 132, + 79, + 471, + 127.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 132, + 127.0, + 471, + 175.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 132, + 175.0, + 471, + 223.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 233, + 505, + 277 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "score": 1.0, + "content": "Figure 12: (a) Example graph of 100 variables (best viewed electronically). 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With this gradient estimator, we experience that as little as 4 graph samples are", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "sufficient to obtain a reasonable gradient variance. Hence, it is possible to perform more gradient", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 149, + 515 + ], + "score": 1.0, + "content": "updates of", + "type": "text" + }, + { + "bbox": [ + 150, + 503, + 156, + 512 + ], + "score": 0.79, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 502, + 485, + 515 + ], + "score": 1.0, + "content": "in the same computation time. Note that this is estimator not efficient when training", + "type": "text" + }, + { + "bbox": [ + 485, + 504, + 494, + 514 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 186, + 527 + ], + "score": 1.0, + "content": "we require different", + "type": "text" + }, + { + "bbox": [ + 187, + 514, + 208, + 525 + ], + "score": 0.89, + "content": "C _ { - i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 513, + 281, + 527 + ], + "score": 1.0, + "content": "samples for every", + "type": "text" + }, + { + "bbox": [ + 281, + 514, + 286, + 523 + ], + "score": 0.58, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 513, + 506, + 527 + ], + "score": 1.0, + "content": ". In experiments, we alternate the standard graph fitting", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 525, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 183, + 536 + ], + "score": 1.0, + "content": "step with this pure", + "type": "text" + }, + { + "bbox": [ + 184, + 525, + 190, + 534 + ], + "score": 0.82, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 525, + 506, + 536 + ], + "score": 1.0, + "content": "-training stage. We want to emphasize that this approach can also be used for", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "small graphs obtaining similar results as in Table 1. However, it is less needed because the orientation", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 547, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 558 + ], + "score": 1.0, + "content": "parameters are more frequently updated in the first place. Such an approach is not possible for the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 556, + 471, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 471, + 569 + ], + "score": 1.0, + "content": "baselines, SDI and DCDI, because they do not model the orientation as a separate variable.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 403, + 506, + 569 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "Hyperparameters To experiment with large graphs, we mostly keep to the same hyperparameters", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "as reported in Section C.1. However, all methods showed to gain by a small hyperparameter search.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "For SDI and ENCO, we increase the number of distribution fitting iterations as the neural networks", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 617, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 462, + 628 + ], + "score": 1.0, + "content": "need to model a larger set of possible parents. We also increase the learning rate of", + "type": "text" + }, + { + "bbox": [ + 462, + 618, + 471, + 628 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 617, + 506, + 628 + ], + "score": 1.0, + "content": "to 2e-2.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "However, while SDI reaches better performance with the increased learning rate at epoch 30, it", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "showed to perform worse when training for longer. This indicates that high learning rates can lead to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "local minima in SDI. Additionally, we noticed that a slightly higher sparsity regularizer improved", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "convergence speed for ENCO while SDI did not improve with a higher sparsity regularizer. Table 6", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "shows a hyperparameter overview of ENCO on large-scale graphs, and Figure 12b the learning curve", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 682, + 214, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 214, + 694 + ], + "score": 1.0, + "content": "on graphs of 1, 000 nodes.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 583, + 506, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "For DCDI, we noticed that the hyperparameters around the Lagrangian constraint needed to be", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "carefully fine-tuned. The Lagrangian constraint can reach values larger than possible to represent", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "with double, and starts with 8e216 for graphs of 1, 000 nodes. Following Brouillard et al. (2020), we", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "normalize the constraint by the value after initialization, which gives us a more reasonable value to", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 364, + 107 + ], + "score": 1.0, + "content": "start learning. 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The", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 239, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 192, + 140 + ], + "score": 1.0, + "content": "best value found was", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 192, + 127, + 235, + 138 + ], + "score": 0.9, + "content": "\\mu _ { 0 } = 1 \\mathrm { e } { - } 7", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 235, + 126, + 239, + 140 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 698, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "normalize the constraint by the value after initialization, which gives us a more reasonable value to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 364, + 107 + ], + "score": 1.0, + "content": "start learning. We performed another hyperparameter search on", + "type": "text" + }, + { + "bbox": [ + 364, + 96, + 376, + 105 + ], + "score": 0.85, + "content": "\\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 93, + 506, + 107 + ], + "score": 1.0, + "content": ", noticed however that it did not", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "have a major impact. In the run time of ENCO, DCDI just starts to increase the weighting factor of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "the augmented Lagrangian while the DAG constraint is lower than 1e-10 for the smallest graph. The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 239, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 192, + 140 + ], + "score": 1.0, + "content": "best value found was", + "type": "text" + }, + { + "bbox": [ + 192, + 127, + 235, + 138 + ], + "score": 0.9, + "content": "\\mu _ { 0 } = 1 \\mathrm { e } { - } 7", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 126, + 239, + 140 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "table", + "bbox": [ + 127, + 181, + 483, + 365 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 149, + 504, + 172 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 149, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 506, + 162 + ], + "score": 1.0, + "content": "Table 6: Hyperparameter overview of ENCO for the scalability experiments presented in Table 7.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 160, + 402, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 402, + 172 + ], + "score": 1.0, + "content": "Numbers in the center represent that we use the same value for all graphs.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "table_body", + "bbox": [ + 127, + 181, + 483, + 365 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 181, + 483, + 365 + ], + "spans": [ + { + "bbox": [ + 127, + 181, + 483, + 365 + ], + "score": 0.985, + "html": "
Hyperparameters100 nodes200 nodes400 nodes1000 nodes
Distribution model2 layers, hidden size 64,LeakyReLU(α = 0.1)
Batch size -model12812812864
Learning rate - model
Distribution fitting iterations F20005e-3 2000 10040004000
Graph fitting iterations G
Graph samples100
Learning rate - γ2e-2 1e-1
Learning rate - 0
0-training iterations1000 1000
0-training graph samples2000 2+2
{0.002, 0.004,0.01}
Sparsity regularizer Xsparse
Number of epochs (max.)30
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Graph size Max Runtime†100 15mins200 45mins400 2.5hrs1000 9hrs
DCDI (Brouillard et al., 2020)583.1 (±21.8)1399.0 (±67.5)4761.2 (±303.4)00M
SDI (Ke et al., 2019)35.7(±2.9)100.9 (±7.6)356.3 (±11.5)1314.4 (±58.5)
ENCO (Ours)0.0 (±0.0)0.0 (±0.0)0.0 (±0.0)0.2 (±0.42)
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Hyperparameters100 nodes200 nodes400 nodes1000 nodes
Distribution model2 layers, hidden size 64,LeakyReLU(α = 0.1)
Batch size -model12812812864
Learning rate - model
Distribution fitting iterations F20005e-3 2000 10040004000
Graph fitting iterations G
Graph samples100
Learning rate - γ2e-2 1e-1
Learning rate - 0
0-training iterations1000 1000
0-training graph samples2000 2+2
{0.002, 0.004,0.01}
Sparsity regularizer Xsparse
Number of epochs (max.)30
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Graph size Max Runtime†100 15mins200 45mins400 2.5hrs1000 9hrs
DCDI (Brouillard et al., 2020)583.1 (±21.8)1399.0 (±67.5)4761.2 (±303.4)00M
SDI (Ke et al., 2019)35.7(±2.9)100.9 (±7.6)356.3 (±11.5)1314.4 (±58.5)
ENCO (Ours)0.0 (±0.0)0.0 (±0.0)0.0 (±0.0)0.2 (±0.42)
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MetricsENCO (10k interv./100k observ.)ENCO (512 interv./5k observ.)ENCO (200 interv./5k observ.)
SHD0.0 (±0.0)1.24 (±1.33)4.12 (±1.86)
Confounder recall96.8%(±9.5%)93.6% (±13.8%)92.0% (±11.5%)
Confounder precision100.0% (±0.0%)96.5%(±7.1%)83.8% (±16.4%)
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Right: Plotting the average score", + "type": "text" + }, + { + "bbox": [ + 394, + 402, + 438, + 415 + ], + "score": 0.95, + "content": "\\mathrm { l c } ( X _ { i } , \\bar { X _ { j } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "for confounders", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 107, + 414, + 174, + 426 + ], + "score": 0.91, + "content": "X _ { i } \\left. X _ { l } \\right. X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "in the true causal graph (orange) and maximum score of any other node pair (blue).", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 495, + 438 + ], + "score": 1.0, + "content": "The plot shows the detection of latent confounders in ENCO is not sensitive to the specific value of", + "type": "text" + }, + { + "bbox": [ + 495, + 427, + 501, + 434 + ], + "score": 0.77, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 424, + 506, + 438 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + } + ], + "index": 11.0 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "Baselines None of our previous continuous optimization baselines, i.e., SDI and DCDI, are able to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "deal with latent confounders. 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To simulate having interventions on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "fewer variables, we randomly sample a subset of variables for which we include the interventional", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "data, and remove for all others. 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MetricsENCO (10k interv./100k observ.)ENCO (512 interv./5k observ.)ENCO (200 interv./5k observ.)
SHD0.0 (±0.0)1.24 (±1.33)4.12 (±1.86)
Confounder recall96.8%(±9.5%)93.6% (±13.8%)92.0% (±11.5%)
Confounder precision100.0% (±0.0%)96.5%(±7.1%)83.8% (±16.4%)
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The dots on the left and right represent even-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 210, + 416 + ], + "score": 1.0, + "content": "tual (observed) parents of", + "type": "text" + }, + { + "bbox": [ + 210, + 403, + 223, + 414 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 402, + 241, + 416 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 241, + 403, + 254, + 415 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 402, + 393, + 416 + ], + "score": 1.0, + "content": ". Right: Plotting the average score", + "type": "text" + }, + { + "bbox": [ + 394, + 402, + 438, + 415 + ], + "score": 0.95, + "content": "\\mathrm { l c } ( X _ { i } , \\bar { X _ { j } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "for confounders", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 107, + 414, + 174, + 426 + ], + "score": 0.91, + "content": "X _ { i } \\left. X _ { l } \\right. X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "in the true causal graph (orange) and maximum score of any other node pair (blue).", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 495, + 438 + ], + "score": 1.0, + "content": "The plot shows the detection of latent confounders in ENCO is not sensitive to the specific value of", + "type": "text" + }, + { + "bbox": [ + 495, + 427, + 501, + 434 + ], + "score": 0.77, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 424, + 506, + 438 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + } + ], + "index": 11.0 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "Baselines None of our previous continuous optimization baselines, i.e., SDI and DCDI, are able to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "deal with latent confounders. To the best of our knowledge, other methods that are able to handle", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "score": 1.0, + "content": "latent confounders commonly take assumptions that do not hold in our experimental setup. Further,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "most methods are able to deal with latent confounders in the sense that they obtain the correct", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "results despite latent confounding being present. However, in our case, we explicitly predict latent", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 517, + 293, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 293, + 529 + ], + "score": 1.0, + "content": "confounders which is a different task by itself.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 461, + 506, + 529 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 647 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "Hyperparameters To show that we can perform latent confounder detection without specific", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 556, + 504, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 504, + 568 + ], + "score": 1.0, + "content": "hyperparameters, we use the same hyperparameters as for the experiment on the previous graph", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 103, + 562, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 103, + 566, + 294, + 585 + ], + "score": 1.0, + "content": "structures (see Appendix C.3). To record γ(I)ij", + "type": "text" + }, + { + "bbox": [ + 290, + 562, + 308, + 589 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 308, + 567, + 328, + 583 + ], + "score": 0.92, + "content": "\\gamma _ { i j } ^ { ( O ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 562, + 506, + 589 + ], + "score": 1.0, + "content": "separately, we use separate first and second", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 581, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 506, + 593 + ], + "score": 1.0, + "content": "order momentum parameters in the Adam optimizer. We plot in Figure 13b the latent confounder", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 135, + 604 + ], + "score": 1.0, + "content": "scores", + "type": "text" + }, + { + "bbox": [ + 135, + 592, + 179, + 604 + ], + "score": 0.94, + "content": "\\operatorname { l c } ( X _ { i } , X _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "calculated based on Equation 8. We see that the score converges close to 1 for", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 603, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 615 + ], + "score": 1.0, + "content": "pairs with a latent confounder, and for all other, it converges to 0. This verifies our motivation of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 614, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 625 + ], + "score": 1.0, + "content": "the score function discussed in Section 3.5, and also shows that the method is not sensitive to the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 625, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 213, + 637 + ], + "score": 1.0, + "content": "threshold hyperparameter", + "type": "text" + }, + { + "bbox": [ + 213, + 627, + 220, + 635 + ], + "score": 0.75, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 625, + 271, + 637 + ], + "score": 1.0, + "content": ". We choose", + "type": "text" + }, + { + "bbox": [ + 272, + 625, + 305, + 635 + ], + "score": 0.89, + "content": "\\tau = 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 625, + 505, + 637 + ], + "score": 1.0, + "content": "which was slightly higher than the highest value", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 635, + 321, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 321, + 648 + ], + "score": 1.0, + "content": "recorded for any other pair at early stages of training.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26, + "bbox_fs": [ + 103, + 545, + 506, + 648 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 666, + 301, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 302, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 302, + 677 + ], + "score": 1.0, + "content": "C.4 INTERVENTIONS ON FEWER VARIABLES", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "Datasets We perform the experiments of interventions on fewer variables on the same graphs and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "datasets as used for the initial synthetic graphs (see Section C.1). To simulate having interventions on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "fewer variables, we randomly sample a subset of variables for which we include the interventional", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "data, and remove for all others. The sampled variables are the same for both ENCO and DCDI, and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "differ across graphs. 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The dataset size is the same as before, namely 200 samples per intervention and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 211, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 211, + 105 + ], + "score": 1.0, + "content": "5k observational datasets.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 117, + 505, + 238 + ], + "lines": [ + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "ENCO for partial intervention sets While the theoretical guarantees for convergence to an acyclic", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 140 + ], + "score": 1.0, + "content": "graph apply when interventions on all variables are possible, it is straightforward to extend the ENCO", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 139, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 505, + 152 + ], + "score": 1.0, + "content": "algorithm to support partial interventions as well. Normally, in the graph fitting stage, we sample", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "one intervention at a time. We can, thus, simply restrict the sampling only to the interventions that", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 159, + 504, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 491, + 176 + ], + "score": 1.0, + "content": "are possible (or provided in the dataset). In this case, we update the orientation parameters", + "type": "text" + }, + { + "bbox": [ + 491, + 161, + 504, + 173 + ], + "score": 0.88, + "content": "\\theta _ { i j }", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 376, + 184 + ], + "score": 1.0, + "content": "of only those edges that connect to an intervened variable, either", + "type": "text" + }, + { + "bbox": [ + 376, + 172, + 389, + 183 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 172, + 402, + 184 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 403, + 172, + 416, + 184 + ], + "score": 0.9, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 172, + 506, + 184 + ], + "score": 1.0, + "content": ", as before. All other", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "orientation parameters would remain unchanged throughout the training, since their gradients rely on", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "score": 1.0, + "content": "interventions missing from the dataset. Instead, we extend the gradient estimator in Equation 4 to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "not be exclusive to adjacent interventions, but include interventions on all variables. 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This skeleton", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 378, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 506, + 389 + ], + "score": 1.0, + "content": "can be used as a prior, and prevents false positive edges between conditionally independent variables.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 400, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "score": 1.0, + "content": "Hyperparameters We reuse the hyperparameters of the experiments on the synthetic graph except", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 411, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 299, + 426 + ], + "score": 1.0, + "content": "that we use a slighly smaller sparsity regularizer,", + "type": "text" + }, + { + "bbox": [ + 299, + 412, + 360, + 424 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 411, + 506, + 426 + ], + "score": 1.0, + "content": ", and a weight decay of 4e-5. 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For DCDI, we observed that a higher regularization term of", + "type": "text" + }, + { + "bbox": [ + 433, + 434, + 466, + 444 + ], + "score": 0.89, + "content": "\\lambda = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "obtained", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 413, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 413, + 458 + ], + "score": 1.0, + "content": "best performance. 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This still represents an", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 108, + 306, + 504, + 317 + ], + "spans": [ + { + "bbox": [ + 108, + 306, + 504, + 317 + ], + "score": 1.0, + "content": "unbiased gradient estimator since in the derivation of the estimator, we excluded interventions on", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 317, + 253, + 329 + ], + "spans": [ + { + "bbox": [ + 107, + 317, + 253, + 329 + ], + "score": 1.0, + "content": "other variables only to reduce noise.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 294, + 505, + 329 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 333, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 106, + 333, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 345 + ], + "score": 1.0, + "content": "ENCO has been designed under the assumption that interventional data is provided. When we", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "have interventional data on only a very small subset of variables, we might not optimally use the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 356, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 505, + 367 + ], + "score": 1.0, + "content": "information that is provided by the observational data. To overcome this issue, we can run a causal", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "discovery method that solely work on observational data and return an undirected graph. This skeleton", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 378, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 506, + 389 + ], + "score": 1.0, + "content": "can be used as a prior, and prevents false positive edges between conditionally independent variables.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 333, + 506, + 389 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 400, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "score": 1.0, + "content": "Hyperparameters We reuse the hyperparameters of the experiments on the synthetic graph except", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 411, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 299, + 426 + ], + "score": 1.0, + "content": "that we use a slighly smaller sparsity regularizer,", + "type": "text" + }, + { + "bbox": [ + 299, + 412, + 360, + 424 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 411, + 506, + 426 + ], + "score": 1.0, + "content": ", and a weight decay of 4e-5. For the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 416, + 436 + ], + "score": 1.0, + "content": "orientation parameters without adjacent intervention, we use a learning rate of", + "type": "text" + }, + { + "bbox": [ + 416, + 423, + 448, + 434 + ], + "score": 0.9, + "content": "0 . 1 \\cdot \\ln _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "which is 1e-3", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 433, + 446 + ], + "score": 1.0, + "content": "for this experiment. For DCDI, we observed that a higher regularization term of", + "type": "text" + }, + { + "bbox": [ + 433, + 434, + 466, + 444 + ], + "score": 0.89, + "content": "\\lambda = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "obtained", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 413, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 413, + 458 + ], + "score": 1.0, + "content": "best performance. All other hyperparameters are the same as in Section C.1.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 400, + 506, + 458 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 468, + 356, + 479 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 358, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 358, + 480 + ], + "score": 1.0, + "content": "Results For additional experimental results, see Section D.2.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 468, + 358, + 480 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 493, + 297, + 504 + ], + "lines": [ + { + "bbox": [ + 107, + 493, + 299, + 505 + ], + "spans": [ + { + "bbox": [ + 107, + 493, + 299, + 505 + ], + "score": 1.0, + "content": "C.5 REAL-WORLD INSPIRED EXPERIMENTS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 505, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "Datasets We perform experiments on a collection of causal graphs from the Bayesian Network", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "Repository (BnLearn) (Scutari, 2010). The repository contains graphs inspired by real-world appli-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "cations that are used as benchmarks in literature. We chose the graphs to reflect a variety of sizes", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "and different challenges (rare events, deterministic variables, etc.). The chosen graphs are cancer", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "(Korb & Nicholson, 2010), earthquake (Korb & Nicholson, 2010), asia (Lauritzen & Spiegelhalter,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "1988), sachs (Sachs et al., 2005), child (Spiegelhalter & Cowell, 1992), alarm (Beinlich et al., 1989),", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 579, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 506, + 591 + ], + "score": 1.0, + "content": "diabetes (Andreassen et al., 1991), and pigs (Scutari, 2010). The graphs have been downloaded", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "from the BnLearn website5. For the small graphs, we have used a dataset size of 50k observational", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "samples and 512 samples per intervention. This is a larger dataset size than for the synthetic graph", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "because many edges in the real-world graphs have very small causal effects that cannot be recovered", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "from limited data, and the goal of the experiment was to show that the convergence conditions also", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "hold on real-world graphs. Hence, we need more observational and interventional samples. The", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "results with a smaller dataset size, i.e. 5k observational and 200 interventional samples as before, are", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "shown in Table 9. For the large graphs, we follow the dataset size for the scalability experiments (see", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 667, + 162, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 162, + 679 + ], + "score": 1.0, + "content": "Section C.2).", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 513, + 506, + 679 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 690, + 504, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "Hyperparameters We reuse most of the hyperparameters of the previous experiments. For all", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 701, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 506, + 714 + ], + "score": 1.0, + "content": "graphs less than 100 nodes, we use the hyperparameters of Appendix C.1, i.e. the synthetic graphs of", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 690, + 506, + 714 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 112, + 503, + 169 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 503, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 78, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 505, + 93 + ], + "score": 1.0, + "content": "Table 9: Results on graphs from the BnLearn library measured in structural hamming distance (lower", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 385, + 103 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 385, + 103 + ], + "score": 1.0, + "content": "is better). Results are averaged over 5 seeds with standard deviations.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 112, + 503, + 169 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 112, + 503, + 169 + ], + "spans": [ + { + "bbox": [ + 108, + 112, + 503, + 169 + ], + "score": 0.978, + "html": "
Datasetcancer (5 nodes)earthquake (5 nodes)asia (8 nodes)sachs (11 nodes)child (20 nodes)alarm (37 nodes)diabetes (413 nodes)pigs (441 nodes)
SDI DCDI3.0 (±0.0) 4.0 (±0.0)0.4(±0.5)4.0 (±0.0)7.0 (±0.0)11.2 (±0.4)24.4(±1.7)422.4(±8.7)18.0 (±1.6)
ENCO0.0 (±0.0)2.0 (±0.0) 0.0 (±0.0)5.0 (±0.0) 0.0 (±0.0)5.4(±2.1) 0.0 (±0.0)8.4(±0.7) 0.0 (±0.0)30.0 (±4.2) 1.0 (±0.0)- 2.0 (±0.0)- 0.0 (±0.0)
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Datasetcancer (5 nodes)earthquake (5 nodes)asia (8 nodes)sachs (11 nodes)child (20 nodes)alarm (37 nodes)
SDI3.0 (±0.0)0.4 (±0.5)4.6 (±0.5)8.4(±0.5)12.4 (±0.9)26.6 (±1.1)
DCDI4.0 (±0.0)2.0 (±0.0)4.4 (±0.7)7.2 (±2.4)9.8 (±0.6)31.4(±0.7)
ENCO1.2 (±0.4)0.4(±0.9)1.4(±0.5)0.4(±0.5)0.8 (±1.1)11.4(±1.8)
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For ENCO, we used a slightly smaller regularizer,", + "type": "text" + }, + { + "bbox": [ + 355, + 322, + 416, + 334 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 320, + 507, + 335 + ], + "score": 1.0, + "content": ", for the small graphs,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 331, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 180, + 347 + ], + "score": 1.0, + "content": "and a larger one,", + "type": "text" + }, + { + "bbox": [ + 180, + 333, + 239, + 344 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 331, + 506, + 347 + ], + "score": 1.0, + "content": ", for the large graphs. Due to the large amount of deterministic", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "variables, ENCO tends to predict more false positives in the beginning before removing them one by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 354, + 507, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 310, + 368 + ], + "score": 1.0, + "content": "one. For SDI, we also found a smaller regularizer,", + "type": "text" + }, + { + "bbox": [ + 310, + 355, + 366, + 366 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 354, + 507, + 368 + ], + "score": 1.0, + "content": ", to work best for the small graphs.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "However, in line with the results of Ke et al. (2019), SDI was not able to detect all edges. Even lower", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 376, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 388 + ], + "score": 1.0, + "content": "regularizers showed to perform considerably worse on the child dataset, while minor improvements", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 332, + 400 + ], + "score": 1.0, + "content": "were made on the small graphs. Hence, we settled for", + "type": "text" + }, + { + "bbox": [ + 333, + 388, + 390, + 399 + ], + "score": 0.89, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 387, + 506, + 400 + ], + "score": 1.0, + "content": ". In terms of run time, both", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 398, + 405, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 405, + 411 + ], + "score": 1.0, + "content": "methods used 100 epochs for the small graphs and 50 for the large graphs.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "Results The results including standard deviations can be found in Table 9. The low standard", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "deviation for ENCO shows that the approach is stable across seeds, even for large graphs. SDI has a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "zero standard deviation for a few graphs. In those cases, SDI converged to the same graph across", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 454, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 506, + 467 + ], + "score": 1.0, + "content": "seeds, but not necessarily the correct graph. We have also applied DCDI (Brouillard et al., 2020) to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "the real-world datasets and report the results in Table 9 and 10. DCDI performs relatively similar", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 367, + 489 + ], + "score": 1.0, + "content": "to SDI, making a few more mistakes on the very small graphs", + "type": "text" + }, + { + "bbox": [ + 367, + 477, + 390, + 488 + ], + "score": 0.81, + "content": "( < 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "nodes) while being slightly", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "better on sachs and child. Nonetheless, ENCO outperforms DCDI on all graphs. We do not", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "score": 1.0, + "content": "report results of DCDI on the largest graphs, diabetes and pigs, because it ran out of memory", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "for diabetes (larger number of max. categories per variable) and did not converge within the same", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 520, + 445, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 445, + 533 + ], + "score": 1.0, + "content": "time limitations as SDI and ENCO (see Section 4.3 for a comparison on scalability).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5 + } + ], + "page_idx": 42, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "43", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 112, + 503, + 169 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 503, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 78, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 505, + 93 + ], + "score": 1.0, + "content": "Table 9: Results on graphs from the BnLearn library measured in structural hamming distance (lower", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 385, + 103 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 385, + 103 + ], + "score": 1.0, + "content": "is better). Results are averaged over 5 seeds with standard deviations.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 112, + 503, + 169 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 112, + 503, + 169 + ], + "spans": [ + { + "bbox": [ + 108, + 112, + 503, + 169 + ], + "score": 0.978, + "html": "
Datasetcancer (5 nodes)earthquake (5 nodes)asia (8 nodes)sachs (11 nodes)child (20 nodes)alarm (37 nodes)diabetes (413 nodes)pigs (441 nodes)
SDI DCDI3.0 (±0.0) 4.0 (±0.0)0.4(±0.5)4.0 (±0.0)7.0 (±0.0)11.2 (±0.4)24.4(±1.7)422.4(±8.7)18.0 (±1.6)
ENCO0.0 (±0.0)2.0 (±0.0) 0.0 (±0.0)5.0 (±0.0) 0.0 (±0.0)5.4(±2.1) 0.0 (±0.0)8.4(±0.7) 0.0 (±0.0)30.0 (±4.2) 1.0 (±0.0)- 2.0 (±0.0)- 0.0 (±0.0)
", + "type": "table", + "image_path": "72a6102464d0e8bf5cd872554e8076638b6a96433def9c19f685ea4e9c6f32e5.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 108, + 112, + 503, + 131.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 108, + 131.0, + 503, + 150.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 108, + 150.0, + 503, + 169.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "table", + "bbox": [ + 123, + 210, + 488, + 280 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 179, + 503, + 202 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 178, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 192 + ], + "score": 1.0, + "content": "Table 10: Results on graphs from the BnLearn library measured in structural hamming distance", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 190, + 396, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 198, + 203 + ], + "score": 1.0, + "content": "(lower is better), using", + "type": "text" + }, + { + "bbox": [ + 199, + 191, + 210, + 201 + ], + "score": 0.46, + "content": "5 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 190, + 396, + 203 + ], + "score": 1.0, + "content": "observational and 200 interventional samples.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "table_body", + "bbox": [ + 123, + 210, + 488, + 280 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 123, + 210, + 488, + 280 + ], + "spans": [ + { + "bbox": [ + 123, + 210, + 488, + 280 + ], + "score": 0.982, + "html": "
Datasetcancer (5 nodes)earthquake (5 nodes)asia (8 nodes)sachs (11 nodes)child (20 nodes)alarm (37 nodes)
SDI3.0 (±0.0)0.4 (±0.5)4.6 (±0.5)8.4(±0.5)12.4 (±0.9)26.6 (±1.1)
DCDI4.0 (±0.0)2.0 (±0.0)4.4 (±0.7)7.2 (±2.4)9.8 (±0.6)31.4(±0.7)
ENCO1.2 (±0.4)0.4(±0.9)1.4(±0.5)0.4(±0.5)0.8 (±1.1)11.4(±1.8)
", + "type": "table", + "image_path": "400ff3be94bec1fdda5b4dd68128b580f85a35614f7cc57197d367f9e8edd95e.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 123, + 210, + 488, + 233.33333333333334 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 123, + 233.33333333333334, + 488, + 256.6666666666667 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 123, + 256.6666666666667, + 488, + 280.0 + ], + "spans": [], + "index": 9 + } + ] + } + ], + "index": 6.75 + }, + { + "type": "text", + "bbox": [ + 106, + 299, + 506, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 298, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 312 + ], + "score": 1.0, + "content": "25 nodes. For all graphs larger than 100 nodes, we use the hyperparameters of Appendix C.2, i.e. the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 311, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 323 + ], + "score": 1.0, + "content": "large-scale graphs. One exception is that we allow the fine-tuning of the regularizer parameter for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 320, + 507, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 320, + 354, + 335 + ], + "score": 1.0, + "content": "both sets. For ENCO, we used a slightly smaller regularizer,", + "type": "text" + }, + { + "bbox": [ + 355, + 322, + 416, + 334 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 320, + 507, + 335 + ], + "score": 1.0, + "content": ", for the small graphs,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 331, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 180, + 347 + ], + "score": 1.0, + "content": "and a larger one,", + "type": "text" + }, + { + "bbox": [ + 180, + 333, + 239, + 344 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 331, + 506, + 347 + ], + "score": 1.0, + "content": ", for the large graphs. Due to the large amount of deterministic", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "variables, ENCO tends to predict more false positives in the beginning before removing them one by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 354, + 507, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 310, + 368 + ], + "score": 1.0, + "content": "one. For SDI, we also found a smaller regularizer,", + "type": "text" + }, + { + "bbox": [ + 310, + 355, + 366, + 366 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 354, + 507, + 368 + ], + "score": 1.0, + "content": ", to work best for the small graphs.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "However, in line with the results of Ke et al. (2019), SDI was not able to detect all edges. Even lower", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 376, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 388 + ], + "score": 1.0, + "content": "regularizers showed to perform considerably worse on the child dataset, while minor improvements", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 332, + 400 + ], + "score": 1.0, + "content": "were made on the small graphs. Hence, we settled for", + "type": "text" + }, + { + "bbox": [ + 333, + 388, + 390, + 399 + ], + "score": 0.89, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 387, + 506, + 400 + ], + "score": 1.0, + "content": ". In terms of run time, both", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 398, + 405, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 405, + 411 + ], + "score": 1.0, + "content": "methods used 100 epochs for the small graphs and 50 for the large graphs.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 298, + 507, + 411 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "Results The results including standard deviations can be found in Table 9. The low standard", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "deviation for ENCO shows that the approach is stable across seeds, even for large graphs. SDI has a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "zero standard deviation for a few graphs. In those cases, SDI converged to the same graph across", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 454, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 506, + 467 + ], + "score": 1.0, + "content": "seeds, but not necessarily the correct graph. We have also applied DCDI (Brouillard et al., 2020) to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "the real-world datasets and report the results in Table 9 and 10. DCDI performs relatively similar", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 367, + 489 + ], + "score": 1.0, + "content": "to SDI, making a few more mistakes on the very small graphs", + "type": "text" + }, + { + "bbox": [ + 367, + 477, + 390, + 488 + ], + "score": 0.81, + "content": "( < 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "nodes) while being slightly", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "better on sachs and child. Nonetheless, ENCO outperforms DCDI on all graphs. We do not", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "score": 1.0, + "content": "report results of DCDI on the largest graphs, diabetes and pigs, because it ran out of memory", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "for diabetes (larger number of max. categories per variable) and did not converge within the same", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 520, + 445, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 445, + 533 + ], + "score": 1.0, + "content": "time limitations as SDI and ENCO (see Section 4.3 for a comparison on scalability).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 421, + 506, + 533 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 271, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 273, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 273, + 96 + ], + "score": 1.0, + "content": "D ADDITIONAL EXPERIMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "score": 1.0, + "content": "In this section, we show additional experiments performed as ablation studies of ENCO. First, we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 505, + 130 + ], + "score": 1.0, + "content": "discuss further experiments We then discuss the effect of using our gradient estimators proposed in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "Section 3.4 compared to Bengio et al. (2020). Next, we show experiments on synthetic graphs with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 137, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 153 + ], + "score": 1.0, + "content": "deterministic variables violating faithfulness, and experiments on continuous data with Normalizing", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 163 + ], + "score": 1.0, + "content": "Flows. Finally, we discuss experiments with different causal mechanism functions for generating", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 162, + 394, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 394, + 173 + ], + "score": 1.0, + "content": "synthetic, conditional categorical distributions besides neural networks.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 108, + 187, + 257, + 198 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 259, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 259, + 199 + ], + "score": 1.0, + "content": "D.1 EFFECT OF THE SAMPLE SIZE", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 108, + 207, + 504, + 252 + ], + "lines": [ + { + "bbox": [ + 106, + 208, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 219 + ], + "score": 1.0, + "content": "The number of samples provided as observational and interventional data is crucial for causal structure", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "learning methods since the more data we have, the better we can estimate the underlying causal", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 230, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 506, + 241 + ], + "score": 1.0, + "content": "mechanisms. To gain further insights in the effect of the sample size on ENCO and the compared", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 241, + 425, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 425, + 253 + ], + "score": 1.0, + "content": "baselines, we repeat the experiments of Section 4.2 with different sample sizes.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 264, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "Large sample size First, we use very large sample sizes to find the upper bound performance level", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 276, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 347, + 288 + ], + "score": 1.0, + "content": "that we can expect from each method. For this, we sample", + "type": "text" + }, + { + "bbox": [ + 347, + 276, + 369, + 286 + ], + "score": 0.33, + "content": "1 0 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 276, + 506, + 288 + ], + "score": 1.0, + "content": "observational samples per graph,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "and 10k samples per intervention. We observed that this is sufficient to model most conditional", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "probabilities up to a negligible error. The results are shown in Table 11. We find that, in line with the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 308, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 322 + ], + "score": 1.0, + "content": "theoretical guarantees, ENCO can reliably recover most graphs, only making 0.3 mistakes on average", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "on the full graph. Of the baselines, only DCDI is able to recover the collider graph without errors", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "since its edges can be independently orientated. For all other graphs, DCDI converges to acyclic", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "graphs, but incorrectly orients some edges and predicts false positive edges, while being 8 times", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "slower than ENCO on the same hardware. All other baselines show improved SHD scores than in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 363, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 375 + ], + "score": 1.0, + "content": "Table 1 as well, but are not able to match ENCO’s performance. This shows that, even in the data", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 375, + 386, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 386, + 386 + ], + "score": 1.0, + "content": "limit, ENCO achieves notably better results than concurrent methods.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 108, + 391, + 504, + 413 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "Next, we consider situations where data is very limited. Thereby, we consider two data sample axes:", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 402, + 258, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 258, + 414 + ], + "score": 1.0, + "content": "observational and interventional data.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "Limited interventional data sample sizes We repeat the experiments of Table 1 for ENCO while", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "limiting the sample size per intervention to 20, 50, and 100 (200 before). The observational dataset", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "size of 5000 samples is thereby kept constant. We plot the performance for all graph structures in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "Figure 14. Overall, the decrease of performance with lower interventional sample size is consistent", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "across graph structures. With only 20 samples per intervention, it becomes especially hard to reason", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "about variables with many parents, since the variable’s distribution is determined by many other", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "parents as well. Yet, for four out of the six graphs, we obtain an SHD of less than 1 with 100", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "interventional samples, and less than 6 when only 20 samples are available. In conclusion, ENCO", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 514, + 433, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 433, + 526 + ], + "score": 1.0, + "content": "works well with little interventional data if most variables have a small parent set.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 538, + 505, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "Limited observational data sample sizes Similarly as above, we repeat the experiments of Table 1", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 547, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 563 + ], + "score": 1.0, + "content": "for ENCO but limit the observational sample size to 1000 and 2000 (5000 before) while keeping", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 560, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 572 + ], + "score": 1.0, + "content": "200 samples per interventions. Observational data is important in ENCO for learning the conditional", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 584 + ], + "score": 1.0, + "content": "distributions. For variables with many parents, this becomes more difficult when fewer samples", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "are available, because the input space grows exponentially with the number of parents. Thus, we", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "table", + "bbox": [ + 108, + 655, + 504, + 730 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 612, + 505, + 646 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 611, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 104, + 611, + 506, + 625 + ], + "score": 1.0, + "content": "Table 11: Repeating experiments of Table 1 with large sample sizes (10k samples per intervention,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 127, + 634 + ], + "score": 0.29, + "content": "1 0 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "observational samples). In line with the theoretical guarantees, ENCO can reliably recover five", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 635, + 289, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 289, + 647 + ], + "score": 1.0, + "content": "out of the six graph structures without errors.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + }, + { + "type": "table_body", + "bbox": [ + 108, + 655, + 504, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 655, + 504, + 730 + ], + "spans": [ + { + "bbox": [ + 108, + 655, + 504, + 730 + ], + "score": 0.981, + "html": "
Graph typebidiagchaincolliderfull junglerandom
GIES47.4(±5.2)22.3(±3.5)13.3 (±3.0)152.7 (±12.0)53.9 (±8.9)86.1(±12.0)
IGSP33.0 (±4.2)12.0 (±1.9)23.4(±2.2)264.6 (±7.4)38.6 (±5.7)76.3 (±7.7)
SDI2.1 (±1.5)0.8(±0.9)14.7(±4.0)121.6 (±18.4)1.8 (±1.6)1.8 (±1.9)
DCDI3.7(±1.5)4.0 (±1.3)0.0 (±0.0)2.8(±2.1)1.2 (±1.5)2.2(±1.5)
ENCO (Ours)0.0 (±0.0)0.0 (±0.0)0.0 (±0.0)0.3 (±0.9)0.0 (±0.0)0.0 (±0.0)
", + "type": "table", + "image_path": "f8b0379371cb60451c69e94dd2f8e41d8a8dbe0f2e8f192a543dab4dfd5ac9d6.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 108, + 655, + 504, + 680.0 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 108, + 680.0, + 504, + 705.0 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 108, + 705.0, + 504, + 730.0 + ], + "spans": [], + "index": 44 + } + ] + } + ], + "index": 41.5 + } + ], + "page_idx": 43, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "44", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 271, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 273, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 273, + 96 + ], + "score": 1.0, + "content": "D ADDITIONAL EXPERIMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "score": 1.0, + "content": "In this section, we show additional experiments performed as ablation studies of ENCO. First, we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 505, + 130 + ], + "score": 1.0, + "content": "discuss further experiments We then discuss the effect of using our gradient estimators proposed in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "Section 3.4 compared to Bengio et al. (2020). Next, we show experiments on synthetic graphs with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 137, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 153 + ], + "score": 1.0, + "content": "deterministic variables violating faithfulness, and experiments on continuous data with Normalizing", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 163 + ], + "score": 1.0, + "content": "Flows. Finally, we discuss experiments with different causal mechanism functions for generating", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 162, + 394, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 394, + 173 + ], + "score": 1.0, + "content": "synthetic, conditional categorical distributions besides neural networks.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 106, + 506, + 173 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 187, + 257, + 198 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 259, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 259, + 199 + ], + "score": 1.0, + "content": "D.1 EFFECT OF THE SAMPLE SIZE", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 108, + 207, + 504, + 252 + ], + "lines": [ + { + "bbox": [ + 106, + 208, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 219 + ], + "score": 1.0, + "content": "The number of samples provided as observational and interventional data is crucial for causal structure", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "learning methods since the more data we have, the better we can estimate the underlying causal", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 230, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 506, + 241 + ], + "score": 1.0, + "content": "mechanisms. To gain further insights in the effect of the sample size on ENCO and the compared", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 241, + 425, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 425, + 253 + ], + "score": 1.0, + "content": "baselines, we repeat the experiments of Section 4.2 with different sample sizes.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 106, + 208, + 506, + 253 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 264, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "Large sample size First, we use very large sample sizes to find the upper bound performance level", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 276, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 347, + 288 + ], + "score": 1.0, + "content": "that we can expect from each method. For this, we sample", + "type": "text" + }, + { + "bbox": [ + 347, + 276, + 369, + 286 + ], + "score": 0.33, + "content": "1 0 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 276, + 506, + 288 + ], + "score": 1.0, + "content": "observational samples per graph,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "and 10k samples per intervention. We observed that this is sufficient to model most conditional", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "probabilities up to a negligible error. The results are shown in Table 11. We find that, in line with the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 308, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 322 + ], + "score": 1.0, + "content": "theoretical guarantees, ENCO can reliably recover most graphs, only making 0.3 mistakes on average", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "on the full graph. Of the baselines, only DCDI is able to recover the collider graph without errors", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "since its edges can be independently orientated. For all other graphs, DCDI converges to acyclic", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "graphs, but incorrectly orients some edges and predicts false positive edges, while being 8 times", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "slower than ENCO on the same hardware. All other baselines show improved SHD scores than in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 363, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 375 + ], + "score": 1.0, + "content": "Table 1 as well, but are not able to match ENCO’s performance. This shows that, even in the data", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 375, + 386, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 386, + 386 + ], + "score": 1.0, + "content": "limit, ENCO achieves notably better results than concurrent methods.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 265, + 506, + 386 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 391, + 504, + 413 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "Next, we consider situations where data is very limited. Thereby, we consider two data sample axes:", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 402, + 258, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 258, + 414 + ], + "score": 1.0, + "content": "observational and interventional data.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 106, + 390, + 505, + 414 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "Limited interventional data sample sizes We repeat the experiments of Table 1 for ENCO while", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "limiting the sample size per intervention to 20, 50, and 100 (200 before). The observational dataset", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "size of 5000 samples is thereby kept constant. We plot the performance for all graph structures in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "Figure 14. Overall, the decrease of performance with lower interventional sample size is consistent", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "across graph structures. With only 20 samples per intervention, it becomes especially hard to reason", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "about variables with many parents, since the variable’s distribution is determined by many other", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "parents as well. Yet, for four out of the six graphs, we obtain an SHD of less than 1 with 100", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "interventional samples, and less than 6 when only 20 samples are available. In conclusion, ENCO", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 514, + 433, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 433, + 526 + ], + "score": 1.0, + "content": "works well with little interventional data if most variables have a small parent set.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 426, + 506, + 526 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 538, + 505, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "Limited observational data sample sizes Similarly as above, we repeat the experiments of Table 1", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 547, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 563 + ], + "score": 1.0, + "content": "for ENCO but limit the observational sample size to 1000 and 2000 (5000 before) while keeping", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 560, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 572 + ], + "score": 1.0, + "content": "200 samples per interventions. Observational data is important in ENCO for learning the conditional", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 584 + ], + "score": 1.0, + "content": "distributions. For variables with many parents, this becomes more difficult when fewer samples", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "are available, because the input space grows exponentially with the number of parents. Thus, we", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "would expect the collider and full graph suffer the most from having less observational data, and", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "this is indeed the case as shown by the results in Figure 15. The results of all other graphs are less", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "affected, although interestingly, some become even better with less observational data. For the chain", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 193, + 564 + ], + "score": 0.92, + "content": "X _ { 1 } X _ { 2 } . . . X _ { N }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 193, + 552, + 506, + 565 + ], + "score": 1.0, + "content": ", for instance, we observed that the learned conditional distributions picked", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 333, + 577 + ], + "score": 1.0, + "content": "up spurious correlations among variables, e.g., between", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 333, + 564, + 347, + 575 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 347, + 564, + 365, + 577 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 365, + 564, + 379, + 575 + ], + "score": 0.89, + "content": "X _ { 3 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 380, + 564, + 446, + 577 + ], + "score": 1.0, + "content": "when modeling", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 446, + 564, + 505, + 576 + ], + "score": 0.92, + "content": "p ( X _ { 3 } | X _ { 1 } , X _ { 2 } )", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 288, + 587 + ], + "score": 1.0, + "content": "which are, in the data limit, independent given", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 288, + 575, + 302, + 586 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 302, + 574, + 506, + 587 + ], + "score": 1.0, + "content": ". Since those correlations do not necessarily transfer", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "to the interventional setting, it is easier to spot false positive edges, and we can obtain even better", + "type": "text", + "cross_page": true + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "results than for the larger sample sizes. In conclusion, having sufficient observational data is crucial", + "type": "text", + "cross_page": true + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "score": 1.0, + "content": "in ENCO for graphs with variables that have larger parent sets, while being less important for sparser", + "type": "text", + "cross_page": true + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 619, + 138, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 619, + 138, + 631 + ], + "score": 1.0, + "content": "graphs.", + "type": "text", + "cross_page": true + } + ], + "index": 22 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 537, + 505, + 595 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 655, + 504, + 730 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 612, + 505, + 646 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 611, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 104, + 611, + 506, + 625 + ], + "score": 1.0, + "content": "Table 11: Repeating experiments of Table 1 with large sample sizes (10k samples per intervention,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 127, + 634 + ], + "score": 0.29, + "content": "1 0 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "observational samples). In line with the theoretical guarantees, ENCO can reliably recover five", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 635, + 289, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 289, + 647 + ], + "score": 1.0, + "content": "out of the six graph structures without errors.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + }, + { + "type": "table_body", + "bbox": [ + 108, + 655, + 504, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 655, + 504, + 730 + ], + "spans": [ + { + "bbox": [ + 108, + 655, + 504, + 730 + ], + "score": 0.981, + "html": "
Graph typebidiagchaincolliderfull junglerandom
GIES47.4(±5.2)22.3(±3.5)13.3 (±3.0)152.7 (±12.0)53.9 (±8.9)86.1(±12.0)
IGSP33.0 (±4.2)12.0 (±1.9)23.4(±2.2)264.6 (±7.4)38.6 (±5.7)76.3 (±7.7)
SDI2.1 (±1.5)0.8(±0.9)14.7(±4.0)121.6 (±18.4)1.8 (±1.6)1.8 (±1.9)
DCDI3.7(±1.5)4.0 (±1.3)0.0 (±0.0)2.8(±2.1)1.2 (±1.5)2.2(±1.5)
ENCO (Ours)0.0 (±0.0)0.0 (±0.0)0.0 (±0.0)0.3 (±0.9)0.0 (±0.0)0.0 (±0.0)
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Note the different scale of the y-axis for the six graphs. 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Note the different scale of the y-axis for the six graphs. The structure learning performance", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 486, + 418, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 418, + 498 + ], + "score": 1.0, + "content": "remains good for sparse graphs, and suffers for graphs with larger parent sets.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "would expect the collider and full graph suffer the most from having less observational data, and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "this is indeed the case as shown by the results in Figure 15. The results of all other graphs are less", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "affected, although interestingly, some become even better with less observational data. For the chain", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 193, + 564 + ], + "score": 0.92, + "content": "X _ { 1 } X _ { 2 } . . . X _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 552, + 506, + 565 + ], + "score": 1.0, + "content": ", for instance, we observed that the learned conditional distributions picked", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 333, + 577 + ], + "score": 1.0, + "content": "up spurious correlations among variables, e.g., between", + "type": "text" + }, + { + "bbox": [ + 333, + 564, + 347, + 575 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 564, + 365, + 577 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 365, + 564, + 379, + 575 + ], + "score": 0.89, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 564, + 446, + 577 + ], + "score": 1.0, + "content": "when modeling", + "type": "text" + }, + { + "bbox": [ + 446, + 564, + 505, + 576 + ], + "score": 0.92, + "content": "p ( X _ { 3 } | X _ { 1 } , X _ { 2 } )", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 288, + 587 + ], + "score": 1.0, + "content": "which are, in the data limit, independent given", + "type": "text" + }, + { + "bbox": [ + 288, + 575, + 302, + 586 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 574, + 506, + 587 + ], + "score": 1.0, + "content": ". Since those correlations do not necessarily transfer", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "to the interventional setting, it is easier to spot false positive edges, and we can obtain even better", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "results than for the larger sample sizes. In conclusion, having sufficient observational data is crucial", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "score": 1.0, + "content": "in ENCO for graphs with variables that have larger parent sets, while being less important for sparser", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 619, + 138, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 619, + 138, + 631 + ], + "score": 1.0, + "content": "graphs.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "Limited interventional and observational data sample sizes Finally, we combine the smallest", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "interventional and observational data sample sizes, and also include the results of the previously best", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "baselines, SDI and DCDI, in Table 12. The results of ENCO show the combination of the previous", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "two effects: graphs consisting of variables with small parent sets can still be recovered well by ENCO,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "while errors increase for the collider and full graph. Similar trends are observed for SDI, while DCDI", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "showed a considerable decrease in performance for all graphs. In conclusion, ENCO still works well", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "for graphs with smaller parent sets under a small observational and interventional data regime, and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 721, + 285, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 285, + 733 + ], + "score": 1.0, + "content": "outperforms related baselines in this setting.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5 + } + ], + "page_idx": 44, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 79, + 504, + 235 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 79, + 504, + 235 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 504, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 504, + 235 + ], + "score": 0.973, + "type": "image", + "image_path": "3771db542bf98843e10131cad9fc24eccdab2ac3e95caa4233477b03d95c428d.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 79, + 504, + 131.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 131.0, + 504, + 183.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 183.0, + 504, + 235.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 243, + 505, + 288 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "Figure 14: Results of ENCO for different graph structures under limited interventional data sample", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "score": 1.0, + "content": "size. 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Note the different scale of the y-axis for the six graphs. The structure learning performance", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 486, + 418, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 418, + 498 + ], + "score": 1.0, + "content": "remains good for sparse graphs, and suffers for graphs with larger parent sets.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 505, + 630 + ], + "lines": [], + "index": 17.5, + "bbox_fs": [ + 104, + 520, + 506, + 631 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "Limited interventional and observational data sample sizes Finally, we combine the smallest", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "interventional and observational data sample sizes, and also include the results of the previously best", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "baselines, SDI and DCDI, in Table 12. The results of ENCO show the combination of the previous", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "two effects: graphs consisting of variables with small parent sets can still be recovered well by ENCO,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "while errors increase for the collider and full graph. Similar trends are observed for SDI, while DCDI", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "showed a considerable decrease in performance for all graphs. In conclusion, ENCO still works well", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "for graphs with smaller parent sets under a small observational and interventional data regime, and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 721, + 285, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 285, + 733 + ], + "score": 1.0, + "content": "outperforms related baselines in this setting.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 643, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 123, + 504, + 185 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 506, + 114 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 506, + 93 + ], + "score": 1.0, + "content": "Table 12: Repeating experiments of Table 1 with very small sample sizes (20 samples per intervention,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 505, + 104 + ], + "score": 1.0, + "content": "1k observational samples). Despite the limited data, ENCO can recover graphs with small parent sets", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 400, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 400, + 114 + ], + "score": 1.0, + "content": "reasonably well, while the graphs collider and full suffer for all methods.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 108, + 123, + 504, + 185 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 123, + 504, + 185 + ], + "spans": [ + { + "bbox": [ + 108, + 123, + 504, + 185 + ], + "score": 0.956, + "html": "
Graph typebidiagchaincolliderfull junglerandom
SDI10.9 (±2.7)6.1 (±1.5)22.1 (±1.9)211.0 (±6.2)10.4 (±2.7)22.7 (±7.4)
DCDI30.0 (±4.2)22.0 (±1.5)23.2 (±1.3)185.2(±4.5)25.8(±2.7)40.2 (±8.4)
ENCO (Ours)9.7 (±3.6)5.6 (±1.7)22.7 (±2.1)132.6 (±8.0)8.1(±2.3)18.4(±4.8)
ENCO-acyclic (Ours)2.0(±2.3)2.7(±2.1)22.9 (±2.3)88.4(±6.6)4.1(±2.0)5.3(±2.5)
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Note the different scale of the y-axis for the six graphs. For four out of six graphs, ENCO", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 387, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 505, + 398 + ], + "score": 1.0, + "content": "outperforms DCDI even for as few as 4 interventions, especially when enforcing acyclicity. The", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 397, + 319, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 319, + 408 + ], + "score": 1.0, + "content": "detailed numbers of the results are listed in Table 13.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + } + ], + "index": 8.75 + }, + { + "type": "title", + "bbox": [ + 109, + 432, + 301, + 442 + ], + "lines": [ + { + "bbox": [ + 106, + 431, + 302, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 302, + 444 + ], + "score": 1.0, + "content": "D.2 INTERVENTIONS ON FEWER VARIABLES", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "We have performed the experiments in Section 4.4 using fewer interventions for all six synthetic", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 464, + 507, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 507, + 477 + ], + "score": 1.0, + "content": "graph structures. The results are visualized in Figure 16, and presented in table-form in Table 13.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "From the experiments, we can see that ENCO with interventions on only 4 variables matches or", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "score": 1.0, + "content": "outperforms DCDI with 10 interventions for 4 out of the 6 graph structures (bidiag, chain, full, jungle)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "when enforcing acyclicity. Especially on chain-like graphs such as jungle, ENCO achieves lower", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "score": 1.0, + "content": "SHD scores for the same number of interventions on variables, while DCDI incorrectly orientates", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 505, + 531 + ], + "score": 1.0, + "content": "many edges and predicts false positive edges between children. On the graph collider, we observed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 530, + 504, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 504, + 541 + ], + "score": 1.0, + "content": "a high variance for settings with very few interventions. This is because when we intervene on the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 540, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 553 + ], + "score": 1.0, + "content": "collider node itself, ENCO can deduce the orientation for all edges. Finally, on the graph random,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "we observe that enforcing acyclicity for ENCO reduces the error a lot. This is because incorrectly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 563, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 575 + ], + "score": 1.0, + "content": "orientated edges cause more false positives in this densely connected graph, which are removed with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "the cycles. We include a longer discussion of the limitations of ENCO on fewer interventions in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "score": 1.0, + "content": "Appendix B.4. Still, we conclude that, in practice, ENCO performs competitively to DCDI, even", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 596, + 434, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 434, + 607 + ], + "score": 1.0, + "content": "when very few interventions are provided, and scales better to more interventions.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 107, + 623, + 329, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 623, + 330, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 330, + 635 + ], + "score": 1.0, + "content": "D.3 ABLATION STUDY ON GRADIENT ESTIMATORS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "To analyze the importance of the low-variance gradient estimators in ENCO, we repeat the experiments", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 655, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 666 + ], + "score": 1.0, + "content": "on synthetic graph structure from Section 4.2 where the gradient estimators of ENCO have been", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "replaced by those from Bengio et al. (2020). The results are shown in Table 14. Overall, the scores", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "are very similar with minor differences for the graphs full and bidiag. In comparisons to the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "learning curves in Figure 11, the curves with the gradient estimator of Bengio et al. (2020) are more", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "noisy, with recall and precision jumping up and down. While ENCO easily converged early to the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "correct graphs for all graph types, this model often required the full 30 iterations to reach the optimal", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 721, + 146, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 146, + 734 + ], + "score": 1.0, + "content": "recovery.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5 + } + ], + "page_idx": 45, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 123, + 504, + 185 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 506, + 114 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 506, + 93 + ], + "score": 1.0, + "content": "Table 12: Repeating experiments of Table 1 with very small sample sizes (20 samples per intervention,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 505, + 104 + ], + "score": 1.0, + "content": "1k observational samples). Despite the limited data, ENCO can recover graphs with small parent sets", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 400, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 400, + 114 + ], + "score": 1.0, + "content": "reasonably well, while the graphs collider and full suffer for all methods.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 108, + 123, + 504, + 185 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 123, + 504, + 185 + ], + "spans": [ + { + "bbox": [ + 108, + 123, + 504, + 185 + ], + "score": 0.956, + "html": "
Graph typebidiagchaincolliderfull junglerandom
SDI10.9 (±2.7)6.1 (±1.5)22.1 (±1.9)211.0 (±6.2)10.4 (±2.7)22.7 (±7.4)
DCDI30.0 (±4.2)22.0 (±1.5)23.2 (±1.3)185.2(±4.5)25.8(±2.7)40.2 (±8.4)
ENCO (Ours)9.7 (±3.6)5.6 (±1.7)22.7 (±2.1)132.6 (±8.0)8.1(±2.3)18.4(±4.8)
ENCO-acyclic (Ours)2.0(±2.3)2.7(±2.1)22.9 (±2.3)88.4(±6.6)4.1(±2.0)5.3(±2.5)
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Note the different scale of the y-axis for the six graphs. For four out of six graphs, ENCO", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 387, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 505, + 398 + ], + "score": 1.0, + "content": "outperforms DCDI even for as few as 4 interventions, especially when enforcing acyclicity. The", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 397, + 319, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 319, + 408 + ], + "score": 1.0, + "content": "detailed numbers of the results are listed in Table 13.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + } + ], + "index": 8.75 + }, + { + "type": "title", + "bbox": [ + 109, + 432, + 301, + 442 + ], + "lines": [ + { + "bbox": [ + 106, + 431, + 302, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 302, + 444 + ], + "score": 1.0, + "content": "D.2 INTERVENTIONS ON FEWER VARIABLES", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "We have performed the experiments in Section 4.4 using fewer interventions for all six synthetic", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 464, + 507, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 507, + 477 + ], + "score": 1.0, + "content": "graph structures. The results are visualized in Figure 16, and presented in table-form in Table 13.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "From the experiments, we can see that ENCO with interventions on only 4 variables matches or", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "score": 1.0, + "content": "outperforms DCDI with 10 interventions for 4 out of the 6 graph structures (bidiag, chain, full, jungle)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "when enforcing acyclicity. Especially on chain-like graphs such as jungle, ENCO achieves lower", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "score": 1.0, + "content": "SHD scores for the same number of interventions on variables, while DCDI incorrectly orientates", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 505, + 531 + ], + "score": 1.0, + "content": "many edges and predicts false positive edges between children. On the graph collider, we observed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 530, + 504, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 504, + 541 + ], + "score": 1.0, + "content": "a high variance for settings with very few interventions. This is because when we intervene on the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 540, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 553 + ], + "score": 1.0, + "content": "collider node itself, ENCO can deduce the orientation for all edges. Finally, on the graph random,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "we observe that enforcing acyclicity for ENCO reduces the error a lot. This is because incorrectly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 563, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 575 + ], + "score": 1.0, + "content": "orientated edges cause more false positives in this densely connected graph, which are removed with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "the cycles. We include a longer discussion of the limitations of ENCO on fewer interventions in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "score": 1.0, + "content": "Appendix B.4. Still, we conclude that, in practice, ENCO performs competitively to DCDI, even", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 596, + 434, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 434, + 607 + ], + "score": 1.0, + "content": "when very few interventions are provided, and scales better to more interventions.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 452, + 507, + 607 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 623, + 329, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 623, + 330, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 330, + 635 + ], + "score": 1.0, + "content": "D.3 ABLATION STUDY ON GRADIENT ESTIMATORS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "To analyze the importance of the low-variance gradient estimators in ENCO, we repeat the experiments", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 655, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 666 + ], + "score": 1.0, + "content": "on synthetic graph structure from Section 4.2 where the gradient estimators of ENCO have been", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "replaced by those from Bengio et al. (2020). The results are shown in Table 14. Overall, the scores", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "are very similar with minor differences for the graphs full and bidiag. In comparisons to the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "learning curves in Figure 11, the curves with the gradient estimator of Bengio et al. (2020) are more", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "noisy, with recall and precision jumping up and down. While ENCO easily converged early to the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "correct graphs for all graph types, this model often required the full 30 iterations to reach the optimal", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 721, + 146, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 146, + 734 + ], + "score": 1.0, + "content": "recovery.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 644, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 111, + 504, + 329 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 102 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "score": 1.0, + "content": "Table 13: Detailed results of the experiments with fewer interventions. See Figure 16 for a visualiza-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 187, + 103 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 187, + 103 + ], + "score": 1.0, + "content": "tion and discussion.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 107, + 111, + 504, + 329 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 111, + 504, + 329 + ], + "spans": [ + { + "bbox": [ + 107, + 111, + 504, + 329 + ], + "score": 0.985, + "html": "
Graph typebidiagchaincolliderfull junglerandom
DCDI 4 vars25.8(±2.0)23.6 (±11.3)12.5 (±1.8)143.8 (±10.7)38.5 (±3.2)26.3(±6.2)
DCDI 6 vars24.3(±2.2)14.6 (±2.7)12.5 (±1.9)142.2 (±13.5)32.7(±3.8)23.1 (±7.2)
DCDI8 vars23.5 (±1.4)13.3 (±2.4)12.3 (±2.0)134.8(±8.9)28.8(±7.4)19.1(±5.5)
DCDI 10 vars22.4(±1.1)13.0 (±4.1)10.8 (±3.6)126.2(±4.2)28.0 (±3.2)14.8 (±6.3)
DCDI 15 vars22.0 (±1.9)12.5 (±2.1)11.5 (±3.7)90.2 (±7.1)25.8 (±2.1)12.2 (±5.3)
DCDI 20 vars20.8 (±1.4)11.5 (±1.3)12.0 (±2.9)62.8 (±9.8)19.8 (±3.3)10.2 (±3.0)
DCDI 25 vars16.9 (±2.0)10.1 (±1.1)10.9 (±3.6)21.0 (±4.8)17.9 (±4.1)7.7(±3.2)
ENCO 4 vars29.8(±5.6)19.9 (±3.2)13.9 (±6.6)110.6 (±8.6)30.8 (±6.1)28.5 (±4.3)
ENCO 6 vars25.0(±3.0)16.4(±2.8)10.1 (±6.1)97.5 (±8.5)23.6(±5.1)23.4(±3.1)
ENCO 8 vars20.8(±3.8)13.2 (±2.4)7.0 (±6.3)86.3(±8.9)19.4(±5.0)20.8 (±3.5)
ENCO 10 vars17.7(±4.7)10.3 (±1.6)5.6 (±4.9)77.3(±8.5)15.6 (±3.8)18.7(±2.9)
ENCO 15 vars10.7(±2.3)6.9 (±1.5)3.8(±2.5)52.8(±9.1)9.3 (±2.3)13.6 (±2.8)
ENCO 20 vars5.0 (±2.2)3.0 (±1.5)3.4(±1.9)32.4 (±4.2)3.5 (±1.1)8.8 (±3.1)
ENCO 25 vars2.2 (±1.4)1.7 (±1.3)1.6 (±1.6)9.2 (±3.4)1.7 (±1.3)4.6 (±1.9)
ENCO-acyclic 4 vars20.4(±7.5)10.6 (±4.9)13.9 (±6.7)110.6 (±8.6)19.8 (±6.0)9.0 (±3.9)
ENCO-acyclic 6 vars15.8(±5.9)7.8 (±4.4)10.1(±6.1)97.5 (±8.5)15.6 (±3.7)6.0 (±3.2)
ENCO-acyclic 8 vars11.2 (±5.4)5.4(±3.7)7.0 (±6.3)86.3 (±8.9)12.6 (±4.6)5.0(±3.5)
ENCO-acyclic 10 vars8.5 (±5.8)4.4(±3.4)5.6(±4.9)77.3(±8.5)9.8(±3.9)3.2 (±2.2)
ENCO-acyclic 15 vars2.5(±3.7)3.4(±2.8)3.8(±2.5)52.8 (±9.1)5.1(±2.7)2.1 (±1.7)
ENCO-acyclic 20 vars0.3 (±0.7)0.7(±1.2)3.4(±1.9)32.4 (±4.2)1.7 (±1.5)1.2 (±1.2)
ENCO-acyclic 25 vars0.0 (±0.0)0.0 (±0.0)1.6 (±1.6)5.3(±2.3)0.6 (±1.1)0.2(±0.5)
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Graph typebidiagchaincolliderfull junglerandom
ENCO (Ours)
- Bengio et al. (2020) grads0.1 (±0.4)0.0 (±0.0)0.0 (±0.0)1.9 (±1.5)0.0 (±0.0)0.0 (±0.0)
- Our gradient estimator0.0 (±0.0)0.0 (±0.0)0.0 (±0.0)0.3 (±0.9)0.0 (±0.0)0.0 (±0.0)
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We", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 450, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 461 + ], + "score": 1.0, + "content": "repeated the experiments of Section 4.3 on the graphs with 100 nodes using the gradient estimator of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "Bengio et al. (2020). Within the 30 epochs, the model obtained an SHD of 15.4 on average over 5", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "experiments, which is considerably higher than ENCO with the proposed gradient estimators (0.0).", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 482, + 507, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 507, + 495 + ], + "score": 1.0, + "content": "Still, this is only half of the errors that SDI (Ke et al., 2019) with the same gradient estimator achieved.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 494, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 506, + 506 + ], + "score": 1.0, + "content": "Hence, we can conclude that the proposed gradient estimators are beneficial for ENCO but not strictly", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 505, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 516 + ], + "score": 1.0, + "content": "necessary for small graphs. For large graphs, the low variance of the estimator becomes much more", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 516, + 151, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 151, + 528 + ], + "score": 1.0, + "content": "important.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 540, + 253, + 551 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 254, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 254, + 552 + ], + "score": 1.0, + "content": "D.4 DETERMINISTIC VARIABLES", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 572 + ], + "score": 1.0, + "content": "In contrast to algorithms working on observational data, ENCO does not strictly require the faithful-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 572, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 583 + ], + "score": 1.0, + "content": "ness assumption. Hence, we can apply ENCO to graphs with deterministic variables. Deterministic", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "variables have a distribution that is defined by a one-to-one mapping of its parents’ inputs to an output", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 594, + 338, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 338, + 606 + ], + "score": 1.0, + "content": "value. In other words, we have the following distribution:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 609, + 382, + 623 + ], + "lines": [ + { + "bbox": [ + 229, + 609, + 382, + 623 + ], + "spans": [ + { + "bbox": [ + 229, + 609, + 382, + 623 + ], + "score": 0.92, + "content": "p ( X _ { i } | \\mathsf { p a } ( X _ { i } ) ) = \\mathbb { 1 } \\left[ X _ { i } = f ( \\mathsf { p a } ( X _ { i } ) ) \\right]", + "type": "interline_equation", + "image_path": "e741e3ce03c8a2822d2d1bbc0125bacaf2ffc34942b0a48cf6185ac5070e349e.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 229, + 609, + 382, + 623 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 504, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 134, + 638 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 627, + 142, + 639 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 626, + 491, + 638 + ], + "score": 1.0, + "content": "is an arbitrary function. The difficulty of deterministic variables is that a variable", + "type": "text" + }, + { + "bbox": [ + 491, + 627, + 504, + 638 + ], + "score": 0.86, + "content": "X _ { i }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 252, + 650 + ], + "score": 1.0, + "content": "can be fully replaced by its parents", + "type": "text" + }, + { + "bbox": [ + 253, + 638, + 282, + 650 + ], + "score": 0.88, + "content": "\\operatorname { p a } ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "in any conditional distribution. The only way we can", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 447, + 660 + ], + "score": 1.0, + "content": "identify deterministic variables is from interventional data, where an intervention on", + "type": "text" + }, + { + "bbox": [ + 448, + 649, + 460, + 660 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "breaks the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 660, + 213, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 213, + 672 + ], + "score": 1.0, + "content": "dependency to its parents.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 107, + 677, + 504, + 688 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 504, + 688 + ], + "score": 1.0, + "content": "We have already tested ENCO on deterministic variables in the context of the real-world inspired", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "graphs of Section 4.6. To have a more detailed analysis, we created synthetic graphs following the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "random graph setup with an edge probability of 0.1 and an average of two parents, and maximum", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "of three parents. An example graph is shown in Figure 17. All variables except the leaf nodes", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 321, + 734 + ], + "score": 1.0, + "content": "have deterministic distributions, where the function", + "type": "text" + }, + { + "bbox": [ + 322, + 720, + 365, + 732 + ], + "score": 0.94, + "content": "\\mathsf { \\bar { f } } ( \\mathsf { p a } ( X _ { i } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "is randomly created by picking a", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + } + ], + "page_idx": 46, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 759 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 111, + 504, + 329 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 102 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "score": 1.0, + "content": "Table 13: Detailed results of the experiments with fewer interventions. See Figure 16 for a visualiza-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 187, + 103 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 187, + 103 + ], + "score": 1.0, + "content": "tion and discussion.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 107, + 111, + 504, + 329 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 111, + 504, + 329 + ], + "spans": [ + { + "bbox": [ + 107, + 111, + 504, + 329 + ], + "score": 0.985, + "html": "
Graph typebidiagchaincolliderfull junglerandom
DCDI 4 vars25.8(±2.0)23.6 (±11.3)12.5 (±1.8)143.8 (±10.7)38.5 (±3.2)26.3(±6.2)
DCDI 6 vars24.3(±2.2)14.6 (±2.7)12.5 (±1.9)142.2 (±13.5)32.7(±3.8)23.1 (±7.2)
DCDI8 vars23.5 (±1.4)13.3 (±2.4)12.3 (±2.0)134.8(±8.9)28.8(±7.4)19.1(±5.5)
DCDI 10 vars22.4(±1.1)13.0 (±4.1)10.8 (±3.6)126.2(±4.2)28.0 (±3.2)14.8 (±6.3)
DCDI 15 vars22.0 (±1.9)12.5 (±2.1)11.5 (±3.7)90.2 (±7.1)25.8 (±2.1)12.2 (±5.3)
DCDI 20 vars20.8 (±1.4)11.5 (±1.3)12.0 (±2.9)62.8 (±9.8)19.8 (±3.3)10.2 (±3.0)
DCDI 25 vars16.9 (±2.0)10.1 (±1.1)10.9 (±3.6)21.0 (±4.8)17.9 (±4.1)7.7(±3.2)
ENCO 4 vars29.8(±5.6)19.9 (±3.2)13.9 (±6.6)110.6 (±8.6)30.8 (±6.1)28.5 (±4.3)
ENCO 6 vars25.0(±3.0)16.4(±2.8)10.1 (±6.1)97.5 (±8.5)23.6(±5.1)23.4(±3.1)
ENCO 8 vars20.8(±3.8)13.2 (±2.4)7.0 (±6.3)86.3(±8.9)19.4(±5.0)20.8 (±3.5)
ENCO 10 vars17.7(±4.7)10.3 (±1.6)5.6 (±4.9)77.3(±8.5)15.6 (±3.8)18.7(±2.9)
ENCO 15 vars10.7(±2.3)6.9 (±1.5)3.8(±2.5)52.8(±9.1)9.3 (±2.3)13.6 (±2.8)
ENCO 20 vars5.0 (±2.2)3.0 (±1.5)3.4(±1.9)32.4 (±4.2)3.5 (±1.1)8.8 (±3.1)
ENCO 25 vars2.2 (±1.4)1.7 (±1.3)1.6 (±1.6)9.2 (±3.4)1.7 (±1.3)4.6 (±1.9)
ENCO-acyclic 4 vars20.4(±7.5)10.6 (±4.9)13.9 (±6.7)110.6 (±8.6)19.8 (±6.0)9.0 (±3.9)
ENCO-acyclic 6 vars15.8(±5.9)7.8 (±4.4)10.1(±6.1)97.5 (±8.5)15.6 (±3.7)6.0 (±3.2)
ENCO-acyclic 8 vars11.2 (±5.4)5.4(±3.7)7.0 (±6.3)86.3 (±8.9)12.6 (±4.6)5.0(±3.5)
ENCO-acyclic 10 vars8.5 (±5.8)4.4(±3.4)5.6(±4.9)77.3(±8.5)9.8(±3.9)3.2 (±2.2)
ENCO-acyclic 15 vars2.5(±3.7)3.4(±2.8)3.8(±2.5)52.8 (±9.1)5.1(±2.7)2.1 (±1.7)
ENCO-acyclic 20 vars0.3 (±0.7)0.7(±1.2)3.4(±1.9)32.4 (±4.2)1.7 (±1.5)1.2 (±1.2)
ENCO-acyclic 25 vars0.0 (±0.0)0.0 (±0.0)1.6 (±1.6)5.3(±2.3)0.6 (±1.1)0.2(±0.5)
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Graph typebidiagchaincolliderfull junglerandom
ENCO (Ours)
- Bengio et al. (2020) grads0.1 (±0.4)0.0 (±0.0)0.0 (±0.0)1.9 (±1.5)0.0 (±0.0)0.0 (±0.0)
- Our gradient estimator0.0 (±0.0)0.0 (±0.0)0.0 (±0.0)0.3 (±0.9)0.0 (±0.0)0.0 (±0.0)
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We", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 450, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 461 + ], + "score": 1.0, + "content": "repeated the experiments of Section 4.3 on the graphs with 100 nodes using the gradient estimator of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "Bengio et al. (2020). Within the 30 epochs, the model obtained an SHD of 15.4 on average over 5", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "experiments, which is considerably higher than ENCO with the proposed gradient estimators (0.0).", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 482, + 507, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 507, + 495 + ], + "score": 1.0, + "content": "Still, this is only half of the errors that SDI (Ke et al., 2019) with the same gradient estimator achieved.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 494, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 506, + 506 + ], + "score": 1.0, + "content": "Hence, we can conclude that the proposed gradient estimators are beneficial for ENCO but not strictly", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 505, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 516 + ], + "score": 1.0, + "content": "necessary for small graphs. For large graphs, the low variance of the estimator becomes much more", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 516, + 151, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 151, + 528 + ], + "score": 1.0, + "content": "important.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 438, + 507, + 528 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 540, + 253, + 551 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 254, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 254, + 552 + ], + "score": 1.0, + "content": "D.4 DETERMINISTIC VARIABLES", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 572 + ], + "score": 1.0, + "content": "In contrast to algorithms working on observational data, ENCO does not strictly require the faithful-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 572, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 583 + ], + "score": 1.0, + "content": "ness assumption. Hence, we can apply ENCO to graphs with deterministic variables. Deterministic", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "variables have a distribution that is defined by a one-to-one mapping of its parents’ inputs to an output", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 594, + 338, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 338, + 606 + ], + "score": 1.0, + "content": "value. In other words, we have the following distribution:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 561, + 506, + 606 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 609, + 382, + 623 + ], + "lines": [ + { + "bbox": [ + 229, + 609, + 382, + 623 + ], + "spans": [ + { + "bbox": [ + 229, + 609, + 382, + 623 + ], + "score": 0.92, + "content": "p ( X _ { i } | \\mathsf { p a } ( X _ { i } ) ) = \\mathbb { 1 } \\left[ X _ { i } = f ( \\mathsf { p a } ( X _ { i } ) ) \\right]", + "type": "interline_equation", + "image_path": "e741e3ce03c8a2822d2d1bbc0125bacaf2ffc34942b0a48cf6185ac5070e349e.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 229, + 609, + 382, + 623 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 504, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 134, + 638 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 627, + 142, + 639 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 626, + 491, + 638 + ], + "score": 1.0, + "content": "is an arbitrary function. The difficulty of deterministic variables is that a variable", + "type": "text" + }, + { + "bbox": [ + 491, + 627, + 504, + 638 + ], + "score": 0.86, + "content": "X _ { i }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 252, + 650 + ], + "score": 1.0, + "content": "can be fully replaced by its parents", + "type": "text" + }, + { + "bbox": [ + 253, + 638, + 282, + 650 + ], + "score": 0.88, + "content": "\\operatorname { p a } ( X _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "in any conditional distribution. The only way we can", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 447, + 660 + ], + "score": 1.0, + "content": "identify deterministic variables is from interventional data, where an intervention on", + "type": "text" + }, + { + "bbox": [ + 448, + 649, + 460, + 660 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "breaks the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 660, + 213, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 213, + 672 + ], + "score": 1.0, + "content": "dependency to its parents.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 626, + 505, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 107, + 677, + 504, + 688 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 504, + 688 + ], + "score": 1.0, + "content": "We have already tested ENCO on deterministic variables in the context of the real-world inspired", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "graphs of Section 4.6. To have a more detailed analysis, we created synthetic graphs following the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "random graph setup with an edge probability of 0.1 and an average of two parents, and maximum", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "of three parents. An example graph is shown in Figure 17. All variables except the leaf nodes", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 321, + 734 + ], + "score": 1.0, + "content": "have deterministic distributions, where the function", + "type": "text" + }, + { + "bbox": [ + 322, + 720, + 365, + 732 + ], + "score": 0.94, + "content": "\\mathsf { \\bar { f } } ( \\mathsf { p a } ( X _ { i } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "is randomly created by picking a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "random output category for any pair of input values. We create 10 such graphs and use the same", + "type": "text", + "cross_page": true + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 553, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 564 + ], + "score": 1.0, + "content": "hyperparameter setup as for the synthetic graphs, except that we increase the sparsity regularizer to", + "type": "text", + "cross_page": true + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 107, + 563, + 163, + 574 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 2", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 163, + 562, + 506, + 576 + ], + "score": 1.0, + "content": ". We report the results in Table 15. In line with the results on the real-world graphs,", + "type": "text", + "cross_page": true + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "ENCO is able to recover most graphs with less than two errors. As a baseline, we apply SDI (Ke et al.,", + "type": "text", + "cross_page": true + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 583, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 597 + ], + "score": 1.0, + "content": "2019) and see a significant higher error rate. The method predicts many false positives, including", + "type": "text", + "cross_page": true + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 596, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 608 + ], + "score": 1.0, + "content": "two-variable loops, but was also missing out some true positive edges. We conclude that ENCO also", + "type": "text", + "cross_page": true + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 606, + 251, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 251, + 619 + ], + "score": 1.0, + "content": "works well on deterministic graphs.", + "type": "text", + "cross_page": true + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 677, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 229, + 123, + 381, + 172 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 506, + 113 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "Table 15: Experiments on graphs with deterministic variables. The performance over 10 experiments", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 505, + 103 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 505, + 103 + ], + "score": 1.0, + "content": "is reported in terms of SHD with standard deviation in brackets. ENCO can recover most of the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 239, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 239, + 115 + ], + "score": 1.0, + "content": "graphs with less than two errors.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 229, + 123, + 381, + 172 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 229, + 123, + 381, + 172 + ], + "spans": [ + { + "bbox": [ + 229, + 123, + 381, + 172 + ], + "score": 0.972, + "html": "
Graph typedeterministic
SDI (Ke et al., 2019)20.6 (±3.8)
ENCO (Ours)1.4 (±1.3)
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This can, for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 320, + 506 + ], + "score": 1.0, + "content": "instance, occur when the conditional distribution of", + "type": "text" + }, + { + "bbox": [ + 320, + 495, + 334, + 505 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 494, + 431, + 506 + ], + "score": 1.0, + "content": "is a Dirac delta around", + "type": "text" + }, + { + "bbox": [ + 432, + 495, + 444, + 506 + ], + "score": 0.53, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 494, + 505, + 507 + ], + "score": 0.75, + "content": "\\hat { \\bf \\Phi } _ { 1 } \\colon p ( X _ { 2 } | X _ { 1 } ) =", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 503, + 162, + 519 + ], + "spans": [ + { + "bbox": [ + 107, + 505, + 156, + 518 + ], + "score": 0.91, + "content": "\\delta [ X _ { 2 } = X _ { 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 503, + 162, + 519 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + } + ], + "index": 22.75 + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "random output category for any pair of input values. We create 10 such graphs and use the same", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 553, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 564 + ], + "score": 1.0, + "content": "hyperparameter setup as for the synthetic graphs, except that we increase the sparsity regularizer to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 107, + 563, + 163, + 574 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { s p a r s e } } = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 562, + 506, + 576 + ], + "score": 1.0, + "content": ". We report the results in Table 15. In line with the results on the real-world graphs,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "ENCO is able to recover most graphs with less than two errors. As a baseline, we apply SDI (Ke et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 583, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 597 + ], + "score": 1.0, + "content": "2019) and see a significant higher error rate. The method predicts many false positives, including", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 596, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 608 + ], + "score": 1.0, + "content": "two-variable loops, but was also missing out some true positive edges. We conclude that ENCO also", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 606, + 251, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 251, + 619 + ], + "score": 1.0, + "content": "works well on deterministic graphs.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "Cancellation of paths Besides deterministic nodes, a common example for faithfulness violation", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "is the cancellation of two paths. 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Graph typedeterministic
SDI (Ke et al., 2019)20.6 (±3.8)
ENCO (Ours)1.4 (±1.3)
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We set", + "type": "text" + }, + { + "bbox": [ + 384, + 720, + 489, + 732 + ], + "score": 0.92, + "content": "p ( X _ { 2 } | X _ { 1 } ) = \\delta [ X _ { 2 } = X _ { 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "canceling the two paths, and the remaining distributions are randomly initialized. ENCO reconstructs", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 264, + 361, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 361, + 277 + ], + "score": 1.0, + "content": "the graph without errors, showing that it also works in practice.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 633, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 105, + 134, + 506, + 232 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 506, + 124 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 507, + 92 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 507, + 92 + ], + "score": 1.0, + "content": "Table 16: Experiments on graph with continuous data from Brouillard et al. (2020). The suffix “-G”", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 506, + 104 + ], + "score": 1.0, + "content": "denotes that the neural networks model a Gaussian density, and “-DSF” a two-layer deep sigmoidal", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 505, + 114 + ], + "score": 1.0, + "content": "flow. ENCO outperforms all baselines in this scenario, verifying that ENCO also works on continuous", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 149, + 125 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 149, + 125 + ], + "score": 1.0, + "content": "data well.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 105, + 134, + 506, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 134, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 506, + 232 + ], + "score": 0.985, + "html": "
Graph typeLinearNonlinear with additive noiseNonlinear with non-additive noise
GIES0.6 (±1.3)9.1(±5.3)4.4(±6.1)
IGSP (best)1.9(±1.8)5.3 (±3.0)4.1 (±2.8)
DCDI-G1.3 (±1.9)5.2(±7.5)2.3 (±3.6)
DCDI-DSF0.9 (±1.3)4.2 (±5.6)7.0 (±10.7)
GES + Orientating2.5 (±4.3)12.5 (±9.2)7.9(±7.4)
ENCO-G (ours)0.0(±0.0)0.2 (±0.4)0.1(±0.3)
ENCO-DSF (ours)0.3(±0.7)1.4(±1.4)1.2 (±1.5)
", + "type": "table", + "image_path": "6f18c288307fce9e3fc83bc6123a6ac46eda6966b65848a26e4e896ea23a958e.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 105, + 134, + 506, + 166.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 105, + 166.66666666666666, + 506, + 199.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 105, + 199.33333333333331, + 506, + 231.99999999999997 + ], + "spans": [], + "index": 6 + } + ] + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 504, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "canceling the two paths, and the remaining distributions are randomly initialized. ENCO reconstructs", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 264, + 361, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 361, + 277 + ], + "score": 1.0, + "content": "the graph without errors, showing that it also works in practice.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 108, + 290, + 216, + 301 + ], + "lines": [ + { + "bbox": [ + 106, + 290, + 217, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 217, + 303 + ], + "score": 1.0, + "content": "D.5 CONTINUOUS DATA", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 311, + 505, + 443 + ], + "lines": [ + { + "bbox": [ + 106, + 310, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 324 + ], + "score": 1.0, + "content": "We verify that ENCO works just as well with continuous data by performing the experiments on", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 321, + 507, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 507, + 335 + ], + "score": 1.0, + "content": "datasets from Brouillard et al. (2020) that contained interventions on all variables. In these datasets,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 334, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 504, + 345 + ], + "score": 1.0, + "content": "the graphs consist of 10 variables with an average of one edge per variable, and deploy three different", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "score": 1.0, + "content": "causal mechanisms: linear, nonlinear additive noise models, and nonlinear models with non-additive", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "score": 1.0, + "content": "noise using neural networks. The datasets contain 909 observational samples and 909 samples per", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "intervention. All results of GIES, IGSP, and DCDI have been taken from Brouillard et al. (2020)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "(Appendix C.7, Table 22-24). We follow the setup of Brouillard et al. (2020) and compare two", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 386, + 507, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 507, + 401 + ], + "score": 1.0, + "content": "different neural network setups. First, we use MLPs that model a Gaussian density by predicting a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 295, + 411 + ], + "score": 1.0, + "content": "mean and variance variable (denoted by suffix", + "type": "text" + }, + { + "bbox": [ + 295, + 400, + 304, + 409 + ], + "score": 0.61, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "). The second setup uses normalizing flows, more", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 410, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 422 + ], + "score": 1.0, + "content": "specifically a two-layer deep sigmoidal flow (Huang et al., 2018), which is flexible enough to model", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "more complex distributions (denoted by suffix DSF). The rest of the experimental setup in ENCO is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 432, + 236, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 236, + 444 + ], + "score": 1.0, + "content": "identical to the categorical case.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 448, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 460 + ], + "score": 1.0, + "content": "Results are shown in Table 16, and the observations are the same as with categorical data. ENCO", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "score": 1.0, + "content": "outperforms all other methods in all settings, especially for the more complex distributions. The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "higher error rate for the DSF setup is mostly due to overfitting of the flow models. We conclude that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 480, + 382, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 382, + 494 + ], + "score": 1.0, + "content": "ENCO works as accurately for both continuous and categorical data.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 107, + 507, + 370, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 371, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 371, + 519 + ], + "score": 1.0, + "content": "D.6 SKELETON LEARNING WITH OBSERVATIONAL BASELINE", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "To show the benefit of learning a graph from observational and interventional data jointly, we compare", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "ENCO to a simple observational baseline. This baseline first learns the skeleton of the graph by", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "score": 1.0, + "content": "applying greedy equivalence search (GES) (Chickering, 2002) on the observational data. Then, for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 562, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 573 + ], + "score": 1.0, + "content": "each interventional dataset, we apply GES as well and use those skeletons to orientate the edges of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 394, + 585 + ], + "score": 1.0, + "content": "the original one. This can be done by checking for each undirected edge", + "type": "text" + }, + { + "bbox": [ + 395, + 572, + 425, + 583 + ], + "score": 0.9, + "content": "X - Y", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 572, + 461, + 585 + ], + "score": 1.0, + "content": "whether", + "type": "text" + }, + { + "bbox": [ + 461, + 572, + 495, + 582 + ], + "score": 0.9, + "content": "X Y", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 246, + 595 + ], + "score": 1.0, + "content": "in the skeleton of interventions on", + "type": "text" + }, + { + "bbox": [ + 246, + 583, + 256, + 593 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "or not. As a reference implementation of GES, we have used", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 594, + 414, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 414, + 606 + ], + "score": 1.0, + "content": "the one provided in the Causal Discovery Toolbox (Kalainathan et al., 2020).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "The results on continuous data are shown in Table 16. Since GES assumes linear mechanisms and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "gaussianity of the data, it is unsurprising that it performs better on the linear Gaussian dataset than", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "on the non-linear datasets. However, on all the three datasets, it constitutes the lowest performance", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 642, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 658 + ], + "score": 1.0, + "content": "compared to the other methods, including ENCO. This highlights the benefits of incorporating", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "interventional data in the learning of the skeleton and graph structure. To gain further insights in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "comparison to the constraint-based baseline, we repeat the experiments with smaller sample sizes. The", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "original dataset has 909 samples for observational data and per intervention, and we sub-sample 500", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "and 100 of those respectively for simulating smaller dataset sizes. The results of those experiments", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "can be found in Table 17. It is apparent that the results of GES on the linear dataset get considerably", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "worse with fewer data samples being available, while ENCO-G is able to reconstruct most graphs", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "still without errors. Especially for the small dataset of 100 samples, we noticed that the skeletons", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 39 + } + ], + "page_idx": 48, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 105, + 134, + 506, + 232 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 506, + 124 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 507, + 92 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 507, + 92 + ], + "score": 1.0, + "content": "Table 16: Experiments on graph with continuous data from Brouillard et al. (2020). The suffix “-G”", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 506, + 104 + ], + "score": 1.0, + "content": "denotes that the neural networks model a Gaussian density, and “-DSF” a two-layer deep sigmoidal", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 505, + 114 + ], + "score": 1.0, + "content": "flow. ENCO outperforms all baselines in this scenario, verifying that ENCO also works on continuous", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 149, + 125 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 149, + 125 + ], + "score": 1.0, + "content": "data well.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 105, + 134, + 506, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 134, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 506, + 232 + ], + "score": 0.985, + "html": "
Graph typeLinearNonlinear with additive noiseNonlinear with non-additive noise
GIES0.6 (±1.3)9.1(±5.3)4.4(±6.1)
IGSP (best)1.9(±1.8)5.3 (±3.0)4.1 (±2.8)
DCDI-G1.3 (±1.9)5.2(±7.5)2.3 (±3.6)
DCDI-DSF0.9 (±1.3)4.2 (±5.6)7.0 (±10.7)
GES + Orientating2.5 (±4.3)12.5 (±9.2)7.9(±7.4)
ENCO-G (ours)0.0(±0.0)0.2 (±0.4)0.1(±0.3)
ENCO-DSF (ours)0.3(±0.7)1.4(±1.4)1.2 (±1.5)
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(2020) that contained interventions on all variables. In these datasets,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 334, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 504, + 345 + ], + "score": 1.0, + "content": "the graphs consist of 10 variables with an average of one edge per variable, and deploy three different", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "score": 1.0, + "content": "causal mechanisms: linear, nonlinear additive noise models, and nonlinear models with non-additive", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "score": 1.0, + "content": "noise using neural networks. The datasets contain 909 observational samples and 909 samples per", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "intervention. All results of GIES, IGSP, and DCDI have been taken from Brouillard et al. (2020)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "(Appendix C.7, Table 22-24). We follow the setup of Brouillard et al. (2020) and compare two", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 386, + 507, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 507, + 401 + ], + "score": 1.0, + "content": "different neural network setups. First, we use MLPs that model a Gaussian density by predicting a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 295, + 411 + ], + "score": 1.0, + "content": "mean and variance variable (denoted by suffix", + "type": "text" + }, + { + "bbox": [ + 295, + 400, + 304, + 409 + ], + "score": 0.61, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "). The second setup uses normalizing flows, more", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 410, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 422 + ], + "score": 1.0, + "content": "specifically a two-layer deep sigmoidal flow (Huang et al., 2018), which is flexible enough to model", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "more complex distributions (denoted by suffix DSF). The rest of the experimental setup in ENCO is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 432, + 236, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 236, + 444 + ], + "score": 1.0, + "content": "identical to the categorical case.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 310, + 507, + 444 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 448, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 460 + ], + "score": 1.0, + "content": "Results are shown in Table 16, and the observations are the same as with categorical data. ENCO", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "score": 1.0, + "content": "outperforms all other methods in all settings, especially for the more complex distributions. The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "higher error rate for the DSF setup is mostly due to overfitting of the flow models. We conclude that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 480, + 382, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 382, + 494 + ], + "score": 1.0, + "content": "ENCO works as accurately for both continuous and categorical data.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 449, + 506, + 494 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 507, + 370, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 371, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 371, + 519 + ], + "score": 1.0, + "content": "D.6 SKELETON LEARNING WITH OBSERVATIONAL BASELINE", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "To show the benefit of learning a graph from observational and interventional data jointly, we compare", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "ENCO to a simple observational baseline. This baseline first learns the skeleton of the graph by", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "score": 1.0, + "content": "applying greedy equivalence search (GES) (Chickering, 2002) on the observational data. Then, for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 562, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 573 + ], + "score": 1.0, + "content": "each interventional dataset, we apply GES as well and use those skeletons to orientate the edges of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 394, + 585 + ], + "score": 1.0, + "content": "the original one. This can be done by checking for each undirected edge", + "type": "text" + }, + { + "bbox": [ + 395, + 572, + 425, + 583 + ], + "score": 0.9, + "content": "X - Y", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 572, + 461, + 585 + ], + "score": 1.0, + "content": "whether", + "type": "text" + }, + { + "bbox": [ + 461, + 572, + 495, + 582 + ], + "score": 0.9, + "content": "X Y", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 246, + 595 + ], + "score": 1.0, + "content": "in the skeleton of interventions on", + "type": "text" + }, + { + "bbox": [ + 246, + 583, + 256, + 593 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "or not. As a reference implementation of GES, we have used", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 594, + 414, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 414, + 606 + ], + "score": 1.0, + "content": "the one provided in the Causal Discovery Toolbox (Kalainathan et al., 2020).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 527, + 506, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "The results on continuous data are shown in Table 16. Since GES assumes linear mechanisms and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "gaussianity of the data, it is unsurprising that it performs better on the linear Gaussian dataset than", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "on the non-linear datasets. However, on all the three datasets, it constitutes the lowest performance", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 642, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 658 + ], + "score": 1.0, + "content": "compared to the other methods, including ENCO. This highlights the benefits of incorporating", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "interventional data in the learning of the skeleton and graph structure. To gain further insights in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "comparison to the constraint-based baseline, we repeat the experiments with smaller sample sizes. The", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "original dataset has 909 samples for observational data and per intervention, and we sub-sample 500", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "and 100 of those respectively for simulating smaller dataset sizes. The results of those experiments", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "can be found in Table 17. It is apparent that the results of GES on the linear dataset get considerably", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "worse with fewer data samples being available, while ENCO-G is able to reconstruct most graphs", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "still without errors. Especially for the small dataset of 100 samples, we noticed that the skeletons", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "found by GES on observational data already contained couple of mistakes. This shows that for small", + "type": "text", + "cross_page": true + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "datasets, observational data alone might not be sufficient to find the correct skeleton while by jointly", + "type": "text", + "cross_page": true + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 525, + 496, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 496, + 538 + ], + "score": 1.0, + "content": "learning from observational and interventional data, we can yet find the graph up to minor errors.", + "type": "text", + "cross_page": true + } + ], + "index": 21 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 611, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 133, + 503, + 217 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 125 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 506, + 93 + ], + "score": 1.0, + "content": "Table 17: Experiments on graph with continuous data from Brouillard et al. (2020) with smaller", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 505, + 103 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 505, + 103 + ], + "score": 1.0, + "content": "sample sizes for both observational and interventional datasets (in brackets). ENCO shows to perform", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 102, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 114 + ], + "score": 1.0, + "content": "much better in smaller sample sizes than a skeleton+orientation method, underlining the benefit of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 410, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 410, + 126 + ], + "score": 1.0, + "content": "learning the whole graph from observational and interventional data jointly.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 107, + 133, + 503, + 217 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 133, + 503, + 217 + ], + "spans": [ + { + "bbox": [ + 107, + 133, + 503, + 217 + ], + "score": 0.981, + "html": "
Graph typeLinear GaussianNonlinear with additive noiseNonlinear with non-additive noise
GES + Orientating (909)2.5(±4.3)12.5 (±9.2)7.9 (±7.4)
ENCO-G (ours) (909)0.0 (±0.0)0.2 (±0.4)0.1(±0.3)
GES + Orientating (500)3.2 (±4.3)12.0 (±9.2)9.2 (±7.5)
ENCO-G (ours) (500)0.1(±0.3)0.5(±0.7)0.0 (±0.0)
GES + Orientating (100)6.1(±5.3)10.2 (±6.7)9.0 (±6.5)
ENCO-G (ours) (100)0.2(±0.4)0.9 (±1.1)1.3(±0.8)
", + "type": "table", + "image_path": "b6c9df360aa3a32397619f7222403faab06f77e846889de6723d8e3083fc4510.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 107, + 133, + 503, + 161.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 107, + 161.0, + 503, + 189.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 107, + 189.0, + 503, + 217.0 + ], + "spans": [], + "index": 6 + } + ] + } + ], + "index": 3.25 + }, + { + "type": "table", + "bbox": [ + 107, + 264, + 503, + 353 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 225, + 505, + 259 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 224, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 238 + ], + "score": 1.0, + "content": "Table 18: Comparing structure learning methods in terms of structural hamming distance (SHD) on", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "common graph structures (lower is better), averaged over 25 graphs each. ENCO outperforms all", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 247, + 500, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 500, + 259 + ], + "score": 1.0, + "content": "baselines, and by enforcing acyclicity after training, can recover most graphs with minimal errors.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "table_body", + "bbox": [ + 107, + 264, + 503, + 353 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 107, + 264, + 503, + 353 + ], + "spans": [ + { + "bbox": [ + 107, + 264, + 503, + 353 + ], + "score": 0.979, + "html": "
Graph typebidiagchaincolliderfull junglerandom
GIES33.6 (±7.5)17.5(±7.3)24.0 (±2.9)216.5 (±15.2)33.1(±2.9)57.5(±14.2)
IGSP32.7(±5.1)14.6 (±2.3)23.7(±2.3)253.8(±12.6)35.9 (±5.2)65.4(±8.0)
SDI9.0 (±2.6)3.9 (±2.0)16.1(±2.4)153.9 (±10.3)6.9 (±2.3)10.8(±3.9)
DCDI16.9 (±2.0)10.1(±1.1)10.9 (±3.6)21.0(±4.8)17.9 (±4.1)7.7(±3.2)
GES + Orientating14.8(±2.6)0.5 (±0.7)20.8(±2.4)282.8(±4.2)14.7(±3.1)60.1(±8.9)
ENCO (ours)2.2 (±1.4)1.7 (±1.3)1.6 (±1.6)9.2 (±3.4)1.7 (±1.3)4.6 (±1.9)
ENCO-acyclic (ours)0.0 (±0.0)0.0(±0.0)1.6 (±1.6)5.3(±2.3)0.6 (±1.1)0.2 (±0.5)
", + "type": "table", + "image_path": "2fc308673dcf08b3d77e2f941ec137e506bbaafcf67f49c4e7e4a05dbb57456d.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 107, + 264, + 503, + 293.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 107, + 293.6666666666667, + 503, + 323.33333333333337 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 107, + 323.33333333333337, + 503, + 353.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + } + ], + "index": 9.5 + }, + { + "type": "table", + "bbox": [ + 108, + 403, + 504, + 483 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 361, + 504, + 394 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 106, + 360, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 506, + 374 + ], + "score": 1.0, + "content": "Table 19: Experiments with a different data simulator, introducing independence among parents for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "each variable. Similar to the neural-based synthetic data, ENCO recovers most graphs with a minor", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 383, + 273, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 273, + 396 + ], + "score": 1.0, + "content": "error rate, outperforming other baselines.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "table_body", + "bbox": [ + 108, + 403, + 504, + 483 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 108, + 403, + 504, + 483 + ], + "spans": [ + { + "bbox": [ + 108, + 403, + 504, + 483 + ], + "score": 0.983, + "html": "
Graph typebidiagchaincolliderfull junglerandom
GIES30.7 (±3.1)16.9 (±2.4)18.6 (±2.5)238.6(±4.0)30.1 (±3.5)110.0 (±11.8)
IGSP27.0 (±4.2)14.0 (±2.8)25.0 (±1.4)259.5(±3.5)24.0 (±7.1)112.5 (±4.9)
SDI12.6(±2.7)7.6 (±2.6)2.8 (±1.6)99.0 (±7.5)14.8 (±3.2)36.7 (±4.7)
DCDI15.7 (±1.9)8.4(±1.3)4.7 (±2.5)25.3(±6.2)18.9 (±3.7)9.8(±3.6)
ENCO (ours)0.5 (±0.6)0.4(±0.6)0.9 (±0.8)1.0 (±1.2)1.2 (±1.2)1.9 (±1.8)
ENCO-acyclic (ours)0.0 (±0.0)0.0 (±0.0)0.8 (±0.8)0.2 (±0.4)0.3 (±0.6)0.1(±0.3)
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The results in Table 18 give a similar conclusion as on the continuous data.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 564, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 506, + 578 + ], + "score": 1.0, + "content": "While the baseline attains good scores for chains, it makes considerably more errors on all other", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 575, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 589 + ], + "score": 1.0, + "content": "graph structures than ENCO. This shows that ENCO is much more robust by jointly learning from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 586, + 258, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 258, + 599 + ], + "score": 1.0, + "content": "observational and interventional data.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 106, + 612, + 307, + 623 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 307, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 307, + 624 + ], + "score": 1.0, + "content": "D.7 NON-NEURAL BASED DATA SIMULATORS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 645 + ], + "score": 1.0, + "content": "Using neural networks to generate the simulated data might give SDI, DCDI and ENCO an advantage", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "in our comparisons since they rely on similar neural networks to model the distribution. 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Graph typeLinear GaussianNonlinear with additive noiseNonlinear with non-additive noise
GES + Orientating (909)2.5(±4.3)12.5 (±9.2)7.9 (±7.4)
ENCO-G (ours) (909)0.0 (±0.0)0.2 (±0.4)0.1(±0.3)
GES + Orientating (500)3.2 (±4.3)12.0 (±9.2)9.2 (±7.5)
ENCO-G (ours) (500)0.1(±0.3)0.5(±0.7)0.0 (±0.0)
GES + Orientating (100)6.1(±5.3)10.2 (±6.7)9.0 (±6.5)
ENCO-G (ours) (100)0.2(±0.4)0.9 (±1.1)1.3(±0.8)
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Graph typebidiagchaincolliderfull junglerandom
GIES33.6 (±7.5)17.5(±7.3)24.0 (±2.9)216.5 (±15.2)33.1(±2.9)57.5(±14.2)
IGSP32.7(±5.1)14.6 (±2.3)23.7(±2.3)253.8(±12.6)35.9 (±5.2)65.4(±8.0)
SDI9.0 (±2.6)3.9 (±2.0)16.1(±2.4)153.9 (±10.3)6.9 (±2.3)10.8(±3.9)
DCDI16.9 (±2.0)10.1(±1.1)10.9 (±3.6)21.0(±4.8)17.9 (±4.1)7.7(±3.2)
GES + Orientating14.8(±2.6)0.5 (±0.7)20.8(±2.4)282.8(±4.2)14.7(±3.1)60.1(±8.9)
ENCO (ours)2.2 (±1.4)1.7 (±1.3)1.6 (±1.6)9.2 (±3.4)1.7 (±1.3)4.6 (±1.9)
ENCO-acyclic (ours)0.0 (±0.0)0.0(±0.0)1.6 (±1.6)5.3(±2.3)0.6 (±1.1)0.2 (±0.5)
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Graph typebidiagchaincolliderfull junglerandom
GIES30.7 (±3.1)16.9 (±2.4)18.6 (±2.5)238.6(±4.0)30.1 (±3.5)110.0 (±11.8)
IGSP27.0 (±4.2)14.0 (±2.8)25.0 (±1.4)259.5(±3.5)24.0 (±7.1)112.5 (±4.9)
SDI12.6(±2.7)7.6 (±2.6)2.8 (±1.6)99.0 (±7.5)14.8 (±3.2)36.7 (±4.7)
DCDI15.7 (±1.9)8.4(±1.3)4.7 (±2.5)25.3(±6.2)18.9 (±3.7)9.8(±3.6)
ENCO (ours)0.5 (±0.6)0.4(±0.6)0.9 (±0.8)1.0 (±1.2)1.2 (±1.2)1.9 (±1.8)
ENCO-acyclic (ours)0.0 (±0.0)0.0 (±0.0)0.8 (±0.8)0.2 (±0.4)0.3 (±0.6)0.1(±0.3)
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Still, ENCO", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 215, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 505, + 228 + ], + "score": 1.0, + "content": "only shows minor decreases, having less than one mistake on average for every graph structure when", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 226, + 243, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 243, + 239 + ], + "score": 1.0, + "content": "applying the orientation heuristic.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 138, + 506, + 239 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/dev/ldRb12nMfLQ/ldRb12nMfLQ.md b/parse/dev/ldRb12nMfLQ/ldRb12nMfLQ.md new file mode 100644 index 0000000000000000000000000000000000000000..8060c1927992352605e84167732a0bdd17c72270 --- /dev/null +++ b/parse/dev/ldRb12nMfLQ/ldRb12nMfLQ.md @@ -0,0 +1,228 @@ +# A DEEP CONJUGATE DIRECTION METHOD FOR ITERATIVELY SOLVING LINEAR SYSTEMS + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for partial differential equations. Algorithms for approximating the solution to these systems are often the bottleneck in problems that require their solution, particularly for modern applications that require many millions of unknowns. Indeed, numerical linear algebra techniques have been investigated for many decades to alleviate this computational burden. Recently, data-driven techniques have also shown promise for these problems. Motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error, we design an approach that utilizes a deep neural network to accelerate convergence via data-driven improvement of the search directions. Our method leverages a carefully chosen convolutional network to approximate the action of the inverse of the linear operator up to an arbitrary constant. We train the network using unsupervised learning with a loss function equal to the $L ^ { 2 }$ difference between an input and the system matrix times the network evaluation, where the unspecified constant in the approximate inverse is accounted for. We demonstrate the efficacy of our approach on spatially discretized Poisson equations with millions of degrees of freedom arising in computational fluid dynamics applications. Unlike state-of-the-art learning approaches, our algorithm is capable of reducing the linear system residual to a given tolerance in a small number of iterations, independent of the problem size. Moreover, our method generalizes effectively to various systems beyond those encountered during training. + +# 1 INTRODUCTION + +In this work, we consider sparse linear systems that arise from discrete Poisson equations in incompressible flow applications (Chorin, 1967; Fedkiw et al., 2001; Bridson, 2008). We use the notation + +$$ +\mathbf { A } { \boldsymbol { \mathbf { \mathit { x } } } } = \mathbf { \mathit { b } } +$$ + +where the dimension $n$ of the matrix $A \in \mathbb { R } ^ { n \times n }$ and the vector $b \in \mathbb { R } ^ { n }$ correlate with spatial fidelity of the computational domain. The appropriate numerical linear algebra technique depends on the nature of the problem. Direct solvers that utilize matrix factorizations (QR, Cholesky, etc. Trefethen & Bau (1997)) have optimal approximation error, but their computational cost is $\operatorname { \dot { O } } ( n ^ { 3 } )$ and they typically require dense storage, even for sparse $\pmb { A }$ . Although Fast Fourier Transforms (Nussbaumer, 1981) can be used in limited instances (periodic boundary conditions, etc.), iterative techniques are most commonly adopted for these systems given their sparsity. Many applications with strict performance constraints (e.g., real-time fluid simulation) utilize basic iterations (Jacobi, Gauss-Seidel, successive over relaxation (SOR), etc.) given limited computational budget (Saad, 2003). However, large approximation errors must be tolerated since iteration counts are limited by the performance constraints. This is particularly problematic since the wide elliptic spectrum of these matrices (a condition that worsens with increased spatial fidelity/matrix dimension) leads to poor conditioning and iteration counts. Iterative techniques can achieve sub-quadratic convergence if their iteration count does not grow excessively with problem size $n$ since each iteration generally requires $O ( n )$ floating point operations for sparse matrices. Discrete elliptic operators are typically symmetric positive (semi) definite and the preconditioned conjugate gradients method (PCG) can be used to minimize iteration counts (Saad, 2003; Hestenes & Stiefel, 1952; Stiefel, 1952). Preconditioners $_ { r }$ for PCG must simultaneously: be symmetric positive definite (SPD) (and therefore admit factorization ${ \boldsymbol { P } } = { \boldsymbol { F } } ^ { 2 }$ ), improve the condition number of the preconditioned system $F A F y = F b$ , and be computationally cheap to construct and apply; accordingly, designing specialized preconditioners for particular classes of problems is somewhat of an art. Incomplete Cholesky preconditioners (ICPCG) (Kershaw, 1978) use a sparse approximation to the Cholesky factorization and significantly reduce iteration counts in practice; however, their inherent data dependency prevents efficient parallel implementation. Nonetheless, these are very commonly adopted for Poisson equations arising in incompressible flow (Fedkiw et al., 2001; Bridson, 2008). Multigrid (Brandt, 1977) and domain decomposition (Saad, 2003) preconditioners greatly reduce iterations counts, but they must be updated (with non-trivial cost) each time the problem changes (e.g., in computational domains with time varying boundaries) and/or for different hardware platforms. In general, choice of an optimal preconditioner for discrete elliptic operators is an open area of research. + +Recently, data-driven approaches that leverage deep learning techniques have shown promise for solving linear systems. Various researchers have investigated machine learning estimation of multigrid parameters (Greenfeld et al., 2019; Grebhahn et al., 2016; Luz et al., 2020). Others have developed machine learning methods to estimate preconditioners (Gotz & Anzt, 2018; Stanaityte, 2020; ¨ Ichimura et al., 2020) and initial guesses for iterative methods (Luna et al., 2021; Um et al., 2020; Ackmann et al., 2020). Tompson et al. (2017) and Yang et al. (2016) develop non-iterative machine learning approximations of the inverse of discrete Poisson equations from incompressible flow. We leverage deep learning and develop a novel version of conjugate gradients iterative method for approximating the solution of SPD linear systems which we call the deep conjugate direction method (DCDM). CG iteratively adds $\pmb { A }$ -conjugate search directions while minimizing the matrix norm of the error. We use a convolutional neural network (CNN) as an approximation of the inverse of the matrix in order to generate more efficient search directions. We only ask that our network approximate the inverse up to an unknown scaling since this decreases the degree of nonlinearity and since it does not affect the quality of the search direction (which is scale independent). The network is similar to a preconditioner, but it is not a linear function, and our modified conjugate gradients approach is designed to accommodate this nonlinearity. We use unsupervised learning to train our network with a loss function equal to the $L ^ { 2 }$ difference between an input vector and a scaling of $\pmb { A }$ times the output of our network. To account for this unknown scaling during training, we choose the scale of the output of the network by minimizing the matrix norm of the error. Our approach allows for efficient training and generalization to problems unseen (matrices $\pmb { A }$ and right-hand sides $^ { b }$ ). We benchmark our algorithm using the ubiquitous pressure Poisson equation (discretized on regular voxelized domains) and compare against FluidNet (Tompson et al., 2017), which is the state of the art learning-based method for these types of problems. Unlike the non-iterative approaches of Tompson et al. (2017) and Yang et al. (2016), our method can reduce the linear system residuals arbitrarily. We showcase our approach with examples that have over 16 million degrees of freedom. + +# 2 RELATED WORK + +Several papers have focused on enhancing the solution of linear systems (arising from discretized PDEs) using learning. For instance, Gotz & Anzt (2018) generate sparsity patterns for block-Jacobi ¨ preconditioners using convolutional neural networks, and Stanaityte (2020) use a CNN to predict non-zero patterns for ILU-type preconditioners for the Navier-Stokes equations (though neither work designs fundamentally new preconditioners). Ichimura et al. (2020) develop a neural-network based preconditioner where the network is used to predict approximate Green’s functions (which arise in the analytical solution of certain PDEs) that in turn yield an approximate inverse of the linear system. Hsieh et al. (2019) learn an iterator that solves linear systems, performing competitively with classical solvers like multigrid-preconditioned MINRES (Paige & Saunders, 1975). Luz et al. (2020) and Greenfeld et al. (2019) use machine learning to estimate algebraic multigrid (AMG) parameters. They note that AMG approaches rely most fundamentally on effectively chosen (problem-dependent) prolongation sparse matrices and that numerous methods have attempted to automatically create them from the matrix $\pmb { A }$ . They train a graph neural network to learn (in an unsupervised fashion) a mapping from matrices $\pmb { A }$ to prolongation operators. Grebhahn et al. (2016) note that geometric multigrid solver parameters can be difficult to choose to guarantee parallel performance on different hardware platforms. They use machine learning to create a code generator to help achieve this. + +![](images/a7ac602c2643cff48deff90a7224438f208d4a7100d53c20e1e19240e8cade2b.jpg) +Figure 1: (a) We illustrate a sample flow domain $\Omega \subset ( 0 , 1 ) ^ { 2 }$ (in 2D for ease of illustration) with internal boundaries (blue lines). (b) We voxelize the domain with a regular grid: white cells represent interior/fluid, and blue cells represent boundary conditions. (c) We train using matrix $A ^ { ( 0 , 1 ) ^ { d } }$ from a discretized domain with no interior boundary conditions, where $d$ is the dimension. This creates linear system with $n = ( n _ { c } + 1 ) ^ { d }$ unknowns where $n _ { c }$ is the number of grid cells on each direction. (d) We illustrate the non-zero entries in an example matrix $A ^ { \Omega }$ from the voxelized and labeled (white vs. blue) grid for three example interior cells (green, purple, and brown). Each case illustrates the non-zero entries in the row associated with the example cell. All entries in rows corresponding to boundary/blue cells are zero. + +Several works consider accelerating the solution of linear systems by learning an initial guess that is close to the true solution or otherwise helpful to descent algorithms for finding the true solution. In order to solve the discretized Poisson equation, Luna et al. (2021) accelerate the convergence of GMRES (Saad & Schultz, 1986) with an initial guess that is learned in real-time (i.e., as a simulation code runs) with no prior data. Um et al. (2020) train a network (incorporating differentiable physics, based on the underlying PDEs) in order to produce high-quality initial guesses for a CG solver. In a somewhat similar vein, Ackmann et al. (2020) use a simple feedforward neural network to predict pointwise solution components, which accelerates the conjugate residual method used to solve a relatively simple shallow-water model (a more sophisticated network and loss function are needed to handle more general PDEs and larger-scale problems). + +At least two papers (Ruelmann et al., 2018; Sappl et al., 2019) have sought to learn a mapping between a matrix and an associated sparse approximate inverse. In their investigation, Ruelmann et al. (2018) propose training a neural network using matrix-inverse pairs as training data. Although straightforward to implement, the cost of generating training data, let alone training the network, is prohibitive for large-scale 3D problems. Sappl et al. (2019) seek to learn a mapping between linear system matrices and sparse (banded) approximate inverses. Their loss function is the condition number of the product of the system matrix and the approximate inverse; the minimum value of the condition number is one, which is achieved exactly when an exact inverse is obtained. Although this framework is quite simple, evaluating the condition number of a matrix is asymptotically costly, and in general, the inverse of a sparse matrix can be quite dense. Accordingly, the method is not efficient or accurate enough for the large-scale 3D problems that arise in real-world engineering problems. + +Most relevant to the present work is FluidNet (Tompson et al., 2017). FluidNet develops a highlytailored CNN architecture that is used to predict the solution of a linear projection operation (specifically, for the discrete Poisson equation) given a matrix and right-hand side. The authors demonstrate fluid simulations where the linear solve is replaced by evaluating their network. Because their network is relatively lightweight and is only evaluated once per time step, their simulations run efficiently. However, their design allows the network only one opportunity to reduce the residual for the linear solve; in practice, we observe that FluidNet is able to reduce the residual by no more than about one order of magnitude. However, in computer graphics applications, at least four orders of magnitude in residual reduction are usually required for visual fidelity, while in scientific and engineering applications, practitioners prefer solutions that reduce the residual by eight or more orders of magnitude (i.e., to within machine precision). Accordingly, FluidNet’s lack of convergence stands in stark contrast to classical, convergent methods like CG. Our method resolves this gap. + +# 3 MOTIVATION: INCOMPRESSIBLE FLOW + +We demonstrate the efficacy of our approach with the linear systems that arise in incompressible flow applications. Specifically, we use our algorithm to solve the discrete Poisson equations in + +regular-grid-based discretization of the pressure projection equations that arise in Chorin’s splitting technique (Chorin, 1967) for the inviscid, incompressible Euler equations. These equations are + +$$ +\rho \left( \frac { \partial \pmb { u } } { \partial t } + \frac { \partial \pmb { u } } { \partial \pmb { x } } \pmb { u } \right) + \nabla p = \pmb { f } ^ { e x t } , \qquad \nabla \cdot \pmb { u } = 0 +$$ + +where $\textbf { \em u }$ is fluid velocity, $p$ is pressure, $\rho$ is density, and $f ^ { e x t }$ accounts for external forces like gravity. The equations are assumed at all positions $_ { \textbf { \em x } }$ in the spatial fluid flow domain $\Omega$ and for time $t > 0$ . The left term in Equation 2 enforces conservation of momentum in the absence of viscosity, and the second part enforces incompressibility and conservation of mass. These equations are subject to initial conditions $\rho ( \pmb { x } , 0 ) = \rho ^ { 0 }$ and ${ \pmb u } ( { \pmb x } , 0 ) = { \pmb u } ^ { 0 } ( { \pmb x } )$ as well as boundary conditions ${ \pmb u } ( { \pmb x } , t ) \cdot { \pmb n } ( { \pmb x } ) = u ^ { \partial \Omega } ( { \pmb x } , t )$ on the boundary of the domain ${ \pmb x } \in \partial \Omega$ (where $\textbf { \em n }$ is the unit outward pointing normal at position $_ { \textbf { \em x } }$ on the boundary). Equation 2 is discretized in both time and space. Temporally, we split the advection $\frac { \partial \pmb { u } } { \partial t } + \frac { \partial \pmb { u } } { \partial \pmb { x } } \pmb { u } = 0$ and body forces terms $\rho \frac { \partial \pmb { u } } { \partial t } = \pmb { f } ^ { e x t }$ , and finally enforce incompressibility via the pressure projection $\frac { \partial \pmb { u } } { \partial t } + \frac { 1 } { \rho } \nabla p = \pmb { 0 }$ such that $\nabla \cdot \pmb { u } = 0$ ; this is the standard advection-projection scheme proposed by Chorin (1967). Using finite differences in time, we can summarize this as + +$$ +\begin{array} { l } { \displaystyle \rho ^ { 0 } \left( \frac { \pmb { u } ^ { * } - \pmb { u } ^ { n } } { \Delta t } + \frac { \partial \pmb { u } ^ { n } } { \partial \pmb { x } } \pmb { u } ^ { n } \right) = \pmb { f } ^ { e x t } } \\ { \displaystyle - \nabla \cdot \frac { 1 } { \rho ^ { 0 } } \nabla p ^ { n + 1 } = - \nabla \cdot \pmb { u } ^ { * } } \\ { \displaystyle \qquad - \frac { 1 } { \rho ^ { 0 } } \nabla p ^ { n + 1 } \cdot \pmb { n } = \frac { 1 } { \Delta t } \left( \pmb { u } ^ { \partial \Omega } - \pmb { u } ^ { * } \cdot \pmb { n } \right) . } \end{array} +$$ + +For the spatial discretization, we use a regular marker-and-cell (MAC) grid (Harlow & Welch, 1965) with cubic voxels whereby velocity components are stored on the face of voxel cells, and scalar quantities (e.g., pressure $p$ or density $\rho \mathrm { \hbar }$ ) are stored at voxel centers. We use backward semiLagrangian advection (Fedkiw et al., 2001; Gagniere et al., 2020) for Equation 3. All spatial partial derivatives are approximated using finite differences. Equations 4 and 5 describe the pressure Poisson equation with Neumann conditions on the boundary of the flow domain. We discretize the left hand side of Equation 4 using a standard 7-point finite difference stencil. The right-hand side is discretized using the MAC grid discrete divergence finite difference stencils as well as contributions from the boundary condition terms in Equation 5. We refer the reader to Bridson (2008) for more in-depth implementation details. Equation 5 is discretized by modifying the Poisson stencil to enforce Neumann boundary conditions. We do this using a simple labeling of the voxels in the domain. For simplicity, we assume $\Omega \subset ( 0 , 1 ) ^ { 3 }$ is a subset of the unit cube, potentially with internal boundaries. We label cells in the domain as either liquid or boundary. This simple classification is enough to define the Poisson discretizations (with appropriate Neumann boundary conditions at domain boundaries) that we focus on in the present work; we illustrate the details in Figure 1. We use the following notation to denote the discrete Poisson equations associated with Equations 4–5: + +$$ +\begin{array} { r } { { \cal A } ^ { \Omega } { \boldsymbol x } = { \boldsymbol b } ^ { \nabla \cdot { \boldsymbol u } ^ { * } } + { \boldsymbol b } ^ { u ^ { \partial \Omega } } , } \end{array} +$$ + +where $A ^ { \Omega }$ is the discrete Poisson matrix associated with the voxelized domain, $_ { \textbf { \em x } }$ is the vector of unknown pressure, and $\mathbf { \delta } _ { b } \nabla \cdot \mathbf { u } ^ { * }$ and b u ∂ Ω are the right-hand side terms from Equations 4 and 5, respectively. We define a special case of the matrix involved in this discretization to be the Poisson matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ associated with $\Omega = ( 0 , 1 ) ^ { 3 }$ , i.e., a full fluid domain with no internal boundaries. We use this matrix for training, yet demonstrate that our network generalizes to all other matrices arising from more complicated flow domains. + +# 4 DEEP CONJUGATE DIRECTION METHOD + +We present our method for the deep learning acceleration of iterative approximations to the solution of linear systems of the form seen in Equation 6. We first discuss relevant details of the conjugate gradients (CG) method, particularly line search and $\pmb { A }$ -orthogonal search directions. We then present a deep learning technique for improving the quality of these search directions that ultimately reduces iteration counts required to achieve satisfactory residual reduction. Lastly, we outline the training procedures for our deep convolutional neural network. + +Our approach iteratively improves approximations to the solution $_ { \textbf { \em x } }$ of Equation 6. We build on the method of CG, which requires the matrix $A ^ { \Omega }$ in Equation 6 to be SPD. SPD matrices $A ^ { \Omega }$ give rise to the matrix norm $\| \pmb { y } \| _ { A ^ { \Omega } } = \sqrt { \pmb { y } ^ { T } A ^ { \Omega } \pmb { y } }$ . CG can be derived in terms of iterative line search improvement based on optimality in this norm. That is, an iterate $\pmb { x } _ { k - 1 } \approx \pmb { x }$ is updated in search direction $\scriptstyle d _ { k }$ by a step size $\alpha _ { k }$ that is chosen to minimize the matrix norm of the error: + +$$ +\alpha _ { k } = \underset { \alpha } { \arg \operatorname* { m i n } } \frac { 1 } { 2 } \left\| \pmb { x } - ( \pmb { x } _ { k - 1 } + \alpha \pmb { d } _ { k } ) \right\| _ { \pmb { A } ^ { \Omega } } ^ { 2 } = \frac { { \pmb { r } } _ { k - 1 } ^ { T } { \pmb { d } _ { k } } } { { \pmb { d } } _ { k } ^ { T } { \pmb { A } } ^ { \Omega } { \pmb { d } _ { k } } } , +$$ + +where $\pmb { r } _ { k - 1 } = \pmb { b } - \pmb { A } ^ { \Omega } \pmb { x } _ { k - 1 }$ is the $( k - 1 ) ^ { \mathrm { t h } }$ residual and $^ { b }$ is the right-hand side in Equation 6. Different search directions result in different algorithms. A natural choice is the negative gradient of the matrix norm of the error (evaluated at the current iterate), since this will point in the direction of steepest decrease $\begin{array} { r } { \pmb { d } _ { k } = - \frac { 1 } { 2 } \nabla \left\| \pmb { x } _ { k - 1 } \right\| _ { \pmb { A } ^ { \Omega } } ^ { 2 } = \pmb { r } _ { k - 1 } } \end{array}$ . This is the gradient descent method (GD). Unfortunately, this approach requires many iterations in practice. A more effective strategy is to choose directions that are $\pmb { A }$ -orthogonal (i.e., ${ \bf \Phi } _ { { \bf i } } ^ { T } { \bf A } ^ { \Omega } { \bf d } _ { j } = \mathrm { ~ ~ \dot { ~ } { ~ 0 ~ } ~ }$ for $i \neq j$ ). With this choice, the search directions form a basis for $\mathbb { R } ^ { n }$ so that the initial error can be written as $\begin{array} { r } { \pmb { e } _ { 0 } = \pmb { x } - \pmb { x } _ { 0 } = \sum _ { i = 1 } ^ { n } e _ { i } \pmb { d } _ { i } } \end{array}$ , where $e _ { i }$ are the components of the initial error written in the basis. Furthermore, when the search directions are $\pmb { A }$ -orthogonal, the optimal step sizes $\alpha _ { k }$ at each iteration satisfy + +$$ +\alpha _ { k } = \frac { r _ { k - 1 } ^ { T } d _ { k } } { d _ { k } ^ { T } A ^ { \Omega } d _ { k } } = \frac { d _ { k } ^ { T } A ^ { \Omega } e _ { k - 1 } } { d _ { k } ^ { T } A ^ { \Omega } d _ { k } } = \frac { d _ { k } ^ { T } A ^ { \Omega } \left( \sum _ { i = 1 } ^ { n } e _ { i } d _ { i } - \sum _ { j = 1 } ^ { k - 1 } \alpha _ { j } d _ { j } \right) } { d _ { k } ^ { T } A ^ { \Omega } d _ { k } } = e _ { k } . +$$ + +That is, the optimal step sizes are chosen to precisely eliminate the components of the error on the basis defined by the search directions. Thus, convergence is determined by the (at most $n$ ) non-zero components $e _ { i }$ in the initial error. Although rounding errors prevent this from happening exactly in practice, this property greatly reduces the number of required iterations (Golub & Loan, 2012). + +CG can be viewed as a modification of GD where the search direction is chosen as the component of the residual (equivalently, the negative gradient of the matrix norm of the error) that is $\pmb { A }$ -orthogonal to all previous search directions: + +$$ +\pmb { d } _ { k } = \pmb { r } _ { k - 1 } - \sum _ { i = 1 } ^ { k - 1 } h _ { i k } \pmb { d } _ { i } , \qquad h _ { i k } = \frac { { d } _ { i } ^ { T } { \pmb { A } } ^ { \Omega } \pmb { r } _ { k - 1 } } { { d } _ { i } ^ { T } { \pmb { A } } ^ { \Omega } \pmb { d } _ { i } } . +$$ + +In practice, $h _ { i k } = 0$ for $i < k - 1$ , and this iteration can therefore be performed without the need to store all previous search directions $\mathbf { \mathbf { { \alpha } } } d _ { i }$ and without the need for computing all previous $h _ { i k }$ . + +While the residual is a natural choice for generating $\pmb { A }$ -orthogonal search directions (since it points in the direction of the steepest local decrease), it is not the optimal search direction. If $\scriptstyle d _ { k }$ is parallel to $( A ^ { \Omega } ) ^ { - 1 } r _ { k - 1 }$ , then $\scriptstyle { \mathbf { { \mathit { x } } } } _ { k }$ will be equal to $_ { \textbf { \em x } }$ since $\alpha _ { k }$ (computed from Equation 7) will step directly to the solution. We can see this by considering the residual and its relation to the search direction: + +$$ +\begin{array} { r } { r _ { k } = b - A ^ { \Omega } x _ { k } = b - A ^ { \Omega } x _ { k - 1 } - \alpha _ { k } A ^ { \Omega } d _ { k } = r _ { k - 1 } - \alpha _ { k } A ^ { \Omega } d _ { k } . } \end{array} +$$ + +In light of this, we use deep learning to create an approximation $f ( c , r )$ to $( A ^ { \Omega } ) ^ { - 1 } r$ , where $^ c$ denotes the network weights and biases. This is analogous to using a preconditioner in PCG; however, our network is not SPD (nor even a linear function). We simply use this data-driven approach as our means of generating better search directions $\scriptstyle d _ { k }$ . Furthermore, we only need to approximate a vector parallel to $( A ^ { \Omega } ) ^ { - 1 } r$ since the step size $\alpha _ { k }$ will account for any scaling in practice. In other words, $f ( c , r ) \approx s _ { r } ( A ^ { \Omega } ) ^ { - 1 } r$ , where the scalar $s _ { r }$ is not defined globally; it only depends on $\pmb { r }$ , and the model does not learn it. Lastly, as with CG, we enforce $\pmb { A }$ -orthogonality, yielding search directions + +$$ +\pmb { d } _ { k } = \pmb { f } ( \pmb { c } , \pmb { r } _ { k - 1 } ) - \sum _ { i = 1 } ^ { k - 1 } h _ { i k } \pmb { d } _ { i } , \qquad h _ { i k } = \frac { \pmb { f } ( \pmb { c } , \pmb { r } _ { k - 1 } ) ^ { T } \pmb { A } ^ { \Omega } \pmb { d } _ { i } } { \pmb { d } _ { i } ^ { T } \pmb { A } ^ { \Omega } \pmb { d } _ { i } } . +$$ + +We summarize our approach in Algorithm 1. Note that we introduce the variable $i _ { \mathrm { s t a r t } }$ . To guarantee $\pmb { A }$ -orthogonality between all search directions, we must have $i _ { \mathrm { s t a r t } } = 1$ . However, this requires storing all prior search directions, which can be costly. We found that using $i _ { \mathrm { s t a r t } } = k - 2$ worked nearly as well as $i _ { \mathrm { s t a r t } } = 1$ in practice (in terms of our ability to iteratively reduce the residual of the system). We demonstrate this in Figure 4c. + +# 5 MODEL ARCHITECTURE, DATASETS, AND TRAINING + +Efficient performance of our method requires effective training of our deep convolutional network for weights and biases $^ c$ such that $f ( c , r ) { \overset { \cdot } { \approx } } s _ { r } ( A ^ { \Omega } ) ^ { - 1 } r$ (for arbitrary scalar $s _ { r }$ ). We design a model architecture, loss function, and unsupervised training approach to achieve this. Our approach has modest training requirements and allows for effective residual reduction while generalizing well to problems not seen in the training data. + +# 5.1 LOSS FUNCTION AND UNSUPERVISED LEARNING + +Although we generalize to arbitrary matrices $A ^ { \Omega }$ from Equation 6 that correspond to domains $\Omega \subset ( 0 , 1 ) ^ { 3 }$ that have internal boundaries (see Figure 1), we train using just the matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ from the full cube domain $( 0 , 1 ) ^ { 3 }$ . In contrast, other similar approaches (Tompson et al., 2017; Yang et al., 2016) train using matrices $A ^ { \Omega }$ and right-hand sides $\pmb { b } ^ { \nabla \cdot \pmb { u } ^ { * } } + \overline { { \pmb { b } } } ^ { u ^ { \partial \Omega } }$ that arise from flow in many domains with internal boundaries. We train our network by minimiz + +# Algorithm 1 DCDM + +1: $\pmb { r } _ { 0 } = \pmb { b } - \pmb { A } ^ { \Omega } \pmb { x } _ { 0 }$ +2: $k = 1$ +3: while $\| r _ { k - 1 } \| \ge \epsilon$ do +4: dk = f (c, rk−1∥rk−1∥ ) +5: for $i _ { \mathrm { s t a r t } } \le i < k$ do +6: hik = dT AΩdi dT AΩdi +7: dk-=hikdi +8: end for +9: $\begin{array} { r l } & { \alpha _ { k } = \frac { r _ { k - 1 } ^ { T } d _ { k } } { d _ { k } ^ { T } A ^ { \Omega } d _ { k } } } \\ & { x _ { k } = x _ { k - 1 } + \alpha _ { k } d _ { k } } \\ & { r _ { k } = b - A ^ { \Omega } x _ { k } } \\ & { k = k + 1 } \end{array}$ +10: +11: +12: +13: end while + +ing the L2 difference ∥r−αA(0,1)3 f (c, r)∥2, where α = rT f(c,r)f(c,r)T A(0,1)3 f(c,r) f rom Equation 7. This choice of $\alpha$ accounts for the unknown scaling in the approximation of $f ( c , r )$ to $\left( A ^ { ( 0 , 1 ) ^ { 3 } } \right) ^ { - 1 } r$ . We use an unsupervised approach and train the model by minimizing + +$$ +\begin{array} { r } { \mathrm { L o s s } ( \pmb { f } , \pmb { c } , \mathscr { D } ) = \frac { 1 } { | \mathscr { D } | } \sum _ { r \in \mathscr { D } } \| \pmb { r } - \frac { r ^ { T } f ( \pmb { c } , r ) } { \pmb { f } ( \pmb { c } , r ) ^ { T } \pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \pmb { f } ( \pmb { c } , r ) } \pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \pmb { f } ( \pmb { c } , r ) \| _ { 2 } } \end{array} +$$ + +for given dataset $\mathcal { D }$ consisting of training vectors $b ^ { i }$ . In Algorithm 1, the normalized residuals $\frac { \boldsymbol { r } _ { k } } { \| \boldsymbol { r } _ { k } \| }$ are passed as inputs to the model. Unlike in e.g. FluidNet (Tompson et al., 2017), only the first residual $\frac { r _ { 0 } } { \Vert r _ { 0 } \Vert }$ is directly related to the problem-dependent original right-hand side $^ { b }$ . Hence we consider a broader range of training vectors than those expected in a given problem of interest, e.g., incompressible flows. We observe that generally the residuals $\mathbf { \nabla } r _ { k }$ in Algorithm 1 are skewed to the lower end of the spectrum of the matrix $A ^ { \Omega }$ . Since $A ^ { \Omega }$ is a discretized elliptic operator, lower end modes are of lower frequency of spatial oscillation. We create our training vectors $\mathbf { \boldsymbol { b } } ^ { i } \in \mathcal { D }$ using $m \ll n$ approximate eigenvectors of the training matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ . We use the Rayleigh-Ritz method to create approximate eigenvectors $\pmb q _ { i }$ , $0 \leq i < m$ . This approach allows us to effectively approximate the full spectrum of $A ^ { ( 0 , 1 ) ^ { 3 } }$ without computing the full eigendecomposition, which can be expensive $( { \cal O } ( \bar { n } ^ { 3 } ) )$ at high resolution. We found that using $m \ : = \ : 1 0 0 0 0$ worked well in practice. The Rayleigh-Ritz vectors are orthonormal and satisfy $Q _ { m } ^ { T } A ^ { ( 0 , 1 ) ^ { 3 } } Q _ { m } = \Lambda _ { m }$ , where $\pmb { \Lambda } _ { m }$ is a diagonal matrix with nondecreasing diagonal entries $\lambda _ { i }$ referred to as Ritz values (approximate eigenvalues) and $\pmb { Q } _ { m } = [ \pmb { q } _ { 0 } , \pmb { q } _ { 1 } , \dots , \pmb { q } _ { m - 1 } ] \in \mathbb { R } ^ { n \times m }$ . We pick $\begin{array} { r } { \pmb { b } ^ { i } = \frac { \sum _ { j = 0 } ^ { m - 1 } c _ { j } ^ { i } \pmb { q } _ { j } } { \left\| \sum _ { j = 0 } ^ { m - 1 } c _ { j } ^ { i } \pmb { q } _ { j } \right\| } } \end{array}$ , where the coefficients $c _ { j } ^ { i }$ are picked from a standard normal distribution + +$$ +c _ { j } ^ { i } = \left\{ { \begin{array} { l l } { 0 \cdot \mathcal { N } ( 0 , 1 ) } & { { \mathrm { i f ~ } } \tilde { j } \leq j \leq { \frac { m } { 2 } } + \theta } \\ { \mathcal { N } ( 0 , 1 ) } & { { \mathrm { o t h e r w i s e } } } \end{array} } \right. +$$ + +where $\theta$ is a small number (we used $\theta = 5 0 0$ ), and $\tilde { j }$ is the first index that $\lambda _ { \tilde { j } } ^ { - } > 0$ . This choice creates $9 0 \%$ of $\mathbf { \nabla } _ { b } i$ from the lower end of the spectrum, with the remaining $1 0 \%$ from the higher end. The Riemann-Lebesgue Lemma states the Fourier spectrum of a continuous function will decay at infinity, so this specific choice of $b _ { i }$ ’s is reasonable for the training set. In practice, we also observed that the right-hand sides of the pressure system that arose in flow problems (in the empty domain) tended to be at the lower end of the spectrum. Notably, even though this dataset only uses RayleighRitz vectors from the training matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ , our network can be effectively generalized to flows in irregular domains, e.g., smoke flow past a rotating box and flow past a bunny (see Figure 3). + +We generate the Rayleigh-Ritz vectors by first tridiagonalizing the training matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ with Lanczos iterations (Lanczos, 1950) to form $\pmb { T } ^ { m } = \pmb { Q } _ { m } ^ { L } ^ { T } \pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \pmb { Q } _ { m } ^ { L } \in \mathbb { R } ^ { m \times m }$ . We then diagonalize $\pmb { T } ^ { m } = \hat { \pmb { Q } } ^ { T } \pmb { \Lambda } _ { m } \hat { \pmb { Q } }$ . While costly, we note that this algorithm is performed on the comparably small $m \times m$ matrix $\mathbf { T } ^ { m }$ (rather than on the $A ^ { ( 0 , 1 ) ^ { 3 } } \in \mathbb { R } ^ { n \times n } ,$ ). This yields the Rayleigh-Ritz vectors as the columns of $Q _ { m } = Q _ { m } ^ { L } \hat { Q }$ . The Lanczos vectors are the columns of the matrix $Q _ { m } ^ { L }$ and satisfy a three-term recurrence whereby the next Lanczos vector can be computed from the previous two as + +$$ +\beta _ { j } \pmb { q } _ { j + 1 } ^ { L } = \pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \pmb { q } _ { j } ^ { L } - \beta _ { j - 1 } \pmb { q } _ { j - 1 } ^ { L } - \alpha _ { j } \pmb { q } _ { j } ^ { L } , +$$ + +where $\alpha _ { j }$ and $\beta _ { j }$ are diagonal and subdiagonal entries of $\pmb { T } ^ { k }$ . $\beta _ { j }$ is computed so that $\pmb q _ { j + 1 } ^ { L }$ is a unit vector, and $\alpha _ { j + 1 } = \pmb { q } _ { j + 1 } ^ { T } \pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \pmb { q } _ { j + 1 }$ . We initialize the iteration with a random $\pmb { q } _ { 0 } ^ { L } \in \mathrm { s p a n } ( \pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } )$ . The Lanczos algorithm can be viewed as a modified Gram-Schmidt technique to create an orthonormal basis for the Krylov space associated with $\pmb q _ { 0 } ^ { L }$ and $A ^ { ( 0 , 1 ) ^ { 3 } }$ , and it therefore suffers from rounding error sensitivities manifested as loss of orthonormality with vectors that do not appear in the recurrence. We found that the simple strategy described in Paige (1971) of orthogonalizing each iterate with all previous Lanczos vectors to be sufficient for our training purposes. Dataset creation takes 5–7 hours for a $6 4 ^ { 3 }$ computational grid, and 2–2.5 days for a $\boldsymbol { 1 2 8 ^ { \overline { { 3 } } } }$ grid. + +# 5.2 MODEL ARCHITECTURE + +![](images/048b839895ab4dfb5d59e79b90cfaaf105f450ca0f80073d468f3f23646ad845.jpg) +Figure 2: Model architecture for training with $A ^ { ( 0 , 1 ) ^ { 3 } }$ on a $1 2 8 ^ { 3 }$ grid. + +The internal structure of our CNN architecture for a $1 2 8 ^ { 3 }$ grid is shown in Figure 2. It consists of a series of convolutional layers with residual connections. The upper left of Figure 2 ( $K$ Residual Blocks) shows our use of multiple blocks of residually connected layers. Notably, within each block, the first layer directly affects the last layer with an addition operator.All non-input or output convolutions use a $3 \times 3 \times 3$ filter, and all layers consist of 16 feature maps. In the middle of the first level, a layer is downsampled (via the average pool + +ing operator with $( 2 \times 2 \times 2 )$ pool size) and another set of convolutional layers is applied with residual connection blocks. The last layer in the second level is upscaled and added to the layer that is downsampled. The last layer in the network is dense with a linear activation function. The activation functions in all convolutional layers are ReLU, except for the first convolution, which uses a linear activation function. + +Initially we tried a simple deep feedforward convolutional network with residual connections (motivated by He et al. (2016)). Although such a simple model works well for DCDM, it requires high number of layers, which results in higher training and inference times. We found that creating parallel layers of CNNs with downsampling reduced the number of layers required. In summary, our goal was to first identify the simplest network architecture that provided adequate accuracy for our target problems, and subsequently, we sought to make architectural changes to minimize training and inference time; further optimizations are possible. + +Differing resolutions use differing numbers of convolutions, but the fundamental structure remains the same. More precisely, the number of residual connections is changed for different resolutions. For example, a $6 \dot { 4 } ^ { 3 }$ grid uses one residual block on the left, two on the right on the upper level, and three on the lower level. Furthermore, the weights trained on a lower resolution grid can be used effectively with higher resolutions. Figure 4d shows convergence results for a $2 \mathrm { { 5 6 ^ { 3 } } }$ grid, using a model trained for a $6 4 ^ { 3 }$ grid and a $1 2 \bar { 8 } ^ { 3 }$ grid. The model that we use for $2 5 6 ^ { 3 }$ grids in our final examples was trained on a $1 2 8 ^ { 3 }$ grid; however, as the shown in the figure, even training with a $6 4 ^ { 3 }$ grid allows for efficient residual reduction. Table 1 shows results for three different resolutions, where DCDM uses $6 4 ^ { 3 }$ and $1 2 8 ^ { 3 }$ trained models. This approach makes the number of parameters in the model independent of the spatial fidelity of the problem. + +# 5.3 TRAINING + +Using the procedure explained in Section 5.1, we create the training dataset ${ \mathcal { D } } \in \mathsf { s p a n } ( A ^ { ( 0 , 1 ) ^ { 3 } } ) \cap { \mathcal { S } } ^ { n - 1 }$ of size 20000 generated from 10000 Rayleigh-Ritz vectors. We train our model with TensorFlow (Abadi et al., 2015) on a single NVIDIA RTX A6000 GPU with 48GB memory. Training is done with standard deep learning techniques—more precisely, back-propagation and the ADAM optimizer (Kingma & Ba, 2015) (with starting learning rate 0.0001). Training takes approximately 10 minutes and 1 hour per epoch for grid resolutions $6 4 ^ { 3 }$ and $\mathrm { \dot { 1 } 2 8 ^ { 3 } }$ , respectively. We trained our model for 50 epochs; however, the model from the thirty + +![](images/32fa04638f7e6d3d367c6df56f430f093be0e2e68b4d13ed0903e506ea28d74a.jpg) +Figure 3: DCDM for simulating a variety of incompressible flow examples. Left: smoke plume at $t \ =$ 6.67, 13.33, 20 seconds. Middle: smoke passing bunny at $t = 5 , 1 0 , 1 5$ seconds. Right: smoke passing a spinning box (time-dependent Neumann boundary conditions) at $t = 2 . 6 7 , 6 , 9 . 3 3$ seconds. + +first epoch was optimal for $6 4 ^ { 3 }$ , and the model from the third epoch was optimal for $1 2 8 ^ { 3 }$ + +# 6 RESULTS AND ANALYSIS + +We demonstrate DCDM on three increasingly difficult examples and provide numerical evidence for the efficient convergence of our method. All examples were run on a workstation with dual AMD EPYC 75F3 processors and 512GB RAM. + +Figure 3 showcases DCDM for incompressible smoke simulations. In each simulation, inlet boundary conditions are set in a circular portion of the bottom of the cubic domain, whereby smoke flows around potential obstacles and fills the domain. We show a smoke plume (no obstacles), flow past a complex static geometry (the Stanford bunny), and flow past a dynamic geometry (a rotating cube). Visually plausible and highly-detailed results are achieved for each simulation (see supplementary material for larger videos). The plume example uses a computational grid with resolution $1 2 8 ^ { 3 }$ , while the other two uses grids with resolution $2 5 6 ^ { 3 }$ (representing over 16 million unknowns). For each linear solve, DCDM was run until the residual was reduced by four orders of magnitude. + +Table 1: Timing and iteration comparison for different methods on the bunny example. DCDM- $\{ 6 4 , 1 2 8 \}$ calls a model whose parameters trained over a $\{ 6 4 ^ { 3 } , 1 2 8 ^ { 3 } \}$ grid. All computations are done using only CPUs; model inference does not use GPUs. + +For the bunny example, Figures 4a–b demonstrate how residuals decrease over the course of a linear solve, comparing DCDM with other methods. Figure 4a shows the mean results (with standard deviations) over the course of 400 simulation frames, while in Figure 4b, we illustrate behavior on a particular frame (frame 150). For + +
64 Grid1283 Grid2563 Grid
Methodtrnrtrnrtrnr
DCDM-642.71s1622s27261s58
DCDM-1285.37s1926s24267s44
CG1.77s16826s4651548s1046
Deflated PCG771.6s1173700s27721030s489
+ +FluidNet, we use the implementation provided by fluidnetsc22 (2022). This implementation includes pre-trained models that we use without modification. In both subfigures, it is evident that the FluidNet residual never changes, since the method is not iterative; FluidNet reduces the initial residual by no more than one order of magnitude. On the other hand, with DCDM, we can continually reduce the residual (e.g., by four orders of magnitude) as we apply more iterations of our method, just as with classical CG. In Figure 4b, we also visualize the convergence of three other classical methods, CG, Deflated PCG (Saad et al., 2000), and incomplete Cholesky preconditioned CG (ICPCG)); clearly, DCDM reduces the residual in the fewest number of iterations (e.g., approximately one order of magnitude fewer iterations than ICPCG). Since FluidNet is not an iterative method and lacks a notion of residual reduction, we treat $r _ { 0 }$ for FluidNet as though an initial guess of zero is used (as is done in our solver). + +![](images/49a3a4d6c185b571472c1686368f20fff99b702247e6716738f61bbe29294d57.jpg) +Figure 4: Convergence data for the bunny example. (a) Mean and std. dev. (over all 400 frames in the simulation) of residual reduction during linear solves (with $1 2 8 ^ { 3 }$ and $2 5 6 ^ { 3 }$ grids) using FluidNet (FN) and DCDM. (b) Residual plots with CG, ICPCG, Deflated PCG, FN, and DCDM at frame 150. Dashed and solid lines represent results for $1 2 8 ^ { 3 }$ and $2 5 6 ^ { 3 }$ , respectively. (c) Decrease in residuals with varying degrees of $\pmb { A }$ -orthogonalization $( i _ { s } ~ = ~ i _ { \mathrm { s t a r t } } )$ . (d) Reduction in residuals when the network is trained with a $6 4 ^ { 3 }$ or $1 \bar { 2 } 8 ^ { 3 }$ grid for the $2 5 6 ^ { 3 }$ grid simulation shown in Figure 3 Middle. + +To clarify these results, Table 1 reports convergence statistics for DCDM compared to standard iterative techniques CG and Deflated PCG. For all $6 4 ^ { 3 }$ , $1 2 8 ^ { 3 }$ , and $2 5 6 ^ { 3 }$ grids with the bunny example, we measure the time $t _ { r }$ and the number of iterations $n _ { r }$ required to reduce the initial residual on a particular time step of the simulation by four orders of magnitude. DCDM achieves the desired results in by far the fewest number of iterations at all resolutions. At $2 5 6 ^ { 3 }$ , DCDM performs approximately 6 times faster than CG, suggesting a potentially even wider performance advantage at higher resolutions. Inference is the dominant cost in an iteration of DCDM; the other linear algebra computations in an iteration of DCDM are comparable to those in CG. The nice result of our method is that despite the increased time per iteration, the number of required iterations is reduced so drastically that DCDM materially outperforms classical methods like CG. Although ICPCG successfully reduces number of iterations $^ { 4 \mathrm { ~ b ~ } }$ , we found the runtime to scale prohibitively with grid resolution, so we exclude it from comparison in table 1. Notably, even though Deflated PCG and DCDM are based on approximate Ritz vectors, DCDM performs far better, indicating the value of using a neural network. + +# 7 CONCLUSIONS + +We presented DCDM, incorporating CNNs into a CG-style algorithm that yields efficient, convergent behavior for solving linear systems. Our method is evaluated on linear systems with over 16 million degrees of freedom and converges to a desired tolerance in merely tens of iterations. Furthermore, despite training the underlying network on domains without obstacles, our network is able to successfully predict search directions that enable efficient linear solves on domains with complex and dynamic geometries. Moreover, the training data for our network does not require running fluid simulations or solving linear systems ahead of time; our Rayleigh-Ritz vector approach enables us to quickly generate very large training datasets, unlike approaches seen in other works. + +Our network was designed for and trained exclusively using data related to the discrete Poisson matrix, which likely limits the generalizability of our present model. However, we believe our method is readily applicable to other classes of PDEs (or general problems with graph structure) that give rise to large, sparse, symmetric linear systems. We note that our method is unlikely to work well for matrices that have high computational cost to evaluate $A * x$ (such as dense matrices), since training relies on efficient $\boldsymbol { A } * \boldsymbol { x }$ evaluations. An interesting question to consider is how well our method and current models would apply to discrete Poisson matrices arising from non-uniform grids, e.g., quadtrees or octrees (Losasso et al., 2004). + +REFERENCES +M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mane, R. Monga, S. Moore, D. Murray, C. Olah, ´ M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viegas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng. ´ TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. 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Balcan, and H. Lin (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 6111–6122. Curran Associates, Inc., 2020. URL https://proceedings.neurips.cc/ paper/2020/file/43e4e6a6f341e00671e123714de019a8-Paper.pdf. +C. Yang, X. Yang, and X. Xiao. Data-driven projection method in fluid simulation. Comp Anim Virt Worlds, 27(3-4):415–424, 2016. \ No newline at end of file diff --git a/parse/dev/ldRb12nMfLQ/ldRb12nMfLQ_content_list.json b/parse/dev/ldRb12nMfLQ/ldRb12nMfLQ_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..50170ecac62d78f2acbb0e8435c2a5aa45b81099 --- /dev/null +++ b/parse/dev/ldRb12nMfLQ/ldRb12nMfLQ_content_list.json @@ -0,0 +1,949 @@ +[ + { + "type": "text", + "text": "A DEEP CONJUGATE DIRECTION METHOD FOR ITERATIVELY SOLVING LINEAR SYSTEMS ", + "text_level": 1, + "bbox": [ + 176, + 98, + 730, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 184, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for partial differential equations. Algorithms for approximating the solution to these systems are often the bottleneck in problems that require their solution, particularly for modern applications that require many millions of unknowns. Indeed, numerical linear algebra techniques have been investigated for many decades to alleviate this computational burden. Recently, data-driven techniques have also shown promise for these problems. Motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error, we design an approach that utilizes a deep neural network to accelerate convergence via data-driven improvement of the search directions. Our method leverages a carefully chosen convolutional network to approximate the action of the inverse of the linear operator up to an arbitrary constant. We train the network using unsupervised learning with a loss function equal to the $L ^ { 2 }$ difference between an input and the system matrix times the network evaluation, where the unspecified constant in the approximate inverse is accounted for. We demonstrate the efficacy of our approach on spatially discretized Poisson equations with millions of degrees of freedom arising in computational fluid dynamics applications. Unlike state-of-the-art learning approaches, our algorithm is capable of reducing the linear system residual to a given tolerance in a small number of iterations, independent of the problem size. Moreover, our method generalizes effectively to various systems beyond those encountered during training. ", + "bbox": [ + 232, + 266, + 764, + 585 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 611, + 336, + 627 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work, we consider sparse linear systems that arise from discrete Poisson equations in incompressible flow applications (Chorin, 1967; Fedkiw et al., 2001; Bridson, 2008). We use the notation ", + "bbox": [ + 171, + 642, + 823, + 670 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/8fead0b4e25add8342bddaa4ba897dff29b2070568d204185e0f1c2ab5d7c00e.jpg", + "text": "$$\n\\mathbf { A } { \\boldsymbol { \\mathbf { \\mathit { x } } } } = \\mathbf { \\mathit { b } }\n$$", + "text_format": "latex", + "bbox": [ + 470, + 678, + 529, + 691 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "where the dimension $n$ of the matrix $A \\in \\mathbb { R } ^ { n \\times n }$ and the vector $b \\in \\mathbb { R } ^ { n }$ correlate with spatial fidelity of the computational domain. The appropriate numerical linear algebra technique depends on the nature of the problem. Direct solvers that utilize matrix factorizations (QR, Cholesky, etc. Trefethen & Bau (1997)) have optimal approximation error, but their computational cost is $\\operatorname { \\dot { O } } ( n ^ { 3 } )$ and they typically require dense storage, even for sparse $\\pmb { A }$ . Although Fast Fourier Transforms (Nussbaumer, 1981) can be used in limited instances (periodic boundary conditions, etc.), iterative techniques are most commonly adopted for these systems given their sparsity. Many applications with strict performance constraints (e.g., real-time fluid simulation) utilize basic iterations (Jacobi, Gauss-Seidel, successive over relaxation (SOR), etc.) given limited computational budget (Saad, 2003). However, large approximation errors must be tolerated since iteration counts are limited by the performance constraints. This is particularly problematic since the wide elliptic spectrum of these matrices (a condition that worsens with increased spatial fidelity/matrix dimension) leads to poor conditioning and iteration counts. Iterative techniques can achieve sub-quadratic convergence if their iteration count does not grow excessively with problem size $n$ since each iteration generally requires $O ( n )$ floating point operations for sparse matrices. Discrete elliptic operators are typically symmetric positive (semi) definite and the preconditioned conjugate gradients method (PCG) can be used to minimize iteration counts (Saad, 2003; Hestenes & Stiefel, 1952; Stiefel, 1952). Preconditioners $_ { r }$ for PCG must simultaneously: be symmetric positive definite (SPD) (and therefore admit factorization ${ \\boldsymbol { P } } = { \\boldsymbol { F } } ^ { 2 }$ ), improve the condition number of the preconditioned system $F A F y = F b$ , and be computationally cheap to construct and apply; accordingly, designing specialized preconditioners for particular classes of problems is somewhat of an art. Incomplete Cholesky preconditioners (ICPCG) (Kershaw, 1978) use a sparse approximation to the Cholesky factorization and significantly reduce iteration counts in practice; however, their inherent data dependency prevents efficient parallel implementation. Nonetheless, these are very commonly adopted for Poisson equations arising in incompressible flow (Fedkiw et al., 2001; Bridson, 2008). Multigrid (Brandt, 1977) and domain decomposition (Saad, 2003) preconditioners greatly reduce iterations counts, but they must be updated (with non-trivial cost) each time the problem changes (e.g., in computational domains with time varying boundaries) and/or for different hardware platforms. In general, choice of an optimal preconditioner for discrete elliptic operators is an open area of research. ", + "bbox": [ + 174, + 702, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 284 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Recently, data-driven approaches that leverage deep learning techniques have shown promise for solving linear systems. Various researchers have investigated machine learning estimation of multigrid parameters (Greenfeld et al., 2019; Grebhahn et al., 2016; Luz et al., 2020). Others have developed machine learning methods to estimate preconditioners (Gotz & Anzt, 2018; Stanaityte, 2020; ¨ Ichimura et al., 2020) and initial guesses for iterative methods (Luna et al., 2021; Um et al., 2020; Ackmann et al., 2020). Tompson et al. (2017) and Yang et al. (2016) develop non-iterative machine learning approximations of the inverse of discrete Poisson equations from incompressible flow. We leverage deep learning and develop a novel version of conjugate gradients iterative method for approximating the solution of SPD linear systems which we call the deep conjugate direction method (DCDM). CG iteratively adds $\\pmb { A }$ -conjugate search directions while minimizing the matrix norm of the error. We use a convolutional neural network (CNN) as an approximation of the inverse of the matrix in order to generate more efficient search directions. We only ask that our network approximate the inverse up to an unknown scaling since this decreases the degree of nonlinearity and since it does not affect the quality of the search direction (which is scale independent). The network is similar to a preconditioner, but it is not a linear function, and our modified conjugate gradients approach is designed to accommodate this nonlinearity. We use unsupervised learning to train our network with a loss function equal to the $L ^ { 2 }$ difference between an input vector and a scaling of $\\pmb { A }$ times the output of our network. To account for this unknown scaling during training, we choose the scale of the output of the network by minimizing the matrix norm of the error. Our approach allows for efficient training and generalization to problems unseen (matrices $\\pmb { A }$ and right-hand sides $^ { b }$ ). We benchmark our algorithm using the ubiquitous pressure Poisson equation (discretized on regular voxelized domains) and compare against FluidNet (Tompson et al., 2017), which is the state of the art learning-based method for these types of problems. Unlike the non-iterative approaches of Tompson et al. (2017) and Yang et al. (2016), our method can reduce the linear system residuals arbitrarily. We showcase our approach with examples that have over 16 million degrees of freedom. ", + "bbox": [ + 173, + 291, + 825, + 637 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 656, + 344, + 672 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Several papers have focused on enhancing the solution of linear systems (arising from discretized PDEs) using learning. For instance, Gotz & Anzt (2018) generate sparsity patterns for block-Jacobi ¨ preconditioners using convolutional neural networks, and Stanaityte (2020) use a CNN to predict non-zero patterns for ILU-type preconditioners for the Navier-Stokes equations (though neither work designs fundamentally new preconditioners). Ichimura et al. (2020) develop a neural-network based preconditioner where the network is used to predict approximate Green’s functions (which arise in the analytical solution of certain PDEs) that in turn yield an approximate inverse of the linear system. Hsieh et al. (2019) learn an iterator that solves linear systems, performing competitively with classical solvers like multigrid-preconditioned MINRES (Paige & Saunders, 1975). Luz et al. (2020) and Greenfeld et al. (2019) use machine learning to estimate algebraic multigrid (AMG) parameters. They note that AMG approaches rely most fundamentally on effectively chosen (problem-dependent) prolongation sparse matrices and that numerous methods have attempted to automatically create them from the matrix $\\pmb { A }$ . They train a graph neural network to learn (in an unsupervised fashion) a mapping from matrices $\\pmb { A }$ to prolongation operators. Grebhahn et al. (2016) note that geometric multigrid solver parameters can be difficult to choose to guarantee parallel performance on different hardware platforms. They use machine learning to create a code generator to help achieve this. ", + "bbox": [ + 174, + 688, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/a7ac602c2643cff48deff90a7224438f208d4a7100d53c20e1e19240e8cade2b.jpg", + "image_caption": [ + "Figure 1: (a) We illustrate a sample flow domain $\\Omega \\subset ( 0 , 1 ) ^ { 2 }$ (in 2D for ease of illustration) with internal boundaries (blue lines). (b) We voxelize the domain with a regular grid: white cells represent interior/fluid, and blue cells represent boundary conditions. (c) We train using matrix $A ^ { ( 0 , 1 ) ^ { d } }$ from a discretized domain with no interior boundary conditions, where $d$ is the dimension. This creates linear system with $n = ( n _ { c } + 1 ) ^ { d }$ unknowns where $n _ { c }$ is the number of grid cells on each direction. (d) We illustrate the non-zero entries in an example matrix $A ^ { \\Omega }$ from the voxelized and labeled (white vs. blue) grid for three example interior cells (green, purple, and brown). Each case illustrates the non-zero entries in the row associated with the example cell. All entries in rows corresponding to boundary/blue cells are zero. " + ], + "image_footnote": [], + "bbox": [ + 174, + 99, + 825, + 219 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Several works consider accelerating the solution of linear systems by learning an initial guess that is close to the true solution or otherwise helpful to descent algorithms for finding the true solution. In order to solve the discretized Poisson equation, Luna et al. (2021) accelerate the convergence of GMRES (Saad & Schultz, 1986) with an initial guess that is learned in real-time (i.e., as a simulation code runs) with no prior data. Um et al. (2020) train a network (incorporating differentiable physics, based on the underlying PDEs) in order to produce high-quality initial guesses for a CG solver. In a somewhat similar vein, Ackmann et al. (2020) use a simple feedforward neural network to predict pointwise solution components, which accelerates the conjugate residual method used to solve a relatively simple shallow-water model (a more sophisticated network and loss function are needed to handle more general PDEs and larger-scale problems). ", + "bbox": [ + 173, + 368, + 825, + 507 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "At least two papers (Ruelmann et al., 2018; Sappl et al., 2019) have sought to learn a mapping between a matrix and an associated sparse approximate inverse. In their investigation, Ruelmann et al. (2018) propose training a neural network using matrix-inverse pairs as training data. Although straightforward to implement, the cost of generating training data, let alone training the network, is prohibitive for large-scale 3D problems. Sappl et al. (2019) seek to learn a mapping between linear system matrices and sparse (banded) approximate inverses. Their loss function is the condition number of the product of the system matrix and the approximate inverse; the minimum value of the condition number is one, which is achieved exactly when an exact inverse is obtained. Although this framework is quite simple, evaluating the condition number of a matrix is asymptotically costly, and in general, the inverse of a sparse matrix can be quite dense. Accordingly, the method is not efficient or accurate enough for the large-scale 3D problems that arise in real-world engineering problems. ", + "bbox": [ + 174, + 513, + 825, + 666 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Most relevant to the present work is FluidNet (Tompson et al., 2017). FluidNet develops a highlytailored CNN architecture that is used to predict the solution of a linear projection operation (specifically, for the discrete Poisson equation) given a matrix and right-hand side. The authors demonstrate fluid simulations where the linear solve is replaced by evaluating their network. Because their network is relatively lightweight and is only evaluated once per time step, their simulations run efficiently. However, their design allows the network only one opportunity to reduce the residual for the linear solve; in practice, we observe that FluidNet is able to reduce the residual by no more than about one order of magnitude. However, in computer graphics applications, at least four orders of magnitude in residual reduction are usually required for visual fidelity, while in scientific and engineering applications, practitioners prefer solutions that reduce the residual by eight or more orders of magnitude (i.e., to within machine precision). Accordingly, FluidNet’s lack of convergence stands in stark contrast to classical, convergent methods like CG. Our method resolves this gap. ", + "bbox": [ + 173, + 672, + 825, + 840 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 MOTIVATION: INCOMPRESSIBLE FLOW ", + "text_level": 1, + "bbox": [ + 176, + 863, + 529, + 878 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We demonstrate the efficacy of our approach with the linear systems that arise in incompressible flow applications. Specifically, we use our algorithm to solve the discrete Poisson equations in ", + "bbox": [ + 176, + 895, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "regular-grid-based discretization of the pressure projection equations that arise in Chorin’s splitting technique (Chorin, 1967) for the inviscid, incompressible Euler equations. These equations are ", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c1b355957f1f3592b8d81befad750201620d6f0041dd14ae0eb7b88b47dac275.jpg", + "text": "$$\n\\rho \\left( \\frac { \\partial \\pmb { u } } { \\partial t } + \\frac { \\partial \\pmb { u } } { \\partial \\pmb { x } } \\pmb { u } \\right) + \\nabla p = \\pmb { f } ^ { e x t } , \\qquad \\nabla \\cdot \\pmb { u } = 0\n$$", + "text_format": "latex", + "bbox": [ + 338, + 137, + 658, + 172 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\textbf { \\em u }$ is fluid velocity, $p$ is pressure, $\\rho$ is density, and $f ^ { e x t }$ accounts for external forces like gravity. The equations are assumed at all positions $_ { \\textbf { \\em x } }$ in the spatial fluid flow domain $\\Omega$ and for time $t > 0$ . The left term in Equation 2 enforces conservation of momentum in the absence of viscosity, and the second part enforces incompressibility and conservation of mass. These equations are subject to initial conditions $\\rho ( \\pmb { x } , 0 ) = \\rho ^ { 0 }$ and ${ \\pmb u } ( { \\pmb x } , 0 ) = { \\pmb u } ^ { 0 } ( { \\pmb x } )$ as well as boundary conditions ${ \\pmb u } ( { \\pmb x } , t ) \\cdot { \\pmb n } ( { \\pmb x } ) = u ^ { \\partial \\Omega } ( { \\pmb x } , t )$ on the boundary of the domain ${ \\pmb x } \\in \\partial \\Omega$ (where $\\textbf { \\em n }$ is the unit outward pointing normal at position $_ { \\textbf { \\em x } }$ on the boundary). Equation 2 is discretized in both time and space. Temporally, we split the advection $\\frac { \\partial \\pmb { u } } { \\partial t } + \\frac { \\partial \\pmb { u } } { \\partial \\pmb { x } } \\pmb { u } = 0$ and body forces terms $\\rho \\frac { \\partial \\pmb { u } } { \\partial t } = \\pmb { f } ^ { e x t }$ , and finally enforce incompressibility via the pressure projection $\\frac { \\partial \\pmb { u } } { \\partial t } + \\frac { 1 } { \\rho } \\nabla p = \\pmb { 0 }$ such that $\\nabla \\cdot \\pmb { u } = 0$ ; this is the standard advection-projection scheme proposed by Chorin (1967). Using finite differences in time, we can summarize this as ", + "bbox": [ + 173, + 179, + 825, + 361 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/b9e83fdc2a77df2f7139c2359537bee131588af2939e28a1c636f7c2e1a8f4d9.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\rho ^ { 0 } \\left( \\frac { \\pmb { u } ^ { * } - \\pmb { u } ^ { n } } { \\Delta t } + \\frac { \\partial \\pmb { u } ^ { n } } { \\partial \\pmb { x } } \\pmb { u } ^ { n } \\right) = \\pmb { f } ^ { e x t } } \\\\ { \\displaystyle - \\nabla \\cdot \\frac { 1 } { \\rho ^ { 0 } } \\nabla p ^ { n + 1 } = - \\nabla \\cdot \\pmb { u } ^ { * } } \\\\ { \\displaystyle \\qquad - \\frac { 1 } { \\rho ^ { 0 } } \\nabla p ^ { n + 1 } \\cdot \\pmb { n } = \\frac { 1 } { \\Delta t } \\left( \\pmb { u } ^ { \\partial \\Omega } - \\pmb { u } ^ { * } \\cdot \\pmb { n } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 326, + 363, + 669, + 467 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For the spatial discretization, we use a regular marker-and-cell (MAC) grid (Harlow & Welch, 1965) with cubic voxels whereby velocity components are stored on the face of voxel cells, and scalar quantities (e.g., pressure $p$ or density $\\rho \\mathrm { \\hbar }$ ) are stored at voxel centers. We use backward semiLagrangian advection (Fedkiw et al., 2001; Gagniere et al., 2020) for Equation 3. All spatial partial derivatives are approximated using finite differences. Equations 4 and 5 describe the pressure Poisson equation with Neumann conditions on the boundary of the flow domain. We discretize the left hand side of Equation 4 using a standard 7-point finite difference stencil. The right-hand side is discretized using the MAC grid discrete divergence finite difference stencils as well as contributions from the boundary condition terms in Equation 5. We refer the reader to Bridson (2008) for more in-depth implementation details. Equation 5 is discretized by modifying the Poisson stencil to enforce Neumann boundary conditions. We do this using a simple labeling of the voxels in the domain. For simplicity, we assume $\\Omega \\subset ( 0 , 1 ) ^ { 3 }$ is a subset of the unit cube, potentially with internal boundaries. We label cells in the domain as either liquid or boundary. This simple classification is enough to define the Poisson discretizations (with appropriate Neumann boundary conditions at domain boundaries) that we focus on in the present work; we illustrate the details in Figure 1. We use the following notation to denote the discrete Poisson equations associated with Equations 4–5: ", + "bbox": [ + 173, + 469, + 825, + 693 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/ac399afedaac667ec410278ede5ec8e5a957aa976ca5e09188331dc7a1ba5bb7.jpg", + "text": "$$\n\\begin{array} { r } { { \\cal A } ^ { \\Omega } { \\boldsymbol x } = { \\boldsymbol b } ^ { \\nabla \\cdot { \\boldsymbol u } ^ { * } } + { \\boldsymbol b } ^ { u ^ { \\partial \\Omega } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 419, + 698, + 576, + 718 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $A ^ { \\Omega }$ is the discrete Poisson matrix associated with the voxelized domain, $_ { \\textbf { \\em x } }$ is the vector of unknown pressure, and $\\mathbf { \\delta } _ { b } \\nabla \\cdot \\mathbf { u } ^ { * }$ and b u ∂ Ω are the right-hand side terms from Equations 4 and 5, respectively. We define a special case of the matrix involved in this discretization to be the Poisson matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ associated with $\\Omega = ( 0 , 1 ) ^ { 3 }$ , i.e., a full fluid domain with no internal boundaries. We use this matrix for training, yet demonstrate that our network generalizes to all other matrices arising from more complicated flow domains. ", + "bbox": [ + 174, + 727, + 825, + 816 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 DEEP CONJUGATE DIRECTION METHOD ", + "text_level": 1, + "bbox": [ + 174, + 835, + 540, + 853 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We present our method for the deep learning acceleration of iterative approximations to the solution of linear systems of the form seen in Equation 6. We first discuss relevant details of the conjugate gradients (CG) method, particularly line search and $\\pmb { A }$ -orthogonal search directions. We then present a deep learning technique for improving the quality of these search directions that ultimately reduces iteration counts required to achieve satisfactory residual reduction. Lastly, we outline the training procedures for our deep convolutional neural network. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 825, + 132 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our approach iteratively improves approximations to the solution $_ { \\textbf { \\em x } }$ of Equation 6. We build on the method of CG, which requires the matrix $A ^ { \\Omega }$ in Equation 6 to be SPD. SPD matrices $A ^ { \\Omega }$ give rise to the matrix norm $\\| \\pmb { y } \\| _ { A ^ { \\Omega } } = \\sqrt { \\pmb { y } ^ { T } A ^ { \\Omega } \\pmb { y } }$ . CG can be derived in terms of iterative line search improvement based on optimality in this norm. That is, an iterate $\\pmb { x } _ { k - 1 } \\approx \\pmb { x }$ is updated in search direction $\\scriptstyle d _ { k }$ by a step size $\\alpha _ { k }$ that is chosen to minimize the matrix norm of the error: ", + "bbox": [ + 174, + 138, + 825, + 212 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/794da5f9a8f94a2ca23ac137e3679829e9c6027164f5d341253cdd67dfb2e112.jpg", + "text": "$$\n\\alpha _ { k } = \\underset { \\alpha } { \\arg \\operatorname* { m i n } } \\frac { 1 } { 2 } \\left\\| \\pmb { x } - ( \\pmb { x } _ { k - 1 } + \\alpha \\pmb { d } _ { k } ) \\right\\| _ { \\pmb { A } ^ { \\Omega } } ^ { 2 } = \\frac { { \\pmb { r } } _ { k - 1 } ^ { T } { \\pmb { d } _ { k } } } { { \\pmb { d } } _ { k } ^ { T } { \\pmb { A } } ^ { \\Omega } { \\pmb { d } _ { k } } } ,\n$$", + "text_format": "latex", + "bbox": [ + 312, + 213, + 683, + 251 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\pmb { r } _ { k - 1 } = \\pmb { b } - \\pmb { A } ^ { \\Omega } \\pmb { x } _ { k - 1 }$ is the $( k - 1 ) ^ { \\mathrm { t h } }$ residual and $^ { b }$ is the right-hand side in Equation 6. Different search directions result in different algorithms. A natural choice is the negative gradient of the matrix norm of the error (evaluated at the current iterate), since this will point in the direction of steepest decrease $\\begin{array} { r } { \\pmb { d } _ { k } = - \\frac { 1 } { 2 } \\nabla \\left\\| \\pmb { x } _ { k - 1 } \\right\\| _ { \\pmb { A } ^ { \\Omega } } ^ { 2 } = \\pmb { r } _ { k - 1 } } \\end{array}$ . This is the gradient descent method (GD). Unfortunately, this approach requires many iterations in practice. A more effective strategy is to choose directions that are $\\pmb { A }$ -orthogonal (i.e., ${ \\bf \\Phi } _ { { \\bf i } } ^ { T } { \\bf A } ^ { \\Omega } { \\bf d } _ { j } = \\mathrm { ~ ~ \\dot { ~ } { ~ 0 ~ } ~ }$ for $i \\neq j$ ). With this choice, the search directions form a basis for $\\mathbb { R } ^ { n }$ so that the initial error can be written as $\\begin{array} { r } { \\pmb { e } _ { 0 } = \\pmb { x } - \\pmb { x } _ { 0 } = \\sum _ { i = 1 } ^ { n } e _ { i } \\pmb { d } _ { i } } \\end{array}$ , where $e _ { i }$ are the components of the initial error written in the basis. Furthermore, when the search directions are $\\pmb { A }$ -orthogonal, the optimal step sizes $\\alpha _ { k }$ at each iteration satisfy ", + "bbox": [ + 173, + 252, + 826, + 382 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/27f862ee424cf548433605036964bc38ad83b6328e4fe8ee6ce093dd03b21c48.jpg", + "text": "$$\n\\alpha _ { k } = \\frac { r _ { k - 1 } ^ { T } d _ { k } } { d _ { k } ^ { T } A ^ { \\Omega } d _ { k } } = \\frac { d _ { k } ^ { T } A ^ { \\Omega } e _ { k - 1 } } { d _ { k } ^ { T } A ^ { \\Omega } d _ { k } } = \\frac { d _ { k } ^ { T } A ^ { \\Omega } \\left( \\sum _ { i = 1 } ^ { n } e _ { i } d _ { i } - \\sum _ { j = 1 } ^ { k - 1 } \\alpha _ { j } d _ { j } \\right) } { d _ { k } ^ { T } A ^ { \\Omega } d _ { k } } = e _ { k } .\n$$", + "text_format": "latex", + "bbox": [ + 240, + 383, + 758, + 429 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "That is, the optimal step sizes are chosen to precisely eliminate the components of the error on the basis defined by the search directions. Thus, convergence is determined by the (at most $n$ ) non-zero components $e _ { i }$ in the initial error. Although rounding errors prevent this from happening exactly in practice, this property greatly reduces the number of required iterations (Golub & Loan, 2012). ", + "bbox": [ + 174, + 430, + 823, + 487 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "CG can be viewed as a modification of GD where the search direction is chosen as the component of the residual (equivalently, the negative gradient of the matrix norm of the error) that is $\\pmb { A }$ -orthogonal to all previous search directions: ", + "bbox": [ + 176, + 492, + 825, + 535 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/823b515874f8878d3bc9c886f39ccbd598d942c63f696c3c9a5dc8fad3a847d7.jpg", + "text": "$$\n\\pmb { d } _ { k } = \\pmb { r } _ { k - 1 } - \\sum _ { i = 1 } ^ { k - 1 } h _ { i k } \\pmb { d } _ { i } , \\qquad h _ { i k } = \\frac { { d } _ { i } ^ { T } { \\pmb { A } } ^ { \\Omega } \\pmb { r } _ { k - 1 } } { { d } _ { i } ^ { T } { \\pmb { A } } ^ { \\Omega } \\pmb { d } _ { i } } .\n$$", + "text_format": "latex", + "bbox": [ + 333, + 537, + 665, + 580 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In practice, $h _ { i k } = 0$ for $i < k - 1$ , and this iteration can therefore be performed without the need to store all previous search directions $\\mathbf { \\mathbf { { \\alpha } } } d _ { i }$ and without the need for computing all previous $h _ { i k }$ . ", + "bbox": [ + 173, + 583, + 821, + 612 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "While the residual is a natural choice for generating $\\pmb { A }$ -orthogonal search directions (since it points in the direction of the steepest local decrease), it is not the optimal search direction. If $\\scriptstyle d _ { k }$ is parallel to $( A ^ { \\Omega } ) ^ { - 1 } r _ { k - 1 }$ , then $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { k }$ will be equal to $_ { \\textbf { \\em x } }$ since $\\alpha _ { k }$ (computed from Equation 7) will step directly to the solution. We can see this by considering the residual and its relation to the search direction: ", + "bbox": [ + 173, + 617, + 825, + 674 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/8bc12357b2f542a177c0dbe7c2b250136ff63d91c180c2f209010068301bb834.jpg", + "text": "$$\n\\begin{array} { r } { r _ { k } = b - A ^ { \\Omega } x _ { k } = b - A ^ { \\Omega } x _ { k - 1 } - \\alpha _ { k } A ^ { \\Omega } d _ { k } = r _ { k - 1 } - \\alpha _ { k } A ^ { \\Omega } d _ { k } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 276, + 681, + 720, + 700 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In light of this, we use deep learning to create an approximation $f ( c , r )$ to $( A ^ { \\Omega } ) ^ { - 1 } r$ , where $^ c$ denotes the network weights and biases. This is analogous to using a preconditioner in PCG; however, our network is not SPD (nor even a linear function). We simply use this data-driven approach as our means of generating better search directions $\\scriptstyle d _ { k }$ . Furthermore, we only need to approximate a vector parallel to $( A ^ { \\Omega } ) ^ { - 1 } r$ since the step size $\\alpha _ { k }$ will account for any scaling in practice. In other words, $f ( c , r ) \\approx s _ { r } ( A ^ { \\Omega } ) ^ { - 1 } r$ , where the scalar $s _ { r }$ is not defined globally; it only depends on $\\pmb { r }$ , and the model does not learn it. Lastly, as with CG, we enforce $\\pmb { A }$ -orthogonality, yielding search directions ", + "bbox": [ + 173, + 708, + 825, + 808 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/2f5892c23191745ca45a89c6df6aa816d57d4b70d9d40bae7947ea58ff2fad11.jpg", + "text": "$$\n\\pmb { d } _ { k } = \\pmb { f } ( \\pmb { c } , \\pmb { r } _ { k - 1 } ) - \\sum _ { i = 1 } ^ { k - 1 } h _ { i k } \\pmb { d } _ { i } , \\qquad h _ { i k } = \\frac { \\pmb { f } ( \\pmb { c } , \\pmb { r } _ { k - 1 } ) ^ { T } \\pmb { A } ^ { \\Omega } \\pmb { d } _ { i } } { \\pmb { d } _ { i } ^ { T } \\pmb { A } ^ { \\Omega } \\pmb { d } _ { i } } .\n$$", + "text_format": "latex", + "bbox": [ + 290, + 809, + 707, + 852 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We summarize our approach in Algorithm 1. Note that we introduce the variable $i _ { \\mathrm { s t a r t } }$ . To guarantee $\\pmb { A }$ -orthogonality between all search directions, we must have $i _ { \\mathrm { s t a r t } } = 1$ . However, this requires storing all prior search directions, which can be costly. We found that using $i _ { \\mathrm { s t a r t } } = k - 2$ worked nearly as well as $i _ { \\mathrm { s t a r t } } = 1$ in practice (in terms of our ability to iteratively reduce the residual of the system). We demonstrate this in Figure 4c. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 MODEL ARCHITECTURE, DATASETS, AND TRAINING ", + "text_level": 1, + "bbox": [ + 174, + 102, + 643, + 118 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Efficient performance of our method requires effective training of our deep convolutional network for weights and biases $^ c$ such that $f ( c , r ) { \\overset { \\cdot } { \\approx } } s _ { r } ( A ^ { \\Omega } ) ^ { - 1 } r$ (for arbitrary scalar $s _ { r }$ ). We design a model architecture, loss function, and unsupervised training approach to achieve this. Our approach has modest training requirements and allows for effective residual reduction while generalizing well to problems not seen in the training data. ", + "bbox": [ + 174, + 132, + 612, + 231 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 LOSS FUNCTION AND UNSUPERVISED LEARNING ", + "text_level": 1, + "bbox": [ + 173, + 246, + 558, + 261 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Although we generalize to arbitrary matrices $A ^ { \\Omega }$ from Equation 6 that correspond to domains $\\Omega \\subset ( 0 , 1 ) ^ { 3 }$ that have internal boundaries (see Figure 1), we train using just the matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ from the full cube domain $( 0 , 1 ) ^ { 3 }$ . In contrast, other similar approaches (Tompson et al., 2017; Yang et al., 2016) train using matrices $A ^ { \\Omega }$ and right-hand sides $\\pmb { b } ^ { \\nabla \\cdot \\pmb { u } ^ { * } } + \\overline { { \\pmb { b } } } ^ { u ^ { \\partial \\Omega } }$ that arise from flow in many domains with internal boundaries. We train our network by minimiz", + "bbox": [ + 173, + 271, + 614, + 376 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Algorithm 1 DCDM ", + "text_level": 1, + "bbox": [ + 627, + 138, + 769, + 152 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "1: $\\pmb { r } _ { 0 } = \\pmb { b } - \\pmb { A } ^ { \\Omega } \\pmb { x } _ { 0 }$ \n2: $k = 1$ \n3: while $\\| r _ { k - 1 } \\| \\ge \\epsilon$ do \n4: dk = f (c, rk−1∥rk−1∥ ) \n5: for $i _ { \\mathrm { s t a r t } } \\le i < k$ do \n6: hik = dT AΩdi dT AΩdi \n7: dk-=hikdi \n8: end for \n9: $\\begin{array} { r l } & { \\alpha _ { k } = \\frac { r _ { k - 1 } ^ { T } d _ { k } } { d _ { k } ^ { T } A ^ { \\Omega } d _ { k } } } \\\\ & { x _ { k } = x _ { k - 1 } + \\alpha _ { k } d _ { k } } \\\\ & { r _ { k } = b - A ^ { \\Omega } x _ { k } } \\\\ & { k = k + 1 } \\end{array}$ \n10: \n11: \n12: \n13: end while ", + "bbox": [ + 632, + 156, + 807, + 356 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "ing the L2 difference ∥r−αA(0,1)3 f (c, r)∥2, where α = rT f(c,r)f(c,r)T A(0,1)3 f(c,r) f rom Equation 7. This choice of $\\alpha$ accounts for the unknown scaling in the approximation of $f ( c , r )$ to $\\left( A ^ { ( 0 , 1 ) ^ { 3 } } \\right) ^ { - 1 } r$ . We use an unsupervised approach and train the model by minimizing ", + "bbox": [ + 173, + 377, + 826, + 439 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/bf11442072e401d6ca5ceb5d5c2fe7051c67c9d21204c42168a7c1b30e85f761.jpg", + "text": "$$\n\\begin{array} { r } { \\mathrm { L o s s } ( \\pmb { f } , \\pmb { c } , \\mathscr { D } ) = \\frac { 1 } { | \\mathscr { D } | } \\sum _ { r \\in \\mathscr { D } } \\| \\pmb { r } - \\frac { r ^ { T } f ( \\pmb { c } , r ) } { \\pmb { f } ( \\pmb { c } , r ) ^ { T } \\pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \\pmb { f } ( \\pmb { c } , r ) } \\pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \\pmb { f } ( \\pmb { c } , r ) \\| _ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 446, + 732, + 474 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "for given dataset $\\mathcal { D }$ consisting of training vectors $b ^ { i }$ . In Algorithm 1, the normalized residuals $\\frac { \\boldsymbol { r } _ { k } } { \\| \\boldsymbol { r } _ { k } \\| }$ are passed as inputs to the model. Unlike in e.g. FluidNet (Tompson et al., 2017), only the first residual $\\frac { r _ { 0 } } { \\Vert r _ { 0 } \\Vert }$ is directly related to the problem-dependent original right-hand side $^ { b }$ . Hence we consider a broader range of training vectors than those expected in a given problem of interest, e.g., incompressible flows. We observe that generally the residuals $\\mathbf { \\nabla } r _ { k }$ in Algorithm 1 are skewed to the lower end of the spectrum of the matrix $A ^ { \\Omega }$ . Since $A ^ { \\Omega }$ is a discretized elliptic operator, lower end modes are of lower frequency of spatial oscillation. We create our training vectors $\\mathbf { \\boldsymbol { b } } ^ { i } \\in \\mathcal { D }$ using $m \\ll n$ approximate eigenvectors of the training matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ . We use the Rayleigh-Ritz method to create approximate eigenvectors $\\pmb q _ { i }$ , $0 \\leq i < m$ . This approach allows us to effectively approximate the full spectrum of $A ^ { ( 0 , 1 ) ^ { 3 } }$ without computing the full eigendecomposition, which can be expensive $( { \\cal O } ( \\bar { n } ^ { 3 } ) )$ at high resolution. We found that using $m \\ : = \\ : 1 0 0 0 0$ worked well in practice. The Rayleigh-Ritz vectors are orthonormal and satisfy $Q _ { m } ^ { T } A ^ { ( 0 , 1 ) ^ { 3 } } Q _ { m } = \\Lambda _ { m }$ , where $\\pmb { \\Lambda } _ { m }$ is a diagonal matrix with nondecreasing diagonal entries $\\lambda _ { i }$ referred to as Ritz values (approximate eigenvalues) and $\\pmb { Q } _ { m } = [ \\pmb { q } _ { 0 } , \\pmb { q } _ { 1 } , \\dots , \\pmb { q } _ { m - 1 } ] \\in \\mathbb { R } ^ { n \\times m }$ . We pick $\\begin{array} { r } { \\pmb { b } ^ { i } = \\frac { \\sum _ { j = 0 } ^ { m - 1 } c _ { j } ^ { i } \\pmb { q } _ { j } } { \\left\\| \\sum _ { j = 0 } ^ { m - 1 } c _ { j } ^ { i } \\pmb { q } _ { j } \\right\\| } } \\end{array}$ , where the coefficients $c _ { j } ^ { i }$ are picked from a standard normal distribution ", + "bbox": [ + 173, + 482, + 826, + 720 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/395e1e20cca55da1945267d43b1e898ba5928b72735e7c30cc209c5a7aadea3e.jpg", + "text": "$$\nc _ { j } ^ { i } = \\left\\{ { \\begin{array} { l l } { 0 \\cdot \\mathcal { N } ( 0 , 1 ) } & { { \\mathrm { i f ~ } } \\tilde { j } \\leq j \\leq { \\frac { m } { 2 } } + \\theta } \\\\ { \\mathcal { N } ( 0 , 1 ) } & { { \\mathrm { o t h e r w i s e } } } \\end{array} } \\right.\n$$", + "text_format": "latex", + "bbox": [ + 367, + 724, + 629, + 760 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\theta$ is a small number (we used $\\theta = 5 0 0$ ), and $\\tilde { j }$ is the first index that $\\lambda _ { \\tilde { j } } ^ { - } > 0$ . This choice creates $9 0 \\%$ of $\\mathbf { \\nabla } _ { b } i$ from the lower end of the spectrum, with the remaining $1 0 \\%$ from the higher end. The Riemann-Lebesgue Lemma states the Fourier spectrum of a continuous function will decay at infinity, so this specific choice of $b _ { i }$ ’s is reasonable for the training set. In practice, we also observed that the right-hand sides of the pressure system that arose in flow problems (in the empty domain) tended to be at the lower end of the spectrum. Notably, even though this dataset only uses RayleighRitz vectors from the training matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ , our network can be effectively generalized to flows in irregular domains, e.g., smoke flow past a rotating box and flow past a bunny (see Figure 3). ", + "bbox": [ + 173, + 763, + 825, + 883 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We generate the Rayleigh-Ritz vectors by first tridiagonalizing the training matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ with Lanczos iterations (Lanczos, 1950) to form $\\pmb { T } ^ { m } = \\pmb { Q } _ { m } ^ { L } ^ { T } \\pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \\pmb { Q } _ { m } ^ { L } \\in \\mathbb { R } ^ { m \\times m }$ . We then diagonalize $\\pmb { T } ^ { m } = \\hat { \\pmb { Q } } ^ { T } \\pmb { \\Lambda } _ { m } \\hat { \\pmb { Q } }$ . While costly, we note that this algorithm is performed on the comparably small $m \\times m$ matrix $\\mathbf { T } ^ { m }$ (rather than on the $A ^ { ( 0 , 1 ) ^ { 3 } } \\in \\mathbb { R } ^ { n \\times n } ,$ ). This yields the Rayleigh-Ritz vectors as the columns of $Q _ { m } = Q _ { m } ^ { L } \\hat { Q }$ . The Lanczos vectors are the columns of the matrix $Q _ { m } ^ { L }$ and satisfy a three-term recurrence whereby the next Lanczos vector can be computed from the previous two as ", + "bbox": [ + 174, + 891, + 823, + 925 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 825, + 165 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/74fb81c962121faba4c85f6b417bd9fc92facd780a9f0d7f579191dbfec861ad.jpg", + "text": "$$\n\\beta _ { j } \\pmb { q } _ { j + 1 } ^ { L } = \\pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \\pmb { q } _ { j } ^ { L } - \\beta _ { j - 1 } \\pmb { q } _ { j - 1 } ^ { L } - \\alpha _ { j } \\pmb { q } _ { j } ^ { L } ,\n$$", + "text_format": "latex", + "bbox": [ + 354, + 171, + 642, + 193 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $\\alpha _ { j }$ and $\\beta _ { j }$ are diagonal and subdiagonal entries of $\\pmb { T } ^ { k }$ . $\\beta _ { j }$ is computed so that $\\pmb q _ { j + 1 } ^ { L }$ is a unit vector, and $\\alpha _ { j + 1 } = \\pmb { q } _ { j + 1 } ^ { T } \\pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \\pmb { q } _ { j + 1 }$ . We initialize the iteration with a random $\\pmb { q } _ { 0 } ^ { L } \\in \\mathrm { s p a n } ( \\pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } )$ . The Lanczos algorithm can be viewed as a modified Gram-Schmidt technique to create an orthonormal basis for the Krylov space associated with $\\pmb q _ { 0 } ^ { L }$ and $A ^ { ( 0 , 1 ) ^ { 3 } }$ , and it therefore suffers from rounding error sensitivities manifested as loss of orthonormality with vectors that do not appear in the recurrence. We found that the simple strategy described in Paige (1971) of orthogonalizing each iterate with all previous Lanczos vectors to be sufficient for our training purposes. Dataset creation takes 5–7 hours for a $6 4 ^ { 3 }$ computational grid, and 2–2.5 days for a $\\boldsymbol { 1 2 8 ^ { \\overline { { 3 } } } }$ grid. ", + "bbox": [ + 173, + 200, + 825, + 320 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 MODEL ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 174, + 337, + 383, + 351 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/048b839895ab4dfb5d59e79b90cfaaf105f450ca0f80073d468f3f23646ad845.jpg", + "image_caption": [ + "Figure 2: Model architecture for training with $A ^ { ( 0 , 1 ) ^ { 3 } }$ on a $1 2 8 ^ { 3 }$ grid. " + ], + "image_footnote": [], + "bbox": [ + 189, + 366, + 544, + 522 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The internal structure of our CNN architecture for a $1 2 8 ^ { 3 }$ grid is shown in Figure 2. It consists of a series of convolutional layers with residual connections. The upper left of Figure 2 ( $K$ Residual Blocks) shows our use of multiple blocks of residually connected layers. Notably, within each block, the first layer directly affects the last layer with an addition operator.All non-input or output convolutions use a $3 \\times 3 \\times 3$ filter, and all layers consist of 16 feature maps. In the middle of the first level, a layer is downsampled (via the average pool", + "bbox": [ + 578, + 363, + 823, + 570 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "ing operator with $( 2 \\times 2 \\times 2 )$ pool size) and another set of convolutional layers is applied with residual connection blocks. The last layer in the second level is upscaled and added to the layer that is downsampled. The last layer in the network is dense with a linear activation function. The activation functions in all convolutional layers are ReLU, except for the first convolution, which uses a linear activation function. ", + "bbox": [ + 173, + 570, + 825, + 640 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Initially we tried a simple deep feedforward convolutional network with residual connections (motivated by He et al. (2016)). Although such a simple model works well for DCDM, it requires high number of layers, which results in higher training and inference times. We found that creating parallel layers of CNNs with downsampling reduced the number of layers required. In summary, our goal was to first identify the simplest network architecture that provided adequate accuracy for our target problems, and subsequently, we sought to make architectural changes to minimize training and inference time; further optimizations are possible. ", + "bbox": [ + 174, + 647, + 825, + 746 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Differing resolutions use differing numbers of convolutions, but the fundamental structure remains the same. More precisely, the number of residual connections is changed for different resolutions. For example, a $6 \\dot { 4 } ^ { 3 }$ grid uses one residual block on the left, two on the right on the upper level, and three on the lower level. Furthermore, the weights trained on a lower resolution grid can be used effectively with higher resolutions. Figure 4d shows convergence results for a $2 \\mathrm { { 5 6 ^ { 3 } } }$ grid, using a model trained for a $6 4 ^ { 3 }$ grid and a $1 2 \\bar { 8 } ^ { 3 }$ grid. The model that we use for $2 5 6 ^ { 3 }$ grids in our final examples was trained on a $1 2 8 ^ { 3 }$ grid; however, as the shown in the figure, even training with a $6 4 ^ { 3 }$ grid allows for efficient residual reduction. Table 1 shows results for three different resolutions, where DCDM uses $6 4 ^ { 3 }$ and $1 2 8 ^ { 3 }$ trained models. This approach makes the number of parameters in the model independent of the spatial fidelity of the problem. ", + "bbox": [ + 173, + 751, + 825, + 891 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.3 TRAINING ", + "text_level": 1, + "bbox": [ + 174, + 103, + 285, + 117 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Using the procedure explained in Section 5.1, we create the training dataset ${ \\mathcal { D } } \\in \\mathsf { s p a n } ( A ^ { ( 0 , 1 ) ^ { 3 } } ) \\cap { \\mathcal { S } } ^ { n - 1 }$ of size 20000 generated from 10000 Rayleigh-Ritz vectors. We train our model with TensorFlow (Abadi et al., 2015) on a single NVIDIA RTX A6000 GPU with 48GB memory. Training is done with standard deep learning techniques—more precisely, back-propagation and the ADAM optimizer (Kingma & Ba, 2015) (with starting learning rate 0.0001). Training takes approximately 10 minutes and 1 hour per epoch for grid resolutions $6 4 ^ { 3 }$ and $\\mathrm { \\dot { 1 } 2 8 ^ { 3 } }$ , respectively. We trained our model for 50 epochs; however, the model from the thirty", + "bbox": [ + 174, + 131, + 419, + 381 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/32fa04638f7e6d3d367c6df56f430f093be0e2e68b4d13ed0903e506ea28d74a.jpg", + "image_caption": [ + "Figure 3: DCDM for simulating a variety of incompressible flow examples. Left: smoke plume at $t \\ =$ 6.67, 13.33, 20 seconds. Middle: smoke passing bunny at $t = 5 , 1 0 , 1 5$ seconds. Right: smoke passing a spinning box (time-dependent Neumann boundary conditions) at $t = 2 . 6 7 , 6 , 9 . 3 3$ seconds. " + ], + "image_footnote": [], + "bbox": [ + 450, + 133, + 803, + 265 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "first epoch was optimal for $6 4 ^ { 3 }$ , and the model from the third epoch was optimal for $1 2 8 ^ { 3 }$ ", + "bbox": [ + 173, + 382, + 759, + 396 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 RESULTS AND ANALYSIS ", + "text_level": 1, + "bbox": [ + 176, + 419, + 413, + 434 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We demonstrate DCDM on three increasingly difficult examples and provide numerical evidence for the efficient convergence of our method. All examples were run on a workstation with dual AMD EPYC 75F3 processors and 512GB RAM. ", + "bbox": [ + 174, + 450, + 823, + 492 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figure 3 showcases DCDM for incompressible smoke simulations. In each simulation, inlet boundary conditions are set in a circular portion of the bottom of the cubic domain, whereby smoke flows around potential obstacles and fills the domain. We show a smoke plume (no obstacles), flow past a complex static geometry (the Stanford bunny), and flow past a dynamic geometry (a rotating cube). Visually plausible and highly-detailed results are achieved for each simulation (see supplementary material for larger videos). The plume example uses a computational grid with resolution $1 2 8 ^ { 3 }$ , while the other two uses grids with resolution $2 5 6 ^ { 3 }$ (representing over 16 million unknowns). For each linear solve, DCDM was run until the residual was reduced by four orders of magnitude. ", + "bbox": [ + 173, + 500, + 825, + 611 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 1: Timing and iteration comparison for different methods on the bunny example. DCDM- $\\{ 6 4 , 1 2 8 \\}$ calls a model whose parameters trained over a $\\{ 6 4 ^ { 3 } , 1 2 8 ^ { 3 } \\}$ grid. All computations are done using only CPUs; model inference does not use GPUs. ", + "bbox": [ + 367, + 623, + 823, + 679 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For the bunny example, Figures 4a–b demonstrate how residuals decrease over the course of a linear solve, comparing DCDM with other methods. Figure 4a shows the mean results (with standard deviations) over the course of 400 simulation frames, while in Figure 4b, we illustrate behavior on a particular frame (frame 150). For ", + "bbox": [ + 173, + 619, + 354, + 797 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/3b7dccc2e3d2c2f389bc842cc59d0461cebabdd9b1b1f7f20a9c1b66da69f783.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
64 Grid1283 Grid2563 Grid
Methodtrnrtrnrtrnr
DCDM-642.71s1622s27261s58
DCDM-1285.37s1926s24267s44
CG1.77s16826s4651548s1046
Deflated PCG771.6s1173700s27721030s489
", + "bbox": [ + 370, + 690, + 816, + 796 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "FluidNet, we use the implementation provided by fluidnetsc22 (2022). This implementation includes pre-trained models that we use without modification. In both subfigures, it is evident that the FluidNet residual never changes, since the method is not iterative; FluidNet reduces the initial residual by no more than one order of magnitude. On the other hand, with DCDM, we can continually reduce the residual (e.g., by four orders of magnitude) as we apply more iterations of our method, just as with classical CG. In Figure 4b, we also visualize the convergence of three other classical methods, CG, Deflated PCG (Saad et al., 2000), and incomplete Cholesky preconditioned CG (ICPCG)); clearly, DCDM reduces the residual in the fewest number of iterations (e.g., approximately one order of magnitude fewer iterations than ICPCG). Since FluidNet is not an iterative method and lacks a notion of residual reduction, we treat $r _ { 0 }$ for FluidNet as though an initial guess of zero is used (as is done in our solver). ", + "bbox": [ + 173, + 799, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/49a3a4d6c185b571472c1686368f20fff99b702247e6716738f61bbe29294d57.jpg", + "image_caption": [ + "Figure 4: Convergence data for the bunny example. (a) Mean and std. dev. (over all 400 frames in the simulation) of residual reduction during linear solves (with $1 2 8 ^ { 3 }$ and $2 5 6 ^ { 3 }$ grids) using FluidNet (FN) and DCDM. (b) Residual plots with CG, ICPCG, Deflated PCG, FN, and DCDM at frame 150. Dashed and solid lines represent results for $1 2 8 ^ { 3 }$ and $2 5 6 ^ { 3 }$ , respectively. (c) Decrease in residuals with varying degrees of $\\pmb { A }$ -orthogonalization $( i _ { s } ~ = ~ i _ { \\mathrm { s t a r t } } )$ . (d) Reduction in residuals when the network is trained with a $6 4 ^ { 3 }$ or $1 \\bar { 2 } 8 ^ { 3 }$ grid for the $2 5 6 ^ { 3 }$ grid simulation shown in Figure 3 Middle. " + ], + "image_footnote": [], + "bbox": [ + 171, + 98, + 825, + 275 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 390, + 823, + 417 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "To clarify these results, Table 1 reports convergence statistics for DCDM compared to standard iterative techniques CG and Deflated PCG. For all $6 4 ^ { 3 }$ , $1 2 8 ^ { 3 }$ , and $2 5 6 ^ { 3 }$ grids with the bunny example, we measure the time $t _ { r }$ and the number of iterations $n _ { r }$ required to reduce the initial residual on a particular time step of the simulation by four orders of magnitude. DCDM achieves the desired results in by far the fewest number of iterations at all resolutions. At $2 5 6 ^ { 3 }$ , DCDM performs approximately 6 times faster than CG, suggesting a potentially even wider performance advantage at higher resolutions. Inference is the dominant cost in an iteration of DCDM; the other linear algebra computations in an iteration of DCDM are comparable to those in CG. The nice result of our method is that despite the increased time per iteration, the number of required iterations is reduced so drastically that DCDM materially outperforms classical methods like CG. Although ICPCG successfully reduces number of iterations $^ { 4 \\mathrm { ~ b ~ } }$ , we found the runtime to scale prohibitively with grid resolution, so we exclude it from comparison in table 1. Notably, even though Deflated PCG and DCDM are based on approximate Ritz vectors, DCDM performs far better, indicating the value of using a neural network. ", + "bbox": [ + 173, + 425, + 825, + 619 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 654, + 328, + 670 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We presented DCDM, incorporating CNNs into a CG-style algorithm that yields efficient, convergent behavior for solving linear systems. Our method is evaluated on linear systems with over 16 million degrees of freedom and converges to a desired tolerance in merely tens of iterations. Furthermore, despite training the underlying network on domains without obstacles, our network is able to successfully predict search directions that enable efficient linear solves on domains with complex and dynamic geometries. Moreover, the training data for our network does not require running fluid simulations or solving linear systems ahead of time; our Rayleigh-Ritz vector approach enables us to quickly generate very large training datasets, unlike approaches seen in other works. ", + "bbox": [ + 174, + 694, + 825, + 806 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Our network was designed for and trained exclusively using data related to the discrete Poisson matrix, which likely limits the generalizability of our present model. However, we believe our method is readily applicable to other classes of PDEs (or general problems with graph structure) that give rise to large, sparse, symmetric linear systems. We note that our method is unlikely to work well for matrices that have high computational cost to evaluate $A * x$ (such as dense matrices), since training relies on efficient $\\boldsymbol { A } * \\boldsymbol { x }$ evaluations. 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URL https://proceedings.neurips.cc/ paper/2020/file/43e4e6a6f341e00671e123714de019a8-Paper.pdf. \nC. Yang, X. Yang, and X. Xiao. Data-driven projection method in fluid simulation. Comp Anim Virt Worlds, 27(3-4):415–424, 2016. 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Many applications with strict per-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "formance constraints (e.g., real-time fluid simulation) utilize basic iterations (Jacobi, Gauss-Seidel,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "successive over relaxation (SOR), etc.) given limited computational budget (Saad, 2003). 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Algorithms for approximating the solution to these", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 255, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 255, + 469, + 268 + ], + "score": 1.0, + "content": "systems are often the bottleneck in problems that require their solution, particu-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 266, + 470, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 266, + 470, + 279 + ], + "score": 1.0, + "content": "larly for modern applications that require many millions of unknowns. Indeed,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 278, + 470, + 289 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 470, + 289 + ], + "score": 1.0, + "content": "numerical linear algebra techniques have been investigated for many decades to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 289, + 469, + 300 + ], + "spans": [ + { + "bbox": [ + 142, + 289, + 469, + 300 + ], + "score": 1.0, + "content": "alleviate this computational burden. Recently, data-driven techniques have also", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 299, + 469, + 311 + ], + "spans": [ + { + "bbox": [ + 142, + 299, + 469, + 311 + ], + "score": 1.0, + "content": "shown promise for these problems. Motivated by the conjugate gradients algo-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 310, + 469, + 322 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 469, + 322 + ], + "score": 1.0, + "content": "rithm that iteratively selects search directions for minimizing the matrix norm of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 469, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 469, + 333 + ], + "score": 1.0, + "content": "the approximation error, we design an approach that utilizes a deep neural network", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 332, + 469, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 332, + 469, + 344 + ], + "score": 1.0, + "content": "to accelerate convergence via data-driven improvement of the search directions.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 343, + 470, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 470, + 356 + ], + "score": 1.0, + "content": "Our method leverages a carefully chosen convolutional network to approximate", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 353, + 470, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 353, + 470, + 366 + ], + "score": 1.0, + "content": "the action of the inverse of the linear operator up to an arbitrary constant. We", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 364, + 468, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 365, + 455, + 376 + ], + "score": 1.0, + "content": "train the network using unsupervised learning with a loss function equal to the", + "type": "text" + }, + { + "bbox": [ + 456, + 364, + 468, + 375 + ], + "score": 0.86, + "content": "L ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 375, + 470, + 388 + ], + "spans": [ + { + "bbox": [ + 142, + 375, + 470, + 388 + ], + "score": 1.0, + "content": "difference between an input and the system matrix times the network evaluation,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 387, + 470, + 399 + ], + "spans": [ + { + "bbox": [ + 142, + 387, + 470, + 399 + ], + "score": 1.0, + "content": "where the unspecified constant in the approximate inverse is accounted for. We", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 397, + 469, + 410 + ], + "spans": [ + { + "bbox": [ + 141, + 397, + 469, + 410 + ], + "score": 1.0, + "content": "demonstrate the efficacy of our approach on spatially discretized Poisson equa-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 408, + 470, + 422 + ], + "spans": [ + { + "bbox": [ + 141, + 408, + 470, + 422 + ], + "score": 1.0, + "content": "tions with millions of degrees of freedom arising in computational fluid dynamics", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 420, + 469, + 432 + ], + "spans": [ + { + "bbox": [ + 142, + 420, + 469, + 432 + ], + "score": 1.0, + "content": "applications. Unlike state-of-the-art learning approaches, our algorithm is capable", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 431, + 470, + 443 + ], + "spans": [ + { + "bbox": [ + 142, + 431, + 470, + 443 + ], + "score": 1.0, + "content": "of reducing the linear system residual to a given tolerance in a small number of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 441, + 470, + 455 + ], + "spans": [ + { + "bbox": [ + 141, + 441, + 470, + 455 + ], + "score": 1.0, + "content": "iterations, independent of the problem size. Moreover, our method generalizes", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 452, + 433, + 466 + ], + "spans": [ + { + "bbox": [ + 141, + 452, + 433, + 466 + ], + "score": 1.0, + "content": "effectively to various systems beyond those encountered during training.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 16, + "bbox_fs": [ + 141, + 212, + 470, + 466 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 484, + 206, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 208, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 208, + 500 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 105, + 509, + 504, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "In this work, we consider sparse linear systems that arise from discrete Poisson equations in incom-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "pressible flow applications (Chorin, 1967; Fedkiw et al., 2001; Bridson, 2008). We use the notation", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 509, + 505, + 533 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 288, + 537, + 324, + 548 + ], + "lines": [ + { + "bbox": [ + 288, + 537, + 324, + 548 + ], + "spans": [ + { + "bbox": [ + 288, + 537, + 324, + 548 + ], + "score": 0.9, + "content": "\\mathbf { A } { \\boldsymbol { \\mathbf { \\mathit { x } } } } = \\mathbf { \\mathit { b } }", + "type": "interline_equation", + "image_path": "8fead0b4e25add8342bddaa4ba897dff29b2070568d204185e0f1c2ab5d7c00e.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 288, + 537, + 324, + 548 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 191, + 570 + ], + "score": 1.0, + "content": "where the dimension", + "type": "text" + }, + { + "bbox": [ + 191, + 559, + 198, + 566 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 554, + 252, + 570 + ], + "score": 1.0, + "content": "of the matrix", + "type": "text" + }, + { + "bbox": [ + 253, + 556, + 298, + 567 + ], + "score": 0.92, + "content": "A \\in \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 554, + 357, + 570 + ], + "score": 1.0, + "content": "and the vector", + "type": "text" + }, + { + "bbox": [ + 357, + 556, + 388, + 567 + ], + "score": 0.91, + "content": "b \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 554, + 505, + 570 + ], + "score": 1.0, + "content": "correlate with spatial fidelity", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "of the computational domain. The appropriate numerical linear algebra technique depends on the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "nature of the problem. Direct solvers that utilize matrix factorizations (QR, Cholesky, etc. Trefethen", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 587, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 104, + 587, + 438, + 603 + ], + "score": 1.0, + "content": "& Bau (1997)) have optimal approximation error, but their computational cost is", + "type": "text" + }, + { + "bbox": [ + 439, + 588, + 466, + 601 + ], + "score": 0.93, + "content": "\\operatorname { \\dot { O } } ( n ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 587, + 506, + 603 + ], + "score": 1.0, + "content": "and they", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 295, + 614 + ], + "score": 1.0, + "content": "typically require dense storage, even for sparse", + "type": "text" + }, + { + "bbox": [ + 295, + 600, + 304, + 610 + ], + "score": 0.67, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 600, + 505, + 614 + ], + "score": 1.0, + "content": ". Although Fast Fourier Transforms (Nussbaumer,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "1981) can be used in limited instances (periodic boundary conditions, etc.), iterative techniques are", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 622, + 504, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 504, + 635 + ], + "score": 1.0, + "content": "most commonly adopted for these systems given their sparsity. Many applications with strict per-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "formance constraints (e.g., real-time fluid simulation) utilize basic iterations (Jacobi, Gauss-Seidel,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "successive over relaxation (SOR), etc.) given limited computational budget (Saad, 2003). However,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "large approximation errors must be tolerated since iteration counts are limited by the performance", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "constraints. This is particularly problematic since the wide elliptic spectrum of these matrices (a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 691 + ], + "score": 1.0, + "content": "condition that worsens with increased spatial fidelity/matrix dimension) leads to poor conditioning", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "and iteration counts. Iterative techniques can achieve sub-quadratic convergence if their iteration", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 316, + 712 + ], + "score": 1.0, + "content": "count does not grow excessively with problem size", + "type": "text" + }, + { + "bbox": [ + 316, + 701, + 323, + 709 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 699, + 481, + 712 + ], + "score": 1.0, + "content": "since each iteration generally requires", + "type": "text" + }, + { + "bbox": [ + 481, + 699, + 504, + 711 + ], + "score": 0.92, + "content": "O ( n )", + "type": "inline_equation" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "floating point operations for sparse matrices. Discrete elliptic operators are typically symmetric", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 721, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 734 + ], + "score": 1.0, + "content": "positive (semi) definite and the preconditioned conjugate gradients method (PCG) can be used to", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 493, + 95 + ], + "score": 1.0, + "content": "minimize iteration counts (Saad, 2003; Hestenes & Stiefel, 1952; Stiefel, 1952). Preconditioners", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 494, + 83, + 504, + 93 + ], + "score": 0.77, + "content": "_ { r }", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "for PCG must simultaneously: be symmetric positive definite (SPD) (and therefore admit factoriza-", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 125, + 118 + ], + "score": 1.0, + "content": "tion", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 126, + 104, + 165, + 115 + ], + "score": 0.9, + "content": "{ \\boldsymbol { P } } = { \\boldsymbol { F } } ^ { 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 165, + 103, + 420, + 118 + ], + "score": 1.0, + "content": "), improve the condition number of the preconditioned system", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 420, + 105, + 483, + 116 + ], + "score": 0.92, + "content": "F A F y = F b", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 484, + 103, + 505, + 118 + ], + "score": 1.0, + "content": ", and", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "be computationally cheap to construct and apply; accordingly, designing specialized precondition-", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "ers for particular classes of problems is somewhat of an art. Incomplete Cholesky preconditioners", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "(ICPCG) (Kershaw, 1978) use a sparse approximation to the Cholesky factorization and significantly", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "reduce iteration counts in practice; however, their inherent data dependency prevents efficient par-", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "allel implementation. Nonetheless, these are very commonly adopted for Poisson equations arising", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "in incompressible flow (Fedkiw et al., 2001; Bridson, 2008). Multigrid (Brandt, 1977) and domain", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "score": 1.0, + "content": "decomposition (Saad, 2003) preconditioners greatly reduce iterations counts, but they must be up-", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 206 + ], + "score": 1.0, + "content": "dated (with non-trivial cost) each time the problem changes (e.g., in computational domains with", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "time varying boundaries) and/or for different hardware platforms. In general, choice of an optimal", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 395, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 395, + 226 + ], + "score": 1.0, + "content": "preconditioner for discrete elliptic operators is an open area of research.", + "type": "text", + "cross_page": true + } + ], + "index": 12 + } + ], + "index": 39.5, + "bbox_fs": [ + 104, + 554, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 225 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 493, + 95 + ], + "score": 1.0, + "content": "minimize iteration counts (Saad, 2003; Hestenes & Stiefel, 1952; Stiefel, 1952). Preconditioners", + "type": "text" + }, + { + "bbox": [ + 494, + 83, + 504, + 93 + ], + "score": 0.77, + "content": "_ { r }", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "for PCG must simultaneously: be symmetric positive definite (SPD) (and therefore admit factoriza-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 125, + 118 + ], + "score": 1.0, + "content": "tion", + "type": "text" + }, + { + "bbox": [ + 126, + 104, + 165, + 115 + ], + "score": 0.9, + "content": "{ \\boldsymbol { P } } = { \\boldsymbol { F } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 103, + 420, + 118 + ], + "score": 1.0, + "content": "), improve the condition number of the preconditioned system", + "type": "text" + }, + { + "bbox": [ + 420, + 105, + 483, + 116 + ], + "score": 0.92, + "content": "F A F y = F b", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 103, + 505, + 118 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "be computationally cheap to construct and apply; accordingly, designing specialized precondition-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "ers for particular classes of problems is somewhat of an art. Incomplete Cholesky preconditioners", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "(ICPCG) (Kershaw, 1978) use a sparse approximation to the Cholesky factorization and significantly", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "reduce iteration counts in practice; however, their inherent data dependency prevents efficient par-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "allel implementation. Nonetheless, these are very commonly adopted for Poisson equations arising", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "in incompressible flow (Fedkiw et al., 2001; Bridson, 2008). Multigrid (Brandt, 1977) and domain", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "score": 1.0, + "content": "decomposition (Saad, 2003) preconditioners greatly reduce iterations counts, but they must be up-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 206 + ], + "score": 1.0, + "content": "dated (with non-trivial cost) each time the problem changes (e.g., in computational domains with", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "time varying boundaries) and/or for different hardware platforms. In general, choice of an optimal", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 395, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 395, + 226 + ], + "score": 1.0, + "content": "preconditioner for discrete elliptic operators is an open area of research.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "Recently, data-driven approaches that leverage deep learning techniques have shown promise for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "solving linear systems. Various researchers have investigated machine learning estimation of multi-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 104, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "grid parameters (Greenfeld et al., 2019; Grebhahn et al., 2016; Luz et al., 2020). Others have devel-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "oped machine learning methods to estimate preconditioners (Gotz & Anzt, 2018; Stanaityte, 2020; ¨", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "Ichimura et al., 2020) and initial guesses for iterative methods (Luna et al., 2021; Um et al., 2020;", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "Ackmann et al., 2020). Tompson et al. (2017) and Yang et al. (2016) develop non-iterative machine", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "learning approximations of the inverse of discrete Poisson equations from incompressible flow. We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 321 + ], + "score": 1.0, + "content": "leverage deep learning and develop a novel version of conjugate gradients iterative method for ap-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 104, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "proximating the solution of SPD linear systems which we call the deep conjugate direction method", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 229, + 342 + ], + "score": 1.0, + "content": "(DCDM). CG iteratively adds", + "type": "text" + }, + { + "bbox": [ + 229, + 330, + 239, + 340 + ], + "score": 0.66, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "-conjugate search directions while minimizing the matrix norm of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "the error. We use a convolutional neural network (CNN) as an approximation of the inverse of the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 104, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "matrix in order to generate more efficient search directions. We only ask that our network approxi-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "mate the inverse up to an unknown scaling since this decreases the degree of nonlinearity and since", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "it does not affect the quality of the search direction (which is scale independent). The network is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "similar to a preconditioner, but it is not a linear function, and our modified conjugate gradients ap-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 395, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 104, + 395, + 506, + 409 + ], + "score": 1.0, + "content": "proach is designed to accommodate this nonlinearity. We use unsupervised learning to train our", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 406, + 504, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 272, + 419 + ], + "score": 1.0, + "content": "network with a loss function equal to the", + "type": "text" + }, + { + "bbox": [ + 273, + 406, + 285, + 416 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 406, + 494, + 419 + ], + "score": 1.0, + "content": "difference between an input vector and a scaling of", + "type": "text" + }, + { + "bbox": [ + 494, + 406, + 504, + 416 + ], + "score": 0.68, + "content": "\\pmb { A }", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "times the output of our network. To account for this unknown scaling during training, we choose", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "the scale of the output of the network by minimizing the matrix norm of the error. Our approach", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 411, + 452 + ], + "score": 1.0, + "content": "allows for efficient training and generalization to problems unseen (matrices", + "type": "text" + }, + { + "bbox": [ + 411, + 439, + 421, + 450 + ], + "score": 0.55, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "and right-hand sides", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 107, + 451, + 113, + 460 + ], + "score": 0.33, + "content": "^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "). We benchmark our algorithm using the ubiquitous pressure Poisson equation (discretized on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "regular voxelized domains) and compare against FluidNet (Tompson et al., 2017), which is the state", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "of the art learning-based method for these types of problems. Unlike the non-iterative approaches", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 484, + 504, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 504, + 495 + ], + "score": 1.0, + "content": "of Tompson et al. (2017) and Yang et al. (2016), our method can reduce the linear system residuals", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "arbitrarily. We showcase our approach with examples that have over 16 million degrees of freedom.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 108, + 520, + 211, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 213, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 213, + 535 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 546, + 504, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 504, + 557 + ], + "score": 1.0, + "content": "Several papers have focused on enhancing the solution of linear systems (arising from discretized", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "PDEs) using learning. For instance, Gotz & Anzt (2018) generate sparsity patterns for block-Jacobi ¨", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 104, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "preconditioners using convolutional neural networks, and Stanaityte (2020) use a CNN to predict", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "non-zero patterns for ILU-type preconditioners for the Navier-Stokes equations (though neither", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "work designs fundamentally new preconditioners). Ichimura et al. (2020) develop a neural-network", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "based preconditioner where the network is used to predict approximate Green’s functions (which", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "arise in the analytical solution of certain PDEs) that in turn yield an approximate inverse of the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "linear system. Hsieh et al. (2019) learn an iterator that solves linear systems, performing com-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "petitively with classical solvers like multigrid-preconditioned MINRES (Paige & Saunders, 1975).", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "Luz et al. (2020) and Greenfeld et al. (2019) use machine learning to estimate algebraic multigrid", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "(AMG) parameters. They note that AMG approaches rely most fundamentally on effectively cho-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "sen (problem-dependent) prolongation sparse matrices and that numerous methods have attempted", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 290, + 690 + ], + "score": 1.0, + "content": "to automatically create them from the matrix", + "type": "text" + }, + { + "bbox": [ + 290, + 677, + 300, + 687 + ], + "score": 0.51, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 677, + 506, + 690 + ], + "score": 1.0, + "content": ". 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(2016)", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 104, + 698, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 505, + 713 + ], + "score": 1.0, + "content": "note that geometric multigrid solver parameters can be difficult to choose to guarantee parallel per-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "formance on different hardware platforms. 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Others have devel-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "oped machine learning methods to estimate preconditioners (Gotz & Anzt, 2018; Stanaityte, 2020; ¨", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "Ichimura et al., 2020) and initial guesses for iterative methods (Luna et al., 2021; Um et al., 2020;", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "Ackmann et al., 2020). Tompson et al. (2017) and Yang et al. 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We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 321 + ], + "score": 1.0, + "content": "leverage deep learning and develop a novel version of conjugate gradients iterative method for ap-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 104, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "proximating the solution of SPD linear systems which we call the deep conjugate direction method", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 229, + 342 + ], + "score": 1.0, + "content": "(DCDM). CG iteratively adds", + "type": "text" + }, + { + "bbox": [ + 229, + 330, + 239, + 340 + ], + "score": 0.66, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "-conjugate search directions while minimizing the matrix norm of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "the error. We use a convolutional neural network (CNN) as an approximation of the inverse of the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 104, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "matrix in order to generate more efficient search directions. We only ask that our network approxi-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "mate the inverse up to an unknown scaling since this decreases the degree of nonlinearity and since", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "it does not affect the quality of the search direction (which is scale independent). The network is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "similar to a preconditioner, but it is not a linear function, and our modified conjugate gradients ap-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 395, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 104, + 395, + 506, + 409 + ], + "score": 1.0, + "content": "proach is designed to accommodate this nonlinearity. We use unsupervised learning to train our", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 406, + 504, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 272, + 419 + ], + "score": 1.0, + "content": "network with a loss function equal to the", + "type": "text" + }, + { + "bbox": [ + 273, + 406, + 285, + 416 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 406, + 494, + 419 + ], + "score": 1.0, + "content": "difference between an input vector and a scaling of", + "type": "text" + }, + { + "bbox": [ + 494, + 406, + 504, + 416 + ], + "score": 0.68, + "content": "\\pmb { A }", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "times the output of our network. To account for this unknown scaling during training, we choose", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "the scale of the output of the network by minimizing the matrix norm of the error. Our approach", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 411, + 452 + ], + "score": 1.0, + "content": "allows for efficient training and generalization to problems unseen (matrices", + "type": "text" + }, + { + "bbox": [ + 411, + 439, + 421, + 450 + ], + "score": 0.55, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "and right-hand sides", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 107, + 451, + 113, + 460 + ], + "score": 0.33, + "content": "^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "). We benchmark our algorithm using the ubiquitous pressure Poisson equation (discretized on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "regular voxelized domains) and compare against FluidNet (Tompson et al., 2017), which is the state", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "of the art learning-based method for these types of problems. Unlike the non-iterative approaches", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 484, + 504, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 504, + 495 + ], + "score": 1.0, + "content": "of Tompson et al. (2017) and Yang et al. (2016), our method can reduce the linear system residuals", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "arbitrarily. We showcase our approach with examples that have over 16 million degrees of freedom.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 231, + 506, + 507 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 520, + 211, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 213, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 213, + 535 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 546, + 504, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 504, + 557 + ], + "score": 1.0, + "content": "Several papers have focused on enhancing the solution of linear systems (arising from discretized", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "PDEs) using learning. For instance, Gotz & Anzt (2018) generate sparsity patterns for block-Jacobi ¨", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 104, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "preconditioners using convolutional neural networks, and Stanaityte (2020) use a CNN to predict", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "non-zero patterns for ILU-type preconditioners for the Navier-Stokes equations (though neither", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "work designs fundamentally new preconditioners). Ichimura et al. (2020) develop a neural-network", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "based preconditioner where the network is used to predict approximate Green’s functions (which", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "arise in the analytical solution of certain PDEs) that in turn yield an approximate inverse of the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "linear system. Hsieh et al. (2019) learn an iterator that solves linear systems, performing com-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "petitively with classical solvers like multigrid-preconditioned MINRES (Paige & Saunders, 1975).", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "Luz et al. (2020) and Greenfeld et al. (2019) use machine learning to estimate algebraic multigrid", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "(AMG) parameters. They note that AMG approaches rely most fundamentally on effectively cho-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "sen (problem-dependent) prolongation sparse matrices and that numerous methods have attempted", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 290, + 690 + ], + "score": 1.0, + "content": "to automatically create them from the matrix", + "type": "text" + }, + { + "bbox": [ + 290, + 677, + 300, + 687 + ], + "score": 0.51, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 677, + 506, + 690 + ], + "score": 1.0, + "content": ". They train a graph neural network to learn (in an", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 297, + 700 + ], + "score": 1.0, + "content": "unsupervised fashion) a mapping from matrices", + "type": "text" + }, + { + "bbox": [ + 298, + 688, + 307, + 698 + ], + "score": 0.78, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "to prolongation operators. Grebhahn et al. (2016)", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 104, + 698, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 505, + 713 + ], + "score": 1.0, + "content": "note that geometric multigrid solver parameters can be difficult to choose to guarantee parallel per-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "formance on different hardware platforms. They use machine learning to create a code generator to", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 720, + 178, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 178, + 734 + ], + "score": 1.0, + "content": "help achieve this.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 47, + "bbox_fs": [ + 104, + 546, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 79, + 505, + 174 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 79, + 505, + 174 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 79, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 107, + 79, + 505, + 174 + ], + "score": 0.97, + "type": "image", + "image_path": "a7ac602c2643cff48deff90a7224438f208d4a7100d53c20e1e19240e8cade2b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 79, + 505, + 110.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 110.66666666666667, + 505, + 142.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 142.33333333333334, + 505, + 174.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 182, + 505, + 284 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 301, + 195 + ], + "score": 1.0, + "content": "Figure 1: (a) We illustrate a sample flow domain", + "type": "text" + }, + { + "bbox": [ + 302, + 182, + 349, + 195 + ], + "score": 0.92, + "content": "\\Omega \\subset ( 0 , 1 ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "(in 2D for ease of illustration) with in-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 192, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 207 + ], + "score": 1.0, + "content": "ternal boundaries (blue lines). (b) We voxelize the domain with a regular grid: white cells represent", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 204, + 507, + 220 + ], + "spans": [ + { + "bbox": [ + 104, + 205, + 444, + 220 + ], + "score": 1.0, + "content": "interior/fluid, and blue cells represent boundary conditions. (c) We train using matrix", + "type": "text" + }, + { + "bbox": [ + 445, + 204, + 475, + 217 + ], + "score": 0.9, + "content": "A ^ { ( 0 , 1 ) ^ { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 205, + 507, + 220 + ], + "score": 1.0, + "content": "from a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 218, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 361, + 230 + ], + "score": 1.0, + "content": "discretized domain with no interior boundary conditions, where", + "type": "text" + }, + { + "bbox": [ + 361, + 218, + 367, + 228 + ], + "score": 0.81, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 218, + 505, + 230 + ], + "score": 1.0, + "content": "is the dimension. This creates lin-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 171, + 241 + ], + "score": 1.0, + "content": "ear system with", + "type": "text" + }, + { + "bbox": [ + 171, + 228, + 229, + 241 + ], + "score": 0.93, + "content": "n = ( n _ { c } + 1 ) ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 229, + 299, + 241 + ], + "score": 1.0, + "content": "unknowns where", + "type": "text" + }, + { + "bbox": [ + 299, + 230, + 311, + 240 + ], + "score": 0.86, + "content": "n _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 229, + 505, + 241 + ], + "score": 1.0, + "content": "is the number of grid cells on each direction. (d)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 331, + 252 + ], + "score": 1.0, + "content": "We illustrate the non-zero entries in an example matrix", + "type": "text" + }, + { + "bbox": [ + 332, + 239, + 348, + 250 + ], + "score": 0.88, + "content": "A ^ { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "from the voxelized and labeled (white", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "vs. blue) grid for three example interior cells (green, purple, and brown). Each case illustrates the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 262, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 274 + ], + "score": 1.0, + "content": "non-zero entries in the row associated with the example cell. All entries in rows corresponding to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 272, + 225, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 225, + 285 + ], + "score": 1.0, + "content": "boundary/blue cells are zero.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7 + } + ], + "index": 4.0 + }, + { + "type": "text", + "bbox": [ + 106, + 292, + 505, + 402 + ], + "lines": [ + { + "bbox": [ + 106, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "Several works consider accelerating the solution of linear systems by learning an initial guess that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "is close to the true solution or otherwise helpful to descent algorithms for finding the true solution.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 315, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 326 + ], + "score": 1.0, + "content": "In order to solve the discretized Poisson equation, Luna et al. (2021) accelerate the convergence of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "GMRES (Saad & Schultz, 1986) with an initial guess that is learned in real-time (i.e., as a simulation", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "code runs) with no prior data. Um et al. (2020) train a network (incorporating differentiable physics,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "based on the underlying PDEs) in order to produce high-quality initial guesses for a CG solver. In", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 104, + 358, + 506, + 370 + ], + "score": 1.0, + "content": "a somewhat similar vein, Ackmann et al. (2020) use a simple feedforward neural network to predict", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 369, + 507, + 381 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 507, + 381 + ], + "score": 1.0, + "content": "pointwise solution components, which accelerates the conjugate residual method used to solve a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 380, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 391 + ], + "score": 1.0, + "content": "relatively simple shallow-water model (a more sophisticated network and loss function are needed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 391, + 336, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 336, + 403 + ], + "score": 1.0, + "content": "to handle more general PDEs and larger-scale problems).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 421 + ], + "score": 1.0, + "content": "At least two papers (Ruelmann et al., 2018; Sappl et al., 2019) have sought to learn a mapping", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "between a matrix and an associated sparse approximate inverse. In their investigation, Ruelmann", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "et al. (2018) propose training a neural network using matrix-inverse pairs as training data. Although", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "straightforward to implement, the cost of generating training data, let alone training the network, is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "prohibitive for large-scale 3D problems. Sappl et al. (2019) seek to learn a mapping between linear", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "system matrices and sparse (banded) approximate inverses. Their loss function is the condition", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "number of the product of the system matrix and the approximate inverse; the minimum value of the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 485, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 496 + ], + "score": 1.0, + "content": "condition number is one, which is achieved exactly when an exact inverse is obtained. Although this", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "framework is quite simple, evaluating the condition number of a matrix is asymptotically costly, and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 507, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 519 + ], + "score": 1.0, + "content": "in general, the inverse of a sparse matrix can be quite dense. Accordingly, the method is not efficient", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 518, + 498, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 498, + 530 + ], + "score": 1.0, + "content": "or accurate enough for the large-scale 3D problems that arise in real-world engineering problems.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "Most relevant to the present work is FluidNet (Tompson et al., 2017). FluidNet develops a highly-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "tailored CNN architecture that is used to predict the solution of a linear projection operation (specif-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "ically, for the discrete Poisson equation) given a matrix and right-hand side. The authors demon-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "strate fluid simulations where the linear solve is replaced by evaluating their network. Because their", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "network is relatively lightweight and is only evaluated once per time step, their simulations run ef-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "score": 1.0, + "content": "ficiently. 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Our method resolves this gap.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38.5 + }, + { + "type": "title", + "bbox": [ + 108, + 684, + 324, + 696 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 325, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 325, + 698 + ], + "score": 1.0, + "content": "3 MOTIVATION: INCOMPRESSIBLE FLOW", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We demonstrate the efficacy of our approach with the linear systems that arise in incompressible", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "flow applications. 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(b) We voxelize the domain with a regular grid: white cells represent", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 204, + 507, + 220 + ], + "spans": [ + { + "bbox": [ + 104, + 205, + 444, + 220 + ], + "score": 1.0, + "content": "interior/fluid, and blue cells represent boundary conditions. (c) We train using matrix", + "type": "text" + }, + { + "bbox": [ + 445, + 204, + 475, + 217 + ], + "score": 0.9, + "content": "A ^ { ( 0 , 1 ) ^ { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 205, + 507, + 220 + ], + "score": 1.0, + "content": "from a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 218, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 361, + 230 + ], + "score": 1.0, + "content": "discretized domain with no interior boundary conditions, where", + "type": "text" + }, + { + "bbox": [ + 361, + 218, + 367, + 228 + ], + "score": 0.81, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 218, + 505, + 230 + ], + "score": 1.0, + "content": "is the dimension. This creates lin-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 171, + 241 + ], + "score": 1.0, + "content": "ear system with", + "type": "text" + }, + { + "bbox": [ + 171, + 228, + 229, + 241 + ], + "score": 0.93, + "content": "n = ( n _ { c } + 1 ) ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 229, + 299, + 241 + ], + "score": 1.0, + "content": "unknowns where", + "type": "text" + }, + { + "bbox": [ + 299, + 230, + 311, + 240 + ], + "score": 0.86, + "content": "n _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 229, + 505, + 241 + ], + "score": 1.0, + "content": "is the number of grid cells on each direction. (d)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 331, + 252 + ], + "score": 1.0, + "content": "We illustrate the non-zero entries in an example matrix", + "type": "text" + }, + { + "bbox": [ + 332, + 239, + 348, + 250 + ], + "score": 0.88, + "content": "A ^ { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "from the voxelized and labeled (white", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "vs. blue) grid for three example interior cells (green, purple, and brown). Each case illustrates the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 262, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 274 + ], + "score": 1.0, + "content": "non-zero entries in the row associated with the example cell. All entries in rows corresponding to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 272, + 225, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 225, + 285 + ], + "score": 1.0, + "content": "boundary/blue cells are zero.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7 + } + ], + "index": 4.0 + }, + { + "type": "text", + "bbox": [ + 106, + 292, + 505, + 402 + ], + "lines": [ + { + "bbox": [ + 106, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "Several works consider accelerating the solution of linear systems by learning an initial guess that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "is close to the true solution or otherwise helpful to descent algorithms for finding the true solution.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 315, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 326 + ], + "score": 1.0, + "content": "In order to solve the discretized Poisson equation, Luna et al. (2021) accelerate the convergence of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "GMRES (Saad & Schultz, 1986) with an initial guess that is learned in real-time (i.e., as a simulation", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "code runs) with no prior data. Um et al. (2020) train a network (incorporating differentiable physics,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "based on the underlying PDEs) in order to produce high-quality initial guesses for a CG solver. In", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 104, + 358, + 506, + 370 + ], + "score": 1.0, + "content": "a somewhat similar vein, Ackmann et al. (2020) use a simple feedforward neural network to predict", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 369, + 507, + 381 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 507, + 381 + ], + "score": 1.0, + "content": "pointwise solution components, which accelerates the conjugate residual method used to solve a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 380, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 391 + ], + "score": 1.0, + "content": "relatively simple shallow-water model (a more sophisticated network and loss function are needed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 391, + 336, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 336, + 403 + ], + "score": 1.0, + "content": "to handle more general PDEs and larger-scale problems).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 291, + 507, + 403 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 421 + ], + "score": 1.0, + "content": "At least two papers (Ruelmann et al., 2018; Sappl et al., 2019) have sought to learn a mapping", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "between a matrix and an associated sparse approximate inverse. 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(2019) seek to learn a mapping between linear", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "system matrices and sparse (banded) approximate inverses. Their loss function is the condition", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "number of the product of the system matrix and the approximate inverse; the minimum value of the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 485, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 496 + ], + "score": 1.0, + "content": "condition number is one, which is achieved exactly when an exact inverse is obtained. Although this", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "framework is quite simple, evaluating the condition number of a matrix is asymptotically costly, and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 507, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 519 + ], + "score": 1.0, + "content": "in general, the inverse of a sparse matrix can be quite dense. Accordingly, the method is not efficient", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 518, + 498, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 498, + 530 + ], + "score": 1.0, + "content": "or accurate enough for the large-scale 3D problems that arise in real-world engineering problems.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 406, + 506, + 530 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "Most relevant to the present work is FluidNet (Tompson et al., 2017). FluidNet develops a highly-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "tailored CNN architecture that is used to predict the solution of a linear projection operation (specif-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "ically, for the discrete Poisson equation) given a matrix and right-hand side. The authors demon-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "strate fluid simulations where the linear solve is replaced by evaluating their network. Because their", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "network is relatively lightweight and is only evaluated once per time step, their simulations run ef-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "score": 1.0, + "content": "ficiently. However, their design allows the network only one opportunity to reduce the residual for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "the linear solve; in practice, we observe that FluidNet is able to reduce the residual by no more than", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "about one order of magnitude. However, in computer graphics applications, at least four orders of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "magnitude in residual reduction are usually required for visual fidelity, while in scientific and engi-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "neering applications, practitioners prefer solutions that reduce the residual by eight or more orders of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "magnitude (i.e., to within machine precision). 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Our method resolves this gap.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38.5, + "bbox_fs": [ + 104, + 533, + 506, + 668 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 684, + 324, + 696 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 325, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 325, + 698 + ], + "score": 1.0, + "content": "3 MOTIVATION: INCOMPRESSIBLE FLOW", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We demonstrate the efficacy of our approach with the linear systems that arise in incompressible", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "flow applications. Specifically, we use our algorithm to solve the discrete Poisson equations in", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5, + "bbox_fs": [ + 106, + 709, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "regular-grid-based discretization of the pressure projection equations that arise in Chorin’s splitting", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 487, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 487, + 106 + ], + "score": 1.0, + "content": "technique (Chorin, 1967) for the inviscid, incompressible Euler equations. These equations are", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 109, + 403, + 137 + ], + "lines": [ + { + "bbox": [ + 207, + 109, + 403, + 137 + ], + "spans": [ + { + "bbox": [ + 207, + 109, + 403, + 137 + ], + "score": 0.93, + "content": "\\rho \\left( \\frac { \\partial \\pmb { u } } { \\partial t } + \\frac { \\partial \\pmb { u } } { \\partial \\pmb { x } } \\pmb { u } \\right) + \\nabla p = \\pmb { f } ^ { e x t } , \\qquad \\nabla \\cdot \\pmb { u } = 0", + "type": "interline_equation", + "image_path": "c1b355957f1f3592b8d81befad750201620d6f0041dd14ae0eb7b88b47dac275.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 207, + 109, + 403, + 123.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 207, + 123.0, + 403, + 137.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 142, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 141, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 135, + 155 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 145, + 143, + 153 + ], + "score": 0.77, + "content": "\\textbf { \\em u }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 141, + 215, + 155 + ], + "score": 1.0, + "content": "is fluid velocity,", + "type": "text" + }, + { + "bbox": [ + 216, + 145, + 222, + 155 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 141, + 275, + 155 + ], + "score": 1.0, + "content": "is pressure,", + "type": "text" + }, + { + "bbox": [ + 275, + 145, + 282, + 154 + ], + "score": 0.82, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 141, + 347, + 155 + ], + "score": 1.0, + "content": "is density, and", + "type": "text" + }, + { + "bbox": [ + 348, + 143, + 367, + 154 + ], + "score": 0.93, + "content": "f ^ { e x t }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 141, + 506, + 155 + ], + "score": 1.0, + "content": "accounts for external forces like", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 320, + 166 + ], + "score": 1.0, + "content": "gravity. 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The left term in Equation 2 enforces conservation of momentum in the absence of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "viscosity, and the second part enforces incompressibility and conservation of mass. These equations", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 233, + 199 + ], + "score": 1.0, + "content": "are subject to initial conditions", + "type": "text" + }, + { + "bbox": [ + 234, + 186, + 287, + 199 + ], + "score": 0.93, + "content": "\\rho ( \\pmb { x } , 0 ) = \\rho ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 186, + 306, + 199 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 306, + 186, + 377, + 199 + ], + "score": 0.93, + "content": "{ \\pmb u } ( { \\pmb x } , 0 ) = { \\pmb u } ^ { 0 } ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 186, + 505, + 199 + ], + "score": 1.0, + "content": "as well as boundary conditions", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 196, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 107, + 197, + 220, + 210 + ], + "score": 0.93, + "content": "{ \\pmb u } ( { \\pmb x } , t ) \\cdot { \\pmb n } ( { \\pmb x } ) = u ^ { \\partial \\Omega } ( { \\pmb x } , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 196, + 349, + 210 + ], + "score": 1.0, + "content": "on the boundary of the domain", + "type": "text" + }, + { + "bbox": [ + 350, + 199, + 385, + 208 + ], + "score": 0.89, + "content": "{ \\pmb x } \\in \\partial \\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 196, + 416, + 210 + ], + "score": 1.0, + "content": "(where", + "type": "text" + }, + { + "bbox": [ + 417, + 200, + 425, + 208 + ], + "score": 0.76, + "content": "\\textbf { \\em n }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 196, + 506, + 210 + ], + "score": 1.0, + "content": "is the unit outward", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 219, + 221 + ], + "score": 1.0, + "content": "pointing normal at position", + "type": "text" + }, + { + "bbox": [ + 219, + 211, + 227, + 218 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "on the boundary). Equation 2 is discretized in both time and space.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 217, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 246, + 243 + ], + "score": 1.0, + "content": "Temporally, we split the advection", + "type": "text" + }, + { + "bbox": [ + 246, + 219, + 312, + 243 + ], + "score": 0.93, + "content": "\\frac { \\partial \\pmb { u } } { \\partial t } + \\frac { \\partial \\pmb { u } } { \\partial \\pmb { x } } \\pmb { u } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 217, + 403, + 245 + ], + "score": 1.0, + "content": "and body forces terms", + "type": "text" + }, + { + "bbox": [ + 404, + 219, + 456, + 243 + ], + "score": 0.94, + "content": "\\rho \\frac { \\partial \\pmb { u } } { \\partial t } = \\pmb { f } ^ { e x t }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 223, + 505, + 238 + ], + "score": 1.0, + "content": ", and finally", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 240, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 317, + 266 + ], + "score": 1.0, + "content": "enforce incompressibility via the pressure projection", + "type": "text" + }, + { + "bbox": [ + 317, + 240, + 382, + 265 + ], + "score": 0.93, + "content": "\\frac { \\partial \\pmb { u } } { \\partial t } + \\frac { 1 } { \\rho } \\nabla p = \\pmb { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 245, + 422, + 264 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 422, + 247, + 461, + 257 + ], + "score": 0.9, + "content": "\\nabla \\cdot \\pmb { u } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 245, + 506, + 264 + ], + "score": 1.0, + "content": "; this is the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 262, + 507, + 277 + ], + "spans": [ + { + "bbox": [ + 104, + 262, + 507, + 277 + ], + "score": 1.0, + "content": "standard advection-projection scheme proposed by Chorin (1967). Using finite differences in time,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 275, + 212, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 212, + 286 + ], + "score": 1.0, + "content": "we can summarize this as", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 288, + 410, + 370 + ], + "lines": [ + { + "bbox": [ + 200, + 288, + 410, + 370 + ], + "spans": [ + { + "bbox": [ + 200, + 288, + 410, + 370 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle \\rho ^ { 0 } \\left( \\frac { \\pmb { u } ^ { * } - \\pmb { u } ^ { n } } { \\Delta t } + \\frac { \\partial \\pmb { u } ^ { n } } { \\partial \\pmb { x } } \\pmb { u } ^ { n } \\right) = \\pmb { f } ^ { e x t } } \\\\ { \\displaystyle - \\nabla \\cdot \\frac { 1 } { \\rho ^ { 0 } } \\nabla p ^ { n + 1 } = - \\nabla \\cdot \\pmb { u } ^ { * } } \\\\ { \\displaystyle \\qquad - \\frac { 1 } { \\rho ^ { 0 } } \\nabla p ^ { n + 1 } \\cdot \\pmb { n } = \\frac { 1 } { \\Delta t } \\left( \\pmb { u } ^ { \\partial \\Omega } - \\pmb { u } ^ { * } \\cdot \\pmb { n } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "b9e83fdc2a77df2f7139c2359537bee131588af2939e28a1c636f7c2e1a8f4d9.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 200, + 288, + 410, + 301.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 200, + 301.6666666666667, + 410, + 315.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 200, + 315.33333333333337, + 410, + 329.00000000000006 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 200, + 329.00000000000006, + 410, + 342.66666666666674 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 200, + 342.66666666666674, + 410, + 356.3333333333334 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 200, + 356.3333333333334, + 410, + 370.0000000000001 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 372, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 386 + ], + "score": 1.0, + "content": "For the spatial discretization, we use a regular marker-and-cell (MAC) grid (Harlow & Welch,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 382, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 506, + 396 + ], + "score": 1.0, + "content": "1965) with cubic voxels whereby velocity components are stored on the face of voxel cells, and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 233, + 408 + ], + "score": 1.0, + "content": "scalar quantities (e.g., pressure", + "type": "text" + }, + { + "bbox": [ + 233, + 397, + 240, + 406 + ], + "score": 0.78, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 394, + 284, + 408 + ], + "score": 1.0, + "content": "or density", + "type": "text" + }, + { + "bbox": [ + 284, + 396, + 291, + 406 + ], + "score": 0.67, + "content": "\\rho \\mathrm { \\hbar }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 394, + 505, + 408 + ], + "score": 1.0, + "content": ") are stored at voxel centers. We use backward semi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "Lagrangian advection (Fedkiw et al., 2001; Gagniere et al., 2020) for Equation 3. All spatial partial", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "derivatives are approximated using finite differences. Equations 4 and 5 describe the pressure Pois-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 428, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 439 + ], + "score": 1.0, + "content": "son equation with Neumann conditions on the boundary of the flow domain. We discretize the left", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 439, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 451 + ], + "score": 1.0, + "content": "hand side of Equation 4 using a standard 7-point finite difference stencil. The right-hand side is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "discretized using the MAC grid discrete divergence finite difference stencils as well as contribu-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 461, + 504, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 504, + 472 + ], + "score": 1.0, + "content": "tions from the boundary condition terms in Equation 5. We refer the reader to Bridson (2008) for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "more in-depth implementation details. Equation 5 is discretized by modifying the Poisson stencil", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "to enforce Neumann boundary conditions. We do this using a simple labeling of the voxels in the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 492, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 248, + 506 + ], + "score": 1.0, + "content": "domain. For simplicity, we assume", + "type": "text" + }, + { + "bbox": [ + 248, + 493, + 296, + 505 + ], + "score": 0.93, + "content": "\\Omega \\subset ( 0 , 1 ) ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 492, + 505, + 506 + ], + "score": 1.0, + "content": "is a subset of the unit cube, potentially with internal", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "boundaries. We label cells in the domain as either liquid or boundary. This simple classification", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "is enough to define the Poisson discretizations (with appropriate Neumann boundary conditions at", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "score": 1.0, + "content": "domain boundaries) that we focus on in the present work; we illustrate the details in Figure 1. We", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 537, + 502, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 502, + 550 + ], + "score": 1.0, + "content": "use the following notation to denote the discrete Poisson equations associated with Equations 4–5:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 553, + 353, + 569 + ], + "lines": [ + { + "bbox": [ + 257, + 553, + 353, + 569 + ], + "spans": [ + { + "bbox": [ + 257, + 553, + 353, + 569 + ], + "score": 0.93, + "content": "\\begin{array} { r } { { \\cal A } ^ { \\Omega } { \\boldsymbol x } = { \\boldsymbol b } ^ { \\nabla \\cdot { \\boldsymbol u } ^ { * } } + { \\boldsymbol b } ^ { u ^ { \\partial \\Omega } } , } \\end{array}", + "type": "interline_equation", + "image_path": "ac399afedaac667ec410278ede5ec8e5a957aa976ca5e09188331dc7a1ba5bb7.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 257, + 553, + 353, + 569 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 576, + 505, + 647 + ], + "lines": [ + { + "bbox": [ + 106, + 575, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 135, + 590 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 576, + 151, + 587 + ], + "score": 0.89, + "content": "A ^ { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 575, + 441, + 590 + ], + "score": 1.0, + "content": "is the discrete Poisson matrix associated with the voxelized domain,", + "type": "text" + }, + { + "bbox": [ + 441, + 579, + 449, + 587 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 575, + 505, + 590 + ], + "score": 1.0, + "content": "is the vector", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 585, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 214, + 601 + ], + "score": 1.0, + "content": "of unknown pressure, and", + "type": "text" + }, + { + "bbox": [ + 215, + 588, + 240, + 600 + ], + "score": 0.9, + "content": "\\mathbf { \\delta } _ { b } \\nabla \\cdot \\mathbf { u } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 585, + 285, + 603 + ], + "score": 1.0, + "content": "and b u ∂ Ω", + "type": "text" + }, + { + "bbox": [ + 282, + 590, + 505, + 601 + ], + "score": 1.0, + "content": "are the right-hand side terms from Equations 4 and 5,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "respectively. 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We first discuss relevant details of the conjugate", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 311, + 723 + ], + "score": 1.0, + "content": "gradients (CG) method, particularly line search and", + "type": "text" + }, + { + "bbox": [ + 311, + 711, + 320, + 720 + ], + "score": 0.79, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "-orthogonal search directions. We then present", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 721, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 734 + ], + "score": 1.0, + "content": "a deep learning technique for improving the quality of these search directions that ultimately reduces", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "regular-grid-based discretization of the pressure projection equations that arise in Chorin’s splitting", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 487, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 487, + 106 + ], + "score": 1.0, + "content": "technique (Chorin, 1967) for the inviscid, incompressible Euler equations. These equations are", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 506, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 109, + 403, + 137 + ], + "lines": [ + { + "bbox": [ + 207, + 109, + 403, + 137 + ], + "spans": [ + { + "bbox": [ + 207, + 109, + 403, + 137 + ], + "score": 0.93, + "content": "\\rho \\left( \\frac { \\partial \\pmb { u } } { \\partial t } + \\frac { \\partial \\pmb { u } } { \\partial \\pmb { x } } \\pmb { u } \\right) + \\nabla p = \\pmb { f } ^ { e x t } , \\qquad \\nabla \\cdot \\pmb { u } = 0", + "type": "interline_equation", + "image_path": "c1b355957f1f3592b8d81befad750201620d6f0041dd14ae0eb7b88b47dac275.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 207, + 109, + 403, + 123.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 207, + 123.0, + 403, + 137.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 142, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 141, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 135, + 155 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 145, + 143, + 153 + ], + "score": 0.77, + "content": "\\textbf { \\em u }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 141, + 215, + 155 + ], + "score": 1.0, + "content": "is fluid velocity,", + "type": "text" + }, + { + "bbox": [ + 216, + 145, + 222, + 155 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 141, + 275, + 155 + ], + "score": 1.0, + "content": "is pressure,", + "type": "text" + }, + { + "bbox": [ + 275, + 145, + 282, + 154 + ], + "score": 0.82, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 141, + 347, + 155 + ], + "score": 1.0, + "content": "is density, and", + "type": "text" + }, + { + "bbox": [ + 348, + 143, + 367, + 154 + ], + "score": 0.93, + "content": "f ^ { e x t }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 141, + 506, + 155 + ], + "score": 1.0, + "content": "accounts for external forces like", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 320, + 166 + ], + "score": 1.0, + "content": "gravity. 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The left term in Equation 2 enforces conservation of momentum in the absence of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "viscosity, and the second part enforces incompressibility and conservation of mass. These equations", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 233, + 199 + ], + "score": 1.0, + "content": "are subject to initial conditions", + "type": "text" + }, + { + "bbox": [ + 234, + 186, + 287, + 199 + ], + "score": 0.93, + "content": "\\rho ( \\pmb { x } , 0 ) = \\rho ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 186, + 306, + 199 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 306, + 186, + 377, + 199 + ], + "score": 0.93, + "content": "{ \\pmb u } ( { \\pmb x } , 0 ) = { \\pmb u } ^ { 0 } ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 186, + 505, + 199 + ], + "score": 1.0, + "content": "as well as boundary conditions", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 196, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 107, + 197, + 220, + 210 + ], + "score": 0.93, + "content": "{ \\pmb u } ( { \\pmb x } , t ) \\cdot { \\pmb n } ( { \\pmb x } ) = u ^ { \\partial \\Omega } ( { \\pmb x } , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 196, + 349, + 210 + ], + "score": 1.0, + "content": "on the boundary of the domain", + "type": "text" + }, + { + "bbox": [ + 350, + 199, + 385, + 208 + ], + "score": 0.89, + "content": "{ \\pmb x } \\in \\partial \\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 196, + 416, + 210 + ], + "score": 1.0, + "content": "(where", + "type": "text" + }, + { + "bbox": [ + 417, + 200, + 425, + 208 + ], + "score": 0.76, + "content": "\\textbf { \\em n }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 196, + 506, + 210 + ], + "score": 1.0, + "content": "is the unit outward", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 219, + 221 + ], + "score": 1.0, + "content": "pointing normal at position", + "type": "text" + }, + { + "bbox": [ + 219, + 211, + 227, + 218 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "on the boundary). Equation 2 is discretized in both time and space.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 217, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 246, + 243 + ], + "score": 1.0, + "content": "Temporally, we split the advection", + "type": "text" + }, + { + "bbox": [ + 246, + 219, + 312, + 243 + ], + "score": 0.93, + "content": "\\frac { \\partial \\pmb { u } } { \\partial t } + \\frac { \\partial \\pmb { u } } { \\partial \\pmb { x } } \\pmb { u } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 217, + 403, + 245 + ], + "score": 1.0, + "content": "and body forces terms", + "type": "text" + }, + { + "bbox": [ + 404, + 219, + 456, + 243 + ], + "score": 0.94, + "content": "\\rho \\frac { \\partial \\pmb { u } } { \\partial t } = \\pmb { f } ^ { e x t }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 223, + 505, + 238 + ], + "score": 1.0, + "content": ", and finally", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 240, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 317, + 266 + ], + "score": 1.0, + "content": "enforce incompressibility via the pressure projection", + "type": "text" + }, + { + "bbox": [ + 317, + 240, + 382, + 265 + ], + "score": 0.93, + "content": "\\frac { \\partial \\pmb { u } } { \\partial t } + \\frac { 1 } { \\rho } \\nabla p = \\pmb { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 245, + 422, + 264 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 422, + 247, + 461, + 257 + ], + "score": 0.9, + "content": "\\nabla \\cdot \\pmb { u } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 245, + 506, + 264 + ], + "score": 1.0, + "content": "; this is the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 262, + 507, + 277 + ], + "spans": [ + { + "bbox": [ + 104, + 262, + 507, + 277 + ], + "score": 1.0, + "content": "standard advection-projection scheme proposed by Chorin (1967). Using finite differences in time,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 275, + 212, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 212, + 286 + ], + "score": 1.0, + "content": "we can summarize this as", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 141, + 507, + 286 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 288, + 410, + 370 + ], + "lines": [ + { + "bbox": [ + 200, + 288, + 410, + 370 + ], + "spans": [ + { + "bbox": [ + 200, + 288, + 410, + 370 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle \\rho ^ { 0 } \\left( \\frac { \\pmb { u } ^ { * } - \\pmb { u } ^ { n } } { \\Delta t } + \\frac { \\partial \\pmb { u } ^ { n } } { \\partial \\pmb { x } } \\pmb { u } ^ { n } \\right) = \\pmb { f } ^ { e x t } } \\\\ { \\displaystyle - \\nabla \\cdot \\frac { 1 } { \\rho ^ { 0 } } \\nabla p ^ { n + 1 } = - \\nabla \\cdot \\pmb { u } ^ { * } } \\\\ { \\displaystyle \\qquad - \\frac { 1 } { \\rho ^ { 0 } } \\nabla p ^ { n + 1 } \\cdot \\pmb { n } = \\frac { 1 } { \\Delta t } \\left( \\pmb { u } ^ { \\partial \\Omega } - \\pmb { u } ^ { * } \\cdot \\pmb { n } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "b9e83fdc2a77df2f7139c2359537bee131588af2939e28a1c636f7c2e1a8f4d9.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 200, + 288, + 410, + 301.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 200, + 301.6666666666667, + 410, + 315.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 200, + 315.33333333333337, + 410, + 329.00000000000006 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 200, + 329.00000000000006, + 410, + 342.66666666666674 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 200, + 342.66666666666674, + 410, + 356.3333333333334 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 200, + 356.3333333333334, + 410, + 370.0000000000001 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 372, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 386 + ], + "score": 1.0, + "content": "For the spatial discretization, we use a regular marker-and-cell (MAC) grid (Harlow & Welch,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 382, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 506, + 396 + ], + "score": 1.0, + "content": "1965) with cubic voxels whereby velocity components are stored on the face of voxel cells, and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 233, + 408 + ], + "score": 1.0, + "content": "scalar quantities (e.g., pressure", + "type": "text" + }, + { + "bbox": [ + 233, + 397, + 240, + 406 + ], + "score": 0.78, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 394, + 284, + 408 + ], + "score": 1.0, + "content": "or density", + "type": "text" + }, + { + "bbox": [ + 284, + 396, + 291, + 406 + ], + "score": 0.67, + "content": "\\rho \\mathrm { \\hbar }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 394, + 505, + 408 + ], + "score": 1.0, + "content": ") are stored at voxel centers. We use backward semi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "Lagrangian advection (Fedkiw et al., 2001; Gagniere et al., 2020) for Equation 3. All spatial partial", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "derivatives are approximated using finite differences. Equations 4 and 5 describe the pressure Pois-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 428, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 439 + ], + "score": 1.0, + "content": "son equation with Neumann conditions on the boundary of the flow domain. We discretize the left", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 439, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 451 + ], + "score": 1.0, + "content": "hand side of Equation 4 using a standard 7-point finite difference stencil. The right-hand side is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "discretized using the MAC grid discrete divergence finite difference stencils as well as contribu-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 461, + 504, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 504, + 472 + ], + "score": 1.0, + "content": "tions from the boundary condition terms in Equation 5. We refer the reader to Bridson (2008) for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "more in-depth implementation details. Equation 5 is discretized by modifying the Poisson stencil", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "to enforce Neumann boundary conditions. We do this using a simple labeling of the voxels in the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 492, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 248, + 506 + ], + "score": 1.0, + "content": "domain. For simplicity, we assume", + "type": "text" + }, + { + "bbox": [ + 248, + 493, + 296, + 505 + ], + "score": 0.93, + "content": "\\Omega \\subset ( 0 , 1 ) ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 492, + 505, + 506 + ], + "score": 1.0, + "content": "is a subset of the unit cube, potentially with internal", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "boundaries. We label cells in the domain as either liquid or boundary. This simple classification", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "is enough to define the Poisson discretizations (with appropriate Neumann boundary conditions at", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "score": 1.0, + "content": "domain boundaries) that we focus on in the present work; we illustrate the details in Figure 1. We", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 537, + 502, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 502, + 550 + ], + "score": 1.0, + "content": "use the following notation to denote the discrete Poisson equations associated with Equations 4–5:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 372, + 506, + 550 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 553, + 353, + 569 + ], + "lines": [ + { + "bbox": [ + 257, + 553, + 353, + 569 + ], + "spans": [ + { + "bbox": [ + 257, + 553, + 353, + 569 + ], + "score": 0.93, + "content": "\\begin{array} { r } { { \\cal A } ^ { \\Omega } { \\boldsymbol x } = { \\boldsymbol b } ^ { \\nabla \\cdot { \\boldsymbol u } ^ { * } } + { \\boldsymbol b } ^ { u ^ { \\partial \\Omega } } , } \\end{array}", + "type": "interline_equation", + "image_path": "ac399afedaac667ec410278ede5ec8e5a957aa976ca5e09188331dc7a1ba5bb7.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 257, + 553, + 353, + 569 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 576, + 505, + 647 + ], + "lines": [ + { + "bbox": [ + 106, + 575, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 135, + 590 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 576, + 151, + 587 + ], + "score": 0.89, + "content": "A ^ { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 575, + 441, + 590 + ], + "score": 1.0, + "content": "is the discrete Poisson matrix associated with the voxelized domain,", + "type": "text" + }, + { + "bbox": [ + 441, + 579, + 449, + 587 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 575, + 505, + 590 + ], + "score": 1.0, + "content": "is the vector", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 585, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 214, + 601 + ], + "score": 1.0, + "content": "of unknown pressure, and", + "type": "text" + }, + { + "bbox": [ + 215, + 588, + 240, + 600 + ], + "score": 0.9, + "content": "\\mathbf { \\delta } _ { b } \\nabla \\cdot \\mathbf { u } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 585, + 285, + 603 + ], + "score": 1.0, + "content": "and b u ∂ Ω", + "type": "text" + }, + { + "bbox": [ + 282, + 590, + 505, + 601 + ], + "score": 1.0, + "content": "are the right-hand side terms from Equations 4 and 5,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "respectively. We define a special case of the matrix involved in this discretization to be the Poisson", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 610, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 136, + 626 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 136, + 611, + 166, + 623 + ], + "score": 0.91, + "content": "A ^ { ( 0 , 1 ) ^ { 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 610, + 234, + 626 + ], + "score": 1.0, + "content": "associated with", + "type": "text" + }, + { + "bbox": [ + 234, + 612, + 285, + 625 + ], + "score": 0.93, + "content": "\\Omega = ( 0 , 1 ) ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 610, + 506, + 626 + ], + "score": 1.0, + "content": ", i.e., a full fluid domain with no internal boundaries.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "We use this matrix for training, yet demonstrate that our network generalizes to all other matrices", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 635, + 291, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 291, + 648 + ], + "score": 1.0, + "content": "arising from more complicated flow domains.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 575, + 506, + 648 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 662, + 331, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 661, + 332, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 332, + 677 + ], + "score": 1.0, + "content": "4 DEEP CONJUGATE DIRECTION METHOD", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "We present our method for the deep learning acceleration of iterative approximations to the solution", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "of linear systems of the form seen in Equation 6. We first discuss relevant details of the conjugate", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 311, + 723 + ], + "score": 1.0, + "content": "gradients (CG) method, particularly line search and", + "type": "text" + }, + { + "bbox": [ + 311, + 711, + 320, + 720 + ], + "score": 0.79, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "-orthogonal search directions. We then present", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 721, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 734 + ], + "score": 1.0, + "content": "a deep learning technique for improving the quality of these search directions that ultimately reduces", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 79, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 97 + ], + "score": 1.0, + "content": "iteration counts required to achieve satisfactory residual reduction. Lastly, we outline the training", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 325, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 325, + 105 + ], + "score": 1.0, + "content": "procedures for our deep convolutional neural network.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 687, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 97 + ], + "score": 1.0, + "content": "iteration counts required to achieve satisfactory residual reduction. Lastly, we outline the training", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 325, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 325, + 105 + ], + "score": 1.0, + "content": "procedures for our deep convolutional neural network.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 168 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 377, + 124 + ], + "score": 1.0, + "content": "Our approach iteratively improves approximations to the solution", + "type": "text" + }, + { + "bbox": [ + 378, + 113, + 385, + 120 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 110, + 506, + 124 + ], + "score": 1.0, + "content": "of Equation 6. We build on", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 119, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 290, + 135 + ], + "score": 1.0, + "content": "the method of CG, which requires the matrix", + "type": "text" + }, + { + "bbox": [ + 290, + 121, + 306, + 132 + ], + "score": 0.89, + "content": "A ^ { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 119, + 468, + 135 + ], + "score": 1.0, + "content": "in Equation 6 to be SPD. SPD matrices", + "type": "text" + }, + { + "bbox": [ + 469, + 121, + 484, + 132 + ], + "score": 0.87, + "content": "A ^ { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 119, + 506, + 135 + ], + "score": 1.0, + "content": "give", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 203, + 147 + ], + "score": 1.0, + "content": "rise to the matrix norm", + "type": "text" + }, + { + "bbox": [ + 204, + 132, + 292, + 147 + ], + "score": 0.94, + "content": "\\| \\pmb { y } \\| _ { A ^ { \\Omega } } = \\sqrt { \\pmb { y } ^ { T } A ^ { \\Omega } \\pmb { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 133, + 506, + 147 + ], + "score": 1.0, + "content": ". CG can be derived in terms of iterative line search", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 145, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 375, + 158 + ], + "score": 1.0, + "content": "improvement based on optimality in this norm. That is, an iterate", + "type": "text" + }, + { + "bbox": [ + 375, + 146, + 420, + 157 + ], + "score": 0.91, + "content": "\\pmb { x } _ { k - 1 } \\approx \\pmb { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 145, + 505, + 158 + ], + "score": 1.0, + "content": "is updated in search", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 156, + 452, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 144, + 169 + ], + "score": 1.0, + "content": "direction", + "type": "text" + }, + { + "bbox": [ + 145, + 156, + 156, + 167 + ], + "score": 0.88, + "content": "\\scriptstyle d _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 156, + 213, + 169 + ], + "score": 1.0, + "content": "by a step size", + "type": "text" + }, + { + "bbox": [ + 214, + 158, + 226, + 167 + ], + "score": 0.87, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 156, + 452, + 169 + ], + "score": 1.0, + "content": "that is chosen to minimize the matrix norm of the error:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 169, + 418, + 199 + ], + "lines": [ + { + "bbox": [ + 191, + 169, + 418, + 199 + ], + "spans": [ + { + "bbox": [ + 191, + 169, + 418, + 199 + ], + "score": 0.92, + "content": "\\alpha _ { k } = \\underset { \\alpha } { \\arg \\operatorname* { m i n } } \\frac { 1 } { 2 } \\left\\| \\pmb { x } - ( \\pmb { x } _ { k - 1 } + \\alpha \\pmb { d } _ { k } ) \\right\\| _ { \\pmb { A } ^ { \\Omega } } ^ { 2 } = \\frac { { \\pmb { r } } _ { k - 1 } ^ { T } { \\pmb { d } _ { k } } } { { \\pmb { d } } _ { k } ^ { T } { \\pmb { A } } ^ { \\Omega } { \\pmb { d } _ { k } } } ,", + "type": "interline_equation", + "image_path": "794da5f9a8f94a2ca23ac137e3679829e9c6027164f5d341253cdd67dfb2e112.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 191, + 169, + 418, + 184.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 191, + 184.0, + 418, + 199.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 200, + 506, + 303 + ], + "lines": [ + { + "bbox": [ + 106, + 200, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 134, + 214 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 200, + 227, + 213 + ], + "score": 0.92, + "content": "\\pmb { r } _ { k - 1 } = \\pmb { b } - \\pmb { A } ^ { \\Omega } \\pmb { x } _ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 201, + 254, + 214 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 255, + 201, + 293, + 214 + ], + "score": 0.92, + "content": "( k - 1 ) ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 201, + 347, + 214 + ], + "score": 1.0, + "content": "residual and", + "type": "text" + }, + { + "bbox": [ + 347, + 202, + 354, + 212 + ], + "score": 0.78, + "content": "^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 201, + 506, + 214 + ], + "score": 1.0, + "content": "is the right-hand side in Equation 6.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 212, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 225 + ], + "score": 1.0, + "content": "Different search directions result in different algorithms. A natural choice is the negative gradient", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "of the matrix norm of the error (evaluated at the current iterate), since this will point in the direction", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 234, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 104, + 234, + 190, + 251 + ], + "score": 1.0, + "content": "of steepest decrease", + "type": "text" + }, + { + "bbox": [ + 191, + 234, + 324, + 249 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\pmb { d } _ { k } = - \\frac { 1 } { 2 } \\nabla \\left\\| \\pmb { x } _ { k - 1 } \\right\\| _ { \\pmb { A } ^ { \\Omega } } ^ { 2 } = \\pmb { r } _ { k - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 234, + 506, + 251 + ], + "score": 1.0, + "content": ". This is the gradient descent method (GD).", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 248, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 506, + 260 + ], + "score": 1.0, + "content": "Unfortunately, this approach requires many iterations in practice. A more effective strategy is to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 257, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 104, + 257, + 210, + 271 + ], + "score": 1.0, + "content": "choose directions that are", + "type": "text" + }, + { + "bbox": [ + 210, + 259, + 220, + 268 + ], + "score": 0.81, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 257, + 289, + 271 + ], + "score": 1.0, + "content": "-orthogonal (i.e.,", + "type": "text" + }, + { + "bbox": [ + 289, + 257, + 345, + 271 + ], + "score": 0.93, + "content": "{ \\bf \\Phi } _ { { \\bf i } } ^ { T } { \\bf A } ^ { \\Omega } { \\bf d } _ { j } = \\mathrm { ~ ~ \\dot { ~ } { ~ 0 ~ } ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 257, + 362, + 271 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 362, + 258, + 386, + 270 + ], + "score": 0.88, + "content": "i \\neq j", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 257, + 506, + 271 + ], + "score": 1.0, + "content": "). With this choice, the search", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 268, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 268, + 214, + 284 + ], + "score": 1.0, + "content": "directions form a basis for", + "type": "text" + }, + { + "bbox": [ + 214, + 270, + 228, + 279 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 268, + 390, + 284 + ], + "score": 1.0, + "content": "so that the initial error can be written as", + "type": "text" + }, + { + "bbox": [ + 390, + 269, + 501, + 282 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\pmb { e } _ { 0 } = \\pmb { x } - \\pmb { x } _ { 0 } = \\sum _ { i = 1 } ^ { n } e _ { i } \\pmb { d } _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 268, + 506, + 284 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 280, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 133, + 292 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 282, + 143, + 291 + ], + "score": 0.85, + "content": "e _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 280, + 506, + 292 + ], + "score": 1.0, + "content": "are the components of the initial error written in the basis. Furthermore, when the search", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 291, + 419, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 163, + 303 + ], + "score": 1.0, + "content": "directions are", + "type": "text" + }, + { + "bbox": [ + 163, + 292, + 173, + 301 + ], + "score": 0.8, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 291, + 311, + 303 + ], + "score": 1.0, + "content": "-orthogonal, the optimal step sizes", + "type": "text" + }, + { + "bbox": [ + 311, + 293, + 324, + 302 + ], + "score": 0.86, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 291, + 419, + 303 + ], + "score": 1.0, + "content": "at each iteration satisfy", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 304, + 464, + 340 + ], + "lines": [ + { + "bbox": [ + 147, + 304, + 464, + 340 + ], + "spans": [ + { + "bbox": [ + 147, + 304, + 464, + 340 + ], + "score": 0.93, + "content": "\\alpha _ { k } = \\frac { r _ { k - 1 } ^ { T } d _ { k } } { d _ { k } ^ { T } A ^ { \\Omega } d _ { k } } = \\frac { d _ { k } ^ { T } A ^ { \\Omega } e _ { k - 1 } } { d _ { k } ^ { T } A ^ { \\Omega } d _ { k } } = \\frac { d _ { k } ^ { T } A ^ { \\Omega } \\left( \\sum _ { i = 1 } ^ { n } e _ { i } d _ { i } - \\sum _ { j = 1 } ^ { k - 1 } \\alpha _ { j } d _ { j } \\right) } { d _ { k } ^ { T } A ^ { \\Omega } d _ { k } } = e _ { k } .", + "type": "interline_equation", + "image_path": "27f862ee424cf548433605036964bc38ad83b6328e4fe8ee6ce093dd03b21c48.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 147, + 304, + 464, + 316.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 147, + 316.0, + 464, + 328.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 147, + 328.0, + 464, + 340.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 341, + 504, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 354 + ], + "score": 1.0, + "content": "That is, the optimal step sizes are chosen to precisely eliminate the components of the error on the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 455, + 365 + ], + "score": 1.0, + "content": "basis defined by the search directions. Thus, convergence is determined by the (at most", + "type": "text" + }, + { + "bbox": [ + 456, + 354, + 462, + 362 + ], + "score": 0.65, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 352, + 505, + 365 + ], + "score": 1.0, + "content": ") non-zero", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 363, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 157, + 375 + ], + "score": 1.0, + "content": "components", + "type": "text" + }, + { + "bbox": [ + 157, + 364, + 167, + 374 + ], + "score": 0.85, + "content": "e _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 363, + 506, + 375 + ], + "score": 1.0, + "content": "in the initial error. Although rounding errors prevent this from happening exactly in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 375, + 487, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 487, + 386 + ], + "score": 1.0, + "content": "practice, this property greatly reduces the number of required iterations (Golub & Loan, 2012).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 108, + 390, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "CG can be viewed as a modification of GD where the search direction is chosen as the component of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 402, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 447, + 413 + ], + "score": 1.0, + "content": "the residual (equivalently, the negative gradient of the matrix norm of the error) that is", + "type": "text" + }, + { + "bbox": [ + 448, + 402, + 457, + 412 + ], + "score": 0.77, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 402, + 505, + 413 + ], + "score": 1.0, + "content": "-orthogonal", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 413, + 239, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 239, + 425 + ], + "score": 1.0, + "content": "to all previous search directions:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 426, + 407, + 460 + ], + "lines": [ + { + "bbox": [ + 204, + 426, + 407, + 460 + ], + "spans": [ + { + "bbox": [ + 204, + 426, + 407, + 460 + ], + "score": 0.94, + "content": "\\pmb { d } _ { k } = \\pmb { r } _ { k - 1 } - \\sum _ { i = 1 } ^ { k - 1 } h _ { i k } \\pmb { d } _ { i } , \\qquad h _ { i k } = \\frac { { d } _ { i } ^ { T } { \\pmb { A } } ^ { \\Omega } \\pmb { r } _ { k - 1 } } { { d } _ { i } ^ { T } { \\pmb { A } } ^ { \\Omega } \\pmb { d } _ { i } } .", + "type": "interline_equation", + "image_path": "823b515874f8878d3bc9c886f39ccbd598d942c63f696c3c9a5dc8fad3a847d7.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 426, + 407, + 443.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 204, + 443.0, + 407, + 460.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 462, + 503, + 485 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 153, + 474 + ], + "score": 1.0, + "content": "In practice,", + "type": "text" + }, + { + "bbox": [ + 154, + 462, + 187, + 473 + ], + "score": 0.92, + "content": "h _ { i k } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 461, + 202, + 474 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 202, + 462, + 242, + 473 + ], + "score": 0.91, + "content": "i < k - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 461, + 505, + 474 + ], + "score": 1.0, + "content": ", and this iteration can therefore be performed without the need to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 471, + 472, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 247, + 487 + ], + "score": 1.0, + "content": "store all previous search directions", + "type": "text" + }, + { + "bbox": [ + 247, + 474, + 258, + 484 + ], + "score": 0.88, + "content": "\\mathbf { \\mathbf { { \\alpha } } } d _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 471, + 453, + 487 + ], + "score": 1.0, + "content": "and without the need for computing all previous", + "type": "text" + }, + { + "bbox": [ + 453, + 474, + 468, + 484 + ], + "score": 0.9, + "content": "h _ { i k }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 471, + 472, + 487 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 489, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 316, + 503 + ], + "score": 1.0, + "content": "While the residual is a natural choice for generating", + "type": "text" + }, + { + "bbox": [ + 317, + 491, + 326, + 500 + ], + "score": 0.72, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 489, + 505, + 503 + ], + "score": 1.0, + "content": "-orthogonal search directions (since it points", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 450, + 513 + ], + "score": 1.0, + "content": "in the direction of the steepest local decrease), it is not the optimal search direction. If", + "type": "text" + }, + { + "bbox": [ + 450, + 501, + 462, + 512 + ], + "score": 0.89, + "content": "\\scriptstyle d _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "is parallel", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 117, + 525 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 511, + 171, + 524 + ], + "score": 0.92, + "content": "( A ^ { \\Omega } ) ^ { - 1 } r _ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 511, + 195, + 525 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 195, + 513, + 208, + 523 + ], + "score": 0.87, + "content": "\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 511, + 273, + 525 + ], + "score": 1.0, + "content": "will be equal to", + "type": "text" + }, + { + "bbox": [ + 273, + 514, + 281, + 522 + ], + "score": 0.77, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 511, + 305, + 525 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 306, + 513, + 318, + 523 + ], + "score": 0.87, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "(computed from Equation 7) will step directly", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 523, + 499, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 499, + 535 + ], + "score": 1.0, + "content": "to the solution. We can see this by considering the residual and its relation to the search direction:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 540, + 441, + 555 + ], + "lines": [ + { + "bbox": [ + 169, + 540, + 441, + 555 + ], + "spans": [ + { + "bbox": [ + 169, + 540, + 441, + 555 + ], + "score": 0.89, + "content": "\\begin{array} { r } { r _ { k } = b - A ^ { \\Omega } x _ { k } = b - A ^ { \\Omega } x _ { k - 1 } - \\alpha _ { k } A ^ { \\Omega } d _ { k } = r _ { k - 1 } - \\alpha _ { k } A ^ { \\Omega } d _ { k } . } \\end{array}", + "type": "interline_equation", + "image_path": "8bc12357b2f542a177c0dbe7c2b250136ff63d91c180c2f209010068301bb834.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 169, + 540, + 441, + 555 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 561, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 104, + 561, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 104, + 561, + 367, + 575 + ], + "score": 1.0, + "content": "In light of this, we use deep learning to create an approximation", + "type": "text" + }, + { + "bbox": [ + 368, + 562, + 398, + 574 + ], + "score": 0.93, + "content": "f ( c , r )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 561, + 411, + 575 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 411, + 561, + 451, + 574 + ], + "score": 0.92, + "content": "( A ^ { \\Omega } ) ^ { - 1 } r", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 561, + 482, + 575 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 482, + 564, + 489, + 572 + ], + "score": 0.71, + "content": "^ c", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 561, + 505, + 575 + ], + "score": 1.0, + "content": "de-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "score": 1.0, + "content": "notes the network weights and biases. This is analogous to using a preconditioner in PCG; however,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 583, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 597 + ], + "score": 1.0, + "content": "our network is not SPD (nor even a linear function). We simply use this data-driven approach as our", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 595, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 283, + 608 + ], + "score": 1.0, + "content": "means of generating better search directions", + "type": "text" + }, + { + "bbox": [ + 284, + 595, + 295, + 606 + ], + "score": 0.89, + "content": "\\scriptstyle d _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 595, + 506, + 608 + ], + "score": 1.0, + "content": ". Furthermore, we only need to approximate a vector", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 150, + 619 + ], + "score": 1.0, + "content": "parallel to", + "type": "text" + }, + { + "bbox": [ + 150, + 605, + 190, + 617 + ], + "score": 0.89, + "content": "( A ^ { \\Omega } ) ^ { - 1 } r", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 604, + 266, + 619 + ], + "score": 1.0, + "content": "since the step size", + "type": "text" + }, + { + "bbox": [ + 266, + 607, + 278, + 617 + ], + "score": 0.87, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 604, + 506, + 619 + ], + "score": 1.0, + "content": "will account for any scaling in practice. In other words,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 615, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 107, + 617, + 201, + 629 + ], + "score": 0.92, + "content": "f ( c , r ) \\approx s _ { r } ( A ^ { \\Omega } ) ^ { - 1 } r", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 615, + 274, + 630 + ], + "score": 1.0, + "content": ", where the scalar", + "type": "text" + }, + { + "bbox": [ + 275, + 619, + 285, + 628 + ], + "score": 0.86, + "content": "s _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 615, + 461, + 630 + ], + "score": 1.0, + "content": "is not defined globally; it only depends on", + "type": "text" + }, + { + "bbox": [ + 462, + 619, + 468, + 627 + ], + "score": 0.71, + "content": "\\pmb { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 615, + 506, + 630 + ], + "score": 1.0, + "content": ", and the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 628, + 504, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 329, + 640 + ], + "score": 1.0, + "content": "model does not learn it. Lastly, as with CG, we enforce", + "type": "text" + }, + { + "bbox": [ + 329, + 628, + 339, + 637 + ], + "score": 0.77, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 628, + 504, + 640 + ], + "score": 1.0, + "content": "-orthogonality, yielding search directions", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 641, + 433, + 675 + ], + "lines": [ + { + "bbox": [ + 178, + 641, + 433, + 675 + ], + "spans": [ + { + "bbox": [ + 178, + 641, + 433, + 675 + ], + "score": 0.94, + "content": "\\pmb { d } _ { k } = \\pmb { f } ( \\pmb { c } , \\pmb { r } _ { k - 1 } ) - \\sum _ { i = 1 } ^ { k - 1 } h _ { i k } \\pmb { d } _ { i } , \\qquad h _ { i k } = \\frac { \\pmb { f } ( \\pmb { c } , \\pmb { r } _ { k - 1 } ) ^ { T } \\pmb { A } ^ { \\Omega } \\pmb { d } _ { i } } { \\pmb { d } _ { i } ^ { T } \\pmb { A } ^ { \\Omega } \\pmb { d } _ { i } } .", + "type": "interline_equation", + "image_path": "2f5892c23191745ca45a89c6df6aa816d57d4b70d9d40bae7947ea58ff2fad11.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 178, + 641, + 433, + 652.3333333333334 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 178, + 652.3333333333334, + 433, + 663.6666666666667 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 178, + 663.6666666666667, + 433, + 675.0000000000001 + ], + "spans": [], + "index": 46 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 429, + 689 + ], + "score": 1.0, + "content": "We summarize our approach in Algorithm 1. Note that we introduce the variable", + "type": "text" + }, + { + "bbox": [ + 430, + 677, + 447, + 688 + ], + "score": 0.89, + "content": "i _ { \\mathrm { s t a r t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 677, + 505, + 689 + ], + "score": 1.0, + "content": ". To guarantee", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 107, + 686, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 107, + 688, + 116, + 698 + ], + "score": 0.67, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 686, + 362, + 701 + ], + "score": 1.0, + "content": "-orthogonality between all search directions, we must have", + "type": "text" + }, + { + "bbox": [ + 362, + 688, + 402, + 699 + ], + "score": 0.92, + "content": "i _ { \\mathrm { s t a r t } } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 686, + 505, + 701 + ], + "score": 1.0, + "content": ". However, this requires", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 416, + 712 + ], + "score": 1.0, + "content": "storing all prior search directions, which can be costly. We found that using", + "type": "text" + }, + { + "bbox": [ + 416, + 699, + 471, + 710 + ], + "score": 0.91, + "content": "i _ { \\mathrm { s t a r t } } = k - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "worked", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 175, + 723 + ], + "score": 1.0, + "content": "nearly as well as", + "type": "text" + }, + { + "bbox": [ + 175, + 710, + 210, + 721 + ], + "score": 0.91, + "content": "i _ { \\mathrm { s t a r t } } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "in practice (in terms of our ability to iteratively reduce the residual of the", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 721, + 279, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 279, + 733 + ], + "score": 1.0, + "content": "system). We demonstrate this in Figure 4c.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 49 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 79, + 506, + 105 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 168 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 377, + 124 + ], + "score": 1.0, + "content": "Our approach iteratively improves approximations to the solution", + "type": "text" + }, + { + "bbox": [ + 378, + 113, + 385, + 120 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 110, + 506, + 124 + ], + "score": 1.0, + "content": "of Equation 6. We build on", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 119, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 290, + 135 + ], + "score": 1.0, + "content": "the method of CG, which requires the matrix", + "type": "text" + }, + { + "bbox": [ + 290, + 121, + 306, + 132 + ], + "score": 0.89, + "content": "A ^ { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 119, + 468, + 135 + ], + "score": 1.0, + "content": "in Equation 6 to be SPD. SPD matrices", + "type": "text" + }, + { + "bbox": [ + 469, + 121, + 484, + 132 + ], + "score": 0.87, + "content": "A ^ { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 119, + 506, + 135 + ], + "score": 1.0, + "content": "give", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 203, + 147 + ], + "score": 1.0, + "content": "rise to the matrix norm", + "type": "text" + }, + { + "bbox": [ + 204, + 132, + 292, + 147 + ], + "score": 0.94, + "content": "\\| \\pmb { y } \\| _ { A ^ { \\Omega } } = \\sqrt { \\pmb { y } ^ { T } A ^ { \\Omega } \\pmb { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 133, + 506, + 147 + ], + "score": 1.0, + "content": ". CG can be derived in terms of iterative line search", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 145, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 375, + 158 + ], + "score": 1.0, + "content": "improvement based on optimality in this norm. That is, an iterate", + "type": "text" + }, + { + "bbox": [ + 375, + 146, + 420, + 157 + ], + "score": 0.91, + "content": "\\pmb { x } _ { k - 1 } \\approx \\pmb { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 145, + 505, + 158 + ], + "score": 1.0, + "content": "is updated in search", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 156, + 452, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 144, + 169 + ], + "score": 1.0, + "content": "direction", + "type": "text" + }, + { + "bbox": [ + 145, + 156, + 156, + 167 + ], + "score": 0.88, + "content": "\\scriptstyle d _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 156, + 213, + 169 + ], + "score": 1.0, + "content": "by a step size", + "type": "text" + }, + { + "bbox": [ + 214, + 158, + 226, + 167 + ], + "score": 0.87, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 156, + 452, + 169 + ], + "score": 1.0, + "content": "that is chosen to minimize the matrix norm of the error:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 110, + 506, + 169 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 169, + 418, + 199 + ], + "lines": [ + { + "bbox": [ + 191, + 169, + 418, + 199 + ], + "spans": [ + { + "bbox": [ + 191, + 169, + 418, + 199 + ], + "score": 0.92, + "content": "\\alpha _ { k } = \\underset { \\alpha } { \\arg \\operatorname* { m i n } } \\frac { 1 } { 2 } \\left\\| \\pmb { x } - ( \\pmb { x } _ { k - 1 } + \\alpha \\pmb { d } _ { k } ) \\right\\| _ { \\pmb { A } ^ { \\Omega } } ^ { 2 } = \\frac { { \\pmb { r } } _ { k - 1 } ^ { T } { \\pmb { d } _ { k } } } { { \\pmb { d } } _ { k } ^ { T } { \\pmb { A } } ^ { \\Omega } { \\pmb { d } _ { k } } } ,", + "type": "interline_equation", + "image_path": "794da5f9a8f94a2ca23ac137e3679829e9c6027164f5d341253cdd67dfb2e112.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 191, + 169, + 418, + 184.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 191, + 184.0, + 418, + 199.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 200, + 506, + 303 + ], + "lines": [ + { + "bbox": [ + 106, + 200, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 134, + 214 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 200, + 227, + 213 + ], + "score": 0.92, + "content": "\\pmb { r } _ { k - 1 } = \\pmb { b } - \\pmb { A } ^ { \\Omega } \\pmb { x } _ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 201, + 254, + 214 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 255, + 201, + 293, + 214 + ], + "score": 0.92, + "content": "( k - 1 ) ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 201, + 347, + 214 + ], + "score": 1.0, + "content": "residual and", + "type": "text" + }, + { + "bbox": [ + 347, + 202, + 354, + 212 + ], + "score": 0.78, + "content": "^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 201, + 506, + 214 + ], + "score": 1.0, + "content": "is the right-hand side in Equation 6.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 212, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 225 + ], + "score": 1.0, + "content": "Different search directions result in different algorithms. A natural choice is the negative gradient", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "of the matrix norm of the error (evaluated at the current iterate), since this will point in the direction", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 234, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 104, + 234, + 190, + 251 + ], + "score": 1.0, + "content": "of steepest decrease", + "type": "text" + }, + { + "bbox": [ + 191, + 234, + 324, + 249 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\pmb { d } _ { k } = - \\frac { 1 } { 2 } \\nabla \\left\\| \\pmb { x } _ { k - 1 } \\right\\| _ { \\pmb { A } ^ { \\Omega } } ^ { 2 } = \\pmb { r } _ { k - 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 234, + 506, + 251 + ], + "score": 1.0, + "content": ". This is the gradient descent method (GD).", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 248, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 506, + 260 + ], + "score": 1.0, + "content": "Unfortunately, this approach requires many iterations in practice. A more effective strategy is to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 257, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 104, + 257, + 210, + 271 + ], + "score": 1.0, + "content": "choose directions that are", + "type": "text" + }, + { + "bbox": [ + 210, + 259, + 220, + 268 + ], + "score": 0.81, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 257, + 289, + 271 + ], + "score": 1.0, + "content": "-orthogonal (i.e.,", + "type": "text" + }, + { + "bbox": [ + 289, + 257, + 345, + 271 + ], + "score": 0.93, + "content": "{ \\bf \\Phi } _ { { \\bf i } } ^ { T } { \\bf A } ^ { \\Omega } { \\bf d } _ { j } = \\mathrm { ~ ~ \\dot { ~ } { ~ 0 ~ } ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 257, + 362, + 271 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 362, + 258, + 386, + 270 + ], + "score": 0.88, + "content": "i \\neq j", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 257, + 506, + 271 + ], + "score": 1.0, + "content": "). With this choice, the search", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 268, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 268, + 214, + 284 + ], + "score": 1.0, + "content": "directions form a basis for", + "type": "text" + }, + { + "bbox": [ + 214, + 270, + 228, + 279 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 268, + 390, + 284 + ], + "score": 1.0, + "content": "so that the initial error can be written as", + "type": "text" + }, + { + "bbox": [ + 390, + 269, + 501, + 282 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\pmb { e } _ { 0 } = \\pmb { x } - \\pmb { x } _ { 0 } = \\sum _ { i = 1 } ^ { n } e _ { i } \\pmb { d } _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 268, + 506, + 284 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 280, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 133, + 292 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 282, + 143, + 291 + ], + "score": 0.85, + "content": "e _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 280, + 506, + 292 + ], + "score": 1.0, + "content": "are the components of the initial error written in the basis. Furthermore, when the search", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 291, + 419, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 163, + 303 + ], + "score": 1.0, + "content": "directions are", + "type": "text" + }, + { + "bbox": [ + 163, + 292, + 173, + 301 + ], + "score": 0.8, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 291, + 311, + 303 + ], + "score": 1.0, + "content": "-orthogonal, the optimal step sizes", + "type": "text" + }, + { + "bbox": [ + 311, + 293, + 324, + 302 + ], + "score": 0.86, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 291, + 419, + 303 + ], + "score": 1.0, + "content": "at each iteration satisfy", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 200, + 506, + 303 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 304, + 464, + 340 + ], + "lines": [ + { + "bbox": [ + 147, + 304, + 464, + 340 + ], + "spans": [ + { + "bbox": [ + 147, + 304, + 464, + 340 + ], + "score": 0.93, + "content": "\\alpha _ { k } = \\frac { r _ { k - 1 } ^ { T } d _ { k } } { d _ { k } ^ { T } A ^ { \\Omega } d _ { k } } = \\frac { d _ { k } ^ { T } A ^ { \\Omega } e _ { k - 1 } } { d _ { k } ^ { T } A ^ { \\Omega } d _ { k } } = \\frac { d _ { k } ^ { T } A ^ { \\Omega } \\left( \\sum _ { i = 1 } ^ { n } e _ { i } d _ { i } - \\sum _ { j = 1 } ^ { k - 1 } \\alpha _ { j } d _ { j } \\right) } { d _ { k } ^ { T } A ^ { \\Omega } d _ { k } } = e _ { k } .", + "type": "interline_equation", + "image_path": "27f862ee424cf548433605036964bc38ad83b6328e4fe8ee6ce093dd03b21c48.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 147, + 304, + 464, + 316.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 147, + 316.0, + 464, + 328.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 147, + 328.0, + 464, + 340.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 341, + 504, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 354 + ], + "score": 1.0, + "content": "That is, the optimal step sizes are chosen to precisely eliminate the components of the error on the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 455, + 365 + ], + "score": 1.0, + "content": "basis defined by the search directions. Thus, convergence is determined by the (at most", + "type": "text" + }, + { + "bbox": [ + 456, + 354, + 462, + 362 + ], + "score": 0.65, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 352, + 505, + 365 + ], + "score": 1.0, + "content": ") non-zero", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 363, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 157, + 375 + ], + "score": 1.0, + "content": "components", + "type": "text" + }, + { + "bbox": [ + 157, + 364, + 167, + 374 + ], + "score": 0.85, + "content": "e _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 363, + 506, + 375 + ], + "score": 1.0, + "content": "in the initial error. Although rounding errors prevent this from happening exactly in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 375, + 487, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 487, + 386 + ], + "score": 1.0, + "content": "practice, this property greatly reduces the number of required iterations (Golub & Loan, 2012).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 340, + 506, + 386 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 390, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "CG can be viewed as a modification of GD where the search direction is chosen as the component of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 402, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 447, + 413 + ], + "score": 1.0, + "content": "the residual (equivalently, the negative gradient of the matrix norm of the error) that is", + "type": "text" + }, + { + "bbox": [ + 448, + 402, + 457, + 412 + ], + "score": 0.77, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 402, + 505, + 413 + ], + "score": 1.0, + "content": "-orthogonal", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 413, + 239, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 239, + 425 + ], + "score": 1.0, + "content": "to all previous search directions:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 391, + 505, + 425 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 426, + 407, + 460 + ], + "lines": [ + { + "bbox": [ + 204, + 426, + 407, + 460 + ], + "spans": [ + { + "bbox": [ + 204, + 426, + 407, + 460 + ], + "score": 0.94, + "content": "\\pmb { d } _ { k } = \\pmb { r } _ { k - 1 } - \\sum _ { i = 1 } ^ { k - 1 } h _ { i k } \\pmb { d } _ { i } , \\qquad h _ { i k } = \\frac { { d } _ { i } ^ { T } { \\pmb { A } } ^ { \\Omega } \\pmb { r } _ { k - 1 } } { { d } _ { i } ^ { T } { \\pmb { A } } ^ { \\Omega } \\pmb { d } _ { i } } .", + "type": "interline_equation", + "image_path": "823b515874f8878d3bc9c886f39ccbd598d942c63f696c3c9a5dc8fad3a847d7.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 426, + 407, + 443.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 204, + 443.0, + 407, + 460.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 462, + 503, + 485 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 153, + 474 + ], + "score": 1.0, + "content": "In practice,", + "type": "text" + }, + { + "bbox": [ + 154, + 462, + 187, + 473 + ], + "score": 0.92, + "content": "h _ { i k } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 461, + 202, + 474 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 202, + 462, + 242, + 473 + ], + "score": 0.91, + "content": "i < k - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 461, + 505, + 474 + ], + "score": 1.0, + "content": ", and this iteration can therefore be performed without the need to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 471, + 472, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 247, + 487 + ], + "score": 1.0, + "content": "store all previous search directions", + "type": "text" + }, + { + "bbox": [ + 247, + 474, + 258, + 484 + ], + "score": 0.88, + "content": "\\mathbf { \\mathbf { { \\alpha } } } d _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 471, + 453, + 487 + ], + "score": 1.0, + "content": "and without the need for computing all previous", + "type": "text" + }, + { + "bbox": [ + 453, + 474, + 468, + 484 + ], + "score": 0.9, + "content": "h _ { i k }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 471, + 472, + 487 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 461, + 505, + 487 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 489, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 316, + 503 + ], + "score": 1.0, + "content": "While the residual is a natural choice for generating", + "type": "text" + }, + { + "bbox": [ + 317, + 491, + 326, + 500 + ], + "score": 0.72, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 489, + 505, + 503 + ], + "score": 1.0, + "content": "-orthogonal search directions (since it points", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 450, + 513 + ], + "score": 1.0, + "content": "in the direction of the steepest local decrease), it is not the optimal search direction. If", + "type": "text" + }, + { + "bbox": [ + 450, + 501, + 462, + 512 + ], + "score": 0.89, + "content": "\\scriptstyle d _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "is parallel", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 117, + 525 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 511, + 171, + 524 + ], + "score": 0.92, + "content": "( A ^ { \\Omega } ) ^ { - 1 } r _ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 511, + 195, + 525 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 195, + 513, + 208, + 523 + ], + "score": 0.87, + "content": "\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 511, + 273, + 525 + ], + "score": 1.0, + "content": "will be equal to", + "type": "text" + }, + { + "bbox": [ + 273, + 514, + 281, + 522 + ], + "score": 0.77, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 511, + 305, + 525 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 306, + 513, + 318, + 523 + ], + "score": 0.87, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "(computed from Equation 7) will step directly", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 523, + 499, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 499, + 535 + ], + "score": 1.0, + "content": "to the solution. We can see this by considering the residual and its relation to the search direction:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 489, + 505, + 535 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 540, + 441, + 555 + ], + "lines": [ + { + "bbox": [ + 169, + 540, + 441, + 555 + ], + "spans": [ + { + "bbox": [ + 169, + 540, + 441, + 555 + ], + "score": 0.89, + "content": "\\begin{array} { r } { r _ { k } = b - A ^ { \\Omega } x _ { k } = b - A ^ { \\Omega } x _ { k - 1 } - \\alpha _ { k } A ^ { \\Omega } d _ { k } = r _ { k - 1 } - \\alpha _ { k } A ^ { \\Omega } d _ { k } . } \\end{array}", + "type": "interline_equation", + "image_path": "8bc12357b2f542a177c0dbe7c2b250136ff63d91c180c2f209010068301bb834.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 169, + 540, + 441, + 555 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 561, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 104, + 561, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 104, + 561, + 367, + 575 + ], + "score": 1.0, + "content": "In light of this, we use deep learning to create an approximation", + "type": "text" + }, + { + "bbox": [ + 368, + 562, + 398, + 574 + ], + "score": 0.93, + "content": "f ( c , r )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 561, + 411, + 575 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 411, + 561, + 451, + 574 + ], + "score": 0.92, + "content": "( A ^ { \\Omega } ) ^ { - 1 } r", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 561, + 482, + 575 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 482, + 564, + 489, + 572 + ], + "score": 0.71, + "content": "^ c", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 561, + 505, + 575 + ], + "score": 1.0, + "content": "de-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "score": 1.0, + "content": "notes the network weights and biases. This is analogous to using a preconditioner in PCG; however,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 583, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 597 + ], + "score": 1.0, + "content": "our network is not SPD (nor even a linear function). We simply use this data-driven approach as our", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 595, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 283, + 608 + ], + "score": 1.0, + "content": "means of generating better search directions", + "type": "text" + }, + { + "bbox": [ + 284, + 595, + 295, + 606 + ], + "score": 0.89, + "content": "\\scriptstyle d _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 595, + 506, + 608 + ], + "score": 1.0, + "content": ". Furthermore, we only need to approximate a vector", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 150, + 619 + ], + "score": 1.0, + "content": "parallel to", + "type": "text" + }, + { + "bbox": [ + 150, + 605, + 190, + 617 + ], + "score": 0.89, + "content": "( A ^ { \\Omega } ) ^ { - 1 } r", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 604, + 266, + 619 + ], + "score": 1.0, + "content": "since the step size", + "type": "text" + }, + { + "bbox": [ + 266, + 607, + 278, + 617 + ], + "score": 0.87, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 604, + 506, + 619 + ], + "score": 1.0, + "content": "will account for any scaling in practice. In other words,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 615, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 107, + 617, + 201, + 629 + ], + "score": 0.92, + "content": "f ( c , r ) \\approx s _ { r } ( A ^ { \\Omega } ) ^ { - 1 } r", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 615, + 274, + 630 + ], + "score": 1.0, + "content": ", where the scalar", + "type": "text" + }, + { + "bbox": [ + 275, + 619, + 285, + 628 + ], + "score": 0.86, + "content": "s _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 615, + 461, + 630 + ], + "score": 1.0, + "content": "is not defined globally; it only depends on", + "type": "text" + }, + { + "bbox": [ + 462, + 619, + 468, + 627 + ], + "score": 0.71, + "content": "\\pmb { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 615, + 506, + 630 + ], + "score": 1.0, + "content": ", and the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 628, + 504, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 329, + 640 + ], + "score": 1.0, + "content": "model does not learn it. Lastly, as with CG, we enforce", + "type": "text" + }, + { + "bbox": [ + 329, + 628, + 339, + 637 + ], + "score": 0.77, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 628, + 504, + 640 + ], + "score": 1.0, + "content": "-orthogonality, yielding search directions", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 104, + 561, + 506, + 640 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 641, + 433, + 675 + ], + "lines": [ + { + "bbox": [ + 178, + 641, + 433, + 675 + ], + "spans": [ + { + "bbox": [ + 178, + 641, + 433, + 675 + ], + "score": 0.94, + "content": "\\pmb { d } _ { k } = \\pmb { f } ( \\pmb { c } , \\pmb { r } _ { k - 1 } ) - \\sum _ { i = 1 } ^ { k - 1 } h _ { i k } \\pmb { d } _ { i } , \\qquad h _ { i k } = \\frac { \\pmb { f } ( \\pmb { c } , \\pmb { r } _ { k - 1 } ) ^ { T } \\pmb { A } ^ { \\Omega } \\pmb { d } _ { i } } { \\pmb { d } _ { i } ^ { T } \\pmb { A } ^ { \\Omega } \\pmb { d } _ { i } } .", + "type": "interline_equation", + "image_path": "2f5892c23191745ca45a89c6df6aa816d57d4b70d9d40bae7947ea58ff2fad11.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 178, + 641, + 433, + 652.3333333333334 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 178, + 652.3333333333334, + 433, + 663.6666666666667 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 178, + 663.6666666666667, + 433, + 675.0000000000001 + ], + "spans": [], + "index": 46 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 429, + 689 + ], + "score": 1.0, + "content": "We summarize our approach in Algorithm 1. Note that we introduce the variable", + "type": "text" + }, + { + "bbox": [ + 430, + 677, + 447, + 688 + ], + "score": 0.89, + "content": "i _ { \\mathrm { s t a r t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 677, + 505, + 689 + ], + "score": 1.0, + "content": ". To guarantee", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 107, + 686, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 107, + 688, + 116, + 698 + ], + "score": 0.67, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 686, + 362, + 701 + ], + "score": 1.0, + "content": "-orthogonality between all search directions, we must have", + "type": "text" + }, + { + "bbox": [ + 362, + 688, + 402, + 699 + ], + "score": 0.92, + "content": "i _ { \\mathrm { s t a r t } } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 686, + 505, + 701 + ], + "score": 1.0, + "content": ". However, this requires", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 416, + 712 + ], + "score": 1.0, + "content": "storing all prior search directions, which can be costly. We found that using", + "type": "text" + }, + { + "bbox": [ + 416, + 699, + 471, + 710 + ], + "score": 0.91, + "content": "i _ { \\mathrm { s t a r t } } = k - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "worked", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 175, + 723 + ], + "score": 1.0, + "content": "nearly as well as", + "type": "text" + }, + { + "bbox": [ + 175, + 710, + 210, + 721 + ], + "score": 0.91, + "content": "i _ { \\mathrm { s t a r t } } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "in practice (in terms of our ability to iteratively reduce the residual of the", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 721, + 279, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 279, + 733 + ], + "score": 1.0, + "content": "system). We demonstrate this in Figure 4c.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 677, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 394, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 80, + 395, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 80, + 395, + 96 + ], + "score": 1.0, + "content": "5 MODEL ARCHITECTURE, DATASETS, AND TRAINING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 375, + 183 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 376, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 376, + 118 + ], + "score": 1.0, + "content": "Efficient performance of our method requires effective training of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 376, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 329, + 127 + ], + "score": 1.0, + "content": "our deep convolutional network for weights and biases", + "type": "text" + }, + { + "bbox": [ + 330, + 118, + 336, + 127 + ], + "score": 0.43, + "content": "^ c", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 117, + 376, + 127 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 126, + 377, + 140 + ], + "spans": [ + { + "bbox": [ + 107, + 127, + 199, + 140 + ], + "score": 0.92, + "content": "f ( c , r ) { \\overset { \\cdot } { \\approx } } s _ { r } ( A ^ { \\Omega } ) ^ { - 1 } r", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 126, + 280, + 140 + ], + "score": 1.0, + "content": "(for arbitrary scalar", + "type": "text" + }, + { + "bbox": [ + 280, + 129, + 290, + 138 + ], + "score": 0.83, + "content": "s _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 126, + 377, + 140 + ], + "score": 1.0, + "content": "). 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In Algorithm 1, the normalized residuals", + "type": "text" + }, + { + "bbox": [ + 484, + 384, + 504, + 398 + ], + "score": 0.92, + "content": "\\frac { \\boldsymbol { r } _ { k } } { \\| \\boldsymbol { r } _ { k } \\| }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "score": 1.0, + "content": "are passed as inputs to the model. 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We observe that generally the residuals", + "type": "text" + }, + { + "bbox": [ + 357, + 433, + 369, + 442 + ], + "score": 0.86, + "content": "\\mathbf { \\nabla } r _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 431, + 506, + 444 + ], + "score": 1.0, + "content": "in Algorithm 1 are skewed to the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 440, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 104, + 440, + 276, + 456 + ], + "score": 1.0, + "content": "lower end of the spectrum of the matrix", + "type": "text" + }, + { + "bbox": [ + 276, + 441, + 292, + 452 + ], + "score": 0.89, + "content": "A ^ { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 440, + 326, + 456 + ], + "score": 1.0, + "content": ". 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This choice", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 136, + 632 + ], + "score": 1.0, + "content": "creates", + "type": "text" + }, + { + "bbox": [ + 136, + 619, + 156, + 631 + ], + "score": 0.88, + "content": "9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 620, + 168, + 632 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 168, + 619, + 177, + 630 + ], + "score": 0.87, + "content": "\\mathbf { \\nabla } _ { b } i", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 620, + 401, + 632 + ], + "score": 1.0, + "content": "from the lower end of the spectrum, with the remaining", + "type": "text" + }, + { + "bbox": [ + 401, + 620, + 420, + 631 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "from the higher end.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "score": 1.0, + "content": "The Riemann-Lebesgue Lemma states the Fourier spectrum of a continuous function will decay at", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 642, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 239, + 654 + ], + "score": 1.0, + "content": "infinity, so this specific choice of", + "type": "text" + }, + { + "bbox": [ + 239, + 642, + 248, + 653 + ], + "score": 0.87, + "content": "b _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 642, + 506, + 654 + ], + "score": 1.0, + "content": "’s is reasonable for the training set. 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Notably, even though this dataset only uses Rayleigh-", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 104, + 673, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 673, + 255, + 691 + ], + "score": 1.0, + "content": "Ritz vectors from the training matrix", + "type": "text" + }, + { + "bbox": [ + 255, + 675, + 285, + 687 + ], + "score": 0.91, + "content": "A ^ { ( 0 , 1 ) ^ { 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 673, + 506, + 691 + ], + "score": 1.0, + "content": ", our network can be effectively generalized to flows in", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 687, + 475, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 475, + 700 + ], + "score": 1.0, + "content": "irregular domains, e.g., smoke flow past a rotating box and flow past a bunny (see Figure 3).", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 53.5, + "bbox_fs": [ + 104, + 605, + 506, + 700 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 706, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 704, + 493, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 704, + 441, + 720 + ], + "score": 1.0, + "content": "We generate the Rayleigh-Ritz vectors by first tridiagonalizing the training matrix", + "type": "text" + }, + { + "bbox": [ + 441, + 704, + 470, + 717 + ], + "score": 0.91, + "content": "A ^ { ( 0 , 1 ) ^ { 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 704, + 493, + 720 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 101, + 711, + 509, + 739 + ], + "spans": [ + { + "bbox": [ + 101, + 711, + 279, + 739 + ], + "score": 1.0, + "content": "Lanczos iterations (Lanczos, 1950) to form", + "type": "text" + }, + { + "bbox": [ + 280, + 718, + 417, + 733 + ], + "score": 0.92, + "content": "\\pmb { T } ^ { m } = \\pmb { Q } _ { m } ^ { L } ^ { T } \\pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \\pmb { Q } _ { m } ^ { L } \\in \\mathbb { R } ^ { m \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 711, + 509, + 739 + ], + "score": 1.0, + "content": ". We then diagonalize", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 79, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 177, + 94 + ], + "score": 0.92, + "content": "\\pmb { T } ^ { m } = \\hat { \\pmb { Q } } ^ { T } \\pmb { \\Lambda } _ { m } \\hat { \\pmb { Q } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 177, + 79, + 506, + 97 + ], + "score": 1.0, + "content": ". While costly, we note that this algorithm is performed on the comparably small", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 92, + 507, + 109 + ], + "spans": [ + { + "bbox": [ + 107, + 96, + 138, + 106 + ], + "score": 0.87, + "content": "m \\times m", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 138, + 92, + 168, + 109 + ], + "score": 1.0, + "content": "matrix", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 169, + 96, + 185, + 106 + ], + "score": 0.86, + "content": "\\mathbf { T } ^ { m }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 185, + 92, + 264, + 109 + ], + "score": 1.0, + "content": "(rather than on the", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 265, + 93, + 334, + 106 + ], + "score": 0.92, + "content": "A ^ { ( 0 , 1 ) ^ { 3 } } \\in \\mathbb { R } ^ { n \\times n } ,", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 334, + 92, + 507, + 109 + ], + "score": 1.0, + "content": "). This yields the Rayleigh-Ritz vectors as", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 106, + 507, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 168, + 123 + ], + "score": 1.0, + "content": "the columns of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 169, + 106, + 224, + 120 + ], + "score": 0.94, + "content": "Q _ { m } = Q _ { m } ^ { L } \\hat { Q }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 225, + 106, + 434, + 123 + ], + "score": 1.0, + "content": ". The Lanczos vectors are the columns of the matrix", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 434, + 107, + 451, + 120 + ], + "score": 0.92, + "content": "Q _ { m } ^ { L }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 451, + 106, + 507, + 123 + ], + "score": 1.0, + "content": "and satisfy a", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 118, + 501, + 132 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 501, + 132 + ], + "score": 1.0, + "content": "three-term recurrence whereby the next Lanczos vector can be computed from the previous two as", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 58.5, + "bbox_fs": [ + 101, + 704, + 509, + 739 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 131 + ], + "lines": [ + { + "bbox": [ + 106, + 79, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 177, + 94 + ], + "score": 0.92, + "content": "\\pmb { T } ^ { m } = \\hat { \\pmb { Q } } ^ { T } \\pmb { \\Lambda } _ { m } \\hat { \\pmb { Q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 79, + 506, + 97 + ], + "score": 1.0, + "content": ". While costly, we note that this algorithm is performed on the comparably small", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 92, + 507, + 109 + ], + "spans": [ + { + "bbox": [ + 107, + 96, + 138, + 106 + ], + "score": 0.87, + "content": "m \\times m", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 92, + 168, + 109 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 169, + 96, + 185, + 106 + ], + "score": 0.86, + "content": "\\mathbf { T } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 92, + 264, + 109 + ], + "score": 1.0, + "content": "(rather than on the", + "type": "text" + }, + { + "bbox": [ + 265, + 93, + 334, + 106 + ], + "score": 0.92, + "content": "A ^ { ( 0 , 1 ) ^ { 3 } } \\in \\mathbb { R } ^ { n \\times n } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 92, + 507, + 109 + ], + "score": 1.0, + "content": "). 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In summary, our", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 556, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 571 + ], + "score": 1.0, + "content": "goal was to first identify the simplest network architecture that provided adequate accuracy for our", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 567, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 582 + ], + "score": 1.0, + "content": "target problems, and subsequently, we sought to make architectural changes to minimize training", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 580, + 324, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 324, + 591 + ], + "score": 1.0, + "content": "and inference time; further optimizations are possible.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 513, + 506, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 595, + 505, + 706 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "Differing resolutions use differing numbers of convolutions, but the fundamental structure remains", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "the same. More precisely, the number of residual connections is changed for different resolutions.", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 617, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 168, + 630 + ], + "score": 1.0, + "content": "For example, a", + "type": "text" + }, + { + "bbox": [ + 168, + 617, + 184, + 628 + ], + "score": 0.87, + "content": "6 \\dot { 4 } ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 617, + 506, + 630 + ], + "score": 1.0, + "content": "grid uses one residual block on the left, two on the right on the upper level, and", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "three on the lower level. Furthermore, the weights trained on a lower resolution grid can be used", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 428, + 653 + ], + "score": 1.0, + "content": "effectively with higher resolutions. Figure 4d shows convergence results for a", + "type": "text" + }, + { + "bbox": [ + 429, + 639, + 450, + 650 + ], + "score": 0.86, + "content": "2 \\mathrm { { 5 6 ^ { 3 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "grid, using a", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 188, + 663 + ], + "score": 1.0, + "content": "model trained for a", + "type": "text" + }, + { + "bbox": [ + 189, + 650, + 204, + 661 + ], + "score": 0.87, + "content": "6 4 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 650, + 251, + 663 + ], + "score": 1.0, + "content": "grid and a", + "type": "text" + }, + { + "bbox": [ + 252, + 650, + 272, + 661 + ], + "score": 0.87, + "content": "1 2 \\bar { 8 } ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 650, + 411, + 663 + ], + "score": 1.0, + "content": "grid. The model that we use for", + "type": "text" + }, + { + "bbox": [ + 411, + 650, + 432, + 661 + ], + "score": 0.87, + "content": "2 5 6 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "grids in our final", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 660, + 504, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 215, + 674 + ], + "score": 1.0, + "content": "examples was trained on a", + "type": "text" + }, + { + "bbox": [ + 215, + 661, + 236, + 672 + ], + "score": 0.87, + "content": "1 2 8 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 660, + 488, + 674 + ], + "score": 1.0, + "content": "grid; however, as the shown in the figure, even training with a", + "type": "text" + }, + { + "bbox": [ + 488, + 661, + 504, + 672 + ], + "score": 0.86, + "content": "6 4 ^ { 3 }", + "type": "inline_equation" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "grid allows for efficient residual reduction. 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Left: smoke plume at", + "type": "text" + }, + { + "bbox": [ + 477, + 235, + 495, + 245 + ], + "score": 0.85, + "content": "t \\ =", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 273, + 244, + 496, + 258 + ], + "spans": [ + { + "bbox": [ + 273, + 244, + 496, + 258 + ], + "score": 1.0, + "content": "6.67, 13.33, 20 seconds. Middle: smoke passing bunny", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 273, + 256, + 495, + 268 + ], + "spans": [ + { + "bbox": [ + 273, + 256, + 284, + 268 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 284, + 257, + 338, + 268 + ], + "score": 0.41, + "content": "t = 5 , 1 0 , 1 5", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 256, + 495, + 268 + ], + "score": 1.0, + "content": "seconds. Right: smoke passing a spin-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 273, + 267, + 495, + 279 + ], + "spans": [ + { + "bbox": [ + 273, + 267, + 495, + 279 + ], + "score": 1.0, + "content": "ning box (time-dependent Neumann boundary condi-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 274, + 278, + 414, + 289 + ], + "spans": [ + { + "bbox": [ + 274, + 278, + 309, + 289 + ], + "score": 1.0, + "content": "tions) at", + "type": "text" + }, + { + "bbox": [ + 309, + 278, + 377, + 289 + ], + "score": 0.75, + "content": "t = 2 . 6 7 , 6 , 9 . 3 3", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 278, + 414, + 289 + ], + "score": 1.0, + "content": "seconds.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + } + ], + "index": 26.0 + }, + { + "type": "text", + "bbox": [ + 106, + 303, + 465, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 301, + 464, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 215, + 316 + ], + "score": 1.0, + "content": "first epoch was optimal for", + "type": "text" + }, + { + "bbox": [ + 216, + 302, + 231, + 313 + ], + "score": 0.86, + "content": "6 4 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 301, + 443, + 316 + ], + "score": 1.0, + "content": ", and the model from the third epoch was optimal for", + "type": "text" + }, + { + "bbox": [ + 443, + 302, + 464, + 313 + ], + "score": 0.84, + "content": "1 2 8 ^ { 3 }", + "type": "inline_equation" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 108, + 332, + 253, + 344 + ], + "lines": [ + { + "bbox": [ + 105, + 330, + 255, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 255, + 347 + ], + "score": 1.0, + "content": "6 RESULTS AND ANALYSIS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 357, + 504, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "We demonstrate DCDM on three increasingly difficult examples and provide numerical evidence for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 368, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 380 + ], + "score": 1.0, + "content": "the efficient convergence of our method. All examples were run on a workstation with dual AMD", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 380, + 278, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 278, + 392 + ], + "score": 1.0, + "content": "EPYC 75F3 processors and 512GB RAM.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 396, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "Figure 3 showcases DCDM for incompressible smoke simulations. In each simulation, inlet bound-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "ary conditions are set in a circular portion of the bottom of the cubic domain, whereby smoke flows", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 419, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 431 + ], + "score": 1.0, + "content": "around potential obstacles and fills the domain. 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All computations are done using only", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 226, + 527, + 402, + 538 + ], + "spans": [ + { + "bbox": [ + 226, + 527, + 402, + 538 + ], + "score": 1.0, + "content": "CPUs; model inference does not use GPUs.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 53.5 + }, + { + "type": "text", + "bbox": [ + 106, + 491, + 217, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 218, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 218, + 503 + ], + "score": 1.0, + "content": "For the bunny example,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 501, + 218, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 218, + 513 + ], + "score": 1.0, + "content": "Figures 4a–b demonstrate", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 511, + 218, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 218, + 524 + ], + "score": 1.0, + "content": "how residuals decrease", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 523, + 218, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 218, + 534 + ], + "score": 1.0, + "content": "over the course of a linear", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 534, + 218, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 218, + 546 + ], + "score": 1.0, + "content": "solve, comparing DCDM", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 545, + 218, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 218, + 556 + ], + "score": 1.0, + "content": "with other methods. 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64 Grid1283 Grid2563 Grid
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DCDM-642.71s1622s27261s58
DCDM-1285.37s1926s24267s44
CG1.77s16826s4651548s1046
Deflated PCG771.6s1173700s27721030s489
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Right: smoke passing a spin-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 273, + 267, + 495, + 279 + ], + "spans": [ + { + "bbox": [ + 273, + 267, + 495, + 279 + ], + "score": 1.0, + "content": "ning box (time-dependent Neumann boundary condi-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 274, + 278, + 414, + 289 + ], + "spans": [ + { + "bbox": [ + 274, + 278, + 309, + 289 + ], + "score": 1.0, + "content": "tions) at", + "type": "text" + }, + { + "bbox": [ + 309, + 278, + 377, + 289 + ], + "score": 0.75, + "content": "t = 2 . 6 7 , 6 , 9 . 3 3", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 278, + 414, + 289 + ], + "score": 1.0, + "content": "seconds.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + } + ], + "index": 26.0 + }, + { + "type": "text", + "bbox": [ + 106, + 303, + 465, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 301, + 464, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 215, + 316 + ], + "score": 1.0, + "content": "first epoch was optimal for", + "type": "text" + }, + { + "bbox": [ + 216, + 302, + 231, + 313 + ], + "score": 0.86, + "content": "6 4 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 301, + 443, + 316 + ], + "score": 1.0, + "content": ", and the model from the third epoch was optimal for", + "type": "text" + }, + { + "bbox": [ + 443, + 302, + 464, + 313 + ], + "score": 0.84, + "content": "1 2 8 ^ { 3 }", + "type": "inline_equation" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 301, + 464, + 316 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 332, + 253, + 344 + ], + "lines": [ + { + "bbox": [ + 105, + 330, + 255, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 255, + 347 + ], + "score": 1.0, + "content": "6 RESULTS AND ANALYSIS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 357, + 504, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "We demonstrate DCDM on three increasingly difficult examples and provide numerical evidence for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 368, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 380 + ], + "score": 1.0, + "content": "the efficient convergence of our method. All examples were run on a workstation with dual AMD", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 380, + 278, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 278, + 392 + ], + "score": 1.0, + "content": "EPYC 75F3 processors and 512GB RAM.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 106, + 358, + 505, + 392 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 396, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "Figure 3 showcases DCDM for incompressible smoke simulations. In each simulation, inlet bound-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "ary conditions are set in a circular portion of the bottom of the cubic domain, whereby smoke flows", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 419, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 431 + ], + "score": 1.0, + "content": "around potential obstacles and fills the domain. We show a smoke plume (no obstacles), flow past a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "complex static geometry (the Stanford bunny), and flow past a dynamic geometry (a rotating cube).", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 440, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 454 + ], + "score": 1.0, + "content": "Visually plausible and highly-detailed results are achieved for each simulation (see supplementary", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 104, + 451, + 480, + 464 + ], + "score": 1.0, + "content": "material for larger videos). The plume example uses a computational grid with resolution", + "type": "text" + }, + { + "bbox": [ + 480, + 451, + 501, + 462 + ], + "score": 0.87, + "content": "1 2 8 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 451, + 506, + 464 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 294, + 475 + ], + "score": 1.0, + "content": "while the other two uses grids with resolution", + "type": "text" + }, + { + "bbox": [ + 295, + 462, + 315, + 473 + ], + "score": 0.88, + "content": "2 5 6 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "(representing over 16 million unknowns). For", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 474, + 483, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 483, + 486 + ], + "score": 1.0, + "content": "each linear solve, DCDM was run until the residual was reduced by four orders of magnitude.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5, + "bbox_fs": [ + 104, + 397, + 506, + 486 + ] + }, + { + "type": "text", + "bbox": [ + 225, + 494, + 504, + 538 + ], + "lines": [ + { + "bbox": [ + 225, + 494, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 225, + 494, + 505, + 506 + ], + "score": 1.0, + "content": "Table 1: Timing and iteration comparison for different methods on the", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 223, + 504, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 223, + 504, + 330, + 518 + ], + "score": 1.0, + "content": "bunny example. DCDM-", + "type": "text" + }, + { + "bbox": [ + 331, + 505, + 368, + 517 + ], + "score": 0.81, + "content": "\\{ 6 4 , 1 2 8 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 504, + 506, + 518 + ], + "score": 1.0, + "content": "calls a model whose parameters", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 224, + 514, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 224, + 514, + 282, + 529 + ], + "score": 1.0, + "content": "trained over a", + "type": "text" + }, + { + "bbox": [ + 282, + 515, + 331, + 528 + ], + "score": 0.92, + "content": "\\{ 6 4 ^ { 3 } , 1 2 8 ^ { 3 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 514, + 505, + 529 + ], + "score": 1.0, + "content": "grid. All computations are done using only", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 226, + 527, + 402, + 538 + ], + "spans": [ + { + "bbox": [ + 226, + 527, + 402, + 538 + ], + "score": 1.0, + "content": "CPUs; model inference does not use GPUs.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 53.5, + "bbox_fs": [ + 223, + 494, + 506, + 538 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 491, + 217, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 218, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 218, + 503 + ], + "score": 1.0, + "content": "For the bunny example,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 501, + 218, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 218, + 513 + ], + "score": 1.0, + "content": "Figures 4a–b demonstrate", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 511, + 218, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 218, + 524 + ], + "score": 1.0, + "content": "how residuals decrease", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 523, + 218, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 218, + 534 + ], + "score": 1.0, + "content": "over the course of a linear", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 534, + 218, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 218, + 546 + ], + "score": 1.0, + "content": "solve, comparing DCDM", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 545, + 218, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 218, + 556 + ], + "score": 1.0, + "content": "with other methods. 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64 Grid1283 Grid2563 Grid
Methodtrnrtrnrtrnr
DCDM-642.71s1622s27261s58
DCDM-1285.37s1926s24267s44
CG1.77s16826s4651548s1046
Deflated PCG771.6s1173700s27721030s489
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This implementation", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "includes pre-trained models that we use without modification. In both subfigures, it is evident", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "that the FluidNet residual never changes, since the method is not iterative; FluidNet reduces the", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "initial residual by no more than one order of magnitude. On the other hand, with DCDM, we can", + "type": "text" + } + ], + "index": 69 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "continually reduce the residual (e.g., by four orders of magnitude) as we apply more iterations of", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "our method, just as with classical CG. In Figure 4b, we also visualize the convergence of three other", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "classical methods, CG, Deflated PCG (Saad et al., 2000), and incomplete Cholesky preconditioned", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 105, + 707, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 506, + 724 + ], + "score": 1.0, + "content": "CG (ICPCG)); clearly, DCDM reduces the residual in the fewest number of iterations (e.g.,", + "type": "text" + } + ], + "index": 73 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "approximately one order of magnitude fewer iterations than ICPCG). 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(a) Mean and std. dev. (over all 400 frames in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 236, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 104, + 236, + 357, + 250 + ], + "score": 1.0, + "content": "the simulation) of residual reduction during linear solves (with", + "type": "text" + }, + { + "bbox": [ + 357, + 237, + 378, + 248 + ], + "score": 0.88, + "content": "1 2 8 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 236, + 396, + 250 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 396, + 237, + 416, + 248 + ], + "score": 0.88, + "content": "2 5 6 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 236, + 506, + 250 + ], + "score": 1.0, + "content": "grids) using FluidNet", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "score": 1.0, + "content": "(FN) and DCDM. (b) Residual plots with CG, ICPCG, Deflated PCG, FN, and DCDM at frame 150.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 284, + 271 + ], + "score": 1.0, + "content": "Dashed and solid lines represent results for", + "type": "text" + }, + { + "bbox": [ + 284, + 259, + 305, + 270 + ], + "score": 0.88, + "content": "1 2 8 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 258, + 324, + 271 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 324, + 259, + 344, + 270 + ], + "score": 0.86, + "content": "2 5 6 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 258, + 505, + 271 + ], + "score": 1.0, + "content": ", respectively. (c) Decrease in residuals", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 269, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 104, + 269, + 208, + 283 + ], + "score": 1.0, + "content": "with varying degrees of", + "type": "text" + }, + { + "bbox": [ + 209, + 271, + 218, + 280 + ], + "score": 0.65, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 269, + 296, + 283 + ], + "score": 1.0, + "content": "-orthogonalization", + "type": "text" + }, + { + "bbox": [ + 296, + 271, + 344, + 281 + ], + "score": 0.88, + "content": "( i _ { s } ~ = ~ i _ { \\mathrm { s t a r t } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 269, + 506, + 283 + ], + "score": 1.0, + "content": ". (d) Reduction in residuals when the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 281, + 499, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 208, + 293 + ], + "score": 1.0, + "content": "network is trained with a", + "type": "text" + }, + { + "bbox": [ + 208, + 281, + 224, + 291 + ], + "score": 0.87, + "content": "6 4 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 281, + 236, + 293 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 236, + 281, + 256, + 292 + ], + "score": 0.88, + "content": "1 \\bar { 2 } 8 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 281, + 305, + 293 + ], + "score": 1.0, + "content": "grid for the", + "type": "text" + }, + { + "bbox": [ + 305, + 281, + 326, + 292 + ], + "score": 0.86, + "content": "2 5 6 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 281, + 499, + 293 + ], + "score": 1.0, + "content": "grid simulation shown in Figure 3 Middle.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 309, + 504, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 382, + 321 + ], + "score": 1.0, + "content": "iterative method and lacks a notion of residual reduction, we treat", + "type": "text" + }, + { + "bbox": [ + 383, + 311, + 393, + 320 + ], + "score": 0.84, + "content": "r _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "for FluidNet as though an", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 320, + 321, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 321, + 332 + ], + "score": 1.0, + "content": "initial guess of zero is used (as is done in our solver).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 337, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 350 + ], + "score": 1.0, + "content": "To clarify these results, Table 1 reports convergence statistics for DCDM compared to standard iter-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 347, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 297, + 362 + ], + "score": 1.0, + "content": "ative techniques CG and Deflated PCG. For all", + "type": "text" + }, + { + "bbox": [ + 298, + 348, + 313, + 359 + ], + "score": 0.83, + "content": "6 4 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 347, + 317, + 362 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 317, + 348, + 338, + 359 + ], + "score": 0.84, + "content": "1 2 8 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 347, + 359, + 362 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 359, + 348, + 380, + 359 + ], + "score": 0.87, + "content": "2 5 6 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 347, + 505, + 362 + ], + "score": 1.0, + "content": "grids with the bunny example,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 194, + 371 + ], + "score": 1.0, + "content": "we measure the time", + "type": "text" + }, + { + "bbox": [ + 194, + 360, + 204, + 370 + ], + "score": 0.87, + "content": "t _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 358, + 324, + 371 + ], + "score": 1.0, + "content": "and the number of iterations", + "type": "text" + }, + { + "bbox": [ + 325, + 361, + 336, + 370 + ], + "score": 0.86, + "content": "n _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "required to reduce the initial residual on", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "a particular time step of the simulation by four orders of magnitude. DCDM achieves the desired", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 379, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 104, + 379, + 392, + 395 + ], + "score": 1.0, + "content": "results in by far the fewest number of iterations at all resolutions. At", + "type": "text" + }, + { + "bbox": [ + 392, + 380, + 413, + 392 + ], + "score": 0.85, + "content": "2 5 6 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 379, + 505, + 395 + ], + "score": 1.0, + "content": ", DCDM performs ap-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 393, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 405 + ], + "score": 1.0, + "content": "proximately 6 times faster than CG, suggesting a potentially even wider performance advantage at", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "higher resolutions. Inference is the dominant cost in an iteration of DCDM; the other linear algebra", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 415, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 425 + ], + "score": 1.0, + "content": "computations in an iteration of DCDM are comparable to those in CG. The nice result of our method", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "is that despite the increased time per iteration, the number of required iterations is reduced so drasti-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 437, + 504, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 504, + 448 + ], + "score": 1.0, + "content": "cally that DCDM materially outperforms classical methods like CG. Although ICPCG successfully", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 445, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 445, + 224, + 460 + ], + "score": 1.0, + "content": "reduces number of iterations", + "type": "text" + }, + { + "bbox": [ + 225, + 447, + 239, + 457 + ], + "score": 0.56, + "content": "^ { 4 \\mathrm { ~ b ~ } }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 445, + 506, + 460 + ], + "score": 1.0, + "content": ", we found the runtime to scale prohibitively with grid resolution,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "so we exclude it from comparison in table 1. Notably, even though Deflated PCG and DCDM are", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "based on approximate Ritz vectors, DCDM performs far better, indicating the value of using a neural", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 480, + 144, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 144, + 491 + ], + "score": 1.0, + "content": "network.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 108, + 518, + 201, + 531 + ], + "lines": [ + { + "bbox": [ + 104, + 516, + 203, + 534 + ], + "spans": [ + { + "bbox": [ + 104, + 516, + 203, + 534 + ], + "score": 1.0, + "content": "7 CONCLUSIONS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "We presented DCDM, incorporating CNNs into a CG-style algorithm that yields efficient, conver-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "gent behavior for solving linear systems. Our method is evaluated on linear systems with over 16", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "million degrees of freedom and converges to a desired tolerance in merely tens of iterations. Fur-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "thermore, despite training the underlying network on domains without obstacles, our network is able", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "to successfully predict search directions that enable efficient linear solves on domains with complex", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "and dynamic geometries. Moreover, the training data for our network does not require running fluid", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "simulations or solving linear systems ahead of time; our Rayleigh-Ritz vector approach enables us", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 627, + 455, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 455, + 640 + ], + "score": 1.0, + "content": "to quickly generate very large training datasets, unlike approaches seen in other works.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "Our network was designed for and trained exclusively using data related to the discrete Poisson", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "matrix, which likely limits the generalizability of our present model. However, we believe our", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "method is readily applicable to other classes of PDEs (or general problems with graph structure)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "that give rise to large, sparse, symmetric linear systems. 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DCDM achieves the desired", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 379, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 104, + 379, + 392, + 395 + ], + "score": 1.0, + "content": "results in by far the fewest number of iterations at all resolutions. At", + "type": "text" + }, + { + "bbox": [ + 392, + 380, + 413, + 392 + ], + "score": 0.85, + "content": "2 5 6 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 379, + 505, + 395 + ], + "score": 1.0, + "content": ", DCDM performs ap-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 393, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 405 + ], + "score": 1.0, + "content": "proximately 6 times faster than CG, suggesting a potentially even wider performance advantage at", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "higher resolutions. Inference is the dominant cost in an iteration of DCDM; the other linear algebra", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 415, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 425 + ], + "score": 1.0, + "content": "computations in an iteration of DCDM are comparable to those in CG. The nice result of our method", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "is that despite the increased time per iteration, the number of required iterations is reduced so drasti-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 437, + 504, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 504, + 448 + ], + "score": 1.0, + "content": "cally that DCDM materially outperforms classical methods like CG. 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Our method is evaluated on linear systems with over 16", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "million degrees of freedom and converges to a desired tolerance in merely tens of iterations. Fur-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "thermore, despite training the underlying network on domains without obstacles, our network is able", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "to successfully predict search directions that enable efficient linear solves on domains with complex", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "and dynamic geometries. 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However, we believe our", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "method is readily applicable to other classes of PDEs (or general problems with graph structure)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "that give rise to large, sparse, symmetric linear systems. We note that our method is unlikely to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 379, + 700 + ], + "score": 1.0, + "content": "work well for matrices that have high computational cost to evaluate", + "type": "text" + }, + { + "bbox": [ + 379, + 688, + 403, + 698 + ], + "score": 0.89, + "content": "A * x", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "(such as dense matrices),", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 235, + 712 + ], + "score": 1.0, + "content": "since training relies on efficient", + "type": "text" + }, + { + "bbox": [ + 236, + 699, + 261, + 709 + ], + "score": 0.89, + "content": "\\boldsymbol { A } * \\boldsymbol { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "evaluations. An interesting question to consider is how well", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 711, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "our method and current models would apply to discrete Poisson matrices arising from non-uniform", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 721, + 324, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 324, + 733 + ], + "score": 1.0, + "content": "grids, e.g., quadtrees or octrees (Losasso et al., 2004).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 644, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 103, + 61, + 506, + 737 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 178, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 178, + 96 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 98, + 506, + 114 + ], + "spans": [ + { + "bbox": [ + 104, + 98, + 506, + 114 + ], + "score": 1.0, + "content": "M. 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OptimizerTaskDataModelMetrictTimeMem saved
32-bit AdamWGLUEMultipleRoBERTa-Large88.91Reference
32-bit AdamWGLUEMultipleRoBERTa-Large88.617h0.0 GB
32-bit AdafactorGLUEMultipleRoBERTa-Large88.724h1.3 GB
8-bit AdamWGLUEMultipleRoBERTa-Large88.715h2.0 GB
32-bit MomentumCLSImageNet-1kResNet-5077.11Reference
32-bit MomentumCLSImageNet-1kResNet-5077.1118h0.0 GB
8-bit MomentumCLSImageNet-1kResNet-5077.2116 h0.1 GB
32-bit AdamMTWMT'14+16Transformer29.3Reference
32-bit AdamMTWMT'14+16Transformer29.0126h0.0 GB
32-bit AdafactorMTWMT'14+16Transformer29.0127h0.3 GB
8-bit AdamMTWMT'14+16Transformer29.1115h1.1 GB
32-bit MomentumMoCo v2ImageNet-1kResNet-5067.51Reference
32-bit MomentumMoCo v2ImageNet-1kResNet-5067.330 days0.0 GB
8-bit MomentumMoCo v2ImageNet-1kResNet-5067.428 days0.1 GB
32-bit AdamLMMultipleTransformer-1.5B9.0308 days0.0 GB
32-bit AdafactorLMMultipleTransformer-1.5B8.9316days5.6 GB
8-bit AdamLMMultipleTransformer-1.5B9.0297 days8.5 GB
32-bit AdamLMMultipleGPT3-Medium10.62795 days0.0 GB
32-bit AdafactorLMMultipleGPT3-Medium10.68816 days1.5 GB
8-bit AdamLMMultipleGPT3-Medium10.62761 days1.7 GB
32-bit AdamMasked-LMMultipleRoBERTa-Base3.49101 days0.0 GB
32-bit AdafactorMasked-LMMultipleRoBERTa-Base3.59112 days0.7 GB
8-bit AdamMasked-LMMultipleRoBERTa-Base3.4894 days1.1 GB
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GPU size in GBLargest finetunable Model (parameters)
32-bit Adam8-bit Adam
6RoBERTa-base (110M)RoBERTa-large (355M)
11MT5-small (300M)MT5-base (580M)
24MT5-base (580M)MT5-large (1.2B)
24GPT-2-medium (762M)GPT-2-large (1.5B)
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8-bit Adam1018.6
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8-bit Adam√√√016.4
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1.3B8-bit Adam√100N/A
1.3B8-bit Adam√√8010.9
1.5B32-bit Adam09.0
1.5B8-bit Adam√一√09.0
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ModelMNLIQNLIQQPRTESST-2MRPCCoLASTS-BMean
32-bit Adam90.4094.8592.284.596.4090.167.4193.0388.61
32-bit Adafactor90.3594.7092.285.496.4590.067.6392.9188.71
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MethodRelative Adam ErrorAbsolute Quantization Error
Linear201% ±17%41.2e-10±3.1e-10
Quantile11.9% ± 0.3%8.8e-10±0.9e-10
Inverse Dynamic6.5%± 0.1%4.6e-10±0.4e-10
Dynamic4.8%± 0.4%3.5e-10±1.1e-10
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Constraints + +Anonymous Author(s) + +# Abstract + +1 Many real-world settings involve costs for performing actions; transaction costs +2 in financial systems and fuel costs being common examples. In these settings, +3 performing actions at each time step quickly accumulates costs leading to vastly +4 suboptimal outcomes. Additionally, repeatedly acting produces wear and tear and +5 ultimately, damage. Determining when to act is crucial for achieving successful +6 outcomes and yet, the challenge of efficiently learning to behave optimally when +7 actions incur minimally bounded costs remains unresolved. In this paper, we intro +8 duce a reinforcement learning (RL) framework named Learnable Impulse Control +9 Reinforcement Algorithm (LICRA), for learning to optimally select both when +10 to act and which actions to take when actions incur costs. At the core of LICRA +11 is a nested structure that combines RL and a form of policy known as impulse +12 control which learns to maximise objectives when actions incur costs. We prove +13 that LICRA, which seamlessly adopts any RL method, converges to policies that +14 optimally select when to perform actions and their optimal magnitudes. We then +15 augment LICRA to handle problems in which the agent can perform at most $k < \infty$ +16 actions and more generally, faces a budget constraint. We show LICRA learns the +17 optimal value function and ensures budget constraints are satisfied almost surely. +18 We demonstrate empirically LICRA’s superior performance against benchmark +19 RL methods in OpenAI gym’s Lunar Lander and in Highway environments and a +20 variant of the Merton portfolio problem within finance. + +# 21 1 Introduction + +22 There are many settings in which agents incur costs each time they perform an action. Transaction +23 costs in financial settings [19], fuel expenditure [32], toxicity as a side effect of controlling bacte +24 ria [29] and physical damage produced by repeated action that produces wear and tear are just a +25 few among many examples [13]. In these settings, performing actions at each time step is vastly +26 suboptimal since acting in this way results in prohibitively high costs and undermines the service life +27 of machinery. Minimising wear and tear is an essential attribute to safeguard against failures that can +28 result in catastrophic losses [13]. +29 Reinforcement learning (RL) is a framework that enables autonomous agents to learn complex +30 behaviours from interactions with the environment [30, 11]. Within the standard RL paradigm, +31 determining optimal actions involves making a selection from among many (possibly infinite) actions; +32 a procedure that must be performed at each time-step as the agent decides on an action. In unknown +33 settings, the agent cannot immediately exploit any topological structure of the action set (if any +34 exists). Consequently, learning not to take an action i.e performing a zero or null action, involves +35 expensive optimisation procedures over the entire action set. Since this must be done at each state, +36 this process is vastly inefficient for learning optimal policies when the agent incurs costs for acting. +37 In this paper, we tackle this problem by developing an RL framework for finding both an optimal +38 criterion to determine whether or not to execute actions as well as learning optimal actions. A key +39 component of our framework is a novel combination of RL with a form of policy known as impulse +40 control [22, 19]. This enables the agent to determine the appropriate points to perform an action as +41 well as the optimal action itself. Despite its fundamental importance as a tool for tackling decision +42 problems with costly actions, presently, the use of impulse control within learning contexts (and +43 unknown environments) is unaddressed. +44 We present an RL impulse control framework called LICRA, which, to our knowledge, is the first +45 learning framework for impulse control. To enable learning optimal impulse control policies in +46 unknown environments, we devise a framework that consists of separate RL components for learning +47 when to act and how to act optimally. The resulting framework is a structured two-part learning +48 process which differs from current RL protocols. In LICRA, at each time step, the agent firstly makes +49 a decision whether to act or not leading to a binary decision space $\{ 0 , 1 \}$ (we later show that this +50 is determined by evaluating an easy-to-evaluate criterion which has the value function as its input). +51 The second decision part determines the best action to take. This generates a subdivision of the state +52 space into two regions; one in which the agent performs actions and another in which it does not act +53 at all. This is extremely useful since the agent quickly determines the set of states to not take actions +54 while performing actions only at the subset of states where actions are to be executed. +55 We then establish theory that ensures convergence of LICRA to an optimal policy for such settings. +56 To do this, we give a series of results namely: +57 i) We establish a dynamic programming principle (DPP) for impulse control and show that the optimal +58 value function can be obtained as a limit of a value iterative procedure (Theorem 1) which lays the +59 foundation for an RL approach to impulse control. +60 ii) We extend result i) to a new variant of Q learning which enables the impulse control problem to be +61 solved using our RL method (Theorem 2). +62 iii) We characterise the optimal conditions for performing an action which we reveal to be a simple +63 ‘obstacle condition’ involving the agent’s value function (Prop. 1). Using this, the agent can quickly +64 determine whether or not it should act and if so, then learn what the optimal action is. +65 iv) We then extend the result i) to (linear) function approximators enabling the value function to be +66 parameterised (Theorem 3). +67 iv) In Sec. 6, we extend LICRA to include budgetary constraints so that each action draws from a +68 fixed budget which the agent must stay within. Analogous to the development of i), we establish +69 another DPP from which we derive a Q-learning variant for tackling impulse control with budgetary +70 constraints (Theorem 4). A particular case of a budget constraint is when the number of actions the +71 agent can take over the horizon is capped. +72 Lastly, we perform a set of experiments to validate our theory within the Highway driving simulator +73 and OpenAI’s LunarLander [7]. +74 LICRA confers a series of advantages. As we demonstrate in our experiments, LICRA learns to +75 compute the optimal problems in which the agent faces costs for acting in an efficient way which +76 outperforms leading RL baselines. Second, as demonstrated in Sec. 6 LICRA handles settings in +77 which the agent has a cap the total number of actions it is allowed to execute and more generally, +78 generic budgetary constraints. LICRA is able to accommodate any RL base algorithm unlike various +79 RL methods designed to handle budgetary constraints. + +# 80 2 Related Work + +81 In continuous-time optimal control theory [24], problems in which the agent faces a cost for each +82 action are tackled with a form of policy known as impulse control [22, 19, 2]. In impulse control +83 frameworks, the dynamics of the system are modified through a sequence of discrete actions or bursts +84 chosen at times that the agent chooses to apply the control policy. This distinguishes impulse control +85 models from classical decision methods in which an agent takes actions at each time step while being +86 tasked with the decision of only which action to take. Impulse control models represent appropriate +87 modelling frameworks for financial environments with transaction costs, liquidity risks and economic +88 environments in which players face fixed adjustment costs (e.g. menu costs) [16, 20]. +89 The current setting is intimately related to the optimal stopping problem which widely occurs in +90 finance, economics and computer science [23, 31]. In the optimal stopping problem, the task is to +91 determine a criterion that determines when to arrest the system and receive a terminal reward. In this +92 case, standard RL methods are unsuitable since they require an expensive sweep (through the set +93 of states) to determine the optimal point to arrest the system. The current problem can be viewed +94 as an augmented problem of optimal stopping since the agent must now determine both a sequence +95 of points to perform an action or intervene and their optimal magnitudes — only acting when the +96 cost of action is justified [25]. Adapting RL to tackle optimal stopping problems has been widely +97 studied [31, 4, 9] and applied to a variety of real-world settings within finance [12], radiation therapy +98 [1] and network operating systems [3]. Our work serves as a natural extenstion to RL approaches to +99 optimal stopping to the case in which the agent must decide at which points to take many actions. As +100 with optimal stopping, standard RL methods cannot efficiently tackle this problem since determining +101 whether to perform a 0 action requires a costly sweep through the action space at every state [31]. In +102 [26] the authors introduce “sparse action” with a similar motivation as impulse control. However, +103 the authors treat only the discrete action space case. The authors in [26] do not discuss a broader +104 theoretical framework of dealing with “sparse actions”, and develop purely algorithmic solutions. +105 Additionally, unlike the approach taken in [26], the problem setting we consider is one in which the +106 agent faces a cost for each action - the produces a need for the agent to be selective about where it +107 performs actions (but does not necessarily constrain the magnitude or choice of those actions). + +# 108 3 Preliminaries + +109 Reinforcement Learning (RL). In RL, an agent sequentially selects actions to maximise its expected +110 returns. The underlying problem is typically formalised as an MDP $\langle S , \mathcal { A } , P , R , \gamma \rangle$ where $S \subset \mathbb { R } ^ { p }$ +111 is the set of states, $\mathcal { A } \subset \mathbb { R } ^ { k }$ is the set of actions, $P : \mathcal { S } \times \mathcal { A } \times \mathcal { S } [ 0 , 1 ]$ is a transition probability +112 function describing the system’s dynamics, $R : S \times \mathcal { A } \mathbb { R }$ is the reward function measuring the +113 agent’s performance and the factor $\gamma \in [ 0 , 1 )$ specifies the degree to which the agent’s rewards are +114 discounted over time [30]. At time $t \in { 0 , 1 , \ldots }$ , the system is in state $s _ { t } \in S$ and the agent must +115 choose an action $a _ { t } \in \mathcal A$ which transitions the system to a new state $s _ { t + 1 } \sim P ( \cdot | s _ { t } , a _ { t } )$ and produces +116 a reward $R ( s _ { t } , a _ { t } )$ . A policy $\pi : S \times A \to [ 0 , 1 ]$ is a probability distribution over state-action pairs +117 where $\pi ( a | s )$ represents the probability of selecting action $a \in { \mathcal { A } }$ in state $s \in S$ . The goal of an +118 RL agent is to find a policy ${ \hat { \pi } } \in \Pi$ that maximises its expected returns given by the value function: +119 $\begin{array} { r } { v ^ { \pi } ( s ) = \mathbb { E } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \bar { R } ( s _ { t } , a _ { t } ) | a _ { t } \sim \pi ( \cdot | s _ { t } ) , s _ { 0 } = s ] } \end{array}$ $\begin{array} { r } { Q ( s , a ) = \mathbb { E } [ \sum _ { t = 0 } ^ { \infty } R ( s _ { t } , a _ { t } ) | a _ { 0 } = a , s _ { 0 } = s ] } \end{array}$ $\Pi$ ent’s policy set. The action. +121 We consider a setting in which the agent faces at least some minimal cost for each action it performs. +122 With this, the agent’s task is to maximise: + +$$ +v ^ { \pi } ( s ) = \mathbb { E } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \left\{ \mathcal { R } ( s _ { t } , a _ { t } ) - \mathcal { C } ( s _ { t } , a _ { t } ) \right\} \Big | s _ { 0 } = s \right] , +$$ + +123 where for any state $s \in S$ and any action $a \in { \mathcal { A } }$ , the functions $\mathcal { R }$ and $\mathcal { C }$ are given by $\textstyle { \mathcal { R } } ( s , a ) =$ +124 $R ( s , a ) \mathbf { 1 } _ { a \in \mathcal { A } } + R ( s , 0 ) ( 1 - \mathbf { 1 } _ { a \in \mathcal { A } } )$ where $\mathbf { 1 } _ { a \in \mathcal { A } }$ is the indicator function which is 1 when $a \in { \mathcal { A } }$ and 0 +125 otherwise and ${ \mathcal { C } } ( s , a ) : = c ( s , a ) \mathbf { 1 } _ { a \in { \mathcal { A } } }$ where $c : S \times \mathcal { A } \mathbb { R }$ is a minimally bounded (cost) function1 +126 that introduces a cost each time the agent performs an action. Examples of the cost function is a +127 quasi-linear function of the form $c ( s _ { t } , \bar { a } _ { t } ) \stackrel { } { = } \kappa + f ( a _ { t } )$ where $f : \mathcal { A } \mathbb { R } _ { > 0 }$ and $\kappa$ is a positive real +128 valued constant. Since acting at each time step would incur prohibitively high costs, the agent must +129 be selective when to perform an action. Therefore, in this setting, the agent’s problem is augmented to +130 learning both an optimal policy for its actions and, learning at which states to apply its action policy. +131 Example: Merton Portfolio Problem with Transaction Costs [10]. An investor performs a series +132 of costly portfolio adjustments by buying and selling amounts of different assets within their portfolio. +133 Each investment incurs a fixed minimal cost (also known as transaction costs) which is deducted +134 from the investor’s available cash-flow. The investor’s aim is to maximise their total wealth (the value +135 of the sum of their assets) at some time horizon by adjusting their portfolio of investments. Problems +136 of this kind, portfolio investment problems are of fundamental importance within finance [18]. +37 Example 2. An autonomous vehicle must perform a series of actions to perform a task. Each action +38 draws from its fuel budget. In order to complete its task successfully, during the task, the vehicle +139 must ensure it maintains an available supply. +141 In RL, the agent’s problem involves learning to act at every state including those in which actions do +142 not significantly impact on its total return. While we can add a zero action to the action set $\mathcal { A }$ and +143 apply standard methods, we argue that this may not be the best solution in many situations. We argue +144 the optimal policy has the following form: + +$$ +\begin{array} { r } { \widetilde { \pi } ( \cdot | s ) = \left\{ \begin{array} { l l } { a _ { t } } & { s \in S _ { I } , } \\ { 0 } & { s \notin S _ { I } , } \end{array} \right. } \end{array} +$$ + +145 which implies that we simplify policy learning by determining the set $ { \boldsymbol { S } } _ { I }$ first — the set where we +146 actually need to learn the policy. +147 We now introduce a learning method for producing impulse controls. This enables the agent to learn +148 to select states to perform actions. Therefore, now agent is tasked with learning to act at states that +149 are most important for maximising its total return given the presence of the cost for each action. Now +150 at each state the agent first makes a binary decision to decide to perform an action. +151 Our framework, LICRA consists of two core components: firstly a RL process $\mathfrak { g } : \mathcal { S } \times \{ 0 , 1 \} [ 0 , 1 ]$ +152 and a second RL process $\pi : S \times A \to [ 0 , 1 ]$ . The role of $\mathfrak { g }$ is to determine whether or not an action +153 is to be performed by the policy $\pi$ at a given state $s$ . If activated, the policy $\pi$ determines the action +154 to be selected. Prior to decisions being made, the policy $\pi$ communicates to $\mathfrak { g }$ the action it would +155 take. An important feature of our LICRA is the sequential decision process. In LICRA, the policy +156 $\pi$ first proposes an action $a \in { \mathcal { A } }$ which is observed by the policy $\mathfrak { g }$ . Therefore, the role of $\mathfrak { g }$ is to +157 prevent actions for which the change in expected future rewards does not exceed the costs incurred +158 for taking such actions. By isolating a decision policy over whether an action should be taken or not, +159 the impulse controls mechanism results in a framework in which the problem facing the agent has a +160 markedly reduced decision space (in comparison to a standard RL method). Crucially, the agent must +161 compute optimal actions at only a subset of states which are chosen by the policy g. Below is the +162 pseudocode for LICRA, we provide full details of the code in Sec. 9 of the Appendix. + +# Algorithm 1: Learnable Impulse Control Reinforcement Algorithm (LICRA) + +1: Input: Stepsize $\alpha$ , batch size $B$ , episodes $K$ , steps per episode $T$ , mini-epochs $e$ +2: Initialise: Policy network (acting) $\pi$ , Policy network (switching) $\mathfrak { g }$ , +Critic network (acting $) V _ { \pi }$ ,Critic network (switching $) V _ { { \mathfrak { g } } }$ +3: Given reward objective function, $R$ , initialise Rollout Buffers $B _ { \pi }$ , $B _ { \mathfrak { g } }$ (use Replay Buffer for +SAC) +4: for $N _ { e p i s o d e s }$ do +5: Reset state $s _ { 0 }$ , Reset Rollout Buffers $B _ { \pi }$ , $B _ { \mathfrak { g } }$ +6: for $t = 0 , 1 , \ldots$ do +7: Sample $a _ { t } \sim \pi ( \cdot | s _ { t } )$ +8: Sample $g _ { t } \sim \mathfrak { g } ( \cdot | s _ { t } )$ +9: if $g _ { t } = 0$ then +10: Apply $a _ { t }$ so $s _ { t + 1 } \sim P ( \cdot | a _ { t } , s _ { t } ) .$ , +11: Receive rewards $r _ { t } = \mathcal { R } ( s _ { t } , a _ { t } )$ +12: Store $( s _ { t } , a _ { t } , s _ { t + 1 } , r _ { t } )$ in $B _ { \pi }$ +13: else +14: Apply the null action so $s _ { t + 1 } \sim P ( \cdot | 0 , s _ { t } )$ , +15: Receive rewards $r _ { t } = \mathcal { R } ( s _ { t } , 0 )$ . +16: end if +17: Store $( s _ { t } , g _ { t } , s _ { t + 1 } , r _ { t } )$ in $B _ { \mathfrak { g } }$ +18: end for +19: // Learn the individual policies +20: Update policy $\pi$ and critic $V _ { \pi }$ networks using $\mathfrak { B } _ { \pi }$ +21: Update policy $\mathfrak { g }$ and critic $V _ { \mathfrak { g } }$ networks using ${ \mathfrak { B } } _ { { \mathfrak { g } } }$ +22: end for +165 Although action space cardinality does not change there are still benefits of using impulse control +166 mechanism. This mechanism forces the agent to first determine the set of states to perform actions +167 only then determine the optimal actions at these states. An important fact to note is that the decision +168 space for the determining whether or not to execute an action is $S \times \{ 0 , 1 \}$ i.e at each state it makes a +169 binary decision. Consequently, the learning process for aspect is much quicker than a policy which +170 must optimise over a decision space which is $| { \cal S } | | { \cal A } |$ (choosing an action from its action space at +171 every state). This results in the agent rapidly learning which states to focus on to learn which actions +172 to perform. In the case of $\pi$ with a continuous action space again the impulse control mechanism +173 does not change the cardinality of the action space. However, if the set ${ \cal S } / { \cal S } _ { I }$ , where the optimal +174 policy chooses 0, is large enough, then again it can be more efficient to learn $\mathfrak { g }$ first and only then +175 learn $\pi$ (we later validate this claim empirically, see Sec. 11.2), +176 In Sec. 5, we prove the convergence properties of LICRA. LICRA consists of two independent +177 procedures: a learning process for the policy $\pi$ and simultaneously, a learning process for the impulse +178 policy $\mathfrak { g }$ which determines at which states to perform an action. In our implementation, we used +179 proximal policy optimisation (PPO) [27] for the policy $\pi$ and for the impulse policy $\mathfrak { g }$ , whose action +180 set consists of two actions (intervene or do not intervene) we used a soft actor critic (SAC) process +181 [14] LICRA is a plug $\&$ play framework which enables these RL components to be replaced with any +182 RL algorithm of choice. + +# 183 5 Convergence and Optimality of LICRA + +184 A key aspect of our framework is the presence of two RL processes that make decisions in a sequential +185 order. In order to determine when to act the policy $\mathfrak { g }$ must learn the states to allow the policy $\pi$ to +186 perform an action which the policy $\pi$ must learn to select optimal actions whenever it is allowed to +187 execute an action. +188 In this section, we prove that LICRA converges to an optimal solution of the system. Central to +189 LICRA is a Q-learning type method which is adapted to handle RL settings in which the agent must +190 also learn when to act. We then extend the result to allow for (linear) function approximators. We +191 provide a result that shows the optimal intervention times are characterised by an ‘obstacle condition’ +192 which can be evaluated online therefore allowing the $\mathfrak { g }$ . + +Given a function $Q \because S \times \mathcal { A } \to \mathbb { R } , \quad \forall \pi , \pi ^ { \prime } \in \Pi$ and $\forall \ S , \ S ^ { \prime } , \quad \forall s _ { \tau _ { k } } \quad \in \quad S$ , we define the intervention operator $\mathcal { M } ^ { \pi , { \mathfrak { g } } }$ by $\begin{array} { r } { \mathcal { M } ( ^ { \pi , \oplus } Q ^ { \pi ^ { \prime } , \oplus ^ { \prime } } ( s _ { \pi _ { k } } , a _ { \pi _ { k } } ) : = { \mathcal R } ( s _ { \pi _ { k } } , a _ { \pi _ { k } } ) - c ( s _ { \pi _ { k } } , a _ { \pi _ { k } } ) + } \end{array}$ $\begin{array} { r } { \gamma \sum _ { s ^ { \prime } \in S } P ( s ^ { \prime } ; a _ { \tau _ { k } } , s ) v ^ { \pi ^ { \prime } , \mathfrak { g ^ { \prime } } } ( s ^ { \prime } ) \| a _ { \tau _ { k } } \sim \pi ( \cdot | s _ { \tau _ { k } } ) , } \end{array}$ , where $\tau _ { k }$ is an intervention time. + +196 The interpretation of $\mathcal { M }$ is the following: suppose that the agent is using the policy $\pi$ and at time $\tau _ { k }$ +197 the system is at a state $s _ { \tau _ { k } }$ and the agent performs an action $a _ { \tau _ { k } } \sim \pi ( \cdot | s _ { \tau _ { k } } )$ . A cost of $c ( s _ { \tau _ { k } } , a _ { \tau _ { k } } )$ +198 is then incurred by the agent and the system transitions to $s ^ { \prime } \sim P ( \cdot ; a _ { \tau _ { k } } , s _ { \tau _ { k } } )$ . Lastly, recall $v ^ { \pi , { \mathfrak { g } } }$ is +199 the agent value function under the policy pair $( \pi , g )$ . Therefore, the quantity ${ \mathcal { M } } Q ^ { \pi , { \mathfrak { g } } }$ measures the +200 expected future stream of rewards after an immediate intervention minus the cost of intervention. +201 This object plays a crucial role in the LICRA framework which as we later discuss, exploits the cost +202 structure of the problem to determine when the agent should perform an intervention. + +Given a function 203 $v ^ { n , \ell } : S \mathbb { \quad }$ , we define the Bellman operator $\boldsymbol { \cdot }$ , by: + +204 + +$$ +T v ^ { \pi , \theta } ( s ) : = \operatorname* { m a x } \Big \{ \mathcal { N } ^ { ( \pi , \theta ) } Q ^ { \pi , \theta } ( s , a ) , \mathcal { R } ( s , 0 ) + \gamma \sum _ { s ^ { \prime } \in S } P ( s ^ { \prime } ; 0 , s ) v ^ { \pi , \theta } ( s ^ { \prime } ) \Big \} , \qquad \forall s \in \mathcal { S } . +$$ + +205 + +206 The Bellman operator captures the nested sequential structure of the LICRA algorithm. In particular, +207 the structure in (3) consists of an inner structure which consists of two terms: the first term is the +208 expected future return given an action is taken at the current state under the policy $\pi$ . The second term +209 is the expected future return given no action is taken at the current state. Lastly, the outer structure is +210 an optimisation which compares the expected return of the two possibilities and selects the maximum. + +11 Our first result proves $T$ is a contraction operator in particular, the following bound holds: + +212 Lemma 1 The Bellman operator $T$ is a contraction, that is the following bound holds: + +$$ +\begin{array} { r } { \| T v - T v ^ { \prime } \| \leq \gamma \left. v - v ^ { \prime } \right. , } \end{array} +$$ + +where 13 $v , v ^ { \prime }$ are elements of a finite normed vector space. We can now state our first main result: + +Theorem 1 Given any $v ^ { \pi , \mathfrak { g } } : \mathcal { S } \times \mathcal { A } \mathbb { R } ,$ , the optimal value function is given by $\operatorname* { l i m } _ { k \to \infty } T ^ { k } v ^ { \pi , \mathfrak { g } } =$ $\operatorname* { m a x } _ { \hat { \pi } , \hat { g } \in \Pi } v ^ { \hat { \pi } , \hat { g } } = v ^ { \pi ^ { \star } , g ^ { \star } }$ where $( \pi ^ { \star } , g ^ { \star } )$ is the optimal policy pair. + +The result of Theorem 1 enables the solution to the agent’s impulse control problem to be determined using a value iteration procedure. Moreover, Theorem 1 enables a Q-learning approach [6] for finding the solution to the agent’s problem. + +219 Theorem 2 Consider the following $Q$ learning variant: + +$$ ++ \alpha _ { t } ( s _ { t } , a _ { t } ) \left[ \operatorname* { m a x } \left\{ \mathcal { M } ^ { \pi , \mathfrak { g } } Q _ { t } ( s _ { t } , a _ { t } ) , \mathcal { R } ( s _ { t } , 0 ) + \gamma _ { a ^ { \prime } \in A } ^ { \operatorname* { m a x } } Q _ { t } ( s _ { t + 1 } , a ^ { \prime } ) \right\} - Q _ { t } ( s _ { t } , a _ { t } ) \right] , +$$ + +then 220 $Q _ { t }$ converges to $Q ^ { \star }$ with probability 1, where $s _ { t } , s _ { t + 1 } \in S$ and $a _ { t } \in \mathcal A$ . + +21 We now extend the result to (linear) function approximators: + +Theorem 3 Given a set of linearly independent basis functions $\Phi = \{ \phi _ { 1 } , \ldots , \phi _ { p } \}$ with $\phi _ { k } \in L _ { 2 } , \forall k$ . LICRA converges to a limit point $r ^ { \star } \in \mathbb { R } ^ { p }$ which is the unique solution to $\Pi \mathfrak { F } ( \Phi r ^ { \star } ) = \Phi r ^ { \star }$ where $\mathfrak { F } v : = \mathcal { R } + \gamma P \operatorname* { m a x } \{ \mathcal { M } v , v \}$ . Moreover, $r ^ { \star }$ satisfies: $\| \Phi r ^ { \star } - Q ^ { \star } \| \leq ( 1 - \gamma ^ { 2 } ) ^ { - 1 / 2 } \| \Pi Q ^ { \star } - Q ^ { \star } \|$ . + +The theorem establishes the convergence of LICRA to a stable point with the use of linear function approximators. The second statement bounds the proximity of the convergence point by the smallest approximation error that can be achieved given the choice of basis functions. + +Having constructed a procedure to find the optimal agent’s optimal value function, we now seek to determine the conditions when an intervention should be performed. Let us denote by $\{ \tau _ { k } \} _ { k \ge 0 }$ the points at which the agent decides to act or intervention times, so for example if the agent chooses to perform an action at state $s _ { 6 }$ and again at state $s _ { 8 }$ , then $\tau _ { 1 } = 6$ and $\tau _ { 2 } = 8$ . The following result characterises the optimal intervention policy $\mathfrak { g }$ and the optimal times $\{ \tau _ { k } \} _ { k \ge 0 }$ . + +Proposition 1 The policy g is given by: $\mathfrak { g } ( s _ { t } ) = H ( \mathcal { M } ^ { \pi , \mathfrak { g } } Q ^ { \pi , \mathfrak { g } } - Q ^ { \pi , \mathfrak { g } } ) ( s _ { t } , a _ { t } ) , \forall s _ { t } \in \mathcal { S } _ { \mathrm { ~ \Gamma ~ } }$ , where $Q ^ { \pi , { \mathfrak { g } } }$ is the solution in Theorem $\cdot$ , $\mathcal { M }$ is the intervention operator and $H$ is the Heaviside function, moreover the intervention times are $\tau _ { k } = \operatorname* { i n f } \{ \tau > \tau _ { k - 1 } | \mathcal { M } ^ { \pi , { \mathfrak { g } } } Q ^ { \pi , { \mathfrak { g } } } = Q ^ { \pi , { \mathfrak { g } } } \}$ . + +Prop. 1 characterises the (categorical) distribution $\mathfrak { g }$ . Moreover, given the function $Q$ , the times $\{ \tau _ { k } \}$ can be determined by evaluating if $\mathcal { M } Q = Q$ holds. + +A key aspect of Prop. 1 is that it exploits the cost structure of the problem to determine when the agent should perform an intervention. In particular, the equality $\mathcal { M } Q = Q$ implies that performing an action and incurring a cost for doing so is optimal. + +# 6 Budget Augmented LICRA via State Augmentation + +We now tackle the problem of RL with a budget. To do this, we combine the above impulse control technology with state augmentation technique proposed in [28] The mathematical formulation of the problem is now given by the following for any $s \in S$ : + +$$ +\operatorname* { m a x } _ { \pi \in \Pi , g } ~ \upsilon ^ { \pi , \mathfrak { g } } ( s ) \ \mathrm { ~ s . ~ t . ~ } n - \sum _ { t = 0 } ^ { \infty } \sum _ { k \geq 1 } \delta _ { \tau _ { k } } ^ { t } \geq 0 , +$$ + +where $n \in \mathbb N$ is a fixed value that represents the maximum number of allowed interventions and $\sum k > 1 8 _ { T k } ^ { t }$ is equal to one if an impulse was applied at time $t$ and zero if it was not. In order to avoid dealing with a constrained MDP, we propose to introduce a new variable $z _ { t }$ tracking the remaining number of impulses: $\begin{array} { r } { z _ { t } = n - \sum _ { i = 0 } ^ { t - 1 } \sum _ { k \geq 1 } \delta _ { \tau _ { k } } ^ { i } } \end{array}$ . We treat $z _ { t }$ as another state and augment the state-space resulting in the transition $\widetilde { \mathcal P }$ : + +$$ +s _ { t + 1 } \sim P ( \cdot | s _ { t } , a _ { t } ) , \qquad z _ { t + 1 } = z _ { t } - \sum _ { k \geq 1 } \delta _ { \tau _ { k } } ^ { t } , \quad z _ { 0 } = n . +$$ + +In order to avoid violations, we reshape the reward as follows: 252 $\begin{array} { r } { \widetilde { \mathcal { R } } ( s _ { t } , z _ { t } , a _ { t } ) = \left\{ \begin{array} { l l } { \mathcal { R } ( s _ { t } , a _ { t } ) } & { z _ { t } \ge 0 } \\ { - \Delta } & { z _ { t } < 0 } \end{array} \right. } \end{array}$ ,, 253 where $\Delta > 0$ is a large enough hyper-parameter ensuring that there are no safety violations. To 254 summarise we aim to solve the following problem: + +$$ +v ^ { \pi , \mathfrak { g } } ( s , z ) = \mathbb { E } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \widetilde { \mathcal { R } } ( s _ { t } , z _ { t } , a _ { t } ) | a _ { t } \sim \pi ( \cdot | s _ { t } , z _ { t } ) \right] , +$$ + +where the policy now depends on the variable $z _ { t }$ . Note that $\widetilde { \mathcal P }$ in Equation 6 is a Markov process and, the rewards $\widetilde { \mathcal { R } }$ are bounded, as long as the rewards $\mathcal { R }$ are bounded. Therefore, we can apply directly the results for impulse control to this case as well. We denote the augmented MDP by $\widetilde { \mathcal { M } } = \langle \overset { \cdot } { S } \times \mathcal { Z } , \overset { \cdot } { A } , \widetilde { \mathcal { P } } , \widetilde { R } , \gamma \rangle$ , where $\mathcal { Z }$ is the space of the augmented state. We have the following. + +Theorem 4 Consider the MDP $\widetilde { \mathcal { M } }$ for the problem 7, then: + +a) The Bellman equation holds, i.e. there exists a function $\tilde { v } ^ { * , \pi , 9 }$ s.th. $\begin{array} { r l } { \tilde { v } ^ { * , \pi , \mathfrak { g } } ( s , z ) } & { { } = } \end{array}$ $\operatorname* { m a x } _ { \mathbf { a } \in \mathcal { A } } \Big ( \widetilde { \mathcal { R } } ( s , z , \pmb { a } ) + \gamma \mathbb { E } _ { s ^ { \prime } , z ^ { \prime } \sim \mathcal { P } } \left[ \widetilde { v } ^ { * , \pi , \mathfrak { g } } ( s ^ { \prime } , z ^ { \prime } ) \right] \Big )$ , where the optimal policy for $\widetilde { \mathcal { M } }$ has the form $\pi ^ { * } ( \cdot | s , z )$ ; + +$b$ ) Given a ${ \widetilde { v } } : { \mathcal { S } } \times { \mathcal { Z } } \to \mathbb { R } ,$ , the stable point solution for $\widetilde { \mathcal { M } }$ is a given by $\operatorname* { l i m } _ { k \infty } \tilde { T } ^ { k } \widetilde { v } ^ { \pi , g } = \operatorname* { m a x } _ { \hat { \pi } \in \Pi , \hat { g } } \hat { \pi } ^ { \hat { \pi } , \hat { g } } =$ $\widetilde { v } ^ { * , \pi , { \mathfrak { g } } ^ { * } }$ , where $( \pi ^ { * } , { \mathfrak { g } } ^ { * } )$ is an optimal policy of $\widetilde { \mathcal { M } }$ and $\tilde { T }$ is the Bellman operator of $\widetilde { \mathcal { M } }$ . + +The result has several important implications. The first is that we can use a modified version of LICRA to obtain the solution of the problem while guaranteeing convergence (under standard assumptions). Secondly, our state augmentation procedure admits a Markovian representation of the optimal policy. + +# 69 7 Experiments + +270 We will now study empirically the performance of the LICRA framework. In experiments, we use +271 different instances of LICRA, one where both policies are trained using PPO update (referred to +272 as LICRA_PPO) and one where the policy deciding whether to act is trained using SAC and the +273 other policy trained with PPO (referred to as LICRA_SAC). We have benchmarked both of these +274 algorithms together with common baselines on environments, where it would be natural to introduce +275 the concept of the cost associated with actions. We lastly performed a series of ablation studies which +276 test LICRA’s ability to handle different cost functions including the case when $c ( s , a ) \equiv 0$ which we +277 defer to the Appendix which also contains further experiment details. +278 Merton’s Portfolio Problem with Transaction Costs. Merton Investment Problem in which the +279 investor faces transaction costs [10] is a well-known problem within finance. In our environment, the +280 agent can decide to move its wealth between a risky asset and a risk-free asset. The agent receives a +281 reward only at the final step, equal to the utility of the portfolio with a risk aversion factor equal to +282 0.5. If the final wealth of risky asset is $s _ { T }$ and final wealth of risk-free asset is $c _ { T }$ , then the agent will +283 receive a reward of $u ( x ) = \hat { 2 \sqrt { s _ { T } + c _ { T } } }$ . The wealth evolves according to the following SDE: + +![](images/c36f4d4fc355a612aa0817f310a3b877c4df8280a953e0bbdade420829da0b36.jpg) +Figure 1: Training results in Merton investment problem for PPO style algorithms. + +$$ +d W _ { t } = ( r + p _ { t } ( \mu - r ) ) W _ { t } + W _ { t } p _ { t } \sigma d B _ { t } +$$ + +![](images/e4f0e42faa34e8a85dc4690b08f54b2a9dca40e02107c30188aa1c25e63de654.jpg) +Figure 2: a) Drive Environment. b) Training results in drive environment. + +284 where $W _ { t }$ is the current wealth and the state variable, $d B _ { t }$ is an increment of Brownian motion and $p _ { t }$ +285 is the proportion of wealth invested in the risky asset. We set the risk-free return $r = 0 . 0 1$ , risky asset +286 return $\mu = 0 . 0 5$ and volatility $\sigma = 1$ . We discretise the action space so that at each step the agent has +287 three actions available: move $10 \%$ of risky asset wealth to the risk-free asset, move $10 \%$ of risk-free +288 asset wealth to the risky asset or do nothing. Each time the agent moves the assets, it incurs a cost +289 of 1 i.e. a transaction fee. The agent can act after a time interval of 0.01 seconds and the episode +290 ends after 75 steps. The results of training are shown in Fig. 1 which clearly demonstrates that +291 LICRA_PPO finds a better policy than standard PPO. Also comparing the variance among different +292 seeds, we can see that LICRA_PPO is a much more stable algorithm than the other two. +293 Driving Environment Fuel Rationing. We studied an autonomous driving scenario where fuel +294 efficient driving is a priority. One of the main components of fuel-efficient driving is controlled usage +295 of acceleration and braking, in the sense that 1) the amount of acceleration and braking should be +296 limited 2) if accelerations should be performed slowly and gently. We believe this is a problem where +297 LICRA should thrive as the impulse control agent can learn to restrict the amount of acceleration +298 and braking in the presence of other cars and choose when to allow the car to decelerate naturally. +299 We used the highway-env [17] environment on a highway task (see Fig (2. a)) where the green +300 vehicle is our controlled vehicle and the goal is to avoid crashing into other vehicles whilst driving +301 at a reasonable speed. We add a cost function into the reward term dependent on the continuous +302 acceleration action, $C ( a _ { t } ) = K + a _ { t } ^ { 2 }$ , where $K > 0$ is a fixed constant cost of taking any action, +303 and $a _ { t } \in [ - 1 , 1 ]$ , with larger values of acceleration or braking being penalised more. The results are +304 presented in Fig. (2.b). Notably, LICRA is able to massively outperform the baselines, especially our +305 safety specific baselines which struggle to deal with the cost function associated with the environment. +306 We believe one reason for the success of LICRA is that it is far easier for it to utilise the null action +307 of zero acceleration/braking than the other algorithms, whilst all the algorithms have a guaranteed +308 cost at every time step whilst not gaining a sizeable reward to counter the cost. + +![](images/5bec575b374d2d9a07b6663aca5f30e26bc81290ee187b6eb4033c9c9f99603c.jpg) +Figure 3: a) The lander must land on the pad between two flags. . b) Training results in Lunar Lander. Lunar Lander Environment. We tested the ability of LICRA to perform in environment that simulate real-world physical dynamics. We tested LICRA’s performance the Lunar Lander environment in OpenAI gym [7] which we adjusted to incorporate minimal bounded costs in the reward definition. In this environment, the agent is required to maintain both a good posture mid-air and reach the landing pad as quickly as possible. The reward function is given by: + +$$ +\begin{array} { r } { \mathrm { R e w a r d } \left( s _ { t } \right) = 3 * \left( 1 - \mathbf { 1 } _ { d _ { t } - d _ { t - 1 } = 0 } \right) - 3 * \left( 1 - \mathbf { 1 } _ { v _ { t } - v _ { t - 1 } = 0 } \right) - 3 * \left( 1 - \mathbf { 1 } _ { \omega _ { t } - \omega _ { t - 1 } = 0 } \right) } \\ { - 0 . 0 3 * \mathrm { F u e l S p e n t } ( s _ { t } ) - 1 0 * \left( v _ { t } - v _ { t - 1 } \right) - 1 0 * \left( \omega _ { t } - \omega _ { t - 1 } \right) + 1 0 0 * \mathrm { h a s L a n d e d } \left( s _ { t } \right) } \end{array} +$$ + +309 where $d _ { t }$ is the distance to the landing pad, $v _ { t }$ is the velocity of the agent, and $\omega _ { t }$ is the angular +310 velocity of the agent at time $t$ . $\mathbf { 1 } _ { X }$ is the indicator function of taking actions, which is 1 when the +311 statement $X$ is true and 0 when $X$ is false. Considering the limited fuel budget, we assume that +312 we have a fixed cost for each action taken by the agent here, and doing nothing brings no cost. Then, +313 to describe the goal of the game, we define the function of the final status by hasLanded(), which is +314 0 when not landing; 1 when the agent has landed softly on the landing pad; and $- 1$ when the lander +315 runs out of fuel or loses contact with the pad on landing. The reward function rewards the agent +316 for reducing its distance to the landing pad, decreasing its speed to land smoothly and keeping the +317 angular speed at a minimum to prevent rolling. Additionally, it penalises the agent for running out +318 of fuel and deters the agent from taking off again after landing. +319 By introducing a reward function with minimally bounded costs, our goal was to test if LICRA can +320 exploit the optimal policy. In Fig. 3, we observe that the LICRA agent outperforms all the baselines, +321 both in terms of sample efficiency and average test return (total rewards at each timestep). We also +322 observe that LICRA enables more stable training than PPO, PPO-Lagrangian and CPO. +323 Ablation Study 1. Prioritisation of Most Important Actions. We next tested LICRA’s ability to +324 prioritise where it performs actions when the necessity to act varies significantly between states. To +325 test this, we modified the Drive Environment to now consist of a single lane, a start state and a goal +326 state start (at the end) where there is a reward. With no acceleration, the vehicle decreases velocity. +327 To reach the goal, the agent must apply an acceleration $a _ { t } \in [ - 1 , 1 ]$ . Each acceleration $a _ { t }$ incurs +328 a cost $C ( \boldsymbol { a } _ { t } )$ as defined above. At zones $k = 1 , 2 , 3$ of the lane, if the vehicle is travelling below a +329 velocity $v _ { m i n }$ , it is penalised by a strictly negative cost $c _ { k }$ where $c _ { 1 } < c _ { 2 } < c _ { 3 }$ . As shown in Fig. 4, +330 when the intervention cost increases i.e. when $K \infty$ , LICRA successfully prioritises the highest +penalty zones to avoid incurring large costs. + +![](images/2ba4542911b4913c1e9265c3276b917ff90a5a40da65ab950216d84702acce94.jpg) +Figure 4: Results for Ablation Study 1. Heatmaps display the number of times the agent drives below $v _ { m i n }$ in the penalty zones. Violation 1 refers to the lowest cost zone, whilst Violation 3 refers to the largest cost zone. $K$ refers to the fixed cost for taking an action. + +# 8 Conclusion + +We presented a novel method to tackle the problem of learning how to select when to act in addition to learning which actions to execute. Our framework, which is a general tool for tackling problems of this kind seamlessly adopts RL algorithms enabling them to efficiently tackle problems in which the agent must be selective about when it executes actions. This is of fundamental importance in practical settings where performing many actions over the horizon can lead to costs and undermine the service life of machinery. We demonstrated that our solution, LICRA which at its core has a sequential decision structure that first decides whether or not an action ought to be taken under the action policy can solve tasks where the agent faces costs with extreme efficiency as compared to leading reinforcement learning methods. In some tasks, we showed that LICRA is able to solve problems that are unsolvable using current reinforcement learning machinery. 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Citeseer, 2011. +[12] Abderrahim Fathan and Erick Delage. Deep reinforcement learning for optimal stopping with application in financial engineering. arXiv preprint arXiv:2105.08877, 2021. +[13] F Grandt Jr. Damage tolerant design and nondestructive inspection-keys to aircraft airworthiness. Procedia Engineering, 17:236–246, 2011. +[14] Tuomas Haarnoja, Aurick Zhou, Kristian Hartikainen, George Tucker, Sehoon Ha, Jie Tan, Vikash Kumar, Henry Zhu, Abhishek Gupta, Pieter Abbeel, et al. Soft actor-critic algorithms and applications. arXiv preprint arXiv:1812.05905, 2018. +[15] Tommi Jaakkola, Michael I Jordan, and Satinder P Singh. Convergence of stochastic iterative dynamic programming algorithms. In Advances in neural information processing systems, pages 703–710, 1994. +[16] Ralf Korn. Some applications of impulse control in mathematical finance. Mathematical Methods of Operations Research, 50(3):493–518, 1999. +[17] Edouard Leurent. An environment for autonomous driving decision-making. https://github. com/eleurent/highway-env, 2018. +[18] Harry M Markowitz. Foundations of portfolio theory. The journal of finance, 46(2):469–477, 1991. +[19] David Mguni. Duopoly investment problems with minimally bounded adjustment costs. arXiv preprint arXiv:1805.11974, 2018. +[20] David Mguni. Optimal capital injections with the risk of ruin: A stochastic differential game of impulse control and stopping approach. arXiv preprint arXiv:1805.01578, 2018. +[21] David Mguni. Optimal selection of transaction costs in a dynamic principal-agent problem. arXiv preprint arXiv:1805.01062, 2018. +[22] David Mguni. A viscosity approach to stochastic differential games of control and stopping involving impulsive control. arXiv preprint arXiv:1803.11432, 2018. +[23] David Mguni. Cutting your losses: Learning fault-tolerant control and optimal stopping under adverse risk. arXiv preprint arXiv:1902.05045, 2019. +[24] Bernt Øksendal. Stochastic differential equations. In Stochastic differential equations, pages 65–84. Springer, 2003. +[25] Bernt Øksendal and Agnès Sulem. Approximating impulse control by iterated optimal stopping. In Applied Stochastic Control of Jump Diffusions, pages 255–272. Springer, 2019. +[26] Jing-Cheng Pang, Tian Xu, Sheng-Yi Jiang, Yu-Ren Liu, and Yang Yu. Sparsity prior regularized q-learning for sparse action tasks. arXiv preprint arXiv:2105.08666, 2021. +[27] John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. CoRR, abs/1707.06347, 2017. +[28] Aivar Sootla, Alexander I. Cowen-Rivers, Taher Jafferjee, Ziyan Wang, David Mguni, Jun Wang, and Haitham Bou-Ammar. Sauté RL: Almost surely safe reinforcement learning using state augmentation. arXiv preprint arXiv:2202.06558, 2022. +[29] Aivar Sootla, Diego Oyarzún, David Angeli, and Guy-Bart Stan. Shaping pulses to control bistable systems: Analysis, computation and counterexamples. Automatica, 63:254–264, 2016. +[30] Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 2018. +[31] John N Tsitsiklis and Benjamin Van Roy. Optimal stopping of markov processes: Hilbert space theory, approximation algorithms, and an application to pricing high-dimensional financial derivatives. IEEE Transactions on Automatic Control, 44(10):1840–1851, 1999. +[32] Pu Zhao, Yanzhi Wang, Naehyuck Chang, Qi Zhu, and Xue Lin. A deep reinforcement learning framework for optimizing fuel economy of hybrid electric vehicles. In 2018 23rd Asia and South Pacific design automation conference (ASP-DAC), pages 196–202. IEEE, 2018. \ No newline at end of file diff --git a/parse/dev/zkk_7sV6gm8/zkk_7sV6gm8_content_list.json b/parse/dev/zkk_7sV6gm8/zkk_7sV6gm8_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..216f71a26f83e9f64660e49f48383fbb624ea7ea --- /dev/null +++ b/parse/dev/zkk_7sV6gm8/zkk_7sV6gm8_content_list.json @@ -0,0 +1,1116 @@ +[ + { + "type": "text", + "text": "Timing is Everything: Learning to Act Selectively with Costly Actions and Budgetary Constraints ", + "text_level": 1, + "bbox": [ + 174, + 122, + 823, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous Author(s) ", + "bbox": [ + 423, + 226, + 578, + 241 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 276, + 535, + 292 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Many real-world settings involve costs for performing actions; transaction costs \n2 in financial systems and fuel costs being common examples. In these settings, \n3 performing actions at each time step quickly accumulates costs leading to vastly \n4 suboptimal outcomes. Additionally, repeatedly acting produces wear and tear and \n5 ultimately, damage. Determining when to act is crucial for achieving successful \n6 outcomes and yet, the challenge of efficiently learning to behave optimally when \n7 actions incur minimally bounded costs remains unresolved. In this paper, we intro \n8 duce a reinforcement learning (RL) framework named Learnable Impulse Control \n9 Reinforcement Algorithm (LICRA), for learning to optimally select both when \n10 to act and which actions to take when actions incur costs. At the core of LICRA \n11 is a nested structure that combines RL and a form of policy known as impulse \n12 control which learns to maximise objectives when actions incur costs. We prove \n13 that LICRA, which seamlessly adopts any RL method, converges to policies that \n14 optimally select when to perform actions and their optimal magnitudes. We then \n15 augment LICRA to handle problems in which the agent can perform at most $k < \\infty$ \n16 actions and more generally, faces a budget constraint. We show LICRA learns the \n17 optimal value function and ensures budget constraints are satisfied almost surely. \n18 We demonstrate empirically LICRA’s superior performance against benchmark \n19 RL methods in OpenAI gym’s Lunar Lander and in Highway environments and a \n20 variant of the Merton portfolio problem within finance. ", + "bbox": [ + 148, + 308, + 766, + 583 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "21 1 Introduction ", + "text_level": 1, + "bbox": [ + 148, + 608, + 312, + 626 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "22 There are many settings in which agents incur costs each time they perform an action. Transaction \n23 costs in financial settings [19], fuel expenditure [32], toxicity as a side effect of controlling bacte \n24 ria [29] and physical damage produced by repeated action that produces wear and tear are just a \n25 few among many examples [13]. In these settings, performing actions at each time step is vastly \n26 suboptimal since acting in this way results in prohibitively high costs and undermines the service life \n27 of machinery. Minimising wear and tear is an essential attribute to safeguard against failures that can \n28 result in catastrophic losses [13]. \n29 Reinforcement learning (RL) is a framework that enables autonomous agents to learn complex \n30 behaviours from interactions with the environment [30, 11]. Within the standard RL paradigm, \n31 determining optimal actions involves making a selection from among many (possibly infinite) actions; \n32 a procedure that must be performed at each time-step as the agent decides on an action. In unknown \n33 settings, the agent cannot immediately exploit any topological structure of the action set (if any \n34 exists). Consequently, learning not to take an action i.e performing a zero or null action, involves \n35 expensive optimisation procedures over the entire action set. Since this must be done at each state, \n36 this process is vastly inefficient for learning optimal policies when the agent incurs costs for acting. \n37 In this paper, we tackle this problem by developing an RL framework for finding both an optimal \n38 criterion to determine whether or not to execute actions as well as learning optimal actions. A key \n39 component of our framework is a novel combination of RL with a form of policy known as impulse \n40 control [22, 19]. This enables the agent to determine the appropriate points to perform an action as \n41 well as the optimal action itself. Despite its fundamental importance as a tool for tackling decision \n42 problems with costly actions, presently, the use of impulse control within learning contexts (and \n43 unknown environments) is unaddressed. \n44 We present an RL impulse control framework called LICRA, which, to our knowledge, is the first \n45 learning framework for impulse control. To enable learning optimal impulse control policies in \n46 unknown environments, we devise a framework that consists of separate RL components for learning \n47 when to act and how to act optimally. The resulting framework is a structured two-part learning \n48 process which differs from current RL protocols. In LICRA, at each time step, the agent firstly makes \n49 a decision whether to act or not leading to a binary decision space $\\{ 0 , 1 \\}$ (we later show that this \n50 is determined by evaluating an easy-to-evaluate criterion which has the value function as its input). \n51 The second decision part determines the best action to take. This generates a subdivision of the state \n52 space into two regions; one in which the agent performs actions and another in which it does not act \n53 at all. This is extremely useful since the agent quickly determines the set of states to not take actions \n54 while performing actions only at the subset of states where actions are to be executed. \n55 We then establish theory that ensures convergence of LICRA to an optimal policy for such settings. \n56 To do this, we give a series of results namely: \n57 i) We establish a dynamic programming principle (DPP) for impulse control and show that the optimal \n58 value function can be obtained as a limit of a value iterative procedure (Theorem 1) which lays the \n59 foundation for an RL approach to impulse control. \n60 ii) We extend result i) to a new variant of Q learning which enables the impulse control problem to be \n61 solved using our RL method (Theorem 2). \n62 iii) We characterise the optimal conditions for performing an action which we reveal to be a simple \n63 ‘obstacle condition’ involving the agent’s value function (Prop. 1). Using this, the agent can quickly \n64 determine whether or not it should act and if so, then learn what the optimal action is. \n65 iv) We then extend the result i) to (linear) function approximators enabling the value function to be \n66 parameterised (Theorem 3). \n67 iv) In Sec. 6, we extend LICRA to include budgetary constraints so that each action draws from a \n68 fixed budget which the agent must stay within. Analogous to the development of i), we establish \n69 another DPP from which we derive a Q-learning variant for tackling impulse control with budgetary \n70 constraints (Theorem 4). A particular case of a budget constraint is when the number of actions the \n71 agent can take over the horizon is capped. \n72 Lastly, we perform a set of experiments to validate our theory within the Highway driving simulator \n73 and OpenAI’s LunarLander [7]. \n74 LICRA confers a series of advantages. As we demonstrate in our experiments, LICRA learns to \n75 compute the optimal problems in which the agent faces costs for acting in an efficient way which \n76 outperforms leading RL baselines. Second, as demonstrated in Sec. 6 LICRA handles settings in \n77 which the agent has a cap the total number of actions it is allowed to execute and more generally, \n78 generic budgetary constraints. LICRA is able to accommodate any RL base algorithm unlike various \n79 RL methods designed to handle budgetary constraints. ", + "bbox": [ + 147, + 640, + 825, + 736 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 742, + 825, + 853 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 859, + 823, + 901 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 92, + 825, + 146 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 154, + 825, + 305 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 311, + 825, + 340 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 150, + 345, + 825, + 387 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 155, + 388, + 823, + 415 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 416, + 823, + 457 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 156, + 457, + 821, + 484 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 484, + 825, + 553 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 153, + 554, + 823, + 582 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 588, + 825, + 670 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "80 2 Related Work ", + "text_level": 1, + "bbox": [ + 148, + 691, + 321, + 708 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "81 In continuous-time optimal control theory [24], problems in which the agent faces a cost for each \n82 action are tackled with a form of policy known as impulse control [22, 19, 2]. In impulse control \n83 frameworks, the dynamics of the system are modified through a sequence of discrete actions or bursts \n84 chosen at times that the agent chooses to apply the control policy. This distinguishes impulse control \n85 models from classical decision methods in which an agent takes actions at each time step while being \n86 tasked with the decision of only which action to take. Impulse control models represent appropriate \n87 modelling frameworks for financial environments with transaction costs, liquidity risks and economic \n88 environments in which players face fixed adjustment costs (e.g. menu costs) [16, 20]. \n89 The current setting is intimately related to the optimal stopping problem which widely occurs in \n90 finance, economics and computer science [23, 31]. In the optimal stopping problem, the task is to \n91 determine a criterion that determines when to arrest the system and receive a terminal reward. In this \n92 case, standard RL methods are unsuitable since they require an expensive sweep (through the set \n93 of states) to determine the optimal point to arrest the system. The current problem can be viewed \n94 as an augmented problem of optimal stopping since the agent must now determine both a sequence \n95 of points to perform an action or intervene and their optimal magnitudes — only acting when the \n96 cost of action is justified [25]. Adapting RL to tackle optimal stopping problems has been widely \n97 studied [31, 4, 9] and applied to a variety of real-world settings within finance [12], radiation therapy \n98 [1] and network operating systems [3]. Our work serves as a natural extenstion to RL approaches to \n99 optimal stopping to the case in which the agent must decide at which points to take many actions. As \n100 with optimal stopping, standard RL methods cannot efficiently tackle this problem since determining \n101 whether to perform a 0 action requires a costly sweep through the action space at every state [31]. In \n102 [26] the authors introduce “sparse action” with a similar motivation as impulse control. However, \n103 the authors treat only the discrete action space case. The authors in [26] do not discuss a broader \n104 theoretical framework of dealing with “sparse actions”, and develop purely algorithmic solutions. \n105 Additionally, unlike the approach taken in [26], the problem setting we consider is one in which the \n106 agent faces a cost for each action - the produces a need for the agent to be selective about where it \n107 performs actions (but does not necessarily constrain the magnitude or choice of those actions). ", + "bbox": [ + 147, + 724, + 825, + 835 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 842, + 823, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 92, + 826, + 285 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "108 3 Preliminaries ", + "text_level": 1, + "bbox": [ + 142, + 314, + 318, + 330 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "109 Reinforcement Learning (RL). In RL, an agent sequentially selects actions to maximise its expected \n110 returns. The underlying problem is typically formalised as an MDP $\\langle S , \\mathcal { A } , P , R , \\gamma \\rangle$ where $S \\subset \\mathbb { R } ^ { p }$ \n111 is the set of states, $\\mathcal { A } \\subset \\mathbb { R } ^ { k }$ is the set of actions, $P : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { S } [ 0 , 1 ]$ is a transition probability \n112 function describing the system’s dynamics, $R : S \\times \\mathcal { A } \\mathbb { R }$ is the reward function measuring the \n113 agent’s performance and the factor $\\gamma \\in [ 0 , 1 )$ specifies the degree to which the agent’s rewards are \n114 discounted over time [30]. At time $t \\in { 0 , 1 , \\ldots }$ , the system is in state $s _ { t } \\in S$ and the agent must \n115 choose an action $a _ { t } \\in \\mathcal A$ which transitions the system to a new state $s _ { t + 1 } \\sim P ( \\cdot | s _ { t } , a _ { t } )$ and produces \n116 a reward $R ( s _ { t } , a _ { t } )$ . A policy $\\pi : S \\times A \\to [ 0 , 1 ]$ is a probability distribution over state-action pairs \n117 where $\\pi ( a | s )$ represents the probability of selecting action $a \\in { \\mathcal { A } }$ in state $s \\in S$ . The goal of an \n118 RL agent is to find a policy ${ \\hat { \\pi } } \\in \\Pi$ that maximises its expected returns given by the value function: \n119 $\\begin{array} { r } { v ^ { \\pi } ( s ) = \\mathbb { E } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\bar { R } ( s _ { t } , a _ { t } ) | a _ { t } \\sim \\pi ( \\cdot | s _ { t } ) , s _ { 0 } = s ] } \\end{array}$ $\\begin{array} { r } { Q ( s , a ) = \\mathbb { E } [ \\sum _ { t = 0 } ^ { \\infty } R ( s _ { t } , a _ { t } ) | a _ { 0 } = a , s _ { 0 } = s ] } \\end{array}$ $\\Pi$ ent’s policy set. The action. \n121 We consider a setting in which the agent faces at least some minimal cost for each action it performs. \n122 With this, the agent’s task is to maximise: ", + "bbox": [ + 140, + 351, + 825, + 522 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 526, + 825, + 555 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/dbb0ac7205352125e445c169be6d5e1f9f57f8b1f68b460d5971f8dca0f3502a.jpg", + "text": "$$\nv ^ { \\pi } ( s ) = \\mathbb { E } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\left\\{ \\mathcal { R } ( s _ { t } , a _ { t } ) - \\mathcal { C } ( s _ { t } , a _ { t } ) \\right\\} \\Big | s _ { 0 } = s \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 321, + 569, + 674, + 613 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "123 where for any state $s \\in S$ and any action $a \\in { \\mathcal { A } }$ , the functions $\\mathcal { R }$ and $\\mathcal { C }$ are given by $\\textstyle { \\mathcal { R } } ( s , a ) =$ \n124 $R ( s , a ) \\mathbf { 1 } _ { a \\in \\mathcal { A } } + R ( s , 0 ) ( 1 - \\mathbf { 1 } _ { a \\in \\mathcal { A } } )$ where $\\mathbf { 1 } _ { a \\in \\mathcal { A } }$ is the indicator function which is 1 when $a \\in { \\mathcal { A } }$ and 0 \n125 otherwise and ${ \\mathcal { C } } ( s , a ) : = c ( s , a ) \\mathbf { 1 } _ { a \\in { \\mathcal { A } } }$ where $c : S \\times \\mathcal { A } \\mathbb { R }$ is a minimally bounded (cost) function1 \n126 that introduces a cost each time the agent performs an action. Examples of the cost function is a \n127 quasi-linear function of the form $c ( s _ { t } , \\bar { a } _ { t } ) \\stackrel { } { = } \\kappa + f ( a _ { t } )$ where $f : \\mathcal { A } \\mathbb { R } _ { > 0 }$ and $\\kappa$ is a positive real \n128 valued constant. Since acting at each time step would incur prohibitively high costs, the agent must \n129 be selective when to perform an action. Therefore, in this setting, the agent’s problem is augmented to \n130 learning both an optimal policy for its actions and, learning at which states to apply its action policy. \n131 Example: Merton Portfolio Problem with Transaction Costs [10]. An investor performs a series \n132 of costly portfolio adjustments by buying and selling amounts of different assets within their portfolio. \n133 Each investment incurs a fixed minimal cost (also known as transaction costs) which is deducted \n134 from the investor’s available cash-flow. The investor’s aim is to maximise their total wealth (the value \n135 of the sum of their assets) at some time horizon by adjusting their portfolio of investments. Problems \n136 of this kind, portfolio investment problems are of fundamental importance within finance [18]. \n37 Example 2. An autonomous vehicle must perform a series of actions to perform a task. Each action \n38 draws from its fuel budget. In order to complete its task successfully, during the task, the vehicle \n139 must ensure it maintains an available supply. \n141 In RL, the agent’s problem involves learning to act at every state including those in which actions do \n142 not significantly impact on its total return. While we can add a zero action to the action set $\\mathcal { A }$ and \n143 apply standard methods, we argue that this may not be the best solution in many situations. We argue \n144 the optimal policy has the following form: ", + "bbox": [ + 142, + 627, + 825, + 739 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 744, + 825, + 828 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 150, + 829, + 826, + 871 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 121, + 825, + 178 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/80e033ef294b86d5333ac21d90363be2a6475df9099dfe3ede1d62f8e5f0363f.jpg", + "text": "$$\n\\begin{array} { r } { \\widetilde { \\pi } ( \\cdot | s ) = \\left\\{ \\begin{array} { l l } { a _ { t } } & { s \\in S _ { I } , } \\\\ { 0 } & { s \\notin S _ { I } , } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 419, + 185, + 576, + 220 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "145 which implies that we simplify policy learning by determining the set $ { \\boldsymbol { S } } _ { I }$ first — the set where we \n146 actually need to learn the policy. \n147 We now introduce a learning method for producing impulse controls. This enables the agent to learn \n148 to select states to perform actions. Therefore, now agent is tasked with learning to act at states that \n149 are most important for maximising its total return given the presence of the cost for each action. Now \n150 at each state the agent first makes a binary decision to decide to perform an action. \n151 Our framework, LICRA consists of two core components: firstly a RL process $\\mathfrak { g } : \\mathcal { S } \\times \\{ 0 , 1 \\} [ 0 , 1 ]$ \n152 and a second RL process $\\pi : S \\times A \\to [ 0 , 1 ]$ . The role of $\\mathfrak { g }$ is to determine whether or not an action \n153 is to be performed by the policy $\\pi$ at a given state $s$ . If activated, the policy $\\pi$ determines the action \n154 to be selected. Prior to decisions being made, the policy $\\pi$ communicates to $\\mathfrak { g }$ the action it would \n155 take. An important feature of our LICRA is the sequential decision process. In LICRA, the policy \n156 $\\pi$ first proposes an action $a \\in { \\mathcal { A } }$ which is observed by the policy $\\mathfrak { g }$ . Therefore, the role of $\\mathfrak { g }$ is to \n157 prevent actions for which the change in expected future rewards does not exceed the costs incurred \n158 for taking such actions. By isolating a decision policy over whether an action should be taken or not, \n159 the impulse controls mechanism results in a framework in which the problem facing the agent has a \n160 markedly reduced decision space (in comparison to a standard RL method). Crucially, the agent must \n161 compute optimal actions at only a subset of states which are chosen by the policy g. Below is the \n162 pseudocode for LICRA, we provide full details of the code in Sec. 9 of the Appendix. ", + "bbox": [ + 143, + 228, + 825, + 258 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 262, + 825, + 319 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 324, + 826, + 491 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 1: Learnable Impulse Control Reinforcement Algorithm (LICRA) ", + "text_level": 1, + "bbox": [ + 176, + 511, + 686, + 526 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1: Input: Stepsize $\\alpha$ , batch size $B$ , episodes $K$ , steps per episode $T$ , mini-epochs $e$ \n2: Initialise: Policy network (acting) $\\pi$ , Policy network (switching) $\\mathfrak { g }$ , \nCritic network (acting $) V _ { \\pi }$ ,Critic network (switching $) V _ { { \\mathfrak { g } } }$ \n3: Given reward objective function, $R$ , initialise Rollout Buffers $B _ { \\pi }$ , $B _ { \\mathfrak { g } }$ (use Replay Buffer for \nSAC) \n4: for $N _ { e p i s o d e s }$ do \n5: Reset state $s _ { 0 }$ , Reset Rollout Buffers $B _ { \\pi }$ , $B _ { \\mathfrak { g } }$ \n6: for $t = 0 , 1 , \\ldots$ do \n7: Sample $a _ { t } \\sim \\pi ( \\cdot | s _ { t } )$ \n8: Sample $g _ { t } \\sim \\mathfrak { g } ( \\cdot | s _ { t } )$ \n9: if $g _ { t } = 0$ then \n10: Apply $a _ { t }$ so $s _ { t + 1 } \\sim P ( \\cdot | a _ { t } , s _ { t } ) .$ , \n11: Receive rewards $r _ { t } = \\mathcal { R } ( s _ { t } , a _ { t } )$ \n12: Store $( s _ { t } , a _ { t } , s _ { t + 1 } , r _ { t } )$ in $B _ { \\pi }$ \n13: else \n14: Apply the null action so $s _ { t + 1 } \\sim P ( \\cdot | 0 , s _ { t } )$ , \n15: Receive rewards $r _ { t } = \\mathcal { R } ( s _ { t } , 0 )$ . \n16: end if \n17: Store $( s _ { t } , g _ { t } , s _ { t + 1 } , r _ { t } )$ in $B _ { \\mathfrak { g } }$ \n18: end for \n19: // Learn the individual policies \n20: Update policy $\\pi$ and critic $V _ { \\pi }$ networks using $\\mathfrak { B } _ { \\pi }$ \n21: Update policy $\\mathfrak { g }$ and critic $V _ { \\mathfrak { g } }$ networks using ${ \\mathfrak { B } } _ { { \\mathfrak { g } } }$ \n22: end for \n165 Although action space cardinality does not change there are still benefits of using impulse control \n166 mechanism. This mechanism forces the agent to first determine the set of states to perform actions \n167 only then determine the optimal actions at these states. An important fact to note is that the decision \n168 space for the determining whether or not to execute an action is $S \\times \\{ 0 , 1 \\}$ i.e at each state it makes a \n169 binary decision. Consequently, the learning process for aspect is much quicker than a policy which \n170 must optimise over a decision space which is $| { \\cal S } | | { \\cal A } |$ (choosing an action from its action space at \n171 every state). This results in the agent rapidly learning which states to focus on to learn which actions \n172 to perform. In the case of $\\pi$ with a continuous action space again the impulse control mechanism \n173 does not change the cardinality of the action space. However, if the set ${ \\cal S } / { \\cal S } _ { I }$ , where the optimal \n174 policy chooses 0, is large enough, then again it can be more efficient to learn $\\mathfrak { g }$ first and only then \n175 learn $\\pi$ (we later validate this claim empirically, see Sec. 11.2), \n176 In Sec. 5, we prove the convergence properties of LICRA. LICRA consists of two independent \n177 procedures: a learning process for the policy $\\pi$ and simultaneously, a learning process for the impulse \n178 policy $\\mathfrak { g }$ which determines at which states to perform an action. In our implementation, we used \n179 proximal policy optimisation (PPO) [27] for the policy $\\pi$ and for the impulse policy $\\mathfrak { g }$ , whose action \n180 set consists of two actions (intervene or do not intervene) we used a soft actor critic (SAC) process \n181 [14] LICRA is a plug $\\&$ play framework which enables these RL components to be replaced with any \n182 RL algorithm of choice. ", + "bbox": [ + 176, + 529, + 816, + 862 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 90, + 825, + 244 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 250, + 825, + 347 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "183 5 Convergence and Optimality of LICRA ", + "text_level": 1, + "bbox": [ + 143, + 366, + 534, + 383 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "184 A key aspect of our framework is the presence of two RL processes that make decisions in a sequential \n185 order. In order to determine when to act the policy $\\mathfrak { g }$ must learn the states to allow the policy $\\pi$ to \n186 perform an action which the policy $\\pi$ must learn to select optimal actions whenever it is allowed to \n187 execute an action. \n188 In this section, we prove that LICRA converges to an optimal solution of the system. Central to \n189 LICRA is a Q-learning type method which is adapted to handle RL settings in which the agent must \n190 also learn when to act. We then extend the result to allow for (linear) function approximators. We \n191 provide a result that shows the optimal intervention times are characterised by an ‘obstacle condition’ \n192 which can be evaluated online therefore allowing the $\\mathfrak { g }$ . ", + "bbox": [ + 142, + 397, + 825, + 453 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 458, + 825, + 530 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Given a function $Q \\because S \\times \\mathcal { A } \\to \\mathbb { R } , \\quad \\forall \\pi , \\pi ^ { \\prime } \\in \\Pi$ and $\\forall \\ S , \\ S ^ { \\prime } , \\quad \\forall s _ { \\tau _ { k } } \\quad \\in \\quad S$ , we define the intervention operator $\\mathcal { M } ^ { \\pi , { \\mathfrak { g } } }$ by $\\begin{array} { r } { \\mathcal { M } ( ^ { \\pi , \\oplus } Q ^ { \\pi ^ { \\prime } , \\oplus ^ { \\prime } } ( s _ { \\pi _ { k } } , a _ { \\pi _ { k } } ) : = { \\mathcal R } ( s _ { \\pi _ { k } } , a _ { \\pi _ { k } } ) - c ( s _ { \\pi _ { k } } , a _ { \\pi _ { k } } ) + } \\end{array}$ $\\begin{array} { r } { \\gamma \\sum _ { s ^ { \\prime } \\in S } P ( s ^ { \\prime } ; a _ { \\tau _ { k } } , s ) v ^ { \\pi ^ { \\prime } , \\mathfrak { g ^ { \\prime } } } ( s ^ { \\prime } ) \\| a _ { \\tau _ { k } } \\sim \\pi ( \\cdot | s _ { \\tau _ { k } } ) , } \\end{array}$ , where $\\tau _ { k }$ is an intervention time. ", + "bbox": [ + 160, + 534, + 826, + 589 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "196 The interpretation of $\\mathcal { M }$ is the following: suppose that the agent is using the policy $\\pi$ and at time $\\tau _ { k }$ \n197 the system is at a state $s _ { \\tau _ { k } }$ and the agent performs an action $a _ { \\tau _ { k } } \\sim \\pi ( \\cdot | s _ { \\tau _ { k } } )$ . A cost of $c ( s _ { \\tau _ { k } } , a _ { \\tau _ { k } } )$ \n198 is then incurred by the agent and the system transitions to $s ^ { \\prime } \\sim P ( \\cdot ; a _ { \\tau _ { k } } , s _ { \\tau _ { k } } )$ . Lastly, recall $v ^ { \\pi , { \\mathfrak { g } } }$ is \n199 the agent value function under the policy pair $( \\pi , g )$ . Therefore, the quantity ${ \\mathcal { M } } Q ^ { \\pi , { \\mathfrak { g } } }$ measures the \n200 expected future stream of rewards after an immediate intervention minus the cost of intervention. \n201 This object plays a crucial role in the LICRA framework which as we later discuss, exploits the cost \n202 structure of the problem to determine when the agent should perform an intervention. ", + "bbox": [ + 140, + 594, + 826, + 693 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Given a function 203 $v ^ { n , \\ell } : S \\mathbb { \\quad }$ , we define the Bellman operator $\\boldsymbol { \\cdot }$ , by: ", + "bbox": [ + 147, + 698, + 637, + 713 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "204 ", + "bbox": [ + 142, + 722, + 160, + 731 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/1d22f029be7fff5695370a24898dcff685b2ed179b08bdb4d9a1525dd068cfd4.jpg", + "text": "$$\nT v ^ { \\pi , \\theta } ( s ) : = \\operatorname* { m a x } \\Big \\{ \\mathcal { N } ^ { ( \\pi , \\theta ) } Q ^ { \\pi , \\theta } ( s , a ) , \\mathcal { R } ( s , 0 ) + \\gamma \\sum _ { s ^ { \\prime } \\in S } P ( s ^ { \\prime } ; 0 , s ) v ^ { \\pi , \\theta } ( s ^ { \\prime } ) \\Big \\} , \\qquad \\forall s \\in \\mathcal { S } .\n$$", + "text_format": "latex", + "bbox": [ + 230, + 724, + 767, + 756 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "205 ", + "bbox": [ + 142, + 756, + 161, + 766 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "206 The Bellman operator captures the nested sequential structure of the LICRA algorithm. In particular, \n207 the structure in (3) consists of an inner structure which consists of two terms: the first term is the \n208 expected future return given an action is taken at the current state under the policy $\\pi$ . The second term \n209 is the expected future return given no action is taken at the current state. Lastly, the outer structure is \n210 an optimisation which compares the expected return of the two possibilities and selects the maximum. ", + "bbox": [ + 140, + 772, + 826, + 843 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "11 Our first result proves $T$ is a contraction operator in particular, the following bound holds: ", + "bbox": [ + 155, + 848, + 761, + 864 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "212 Lemma 1 The Bellman operator $T$ is a contraction, that is the following bound holds: ", + "bbox": [ + 137, + 875, + 743, + 890 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/cf50390db76541add22697b97f1f8bef43214a39a5b762388d0195cd6f10c1c6.jpg", + "text": "$$\n\\begin{array} { r } { \\| T v - T v ^ { \\prime } \\| \\leq \\gamma \\left. v - v ^ { \\prime } \\right. , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 408, + 895, + 589, + 912 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where 13 $v , v ^ { \\prime }$ are elements of a finite normed vector space. We can now state our first main result: ", + "bbox": [ + 155, + 90, + 800, + 107 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 1 Given any $v ^ { \\pi , \\mathfrak { g } } : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R } ,$ , the optimal value function is given by $\\operatorname* { l i m } _ { k \\to \\infty } T ^ { k } v ^ { \\pi , \\mathfrak { g } } =$ $\\operatorname* { m a x } _ { \\hat { \\pi } , \\hat { g } \\in \\Pi } v ^ { \\hat { \\pi } , \\hat { g } } = v ^ { \\pi ^ { \\star } , g ^ { \\star } }$ where $( \\pi ^ { \\star } , g ^ { \\star } )$ is the optimal policy pair. ", + "bbox": [ + 168, + 114, + 825, + 161 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The result of Theorem 1 enables the solution to the agent’s impulse control problem to be determined using a value iteration procedure. Moreover, Theorem 1 enables a Q-learning approach [6] for finding the solution to the agent’s problem. ", + "bbox": [ + 147, + 170, + 826, + 213 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "219 Theorem 2 Consider the following $Q$ learning variant: ", + "bbox": [ + 142, + 222, + 540, + 238 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/23f8930b1b8b17d1d3a70216119b13eb66fa3f4560ca71ea1ccb26e4c5ed1067.jpg", + "text": "$$\n+ \\alpha _ { t } ( s _ { t } , a _ { t } ) \\left[ \\operatorname* { m a x } \\left\\{ \\mathcal { M } ^ { \\pi , \\mathfrak { g } } Q _ { t } ( s _ { t } , a _ { t } ) , \\mathcal { R } ( s _ { t } , 0 ) + \\gamma _ { a ^ { \\prime } \\in A } ^ { \\operatorname* { m a x } } Q _ { t } ( s _ { t + 1 } , a ^ { \\prime } ) \\right\\} - Q _ { t } ( s _ { t } , a _ { t } ) \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 187, + 257, + 790, + 292 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "then 220 $Q _ { t }$ converges to $Q ^ { \\star }$ with probability 1, where $s _ { t } , s _ { t + 1 } \\in S$ and $a _ { t } \\in \\mathcal A$ . ", + "bbox": [ + 147, + 294, + 673, + 310 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "21 We now extend the result to (linear) function approximators: ", + "bbox": [ + 155, + 319, + 570, + 334 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 3 Given a set of linearly independent basis functions $\\Phi = \\{ \\phi _ { 1 } , \\ldots , \\phi _ { p } \\}$ with $\\phi _ { k } \\in L _ { 2 } , \\forall k$ . LICRA converges to a limit point $r ^ { \\star } \\in \\mathbb { R } ^ { p }$ which is the unique solution to $\\Pi \\mathfrak { F } ( \\Phi r ^ { \\star } ) = \\Phi r ^ { \\star }$ where $\\mathfrak { F } v : = \\mathcal { R } + \\gamma P \\operatorname* { m a x } \\{ \\mathcal { M } v , v \\}$ . Moreover, $r ^ { \\star }$ satisfies: $\\| \\Phi r ^ { \\star } - Q ^ { \\star } \\| \\leq ( 1 - \\gamma ^ { 2 } ) ^ { - 1 / 2 } \\| \\Pi Q ^ { \\star } - Q ^ { \\star } \\|$ . ", + "bbox": [ + 171, + 343, + 826, + 388 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The theorem establishes the convergence of LICRA to a stable point with the use of linear function approximators. The second statement bounds the proximity of the convergence point by the smallest approximation error that can be achieved given the choice of basis functions. ", + "bbox": [ + 171, + 397, + 825, + 440 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Having constructed a procedure to find the optimal agent’s optimal value function, we now seek to determine the conditions when an intervention should be performed. Let us denote by $\\{ \\tau _ { k } \\} _ { k \\ge 0 }$ the points at which the agent decides to act or intervention times, so for example if the agent chooses to perform an action at state $s _ { 6 }$ and again at state $s _ { 8 }$ , then $\\tau _ { 1 } = 6$ and $\\tau _ { 2 } = 8$ . The following result characterises the optimal intervention policy $\\mathfrak { g }$ and the optimal times $\\{ \\tau _ { k } \\} _ { k \\ge 0 }$ . ", + "bbox": [ + 173, + 445, + 825, + 517 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proposition 1 The policy g is given by: $\\mathfrak { g } ( s _ { t } ) = H ( \\mathcal { M } ^ { \\pi , \\mathfrak { g } } Q ^ { \\pi , \\mathfrak { g } } - Q ^ { \\pi , \\mathfrak { g } } ) ( s _ { t } , a _ { t } ) , \\forall s _ { t } \\in \\mathcal { S } _ { \\mathrm { ~ \\Gamma ~ } }$ , where $Q ^ { \\pi , { \\mathfrak { g } } }$ is the solution in Theorem $\\cdot$ , $\\mathcal { M }$ is the intervention operator and $H$ is the Heaviside function, moreover the intervention times are $\\tau _ { k } = \\operatorname* { i n f } \\{ \\tau > \\tau _ { k - 1 } | \\mathcal { M } ^ { \\pi , { \\mathfrak { g } } } Q ^ { \\pi , { \\mathfrak { g } } } = Q ^ { \\pi , { \\mathfrak { g } } } \\}$ . ", + "bbox": [ + 173, + 525, + 825, + 569 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Prop. 1 characterises the (categorical) distribution $\\mathfrak { g }$ . Moreover, given the function $Q$ , the times $\\{ \\tau _ { k } \\}$ can be determined by evaluating if $\\mathcal { M } Q = Q$ holds. ", + "bbox": [ + 163, + 578, + 823, + 608 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "A key aspect of Prop. 1 is that it exploits the cost structure of the problem to determine when the agent should perform an intervention. In particular, the equality $\\mathcal { M } Q = Q$ implies that performing an action and incurring a cost for doing so is optimal. ", + "bbox": [ + 173, + 612, + 825, + 656 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6 Budget Augmented LICRA via State Augmentation ", + "text_level": 1, + "bbox": [ + 158, + 672, + 633, + 691 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We now tackle the problem of RL with a budget. To do this, we combine the above impulse control technology with state augmentation technique proposed in [28] The mathematical formulation of the problem is now given by the following for any $s \\in S$ : ", + "bbox": [ + 169, + 703, + 823, + 746 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/c55bbf1ceb08375a8c470d28ced7b07f37b91614755892322a27a5a1575451ae.jpg", + "text": "$$\n\\operatorname* { m a x } _ { \\pi \\in \\Pi , g } ~ \\upsilon ^ { \\pi , \\mathfrak { g } } ( s ) \\ \\mathrm { ~ s . ~ t . ~ } n - \\sum _ { t = 0 } ^ { \\infty } \\sum _ { k \\geq 1 } \\delta _ { \\tau _ { k } } ^ { t } \\geq 0 ,\n$$", + "text_format": "latex", + "bbox": [ + 370, + 752, + 624, + 791 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $n \\in \\mathbb N$ is a fixed value that represents the maximum number of allowed interventions and $\\sum k > 1 8 _ { T k } ^ { t }$ is equal to one if an impulse was applied at time $t$ and zero if it was not. In order to avoid dealing with a constrained MDP, we propose to introduce a new variable $z _ { t }$ tracking the remaining number of impulses: $\\begin{array} { r } { z _ { t } = n - \\sum _ { i = 0 } ^ { t - 1 } \\sum _ { k \\geq 1 } \\delta _ { \\tau _ { k } } ^ { i } } \\end{array}$ . We treat $z _ { t }$ as another state and augment the state-space resulting in the transition $\\widetilde { \\mathcal P }$ : ", + "bbox": [ + 171, + 792, + 825, + 872 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/6346be3fbe5274a0800604dd10bbe7c0ae289219dc95d9e13ea962d4639c0036.jpg", + "text": "$$\ns _ { t + 1 } \\sim P ( \\cdot | s _ { t } , a _ { t } ) , \\qquad z _ { t + 1 } = z _ { t } - \\sum _ { k \\geq 1 } \\delta _ { \\tau _ { k } } ^ { t } , \\quad z _ { 0 } = n .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 877, + 671, + 910 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In order to avoid violations, we reshape the reward as follows: 252 $\\begin{array} { r } { \\widetilde { \\mathcal { R } } ( s _ { t } , z _ { t } , a _ { t } ) = \\left\\{ \\begin{array} { l l } { \\mathcal { R } ( s _ { t } , a _ { t } ) } & { z _ { t } \\ge 0 } \\\\ { - \\Delta } & { z _ { t } < 0 } \\end{array} \\right. } \\end{array}$ ,, 253 where $\\Delta > 0$ is a large enough hyper-parameter ensuring that there are no safety violations. To 254 summarise we aim to solve the following problem: ", + "bbox": [ + 142, + 89, + 825, + 151 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 159, + 161, + 169 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/f7782e97a7c6f6b939a51b9531f333bd46fa03de6e45a900410f8a587c3fa221.jpg", + "text": "$$\nv ^ { \\pi , \\mathfrak { g } } ( s , z ) = \\mathbb { E } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\widetilde { \\mathcal { R } } ( s _ { t } , z _ { t } , a _ { t } ) | a _ { t } \\sim \\pi ( \\cdot | s _ { t } , z _ { t } ) \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 333, + 160, + 665, + 199 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where the policy now depends on the variable $z _ { t }$ . Note that $\\widetilde { \\mathcal P }$ in Equation 6 is a Markov process and, the rewards $\\widetilde { \\mathcal { R } }$ are bounded, as long as the rewards $\\mathcal { R }$ are bounded. Therefore, we can apply directly the results for impulse control to this case as well. We denote the augmented MDP by $\\widetilde { \\mathcal { M } } = \\langle \\overset { \\cdot } { S } \\times \\mathcal { Z } , \\overset { \\cdot } { A } , \\widetilde { \\mathcal { P } } , \\widetilde { R } , \\gamma \\rangle$ , where $\\mathcal { Z }$ is the space of the augmented state. We have the following. ", + "bbox": [ + 166, + 205, + 826, + 268 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 4 Consider the MDP $\\widetilde { \\mathcal { M } }$ for the problem 7, then: ", + "bbox": [ + 165, + 280, + 565, + 297 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "a) The Bellman equation holds, i.e. there exists a function $\\tilde { v } ^ { * , \\pi , 9 }$ s.th. $\\begin{array} { r l } { \\tilde { v } ^ { * , \\pi , \\mathfrak { g } } ( s , z ) } & { { } = } \\end{array}$ $\\operatorname* { m a x } _ { \\mathbf { a } \\in \\mathcal { A } } \\Big ( \\widetilde { \\mathcal { R } } ( s , z , \\pmb { a } ) + \\gamma \\mathbb { E } _ { s ^ { \\prime } , z ^ { \\prime } \\sim \\mathcal { P } } \\left[ \\widetilde { v } ^ { * , \\pi , \\mathfrak { g } } ( s ^ { \\prime } , z ^ { \\prime } ) \\right] \\Big )$ , where the optimal policy for $\\widetilde { \\mathcal { M } }$ has the form $\\pi ^ { * } ( \\cdot | s , z )$ ; ", + "bbox": [ + 173, + 303, + 825, + 358 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "$b$ ) Given a ${ \\widetilde { v } } : { \\mathcal { S } } \\times { \\mathcal { Z } } \\to \\mathbb { R } ,$ , the stable point solution for $\\widetilde { \\mathcal { M } }$ is a given by $\\operatorname* { l i m } _ { k \\infty } \\tilde { T } ^ { k } \\widetilde { v } ^ { \\pi , g } = \\operatorname* { m a x } _ { \\hat { \\pi } \\in \\Pi , \\hat { g } } \\hat { \\pi } ^ { \\hat { \\pi } , \\hat { g } } =$ $\\widetilde { v } ^ { * , \\pi , { \\mathfrak { g } } ^ { * } }$ , where $( \\pi ^ { * } , { \\mathfrak { g } } ^ { * } )$ is an optimal policy of $\\widetilde { \\mathcal { M } }$ and $\\tilde { T }$ is the Bellman operator of $\\widetilde { \\mathcal { M } }$ . ", + "bbox": [ + 173, + 363, + 825, + 409 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The result has several important implications. The first is that we can use a modified version of LICRA to obtain the solution of the problem while guaranteeing convergence (under standard assumptions). Secondly, our state augmentation procedure admits a Markovian representation of the optimal policy. ", + "bbox": [ + 171, + 416, + 826, + 460 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "69 7 Experiments ", + "text_level": 1, + "bbox": [ + 153, + 478, + 312, + 496 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "270 We will now study empirically the performance of the LICRA framework. In experiments, we use \n271 different instances of LICRA, one where both policies are trained using PPO update (referred to \n272 as LICRA_PPO) and one where the policy deciding whether to act is trained using SAC and the \n273 other policy trained with PPO (referred to as LICRA_SAC). We have benchmarked both of these \n274 algorithms together with common baselines on environments, where it would be natural to introduce \n275 the concept of the cost associated with actions. We lastly performed a series of ablation studies which \n276 test LICRA’s ability to handle different cost functions including the case when $c ( s , a ) \\equiv 0$ which we \n277 defer to the Appendix which also contains further experiment details. \n278 Merton’s Portfolio Problem with Transaction Costs. Merton Investment Problem in which the \n279 investor faces transaction costs [10] is a well-known problem within finance. In our environment, the \n280 agent can decide to move its wealth between a risky asset and a risk-free asset. The agent receives a \n281 reward only at the final step, equal to the utility of the portfolio with a risk aversion factor equal to \n282 0.5. If the final wealth of risky asset is $s _ { T }$ and final wealth of risk-free asset is $c _ { T }$ , then the agent will \n283 receive a reward of $u ( x ) = \\hat { 2 \\sqrt { s _ { T } + c _ { T } } }$ . The wealth evolves according to the following SDE: ", + "bbox": [ + 138, + 508, + 826, + 621 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/c36f4d4fc355a612aa0817f310a3b877c4df8280a953e0bbdade420829da0b36.jpg", + "image_caption": [ + "Figure 1: Training results in Merton investment problem for PPO style algorithms. " + ], + "image_footnote": [], + "bbox": [ + 367, + 631, + 632, + 768 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 801, + 825, + 886 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/41cceaf9456638a4b1e99b4ad0d95e76acc025817d31222486f95548cde55275.jpg", + "text": "$$\nd W _ { t } = ( r + p _ { t } ( \\mu - r ) ) W _ { t } + W _ { t } p _ { t } \\sigma d B _ { t }\n$$", + "text_format": "latex", + "bbox": [ + 362, + 891, + 635, + 909 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/e4f0e42faa34e8a85dc4690b08f54b2a9dca40e02107c30188aa1c25e63de654.jpg", + "image_caption": [ + "Figure 2: a) Drive Environment. b) Training results in drive environment. " + ], + "image_footnote": [], + "bbox": [ + 183, + 88, + 810, + 233 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "284 where $W _ { t }$ is the current wealth and the state variable, $d B _ { t }$ is an increment of Brownian motion and $p _ { t }$ \n285 is the proportion of wealth invested in the risky asset. We set the risk-free return $r = 0 . 0 1$ , risky asset \n286 return $\\mu = 0 . 0 5$ and volatility $\\sigma = 1$ . We discretise the action space so that at each step the agent has \n287 three actions available: move $10 \\%$ of risky asset wealth to the risk-free asset, move $10 \\%$ of risk-free \n288 asset wealth to the risky asset or do nothing. Each time the agent moves the assets, it incurs a cost \n289 of 1 i.e. a transaction fee. The agent can act after a time interval of 0.01 seconds and the episode \n290 ends after 75 steps. The results of training are shown in Fig. 1 which clearly demonstrates that \n291 LICRA_PPO finds a better policy than standard PPO. Also comparing the variance among different \n292 seeds, we can see that LICRA_PPO is a much more stable algorithm than the other two. \n293 Driving Environment Fuel Rationing. We studied an autonomous driving scenario where fuel \n294 efficient driving is a priority. One of the main components of fuel-efficient driving is controlled usage \n295 of acceleration and braking, in the sense that 1) the amount of acceleration and braking should be \n296 limited 2) if accelerations should be performed slowly and gently. We believe this is a problem where \n297 LICRA should thrive as the impulse control agent can learn to restrict the amount of acceleration \n298 and braking in the presence of other cars and choose when to allow the car to decelerate naturally. \n299 We used the highway-env [17] environment on a highway task (see Fig (2. a)) where the green \n300 vehicle is our controlled vehicle and the goal is to avoid crashing into other vehicles whilst driving \n301 at a reasonable speed. We add a cost function into the reward term dependent on the continuous \n302 acceleration action, $C ( a _ { t } ) = K + a _ { t } ^ { 2 }$ , where $K > 0$ is a fixed constant cost of taking any action, \n303 and $a _ { t } \\in [ - 1 , 1 ]$ , with larger values of acceleration or braking being penalised more. The results are \n304 presented in Fig. (2.b). Notably, LICRA is able to massively outperform the baselines, especially our \n305 safety specific baselines which struggle to deal with the cost function associated with the environment. \n306 We believe one reason for the success of LICRA is that it is far easier for it to utilise the null action \n307 of zero acceleration/braking than the other algorithms, whilst all the algorithms have a guaranteed \n308 cost at every time step whilst not gaining a sizeable reward to counter the cost. ", + "bbox": [ + 138, + 276, + 825, + 400 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 401, + 826, + 621 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/5bec575b374d2d9a07b6663aca5f30e26bc81290ee187b6eb4033c9c9f99603c.jpg", + "image_caption": [ + "Figure 3: a) The lander must land on the pad between two flags. . b) Training results in Lunar Lander. Lunar Lander Environment. We tested the ability of LICRA to perform in environment that simulate real-world physical dynamics. We tested LICRA’s performance the Lunar Lander environment in OpenAI gym [7] which we adjusted to incorporate minimal bounded costs in the reward definition. In this environment, the agent is required to maintain both a good posture mid-air and reach the landing pad as quickly as possible. The reward function is given by: " + ], + "image_footnote": [], + "bbox": [ + 184, + 633, + 738, + 776 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/de80a6064355eabf656cffa30e23f700882d4e6652151a389d2602021969d647.jpg", + "text": "$$\n\\begin{array} { r } { \\mathrm { R e w a r d } \\left( s _ { t } \\right) = 3 * \\left( 1 - \\mathbf { 1 } _ { d _ { t } - d _ { t - 1 } = 0 } \\right) - 3 * \\left( 1 - \\mathbf { 1 } _ { v _ { t } - v _ { t - 1 } = 0 } \\right) - 3 * \\left( 1 - \\mathbf { 1 } _ { \\omega _ { t } - \\omega _ { t - 1 } = 0 } \\right) } \\\\ { - 0 . 0 3 * \\mathrm { F u e l S p e n t } ( s _ { t } ) - 1 0 * \\left( v _ { t } - v _ { t - 1 } \\right) - 1 0 * \\left( \\omega _ { t } - \\omega _ { t - 1 } \\right) + 1 0 0 * \\mathrm { h a s L a n d e d } \\left( s _ { t } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 205, + 876, + 789, + 915 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "309 where $d _ { t }$ is the distance to the landing pad, $v _ { t }$ is the velocity of the agent, and $\\omega _ { t }$ is the angular \n310 velocity of the agent at time $t$ . $\\mathbf { 1 } _ { X }$ is the indicator function of taking actions, which is 1 when the \n311 statement $X$ is true and 0 when $X$ is false. Considering the limited fuel budget, we assume that \n312 we have a fixed cost for each action taken by the agent here, and doing nothing brings no cost. Then, \n313 to describe the goal of the game, we define the function of the final status by hasLanded(), which is \n314 0 when not landing; 1 when the agent has landed softly on the landing pad; and $- 1$ when the lander \n315 runs out of fuel or loses contact with the pad on landing. The reward function rewards the agent \n316 for reducing its distance to the landing pad, decreasing its speed to land smoothly and keeping the \n317 angular speed at a minimum to prevent rolling. Additionally, it penalises the agent for running out \n318 of fuel and deters the agent from taking off again after landing. \n319 By introducing a reward function with minimally bounded costs, our goal was to test if LICRA can \n320 exploit the optimal policy. In Fig. 3, we observe that the LICRA agent outperforms all the baselines, \n321 both in terms of sample efficiency and average test return (total rewards at each timestep). We also \n322 observe that LICRA enables more stable training than PPO, PPO-Lagrangian and CPO. \n323 Ablation Study 1. Prioritisation of Most Important Actions. We next tested LICRA’s ability to \n324 prioritise where it performs actions when the necessity to act varies significantly between states. To \n325 test this, we modified the Drive Environment to now consist of a single lane, a start state and a goal \n326 state start (at the end) where there is a reward. With no acceleration, the vehicle decreases velocity. \n327 To reach the goal, the agent must apply an acceleration $a _ { t } \\in [ - 1 , 1 ]$ . Each acceleration $a _ { t }$ incurs \n328 a cost $C ( \\boldsymbol { a } _ { t } )$ as defined above. At zones $k = 1 , 2 , 3$ of the lane, if the vehicle is travelling below a \n329 velocity $v _ { m i n }$ , it is penalised by a strictly negative cost $c _ { k }$ where $c _ { 1 } < c _ { 2 } < c _ { 3 }$ . As shown in Fig. 4, \n330 when the intervention cost increases i.e. when $K \\infty$ , LICRA successfully prioritises the highest \npenalty zones to avoid incurring large costs. ", + "bbox": [ + 140, + 90, + 825, + 229 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 236, + 826, + 291 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 291, + 826, + 416 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/2ba4542911b4913c1e9265c3276b917ff90a5a40da65ab950216d84702acce94.jpg", + "image_caption": [ + "Figure 4: Results for Ablation Study 1. Heatmaps display the number of times the agent drives below $v _ { m i n }$ in the penalty zones. Violation 1 refers to the lowest cost zone, whilst Violation 3 refers to the largest cost zone. $K$ refers to the fixed cost for taking an action. " + ], + "image_footnote": [], + "bbox": [ + 285, + 425, + 710, + 574 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "8 Conclusion ", + "text_level": 1, + "bbox": [ + 169, + 645, + 299, + 661 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We presented a novel method to tackle the problem of learning how to select when to act in addition to learning which actions to execute. Our framework, which is a general tool for tackling problems of this kind seamlessly adopts RL algorithms enabling them to efficiently tackle problems in which the agent must be selective about when it executes actions. This is of fundamental importance in practical settings where performing many actions over the horizon can lead to costs and undermine the service life of machinery. We demonstrated that our solution, LICRA which at its core has a sequential decision structure that first decides whether or not an action ought to be taken under the action policy can solve tasks where the agent faces costs with extreme efficiency as compared to leading reinforcement learning methods. In some tasks, we showed that LICRA is able to solve problems that are unsolvable using current reinforcement learning machinery. We envisge that this framework can serve as the basis extensions to different settings including adversarial training for solving a variety of problems within RL. ", + "bbox": [ + 173, + 674, + 825, + 840 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 168, + 859, + 266, + 876 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "[1] Ali Ajdari, Maximilian Niyazi, Nils Henrik Nicolay, Christian Thieke, Robert Jeraj, and Thomas Bortfeld. Towards optimal stopping in radiation therapy. Radiotherapy and Oncology, 134:96– ", + "bbox": [ + 171, + 883, + 823, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "[2] Luluwah Al-Fagih. The british knock-out put option. International Journal of Theoretical and Applied Finance, 18(02):1550008, 2015. \n[3] Juan J Alcaraz, Jose A Ayala-Romero, Javier Vales-Alonso, and Fernando Losilla-López. Online reinforcement learning for adaptive interference coordination. Transactions on Emerging Telecommunications Technologies, 31(10):e4087, 2020. \n[4] Sebastian Becker, Patrick Cheridito, and Arnulf Jentzen. Deep optimal stopping. arXiv preprint arXiv:1804.05394, 2018. \n[5] Albert Benveniste, Michel Métivier, and Pierre Priouret. Adaptive algorithms and stochastic approximations, volume 22. Springer Science & Business Media, 2012. \n[6] Dimitri P Bertsekas. Approximate dynamic programming. 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In these settings,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 92, + 266, + 470, + 279 + ], + "spans": [ + { + "bbox": [ + 92, + 268, + 99, + 277 + ], + "score": 1.0, + "content": "3", + "type": "text" + }, + { + "bbox": [ + 141, + 266, + 470, + 279 + ], + "score": 1.0, + "content": "performing actions at each time step quickly accumulates costs leading to vastly", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 92, + 277, + 469, + 289 + ], + "spans": [ + { + "bbox": [ + 92, + 280, + 99, + 288 + ], + "score": 1.0, + "content": "4", + "type": "text" + }, + { + "bbox": [ + 142, + 277, + 469, + 289 + ], + "score": 1.0, + "content": "suboptimal outcomes. Additionally, repeatedly acting produces wear and tear and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 92, + 288, + 469, + 300 + ], + "spans": [ + { + "bbox": [ + 92, + 290, + 99, + 299 + ], + "score": 1.0, + "content": "5", + "type": "text" + }, + { + "bbox": [ + 142, + 288, + 469, + 300 + ], + "score": 1.0, + "content": "ultimately, damage. Determining when to act is crucial for achieving successful", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 93, + 299, + 469, + 311 + ], + "spans": [ + { + "bbox": [ + 93, + 301, + 99, + 309 + ], + "score": 1.0, + "content": "6", + "type": "text" + }, + { + "bbox": [ + 142, + 299, + 469, + 311 + ], + "score": 1.0, + "content": "outcomes and yet, the challenge of efficiently learning to behave optimally when", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 93, + 309, + 469, + 322 + ], + "spans": [ + { + "bbox": [ + 93, + 312, + 99, + 321 + ], + "score": 1.0, + "content": "7", + "type": "text" + }, + { + "bbox": [ + 142, + 309, + 469, + 322 + ], + "score": 1.0, + "content": "actions incur minimally bounded costs remains unresolved. In this paper, we intro-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 92, + 320, + 470, + 333 + ], + "spans": [ + { + "bbox": [ + 92, + 322, + 99, + 332 + ], + "score": 1.0, + "content": "8", + "type": "text" + }, + { + "bbox": [ + 142, + 320, + 470, + 333 + ], + "score": 1.0, + "content": "duce a reinforcement learning (RL) framework named Learnable Impulse Control", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 92, + 330, + 470, + 344 + ], + "spans": [ + { + "bbox": [ + 92, + 334, + 99, + 343 + ], + "score": 1.0, + "content": "9", + "type": "text" + }, + { + "bbox": [ + 141, + 330, + 470, + 344 + ], + "score": 1.0, + "content": "Reinforcement Algorithm (LICRA), for learning to optimally select both when", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 89, + 342, + 470, + 354 + ], + "spans": [ + { + "bbox": [ + 89, + 344, + 100, + 354 + ], + "score": 1.0, + "content": "10", + "type": "text" + }, + { + "bbox": [ + 141, + 342, + 470, + 354 + ], + "score": 1.0, + "content": "to act and which actions to take when actions incur costs. At the core of LICRA", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 90, + 353, + 469, + 365 + ], + "spans": [ + { + "bbox": [ + 90, + 355, + 99, + 365 + ], + "score": 1.0, + "content": "11", + "type": "text" + }, + { + "bbox": [ + 140, + 353, + 469, + 365 + ], + "score": 1.0, + "content": "is a nested structure that combines RL and a form of policy known as impulse", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 89, + 363, + 469, + 376 + ], + "spans": [ + { + "bbox": [ + 89, + 365, + 100, + 376 + ], + "score": 1.0, + "content": "12", + "type": "text" + }, + { + "bbox": [ + 141, + 363, + 469, + 376 + ], + "score": 1.0, + "content": "control which learns to maximise objectives when actions incur costs. We prove", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 90, + 374, + 469, + 388 + ], + "spans": [ + { + "bbox": [ + 90, + 378, + 99, + 387 + ], + "score": 1.0, + "content": "13", + "type": "text" + }, + { + "bbox": [ + 141, + 374, + 469, + 388 + ], + "score": 1.0, + "content": "that LICRA, which seamlessly adopts any RL method, converges to policies that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 88, + 385, + 470, + 399 + ], + "spans": [ + { + "bbox": [ + 88, + 388, + 100, + 398 + ], + "score": 1.0, + "content": "14", + "type": "text" + }, + { + "bbox": [ + 142, + 385, + 470, + 399 + ], + "score": 1.0, + "content": "optimally select when to perform actions and their optimal magnitudes. We then", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 88, + 396, + 468, + 410 + ], + "spans": [ + { + "bbox": [ + 88, + 399, + 100, + 409 + ], + "score": 1.0, + "content": "15", + "type": "text" + }, + { + "bbox": [ + 141, + 396, + 438, + 410 + ], + "score": 1.0, + "content": "augment LICRA to handle problems in which the agent can perform at most", + "type": "text" + }, + { + "bbox": [ + 438, + 397, + 468, + 407 + ], + "score": 0.89, + "content": "k < \\infty", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 88, + 407, + 469, + 420 + ], + "spans": [ + { + "bbox": [ + 88, + 410, + 100, + 420 + ], + "score": 1.0, + "content": "16", + "type": "text" + }, + { + "bbox": [ + 142, + 407, + 469, + 420 + ], + "score": 1.0, + "content": "actions and more generally, faces a budget constraint. We show LICRA learns the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 90, + 419, + 470, + 431 + ], + "spans": [ + { + "bbox": [ + 90, + 421, + 99, + 430 + ], + "score": 1.0, + "content": "17", + "type": "text" + }, + { + "bbox": [ + 142, + 419, + 470, + 431 + ], + "score": 1.0, + "content": "optimal value function and ensures budget constraints are satisfied almost surely.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 89, + 428, + 470, + 442 + ], + "spans": [ + { + "bbox": [ + 89, + 431, + 100, + 442 + ], + "score": 1.0, + "content": "18", + "type": "text" + }, + { + "bbox": [ + 141, + 428, + 470, + 442 + ], + "score": 1.0, + "content": "We demonstrate empirically LICRA’s superior performance against benchmark", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 90, + 439, + 470, + 453 + ], + "spans": [ + { + "bbox": [ + 90, + 442, + 99, + 452 + ], + "score": 1.0, + "content": "19", + "type": "text" + }, + { + "bbox": [ + 141, + 439, + 470, + 453 + ], + "score": 1.0, + "content": "RL methods in OpenAI gym’s Lunar Lander and in Highway environments and a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 88, + 450, + 365, + 465 + ], + "spans": [ + { + "bbox": [ + 88, + 453, + 100, + 463 + ], + "score": 1.0, + "content": "20", + "type": "text" + }, + { + "bbox": [ + 141, + 450, + 365, + 465 + ], + "score": 1.0, + "content": "variant of the Merton portfolio problem within finance.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 91, + 482, + 191, + 496 + ], + "lines": [ + { + "bbox": [ + 86, + 481, + 192, + 498 + ], + "spans": [ + { + "bbox": [ + 86, + 481, + 192, + 498 + ], + "score": 1.0, + "content": "21 1 Introduction", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 90, + 507, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 89, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 89, + 508, + 99, + 518 + ], + "score": 1.0, + "content": "22", + "type": "text" + }, + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "score": 1.0, + "content": "There are many settings in which agents incur costs each time they perform an action. Transaction", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 89, + 517, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 89, + 520, + 99, + 529 + ], + "score": 1.0, + "content": "23", + "type": "text" + }, + { + "bbox": [ + 105, + 517, + 506, + 529 + ], + "score": 1.0, + "content": "costs in financial settings [19], fuel expenditure [32], toxicity as a side effect of controlling bacte-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 89, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 89, + 531, + 100, + 540 + ], + "score": 1.0, + "content": "24", + "type": "text" + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "ria [29] and physical damage produced by repeated action that produces wear and tear are just a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 89, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 89, + 541, + 99, + 551 + ], + "score": 1.0, + "content": "25", + "type": "text" + }, + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "few among many examples [13]. In these settings, performing actions at each time step is vastly", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 88, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 88, + 552, + 100, + 563 + ], + "score": 1.0, + "content": "26", + "type": "text" + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "suboptimal since acting in this way results in prohibitively high costs and undermines the service life", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 89, + 560, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 89, + 563, + 100, + 573 + ], + "score": 1.0, + "content": "27", + "type": "text" + }, + { + "bbox": [ + 104, + 560, + 506, + 575 + ], + "score": 1.0, + "content": "of machinery. Minimising wear and tear is an essential attribute to safeguard against failures that can", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 90, + 572, + 240, + 585 + ], + "spans": [ + { + "bbox": [ + 90, + 575, + 99, + 583 + ], + "score": 1.0, + "content": "28", + "type": "text" + }, + { + "bbox": [ + 105, + 572, + 240, + 585 + ], + "score": 1.0, + "content": "result in catastrophic losses [13].", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 90, + 588, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 89, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 89, + 591, + 100, + 600 + ], + "score": 1.0, + "content": "29", + "type": "text" + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "Reinforcement learning (RL) is a framework that enables autonomous agents to learn complex", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 89, + 597, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 89, + 601, + 100, + 611 + ], + "score": 1.0, + "content": "30", + "type": "text" + }, + { + "bbox": [ + 105, + 597, + 506, + 613 + ], + "score": 1.0, + "content": "behaviours from interactions with the environment [30, 11]. 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In this paper, we intro-", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 320, + 470, + 333 + ], + "spans": [ + { + "bbox": [ + 92, + 322, + 99, + 332 + ], + "score": 1.0, + "content": "8", + "type": "text" + }, + { + "bbox": [ + 142, + 320, + 470, + 333 + ], + "score": 1.0, + "content": "duce a reinforcement learning (RL) framework named Learnable Impulse Control", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 330, + 470, + 344 + ], + "spans": [ + { + "bbox": [ + 92, + 334, + 99, + 343 + ], + "score": 1.0, + "content": "9", + "type": "text" + }, + { + "bbox": [ + 141, + 330, + 470, + 344 + ], + "score": 1.0, + "content": "Reinforcement Algorithm (LICRA), for learning to optimally select both when", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 342, + 470, + 354 + ], + "spans": [ + { + "bbox": [ + 89, + 344, + 100, + 354 + ], + "score": 1.0, + "content": "10", + "type": "text" + }, + { + "bbox": [ + 141, + 342, + 470, + 354 + ], + "score": 1.0, + "content": "to act and which actions to take when actions incur costs. At the core of LICRA", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 353, + 469, + 365 + ], + "spans": [ + { + "bbox": [ + 90, + 355, + 99, + 365 + ], + "score": 1.0, + "content": "11", + "type": "text" + }, + { + "bbox": [ + 140, + 353, + 469, + 365 + ], + "score": 1.0, + "content": "is a nested structure that combines RL and a form of policy known as impulse", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 363, + 469, + 376 + ], + "spans": [ + { + "bbox": [ + 89, + 365, + 100, + 376 + ], + "score": 1.0, + "content": "12", + "type": "text" + }, + { + "bbox": [ + 141, + 363, + 469, + 376 + ], + "score": 1.0, + "content": "control which learns to maximise objectives when actions incur costs. We prove", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 374, + 469, + 388 + ], + "spans": [ + { + "bbox": [ + 90, + 378, + 99, + 387 + ], + "score": 1.0, + "content": "13", + "type": "text" + }, + { + "bbox": [ + 141, + 374, + 469, + 388 + ], + "score": 1.0, + "content": "that LICRA, which seamlessly adopts any RL method, converges to policies that", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 385, + 470, + 399 + ], + "spans": [ + { + "bbox": [ + 88, + 388, + 100, + 398 + ], + "score": 1.0, + "content": "14", + "type": "text" + }, + { + "bbox": [ + 142, + 385, + 470, + 399 + ], + "score": 1.0, + "content": "optimally select when to perform actions and their optimal magnitudes. We then", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 396, + 468, + 410 + ], + "spans": [ + { + "bbox": [ + 88, + 399, + 100, + 409 + ], + "score": 1.0, + "content": "15", + "type": "text" + }, + { + "bbox": [ + 141, + 396, + 438, + 410 + ], + "score": 1.0, + "content": "augment LICRA to handle problems in which the agent can perform at most", + "type": "text" + }, + { + "bbox": [ + 438, + 397, + 468, + 407 + ], + "score": 0.89, + "content": "k < \\infty", + "type": "inline_equation" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 407, + 469, + 420 + ], + "spans": [ + { + "bbox": [ + 88, + 410, + 100, + 420 + ], + "score": 1.0, + "content": "16", + "type": "text" + }, + { + "bbox": [ + 142, + 407, + 469, + 420 + ], + "score": 1.0, + "content": "actions and more generally, faces a budget constraint. We show LICRA learns the", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 419, + 470, + 431 + ], + "spans": [ + { + "bbox": [ + 90, + 421, + 99, + 430 + ], + "score": 1.0, + "content": "17", + "type": "text" + }, + { + "bbox": [ + 142, + 419, + 470, + 431 + ], + "score": 1.0, + "content": "optimal value function and ensures budget constraints are satisfied almost surely.", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 428, + 470, + 442 + ], + "spans": [ + { + "bbox": [ + 89, + 431, + 100, + 442 + ], + "score": 1.0, + "content": "18", + "type": "text" + }, + { + "bbox": [ + 141, + 428, + 470, + 442 + ], + "score": 1.0, + "content": "We demonstrate empirically LICRA’s superior performance against benchmark", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 439, + 470, + 453 + ], + "spans": [ + { + "bbox": [ + 90, + 442, + 99, + 452 + ], + "score": 1.0, + "content": "19", + "type": "text" + }, + { + "bbox": [ + 141, + 439, + 470, + 453 + ], + "score": 1.0, + "content": "RL methods in OpenAI gym’s Lunar Lander and in Highway environments and a", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 450, + 365, + 465 + ], + "spans": [ + { + "bbox": [ + 88, + 453, + 100, + 463 + ], + "score": 1.0, + "content": "20", + "type": "text" + }, + { + "bbox": [ + 141, + 450, + 365, + 465 + ], + "score": 1.0, + "content": "variant of the Merton portfolio problem within finance.", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + } + ], + "index": 13.5, + "bbox_fs": [ + 88, + 243, + 470, + 465 + ] + }, + { + "type": "title", + "bbox": [ + 91, + 482, + 191, + 496 + ], + "lines": [ + { + "bbox": [ + 86, + 481, + 192, + 498 + ], + "spans": [ + { + "bbox": [ + 86, + 481, + 192, + 498 + ], + "score": 1.0, + "content": "21 1 Introduction", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "index", + "bbox": [ + 90, + 507, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 89, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 89, + 508, + 99, + 518 + ], + "score": 1.0, + "content": "22", + "type": "text" + }, + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "score": 1.0, + "content": "There are many settings in which agents incur costs each time they perform an action. Transaction", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 517, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 89, + 520, + 99, + 529 + ], + "score": 1.0, + "content": "23", + "type": "text" + }, + { + "bbox": [ + 105, + 517, + 506, + 529 + ], + "score": 1.0, + "content": "costs in financial settings [19], fuel expenditure [32], toxicity as a side effect of controlling bacte-", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 89, + 531, + 100, + 540 + ], + "score": 1.0, + "content": "24", + "type": "text" + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "ria [29] and physical damage produced by repeated action that produces wear and tear are just a", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 89, + 541, + 99, + 551 + ], + "score": 1.0, + "content": "25", + "type": "text" + }, + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "few among many examples [13]. In these settings, performing actions at each time step is vastly", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 88, + 552, + 100, + 563 + ], + "score": 1.0, + "content": "26", + "type": "text" + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "suboptimal since acting in this way results in prohibitively high costs and undermines the service life", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 560, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 89, + 563, + 100, + 573 + ], + "score": 1.0, + "content": "27", + "type": "text" + }, + { + "bbox": [ + 104, + 560, + 506, + 575 + ], + "score": 1.0, + "content": "of machinery. Minimising wear and tear is an essential attribute to safeguard against failures that can", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 572, + 240, + 585 + ], + "spans": [ + { + "bbox": [ + 90, + 575, + 99, + 583 + ], + "score": 1.0, + "content": "28", + "type": "text" + }, + { + "bbox": [ + 105, + 572, + 240, + 585 + ], + "score": 1.0, + "content": "result in catastrophic losses [13].", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 89, + 591, + 100, + 600 + ], + "score": 1.0, + "content": "29", + "type": "text" + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "Reinforcement learning (RL) is a framework that enables autonomous agents to learn complex", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 597, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 89, + 601, + 100, + 611 + ], + "score": 1.0, + "content": "30", + "type": "text" + }, + { + "bbox": [ + 105, + 597, + 506, + 613 + ], + "score": 1.0, + "content": "behaviours from interactions with the environment [30, 11]. Within the standard RL paradigm,", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 88, + 612, + 99, + 622 + ], + "score": 1.0, + "content": "31", + "type": "text" + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "determining optimal actions involves making a selection from among many (possibly infinite) actions;", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 88, + 624, + 100, + 633 + ], + "score": 1.0, + "content": "32", + "type": "text" + }, + { + "bbox": [ + 104, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "a procedure that must be performed at each time-step as the agent decides on an action. In unknown", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 89, + 634, + 100, + 645 + ], + "score": 1.0, + "content": "33", + "type": "text" + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "settings, the agent cannot immediately exploit any topological structure of the action set (if any", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 89, + 645, + 100, + 655 + ], + "score": 1.0, + "content": "34", + "type": "text" + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "exists). Consequently, learning not to take an action i.e performing a zero or null action, involves", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 89, + 657, + 100, + 666 + ], + "score": 1.0, + "content": "35", + "type": "text" + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "expensive optimisation procedures over the entire action set. Since this must be done at each state,", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 664, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 89, + 667, + 100, + 676 + ], + "score": 1.0, + "content": "36", + "type": "text" + }, + { + "bbox": [ + 104, + 664, + 505, + 679 + ], + "score": 1.0, + "content": "this process is vastly inefficient for learning optimal policies when the agent incurs costs for acting.", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 681, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 90, + 684, + 100, + 692 + ], + "score": 1.0, + "content": "37", + "type": "text" + }, + { + "bbox": [ + 104, + 681, + 505, + 693 + ], + "score": 1.0, + "content": "In this paper, we tackle this problem by developing an RL framework for finding both an optimal", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 691, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 89, + 695, + 100, + 703 + ], + "score": 1.0, + "content": "38", + "type": "text" + }, + { + "bbox": [ + 105, + 691, + 505, + 705 + ], + "score": 1.0, + "content": "criterion to determine whether or not to execute actions as well as learning optimal actions. A key", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 703, + 505, + 716 + ], + "spans": [ + { + "bbox": [ + 89, + 705, + 100, + 715 + ], + "score": 1.0, + "content": "39", + "type": "text" + }, + { + "bbox": [ + 104, + 703, + 505, + 716 + ], + "score": 1.0, + "content": "component of our framework is a novel combination of RL with a form of policy known as impulse", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 90, + 75, + 100, + 83 + ], + "score": 1.0, + "content": "40", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "control [22, 19]. This enables the agent to determine the appropriate points to perform an action as", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 90, + 86, + 99, + 94 + ], + "score": 1.0, + "content": "41", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "well as the optimal action itself. Despite its fundamental importance as a tool for tackling decision", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 89, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "42", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "problems with costly actions, presently, the use of impulse control within learning contexts (and", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 105, + 268, + 118 + ], + "spans": [ + { + "bbox": [ + 89, + 107, + 100, + 117 + ], + "score": 1.0, + "content": "43", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 105, + 268, + 118 + ], + "score": 1.0, + "content": "unknown environments) is unaddressed.", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 121, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 89, + 124, + 100, + 133 + ], + "score": 1.0, + "content": "44", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 121, + 506, + 135 + ], + "score": 1.0, + "content": "We present an RL impulse control framework called LICRA, which, to our knowledge, is the first", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 88, + 135, + 100, + 144 + ], + "score": 1.0, + "content": "45", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "learning framework for impulse control. To enable learning optimal impulse control policies in", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 142, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 88, + 146, + 100, + 155 + ], + "score": 1.0, + "content": "46", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 142, + 506, + 158 + ], + "score": 1.0, + "content": "unknown environments, we devise a framework that consists of separate RL components for learning", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 153, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 88, + 156, + 99, + 166 + ], + "score": 1.0, + "content": "47", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 153, + 506, + 169 + ], + "score": 1.0, + "content": "when to act and how to act optimally. The resulting framework is a structured two-part learning", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 88, + 168, + 99, + 177 + ], + "score": 1.0, + "content": "48", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "process which differs from current RL protocols. In LICRA, at each time step, the agent firstly makes", + "type": "text", + "cross_page": true + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 88, + 178, + 99, + 188 + ], + "score": 1.0, + "content": "49", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 177, + 380, + 189 + ], + "score": 1.0, + "content": "a decision whether to act or not leading to a binary decision space", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 380, + 177, + 405, + 189 + ], + "score": 0.92, + "content": "\\{ 0 , 1 \\}", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 406, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "(we later show that this", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 88, + 189, + 99, + 199 + ], + "score": 1.0, + "content": "50", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "is determined by evaluating an easy-to-evaluate criterion which has the value function as its input).", + "type": "text", + "cross_page": true + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 88, + 200, + 99, + 210 + ], + "score": 1.0, + "content": "51", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "The second decision part determines the best action to take. This generates a subdivision of the state", + "type": "text", + "cross_page": true + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 89, + 211, + 99, + 221 + ], + "score": 1.0, + "content": "52", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "space into two regions; one in which the agent performs actions and another in which it does not act", + "type": "text", + "cross_page": true + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 89, + 222, + 99, + 232 + ], + "score": 1.0, + "content": "53", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "at all. This is extremely useful since the agent quickly determines the set of states to not take actions", + "type": "text", + "cross_page": true + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 231, + 450, + 243 + ], + "spans": [ + { + "bbox": [ + 89, + 233, + 100, + 242 + ], + "score": 1.0, + "content": "54", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 231, + 450, + 243 + ], + "score": 1.0, + "content": "while performing actions only at the subset of states where actions are to be executed.", + "type": "text", + "cross_page": true + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 246, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 89, + 250, + 100, + 258 + ], + "score": 1.0, + "content": "55", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 103, + 246, + 506, + 261 + ], + "score": 1.0, + "content": "We then establish theory that ensures convergence of LICRA to an optimal policy for such settings.", + "type": "text", + "cross_page": true + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 258, + 290, + 271 + ], + "spans": [ + { + "bbox": [ + 87, + 258, + 290, + 271 + ], + "score": 1.0, + "content": "56 To do this, we give a series of results namely:", + "type": "text", + "cross_page": true + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 89, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "57 i) We establish a dynamic programming principle (DPP) for impulse control and show that the optimal", + "type": "text", + "cross_page": true + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 91, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "58 value function can be obtained as a limit of a value iterative procedure (Theorem 1) which lays the", + "type": "text", + "cross_page": true + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 297, + 310, + 308 + ], + "spans": [ + { + "bbox": [ + 91, + 297, + 310, + 308 + ], + "score": 1.0, + "content": "59 foundation for an RL approach to impulse control.", + "type": "text", + "cross_page": true + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 306, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 91, + 306, + 505, + 320 + ], + "score": 1.0, + "content": "60 ii) We extend result i) to a new variant of Q learning which enables the impulse control problem to be", + "type": "text", + "cross_page": true + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 317, + 278, + 331 + ], + "spans": [ + { + "bbox": [ + 92, + 317, + 278, + 331 + ], + "score": 1.0, + "content": "61 solved using our RL method (Theorem 2).", + "type": "text", + "cross_page": true + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 89, + 331, + 100, + 340 + ], + "score": 1.0, + "content": "62", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "iii) We characterise the optimal conditions for performing an action which we reveal to be a simple", + "type": "text", + "cross_page": true + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 89, + 342, + 100, + 351 + ], + "score": 1.0, + "content": "63", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "‘obstacle condition’ involving the agent’s value function (Prop. 1). Using this, the agent can quickly", + "type": "text", + "cross_page": true + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 350, + 451, + 364 + ], + "spans": [ + { + "bbox": [ + 89, + 353, + 100, + 362 + ], + "score": 1.0, + "content": "64", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 350, + 451, + 364 + ], + "score": 1.0, + "content": "determine whether or not it should act and if so, then learn what the optimal action is.", + "type": "text", + "cross_page": true + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 92, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "65 iv) We then extend the result i) to (linear) function approximators enabling the value function to be", + "type": "text", + "cross_page": true + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 372, + 220, + 385 + ], + "spans": [ + { + "bbox": [ + 93, + 372, + 220, + 385 + ], + "score": 1.0, + "content": "66 parameterised (Theorem 3).", + "type": "text", + "cross_page": true + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 89, + 386, + 100, + 394 + ], + "score": 1.0, + "content": "67", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "iv) In Sec. 6, we extend LICRA to include budgetary constraints so that each action draws from a", + "type": "text", + "cross_page": true + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 89, + 397, + 100, + 406 + ], + "score": 1.0, + "content": "68", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "fixed budget which the agent must stay within. Analogous to the development of i), we establish", + "type": "text", + "cross_page": true + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 89, + 408, + 100, + 417 + ], + "score": 1.0, + "content": "69", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "another DPP from which we derive a Q-learning variant for tackling impulse control with budgetary", + "type": "text", + "cross_page": true + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 417, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 89, + 419, + 100, + 428 + ], + "score": 1.0, + "content": "70", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 417, + 505, + 428 + ], + "score": 1.0, + "content": "constraints (Theorem 4). A particular case of a budget constraint is when the number of actions the", + "type": "text", + "cross_page": true + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 428, + 276, + 439 + ], + "spans": [ + { + "bbox": [ + 89, + 430, + 100, + 439 + ], + "score": 1.0, + "content": "71", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 428, + 276, + 439 + ], + "score": 1.0, + "content": "agent can take over the horizon is capped.", + "type": "text", + "cross_page": true + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 93, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "72 Lastly, we perform a set of experiments to validate our theory within the Highway driving simulator", + "type": "text", + "cross_page": true + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 449, + 237, + 461 + ], + "spans": [ + { + "bbox": [ + 91, + 449, + 237, + 461 + ], + "score": 1.0, + "content": "73 and OpenAI’s LunarLander [7].", + "type": "text", + "cross_page": true + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 466, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 89, + 468, + 100, + 476 + ], + "score": 1.0, + "content": "74", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 466, + 505, + 477 + ], + "score": 1.0, + "content": "LICRA confers a series of advantages. As we demonstrate in our experiments, LICRA learns to", + "type": "text", + "cross_page": true + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 89, + 479, + 100, + 488 + ], + "score": 1.0, + "content": "75", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "compute the optimal problems in which the agent faces costs for acting in an efficient way which", + "type": "text", + "cross_page": true + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 89, + 489, + 100, + 498 + ], + "score": 1.0, + "content": "76", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "outperforms leading RL baselines. Second, as demonstrated in Sec. 6 LICRA handles settings in", + "type": "text", + "cross_page": true + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 89, + 500, + 100, + 510 + ], + "score": 1.0, + "content": "77", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "which the agent has a cap the total number of actions it is allowed to execute and more generally,", + "type": "text", + "cross_page": true + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 89, + 511, + 100, + 521 + ], + "score": 1.0, + "content": "78", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "generic budgetary constraints. LICRA is able to accommodate any RL base algorithm unlike various", + "type": "text", + "cross_page": true + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 520, + 326, + 532 + ], + "spans": [ + { + "bbox": [ + 89, + 522, + 100, + 532 + ], + "score": 1.0, + "content": "79", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 520, + 326, + 532 + ], + "score": 1.0, + "content": "RL methods designed to handle budgetary constraints.", + "type": "text", + "cross_page": true + } + ], + "index": 39, + "is_list_start_line": true + } + ], + "index": 28, + "bbox_fs": [ + 88, + 506, + 506, + 585 + ] + }, + { + "type": "index", + "bbox": [ + 90, + 588, + 505, + 676 + ], + "lines": [], + "index": 35.5, + "bbox_fs": [ + 88, + 589, + 506, + 679 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 91, + 681, + 504, + 714 + ], + "lines": [], + "index": 41, + "bbox_fs": [ + 89, + 681, + 505, + 716 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 90, + 73, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 90, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 90, + 75, + 100, + 83 + ], + "score": 1.0, + "content": "40", + "type": "text" + }, + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "control [22, 19]. This enables the agent to determine the appropriate points to perform an action as", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 90, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 90, + 86, + 99, + 94 + ], + "score": 1.0, + "content": "41", + "type": "text" + }, + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "well as the optimal action itself. Despite its fundamental importance as a tool for tackling decision", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 89, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 89, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "42", + "type": "text" + }, + { + "bbox": [ + 106, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "problems with costly actions, presently, the use of impulse control within learning contexts (and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 89, + 105, + 268, + 118 + ], + "spans": [ + { + "bbox": [ + 89, + 107, + 100, + 117 + ], + "score": 1.0, + "content": "43", + "type": "text" + }, + { + "bbox": [ + 106, + 105, + 268, + 118 + ], + "score": 1.0, + "content": "unknown environments) is unaddressed.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 89, + 122, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 89, + 121, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 89, + 124, + 100, + 133 + ], + "score": 1.0, + "content": "44", + "type": "text" + }, + { + "bbox": [ + 104, + 121, + 506, + 135 + ], + "score": 1.0, + "content": "We present an RL impulse control framework called LICRA, which, to our knowledge, is the first", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 88, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 88, + 135, + 100, + 144 + ], + "score": 1.0, + "content": "45", + "type": "text" + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "learning framework for impulse control. To enable learning optimal impulse control policies in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 88, + 142, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 88, + 146, + 100, + 155 + ], + "score": 1.0, + "content": "46", + "type": "text" + }, + { + "bbox": [ + 105, + 142, + 506, + 158 + ], + "score": 1.0, + "content": "unknown environments, we devise a framework that consists of separate RL components for learning", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 88, + 153, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 88, + 156, + 99, + 166 + ], + "score": 1.0, + "content": "47", + "type": "text" + }, + { + "bbox": [ + 104, + 153, + 506, + 169 + ], + "score": 1.0, + "content": "when to act and how to act optimally. The resulting framework is a structured two-part learning", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 88, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 88, + 168, + 99, + 177 + ], + "score": 1.0, + "content": "48", + "type": "text" + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "process which differs from current RL protocols. In LICRA, at each time step, the agent firstly makes", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 88, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 88, + 178, + 99, + 188 + ], + "score": 1.0, + "content": "49", + "type": "text" + }, + { + "bbox": [ + 105, + 177, + 380, + 189 + ], + "score": 1.0, + "content": "a decision whether to act or not leading to a binary decision space", + "type": "text" + }, + { + "bbox": [ + 380, + 177, + 405, + 189 + ], + "score": 0.92, + "content": "\\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "(we later show that this", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 88, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 88, + 189, + 99, + 199 + ], + "score": 1.0, + "content": "50", + "type": "text" + }, + { + "bbox": [ + 104, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "is determined by evaluating an easy-to-evaluate criterion which has the value function as its input).", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 88, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 88, + 200, + 99, + 210 + ], + "score": 1.0, + "content": "51", + "type": "text" + }, + { + "bbox": [ + 106, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "The second decision part determines the best action to take. This generates a subdivision of the state", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 89, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 89, + 211, + 99, + 221 + ], + "score": 1.0, + "content": "52", + "type": "text" + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "space into two regions; one in which the agent performs actions and another in which it does not act", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 89, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 89, + 222, + 99, + 232 + ], + "score": 1.0, + "content": "53", + "type": "text" + }, + { + "bbox": [ + 104, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "at all. This is extremely useful since the agent quickly determines the set of states to not take actions", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 89, + 231, + 450, + 243 + ], + "spans": [ + { + "bbox": [ + 89, + 233, + 100, + 242 + ], + "score": 1.0, + "content": "54", + "type": "text" + }, + { + "bbox": [ + 104, + 231, + 450, + 243 + ], + "score": 1.0, + "content": "while performing actions only at the subset of states where actions are to be executed.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 91, + 247, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 89, + 246, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 89, + 250, + 100, + 258 + ], + "score": 1.0, + "content": "55", + "type": "text" + }, + { + "bbox": [ + 103, + 246, + 506, + 261 + ], + "score": 1.0, + "content": "We then establish theory that ensures convergence of LICRA to an optimal policy for such settings.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 87, + 258, + 290, + 271 + ], + "spans": [ + { + "bbox": [ + 87, + 258, + 290, + 271 + ], + "score": 1.0, + "content": "56 To do this, we give a series of results namely:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 92, + 274, + 505, + 307 + ], + "lines": [ + { + "bbox": [ + 89, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 89, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "57 i) We establish a dynamic programming principle (DPP) for impulse control and show that the optimal", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 91, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 91, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "58 value function can be obtained as a limit of a value iterative procedure (Theorem 1) which lays the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 91, + 297, + 310, + 308 + ], + "spans": [ + { + "bbox": [ + 91, + 297, + 310, + 308 + ], + "score": 1.0, + "content": "59 foundation for an RL approach to impulse control.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 95, + 308, + 504, + 329 + ], + "lines": [ + { + "bbox": [ + 91, + 306, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 91, + 306, + 505, + 320 + ], + "score": 1.0, + "content": "60 ii) We extend result i) to a new variant of Q learning which enables the impulse control problem to be", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 92, + 317, + 278, + 331 + ], + "spans": [ + { + "bbox": [ + 92, + 317, + 278, + 331 + ], + "score": 1.0, + "content": "61 solved using our RL method (Theorem 2).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 91, + 330, + 504, + 362 + ], + "lines": [ + { + "bbox": [ + 89, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 89, + 331, + 100, + 340 + ], + "score": 1.0, + "content": "62", + "type": "text" + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "iii) We characterise the optimal conditions for performing an action which we reveal to be a simple", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 89, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 89, + 342, + 100, + 351 + ], + "score": 1.0, + "content": "63", + "type": "text" + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "‘obstacle condition’ involving the agent’s value function (Prop. 1). Using this, the agent can quickly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 89, + 350, + 451, + 364 + ], + "spans": [ + { + "bbox": [ + 89, + 353, + 100, + 362 + ], + "score": 1.0, + "content": "64", + "type": "text" + }, + { + "bbox": [ + 104, + 350, + 451, + 364 + ], + "score": 1.0, + "content": "determine whether or not it should act and if so, then learn what the optimal action is.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 96, + 362, + 503, + 384 + ], + "lines": [ + { + "bbox": [ + 92, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 92, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "65 iv) We then extend the result i) to (linear) function approximators enabling the value function to be", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 93, + 372, + 220, + 385 + ], + "spans": [ + { + "bbox": [ + 93, + 372, + 220, + 385 + ], + "score": 1.0, + "content": "66 parameterised (Theorem 3).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 90, + 384, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 89, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 89, + 386, + 100, + 394 + ], + "score": 1.0, + "content": "67", + "type": "text" + }, + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "iv) In Sec. 6, we extend LICRA to include budgetary constraints so that each action draws from a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 89, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 89, + 397, + 100, + 406 + ], + "score": 1.0, + "content": "68", + "type": "text" + }, + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "fixed budget which the agent must stay within. Analogous to the development of i), we establish", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 89, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 89, + 408, + 100, + 417 + ], + "score": 1.0, + "content": "69", + "type": "text" + }, + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "another DPP from which we derive a Q-learning variant for tackling impulse control with budgetary", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 89, + 417, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 89, + 419, + 100, + 428 + ], + "score": 1.0, + "content": "70", + "type": "text" + }, + { + "bbox": [ + 106, + 417, + 505, + 428 + ], + "score": 1.0, + "content": "constraints (Theorem 4). A particular case of a budget constraint is when the number of actions the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 89, + 428, + 276, + 439 + ], + "spans": [ + { + "bbox": [ + 89, + 430, + 100, + 439 + ], + "score": 1.0, + "content": "71", + "type": "text" + }, + { + "bbox": [ + 105, + 428, + 276, + 439 + ], + "score": 1.0, + "content": "agent can take over the horizon is capped.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 94, + 439, + 504, + 461 + ], + "lines": [ + { + "bbox": [ + 93, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 93, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "72 Lastly, we perform a set of experiments to validate our theory within the Highway driving simulator", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 91, + 449, + 237, + 461 + ], + "spans": [ + { + "bbox": [ + 91, + 449, + 237, + 461 + ], + "score": 1.0, + "content": "73 and OpenAI’s LunarLander [7].", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 90, + 466, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 89, + 466, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 89, + 468, + 100, + 476 + ], + "score": 1.0, + "content": "74", + "type": "text" + }, + { + "bbox": [ + 105, + 466, + 505, + 477 + ], + "score": 1.0, + "content": "LICRA confers a series of advantages. As we demonstrate in our experiments, LICRA learns to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 89, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 89, + 479, + 100, + 488 + ], + "score": 1.0, + "content": "75", + "type": "text" + }, + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "compute the optimal problems in which the agent faces costs for acting in an efficient way which", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 89, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 89, + 489, + 100, + 498 + ], + "score": 1.0, + "content": "76", + "type": "text" + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "outperforms leading RL baselines. Second, as demonstrated in Sec. 6 LICRA handles settings in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 89, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 89, + 500, + 100, + 510 + ], + "score": 1.0, + "content": "77", + "type": "text" + }, + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "which the agent has a cap the total number of actions it is allowed to execute and more generally,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 89, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 89, + 511, + 100, + 521 + ], + "score": 1.0, + "content": "78", + "type": "text" + }, + { + "bbox": [ + 104, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "generic budgetary constraints. LICRA is able to accommodate any RL base algorithm unlike various", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 89, + 520, + 326, + 532 + ], + "spans": [ + { + "bbox": [ + 89, + 522, + 100, + 532 + ], + "score": 1.0, + "content": "79", + "type": "text" + }, + { + "bbox": [ + 105, + 520, + 326, + 532 + ], + "score": 1.0, + "content": "RL methods designed to handle budgetary constraints.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "title", + "bbox": [ + 91, + 548, + 197, + 561 + ], + "lines": [ + { + "bbox": [ + 87, + 547, + 199, + 563 + ], + "spans": [ + { + "bbox": [ + 87, + 547, + 199, + 563 + ], + "score": 1.0, + "content": "80 2 Related Work", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 90, + 574, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 89, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 89, + 576, + 99, + 586 + ], + "score": 1.0, + "content": "81", + "type": "text" + }, + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "In continuous-time optimal control theory [24], problems in which the agent faces a cost for each", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 89, + 584, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 89, + 587, + 100, + 597 + ], + "score": 1.0, + "content": "82", + "type": "text" + }, + { + "bbox": [ + 105, + 584, + 506, + 597 + ], + "score": 1.0, + "content": "action are tackled with a form of policy known as impulse control [22, 19, 2]. In impulse control", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 89, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 89, + 598, + 100, + 608 + ], + "score": 1.0, + "content": "83", + "type": "text" + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "frameworks, the dynamics of the system are modified through a sequence of discrete actions or bursts", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 88, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 88, + 609, + 100, + 618 + ], + "score": 1.0, + "content": "84", + "type": "text" + }, + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "chosen at times that the agent chooses to apply the control policy. This distinguishes impulse control", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 88, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 88, + 620, + 100, + 629 + ], + "score": 1.0, + "content": "85", + "type": "text" + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "models from classical decision methods in which an agent takes actions at each time step while being", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 88, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 88, + 631, + 100, + 640 + ], + "score": 1.0, + "content": "86", + "type": "text" + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "tasked with the decision of only which action to take. 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At time", + "type": "text" + }, + { + "bbox": [ + 252, + 335, + 302, + 345 + ], + "score": 0.88, + "content": "t \\in { 0 , 1 , \\ldots }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 334, + 394, + 347 + ], + "score": 1.0, + "content": ", the system is in state", + "type": "text" + }, + { + "bbox": [ + 394, + 334, + 424, + 345 + ], + "score": 0.91, + "content": "s _ { t } \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "and the agent must", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 86, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 86, + 347, + 99, + 356 + ], + "score": 1.0, + "content": "115", + "type": "text" + }, + { + "bbox": [ + 105, + 344, + 174, + 358 + ], + "score": 1.0, + "content": "choose an action", + "type": "text" + }, + { + "bbox": [ + 174, + 345, + 204, + 356 + ], + "score": 0.91, + "content": "a _ { t } \\in \\mathcal A", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 344, + 375, + 358 + ], + "score": 1.0, + "content": "which transitions the system to a new state", + "type": "text" + }, + { + "bbox": [ + 375, + 345, + 449, + 357 + ], + "score": 0.92, + "content": "s _ { t + 1 } \\sim P ( \\cdot | s _ { t } , a _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "and produces", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 86, + 355, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 86, + 358, + 99, + 367 + ], + "score": 1.0, + "content": "116", + "type": "text" + }, + { + "bbox": [ + 104, + 355, + 144, + 370 + ], + "score": 1.0, + "content": "a reward", + "type": "text" + }, + { + "bbox": [ + 144, + 356, + 181, + 368 + ], + "score": 0.9, + "content": "R ( s _ { t } , a _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 355, + 223, + 370 + ], + "score": 1.0, + "content": ". 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The goal of an", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 86, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 86, + 379, + 100, + 390 + ], + "score": 1.0, + "content": "118", + "type": "text" + }, + { + "bbox": [ + 105, + 377, + 218, + 390 + ], + "score": 1.0, + "content": "RL agent is to find a policy", + "type": "text" + }, + { + "bbox": [ + 219, + 378, + 246, + 388 + ], + "score": 0.89, + "content": "{ \\hat { \\pi } } \\in \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "that maximises its expected returns given by the value function:", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 384, + 508, + 419 + ], + "spans": [ + { + "bbox": [ + 86, + 390, + 100, + 412 + ], + "score": 1.0, + "content": "119", + "type": "text" + }, + { + "bbox": [ + 107, + 388, + 316, + 401 + ], + "score": 0.86, + "content": "\\begin{array} { r } { v ^ { \\pi } ( s ) = \\mathbb { E } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\bar { R } ( s _ { t } , a _ { t } ) | a _ { t } \\sim \\pi ( \\cdot | s _ { t } ) , s _ { 0 } = s ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 401, + 393, + 414 + ], + "score": 0.89, + "content": "\\begin{array} { r } { Q ( s , a ) = \\mathbb { E } [ \\sum _ { t = 0 } ^ { \\infty } R ( s _ { t } , a _ { t } ) | a _ { 0 } = a , s _ { 0 } = s ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 389, + 354, + 399 + ], + "score": 0.63, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 384, + 508, + 419 + ], + "score": 1.0, + "content": "ent’s policy set. The action.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 90, + 417, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 87, + 416, + 507, + 429 + ], + "spans": [ + { + "bbox": [ + 87, + 416, + 507, + 429 + ], + "score": 1.0, + "content": "121 We consider a setting in which the agent faces at least some minimal cost for each action it performs.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 87, + 427, + 275, + 441 + ], + "spans": [ + { + "bbox": [ + 87, + 427, + 275, + 441 + ], + "score": 1.0, + "content": "122 With this, the agent’s task is to maximise:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 451, + 413, + 486 + ], + "lines": [ + { + "bbox": [ + 197, + 451, + 413, + 486 + ], + "spans": [ + { + "bbox": [ + 197, + 451, + 413, + 486 + ], + "score": 0.93, + "content": "v ^ { \\pi } ( s ) = \\mathbb { E } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\left\\{ \\mathcal { R } ( s _ { t } , a _ { t } ) - \\mathcal { C } ( s _ { t } , a _ { t } ) \\right\\} \\Big | s _ { 0 } = s \\right] ,", + "type": "interline_equation", + "image_path": "dbb0ac7205352125e445c169be6d5e1f9f57f8b1f68b460d5971f8dca0f3502a.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 197, + 451, + 413, + 468.5 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 197, + 468.5, + 413, + 486.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 497, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 86, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 86, + 500, + 100, + 509 + ], + "score": 1.0, + "content": "123", + "type": "text" + }, + { + "bbox": [ + 105, + 497, + 187, + 510 + ], + "score": 1.0, + "content": "where for any state", + "type": "text" + }, + { + "bbox": [ + 188, + 498, + 215, + 508 + ], + "score": 0.9, + "content": "s \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 497, + 280, + 510 + ], + "score": 1.0, + "content": "and any action", + "type": "text" + }, + { + "bbox": [ + 280, + 498, + 308, + 508 + ], + "score": 0.91, + "content": "a \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 497, + 369, + 510 + ], + "score": 1.0, + "content": ", the functions", + "type": "text" + }, + { + "bbox": [ + 369, + 498, + 379, + 507 + ], + "score": 0.84, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 497, + 398, + 510 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 398, + 498, + 405, + 507 + ], + "score": 0.81, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 497, + 460, + 510 + ], + "score": 1.0, + "content": "are given by", + "type": "text" + }, + { + "bbox": [ + 461, + 497, + 505, + 509 + ], + "score": 0.9, + "content": "\\textstyle { \\mathcal { R } } ( s , a ) =", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 86, + 507, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 86, + 510, + 100, + 521 + ], + "score": 1.0, + "content": "124", + "type": "text" + }, + { + "bbox": [ + 106, + 509, + 243, + 520 + ], + "score": 0.89, + "content": "R ( s , a ) \\mathbf { 1 } _ { a \\in \\mathcal { A } } + R ( s , 0 ) ( 1 - 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An investor performs a series", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 86, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 86, + 603, + 100, + 613 + ], + "score": 1.0, + "content": "132", + "type": "text" + }, + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "of costly portfolio adjustments by buying and selling amounts of different assets within their portfolio.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 86, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 86, + 614, + 100, + 624 + ], + "score": 1.0, + "content": "133", + "type": "text" + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "Each investment incurs a fixed minimal cost (also known as transaction costs) which is deducted", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 86, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 86, + 625, + 100, + 635 + ], + "score": 1.0, + "content": "134", + "type": "text" + }, + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "from the investor’s available cash-flow. 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Each action", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 90, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 90, + 669, + 100, + 677 + ], + "score": 1.0, + "content": "38", + "type": "text" + }, + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "draws from its fuel budget. 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The action.", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 416, + 507, + 429 + ], + "spans": [ + { + "bbox": [ + 87, + 416, + 507, + 429 + ], + "score": 1.0, + "content": "121 We consider a setting in which the agent faces at least some minimal cost for each action it performs.", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 427, + 275, + 441 + ], + "spans": [ + { + "bbox": [ + 87, + 427, + 275, + 441 + ], + "score": 1.0, + "content": "122 With this, the agent’s task is to maximise:", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + } + ], + "index": 20, + "bbox_fs": [ + 85, + 279, + 508, + 419 + ] + }, + { + "type": "index", + "bbox": [ + 90, + 417, + 505, + 440 + ], + "lines": [], + "index": 26.5, + "bbox_fs": [ + 87, + 416, + 507, + 441 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 451, + 413, + 486 + ], + "lines": [ + { + "bbox": [ + 197, + 451, + 413, + 486 + ], + "spans": [ + { + "bbox": [ + 197, + 451, + 413, + 486 + ], + "score": 0.93, + "content": "v ^ { \\pi } ( s ) = \\mathbb { E } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\left\\{ \\mathcal { R } ( s _ { t } , a _ { t } ) - 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Therefore, in this setting, the agent’s problem is augmented to", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 572, + 507, + 588 + ], + "spans": [ + { + "bbox": [ + 86, + 575, + 100, + 586 + ], + "score": 1.0, + "content": "130", + "type": "text" + }, + { + "bbox": [ + 104, + 572, + 507, + 588 + ], + "score": 1.0, + "content": "learning both an optimal policy for its actions and, learning at which states to apply its action policy.", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 591, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 86, + 592, + 99, + 602 + ], + "score": 1.0, + "content": "131", + "type": "text" + }, + { + "bbox": [ + 106, + 591, + 505, + 602 + ], + "score": 1.0, + "content": "Example: Merton Portfolio Problem with Transaction Costs [10]. An investor performs a series", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 86, + 603, + 100, + 613 + ], + "score": 1.0, + "content": "132", + "type": "text" + }, + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "of costly portfolio adjustments by buying and selling amounts of different assets within their portfolio.", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 86, + 614, + 100, + 624 + ], + "score": 1.0, + "content": "133", + "type": "text" + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "Each investment incurs a fixed minimal cost (also known as transaction costs) which is deducted", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 86, + 625, + 100, + 635 + ], + "score": 1.0, + "content": "134", + "type": "text" + }, + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "from the investor’s available cash-flow. The investor’s aim is to maximise their total wealth (the value", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 634, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 86, + 636, + 100, + 646 + ], + "score": 1.0, + "content": "135", + "type": "text" + }, + { + "bbox": [ + 106, + 634, + 505, + 646 + ], + "score": 1.0, + "content": "of the sum of their assets) at some time horizon by adjusting their portfolio of investments. Problems", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 645, + 486, + 658 + ], + "spans": [ + { + "bbox": [ + 86, + 646, + 100, + 657 + ], + "score": 1.0, + "content": "136", + "type": "text" + }, + { + "bbox": [ + 105, + 645, + 486, + 658 + ], + "score": 1.0, + "content": "of this kind, portfolio investment problems are of fundamental importance within finance [18].", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 90, + 658, + 103, + 667 + ], + "score": 1.0, + "content": "37", + "type": "text" + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "Example 2. An autonomous vehicle must perform a series of actions to perform a task. Each action", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 90, + 669, + 100, + 677 + ], + "score": 1.0, + "content": "38", + "type": "text" + }, + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "draws from its fuel budget. In order to complete its task successfully, during the task, the vehicle", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 677, + 286, + 690 + ], + "spans": [ + { + "bbox": [ + 89, + 677, + 286, + 690 + ], + "score": 1.0, + "content": "139 must ensure it maintains an available supply.", + "type": "text" + } + ], + "index": 46, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 95, + 505, + 110 + ], + "spans": [ + { + "bbox": [ + 86, + 98, + 99, + 108 + ], + "score": 1.0, + "content": "141", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 95, + 505, + 110 + ], + "score": 1.0, + "content": "In RL, the agent’s problem involves learning to act at every state including those in which actions do", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 108, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 86, + 109, + 100, + 119 + ], + "score": 1.0, + "content": "142", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 108, + 477, + 119 + ], + "score": 1.0, + "content": "not significantly impact on its total return. While we can add a zero action to the action set", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 478, + 108, + 487, + 117 + ], + "score": 0.76, + "content": "\\mathcal { A }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 487, + 108, + 505, + 119 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 118, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 86, + 120, + 100, + 130 + ], + "score": 1.0, + "content": "143", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 118, + 505, + 132 + ], + "score": 1.0, + "content": "apply standard methods, we argue that this may not be the best solution in many situations. We argue", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 129, + 277, + 141 + ], + "spans": [ + { + "bbox": [ + 86, + 131, + 100, + 140 + ], + "score": 1.0, + "content": "144", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 129, + 277, + 141 + ], + "score": 1.0, + "content": "the optimal policy has the following form:", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + } + ], + "index": 33.5, + "bbox_fs": [ + 85, + 497, + 507, + 588 + ] + }, + { + "type": "index", + "bbox": [ + 87, + 590, + 505, + 656 + ], + "lines": [], + "index": 40.5, + "bbox_fs": [ + 86, + 591, + 505, + 658 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 92, + 657, + 506, + 690 + ], + "lines": [], + "index": 45, + "bbox_fs": [ + 89, + 656, + 505, + 690 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 86, + 96, + 505, + 141 + ], + "lines": [ + { + "bbox": [ + 86, + 95, + 505, + 110 + ], + "spans": [ + { + "bbox": [ + 86, + 98, + 99, + 108 + ], + "score": 1.0, + "content": "141", + "type": "text" + }, + { + "bbox": [ + 105, + 95, + 505, + 110 + ], + "score": 1.0, + "content": "In RL, the agent’s problem involves learning to act at every state including those in which actions do", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 86, + 108, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 86, + 109, + 100, + 119 + ], + "score": 1.0, + "content": "142", + "type": "text" + }, + { + "bbox": [ + 105, + 108, + 477, + 119 + ], + "score": 1.0, + "content": "not significantly impact on its total return. 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We argue", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 86, + 129, + 277, + 141 + ], + "spans": [ + { + "bbox": [ + 86, + 131, + 100, + 140 + ], + "score": 1.0, + "content": "144", + "type": "text" + }, + { + "bbox": [ + 106, + 129, + 277, + 141 + ], + "score": 1.0, + "content": "the optimal policy has the following form:", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 147, + 353, + 175 + ], + "lines": [ + { + "bbox": [ + 257, + 147, + 353, + 175 + ], + "spans": [ + { + "bbox": [ + 257, + 147, + 353, + 175 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\widetilde { \\pi } ( \\cdot | s ) = \\left\\{ \\begin{array} { l l } { a _ { t } } & { s \\in S _ { I } , } \\\\ { 0 } & { s \\notin S _ { I } , } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "80e033ef294b86d5333ac21d90363be2a6475df9099dfe3ede1d62f8e5f0363f.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 257, + 147, + 353, + 175 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 88, + 181, + 505, + 205 + ], + "lines": [ + { + "bbox": [ + 84, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 84, + 181, + 390, + 195 + ], + "score": 1.0, + "content": "145 which implies that we simplify policy learning by determining the set", + "type": "text" + }, + { + "bbox": [ + 390, + 182, + 402, + 193 + ], + "score": 0.88, + "content": " { \\boldsymbol { S } } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 181, + 505, + 195 + ], + "score": 1.0, + "content": "first — the set where we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 84, + 192, + 238, + 206 + ], + "spans": [ + { + "bbox": [ + 84, + 192, + 238, + 206 + ], + "score": 1.0, + "content": "146 actually need to learn the policy.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 87, + 208, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 86, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 86, + 210, + 99, + 220 + ], + "score": 1.0, + "content": "147", + "type": "text" + }, + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "We now introduce a learning method for producing impulse controls. This enables the agent to learn", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 86, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 86, + 222, + 99, + 232 + ], + "score": 1.0, + "content": "148", + "type": "text" + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "to select states to perform actions. Therefore, now agent is tasked with learning to act at states that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 86, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 86, + 233, + 99, + 242 + ], + "score": 1.0, + "content": "149", + "type": "text" + }, + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "are most important for maximising its total return given the presence of the cost for each action. Now", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 86, + 241, + 438, + 254 + ], + "spans": [ + { + "bbox": [ + 86, + 244, + 99, + 253 + ], + "score": 1.0, + "content": "150", + "type": "text" + }, + { + "bbox": [ + 105, + 241, + 438, + 254 + ], + "score": 1.0, + "content": "at each state the agent first makes a binary decision to decide to perform an action.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 86, + 257, + 506, + 389 + ], + "lines": [ + { + "bbox": [ + 86, + 257, + 504, + 271 + ], + "spans": [ + { + "bbox": [ + 86, + 260, + 99, + 270 + ], + "score": 1.0, + "content": "151", + "type": "text" + }, + { + "bbox": [ + 105, + 257, + 412, + 271 + ], + "score": 1.0, + "content": "Our framework, LICRA consists of two core components: firstly a RL process", + "type": "text" + }, + { + "bbox": [ + 413, + 258, + 504, + 270 + ], + "score": 0.93, + "content": "\\mathfrak { g } : \\mathcal { S } \\times \\{ 0 , 1 \\} [ 0 , 1 ]", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 86, + 271, + 99, + 280 + ], + "score": 1.0, + "content": "152", + "type": "text" + }, + { + "bbox": [ + 105, + 268, + 208, + 282 + ], + "score": 1.0, + "content": "and a second RL process", + "type": "text" + }, + { + "bbox": [ + 208, + 269, + 286, + 281 + ], + "score": 0.92, + "content": "\\pi : S \\times A \\to [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 268, + 336, + 282 + ], + "score": 1.0, + "content": ". 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Crucially, the agent must", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 86, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 86, + 370, + 99, + 379 + ], + "score": 1.0, + "content": "161", + "type": "text" + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "compute optimal actions at only a subset of states which are chosen by the policy g. 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This mechanism forces the agent to first determine the set of states to perform actions", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 86, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "167", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "only then determine the optimal actions at these states. 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This results in the agent rapidly learning which states to focus on to learn which actions", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 85, + 151, + 100, + 162 + ], + "score": 1.0, + "content": "172", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 149, + 213, + 162 + ], + "score": 1.0, + "content": "to perform. In the case of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 213, + 151, + 221, + 159 + ], + "score": 0.71, + "content": "\\pi", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 221, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "with a continuous action space again the impulse control mechanism", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 85, + 162, + 100, + 172 + ], + "score": 1.0, + "content": "173", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 160, + 400, + 172 + ], + "score": 1.0, + "content": "does not change the cardinality of the action space. 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We", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 86, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 86, + 398, + 100, + 409 + ], + "score": 1.0, + "content": "191", + "type": "text" + }, + { + "bbox": [ + 105, + 396, + 505, + 409 + ], + "score": 1.0, + "content": "provide a result that shows the optimal intervention times are characterised by an ‘obstacle condition’", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 406, + 330, + 422 + ], + "spans": [ + { + "bbox": [ + 86, + 410, + 100, + 419 + ], + "score": 1.0, + "content": "192", + "type": "text" + }, + { + "bbox": [ + 104, + 406, + 318, + 422 + ], + "score": 1.0, + "content": "which can be evaluated online therefore allowing the", + "type": "text" + }, + { + "bbox": [ + 319, + 410, + 325, + 420 + ], + "score": 0.61, + "content": "\\mathfrak { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 406, + 330, + 422 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 98, + 423, + 506, + 467 + ], + "lines": [ + { + "bbox": [ + 106, + 422, + 507, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 187, + 437 + ], + "score": 1.0, + "content": "Given a function", + "type": "text" + }, + { + "bbox": [ + 187, + 424, + 356, + 436 + ], + "score": 0.68, + "content": "Q \\because S \\times \\mathcal { A } \\to \\mathbb { R } , \\quad \\forall \\pi , \\pi ^ { \\prime } \\in \\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 422, + 380, + 437 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 381, + 424, + 465, + 437 + ], + "score": 0.74, + "content": "\\forall \\ S , \\ S ^ { \\prime } , \\quad \\forall s _ { \\tau _ { k } } \\quad \\in \\quad S", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 422, + 507, + 437 + ], + "score": 1.0, + "content": ", we de-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 433, + 507, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 234, + 452 + ], + "score": 1.0, + "content": "fine the intervention operator", + "type": "text" + }, + { + "bbox": [ + 234, + 439, + 259, + 447 + ], + "score": 0.84, + "content": "\\mathcal { M } ^ { \\pi , { \\mathfrak { g } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 433, + 276, + 452 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 277, + 436, + 507, + 450 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathcal { M } ( ^ { \\pi , \\oplus } Q ^ { \\pi ^ { \\prime } , \\oplus ^ { \\prime } } ( s _ { \\pi _ { k } } , a _ { \\pi _ { k } } ) : = { \\mathcal R } ( s _ { \\pi _ { k } } , a _ { \\pi _ { k } } ) - c ( s _ { \\pi _ { k } } , a _ { \\pi _ { k } } ) + } \\end{array}", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 449, + 434, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 297, + 469 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\gamma \\sum _ { s ^ { \\prime } \\in S } P ( s ^ { \\prime } ; a _ { \\tau _ { k } } , s ) v ^ { \\pi ^ { \\prime } , \\mathfrak { g ^ { \\prime } } } ( s ^ { \\prime } ) \\| a _ { \\tau _ { k } } \\sim \\pi ( \\cdot | s _ { \\tau _ { k } } ) , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 449, + 326, + 466 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 327, + 453, + 337, + 464 + ], + "score": 0.82, + "content": "\\tau _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 449, + 434, + 466 + ], + "score": 1.0, + "content": "is an intervention time.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 86, + 471, + 506, + 549 + ], + "lines": [ + { + "bbox": [ + 86, + 471, + 504, + 486 + ], + "spans": [ + { + "bbox": [ + 86, + 474, + 100, + 483 + ], + "score": 1.0, + "content": "196", + "type": "text" + }, + { + "bbox": [ + 105, + 471, + 191, + 486 + ], + "score": 1.0, + "content": "The interpretation of", + "type": "text" + }, + { + "bbox": [ + 191, + 473, + 204, + 482 + ], + "score": 0.76, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 471, + 438, + 486 + ], + "score": 1.0, + "content": "is the following: suppose that the agent is using the policy", + "type": "text" + }, + { + "bbox": [ + 438, + 474, + 446, + 482 + ], + "score": 0.74, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 471, + 493, + 486 + ], + "score": 1.0, + "content": "and at time", + "type": "text" + }, + { + "bbox": [ + 493, + 474, + 504, + 483 + ], + "score": 0.8, + "content": "\\tau _ { k }", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 86, + 480, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 86, + 484, + 100, + 494 + ], + "score": 1.0, + "content": "197", + "type": "text" + }, + { + "bbox": [ + 104, + 480, + 201, + 498 + ], + "score": 1.0, + "content": "the system is at a state", + "type": "text" + }, + { + "bbox": [ + 201, + 485, + 214, + 495 + ], + "score": 0.81, + "content": "s _ { \\tau _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 480, + 353, + 498 + ], + "score": 1.0, + "content": "and the agent performs an action", + "type": "text" + }, + { + "bbox": [ + 353, + 483, + 415, + 495 + ], + "score": 0.89, + "content": "a _ { \\tau _ { k } } \\sim \\pi ( \\cdot | s _ { \\tau _ { k } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 480, + 460, + 498 + ], + "score": 1.0, + "content": ". 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In particular,", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 624, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 85, + 625, + 100, + 635 + ], + "score": 1.0, + "content": "207", + "type": "text" + }, + { + "bbox": [ + 106, + 624, + 505, + 635 + ], + "score": 1.0, + "content": "the structure in (3) consists of an inner structure which consists of two terms: the first term is the", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 85, + 636, + 100, + 646 + ], + "score": 1.0, + "content": "208", + "type": "text" + }, + { + "bbox": [ + 105, + 635, + 426, + 647 + ], + "score": 1.0, + "content": "expected future return given an action is taken at the current state under the policy", + "type": "text" + }, + { + "bbox": [ + 427, + 640, + 433, + 645 + ], + "score": 0.84, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 635, + 505, + 647 + ], + "score": 1.0, + "content": ". The second term", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 646, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 85, + 647, + 100, + 657 + ], + "score": 1.0, + "content": "209", + "type": "text" + }, + { + "bbox": [ + 105, + 646, + 506, + 658 + ], + "score": 1.0, + "content": "is the expected future return given no action is taken at the current state. Lastly, the outer structure is", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 656, + 507, + 669 + ], + "spans": [ + { + "bbox": [ + 86, + 659, + 100, + 668 + ], + "score": 1.0, + "content": "210", + "type": "text" + }, + { + "bbox": [ + 105, + 656, + 507, + 669 + ], + "score": 1.0, + "content": "an optimisation which compares the expected return of the two possibilities and selects the maximum.", + "type": "text" + } + ], + "index": 46, + "is_list_start_line": true + } + ], + "index": 44, + "bbox_fs": [ + 85, + 613, + 507, + 669 + ] + }, + { + "type": "text", + "bbox": [ + 95, + 672, + 466, + 685 + ], + "lines": [ + { + "bbox": [ + 92, + 672, + 468, + 686 + ], + "spans": [ + { + "bbox": [ + 92, + 672, + 196, + 686 + ], + "score": 1.0, + "content": "11 Our first result proves", + "type": "text" + }, + { + "bbox": [ + 196, + 674, + 204, + 683 + ], + "score": 0.83, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 672, + 468, + 686 + ], + "score": 1.0, + "content": "is a contraction operator in particular, the following bound holds:", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47, + "bbox_fs": [ + 92, + 672, + 468, + 686 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 693, + 455, + 705 + ], + "lines": [ + { + "bbox": [ + 84, + 692, + 456, + 706 + ], + "spans": [ + { + "bbox": [ + 84, + 692, + 242, + 706 + ], + "score": 1.0, + "content": "212 Lemma 1 The Bellman operator", + "type": "text" + }, + { + "bbox": [ + 243, + 695, + 250, + 703 + ], + "score": 0.85, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 692, + 456, + 706 + ], + "score": 1.0, + "content": "is a contraction, that is the following bound holds:", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48, + "bbox_fs": [ + 84, + 692, + 456, + 706 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 709, + 361, + 723 + ], + "lines": [ + { + "bbox": [ + 250, + 709, + 361, + 723 + ], + "spans": [ + { + "bbox": [ + 250, + 709, + 361, + 723 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\| T v - T v ^ { \\prime } \\| \\leq \\gamma \\left. v - v ^ { \\prime } \\right. , } \\end{array}", + "type": "interline_equation", + "image_path": "cf50390db76541add22697b97f1f8bef43214a39a5b762388d0195cd6f10c1c6.jpg" + } + ] + } + ], + "index": 49, + "virtual_lines": [ + { + "bbox": [ + 250, + 709, + 361, + 723 + ], + "spans": [], + "index": 49 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 95, + 72, + 490, + 85 + ], + "lines": [ + { + "bbox": [ + 92, + 72, + 489, + 86 + ], + "spans": [ + { + "bbox": [ + 92, + 72, + 133, + 86 + ], + "score": 1.0, + "content": "where 13", + "type": "text" + }, + { + "bbox": [ + 133, + 73, + 151, + 84 + ], + "score": 0.93, + "content": "v , v ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 72, + 489, + 86 + ], + "score": 1.0, + "content": "are elements of a finite normed vector space. We can now state our first main result:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 103, + 91, + 505, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 90, + 505, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 201, + 105 + ], + "score": 1.0, + "content": "Theorem 1 Given any", + "type": "text" + }, + { + "bbox": [ + 202, + 92, + 281, + 103 + ], + "score": 0.9, + "content": "v ^ { \\pi , \\mathfrak { g } } : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 90, + 442, + 105 + ], + "score": 1.0, + "content": ", the optimal value function is given by", + "type": "text" + }, + { + "bbox": [ + 442, + 91, + 505, + 109 + ], + "score": 0.91, + "content": "\\operatorname* { l i m } _ { k \\to \\infty } T ^ { k } v ^ { \\pi , \\mathfrak { g } } =", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 106, + 354, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 108, + 186, + 128 + ], + "score": 0.85, + "content": "\\operatorname* { m a x } _ { \\hat { \\pi } , \\hat { g } \\in \\Pi } v ^ { \\hat { \\pi } , \\hat { g } } = v ^ { \\pi ^ { \\star } , g ^ { \\star } }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 106, + 214, + 124 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 214, + 109, + 248, + 122 + ], + "score": 0.91, + "content": "( \\pi ^ { \\star } , g ^ { \\star } )", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 106, + 354, + 124 + ], + "score": 1.0, + "content": "is the optimal policy pair.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 90, + 135, + 506, + 169 + ], + "lines": [ + { + "bbox": [ + 105, + 135, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 505, + 147 + ], + "score": 1.0, + "content": "The result of Theorem 1 enables the solution to the agent’s impulse control problem to be determined", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 145, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 104, + 145, + 505, + 159 + ], + "score": 1.0, + "content": "using a value iteration procedure. Moreover, Theorem 1 enables a Q-learning approach [6] for finding", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 157, + 249, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 249, + 169 + ], + "score": 1.0, + "content": "the solution to the agent’s problem.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 87, + 176, + 331, + 189 + ], + "lines": [ + { + "bbox": [ + 85, + 175, + 331, + 190 + ], + "spans": [ + { + "bbox": [ + 85, + 175, + 251, + 190 + ], + "score": 1.0, + "content": "219 Theorem 2 Consider the following", + "type": "text" + }, + { + "bbox": [ + 252, + 177, + 260, + 188 + ], + "score": 0.74, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 175, + 331, + 190 + ], + "score": 1.0, + "content": "learning variant:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 204, + 484, + 232 + ], + "lines": [ + { + "bbox": [ + 115, + 204, + 484, + 232 + ], + "spans": [ + { + "bbox": [ + 115, + 204, + 484, + 232 + ], + "score": 0.87, + "content": "+ \\alpha _ { t } ( s _ { t } , a _ { t } ) \\left[ \\operatorname* { m a x } \\left\\{ \\mathcal { M } ^ { \\pi , \\mathfrak { g } } Q _ { t } ( s _ { t } , a _ { t } ) , \\mathcal { R } ( s _ { t } , 0 ) + \\gamma _ { a ^ { \\prime } \\in A } ^ { \\operatorname* { m a x } } Q _ { t } ( s _ { t + 1 } , a ^ { \\prime } ) \\right\\} - Q _ { t } ( s _ { t } , a _ { t } ) \\right] ,", + "type": "interline_equation", + "image_path": "23f8930b1b8b17d1d3a70216119b13eb66fa3f4560ca71ea1ccb26e4c5ed1067.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 115, + 204, + 484, + 213.33333333333334 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 115, + 213.33333333333334, + 484, + 222.66666666666669 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 115, + 222.66666666666669, + 484, + 232.00000000000003 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 90, + 233, + 412, + 246 + ], + "lines": [ + { + "bbox": [ + 87, + 232, + 412, + 247 + ], + "spans": [ + { + "bbox": [ + 87, + 232, + 126, + 247 + ], + "score": 1.0, + "content": "then 220", + "type": "text" + }, + { + "bbox": [ + 126, + 234, + 138, + 245 + ], + "score": 0.88, + "content": "Q _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 232, + 192, + 247 + ], + "score": 1.0, + "content": "converges to", + "type": "text" + }, + { + "bbox": [ + 192, + 234, + 206, + 245 + ], + "score": 0.89, + "content": "Q ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 232, + 309, + 247 + ], + "score": 1.0, + "content": "with probability 1, where", + "type": "text" + }, + { + "bbox": [ + 310, + 234, + 361, + 245 + ], + "score": 0.9, + "content": "s _ { t } , s _ { t + 1 } \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 232, + 379, + 247 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 380, + 234, + 410, + 244 + ], + "score": 0.89, + "content": "a _ { t } \\in \\mathcal A", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 232, + 412, + 247 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 95, + 253, + 349, + 265 + ], + "lines": [ + { + "bbox": [ + 92, + 252, + 350, + 267 + ], + "spans": [ + { + "bbox": [ + 92, + 252, + 350, + 267 + ], + "score": 1.0, + "content": "21 We now extend the result to (linear) function approximators:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 105, + 272, + 506, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 357, + 285 + ], + "score": 1.0, + "content": "Theorem 3 Given a set of linearly independent basis functions", + "type": "text" + }, + { + "bbox": [ + 358, + 272, + 432, + 285 + ], + "score": 0.94, + "content": "\\Phi = \\{ \\phi _ { 1 } , \\ldots , \\phi _ { p } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 272, + 452, + 285 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 452, + 273, + 503, + 284 + ], + "score": 0.88, + "content": "\\phi _ { k } \\in L _ { 2 } , \\forall k", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 272, + 505, + 285 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 282, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 243, + 297 + ], + "score": 1.0, + "content": "LICRA converges to a limit point", + "type": "text" + }, + { + "bbox": [ + 243, + 284, + 278, + 294 + ], + "score": 0.91, + "content": "r ^ { \\star } \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 282, + 407, + 297 + ], + "score": 1.0, + "content": "which is the unique solution to", + "type": "text" + }, + { + "bbox": [ + 408, + 284, + 476, + 295 + ], + "score": 0.91, + "content": "\\Pi \\mathfrak { F } ( \\Phi r ^ { \\star } ) = \\Phi r ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 282, + 505, + 297 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 294, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 227, + 308 + ], + "score": 0.88, + "content": "\\mathfrak { F } v : = \\mathcal { R } + \\gamma P \\operatorname* { m a x } \\{ \\mathcal { M } v , v \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 294, + 275, + 310 + ], + "score": 1.0, + "content": ". Moreover,", + "type": "text" + }, + { + "bbox": [ + 276, + 297, + 286, + 306 + ], + "score": 0.84, + "content": "r ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 294, + 325, + 310 + ], + "score": 1.0, + "content": "satisfies:", + "type": "text" + }, + { + "bbox": [ + 326, + 295, + 501, + 308 + ], + "score": 0.89, + "content": "\\| \\Phi r ^ { \\star } - Q ^ { \\star } \\| \\leq ( 1 - \\gamma ^ { 2 } ) ^ { - 1 / 2 } \\| \\Pi Q ^ { \\star } - Q ^ { \\star } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 294, + 505, + 310 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 105, + 315, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 328 + ], + "score": 1.0, + "content": "The theorem establishes the convergence of LICRA to a stable point with the use of linear function", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "approximators. The second statement bounds the proximity of the convergence point by the smallest", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 338, + 416, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 416, + 349 + ], + "score": 1.0, + "content": "approximation error that can be achieved given the choice of basis functions.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "Having constructed a procedure to find the optimal agent’s optimal value function, we now seek to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 363, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 453, + 378 + ], + "score": 1.0, + "content": "determine the conditions when an intervention should be performed. Let us denote by", + "type": "text" + }, + { + "bbox": [ + 454, + 365, + 489, + 377 + ], + "score": 0.93, + "content": "\\{ \\tau _ { k } \\} _ { k \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 363, + 505, + 378 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "points at which the agent decides to act or intervention times, so for example if the agent chooses", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 222, + 399 + ], + "score": 1.0, + "content": "to perform an action at state", + "type": "text" + }, + { + "bbox": [ + 222, + 388, + 233, + 398 + ], + "score": 0.84, + "content": "s _ { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 386, + 306, + 399 + ], + "score": 1.0, + "content": "and again at state", + "type": "text" + }, + { + "bbox": [ + 306, + 388, + 317, + 398 + ], + "score": 0.84, + "content": "s _ { 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 386, + 340, + 399 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 340, + 387, + 369, + 398 + ], + "score": 0.91, + "content": "\\tau _ { 1 } = 6", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 386, + 387, + 399 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 387, + 387, + 416, + 397 + ], + "score": 0.9, + "content": "\\tau _ { 2 } = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 386, + 506, + 399 + ], + "score": 1.0, + "content": ". The following result", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 395, + 422, + 413 + ], + "spans": [ + { + "bbox": [ + 104, + 395, + 285, + 413 + ], + "score": 1.0, + "content": "characterises the optimal intervention policy", + "type": "text" + }, + { + "bbox": [ + 286, + 399, + 292, + 409 + ], + "score": 0.76, + "content": "\\mathfrak { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 395, + 382, + 413 + ], + "score": 1.0, + "content": "and the optimal times", + "type": "text" + }, + { + "bbox": [ + 382, + 398, + 417, + 410 + ], + "score": 0.91, + "content": "\\{ \\tau _ { k } \\} _ { k \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 395, + 422, + 413 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 104, + 415, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 272, + 431 + ], + "score": 1.0, + "content": "Proposition 1 The policy g is given by:", + "type": "text" + }, + { + "bbox": [ + 272, + 417, + 474, + 429 + ], + "score": 0.85, + "content": "\\mathfrak { g } ( s _ { t } ) = H ( \\mathcal { M } ^ { \\pi , \\mathfrak { g } } Q ^ { \\pi , \\mathfrak { g } } - Q ^ { \\pi , \\mathfrak { g } } ) ( s _ { t } , a _ { t } ) , \\forall s _ { t } \\in \\mathcal { S } _ { \\mathrm { ~ \\Gamma ~ } }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 415, + 506, + 431 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 426, + 507, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 127, + 440 + ], + "score": 0.87, + "content": "Q ^ { \\pi , { \\mathfrak { g } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 426, + 236, + 441 + ], + "score": 1.0, + "content": "is the solution in Theorem", + "type": "text" + }, + { + "bbox": [ + 236, + 429, + 242, + 438 + ], + "score": 0.27, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 426, + 246, + 441 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 246, + 429, + 259, + 438 + ], + "score": 0.67, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 426, + 390, + 441 + ], + "score": 1.0, + "content": "is the intervention operator and", + "type": "text" + }, + { + "bbox": [ + 391, + 429, + 401, + 438 + ], + "score": 0.81, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 426, + 507, + 441 + ], + "score": 1.0, + "content": "is the Heaviside function,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 437, + 419, + 452 + ], + "spans": [ + { + "bbox": [ + 104, + 437, + 251, + 452 + ], + "score": 1.0, + "content": "moreover the intervention times are", + "type": "text" + }, + { + "bbox": [ + 251, + 439, + 415, + 451 + ], + "score": 0.93, + "content": "\\tau _ { k } = \\operatorname* { i n f } \\{ \\tau > \\tau _ { k - 1 } | \\mathcal { M } ^ { \\pi , { \\mathfrak { g } } } Q ^ { \\pi , { \\mathfrak { g } } } = Q ^ { \\pi , { \\mathfrak { g } } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 437, + 419, + 452 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 100, + 458, + 504, + 482 + ], + "lines": [ + { + "bbox": [ + 104, + 456, + 504, + 473 + ], + "spans": [ + { + "bbox": [ + 104, + 456, + 305, + 473 + ], + "score": 1.0, + "content": "Prop. 1 characterises the (categorical) distribution", + "type": "text" + }, + { + "bbox": [ + 305, + 460, + 311, + 470 + ], + "score": 0.42, + "content": "\\mathfrak { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 456, + 432, + 473 + ], + "score": 1.0, + "content": ". 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In particular, the equality", + "type": "text" + }, + { + "bbox": [ + 365, + 497, + 406, + 509 + ], + "score": 0.92, + "content": "\\mathcal { M } Q = Q", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "implies that performing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 508, + 321, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 321, + 520 + ], + "score": 1.0, + "content": "an action and incurring a cost for doing so is optimal.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 97, + 533, + 388, + 548 + ], + "lines": [ + { + "bbox": [ + 104, + 532, + 390, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 532, + 390, + 551 + ], + "score": 1.0, + "content": "6 Budget Augmented LICRA via State Augmentation", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 104, + 557, + 504, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "score": 1.0, + "content": "We now tackle the problem of RL with a budget. 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We can now state our first main result:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 92, + 72, + 489, + 86 + ] + }, + { + "type": "text", + "bbox": [ + 103, + 91, + 505, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 90, + 505, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 201, + 105 + ], + "score": 1.0, + "content": "Theorem 1 Given any", + "type": "text" + }, + { + "bbox": [ + 202, + 92, + 281, + 103 + ], + "score": 0.9, + "content": "v ^ { \\pi , \\mathfrak { g } } : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 90, + 442, + 105 + ], + "score": 1.0, + "content": ", the optimal value function is given by", + "type": "text" + }, + { + "bbox": [ + 442, + 91, + 505, + 109 + ], + "score": 0.91, + "content": "\\operatorname* { l i m } _ { k \\to \\infty } T ^ { k } v ^ { \\pi , \\mathfrak { g } } =", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 106, + 354, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 108, + 186, + 128 + ], + "score": 0.85, + "content": "\\operatorname* { m a x } _ { \\hat { \\pi } , \\hat { g } \\in \\Pi } v ^ { \\hat { \\pi } , \\hat { g } } = v ^ { \\pi ^ { \\star } , g ^ { \\star } }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 106, + 214, + 124 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 214, + 109, + 248, + 122 + ], + "score": 0.91, + "content": "( \\pi ^ { \\star } , g ^ { \\star } )", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 106, + 354, + 124 + ], + "score": 1.0, + "content": "is the optimal policy pair.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 90, + 505, + 128 + ] + }, + { + "type": "text", + "bbox": [ + 90, + 135, + 506, + 169 + ], + "lines": [ + { + "bbox": [ + 105, + 135, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 505, + 147 + ], + "score": 1.0, + "content": "The result of Theorem 1 enables the solution to the agent’s impulse control problem to be determined", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 145, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 104, + 145, + 505, + 159 + ], + "score": 1.0, + "content": "using a value iteration procedure. Moreover, Theorem 1 enables a Q-learning approach [6] for finding", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 157, + 249, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 249, + 169 + ], + "score": 1.0, + "content": "the solution to the agent’s problem.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 104, + 135, + 505, + 169 + ] + }, + { + "type": "text", + "bbox": [ + 87, + 176, + 331, + 189 + ], + "lines": [ + { + "bbox": [ + 85, + 175, + 331, + 190 + ], + "spans": [ + { + "bbox": [ + 85, + 175, + 251, + 190 + ], + "score": 1.0, + "content": "219 Theorem 2 Consider the following", + "type": "text" + }, + { + "bbox": [ + 252, + 177, + 260, + 188 + ], + "score": 0.74, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 175, + 331, + 190 + ], + "score": 1.0, + "content": "learning variant:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 85, + 175, + 331, + 190 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 204, + 484, + 232 + ], + "lines": [ + { + "bbox": [ + 115, + 204, + 484, + 232 + ], + "spans": [ + { + "bbox": [ + 115, + 204, + 484, + 232 + ], + "score": 0.87, + "content": "+ \\alpha _ { t } ( s _ { t } , a _ { t } ) \\left[ \\operatorname* { m a x } \\left\\{ \\mathcal { M } ^ { \\pi , \\mathfrak { g } } Q _ { t } ( s _ { t } , a _ { t } ) , \\mathcal { R } ( s _ { t } , 0 ) + \\gamma _ { a ^ { \\prime } \\in A } ^ { \\operatorname* { m a x } } Q _ { t } ( s _ { t + 1 } , a ^ { \\prime } ) \\right\\} - Q _ { t } ( s _ { t } , a _ { t } ) \\right] ,", + "type": "interline_equation", + "image_path": "23f8930b1b8b17d1d3a70216119b13eb66fa3f4560ca71ea1ccb26e4c5ed1067.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 115, + 204, + 484, + 213.33333333333334 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 115, + 213.33333333333334, + 484, + 222.66666666666669 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 115, + 222.66666666666669, + 484, + 232.00000000000003 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 90, + 233, + 412, + 246 + ], + "lines": [ + { + "bbox": [ + 87, + 232, + 412, + 247 + ], + "spans": [ + { + "bbox": [ + 87, + 232, + 126, + 247 + ], + "score": 1.0, + "content": "then 220", + "type": "text" + }, + { + "bbox": [ + 126, + 234, + 138, + 245 + ], + "score": 0.88, + "content": "Q _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 232, + 192, + 247 + ], + "score": 1.0, + "content": "converges to", + "type": "text" + }, + { + "bbox": [ + 192, + 234, + 206, + 245 + ], + "score": 0.89, + "content": "Q ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 232, + 309, + 247 + ], + "score": 1.0, + "content": "with probability 1, where", + "type": "text" + }, + { + "bbox": [ + 310, + 234, + 361, + 245 + ], + "score": 0.9, + "content": "s _ { t } , s _ { t + 1 } \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 232, + 379, + 247 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 380, + 234, + 410, + 244 + ], + "score": 0.89, + "content": "a _ { t } \\in \\mathcal A", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 232, + 412, + 247 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 87, + 232, + 412, + 247 + ] + }, + { + "type": "text", + "bbox": [ + 95, + 253, + 349, + 265 + ], + "lines": [ + { + "bbox": [ + 92, + 252, + 350, + 267 + ], + "spans": [ + { + "bbox": [ + 92, + 252, + 350, + 267 + ], + "score": 1.0, + "content": "21 We now extend the result to (linear) function approximators:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 92, + 252, + 350, + 267 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 272, + 506, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 357, + 285 + ], + "score": 1.0, + "content": "Theorem 3 Given a set of linearly independent basis functions", + "type": "text" + }, + { + "bbox": [ + 358, + 272, + 432, + 285 + ], + "score": 0.94, + "content": "\\Phi = \\{ \\phi _ { 1 } , \\ldots , \\phi _ { p } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 272, + 452, + 285 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 452, + 273, + 503, + 284 + ], + "score": 0.88, + "content": "\\phi _ { k } \\in L _ { 2 } , \\forall k", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 272, + 505, + 285 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 282, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 243, + 297 + ], + "score": 1.0, + "content": "LICRA converges to a limit point", + "type": "text" + }, + { + "bbox": [ + 243, + 284, + 278, + 294 + ], + "score": 0.91, + "content": "r ^ { \\star } \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 282, + 407, + 297 + ], + "score": 1.0, + "content": "which is the unique solution to", + "type": "text" + }, + { + "bbox": [ + 408, + 284, + 476, + 295 + ], + "score": 0.91, + "content": "\\Pi \\mathfrak { F } ( \\Phi r ^ { \\star } ) = \\Phi r ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 282, + 505, + 297 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 294, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 227, + 308 + ], + "score": 0.88, + "content": "\\mathfrak { F } v : = \\mathcal { R } + \\gamma P \\operatorname* { m a x } \\{ \\mathcal { M } v , v \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 294, + 275, + 310 + ], + "score": 1.0, + "content": ". Moreover,", + "type": "text" + }, + { + "bbox": [ + 276, + 297, + 286, + 306 + ], + "score": 0.84, + "content": "r ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 294, + 325, + 310 + ], + "score": 1.0, + "content": "satisfies:", + "type": "text" + }, + { + "bbox": [ + 326, + 295, + 501, + 308 + ], + "score": 0.89, + "content": "\\| \\Phi r ^ { \\star } - Q ^ { \\star } \\| \\leq ( 1 - \\gamma ^ { 2 } ) ^ { - 1 / 2 } \\| \\Pi Q ^ { \\star } - Q ^ { \\star } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 294, + 505, + 310 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 272, + 505, + 310 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 315, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 328 + ], + "score": 1.0, + "content": "The theorem establishes the convergence of LICRA to a stable point with the use of linear function", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "approximators. The second statement bounds the proximity of the convergence point by the smallest", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 338, + 416, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 416, + 349 + ], + "score": 1.0, + "content": "approximation error that can be achieved given the choice of basis functions.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 315, + 505, + 349 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "Having constructed a procedure to find the optimal agent’s optimal value function, we now seek to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 363, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 453, + 378 + ], + "score": 1.0, + "content": "determine the conditions when an intervention should be performed. Let us denote by", + "type": "text" + }, + { + "bbox": [ + 454, + 365, + 489, + 377 + ], + "score": 0.93, + "content": "\\{ \\tau _ { k } \\} _ { k \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 363, + 505, + 378 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "points at which the agent decides to act or intervention times, so for example if the agent chooses", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 222, + 399 + ], + "score": 1.0, + "content": "to perform an action at state", + "type": "text" + }, + { + "bbox": [ + 222, + 388, + 233, + 398 + ], + "score": 0.84, + "content": "s _ { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 386, + 306, + 399 + ], + "score": 1.0, + "content": "and again at state", + "type": "text" + }, + { + "bbox": [ + 306, + 388, + 317, + 398 + ], + "score": 0.84, + "content": "s _ { 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 386, + 340, + 399 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 340, + 387, + 369, + 398 + ], + "score": 0.91, + "content": "\\tau _ { 1 } = 6", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 386, + 387, + 399 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 387, + 387, + 416, + 397 + ], + "score": 0.9, + "content": "\\tau _ { 2 } = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 386, + 506, + 399 + ], + "score": 1.0, + "content": ". The following result", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 395, + 422, + 413 + ], + "spans": [ + { + "bbox": [ + 104, + 395, + 285, + 413 + ], + "score": 1.0, + "content": "characterises the optimal intervention policy", + "type": "text" + }, + { + "bbox": [ + 286, + 399, + 292, + 409 + ], + "score": 0.76, + "content": "\\mathfrak { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 395, + 382, + 413 + ], + "score": 1.0, + "content": "and the optimal times", + "type": "text" + }, + { + "bbox": [ + 382, + 398, + 417, + 410 + ], + "score": 0.91, + "content": "\\{ \\tau _ { k } \\} _ { k \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 395, + 422, + 413 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 353, + 506, + 413 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 104, + 415, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 272, + 431 + ], + "score": 1.0, + "content": "Proposition 1 The policy g is given by:", + "type": "text" + }, + { + "bbox": [ + 272, + 417, + 474, + 429 + ], + "score": 0.85, + "content": "\\mathfrak { g } ( s _ { t } ) = H ( \\mathcal { M } ^ { \\pi , \\mathfrak { g } } Q ^ { \\pi , \\mathfrak { g } } - Q ^ { \\pi , \\mathfrak { g } } ) ( s _ { t } , a _ { t } ) , \\forall s _ { t } \\in \\mathcal { S } _ { \\mathrm { ~ \\Gamma ~ } }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 415, + 506, + 431 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 426, + 507, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 127, + 440 + ], + "score": 0.87, + "content": "Q ^ { \\pi , { \\mathfrak { g } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 426, + 236, + 441 + ], + "score": 1.0, + "content": "is the solution in Theorem", + "type": "text" + }, + { + "bbox": [ + 236, + 429, + 242, + 438 + ], + "score": 0.27, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 426, + 246, + 441 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 246, + 429, + 259, + 438 + ], + "score": 0.67, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 426, + 390, + 441 + ], + "score": 1.0, + "content": "is the intervention operator and", + "type": "text" + }, + { + "bbox": [ + 391, + 429, + 401, + 438 + ], + "score": 0.81, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 426, + 507, + 441 + ], + "score": 1.0, + "content": "is the Heaviside function,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 437, + 419, + 452 + ], + "spans": [ + { + "bbox": [ + 104, + 437, + 251, + 452 + ], + "score": 1.0, + "content": "moreover the intervention times are", + "type": "text" + }, + { + "bbox": [ + 251, + 439, + 415, + 451 + ], + "score": 0.93, + "content": "\\tau _ { k } = \\operatorname* { i n f } \\{ \\tau > \\tau _ { k - 1 } | \\mathcal { M } ^ { \\pi , { \\mathfrak { g } } } Q ^ { \\pi , { \\mathfrak { g } } } = Q ^ { \\pi , { \\mathfrak { g } } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 437, + 419, + 452 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 415, + 507, + 452 + ] + }, + { + "type": "text", + "bbox": [ + 100, + 458, + 504, + 482 + ], + "lines": [ + { + "bbox": [ + 104, + 456, + 504, + 473 + ], + "spans": [ + { + "bbox": [ + 104, + 456, + 305, + 473 + ], + "score": 1.0, + "content": "Prop. 1 characterises the (categorical) distribution", + "type": "text" + }, + { + "bbox": [ + 305, + 460, + 311, + 470 + ], + "score": 0.42, + "content": "\\mathfrak { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 456, + 432, + 473 + ], + "score": 1.0, + "content": ". Moreover, given the function", + "type": "text" + }, + { + "bbox": [ + 432, + 459, + 441, + 470 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 456, + 483, + 473 + ], + "score": 1.0, + "content": ", the times", + "type": "text" + }, + { + "bbox": [ + 484, + 458, + 504, + 470 + ], + "score": 0.91, + "content": "\\{ \\tau _ { k } \\}", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 469, + 317, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 246, + 482 + ], + "score": 1.0, + "content": "can be determined by evaluating if", + "type": "text" + }, + { + "bbox": [ + 247, + 470, + 289, + 481 + ], + "score": 0.92, + "content": "\\mathcal { M } Q = Q", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 469, + 317, + 482 + ], + "score": 1.0, + "content": "holds.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 456, + 504, + 482 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 485, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "A key aspect of Prop. 1 is that it exploits the cost structure of the problem to determine when the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 364, + 510 + ], + "score": 1.0, + "content": "agent should perform an intervention. In particular, the equality", + "type": "text" + }, + { + "bbox": [ + 365, + 497, + 406, + 509 + ], + "score": 0.92, + "content": "\\mathcal { M } Q = Q", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "implies that performing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 508, + 321, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 321, + 520 + ], + "score": 1.0, + "content": "an action and incurring a cost for doing so is optimal.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 485, + 505, + 520 + ] + }, + { + "type": "title", + "bbox": [ + 97, + 533, + 388, + 548 + ], + "lines": [ + { + "bbox": [ + 104, + 532, + 390, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 532, + 390, + 551 + ], + "score": 1.0, + "content": "6 Budget Augmented LICRA via State Augmentation", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 104, + 557, + 504, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "score": 1.0, + "content": "We now tackle the problem of RL with a budget. To do this, we combine the above impulse control", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "technology with state augmentation technique proposed in [28] The mathematical formulation of the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 579, + 324, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 294, + 592 + ], + "score": 1.0, + "content": "problem is now given by the following for any", + "type": "text" + }, + { + "bbox": [ + 294, + 580, + 318, + 590 + ], + "score": 0.89, + "content": "s \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 579, + 324, + 592 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 557, + 505, + 592 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 596, + 382, + 627 + ], + "lines": [ + { + "bbox": [ + 227, + 596, + 382, + 627 + ], + "spans": [ + { + "bbox": [ + 227, + 596, + 382, + 627 + ], + "score": 0.95, + "content": "\\operatorname* { m a x } _ { \\pi \\in \\Pi , g } ~ \\upsilon ^ { \\pi , \\mathfrak { g } } ( s ) \\ \\mathrm { ~ s . ~ t . ~ } n - \\sum _ { t = 0 } ^ { \\infty } \\sum _ { k \\geq 1 } \\delta _ { \\tau _ { k } } ^ { t } \\geq 0 ,", + "type": "interline_equation", + "image_path": "c55bbf1ceb08375a8c470d28ced7b07f37b91614755892322a27a5a1575451ae.jpg" + } + ] + } + ], + "index": 35.5, + "virtual_lines": [ + { + "bbox": [ + 227, + 596, + 382, + 611.5 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 227, + 611.5, + 382, + 627.0 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 628, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 134, + 639 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 629, + 163, + 639 + ], + "score": 0.88, + "content": "n \\in \\mathbb N", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "is a fixed value that represents the maximum number of allowed interventions and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 637, + 507, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 147, + 653 + ], + "score": 0.9, + "content": "\\sum k > 1 8 _ { T k } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 637, + 342, + 654 + ], + "score": 1.0, + "content": "is equal to one if an impulse was applied at time", + "type": "text" + }, + { + "bbox": [ + 342, + 641, + 347, + 650 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 637, + 507, + 654 + ], + "score": 1.0, + "content": "and zero if it was not. In order to avoid", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 651, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 400, + 664 + ], + "score": 1.0, + "content": "dealing with a constrained MDP, we propose to introduce a new variable", + "type": "text" + }, + { + "bbox": [ + 401, + 653, + 410, + 662 + ], + "score": 0.83, + "content": "z _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 651, + 505, + 664 + ], + "score": 1.0, + "content": "tracking the remaining", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 103, + 660, + 507, + 681 + ], + "spans": [ + { + "bbox": [ + 103, + 660, + 196, + 681 + ], + "score": 1.0, + "content": "number of impulses:", + "type": "text" + }, + { + "bbox": [ + 197, + 662, + 308, + 678 + ], + "score": 0.92, + "content": "\\begin{array} { r } { z _ { t } = n - \\sum _ { i = 0 } ^ { t - 1 } \\sum _ { k \\geq 1 } \\delta _ { \\tau _ { k } } ^ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 660, + 354, + 681 + ], + "score": 1.0, + "content": ". We treat", + "type": "text" + }, + { + "bbox": [ + 354, + 666, + 363, + 675 + ], + "score": 0.83, + "content": "z _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 660, + 507, + 681 + ], + "score": 1.0, + "content": "as another state and augment the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 678, + 268, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 255, + 690 + ], + "score": 1.0, + "content": "state-space resulting in the transition", + "type": "text" + }, + { + "bbox": [ + 255, + 678, + 264, + 689 + ], + "score": 0.88, + "content": "\\widetilde { \\mathcal P }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 679, + 268, + 690 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 103, + 627, + 507, + 690 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 695, + 411, + 721 + ], + "lines": [ + { + "bbox": [ + 200, + 695, + 411, + 721 + ], + "spans": [ + { + "bbox": [ + 200, + 695, + 411, + 721 + ], + "score": 0.91, + "content": "s _ { t + 1 } \\sim P ( \\cdot | s _ { t } , a _ { t } ) , \\qquad z _ { t + 1 } = z _ { t } - \\sum _ { k \\geq 1 } \\delta _ { \\tau _ { k } } ^ { t } , \\quad z _ { 0 } = n .", + "type": "interline_equation", + "image_path": "6346be3fbe5274a0800604dd10bbe7c0ae289219dc95d9e13ea962d4639c0036.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 200, + 695, + 411, + 721 + ], + "spans": [], + "index": 42 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 87, + 71, + 505, + 120 + ], + "lines": [ + { + "bbox": [ + 83, + 70, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 83, + 74, + 354, + 93 + ], + "score": 1.0, + "content": "In order to avoid violations, we reshape the reward as follows: 252", + "type": "text" + }, + { + "bbox": [ + 354, + 71, + 500, + 97 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\widetilde { \\mathcal { R } } ( s _ { t } , z _ { t } , a _ { t } ) = \\left\\{ \\begin{array} { l l } { \\mathcal { R } ( s _ { t } , a _ { t } ) } & { z _ { t } \\ge 0 } \\\\ { - \\Delta } & { z _ { t } < 0 } \\end{array} \\right. } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 70, + 505, + 98 + ], + "score": 1.0, + "content": ",,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 83, + 96, + 506, + 109 + ], + "spans": [ + { + "bbox": [ + 83, + 96, + 134, + 109 + ], + "score": 1.0, + "content": "253 where", + "type": "text" + }, + { + "bbox": [ + 134, + 97, + 164, + 107 + ], + "score": 0.9, + "content": "\\Delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 96, + 506, + 109 + ], + "score": 1.0, + "content": "is a large enough hyper-parameter ensuring that there are no safety violations. To", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 86, + 108, + 311, + 120 + ], + "spans": [ + { + "bbox": [ + 86, + 108, + 311, + 120 + ], + "score": 1.0, + "content": "254 summarise we aim to solve the following problem:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 86, + 126, + 99, + 134 + ], + "lines": [ + { + "bbox": [ + 86, + 126, + 99, + 134 + ], + "spans": [], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 127, + 407, + 158 + ], + "lines": [ + { + "bbox": [ + 204, + 127, + 407, + 158 + ], + "spans": [ + { + "bbox": [ + 204, + 127, + 407, + 158 + ], + "score": 0.94, + "content": "v ^ { \\pi , \\mathfrak { g } } ( s , z ) = \\mathbb { E } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\widetilde { \\mathcal { R } } ( s _ { t } , z _ { t } , a _ { t } ) | a _ { t } \\sim \\pi ( \\cdot | s _ { t } , z _ { t } ) \\right] ,", + "type": "interline_equation", + "image_path": "f7782e97a7c6f6b939a51b9531f333bd46fa03de6e45a900410f8a587c3fa221.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 127, + 407, + 142.5 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 204, + 142.5, + 407, + 158.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 102, + 163, + 506, + 213 + ], + "lines": [ + { + "bbox": [ + 106, + 162, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 296, + 177 + ], + "score": 1.0, + "content": "where the policy now depends on the variable", + "type": "text" + }, + { + "bbox": [ + 297, + 165, + 306, + 175 + ], + "score": 0.83, + "content": "z _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 163, + 352, + 177 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 352, + 162, + 361, + 174 + ], + "score": 0.87, + "content": "\\widetilde { \\mathcal P }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 163, + 506, + 177 + ], + "score": 1.0, + "content": "in Equation 6 is a Markov process", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 175, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 176, + 190 + ], + "score": 1.0, + "content": "and, the rewards", + "type": "text" + }, + { + "bbox": [ + 176, + 175, + 186, + 187 + ], + "score": 0.86, + "content": "\\widetilde { \\mathcal { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 176, + 335, + 190 + ], + "score": 1.0, + "content": "are bounded, as long as the rewards", + "type": "text" + }, + { + "bbox": [ + 336, + 177, + 346, + 187 + ], + "score": 0.83, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 176, + 505, + 190 + ], + "score": 1.0, + "content": "are bounded. Therefore, we can apply", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 187, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 201 + ], + "score": 1.0, + "content": "directly the results for impulse control to this case as well. We denote the augmented MDP by", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 199, + 494, + 214 + ], + "spans": [ + { + "bbox": [ + 107, + 199, + 214, + 213 + ], + "score": 0.93, + "content": "\\widetilde { \\mathcal { M } } = \\langle \\overset { \\cdot } { S } \\times \\mathcal { Z } , \\overset { \\cdot } { A } , \\widetilde { \\mathcal { P } } , \\widetilde { R } , \\gamma \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 200, + 245, + 214 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 245, + 201, + 254, + 211 + ], + "score": 0.82, + "content": "\\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 200, + 494, + 214 + ], + "score": 1.0, + "content": "is the space of the augmented state. We have the following.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 101, + 222, + 346, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 221, + 347, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 235, + 237 + ], + "score": 1.0, + "content": "Theorem 4 Consider the MDP", + "type": "text" + }, + { + "bbox": [ + 236, + 221, + 249, + 235 + ], + "score": 0.79, + "content": "\\widetilde { \\mathcal { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 221, + 347, + 237 + ], + "score": 1.0, + "content": "for the problem 7, then:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 240, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 104, + 237, + 504, + 254 + ], + "spans": [ + { + "bbox": [ + 104, + 237, + 380, + 254 + ], + "score": 1.0, + "content": "a) The Bellman equation holds, i.e. there exists a function", + "type": "text" + }, + { + "bbox": [ + 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} ^ { \\hat { \\pi } , \\hat { g } } =", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 308, + 460, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 135, + 321 + ], + "score": 0.87, + "content": "\\widetilde { v } ^ { * , \\pi , { \\mathfrak { g } } ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 308, + 167, + 324 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 167, + 311, + 201, + 323 + ], + "score": 0.91, + "content": "( \\pi ^ { * } , { \\mathfrak { g } } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 308, + 295, + 324 + ], + "score": 1.0, + "content": "is an optimal policy of", + "type": "text" + }, + { + "bbox": [ + 295, + 308, + 308, + 321 + ], + "score": 0.84, + "content": "\\widetilde { \\mathcal { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 308, + 326, + 324 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": 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\\Delta } & { z _ { t } < 0 } \\end{array} \\right. } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 70, + 505, + 98 + ], + "score": 1.0, + "content": ",,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 83, + 96, + 506, + 109 + ], + "spans": [ + { + "bbox": [ + 83, + 96, + 134, + 109 + ], + "score": 1.0, + "content": "253 where", + "type": "text" + }, + { + "bbox": [ + 134, + 97, + 164, + 107 + ], + "score": 0.9, + "content": "\\Delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 96, + 506, + 109 + ], + "score": 1.0, + "content": "is a large enough hyper-parameter ensuring that there are no safety violations. To", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 86, + 108, + 311, + 120 + ], + "spans": [ + { + "bbox": [ + 86, + 108, + 311, + 120 + ], + "score": 1.0, + "content": "254 summarise we aim to solve the following problem:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 83, + 70, + 506, + 120 + ] + }, + { + "type": "text", + "bbox": [ + 86, + 126, + 99, + 134 + ], + "lines": [ + { + "bbox": [ + 86, + 126, + 99, + 134 + ], + "spans": [], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 86, + 126, + 99, + 134 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 127, + 407, + 158 + ], + "lines": [ + { + "bbox": [ + 204, + 127, + 407, + 158 + ], + "spans": [ + { + "bbox": [ + 204, + 127, + 407, + 158 + ], + "score": 0.94, + "content": "v ^ { \\pi , \\mathfrak { g } } ( s , z ) = \\mathbb { E } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\widetilde { \\mathcal { R } } ( s _ { t } , z _ { t } , a _ { t } ) | a _ { t } \\sim \\pi ( \\cdot | s _ { t } , z _ { t } ) \\right] ,", + "type": "interline_equation", + "image_path": "f7782e97a7c6f6b939a51b9531f333bd46fa03de6e45a900410f8a587c3fa221.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 127, + 407, + 142.5 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 204, + 142.5, + 407, + 158.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 102, + 163, + 506, + 213 + ], + "lines": [ + { + "bbox": [ + 106, + 162, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 296, + 177 + ], + "score": 1.0, + "content": "where the policy now depends on the variable", + "type": "text" + }, + { + "bbox": [ + 297, + 165, + 306, + 175 + ], + "score": 0.83, + "content": "z _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 163, + 352, + 177 + ], + "score": 1.0, + "content": ". 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We denote the augmented MDP by", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 199, + 494, + 214 + ], + "spans": [ + { + "bbox": [ + 107, + 199, + 214, + 213 + ], + "score": 0.93, + "content": "\\widetilde { \\mathcal { M } } = \\langle \\overset { \\cdot } { S } \\times \\mathcal { Z } , \\overset { \\cdot } { A } , \\widetilde { \\mathcal { P } } , \\widetilde { R } , \\gamma \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 200, + 245, + 214 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 245, + 201, + 254, + 211 + ], + "score": 0.82, + "content": "\\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 200, + 494, + 214 + ], + "score": 1.0, + "content": "is the space of the augmented state. 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The first is that we can use a modified version of LICRA", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 506, + 354 + ], + "score": 1.0, + "content": "to obtain the solution of the problem while guaranteeing convergence (under standard assumptions).", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 352, + 507, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 507, + 367 + ], + "score": 1.0, + "content": "Secondly, our state augmentation procedure admits a Markovian representation of the optimal policy.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 331, + 507, + 367 + ] + }, + { + "type": "title", + "bbox": [ + 94, + 379, + 191, + 393 + ], + "lines": [ + { + "bbox": [ + 90, + 376, + 192, + 397 + ], + "spans": [ + { + "bbox": [ + 90, + 376, + 192, + 397 + ], + "score": 1.0, + "content": "69 7 Experiments", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "index", + "bbox": [ + 85, + 403, + 506, + 492 + ], + "lines": [ + { + "bbox": [ + 86, + 403, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 86, + 406, + 99, + 415 + ], + "score": 1.0, + "content": "270", + "type": "text" + }, + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "score": 1.0, + "content": "We will now study empirically the performance of the LICRA framework. In experiments, we use", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 86, + 417, + 99, + 428 + ], + "score": 1.0, + "content": "271", + "type": "text" + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "different instances of LICRA, one where both policies are trained using PPO update (referred to", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 425, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 86, + 426, + 100, + 438 + ], + "score": 1.0, + "content": "272", + "type": "text" + }, + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "score": 1.0, + "content": "as LICRA_PPO) and one where the policy deciding whether to act is trained using SAC and the", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 437, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 86, + 439, + 99, + 448 + ], + "score": 1.0, + "content": "273", + "type": "text" + }, + { + "bbox": [ + 106, + 437, + 506, + 449 + ], + "score": 1.0, + "content": "other policy trained with PPO (referred to as LICRA_SAC). We have benchmarked both of these", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 86, + 450, + 99, + 459 + ], + "score": 1.0, + "content": "274", + "type": "text" + }, + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "algorithms together with common baselines on environments, where it would be natural to introduce", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 459, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 86, + 461, + 99, + 470 + ], + "score": 1.0, + "content": "275", + "type": "text" + }, + { + "bbox": [ + 105, + 459, + 505, + 470 + ], + "score": 1.0, + "content": "the concept of the cost associated with actions. We lastly performed a series of ablation studies which", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 86, + 472, + 99, + 481 + ], + "score": 1.0, + "content": "276", + "type": "text" + }, + { + "bbox": [ + 105, + 469, + 417, + 482 + ], + "score": 1.0, + "content": "test LICRA’s ability to handle different cost functions including the case when", + "type": "text" + }, + { + "bbox": [ + 417, + 469, + 464, + 482 + ], + "score": 0.93, + "content": "c ( s , a ) \\equiv 0", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "which we", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 481, + 385, + 493 + ], + "spans": [ + { + "bbox": [ + 86, + 481, + 100, + 493 + ], + "score": 1.0, + "content": "277", + "type": "text" + }, + { + "bbox": [ + 106, + 481, + 385, + 493 + ], + "score": 1.0, + "content": "defer to the Appendix which also contains further experiment details.", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 86, + 637, + 99, + 646 + ], + "score": 1.0, + "content": "278", + "type": "text" + }, + { + "bbox": [ + 105, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "Merton’s Portfolio Problem with Transaction Costs. Merton Investment Problem in which the", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 85, + 648, + 100, + 658 + ], + "score": 1.0, + "content": "279", + "type": "text" + }, + { + "bbox": [ + 105, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "investor faces transaction costs [10] is a well-known problem within finance. In our environment, the", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 86, + 659, + 99, + 668 + ], + "score": 1.0, + "content": "280", + "type": "text" + }, + { + "bbox": [ + 105, + 657, + 506, + 669 + ], + "score": 1.0, + "content": "agent can decide to move its wealth between a risky asset and a risk-free asset. 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Each time the agent moves the assets, it incurs a cost", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 85, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 85, + 275, + 100, + 285 + ], + "score": 1.0, + "content": "289", + "type": "text" + }, + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "of 1 i.e. a transaction fee. The agent can act after a time interval of 0.01 seconds and the episode", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 86, + 284, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 86, + 287, + 100, + 296 + ], + "score": 1.0, + "content": "290", + "type": "text" + }, + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "score": 1.0, + "content": "ends after 75 steps. The results of training are shown in Fig. 1 which clearly demonstrates that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 85, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 85, + 297, + 100, + 307 + ], + "score": 1.0, + "content": "291", + "type": "text" + }, + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "LICRA_PPO finds a better policy than standard PPO. 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We studied an autonomous driving scenario where fuel-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 85, + 326, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 85, + 330, + 100, + 340 + ], + "score": 1.0, + "content": "294", + "type": "text" + }, + { + "bbox": [ + 105, + 326, + 506, + 343 + ], + "score": 1.0, + "content": "efficient driving is a priority. 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The results of training are shown in Fig. 1 which clearly demonstrates that", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 85, + 297, + 100, + 307 + ], + "score": 1.0, + "content": "291", + "type": "text" + }, + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "LICRA_PPO finds a better policy than standard PPO. 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We studied an autonomous driving scenario where fuel-", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 326, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 85, + 330, + 100, + 340 + ], + "score": 1.0, + "content": "294", + "type": "text" + }, + { + "bbox": [ + 105, + 326, + 506, + 343 + ], + "score": 1.0, + "content": "efficient driving is a priority. 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In Fig. 3, we observe that the LICRA agent outperforms all the baselines,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 86, + 211, + 99, + 221 + ], + "score": 1.0, + "content": "321", + "type": "text" + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "both in terms of sample efficiency and average test return (total rewards at each timestep). We also", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 86, + 220, + 457, + 232 + ], + "spans": [ + { + "bbox": [ + 86, + 222, + 100, + 231 + ], + "score": 1.0, + "content": "322", + "type": "text" + }, + { + "bbox": [ + 105, + 220, + 457, + 232 + ], + "score": 1.0, + "content": "observe that LICRA enables more stable training than PPO, PPO-Lagrangian and CPO.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 86, + 231, + 506, + 330 + ], + "lines": [ + { + "bbox": [ + 86, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 86, + 233, + 99, + 242 + ], + "score": 1.0, + "content": "323", + "type": "text" + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "Ablation Study 1. Prioritisation of Most Important Actions. 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In Fig. 3, we observe that the LICRA agent outperforms all the baselines,", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 86, + 211, + 99, + 221 + ], + "score": 1.0, + "content": "321", + "type": "text" + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "both in terms of sample efficiency and average test return (total rewards at each timestep). We also", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 220, + 457, + 232 + ], + "spans": [ + { + "bbox": [ + 86, + 222, + 100, + 231 + ], + "score": 1.0, + "content": "322", + "type": "text" + }, + { + "bbox": [ + 105, + 220, + 457, + 232 + ], + "score": 1.0, + "content": "observe that LICRA enables more stable training than PPO, PPO-Lagrangian and CPO.", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 86, + 233, + 99, + 242 + ], + "score": 1.0, + "content": "323", + "type": "text" + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "Ablation Study 1. Prioritisation of Most Important Actions. We next tested LICRA’s ability to", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 86, + 244, + 99, + 253 + ], + "score": 1.0, + "content": "324", + "type": "text" + }, + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "prioritise where it performs actions when the necessity to act varies significantly between states. To", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 253, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 86, + 255, + 99, + 264 + ], + "score": 1.0, + "content": "325", + "type": "text" + }, + { + "bbox": [ + 105, + 253, + 505, + 264 + ], + "score": 1.0, + "content": "test this, we modified the Drive Environment to now consist of a single lane, a start state and a goal", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 85, + 265, + 100, + 275 + ], + "score": 1.0, + "content": "326", + "type": "text" + }, + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "state start (at the end) where there is a reward. 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At zones", + "type": "text" + }, + { + "bbox": [ + 271, + 286, + 316, + 297 + ], + "score": 0.91, + "content": "k = 1 , 2 , 3", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "of the lane, if the vehicle is travelling below a", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 296, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 86, + 299, + 99, + 308 + ], + "score": 1.0, + "content": "329", + "type": "text" + }, + { + "bbox": [ + 105, + 296, + 141, + 309 + ], + "score": 1.0, + "content": "velocity", + "type": "text" + }, + { + "bbox": [ + 141, + 298, + 162, + 308 + ], + "score": 0.88, + "content": "v _ { m i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 296, + 327, + 309 + ], + "score": 1.0, + "content": ", it is penalised by a strictly negative cost", + "type": "text" + }, + { + "bbox": [ + 328, + 298, + 338, + 307 + ], + "score": 0.85, + "content": "c _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 296, + 366, + 309 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 367, + 297, + 420, + 307 + ], + "score": 0.91, + "content": "c _ { 1 } < c _ { 2 } < c _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 296, + 506, + 309 + ], + "score": 1.0, + "content": ". 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Heatmaps display the number of times the agent drives below", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 476, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 128, + 487 + ], + "score": 0.86, + "content": "v _ { m i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 476, + 506, + 488 + ], + "score": 1.0, + "content": "in the penalty zones. Violation 1 refers to the lowest cost zone, whilst Violation 3 refers to the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 487, + 363, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 178, + 499 + ], + "score": 1.0, + "content": "largest cost zone.", + "type": "text" + }, + { + "bbox": [ + 178, + 487, + 189, + 496 + ], + "score": 0.76, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 487, + 363, + 499 + ], + "score": 1.0, + "content": "refers to the fixed cost for taking an action.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 104, + 511, + 183, + 524 + ], + "lines": [ + { + "bbox": [ + 104, + 509, + 185, + 527 + ], + "spans": [ + { + "bbox": [ + 104, + 509, + 185, + 527 + ], + "score": 1.0, + "content": "8 Conclusion", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "We presented a novel method to tackle the problem of learning how to select when to act in addition", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "to learning which actions to execute. 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This is of fundamental importance in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "practical settings where performing many actions over the horizon can lead to costs and undermine", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 590, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 602 + ], + "score": 1.0, + "content": "the service life of machinery. We demonstrated that our solution, LICRA which at its core has a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "sequential decision structure that first decides whether or not an action ought to be taken under the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "action policy can solve tasks where the agent faces costs with extreme efficiency as compared to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "leading reinforcement learning methods. In some tasks, we showed that LICRA is able to solve", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "problems that are unsolvable using current reinforcement learning machinery. We envisge that this", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "framework can serve as the basis extensions to different settings including adversarial training for", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 655, + 271, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 271, + 667 + ], + "score": 1.0, + "content": "solving a variety of problems within RL.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 535, + 506, + 667 + ] + }, + { + "type": "title", + "bbox": [ + 103, + 681, + 163, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 680, + 165, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 165, + 696 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 105, + 700, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 111, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 111, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "[1] Ali Ajdari, Maximilian Niyazi, Nils Henrik Nicolay, Christian Thieke, Robert Jeraj, and Thomas", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 127, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 127, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "Bortfeld. 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