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+# Meta-sketch: A Neural Data Structure for Estimating Item Frequencies of Data Streams
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 To estimate item frequencies of data streams with limited space, sketches are widely   
+2 used in real applications, including real-time web analytics, network monitoring,   
+3 and self-driving. Sketches can be viewed as a model which maps the identifier of a   
+4 stream item to the corresponding frequency domain. Starting from the premise, we   
+5 envision a neural data structure, which we term the meta-sketch, to go beyond the   
+6 basic structure of conventional sketches. The meta-sketch learns basic sketching   
+7 abilities from meta-tasks constituted with synthetic datasets following Zipf distribu  
+8 tions in the pre-training phase, and can be fast adapted to real (skewed) distributions   
+9 in the adaption phase. Extensive experiments demonstrate the performance gains   
+10 of the meta-sketch and offer insights into our proposals.
+
+# 11 1 Introduction
+
+12 Estimating item frequency is a basic topic in data stream processing, which finds applications in   
+13 the fields of networking, databases, and machine learning, such as real-time data analyzing [1–4],   
+14 network traffic monitoring [5–7], natural language processing [8] and search ranking [9]. Towards   
+15 infinite data streams, a common class of solutions [10–15] use a compact structure taking sublinear   
+16 space for counting the number of occurrences of each stream item, called the sketch.   
+17 Under the prevalent evidence of skewed distributions in data streams, basic sketches achieve the space   
+18 compactness by hashing and approximately aggregating stream items. Basic sketches, including   
+19 CM-sketch [10], C-sketch [11] and CU-sketch [12], use a 2D array of counters as the core structure.   
+20 To optimize the sketching performance, there arise augmented sketches [13,14], which attach filters to   
+21 basic sketches, to capture the preliminary patterns of skewed distributions (e.g., high/low-frequency   
+22 items). By separately maintaining the filtered high/low-frequency items, augmented sketches strive   
+23 to eliminate the estimation error incurred by hash collisions between the high- and low-frequency   
+24 items. Further, learned augmented sketches [15] improve the filters of the augmented sketches by   
+25 memorizing short-term high/low-frequency items via a pre-trained neural network (NN in short)   
+26 classifier. But it is not clear how the pre-trained NN can be adapted to dynamic streaming scenarios,   
+27 where the correspondence between items and frequencies varies. In a nutshell, sketches are structures   
+28 compactly summarizing stream distributions to count item frequencies with limited space budgets.   
+29 From the retrospective analysis of sketches, an observation can be drawn that the evolution of   
+30 sketches conforms with the exploitation of data distributions. It is thus a natural evolution to consider   
+31 a sketch that generally and automatically captures more distribution patterns with limited space   
+32 budgets. In this paper, we envision a novel neural sketch, called the meta-sketch, with techniques   
+33 of meta-learning and memory-augmented neural networks. The meta-sketch learns the sketching   
+34 abilities from automatically generated meta-tasks. Depending on the types of meta-tasks, we study   
+35 two versions of the meta-sketch, called basic and advanced meta-sketches.   
+36 The basic meta-sketch implements the simulation of basic sketches, through the training process with   
+37 basic meta-tasks following Zipf distributions, which are prevalent in the scenes of real data streams [16–   
+38 20]. The advanced meta-sketch extends the basic version to fast adapt to the specific runtime of stream   
+39 processing, through the training with adaptive meta-tasks, which are generated by online sampling   
+40 of real data streams. Our work follows a typical setting where the distribution of item frequencies   
+41 follows a skewed distribution, but the correspondence between items and frequencies varies. For   
+42 example, in software-defined networks (SDN), sketches are deployed to programmable switches to   
+43 collect per-flow statistics, where IP packets follow heavy-tailed distributions [15, 21]. In distributed   
+44 databases, it gives advances to collect statistics of data shards to optimize data placement and   
+45 query caching, where query phrases follow approximate Zipf distributions [15]. Given that the item   
+46 population follows a specific distribution, the local distributions, i.e., item-frequency correspondences   
+47 on shards or flows, are different. Instead of retraining learned augmented sketches on each local   
+48 distribution, the advanced-sketch can be quickly adapted to different local distributions once trained.   
+49 As a member of the neural data structure family [15, 22–24], the meta-sketch significantly differs   
+50 from conventional sketches, in terms of the structure and working mechanism. The meta-sketch   
+51 utilizes NN’s powerful encoding/decoding capabilities to perceive data distributions and express   
+52 and compress explicit or implicit information to retrieve item frequencies with better accuracies.   
+53 Meanwhile, the meta-sketch is differentiable to fully perceive frequency patterns for self-optimization.   
+54 Our contributions are as follows. 1) We propose the meta-sketch, the first neural data structure for the   
+55 problem of item frequency estimation, based on meta-learning. 2) The basic meta-sketch acquires   
+56 sketching abilities by learning from synthetic datasets, and outperforms basic sketches in real datasets.   
+57 The advanced meta-sketch automatically encompasses the ability analogous to the auxiliary structures   
+58 deliberately devised in (learned) augmented sketches, yet yielding better accuracies and robustness   
+59 when adapted to dynamic scenes. 3) Through extensive empirical studies on real and synthetic   
+60 datasets, we evaluate our proposed meta-sketches and analyze the mechanism of major modules.
+
+![](images/5cab66ddbcd705cb1ea9c63fdc83ce3fca981a2116fd4274ddddc19ffd5865f3.jpg)  
+Figure 1: The Framework of the Meta-sketch
+
+# 61 2 Meta-sketch Structure
+
+# 2.1 Preliminaries
+
+We consider a standard data stream scenario [19]. Suppose a data stream $\boldsymbol { S } _ { N } : \{ e _ { 1 } , . . . , e _ { N } \}$ with $N$ items and $n$ distinct items. Each item $e _ { i } \in S _ { N }$ takes a value from the item domain $\mathbb { X } = \{ x _ { 1 } , . . . , x _ { n } \}$ where $x _ { i } \neq x _ { j }$ . The frequency $f _ { i }$ is equal to the number of times that item $x _ { i }$ appears in $\mathcal { S } _ { N }$ .
+
+66 To leverage learning techniques for item frequency estimation, a naïve way is to train a NN model   
+67 (e.g., MLP/LSTM) that learns/memorizes the mapping relationship between items and frequencies   
+68 with multiple training iterations, similar to [15, 22, 24]. However, it violates the typical setting of   
+69 stream processing where item observations are transient and are therefore handled in one pass [18].   
+70 More, the costly procedure has to be repeated from the scratch for a new data stream. Inspired by the   
+71 meta-bloom filter [23], we consider a case of one-shot learning (fitting for one-pass stream processing)   
+72 by using meta-learning [25, 26] and memory-augmented networks [27, 28]. Meta-learning employs   
+73 sampled meta-tasks to learn the ability to solve a class of domain tasks rather than memorizing patterns   
+74 for a specific task. The memory-augmented networks incorporate external memories into NN models,   
+75 significantly enhancing the potentials of NN models with more learnable parameters. Meanwhile, it   
+76 performs efficient and explicit operations (i.e., reading and storing) for external memories, allowing   
+77 NN models to process information similarly to conventional data structures.   
+78 The framework of the meta-sketch consists of 4 functional modules, Embedding $( \mathcal { F } _ { E } )$ , Sparse   
+79 addressing $( \mathcal { F } _ { S a } )$ , Compressed storage matrix $( M )$ , and Decoding $( \mathcal { F } _ { d e c } )$ , as shown in Figure 1. Like   
+80 traditional sketches, the meta-sketch encodes and memorizes online stream items in one pass, and   
+81 answers queries by decoding corresponding item-frequency information from the structure.   
+82 Thus, we define 2 operations, Store and Query. Specifically, the Store operation first passes each   
+83 incoming stream item to $\mathcal { F } _ { E }$ for the embedding representation, and then writes the embedding vector   
+84 into $M$ , according to the address derived by ${ \mathcal { F } } _ { S a }$ . When estimating the frequency of an item, the   
+5 Query operation calculates the item’s address in $M$ via ${ \mathcal { F } } _ { S a }$ , reads the corresponding information   
+6 vector from $M$ , and decodes the item frequency by $\mathcal { F } _ { d e c }$ from the retrieved information vector .
+
+# 2.2 Modules
+
+Embedding. The module $\mathcal { F } _ { E }$ has two purposes: 1) performing representational transformation for an incoming item $e _ { i }$ and mapping it into a dense embedding vector $z _ { i }$ that holds implicit features about item-frequency distributions and serves as the basis for identifying stream items; 2) decoupling the embedding vector $z _ { i }$ to obtain a refined vector $r _ { i }$ , which is used to derive the address for reading/writing on the compressed storage matrix $M$ .
+
+Accordingly, $\mathcal { F } _ { E }$ consists of the embedding network $g _ { e m b }$ and the address network $g _ { a d d }$ . We assume that an item $e _ { i } \in S _ { N }$ is numerically encoded for the unique identification, following the conventions of stream processing [18, 19]. Thus, we have $z _ { i } , r _ { i } \gets \mathcal { F } _ { E } ( e _ { i } )$ , where $z _ { i } \gets g _ { e m b } ( e _ { i } )$ and $r _ { i } \gets$ $g _ { a d d } ( z _ { i } )$ . Here, $z _ { i } \in \mathbb { R } ^ { l _ { z } }$ is an embedding vector of length $l _ { z }$ , and $r _ { i } \in \mathbb { R } ^ { l _ { r } }$ is a refined vector of length $l _ { r }$ . The vector $z _ { i }$ serves multiple intents: 1) it makes a basis for deriving the address of an item in $\mathcal { F } _ { S a } ; 2 )$ it serves as the compressed vector of an item written into $M ; 3 )$ ) it works as a partial input of $\mathcal { F } _ { d e c }$ for decoding the item frequency; 4) it also plays the role of perceiving/compressing patterns of a specific frequency distribution, as discussed in Section 5. In addition, to enhance the addressing functionality and eliminate other interference factors, we decouple $z _ { i }$ to generate a refined vector $r _ { i }$ instead of using $z _ { i }$ directly for the addressing.
+
+103 Sparse addressing. The module ${ \mathcal { F } } _ { S a }$ aims to derive the address $a _ { i }$ for storing the embedding vector   
+104 $z _ { i }$ into the storage matrix: $a _ { i } \gets \mathcal { F } _ { S a } ( r _ { i } )$ . In terms of functionality, ${ \mathcal { F } } _ { S a }$ is analogous to the hash   
+105 functions of traditional sketches, except that ${ \mathcal { F } } _ { S a }$ is parameterized and differentiable. Specifically,   
+106 the addressing of the meta-sketch is done via a 3D addressing matrix $A$ of parameters to be learned   
+107 and a sparse SoftMax function: $a _ { i }  S p a r s e M a x ( r _ { i } ^ { T } A )$ , where $A \in \mathbb { R } ^ { d _ { 1 } ^ { \bullet } \times l _ { r } \times d _ { 2 } }$ . Then, the batch   
+108 matrix multiplication of $A$ and the transpose of $r _ { i }$ results in the addressing vector $a _ { i } \in \mathbb { R } ^ { d _ { 1 } \times 1 \times d _ { 2 } }$ .   
+109 The setting of $d _ { 1 }$ and $d _ { 2 }$ determines the size of address space for storing the embedding vectors.   
+110 Typical addressing methods [23, 28] use a 2D matrix $( l _ { r } \times d _ { 2 } )$ for recording the mapping of an   
+111 embedding vector to a slot ( $d _ { 2 }$ is the number of slots). In contrast, we add one more dimension $d _ { 1 }$   
+112 to simulate the multi-hash setting of traditional sketches, in view of that a 2D addressing matrix   
+113 can reach a differentiable simulation of a hash function [23, 24]. Matrix $A$ simulates multiple hash   
+114 functions, yielding robust frequency decoding and the rationality of the learning optimization. Note   
+115 that each 2D slice $A ^ { * }$ of $A$ is stacked from $d _ { 2 }$ -unit vectors $b _ { i } \in \mathbb { R } ^ { l _ { r } }$ by normalizing the parameters   
+116 of $A$ at each gradient update of the training process. Normalized $A$ can avoid overflowing when   
+117 compressing its size by reducing data precisions and enhance the interpretability (see Section 5).   
+118 In addition, we utilize sparse SoftMax [29, 30] instead of SoftMax to normalize the address $a _ { i }$   
+119 It brings the following benefits by constraining some bits of $a _ { i }$ to zero, which 1) promotes quick   
+120 derivation during the back-propagation; 2) reduces the overhead of storage matrix accessing by   
+121 skipping the slots of $M$ corresponding to the $\mathbf { \vec { \Delta } } ^ { 6 } 0 ^ { 9 }$ bits of $a _ { i }$ ; 3) leads to de-noising with the vector   
+122 compression.   
+123 Compressed storage matrix. We use a matrix $M \in \mathbb { R } ^ { d _ { 1 } \times l _ { z } \times d _ { 2 } } \ .$ 1 to store an embedding vector   
+124 $z _ { i } \in \bar { \mathbb { R } } ^ { l _ { z } }$ in accordance to its address $a _ { i } \in \mathbb { R } ^ { d _ { 1 } \times 1 \times d _ { 2 } }$ . The functionality of $M$ is similar to the 2D   
+125 array of counters in traditional sketches, yet yielding better capabilities in the storage compression.   
+126 Traditional sketches store item counts. Differently, $M$ stores embedding vectors, which have richer   
+27 information compression capabilities, due to the diversity of value changes on different bits.
+
+Decoding. Given a query item $x _ { i }$ , the module $\mathcal { F } _ { d e c }$ , consisting of one NN component $g _ { d e c }$ , decodes the information corresponding to $x _ { i }$ , in order to obtain the estimated frequency $\hat { f } _ { i }$ . The vector fed into $g _ { d e c }$ is the concatenation of vector $\{ M \ominus a _ { i } \}$ , vector $z _ { i }$ , and the current number of items (i.e., $N _ { , }$ ) recorded in a counter, $\hat { f } _ { i }  g _ { d e c } ( \{ M \ominus a _ { i } \} , z _ { i } , N )$ . The operator $\ominus$ refers to the reading operation for the storage matrix. The basic form of $\ominus$ gives the operation as $M \ominus a _ { i } = M a _ { i } ^ { T 2 }$ [27, 28]. For optimization, we consider two optimized forms of $\ominus$ , inspired by the “count-min” mechanism of the CM-sketch. The first one gives the minimum value of each row in $M a _ { i } ^ { T }$ , aiming to remove the noise of other items. The second one gives the minimum value of each row in $\begin{array} { r } { M a _ { i } ^ { T } \circ \frac { 1 } { z _ { i } } } \end{array}$ 1 , a normalized
+
+136 form of $M a _ { i } ^ { T }$ . Here, $\circ$ denotes the Hadamard product, and $z _ { i }$ requires broadcast operations to comply   
+137 with its requirements. So, $\{ M \ominus a _ { i } \}$ refers to the concatenation of vectors generated by the basic   
+138 form and the two optimized forms. Please refer to supplement materials for more details.
+
+# 2.3 Operations
+
+Operation Store is performed by feeding an incoming item $e _ { i }$ to $\mathcal { F } _ { E }$ and ${ \mathcal { F } } _ { S a }$ to obtain embedding vector $z _ { i }$ and address $a _ { i }$ , and then additively writing $z _ { i }$ to $M$ , weighted by $a _ { i }$ : $M \gets M + z _ { i } a _ { i }$ . Here, other writing types [23, 26–28] can also be employed, but simple additive writing is more efficient and allows to compute gradients in parallel [23]. In addition, additive writing also allows to define an optional Delete operation for the meta-sketch (see the supplement materials).
+
+Operation Query estimates the frequency of a given query item $x _ { i }$ . First, $z _ { i }$ and $a _ { i }$ are obtained, similar to that of operation Store. Then, the vectors $\{ M \ominus a _ { i } \}$ are retrieved from $M$ and $N$ can be easily obtained by a small counter. Finally, $\{ M \ominus a _ { i } \}$ , $z _ { i }$ and $N$ are jointly fed into $g _ { d e c }$ to get the estimated frequency ${ \hat { f } } _ { i }$ of $x _ { i }$ as the returned value. The two operations are shown in Algorithm 1.
+
+# Algorithm 1: Operations
+
+1 Operation Store $( e _ { i }$ , M):   
+2 zi, ri ← FE (ei) ;   
+3 ai ← FSa(ri);   
+4 $M \gets M + z _ { i } a _ { i }$ ;   
+9 5 Operation Query $( x _ { i } , M , N )$ :   
+6 $z _ { i } , r _ { i } \gets \mathcal { F } _ { E } ( x _ { i } )$ ;   
+7 $a _ { i } \gets \mathcal { F } _ { S a } ( r _ { i } )$ ;   
+8 $\hat { f } _ { i } \gets \mathcal { F } _ { d e c } ( \{ M \ominus a _ { i } \} , z _ { i } , N )$ ;   
+9 return ${ \hat { f } } _ { i }$ ;
+
+# Algorithm 2: Training Framework
+
+Data: Meta-sketch with all learnable parameters $\theta$ , Meta-task sampler $R$ ;   
+1 while $_ i$ not reach max training steps do   
+2 Sample a meta-task $t _ { i } : \{ s _ { i } , q _ { i } \} \sim R$ and count $N$ ;   
+3 for e(i)j ∈ $e _ { j } ^ { ( i ) } \in s _ { i }$ do Store $( e _ { j } ^ { ( i ) }$ , $M )$ ; end   
+4 for x(ij 7 $f _ { j } ^ { ( i ) } \in q _ { i }$ do fˆ(i ) ← Query(x(i)j , $M , N )$ ; $\mathcal { L } + = \mathrm { L o s s F u n } ( f _ { j } ^ { ( i ) } , \hat { f } _ { j } ^ { ( i ) } ) ;$ ;   
+5 Backprop through: $d \mathcal { L } / d \theta$ and update parameters: $\theta \gets \mathrm { O p t i m i z e r } ( \theta , d \mathcal { L } / d \theta )$ ;   
+6 Normalize A;   
+7 Clear $M$ ;   
+8 end
+
+# 150 3 Meta-sketch training
+
+# 3.1 Training Framework
+
+The meta-sketch employs an efficient one-shot meta-training method [31]. The training process thus contains two phases, pre-training and adaption phases. In the pre-training phase, the meta-sketch learns an initial set of module parameters, including $g _ { e m b }$ , $g _ { a d d }$ , $A$ , and $g _ { d e c }$ . The pre-training goes offline across training units, i.e., basic meta-tasks, to acquire the ability of stream frequency estimation. Then, in the adaption phase, the pre-trained meta-sketch goes fast across a set of lightweighted training units, i.e., adaptive meta-tasks, to quickly acquire the task-specific knowledge, i.e., parameters for sketching real data streams at runtime.
+
+159 The training units, i.e., meta-tasks, are crucial for both phases. The training process of the meta-sketch   
+160 on a single meta-task is equivalent to simulating storing and querying an instance of data streams   
+161 while computing the estimation error to optimize the learnable parameters. Thus, a meta-task $t _ { i }$   
+162 163 consists oas an inst a store setce of data $s _ { i }$ (also eams, ) and a, where ery set is the $q _ { i }$ . The store set mber of strea $s _ { i }$ can betems in ewed. The   
+$s _ { i } : \{ e _ { 1 } ^ { ( i ) } , . . . , e _ { N _ { i } } ^ { ( i ) } \}$ $N _ { i }$ $s _ { i }$   
+164 $q _ { i }$ ican be represented by a set of items from the stream instance with paired frequencies in   
+165 the store set $s _ { i }$ , formally, $q _ { i } : \{ ( x _ { 1 } ^ { ( i ) } : f _ { 1 } ^ { ( i ) } ) , . . . , ( x _ { n _ { i } } ^ { ( i ) } : f _ { n _ { i } } ^ { ( i ) } ) \}$ , where $n _ { i }$ is the number of distinct items   
+166 in $s _ { i }$ . In this work, we define two types of meta-tasks, basic (Section 3.2) and adaptive (Section 3.3)   
+167 meta-tasks, corresponding to the pre-training and adaption phases, respectively.   
+168 The two training phases, that are based on different types of meta-tasks, follow the same training   
+169 framework, as shown in Algorithm 2, except for the sampler and initial parameters. To optimize on   
+170 reducing both absolute and relative frequency estimation errors3, we devise an adaptive hybrid loss   
+171 function [32] for the meta-sketch: $\begin{array} { r } { \frac { 1 } { 2 \sigma _ { 1 } ^ { 2 } } ( f _ { i } - \hat { f } _ { i } ) ^ { 2 } + \frac { 1 } { 2 \sigma _ { 2 } ^ { 2 } } | f _ { i } - \hat { f } _ { i } | / f _ { i } + l o g \sigma _ { 1 } \sigma _ { 2 } } \end{array}$ , where $\sigma _ { 1 }$ and $\sigma _ { 2 }$ are   
+172 learned parameters, and $f _ { i }$ and $\hat { f } _ { i }$ are the true and estimated frequencies of item $x _ { i }$ , respectively.   
+174 In the pre-training phase, basic meta-tasks should make the meta-sketch to simulate traditional   
+175 sketches and preserve certain generality without relying too much on the patterns of specific distribu  
+176 tions (Section 5). Therefore, we generate meta-tasks based on the Zipf distribution, which is found to   
+177 be prevalent in real scenes of data streams [16–20].   
+178 A meta-task is essentially a data stream instance with item size $n$ , which can be determined by the   
+179 total number of items $N$ and the relative frequency distribution $p$ . Alternatively, we can generate   
+180 meta-tasks by presupposing different $n$ , $\bar { f }$ and $p$ , where $\bar { f }$ is the frequency mean, since $\scriptstyle { \bar { N } } = { \bar { f } } \times n$   
+181 Thus, basic meta-task generation is based on a sampler $R : \{ I , L , P \}$ , as follows.   
+182 An item pool $I$ is a subset of the item domain $\mathbb { X }$ . The cardinality of $I$ is in relevance to the   
+183 identification capability of the meta-sketch. If the item domain is known a-priori, it can be directly   
+184 taken as the item pool. Otherwise, in applications where the item domain is only partially known or   
+185 even unknown, the item pool can be constructed by sampling from the historical records. Even in the   
+186 case that the item pool does not completely cover the item domain, the “missing” item can still be   
+187 identified, due to the homogeneity of the domain-specific embedding space, given that the number of   
+188 distinct items does not meet the item pool capacity $| I |$ .
+
+A frequency mean range $L$ is the range for the frequency mean $\bar { f }$ . One can get the value of $\bar { f }$ by statistics of each sampled stream instance and extract the minimum and maximum $\bar { f } \mathbf { s }$ to build $L$ .
+
+91 A distribution pool $P$ consists of many instances generated according to different parameters of   
+92 relative frequency distributions. In this paper, we consider a family of Zipf distributions [33] with   
+93 varied parameter $\alpha$ , as the base for constructing $P$ . $\alpha$ can be selected from a wide range to have a   
+94 good coverage of different distributions.   
+195 Notice that the meta-tasks are for the meta-sketch to learn the sketching ability, instead of spoon  
+196 feeding the meta-sketch to mechanically memorize the parameters of $R$ . It means that the trained   
+197 meta-sketch has the generalization ability to handle the case not covered in $R$ (see Section 4.2).   
+198 The generation of a meta-task $t _ { i }$ can be done based on sampler $R$ , as follows. We first randomly   
+199 sample a subset of $n _ { i }$ items from $I$ , and a frequency mean ${ \hat { f } } _ { i } \in L$ . Then, we sample a distribution   
+200 instance $p _ { i } ~ \in ~ P$ and make the $n _ { i }$ items’ frequencies conform to $p _ { i }$ and ${ \bar { f } } _ { i }$ . For example, the   
+201 frequencies of $n _ { i }$ items can be set as $n _ { i } \times \bar { f } _ { i } \times \bar { p } _ { i }$ , where $p _ { i } \sim Z i p f ( \alpha )$ is a random variable. The   
+202 above steps are repeated until the store set $s _ { i }$ and query set $q _ { i }$ are built.
+
+# 203 3.3 Adaptive Meta-task Generation
+
+While processing real data streams, we can get the item set $I _ { r }$ and its distribution $p _ { r }$ by online sampling. $I _ { r }$ and $p _ { r }$ are then used for generating the set of adaptive meta-tasks. For each adaptive meta-task, an item subset is sampled from $I _ { r }$ , and the relative frequency corresponding to each item is sampled from $p _ { r }$ . The process is similar to the generation of basic meta-tasks. The only difference from basic meta-task generation is that, there is no distribution pool anymore, because the real data stream is unique. Also, we intentionally randomize the correspondence between an item and its real relative frequency on the original data records. It is equivalent to constructing meta-tasks where the item frequencies dynamically change. For example, the frequency of an item may first increase, then suddenly drop [21]. With adaptive meta-tasks, the meta-sketch learns to quickly adapt to the distribution $p _ { r }$ , while being flexible against the item frequency change. The detailed algorithms of generating basic/adaptive meta-tasks are shown in supplement materials.
+
+# 4 Experiments
+
+# 4.1 Basic Setup
+
+Dataset. We use two real datasets. Word-query is a streaming record of search queries, where each query contains multiple words (e.g., “News today”) [15]. IP-trace consists of IP packets, where each packet is identified by a unique source/destination address pair (e.g., 192.168.1.1/12.13.41.4) [21]. We assume that query phrases and IP addresses are numerically encoded, similar to [15].
+
+Table 1: Results of Basic Meta-sketch $( T _ { r } )$   
+
+<table><tr><td></td><td colspan="5">Word-query</td><td colspan="4">IP-trace</td></tr><tr><td>Method</td><td>Metrics</td><td>n=5K, B=9KB</td><td>n=10K, B=11KB</td><td>n=20K, B=13KB</td><td>n=40K, B=15KB</td><td>n=5K, B=9KB</td><td>n=10K, B=11KB</td><td>n=20K, B=13KB</td><td>n=40K, B=15KB</td></tr><tr><td>Basic MS</td><td>ARE</td><td>12.3</td><td>14.74</td><td>10.98</td><td>13.79</td><td>3.00</td><td>1.51</td><td>2.97</td><td>1.13</td></tr><tr><td rowspan="2">CS</td><td>AAE</td><td>31.54</td><td>38.54</td><td>40.63</td><td>53.67</td><td>5.57</td><td>5.01</td><td>6.94</td><td>5.56</td></tr><tr><td>ARE</td><td>32.94</td><td>57.97</td><td>98.01</td><td>162.43</td><td>6.08</td><td>9.94</td><td>15.57</td><td>24.49</td></tr><tr><td rowspan="2">CMS</td><td>AAE</td><td>57.54</td><td>101.44</td><td>172.44</td><td>282.59</td><td>10.42</td><td>16.82</td><td>26.46</td><td>41.91</td></tr><tr><td>ARE</td><td>21.34</td><td>48.33</td><td>111.82</td><td>239.11</td><td>8.12</td><td>16.07</td><td>32.77</td><td>65.19</td></tr><tr><td></td><td>AAE</td><td>38.04</td><td>84.62</td><td>195.61</td><td>416.01</td><td>13.67</td><td>27.39</td><td>55.29</td><td>110.65</td></tr></table>
+
+Table 2: Results of Basic Meta-sketch $( T _ { s } )$ )   
+
+<table><tr><td>Method</td><td>Metrics</td><td colspan="3">n=5K,B=9KB</td><td colspan="3">n=10K,B=11KB</td><td colspan="3">n=20K,B=13KB</td><td colspan="3">n=40K,B=15KB</td></tr><tr><td></td><td></td><td>0.5</td><td>1.1</td><td>1.5</td><td>0.5</td><td>1.1</td><td>1.5</td><td>0.5</td><td>1.1</td><td>1.5</td><td>0.5</td><td>1.1</td><td>1.5</td></tr><tr><td>Basic MS</td><td>ARE</td><td>0.43</td><td>1.05</td><td>2.63</td><td>0.73</td><td>3.25</td><td>3.14</td><td>0.47</td><td>1.67</td><td>1.35</td><td>0.43</td><td>2.58</td><td>9.65</td></tr><tr><td>(Word-query)</td><td>AAE</td><td>24</td><td>17.72</td><td>8.93</td><td>3124</td><td>27.02</td><td>9.41</td><td>27.29</td><td>22.19</td><td>三</td><td>25.04</td><td>26.95</td><td>19.87</td></tr><tr><td>Basic MS</td><td>ARE</td><td>0.59</td><td>2.27</td><td>9.38</td><td>0.73</td><td>0.86</td><td>1.02</td><td>0.72</td><td>1.73</td><td>7.52</td><td>0.73</td><td>0.79</td><td>233</td></tr><tr><td>(IP-trace)</td><td>AAE</td><td>26.45</td><td>21.49</td><td>14.73</td><td>38.33</td><td>19.32</td><td></td><td>35.48</td><td>22.28</td><td>15.74</td><td>39.57</td><td>21.75</td><td>14.06</td></tr><tr><td rowspan="3">CS</td><td>ARE</td><td>1.98</td><td>6.72</td><td>10.99</td><td>2.7</td><td>12.12</td><td>16.9</td><td>3.73</td><td>20.8</td><td>27.46</td><td>5.17</td><td>37.96</td><td>43.76</td></tr><tr><td>AAE</td><td>74.96</td><td>47.98</td><td>15.89</td><td>102.05</td><td>75.83</td><td>23.8</td><td>140.65</td><td>118.29</td><td>38.7</td><td>194.32</td><td>198.4</td><td>59.96</td></tr><tr><td>ARE</td><td>4.96</td><td>7.52</td><td>5.47</td><td>9.27</td><td>15.85</td><td>9.44</td><td>17.29</td><td>32.7</td><td>16.38</td><td>32.24</td><td>66.35</td><td>27.89</td></tr><tr><td>CMS</td><td>AAE</td><td>187.52</td><td>53.81</td><td>8.17</td><td>350.08</td><td>99.82</td><td>13.58</td><td>651.63</td><td>185.54</td><td>22.88</td><td>1213.38</td><td>347.32</td><td>38.18</td></tr></table>
+
+Baseline. We hereby evaluate the basic and advanced meta-sketches. From now on, we use MS to represent the term meta-sketch for brevity. We compare basic MS (after the pre-training phase) with CM-sketch (CMS) and C-sketch (CS). We compare the advanced MS (after the adaptation phase) with learned augmented sketch (LS) and cold filter (CF), which are two variants of CM/C sketches with auxiliary structures. According to the default setting [10, 11], the number of hash functions for all sketches is 3. We adopt two commonly accepted metrics for evaluating the accuracies of stream frequency estimation, AAE and $\mathrm { A R E } ^ { 3 }$ .
+
+Parameters. We implement $g _ { e m b }$ or $g _ { a d d }$ in MLP with 2-layers of sizes 128 and 48, followed by batch normalization, and $g _ { d e c }$ in an MLP with 3-layers of 256 with residual connections. We use the relu function for layer connections. The space budget $B$ is spent on storing $M$ , the same as the setting in neural data structures [23]. Other modules, like hashing libraries, are commonly accepted as reusable and amortizable resources for multi-deployment of sketches [21, 23]. Note that due to space limitations, the details and methods of parameter settings of $M ( A )$ , the ablation experiments and some parameter discussions are shown in the supporting material.
+
+# 4.2 Basic Meta-sketch
+
+Settings. For each dataset, we train the basic MSs under 4 item pools with $\{ 5 K , 1 0 K , 2 0 K , 4 0 K \}$ different items, respectively. The meta-task sampler are with Zipf distributions. We build the distribution pools set with $\alpha \in [ 0 . 8 , 1 . 3 ]$ and set frequency mean range $L = [ 5 0 , 5 0 0 ]$ . For basic meta-sketch training, the default maximum number of training steps $\phi$ is 5 million, the learning rate is 0.0001, and the Adam optimizer is used. For evaluation, we consider two types of tasks, $T _ { r }$ and $T _ { s }$ . $T _ { r }$ are directly obtained by random sampling on two real data streams with different values of $n$ i.e., the number of distinct items. Note that the frequency distributions of $T _ { r }$ are not necessarily obey Zipf distributions. $T _ { s }$ are the synthetic tasks, where the item frequency follows the Zipf distribution with $\alpha \in \{ 0 . 5 , 1 . 1 , 1 . 5 \}$ . To evaluate the generability and stability of basic MS, both $T _ { s } ( 0 . 5 )$ and $T _ { s } ( 1 . 5 )$ ’s distributions are not covered by the distribution pool of the meta-task samplers.
+
+Performance. Table 1 shows the performance of all competitors based on real dataset $T _ { r }$ . It shows that the basic MS outperforms traditional basic sketches, i.e., CMS and CS, on all testing cases. For example,the results on IP-trace show that, when $\scriptstyle n = 4 0 \mathrm { K }$ and $B { = } 1 5 \mathrm { K B }$ , the ARE of basic MS is 1.13, while AREs of CMS and CS are 65.19 and 24.49, respectively. The advantage of meta-sketch is significant when testing on $T _ { s }$ with different $\alpha \mathbf { s }$ , as shown in Table 2. Note that we use random choices to simulate the ideal hash functions for traditional sketches like [15], so that CS and CMS have the same result on test tasks with the same $\alpha$ in both datasets.
+
+253 We show the trend of ARE w.r.t. the space budget, in Figure 2 ( $T _ { r }$ , $n { = } 5 \mathrm { K }$ , Word-query). Compared to   
+254 the dramatic performance degrading of traditional sketches, basic MS holds stable performance. We   
+255 show that the trend of ARE w.r.t. the number of distinct items in Figure 3 $T _ { r }$ , $B { = } 9 { \mathrm { K B } }$ , Word-query).   
+256 Compared to traditional sketches, the ARE of basic MS increases sub-linearly w.r.t. the value of $n$   
+257 Note that AAE has similar results for the above experiments, see the supplement materials.   
+258 Generalization. We test the generality of basic MS to new items that are not in the item pool of   
+259 the meta-task sampler in Figure 4(a). We make the experiments $_ { \it { n } = 5 \mathrm { { K } } }$ , $B { = } 9 \mathrm { K B }$ , Word-query)   
+260 by replacing some items in $T _ { r }$ with new items, and vary the fraction of new items to observe   
+261 the trend of the performance. It shows that the ARE/AAE moderately increases w.r.t. the ratio   
+262 of new items. The performance is acceptable considering the fact that the item domain is often   
+263 stable in practical applications. We then test the generality of meta-sketches to varied frequency   
+264 means that are not in range $L$ of the meta-task sampler, as shown in Figure 4(b). The experiment   
+265 $_ { \mathrm { { \it ~ \Omega } } } \mathrm { { \it { n } = } } 5 \mathrm { { K } }$ , $B { = } 9 \mathrm { K B }$ , Word-query) is done by sampling a series of $T _ { s }$ tasks with frequency means in   
+266 $\{ 5 0 0 , 5 K , 5 0 K , 5 0 0 K , 5 0 0 0 K \}$ . It shows that as the mean of the true frequencies increases, the   
+267 estimated frequencies of the meta-sketch increase linearly, so that the ARE keeps stable.
+
+![](images/273c358ecb0e44ea9148d4a4a9ecebbb1e6ae0e4e13880455e100dded29bf51c.jpg)  
+Figure 2: ARE w.r.t. $B$
+
+![](images/7a983ba6aed4f129823b48f94344d29d582389e050a05e2c5fdabba1feeaa01a.jpg)  
+Figure 3: ARE w.r.t. n
+
+![](images/eccb4d68e053dfab3c8b7424866934ab319fe055d727c787fe6a6a21f60f4423.jpg)  
+Figure 4: Generality of Meta-sketch
+
+Table 3: Results of Advanced Meta-sketch   
+
+<table><tr><td>Method</td><td>Metrics</td><td colspan="4">Word-query</td><td colspan="4">IP-trace</td></tr><tr><td></td><td></td><td>n=5K B=9KB</td><td>n=10K, B=11KB</td><td>n=20K B=13KB</td><td>n=40K B=15KB</td><td>n=5K B=9KB</td><td>n=10K B=11KB</td><td>n=20K B=13KB</td><td>n=40K B=15KB</td></tr><tr><td>Advanced</td><td>ARE</td><td>3.05</td><td>2.83</td><td>4.06</td><td>5.20</td><td>0.87</td><td>0.89</td><td>1.38</td><td>2.29</td></tr><tr><td rowspan="2">MS</td><td>AAE</td><td>21.42</td><td>26.11</td><td>35.00</td><td>43.81</td><td>3.77</td><td>4.46</td><td>5.13</td><td>6.55</td></tr><tr><td>ARE</td><td>3.58</td><td>14.53</td><td>141.70</td><td>1127.11</td><td>0.85</td><td>2.74</td><td>4.20</td><td>16.71</td></tr><tr><td rowspan="2">CF90</td><td>AAE</td><td>21.13</td><td>59.18</td><td>381.63</td><td>2217.28</td><td>1.32</td><td>3.01</td><td>7.71</td><td>31.20</td></tr><tr><td>ARE</td><td>7.95</td><td>29.02</td><td>139.87</td><td>541.37</td><td>1.51</td><td>3.10</td><td>8.95</td><td>46.79</td></tr><tr><td rowspan="2">CF70 CF40</td><td>AAE</td><td>29.02</td><td>76.58</td><td>295.63</td><td>970.94</td><td>2.57</td><td>5.51</td><td>16.83</td><td>82.84</td></tr><tr><td>ARE</td><td>91.16</td><td>138.64</td><td>244.24</td><td>407.83</td><td>12.62</td><td>33.50</td><td>103.76</td><td>155.61</td></tr><tr><td rowspan="2"></td><td>AAE</td><td>174.86</td><td>252.22</td><td>421.85</td><td>693.47</td><td>24.16</td><td>60.79</td><td>175.14</td><td>279.72</td></tr><tr><td>ARE</td><td>20.52</td><td>48.69</td><td>111.85</td><td>266.50</td><td>8.34</td><td>17.09</td><td>35.22</td><td>77.79</td></tr><tr><td rowspan="2">LCMS(1%) LCS(1%)</td><td>AAE</td><td>37.80</td><td>81.93</td><td>194.15</td><td>451.28</td><td>13.72</td><td>28.39</td><td>59.10</td><td>129.86</td></tr><tr><td>ARE</td><td>25.53</td><td>40.84</td><td>67.21</td><td>104.54</td><td>5.20</td><td>7.80</td><td>11.33</td><td>17.12</td></tr><tr><td rowspan="2"></td><td>AAE</td><td>44.53</td><td>78.17</td><td>122.57</td><td>180.56</td><td>8.78</td><td>13.10</td><td>18.97</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>28.38</td></tr></table>
+
+# 4.3 Advanced Meta-sketch
+
+Settings. The generation of adaptive meta-tasks is similar to that of basic meta-tasks (Section 3.2), except that each item pool reads real frequency distributions for the adaption as described in Section 3.3. In the adaption phase, the maximum number of training steps is $0 . 0 0 2 * \phi$ .
+
+Performance. Table 3 compares the performance of advanced MS with traditional sketches and their variants, LS and CF, on real dataset $T _ { r }$ . We implement two LSs according to [15], learned CM-sketch (LCMS) and learned C-sketch (LCS), following the default setting that (top $1 \%$ ) high-frequency items are separately stored. For CF, we follow the parameter setting in [14], and use CF40, CF70, and CF90 for setting the filter percentages to $40 \%$ , $70 \%$ , and $90 \%$ of the total size, respectively. It shows that the advanced MS achieves a better performance than LSs and CFs. Also, AAE/ARE of advanced MS increases more moderately w.r.t. the number of distinct items $n$ , compared to its competitors.
+
+Furthermore, we compare the performance of the advanced MS and the LS under dynamic streaming scenarios, as shown in Figure 5. We select a set of $T _ { r }$ $\scriptstyle ( n = 5 \mathbf { K } , B = 9 \mathbf { K } \mathbf { B }$ ,Word-query), and gradually shuffle the correspondence between items and frequencies. Here, the shuffle ratio is increased from 0 to $100 \%$ . It shows that the average ARE of advanced MS only slightly fluctuates between 3.26 and 4.0, and the average AAE is in the range of 21.28 and 21.68. In contrast, AAE of LCS or LCMS starts above 37, and increase significantly w.r.t. the increase of the shuffle ratio. Actually, the classifier of LS tends to incur more errors due to the gradual shift of high- and low-frequency items, resulting in an increased number of hash collisions, thus deteriorating the estimation accuracy.
+
+# 87 5 Analysis
+
+The meta-sketch is trained based on meta-tasks, consisting of various stream distributions. We expected that the meta-sketch can learn the ability to sketch item frequencies. Somehow, it is unavoidable that the meta-sketch’s ability is limited by patterns of given meta-tasks. Thus, setting up the two training phases benefits the balance of the trade-offs. In the pre-training phase, we select the most representative Zipf distribution to form basic meta-tasks, making the basic meta-sketch adaptable to a wide range of data streams. In the adaptation phase, we sample adaptive meta-tasks from raw data streams to make the advanced meta-sketch more specialized. Next, we analyze the
+
+![](images/d0d131b9a2a79cf8bf7e61fd7ccfb428bf42fe147e8275ccfd8b44db70bdb7eb.jpg)  
+Figure 5: Learned sketch vs. Meta-sketch   
+Figure 6: $| r |$ and $| z |$ w.r.t. Sparsity of $a$
+
+![](images/0779b9509a5fe768b70c87bfd9beb5d0fa863c31f8781ba7defec798cbf20906.jpg)  
+Figure 7: Three addressing matrices
+
+![](images/043316aa344876317461ed7124d4cba5dcd0936958f3ed12872e46f4239441de.jpg)  
+Figure 8: The sparsity of embedding vectors
+
+295 working mechanism of the three modules of the meta-sketch as well as their roles in acquiring the   
+296 two abilities.
+
+Sparse Addressing Module. We take a 2D slice $A ^ { * }$ (size is $l _ { r } \times d _ { 2 } ,$ ) of the $A$ matrix to analyze the process of a refined vector $r$ getting addressing $a$ through this module. First, we have $a \gets$ $\bar { S p a r s e M a x } ( r ^ { T } A ^ { * } ) \Rightarrow a  \bar { S p a r s e M a x } ( \langle r \cdot b _ { 1 } , \bar { r } \cdot b _ { 2 } , . . . , \bar { r } \cdot b _ { d _ { 2 } } \rangle )$ . Since $b _ { i }$ are unit vectors, we can get $a \gets S p a r s e M a x ( | r | c )$ , $c { = } \langle c o s \theta _ { 1 } , c o s \theta _ { 2 } { , } . . . , c o s \theta _ { d _ { 2 } } \rangle$ , where $\theta _ { i }$ is the angle between $r$ and $b _ { i }$ . We continue to transform the form to get addressing $\begin{array} { r } { a \gets S p a r s e g e n ( c ; u ; \frac { | r | - 1 } { | r | } ) } \end{array}$ ; |r|−1|r| ) [30], where u is a component-wise transformation function applied on $c$ . in this paper, we set $u ( c ) { = } c$ .
+
+Based on the principle of Sparsegen [30], $| r |$ mainly affects the sparsity (i.e., the proportion of non-zero bits in the vector) of $a$ during training process, while $c$ determines the positions and values of non-sparse bits. The Figure 6 shows a strong correlation between the average $| r |$ and the sparsity of $a$ during training from scratch ( $_ { \mathrm { \Delta } n = 5 \mathrm { K } }$ , $B { = } 9 \mathrm { K B }$ , Word-query, Basic MS). Since the embedding vector $z$ does not directly participate in the addressing process, the average $| z |$ remains stable. Further, we observe that the sparsity of $a$ will eventually converge to around 1, which means that each item is generally stored in a slot corresponding to the refined vector $r$ and the unit vector in $A ^ { * }$ with the maximum cosine similarity.
+
+Therefore, the role of $A ^ { * }$ is to map refined vectors to the addressing vectors. The $d _ { 2 }$ unit vectors in $A ^ { * }$ are the reference standard for mapping, which is equivalent to the mutually exclusive $d _ { 2 }$ -divisions of the refined vector space. Follow this point, we construct two matrices $K ^ { * }$ and $R ^ { * }$ of the same size as $A ^ { * }$ . Among them, the $d _ { 2 }$ unit vectors in $K ^ { * }$ come from the cluster centers of the sampled refined vectors. To achieve mutually exclusive division, we perform Kmeans clustering with $K = d _ { 2 }$ and Cosine similarity criterion. Then, we normalize the resulting $d _ { 2 }$ cluster centers and stack them as $K ^ { * }$ In contrast, the unit vectors in $R ^ { * }$ are entirely randomly generated.
+
+Figure 7 (a) shows the results of replacing $A ^ { * }$ on the trained meta-sketch with $K ^ { * }$ and $R ^ { * }$ . The meta-sketch with $R ^ { * }$ shows the worst performance, but the performance of the meta-sketch with $K ^ { * }$ is close to the original $A ^ { * }$ . Furthermore, We count the number of items mapped in every slot of $A ^ { * }$ , $K ^ { * }$ , $R ^ { * }$ and show their standard deviation in Figure 7 (b). The standard deviation of $R ^ { * }$ is much higher than $A ^ { * }$ and $K ^ { * }$ , and a better meta-sketch tends to store items more evenly in each slot. Thus, The addressing module simulates the traditional sketch mechanism. Its principal function is to store the embedding vectors of items as evenly as possible in multiple memory slots, and an item is written to only one slot.
+
+Embedding Module. The major source of conflicts in the meta-sketch is the stacking of different embedding vectors in a single slot. Thus, the sparsity of the embedding vector becomes an important indicator to determine the degree of conflicts. Figure 8 shows the relation between the sparsity of embedding vectors and the stream distributions $_ { \mathrm { { \it ~ \Omega } } } \mathrm { { n = 5 K } }$ , $B { = } 9 \mathrm { K B }$ , Word-query, advanced MS). We select the meta-tasks under Zipf, Triangular, and Uniform distributions with different skewness levels (the definition of skewness and corresponding distribution parameters are shown in the supplement materials). The results show that the sparsity of the embedding vector is positively proportional to
+
+![](images/5620e00a605ae8e5f51d00f04f6f4a362ff0bcfb931112fd394003077a6bbd5a.jpg)  
+Figure 9: Generality w.r.t. Decoding module
+
+![](images/a9e757a406ba1125a8bff13effa28389881a2171b22e5518d37206404778e386.jpg)  
+Figure 10: Unstable case vs. Stable case
+
+333 the skewness of a distribution. Therefore, we speculate that the meta-sketch memorizes the pattern   
+334 information of the distribution being adapted by self-tuning the sparsity of embedding vectors.
+
+Decoding Module. The decoding module, as the deepest NNs in the meta-sketch, integrates various information to predict the item frequency and achieves generalization ability. To verify this, we adapt the advanced MS $_ { \it { n } = 5 \mathrm { { K } } }$ , $B { = } 9 \mathrm { K B }$ , Word-query) to a special adaptive meta-task. The meta-task was sampled from the real data stream but with a fixed item size (5000) and frequency mean (250). Meanwhile, we do not change the correspondence between items and frequencies. Such meta-task forces the meta-sketch to pay more attention to the fixed patterns and thus limit its generalization.
+
+Thus, we train the advanced MS with (or without) freezing the decoding module parameters based on the above meta-task. Figure 9 (a) shows the performance changes of the three models (advanced MS as baseline) on the evaluation tasks $( T _ { r } )$ of different item sizes. Without the frozen decoding module, the meta-sketch loses generalization ability at extended item sizes other than 5000. On the contrary, the meta-sketch with the frozen decoding module still retains the generalization ability and further utilizes the data stream pattern compared to the advanced MS, achieving the best performance. Similarly, as shown in Figure 9 (b), the meta-sketch without the frozen decoding module also loses a certain generalization ability in terms of frequency mean.
+
+349 Actually, the above meta-task (termed as the stable case) can be viewed as a special case of an ordinary   
+350 adaptive meta-task (termed as the unstable case). As a matter of fact, augmented sketches utilize   
+351 frequency patterns similar to the stable case. For example, the learned augmented sketch memorizes   
+352 (relatively) stable correspondence between items and frequencies, for filtering high-frequency items.   
+353 To understand the meta-sketch’s self-optimizing mechanism from the unstable case to the stable case,   
+354 we analyze the storage of high/low-frequency items between multiple slots and a single slot in the   
+355 memory. In Figure 10 (a), we show density heat-maps of low-frequency (below the top $20 \%$ high   
+356 frequencies) items, stored by meta-sketches of stable and unstable cases on a 2D slice $d _ { 1 } = 2 \AA$ ) of the   
+357 storage matrix $M$ , where the $\mathbf { X }$ -axis is the index of slots. The two heat-maps show that the meta-sketch   
+358 under the stable case can store the low-frequency items concentratedly in some slots to avoid the   
+359 conflicts with high-frequency items. Interestingly, the meta-sketch does not intentionally do this like   
+360 augmented sketches. Instead, it is achieved by self-optimization during the training. Furthermore,   
+361 Figure 10 (b) shows the relation between the sparsity of the embedding vector of items stored in a   
+362 single slot and the frequency order, where the $x$ -axis represents the frequencies in the ascending order.   
+363 We speculate that the meta-sketch autonomously adjusts the sparsity of the embedding vector within   
+364 a single slot in the stable case, so that the high/low-frequency items are automatically separated.
+
+# 365 6 Conclusion
+
+In this paper, we propose a neural data structure, called the meta-sketch, for estimating item frequencies in data streams. Unlike traditional sketches, the meta-sketch utilizes meta-learning and memory-augmented neural networks. The meta-sketch is pre-trained with Zipf distributions and can be fast adapted to specific runtime streams. We study a series of techniques for constructing the meta-sketch. We also devise the generation of basic and adaptive meta-tasks corresponding to the pre-training and adaption phases, respectively. Extensive empirical studies on real datasets are done to evaluate our proposals. In the future, it is interesting to extend our proposal to other sketching tasks that are supported by traditional sketches.
+
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+[22] Tim Kraska, Alex Beutel, Ed H Chi, Jeffrey Dean, and Neoklis Polyzotis. The case for learned index structures. In SIGMOD, pages 489–504, 2018.   
+[23] Jack Rae, Sergey Bartunov, and Timothy Lillicrap. Meta-learning neural bloom filters. In ICML, pages 5271–5280. PMLR, 2019.   
+[24] Michael Mitzenmacher. A model for learned bloom filters and related structures. arXiv preprint arXiv:1802.00884, 2018.   
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+
+# 7 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] Compared with traditional data structures,neural data structures are usually relative weak in term of time latency. In future research, We need to study and reduce time cost of meta-sketches’ operation or disign a framework to get a huge throughput utilizing parallel algebraic operations as a remedy.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] There is no negative societal impacts of my work, since it is foundational research.   
+(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? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+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] We give the details of the implementation as much as possible,but some of them will put into appendix.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We visualize the difference in line graphs by drawing shadows, which includes various of comparative experiments with all type of metasketches. But due to the huge amount of data ,error bars of table are not included. See section 4 and 5.
+
+(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] All of our experiments are implemented in python and run at a NVIDIA DGX workstation with CPU E5-2698 (2.20GHz, 20 cores), and 4 NVIDIA V100 GPUs (5120 CUDA cores and 16GB GPU memory on each GPU).
+
+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? [N/A]   
+(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...
+
+(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]

parse/dev/-kS21GWVJU/-kS21GWVJU_content_list.json

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+        "type": "text",
+        "text": "Meta-sketch: A Neural Data Structure for Estimating Item Frequencies of Data Streams ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "text": "1 To estimate item frequencies of data streams with limited space, sketches are widely   \n2 used in real applications, including real-time web analytics, network monitoring,   \n3 and self-driving. Sketches can be viewed as a model which maps the identifier of a   \n4 stream item to the corresponding frequency domain. Starting from the premise, we   \n5 envision a neural data structure, which we term the meta-sketch, to go beyond the   \n6 basic structure of conventional sketches. The meta-sketch learns basic sketching   \n7 abilities from meta-tasks constituted with synthetic datasets following Zipf distribu  \n8 tions in the pre-training phase, and can be fast adapted to real (skewed) distributions   \n9 in the adaption phase. Extensive experiments demonstrate the performance gains   \n10 of the meta-sketch and offer insights into our proposals. ",
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+    {
+        "type": "text",
+        "text": "11 1 Introduction ",
+        "text_level": 1,
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+        "text": "12 Estimating item frequency is a basic topic in data stream processing, which finds applications in   \n13 the fields of networking, databases, and machine learning, such as real-time data analyzing [1–4],   \n14 network traffic monitoring [5–7], natural language processing [8] and search ranking [9]. Towards   \n15 infinite data streams, a common class of solutions [10–15] use a compact structure taking sublinear   \n16 space for counting the number of occurrences of each stream item, called the sketch.   \n17 Under the prevalent evidence of skewed distributions in data streams, basic sketches achieve the space   \n18 compactness by hashing and approximately aggregating stream items. Basic sketches, including   \n19 CM-sketch [10], C-sketch [11] and CU-sketch [12], use a 2D array of counters as the core structure.   \n20 To optimize the sketching performance, there arise augmented sketches [13,14], which attach filters to   \n21 basic sketches, to capture the preliminary patterns of skewed distributions (e.g., high/low-frequency   \n22 items). By separately maintaining the filtered high/low-frequency items, augmented sketches strive   \n23 to eliminate the estimation error incurred by hash collisions between the high- and low-frequency   \n24 items. Further, learned augmented sketches [15] improve the filters of the augmented sketches by   \n25 memorizing short-term high/low-frequency items via a pre-trained neural network (NN in short)   \n26 classifier. But it is not clear how the pre-trained NN can be adapted to dynamic streaming scenarios,   \n27 where the correspondence between items and frequencies varies. In a nutshell, sketches are structures   \n28 compactly summarizing stream distributions to count item frequencies with limited space budgets.   \n29 From the retrospective analysis of sketches, an observation can be drawn that the evolution of   \n30 sketches conforms with the exploitation of data distributions. It is thus a natural evolution to consider   \n31 a sketch that generally and automatically captures more distribution patterns with limited space   \n32 budgets. In this paper, we envision a novel neural sketch, called the meta-sketch, with techniques   \n33 of meta-learning and memory-augmented neural networks. The meta-sketch learns the sketching   \n34 abilities from automatically generated meta-tasks. Depending on the types of meta-tasks, we study   \n35 two versions of the meta-sketch, called basic and advanced meta-sketches.   \n36 The basic meta-sketch implements the simulation of basic sketches, through the training process with   \n37 basic meta-tasks following Zipf distributions, which are prevalent in the scenes of real data streams [16–   \n38 20]. The advanced meta-sketch extends the basic version to fast adapt to the specific runtime of stream   \n39 processing, through the training with adaptive meta-tasks, which are generated by online sampling   \n40 of real data streams. Our work follows a typical setting where the distribution of item frequencies   \n41 follows a skewed distribution, but the correspondence between items and frequencies varies. For   \n42 example, in software-defined networks (SDN), sketches are deployed to programmable switches to   \n43 collect per-flow statistics, where IP packets follow heavy-tailed distributions [15, 21]. In distributed   \n44 databases, it gives advances to collect statistics of data shards to optimize data placement and   \n45 query caching, where query phrases follow approximate Zipf distributions [15]. Given that the item   \n46 population follows a specific distribution, the local distributions, i.e., item-frequency correspondences   \n47 on shards or flows, are different. Instead of retraining learned augmented sketches on each local   \n48 distribution, the advanced-sketch can be quickly adapted to different local distributions once trained.   \n49 As a member of the neural data structure family [15, 22–24], the meta-sketch significantly differs   \n50 from conventional sketches, in terms of the structure and working mechanism. The meta-sketch   \n51 utilizes NN’s powerful encoding/decoding capabilities to perceive data distributions and express   \n52 and compress explicit or implicit information to retrieve item frequencies with better accuracies.   \n53 Meanwhile, the meta-sketch is differentiable to fully perceive frequency patterns for self-optimization.   \n54 Our contributions are as follows. 1) We propose the meta-sketch, the first neural data structure for the   \n55 problem of item frequency estimation, based on meta-learning. 2) The basic meta-sketch acquires   \n56 sketching abilities by learning from synthetic datasets, and outperforms basic sketches in real datasets.   \n57 The advanced meta-sketch automatically encompasses the ability analogous to the auxiliary structures   \n58 deliberately devised in (learned) augmented sketches, yet yielding better accuracies and robustness   \n59 when adapted to dynamic scenes. 3) Through extensive empirical studies on real and synthetic   \n60 datasets, we evaluate our proposed meta-sketches and analyze the mechanism of major modules. 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+        "image_caption": [
+            "Figure 1: The Framework of the Meta-sketch "
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+        "text": "61 2 Meta-sketch Structure ",
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+        "text": "2.1 Preliminaries ",
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+        "text": "We consider a standard data stream scenario [19]. Suppose a data stream $\\boldsymbol { S } _ { N } : \\{ e _ { 1 } , . . . , e _ { N } \\}$ with $N$ items and $n$ distinct items. Each item $e _ { i } \\in S _ { N }$ takes a value from the item domain $\\mathbb { X } = \\{ x _ { 1 } , . . . , x _ { n } \\}$ where $x _ { i } \\neq x _ { j }$ . The frequency $f _ { i }$ is equal to the number of times that item $x _ { i }$ appears in $\\mathcal { S } _ { N }$ . ",
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+        "text": "66 To leverage learning techniques for item frequency estimation, a naïve way is to train a NN model   \n67 (e.g., MLP/LSTM) that learns/memorizes the mapping relationship between items and frequencies   \n68 with multiple training iterations, similar to [15, 22, 24]. However, it violates the typical setting of   \n69 stream processing where item observations are transient and are therefore handled in one pass [18].   \n70 More, the costly procedure has to be repeated from the scratch for a new data stream. Inspired by the   \n71 meta-bloom filter [23], we consider a case of one-shot learning (fitting for one-pass stream processing)   \n72 by using meta-learning [25, 26] and memory-augmented networks [27, 28]. Meta-learning employs   \n73 sampled meta-tasks to learn the ability to solve a class of domain tasks rather than memorizing patterns   \n74 for a specific task. The memory-augmented networks incorporate external memories into NN models,   \n75 significantly enhancing the potentials of NN models with more learnable parameters. Meanwhile, it   \n76 performs efficient and explicit operations (i.e., reading and storing) for external memories, allowing   \n77 NN models to process information similarly to conventional data structures.   \n78 The framework of the meta-sketch consists of 4 functional modules, Embedding $( \\mathcal { F } _ { E } )$ , Sparse   \n79 addressing $( \\mathcal { F } _ { S a } )$ , Compressed storage matrix $( M )$ , and Decoding $( \\mathcal { F } _ { d e c } )$ , as shown in Figure 1. Like   \n80 traditional sketches, the meta-sketch encodes and memorizes online stream items in one pass, and   \n81 answers queries by decoding corresponding item-frequency information from the structure.   \n82 Thus, we define 2 operations, Store and Query. Specifically, the Store operation first passes each   \n83 incoming stream item to $\\mathcal { F } _ { E }$ for the embedding representation, and then writes the embedding vector   \n84 into $M$ , according to the address derived by ${ \\mathcal { F } } _ { S a }$ . When estimating the frequency of an item, the   \n5 Query operation calculates the item’s address in $M$ via ${ \\mathcal { F } } _ { S a }$ , reads the corresponding information   \n6 vector from $M$ , and decodes the item frequency by $\\mathcal { F } _ { d e c }$ from the retrieved information vector . ",
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+        "text": "Embedding. The module $\\mathcal { F } _ { E }$ has two purposes: 1) performing representational transformation for an incoming item $e _ { i }$ and mapping it into a dense embedding vector $z _ { i }$ that holds implicit features about item-frequency distributions and serves as the basis for identifying stream items; 2) decoupling the embedding vector $z _ { i }$ to obtain a refined vector $r _ { i }$ , which is used to derive the address for reading/writing on the compressed storage matrix $M$ . ",
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+        "text": "Accordingly, $\\mathcal { F } _ { E }$ consists of the embedding network $g _ { e m b }$ and the address network $g _ { a d d }$ . We assume that an item $e _ { i } \\in S _ { N }$ is numerically encoded for the unique identification, following the conventions of stream processing [18, 19]. Thus, we have $z _ { i } , r _ { i } \\gets \\mathcal { F } _ { E } ( e _ { i } )$ , where $z _ { i } \\gets g _ { e m b } ( e _ { i } )$ and $r _ { i } \\gets$ $g _ { a d d } ( z _ { i } )$ . Here, $z _ { i } \\in \\mathbb { R } ^ { l _ { z } }$ is an embedding vector of length $l _ { z }$ , and $r _ { i } \\in \\mathbb { R } ^ { l _ { r } }$ is a refined vector of length $l _ { r }$ . The vector $z _ { i }$ serves multiple intents: 1) it makes a basis for deriving the address of an item in $\\mathcal { F } _ { S a } ; 2 )$ it serves as the compressed vector of an item written into $M ; 3 )$ ) it works as a partial input of $\\mathcal { F } _ { d e c }$ for decoding the item frequency; 4) it also plays the role of perceiving/compressing patterns of a specific frequency distribution, as discussed in Section 5. In addition, to enhance the addressing functionality and eliminate other interference factors, we decouple $z _ { i }$ to generate a refined vector $r _ { i }$ instead of using $z _ { i }$ directly for the addressing. ",
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+        "text": "103 Sparse addressing. The module ${ \\mathcal { F } } _ { S a }$ aims to derive the address $a _ { i }$ for storing the embedding vector   \n104 $z _ { i }$ into the storage matrix: $a _ { i } \\gets \\mathcal { F } _ { S a } ( r _ { i } )$ . In terms of functionality, ${ \\mathcal { F } } _ { S a }$ is analogous to the hash   \n105 functions of traditional sketches, except that ${ \\mathcal { F } } _ { S a }$ is parameterized and differentiable. Specifically,   \n106 the addressing of the meta-sketch is done via a 3D addressing matrix $A$ of parameters to be learned   \n107 and a sparse SoftMax function: $a _ { i }  S p a r s e M a x ( r _ { i } ^ { T } A )$ , where $A \\in \\mathbb { R } ^ { d _ { 1 } ^ { \\bullet } \\times l _ { r } \\times d _ { 2 } }$ . Then, the batch   \n108 matrix multiplication of $A$ and the transpose of $r _ { i }$ results in the addressing vector $a _ { i } \\in \\mathbb { R } ^ { d _ { 1 } \\times 1 \\times d _ { 2 } }$ .   \n109 The setting of $d _ { 1 }$ and $d _ { 2 }$ determines the size of address space for storing the embedding vectors.   \n110 Typical addressing methods [23, 28] use a 2D matrix $( l _ { r } \\times d _ { 2 } )$ for recording the mapping of an   \n111 embedding vector to a slot ( $d _ { 2 }$ is the number of slots). In contrast, we add one more dimension $d _ { 1 }$   \n112 to simulate the multi-hash setting of traditional sketches, in view of that a 2D addressing matrix   \n113 can reach a differentiable simulation of a hash function [23, 24]. Matrix $A$ simulates multiple hash   \n114 functions, yielding robust frequency decoding and the rationality of the learning optimization. Note   \n115 that each 2D slice $A ^ { * }$ of $A$ is stacked from $d _ { 2 }$ -unit vectors $b _ { i } \\in \\mathbb { R } ^ { l _ { r } }$ by normalizing the parameters   \n116 of $A$ at each gradient update of the training process. Normalized $A$ can avoid overflowing when   \n117 compressing its size by reducing data precisions and enhance the interpretability (see Section 5).   \n118 In addition, we utilize sparse SoftMax [29, 30] instead of SoftMax to normalize the address $a _ { i }$   \n119 It brings the following benefits by constraining some bits of $a _ { i }$ to zero, which 1) promotes quick   \n120 derivation during the back-propagation; 2) reduces the overhead of storage matrix accessing by   \n121 skipping the slots of $M$ corresponding to the $\\mathbf { \\vec { \\Delta } } ^ { 6 } 0 ^ { 9 }$ bits of $a _ { i }$ ; 3) leads to de-noising with the vector   \n122 compression.   \n123 Compressed storage matrix. We use a matrix $M \\in \\mathbb { R } ^ { d _ { 1 } \\times l _ { z } \\times d _ { 2 } } \\ .$ 1 to store an embedding vector   \n124 $z _ { i } \\in \\bar { \\mathbb { R } } ^ { l _ { z } }$ in accordance to its address $a _ { i } \\in \\mathbb { R } ^ { d _ { 1 } \\times 1 \\times d _ { 2 } }$ . The functionality of $M$ is similar to the 2D   \n125 array of counters in traditional sketches, yet yielding better capabilities in the storage compression.   \n126 Traditional sketches store item counts. Differently, $M$ stores embedding vectors, which have richer   \n27 information compression capabilities, due to the diversity of value changes on different bits. ",
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+        "text": "Decoding. Given a query item $x _ { i }$ , the module $\\mathcal { F } _ { d e c }$ , consisting of one NN component $g _ { d e c }$ , decodes the information corresponding to $x _ { i }$ , in order to obtain the estimated frequency $\\hat { f } _ { i }$ . The vector fed into $g _ { d e c }$ is the concatenation of vector $\\{ M \\ominus a _ { i } \\}$ , vector $z _ { i }$ , and the current number of items (i.e., $N _ { , }$ ) recorded in a counter, $\\hat { f } _ { i }  g _ { d e c } ( \\{ M \\ominus a _ { i } \\} , z _ { i } , N )$ . The operator $\\ominus$ refers to the reading operation for the storage matrix. The basic form of $\\ominus$ gives the operation as $M \\ominus a _ { i } = M a _ { i } ^ { T 2 }$ [27, 28]. For optimization, we consider two optimized forms of $\\ominus$ , inspired by the “count-min” mechanism of the CM-sketch. The first one gives the minimum value of each row in $M a _ { i } ^ { T }$ , aiming to remove the noise of other items. The second one gives the minimum value of each row in $\\begin{array} { r } { M a _ { i } ^ { T } \\circ \\frac { 1 } { z _ { i } } } \\end{array}$ 1 , a normalized ",
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+        "text": "136 form of $M a _ { i } ^ { T }$ . Here, $\\circ$ denotes the Hadamard product, and $z _ { i }$ requires broadcast operations to comply   \n137 with its requirements. So, $\\{ M \\ominus a _ { i } \\}$ refers to the concatenation of vectors generated by the basic   \n138 form and the two optimized forms. Please refer to supplement materials for more details. ",
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+        "text": "Operation Store is performed by feeding an incoming item $e _ { i }$ to $\\mathcal { F } _ { E }$ and ${ \\mathcal { F } } _ { S a }$ to obtain embedding vector $z _ { i }$ and address $a _ { i }$ , and then additively writing $z _ { i }$ to $M$ , weighted by $a _ { i }$ : $M \\gets M + z _ { i } a _ { i }$ . Here, other writing types [23, 26–28] can also be employed, but simple additive writing is more efficient and allows to compute gradients in parallel [23]. In addition, additive writing also allows to define an optional Delete operation for the meta-sketch (see the supplement materials). ",
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+        "type": "text",
+        "text": "Operation Query estimates the frequency of a given query item $x _ { i }$ . First, $z _ { i }$ and $a _ { i }$ are obtained, similar to that of operation Store. Then, the vectors $\\{ M \\ominus a _ { i } \\}$ are retrieved from $M$ and $N$ can be easily obtained by a small counter. Finally, $\\{ M \\ominus a _ { i } \\}$ , $z _ { i }$ and $N$ are jointly fed into $g _ { d e c }$ to get the estimated frequency ${ \\hat { f } } _ { i }$ of $x _ { i }$ as the returned value. The two operations are shown in Algorithm 1. ",
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+        "type": "text",
+        "text": "Algorithm 1: Operations ",
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+        "text": "1 Operation Store $( e _ { i }$ , M):   \n2 zi, ri ← FE (ei) ;   \n3 ai ← FSa(ri);   \n4 $M \\gets M + z _ { i } a _ { i }$ ;   \n9 5 Operation Query $( x _ { i } , M , N )$ :   \n6 $z _ { i } , r _ { i } \\gets \\mathcal { F } _ { E } ( x _ { i } )$ ;   \n7 $a _ { i } \\gets \\mathcal { F } _ { S a } ( r _ { i } )$ ;   \n8 $\\hat { f } _ { i } \\gets \\mathcal { F } _ { d e c } ( \\{ M \\ominus a _ { i } \\} , z _ { i } , N )$ ;   \n9 return ${ \\hat { f } } _ { i }$ ; ",
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+        "type": "text",
+        "text": "Algorithm 2: Training Framework ",
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+        "text": "Data: Meta-sketch with all learnable parameters $\\theta$ , Meta-task sampler $R$ ;   \n1 while $_ i$ not reach max training steps do   \n2 Sample a meta-task $t _ { i } : \\{ s _ { i } , q _ { i } \\} \\sim R$ and count $N$ ;   \n3 for e(i)j ∈ $e _ { j } ^ { ( i ) } \\in s _ { i }$ do Store $( e _ { j } ^ { ( i ) }$ , $M )$ ; end   \n4 for x(ij 7 $f _ { j } ^ { ( i ) } \\in q _ { i }$ do fˆ(i ) ← Query(x(i)j , $M , N )$ ; $\\mathcal { L } + = \\mathrm { L o s s F u n } ( f _ { j } ^ { ( i ) } , \\hat { f } _ { j } ^ { ( i ) } ) ;$ ;   \n5 Backprop through: $d \\mathcal { L } / d \\theta$ and update parameters: $\\theta \\gets \\mathrm { O p t i m i z e r } ( \\theta , d \\mathcal { L } / d \\theta )$ ;   \n6 Normalize A;   \n7 Clear $M$ ;   \n8 end ",
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+        "type": "text",
+        "text": "150 3 Meta-sketch training ",
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+        "type": "text",
+        "text": "3.1 Training Framework ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The meta-sketch employs an efficient one-shot meta-training method [31]. The training process thus contains two phases, pre-training and adaption phases. In the pre-training phase, the meta-sketch learns an initial set of module parameters, including $g _ { e m b }$ , $g _ { a d d }$ , $A$ , and $g _ { d e c }$ . The pre-training goes offline across training units, i.e., basic meta-tasks, to acquire the ability of stream frequency estimation. Then, in the adaption phase, the pre-trained meta-sketch goes fast across a set of lightweighted training units, i.e., adaptive meta-tasks, to quickly acquire the task-specific knowledge, i.e., parameters for sketching real data streams at runtime. ",
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+        "type": "text",
+        "text": "159 The training units, i.e., meta-tasks, are crucial for both phases. The training process of the meta-sketch   \n160 on a single meta-task is equivalent to simulating storing and querying an instance of data streams   \n161 while computing the estimation error to optimize the learnable parameters. Thus, a meta-task $t _ { i }$   \n162 163 consists oas an inst a store setce of data $s _ { i }$ (also eams, ) and a, where ery set is the $q _ { i }$ . The store set mber of strea $s _ { i }$ can betems in ewed. The   \n$s _ { i } : \\{ e _ { 1 } ^ { ( i ) } , . . . , e _ { N _ { i } } ^ { ( i ) } \\}$ $N _ { i }$ $s _ { i }$   \n164 $q _ { i }$ ican be represented by a set of items from the stream instance with paired frequencies in   \n165 the store set $s _ { i }$ , formally, $q _ { i } : \\{ ( x _ { 1 } ^ { ( i ) } : f _ { 1 } ^ { ( i ) } ) , . . . , ( x _ { n _ { i } } ^ { ( i ) } : f _ { n _ { i } } ^ { ( i ) } ) \\}$ , where $n _ { i }$ is the number of distinct items   \n166 in $s _ { i }$ . In this work, we define two types of meta-tasks, basic (Section 3.2) and adaptive (Section 3.3)   \n167 meta-tasks, corresponding to the pre-training and adaption phases, respectively.   \n168 The two training phases, that are based on different types of meta-tasks, follow the same training   \n169 framework, as shown in Algorithm 2, except for the sampler and initial parameters. To optimize on   \n170 reducing both absolute and relative frequency estimation errors3, we devise an adaptive hybrid loss   \n171 function [32] for the meta-sketch: $\\begin{array} { r } { \\frac { 1 } { 2 \\sigma _ { 1 } ^ { 2 } } ( f _ { i } - \\hat { f } _ { i } ) ^ { 2 } + \\frac { 1 } { 2 \\sigma _ { 2 } ^ { 2 } } | f _ { i } - \\hat { f } _ { i } | / f _ { i } + l o g \\sigma _ { 1 } \\sigma _ { 2 } } \\end{array}$ , where $\\sigma _ { 1 }$ and $\\sigma _ { 2 }$ are   \n172 learned parameters, and $f _ { i }$ and $\\hat { f } _ { i }$ are the true and estimated frequencies of item $x _ { i }$ , respectively.   \n174 In the pre-training phase, basic meta-tasks should make the meta-sketch to simulate traditional   \n175 sketches and preserve certain generality without relying too much on the patterns of specific distribu  \n176 tions (Section 5). Therefore, we generate meta-tasks based on the Zipf distribution, which is found to   \n177 be prevalent in real scenes of data streams [16–20].   \n178 A meta-task is essentially a data stream instance with item size $n$ , which can be determined by the   \n179 total number of items $N$ and the relative frequency distribution $p$ . Alternatively, we can generate   \n180 meta-tasks by presupposing different $n$ , $\\bar { f }$ and $p$ , where $\\bar { f }$ is the frequency mean, since $\\scriptstyle { \\bar { N } } = { \\bar { f } } \\times n$   \n181 Thus, basic meta-task generation is based on a sampler $R : \\{ I , L , P \\}$ , as follows.   \n182 An item pool $I$ is a subset of the item domain $\\mathbb { X }$ . The cardinality of $I$ is in relevance to the   \n183 identification capability of the meta-sketch. If the item domain is known a-priori, it can be directly   \n184 taken as the item pool. Otherwise, in applications where the item domain is only partially known or   \n185 even unknown, the item pool can be constructed by sampling from the historical records. Even in the   \n186 case that the item pool does not completely cover the item domain, the “missing” item can still be   \n187 identified, due to the homogeneity of the domain-specific embedding space, given that the number of   \n188 distinct items does not meet the item pool capacity $| I |$ . ",
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+        "type": "text",
+        "text": "A frequency mean range $L$ is the range for the frequency mean $\\bar { f }$ . One can get the value of $\\bar { f }$ by statistics of each sampled stream instance and extract the minimum and maximum $\\bar { f } \\mathbf { s }$ to build $L$ . ",
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+        "text": "91 A distribution pool $P$ consists of many instances generated according to different parameters of   \n92 relative frequency distributions. In this paper, we consider a family of Zipf distributions [33] with   \n93 varied parameter $\\alpha$ , as the base for constructing $P$ . $\\alpha$ can be selected from a wide range to have a   \n94 good coverage of different distributions.   \n195 Notice that the meta-tasks are for the meta-sketch to learn the sketching ability, instead of spoon  \n196 feeding the meta-sketch to mechanically memorize the parameters of $R$ . It means that the trained   \n197 meta-sketch has the generalization ability to handle the case not covered in $R$ (see Section 4.2).   \n198 The generation of a meta-task $t _ { i }$ can be done based on sampler $R$ , as follows. We first randomly   \n199 sample a subset of $n _ { i }$ items from $I$ , and a frequency mean ${ \\hat { f } } _ { i } \\in L$ . Then, we sample a distribution   \n200 instance $p _ { i } ~ \\in ~ P$ and make the $n _ { i }$ items’ frequencies conform to $p _ { i }$ and ${ \\bar { f } } _ { i }$ . For example, the   \n201 frequencies of $n _ { i }$ items can be set as $n _ { i } \\times \\bar { f } _ { i } \\times \\bar { p } _ { i }$ , where $p _ { i } \\sim Z i p f ( \\alpha )$ is a random variable. The   \n202 above steps are repeated until the store set $s _ { i }$ and query set $q _ { i }$ are built. ",
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+        "type": "text",
+        "text": "203 3.3 Adaptive Meta-task Generation ",
+        "text_level": 1,
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+        "text": "While processing real data streams, we can get the item set $I _ { r }$ and its distribution $p _ { r }$ by online sampling. $I _ { r }$ and $p _ { r }$ are then used for generating the set of adaptive meta-tasks. For each adaptive meta-task, an item subset is sampled from $I _ { r }$ , and the relative frequency corresponding to each item is sampled from $p _ { r }$ . The process is similar to the generation of basic meta-tasks. The only difference from basic meta-task generation is that, there is no distribution pool anymore, because the real data stream is unique. Also, we intentionally randomize the correspondence between an item and its real relative frequency on the original data records. It is equivalent to constructing meta-tasks where the item frequencies dynamically change. For example, the frequency of an item may first increase, then suddenly drop [21]. With adaptive meta-tasks, the meta-sketch learns to quickly adapt to the distribution $p _ { r }$ , while being flexible against the item frequency change. The detailed algorithms of generating basic/adaptive meta-tasks are shown in supplement materials. ",
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+        "type": "text",
+        "text": "4 Experiments ",
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+        "text": "4.1 Basic Setup",
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+        "text": "Dataset. We use two real datasets. Word-query is a streaming record of search queries, where each query contains multiple words (e.g., “News today”) [15]. IP-trace consists of IP packets, where each packet is identified by a unique source/destination address pair (e.g., 192.168.1.1/12.13.41.4) [21]. We assume that query phrases and IP addresses are numerically encoded, similar to [15]. ",
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+        "img_path": "images/c9cf5fd95ce3068e592388c3bd8052c96c85108167ae82aaf993e6d4f2c833ae.jpg",
+        "table_caption": [
+            "Table 1: Results of Basic Meta-sketch $( T _ { r } )$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"5\">Word-query</td><td colspan=\"4\">IP-trace</td></tr><tr><td>Method</td><td>Metrics</td><td>n=5K, B=9KB</td><td>n=10K, B=11KB</td><td>n=20K, B=13KB</td><td>n=40K, B=15KB</td><td>n=5K, B=9KB</td><td>n=10K, B=11KB</td><td>n=20K, B=13KB</td><td>n=40K, B=15KB</td></tr><tr><td>Basic MS</td><td>ARE</td><td>12.3</td><td>14.74</td><td>10.98</td><td>13.79</td><td>3.00</td><td>1.51</td><td>2.97</td><td>1.13</td></tr><tr><td rowspan=\"2\">CS</td><td>AAE</td><td>31.54</td><td>38.54</td><td>40.63</td><td>53.67</td><td>5.57</td><td>5.01</td><td>6.94</td><td>5.56</td></tr><tr><td>ARE</td><td>32.94</td><td>57.97</td><td>98.01</td><td>162.43</td><td>6.08</td><td>9.94</td><td>15.57</td><td>24.49</td></tr><tr><td rowspan=\"2\">CMS</td><td>AAE</td><td>57.54</td><td>101.44</td><td>172.44</td><td>282.59</td><td>10.42</td><td>16.82</td><td>26.46</td><td>41.91</td></tr><tr><td>ARE</td><td>21.34</td><td>48.33</td><td>111.82</td><td>239.11</td><td>8.12</td><td>16.07</td><td>32.77</td><td>65.19</td></tr><tr><td></td><td>AAE</td><td>38.04</td><td>84.62</td><td>195.61</td><td>416.01</td><td>13.67</td><td>27.39</td><td>55.29</td><td>110.65</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/22c1d7205425d3cf39b2b5c2c0f022e3602d8aedfb63f727ae6a47f6dabf9561.jpg",
+        "table_caption": [
+            "Table 2: Results of Basic Meta-sketch $( T _ { s } )$ ) "
+        ],
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+        "table_body": "<table><tr><td>Method</td><td>Metrics</td><td colspan=\"3\">n=5K,B=9KB</td><td colspan=\"3\">n=10K,B=11KB</td><td colspan=\"3\">n=20K,B=13KB</td><td colspan=\"3\">n=40K,B=15KB</td></tr><tr><td></td><td></td><td>0.5</td><td>1.1</td><td>1.5</td><td>0.5</td><td>1.1</td><td>1.5</td><td>0.5</td><td>1.1</td><td>1.5</td><td>0.5</td><td>1.1</td><td>1.5</td></tr><tr><td>Basic MS</td><td>ARE</td><td>0.43</td><td>1.05</td><td>2.63</td><td>0.73</td><td>3.25</td><td>3.14</td><td>0.47</td><td>1.67</td><td>1.35</td><td>0.43</td><td>2.58</td><td>9.65</td></tr><tr><td>(Word-query)</td><td>AAE</td><td>24</td><td>17.72</td><td>8.93</td><td>3124</td><td>27.02</td><td>9.41</td><td>27.29</td><td>22.19</td><td>三</td><td>25.04</td><td>26.95</td><td>19.87</td></tr><tr><td>Basic MS</td><td>ARE</td><td>0.59</td><td>2.27</td><td>9.38</td><td>0.73</td><td>0.86</td><td>1.02</td><td>0.72</td><td>1.73</td><td>7.52</td><td>0.73</td><td>0.79</td><td>233</td></tr><tr><td>(IP-trace)</td><td>AAE</td><td>26.45</td><td>21.49</td><td>14.73</td><td>38.33</td><td>19.32</td><td></td><td>35.48</td><td>22.28</td><td>15.74</td><td>39.57</td><td>21.75</td><td>14.06</td></tr><tr><td rowspan=\"3\">CS</td><td>ARE</td><td>1.98</td><td>6.72</td><td>10.99</td><td>2.7</td><td>12.12</td><td>16.9</td><td>3.73</td><td>20.8</td><td>27.46</td><td>5.17</td><td>37.96</td><td>43.76</td></tr><tr><td>AAE</td><td>74.96</td><td>47.98</td><td>15.89</td><td>102.05</td><td>75.83</td><td>23.8</td><td>140.65</td><td>118.29</td><td>38.7</td><td>194.32</td><td>198.4</td><td>59.96</td></tr><tr><td>ARE</td><td>4.96</td><td>7.52</td><td>5.47</td><td>9.27</td><td>15.85</td><td>9.44</td><td>17.29</td><td>32.7</td><td>16.38</td><td>32.24</td><td>66.35</td><td>27.89</td></tr><tr><td>CMS</td><td>AAE</td><td>187.52</td><td>53.81</td><td>8.17</td><td>350.08</td><td>99.82</td><td>13.58</td><td>651.63</td><td>185.54</td><td>22.88</td><td>1213.38</td><td>347.32</td><td>38.18</td></tr></table>",
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+        "text": "Baseline. We hereby evaluate the basic and advanced meta-sketches. From now on, we use MS to represent the term meta-sketch for brevity. We compare basic MS (after the pre-training phase) with CM-sketch (CMS) and C-sketch (CS). We compare the advanced MS (after the adaptation phase) with learned augmented sketch (LS) and cold filter (CF), which are two variants of CM/C sketches with auxiliary structures. According to the default setting [10, 11], the number of hash functions for all sketches is 3. We adopt two commonly accepted metrics for evaluating the accuracies of stream frequency estimation, AAE and $\\mathrm { A R E } ^ { 3 }$ . ",
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+        "text": "Parameters. We implement $g _ { e m b }$ or $g _ { a d d }$ in MLP with 2-layers of sizes 128 and 48, followed by batch normalization, and $g _ { d e c }$ in an MLP with 3-layers of 256 with residual connections. We use the relu function for layer connections. The space budget $B$ is spent on storing $M$ , the same as the setting in neural data structures [23]. Other modules, like hashing libraries, are commonly accepted as reusable and amortizable resources for multi-deployment of sketches [21, 23]. Note that due to space limitations, the details and methods of parameter settings of $M ( A )$ , the ablation experiments and some parameter discussions are shown in the supporting material. ",
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+        "type": "text",
+        "text": "4.2 Basic Meta-sketch ",
+        "text_level": 1,
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+        "text": "Settings. For each dataset, we train the basic MSs under 4 item pools with $\\{ 5 K , 1 0 K , 2 0 K , 4 0 K \\}$ different items, respectively. The meta-task sampler are with Zipf distributions. We build the distribution pools set with $\\alpha \\in [ 0 . 8 , 1 . 3 ]$ and set frequency mean range $L = [ 5 0 , 5 0 0 ]$ . For basic meta-sketch training, the default maximum number of training steps $\\phi$ is 5 million, the learning rate is 0.0001, and the Adam optimizer is used. For evaluation, we consider two types of tasks, $T _ { r }$ and $T _ { s }$ . $T _ { r }$ are directly obtained by random sampling on two real data streams with different values of $n$ i.e., the number of distinct items. Note that the frequency distributions of $T _ { r }$ are not necessarily obey Zipf distributions. $T _ { s }$ are the synthetic tasks, where the item frequency follows the Zipf distribution with $\\alpha \\in \\{ 0 . 5 , 1 . 1 , 1 . 5 \\}$ . To evaluate the generability and stability of basic MS, both $T _ { s } ( 0 . 5 )$ and $T _ { s } ( 1 . 5 )$ ’s distributions are not covered by the distribution pool of the meta-task samplers. ",
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+        "type": "text",
+        "text": "Performance. Table 1 shows the performance of all competitors based on real dataset $T _ { r }$ . It shows that the basic MS outperforms traditional basic sketches, i.e., CMS and CS, on all testing cases. For example,the results on IP-trace show that, when $\\scriptstyle n = 4 0 \\mathrm { K }$ and $B { = } 1 5 \\mathrm { K B }$ , the ARE of basic MS is 1.13, while AREs of CMS and CS are 65.19 and 24.49, respectively. The advantage of meta-sketch is significant when testing on $T _ { s }$ with different $\\alpha \\mathbf { s }$ , as shown in Table 2. Note that we use random choices to simulate the ideal hash functions for traditional sketches like [15], so that CS and CMS have the same result on test tasks with the same $\\alpha$ in both datasets. ",
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+        "text": "253 We show the trend of ARE w.r.t. the space budget, in Figure 2 ( $T _ { r }$ , $n { = } 5 \\mathrm { K }$ , Word-query). Compared to   \n254 the dramatic performance degrading of traditional sketches, basic MS holds stable performance. We   \n255 show that the trend of ARE w.r.t. the number of distinct items in Figure 3 $T _ { r }$ , $B { = } 9 { \\mathrm { K B } }$ , Word-query).   \n256 Compared to traditional sketches, the ARE of basic MS increases sub-linearly w.r.t. the value of $n$   \n257 Note that AAE has similar results for the above experiments, see the supplement materials.   \n258 Generalization. We test the generality of basic MS to new items that are not in the item pool of   \n259 the meta-task sampler in Figure 4(a). We make the experiments $_ { \\it { n } = 5 \\mathrm { { K } } }$ , $B { = } 9 \\mathrm { K B }$ , Word-query)   \n260 by replacing some items in $T _ { r }$ with new items, and vary the fraction of new items to observe   \n261 the trend of the performance. It shows that the ARE/AAE moderately increases w.r.t. the ratio   \n262 of new items. The performance is acceptable considering the fact that the item domain is often   \n263 stable in practical applications. We then test the generality of meta-sketches to varied frequency   \n264 means that are not in range $L$ of the meta-task sampler, as shown in Figure 4(b). The experiment   \n265 $_ { \\mathrm { { \\it ~ \\Omega } } } \\mathrm { { \\it { n } = } } 5 \\mathrm { { K } }$ , $B { = } 9 \\mathrm { K B }$ , Word-query) is done by sampling a series of $T _ { s }$ tasks with frequency means in   \n266 $\\{ 5 0 0 , 5 K , 5 0 K , 5 0 0 K , 5 0 0 0 K \\}$ . It shows that as the mean of the true frequencies increases, the   \n267 estimated frequencies of the meta-sketch increase linearly, so that the ARE keeps stable. ",
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+        "image_caption": [
+            "Figure 2: ARE w.r.t. $B$ "
+        ],
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+        "image_caption": [
+            "Figure 3: ARE w.r.t. n "
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+        "image_caption": [
+            "Figure 4: Generality of Meta-sketch "
+        ],
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+    {
+        "type": "table",
+        "img_path": "images/76fc34e16850986ff5afa3cd47b5a92db3f23d0d4f6b821e3be0a3f0926cefb3.jpg",
+        "table_caption": [
+            "Table 3: Results of Advanced Meta-sketch "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Metrics</td><td colspan=\"4\">Word-query</td><td colspan=\"4\">IP-trace</td></tr><tr><td></td><td></td><td>n=5K B=9KB</td><td>n=10K, B=11KB</td><td>n=20K B=13KB</td><td>n=40K B=15KB</td><td>n=5K B=9KB</td><td>n=10K B=11KB</td><td>n=20K B=13KB</td><td>n=40K B=15KB</td></tr><tr><td>Advanced</td><td>ARE</td><td>3.05</td><td>2.83</td><td>4.06</td><td>5.20</td><td>0.87</td><td>0.89</td><td>1.38</td><td>2.29</td></tr><tr><td rowspan=\"2\">MS</td><td>AAE</td><td>21.42</td><td>26.11</td><td>35.00</td><td>43.81</td><td>3.77</td><td>4.46</td><td>5.13</td><td>6.55</td></tr><tr><td>ARE</td><td>3.58</td><td>14.53</td><td>141.70</td><td>1127.11</td><td>0.85</td><td>2.74</td><td>4.20</td><td>16.71</td></tr><tr><td rowspan=\"2\">CF90</td><td>AAE</td><td>21.13</td><td>59.18</td><td>381.63</td><td>2217.28</td><td>1.32</td><td>3.01</td><td>7.71</td><td>31.20</td></tr><tr><td>ARE</td><td>7.95</td><td>29.02</td><td>139.87</td><td>541.37</td><td>1.51</td><td>3.10</td><td>8.95</td><td>46.79</td></tr><tr><td rowspan=\"2\">CF70 CF40</td><td>AAE</td><td>29.02</td><td>76.58</td><td>295.63</td><td>970.94</td><td>2.57</td><td>5.51</td><td>16.83</td><td>82.84</td></tr><tr><td>ARE</td><td>91.16</td><td>138.64</td><td>244.24</td><td>407.83</td><td>12.62</td><td>33.50</td><td>103.76</td><td>155.61</td></tr><tr><td rowspan=\"2\"></td><td>AAE</td><td>174.86</td><td>252.22</td><td>421.85</td><td>693.47</td><td>24.16</td><td>60.79</td><td>175.14</td><td>279.72</td></tr><tr><td>ARE</td><td>20.52</td><td>48.69</td><td>111.85</td><td>266.50</td><td>8.34</td><td>17.09</td><td>35.22</td><td>77.79</td></tr><tr><td rowspan=\"2\">LCMS(1%) LCS(1%)</td><td>AAE</td><td>37.80</td><td>81.93</td><td>194.15</td><td>451.28</td><td>13.72</td><td>28.39</td><td>59.10</td><td>129.86</td></tr><tr><td>ARE</td><td>25.53</td><td>40.84</td><td>67.21</td><td>104.54</td><td>5.20</td><td>7.80</td><td>11.33</td><td>17.12</td></tr><tr><td rowspan=\"2\"></td><td>AAE</td><td>44.53</td><td>78.17</td><td>122.57</td><td>180.56</td><td>8.78</td><td>13.10</td><td>18.97</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>28.38</td></tr></table>",
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+        "type": "text",
+        "text": "4.3 Advanced Meta-sketch ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Settings. The generation of adaptive meta-tasks is similar to that of basic meta-tasks (Section 3.2), except that each item pool reads real frequency distributions for the adaption as described in Section 3.3. In the adaption phase, the maximum number of training steps is $0 . 0 0 2 * \\phi$ . ",
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+    {
+        "type": "text",
+        "text": "Performance. Table 3 compares the performance of advanced MS with traditional sketches and their variants, LS and CF, on real dataset $T _ { r }$ . We implement two LSs according to [15], learned CM-sketch (LCMS) and learned C-sketch (LCS), following the default setting that (top $1 \\%$ ) high-frequency items are separately stored. For CF, we follow the parameter setting in [14], and use CF40, CF70, and CF90 for setting the filter percentages to $40 \\%$ , $70 \\%$ , and $90 \\%$ of the total size, respectively. It shows that the advanced MS achieves a better performance than LSs and CFs. Also, AAE/ARE of advanced MS increases more moderately w.r.t. the number of distinct items $n$ , compared to its competitors. ",
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+        "type": "text",
+        "text": "Furthermore, we compare the performance of the advanced MS and the LS under dynamic streaming scenarios, as shown in Figure 5. We select a set of $T _ { r }$ $\\scriptstyle ( n = 5 \\mathbf { K } , B = 9 \\mathbf { K } \\mathbf { B }$ ,Word-query), and gradually shuffle the correspondence between items and frequencies. Here, the shuffle ratio is increased from 0 to $100 \\%$ . It shows that the average ARE of advanced MS only slightly fluctuates between 3.26 and 4.0, and the average AAE is in the range of 21.28 and 21.68. In contrast, AAE of LCS or LCMS starts above 37, and increase significantly w.r.t. the increase of the shuffle ratio. Actually, the classifier of LS tends to incur more errors due to the gradual shift of high- and low-frequency items, resulting in an increased number of hash collisions, thus deteriorating the estimation accuracy. ",
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+    {
+        "type": "text",
+        "text": "87 5 Analysis ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The meta-sketch is trained based on meta-tasks, consisting of various stream distributions. We expected that the meta-sketch can learn the ability to sketch item frequencies. Somehow, it is unavoidable that the meta-sketch’s ability is limited by patterns of given meta-tasks. Thus, setting up the two training phases benefits the balance of the trade-offs. In the pre-training phase, we select the most representative Zipf distribution to form basic meta-tasks, making the basic meta-sketch adaptable to a wide range of data streams. In the adaptation phase, we sample adaptive meta-tasks from raw data streams to make the advanced meta-sketch more specialized. Next, we analyze the ",
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+        "image_caption": [
+            "Figure 5: Learned sketch vs. Meta-sketch ",
+            "Figure 6: $| r |$ and $| z |$ w.r.t. Sparsity of $a$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "image_caption": [
+            "Figure 7: Three addressing matrices "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "img_path": "images/043316aa344876317461ed7124d4cba5dcd0936958f3ed12872e46f4239441de.jpg",
+        "image_caption": [
+            "Figure 8: The sparsity of embedding vectors "
+        ],
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+        "type": "text",
+        "text": "295 working mechanism of the three modules of the meta-sketch as well as their roles in acquiring the   \n296 two abilities. ",
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+        "type": "text",
+        "text": "Sparse Addressing Module. We take a 2D slice $A ^ { * }$ (size is $l _ { r } \\times d _ { 2 } ,$ ) of the $A$ matrix to analyze the process of a refined vector $r$ getting addressing $a$ through this module. First, we have $a \\gets$ $\\bar { S p a r s e M a x } ( r ^ { T } A ^ { * } ) \\Rightarrow a  \\bar { S p a r s e M a x } ( \\langle r \\cdot b _ { 1 } , \\bar { r } \\cdot b _ { 2 } , . . . , \\bar { r } \\cdot b _ { d _ { 2 } } \\rangle )$ . Since $b _ { i }$ are unit vectors, we can get $a \\gets S p a r s e M a x ( | r | c )$ , $c { = } \\langle c o s \\theta _ { 1 } , c o s \\theta _ { 2 } { , } . . . , c o s \\theta _ { d _ { 2 } } \\rangle$ , where $\\theta _ { i }$ is the angle between $r$ and $b _ { i }$ . We continue to transform the form to get addressing $\\begin{array} { r } { a \\gets S p a r s e g e n ( c ; u ; \\frac { | r | - 1 } { | r | } ) } \\end{array}$ ; |r|−1|r| ) [30], where u is a component-wise transformation function applied on $c$ . in this paper, we set $u ( c ) { = } c$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Based on the principle of Sparsegen [30], $| r |$ mainly affects the sparsity (i.e., the proportion of non-zero bits in the vector) of $a$ during training process, while $c$ determines the positions and values of non-sparse bits. The Figure 6 shows a strong correlation between the average $| r |$ and the sparsity of $a$ during training from scratch ( $_ { \\mathrm { \\Delta } n = 5 \\mathrm { K } }$ , $B { = } 9 \\mathrm { K B }$ , Word-query, Basic MS). Since the embedding vector $z$ does not directly participate in the addressing process, the average $| z |$ remains stable. Further, we observe that the sparsity of $a$ will eventually converge to around 1, which means that each item is generally stored in a slot corresponding to the refined vector $r$ and the unit vector in $A ^ { * }$ with the maximum cosine similarity. ",
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+    {
+        "type": "text",
+        "text": "Therefore, the role of $A ^ { * }$ is to map refined vectors to the addressing vectors. The $d _ { 2 }$ unit vectors in $A ^ { * }$ are the reference standard for mapping, which is equivalent to the mutually exclusive $d _ { 2 }$ -divisions of the refined vector space. Follow this point, we construct two matrices $K ^ { * }$ and $R ^ { * }$ of the same size as $A ^ { * }$ . Among them, the $d _ { 2 }$ unit vectors in $K ^ { * }$ come from the cluster centers of the sampled refined vectors. To achieve mutually exclusive division, we perform Kmeans clustering with $K = d _ { 2 }$ and Cosine similarity criterion. Then, we normalize the resulting $d _ { 2 }$ cluster centers and stack them as $K ^ { * }$ In contrast, the unit vectors in $R ^ { * }$ are entirely randomly generated. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 7 (a) shows the results of replacing $A ^ { * }$ on the trained meta-sketch with $K ^ { * }$ and $R ^ { * }$ . The meta-sketch with $R ^ { * }$ shows the worst performance, but the performance of the meta-sketch with $K ^ { * }$ is close to the original $A ^ { * }$ . Furthermore, We count the number of items mapped in every slot of $A ^ { * }$ , $K ^ { * }$ , $R ^ { * }$ and show their standard deviation in Figure 7 (b). The standard deviation of $R ^ { * }$ is much higher than $A ^ { * }$ and $K ^ { * }$ , and a better meta-sketch tends to store items more evenly in each slot. Thus, The addressing module simulates the traditional sketch mechanism. Its principal function is to store the embedding vectors of items as evenly as possible in multiple memory slots, and an item is written to only one slot. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Embedding Module. The major source of conflicts in the meta-sketch is the stacking of different embedding vectors in a single slot. Thus, the sparsity of the embedding vector becomes an important indicator to determine the degree of conflicts. Figure 8 shows the relation between the sparsity of embedding vectors and the stream distributions $_ { \\mathrm { { \\it ~ \\Omega } } } \\mathrm { { n = 5 K } }$ , $B { = } 9 \\mathrm { K B }$ , Word-query, advanced MS). We select the meta-tasks under Zipf, Triangular, and Uniform distributions with different skewness levels (the definition of skewness and corresponding distribution parameters are shown in the supplement materials). The results show that the sparsity of the embedding vector is positively proportional to ",
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+    {
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+        "img_path": "images/5620e00a605ae8e5f51d00f04f6f4a362ff0bcfb931112fd394003077a6bbd5a.jpg",
+        "image_caption": [
+            "Figure 9: Generality w.r.t. Decoding module "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/a9e757a406ba1125a8bff13effa28389881a2171b22e5518d37206404778e386.jpg",
+        "image_caption": [
+            "Figure 10: Unstable case vs. Stable case "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "333 the skewness of a distribution. Therefore, we speculate that the meta-sketch memorizes the pattern   \n334 information of the distribution being adapted by self-tuning the sparsity of embedding vectors. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Decoding Module. The decoding module, as the deepest NNs in the meta-sketch, integrates various information to predict the item frequency and achieves generalization ability. To verify this, we adapt the advanced MS $_ { \\it { n } = 5 \\mathrm { { K } } }$ , $B { = } 9 \\mathrm { K B }$ , Word-query) to a special adaptive meta-task. The meta-task was sampled from the real data stream but with a fixed item size (5000) and frequency mean (250). Meanwhile, we do not change the correspondence between items and frequencies. Such meta-task forces the meta-sketch to pay more attention to the fixed patterns and thus limit its generalization. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Thus, we train the advanced MS with (or without) freezing the decoding module parameters based on the above meta-task. Figure 9 (a) shows the performance changes of the three models (advanced MS as baseline) on the evaluation tasks $( T _ { r } )$ of different item sizes. Without the frozen decoding module, the meta-sketch loses generalization ability at extended item sizes other than 5000. On the contrary, the meta-sketch with the frozen decoding module still retains the generalization ability and further utilizes the data stream pattern compared to the advanced MS, achieving the best performance. Similarly, as shown in Figure 9 (b), the meta-sketch without the frozen decoding module also loses a certain generalization ability in terms of frequency mean. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "349 Actually, the above meta-task (termed as the stable case) can be viewed as a special case of an ordinary   \n350 adaptive meta-task (termed as the unstable case). As a matter of fact, augmented sketches utilize   \n351 frequency patterns similar to the stable case. For example, the learned augmented sketch memorizes   \n352 (relatively) stable correspondence between items and frequencies, for filtering high-frequency items.   \n353 To understand the meta-sketch’s self-optimizing mechanism from the unstable case to the stable case,   \n354 we analyze the storage of high/low-frequency items between multiple slots and a single slot in the   \n355 memory. In Figure 10 (a), we show density heat-maps of low-frequency (below the top $20 \\%$ high   \n356 frequencies) items, stored by meta-sketches of stable and unstable cases on a 2D slice $d _ { 1 } = 2 \\AA$ ) of the   \n357 storage matrix $M$ , where the $\\mathbf { X }$ -axis is the index of slots. The two heat-maps show that the meta-sketch   \n358 under the stable case can store the low-frequency items concentratedly in some slots to avoid the   \n359 conflicts with high-frequency items. Interestingly, the meta-sketch does not intentionally do this like   \n360 augmented sketches. Instead, it is achieved by self-optimization during the training. Furthermore,   \n361 Figure 10 (b) shows the relation between the sparsity of the embedding vector of items stored in a   \n362 single slot and the frequency order, where the $x$ -axis represents the frequencies in the ascending order.   \n363 We speculate that the meta-sketch autonomously adjusts the sparsity of the embedding vector within   \n364 a single slot in the stable case, so that the high/low-frequency items are automatically separated. ",
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+    {
+        "type": "text",
+        "text": "365 6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            150,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a neural data structure, called the meta-sketch, for estimating item frequencies in data streams. Unlike traditional sketches, the meta-sketch utilizes meta-learning and memory-augmented neural networks. The meta-sketch is pre-trained with Zipf distributions and can be fast adapted to specific runtime streams. We study a series of techniques for constructing the meta-sketch. We also devise the generation of basic and adaptive meta-tasks corresponding to the pre-training and adaption phases, respectively. Extensive empirical studies on real datasets are done to evaluate our proposals. In the future, it is interesting to extend our proposal to other sketching tasks that are supported by traditional sketches. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            160,
+            854,
+            266,
+            869
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "[1] Tobias Weller. Compromised account detection based on clickstream data. In WWW, pages 819–823, 2018. ",
+        "bbox": [
+            171,
+            883,
+            821,
+            911
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+        "page_idx": 8
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+    {
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In SIGMOD, 2013.   \n[8] Amit Goyal, Hal Daumé III, and Graham Cormode. Sketch algorithms for estimating point queries in NLP. In EMNLP-CoNLL 2012, July 12-14, 2012, Jeju Island, Korea, pages 1093– 1103. ACL, 2012.   \n[9] Fabon Dzogang, Thomas Lansdall-Welfare, Saatviga Sudhahar, and Nello Cristianini. Scalable preference learning from data streams. In WWW 2015, Florence, Italy, May 18-22, 2015 - Companion Volume, pages 885–890. ACM, 2015.   \n[10] Graham Cormode and S. Muthukrishnan. An improved data stream summary: the count-min sketch and its applications. J. Algorithms, 55(1):58–75, 2005.   \n[11] Moses Charikar, Kevin C. Chen, and Martin Farach-Colton. Finding frequent items in data streams. In ICALP, pages 693–703, 2002.   \n[12] Cristian Estan and George Varghese. New directions in traffic measurement and accounting. In SIGCOMM, pages 323–336, 2002.   \n[13] Pratanu Roy, Arijit Khan, and Gustavo Alonso. Augmented sketch: Faster and more accurate stream processing. In SIGMOD, pages 1449–1463, 2016.   \n[14] Yang Zhou, Tong Yang, Jie Jiang, Bin Cui, Minlan Yu, Xiaoming Li, and Steve Uhlig. Cold filter: A meta-framework for faster and more accurate stream processing. In SIGMOD, pages 741–756, 2018.   \n[15] Chen-Yu Hsu, Piotr Indyk, Dina Katabi, and Ali Vakilian. Learning-based frequency estimation algorithms. In ICLR, 2019.   \n[16] Taiwo Kolajo, Olawande J. Daramola, and Ayodele Ariyo Adebiyi. Big data stream analysis: a systematic literature review. J. Big Data, 6:47, 2019.   \n[17] Xue-Qiang Zeng and Guo-Zheng Li. Incremental partial least squares analysis of big streaming data. Pattern Recognition, 47(11):3726–3735, 2014.   \n[18] Brian Babcock, Shivnath Babu, Mayur Datar, Rajeev Motwani, and Jennifer Widom. Models and issues in data stream systems. In PODS, pages 1–16, 2002.   \n[19] Graham Cormode, Minos N. Garofalakis, Peter J. Haas, and Chris Jermaine. Synopses for massive data: Samples, histograms, wavelets, sketches. Found. Trends Databases, 4(1-3):1–294, 2012.   \n[20] M. S. B. PhridviRaja and C. V. GuruRao. Data mining : past present and future - a typical survey on data streams. CoRR, abs/1605.01429, 2016.   \n[21] Lu Tang, Qun Huang, and Patrick P. C. Lee. Mv-sketch: A fast and compact invertible sketch for heavy flow detection in network data streams. In INFOCOM, pages 2026–2034. IEEE, 2019.   \n[22] Tim Kraska, Alex Beutel, Ed H Chi, Jeffrey Dean, and Neoklis Polyzotis. The case for learned index structures. In SIGMOD, pages 489–504, 2018.   \n[23] Jack Rae, Sergey Bartunov, and Timothy Lillicrap. Meta-learning neural bloom filters. In ICML, pages 5271–5280. PMLR, 2019.   \n[24] Michael Mitzenmacher. A model for learned bloom filters and related structures. arXiv preprint arXiv:1802.00884, 2018.   \n[25] Timothy M. Hospedales, Antreas Antoniou, Paul Micaelli, and Amos J. Storkey. Meta-learning in neural networks: A survey. CoRR, abs/2004.05439, 2020.   \n[26] Adam Santoro, Sergey Bartunov, Matthew Botvinick, Daan Wierstra, and Timothy Lillicrap. Meta-learning with memory-augmented neural networks. In ICML, pages 1842–1850. PMLR, 2016.   \n[27] Alex Graves, Greg Wayne, and Ivo Danihelka. Neural turing machines. arXiv preprint arXiv:1410.5401, 2014.   \n[28] Alex Graves, Greg Wayne, Malcolm Reynolds, Tim Harley, Ivo Danihelka, Agnieszka GrabskaBarwinska, Sergio Gómez Colmenarejo, Edward Grefenstette, Tiago Ramalho, John Agapiou, ´ et al. Hybrid computing using a neural network with dynamic external memory. Nature, 538(7626):471–476, 2016.   \n[29] Andre Martins and Ramon Astudillo. From softmax to sparsemax: A sparse model of attention and multi-label classification. In ICML, pages 1614–1623. PMLR, 2016.   \n[30] Anirban Laha, Saneem Ahmed Chemmengath, Priyanka Agrawal, Mitesh Khapra, Karthik Sankaranarayanan, and Harish G Ramaswamy. On controllable sparse alternatives to softmax. NIPS, 31, 2018.   \n[31] Oriol Vinyals, Charles Blundell, Tim Lillicrap, Koray Kavukcuoglu, and Daan Wierstra. Matching networks for one shot learning. In Daniel D. Lee, Masashi Sugiyama, Ulrike von Luxburg, Isabelle Guyon, and Roman Garnett, editors, NIPS, pages 3630–3638, 2016.   \n[32] Alex Kendall, Yarin Gal, and Roberto Cipolla. Multi-task learning using uncertainty to weigh losses for scene geometry and semantics. In CVPR, pages 7482–7491, 2018. ",
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+        "text": "[33] Lada A. Adamic. Zipf, power-laws, and pareto- a ranking tutorial. ",
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parse/dev/2PSrjVtj6gU/2PSrjVtj6gU.md

@@ -0,0 +1,404 @@
+# GRAPH ATTENTION MULTI-LAYER PERCEPTRON
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Recently, graph neural networks (GNNs) have achieved a stride of success in many graph-based applications. However, most GNNs suffer from a critical issue: representation learned is constructed based on a fixed $k$ -hop neighborhood and insensitive to individual needs for each node, which greatly hampers the performance of GNNs. To satisfy the unique needs of each node, we propose a new architecture – Graph Attention Multi-Layer Perceptron (GAMLP). This architecture combines multi-scale knowledge and learns to capture the underlying correlations between different scales of knowledge with two novel attention mechanisms: Recursive attention and Jumping Knowledge (JK) attention. Instead of using node feature only, the knowledge within node labels is also exploited to reinforce the performance of GAMLP. Extensive experiments on 12 real-world datasets demonstrate that GAMLP achieves state-of-the-art performance while enjoying high scalability and efficiency.
+
+# 1 INTRODUCTION
+
+Graph Neural Networks (GNNs) generalize convolutional neural networks to graph-structured data and have achieved great success in a wide range of tasks, including node classification, link prediction, and recommendation. (Kipf & Welling, 2016; Hamilton et al., 2017; Bo et al., 2020; Cui et al., 2020; Fan et al., 2019). Through stacking $K$ graph convolution layers, GNNs learn node representations by utilizing information from the $K$ -hop neighborhood and thus enhance the performance by getting more unlabeled nodes involved in the training process. In such a GNN model, the nodes within the $K$ -hop neighborhood of a specific node are called this node’s Receptive Field (RF). As the size of RF grows exponentially to the number of GNN layers, the rapidly expanding RF incurs high computation and memory costs in a single machine. Besides, even in a distributed environment, GNN has to pull a great number of neighboring node features to compute the representation of each node, leading to high communication cost (Zheng et al., 2020).
+
+Many recent advancements towards scalable GNNs are based on model simplification. For example, Simplified GCN (SGC) (Wu et al., 2019) decouples the feature propagation and the non-linear transformation process, and the former is executed during pre-processing. Unlike the sampling-based methods (Hamilton et al., 2017), which still need feature propagation during each training epoch, this time-consuming process in SGC is only executed once, and only the nodes of the training set are involved in the training process. As a result, SGC is computation and memory-efficient in a single machine and scalable in distributed settings since it does not require each machine to fetch neighboring node features during the model training process. Despite the high efficiency and scalability, SGC simply preserves a fixed RF for all the nodes by assigning them the same feature propagation depth. Such a fixed propagation mechanism in SGC disables its ability to exploit knowledge within neighborhoods of different sizes.
+
+Lines of other simplified models have been proposed to learn better node representations exploiting multi-scale knowledge. SIGN (Frasca et al., 2020) proposes to concatenate all the propagated features without information loss, while $\mathrm { S ^ { 2 } G C }$ (Zhu & Koniusz, 2021) averages all these propagated features to generate the combined feature. Although multi-scale knowledge is considered, the importance and correlations between multiple scales are ignored. Being the first attempt to explore the correlations between different scales of knowledge, GBP (Chen et al., 2020b) adopts a heuristic constant decay factor for the weighted average for propagated features at different propagation steps. Motivated by Personalized PageRank, the large-scale features has a higher risk of over-smoothing, and they will contribute less to the combination in GBP.
+
+![](images/ce549669ff8e14edce795001721a899c92d9b43fcffe2c7d9fc54aaf98ef9990.jpg)  
+Figure 1: (Left) Test accuracy of SGC on 20 randomly sampled nodes of Citeseer. The X-axis is the node id, and Y-axis is the propagation steps (layers). The color from white to blue represents the ratio of being predicted correctly in 50 different runs. (Right) The local graph structures for two nodes in different regions; the node in the dense region has larger RF within two iterations of propagation.
+
+Unfortunately, the coarse-grained, layer-wise combination prevents these methods from unleashing their full potential. As shown in Figure 1(a), different nodes require different propagation steps to achieve optimal predictive accuracy. Besides, assigning the same weight distribution to propagated features along with propagation depth to all the nodes may be unsuitable due to the inconsistent RF expansion speed shown in Figure 1(b). However, nodes in most existing GNNs are restricted to a fixed-hop neighborhood and insensitive to the actual demands of different nodes. This imperfection either makes that long-range dependencies cannot be fully leveraged due to limited hops/layers or loses local information by introducing many irrelevant nodes into the receptive fields for many nodes when increasing the number of propagation depth (Chen et al., 2020a; Li et al., 2018; Xu et al., 2018).
+
+The above observations motivate us to explicitly learn the importance and correlation of multiscale knowledge in a node-adaptive manner. To this end, we develop a new architecture – Graph Attention Multi-Layer Perceptron (GAMLP) – that could automatically exploit the knowledge over different neighborhoods at the granularity of nodes. GAMLP achieves this by introducing two novel attention mechanisms: Recursive attention and Jumping Knowledge (JK) attention. These two attention mechanisms can capture the complex correlations between propagated features at different propagation depths in a node-adaptive manner. Consequently, our architecture has the same benefits as the existing simplified and scalable GNN models while providing much better performance derived from its ability to utilizes a node-adaptive receptive field. Moreover, the proposed attention mechanisms can be applied to both node features and labels over neighborhoods with different sizes. By combining these two categories of information together, GAMLP could achieve the best of both worlds in terms of accuracy.
+
+Our contributions are as follows: (1) New perspective. To the best of our knowledge, we are the first to explore both node-adaptive feature and label propagation schemes for scalable GNNs. (2) Novel method. We propose GAMLP, a scalable, efficient, and deep graph model. (3) State-of-the-art performance. Experimental results demonstrate that GAMLP achieves state-of-theart performance on 12 benchmark datasets while maintains high scalability and efficiency. In particular, GAMLP outperforms the competitive baseline GraphSAINT (Zeng et al., 2020) in terms of accuracy by a margin of $0 . 4 2 \%$ , $3 . 0 2 \%$ and $0 . 4 4 \%$ on PPI, Flickr, and Reddit datasets under the inductive setting, while achieving up to $4 5 \times$ training speedups in the large ogbn-products dataset. Remarkably, under the transductive setting in large OGB datasets, the accuracy of GAMLP exceeds the current state-of-the-art method by $1 . 0 { \bar { 3 } } \%$ and $\mathbf { \bar { 1 . 3 2 \% } }$ on the ogbn-products and ogbn-papers100M datasets, respectively.
+
+# 2 PRELIMINARIES
+
+# 2.1 PROBLEM FORMULATION
+
+We consider an undirected graph $\mathcal { G } = ( \nu , \mathcal { E } )$ with $| \nu | = n$ nodes, $| { \mathcal { E } } | = m$ edges, and $c$ different node classes. We denote by $\mathbf { A }$ the adjacency matrix of $\mathcal { G }$ , weighted or not. Nodes can possibly have features vector of size $f$ , stacked up in an $n \times f$ matrix $\mathbf { X }$ $\mathbf { \bar { \Phi } } _ { \cdot } \mathbf { \bar { D } } = \operatorname { d i a g } \left( d _ { 1 } , d _ { 2 } , \cdot \cdot \cdot , \mathbf { \bar { \Phi } } _ { \cdot } d _ { N } \right) \in \mathbf { \bar { \Phi } } _ { \mathbb { R } ^ { n \times n } }$ denotes the degree matrix of $\mathbf { A }$ , where $\begin{array} { r } { d _ { i } = \sum _ { v _ { j } \in \mathcal { V } } \mathbf { A } _ { i j } } \end{array}$ is the degree of node $v _ { i }$ . Suppose $\mathcal { V } _ { l }$ is the labeled set, and our goal is to predict the labels for nodes in the unlabeled set $\nu _ { u }$ with the supervision of $\nu _ { l }$ .
+
+# 2.2 SCALABLE GNNS
+
+Sampling. A commonly used method to tackle the scalability issue (i.e., the recursive neighborhood expansion) in GNN is sampling. As a node-wise sampling method, GraphSAGE (Hamilton et al., 2017) randomly samples a fixed-size set of neighbors for computation in each mini-batch. VRGCN (Chen et al., 2018a) analyzes the variance reduction, and it reduces the size of samples with additional memory cost. For the layer-wise sampling, Fast-GCN (Chen et al., 2018b) samples a fixed number of nodes at each layer, and ASGCN (Huang et al., 2018) proposes the adaptive layer-wise sampling with better variance control. In the graph level, Cluster-GCN (Chiang et al., 2019) firstly clusters the nodes and then samples the nodes in the clusters, and GraphSAINT (Zeng et al., 2020) directly samples a subgraph for mini-batch training. Orthogonal to model simplification, sampling has already been widely used in many GNNs and GNN systems (Zheng et al., 2020; Zhu et al., 2019; Fey & Lenssen, 2019). However, these sampling-based GNNs are imperfect because they still face high communication costs, and the sampling quality highly influences the model performance.
+
+Graph-wise Propagation. Recently studies have observed that non-linear feature transformation contributes little to the performance of the GNNs as compared to feature propagation. Thus, a new direction recently emerging for scalable GNN is based on the simplified GCN (SGC) (Wu et al., 2019), which successively removes nonlinearities and collapsing weight matrices between consecutive layers. This reduces GNNs into a linear model operating on $K$ -layers propagated features:
+
+$$
+\mathbf { X } ^ { ( K ) } = \hat { \mathbf { A } } ^ { K } \mathbf { X } ^ { ( 0 ) } , \qquad \mathbf { Y } = \mathrm { s o f t m a x } ( \Theta \mathbf { X } ^ { ( K ) } ) ,
+$$
+
+where $\mathbf { X } ^ { ( 0 ) } = \mathbf { X }$ , $\mathbf { X } ^ { ( K ) }$ is the $K$ -layers propagated feature, and $\hat { \mathbf { A } } = \widetilde { \mathbf { D } } ^ { r - 1 } \widetilde { \mathbf { A } } \widetilde { \mathbf { D } } ^ { - r }$ . By setting $r =$ 0.5, 1 and 0, $\hat { \bf A }$ represents the symmetric normalization adjacency matrix $\widetilde { \mathbf { D } } ^ { - 1 / 2 } \widetilde { \mathbf { A } } \widetilde { \mathbf { D } } ^ { - \mathrm { i } / 2 }$ (Klicpera et al., 2019), the transition probability matrix $\widetilde { \mathbf { A } } \widetilde { \mathbf { D } } ^ { - 1 }$ (Zeng et al., 2020), or the reverse transition probability matrix $\widetilde { \bf D } ^ { - 1 } \widetilde { \bf A }$ ( $\mathrm { X u }$ et al., 2018), respectively. As the propagated features $\mathbf { X } ^ { ( K ) }$ can be precomputed, SGC is more scalable and efficient for the large graph. However, such graph-wise propagation restricts the same propagation steps and a fixed RF for each node. Therefore, some nodes’ features may be over-smoothed or under-smoothed due to the inconsistent RF expansion speed, leading to non-optimal performance.
+
+Layer-wise Propagation. Following SGC, some recent methods adopt layer-wise propagation to combine the features with different propagation layers. SIGN (Frasca et al., 2020) proposes to concatenate the propagated features at different propagation depth after simple linear transformation: $[ \mathbf { X } ^ { ( 0 ) } \mathbf { W } _ { 0 } , \mathbf { X } ^ { ( 1 ) } \mathbf { W } _ { 1 } , . . . , \mathbf { X } ^ { ( K ) } \mathbf { W } _ { K } ]$ . $\mathrm { { S ^ { 2 } G C } }$ (Zhu & Koniusz, 2021) proposes the simple spectral graph convolution to average the propagated features in different iterations as $\mathbf { X } ^ { ( K ) } = \sum _ { l = 0 } ^ { K } \hat { \mathbf { A } } ^ { l } \mathbf { X } ^ { ( 0 ) }$ . In addition, GBP (Chen et al., 2020b) further improves the combination process by weighted averaging $\mathbf { X } ^ { ( K ) } = \sum _ { l = 0 } ^ { K } w _ { l } \hat { \mathbf { A } } ^ { l } \mathbf { X } ^ { ( 0 ) }$ with the layer weight $w _ { l } = \beta { \left( 1 - \beta \right) } ^ { l }$ . Similar to these works, we also use a linear model for higher training scalability. The difference lies in that we consider the propagation process from a node-wise perspective and each node in GAMLP has a personalized combination of different steps of the propagated features.
+
+# 2.3 LABEL UTILIZATION ON GNNS.
+
+Labels of training nodes are conventionally only used as supervision signals in loss functions in most graph learning methods. However, there also exist some graph learning methods that directly exploit the labels of training nodes. Among them, the label propagation algorithm (Zhu & Ghahramani, 2002) is the most well-known one. It simply regards the partially observed label matrix $\mathbf { Y } \in \mathbb { R } ^ { N \times C }$ as input features for nodes in the graph and propagates the input features through the graph structure, where $C$ is the number of candidate classes. UniMP (Shi et al., 2020) proposes to map the partially observed label matrix $\mathbf { Y }$ to the dimension of the node feature matrix $\mathbf { X }$ and add these two matrices together as the new input feature. To fight against the label leakage problem, UniMP further randomly masks the training nodes during every training epoch.
+
+Instead of using only the hard training labels, Correct & Smooth (Huang et al., 2020) first trains a simple model such as an MLP and gets this model’s predicted soft labels for unlabeled nodes. Then, it propagates the learning errors on the labeled nodes to connected nodes and smooths the output in a
+
+![](images/62d7673bc40aa87230cac6510d4d33f63d305d75d7fd4451adc76c0d3dd60a5d.jpg)  
+Figure 2: Overview of the proposed GAMLP, including (1) feature and label propagation, (2) combine the propagated features and labels with RF attention, and (3) MLP training. Note that both the feature and label propagation can be pre-processed.
+
+Personalized PageRank manner like APPNP (Klicpera et al., 2019). Besides, SLE (Sun & Wu, 2021) decouples the label utilization procedure in UniMP, and executes the propagation in advance. Unlike UniMP, “label reuse” (Wang et al., 2021) concatenates the partially observed label matrix $\mathbf { Y }$ with the node feature matrix $\mathbf { X }$ to form the new input matrix. Concretely, it fills the missing elements in the partially observed label matrix $\mathbf { Y }$ with the soft label predicted by the model, and this newly generated $\mathbf { Y } ^ { \prime }$ is again concatenated with $\mathbf { X }$ and then fed into the model to generate new predictions.
+
+# 3 GRAPH ATTENTION MULTI-LAYER PERCEPTRON
+
+# 3.1 ARCHITECTURE OVERVIEW
+
+As shown in Fig. 2, GAMLP decomposes the end-to-end GNN training into three parts: feature and label propagation, feature and label combination with RF attention, and the MLP training. As the feature and label propagation is pre-processed only once, and MLP training is efficient and salable, we can easily scale GAMLP to large graphs. Besides, with the RF attention, each node in GAMLP can adaptively get the suitable combination weights for propagated features and labels under different receptive fields, thus boosting model performance.
+
+# 3.2 NODE-WISE FEATURE AND LABEL PROPAGATION
+
+Node-wise Feature Propagation. We separate the essential operation of GNNs — feature propagation by removing the neural network $\Theta$ and nonlinear activation $\delta$ for feature transformation. Specifically, we construct a parameter-free $K$ -step feature propagation as:
+
+$$
+\mathbf { X } ^ { ( k ) }  \hat { \mathbf { A } } \mathbf { X } ^ { ( k - 1 ) } , \forall k = 1 , \ldots , K ,
+$$
+
+where $\mathbf { X } ^ { ( k ) }$ contains the features of a fixed RF: the node itself and its $k$ -hop neighborhoods.
+
+After $K$ -step feature propagation shown in E.q. 2, we correspondingly get a list of propagated features under different propagation steps: $[ \mathbf { X } ^ { ( 0 ) } , \mathbf { X } ^ { ( 1 ) } , \mathbf { X } ^ { ( k ) } , . . . , \mathbf { X } ^ { ( K ) } ]$ . For a node-wise propagation, we propose to average these propagated features in a weighted manner:
+
+$$
+\mathbf { H } _ { \mathbf { X } } = \sum _ { k = 0 } ^ { K } \mathbf { W } _ { k } \mathbf { X } ^ { ( k ) } ,
+$$
+
+where $\mathbf { W } _ { k } = D i a g ( \eta _ { k } ) \in \mathbb { R } ^ { n \times n }$ is the diagonal matrix derived from vector $\eta _ { k }$ , and $\eta _ { k } \in \mathbb { R } ^ { n }$ is a vector derived from vector $\eta _ { k } [ i ] = w _ { i } ( k ) , 1 \le i \le n$ , and $w _ { i } ( k )$ measures the importance of the $k$ -step propagated feature for node $v _ { i }$ .
+
+Node-wise Label Propagation. We use a scalable and node-adaptive way to take advantage of the node labels of the training set. Concretely, the label embedding matrix $\mathbf { Y } \in \mathbb { R } ^ { n \times c } ( \mathbf { Y } ^ { ( \bar { 0 } ) } )$ is propagated with the normalized adjacency matrix $\hat { \bf A }$ :
+
+$$
+\mathbf { Y } ^ { ( l ) }  \hat { \mathbf { A } } \mathbf { Y } ^ { ( l - 1 ) } , \forall l = 1 , \dots , L ,
+$$
+
+After $L$ -step label propagation, we get a list of propagated labels under different propagation steps: $[ \mathbf { Y } ^ { ( 0 ) } , \mathbf { Y } ^ { ( 1 ) } , \mathbf { Y } ^ { ( 2 ) } , . . . , \mathbf { Y } ^ { ( L ) } ]$ . Generally, the propagated label $\mathbf { Y } ^ { ( l ) }$ is closer to the original label matrix $\mathbf { Y } ^ { ( 0 ) }$ with smaller propagation step $l$ , and thus face a higher risk of data leakage problem if it is directly used as the model input. We propose last residual connection to solve this problem.
+
+Definition 3.1 (Last Residual Connection). Given the propagation step $l$ , and a list of propagated labels: $[ \mathbf { Y } ^ { ( 0 ) } , \dot { \mathbf { Y } } ^ { ( 1 ) } , \mathbf { Y } ^ { ( 2 ) } , . . . , \mathbf { Y } ^ { ( L ) } ]$ , we smooth each label $\dot { \mathbf { Y } } ^ { ( \tilde { l } ) }$ with the smoothed label ${ \bf Y } ^ { ( L ) }$ :
+
+$$
+\hat { \mathbf { Y } } ^ { ( l ) } \gets ( 1 - \alpha _ { l } ) \mathbf { Y } ^ { ( l ) } + \alpha _ { l } \mathbf { Y } ^ { ( L ) } , l = 1 , \dots , L ,
+$$
+
+where $\begin{array} { r } { \alpha _ { l } = \cos \left( \frac { \pi l } { 2 L } \right) } \end{array}$ controls the proportion of $\mathbf { Y } ^ { ( L ) }$ in the $l$ -step propagated label.
+
+Similar to the node-wise feature propagation introduced in Sec. 3.2, we propose to average these propagated labels in a weighted manner:
+
+$$
+\mathbf { H } _ { \mathbf { Y } } = \sum _ { l = 0 } ^ { L } \hat { \mathbf { W } } _ { l } \hat { \mathbf { Y } } ^ { ( l ) } .
+$$
+
+# 3.3 NODE-ADAPTIVE ATTENTION MECHANISMS
+
+To satisfy different RF requirements for each node, we introduce two RF attention mechanisms to get $w _ { i } ( k )$ . Note that these attention mechanisms can be used in both the feature and label propagation, and we introduce them from a feature perspective here. To apply them for node-wise label propagation, we only need to replace the feature $\mathbf { X } _ { i }$ in Eq. 7 and Eq. 8 with the label $\mathbf { Y } _ { i }$ .
+
+Definition 3.2 (Recursive Attention). At each propagation step $l$ , suppose $s \in \mathbb { R } ^ { d }$ is a learnable parameter vector, we recursively measure the feature information gain compared with the previous combined feature as:
+
+$$
+\widetilde { \mathbf { X } } _ { i } ^ { ( l ) } = \mathbf { X } _ { i } ^ { ( l ) } \parallel \sum _ { k = 0 } ^ { l - 1 } w _ { i } ( k ) \mathbf { X } _ { i } ^ { ( k ) } , \quad \widetilde { w } _ { i } ( l ) = \delta ( \widetilde { \mathbf { X } } _ { i } ^ { ( l ) } \cdot s ) , \quad w _ { i } ( l ) = e ^ { \widetilde { w } _ { i } ( l ) } / \sum _ { k = 0 } ^ { K } e ^ { \widetilde { w } _ { i } ( k ) } .
+$$
+
+As Xe (l−1)i ∈ combines the graph information under different propagation steps and RF, large proportion of the information in $\widetilde { \mathbf { X } } _ { i } ^ { ( l ) }$ may have already existed in $\begin{array} { r } { \sum _ { k = 0 } ^ { l - 1 } w _ { i } ( k ) \mathbf { X } _ { i } ^ { ( k ) } } \end{array}$ , leading to small information gain. A larger $w _ { i } ( l )$ indicates the feature $\mathbf { X } _ { i } ^ { ( l ) }$ is more important to the current state of node $v _ { i }$ since combining $\widetilde { \mathbf X } _ { i } ^ { ( l ) }$ will introduce higher information gain.
+
+Jumping Knowledge Network (JK-Net) (Xu et al., 2018) adopts layer aggregation to combine the node embeddings of different GCN layers, and thus it can leverage the propagated nodes’ information with different RF. Motivated by JK-Net, we propose to guide the feature combination process with the model prediction trained on all the propagated features. Concretely, GAMLP with JK attention includes two branches: the concatenated JK branch and the attention-based combination branch.
+
+Definition 3.3 (JK Attention). Given the MLP prediction of the JK branch as $\mathbf { E } _ { i } = M L P ( \mathbf { X } _ { i } ^ { ( 1 ) } \mid \mid$ $\mathbf { X } _ { i } ^ { ( 2 ) } \parallel . . . \parallel \mathbf { X } _ { i } ^ { ( K ) } ) \in \mathbb { R } ^ { K f }$ , the combination weight is defined as:
+
+$$
+\widetilde { \mathbf { X } } _ { i } ^ { ( l ) } = \mathbf { X } _ { i } ^ { ( l ) } \parallel \mathbf { E } _ { i } , \quad \widetilde { w } _ { i } ( l ) = \delta ( \widetilde { \mathbf { X } } _ { i } ^ { ( l ) } \cdot s ) , \quad w _ { i } ( l ) = e ^ { \widetilde { w } _ { i } ( l ) } / \sum _ { k = 0 } ^ { K } e ^ { \widetilde { w } _ { i } ( k ) } .
+$$
+
+The JK branch aims to create a multi-scale feature representation for each node, which helps the attention mechanism learn the weight $w _ { i } ( k )$ . The learned weights are then fed into the attentionbased combination branch to generate each node’s refined attention feature representation. As the training process continues, the attention-based combination branch will gradually emphasize those neighborhood regions that are more helpful to the target nodes. The JK attention can model a wider neighborhood while enhancing correlations, bringing a better feature representation for each node.
+
+# 3.4 MODEL TRAINING
+
+Both the combined feature $\mathbf { H } _ { \mathbf { X } }$ and combined label $\mathbf { H } _ { \mathbf { Y } }$ are transformed with MLP, and then be added to get the final output embedding:
+
+$$
+\widetilde { \mathbf { H } } = \mathbf { M } \mathbf { L } \mathbf { P } ( \mathbf { H } _ { \mathbf { X } } ) + \beta \mathbf { M } \mathbf { L } \mathbf { P } ( \mathbf { H } _ { \mathbf { Y } } ) ,
+$$
+
+where $\beta$ is a hyper-parameter that measures the importance of the combined label. For example, some graphs have good features but low-quality labels (e.g., label noise or low label rate), and we should decrease $\beta$ so that more attention is paid to the graph features.
+
+We adopt the Cross-Entropy (CE) measurement between the predicted softmax outputs and the one-hot ground-truth label distributions as the objective function:
+
+$$
+\mathcal { L } _ { C E } = - \sum _ { i \in \mathcal { V } _ { l } } \sum _ { j } \mathbf { Y } _ { i j } \log ( \mathrm { s o f t m a x } ( \widetilde { \mathbf { H } } ) _ { i j } ) ,
+$$
+
+where $\mathbf { Y } _ { i }$ is the one-hot label indicator vector.
+
+# 3.5 PROPERTIES OF GAMLP
+
+High Efficiency and Scalability. Compared with the previous GNNs (e.g., GCN and GraphSAGE), our proposed GAMLP only need to do the feature and label propagation only once. Suppose $P$ and $Q$ are the number of layers in MLP trained with feature and labels, and $k$ is the sampled nodes, the time complexity of GAMLP is $\mathcal { O } ( P n f ^ { 2 } + Q n c ^ { 2 } )$ , which is smaller than the complexity of GraphSAGE (i.e., $\mathcal { O } ( k ^ { \bar { K } _ { n } } f ^ { 2 } ) )$ . Besides, it also cost less memory than the sampling-based GNNs, and thus can scale to a larger graph in a single machine. Notably, like other simplified GNNs (i.e., SGC and SIGN), GAMLP can pre-compute the propagated features and labels only once. It doesn’t need to pull the intermediate representation of other nodes during the MLP training. Therefore, it can also be well adapted to the distributed environment. Further details can be found in Appendix A.3.
+
+Deep propagation. With our recursive and JK attention, GAMLP can support large propagation depth without the over-smoothing issue since each node can get the node personalized combination weights for different propagated features and labels according to its demand. Such characteristic is essential for sparse graph, i.e., sparse labels, edges, and features. For example, a graph with a low label rate or edge rate can increase the propagation depth to spread the label supervision over the full graph. Each node can utilize the high-order graph structure information with deep propagation and then boost the node classification performance. Further details is in Appendix B.2.
+
+# 3.6 RELATION WITH CURRENT METHODS
+
+GAMLP vs. GBP. Both GAMLP and GBP propose to combine the propagated features under different propagation steps. However, GBP adopts a layer-wise propagation scheme and ignores the inconsistent receptive field expansion speed for different nodes. As the optimal propagation steps and smoothing levels of different nodes are different, some nodes may face the over-smoothing issue, even propagating the same step. GAMLP considers the feature and label propagation in a more fine-grained node perspective.
+
+GAMLP vs. GAT. Each node in a GAT layer learns to weighted combine the embedding (or feature) of its neighborhoods with an attention mechanism, and the attention weights are measured by the local information in a fixed RF – the node itself and its direct neighbors. Different from the attention mechanism in GAT, GAMLP considers more global information under different RF.
+
+GAMLP vs. JK-Net. Motivated by JK-Net, GAMLP with JK attention concatenate the propagated features under different propagation steps. However, the model prediction based on the concatenated feature is just used as a reference vector for the attention-based combination branch in GAMLP rather than the final results. Compared with JK-Net, GAMLP with JK attention is more effective in alleviating the over-smoothing and scalability issue that deep architecture introduces.
+
+GAMLP vs. SAGN. SAGN also proposes to do node-specific propagation in GNN. Concretely, SAGN learns the node-specific attention weights with the original node feature. Unlike SAGN, GAMLP adopts two attention mechanisms to learn the interactions between the propagated features over different sizes of receptive fields. Besides, node-wise label propagation is also employed in GAMLP for better utilization of node labels.
+
+# 4 EXPERIMENTS
+
+In this section, we verify the effectiveness of GAMLP on 12 real-world graph datasets under both the transductive and inductive settings. We aim to answer the following four questions. Q1: Can GAMLP outperform the state-of-the-art GNN methods? Q2: If so, where does the performance gain of GAMLP come from? Q3: How about the efficiency of GAMLP compared with current GNN methods? Q4: How does GAMLP perform when applied to highly sparse graphs (i.e., given few edges and low label rate)? More experimental results about the heterogeneous graph, propagation depth and interpretability can be found in Appendix B.
+
+Table 1: Performance comparison on seven transductive datasets.   
+
+<table><tr><td>Methods</td><td>Cora</td><td>Citeseer</td><td>PubMed</td><td>Amazon Computer</td><td>Amazon Photo</td><td>Coauthor CS</td><td>Coauthor Physics</td></tr><tr><td>GCN</td><td>81.8±0.5</td><td>70.8±0.5</td><td>79.3±0.7</td><td>82.4±0.4</td><td>91.2±0.6</td><td>90.7±0.2</td><td>92.7±1.1</td></tr><tr><td>GAT</td><td>83.0±0.7</td><td>72.5±0.7</td><td>79.0±0.3</td><td>80.1±0.6</td><td>90.8±1.0</td><td>87.4±0.2</td><td>90.2±1.4</td></tr><tr><td>JK-Net</td><td>81.8±0.5</td><td>70.7±0.7</td><td>78.8±0.7</td><td>82.0±0.6</td><td>91.9±0.7</td><td>89.5±0.6</td><td>92.5±0.4</td></tr><tr><td>ResGCN</td><td>82.2±0.6</td><td>70.8±0.7</td><td>78.3±0.6</td><td>81.1±0.7</td><td>91.3±0.9</td><td>87.9±0.6</td><td>92.2±1.5</td></tr><tr><td>APPNP</td><td>83.3±0.5</td><td>71.8±0.5</td><td>80.1±0.2</td><td>81.7±0.3</td><td>91.4±0.3</td><td>92.1±0.4</td><td>92.8±0.9</td></tr><tr><td>AP-GCN</td><td>83.4±0.3</td><td>71.3±0.5</td><td>79.7±0.3</td><td>83.7±0.6</td><td>92.1±0.3</td><td>91.6±0.7</td><td>93.1±0.9</td></tr><tr><td>SGC</td><td>81.0±0.2</td><td>71.3±0.5</td><td>78.9±0.5</td><td>82.2±0.9</td><td>91.6±0.7</td><td>90.3±0.5</td><td>91.7±1.1</td></tr><tr><td>SIGN</td><td>82.1±0.3</td><td>72.4±0.8</td><td>79.5±0.5</td><td>83.1±0.8</td><td>91.7±0.7</td><td>91.9±0.3</td><td>92.8±0.8</td></tr><tr><td>S²GC</td><td>82.7±0.3</td><td>73.0±0.2</td><td>79.9±0.3</td><td>83.1±0.7</td><td>91.6±0.6</td><td>91.6±0.6</td><td>93.1±0.8</td></tr><tr><td>GBP</td><td>83.9±0.7</td><td>72.9±0.5</td><td>80.6±0.4</td><td>83.5±0.8</td><td>92.1±0.8</td><td>92.3±0.4</td><td>93.3±0.7</td></tr><tr><td>UNIMP</td><td>82.6±0.4</td><td>72.5±0.9</td><td>80.1±0.5</td><td>83.9±0.8</td><td>92.0±1.1</td><td>92.4±0.3</td><td>93.5±0.8</td></tr><tr><td>GAMLP(JK)</td><td>84.3±0.8</td><td>74.6±0.4</td><td>80.7±0.4</td><td>84.5±0.7</td><td>92.8±0.7</td><td>92.6±0.5</td><td>93.6±1.0</td></tr><tr><td>GAMLP(R)</td><td>83.9±0.6</td><td>73.9±0.6</td><td>80.8±0.5</td><td>84.2±0.5</td><td>92.6±0.8</td><td>92.8±0.7</td><td>93.2±1.0</td></tr></table>
+
+# 4.1 EXPERIMENTAL SETUP
+
+Datasets. We evaluate the predictive accuracy of GAMLP under both transductive and inductive settings. For transductive settings, we conduct experiments on nine transductive datasets: three citation network datasets (Cora, Citeseer, PubMed) (Sen et al., 2008), two user-item datasets (Amazon Computer, Amazon Photo), two co-author datasets (Coauthor CS, Coauthor Physics) (Shchur et al., 2018), and two OGB datasets (ogbn-products, ogbn-papers100M) (Hu et al., 2021). For inductive settings, we perform the comparison experiments on three inductive datasets: PPI, Flickr, and Reddit (Zeng et al., 2019). The statistics about these datasets can be found in Table 10 in Appendix C.1.
+
+Baselines. Under the transductive setting, we compare GAMLP with the following representative baseline methods: GCN (Kipf & Welling, 2016), GAT (Velickovi ˇ c et al., 2017), JK-Net (Xu et al., ´ 2018), ResGCN (Li et al., 2019), APPNP (Klicpera et al., 2018), AP-GCN (Spinelli et al., 2020), SGC (Wu et al., 2019), SIGN (Frasca et al., 2020), $\mathrm { { S ^ { 2 } G C } }$ (Zhu & Koniusz, 2021), and GBP (Chen et al., 2020b). For the comparison in the OGB datasets, we choose the top-performing methods from the OGB leaderboard along with their accuracy results. Under the inductive setting, we choose following representative methods: SGC (Wu et al., 2019), GraphSAGE (Hamilton et al., 2017), Cluster-GCN (Chiang et al., 2019), and GraphSAINT (Zeng et al., 2019).
+
+In addition, two variants of GAMLP are tested in the evaluation: GAMLP(JK) and GAMLP(R). “JK” and “R” stand for adopting “JK attention” and “Recursive attention” for the node-adaptive attention mechanism, respectively.
+
+Table 2: Performance comparison on the ogbnproducts dataset.   
+
+<table><tr><td>Methods</td><td>Val Accuracy</td><td>Test Accuracy</td></tr><tr><td>GCN</td><td>92.00±0.03</td><td>75.64±0.21</td></tr><tr><td>SGC</td><td>92.13±0.02</td><td>75.87±0.14</td></tr><tr><td>GraphSAGE</td><td>92.24±0.07</td><td>78.50±0.14</td></tr><tr><td>GraphSAINT</td><td>92.52±0.13</td><td>80.27±0.26</td></tr><tr><td>GBP</td><td>92.82±0.10</td><td>80.48±0.05</td></tr><tr><td>SIGN</td><td>92.99±0.04</td><td>80.52±0.16</td></tr><tr><td>DeeperGCN</td><td>92.38±0.09</td><td>80.98±0.20</td></tr><tr><td>UniMP</td><td>93.08±0.17</td><td>82.56±0.31</td></tr><tr><td>SAGN</td><td>93.09±0.04</td><td>81.20±0.07</td></tr><tr><td>SAGN+0-SLE</td><td>93.27±0.04</td><td>83.29±0.18</td></tr><tr><td>GAMLP(JK)</td><td>93.19±0.03</td><td>83.54±0.25</td></tr><tr><td>GAMLP(R)</td><td>93.11±0.05</td><td>83.59±0.09</td></tr></table>
+
+Table 3: Performance comparison on the ogbnpapers100M dataset.   
+
+<table><tr><td>Methods</td><td>Val Accuracy</td><td>Test Accuracy</td></tr><tr><td>SGC</td><td>66.48±0.20</td><td>63.29±0.19</td></tr><tr><td>SIGN</td><td>69.32±0.06</td><td>65.68±0.06</td></tr><tr><td>SIGN-XL</td><td>69.84±0.06</td><td>66.06±0.19</td></tr><tr><td>SAGN</td><td>70.34±0.99</td><td>66.75±0.84</td></tr><tr><td>SAGN+0-SLE</td><td>71.06±0.08</td><td>67.55±0.15</td></tr><tr><td>GAMLP(JK)</td><td>71.92±0.04</td><td>68.07±0.10</td></tr><tr><td>GAMLP(R)</td><td>71.21±0.03</td><td>67.46±0.02</td></tr></table>
+
+Table 4: Performance comparison on three inductive datasets.   
+
+<table><tr><td>Methods</td><td>PPI</td><td>Flickr</td><td>Reddit</td></tr><tr><td>SGC</td><td>65.7±0.01</td><td>50.2±0.12</td><td>94.9±0.00</td></tr><tr><td>GraphSAGE</td><td>61.2±0.05</td><td>50.1±0.13</td><td>95.4±0.01</td></tr><tr><td>Cluster-GCN</td><td>99.2±0.04</td><td>48.1±0.05</td><td>95.7±0.00</td></tr><tr><td>GraphSAINT</td><td>99.4±0.03</td><td>51.1±0.10</td><td>96.6±0.01</td></tr><tr><td>GAMLP(JK)</td><td>99.82±0.01</td><td>54.12±0.01</td><td>97.04±0.01</td></tr><tr><td>GAMLP(R)</td><td>99.66±0.01</td><td>53.12±0.00</td><td>96.62±0.01</td></tr></table>
+
+# 4.2 END-TO-END COMPARISON
+
+Transductive Performance. To answer Q1, we report the transductive performance of GAMLP in Tables 1, 2, and 3. We observe that both variants of GAMLP outperform all the baseline methods on almost all the datasets. For example, on the small Citeseer dataset, GAMLP(JK) outperforms the state-of-the-art method $\mathrm { { S ^ { 2 } G C } }$ by a large margin of $1 . 6 \%$ ; on the medium-sized dataset Amazon Computers, the predictive accuracy of GAMLP (JK) exceeds the one of the state-of-the-art method GBP by $1 . 0 \%$ ; on the two large OGB datasets, GAMLP takes the lead by $1 . 0 3 \%$ and $1 . 3 2 \%$ on ogbn-products and ogbn-papers100M, respectively. Furthermore, the experimental results illustrate that the contest between the two variants of GAMLP is not a one-horse race, which suggests that these two different attention mechanisms both have their irreplaceable sense in some ways.
+
+Inductive Performance. We also evaluate GAMLP under the inductive setting. The experiment results in Table 4 show that GAMLP consistently outperforms all the baseline methods. The leading advantage of GAMLP(JK) over SOTA inductive method – GraphSAINT is more than $3 . 0 \%$ on the widely-used dataset – Filckr. The impressive performance of GAMLP under the inductive setting illustrates that GAMLP is alpowerful in predicting the properties of unseen nodes.
+
+Table 5: Ablation study on label utilization.   
+
+<table><tr><td>Methods</td><td>Val Accuracy</td><td>Test Accuracy</td></tr><tr><td>GAMLP(R)</td><td>93.11±0.05</td><td>83.59±0.05</td></tr><tr><td>-no_label</td><td>92.29±0.06</td><td>81.43±0.18</td></tr><tr><td>-plain_label</td><td>92.53±0.21</td><td>81.12±0.45</td></tr><tr><td>-uniform</td><td>92.72±0.15</td><td>81.28±0.93</td></tr></table>
+
+Table 6: Ablation study on reference vector.   
+
+<table><tr><td>Methods</td><td>Val Accuracy</td><td>Test Accuracy</td></tr><tr><td>GAMLP(JK)</td><td>82.5±0.5</td><td>80.7±0.4</td></tr><tr><td>-origin_feature</td><td>82.2±0.4</td><td>80.5±0.4</td></tr><tr><td>-normal_noise</td><td>81.8±0.4</td><td>79.8±0.5</td></tr><tr><td>-no_reference</td><td>81.5±0.5</td><td>79.9±0.3</td></tr></table>
+
+# 4.3 ABLATION STUDY
+
+To answer Q2, we focus on two modules in GAMLP: (1) label utilization; (2) attention mechanism in the node-wise propagation. For the second one, we evaluate the effects of different choices for reference vectors in the JK attention.
+
+Label Utilization. In this part, we evaluate whether adding last residual connection and making use of training labels really help or not. The predictive accuracy of GAMLP(R) is evaluated on the ogbn-products dataset along with its three variants: “-no_label”, “-plain_label”, and “-uniform”, which stands for not using labels, removing last residual connections, and replacing last residual connections with uniform distributions, respectively. The experimental results in Table 5 show that utilizing labels brings huge performance gain to GAMLP: from $8 1 . 4 3 \%$ to $8 3 . 5 9 \%$ . The performance drop from removing the last residual connections (“-plain_label” in Table 5) is significant since directly adopting the raw training labels leads to the overfitting issue. The fact that “-uniform” performs worse than “-no_label” illustrates that intuitively fusing the original label distribution with the uniform distribution would harm the predictive accuracy. It further demonstrates the effectiveness of our proposed last residual connections.
+
+Reference Vector in Attention Mechanism. In this part, we study the role of the reference vector (originally set as the concatenated features from different propagation steps) in our proposed JK attention. We evaluate the three variants of GAMLP(JK): “-origin_feature”, “-normal_noise”, and “-no_reference”, which changes the reference vector to the original node feature, noise from the normal distribution, and nothing, respectively. The predictive accuracy of each variant on the PubMed dataset is reported in Table 6. The experimental results show that our original choice of the reference vector is the best among itself and its three variants. The superiority of the concatenated features from different propagation steps comes from the fact that it allows the model to capture the interactions between the propagated features over the receptive fields with different sizes.
+
+Table 7: Efficiency comparison on the ogbn-products dataset.   
+
+<table><tr><td>Methods</td><td>SGC</td><td>SIGN</td><td>GAMLP(JK)</td><td>GAMLP(R)</td><td>GraphSAINT</td><td>Cluster-GCN</td></tr><tr><td>Training time</td><td>1.0</td><td>4.0</td><td>8.0</td><td>9.3</td><td>364</td><td>503</td></tr><tr><td>Test accuracy</td><td>75.87</td><td>80.52</td><td>83.54</td><td>83.59</td><td>79.08</td><td>78.97</td></tr></table>
+
+![](images/60b26cb2f146ebaf212df7df59a70cd9b9815012880c1e774b16d614e3160772.jpg)  
+Figure 3: Test accuracy on PubMed dataset under different levels of label and edge sparsity.
+
+# 4.4 EFFICIENCY COMPARISON
+
+To answer Q3, we evaluate the efficiency of each method on the ogbn-products dataset. We compare the efficiency of GAMLP with sampling-based GraphSAINT and Cluster-GCN, graph-wisepropagation-based SGC, and layer-wise-propagation-based SIGN. Table 7 illustrates the relative training time of each compared method along with its predictive accuracy. The training time of SGC is set to 1 as reference. We observe that (1) sampling-based methods (e.g., GraphSAINT) consume much more time than graph/layer-wise-propagation based methods (e.g., SGC, SIGN) due to the high computation cost introduced by the sampling process; (2) the two variants of GAMLP achieve the best predictive accuracy while requiring comparable training time with SGC.
+
+# 4.5 EXPERIMENTS ON SPARSE GRAPHS
+
+To answer Q4, we conduct experiments to evaluate the predictive accuracy of GAMLP when faced with edge and label sparsity problems, where the number of edges and training labels are highly scarce. We randomly remove a fixed percentage of edges from the original graph to simulate the edge sparsity problem. The removed edges are exactly the same for all the compared methods. Besides, we enumerate the number of training nodes per class from 1 to 20 to evaluate the performance of GAMLP given different levels of label sparsity. The experimental results in Figure 3 show that GAMLP consistently outperforms all the baselines when faced with different levels of edge and label sparsity. This experiment further demonstrates the effectiveness of our proposed node-wise propagation scheme. The node-wise propagation enables GAMLP to better capture long-range dependencies, which is crucial when applying GNN methods to highly sparse graphs.
+
+# 5 CONCLUSION
+
+We presented Graph Attention Multilayer Perceptron (GAMLP), a scalable, efficient, and deep graph model based on receptive field attention. GAMLP introduced two new attention mechanisms: recursive attention and JK attention, which enables to learn the representations over RF with different sizes in a node-adaptive manner. Extensive experiments on 12 graph datasets verified the effectiveness of the proposed method. GAMLP moves forward the performance boundary of scalable GNNs, especially on large-scale graphs. This initial attempt also motivates several interesting future directions: (1) exploring other attention mechanisms and (2) studying the mechanisms on heterogeneous graphs.
+
+# 6 REPRODUCIBILITY STATEMENT
+
+The source code of GAMLP can be found in Anonymous Github (https://anonymous.4open. science/r/ICLR-GAMLP). To ensure reproducibility, we have provided the overview of datasets and baselines in Section 4.1 and Table 10 in Appendix C.1. The detailed hyperparameter settings for our GAMLP can be found in Appendix C.2. Our experimental environment is presented in Appendix C.1, and please refer to “README.md” in the Github repository for more details.
+
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+
+![](images/d7ef11ec87782651e0ec931f33eee529b59ac4904f874fb1fb3649c11687c823.jpg)  
+Figure 4: The architecture of GAMLP with JK Attention.
+
+# A MORE DETAILS ABOUT GAMLP
+
+A.1 AN EXAMPLE OF JK ATTENTION
+
+Fig. 4 provides a more zoomed-in look of JK attention, one of the two node-adaptive attention mechanisms we proposed. The propagated features are concatenated and then fed into an MLP to map the concatenated feature to the hidden dimension of the model. The mapped feature is then set as the reference vector of the following attention mechanism, where a linear layer is adopted to calculate the combination weight for propagated features at different propagation steps. The propagated features are then multiplied with the corresponding combination weight, and the summed results are fed into another MLP to generate final predictions.
+
+# A.2 COMPARISON BETWEEN THE LABEL USAGE IN UNIMP AND GAMLP
+
+(1) The label usage in UniMP is coupled with the training process, making it hard to scale to large graphs. While GAMLP decouples the label usage from the training process, the label propagation process can be executed as preprocessing.
+
+(2) The label propagation steps in UniMP are restricted to the same number of model layers. And if the number of model layers becomes large, UniMP will encounter the efficiency and scalability issues even on relatively small graphs. While the label propagation steps in GAMLP can be quite large since the label propagation is performed as preprocessing.
+
+(3) Both UniMP and GAMLP propose approaches to fight against the label leakage issue. However, the random masking in UniMP has to be executed in each training epoch, while the last residual connection (composed of simple matrix addition) in GAMLP just needs to be executed once during preprocessing. Thus, UniMP consumes more resources than GAMLP to fight the label leakage issue.
+
+# A.3 COMPLEXITY ANALYSIS
+
+Table 8 provides a detailed asymptotic complexity comparison between GAMLP and representative scalable GNN methods. During preprocessing, the time cost of clustering in Cluster-GCN is $\mathcal { O } ( m )$ and the time complexity of most linear models is $\mathcal { O } ( K m f )$ . Besides, GAMLP has an extra time cost $\mathcal { O } ( L m c )$ for the propagation of training labels. GBP takes advantage of Monte-Carlo method and conducts this process approximately with a bound of $\begin{array} { r } { \mathcal { O } ( K n f + K \frac { \sqrt { m \log n } } { \varepsilon } ) } \end{array}$ , where $\varepsilon$ is a error threshold. Compared with sampling-based GNNs, graph/layer/node-wise-propagation-based models usually have smaller training and inference time complexity. Memory complexity is a crucial factor in large-scale graph learning as it fundamentally determines whether it is possible to adopt the method. Compared with SIGN, both GBP and GAMLP do not need to store smoothed features at different propagation steps, and the memory complexity can be reduced from $\mathcal { O } ( b L f )$ to $\mathcal { O } ( b f )$ .
+
+Table 8: Algorithm analysis for existing scalable GNNs. $n , m , c ,$ , and $f$ are the number of nodes, edges, classes, and feature dimensions, respectively. $b$ is the batch size, and $k$ refers to the number of sampled nodes. $K$ and $L$ corresponds to the number of times we aggregate features and labels respectively. Besides, $P$ and $Q$ are the number of layers in MLP classifiers trained with features and labels respectively.   
+
+<table><tr><td>Type</td><td>Method</td><td>Pre-processing</td><td>Training</td><td>Memory</td></tr><tr><td rowspan="3">Sampling</td><td>GraphSAGE</td><td>=</td><td>O(kKnf²)</td><td>O(bkK f+Kf²)</td></tr><tr><td>FastGCN</td><td></td><td>O(kKnf2)</td><td>O(bkKf+Kf²)</td></tr><tr><td>Cluster-GCN</td><td>0(m)</td><td>O(Pmf+Pnf²)</td><td>O(bKf+Kf²)</td></tr><tr><td>Graph-wise propagation</td><td>SGC</td><td>O(Kmf)</td><td>O（nf²)</td><td>O(bf+f²)</td></tr><tr><td rowspan="3">Layer-wise propagation</td><td>SIGN</td><td>O(Kmf)</td><td>O(Pnf2)</td><td>O(bLf+Pf²)</td></tr><tr><td>S²GC</td><td>O(Kmf)</td><td>O(nf2)</td><td>O(bf+f²)</td></tr><tr><td>GBP</td><td>O(Knf+KVmlgn)</td><td>O(Pnf²)</td><td>O(bf+Pf²)</td></tr><tr><td>Node-wise propagation</td><td>GAMLP</td><td>O(Kmf+Lmc)</td><td>O(Pnf² +Qnc²)</td><td>O(bf+Pf²+Qc²）</td></tr></table>
+
+Table 9: Test accuracy on ogbn-mag dataset.   
+
+<table><tr><td>Methods</td><td>Validation Accuracy</td><td>Test Accuracy</td></tr><tr><td>R-GCN</td><td>40.84±0.41</td><td>39.77±0.46</td></tr><tr><td>SIGN</td><td>40.68±0.10</td><td>40.46±0.12</td></tr><tr><td>HGT</td><td>49.84±0.47</td><td>49.27±0.61</td></tr><tr><td>R-GSN</td><td>51.82±0.41</td><td>50.32±0.37</td></tr><tr><td>HGConv</td><td>53.00±0.18</td><td>50.45±0.17</td></tr><tr><td>R-HGNN</td><td>53.61±0.22</td><td>52.04±0.26</td></tr><tr><td>NARS</td><td>53.72±0.09</td><td>52.40±0.16</td></tr><tr><td>NARS-GAMLP</td><td>55.52±0.08</td><td>54.01±0.21</td></tr></table>
+
+# B ADDITIONAL EXPERIMENTS
+
+# B.1 EXPERIMENTS ON OGBN-MAG
+
+Compared Baselines. Ogbn-mag dataset is a heterogeneous graph consists of 1,939,743 nodes and 21,111,007 edges of different types. For comparison, we choose eight baseline methods from the OGB ogbn-mag leaderboard: R-GCN (Schlichtkrull et al., 2018), SIGN (Frasca et al., 2020), HGT (Hu et al., 2020), R-GSN (Wu et al., 2021), HGConv (Yu et al., 2020a), R-HGNN (Yu et al., 2021), and NARS (Yu et al., 2020b).
+
+Adapt GAMLP to Heterogeneous Graphs. In its original design, GAMLP does not support training on heterogeneous graphs. Here we imitate the model design of NARS to adapt GAMLP to heterogeneous graphs.
+
+First, we sample subgraphs from the original heterogeneous graphs according to relation types and regard the subgraph as a homogeneous graph although it may have different kinds of nodes and edges. Then, on each subgraph, the propagated features of different steps are generated. The propagated features of the same propagation step across different subgraphs are aggregated using 1-d convolution. After that, aggregated features of different steps are fed into our GAMLP to get the final results. This variant of our GAMLP is called NARS-GAMLP as it mimics the design of NARS.
+
+As ogbn-mag dataset only contains node features for “paper” nodes, we here adopt the ComplEx algorithm (Trouillon et al., 2017) to generate features for other nodes.
+
+Experiment Results. We report the validation and test accuracy of our proposed GAMLP on the ogbn-mag dataset in Table 9. It can be seen from the results that NARS-GAMLP achieves great performance on the heterogeneous graph ogbn-mag, outperforming the performance of the strongest single model baseline NARS by a large margin of $1 . 6 1 \%$ .
+
+![](images/f1ba10bc33b33c83e75d03d73546e6bd2a8de4e08b977541c38bd2a6524f2523.jpg)  
+Figure 5: Test accuracy when the propagation depth increases from 10 to 100.
+
+![](images/d5b642a4acc2caa43e1d152800df9623c7cfe56b5240b4086743fe9f9048d868.jpg)  
+Figure 6: The average attention weights of propagated features of different steps on 60 randomly selected nodes from ogbn-products.
+
+# B.2 DEEP PROPAGATION IS POSSIBLE
+
+Equipped with the learnable node-wise propagation scheme, our GAMLP can still maintain high predictive accuracy even when the propagation depth is over 50. Here, we evaluate the predictive accuracy of our proposed GAMLP(JK) at propagation depth 10, 30, 50, 80, 100 on the PubMed dataset. The performance of JK-Net and SGC are also reported as baselines. The experimental results in Fig. 5 show that even at propagation depth equals 100, the predictive accuracy of our GAMLP(JK) still exceeds $8 0 . 0 \%$ , higher than the predictive accuracy of most baselines in Table 1. At the same time, the predictive accuracy of SGC and JK-Net both drops rapidly when propagation depth increases from 10 to 100.
+
+# B.3 INTERPRETABILITY OF THE ATTENTION MECHANISM
+
+GAMLP can adaptively and effectively combine multi-scale propagated features for each node. To demonstrate this, Fig. 6 shows the average attention weights of propagated features of GAMLP(JK) according to the number of steps and degrees of input nodes, where the maximum step is 6. In this experiment, we randomly select 20 nodes for each degree range (1-4, 5-8, 9-12) and plot the relative weight based on the maximum value. We get two observations from the heat map: 1) The 1-step and 2-step propagated features are always of great importance, which shows that GAMLP captures the local information as those widely 2-layer methods do; 2) The weights of propagated features with larger steps drop faster as the degree grows, indicating that our attention mechanism could prevent high-degree nodes from including excessive irrelevant nodes, leading to over-smoothing. From the two observations, we conclude that GAMLP can identify the different RF demands of nodes and explicitly weight each propagated feature.
+
+Table 10: Overview of the Datasets   
+
+<table><tr><td>Dataset</td><td>#Nodes</td><td>#Features</td><td>#Edges</td><td>#Classes</td><td>#Train/Val/Test</td><td>Task type</td><td>Description</td></tr><tr><td>Cora</td><td>2,708</td><td>1,433</td><td>5,429</td><td>7</td><td>140/500/1000</td><td>Transductive</td><td>citation network</td></tr><tr><td>Citeseer</td><td>3,327</td><td>3,703</td><td>4,732</td><td>6</td><td>120/500/1000</td><td>Transductive</td><td>citation network</td></tr><tr><td>Pubmed</td><td>19,717</td><td>500</td><td>44,338</td><td>3</td><td>60/500/1000</td><td>Transductive</td><td>citation network</td></tr><tr><td>Amazon Computer</td><td>13,381</td><td>767</td><td>245,778</td><td>10</td><td>200/300/12881</td><td>Transductive</td><td>co-purchase graph</td></tr><tr><td>Amazon Photo</td><td>7,487</td><td>745</td><td>119,043</td><td>8</td><td>160/240/7,087</td><td>Transductive</td><td>co-purchase graph</td></tr><tr><td>Coauthor CS</td><td>18,333</td><td>6,805</td><td>81,894</td><td>15</td><td>300/450/17,583</td><td>Transductive</td><td>co-authorship graph</td></tr><tr><td>Coauthor Physics</td><td>34,493</td><td>8,415</td><td>247,962</td><td>5</td><td>100/150/34,243</td><td>Transductive</td><td>co-authorship graph</td></tr><tr><td>ogbn-products</td><td>2,449,029</td><td>100</td><td>61,859,140</td><td>47</td><td>196k/49k/2204k</td><td>Transductive</td><td>co-purchase graph</td></tr><tr><td>ogbn-papers100M</td><td>111,059,956</td><td>128</td><td>1,615,685,872</td><td>172</td><td>1207k/125k/214k</td><td>Transductive</td><td>citation network</td></tr><tr><td>ogbn-mag</td><td>1,939,743</td><td>128</td><td>21,111,007</td><td>349</td><td>626k/66k/37k</td><td>Transductive</td><td>citation network</td></tr><tr><td>PPI</td><td>56,944</td><td>50</td><td>818,716</td><td>121</td><td>45k/6k/6k</td><td>Inductive</td><td>protein interactions network</td></tr><tr><td>Flickr</td><td>89,250</td><td>500</td><td>899,756</td><td>7</td><td>44k/22k/22k</td><td>Inductive</td><td>image network</td></tr><tr><td>Reddit</td><td>232.965</td><td>602</td><td>11,606,919</td><td>41</td><td>155k/23k/54k</td><td>Inductive</td><td>social network</td></tr></table>
+
+Table 11: Ablation study of choices for $\alpha _ { l }$ on the ogbn-products dataset.   
+B.4 CHOICES FOR $\alpha _ { l }$ IN THE LAST RESIDUAL CONNECTION   
+
+<table><tr><td>Choices</td><td>Test Accuracy</td></tr><tr><td>Fixed weight</td><td>82.56±0.43</td></tr><tr><td>Linear-decreasing weight</td><td>82.72±0.93</td></tr><tr><td>Cosine function</td><td>83.59±0.05</td></tr></table>
+
+Our first choice for the that GAMLP still enco $\alpha _ { l }$ in the last residual connection module is ters the over-fitting issue on some datas $\begin{array} { r } { \alpha _ { l } = \frac { L - l } { L } } \end{array}$ . However, we find we instead choose $\begin{array} { r } { \alpha _ { l } = \cos ( \frac { \pi l } { 2 L } ) } \end{array}$ to give more penalties to labels at large propagation steps. We provide the performance comparison on the ogbn-products dataset in Table 11. Three weighting schemes for the last residual connection module are tested: "Cosine function" stands for $\begin{array} { r } { \alpha _ { l } = \cos ( \frac { \pi l } { 2 L } ) } \end{array}$ , the one in GAMLP; "Linear-decreasing weight" stands for $\begin{array} { r } { \alpha _ { l } = \frac { L - l } { L } } \end{array}$ ; and "Fixed weight" stands for $\alpha _ { l } = 0 . 7$ . Table 11 shows that the weighting scheme GAMLP adopts, $\begin{array} { r } { \alpha _ { l } = \cos ( \frac { \pi l } { 2 L } ) } \end{array}$ , outperforms the other two options.
+
+# C DETAILED EXPERIMENT SETUP
+
+# C.1 EXPERIMENT ENVIRONMENT
+
+We provide detailed information about the datasets we adopted during the experiment in Table 10. To alleviate the influence of randomness, we repeat each method ten times and report the mean performance and the standard deviations. For the largest ogbn-papers100M dataset, we run each method five times instead. The experiments are conducted on a machine with Intel(R) Xeon(R) Platinum 8255C $\mathrm { P U } @ 2 . 5 0 \mathrm { G H z }$ , and a single Tesla V100 GPU with 32GB GPU memory. The operating system of the machine is Ubuntu 16.04. As for software versions, we use Python 3.6, Pytorch 1.7.1, and CUDA 10.1. The hyper-parameters in each baseline are set according to the original paper if available. Please refer to Appendix C.2 for the detailed hyperparameter settings for our GAMLP.
+
+# C.2 DETAILED HYPERPARAMETERS
+
+We provide the detailed hyperparameter setting on GAMLP in Table 12, 13 and 14 to help reproduce the results. To reproduce the experimental results of GAMLP, just follow the same hyperparameter setting yet only run the first stage.
+
+Table 12: Detailed hyperparameter setting on OGB datasets.   
+
+<table><tr><td rowspan=1 colspan=1>Datasets</td><td rowspan=1 colspan=1>attention type</td><td rowspan=1 colspan=1>hidden size</td><td rowspan=1 colspan=1>num layer in JK</td><td rowspan=1 colspan=1>num layer</td><td rowspan=1 colspan=1>activation</td></tr><tr><td rowspan=1 colspan=1>ogb-products</td><td rowspan=1 colspan=1>Recursive</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>/</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>leakyrelu,a=0.2</td></tr><tr><td rowspan=1 colspan=1>ogb-papers100M</td><td rowspan=1 colspan=1>JK</td><td rowspan=1 colspan=1>1280</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>sigmoid</td></tr><tr><td rowspan=1 colspan=1>ogb-mag</td><td rowspan=1 colspan=1>JK</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>leaky relu,a=0.2</td></tr></table>
+
+Table 13: Detailed hyperparameter setting on OGB datasets.   
+
+<table><tr><td rowspan=1 colspan=1>Datasets</td><td rowspan=1 colspan=1>hops</td><td rowspan=1 colspan=1>hops for label</td><td rowspan=1 colspan=1>input dropout</td><td rowspan=1 colspan=1> attention dropout</td><td rowspan=1 colspan=1>dropout</td></tr><tr><td rowspan=1 colspan=1>ogb-products</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>0.2</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.5</td></tr><tr><td rowspan=1 colspan=1>ogb-papers100M</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.5</td></tr><tr><td rowspan=1 colspan=1>ogb-mag</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0.5</td></tr></table>
+
+Table 14: Detailed hyperparameter setting on OGB datasets.   
+
+<table><tr><td rowspan=1 colspan=1>Datasets</td><td rowspan=1 colspan=1>beta</td><td rowspan=1 colspan=1>patience</td><td rowspan=1 colspan=1>lr</td><td rowspan=1 colspan=1>batch size</td><td rowspan=1 colspan=1>epochs</td></tr><tr><td rowspan=1 colspan=1>ogb-products</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>300</td><td rowspan=1 colspan=1>0.001</td><td rowspan=1 colspan=1>50000</td><td rowspan=1 colspan=1>400</td></tr><tr><td rowspan=1 colspan=1>ogb-papers100M</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>60</td><td rowspan=1 colspan=1>0.0001</td><td rowspan=1 colspan=1>5000</td><td rowspan=1 colspan=1>400</td></tr><tr><td rowspan=1 colspan=1>ogb-mag</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>0.001</td><td rowspan=1 colspan=1>10000</td><td rowspan=1 colspan=1>400</td></tr></table>

parse/dev/ARtBIBAmNR/ARtBIBAmNR.md

@@ -0,0 +1,562 @@
+# Visually Guided Generative Text-Layout Pre-training for Document Intelligence
+
+Zhiming Mao1,2, Haoli Bai3, Lu Hou3, Jiansheng Wei3, Xin Jiang3, Kam-Fai Wong1,2, Qun Liu3 1The Chinese University of Hong Kong, Hong Kong, China 2MoE Key Laboratory of High Confidence Software Technologies, China 3Noah’s Ark Lab, Huawei Technologies 1,2{zmmao,kfwong}@se.cuhk.edu.hk 3{baihaoli, houlu3, weijiansheng, Jiang.Xin, qun.liu}@huawei.com
+
+# Abstract
+
+Prior study shows that pre-training techniques can boost the performance of visual document understanding (VDU), which typically requires models to gain abilities to perceive and reason both document texts and layouts (e.g., locations of texts and table-cells). To this end, we propose visually guided generative text-layout pre-training, named ViTLP. Given a document image, the model optimizes hierarchical language and layout modeling objectives to generate the interleaved text and layout sequence. In addition, to address the limitation of processing long documents by Transformers, we introduce a straightforward yet effective multisegment generative pre-training scheme, facilitating ViTLP to process word-intensive documents of any length. ViTLP can function as a native OCR model to localize and recognize texts of document images. Besides, ViTLP can be effectively applied to various downstream VDU tasks. Extensive experiments show that ViTLP achieves superior performance over existing baselines on benchmark VDU tasks, including information extraction, document classification, and document question answering.1
+
+# 1 Introduction
+
+Processing and reasoning document images with dense texts (e.g., scanned PDF files, digital forms, and spreadsheets) is a persistent yet challenging task for the research community and industry (Katti et al., 2018; Majumder et al., 2020; Li et al., 2021a). Advances in multimodal pre-training substantially improve the performance of visual document understanding (VDU) (Xu et al., 2020, 2021; Gu et al., 2021; Appalaraju et al., 2021; Wang et al., 2022a). These pre-training methods typically take multimodal inputs of given document images: i) visual features, ii) document OCR texts, and iii) spatial layouts of document elements, e.g., 2D coordinates of texts and table-cells. Among these elements, spatial layout information plays an essential role in connecting visual and textual features, as well as developing a thorough understanding of document structures (Lee et al., 2022; Wang et al., 2022b).
+
+![](images/389fdf4f9aace51d1d1edef09e7532c72ec7695d4e9b2986f6f0fdd98f0f553a.jpg)  
+Figure 1: An overview workflow of the proposed ViTLP. Given a document image as input, ViTLP can generate sequences of text and layout (i.e., word bounding boxes) for various VDU tasks with task-specific prefixes.
+
+Though effective, the performance of most existing VDU approaches relies heavily on the OCR pipelines, because the pre-processed OCR texts and corresponding 2D coordinates are used as intermediate inputs to pre-trained VDU models. The external OCR pipelines may produce incorrect or incomplete recognition results, which cannot be jointly optimized by the gradient back from VDU models. Another research line (Kim et al., 2022; Lee et al., 2023b) explores pre-training VDU models solely based on image inputs. Despite no OCR errors introduced, these methods focus on understanding texts from raw document images but dismiss layout information modeling. Since the spatial information contained in layout locations is not exploited, it may hinder the models from understanding complex document structures, especially for documents containing nested paragraphs, forms, and tables.
+
+In this work, we propose Visually guided generative Text-Layout Pre-training (ViTLP) to jointly model text and layout information from document images. As shown in Figure 1, ViTLP can localize, recognize, and understand visual document texts given the input document image and task prefixes. To achieve this goal, ViTLP is pre-trained to generate unified text-layout sequences from document images. Since natively generating text and layout tokens in a flattened sequence is token-inefficient (see Sec. 2.1), we introduce hierarchical generation modules to achieve both effective and efficient text-layout sequence generation. To the best of our knowledge, ViTLP is the first attempt to learn OCR (i.e., text localization and recognition) and VDU (i.e., document understanding) abilities in a unified generative text-layout pre-training framework.
+
+Besides, ViTLP is designed to handle long documents with intensive texts. Long document processing is ubiquitous in real-world scenarios. However, existing pre-trained models are constrained to certain token limits of input sequences. For instance, LayoutLMv2 (Xu et al., 2021) accepts the maximum inputs of 512 word tokens using a BERTstructure encoder. In both pre-training and finetuning, the exceeded text tokens are truncated, leading to incomplete document information modeling. To tackle this issue, we introduce a multi-segment pre-training scheme which divides the target textlayout sequence into consecutive segments to perform generative pre-training. Given that the full document information is already encoded in visual representations, ViTLP employs the suffix tokens from previous segments as prefix prompts to generate next-segment tokens. This multi-segment pre-training scheme further enables ViTLP to process documents of arbitrary length in fine-tuning. Notably, our multi-segment generation scheme retains the intact transformer architecture. Thus, it is more feasible than other long-document modeling workarounds, e.g., sparse attention (Beltagy et al., 2020) and memory modules (Bulatov et al., 2022), which need to modify the Transformer architecture and may affect the capacity of pre-trained models.
+
+We evaluate the performance of ViTLP on a variety of OCR and VDU tasks. Empirical results demonstrate that ViTLP can generally achieve superior performance on both OCR and VDU tasks. For instance, ViTLP achieves the 95.16 F1 score on CORD information extraction and $9 5 . 2 8 \%$ accuracy on RVL-CDIP document classification, both of which surpass existing approaches. Meanwhile, as ViTLP can provide regions of interest (ROI) for the generated text, this also helps in some VDU tasks (e.g., document question answering) to be more explainable and reliable.
+
+# 2 Approach
+
+# 2.1 Problem Formulation
+
+We study multimodal pre-training for visual document modeling. As widely studied (Xu et al., 2020, 2021; Appalaraju et al., 2021; Li et al., 2021b; Powalski et al., 2021; Wang et al., 2022a; Huang et al., 2022; Wang et al., 2022b), document images $\mathbf { V }$ , texts $\mathbf { T }$ , and layouts $\mathbf { L }$ are three fundamental modalities for visual document modeling.
+
+Unified Text-Layout Generation We cast the pre-training objective on visual documents as textlayout sequence (i.e., $\{ \mathbf { T } ; \mathbf { L } \} )$ generation conditioned on document images $\mathbf { V }$ . The document texts $\mathbf { T }$ are represented as word-token sequences. The layouts L, following prior studies (Xu et al., 2020, 2021), can be represented by location bounding boxes of words. Instead of generating two separate sequences of $\mathbf { T }$ and $\mathbf { L }$ , ViTLP generates the texts with corresponding layout locations in a sequence of interleaved text-layout tokens, which facilitates compact multimodal interaction between texts and layouts. For the $i$ -th word of a document, its textlayout tokens $\{ \mathbf { T } ; \mathbf { L } \} _ { i }$ are represented as
+
+$$
+\{ \mathbf { T } ; \mathbf { L } \} _ { i } = \big \{ \{ w \} _ { i } , \{ z _ { x 1 } , z _ { y 1 } , z _ { x 2 } , z _ { y 2 } \} _ { i } \big \} ,
+$$
+
+where $\{ { \pmb w } \} _ { i }$ denotes the BPE tokens (Radford et al., 2018) of the $i$ -th word, $\{ z _ { x 1 } , z _ { y 1 } , z _ { x 2 } , z _ { y 2 } \} _ { i } \in \mathbb { Z } _ { + } ^ { 4 }$ are the corresponding left-top and right-bottom bounding box coordinates. Given a document with $N$ words, the objective is to maximize the likelihood function $\log p ( \mathbf { T } ; \mathbf { L } | \mathbf { V } )$ which can be decomposed as autoregressive text and layout modeling:
+
+$$
+\begin{array} { r } { \log p ( \mathbf { T } ; \mathbf { L } | \mathbf { V } ) = \displaystyle \sum _ { i = 1 } ^ { N } \underbrace { \log p ( \mathbf { T } _ { i } | \mathbf { T } _ { < i } , \mathbf { L } _ { < i } , \mathbf { V } ) } _ { \mathrm { T e x t - m o d e l i n g ~ T e r m } } } \\ { + \underbrace { \log p ( \mathbf { L } _ { i } | \mathbf { T } _ { \leq i } , \mathbf { L } _ { < i } , \mathbf { V } ) } _ { \mathrm { L a y o u t - m o d e l i n g ~ T e r m } } . \quad ( } \end{array}
+$$
+
+Note that Eq. (2) shares similar ideas with Chen et al. (2021), where word and bounding box generation can be formulated as language modeling on a unified text-layout sequence. However, it is in fact nontrivial to generate sequences as in Eq. (1), because real-world documents commonly contain intensive texts, generating each word followed by four coordinate tokens in a long flattened sequence is especially token-inefficient. This would bring prohibitive computational and space overhead2 to the Transformer-based text-layout decoder.
+
+![](images/fc75cfba6ca2ffb4c1e1bae99ca1696db113e59f633760675ebe3db9f2e40739.jpg)  
+Figure 2: Overview of the ViTLP architecture. ViTLP is a generative pre-training model that performs autoregressive text-layout modeling conditioned on visual document inputs. ViTLP adopts hierarchical decoder heads to generate target text-layout sequences in a global-to-local manner. The segment mode tokens $\in \{ \left[ \mathrm { B O S } \right] , \left[ \mathrm { C O N T } \right] \}$ prompt the beginning and continuous modes of generation, respectively.
+
+# 2.2 Model Architecture
+
+The architecture of ViTLP is shown in Figure 2. ViTLP employs an encoder-decoder framework to encode document images $\mathbf { V }$ and generate target text-layout sequences $\{ \mathbf { T } ; \mathbf { L } \}$ . Specifically, given an input document image $\mathbf { V }$ , ViTLP employs a vision transformer (ViT) (Dosovitskiy et al., 2020) to learn visual representations $\mathbf { H } ^ { V } \in \mathbb { R } ^ { | V | \times d }$ , where $| V |$ is the ViT patch number and $d$ is the hidden size. The decoder receives the visual representations $\mathbf { H } ^ { V }$ and generates the unified text-layout sequence $\{ \mathbf { T } ; \mathbf { L } \}$ . To address the token-inefficiency issue discussed in Sec. 2.1, we design the globalto-local text-layout generation process as follows.
+
+# 2.2.1 Global Text-Layout Modeling
+
+Instead of directly generating the text-layout sequence as in Eq. (1), we first replace the bounding box coordinates $\{ z _ { x 1 } , z _ { y 1 } , z _ { x 2 } , z _ { y 2 } \}$ with a generic layout location token $\hat { w } = [ \mathrm { L O C } ]$ . This integrates the mixed text-layout sequence $\{ \mathbf { T } ; \mathbf { L } \}$ to unified language modeling. Given the original vocabulary $\nu$ , the global text-layout sequence $\hat { \mathbf { T } }$ derives from the augmented vocabulary $\hat { \mathcal { V } } = \mathcal { V } \cup \ [ \mathrm { L O C } ]$ . The layout token embeddings $\mathrm { E } _ { [ \mathrm { L O C ] } }$ are computed as
+
+$$
+{ \bf E } _ { \left[ \mathrm { L O C } \right] } = \left[ { \mathrm E } _ { x } { \left( z _ { x 1 } \right) } , { \mathrm E } _ { y } { \left( z _ { y 1 } \right) } , { \mathrm E } _ { x } { \left( z _ { x 2 } \right) } , { \mathrm E } _ { y } { \left( z _ { y 2 } \right) } \right] ,
+$$
+
+where $\operatorname { E } _ { x } ( \cdot ) \in \mathbb { R } ^ { \frac { d } { 4 } }$ and $\operatorname { E } _ { y } ( \cdot ) \in \mathbb { R } ^ { \frac { d } { 4 } }$ denote the $\mathbf { X }$ - and y-axis spatial embeddings. Besides, the word tokens are embedded by $\operatorname { E } _ { w } ( \cdot ) \in \mathbb { R } ^ { d }$ . Given a document of $N$ words and the corresponding bounding boxes, the text-layout input embeddings are represented as ${ \bf H } ^ { T L } = \{ { \mathrm { E } } _ { w } , { \bar { \mathrm { E } } } _ { [ \bar { \mathrm { L O C } } ] } \} \in \mathbb { R } ^ { | \bar { \hat { \bf T } } | \times d }$ .
+
+The ViTLP text-layout decoder performs multimodal interaction among visual, textual, and layout information via the Transformer cross-attention
+
+$$
+\mathbf { H } ^ { V T L } = \mathrm { T r a n s f o r m e r - D e c o d e r } ( \mathbf { H } ^ { V } , \mathbf { H } ^ { T L } ) .
+$$
+
+For the $i$ -th target token $\hat { \mathbf { T } } _ { i }$ , the multimodal decoder output $\mathbf { H } _ { i } ^ { V T L }$ is fed to a linear language modeling (LM) head with the softmax function to compute the conditional generative probability
+
+$$
+p ( \hat { \mathbf { T } } _ { i } | \hat { \mathbf { T } } _ { < i } , \mathbf { V } ) = \operatorname { S o f t m a x } \big ( \operatorname { L i n e a r } ( \mathbf { H } _ { i } ^ { V T L } ) \big ) .
+$$
+
+With the generic layout token [LOC] incorporated, the text-modeling term in Eq. (2) is expressed as
+
+$$
+\mathcal { L } _ { \mathrm { g l o b a l - t e x t } } = - \frac { 1 } { | \hat { \mathbf { T } } | } \sum _ { i = 1 } ^ { | \hat { \mathbf { T } } | } \log p ( \hat { \mathbf { T } } _ { i } | \hat { \mathbf { T } } _ { < i } , \mathbf { V } ) .
+$$
+
+# 2.2.2 Local Layout Modeling
+
+Local layout modeling aims to generate specific layout locations for each generic layout token [LOC].
+
+To capture the spatial relation among coordinates, we employ a simple sequential MLP layout head3 to decode the short sequence of four layout coordinate tokens from the last hidden state of [LOC]. For notation simplicity, we denote $\{ \mathbf { L } _ { i , j } \} _ { j = 1 } ^ { 4 } =$ $\{ z _ { x 1 } , z _ { y 1 } , z _ { x 2 } , z _ { y 2 } \} _ { i }$ as the corresponding layout coordinates of the [LOC] token at the $i$ -th position, and its generative probability is modeled as $p ( \mathbf { L } _ { i , j } | \mathbf { L } _ { i , < j } , \hat { \mathbf { T } } _ { \leq i } , \mathbf { V } ) = \operatorname { S o f t m a x } \big ( \operatorname { M L P } ( \mathbf { H } _ { i , < j } ) \big ) ,$ where $\mathbf { H } _ { i , 0 } = \mathbf { H } _ { i } ^ { V T L }$ is selected from the learned multimodal representations where $\hat { \mathbf { T } } _ { i } = ~ [ \mathrm { L O C } ]$ . Here, we denote the index set of [LOC] tokens as $\mathcal { S } _ { L } = \left\{ i : \hat { \mathbf { T } } _ { i } = \left. [ \mathbb { L } 0 \mathrm { C } ] \right| i = 1 , 2 , . . . , | \hat { \mathbf { T } } | \right\}$ . The layout-modeling term in Eq. (2) is expressed as
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { \mathrm { l o c a l - l a y o u t } } = - \log p ( \mathbf { L } _ { i } | \hat { \mathbf { T } } _ { \leq i } , \mathbf { L } _ { < i } , \mathbf { V } ) } \\ & { = - \frac { 1 } { 4 | S _ { L } | } \displaystyle \sum _ { i \in S _ { L } } \sum _ { j = 1 } ^ { 4 } \log p ( \mathbf { L } _ { i , j } | \mathbf { L } _ { i , < j } , \hat { \mathbf { T } } _ { \leq i } , \mathbf { V } ) . } \end{array}
+$$
+
+In summary, with the global and local text-layout modeling in a hierarchy, the original pre-training objective in Eq. (2) evolves to
+
+$$
+\mathcal { L } = \mathcal { L } _ { \mathrm { g l o b a l - t e x t } } + \mathcal { L } _ { \mathrm { l o c a l - l a y o u t } } .
+$$
+
+The global-to-local generation process aims to be effective and efficient for text-layout modeling. On effectiveness, the interleaved text-layout sequence modeling enables compact interaction between text and layout inputs, which can effectively fuse the information of text and layout modalities. On efficiency, suppose that the average BPE tokens of a document word are $| w |$ , and the compression ratio of the text-layout sequence is $\frac { | \boldsymbol { w } | + 1 } { | \boldsymbol { w } | + 4 }$ , i.e., four coordinate tokens are compressed to one. In our experiment datasets, the compression ratio is 0.48.
+
+# 2.3 Multi-segment Pre-training Scheme
+
+Documents are usually intensive in text and layout, and it would be computationally intractable to fit the entire sequence into ViTLP. To process documents with arbitrary length, we propose a multisegment pre-training scheme that divides the long sequence into multiple segments for generation. Since a document image already contains all necessary information of the text and layout, long document modeling is feasible based on the visual representations and generation history context.
+
+Given the maximum sequence length of the decoder as $M$ , we first divide the text-layout sequence into $K$ segmented sequences $\{ \mathbf { S } _ { i } \} _ { i = 1 } ^ { K }$ . The beginning segment $\mathbf { S } _ { 1 }$ contains $M$ tokens to be generated, and the continuous segment $\mathbf { S } _ { i > 1 }$ contains $\alpha _ { p } \cdot M$ prefix tokens and $( 1 - \alpha _ { p } ) \cdot M$ tokens to be generated. Here, $\alpha _ { p }$ is the pre-defined prefix ratio. The overall generation process comprises beginning and continuous modes.
+
+Beginning Generation Mode. In this mode, we prepend a special mode token [BOS] to the beginning sequence $\mathbf { S } _ { 1 }$ . The model then follows the objective in Eq. (5) to generate the first $M$ tokens.
+
+Continuous Generation Mode. For the continuous segments $\mathbf { S } _ { i > 1 }$ , we prepend a special mode token [CONT] to the input sequence. $| { P } | = \alpha _ { p } { \cdot } M$ prefix tokens are prepended to the input sequence. The $| P |$ prefix tokens of segmented sequence $\mathbf { S } _ { i }$ come from the $| P |$ suffix tokens of the previous segmented sequence $\mathbf { S } _ { i - 1 }$ . These prefix tokens serve as the prompt of generation history context guiding the decoder to generate subsequent tokens from arbitrary locations of a document. The special token [EOS] is appended to the last segmented sequence $\mathbf { S } _ { K }$ to signal the end of generation.
+
+Segmentation in Pre-training and Fine-tuning. In pre-training, the segmented sequences of a long document are randomly scattered into different data batches. In this way, ViTLP learns to model the complete textual and layout information of a document, conditioned on different prefix history-token contexts. As for fine-tuning (and inference), ViTLP can also apply the multi-segment scheme to process long input/output text sequences, which is consistent with the pre-training phase. For example, OCR and information extraction on long document texts can be processed segment by segment.
+
+# 2.4 Applications of ViTLP
+
+# 2.4.1 OCR Text Localization and Recognition
+
+Text localization and recognition (Li et al., 2023) are two fundamental functions of OCR engines. As ViTLP is pre-trained to generate texts and layouts (i.e., 2D bounding boxes) sequences from document images, it naturally supports text localization and recognition by means of zero-shot or finetuning on downstream OCR datasets. Thus ViTLP can be applied as an alternative to OCR engines.
+
+# 2.4.2 Downstream VDU Tasks
+
+Information Extraction. The information extraction task is formulated as sequential tokenlabeling on the target texts given document images. Following BART (Lewis et al., 2020), we feed the decoder’s final hidden states of a target word (with the layout token [LOC]) to a linear classifier which outputs the token-level classification label.
+
+Document Classification. Given an input document image to the encoder, we feed a special prefix token [DOC_CLS] as input to the decoder, and output the corresponding label token.
+
+Document Question Answering. Different from discriminative VDU models that perform extractive QA on pre-processed OCR results, ViTLP directly generates answers with the referring 2D layout locations in images. In this way, ViTLP can also provide explainable grounding region of interest (ROI) to justify the generated answers.
+
+# 3 Experiments
+
+# 3.1 Experiment Configurations
+
+Implementation Details We implement ViTLP with a 12-layer ViT (Dosovitskiy et al., 2020) image encoder and a 6-layer text-layout decoder. The Transformer hidden size is $d = 7 6 8$ with 12 attention heads. In both pre-training and fine-tuning phases, the input image resolution is $1 6 0 0 \times 1 2 8 0$ with $3 2 \times 3 2$ ViT patch size, and the decoder segmented sequence length is $M = 1 0 2 4$ . Following LayoutLMv2 (Xu et al., 2021), the layout location coordinates are normalized into discrete bins of $\left[ 0 , 1 0 0 0 \right]$ , resulting that the vocabulary size of RNN layout-head is 1001. The multi-segment prefix ratio is set as $\alpha _ { p } = 0 . 2 5$ . We use the AdamW optimizer (Loshchilov and Hutter, 2019) to train ViTLP in 200K steps, with the batch size of 384 and initial learning rate of $2 e$ -4.
+
+Pre-training Data Following previous work (Xu et al., 2021), we use IIT-CDIP Test Collection 1.0 (Lewis et al., 2006) containing 11M document images for pre-training. Five supplement datasets with $2 . 5 \mathrm { M }$ document images are also added to augment the diversity of pre-training data, including IAM (Marti and Bunke, 2002), SciTSR (Chi et al., 2019), PubLayNet (Zhong et al., 2019), DocBank (Li et al., 2020), and SynthDog (Kim et al., 2022). We use our internal OCR tool4 to extract words with location coordinates from the IIT-CDIP and PubLayNet images. Words with locations are provided in IAM, SciTSR, and DocBank. Following
+
+Kim et al. (2022), we generate 2M synthetic document images with text and layout annotations. Detailed data statistics are provided in Appendix A.1.
+
+Evaluation Tasks We evaluate ViTLP on two types of document processing tasks: 1) perception tasks of document OCR and 2) cognition tasks of visual document understanding (VDU).
+
+For OCR evaluation, we conduct two benchmark OCR sub-tasks, i.e., document text localization and recognition. We evaluate model performance on SROIE competition5 Task #1 for text localization and Task #2 for text recognition. The text localization task is evaluated by DetEval protocol (Wolf and Jolion, 2006) which calculates the precision, recall, and F1 based on the area of overlapping regions between model predictions and groundtruth text coordinates. Given the model output and ground-truth words, the text recognition task evaluates the word-level precision, recall, and F1 based on exact word matches.
+
+For VDU evaluation, we conduct three document understanding tasks. 1) Form Understanding. Given a document image and its word entities, it is a sequential labeling task to predict the BIO tags for each entity. We use FUNSD (Jaume et al., 2019) which contains 199 scanned forms, and the entities are labeled in four categories: Header, Question, Answer, and Other. FUNSD is divided into 149 images for training and 50 for testing. We report entity-level F1 as the evaluation score. 2) Receipt Understanding. We use CORD (Park et al., 2019) containing 800 training and 100 testing images of real-world receipts. The receipt entities are labeled in 30 categories. We use entity-level F1 for evaluation. 3) Document Classification. We conduct experiments on the RVL-CDIP dataset (Harley et al., 2015) containing 400K scanned documents in 16 classes. We adopt classification accuracy as the evaluation metric. For the sequence labeling tasks on FUNSD and CORD, we perform multi-segment fine-tuning on those samples whose entity-word sequences exceed the maximum decoder sequence length. This differs from previous work that truncates the input sequences into certain tokens, e.g., 512 tokens in LayoutLM (Xu et al., 2020).
+
+Furthermore, we perform generative question answering on the DocVQA dataset (Mathew et al., 2021). DocVQA consists of over 12K document images with 50K question-answer pairs. Since the answer word locations are not provided in the training data, we use an OCR tool to obtain the layout coordinates of document texts and then derive the answer word locations with heuristic text matching. In this way, we can feed the answers with layout coordinates to ViTLP for DocVQA fine-tuning.
+
+# 3.2 OCR Evaluation Results
+
+We compare ViTLP with representative OCR baselines on the SROIE 2019 benchmark (Huang et al., 2019). The text localization baselines include CRAFT (Baek et al., 2019), YOLO-v3 (Redmon and Farhadi, 2018), and CTPN (Tian et al., 2016). The text recognition baselines include BiLSTM-CTC (Graves et al., 2006), BiLSTMResNet, CTPN-CRNN (Shi et al., 2015), UNetCRNN (Ronneberger et al., 2015), and TrOCR (Li et al., 2023). Unlike the conventional OCR models that first perform text localization and then use the localized text-regions to perform text recognition, ViTLP unifies text localization and recognition in a Transformer decoder and does not need ground truth text-region inputs in the recognition task.
+
+Table 1 shows the OCR evaluation performance. ViTLP outperforms most baseline methods on both localization and recognition tasks. ViTLP underperforms TrOCR, given that TrOCR is a strong pre-trained model for two-stage OCR text recognition, while ViTLP performs text localization and recognition in one stage. Note that the SROIE training samples are few, i.e., only 626 images, and the provided input text coordinates are at textlinelevel, which are different from our word-level pretraining input format and thus render it challenging to fine-tune our model. Nonetheless, ViTLP can still achieve comparable performance by finetuning on the limited samples without extra data augmentation (Li et al., 2023) and adapting to output the textline coordinates that have never met in the pre-training phase. We also provide demonstrative zero-shot OCR samples in Appendix C.
+
+# 3.3 VDU Evaluation Results
+
+We compare ViTLP with competitive pre-training baselines including i) general pre-trained language model, RoBERTa (Liu et al., 2019), ii) disriminative VDU models: LayoutLM (Xu et al., 2020), SelfDoc (Li et al., 2021b), TILT (Powalski et al., 2021), LayoutLMv2 (Xu et al., 2021), and iii) generative VDU model, DONUT (Kim et al., 2022). Table 2 shows the VDU evaluation performance.
+
+Table 1: OCR text localization and recognition results on SORIE 2019. †TrOCR uses the ground-truth cropped image regions as inputs, whereas ViTLP performs text localization and recognition in one stage. All scores are reported in percentage.   
+
+<table><tr><td colspan="4">Text Localization</td></tr><tr><td>Method</td><td>Area-Precision</td><td>Area-Recall</td><td>Area-F1</td></tr><tr><td>CRAFT</td><td>62.73</td><td>59.94</td><td>61.31</td></tr><tr><td>YOLO-v3</td><td>77.29</td><td>79.32</td><td>78.29</td></tr><tr><td>CTPN</td><td>81.14</td><td>87.23</td><td>84.07</td></tr><tr><td>ViTLP</td><td>90.12</td><td>90.77</td><td>90.45</td></tr><tr><td colspan="4"></td></tr><tr><td>Method</td><td>Word-Precision</td><td>TextRecognition Word-Recall</td><td>Word-F1</td></tr><tr><td>CTPN-CRNN</td><td>35.75</td><td>63.89</td><td>45.85</td></tr><tr><td>BiLSTM-ResNet</td><td>74.05</td><td>77.81</td><td>75.88</td></tr><tr><td>BiLSTM-CTC</td><td>83.38</td><td>87.37</td><td>85.33</td></tr><tr><td>UNet-CRNN</td><td>85.77</td><td>86.48</td><td>86.12</td></tr><tr><td>TrOCR†</td><td>95.89</td><td>95.74</td><td>95.82</td></tr><tr><td>ViTLP</td><td>90.61</td><td>90.72</td><td>90.66</td></tr></table>
+
+Information Extraction In terms of the entitylevel F1 scores on FUNSD, our model surpasses all baseline models by a clear margin, e.g., $+ 3 . 9 1$ points over LayoutLMv2. This result indicates that ViTLP can develop a thorough understanding of form structures in document images. In terms of CORD, ViTLP also achieves the highest F1 score. Note that ViTLP significantly outperforms DONUT, proving that explicit layout modeling is also necessary like language modeling for generative VDU models. Concretely, for the receipt images of CORD, entities with the same semantic label <menu.price> are always located in the same rightmost column of the receipt, sharing closed horizontal coordinates. Modeling the layout information could help generative VDU models better understand the document structures.
+
+Document Classification As shown in Table 2, ViTLP achieves competitive performance on classification accuracy. We find that the performance of ViTLP, TILT, LayoutLMv2, and DONUT are nearly matched. This may be because document classification is a coarse-grained task, where the vision modality plays the most important role compared to texts and layouts. Regarding similar classification accuracy, OCR-free models, i.e., ViTLP and DONUT, are more flexible because no preprocessed OCR results are needed.
+
+# 3.4 Further Discussions
+
+# 3.4.1 Ablation Study
+
+We conduct a comprehensive ablation study to investigate the effect of hierarchical text-layout modeling and multi-segment pre-training scheme. We compare ViTLP with three variants: i) pre-training with the language modeling objective only, without the layout modeling objective; ii) truncating the long input sequences, without the multi-segment pre-training scheme; iii) generating four location tokens for each word in a flatten long sequence, without hierarchical text-layout modeling.
+
+Table 2: Multiple VDU evaluation tasks on form understanding (FUNSD), receipt understanding (CORD), and document classification (RVL-CDIP). “Maximum Doc-Length” stands for the maximum input tokens of a document that the pre-trained models can process.   
+
+<table><tr><td>Pre-trained Model</td><td># Param.</td><td>Maximum Doc-Length</td><td>FUNSD (F1↑)</td><td>CORD (F1↑)</td><td>RVL-CDIP (Acc↑)</td></tr><tr><td>RoBERTabase (Liu et al., 2019)</td><td>125M</td><td>512</td><td>66.48</td><td>93.54</td><td>90.06</td></tr><tr><td>LayoutLMbase (Xu et al., 2020)</td><td>160M</td><td>512</td><td>79.27</td><td>1</td><td>94.42</td></tr><tr><td>SelfDoc (Li et al., 2021b)</td><td>137M</td><td>1024</td><td>83.36</td><td>1</td><td>93.81</td></tr><tr><td>TILTbase (Powalski et al., 2021)</td><td>230M</td><td>512</td><td>1</td><td>95.11</td><td>95.25</td></tr><tr><td>LayoutLMv2base (Xu et al., 2021)</td><td>200M</td><td>512</td><td>82.76</td><td>94.95</td><td>95.25</td></tr><tr><td>DONUT (Kim et al., 2022)</td><td>259M</td><td>1536</td><td>1</td><td>84.10</td><td>95.30</td></tr><tr><td>ViTLP</td><td>253M</td><td>unlimited</td><td>86.67</td><td>95.16</td><td>95.28</td></tr></table>
+
+Table 3: Ablation variant performance on the information extraction tasks.   
+
+<table><tr><td>Ablation Model</td><td>FUNSD (F1)</td><td>CORD (F1)</td></tr><tr><td>w/o layout modeling</td><td>81.33</td><td>91.73</td></tr><tr><td>w/o multi-segment training</td><td>85.86</td><td>94.96</td></tr><tr><td>w/o hierarchical modeling</td><td>85.58</td><td>95.04</td></tr><tr><td>ViTLP</td><td>86.87</td><td>95.16</td></tr></table>
+
+Table 3 displays the ablation performance. We observe that removing layout modeling leads to a substantial performance drop, i.e., 5.54 and 3.37 F1 drops on FUNSD and CORD. The results suggest that generative pre-training on the layout modality can enhance the document understanding ability of VDU models. Besides, truncating long document inputs without the multi-segment pre-training strategy leads to suboptimal performance. We believe that the multi-segment pre-training scheme enables ViTLP to model complete text and layout tokens of the pre-training corpora, which benefits pre-trained model performance. We can see that hierarchical text-layout modeling also brings incremental performance gain, implying its effectiveness in textlayout information fusion.
+
+# 3.4.2 Generative Document VQA
+
+Performance and Analysis Table 4 shows the DocVQA performance. ViTLP underperforms
+
+Table 4: Average Normalized Levenshtein Similarity (ANLS) between the generated answers and groundtruth answers. † denotes our re-implementation with the official codebase and our pre-training data.   
+
+<table><tr><td>Generative Pre-trained Model</td><td>ANLS ↑</td></tr><tr><td>DONUT (Kim et al., 2022)</td><td>67.5</td></tr><tr><td>DONUTt (official code + our OCR data)</td><td>62.2</td></tr><tr><td>ViTLP</td><td>61.3</td></tr></table>
+
+DONUT by 6.2 ANLS scores. We analyze that the performance gap probably dues to two reasons: (1) As compared in Davis et al. (2022), pre-training data quality would significantly affect the downstream DocVQA task performance. This is because, different from discriminative models that perform extractive QA upon given OCR of document spans, generative models themselves need to develop strong language modeling abilities to organize accurate output answer words, which requires clean and diverse data for LM pre-training. We have benchmarked the official DONUT codebase with our pre-training data. The ANLS score frustratingly decreases from 67.5 to 62.2, indicating the pre-training data quality needs to be improved. (2) On the other hand, the DocVQA questions may not appear in document images. To keep consistent with the pre-training sequence format, we append the dummy coordinates $\{ 0 , 0 , 0 , 0 \}$ after each question word. This input text-location discrepancy between pre-training and DocVQA fine-tuning may also affect the VQA task performance.
+
+Explainable DocVQA with ROI Localization. Albeit ViTLP underperforms the strong baseline DONUT, it acquires localization capacity to ground the answers, which is unprecedented to prior work. Benefiting from the layout localization capacity learned in the pre-training phase, ViTLP can output the region of interest (ROI) with the generated answers, as presented in Figure 3. The visualized ROI localization can help users justify the model output answers, making the generative questionanswering process more explainable and reliable. A potential application is to use ViTLP as a semiautomatic annotator to construct large-scale document answering datasets, where human annotators can easily justify the annotations according to the visualized ROI outputs.
+
+![](images/d5b191d3a70f221c76499476a34b28389b5f56e18e90f0e1cbf34b6f589699ba.jpg)  
+Figure 3: Visualization on ViTLP output answers for DocVQA. The ViTLP output sequences consist of answer words (in blue) and corresponding location coordinates (in red). For comprehensive visualization, we draw the region of interest (ROI) referring to the output location coordinates on the image.
+
+# 4 Related Work
+
+Visual document processing with multimodal pretraining has been widely studied recently. Depending on the pre-processing of documents, existing VDU works can be generally divided into two strands of research as listed below.
+
+OCR-based Methods. Most existing VDU efforts (Xu et al., 2020, 2021; Huang et al., 2022; Peng et al., 2022; Li et al., 2021b; Bai et al., 2023) adopt OCR tools to localize and recognize document layouts and texts, and then feed them to the multimodal pre-trained models (Xu et al., 2021; Peng et al., 2022; Appalaraju et al., 2021; Li et al., 2021a). These methods usually involve multiple multimodal pre-training objectives over the vision, text, and layout. For instance, document words (Xu et al., 2020, 2021), textlines (Bai et al., 2023), regions (Li et al., 2021b; Wang et al., 2022b) are rich in document structure information to align visual features with text embeddings. Though promising, such pipeline suffers from heavy OCR preprocessing. Moreover, OCR errors can easily lead to incorrect results for downstream tasks such as document question answering (Kim et al., 2022).
+
+OCR-free Methods. There are few recent studies (Kim et al., 2022; Davis et al., 2022; Lee et al., 2023b) that jointly consider text recognition and understanding without OCR tools. For instance, Kim et al. (2022) takes document images as input to the network without prerequisite OCRs and conducts visual language pre-training. Lee et al. (2023b) further improves the pre-training objectives over various collected visual document corpus.
+
+Our research lies within the OCR-free branch. Different from existing works, we first study to learning both document text and layout information from the input images. Our empirical results also verify that layout information not only enhances the learned representations for downstream VDU tasks but also makes the generation interpretable.
+
+# 5 Conclusion
+
+The paper proposes the visually guided generative text-layout pre-training (ViTLP) approach to boost the performance of visual document processing. The ViTLP model optimizes hierarchical language and layout modeling objectives to generate a mixed target sequence of texts and layouts. ViTLP can also serve as an OCR tool to locate and recognize texts of document images. Besides, the proposed approach can handle documents with arbitrary lengths via multi-segment generative pretraining. Experiments show that ViTLP achieves superior performance compared to existing counterparts on various VDU tasks.
+
+# Limitations
+
+Our community has entered the era of large language models (LLMs) (OpenAI, 2023; Chen et al., 2023). However, regarding the model size, ViTLP is still a rather small-scale pre-trained model, which limits its potential to become an interactive and generalized document AI assistant. In future work, we plan to explore two paths: 1) scaling up ViTLP with more parameters and training data, extending it to a more powerful foundation document model; 2) integrating ViTLP’s text-layout aligned image encoder with open-source LLMs and instruction tuning (Zhu et al., 2023; Liu et al., 2023) to construct an interactive document AI system.
+
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+
+Table 5: Pre-training dataset statistics.   
+
+<table><tr><td>Dataset</td><td>Size</td><td>Proportion</td><td>Document Type</td></tr><tr><td>IIT-CDIP</td><td>10,816,672</td><td>81.89%</td><td>Scanned Document</td></tr><tr><td>SynthDog</td><td>2,000,000</td><td>15.14%</td><td>Synthetic Document</td></tr><tr><td>PublayNet</td><td>261,076</td><td>1.98%</td><td>Scientific Paper</td></tr><tr><td>DocBank</td><td>125,815</td><td>0.95%</td><td>Arxiv Paper</td></tr><tr><td>IAM</td><td>1,198</td><td>0.01%</td><td>Hand Written</td></tr><tr><td>SciTSR</td><td>3,536</td><td>0.03%</td><td>Figure and Table</td></tr></table>
+
+# A Experiment Details
+
+# A.1 Pre-training Data Statistics
+
+Table 5 shows the pre-training data statistics. Following previous work, e.g., LayoutLMv2 (Xu et al., 2021) and FormNetv2 (Lee et al., 2023a), we use 11M IIT-CDIP document images as the main pretraining data. Besides, we follow Kim et al. (2022) and Davis et al. (2022) to include 2M machinerendered synthetic documents for generative pretraining. Specifically, we adapt the official SynthDog generator6 to generate synthetic document images with text and layout metadata. The other four corpora, i.e., IAM, SciTSR, PublayNet, and DocBank, account for only $\sim 3 \%$ pre-training data whereby we aim to improve the diversity of pretraining document types.
+
+The distribution of document sequence length is displayed in Figure 4. The number of text-layout sequence tokens follows a long-tailed distribution: there exist few long documents with the sequence length ranging from 1024 to 3072. This brings a trade-off to pre-training. With a relatively short sequence length (e.g., 512 tokens in LayoutLM), language modeling on long documents is incomplete, as the sequence tokens are truncated and wasted. However, with a relatively long sequence length (e.g., 3072), the GPU computation and memory overload would become prohibitive, which further forbids large batch sizes for better performance.7 The multi-segment pre-training scheme can circumvent this bitter trade-off. Notably, the multisegment processing scheme can be directly applied to long document fine-tuning (and inference). For example in the OCR and sequence labeling tasks, ViTLP also employs the multi-segment scheme to process the long documents by multiple segments with prefix context tokens.
+
+![](images/e0ed7013b03c45c714c106180fa3b772ae3250fbcc56d7b8f6dddbee48bf1f61.jpg)  
+Figure 4: Distribution of document sequence length.
+
+# B Implementation Details of Sequential Layout Head
+
+Given that multimodal interaction is learned by the stacked Transformer text-layout decoder layers, the LM and layout heads hereby function as a prober to output the next word and coordinate predictions. As introduced in $\mathrm { S e c } 2 . 2 . 2$ , the layout head predicts output probability $\mathrm { P r o b } ( \mathbf { L } _ { i , j } )$ of the four coordinates $\{ { \bf L } _ { i , j } \} _ { j = 1 } ^ { 4 } = \{ z _ { x 1 } , z _ { y 1 } , z _ { x 2 } , z _ { y 2 } \} _ { i }$ based on the $i$ -th global [LOC] token’s final hidden state $\mathbf { H } _ { i , 0 } = \mathbf { H } _ { i } ^ { V T L } \in \mathbb { R } ^ { d }$ as follows.
+
+$$
+\left\{ \begin{array} { l l } { \mathbf { H } _ { i , 1 } = \mathrm { G E L U } \big ( \mathbf { W } _ { s } \mathbf { H } _ { i , 0 } \big ) } \\ { \mathbf { H } _ { i , 2 } = \mathrm { G E L U } \big ( \mathbf { W } _ { s } \mathbf { H } _ { i , 1 } + \mathrm { E } _ { x } ^ { ' } ( \mathbf { L } _ { i , 1 } ) \big ) } \\ { \mathbf { H } _ { i , 3 } = \mathrm { G E L U } \big ( \mathbf { W } _ { s } \mathbf { H } _ { i , 2 } + \mathrm { E } _ { y } ^ { ' } ( \mathbf { L } _ { i , 2 } ) \big ) } \\ { \mathbf { H } _ { i , 4 } = \mathrm { G E L U } \big ( \mathbf { W } _ { s } \mathbf { H } _ { i , 3 } + \mathrm { E } _ { x } ^ { ' } ( \mathbf { L } _ { i , 3 } ) \big ) } \end{array} \right.
+$$
+
+$$
+\mathrm { P r o b } ( \mathbf { L } _ { i , j } ) = \mathrm { S o f t m a x } \left( \mathbf { W } _ { L } \mathbf { H } _ { i , j } \right)
+$$
+
+The coordinate tokens are quantized into a discrete range of [0, 1000], making the layout-token vocabulary size of $| L | = 1 0 0 1$ . The layout head parameters are lightweight including a hidden matrix $\mathbf { W } _ { s } \in \mathbb { R } ^ { d \times d }$ , two embeddings $\bar { \mathrm { E } } _ { x } ^ { \prime } ( \cdot ) \in \mathbb { R } ^ { d }$ and $\operatorname { E } _ { y } ^ { ' } ( \cdot ) \in \mathbb { R } ^ { d }$ , and a linear projection $\dot { \mathbf { W } } _ { L } \in \mathbb { R } ^ { | L | \times d }$ We use the same GELU activation (Hendrycks and Gimpel, 2016) as in the Transformer layers. The layout head works similar to a vanilla RNN, as each coordinate decoding step also considers the information of previous coordinates. Compared to naively using four independent linear heads, the sequential layout head can capture the spatial relation among the output coordinates (e.g., $x _ { 1 } < x _ { 2 }$ and $y _ { 1 } < y _ { 2 } ,$ ), bootstrapping more accurate coordinate prediction in inference.
+
+# C Qualitative Cases of ViTLP Document OCR Functionality
+
+As shown in Figure 5 to 7, ViTLP demonstrates its functionality on zero-shot document OCR. ViTLP outputs the word-level OCR results including texts and bounding boxes.
+
+![](images/c585d282d12b05a7f87b3cef1402be24957f5f314678aae38600288fc23dab6e.jpg)
+
+# Introduction
+
+![](images/0c513f772ee20c1e61c8145519105c4b5a756a85ecca0b92fe882a91bacc6399.jpg)  
+Figure 5: ViTLP OCR results on a paper. For comprehensive visualization, we draw the output texts (in blue) and bounding boxes (in red) according to the OCR outputs.
+
+"Attention"，[386,47，456,77]   
+"Is"，[465,47,493,77]   
+"ALl"，[502,47,550,77]   
+"You"，[559,47,619,77]   
+“Need”，[628,47,708,77]   
+"Ashish"，[139,174，201,191]   
+“Vaswani\*”，[206，174,290,191]   
+"Noam"，[365，174，420，191]   
+"Shazeer\*"，[425,174,505,191]   
+“Niki”，[578,174,617,191]   
+"Parmar\*",[622,174,701,191]   
+“Jakob"，[761,174,815,191]   
+"Uszk0reit\*"，[821，174，915，191]   
+"Google"，[152,191,215,207]   
+"Brain",[221,191,268,207]   
+"Gogle"，[373,191,436,207]   
+"Brain",[441,191,489,207]   
+"Google"，[562,191,625,207]   
+“Gesearch",[6,0，11,78,207]   
+"Resgle"，[76,191,,23,207]   
+“Research",[829,191,907,207]   
+"avaswani@google.com",[104，207,317,223]   
+"noam@qooqLe.com"，[346，287，515,223]   
+"nikipl@google.com”，[545，207,725，223]   
+"usg0oge.e,[755,27，9,2]   
+“Lilon"，[162,246,212,264]   
+  
+"Aidan"，[386,246,442,264]   
+"N."，[447,246,468,264]   
+"Gomez\*"，[473,246，546，264]   
+"\u2020"，[552,244,560,260]   
+"Lukasz"，[696,246,762,264]   
+"Kaiser\*"，[767,246，836,264]   
+"Google"，[142,263,285，280]   
+"Research"，[210，263，288,288]   
+"University”，377,263,466,280]   
+"of”，[471,263,489,280]   
+"Toronto"，[494,263,561,280]   
+"Google"，[704,263,767,280]   
+"Brain",[772,263,820,280]   
+"lLion@google.com"，[126，279,305，295]   
+"aidan@cs.toronto.edu”，[357,279,588,295]   
+"Lukaszkaiser@google.com”，[634，279,890,295]   
+"IlLia”，[429,318,465，336]   
+"Polosukhin\*”，[471,318,581,336]   
+"u2021，[587,316,595，333]   
+"illia.polosukhin@gmail.com"，[364，336，653,352]   
+“Abstract”，[460,391,555,413]   
+"The"，[162,426，196,443]   
+"dominant"，[202，426，284,443]   
+"sequence”，[290，426，369,443]   
+"transduction"，[375，426，482,443]   
+"dels”，[488,426,，551,443]   
+"are"，[556，426，583,443]   
+"based"、[589，426，639，443]   
+"on"，[644，426，666，443]   
+"complex"，[672，426,746,443]   
+"recurrent"，[752,426,830，44]   
+"r"，[836，426,855，443]   
+"convolutional",[162,443,280,460]   
+"neural"，[286，443,341,460]   
+"networks"，[347，443，426,460]   
+“that"，[432，443，465,460]   
+"include”，[471,443,535,460]   
+"an”，[541,443,562,466]   
+"encoder"，[568,443，637,460]   
+"and"，[643，443,675,468]   
+"an,[1,44,,91,44]   
+"decoder.",[697,443,770,460]   
+“The"，[779,443,813,460]   
+"best"，[819,443,854,460]   
+"performing"，[162,459,260,475]   
+"odels，[266，459,，30，475]   
+"also"，[336，459,372,475]   
+"aonnect",[378,459,46，475]   
+"the"，[453,459,480,475]   
+"encoder”,[486,，459,55,475]   
+"and"，[562,459,594,475]   
+"decoder”，[600,459,669,475]   
+"through”，[676,459,744,475]   
+“"an",[75,,459,771,475]   
+"attention"，[778,459,854,475]   
+"mechanism."，[162,475,267,491]   
+"We"，[279，475,388，491]   
+"propose"，[315,475，384,491]   
+"a"，[391,475,401,491]   
+"new”，[408,475,444,491]   
+"simple",[451,475,509,491]   
+"network"，[516,475,587,491]   
+"architecture,"，[594，475,703,491]   
+"the"，[710,475，737,491]   
+"Transformer,"，[744,475,856,491]   
+"based”,[162,491,210,507]   
+"solely"，[215,491,264,507]   
+"on"，[269,491,290,507]   
+"attention"，[295,491,367,507]   
+"mechaniss,，[372，41,40,5]   
+“dispensing"，[484,491,574,507]   
+"with"，[579,491,615,507]   
+"recurrence"，[62e,491,788,507]   
+"and"，[713,491,743,507]   
+"convolutions",[747,491,854,507]   
+"entirely."，[162,506,233,523]   
+"Experiments"，[245,506,355,523]   
+"on"，[363，506，385，523]   
+"two"，[392,586,424,523]   
+"machine"，[431,506,505,523]   
+"translation",[512,506,604,523]   
+"tasks"，[611,506,655,523]   
+"ow"，[662,506,708,5]   
+"these”，[715,506,760,523]   
+"models，[767,586,830,523]   
+"to"，[837,506,854,523]   
+"be"，[162,522,182,538]   
+"superior"，[188,522,258,538]   
+"in”，[264,522,280,,58]   
+"quality"，[286,522,346,538]   
+"while”，[351,522,398，538]   
+"being"，[404,522,451,538]   
+"e，[47,,   
+"parallelizable"，[506,522,622,538]   
+"and"，[628，522,659,538]   
+"requiring”，[664，522,743,538]   
+"significantly”，[748,522,854,538]   
+"less"，[162,538,195,554]   
+"time"，[201,538,240,554]   
+"to"，[246,538,263,554]   
+"train."，[269,538,315,554]   
+"0ur”，[325,538,359,554]   
+"odel"，[365,538,419,554]   
+"achieves"，[426，538，500,554]   
+“28.4"，[586，538,544,554]   
+"BLEU”，[550,538，607,554]   
+"on"，[613,538,635,554]   
+“the，[641,538,668,554]   
+"WMT"，[675，538,727,554]   
+"2014"，[733,538,778,554]   
+"English-",[784,538,856,554]   
+"to-German”，[162,554,256，570]   
+"translation"，[263，554，355,570]   
+"task,"，[362,554,402,570]   
+"inproving”，[409,554,500,570]   
+"over"，[587,554,545,570]   
+"the"，[552,554,578,570]   
+"existing"，[585,554,653,570]   
+"best"，[660,554,696,570]   
+"results,"，[703,554,764,570]   
+"including”，[772,554,854,570]   
+"ensembles,"，[162,570,254,586]   
+"by”，[259,570，28,586   
+"over",[285,570,322,586]   
+"2"，[326，570，336,586]   
+"BLEU."，[341，570，401，586]   
+"On"，[406,570,431,586]   
+"the"，[436,570,462,586]   
+"WMT",[467,570,517,586]   
+"2014"，[522,570,563,586]   
+"English-to-French"，[568,57e,720,586]   
+"translation"，[725，570，813，586]   
+"task,"，[818,570,856,586]   
+"our"，[162,586,190,602]   
+"model",[195,586,247,602]   
+"establishes",[252,586,342,602]   
+"a"，[347,586,357,602]   
+"new”，[362,586,396,602]   
+"single-model",[401,586,510,602]   
+"state−of-the-art"，[515,，586,639,602]   
+"BLEU"，[644，586，698，602]   
+"score"，[703，586，747，602]   
+"of",[752,586,770,602]   
+"41."，[775,586,81,62]   
+"after",[816,586,855,602]   
+"training”，[162,601,230，61]   
+"for"， [236,601,261,617]   
+"3.5"，[267,601,293,617]   
+"days",[299,601,339,617]   
+"on"，[345,601,366,617]   
+"eight",[372,601,415,617]   
+"Gus,",[421,601,478,617]   
+"a"，[484,601,494，617]   
+"snall",[500,601,546,617]   
+"fraction",[552,601,620,617]   
+"of"，[626,601,643,617]   
+"the"，[649,601,676,617]   
+"training"，[682,601,749,617]   
+"costs"，[755,601, 798,617]   
+"of"，[804,601,822,617]   
+"the"，[827,601,854,617]   
+"best"，[162,617,197,633]   
+"models"，[202,617,263,633]   
+"from"，[269,617,310,633]   
+"the"，[315,617,341,633]   
+"literature.",[346,617,428,633]   
+"1”，[85,657,98,679]   
+"Introduction"，[124,657,263,679]   
+"Recurrent"，[85,694,171,711]   
+"neural"，[177,694,231,711]   
+"networks,"，[237,694,321,711]   
+"leng"，[327,694,66,7]   
+"short-term"，[372,694，462,711]
+
+"n "a "e "de "r
+
+"memory"，[468,694，540，711] "[12]"，[546,694,581,711] "and"，[587,694,618,711] "gated"，[624，694，671，711] "recurrent”,[677,694,755,711] “[7]"，[761,694,785,711] "neural",[791,694,845，711] "networks"，[851,694,930,711] "in"，[85,710,102,727] "particular,",[108,710,195,727] "have"，[200,710,240,727] "been"，[246，710,287,727] "firmly"，[292,710,345,727] "established"，[350,710,446,727] "as”，[452,710,469,727] "state"，[475,710,514,727] "of"，[520,710,538，727] "the"，[543,710,570,727] "art"，[575,710,598,727] "approaches"，[603,710,700,727] "in”，[706,710,722,727] "sequence"，[728,710,807,727] "modeling"，[812,710,894,727] "and"，[899,718,930,727] "transduction”，[85,726，193,743] "problems"，[199,726,280,743] "such"，[286，726,326,743] "as"，[332,726,351,743] "language"，[357，726，435,743] "modeling"，[441,726，523,743] "and"，[529,726，561,743] "machine”，[567,726,,640,743] "translation"，[647,726，738,743] "[29,"，[745，726，778,743] "2,"，[785，726,81,74] "5]."，[808,726,830,743] "Numerous",[848，726，930,743] "efforts"，[85,742,139,759] "have"，[144，742,183,759] "since"，[187,742,230,759] "continued”，[234,742,317,759] "to"，[321,742,337,759] "push",[342,742,381,759] "the"，[386,742,411,759] "boundaries"，[416,742,587,759] "of"，[512,742,529,759] "recurrent"，[534,742,609,759] "language"，[614，742,689,759] "models"，[693,742,753,759] "and"，[758，742，788，759] "encoder-decoder".[793,742,930,759] "architectures",[85,758,194,775] "[31,"，[199,758,233,775] "21,"，[238,758,265,775] "13]."，[271,758,303,775] "\*Equal"，[111,781,164,797] "contribution.",[169,781,265,797] “Listing"，[27e,781,324,797] "order"，[329，781，368，797] "is"，[373,781,385,797] "random."，[390,781,452,797] "Jakob"，[459,781,502,797] "proposed"，[507,781,576,797] "replacing”，[580,781,650,797] "RNs",[655,781,702,797] "with"，[707,781,740,797] "self-attention",[745,781,844,797] "and"，[849,781，876，797] "started"，[88,781,931,797] "the"，[85,796,109,812] "effort",[114,796,156,812] "to”，[161,796,176,812] "evaluate”，[181,796,244,812] "this",[249,796,277,812] "idea."，[281.796.318，812] "Ashish,"，[324,796,383,812] "with"，[388，796,422,812] "Illia,"，[426,796,462,812] "designed",[467,796,536,812] "and”，[540,796，568,812] "implemented",[573,796,674,812] the"，[678，796,702,812] "first",[707,796,737,812] "Transformer",[742,796,838,812] "models"，[842,796,898,812] "and"，[902,796，930,812] "has"，[85,810,111,826] "been"，[115,810,151,826] "crucially"，[155,810,220,826] "nvled，224,1，2,6] "in"，[293,810,308,826] "every"，[312,810,353,826] "aspect"，[357,810,484，826] "of"，[408,810,424,826] "this",[428,810,455,826] "work.",[459,810,583,826] "Noam"，[588，810,554,826] "proposed"，[559,810,628,826] "scaled"，[632,810,679,826] "dot-product"，[683，810，770，826]
+
+"dot-product", [683, 81e, 770,826] "attention,"，[775,810,845,826] "multi-head”，[849,810,930,826] "attention"，[85，825，153，841] "and"，[158,825，186,841] "the”，[191,825,214,841] "parameter–free"，[219,825,333,841] "position"，[338,825,400,841] "representation"，[405,825，514,841] "and”，[519,825，547,841] "became"，[552,825,610,841] "the”，[615，825,639,841] "other"，[644,825,684,841] "person"，[689,825,740,841] "involved", [745, 825, 811, 841] "in"，[816,825,831,841] "nearly", [836,825,884,841] "every"，[889,825,930,841] "etail,"，[85,839,133,855] “Niki"，[138,839,173,855] "designed,",[178,839,250,855] "inplemented,"，[255，839，359,855] "tuned"，[364,839,487,855] "and"，[412,839,439,855] "evaluated", [444, 839, 516,855] "countless"，[521,839,592,855] 5 "in"，[714,839,729,855] "our"，[734,839,759,855] "original",[764,839,823,855] "codebase"，[828,839,898,855] "and"，[983,839,930,855] "tensor2tensor."，[85,854,192,869] “Lilon"，[197,854,239,869] "also”，[243,854，274,,869] "experimented"，[278,854，382,869] "with"，[386,854,420,869] "novel",[425,854，467,869] "model",[471,854,519,869] "variants,",[524,854,588,869] "was"，[593,854，622,869] "responsible”，[627,854,713,869] "rerp,718，854，70,9] "our"，[745,854,770,869] "initial",[775,854,819,869] "codebase,"，[824,854，898,869] "and"，[903,854,930,869] "efficient"，[85,869,147,884] "inference"，[151,869,221,884] "and"，[225，869,253,884] "visualizations,",[257,869,365,884] “Lukasz"，[370，869,424,884] "and"，[429,869,456,884] “Aidan”,[461,869,586,884] "spent"，[511,869,551,884] "countess",555,86,65，84] “ong,[629,869,663,884 "days"，[667,869,701,884] "designing"，[706,869,779,884] "various”，[783,869,838,884] "parts"，[842,869,879,884] "of"，[883,869,899,884] "and",[903,869,930,884] "inplementing”，[85,883,190,898] "tensertesr,"，[4,83,299,898] "replacing"，[383,883,374,898] "p，378,3，43,] "earlier"，[407,883,455,898] "codebase,"，[460,883,533,898] "greatly"，[537,883,590,898] "improving"，[594,883,672,898] "results",[676,883,725,898] "and"，[730，883，757，898] "massively"，[761,883,836,898] "accelerating"，[840,883,930,898] "our"，[85,898,111,913] "research.",[116,898,184,913] "\u2828Work"，[112,911,162,928] "performed",， [167, 913, 247, 928] "while"，[252,913,294,928] "at"，[299,913,313,928] "Google"，[317,913,374,928] "Brain.",[378,913,426,928] \u2021Wr，[112,97,162,44] "performed",[167,929,247,944] "while"，[252,929,294,944] "at"，[299,929,313,944] "Google"，[317,929,374,944] "Research."，[378,929,453,944] "3”，[86,967,94,983] "1st”,[98,967,118,983] "Conference",[123,967,210,983] "on"，[215,967,235,983] “Neural",[239,967,292,983] "Information",[296,967,388,983] "Processing"，[392,967,475,983] "Systems",[480, 967, 543,983] "(NIPS"，[548,967,596,983] "2017),"，[601,967,650,983] "Long"，[655,967,695,983] “Beach,"，[700,967,753,983] "CA,"，[758,967,789,983] "USA.",[794,967,836,983]
+
+![](images/46945bd1f66a190198f5e7e3ab0b8c3b544616c685093bfb69337f197497fba5.jpg)
+
+![](images/c74fe916201edc3c2275720271f610373e0e000468e8d16494929d5c540cfed7.jpg)  
+Figure 7: ViTLP OCR results on a webpage. For comprehensive visualization, we draw the output texts (in blue) and bounding boxes (in red) according to the beneath ViTLP OCR outputs.

parse/dev/VwRFJi9crEH/VwRFJi9crEH_content_list.json

@@ -0,0 +1,1061 @@
+[
+    {
+        "type": "text",
+        "text": "Personalized Subgraph Federated Learning ",
+        "text_level": 1,
+        "bbox": [
+            232,
+            122,
+            764,
+            147
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
+            200,
+            580,
+            256
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            292,
+            535,
+            309
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 In real-world scenarios, subgraphs of a larger global graph may be distributed   \n2 across multiple devices or institutions, and only locally accessible due to privacy re  \n3 strictions, although there may be links between them. Recently proposed subgraph   \n4 Federated Learning (FL) methods deal with those missing links across private local   \n5 subgraphs while distributively training Graph Neural Networks (GNNs) on them.   \n6 However, they have overlooked the inevitable heterogeneity among subgraphs,   \n7 caused by subgraphs comprising different parts of a global graph. For example,   \n8 a subgraph may belong to one of the communities within the larger global graph.   \n9 A naive subgraph FL in such a case will collapse incompatible knowledge from   \n10 local GNN models trained on heterogeneous graph distributions. To overcome   \n11 such a limitation, we introduce a new subgraph FL problem, personalized subgraph   \n12 FL, which focuses on the joint improvement of the interrelated local GNN models   \n13 rather than learning a single global GNN model, and propose a novel framework,   \n14 FEDerated Personalized sUBgraph learning (FED-PUB), to tackle it. A crucial   \n15 challenge in personalized subgraph FL is that the server does not know which   \n16 subgraph each client has. FED-PUB thus utilizes functional embeddings of the   \n17 local GNNs using random graphs as inputs to compute similarities between them,   \n18 and use them to perform weighted averaging for server-side aggregation. Further,   \n19 it learns a personalized sparse mask at each client to select and update only the   \n20 subgraph-relevant subset of the aggregated parameters. We validate FED-PUB for   \n21 its subgraph FL performance on six datasets, considering both non-overlapping   \n22 and overlapping subgraphs, on which ours largely outperforms relevant baselines. ",
+        "bbox": [
+            147,
+            324,
+            766,
+            628
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "23 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            655,
+            312,
+            671
+        ],
+        "page_idx": 0
+    },
+    {
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+        "text": "24 A graph, which defines the relationships among instances, can model a wide range of structured data   \n25 including social [7], co-purchasing [23], and collaboration networks [36]. Most of the previous works   \n26 on graph representation learning focus on a single graph, whose nodes and edges collected from   \n27 multiple sources are stored in a central server. For instance, in a social network platform, every user,   \n28 with his/her social networks, contributes to creating a giant network consisting of all users and their   \n29 connections. However, in some practical scenarios, each user/institution collects its own private graph,   \n30 which is only locally accessible due to privacy restrictions. For instance, as described in Zhang et al.   \n31 [45], each hospital may have its own patient interaction network to track their physical contacts or   \n32 co-diagnosis of a disease, however, such a graph may not be shared with others. An obvious challenge   \n33 for such a scenario is how to deal with potentially missing edges between subgraphs [42, 45] that are   \n34 not captured by individual data owners, that may carry important information (See Figure 1 (A)).   \n35 How can we then collaboratively train, without sharing actual data, a neural network with its subgraphs   \n36 distributed across multiple participants (i.e., clients) over different devices or institutions? The most   \n37 straightforward way is to perform Federated Learning (FL) with Graph Neural Networks (GNNs). In   \n38 particular, in such an FL framework, each client will individually train a local GNN on the private   \n39 local data, while a central server aggregates the locally updated GNN weights from multiple clients   \n40 into one, and then transmits it back to the clients. Recent subgraph FL methods work in such a   \n41 manner [42, 45] while additionally tackling the problem of missing edges between subgraphs. This is   \n42 done as illustrated in Figure 1 (B), where the local subgraph is expanded either by exactly augmenting   \n43 the relevant nodes from the other subgraphs at the other clients [42], or by estimating the nodes using   \n44 the node information in the other subgraphs [45]. However, such sharing of node information may   \n45 compromise data privacy and can incur high communication costs.   \n46 Also, there exists a more important challenge that has been overlooked by the existing subgraph FL   \n47 methods. We observe that they suffer from large performance degeneration (See Figure 1 right), due   \n48 to a lack of consideration of the heterogeneity among the subgraphs, which is natural since subgraphs   \n49 comprise different parts of a global graph. Notably, there could be multiple communities within   \n50 a global graph, each of which is formed by a group of densely connected subgraphs with similar   \n51 characteristics (Figure 1 (A)). For example, some of patient networks from hospitals can be grouped   \n52 by their specialized sectors according to the disease categories, namely psychiatric or ophthalmology.   \n53 Motivated by this challenge, we introduce a novel problem of personalized subgraph FL, whose goal   \n54 is the joint improvement of interrelated local models trained on the interconnected local subgraphs,   \n55 for instance, subgraphs belonging to the same community (See Figure 1 (C)), by sharing weights   \n56 among them. However, tackling personalized subgraph FL is challenging, since we do not know   \n57 which subgraph each client has, due to their local accessibility. To resolve this issue, we use functional   \n58 embeddings of GNNs on random graphs to obtain similarity scores between two local GNNs, inspired   \n59 by a work for neural network search that effectively represents entire neural networks in the vector   \n60 space [17], and then use them to perform weighted averaging of the model weights at the server.   \n61 However, the similarity scores only tell how relevant each local model from the other clients is, but   \n62 not which of the parameters are relevant. Thus we further learn and apply personalized sparse masks   \n63 on the local GNN at each client to obtain only the subnetwork, relevant for the local subgraph. We   \n64 refer to this subgraph FL framework as FEDerated Personalized sUBgraph learning (FED-PUB).   \n65 We extensively validate our FED-PUB on six different datasets with varying numbers of clients,   \n66 under both overlapping and disjoint subgraph FL scenarios. The experimental results show that ours   \n67 significantly outperforms relevant baselines. Further analysis shows that our method can discover   \n68 community structures among subgraphs, and the subgraph-specific masking localizes the knowledge   \n69 with respect to subgraphs belonging to each community. Our main contributions are as follows: ",
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+            "Figure 1: (A) An illustration of local subgraphs distributed across multiple participants with overlapping nodes, missing edges and community structures between subgraphs. (B) Existing subgraph FL methods [42, 45] expand the local subgraphs to tackle the missing edge problem, but collapse incompatible knowledge from heterogeneous subgraphs. (C) Our personalized subgraph FL focuses on the joint improvement of local models working on interrelated subgraphs, such as ones within the same community, by selectively sharing knowledge across them. (Right:) Knowledge collapse results, where local models belonging to two small communities $\\mathbf { C o m m 1 }$ and 2) suffer from large performance degeneration by existing subgraph FL (e.g., FedGNN [42] and FedSage $^ +$ [45]). A personalized FL method, FedPer [2] also underperforms ours since it only focuses on individual model’s improvement without sharing local personalization layers between similar subgraphs. "
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+        "text": "• We introduce a novel problem of personalized subgraph FL, which aims at collaborative improvements of the related local models (e.g. subgraphs belonging to the same community), which has been relatively overlooked by previous works on graph and subgraph FL. ",
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+        "text": "• We propose a novel framework for personalized subgraph FL, which performs weighted averaging of the local model parameters based on their functional similarities obtained without accessing the data, and learns sparse masks to select only the relevant subnetworks for the given subgraphs. ",
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+        "text": "76 • We validate our personalized subgraph FL framework on six real-world datasets under two different   \n77 settings, demonstrating its effectiveness over existing subgraph FL baselines. ",
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+        "text": "78 2 Related Work ",
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+        "text": "Graph Neural Networks Graph representation learning with Graph Neural Networks (GNNs) [10, 48, 43, 18, 3], which aims to learn the representations of the nodes, the edges, and the entire graph, is an extensively studied topic. Most existing GNNs under the message passing scheme [8] iteratively represent a node by aggregating features from its neighboring nodes as well as itself. For example, Graph Convolutional Network (GCN) [22] approximates the spectral graph convolutions [12], yielding a mean aggregation over neighboring nodes. Similarly, for each node, GraphSAGE [11] aggregates the features from its neighbors to update the node representation. Such advances in GNNs have led to successes on node and link prediction tasks [22, 47]. However, they are not directly applicable to real-world systems with locally distributed graphs, where graphs from different sources are not shared across participants, which gives rise to federated learning approaches to train GNNs. ",
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+        "text": "Federated Learning Federated Learning (FL) [32, 41, 19, 24], aiming to learn a model by aggregating model weights trained on local data, is an essential approach for our distributed subgraph learning problem. To mention a few, FedAvg [32] locally trains a model for each client and then transmits the trained model to a server, while the server aggregates the model weights from local clients and then sends the aggregated model back to them. However, since the locally collected data from different clients may largely vary, heterogeneity is a crucial issue. To tackle this, FedProx [25] proposes the regularization term that minimizes the weight differences between local and global models, which prevents the model from diverging by overfitting to the local training data. However, when the local data is extremely heterogeneous, it is more appropriate to collaboratively train a personalized model for each client rather than learning a single global model [2, 30, 26, 46, 6]. FedPer [2] is such a personalized FL method, which shares only the base layers while having local personalized layers for each client, to keep the local knowledge. Unlike the commonly studied image and text data, graph-structured data is defined by connections between instances, and consequently introduces additional challenges: missing edges and shared nodes between private subgraphs. Note that, regarding architectures, there is literature [29, 27, 38, 49] that leverages outputs of neural networks for predicting/minimizing outputs across different client models; however, we use functional outputs of neural networks to identify interconnected subgraphs, thus ours differs from them methodologically. ",
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+        "text": "Graph Federated Learning Few recent studies propose to use the FL framework to collaboratively train GNNs without sharing graph data [13], which can be broadly classified into subgraph- and graph-level methods. Graph-level FL methods assume that different clients have completely disjoint graphs (e.g., molecular graphs), and recent works [44, 14] focus on the heterogeneity among non-IID graphs (i.e., difference in graph labels across various clients). In contrast to graph-level FL methods that have similar challenges to general FL scenarios, the subgraph-level FL problem we target has a unique graph-structural challenge, that there exist missing yet probable links between subgraphs, since a subgraph is a part of a larger global graph. To deal with such a missing link problem among subgraphs, existing methods [42, 45] augment the nodes by requesting the node information in the other subgraphs, and then connecting the existing nodes with the augmented ones. However, this scheme could compromise data privacy constraints, and also increases communication overhead across clients. Unlike existing subgraph FL that focuses on the problem of missing links, our subgraph FL method tackles the problem with a completely different perspective, focusing on discovering subgraph communities [35, 9, 34], which are groups of densely connected subgraphs. ",
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+        "type": "text",
+        "text": "3 Personalized Subgraph Federated Learning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We provide the general descriptions of Graph Neural Networks (GNNs) and Federated Learning (FL), and then define our novel problem of personalized subgraph FL lying at the intersection of them. ",
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+        "text": "Graph Neural Networks A graph $\\mathcal { G } = ( \\nu , \\mathcal { E } )$ consists of a set of nodes $\\nu$ with $n$ elements and a set of edges $\\mathcal { E }$ with $m$ elements along with its node feature matrix $\\ b { X } \\in \\mathbb { R } ^ { n \\times d }$ , where each column represents a $d$ -dimensional feature for each node. Further, $( u , v ) \\in \\mathcal { E }$ represents an edge from a node $u$ to a node $v$ . Then, given the graph, Graph Neural Networks (GNNs) [8, 10] generally represent each node based on features from its neighbors as well as itself, formally defined as follows: ",
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+        "text": "$$\n\\pmb { H } _ { v } ^ { ( l + 1 ) } = \\mathrm { U P D A T E } ^ { ( l ) } \\left( \\pmb { H } _ { v } ^ { ( l ) } , \\mathrm { A G G R E G A T E } ^ { ( l ) } \\left( \\left\\{ \\pmb { H } _ { u } ^ { ( l ) } : \\forall u \\in \\pmb { \\mathcal { N } } ( v ) \\right\\} \\right) \\right) ,\n$$",
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+        "text": "128 where $\\pmb { H } _ { v } ^ { ( l ) }$ is the feature matrix for node $v$ at $l$ -th layer, $\\mathcal { N } ( v )$ denotes a set of adjacent nodes of   \n129 node $v$ : $\\mathcal { N } ( v ) = \\{ u \\in \\mathcal { V } \\mid ( u , v ) \\in \\mathcal { E } \\}$ , AGGREGATE aggregates the features of $v$ ’s neighbors, and   \n130 UPDATE updates the node $v$ ’s representation given its previous representation and the aggregated   \n131 representations from the neighbors. $H ^ { ( 1 ) }$ is initialized as input node features $\\boldsymbol { X }$ .   \n132 Federated Learning The objective of Federated Learning (FL) is to collaboratively train a model   \n133 with local private data.accessible from others: $\\mathcal { D } _ { k } = \\{ \\boldsymbol { X } _ { i } , \\boldsymbol { y } _ { i } \\} _ { i = 1 } ^ { N _ { k } }$ have , whe $K$ $X _ { i }$ rticipants with locis a data instance, $\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { y } } _ { i } }$ collected data that is notis its corresponding class   \n135 label, and is the number of data instances at $k$ -th client. Then, for decentralized training with   \n136 local data, a popular $\\mathrm { F L }$ algorithm, FedAvg [32], works as the following three steps: ",
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+        "text": "1. (Initialization) At the initial communication round $r = 0$ , the central server first selects $K$ clients that are available for training, and initializes their local model parameters as the global parameter $\\bar { \\pmb \\theta }$ , represented as follows: $\\bar { \\pmb { \\theta } _ { k } ^ { ( 0 ) } }  \\bar { \\pmb { \\theta } } ^ { ( 0 ) } \\forall k$ , where $\\theta _ { k } ^ { ( 0 ) }$ is the parameters for $k$ -th client. ",
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+        "text": "2. (Local Updates) Each active local model performs training on private local data $\\mathcal { D } _ { k }$ to minimize the task loss $\\mathcal { L } ( \\mathcal { D } _ { k } ; \\boldsymbol { \\theta } _ { k } ^ { ( 0 ) } )$ , consequently updating the parameters $\\pmb { \\theta } _ { k } ^ { ( 1 ) }  \\pmb { \\theta } _ { k } ^ { ( 0 ) } - \\eta \\nabla \\mathcal { L }$ . ",
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+        "text": "3. (Global Aggregation) After local training, the server aggregates the locally learned knowledge with respect to the number of training instances, i.e., $\\begin{array} { r } { \\bar { \\pmb { \\theta } } ^ { ( 1 ) }  \\frac { N _ { k } } { N } \\sum _ { k = 1 } ^ { K } \\pmb { \\theta } _ { k } ^ { ( 1 ) } } \\end{array}$ with $\\begin{array} { r } { N = \\sum _ { k } N _ { k } } \\end{array}$ and distributes the updated global parameters $\\bar { \\theta } ^ { ( 1 ) }$ to the local clients selected at the next round. ",
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+        "type": "text",
+        "text": "This $\\mathrm { F L }$ algorithm iterates between Step 2 and 3 until reaching the final round $R$ ",
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+        "text": "146 Challenges in Subgraph FL While the above FL works well on image and text data, due to the   \n147 unique structure of graphs, there exist nontrivial challenges for applying this FL scheme to graph  \n148 structured data. In particular, unlike with an image domain where each instance $X _ { i }$ is independent   \n149 from the other images, each node $v$ in a graph is always influenced by its relationships to adjacent   \n150 nodes $\\mathcal { N } ( v )$ . Moreover, a local graph $G _ { i }$ could be a subgraph of a larger global graph $\\mathcal { G }$ : $G _ { i } \\subseteq { \\mathcal { G } }$ . In   \n151 such a case, there could be missing edges between local subgraphs in two different clients: $( u , v )$   \n152 with $u \\in \\mathcal V _ { i }$ and $v \\in \\mathcal { V } _ { j }$ for clients $i$ and $j$ , respectively. To tackle this missing edge problem, few   \n153 existing subgraph FL methods [42, 45] estimate the nodes from a local subgraph $G _ { k }$ based on the   \n154 node information from the subgraphs at other clients $G _ { i } \\forall i \\neq k$ , and then extend the existing nodes   \n155 with the estimated ones. However, this augmentation scheme incurs high communication costs as it   \n156 requires sharing node information across clients, which may also violate data privacy constraints [1].   \n157 Yet, there exists another issue that makes subgraph $\\mathrm { F L }$ even more challenging. Assume that we have   \n158 a global graph consisting of all the subgraphs. Then, there exists communities of such subgraphs [35,   \n159 9, 34], where subgraphs within the same community are more densely connected to each other   \n160 than subgraphs outside the community. Formally, a global graph $\\mathcal { G }$ can be decomposed into $T$   \n161 different communities: $C _ { i } \\subseteq \\mathcal { G } \\forall i = 1 , . . . , T$ , where $i$ -th community $C _ { i } = ( \\mathcal { V } _ { i } , \\mathcal { E } _ { i } )$ consists of   \n162 densely connected nodes. Then, in a subgraph FL problem, each client has a local subgraph $G _ { j }$ that   \n163 belongs to at least a single community1: $\\begin{array} { r } { C _ { i } = \\bigcup _ { j = 1 } ^ { J } G _ { j } } \\end{array}$ . Note that, based on the theory of network   \n164 homophily [33], such connected subgraphs within the same community have similar properties, while   \n165 subgraphs in two opposite communities are not. Such distributional heterogeneity across communities   \n166 may lead a naive FL algorithm to collapse incompatible knowledge across different communities.   \n167 Personalized Subgraph FL To prevent the above knowledge collapse issue, we aim to personalize   \n168 the subgraph FL algorithm by performing weighted averaging of the local model parameters at   \n169 the server, rather than learning a single set of global parameters; thereby capturing the subgraph   \n170 community structures among interrelated subgraphs. Formally, the objective of existing subgraph   \n171 FL [42, 45, 28] is as follows: minθ $\\textstyle \\sum _ { G _ { i } \\subseteq \\mathcal { G } } { \\mathcal { L } } ( G _ { i } ; \\theta )$ . However, a major drawback of such a scheme   \n172 is that, since the subgraphs in two different communities with sparse connections are extremely   \n173 heterogeneous due to network homophily [33], finding a universal set of parameters (i.e., $\\pmb \\theta$ ) that work   \n174 on all tasks will result in finding a suboptimal parameter set. To address such limitations of existing   \n175 subgraph FL, we formulate a novel problem of personalized subgraph FL, formalized as follows: ",
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+        "text": "$$\n\\operatorname* { m i n } _ { ( \\theta _ { i } ) } \\sum _ { G _ { i } \\subseteq \\mathcal { G } } \\mathcal { L } ( G _ { i } ; \\theta _ { i } ) , \\theta _ { i } \\gets \\sum _ { j = 1 } ^ { K } \\alpha _ { i j } \\theta _ { j } \\mathrm { ~ w i t h ~ } \\alpha _ { i k } \\gg \\alpha _ { i l } \\mathrm { ~ f o r ~ } G _ { k } \\subseteq C \\mathrm { ~ a n d ~ } G _ { l } \\not \\subseteq C ,\n$$",
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+        "text": "176 where $\\theta _ { i }$ is the weight for subgraph $G _ { i }$ belonging to community $C$ , and $\\alpha _ { i j }$ is the coefficient for   \n177 weight aggregation which we will specify in Section 4.1. This formulation promotes the collaborative   \n178 learning across multiple local models that work on the interrelated subgraphs that belong to the same   \n179 community, by assigning larger weights on them. ",
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+        "image_caption": [
+            "Figure 2: (A) Two communities, each of which consists of one/two subgraphs. (B) Client Similarity Matching: we forward randomly generated graphs to models $f ( \\bar { G } ; \\pmb { \\theta } )$ , and then obtain the functional embeddings of them $\\tilde { h }$ , which are then used to estimate the similarities between subgraphs. The similarities are used in the weight aggregation, resulting in the personalized model weights $\\bar { \\pmb \\theta }$ . (C) Weight Masking: the transmitted weights from the server to clients $\\bar { \\pmb \\theta }$ are masked and shifted by local masks for localization to the local subgraph distribution. "
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+        "type": "text",
+        "text": "4 Federated Personalized Subgraph Learning (FED-PUB) Framework ",
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+        "text": "Our goal of personalized subgraph FL is to jointly improve the local models trained on the interconnected local subgraphs forming the community structures. To this end, we propose to compute subgraph similarity scores for detecting communities, and to mask subgraph-irrelevant weights. ",
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+        "text": "4.1 Subgraph Similarity Estimation for Detecting Subgraph Community ",
+        "text_level": 1,
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+        "text": "We aim to reflect the community structure consisting of a group of densely connected subgraphs, by sharing more weights among subgraphs in the same community, as formalized in equation 2. Due to network homophily where similar instances in the graph are more associated with each other [33], the subgraphs within the same community should have similar properties. Therefore, if one can measure the subgraph similarities, we can group the similar ones into the community. However, measuring the similarity between local subgraphs is challenging since we do not know which subgraph each client has due to local accessibility. How can we then compute subgraph similarities, without accessing them? To this end, we aim to approximate the subgraph similarity at local clients using auxiliary information obtained from the local GNN models that work on the subgraphs. ",
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+        "type": "text",
+        "text": "Subgraph Similarity Estimation with Model Parameters For measuring the similarity between subgraphs at each client, without accessing them, we may use the model parameters as proxies, as follows: $S ( i , j ) = ( \\pmb { \\theta } _ { i } \\cdot \\pmb { \\theta } _ { j } ) / ( \\lVert \\pmb { \\theta } _ { i } \\rVert \\lVert \\pmb { \\theta } _ { j } \\rVert )$ , where $\\pmb \\theta$ is a flattened parameter into the vector, and $S$ is a similarity measure. This may sound reasonable since the GNN model trained on the subgraph will embed its knowledge into its parameters. However, this scheme has a notable drawback that similarity measured in the high-dimensional parameter space is not meaningful due to the curse of dimensionality [4], and that the cost of calculating the similarity between parameters grows rapidly as the model size increases (See Figure 3). ",
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+        "image_caption": [
+            "Figure 3: Effectiveness (top) and efficiencies (bottom) of different similarity measurements. "
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+        "text": "205 Subgraph Similarity Estimation with Functional Embedding To tackle the limitations of using   \n206 parameter distance, we propose to measure the functional similarity of neural networks by feeding the   \n207 same input to every local client and then calculating the similarities using their outputs, inspired by a   \n208 work for neural network search [17]. The main intuition is that we can consider the transformation   \n209 defined with a neural network as a function, and we measure the functional similarity of two networks   \n210 by the distance of their outputs for the same input. However, unlike the previous work [17] that   \n211 tackles image classification, which uses Gaussian noises as inputs, we use random graphs as inputs   \n212 as we work with GNNs. Formally, let $\\tilde { G } = ( \\tilde { \\mathcal { V } } , \\tilde { \\mathcal { E } } )$ be a random community graph obtained from a   \n213 stochastic block model [15], where subgraphs within the community have more edges between them   \n214 than edges across the communities. Further, $\\tilde { \\nu }$ is randomly initialized from the normal distribution.   \n215 Then, the similarity between two functions defined by GNNs at clients $i$ and $j$ is defined as follows: ",
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+    {
+        "type": "equation",
+        "img_path": "images/9e4923195d4eb69bd01e7013fde1fee11d2ca7986b3b605e8046072b613100f7.jpg",
+        "text": "$$\nS ( i , j ) = \\frac { \\tilde { h } _ { i } \\cdot \\tilde { h } _ { j } } { \\| \\tilde { h } _ { i } \\| \\| \\tilde { h } _ { j } \\| } , \\quad \\tilde { h } _ { i } = \\mathtt { A V G } ( f ( \\tilde { G } ; \\theta _ { i } ) ) \\mathrm { ~ a n d ~ } \\tilde { h } _ { j } = \\mathtt { A V G } ( f ( \\tilde { G } ; \\theta _ { j } ) ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where 216 $\\tilde { h }$ is the averaged output of all node embeddings for input $\\tilde { G }$ with AVG operation to reduce the 217 dimensionality of the output from $n \\times d$ to $d$ , for $n$ nodes with $d$ -dimensional node features. ",
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+        "type": "text",
+        "text": "218 Personalized Weight Aggregation based on Subgraph Similarity With equation 3, the remaining   \n219 step is then to share the model weights between models working on similar subgraphs belonging to   \n220 the same community. However, entirely ignoring the model parameters from different communities   \n221 may result in exploiting only the local objective while ignoring globally useful weights, which may   \n222 result in performance degeneration. Therefore, we perform weighted averaging of all the local models   \n223 from the other clients based on their functional (subgraph) similarities, as follows (Figure 2 (B)): ",
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+    {
+        "type": "equation",
+        "img_path": "images/2e2cb81754bff352846cf26b0668c050d072ee5a47304ba4a7c68e640050cb60.jpg",
+        "text": "$$\n\\bar { \\theta } _ { i }  \\sum _ { j } \\alpha _ { i j } \\cdot \\theta _ { j } , \\quad \\alpha _ { i j } = \\frac { \\exp ( \\tau \\cdot S ( i , j ) ) } { \\sum _ { k } \\exp ( \\tau \\cdot S ( i , k ) ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "224 where $\\alpha _ { i j }$ is a normalized similarity between clients $i$ and $j$ , and $\\tau$ is a hyperparameter for scaling   \n225 the unnormalized similarity score. Note that increasing the value of $\\tau$ (e.g., 10) will result in model   \n226 averaging done almost exclusively among subgraphs detected as belonging to the same community. ",
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+        "type": "text",
+        "text": "This personalized scheme handles two challenges in subgraph FL. First, in contrast to the global weight aggregation scheme which easily collapses the knowledge from heterogeneous communities into a single model, our subgraph $\\mathrm { F L }$ allows the models belonging to different communities to obtain model weights that are beneficial for each community. Also, the missing edges between subgraphs that have been explicitly handled by previous works [42, 45] could be also implicitly considered by assigning larger weights to models within the same community (See Figure 10). This also enhances data privacy while minimizing the communication costs between probably linked subgraphs. ",
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+    {
+        "type": "text",
+        "text": "4.2 Adaptive Weight Masking for Selecting Subgraph-Relevant Parameters ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "With the previous similarity matching scheme, we can effectively group GNN models that belong to the same community, thus preventing the collapsing of irrelevant knowledge from other communities. However, the scalar weighting scheme only considers how much each local model from other clients is relevant for the subgraph task, but not which parameters are relevant. Thus we propose a scheme to select only the relevant parameters from the aggregated model weights transmitted from the server. ",
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+        "type": "text",
+        "text": "Personalized Parameter Masking We perform selective training and updating of the aggregated parameters by modulating and shifting them, using sparse local masks. Formally, let $\\pmb { \\mu } _ { k }$ be a local mask for a client $k$ . Then, our local model weight is obtained by modulating the weights from the server, as follows: $\\mathbf { \\theta } _ { k } = \\bar { \\theta } _ { \\underline { { k } } } \\odot \\mu _ { k }$ , where $\\odot$ is an element-wise multiplication operation between the globally given weight $\\bar { \\pmb { \\theta } } _ { k }$ and the local mask $\\pmb { \\mu } _ { k }$ . Note that the local mask is a free variable and is not shared across clients. Also, we initialize $\\pmb { \\mu } _ { k }$ as ones, in order to start training with the globally initialized model parameters without modification. We then further promote sparsity on the mask, which brings two key advantages. First, we can transmit only the partial parameters, that have not been sparsified at the client to the server rather than sending all parameters, thus reducing the communication costs. Moreover, if local masks are sufficiently sparse, the local models can be trained faster, given that zero-skipping operations are supported (Figure 2 (C)). To take these benefits in sparsity, we use $L _ { 1 }$ regularizer on $\\pmb { \\mu } _ { k }$ when performing local optimization, as shown in equation 5. ",
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+    {
+        "type": "text",
+        "text": "Preventing Local Divergence with Proximal Term As masks are trained only with limited local data without parameter sharing, they may be easily overfitted to the training instances in each client. To alleviate this issue, we adopt the proximal term proposed in Li et al. [25] that regularizes the locally updated models $\\pmb { \\theta } _ { k }$ to be closer to the globally given model $\\bar { \\theta } _ { k }$ , therefore, preventing the model from extremely drifting to the local training distribution. To sum up, at $k$ -th client, our objective function including sparsity and proximal terms with $L _ { 1 }$ and $L _ { 2 }$ losses is denoted as follows: ",
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+        "type": "equation",
+        "img_path": "images/c6195bad9517f55c6796a35b1a15db6e10bdaae2d942fde15ea5ae1570796634.jpg",
+        "text": "$$\n\\begin{array} { r } { \\operatorname* { m i n } _ { ( \\theta _ { k } , \\mu _ { k } ) } \\mathcal { L } ( \\mathcal { D } _ { k } ; \\theta _ { k } , \\mu _ { k } ) + \\lambda _ { 1 } \\| \\pmb { \\mu } _ { k } \\| _ { 1 } + \\lambda _ { 2 } \\| \\pmb { \\theta } _ { k } - \\bar { \\pmb { \\theta } } _ { k } \\| _ { 2 } ^ { 2 } , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { L }$ is the conventional cross-entropy loss function, and $\\lambda _ { 1 }$ and $\\lambda _ { 2 }$ are scaling hyper-parameters. ",
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+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We now experimentally validate our FED-PUB on six different datasets under both the overlapping and disjoint subgraph scenarios with varying client numbers, with node classification tasks. ",
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+    {
+        "type": "text",
+        "text": "5.1 Experimental Setups ",
+        "text_level": 1,
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+        "text": "Datasets Following the setup from Zhang et al. [45], we construct the distributed subgraphs from the benchmark dataset by dividing it into the number of participants, each of which has a subgraph that is a part of an original graph. Specifically, we use six datasets: Cora, CiteSeer, Pubmed and ogbn-arxiv for citation graphs [39, 16]; Computer and Photo for product graphs [31, 40]. We then divide the original graph into multiple subgraphs using the METIS graph partitioning algorithm [20]. Note that, unlike the Louvain algorithm [5] presented in Zhang et al. [45] that requires to further merge partitioned subgraphs into particular numbers of subgraphs since it cannot specify the number of subsets (i.e., clients for FL), the METIS algorithm can specify the number of subsets, thus making more reasonable experimental settings in subgraph FL (See Section C.2 of the supplementary file). For the non-overlapping scenario where there are no duplicate nodes between subgraphs, we use the output from the METIS as it provides the non-overlapping partitions. Meanwhile, for the overlapping scenario where nodes are duplicated among subgraphs, we randomly sample the subgraphs multiple times from the partitioned graph. For more details, please see Section B of the supplementary file. ",
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+    {
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+        "img_path": "images/e263c7ef0b664c9ea8fadbe4b7bff82da8185b938cb81675d896cfeaecca53c1.jpg",
+        "image_caption": [
+            "Figure 4: Convergence plots for the overlapping node scenario. We visualize the test accuracy curves for all six datasets corresponding to Table 1, over 100 communication rounds with 10 clients. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Baselines 1) FedAvg [32] and 2) FedProx [25]: The most popular FL baselines. 3) FedPer [2]: A personalized FL baseline without sharing personalized layers. 4) FedGNN [42] and 5) FedSage+ [45]: Subgraph FL baselines which we mainly target. 6) GCFL [44]: A graph FL baseline which learns completely disjoint graphs as in clustered FL [37], adopted for subgraph FL. 7) Local: A baseline without sharing weights with other clients. 8) FED-PUB: Our personalized subgraph FL including subgraph similarity matching and weight masking. See Section B of the supplementary file for details. ",
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+        "type": "text",
+        "text": "Implementation Details We set the GCN [22] with two layers as the base GNN for all models. We perform federated learning over 100 communication rounds for Cora, CiteSeer and Pubmed datasets, while 200 rounds for Computer, Photo and arxiv datasets, considering the size of datasets. The local training epoch is selected in the range of $\\{ 1 , 2 , 3 \\}$ depending on the dataset size (e.g., Computer is three while CiteSeer is one)2. We use the Adam optimizer [21] for model optimization. We then measure the node classification accuracy on subgraphs at the client-side, and then average the performance across clients. We provide further details in Section B of the supplementary file. ",
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+    {
+        "type": "text",
+        "text": "5.2 Experimental Results ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Main Results Table 1 shows the node classification performance under the overlapping subgraph scenario, in which our FED-PUB statistically $( p > 0 . 0 5 )$ significantly outperforms all the baselines. In particular, while FedGNN and FedSage+ are two pioneer works for the subgraph FL problem, they significantly underperform personalized FL methods including ours, especially at the larger number of clients. This is even surprising as they share node information between clients for handling the missing edge problem, yet we suppose such inferior performance comes from naive averaging of local weights without consideration of community structures. While personalized FL baselines including FedPer and GCFL show decent performance by alleviating the knowledge collapse between subgraphs with local parameters or clustering, they still largely underperform ours as they are not concerned with the aggregation between similar subgraphs that form a community (i.e., GCFL uses a bi-partitioning scheme where it iteratively divides a group of subgraphs within the same community ",
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+    {
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+        "img_path": "images/ac068ddc412ef8fa2aa6190e1bb3cc9e0145d2e424ef07b4d6dd899eab28f30c.jpg",
+        "image_caption": [
+            "Figure 5: Convergence plots for the non-overlapping node scenario. We visualize the test accuracy curves for all six datasets corresponding to Table 2, over 100 communication rounds with 10 clients. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "301 into two disjoint sets). We then further conduct the experiments on the disjoint subgraph scenarios   \n302 (non-overlapping scenario), where nodes are not overlapped between subgraphs, which makes the   \n303 subgraph FL problem more heterogeneous. As shown in Table 2, FED-PUB consistently outperforms   \n304 all existing baselines in such a challenging scenario, demonstrating the efficacy of ours. ",
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+    {
+        "type": "text",
+        "text": "Fast Local Convergence As shown in Figure 4 and 5, our FED-PUB converges rapidly, compared against baselines including personalized FL models. We conjecture that this is because, not only ours accurately identifies subgraphs forming the community and then shares weights largely across them for promoting the joint improvement of them, but also masking subgraph-irrelevant weights received from the server for localization to local subgraphs, demonstrated in the next two paragraphs. ",
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+    {
+        "type": "text",
+        "text": "310 Accurate Community Detection We aim to show whether FED-PUB accurately groups subgraphs   \n311 comprising a community during weight aggregation. If two different subgraphs have many missing   \n312 edges or have similar label distributions, we usually regard those two as within the same commu  \n313 nity [35, 9, 34]. Thereby, as shown in Figure 6 (a) and (b), there are four different communities by the   \n314 interval of five, and the last two communities further comprise a larger community. Then, as shown   \n315 in Figure 6 (c) and (d), FED-PUB detects obvious four communities at the first few rounds, and then   \n316 captures the larger yet somewhat less-obvious community consisting of two smaller communities.   \n317 Ablation Study To analyze the contribution of each component, we conduct the ablation studies.   \n318 As shown in Figure 7, we observe that each of our subgraph similarity matching and weight masking   \n319 significantly improves the performances from the naive FedAvg, while the performance is much   \n320 improved when using both together. However, the benefit from each component is different across   \n321 overlapping and non-overlapping scenarios. In particular, in the former scenario where a group of   \n322 highly overlapped subgraphs usually comprise a community, similarity matching for community   \n323 detection is more beneficial since capturing the community would promote the joint improvement of   \n324 subgraphs belonging to the same community. However, in the non-overlapping scenarios, subgraphs   \n325 within the same community become lesser similar, thus selectively using the aggregated model   \n326 weights from the server with personalized weight masks improves the performance a lot.   \n327 Communication Efficiency Another notable advantage of using the sparse masks is that we can   \n328 reduce the communication costs at every FL round, as well as the model size for faster training, which   \n329 we demonstrate in Table 8. In particular, Table 8 shows that existing subgraph FL methods require   \n330 more than two times larger communications costs, measured by adding both the client-to-server and   \n331 server-to-client costs, compared against the naive FedAvg, since they require to transfer additional   \n332 node information between clients for estimating the probable nodes on the subgraphs. Contrarily, our   \n333 FED-PUB has significantly lower communication costs and lower model sizes by using the sparse   \n334 masks on the model weights: transmitting and training with only the partial parameters not sparsified   \n335 at the client. Further, as shown in ours variants in Table 8, we can manage the trade-off between the   \n336 model sparsity and the performance by controlling the hyperparameter for sparsity regularization, $\\lambda _ { 1 }$   \n337 Varying Local Epochs As shown in Figure 9, when we increase the number of communication   \n338 rounds and the local steps, the model diverges to the local subgraphs (i.e., overfitting), due to the   \n339 small number of training instances and the direct connection between training and test nodes: struggle   \n340 to generalize to the test instances. However, our model with the proximal term in equation 5 alleviates   \n341 this issue, therefore, maintaining the highest local performance. Notably, the performance with five   \n342 local epochs is inferior to the performance of one epoch, which indicates that increasing the local   \n343 epochs does not always bring advantages and properly tuning them is important for subgraph FL. ",
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+        "img_path": "images/d50d23a02a73932c5018e430df053078c834c13bc32fa335fda5ed03911737d2.jpg",
+        "image_caption": [
+            "Figure 6: The heatmaps of the community structure on the overlap- Figure 7: Ablation studies of our FEDping node scenario with Cora (20 clients). Dark color indicates lots PUB on both the overlapping (a) and nonof missing edges between subgraphs (a) or high similarities in labels overlapping (b) subgraph scenarios, on (b). (c) and (d) are functional similarities captured by our FED-PUB. the Cora dataset. "
+        ],
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+    {
+        "type": "table",
+        "img_path": "images/9a4f652b64239622f4a492e75061031224b95ba614ba16f407afd1440ca51cad.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Acc.[%]</td><td>Model Size[%]</td><td>Cost[%]</td></tr><tr><td>FedAvg</td><td>76.48 ± 0.36</td><td>100.00 ±0.00</td><td>100.00 ± 0.00</td></tr><tr><td>FedGNN</td><td>70.63±0.83</td><td>100.00 ± 0.00</td><td>214.94 ± 0.00</td></tr><tr><td>FedSage+</td><td>77.52 ±0.46</td><td>100.00 ±0.00</td><td>276.84 ± 0.00</td></tr><tr><td>GCFL</td><td>78.84 ±0.26</td><td>100.00 ±0.00</td><td>100.00 ±0.00</td></tr><tr><td>Ours (λ1=9e-1)</td><td>77.36± 0.99</td><td>25.13±0.34</td><td>37.70 ± 0.56</td></tr><tr><td>Ours (λi=7e-1)</td><td>79.46 ± 0.41</td><td>42.59 ± 1.33</td><td>63.89 ± 1.99</td></tr><tr><td>Ours (λ1=5e-1)</td><td>79.89 ±0.12</td><td>57.07±0.52</td><td>85.61 ±0.78</td></tr></table>",
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+    {
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+        "image_caption": [
+            "Figure 8: Analysis on efficiencies of communication costs and model sizes. ",
+            "Figure 9: Varying the local epochs with accuracy curves. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "img_path": "images/8e16664a83a34cb86e5b057dc70e981031573146658664e253b23cde5859065f.jpg",
+        "image_caption": [
+            "Figure 10: Performance on neighboring subgraphs. "
+        ],
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+    {
+        "type": "text",
+        "text": "Handling Missing Edges To measure whether FED-PUB can handle the missing edge problem: information is not shared between two neighboring subgraphs due to the missing edges, we use the local model trained on the local subgraph for evaluating the performance on its neighboring subgraph, in which the local subgraph has the most missing edges to its neighboring subgraph. Specifically, in Figure 10, (Neighbor) denotes the subgraph performance evaluated by its neighbor model, while (Local) denotes the subgraph performance from its own local model. Then, the high performance on (Neighbor) measure means two associated subgraphs share meaningful knowledge without having explicit edges between them, thereby solving the missing edge problem. Note that, existing subgraph FL explicitly augments the nodes and edges for capturing the potential information flow over the missing edges between subgraphs, while ours implicitly shares weights a lot across similar subgraphs within the same community. Figure 10 shows that ours achieves the significantly superior performance on the neighboring subgraph problem against subgraph FL baselines, which confirms that ours has an advantage on the missing edge problem by meaningfully sharing knowledge between two subgraphs having potentially missing edges, without explicitly estimating them. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "359 We introduced a novel problem of personalized subgraph FL, which focuses on the joint improvement   \n360 of local GNNs working on interrelated subgraphs (e.g. subgraphs belonging to the same community),   \n361 by selectively utilizing knowledge from other models. The proposed personalized subgraph FL is   \n362 highly challenging due to 1) difficulty of computing similarities between local subgraphs that are   \n363 only locally accessible, and 2) knowledge collapse among local models that work on heterogeneous   \n364 subgraphs during weight aggregation. To this end, we proposed a novel personalized subgraph FL   \n365 framework, referred to as FEDerated Personalized sUBgraph learning (FED-PUB), which computes   \n366 the similarities across subgraphs using functional embeddings of their local GNNs on random graphs,   \n367 and uses them to perform a weighted average of the local models for each client. Further, we mask out   \n368 globally given weights to focus on only the relevant subnetwork for each client (or community). We   \n369 extensively validated our framework on multiple benchmark datasets with both overlapping and non  \n370 overlapping subgraphs, on which our FED-PUB significantly outperforms relevant baselines. Further   \n371 analyses show the effectiveness of the subgraph similarity matching for detecting the community   \n372 structures, as well as the weight masking for tackling the subgraph heterogeneity. We provide the   \n373 limitations and potential societal impacts of our work in Section D of the supplementary file. ",
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parse/dev/sc7bBHAmcN/sc7bBHAmcN.md

@@ -0,0 +1,329 @@
+# Understanding and Extending Subgraph GNNs by Rethinking Their Symmetries
+
+Fabrizio Frasca∗ Imperial College London & Twitter ffrasca@twitter.com
+
+Beatrice Bevilacqua∗ Purdue University bbevilac@purdue.edu
+
+Michael M. Bronstein University of Oxford & Twitter mbronstein@twitter.com
+
+Haggai Maron NVIDIA Research hmaron@nvidia.com
+
+# Abstract
+
+Subgraph GNNs are a recent class of expressive Graph Neural Networks (GNNs) which model graphs as collections of subgraphs. So far, the design space of possible Subgraph GNN architectures as well as their basic theoretical properties are still largely unexplored. In this paper, we study the most prominent form of subgraph methods, which employs node-based subgraph selection policies such as ego-networks or node marking and deletion. We address two central questions: (1) What is the upper-bound of the expressive power of these methods? and (2) What is the family of equivariant message passing layers on these sets of subgraphs?. Our first step in answering these questions is a novel symmetry analysis which shows that modelling the symmetries of node-based subgraph collections requires a significantly smaller symmetry group than the one adopted in previous works. This analysis is then used to establish a link between Subgraph GNNs and Invariant Graph Networks (IGNs). We answer the questions above by first bounding the expressive power of subgraph methods by 3-WL, and then proposing a general family of message-passing layers for subgraph methods that generalises all previous node-based Subgraph GNNs. Finally, we design a novel Subgraph GNN dubbed SUN, which theoretically unifies previous architectures while providing better empirical performance on multiple benchmarks.
+
+# 1 Introduction
+
+Message Passing Neural Networks (MPNNs) are arguably the most commonly used version of Graph Neural Networks (GNNs). The limited expressive power of MPNNs [36, 55] has led to a plethora of works aimed at designing expressive GNNs while maintaining the simplicity and scalability of MPNNs [11, 39, 49, 30]. Several recent studies have proposed a new class of such architectures [14, 59, 7, 61, 43, 42], dubbed Subgraph GNNs, which apply MPNNs to collections (‘bags’) of subgraphs extracted from the original input graph and then aggregate the resulting representations. Subgraphs are selected according to a predefined policy; in the most popular ones, each subgraph is tied to a specific node in the original graph, for example by deleting it or extracting its local ego-network. Subgraph GNNs have demonstrated outstanding empirical performance, with state-of-the-art results on popular benchmarks like the ZINC molecular property prediction [61, 7].
+
+While offering great promise, it is fair to say that we still lack a full understanding of Subgraph GNNs. Firstly, on the theoretical side, it is known that subgraph methods are strictly stronger than the Weisfeiler-Leman (WL) test [54, 39], but an upper-bound on their expressive power is generally unknown. Secondly, on a more practical level, Subgraph GNN architectures differ considerably in the way information is aggregated and shared across the subgraphs, and an understanding of the possible aggregation and sharing rules is missing. Both aspects are important: an understanding of the former can highlight the limitations of emerging architectures, a study of the latter paves the way for improved Subgraph GNNs.
+
+Main contributions. The goal of this paper is to provide a deeper understanding of node-based Subgraph GNNs in light of the two aforementioned aspects. The main theoretical tool underpinning our contributions is a novel analysis of the symmetry group that acts on the sets of subgraphs. While several previous approaches [43, 14, 7] have (often implicitly) assumed that a subgraph architecture should be equivariant to independent node and subgraph permutations, we leverage the fact that nodebased policies induce an inherent bijection between the subgraphs and the nodes. This observation allows us to align the two groups and model the symmetry with a single (smaller) permutation group that acts on nodes and subgraphs jointly. Other works [61, 56, 59] have (again, implicitly) recognised such node-subgraph correspondence but without studying the implications on the symmetry group, and resorting, as a result, to a partial and heuristic choice of equivariant operations.
+
+The use of this stricter symmetry group raises a fruitful connection with $k$ -order Invariant Graph Networks (k-IGNs) [33, 32], a well studied family of architectures for processing graphs and hypergraphs designed to be equivariant to the same symmetry group. This connection allows us to transfer and reinterpret previous results on IGNs to our Subgraph GNN setup. As our first contribution we show that the expressive power of Subgraph GNNs with node-based policies is bounded by that of the 3-WL test. This is shown by proving that all previous Subgraph GNNs can be implemented by a 3-IGN and by leveraging the fact that the expressive power of these models is bounded by 3-WL [21, 5].
+
+Our second contribution is the proposal of a general layer formulation for Subgraph GNNs, based on the observation that these methods maintain an $n \times n$ representation of $n$ subgraphs with $n$ nodes, following the same symmetry structure of 2-IGNs (same permutation applied to both rows and columns of this representation). We propose a novel extension of 2-IGNs capturing both local (message-passing-like) and global operations. This extension easily recovers previous methods facilitating their comparison. Also, we present a number of new operations that previous methods did not implement. We build upon these observations to devise a new Subgraph GNN dubbed SUN, (Subgraph Union Network). We prove that SUN generalises all previous node-based Subgraph GNNs and we empirically compare it to these methods, showing it can outperform them.
+
+# 2 Previous and related work
+
+Expressive power of GNNs. The expressive power of GNNs is a central research focus since it was realised that message-passing type GNNs are constrained by the expressivity of the WL isomorphism test [36, 55]. Other than the aforementioned subgraph-based methods, numerous approaches for more powerful GNNs have been proposed, including positional and structural encodings [1, 45, 11, 17, 28, 31], higher-order message-passing schemes [36, 38, 10, 9], equivariant models [24, 33, 32, 53, 15, 51, 40]. We refer readers to the recent survey by Morris et al. [39] for additional details. Finally we note that, in a related and concurrent work, Qian et al. [46] propose a theoretical framework to study the expressive power of subgraph-based GNNs by relating them to the $\mathtt { k - W L }$ hierarchy, and explore how to sample subgraphs in a data-driven fashion.
+
+Invariant graph networks. IGNs were recently introduced in a series of works by Maron et al. [33, 32, 34] as an alternative to MPNNs for processing graph and hyper-graph data. For $k \geq 2$ , $\mathtt { k }$ -IGNs represent hyper-graphs with hyper-edges up to size $k$ with $k$ -order tensor $\bar { \boldsymbol { y } } \in \mathbb { R } ^ { n ^ { k } }$ , where each entry holds information about a specific hyper-edge. On these they apply linear $S _ { n }$ -equivariant layers interspersed with pointwise nonlinearities. These models have been thoroughly studied in terms of: (i) their expressive power; (ii) the space of their equivariant linear layers. As for (i), IGNs were shown to have exactly the same graph separation power as the $\mathtt { k - W L }$ graph isomorphism test [32, 5, 21] and, for sufficiently large $k$ , to have a universal approximation property w.r.t. $S _ { n }$ -invariant and equivariant functions [34, 26, 47]. Concerning (ii), the work in [33] completely characterised the space of linear layers equivariant to $S _ { n }$ from $\mathbb { R } ^ { n ^ { k } }$ to $\mathbb { R } ^ { n ^ { k ^ { \prime } } }$ : the authors derived a basis of bel $1 ( k + k ^ { \prime } )$ linear operators consisting of indicator tensors of equality patterns over the multi-index set $\{ 1 , . . . , n \} ^ { k + k ^ { \prime } } = [ n ] ^ { k + k ^ { \prime } }$ . Albooyeh et al. [2] showed these layers can be (re-)written as sums of pooling-broadcasting operations between elements of $\mathcal { V }$ indexed by the orbits 2 of the action of $S _ { n }$ on $[ n ] ^ { k }$ and $[ n ] ^ { k ^ { \prime } }$ . Take, e.g., $k = k ^ { \prime } = 2$ . In this case there are only two orbits: $\{ i , i \} , i \in [ n ]$ corresponding to on-diagonal terms, and $\{ i , j \} , i \neq j \in [ n ] ,$ , off-diagonal terms. According to Albooyeh et al. [2] any equivariant linear layer $L : \mathbb { R } ^ { n ^ { 2 } } \to \mathbb { R } ^ { n ^ { 2 } }$ can be represented as a composition of pooling and broadcasting operations on the elements indexed by these orbits. One example is the linear map that sums the on-diagonal elements and broadcasts the result to the off-diagonal ones: $\begin{array} { r } { L ( \mathcal { Y } ) _ { i j } = \sum _ { k } \mathcal { Y } _ { k k } } \end{array}$ for $i \neq j$ , 0 otherwise. See Appendix B, for additional details. These results particularly important as they underpin most of our theoretical derivations. Lastly, a more comprehensive coverage of IGNs can be found in [39].
+
+Subgraph GNNs. Despite motivated by diverse premises, a collection of concurrent methods share the overarching design whereby graphs are modelled through the application of a GNN to their subgraphs. Bevilacqua et al. [7] first explicitly formulated the concept of bags of subgraphs generated by a predefined policy and studied layers to process them in an equivariant manner: the same GNN can encode each subgraph independently (DS-GNN), or information can be shared between these computations in view of the alignment of nodes across the bag [35] (DSS-GNN). Building upon the Reconstruction Conjecture [25, 52], Reconstruction GNNs [14] obtain node-deleted subgraphs, process them with a GNN and then aggregate the resulting representations by means of a set model. Nested GNNs [59] and GNN-As-Kernel models (GNN-AK) [61] shift their computation from rooted subtrees to rooted subgraphs, effectively representing nodes by means of GNNs applied to their enclosing ego-networks. Similarly to DSS-GNNs [7], GNN-AK models may feature information sharing modules aggregating node representations across subgraphs. ID-GNNs [56] also process ego-network subgraphs, but their roots are ‘marked’ so to specifically alter the exchange of messages involving them. Intuitively, the use of subgraphs implicitly breaks those local symmetries which determine the notorious expressiveness bottleneck of MPNNs. We note that other works can be interpreted as Subgraph GNNs, including those by Papp et al. [43], Papp and Wattenhofer [42].
+
+# 3 Node-based Subgraph GNNs
+
+Notation. Let $G = ( A , X )$ be a member of the family $\mathcal { G }$ of node-attributed, undirected, finite, simple graphs3. The adjacency matrix $A \in \mathbb { R } ^ { n \times n }$ represents $G$ ’s edge set $E$ over its set of $n$ nodes $V$ . The feature matrix $\dot { X } \in \mathbb { R } ^ { \dot { n } \times d }$ gathers the node features; we denote by $x _ { j } \in \mathbb { R } ^ { d \times 1 }$ the features of node $j$ corresponding to the $j$ -th row of $X$ . $B _ { G }$ is used to denote a multiset (bag) of $m$ subgraphs of $G$ . Adjacency and feature matrices for subgraphs in $B _ { G }$ are arranged in tensors $\mathcal { A } \in \mathbb { R } ^ { m \times n \times n }$ and $\mathcal { X } \in \mathbb { R } ^ { m \times n \times d }$ . Superscript $i , ( t )$ refers to representations on subgraph $i$ at the $t$ -th layer of a stacking, as in xi,j $x _ { j } ^ { i , ( t ) }$ . Finally, we denote $[ n ] = \{ 1 , \dots , n \}$ . All proofs are deferred to Appendices $\mathbf { B }$ and D.
+
+Formalising Subgraph GNNs. Subgraph GNNs compute a representation of $G \in { \mathcal { G } }$ as
+
+$$
+( A , X ) \mapsto \left( \mu \circ \rho \circ S \circ \pi \right) ( A , X ) .
+$$
+
+Here, $\pi : G \mapsto \{ G ^ { 1 } , . . . , G ^ { m } \} = \{ ( A ^ { 1 } , X ^ { 1 } ) , . . . , ( A ^ { m } , X ^ { m } ) \} = B _ { G } ^ { ( 0 ) }$ is a selection policy generating a bag of subgraphs from $G$ ; $\begin{array} { r } { S = L _ { T } \circ \dots \circ L _ { 1 } : B _ { G } ^ { ( 0 ) } \mapsto B _ { G } ^ { ( T ) } } \end{array}$ is a stacking of (node- and subgraph-) permutation equivariant layers; $\rho : ( G , B _ { G } ^ { ( T ) } ) \mapsto x _ { G }$ is a permutation invariant pooling function, $\mu$ is an MLP. The layers in $s$ comprise a base-encoder in the form of a GNN applied to subgraphs; throughout this paper, we assume it to be a $1 - \mathrm { { W L } }$ maximally expressive MPNN such as the one in Morris et al. [36]. Subgraph GNNs differ in the implementation of $\pi$ , $s$ and, in some cases, $\rho$ . For example, in (n-1)-Reconstruction GNNs [14], $\pi$ selects node-deleted subgraphs and $s$ applies a Siamese MPNN to each subgraph independently. To exemplify the variability in $s$ , DSS-GNN [7] extends this method with cross-subgraph node and connectivity aggregation. More details are on how currently known Subgraph GNNs are captured by Equation (1) can be found in Appendix A.
+
+Node-based selection policies. In this work, we focus on a specific family of node-based subgraph selection policies, wherein every subgraph is associated with a unique node in the graph. Formally, we call a subgraph selection policy node-based if it is of the form $\bar { \pi ( G ) } = \{ f ( G , v ) \} _ { v \in V }$ , for some selection function $f ( G , v )$ that takes a graph $G$ and a node $v$ as inputs and outputs a subgraph $G ^ { v }$ . In the following, we refer to $v$ as the root of subgraph $G ^ { v }$ . We require $f$ to be a bijection and we note that such policies produce $m = n$ different subgraphs. Amongst the most common examples are node-deletion (ND), node-marking (NM), and ego-networks (EGO) policies. For input graph $G$ , $f _ { \mathrm { N D } } ( G , v )$ removes node $v$ and the associated connectivity; $f _ { \mathrm { N M } } ( G , v )$ adds a special ‘mark’ attribute to $v$ ’s features (with no connectivity alterations), and $f _ { \mathrm { E G O } ( h ) } ( G , v )$ returns the subgraph induced by the $h$ -hop-neighbourhood around the root $v$ . EGO policies can be ‘marked’: $f _ { \mathrm { E G O + } ( h ) } ( G , v )$ extracts the $h$ -hop ego-net around $v$ and marks this node as done by $f _ { \mathrm { N M } }$ . For convenience, we denote the class of such node-based selection policies by $\Pi$ :
+
+![](images/3ce1fcf4bad468506f6ba5f0f71244f25641e307936926f5643bede808439499.jpg)  
+Figure 1: Symmetries of bags of subgraphs (left) and corresponding function space diagrams (right). In ESAN [7] symmetries are modelled as a direct product of node and subgraph permutation groups; however, node-based policies enable the use of one single permutation group, the same as in 3-IGNs. 3-IGNs are less constrained, thus more expressive than ESAN and other Subgraph GNNs. See diagram on the right and formal statement in Section 5.
+
+Definition 1 (Known node-based selection policies $\Pi$ ). Let $\Sigma$ be the set of all node-based subgraph selection policies operating on $\mathcal { G }$ . Class $\Pi \subset \Sigma$ collects the node-based policies node-deletion $( N D )$ , node-marking (NM), ego-nets $( E G O )$ and marked ego-nets $( E G O + )$ of any depth: $\Pi =$ $\{ \pi _ { \mathrm { N D } } , \pi _ { \mathrm { N M } } , \pi _ { \mathrm { E G O } ( h ) } , \pi _ { \mathrm { E G O } + ( h ) } \mid h > 0 \}$ .
+
+Node-based Subgraph GNNs are those Subgraph GNNs which, implicitly or explicitly, process bags generated by node-based policies. We group known formulations in the following family:
+
+Definition 2 (Known node-based Subgraph GNNs $\Upsilon$ ). Let Ξ be the set of all node-based Subgraph GNNs. Class $\Upsilon \subset \Xi$ collects known Subgraph GNNs when equipped with 1-WL base-encoders: $\Upsilon = \lbrace ( \mathtt { n - l } )$ -Reconstr.GNN, GNN-AK, GNN-AK-ctx, NGNN, ID-GNN, DS- $\mathrm { G N N } _ { \Pi }$ , $\mathrm { D S S - G N N _ { \Pi } } \big \}$ $\mathrm { D S - G N N _ { \Pi } }$ , DSS-GNNΠ refer to DS- and DSS-GNN models equipped with any $\pi \in \Pi$ .
+
+Importantly, all these methods apply MPNNs to subgraphs of the original graph, but differ in the way information is shared between subgraphs/nodes. In all cases, their expressive power is strictly larger than 1-WL, but an upper-bound is currently unknown.
+
+# 4 Symmetries of node-based subgraph selection policies
+
+In an effort to characterise the representational power of node-based Subgraph GNNs, we first study the symmetry group of the objects they process: ‘bags of subgraphs’ represented as tensors $( { \mathcal { A } } , { \mathcal { X } } ) \in$ $\mathbb { R } ^ { m \times n \times n } \times \mathbb { R } ^ { m \times n \times d }$ , assuming $n$ nodes across $m$ subgraphs. Previous approaches [14, 7, 43] used two permutation groups: one copy of the symmetric group $S _ { n }$ models node permutations, while another copy $S _ { m }$ models subgraph permutations in the bag. These two were combined by a group product4 acting independently on the nodes and subgraphs in $( { \mathcal { A } } , { \mathcal { X } } )$ . For example, Bevilacqua et al. [7] model the symmetry as:
+
+$$
+\begin{array} { r } { ( ( \tau , \mathbf { \mu } ) \cdot \mathcal { A } ) _ { i j k } = \mathcal { A } _ { \tau ^ { - 1 } ( i ) } \cdot \mathbf { \mu } ^ { - 1 } ( j ) \cdot \mathbf { \mu } ^ { - 1 } ( k ) , \left( ( \tau , \mathbf { \mu } ) \cdot \mathcal { X } \right) _ { i j l } = \mathcal { X } _ { \tau ^ { - 1 } ( i ) } \cdot \mathbf { \mu } ^ { - 1 } ( j ) l , \left( \tau , \mathbf { \mu } \right) \in S _ { m } \times S _ { m } } \end{array}
+$$
+
+Our contributions stem from the following crucial observation: When using node-based policies, the subgraphs in $( { \mathcal { A } } , { \mathcal { X } } )$ can be ordered consistently with the nodes by leveraging the bijection $f : v \mapsto G _ { v }$ characterising this policy class. In other words, $f$ suggests a node-subgraph alignment inducing a new structure on $( { \mathcal { A } } , { \mathcal { X } } )$ , whereby the subgraph order is not independent of that of nodes anymore. Importantly, this new structure is preserved only by those permutations operating identically on both nodes and subgraphs. Following this observation, the symmetry of a node-based bag of subgraphs is modelled more accurately using only one single permutation group $S _ { n }$ jointly acting on both nodes and subgraphs:
+
+![](images/04acf2192cc176656321572089bb029c8511160c3d7adc097bc0ac53adfc0449.jpg)  
+Figure 2: Depiction of cubed tensor $\mathcal { V }$ , its orbit-induced partitioning and the related semantics when $\mathcal { V }$ is interpreted as a bag of node-based subgraphs, $n = 5$ . Elements in the same partition are depicted with the same colour. Left: the whole tensor. Middle and right: sections; elements in purple and green constitute sub-tensor $\mathcal { X }$ , the remaining ones sub-tensor $\mathcal { A }$ .
+
+$$
+( \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } ^ { - 1 } ( i ) \mathbf { \partial } \cdot \mathbf { \partial } ^ { - 1 } ( k ) , \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial } \cdot \mathbf { \partial }
+$$
+
+It should be noted that $S _ { n }$ is significantly smaller than $S _ { n } \times S _ { n } ^ { \phantom { \dagger } 5 }$ . Informally, the latter group contains many permutations which are not in the former: those acting differently on nodes and subgraphs and, thus, not preserving the new structure of $( { \mathcal { A } } , { \mathcal { X } } )$ . Since they are restricted by a smaller set of equivariance constraints, we expect GNNs designed to be equivariant to $S _ { n }$ to be be more expressive than those equivariant to the larger groups considered by previous works [34] (see Figure 1).
+
+The insight we obtain from Equation (3) is profound: it reveals that the symmetry structure of $\mathcal { A }$ exactly matches the symmetries of third-order tensors used by 3-IGNs, and similarly, that the symmetry structure for $\mathcal { X }$ matches the symmetries of second-order tensors used by 2-IGNs. In the following, we will make use of this insight and the fact that IGNs are well-studied objects to prove an upper-bound on the expressive power of Subgraph GNNs and to design principled extensions to these models. We remark that bags of node-based subgraphs can also be represented as tensors $\mathcal { V } \in \mathbb { R } ^ { n ^ { 3 } \times d }$ , the same objects on which 3-IGNs operate. Here, $\mathcal { X }$ is embedded in the main diagonal plane of $\mathcal { V }$ , $\mathcal { A }$ in its remaining entries. Within this context, it is informative to study the semantics of the 5 orbits induced by the action of $S _ { n }$ on $\mathcal { V }$ ’s multi-index set $[ n ] ^ { 3 }$ : each of these uniquely identify root nodes, non-root nodes, edges to and from root nodes as well as edges between non-root nodes (see Figure 2 and additional details in Appendix B.1.1). We build upon this observation, along with the layer construction by Albooyeh et al. [2], to prove many of the results presented in the following.
+
+# 5 A representational bound for Subgraph GNNs
+
+In this section we prove that the expressive power of known node-based Subgraph GNNs is bounded by $3 - \mathrm { { W L } }$ by showing that they can be implemented by 3-IGNs, which have the same expressive power as $3 - \mathbb { W } \mathbb { L }$ . Underpinning the possibility of IGNs to upper-bound a certain Subgraph GNN $\mathcal { N }$ in its expressive power is the ability of IGNs to (i) implement $\mathcal { N }$ ’s subgraph selection policy $( \pi )$ and (ii) implement $\mathcal { N }$ ’s (generalised) message-passing and pooling equations $( \mu \circ \rho \circ S )$ . This would ensure that whenever $\mathcal { N }$ assigns distinct representations to two non-isomorphic graphs, an IGN implementing $\mathcal { N }$ would do the same. We start by introducing a recurring, useful concept.
+
+Definition 3 (“implements”). Let $f : D _ { f } \to C _ { f }$ , $g : D _ { g } \to C _ { g }$ be two functions and such that $D _ { g } \subseteq D _ { f } , C _ { g } \subseteq C _ { f }$ . We say $f$ implements $g$ (and write $f \cong g ,$ ) when $\forall x \in D _ { g } , f ( x ) = g ( x )$ .
+
+Our first result shows that 3-IGNs can implement the selection policies in class $\Pi$ (Definition 1), which, to the best of our knowledge, represent all known node-based policies utilised by previously proposed Subgraph GNNs.
+
+Lemma 4 (3-IGNs implement known node-based selection policies). For any $\pi \in \Pi$ there exists a stacking of 3-IGN layers $\mathcal { M } _ { \pi }$ s.t. $\mathcal { M } _ { \pi } \cong \pi$ .
+
+Intuitively, 3-IGNs start from a $\mathbb { R } ^ { n ^ { 2 } }$ representation of $G$ and, first, move to a $\mathbb { R } ^ { n ^ { 3 } }$ tensor ‘copying’ this latter along its first (subgraph) dimension. This is realised via an appropriate broadcast operation. Then, they proceed by adding a ‘mark’ to the features of some nodes and/or by nullifying elements corresponding to some edges. We refer readers to Figure 2 and Appendix B.1.2 for additional details on how nodes in each subgraph are represented in 3-IGNs. Next, we show 3-IGNs can implement layers of any model $\in \Upsilon$ .
+
+Lemma 5 (3-IGNs implement Subgraph GNN layers). Let $G _ { 1 } , G _ { 2 }$ be two graphs in $\mathcal { G }$ and $\mathcal { N }$ a model in family be bags of subg $\Upsilon$ equipped with Morris et al. [36] messagehs in the input of some intermediate layer asin g base-encoders. Let . Then there exists a $B _ { 1 } ^ { ( t ) } , B _ { 2 } ^ { ( t ) }$ $L$ $\mathcal { N }$ 3-IGN layers $\mathcal { M } _ { L }$ for which $\begin{array} { r } { \dot { \mathcal { M } } _ { L } ( { B } _ { i } ^ { ( t ) } ) = { B } _ { i } ^ { ( t + 1 ) } = { L } ( { B } _ { i } ^ { ( t ) } ) } \end{array}$ for $i = 1 , 2$ .
+
+Lemmas 4 and 5 allow us to upper-bound the expressive power of all known instances of node-based Subgraph GNNs by that of 3-IGNs:
+
+Theorem 6 (3-IGNs upper-bound node-based Subgraph GNNs). For any pair of non-isomorphic graphs $G _ { 1 } , G _ { 2 }$ in family $\mathcal { G }$ and Subgraph GNN model $\mathcal { N } \in \Upsilon$ equipped with Morris et al. [36] message-passing base-encoders, if there exists weights $\Theta$ such that $G _ { 1 }$ , $G _ { 2 }$ are distinguished by instance $\mathcal { N } _ { \Theta }$ , then there exist weights Ω for a 3-IGN instance $\mathcal { M } _ { \Omega }$ such that $G _ { 1 } , G _ { 2 }$ are distinguished by $\mathcal { M } _ { \Omega }$ as well.
+
+Theorem 6 has profound consequences in the characterisation of the expressive power of node-based Subgraph GNNs, as we show in the following
+
+Corollary 7 (3-WL upper-bounds node-based Subgraph GNNs). Let $G _ { 1 } , G _ { 2 } \ \in \ { \mathcal { G } }$ be two nonisomorphic graphs and $\mathcal { N } _ { \Theta } \in \Upsilon$ one instance of model $\mathcal { N }$ with weights $\Theta$ . If $\mathcal { N } _ { \Theta }$ distinguishes $G _ { 1 } , G _ { 2 }$ , then the 3-WL algorithm does so as well.
+
+Proof idea: If there is a pair of graphs undistinguishable by 3-WL, but for which there exists a Subgraph GNN separating them, there must exists a 3-IGN separating these (Theorem 6). This is in contradiction with the result by Geerts [21], Azizian and Lelarge $[ 5 ] ^ { 6 }$ .
+
+# 6 A design space for Subgraph GNNs
+
+As discussed, different formulations of Subgraph GNNs differ primarily in the specific rules for updating node representations across subgraphs. However, until now it is not clear whether existing rules exhaust all the possible equivariant options. We devote this section to a systematic characterisation of the ‘layer space’ of Subgraph GNNs.
+
+In the spirit of the previous Section 5, where we “embedded” Subgraph GNNs in 3-IGNs, one option would be to consider all $\mathsf { b e l l } ( 6 ) = 2 0 3$ linear equivariant operations prescribed by this formalism. However, this choice would be problematic for three main reasons: (i) This layer space is too vast to be conveniently explored; (ii) It includes operations involving $\mathcal { O } ( n ^ { 3 } )$ space complexity, impractical in most applications; (iii) The linear IGN basis does not directly support local message passing, a key operation in subgraph methods. Following previous Subgraph GNN variants, which use $\mathcal { O } ( { \bar { n } } ^ { 2 } )$ storage for the representation of $n$ nodes in $n$ subgraphs, we set the desideratum of $\mathcal { O } ( n ^ { 2 } )$ memory complexity as our main constraint, and use this restriction to reduce the design space. Precisely, we are interested in modelling $S _ { n }$ -equivariant transformations on the subgraph-node tensor $\mathcal { X }$ .
+
+# 6.1 Extended 2-IGNs
+
+As we have already observed in Equation 3 in Section 4, such a second order tensor $\mathcal { X }$ abides by the same symmetry structure of 2-IGNs. We therefore gain intuition from the characterisation of linear equivariant mappings as introduced by Maron et al. [33], and propose an extension of this formalism.
+
+![](images/7462b6faa7624521d59d4391ded56346740f6471db88f1502e7e2d6cbc3e92d5.jpg)  
+Figure 3: Comparison of aggregation and update rules in Subgraph GNNs, illustrated on an $n \times n$ matrix ( $\scriptstyle { n }$ subgraphs with $n$ nodes). Top row: off-diagonal updates; bottom row: diagonal (root node) updates. Each colour represents a different parameter. Full squares: global sum pooling; triangles: local pooling; two triangles: both local and global pooling. See Appendix C for more details.
+
+2-IGN layer space. A 2-IGN layer $L _ { \Theta }$ updates $\mathcal { X } \in \mathbb { R } ^ { n \times n \times d }$ as $\mathcal { X } ^ { ( t + 1 ) } = L _ { \Theta } \big ( \mathcal { X } ^ { ( t ) } \big )$ by applying a specific transformation to on- $( x _ { i } ^ { i } )$ and off-diagonal terms $( x _ { j } ^ { i } , i \neq j )$ :
+
+$$
+\begin{array} { r l } & { x _ { i } ^ { i , ( t + 1 ) } { = } v _ { \theta _ { 1 } } \big ( x _ { i } ^ { i , ( t ) } , \underset { j } { \sum } x _ { j } ^ { j , ( t ) } , \underset { j \neq i } { \prod } x _ { j } ^ { i , ( t ) } , \underset { h \neq i } { \prod } x _ { i } ^ { h , ( t ) } , \underset { h \neq j } { \prod } x _ { j } ^ { h , ( t ) } \big ) } \\ & { x _ { i } ^ { k , ( t + 1 ) } { = } v _ { \theta _ { 2 } } \big ( x _ { i } ^ { k , ( t ) } , x _ { k } ^ { i , ( t ) } , \underset { h \neq j } { \prod } x _ { j } ^ { h , ( t ) } , \underset { h \neq i } { \prod } x _ { i } ^ { h , ( t ) } , \underset { j \neq k } { \prod } x _ { j } ^ { k , ( t ) } , \underset { j \neq i } { \prod } x _ { j } ^ { i , ( t ) } , \underset { h \neq k } { \prod } x _ { k } ^ { h , ( t ) } , x _ { k } ^ { k , ( t ) } , x _ { k } ^ { i , ( t ) } , x _ { i } ^ { i , ( t ) } , \underset { j } { \prod } x _ { j } ^ { j , ( t ) } \big ) } \end{array}
+$$
+
+Here, $\boxed { \begin{array} { r l } \end{array} }$ indicates a permutation invariant aggregation function, $v _ { \theta _ { 1 } } , v _ { \theta _ { 2 } }$ apply a specific $d \times d ^ { \prime }$ linear transformation to each input term and sum the outputs including bias terms.
+
+ReIGN(2) layer space. As 2-IGN layers are linear, the authors advocate setting $\sqcap \equiv \Sigma$ , performing pooling as global summation. Here, we extend this formulation to additionally include different local aggregation schemes. In this new extended formalism, entry $x _ { i } ^ { k }$ represents node $i$ in subgraph $k$ ; accordingly, each aggregation in Equation 4 can be also performed locally, i.e. extending only over $\ddot { \iota }$ ’s neighbours, as prescribed by the connectivity of subgraph $k$ or of the original input graph. As an example, when updating entry $x _ { i } ^ { k , ( t ) }$ , term □j̸=k xk,j is expanded as $\begin{array} { r } { \Big ( \bigtriangledown _ { j \neq k } x _ { j } ^ { k , ( t ) } , \bigtriangledown _ { j \sim _ { k } i } x _ { j } ^ { k , ( t ) } , \bigtriangledown _ { j \sim i } x _ { j } ^ { k , ( t ) } . } \end{array}$ , with $\sim _ { k }$ denoting adjacency in subgraph $k$ , and $\sim$ that in the original graph connectivity. Each term in the expansion is associated with a specific learnable linear transformation. We report a full list of pooling operations in Appendix D, Table 3. These local pooling operations allow to readily recover sparse message passing, which constitutes the main computational primitive of all popular (Subgraph) GNNs. Other characteristic Subgraph GNN operations are also recovered by this formalism: for example, $\textstyle \prod _ { h \neq i } x _ { i } ^ { h , ( t ) }$ operates global pooling of node $i$ ’s representations across subgraphs, as previously introduced in Bevilacqua et al. [7], Zhao et al. [61]. We also note that additional, novel, operations are supported, e.g. the transpose $x _ { k } ^ { i , ( t ) }$ . We generally refer to this framework as ReIGN(2) (“Rethought 2-IGN”).
+
+ReIGN(2) architectures. ReIGN(2) induces (linear) layers in the same form of Equation 4, but where $\boxed { \begin{array} { r l } \end{array} }$ terms are expanded to both local and global operations, as explained. These layers can operate on any bag generated by a node-based selection policy $\bar { \pi }$ , and can be combined together in ReIGN(2) stacks of the form $\mathcal S _ { \mathcal R } = L ^ { ( T ) } \circ \sigma \circ L ^ { ( T - 1 ) } \circ \sigma \circ . . . \circ \sigma \circ L ^ { ( 1 ) } ,$ , where $\sigma$ ’s are pointwise nonlinearities and $L$ ’s are ReIGN(2) layers. This allows us to define ReIGN(2) models as Subgraph GNNs in the form of Equation 1, where $s$ is a ReIGN(2) layer stacking and $\pi$ is node-based: $\mathcal { R } _ { \bar { \pi } } = \mu \circ \rho \circ S _ { \mathcal { R } } \circ \bar { \pi }$ .
+
+More generally, ReIGN(2) induces a ‘layer space’ for node-based Subgraph GNNs: the expanded terms in its update equations represent a pool of atomic operations that can be selected and combined to define new equivariant layers. Compared to that of 3-IGNs, this space is of tractable size, yet it recovers previously proposed Subgraph GNNs and allows to define novel interesting variants.
+
+Recovering previous Subgraph GNNs. The following result states that the ReIGN(2) generalises all known subgraph methods in $\Upsilon$ , as their layers are captured by a ReIGN(2) stacking.
+
+Theorem 8 (ReIGN(2) implements node-based Subgraph GNNs). Let N be a model in family $\Upsilon$ equipped with Morris et al. [36] message-passing base-encoders. For any instance $\mathcal { N } _ { \Theta }$ , there exists ReIGN(2) instance $\mathcal { R } _ { \Omega }$ such that $\mathcal { R } _ { \Omega } \cong \mathcal { N } _ { \Theta }$ .
+
+This shows that known methods are generalised without resorting to the $\mathcal { O } ( n ^ { 3 } )$ computational complexity of 3-IGNs. Figure 3 illustrates the aggregation and sharing rules used by previous Subgraph GNNs to update root and non-root nodes, and compare them with those of ReIGN(2) and 2-IGNs. We visualise these on the subgraph-node sub-tensor gathering node representations across subgraphs; here, root nodes occupy the main diagonal, non-root nodes all the remaining off-diagonal entries. As for to the 2-IGN Equations 4, the elements in these two partitions may be updated differently, so we depict them separately in, respectively, the bottom and top rows. In each depiction we colour elements depending on the set of weights parameterising their contribution in the update process, with two main specifications: (i) Elements sharing the same colour are pooled together; (ii) Triangles indicate such pooling is performed locally based on the subgraph connectivity at hand (two triangles indicate both local and global pooling ops are performed). E.g., note how DS-GNN equivalently updates the representations of root and non-root nodes via the same (local) messagepassing layer (triangles, yellow, leftmost picture). By illustrating how ReIGN(2) generalises previous node-based methods, this figure is to be interpreted as visual support for the Proof of Theorem 8 (see Appendix D). Additional details and discussions on Figure 3 are found in Appendix C.
+
+Notably, as methods in $\Upsilon$ have been shown to be strictly stronger than 2-WL [7, 14, 61, 59, 56], Theorem 8 implies the same lower bound for ReIGN(2). Nevertheless, when employing policies in $\Pi$ and 3-IGN-computable invariant pooling functions $\rho$ (as those used by models in $\Upsilon$ ), ReIGN(2)s are upper-bounded by 3-IGNs:
+
+Proposition 9 (3-IGNs implement ReIGN(2)). For any pair of non-isomorphic graphs $G _ { 1 } , G _ { 2 }$ in family $\mathcal { G }$ , if there exist policy $\bar { \pi } \in \Pi$ , parameters $\Theta$ and 3-IGN-computable invariant pooling function $\rho$ such that the ReIGN(2) instance $\mathcal { R } _ { \rho , \Theta , \bar { \pi } }$ distinguishes $G _ { 1 }$ , $G _ { 2 }$ , then there exist weights $\Omega$ for $a$ 3-IGN instance $\mathcal { M } _ { \Omega }$ such that $G _ { 1 } , G _ { 2 }$ are distinguished by $\mathcal { M } _ { \Omega }$ as well.
+
+This proposition entails an upper-bound on the expressive power of ReIGN(2).
+
+Corollary 10 (3-WL upper-bounds ReIGN(2)). The expressive power of a ReIGN(2) model with policy $\pi \in \Pi$ and 3-IGN-computable invariant pooling function $\rho$ is upper-bounded by 3-WL.
+
+We note that there may be layers equivariant to $S _ { n }$ over $\mathbb { R } ^ { n ^ { 2 } }$ not captured by ReIGN(2). Yet, previously proposed Subgraph GNN layers do not exhaust the ReIGN(2) design space, which remains largely unexplored. One, amongst possible novel constructions, is introduced next.
+
+# 6.2 A unifying architecture: Subgraph Union Networks
+
+We now show how the ReIGN(2) layer space can guide the design of novel, expressive, Subgraph GNNs. Our present endeavour is to conceive a computationally tractable architecture subsuming known node-based models: in virtue of this latter desideratum, we will dub this architecture “Subgraph Union Network” (SUN). To design the base equivariant layer for SUN, we select and combine specific aggregation terms suggested by the ReIGN(2) framework:
+
+$$
+\begin{array} { l } { { \displaystyle x _ { i } ^ { i , ( t + 1 ) } = \sigma \Big ( v _ { \theta _ { 1 } } \big ( x _ { i } ^ { i , ( t ) } , \sum _ { j \sim i } x _ { j } ^ { i , ( t ) } , \sum _ { j } x _ { j } ^ { i , ( t ) } , \sum _ { h } x _ { i } ^ { h , ( t ) } , \sum _ { j \sim i } \sum _ { h } x _ { j } ^ { h , ( t ) } \big ) \Big ) } } \\ { { \displaystyle x _ { i } ^ { k , ( t + 1 ) } = \sigma \Big ( v _ { \theta _ { 2 } } \big ( x _ { i } ^ { k , ( t ) } , \sum _ { j \sim k i } x _ { j } ^ { k , ( t ) } , x _ { i } ^ { i , ( t ) } , x _ { k } ^ { k , ( t ) } , \sum _ { j } x _ { j } ^ { k , ( t ) } , \sum _ { h } x _ { i } ^ { h , ( t ) } , \sum _ { j \sim i } \sum _ { h } x _ { j } ^ { h , ( t ) } \big ) \Big ) } } \end{array}
+$$
+
+where $\upsilon$ ’s sum their inputs after applying a specific linear transformations to each term. One of the novel features of SUN is that roots are transformed by a different set of parameters $( \theta _ { 1 } )$ than the other nodes 7 ( $\ \theta _ { 2 }$ , see Figure 2). In practice, the first and last two terms in each one of Equations (5) and (6) can be processed by maximally expressive MPNNs [36, 55], the remaining terms by MLPs. We test these variants in our experiments, with their formulations in Appendix G. SUN remains an instantiation of the ReIGN(2) framework:
+
+Proposition 11 (A ReIGN(2) stacking implements SUN layers). For any SUN layer $L$ defined according to Equations 5 and 6, there exists a ReIGN(2) layer stacking $ { \boldsymbol { S } } _ { L }$ , such that $S _ { L } \cong L$ .
+
+Table 1: Test mean MAE on the Counting Substructures and ZINC-12k datasets. All Subgraph GNNs employ a GIN base-encoder. †This version of GNN-AK $^ +$ does not follow the standard evaluation procedure.   
+
+<table><tr><td>Method</td><td>ZINC (MAE ↓)</td></tr><tr><td>GCN[27]</td><td>0.321 ± 0.009</td></tr><tr><td>GIN [55]</td><td>0.163 ± 0.004</td></tr><tr><td>PNA[13]</td><td>0.133 ± 0.011</td></tr><tr><td>GSN[11] CIN [9]</td><td>0.101 ± 0.010</td></tr><tr><td></td><td>0.079 ± 0.006</td></tr><tr><td>NGNN [59] DS-GNN (EGO) [7]</td><td>0.111 ± 0.003</td></tr><tr><td>DS-GNN (EGO+) [7]</td><td>0.115 ± 0.004</td></tr><tr><td></td><td>0.105 ±0.003</td></tr><tr><td>DSS-GNN (EGO) [7]</td><td>0.099 ±0.003</td></tr><tr><td>DSS-GNN (EGO+) [7]</td><td>0.097 ± 0.006</td></tr><tr><td>GNN-AK[61]</td><td>0.105 ± 0.010</td></tr><tr><td>GNN-AK-CTX [61]</td><td>0.093±0.002</td></tr><tr><td>GNN-AK+ [61]†</td><td>0.086± ???</td></tr><tr><td>GNN-AK+ [61]</td><td>0.091 ± 0.011</td></tr><tr><td>SUN (EGO)</td><td>0.083 ±0.003</td></tr><tr><td>SUN (EGO+)</td><td>0.084 ±0.002</td></tr></table>
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">Counting Substructures (MAE ↓)</td></tr><tr><td>Triangle</td><td>Tailed Tri.</td><td>Star</td><td>4-Cycle</td></tr><tr><td>GCN [27]</td><td>0.4186</td><td>0.3248</td><td>0.1798</td><td>0.2822</td></tr><tr><td>GIN [55]</td><td>0.3569</td><td>0.2373</td><td>0.0224</td><td>0.2185</td></tr><tr><td>PNA [13]</td><td>0.3532</td><td>0.2648</td><td>0.1278</td><td>0.2430</td></tr><tr><td>PPGN [32]</td><td>0.0089</td><td>0.0096</td><td>0.0148</td><td>0.0090</td></tr><tr><td>GNN-AK [61]</td><td>0.0934</td><td>0.0751</td><td>0.0168</td><td>0.0726</td></tr><tr><td>GNN-AK-CTX [61]</td><td>0.0885</td><td>0.0696</td><td>0.0162</td><td>0.0668</td></tr><tr><td>GNN-AK+[61]</td><td>0.0123</td><td>0.0112</td><td>0.0150</td><td>0.0126</td></tr><tr><td>SUN (EGO)</td><td>0.0092</td><td>0.0105</td><td>0.0064</td><td>0.0140</td></tr><tr><td>SUN (EGO+)</td><td>0.0079</td><td>0.0080</td><td>0.0064</td><td>0.0105</td></tr></table>
+
+Finally, we show that a stacking of SUN layers can implement any layer of known node-based Subgraph Networks, making this model a principled generalisation thereof.
+
+Proposition 12 (A SUN stacking implements known Subgraph GNN layers). Let N be a model in family $\Upsilon$ employing Morris et al. [36] as a message-passing base-encoder. Then, for any layer L in $\mathcal { N }$ , there exists a stacking of SUN layers $S _ { L }$ such that $S _ { L } \cong L$ .
+
+Beyond SUN. As it can be seen in Figure 3, SUN does not use all possible operations in the ReIGN(2) framework. Notably, two interesting operations that are not a part of SUN are: (i) The ‘transpose’: $x _ { i } ^ { k } = v _ { \theta } ( x _ { k } ^ { i } )$ , which shares information between the $i$ -th node in the $k$ -th subgraph and the $k$ -th node in the $i$ -th subgraph; (ii) Local vertical pooling $\begin{array} { r } { x _ { i } ^ { k } = v _ { \theta } ( \sum _ { h \sim i } x _ { i } ^ { h } ) } \end{array}$ . The exploration of these and other operations is left to future work.
+
+# 7 Experiments
+
+We experimentally validate the effectiveness of one ReIGN(2) instantiation, comparing SUN to previously proposed Subgraph $\mathrm { G N N s } ^ { 8 }$ . We seek to verify whether its theoretical representational power practically enables superior accuracy in expressiveness tasks and real-world benchmarks. Concurrently, we pay attention to the generalisation ability of models in comparison. SUN layers are less constrained in their weight sharing pattern, resulting in a more complex model. As this is traditionally associated with inferior generalisation abilities in low data regimes, we deem it important to additionally assess this aspect. Our code is also available.9
+
+Synthetic. Counting substructures and regressing graph topological features are notoriously hard tasks for GNNs [12, 17, 13]. We test the representational ability of SUN on common benchmarks of this kind [12, 13]. Table 1 reports results on the substructure counting suite, on which SUN attains state-of-the-art results in 3 out of 4 tasks. Additional results on the regression of global, structural properties are reported in Appendix G.
+
+Real-world. On the molecular ZINC-12k benchmark (constrained solubility regression) [50, 22, 16], SUN exhibits best performance amongst all domain-agnostic GNNs under the $5 0 0 \mathrm { k }$ parameter budget, including other Subgraph GNNs (see Table 1). A similar trend is observed on the large-scale Molhiv dataset from the OGB [23] (inhibition of HIV replication). Results are in Table 2. Remarkably, on both datasets, SUN either outperforms or approaches HIMP [19], GSN [11] and CIN [9], GNNs which explicitly model rings. We experiment on smaller-scale TUDatasets [37] in Appendix G, where we also compare selection policies.
+
+![](images/c7906ab21719b791330f9739760bfe956c7d9b4463d1306fa25bab6a3a493636.jpg)  
+Figure 4: Generalisation capabilities of Subgraph GNNs in the counting prediction task (Figures 4a and 4b) and in the ZINC-12k dataset (Figure 4c).
+
+Table 2: Test results for OGB dataset. GIN base-encoder for each Subgraph GNN.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>0GBG-MOLHIVROC-AUC(%)</td></tr><tr><td rowspan=1 colspan=1>GCN [27]GIN [55]PNA [13]DGN [6]HIMP [19]GSN[11]CIN [9]</td><td rowspan=1 colspan=1>76.06±0.9775.58±1.4079.05±1.3279.70±0.9778.80±0.8280.39±0.9080.94±0.57</td></tr><tr><td rowspan=1 colspan=1>RECONSTR.GNN[14]DS-GNN (EGO+) [7]DSS-GNN (EGO+) [7]GNN-AK+ [61]</td><td rowspan=1 colspan=1>76.32±1.4077.40±2.1976.78±1.6679.61±1.19</td></tr><tr><td rowspan=1 colspan=1>SUN (EGO+)</td><td rowspan=1 colspan=1>80.03±0.55</td></tr></table>
+
+Generalisation from limited data. In this set of experiments we compare the test performance of Subgraph GNNs when trained on increasing fractions of the available training data. Each architecture is selected by tuning the hyperparameters with the entire training and validation sets. We run this experiment on the 4-cycle counting task and the real-world ZINC-12k. We illustrate results in Figures 4a to 4c. Except for a short initial phase in the EGO policy, SUN generalises better than other Subgraph GNNs on cycle-counting. On ZINC-12k, SUN outruns DSS-, DS-GNN and GNN-AK variants from, respectively, 20, 30 and $4 0 \%$ of the samples. These results demonstrate that SUN’s expressiveness is not at the expense of sample efficiency, suggesting that its modelled symmetries guarantee strong representational power while retaining important inductive biases for learning on graphs.
+
+# 8 Conclusions
+
+Our work unifies, extends, and analyses the emerging class of Subgraph GNNs. Notably, we demonstrated that the expressive power of these methods is bounded by 3-WL. Towards a systematic study of models whose expressivity lies between 1- and 3-WL, we proposed a new family of layers for the class of Subgraph GNNs and, unlike most previous works on the expressive power of GNNs, we also investigated the generalisation abilities of these models, for which SUN shows considerable improvement. Appendix E lists several directions for future work, including an extension of our work to higher-order node-based policies.
+
+Societal impact. We do not envision any negative, immediate societal impact originating from our theoretical results, which represent most of our contribution. Experimentally, our model has shown promising results on molecular property prediction tasks and strong generalisation ability in low-data regimes. This leads us to believe our work may contribute to positively impactful pharmaceutical research, such as drug discovery [20, 3].
+
+# Acknowledgments and Disclosure of Funding
+
+The authors are grateful to Joshua Southern, Davide Eynard, Maria Gorinova, Guadalupe Gonzalez, Katarzyna Janocha for valuable feedback on early versions of the manuscript. They would like to thank Bruno Ribeiro and Or Litany for helpful discussions, Giorgos Bouritsas for constructive conversations about the generalisation experiments and, in particular, Marco Ciccone for the precious exchange on sharpness-aware optimisation and Neapolitan pizza. MB is supported in part by ERC Consolidator grant no 724228 (LEMAN). No competing interests are declared.
+
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+
+# 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] We discussed limitations of several previous works, as well as our own model, throughout the paper as our main contribution.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 8.   
+(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] (b) Did you include complete proofs of all theoretical results? [Yes] See Appendices B and D.
+
+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] See Section 7 and Appendix G.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix G.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We report the standard deviation computed over multiple seeds for experiments on ZINC12k (Table 1), ogbg-molhiv (Table 2) and on all generalisation experiments (Figures 4a to 4c). We report the standard deviation for the “Counting Substructures” experiments (Table 1) in Appendix G.   
+(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 Appendix G.
+
+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? [Yes] See Appendix G.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(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...
+
+(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]

parse/dev/sc7bBHAmcN/sc7bBHAmcN_content_list.json

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+        "type": "text",
+        "text": "Understanding and Extending Subgraph GNNs by Rethinking Their Symmetries ",
+        "text_level": 1,
+        "bbox": [
+            196,
+            122,
+            805,
+            172
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Fabrizio Frasca∗ Imperial College London & Twitter ffrasca@twitter.com ",
+        "bbox": [
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+            267
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Beatrice Bevilacqua∗ Purdue University bbevilac@purdue.edu ",
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+        ],
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+    {
+        "type": "text",
+        "text": "Michael M. Bronstein University of Oxford & Twitter mbronstein@twitter.com ",
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+            477,
+            330
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Haggai Maron NVIDIA Research hmaron@nvidia.com ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            367,
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+            382
+        ],
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+    },
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+        "text": "Subgraph GNNs are a recent class of expressive Graph Neural Networks (GNNs) which model graphs as collections of subgraphs. So far, the design space of possible Subgraph GNN architectures as well as their basic theoretical properties are still largely unexplored. In this paper, we study the most prominent form of subgraph methods, which employs node-based subgraph selection policies such as ego-networks or node marking and deletion. We address two central questions: (1) What is the upper-bound of the expressive power of these methods? and (2) What is the family of equivariant message passing layers on these sets of subgraphs?. Our first step in answering these questions is a novel symmetry analysis which shows that modelling the symmetries of node-based subgraph collections requires a significantly smaller symmetry group than the one adopted in previous works. This analysis is then used to establish a link between Subgraph GNNs and Invariant Graph Networks (IGNs). We answer the questions above by first bounding the expressive power of subgraph methods by 3-WL, and then proposing a general family of message-passing layers for subgraph methods that generalises all previous node-based Subgraph GNNs. Finally, we design a novel Subgraph GNN dubbed SUN, which theoretically unifies previous architectures while providing better empirical performance on multiple benchmarks. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Message Passing Neural Networks (MPNNs) are arguably the most commonly used version of Graph Neural Networks (GNNs). The limited expressive power of MPNNs [36, 55] has led to a plethora of works aimed at designing expressive GNNs while maintaining the simplicity and scalability of MPNNs [11, 39, 49, 30]. Several recent studies have proposed a new class of such architectures [14, 59, 7, 61, 43, 42], dubbed Subgraph GNNs, which apply MPNNs to collections (‘bags’) of subgraphs extracted from the original input graph and then aggregate the resulting representations. Subgraphs are selected according to a predefined policy; in the most popular ones, each subgraph is tied to a specific node in the original graph, for example by deleting it or extracting its local ego-network. Subgraph GNNs have demonstrated outstanding empirical performance, with state-of-the-art results on popular benchmarks like the ZINC molecular property prediction [61, 7]. ",
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+        "type": "text",
+        "text": "While offering great promise, it is fair to say that we still lack a full understanding of Subgraph GNNs. Firstly, on the theoretical side, it is known that subgraph methods are strictly stronger than the Weisfeiler-Leman (WL) test [54, 39], but an upper-bound on their expressive power is generally unknown. Secondly, on a more practical level, Subgraph GNN architectures differ considerably in the way information is aggregated and shared across the subgraphs, and an understanding of the possible aggregation and sharing rules is missing. Both aspects are important: an understanding of the former can highlight the limitations of emerging architectures, a study of the latter paves the way for improved Subgraph GNNs. ",
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+        "text": "Main contributions. The goal of this paper is to provide a deeper understanding of node-based Subgraph GNNs in light of the two aforementioned aspects. The main theoretical tool underpinning our contributions is a novel analysis of the symmetry group that acts on the sets of subgraphs. While several previous approaches [43, 14, 7] have (often implicitly) assumed that a subgraph architecture should be equivariant to independent node and subgraph permutations, we leverage the fact that nodebased policies induce an inherent bijection between the subgraphs and the nodes. This observation allows us to align the two groups and model the symmetry with a single (smaller) permutation group that acts on nodes and subgraphs jointly. Other works [61, 56, 59] have (again, implicitly) recognised such node-subgraph correspondence but without studying the implications on the symmetry group, and resorting, as a result, to a partial and heuristic choice of equivariant operations. ",
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+        "type": "text",
+        "text": "The use of this stricter symmetry group raises a fruitful connection with $k$ -order Invariant Graph Networks (k-IGNs) [33, 32], a well studied family of architectures for processing graphs and hypergraphs designed to be equivariant to the same symmetry group. This connection allows us to transfer and reinterpret previous results on IGNs to our Subgraph GNN setup. As our first contribution we show that the expressive power of Subgraph GNNs with node-based policies is bounded by that of the 3-WL test. This is shown by proving that all previous Subgraph GNNs can be implemented by a 3-IGN and by leveraging the fact that the expressive power of these models is bounded by 3-WL [21, 5]. ",
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+        "type": "text",
+        "text": "Our second contribution is the proposal of a general layer formulation for Subgraph GNNs, based on the observation that these methods maintain an $n \\times n$ representation of $n$ subgraphs with $n$ nodes, following the same symmetry structure of 2-IGNs (same permutation applied to both rows and columns of this representation). We propose a novel extension of 2-IGNs capturing both local (message-passing-like) and global operations. This extension easily recovers previous methods facilitating their comparison. Also, we present a number of new operations that previous methods did not implement. We build upon these observations to devise a new Subgraph GNN dubbed SUN, (Subgraph Union Network). We prove that SUN generalises all previous node-based Subgraph GNNs and we empirically compare it to these methods, showing it can outperform them. ",
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+        "type": "text",
+        "text": "2 Previous and related work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Expressive power of GNNs. The expressive power of GNNs is a central research focus since it was realised that message-passing type GNNs are constrained by the expressivity of the WL isomorphism test [36, 55]. Other than the aforementioned subgraph-based methods, numerous approaches for more powerful GNNs have been proposed, including positional and structural encodings [1, 45, 11, 17, 28, 31], higher-order message-passing schemes [36, 38, 10, 9], equivariant models [24, 33, 32, 53, 15, 51, 40]. We refer readers to the recent survey by Morris et al. [39] for additional details. Finally we note that, in a related and concurrent work, Qian et al. [46] propose a theoretical framework to study the expressive power of subgraph-based GNNs by relating them to the $\\mathtt { k - W L }$ hierarchy, and explore how to sample subgraphs in a data-driven fashion. ",
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+        "type": "text",
+        "text": "Invariant graph networks. IGNs were recently introduced in a series of works by Maron et al. [33, 32, 34] as an alternative to MPNNs for processing graph and hyper-graph data. For $k \\geq 2$ , $\\mathtt { k }$ -IGNs represent hyper-graphs with hyper-edges up to size $k$ with $k$ -order tensor $\\bar { \\boldsymbol { y } } \\in \\mathbb { R } ^ { n ^ { k } }$ , where each entry holds information about a specific hyper-edge. On these they apply linear $S _ { n }$ -equivariant layers interspersed with pointwise nonlinearities. These models have been thoroughly studied in terms of: (i) their expressive power; (ii) the space of their equivariant linear layers. As for (i), IGNs were shown to have exactly the same graph separation power as the $\\mathtt { k - W L }$ graph isomorphism test [32, 5, 21] and, for sufficiently large $k$ , to have a universal approximation property w.r.t. $S _ { n }$ -invariant and equivariant functions [34, 26, 47]. Concerning (ii), the work in [33] completely characterised the space of linear layers equivariant to $S _ { n }$ from $\\mathbb { R } ^ { n ^ { k } }$ to $\\mathbb { R } ^ { n ^ { k ^ { \\prime } } }$ : the authors derived a basis of bel $1 ( k + k ^ { \\prime } )$ linear operators consisting of indicator tensors of equality patterns over the multi-index set $\\{ 1 , . . . , n \\} ^ { k + k ^ { \\prime } } = [ n ] ^ { k + k ^ { \\prime } }$ . Albooyeh et al. [2] showed these layers can be (re-)written as sums of pooling-broadcasting operations between elements of $\\mathcal { V }$ indexed by the orbits 2 of the action of $S _ { n }$ on $[ n ] ^ { k }$ and $[ n ] ^ { k ^ { \\prime } }$ . Take, e.g., $k = k ^ { \\prime } = 2$ . In this case there are only two orbits: $\\{ i , i \\} , i \\in [ n ]$ corresponding to on-diagonal terms, and $\\{ i , j \\} , i \\neq j \\in [ n ] ,$ , off-diagonal terms. According to Albooyeh et al. [2] any equivariant linear layer $L : \\mathbb { R } ^ { n ^ { 2 } } \\to \\mathbb { R } ^ { n ^ { 2 } }$ can be represented as a composition of pooling and broadcasting operations on the elements indexed by these orbits. One example is the linear map that sums the on-diagonal elements and broadcasts the result to the off-diagonal ones: $\\begin{array} { r } { L ( \\mathcal { Y } ) _ { i j } = \\sum _ { k } \\mathcal { Y } _ { k k } } \\end{array}$ for $i \\neq j$ , 0 otherwise. See Appendix B, for additional details. These results particularly important as they underpin most of our theoretical derivations. Lastly, a more comprehensive coverage of IGNs can be found in [39]. ",
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+        "type": "text",
+        "text": "Subgraph GNNs. Despite motivated by diverse premises, a collection of concurrent methods share the overarching design whereby graphs are modelled through the application of a GNN to their subgraphs. Bevilacqua et al. [7] first explicitly formulated the concept of bags of subgraphs generated by a predefined policy and studied layers to process them in an equivariant manner: the same GNN can encode each subgraph independently (DS-GNN), or information can be shared between these computations in view of the alignment of nodes across the bag [35] (DSS-GNN). Building upon the Reconstruction Conjecture [25, 52], Reconstruction GNNs [14] obtain node-deleted subgraphs, process them with a GNN and then aggregate the resulting representations by means of a set model. Nested GNNs [59] and GNN-As-Kernel models (GNN-AK) [61] shift their computation from rooted subtrees to rooted subgraphs, effectively representing nodes by means of GNNs applied to their enclosing ego-networks. Similarly to DSS-GNNs [7], GNN-AK models may feature information sharing modules aggregating node representations across subgraphs. ID-GNNs [56] also process ego-network subgraphs, but their roots are ‘marked’ so to specifically alter the exchange of messages involving them. Intuitively, the use of subgraphs implicitly breaks those local symmetries which determine the notorious expressiveness bottleneck of MPNNs. We note that other works can be interpreted as Subgraph GNNs, including those by Papp et al. [43], Papp and Wattenhofer [42]. ",
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+        "type": "text",
+        "text": "3 Node-based Subgraph GNNs ",
+        "text_level": 1,
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+        "text": "Notation. Let $G = ( A , X )$ be a member of the family $\\mathcal { G }$ of node-attributed, undirected, finite, simple graphs3. The adjacency matrix $A \\in \\mathbb { R } ^ { n \\times n }$ represents $G$ ’s edge set $E$ over its set of $n$ nodes $V$ . The feature matrix $\\dot { X } \\in \\mathbb { R } ^ { \\dot { n } \\times d }$ gathers the node features; we denote by $x _ { j } \\in \\mathbb { R } ^ { d \\times 1 }$ the features of node $j$ corresponding to the $j$ -th row of $X$ . $B _ { G }$ is used to denote a multiset (bag) of $m$ subgraphs of $G$ . Adjacency and feature matrices for subgraphs in $B _ { G }$ are arranged in tensors $\\mathcal { A } \\in \\mathbb { R } ^ { m \\times n \\times n }$ and $\\mathcal { X } \\in \\mathbb { R } ^ { m \\times n \\times d }$ . Superscript $i , ( t )$ refers to representations on subgraph $i$ at the $t$ -th layer of a stacking, as in xi,j $x _ { j } ^ { i , ( t ) }$ . Finally, we denote $[ n ] = \\{ 1 , \\dots , n \\}$ . All proofs are deferred to Appendices $\\mathbf { B }$ and D. ",
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+        "type": "text",
+        "text": "Formalising Subgraph GNNs. Subgraph GNNs compute a representation of $G \\in { \\mathcal { G } }$ as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/f17b656f2e3e574547dbfce57cd2a8a49ccdfbe05e5061a8f2f4d509c0abee36.jpg",
+        "text": "$$\n( A , X ) \\mapsto \\left( \\mu \\circ \\rho \\circ S \\circ \\pi \\right) ( A , X ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Here, $\\pi : G \\mapsto \\{ G ^ { 1 } , . . . , G ^ { m } \\} = \\{ ( A ^ { 1 } , X ^ { 1 } ) , . . . , ( A ^ { m } , X ^ { m } ) \\} = B _ { G } ^ { ( 0 ) }$ is a selection policy generating a bag of subgraphs from $G$ ; $\\begin{array} { r } { S = L _ { T } \\circ \\dots \\circ L _ { 1 } : B _ { G } ^ { ( 0 ) } \\mapsto B _ { G } ^ { ( T ) } } \\end{array}$ is a stacking of (node- and subgraph-) permutation equivariant layers; $\\rho : ( G , B _ { G } ^ { ( T ) } ) \\mapsto x _ { G }$ is a permutation invariant pooling function, $\\mu$ is an MLP. The layers in $s$ comprise a base-encoder in the form of a GNN applied to subgraphs; throughout this paper, we assume it to be a $1 - \\mathrm { { W L } }$ maximally expressive MPNN such as the one in Morris et al. [36]. Subgraph GNNs differ in the implementation of $\\pi$ , $s$ and, in some cases, $\\rho$ . For example, in (n-1)-Reconstruction GNNs [14], $\\pi$ selects node-deleted subgraphs and $s$ applies a Siamese MPNN to each subgraph independently. To exemplify the variability in $s$ , DSS-GNN [7] extends this method with cross-subgraph node and connectivity aggregation. More details are on how currently known Subgraph GNNs are captured by Equation (1) can be found in Appendix A. ",
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+        "type": "text",
+        "text": "Node-based selection policies. In this work, we focus on a specific family of node-based subgraph selection policies, wherein every subgraph is associated with a unique node in the graph. Formally, we call a subgraph selection policy node-based if it is of the form $\\bar { \\pi ( G ) } = \\{ f ( G , v ) \\} _ { v \\in V }$ , for some selection function $f ( G , v )$ that takes a graph $G$ and a node $v$ as inputs and outputs a subgraph $G ^ { v }$ . In the following, we refer to $v$ as the root of subgraph $G ^ { v }$ . We require $f$ to be a bijection and we note that such policies produce $m = n$ different subgraphs. Amongst the most common examples are node-deletion (ND), node-marking (NM), and ego-networks (EGO) policies. For input graph $G$ , $f _ { \\mathrm { N D } } ( G , v )$ removes node $v$ and the associated connectivity; $f _ { \\mathrm { N M } } ( G , v )$ adds a special ‘mark’ attribute to $v$ ’s features (with no connectivity alterations), and $f _ { \\mathrm { E G O } ( h ) } ( G , v )$ returns the subgraph induced by the $h$ -hop-neighbourhood around the root $v$ . EGO policies can be ‘marked’: $f _ { \\mathrm { E G O + } ( h ) } ( G , v )$ extracts the $h$ -hop ego-net around $v$ and marks this node as done by $f _ { \\mathrm { N M } }$ . For convenience, we denote the class of such node-based selection policies by $\\Pi$ : ",
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+        "type": "image",
+        "img_path": "images/3ce1fcf4bad468506f6ba5f0f71244f25641e307936926f5643bede808439499.jpg",
+        "image_caption": [
+            "Figure 1: Symmetries of bags of subgraphs (left) and corresponding function space diagrams (right). In ESAN [7] symmetries are modelled as a direct product of node and subgraph permutation groups; however, node-based policies enable the use of one single permutation group, the same as in 3-IGNs. 3-IGNs are less constrained, thus more expressive than ESAN and other Subgraph GNNs. See diagram on the right and formal statement in Section 5. "
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+        "text": "Definition 1 (Known node-based selection policies $\\Pi$ ). Let $\\Sigma$ be the set of all node-based subgraph selection policies operating on $\\mathcal { G }$ . Class $\\Pi \\subset \\Sigma$ collects the node-based policies node-deletion $( N D )$ , node-marking (NM), ego-nets $( E G O )$ and marked ego-nets $( E G O + )$ of any depth: $\\Pi =$ $\\{ \\pi _ { \\mathrm { N D } } , \\pi _ { \\mathrm { N M } } , \\pi _ { \\mathrm { E G O } ( h ) } , \\pi _ { \\mathrm { E G O } + ( h ) } \\mid h > 0 \\}$ . ",
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+        "type": "text",
+        "text": "Node-based Subgraph GNNs are those Subgraph GNNs which, implicitly or explicitly, process bags generated by node-based policies. We group known formulations in the following family: ",
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+    {
+        "type": "text",
+        "text": "Definition 2 (Known node-based Subgraph GNNs $\\Upsilon$ ). Let Ξ be the set of all node-based Subgraph GNNs. Class $\\Upsilon \\subset \\Xi$ collects known Subgraph GNNs when equipped with 1-WL base-encoders: $\\Upsilon = \\lbrace ( \\mathtt { n - l } )$ -Reconstr.GNN, GNN-AK, GNN-AK-ctx, NGNN, ID-GNN, DS- $\\mathrm { G N N } _ { \\Pi }$ , $\\mathrm { D S S - G N N _ { \\Pi } } \\big \\}$ $\\mathrm { D S - G N N _ { \\Pi } }$ , DSS-GNNΠ refer to DS- and DSS-GNN models equipped with any $\\pi \\in \\Pi$ . ",
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+        "type": "text",
+        "text": "Importantly, all these methods apply MPNNs to subgraphs of the original graph, but differ in the way information is shared between subgraphs/nodes. In all cases, their expressive power is strictly larger than 1-WL, but an upper-bound is currently unknown. ",
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+        "type": "text",
+        "text": "4 Symmetries of node-based subgraph selection policies ",
+        "text_level": 1,
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+        "text": "In an effort to characterise the representational power of node-based Subgraph GNNs, we first study the symmetry group of the objects they process: ‘bags of subgraphs’ represented as tensors $( { \\mathcal { A } } , { \\mathcal { X } } ) \\in$ $\\mathbb { R } ^ { m \\times n \\times n } \\times \\mathbb { R } ^ { m \\times n \\times d }$ , assuming $n$ nodes across $m$ subgraphs. Previous approaches [14, 7, 43] used two permutation groups: one copy of the symmetric group $S _ { n }$ models node permutations, while another copy $S _ { m }$ models subgraph permutations in the bag. These two were combined by a group product4 acting independently on the nodes and subgraphs in $( { \\mathcal { A } } , { \\mathcal { X } } )$ . For example, Bevilacqua et al. [7] model the symmetry as: ",
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+        "img_path": "images/9eb54401952957c418d21a5e84b0e1e6953a91ad9a4bc1f23ccd54bef8ad17d7.jpg",
+        "text": "$$\n\\begin{array} { r } { ( ( \\tau , \\mathbf { \\mu } ) \\cdot \\mathcal { A } ) _ { i j k } = \\mathcal { A } _ { \\tau ^ { - 1 } ( i ) } \\cdot \\mathbf { \\mu } ^ { - 1 } ( j ) \\cdot \\mathbf { \\mu } ^ { - 1 } ( k ) , \\left( ( \\tau , \\mathbf { \\mu } ) \\cdot \\mathcal { X } \\right) _ { i j l } = \\mathcal { X } _ { \\tau ^ { - 1 } ( i ) } \\cdot \\mathbf { \\mu } ^ { - 1 } ( j ) l , \\left( \\tau , \\mathbf { \\mu } \\right) \\in S _ { m } \\times S _ { m } } \\end{array}\n$$",
+        "text_format": "latex",
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+        "text": "Our contributions stem from the following crucial observation: When using node-based policies, the subgraphs in $( { \\mathcal { A } } , { \\mathcal { X } } )$ can be ordered consistently with the nodes by leveraging the bijection $f : v \\mapsto G _ { v }$ characterising this policy class. In other words, $f$ suggests a node-subgraph alignment inducing a new structure on $( { \\mathcal { A } } , { \\mathcal { X } } )$ , whereby the subgraph order is not independent of that of nodes anymore. Importantly, this new structure is preserved only by those permutations operating identically on both nodes and subgraphs. Following this observation, the symmetry of a node-based bag of subgraphs is modelled more accurately using only one single permutation group $S _ { n }$ jointly acting on both nodes and subgraphs: ",
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+        "image_caption": [
+            "Figure 2: Depiction of cubed tensor $\\mathcal { V }$ , its orbit-induced partitioning and the related semantics when $\\mathcal { V }$ is interpreted as a bag of node-based subgraphs, $n = 5$ . Elements in the same partition are depicted with the same colour. Left: the whole tensor. Middle and right: sections; elements in purple and green constitute sub-tensor $\\mathcal { X }$ , the remaining ones sub-tensor $\\mathcal { A }$ . "
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+        "img_path": "images/96a92ec538c88198f40c70251f4d9aa42877de1946419d38b1addfa5f3ef1adf.jpg",
+        "text": "$$\n( \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } ^ { - 1 } ( i ) \\mathbf { \\partial } \\cdot \\mathbf { \\partial } ^ { - 1 } ( k ) , \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial } \\cdot \\mathbf { \\partial }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "It should be noted that $S _ { n }$ is significantly smaller than $S _ { n } \\times S _ { n } ^ { \\phantom { \\dagger } 5 }$ . Informally, the latter group contains many permutations which are not in the former: those acting differently on nodes and subgraphs and, thus, not preserving the new structure of $( { \\mathcal { A } } , { \\mathcal { X } } )$ . Since they are restricted by a smaller set of equivariance constraints, we expect GNNs designed to be equivariant to $S _ { n }$ to be be more expressive than those equivariant to the larger groups considered by previous works [34] (see Figure 1). ",
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+        "type": "text",
+        "text": "The insight we obtain from Equation (3) is profound: it reveals that the symmetry structure of $\\mathcal { A }$ exactly matches the symmetries of third-order tensors used by 3-IGNs, and similarly, that the symmetry structure for $\\mathcal { X }$ matches the symmetries of second-order tensors used by 2-IGNs. In the following, we will make use of this insight and the fact that IGNs are well-studied objects to prove an upper-bound on the expressive power of Subgraph GNNs and to design principled extensions to these models. We remark that bags of node-based subgraphs can also be represented as tensors $\\mathcal { V } \\in \\mathbb { R } ^ { n ^ { 3 } \\times d }$ , the same objects on which 3-IGNs operate. Here, $\\mathcal { X }$ is embedded in the main diagonal plane of $\\mathcal { V }$ , $\\mathcal { A }$ in its remaining entries. Within this context, it is informative to study the semantics of the 5 orbits induced by the action of $S _ { n }$ on $\\mathcal { V }$ ’s multi-index set $[ n ] ^ { 3 }$ : each of these uniquely identify root nodes, non-root nodes, edges to and from root nodes as well as edges between non-root nodes (see Figure 2 and additional details in Appendix B.1.1). We build upon this observation, along with the layer construction by Albooyeh et al. [2], to prove many of the results presented in the following. ",
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+        "type": "text",
+        "text": "5 A representational bound for Subgraph GNNs ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In this section we prove that the expressive power of known node-based Subgraph GNNs is bounded by $3 - \\mathrm { { W L } }$ by showing that they can be implemented by 3-IGNs, which have the same expressive power as $3 - \\mathbb { W } \\mathbb { L }$ . Underpinning the possibility of IGNs to upper-bound a certain Subgraph GNN $\\mathcal { N }$ in its expressive power is the ability of IGNs to (i) implement $\\mathcal { N }$ ’s subgraph selection policy $( \\pi )$ and (ii) implement $\\mathcal { N }$ ’s (generalised) message-passing and pooling equations $( \\mu \\circ \\rho \\circ S )$ . This would ensure that whenever $\\mathcal { N }$ assigns distinct representations to two non-isomorphic graphs, an IGN implementing $\\mathcal { N }$ would do the same. We start by introducing a recurring, useful concept. ",
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+        "type": "text",
+        "text": "Definition 3 (“implements”). Let $f : D _ { f } \\to C _ { f }$ , $g : D _ { g } \\to C _ { g }$ be two functions and such that $D _ { g } \\subseteq D _ { f } , C _ { g } \\subseteq C _ { f }$ . We say $f$ implements $g$ (and write $f \\cong g ,$ ) when $\\forall x \\in D _ { g } , f ( x ) = g ( x )$ . ",
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+    {
+        "type": "text",
+        "text": "Our first result shows that 3-IGNs can implement the selection policies in class $\\Pi$ (Definition 1), which, to the best of our knowledge, represent all known node-based policies utilised by previously proposed Subgraph GNNs. ",
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+        "type": "text",
+        "text": "Lemma 4 (3-IGNs implement known node-based selection policies). For any $\\pi \\in \\Pi$ there exists a stacking of 3-IGN layers $\\mathcal { M } _ { \\pi }$ s.t. $\\mathcal { M } _ { \\pi } \\cong \\pi$ . ",
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+        "type": "text",
+        "text": "Intuitively, 3-IGNs start from a $\\mathbb { R } ^ { n ^ { 2 } }$ representation of $G$ and, first, move to a $\\mathbb { R } ^ { n ^ { 3 } }$ tensor ‘copying’ this latter along its first (subgraph) dimension. This is realised via an appropriate broadcast operation. Then, they proceed by adding a ‘mark’ to the features of some nodes and/or by nullifying elements corresponding to some edges. We refer readers to Figure 2 and Appendix B.1.2 for additional details on how nodes in each subgraph are represented in 3-IGNs. Next, we show 3-IGNs can implement layers of any model $\\in \\Upsilon$ . ",
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+    {
+        "type": "text",
+        "text": "Lemma 5 (3-IGNs implement Subgraph GNN layers). Let $G _ { 1 } , G _ { 2 }$ be two graphs in $\\mathcal { G }$ and $\\mathcal { N }$ a model in family be bags of subg $\\Upsilon$ equipped with Morris et al. [36] messagehs in the input of some intermediate layer asin g base-encoders. Let . Then there exists a $B _ { 1 } ^ { ( t ) } , B _ { 2 } ^ { ( t ) }$ $L$ $\\mathcal { N }$ 3-IGN layers $\\mathcal { M } _ { L }$ for which $\\begin{array} { r } { \\dot { \\mathcal { M } } _ { L } ( { B } _ { i } ^ { ( t ) } ) = { B } _ { i } ^ { ( t + 1 ) } = { L } ( { B } _ { i } ^ { ( t ) } ) } \\end{array}$ for $i = 1 , 2$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Lemmas 4 and 5 allow us to upper-bound the expressive power of all known instances of node-based Subgraph GNNs by that of 3-IGNs: ",
+        "bbox": [
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+        "type": "text",
+        "text": "Theorem 6 (3-IGNs upper-bound node-based Subgraph GNNs). For any pair of non-isomorphic graphs $G _ { 1 } , G _ { 2 }$ in family $\\mathcal { G }$ and Subgraph GNN model $\\mathcal { N } \\in \\Upsilon$ equipped with Morris et al. [36] message-passing base-encoders, if there exists weights $\\Theta$ such that $G _ { 1 }$ , $G _ { 2 }$ are distinguished by instance $\\mathcal { N } _ { \\Theta }$ , then there exist weights Ω for a 3-IGN instance $\\mathcal { M } _ { \\Omega }$ such that $G _ { 1 } , G _ { 2 }$ are distinguished by $\\mathcal { M } _ { \\Omega }$ as well. ",
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+    {
+        "type": "text",
+        "text": "Theorem 6 has profound consequences in the characterisation of the expressive power of node-based Subgraph GNNs, as we show in the following ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Corollary 7 (3-WL upper-bounds node-based Subgraph GNNs). Let $G _ { 1 } , G _ { 2 } \\ \\in \\ { \\mathcal { G } }$ be two nonisomorphic graphs and $\\mathcal { N } _ { \\Theta } \\in \\Upsilon$ one instance of model $\\mathcal { N }$ with weights $\\Theta$ . If $\\mathcal { N } _ { \\Theta }$ distinguishes $G _ { 1 } , G _ { 2 }$ , then the 3-WL algorithm does so as well. ",
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+    {
+        "type": "text",
+        "text": "Proof idea: If there is a pair of graphs undistinguishable by 3-WL, but for which there exists a Subgraph GNN separating them, there must exists a 3-IGN separating these (Theorem 6). This is in contradiction with the result by Geerts [21], Azizian and Lelarge $[ 5 ] ^ { 6 }$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "6 A design space for Subgraph GNNs ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As discussed, different formulations of Subgraph GNNs differ primarily in the specific rules for updating node representations across subgraphs. However, until now it is not clear whether existing rules exhaust all the possible equivariant options. We devote this section to a systematic characterisation of the ‘layer space’ of Subgraph GNNs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the spirit of the previous Section 5, where we “embedded” Subgraph GNNs in 3-IGNs, one option would be to consider all $\\mathsf { b e l l } ( 6 ) = 2 0 3$ linear equivariant operations prescribed by this formalism. However, this choice would be problematic for three main reasons: (i) This layer space is too vast to be conveniently explored; (ii) It includes operations involving $\\mathcal { O } ( n ^ { 3 } )$ space complexity, impractical in most applications; (iii) The linear IGN basis does not directly support local message passing, a key operation in subgraph methods. Following previous Subgraph GNN variants, which use $\\mathcal { O } ( { \\bar { n } } ^ { 2 } )$ storage for the representation of $n$ nodes in $n$ subgraphs, we set the desideratum of $\\mathcal { O } ( n ^ { 2 } )$ memory complexity as our main constraint, and use this restriction to reduce the design space. Precisely, we are interested in modelling $S _ { n }$ -equivariant transformations on the subgraph-node tensor $\\mathcal { X }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6.1 Extended 2-IGNs ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As we have already observed in Equation 3 in Section 4, such a second order tensor $\\mathcal { X }$ abides by the same symmetry structure of 2-IGNs. We therefore gain intuition from the characterisation of linear equivariant mappings as introduced by Maron et al. [33], and propose an extension of this formalism. ",
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+    {
+        "type": "image",
+        "img_path": "images/7462b6faa7624521d59d4391ded56346740f6471db88f1502e7e2d6cbc3e92d5.jpg",
+        "image_caption": [
+            "Figure 3: Comparison of aggregation and update rules in Subgraph GNNs, illustrated on an $n \\times n$ matrix ( $\\scriptstyle { n }$ subgraphs with $n$ nodes). Top row: off-diagonal updates; bottom row: diagonal (root node) updates. Each colour represents a different parameter. Full squares: global sum pooling; triangles: local pooling; two triangles: both local and global pooling. See Appendix C for more details. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "2-IGN layer space. A 2-IGN layer $L _ { \\Theta }$ updates $\\mathcal { X } \\in \\mathbb { R } ^ { n \\times n \\times d }$ as $\\mathcal { X } ^ { ( t + 1 ) } = L _ { \\Theta } \\big ( \\mathcal { X } ^ { ( t ) } \\big )$ by applying a specific transformation to on- $( x _ { i } ^ { i } )$ and off-diagonal terms $( x _ { j } ^ { i } , i \\neq j )$ : ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "equation",
+        "img_path": "images/289b866280828ea3f27c1cb4386a87140165adcad4fddb4cce9193c0d3d9065c.jpg",
+        "text": "$$\n\\begin{array} { r l } & { x _ { i } ^ { i , ( t + 1 ) } { = } v _ { \\theta _ { 1 } } \\big ( x _ { i } ^ { i , ( t ) } , \\underset { j } { \\sum } x _ { j } ^ { j , ( t ) } , \\underset { j \\neq i } { \\prod } x _ { j } ^ { i , ( t ) } , \\underset { h \\neq i } { \\prod } x _ { i } ^ { h , ( t ) } , \\underset { h \\neq j } { \\prod } x _ { j } ^ { h , ( t ) } \\big ) } \\\\ & { x _ { i } ^ { k , ( t + 1 ) } { = } v _ { \\theta _ { 2 } } \\big ( x _ { i } ^ { k , ( t ) } , x _ { k } ^ { i , ( t ) } , \\underset { h \\neq j } { \\prod } x _ { j } ^ { h , ( t ) } , \\underset { h \\neq i } { \\prod } x _ { i } ^ { h , ( t ) } , \\underset { j \\neq k } { \\prod } x _ { j } ^ { k , ( t ) } , \\underset { j \\neq i } { \\prod } x _ { j } ^ { i , ( t ) } , \\underset { h \\neq k } { \\prod } x _ { k } ^ { h , ( t ) } , x _ { k } ^ { k , ( t ) } , x _ { k } ^ { i , ( t ) } , x _ { i } ^ { i , ( t ) } , \\underset { j } { \\prod } x _ { j } ^ { j , ( t ) } \\big ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Here, $\\boxed { \\begin{array} { r l } \\end{array} }$ indicates a permutation invariant aggregation function, $v _ { \\theta _ { 1 } } , v _ { \\theta _ { 2 } }$ apply a specific $d \\times d ^ { \\prime }$ linear transformation to each input term and sum the outputs including bias terms. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "ReIGN(2) layer space. As 2-IGN layers are linear, the authors advocate setting $\\sqcap \\equiv \\Sigma$ , performing pooling as global summation. Here, we extend this formulation to additionally include different local aggregation schemes. In this new extended formalism, entry $x _ { i } ^ { k }$ represents node $i$ in subgraph $k$ ; accordingly, each aggregation in Equation 4 can be also performed locally, i.e. extending only over $\\ddot { \\iota }$ ’s neighbours, as prescribed by the connectivity of subgraph $k$ or of the original input graph. As an example, when updating entry $x _ { i } ^ { k , ( t ) }$ , term □j̸=k xk,j is expanded as $\\begin{array} { r } { \\Big ( \\bigtriangledown _ { j \\neq k } x _ { j } ^ { k , ( t ) } , \\bigtriangledown _ { j \\sim _ { k } i } x _ { j } ^ { k , ( t ) } , \\bigtriangledown _ { j \\sim i } x _ { j } ^ { k , ( t ) } . } \\end{array}$ , with $\\sim _ { k }$ denoting adjacency in subgraph $k$ , and $\\sim$ that in the original graph connectivity. Each term in the expansion is associated with a specific learnable linear transformation. We report a full list of pooling operations in Appendix D, Table 3. These local pooling operations allow to readily recover sparse message passing, which constitutes the main computational primitive of all popular (Subgraph) GNNs. Other characteristic Subgraph GNN operations are also recovered by this formalism: for example, $\\textstyle \\prod _ { h \\neq i } x _ { i } ^ { h , ( t ) }$ operates global pooling of node $i$ ’s representations across subgraphs, as previously introduced in Bevilacqua et al. [7], Zhao et al. [61]. We also note that additional, novel, operations are supported, e.g. the transpose $x _ { k } ^ { i , ( t ) }$ . We generally refer to this framework as ReIGN(2) (“Rethought 2-IGN”). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "ReIGN(2) architectures. ReIGN(2) induces (linear) layers in the same form of Equation 4, but where $\\boxed { \\begin{array} { r l } \\end{array} }$ terms are expanded to both local and global operations, as explained. These layers can operate on any bag generated by a node-based selection policy $\\bar { \\pi }$ , and can be combined together in ReIGN(2) stacks of the form $\\mathcal S _ { \\mathcal R } = L ^ { ( T ) } \\circ \\sigma \\circ L ^ { ( T - 1 ) } \\circ \\sigma \\circ . . . \\circ \\sigma \\circ L ^ { ( 1 ) } ,$ , where $\\sigma$ ’s are pointwise nonlinearities and $L$ ’s are ReIGN(2) layers. This allows us to define ReIGN(2) models as Subgraph GNNs in the form of Equation 1, where $s$ is a ReIGN(2) layer stacking and $\\pi$ is node-based: $\\mathcal { R } _ { \\bar { \\pi } } = \\mu \\circ \\rho \\circ S _ { \\mathcal { R } } \\circ \\bar { \\pi }$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "More generally, ReIGN(2) induces a ‘layer space’ for node-based Subgraph GNNs: the expanded terms in its update equations represent a pool of atomic operations that can be selected and combined to define new equivariant layers. Compared to that of 3-IGNs, this space is of tractable size, yet it recovers previously proposed Subgraph GNNs and allows to define novel interesting variants. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Recovering previous Subgraph GNNs. The following result states that the ReIGN(2) generalises all known subgraph methods in $\\Upsilon$ , as their layers are captured by a ReIGN(2) stacking. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 8 (ReIGN(2) implements node-based Subgraph GNNs). Let N be a model in family $\\Upsilon$ equipped with Morris et al. [36] message-passing base-encoders. For any instance $\\mathcal { N } _ { \\Theta }$ , there exists ReIGN(2) instance $\\mathcal { R } _ { \\Omega }$ such that $\\mathcal { R } _ { \\Omega } \\cong \\mathcal { N } _ { \\Theta }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This shows that known methods are generalised without resorting to the $\\mathcal { O } ( n ^ { 3 } )$ computational complexity of 3-IGNs. Figure 3 illustrates the aggregation and sharing rules used by previous Subgraph GNNs to update root and non-root nodes, and compare them with those of ReIGN(2) and 2-IGNs. We visualise these on the subgraph-node sub-tensor gathering node representations across subgraphs; here, root nodes occupy the main diagonal, non-root nodes all the remaining off-diagonal entries. As for to the 2-IGN Equations 4, the elements in these two partitions may be updated differently, so we depict them separately in, respectively, the bottom and top rows. In each depiction we colour elements depending on the set of weights parameterising their contribution in the update process, with two main specifications: (i) Elements sharing the same colour are pooled together; (ii) Triangles indicate such pooling is performed locally based on the subgraph connectivity at hand (two triangles indicate both local and global pooling ops are performed). E.g., note how DS-GNN equivalently updates the representations of root and non-root nodes via the same (local) messagepassing layer (triangles, yellow, leftmost picture). By illustrating how ReIGN(2) generalises previous node-based methods, this figure is to be interpreted as visual support for the Proof of Theorem 8 (see Appendix D). Additional details and discussions on Figure 3 are found in Appendix C. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Notably, as methods in $\\Upsilon$ have been shown to be strictly stronger than 2-WL [7, 14, 61, 59, 56], Theorem 8 implies the same lower bound for ReIGN(2). Nevertheless, when employing policies in $\\Pi$ and 3-IGN-computable invariant pooling functions $\\rho$ (as those used by models in $\\Upsilon$ ), ReIGN(2)s are upper-bounded by 3-IGNs: ",
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+    {
+        "type": "text",
+        "text": "Proposition 9 (3-IGNs implement ReIGN(2)). For any pair of non-isomorphic graphs $G _ { 1 } , G _ { 2 }$ in family $\\mathcal { G }$ , if there exist policy $\\bar { \\pi } \\in \\Pi$ , parameters $\\Theta$ and 3-IGN-computable invariant pooling function $\\rho$ such that the ReIGN(2) instance $\\mathcal { R } _ { \\rho , \\Theta , \\bar { \\pi } }$ distinguishes $G _ { 1 }$ , $G _ { 2 }$ , then there exist weights $\\Omega$ for $a$ 3-IGN instance $\\mathcal { M } _ { \\Omega }$ such that $G _ { 1 } , G _ { 2 }$ are distinguished by $\\mathcal { M } _ { \\Omega }$ as well. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This proposition entails an upper-bound on the expressive power of ReIGN(2). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Corollary 10 (3-WL upper-bounds ReIGN(2)). The expressive power of a ReIGN(2) model with policy $\\pi \\in \\Pi$ and 3-IGN-computable invariant pooling function $\\rho$ is upper-bounded by 3-WL. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We note that there may be layers equivariant to $S _ { n }$ over $\\mathbb { R } ^ { n ^ { 2 } }$ not captured by ReIGN(2). Yet, previously proposed Subgraph GNN layers do not exhaust the ReIGN(2) design space, which remains largely unexplored. One, amongst possible novel constructions, is introduced next. ",
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+    {
+        "type": "text",
+        "text": "6.2 A unifying architecture: Subgraph Union Networks ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "We now show how the ReIGN(2) layer space can guide the design of novel, expressive, Subgraph GNNs. Our present endeavour is to conceive a computationally tractable architecture subsuming known node-based models: in virtue of this latter desideratum, we will dub this architecture “Subgraph Union Network” (SUN). To design the base equivariant layer for SUN, we select and combine specific aggregation terms suggested by the ReIGN(2) framework: ",
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+    {
+        "type": "equation",
+        "img_path": "images/bc35102a7d7121b6a68124e096be0b39f7eb18e0fa207f526dc94943d2ccd4d0.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle x _ { i } ^ { i , ( t + 1 ) } = \\sigma \\Big ( v _ { \\theta _ { 1 } } \\big ( x _ { i } ^ { i , ( t ) } , \\sum _ { j \\sim i } x _ { j } ^ { i , ( t ) } , \\sum _ { j } x _ { j } ^ { i , ( t ) } , \\sum _ { h } x _ { i } ^ { h , ( t ) } , \\sum _ { j \\sim i } \\sum _ { h } x _ { j } ^ { h , ( t ) } \\big ) \\Big ) } } \\\\ { { \\displaystyle x _ { i } ^ { k , ( t + 1 ) } = \\sigma \\Big ( v _ { \\theta _ { 2 } } \\big ( x _ { i } ^ { k , ( t ) } , \\sum _ { j \\sim k i } x _ { j } ^ { k , ( t ) } , x _ { i } ^ { i , ( t ) } , x _ { k } ^ { k , ( t ) } , \\sum _ { j } x _ { j } ^ { k , ( t ) } , \\sum _ { h } x _ { i } ^ { h , ( t ) } , \\sum _ { j \\sim i } \\sum _ { h } x _ { j } ^ { h , ( t ) } \\big ) \\Big ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $\\upsilon$ ’s sum their inputs after applying a specific linear transformations to each term. One of the novel features of SUN is that roots are transformed by a different set of parameters $( \\theta _ { 1 } )$ than the other nodes 7 ( $\\ \\theta _ { 2 }$ , see Figure 2). In practice, the first and last two terms in each one of Equations (5) and (6) can be processed by maximally expressive MPNNs [36, 55], the remaining terms by MLPs. We test these variants in our experiments, with their formulations in Appendix G. SUN remains an instantiation of the ReIGN(2) framework: ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Proposition 11 (A ReIGN(2) stacking implements SUN layers). For any SUN layer $L$ defined according to Equations 5 and 6, there exists a ReIGN(2) layer stacking $ { \\boldsymbol { S } } _ { L }$ , such that $S _ { L } \\cong L$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/a00eed868f26b77cddff0107be8e4138d0c4c7ce131d6f181707e377529e6829.jpg",
+        "table_caption": [
+            "Table 1: Test mean MAE on the Counting Substructures and ZINC-12k datasets. All Subgraph GNNs employ a GIN base-encoder. †This version of GNN-AK $^ +$ does not follow the standard evaluation procedure. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>ZINC (MAE ↓)</td></tr><tr><td>GCN[27]</td><td>0.321 ± 0.009</td></tr><tr><td>GIN [55]</td><td>0.163 ± 0.004</td></tr><tr><td>PNA[13]</td><td>0.133 ± 0.011</td></tr><tr><td>GSN[11] CIN [9]</td><td>0.101 ± 0.010</td></tr><tr><td></td><td>0.079 ± 0.006</td></tr><tr><td>NGNN [59] DS-GNN (EGO) [7]</td><td>0.111 ± 0.003</td></tr><tr><td>DS-GNN (EGO+) [7]</td><td>0.115 ± 0.004</td></tr><tr><td></td><td>0.105 ±0.003</td></tr><tr><td>DSS-GNN (EGO) [7]</td><td>0.099 ±0.003</td></tr><tr><td>DSS-GNN (EGO+) [7]</td><td>0.097 ± 0.006</td></tr><tr><td>GNN-AK[61]</td><td>0.105 ± 0.010</td></tr><tr><td>GNN-AK-CTX [61]</td><td>0.093±0.002</td></tr><tr><td>GNN-AK+ [61]†</td><td>0.086± ???</td></tr><tr><td>GNN-AK+ [61]</td><td>0.091 ± 0.011</td></tr><tr><td>SUN (EGO)</td><td>0.083 ±0.003</td></tr><tr><td>SUN (EGO+)</td><td>0.084 ±0.002</td></tr></table>",
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+        "type": "table",
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+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">Counting Substructures (MAE ↓)</td></tr><tr><td>Triangle</td><td>Tailed Tri.</td><td>Star</td><td>4-Cycle</td></tr><tr><td>GCN [27]</td><td>0.4186</td><td>0.3248</td><td>0.1798</td><td>0.2822</td></tr><tr><td>GIN [55]</td><td>0.3569</td><td>0.2373</td><td>0.0224</td><td>0.2185</td></tr><tr><td>PNA [13]</td><td>0.3532</td><td>0.2648</td><td>0.1278</td><td>0.2430</td></tr><tr><td>PPGN [32]</td><td>0.0089</td><td>0.0096</td><td>0.0148</td><td>0.0090</td></tr><tr><td>GNN-AK [61]</td><td>0.0934</td><td>0.0751</td><td>0.0168</td><td>0.0726</td></tr><tr><td>GNN-AK-CTX [61]</td><td>0.0885</td><td>0.0696</td><td>0.0162</td><td>0.0668</td></tr><tr><td>GNN-AK+[61]</td><td>0.0123</td><td>0.0112</td><td>0.0150</td><td>0.0126</td></tr><tr><td>SUN (EGO)</td><td>0.0092</td><td>0.0105</td><td>0.0064</td><td>0.0140</td></tr><tr><td>SUN (EGO+)</td><td>0.0079</td><td>0.0080</td><td>0.0064</td><td>0.0105</td></tr></table>",
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+        "type": "text",
+        "text": "Finally, we show that a stacking of SUN layers can implement any layer of known node-based Subgraph Networks, making this model a principled generalisation thereof. ",
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+        "text": "Proposition 12 (A SUN stacking implements known Subgraph GNN layers). Let N be a model in family $\\Upsilon$ employing Morris et al. [36] as a message-passing base-encoder. Then, for any layer L in $\\mathcal { N }$ , there exists a stacking of SUN layers $S _ { L }$ such that $S _ { L } \\cong L$ . ",
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+        "type": "text",
+        "text": "Beyond SUN. As it can be seen in Figure 3, SUN does not use all possible operations in the ReIGN(2) framework. Notably, two interesting operations that are not a part of SUN are: (i) The ‘transpose’: $x _ { i } ^ { k } = v _ { \\theta } ( x _ { k } ^ { i } )$ , which shares information between the $i$ -th node in the $k$ -th subgraph and the $k$ -th node in the $i$ -th subgraph; (ii) Local vertical pooling $\\begin{array} { r } { x _ { i } ^ { k } = v _ { \\theta } ( \\sum _ { h \\sim i } x _ { i } ^ { h } ) } \\end{array}$ . The exploration of these and other operations is left to future work. ",
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+        "type": "text",
+        "text": "7 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We experimentally validate the effectiveness of one ReIGN(2) instantiation, comparing SUN to previously proposed Subgraph $\\mathrm { G N N s } ^ { 8 }$ . We seek to verify whether its theoretical representational power practically enables superior accuracy in expressiveness tasks and real-world benchmarks. Concurrently, we pay attention to the generalisation ability of models in comparison. SUN layers are less constrained in their weight sharing pattern, resulting in a more complex model. As this is traditionally associated with inferior generalisation abilities in low data regimes, we deem it important to additionally assess this aspect. Our code is also available.9 ",
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+        "type": "text",
+        "text": "Synthetic. Counting substructures and regressing graph topological features are notoriously hard tasks for GNNs [12, 17, 13]. We test the representational ability of SUN on common benchmarks of this kind [12, 13]. Table 1 reports results on the substructure counting suite, on which SUN attains state-of-the-art results in 3 out of 4 tasks. Additional results on the regression of global, structural properties are reported in Appendix G. ",
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+        "type": "text",
+        "text": "Real-world. On the molecular ZINC-12k benchmark (constrained solubility regression) [50, 22, 16], SUN exhibits best performance amongst all domain-agnostic GNNs under the $5 0 0 \\mathrm { k }$ parameter budget, including other Subgraph GNNs (see Table 1). A similar trend is observed on the large-scale Molhiv dataset from the OGB [23] (inhibition of HIV replication). Results are in Table 2. Remarkably, on both datasets, SUN either outperforms or approaches HIMP [19], GSN [11] and CIN [9], GNNs which explicitly model rings. We experiment on smaller-scale TUDatasets [37] in Appendix G, where we also compare selection policies. ",
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+    {
+        "type": "image",
+        "img_path": "images/c7906ab21719b791330f9739760bfe956c7d9b4463d1306fa25bab6a3a493636.jpg",
+        "image_caption": [
+            "Figure 4: Generalisation capabilities of Subgraph GNNs in the counting prediction task (Figures 4a and 4b) and in the ZINC-12k dataset (Figure 4c). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/7d5356860249deb9de19b8a4b7933490509c6d7d4bf7fd6692006c79ea4eaf8b.jpg",
+        "table_caption": [
+            "Table 2: Test results for OGB dataset. GIN base-encoder for each Subgraph GNN. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>0GBG-MOLHIVROC-AUC(%)</td></tr><tr><td rowspan=1 colspan=1>GCN [27]GIN [55]PNA [13]DGN [6]HIMP [19]GSN[11]CIN [9]</td><td rowspan=1 colspan=1>76.06±0.9775.58±1.4079.05±1.3279.70±0.9778.80±0.8280.39±0.9080.94±0.57</td></tr><tr><td rowspan=1 colspan=1>RECONSTR.GNN[14]DS-GNN (EGO+) [7]DSS-GNN (EGO+) [7]GNN-AK+ [61]</td><td rowspan=1 colspan=1>76.32±1.4077.40±2.1976.78±1.6679.61±1.19</td></tr><tr><td rowspan=1 colspan=1>SUN (EGO+)</td><td rowspan=1 colspan=1>80.03±0.55</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Generalisation from limited data. In this set of experiments we compare the test performance of Subgraph GNNs when trained on increasing fractions of the available training data. Each architecture is selected by tuning the hyperparameters with the entire training and validation sets. We run this experiment on the 4-cycle counting task and the real-world ZINC-12k. We illustrate results in Figures 4a to 4c. Except for a short initial phase in the EGO policy, SUN generalises better than other Subgraph GNNs on cycle-counting. On ZINC-12k, SUN outruns DSS-, DS-GNN and GNN-AK variants from, respectively, 20, 30 and $4 0 \\%$ of the samples. These results demonstrate that SUN’s expressiveness is not at the expense of sample efficiency, suggesting that its modelled symmetries guarantee strong representational power while retaining important inductive biases for learning on graphs. ",
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+    {
+        "type": "text",
+        "text": "8 Conclusions ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our work unifies, extends, and analyses the emerging class of Subgraph GNNs. Notably, we demonstrated that the expressive power of these methods is bounded by 3-WL. Towards a systematic study of models whose expressivity lies between 1- and 3-WL, we proposed a new family of layers for the class of Subgraph GNNs and, unlike most previous works on the expressive power of GNNs, we also investigated the generalisation abilities of these models, for which SUN shows considerable improvement. Appendix E lists several directions for future work, including an extension of our work to higher-order node-based policies. ",
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+    {
+        "type": "text",
+        "text": "Societal impact. We do not envision any negative, immediate societal impact originating from our theoretical results, which represent most of our contribution. Experimentally, our model has shown promising results on molecular property prediction tasks and strong generalisation ability in low-data regimes. This leads us to believe our work may contribute to positively impactful pharmaceutical research, such as drug discovery [20, 3]. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The authors are grateful to Joshua Southern, Davide Eynard, Maria Gorinova, Guadalupe Gonzalez, Katarzyna Janocha for valuable feedback on early versions of the manuscript. They would like to thank Bruno Ribeiro and Or Litany for helpful discussions, Giorgos Bouritsas for constructive conversations about the generalisation experiments and, in particular, Marco Ciccone for the precious exchange on sharpness-aware optimisation and Neapolitan pizza. MB is supported in part by ERC Consolidator grant no 724228 (LEMAN). No competing interests are declared. ",
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+        "type": "text",
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parse/dev/wiBEFdAvl8L/wiBEFdAvl8L.md

@@ -0,0 +1,309 @@
+# GLIPv2: Unifying Localization and VL Understanding
+
+Haotian Zhang∗1†, Pengchuan Zhang $\ast 2 \dagger \spadesuit$ , Xiaowei $\mathbf { H } \mathbf { u } ^ { 3 }$ , Yen-Chun Chen3, Liunian Harold Li4† Xiyang $\mathbf { D a i } ^ { 3 }$ , Lijuan Wang3, Lu Yuan3, Jenq-Neng Hwang1, Jianfeng Gao3
+
+1University of Washington, 2Meta AI, 3Microsoft, 4UCLA {haotiz,hwang}@uw.edu,pengchuanzhang@fb.com,liunian.harold.li@cs.ucla.edu, {Xiaowei.Hu,Yen-Chun.Chen,Xiyang.Dai,lijuanw,luyuan,jfgao}@microsoft.com
+
+# Abstract
+
+We present GLIPv2, a grounded VL understanding model, that serves both localization tasks (e.g., object detection, instance segmentation) and Vision-Language (VL) understanding tasks (e.g., VQA, image captioning). GLIPv2 elegantly unifies localization pre-training and Vision-Language Pre-training (VLP) with three pre-training tasks: phrase grounding as a VL reformulation of the detection task, region-word contrastive learning as a novel region-word level contrastive learning task, and the masked language modeling. This unification not only simplifies the previous multi-stage VLP procedure but also achieves mutual benefits between localization and understanding tasks. Experimental results show that a single GLIPv2 model (all model weights are shared) achieves near SoTA performance on various localization and understanding tasks. The model also shows (1) strong zero-shot and few-shot adaption performance on open-vocabulary object detection tasks and (2) superior grounding capability on VL understanding tasks. Code is released at https://github.com/microsoft/GLIP.
+
+# 1 Introduction
+
+Recently, a general interest arises in building general-purpose vision systems [21, 24, 56, 42], also called vision foundation models [6, 57], that solve various vision tasks simultaneously, such as image classification [30], object detection [39], and Visual-Language (VL) understanding [3, 11, 27]. Of particular interest, is the unification between localization tasks (e.g., object detection [39] and segmentation [8, 20]) and VL understanding tasks (e.g., VQA [3] and image captioning [11]). Localization pre-training benefits VL tasks [1, 59], and the “localization- $\mathrm { . > V L P ^ { \prime } }$ two-stage pretraining procedure [41, 49, 13, 48, 34, 32, 61, 37, 35] is the common practice in VL community. A long-standing challenge is the unification of localization and understanding, which aims at mutual benefit between these two kinds of tasks, simplified pre-training procedure, and reduced pre-training cost.
+
+However, these two kinds of tasks appear to be dramatically different: localization tasks are visiononly and require fine-grained output (e.g., bounding boxes or pixel masks), while VL understanding tasks emphasize fusion between two modalities and require high-level semantic outputs (e.g., answers or captions).
+
+[21, 24, 56] have made early attempts at unifying these tasks in a straightforward multi-task manner, where a low-level visual encoder is shared across tasks, and two separate high-level branches are designed for localization and VL understanding, respectively. The localization tasks are still vision-only and do not benefit from the rich semantics in vision-language data. As a result, such unified models see the marginal mutual benefit or even performance degradation [24] compared with task-specific models.
+
+In this paper, we identify “VL grounding” as a “meta”-capability for localization and understanding capabilities. VL grounding involves not only understanding an input sentence but also localizing the mentioned entities in the image (see an example in Figure 1). We build a grounded VL understanding model (GLIPv2) as a unified model for localization and VL understanding tasks.
+
+![](images/adbfa03373afd92f159bb9702b1f0cfe1bcc88f56f93dcf4b3e7468790aeb8e9.jpg)  
+Figure 1: Left: GLIPv2, a pre-trained grounded VL understanding model, unifies various localization and VL understanding tasks. These two kinds of tasks mutually benefit each other, and enables new capabilities such as language-guided detection/segmentation and grounded VQA/captioning. Right: Additional examples from ODinW (detection), LVIS (segmentation), VQA, COCO Captioning.
+
+Localization ${ \bf \Pi } + { \bf \delta V L }$ understanding $=$ grounded VL understanding. Localization tasks involve both localization and semantic classification, where classification can be cast as a VL understanding problem using the classification-to-matching trick (Section 3.1). Therefore, we reformulate localization tasks as VL grounding tasks, in which the language input is a synthesized sentence as the concatenation of category names [36]. Localization data are turned into VL grounding data, accordingly. The massive VL understanding data (image-text pairs) can be easily turned into VL grounding data in a self-training manner [36]. Therefore, GLIPv2 has a unified pre-training process: all task data are turned into grounding data and GLIPv2 is pre-trained to perform grounded VL understanding.
+
+A stronger VL grounding task: inter-image region-word contrastive learning. GLIP [36] proposes the phrase grounding task as its pre-training task, which we argue is an easy task and does not fully utilize data information. For example, in the VL grounding task in Figure 1, the phrase grounding task only requires the model to match a given image region to one of the three phrases in the text input, i.e., “green, pink striped, or plain white umbrella?”. This 1-in-3 choice is very easy, only requires color understanding, but loses lots of information in this grounding data: the umbrellas are not any other colors, like black, yellow, etc; objects in those regions are umbrellas but not any other categories, like car, bike, etc. From a contrastive learning view, this phrase grounding task only has two negatives. More negatives can be created from this annotation and thus enable stronger contrastive learning. In GLIPv2, we introduce the novel inter-image region-word contrastive learning task, which leverages phrases from other sentences in the same batch as potential negatives, as another much stronger VL grounding task. This new region-word contrastive loss enables GLIPv2 to learn more discriminative region-word features and demonstrates improvements over all downstream tasks.
+
+GLIPv2 achieves mutual benefit between localization and VL understanding. 1) Experimental results (Table 2) show that a single GLIPv2 model (all model weights are shared) achieves near SoTA performance on various localization and understanding tasks. 2) Thanks to semantic-rich annotations from the image-text data, GLIPv2 shows superior zero-shot and few-shot transfer learning ability to open-world object detection and instance segmentation tasks, evaluated on the LVIS dataset and the "Object Detection in the Wild (ODinW)" benchmark. 3) GLIPv2 enables language-guided detection and segmentation ability, and achieves new SoTA performance on the Flick30K-entities phrase grounding and PhraseCut referring image segmentation tasks. 4) Inherently a grounding model, GLIPv2 leads to VL understanding models with strong grounding ability, which are self-explainable and easy to debug. For example, GLIPv2, when GLIPv2 is finetuned on VQA, it can answer questions while localizing mentioned entities (see Figure 1 and Section 4.4).
+
+# 2 Related Work
+
+Localization models. Traditionally, localization tasks such as object detection and segmentation are single-modality and output bounding boxes or pixel masks [45, 38, 23, 14, 46, 10, 9]. One challenge of these single-modality models lies in generalization to rare and novel concepts: it is hard to collect localization data that cover many rare categories [20]. A long line of research focuses on this generalization problem, under the name of zero-shot [4, 62, 7, 63], weakly-supervised [18, 5, 52], or open-vocabulary [58, 19] localization. Built upon MDETR [25] and GLIP [36], GLIPv2 converts localization tasks into a grounded vision-language task using the classification-to-matching trick (Section 3). Thus GLIPv2 can learn from the semantic-rich vision-language data and shows strong performance on open-vocabulary localization tasks.
+
+Vision-language understanding models. Vision-language (VL) understanding tasks such as VQA [3], image captioning [11], and image-text retrieval [26] involve understanding visual semantics and how they are expressed in natural language. Many VL models (e.g., BUTD) [2, 59] rely on a pre-trained localization model as their visual encoder; the downside is the pro-longed “localization- $\mathrm { . > V L P ^ { \prime } }$ pre-training pipeline [41, 49, 13, 48, 34, 32, 61, 37, 35]. In contrast, GLIPv2 simplifies the pre-training pipeline and enables grounded VL understanding for better interpretability (Section 4.4).
+
+Unifying localization and understanding. [21, 24, 56] made pioneering efforts in unifying localization and understanding. However, localization tasks are still treated as single-modality tasks, while VL tasks involve two modalities. The unification is achieved via straightforward multi-tasking: a low-level visual encoder is shared across tasks and two separate branches are designed for localization and VL understanding. Such unified models do not bring evident mutual benefit and often underperform task-specific models. In contrast, GLIPv2 identifies grounded VL understanding as a meta-task for localization and understanding. The task unification brings architecture unification: the unified grounded VL understanding model empowers a localization branch with VL capacity, arriving at a unified branch that excels at both tasks.
+
+GLIPv2 vs GLIP. 1) GLIP shows that grounded pre-training improves localization. GLIPv2 further shows grounded pre-training improves VL understanding and thus leads to a unified model for localization and VL understanding. 2) GLIPv2 introduces the inter-image region-word contrastive loss, which is another and stronger grounding task than the pre-training task in GLIP. The proposed loss can be viewed as a region-word level generalization of the prevalent image-level contrastive learning [33, 44, 55]. 3) GLIPv2 outperforms GLIP on all benchmarks with the same pre-training data.
+
+# 3 GLIPv2: Unifying Localization and VL Understanding
+
+Based on the reformulation of object detection as a generalized phrase grounding task in GLIP [36], we unify both localization and VL understanding tasks as grounded vision-language tasks. A grounded vision-language task takes both image and text as inputs, and outputs region-level understanding results (e.g., detection, segmentation) and/or image-level understanding results with associated grounding/localization information (e.g., VQA, image captioning). We will present the unified grounded VL formulation and architecture in Section 3.1, the pre-training losses in Section 3.2, and transfer to downstream tasks in Section 3.3.
+
+# 3.1 A Unified VL Formulation and Architecture
+
+At the center of GLIPv2’s unified formulation is the classification-to-matching trick, which reformulates any task-specific fixed-vocab classification problem as an task-agnostic open-vocabulary vision-language matching problem. The best example is the reformulation of image classification as image-text matching in CLIP [44], which enables the model to learn from raw image-text data directly, and achieves strong zero-shot results on open-vocabulary classification tasks. In GLIPv2, we replace every semantic classification linear layer in traditional single-modality vision models with a vision-language matching dot-product layer.
+
+As illustrated in Figure 1, GLIPv2’s unified VL architecture is based on the generic architecture we term Architecture $\mathbf { I I }$ . It consists of a dual encoder, denoted as $\operatorname { E n c } _ { V }$ and $\mathrm { E n c } _ { L }$ , and a fusion encoder, denoted as $\mathtt { E n c } _ { V L }$ . The model takes an image-text pair (Img, Text) as input, and extract visual and text features as below:
+
+$$
+\begin{array} { r } { \dot { O } = \mathrm { E n c } _ { V } ( \mathrm { I m g } ) , \quad \mathring { P } = \mathrm { E n c } _ { L } ( \mathrm { T e x t } ) , \quad O , P = \mathrm { E n c } _ { V L } ( \mathring { O } , \mathring { P } ) , } \end{array}
+$$
+
+where $( \mathring { O } , \mathring { P } )$ and $( O , P )$ denote the image/text features before and after VL fusion, respectively.
+
+Vision-Language understanding tasks. Arch $\mathbf { I I }$ is the most popular model architecture for VL understanding tasks. Given the cross-modality fused representations $O$ and $P$ , it is straightforward to add lightweight task-specific heads for various VL tasks. For example, GLIPv2 adds a two-layer MLP on top of text features $P$ as the masked language modeling (MLM) head, to perform the MLM pre-training. We provide model details of VQA and image captioning in Section 3.3.
+
+(Language-guided) object detection and phrase grounding. Following GLIP [36], GLIPv2 uses the classification-to-matching trick to unify detection and grounding. More specifically, for detection, we simply replace the class logits $S _ { \mathrm { c l s } } = \dot { O } W ^ { T }$ , where $W$ is the weight matrix of the box classifier, with a task-agnostic region-word similarity logits $S _ { \mathrm { g r o u n d } } = O P ^ { T }$ , where text features $P$ are label embeddings from a task-agnostic language encoder. As shown in Figure 1, object detection and phrase grounding share the same input/output format and model architecture. See GLIP [36] for more details. Their only difference is the input text format: (1) for object detection, the text input is a string of concatenated candidate object labels; (2) for phrase grounding, the text input is a natural language sentence. We refer to GLIP [36] for more details.
+
+(Language-guided) instance segmentation and referring image segmentation. Given the object detection results, an instance segmentation head is added to classify each pixel within the box into a semantic class. Again, GLIPv2 uses the classification-to-matching trick to produce a unified instance segmentation head for the standard instance segmentation tasks and the referring image segmentation tasks and leverage both types of data for its pre-training. This classification-to-matching trick can also apply to many other semantic classification heads in single modality CV models (e.g., semantic segmentation) and thus transfers them to language-guided CV models.
+
+# 3.2 GLIPv2 Pre-training
+
+The GLIPv2 is pre-trained with three pre-training losses: phrase grounding loss $\mathcal { L } _ { \mathrm { g r o u n d } }$ from a vision-language reformulation of the object detection task, region-word contrastive loss ${ \mathcal { L } } _ { \mathrm { { i n t e r } } }$ from a novel region-word level contrastive learning task, and the standard masked language modeling loss ${ \mathcal { L } } _ { \mathrm { m l m } }$ proposed in BERT [16].
+
+$$
+{ \mathcal { L } } _ { \mathrm { G L I P v 2 } } = \underbrace { { \mathcal { L } } _ { \mathrm { l o c } } + { \mathcal { L } } _ { \mathrm { i n t r a } } } _ { { \mathcal { L } } _ { \mathrm { g r o u n d } } } + { \mathcal { L } } _ { \mathrm { i n t e r } } + { \mathcal { L } } _ { \mathrm { m l m } }
+$$
+
+Similar to losses in detection tasks, the grounding loss $\mathcal { L } _ { \mathrm { g r o u n d } }$ has two parts: the localization loss $\mathcal { L } _ { \mathrm { l o c } }$ trains localization heads with bounding-box supervision, e.g., RPN loss, box regression loss and/or centerness loss [50]; the intra-image region-word alignment loss ${ \mathcal { L } } _ { \mathrm { { i n t r a } } }$ is essentially the semantic classification/retrieval loss for each region.
+
+Intra-image region-word alignment loss. Given one image-text pair (Img, Text), we obtain the image and text features after cross-modality fusion $O$ and $P$ . The Intra-image region-word alignment loss is computed by
+
+$$
+\mathcal { L } _ { \mathrm { i n t r a } } = l o s s ( O P ^ { T } ; T ) ,
+$$
+
+where $O P ^ { T }$ is the similarity score between image regions and word tokens, and $T$ is the target affinity matrix determined by the ground-truth annotations. The loss function loss is typically a cross-entropy loss for two-stage detectors [46] and a focal loss [38] for one-stage detectors.
+
+However, as discussed in Section 1, this intra-image region-word contrastive learning is rather weak in the sense of contrastive learning, due to the limited number of phrases that can one caption can contain. GLIP [36] alleviates this problem by appending a few negative sentences to form a longer text input with more (negative) phrases. However, constrained by the maximal length of text tokens (256 in GLIP and GLIPv2), only a few negative sentences can be added and the number of negative phrases remains in the order of $1 0 \mathrm { { ^ { \circ } s } }$ . This small-negative-example problem also exists in detection data [36] when the input text cannot include all class names in a detection dataset, e.g., Objects365.
+
+Inter-image region-word contrastive loss. In GLIPv2, we propose using phrases from other imagetext pairs in the same batch as negative examples, which effectively increases the number of negative examples to the order of 1000’s, with nearly negligible additional computational cost.
+
+As in (1), given a batch of image-text pairs $( \mathrm { I m } { \bf g } ^ { i } , \mathrm { T e x t } ^ { i } ) _ { i = 1 } ^ { B }$ and their ground-truth annotations $( T ^ { i } ) _ { i = 1 } ^ { B }$ , the model produces the image and text features before and after VL fusion, denoted as $( \mathring { O } ^ { i } , \mathring { P } ^ { i } ) _ { i = 1 } ^ { B }$ and $( O ^ { i } , P ^ { i } ) _ { i = 1 } ^ { B }$ , respectively. Then as illustrated in Figure 2 (Left), a batch-wise similarity matrix Sbatchground and a batch-wise target affinity matrix $T ^ { \mathrm { b a t c h } }$ are constructed by considering all the image regions and text phrases across this batch. Their $( i , j )$ ’th blocks are obtained as below:
+
+$$
+S _ { \mathrm { g r o u n d } } ^ { \mathrm { b a t c h } } [ i , j ] = \bar { O } ^ { i } ( \bar { P } ^ { j } ) ^ { T } , \quad T ^ { \mathrm { b a t c h } } [ i , j ] = \left\{ \begin{array} { l l } { T ^ { i } , } & { \mathrm { i f ~ } i = j } \\ { \mathrm { o b t a i n e d ~ b y ~ l a b e l ~ p r o p a g a t i o n , } } & { \mathrm { o t h e r w i s e . } } \end{array} \right.
+$$
+
+The inter-image region-word contrastive loss is then defined as the standard bi-directional contrastive loss applied on all image regions and phrases in this batch:
+
+${ \mathrm {  ~ \dot { \ z } _ { \mathrm { { i n t e r } } } = \ c r o s s \_ e n t r o p y \mit \mathrm { \mit \mathrm { \ l o s s } ( \cal S \mit _ { \mathrm { { g r o u n d } } } ^ { \mathrm { b a t c h } } , \mit T \mathrm { ^ { b a t c h } , \ a x i s = 0 ) + \ c r o s s \mit \mathrm {  _ { - } \ e n t r o p y \mit \mathrm { \mit { \mit \mathrm { \ l o s s } ( \cal S \mit _ { \mathrm { { g r o u n d } } } ^ { \mathrm { b a t c h } } , \mathrm { T \mathrm { ^ { b a t c h } , \ a x i s = 1 ) \mit . } } } } } } } } } $ (5)
+
+Compared with that in the inter-image contrastive loss (3), the number of negatives is multiplied by batch size $B$ in this inter-image contrastive loss (5). We elaborate two important details in (4). (1) GLIPv2 uses the image text features $( \mathring { O } ^ { i } , \mathring { P } ^ { i } ) _ { i = 1 } ^ { B }$ before VL fusion, not $( O ^ { i } , P ^ { i } ) _ { i = 1 } ^ { B }$ after VL fusion, to compute the batch-wise similarity matrix in the inter-image contrastive loss (4). Otherwise, the image and text features after VL fusion would have seen the paired information (1), and thus the model can easily rule out the negatives from misaligned images/texts. (2) We cannot simply assign all regions and texts from unpaired image-text as negative pairs, as done in the standard contrastive loss in CLIP [44]. Instead, we determine the off-diagonal blocks in the target affinity matrix $T ^ { \mathrm { b a t c h } }$ by label propagation. For example, as illustrated in Figure 2 (Left), if a region is annotated as “person”, it should be a positive pair with all “person” phrases in detection-type texts. We do not propagate positives to grounding-type texts (natural sentences) because phrases in sentences carry contexts that are unique to that image-sentence pair.
+
+Pre-training with both detection and paired-image-text data. GLIPv2 pre-training data is in the image-text-target triplet format (Img, Text, $T$ ), where the target affinity matrix $T$ contains the box-label localization annotations. We also use massive image-text pair data (Img, Text) to pre-train GLIPv2, by generating grounding boxes $\hat { T }$ for phrases in the text with the GLIP pre-trained model from [36]. The human-annotated OD/grounding data provides high-fidelity localization supervision, while the massive image-text data greatly improves the concept diversity for GLIPv2.
+
+Second-stage pre-training of the segmentation head. GLIPv2 performs a second-stage pre-training of the language-guided segmentation head on both instance segmentation and image referring segmentation data, while fixing all other parts of the model.
+
+# 3.3 Transfer GLIPv2 to Localization and VL Tasks
+
+We introduce two ways to easily transfer GLIPv2 to various downstream tasks. In addition, GLIPv2 can perform conventional VL tasks (e.g., VQA) along with localization, effectively making every task we consider a “grounded VL understanding” task.
+
+One model architecture for all. GLIPv2 can be transferred to downstream tasks by fine-tuning the model with an (optional) task-specific head. 1) For detection and segmentation tasks, no task-specific head is needed as the pre-training architecture can inherently perform detection and segmentation. 2) For $V L$ tasks: for VQA, a classification head is added on top of the hidden representation of the start-of-sequence token; for caption generation, we train with a unidirectional language modeling loss, which maximizes the likelihood of the next word given context. We use a unidirectional attention mask and prevent the image part from attending to the text in the fusion layers.
+
+One set of weights for all. There is a growing interest in developing models that can be transferred to various tasks while only changing the least amount of parameters to save training time and storage cost [47, 31]. Following GLIP, GLIPv2 can be transferred to localization tasks in a zero-shot or a prompt-tuning setting (Section 4.2). One single GLIPv2 model can serve various tasks, where each task only keeps few or no parameters. Of particular interest is the prompt tuning setting. For a certain localization task, the text prompt is the same for all input images; thus, we could directly tune $\mathring { P }$ , a small prompt embedding matrix, to adapt GLIPv2 to new tasks. Prompt tuning in a deep-fused model such as GLIPv2 is different from the conventional linear probing/prompt tuning setting [53, 44, 60] in shallow-interacting vision models such as CLIP. The latter can also be viewed as only tuning a small prompt/softmax embedding $P$ ; however, tuning $P$ only affects the very last layer of the model while the visual representation is still frozen. In contrast, GLIP/GLIPv2’s visual representation is conditioned on the prompt embedding $\mathring { P }$ ; tuning $\mathring { P }$ changes the text, visual, as well as fused embeddings. As a result, prompt tuning in GLIPv2 is highly effective, often matching the performance of fine-tuning (see Table 2). This is in contrast to the common observation in CV that linear probing lags behind fine-tuning by a large gap [22].
+
+![](images/a2126e3d117d3e41b61f4a05e16f3c1c31302f73ea9c4df2bc4eaaa3b313fdd2.jpg)  
+Figure 2: GLIPv2 pre-training losses: the intra-image alignment loss ${ \mathcal { L } } _ { \mathrm { { i n t r a } } }$ (right) takes features after VL fusion and compute loss over region-word pairs within each image-text pair; the inter-image contrastive loss (left) ${ \mathcal { L } } _ { \mathrm { { i n t e r } } }$ takes features before VL fusion and computes loss over all region-word pairs across a batch of image-text pairs. Label propagation is used to determine the off-diagonal blocks of the ${ \mathcal { L } } _ { \mathrm { { i n t e r } } }$ target matrix (4).
+
+Grounded VL understanding. GLIPv2 also enables grounded VL understanding, where we retain the ability to perform grounding when fine-tuning the model to a downstream VL task. This increases the interpretability of the model. Specifically, we first turn the VL data of the downstream task into grounded VL data using a pre-trained GLIP model. Then we train the model with both the downstream task head and grounding head. For VQA, the model is trained to predict the answer and ground entities in the question as well as the implied entity in the answer; for captioning, the model is trained to predict the next word given the context and ground the current decoded word. By tuning localization tasks into a grounded VL task and augmenting VL tasks with grounding ability, we effectively turn every task into a grounded VL understanding task (see examples in Figure 1).
+
+# 4 Experiments
+
+In this section, we show that GLIPv2 serves as a performant and easy-to-deploy general-purpose vision system. 1) One Model Architecture for All (Section 4.1). GLIPv2 can be directly fine-tuned to both localization and VL understanding tasks with minimal architecture change. It achieves performance on par with SOTA models with specialized architectures. 2) One Model Weight for All (Section 4.2). GLIPv2 can be transferred to localization tasks in a zero-shot manner with zero parameter update; with prompt tuning, a single GLIPv2 model can achieve comparable performance with fully fine-tuned settings on both localization and understanding tasks.
+
+<table><tr><td>Model</td><td>Model Type</td><td>COCO-Det (test-dev)</td><td>ODinW (test)</td><td>LVIS (minival)</td><td>COCO-Mask (test-dev)</td><td>Flickr30K PhraseCut (test)</td><td>(test)</td><td>VQA (test-dev/test-std) (Karpathy-test)</td><td>Captioning</td></tr><tr><td>Mask R-CNN [23]</td><td rowspan="4"></td><td>39.8</td><td>-</td><td>33.3/-</td><td>-/37.1</td><td>=</td><td>=</td><td></td><td></td></tr><tr><td>DETR[9]</td><td>42.0</td><td></td><td>17.8/-</td><td>=</td><td></td><td>=</td><td></td><td></td></tr><tr><td>DyHead-T[15]</td><td>49.7</td><td>60.8</td><td>=</td><td>=</td><td>=</td><td></td><td></td><td></td></tr><tr><td>DyHead-L [15]</td><td>60.3*</td><td>=</td><td>=</td><td></td><td></td><td>=</td><td></td><td></td></tr><tr><td>VisualBERT[34]</td><td rowspan="3">Understanding</td><td>=</td><td></td><td></td><td></td><td>71.33</td><td>=</td><td>70.8/71.0</td><td>=</td></tr><tr><td>UNITER[12]</td><td></td><td></td><td></td><td></td><td>=</td><td>=</td><td>73.8/74.0</td><td>=</td></tr><tr><td>VinVL[59]</td><td></td><td></td><td></td><td></td><td>=</td><td></td><td>76.5/76.6</td><td>130.8</td></tr><tr><td>GPV[21]</td><td rowspan="3">Localization &amp;</td><td></td><td></td><td>=</td><td></td><td>·</td><td>=</td><td>62.5/-</td><td>102.3</td></tr><tr><td>UniT[24]</td><td>42.3</td><td>=</td><td></td><td></td><td></td><td></td><td>67.6/-</td><td>=</td></tr><tr><td>MDETR[25]</td><td>=</td><td>=</td><td>24.2/-</td><td>=</td><td>84.3</td><td>53.7</td><td>70.6 /70.6</td><td>■</td></tr><tr><td>Unicorn [56]</td><td rowspan="3">Understanding Localization &amp;</td><td>=</td><td>■</td><td>=</td><td></td><td>80.4</td><td>=</td><td>69.2/69.4</td><td>119.1</td></tr><tr><td>GLIP-T[36]</td><td>55.2</td><td>64.9</td><td>=</td><td>=</td><td>85.7</td><td>-</td><td>=</td><td>-</td></tr><tr><td>GLIP-L [36]</td><td>Understanding 61.5*</td><td>68.9</td><td>=</td><td>=</td><td>87.1</td><td>-</td><td>=</td><td>-</td></tr><tr><td>GLIPv2-T(Ours)</td><td>Localization</td><td>55.5</td><td>66.5</td><td>50.6/41.4</td><td>53.5/42.0</td><td>86.5</td><td>59.4</td><td>71.6/71.8</td><td>122.1</td></tr><tr><td>GLIPv2-B (Ours)</td><td>&amp;</td><td>58.8</td><td>69.4</td><td>57.3/46.2</td><td>59.0/45.8</td><td>87.5</td><td>61.3</td><td>73.1/73.3</td><td>128.5</td></tr><tr><td>GLIPv2-H(Ours)</td><td>Understanding</td><td>60.6 (62.4*)</td><td>70.4</td><td>59.8 / 48.8</td><td>59.8 / 48.9</td><td>87.7</td><td>61.3</td><td>74.6/74.8</td><td>131.0</td></tr></table>
+
+Table 1: One model architecture results. For COCO-Det test-dev, \* indicates multi-scale evaluation. For LVIS, we report the numbers for both bbox and segm on minival to avoid data contamination due to the pre-training. For Flickr30K test, we report the metric under R@1. For COCO-Mask, we also report both bbox and segm on test-dev.
+
+Following GLIP [36], we adopt Swin Transformer [40] as the image encoder $\operatorname { E n c } _ { V }$ , text transformers [51, 44] as the text encoder $\mathrm { E n c } _ { L }$ , Dynamic Head [15] with language-aware deep fusion [36] as the fusion encoder $\mathtt { E n c } _ { V L }$ , and Hourglass network [43] as instance segmentation head feature extractor. We train GLIPv2 at three scales: GLIPv2-T, GLIPv2-B, and GLIPv2-H.
+
+GLIPv2-T has the same model config and initialization as GLIP-T: Swin-Tiny and BERT-Base as the dual encoder. The model is pre-trained on the following data: 1) O365, 2) GoldG as in GLIP-T (C), and 3) Cap4M, 4M image-text pairs collected from the web with boxes generated by GLIP-T [36]. GLIPv2-B/GLIPv2-H are based on Swin-Base/Swin-Huge and the pre-layernorm text transformer [17] as dual encoder, and are initialized from the UniCL [55] checkpoints. We observe much stabler training with GPT-type pre-layernorm transformer [17] than BERT-type post-layernorm transformer. The training data contain: 1) FiveODs (2.78M data) 1; 2) GoldG as in MDETR [25]; and 3) C $\mathrm { C 1 5 M + S B U }$ , 16M public image-text data with generated boxes by GLIP-L [36]. Segmentation heads of GLIPv2 models are pre-trained on COCO, LVIS [20] and PhraseCut [54], with all other model parameters are frozen.
+
+Note All datasets above were collected by the creators (cited) and consent for any personally identifiable information (PII) was ascertained by the authors where necessary. Due to limited space, we refer to supplementary for details of training recipes and hyper-parameters.
+
+# 4.1 One Model Architecture for All
+
+We compare GLIPv2 to existing object detection and vision-language pre-training methods on a wide range of tasks. We fine-tune the model on 8 different downstream tasks and report the performance in Table 1. We make the following observations.
+
+GLIPv2 v.s. specialized Localization methods. GLIPv2 outperforms previous localization models on generalization to both common and rare classes and domains with a single model architecture and pre-training stage. 1) OD on common categories (COCO-Det), GLIPv2-T achieves 5.8 improvement compared to the standard DyHead-T trained on O365 (55.5 v.s. 49.7). GLIPv2-H reaches $6 2 . 4 \mathrm { A P }$ on test-dev, and surpass the performance of the previous SoTA model GLIP-L. 2) OD on rare / unseen categories (LVIS), GLIPv2-T outperforms a supervised MDETR on the bbox by a great margin (59.8 v.s. 24.2). 3) Generalization to diverse real-word tasks (ODinw), GLIPv2-T (55.5) performs better than original GLIP-T (64.9) on the average of 13 public datasets; GLIPv2-B outperforms GLIP-L by 0.5 AP. 4) Instance segmentation (COCO-Mask & PhraseCut), for traditional instance segmentation (i.e., COCO-Mask), GLIPv2-H outperforms the well-known Mask R-CNN by a great margin on segm.
+
+Table 2: One set of weights results v.s. Original GLIP. \* indicates multi-scale evaluation. Numbers in red clearly points out the difference between the prompt tuning and full fine-tuning results (see Table 1). Numbers in gray mean that they are not in zero-shot manner. $\dagger$ : these two numbers are artificially high due to some overlap between COCO-minival and VisualGenome-train.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">Direct Evaluation</td><td colspan="5">Prompt Tuning</td></tr><tr><td>COCO-Mask (minival)</td><td>ODinW (test)</td><td>LVIS-Det (minival)</td><td>Flickr30K (minival)</td><td>COCO-Det (test-dev)</td><td>ODinW (test)</td><td>LVIS (minival)</td><td>COCO-Mask (test-dev)</td><td>PhraseCut (test)</td></tr><tr><td>GLIP-T</td><td>46.6/-</td><td>46.5</td><td>26.0</td><td>85.7</td><td>二</td><td>46.5</td><td>-</td><td>-</td><td>-</td></tr><tr><td>GLIP-L</td><td>49.8/-</td><td>52.1</td><td>37.3</td><td>87.1</td><td>58.8</td><td>67.9</td><td>=</td><td>=</td><td>-</td></tr><tr><td>GLIPv2-T</td><td>47.3/35.7</td><td>48.5</td><td>29.0</td><td>86.0</td><td>53.4 (-2.1)</td><td>64.8 (-1.7)</td><td>49.3 / 34.8 (-13/-6.6)</td><td>53.2 / 41.2 (-0.3/-0.8)</td><td>49.4</td></tr><tr><td>GLIPv2-B</td><td>61.9†/43.4</td><td>54.2</td><td>48.5</td><td>87.2</td><td>59.0 (+0.2)</td><td>67.3 (-2.1)</td><td>56.8 / 41.7 (-0.5/-4.5)</td><td>58.8 / 44.9 (-0.2/-0.9)</td><td>55.9</td></tr><tr><td>GLIPv2-H</td><td>64.1/47.4</td><td>55.5</td><td>50.1</td><td>87.7</td><td>60.2 /61.9* (-0.4 /-0.5)</td><td>69.1 (-1.3)</td><td>59.2 / 43.2 (-0.6/-5.7)</td><td>59.8 / 47.2 (-0.0/-1.7)</td><td>56.1</td></tr></table>
+
+![](images/5f74b465df4d065f84ce4d92f60d2f682c7a078495dd46741bd6fcf8dd804923.jpg)  
+Figure 3: Data efficiency of GLIPv2 on ODinW. The $\mathbf { X }$ -axis is the amount of task-specific data, from zero-shot to all data. Y-axis is the average AP across 13 datasets.
+
+Table 3: Zero-shot, prompt tuning, and full finetuning performance on ODinW. GLIPv2 models exhibit superior data efficiency.   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Zero-Shot 0</td><td colspan="5">Prompt Tuning /Fine Tuning</td></tr><tr><td>1</td><td>3</td><td>5</td><td>10</td><td>All</td></tr><tr><td rowspan="2">DyHead-T 0365 [36]</td><td rowspan="2">-</td><td>=</td><td></td><td>-</td><td>-</td><td>-</td></tr><tr><td>33.8</td><td>43.6</td><td>46.4</td><td>50.8</td><td>60.8</td></tr><tr><td rowspan="2">Lloc + Lintra (GLIP-T)</td><td rowspan="2">46.5</td><td>49.9</td><td>53.7</td><td>55.5</td><td>56.6</td><td>62.4</td></tr><tr><td>51.3</td><td>54.9</td><td>56.4</td><td>58.4</td><td>64.9</td></tr><tr><td rowspan="2">Lloc + Lintra +Linter</td><td rowspan="2">48.4</td><td>52.1</td><td>55.6</td><td>56.7</td><td>58.3</td><td>62.9</td></tr><tr><td>51.4</td><td>55.3</td><td>56.6</td><td>59.5</td><td>66.3</td></tr><tr><td rowspan="2">Lloc +Lintra +Linter +Cmm</td><td rowspan="2">48.5</td><td>52.4</td><td>55.6</td><td>57.4</td><td>58.8</td><td>64.8</td></tr><tr><td>52.8</td><td>55.6</td><td>57.4</td><td>59.7</td><td>66.5</td></tr></table>
+
+For language-guided segmentation (i.e., PhraseCut), compared to MDETR, GLIPv2-T achieves an improvement of 5.7 mask AP.
+
+GLIPv2 v.s. specialized VL Understanding methods. GLIPv2 rivals with SoTA specialized models for VL tasks. 1) For VQA, GLIPv2 outperforms VisualBERT and UNITER and approaches the previous SoTA model VinVL. 2) For Captioning, the best GLIPv2 even surpasses VinVL (VinVL and GLIPv2 are not trained with CIDEr optimization).
+
+GLIPv2 v.s. localization and VL models. Prior works such GPV, UniT and Unicorn have also explored unifying localization and VL models (see a discussion in Section 2). GLIPv2 outperforms all previous systems on both localization and VL tasks. For the best GLIPv2-H, it outperforms the UniT by a great margin (18.3 AP) on COCO object detection tasks. Meanwhile, it also surpasses UniT’s performance on VQA by 6.9 points and GPV’s peformance on Image Captioning as well.
+
+Takeaway. Most notably, GLIPv2 outperforms previous “unified” models (GPV, UniT, MDETR, Unicorn) by a large margin. This is the first time that a single model architecture could achieve near SoTA performance on both localization and understanding. In contrast, in prior work, there exists certain trade-off between localization and understanding: models that aim to achieve high understanding performance tend to have lower localization performance (e.g., UNiT’s detection performance is limited to the DETR [9] architecture), as it is not trivial to merge a SoTA localization branch and a SoTA VL branch into a single model.
+
+# 4.2 One Set of Model Parameters for All
+
+GLIPv2 is pre-trained to perform grounding; thus it can be transferred to various localization tasks with changing zero or few parameters. We evaluate GLIPv2 under two such settings: 1) direct evaluation, where we transfer the model “as is” without any parameter change, and 2) prompt tuning, where only the prompt embedding is tuned for specific tasks (Section 3.3).
+
+Direct evaluation. The pre-trained GLIPv2 can be directly evaluated on any object detection task (by concatenating the object categories into a text prompt) and visual grounding task without any further tuning. We evaluate the models on four localization tasks: COCO, ODinW, LVIS, and Flickr30, and their results are presented in Table 2. Note that for GLIPv2-B and GLIPv2-H, the training sets of Flick30K and LVIS are present in the pre-training data. Thus, reported numbers on these metrics are not zero-shot evaluation (we have marked them gray). For all other evaluation results, the models are evaluated in zero-shot settings without any further tuning.
+
+GLIPv2 can be effortlessly transferred to different localization tasks without further tuning. 1) For COCO, GLIPv2-T achieves a zero-shot performance of 47.3 without seeing any COCO training images. This surpasses well-established supervised systems (e.g., Mask R-CNN) and also outperforms GLIP-T by 0.7 AP. 2) For ODinW, GLIPv2 also shows strong zero-shot performance. GLIPv2-T (48.5) surpasses the GLIP-T (46.5). Meanwhile, the zero-shot performance of GLIPv2-B and GLIPv2- H even surpasses the 10-shot tuning performance of DyHead-T (to be introduced in Figure 3). 3) For LVIS, GLIPv2-T achieves a 3 AP improvement performance compared to the GLIP-T. 4) For Flickr30K, GLIPv2-B achieves even higher number (87.2) compared to original GLIP-L (87.1).
+
+Prompt Tuning. Following GLIP, GLIPv2 supports efficient prompt tuning: the visual representation is heavily conditioned on the text representation due to the deep fusion block (Section 3.3); thus we could fine-tune only the prompt embedding for each task but still maintain high performance.
+
+Prompt tuning $G L I P \nu 2$ achieves similar performance as full fine-tuning. When comparing the performance of each task in Table 1 and 2 at the same time, for GLIPv2, prompt tuning performance almost matches the one model architecture results on localization tasks, without changing any of the grounding model parameters.
+
+# 4.3 GLIPv2 as a Strong Few-Shot Learner
+
+We demonstrate GLIPv2’s performance on ODinW datasets with respect to different amounts of training data in Figure 3. The performance improvement between GLIPv2-T and GLIP-T exhibits more superior data efficiency for prompt tuning. We compare with the SoTA detector DyHead-T, pre-trained on Objects365 in Table 3. It can be seen that a zero-shot GLIPv2-T (48.5) outperforms a outperforms 5-shot DyHead-T (46.4) while the performance of one-shot GLIPv2-H (61.3) surpasses a all-shot fully supervised DyHead-T (60.8).
+
+# 4.4 Analysis
+
+Pre-training losses Table 4 shows the performance of the downstream tasks with different variants of our method. Compared to the GLIP pre-training tasks with only intra-image region-word contrastive loss (Row 3), adding inter-image word-region loss (Row 5) substantially improves the pre-trained model performance across all the object detection tasks (COCO, ODinW, and LVIS) on both zero-shot and fine-tuned manner. Consistent with common observations from most VL understanding methods, adding MLM loss (Row4) benefits for learning the representation for understanding tasks (Flick30k, VQA, and Captioning). Furthermore, using all three losses together at the 1st stage pre-training and doing the 2nd stage pre-training without MLM on OD and GoldG data, GLIPv2 (Row6) can perform well on both the localization and VL understanding tasks.
+
+An additional stage of pre-training is applied for small models (GLIPv2-T and GLIPv2-B) due to limited model capacity. In order to achieve higher performance on both localization and understanding tasks, we find that including all data (even with some noise) and MLM loss in the first stage of pre-training will benefit the model for learning a better representation of both localization and understanding capability. Since the OD tasks require the model with more accurate localization ability, in our 2nd stage of pre-training, we decide to eliminate the MLM loss. The large model (GLIPv2-H) does not need this additional stage because it has enough capacity to learn both wordregion alignment and MLM together in a single stage.
+
+Pre-training data Table 5 reports the last checkpoint results on GLIPv2 when we do the scaling up of pre-training data. As more weak image-text pair data (Cap) is involved in our training, it benefits both standard/in-domain (i.e., COCO, Flickr30K) and large-domain gap (i.e., ODinW, LVIS) tasks. We also show that by adding the inter-image region-word contrastive helps when we are fixing the data at the same scale. For large-domain gap tasks, adding the inter-image region-word contrastive
+
+<table><tr><td>Row Model</td><td></td><td>COCO</td><td>ODinW</td><td>LVIS|Flickr30K VQA Captioning</td><td></td><td></td><td></td></tr><tr><td>1</td><td>No pre-train</td><td>-/50.6</td><td>-/60.8</td><td>1</td><td>1</td><td>64.6</td><td>111.5</td></tr><tr><td>2</td><td>+Lmlm</td><td>-/48.5</td><td>-/37.4</td><td>1</td><td>1</td><td>64.6</td><td>110.9</td></tr><tr><td>3</td><td>+ Lloc+Lintra</td><td></td><td>46.6/55.2 46.5/64.9</td><td>26.0</td><td>85.7</td><td>69.4</td><td>119.7</td></tr><tr><td>4</td><td>+Lloc+Lintra+Lmlm</td><td></td><td>47.0/55.2 47.6/66.2</td><td>28.5</td><td>86.5</td><td>69.8</td><td>120.7</td></tr><tr><td>5</td><td>+Lloc+Lintra+Linter</td><td></td><td>47.1/55.4 48.4/66.3</td><td>28.6</td><td>85.8</td><td>68.7</td><td>120.4</td></tr><tr><td>6</td><td>+Lloc+Lintra+Linter+Lmlm</td><td></td><td>47.3/55.5 48.5/66.5</td><td>29.0</td><td>86.3</td><td>70.7</td><td>122.1</td></tr></table>
+
+Table 4: Pre-training losses on Tiny-scale model. Involving intra-image region-word alignment loss $\mathcal { L } _ { \mathrm { { i n t r a } } }$ , inter-image region-word contrastive loss ${ \mathcal { L } } _ { \mathrm { i n t e r } }$ and MLM loss ${ \mathcal { L } } _ { \mathrm { m l m } }$ will benefit both localization and understanding tasks.
+
+Table 5: Pre-train data scale up on Base-scale model. Results are reported at the last checkpoint. See supplementary for results at all checkpoints.   
+
+<table><tr><td>Linter</td><td>Pre-train Data</td><td>CoCo</td><td>ODinW</td><td>LVIS</td><td>Flick30K</td></tr><tr><td>X</td><td>0365,GoldG</td><td>48.06</td><td>43.14</td><td>25.6</td><td>84.36</td></tr><tr><td></td><td>0365,GoldG</td><td>48.59</td><td>42.64</td><td>26.9</td><td>83.90</td></tr><tr><td></td><td>0365,GoldG,Cap4M</td><td>48.21</td><td>51.35</td><td>34.2</td><td>85.56</td></tr><tr><td>X</td><td>0365,GoldG,Cap4M</td><td>48.79</td><td>52.70</td><td>35.0</td><td>85.50</td></tr><tr><td>×</td><td>0365,GoldG,Cap12M</td><td>48.50</td><td>49.32</td><td>35.5</td><td>85.79</td></tr><tr><td>√</td><td>0365,GoldG,Capl2M</td><td>49.26</td><td>53.15</td><td>36.6</td><td>85.84</td></tr></table>
+
+Table 6: GLIPv2 can perform captioning and grounding at the same time (a.k.a., grounded VL understanding).   
+
+<table><tr><td>Model</td><td>COCO Caption B4 CIDEr</td><td>SPICE</td><td>R@1 R@5</td><td>Flickr30K Grounding</td><td>R@10°</td></tr><tr><td>GLIPv2-T</td><td>36.5</td><td>119.8 21.6</td><td>80.8</td><td>94.4</td><td>96.5</td></tr><tr><td>GLIPv2-B</td><td>37.4</td><td>123.0 21.9</td><td>81.0</td><td>94.5</td><td>96.5</td></tr></table>
+
+loss will further boost the model to learn better representation. For more detailed scaling-up effects on various tasks under all the checkpoints for GLIP and GLIPv2, refer to Appendix.
+
+Note that the (Img, Text, $T$ ) data used in GLIPv2 pre-training can be just human-annotated data (Row1&2 in Table 5), with which GLIPv2 pre-training does not involve any pseudo data from a pre-trained grounding/localization model. In order to achieve the best performance, GLIPv2 uses image-text pair data with pseudo boxes (Cap) from a pre-trained GLIP model (Row3-6 in Table 4), which is trained with the same "grounded VL understanding" task but just with smaller data.
+
+Grounded Vision-Language Understanding GLIPv2 can be trained to perform a VL task and grounding at the same time (Section 3.3). We denote such an ability as grounded VL understanding. In Figure 1, we showcase grounded predictions of GLIPv2 on VQA and COCO captions. We also conduct quantitative evaluations (Table 6). The model achieves strong performance for both VL understanding (on COCO Caption) and localization (on Flickr30K Grounding). Such an ability to produce high-level semantic outputs (i.e., answers and captions) and supporting localization results is another appealing trait of GLIPv2, as potential users can have a better understanding of the model behaviour. See more detailed analysis and qualitative examples in the Appendix.
+
+# 5 Conclusion and Social Impacts
+
+This paper proposes GLIPv2, a unified framework for VL representation learning that serves both localization tasks and VL understanding tasks. We experimentally verify the effectiveness of the unified model and the novel region-word contrastive learning. Compared to existing methods, GLIPv2 achieves competitive near SoTA performance on various localization and understanding tasks. However, additional analysis of the data and the model is necessary before deploying it in practice since large-scale web data may contain unintended private information, unsuitable images/text, or some bias leakage. Further investigation may be needed for web data due to the above issues.
+
+# 6 Acknowledgement
+
+We thank anonymous reviewers for their comments and suggestions. Additional thanks go to the Microsoft Research Horizontal AI Team and Microsoft Alexander Multi-modal Team for providing computer resources for large-scale training. The baseline models used in our experiments are based on the open-source code released in the GitHub repository; we acknowledge all the authors who made their code public, which tremendously accelerates our project progress.
+
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+
+# Checklist
+
+The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example:
+
+• Did you include the license to the code and datasets? [Yes]   
+• Did you include the license to the code and datasets? [No] The code and the data are proprietary.   
+• Did you include the license to the code and datasets? [N/A]
+
+Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below.
+
+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]   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
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+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes]
+
+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)? [Yes]   
+(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]
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+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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+5. If you used crowdsourcing or conducted research with human subjects...
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