Add Batch 80fb586c-5d7e-46fc-bc1e-32ee2ce0f981 data

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2023/GoBigger_ A Scalable Platform for Cooperative-Competitive Multi-Agent Interactive Simulation/a8eca6cf-55fd-4234-ae59-c025bf97ab26_origin.pdf

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2023/GoBigger_ A Scalable Platform for Cooperative-Competitive Multi-Agent Interactive Simulation/full.md

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+# GOBIGGER: A SCALABLE PLATFORM FOR COOPERATIVE-COMPETITIVE MULTI-AGENT REINFORCEMENT LEARNING
+
+Ming Zhang $^{1*}$ , Shenghan Zhang $^{1*}$ , Zhenjie Yang $^{1*}$ , Lekai Chen $^{1*}$ , Jinliang Zheng $^{1}$ , Chao Yang $^{2}$ , Chuming Li $^{1}$ , Hang Zhou $^{1}$ , Yazhe Niu $^{2}$ , Yu Liu $^{1,2\dagger}$
+
+$^{1}$ SenseTime Research,  $^{2}$ Shanghai AI Laboratory, Shanghai, China  
+{zhangming, zhangshenghan, yangzhenjie, chenlekai, zhengjinliang, zhouhang2}@sensetime.com  
+{lichuming, niuyazhe, liuyu}@pjlab.org.cn
+
+# ABSTRACT
+
+The emergence of various multi-agent environments has motivated powerful algorithms to explore agents' cooperation or competition. Even though this has greatly promoted the development of multi-agent reinforcement learning (MARL), it is still not enough to support further exploration on the behavior of swarm intelligence between multiple teams, and cooperation between multiple agents due to their limited scalability. To alleviate this, we introduce GoBigger, a scalable platform for cooperative-competition multi-agent interactive simulation. GoBigger is an enhanced environment for the Agar-like game, enabling the simulation of multiple scales of agent intra-team cooperation and inter-team competition. Compared with existing multi-agent simulation environments, our platform supports multi-team games with more than two teams simultaneously, which dramatically expands the diversity of agent cooperation and competition, and can more effectively simulate the swarm intelligent agent behavior. Besides, in GoBigger, the cooperation between the agents in a team can lead to much higher performance. We offer a diverse set of challenging scenarios, built-in bots, and visualization tools for best practices in benchmarking. We evaluate several state-of-the-art algorithms on GoBigger and demonstrate the potential of the environment. We believe this platform can inspire various emerging research directions in MARL, swarm intelligence, and large-scale agent interactive learning. Both GoBigger and its related benchmark are open-sourced. More information could be found at https://github.com/opendilab/GoBigger.
+
+# 1 INTRODUCTION
+
+The swarm behavior of multi-agent systems (MAS) widely exists in nature and human society. In MAS, individual agent pursues their goal and interacts with each other in local areas, following the rules of cooperation or competition, and then the intelligent behavior of the agent group forms the complex collective behaviors. The phenomena of collective behaviors can be found in the flocking birds (Bhattacharya & Vicsek, 2010), molecular motors (Chowdhury, 2006), human crowds (Helbing et al., 2000), and traffic systems (Kanagaraj & Treiber, 2018). To understand and simulate such phenomena, some rule-based models (Castellano et al., 2009) can simulate the swarm behavior in an unconstrained environment with random movement. However, in a complex interactive environment such as intra-cellular molecular motor transport, where the interaction of agents is time-varying and updatable, it is challenging to recover the underlying collective behaviors by manually designing the controllers or rules.
+
+Interactive simulation of multi-agent systems can provide significant convenience for multi-agent learning algorithms. Some existing multi-agent simulation environments mainly focus on the coop-
+
+![](images/118da781b99f527346b62229f2cd4bcd60e51ff0b44226d211495b6c2a8c5302.jpg)  
+Figure 1: New users can follow the given research workflow, while advanced users can customize the configuration of the environment and define new tasks based on GoBigger. The lower left part shows the basic units (balls) and the related actions. The lower right part shows the training in the league with many games going on at the same time, from where cooperative and competitive behaviors of agents can be observed.
+
+eration (Xuan et al., 2001) between agents, such as Bi-DexHands (Chen et al., 2022), Multi-Agent MuJoCo (Peng et al., 2021), PressurePlate McInroe (2022), RWARE (Papoudakis et al., 2021), and Flatland (Mohanty et al., 2020). These environments offer simulation designs for cooperation between agents but do not care about multi-agent competition. With the rapid development of the field of multi-agent reinforcement learning, more and more multi-agent simulation environments focus on both cooperation and competition, including LBF (Papoudakis et al., 2020), MALMO (Johnson et al., 2016), DM Lab (Beattie et al., 2016), DM Lab2D (Beattie et al., 2020), Derk's Gym Noren (2020), PommerMan (Resnick et al., 2018), StarCraft Multi-Agent Challenge (SMAC) (Samvelyan et al., 2019), Google Research Football (Kurach et al., 2019), Multi-Agent Particle Environment (MPE) (Mordatch & Abbeel, 2017), Hide and Seek (Baker et al., 2019), and Neural MMO (Suarez et al., 2021). Agents in these environments are divided into different teams to achieve intra-team cooperation and inter-team competition. However, most of them consist of up to two teams, named  $2 \times N$  mode. That means they can not handle situations where multiple teams cooperate or compete with each other, which is necessary for the research on the swarm behavior of multi-agent systems. Besides, the performance gap caused by different levels of cooperation is not significant in most environments including SMAC, Google Research Football, and NeuralMMO. For the most popular multi-agent environment SMAC, recently SMACv2 (Ellis, 2022) shows that agents can get high performance after dropping teammates' observations. The author uses only the identity and observation of agents in the training phase under QMIX (Rashid et al., 2018) and MAPPO (Yu et al., 2021) to achieve the same performance as that using the global state additionally. And the replay videos of these environments with well-trained agents could not show the complex cooperation. For those environments that emphasize cooperation through game mechanics, the observations of them are too simple that different multi-agent algorithms can easily reach high performance. More details about the various multi-agent simulation environments are shown in Table 1.
+
+In this paper, we propose a scalable platform, GoBigger, aiming to delve into cooperative-competitive multi-agent reinforcement learning for swarm intelligence between multiple teams. Different from the previous multi-agent simulation environment, GoBigger is a scalable environment that enables the simulation of various teams and agents in each team. In other words, in the  $M \times N$  game mode of GoBigger,  $M$  means the number of teams in the environment, and  $N$  means the number of players in each team. This new game mode dramatically expands the way of agent cooperation and competition and can more effectively simulate the swarm intelligent agent behavior. In addition, GoBigger makes agents with intra-team cooperation and inter-team competition achieve higher performance according to the restriction of game mechanics and rules, which is approved in Section 6.1. We offer a diverse set of challenge scenarios in GoBigger for best practices in benchmarking. In most of the given scenarios, each player in a team is controlled by an independent agent that has to act based on only its local observation or all teammates' observations. Meanwhile, GoBigger is a more complex
+
+Table 1: Comparison of GoBigger with other related multi-agent simulation environments. Agents Size denotes agent scale in the interactive environment, where  $M$  and  $N$  mean the number of competitive teams and the agent number in each team, respectively, in the  $M \times  N$  settings. Actions denotes whether the action space is continuous, discrete, or hybrid. Obs denotes whether to support the partial observation. + in Coop and Comp indicates the importance of cooperation and competition. - means there is no cooperation or competition in the environment.  
+
+<table><tr><td></td><td>Agents Size</td><td>Actions</td><td>Obs</td><td>Coop</td><td>Comp</td></tr><tr><td>SMAC (Samvelyan et al., 2019)</td><td>2 × N</td><td>Discrete</td><td>Partial</td><td>+</td><td>+++</td></tr><tr><td>GRF (Kurach et al., 2019)</td><td>2 × N</td><td>Discrete</td><td>Partial</td><td>+</td><td>+++</td></tr><tr><td>Bi-DexHands (Chen et al., 2022)</td><td>1 × N</td><td>Discrete</td><td>Full</td><td>+++</td><td>—</td></tr><tr><td>MPE (Mordatch &amp; Abbeel, 2017)</td><td>2 × N</td><td>Hybrid</td><td>Full</td><td>++</td><td>++</td></tr><tr><td>Hide-and-Seek (Baker et al., 2019)</td><td>2 × N</td><td>Discrete</td><td>Partial</td><td>++</td><td>++</td></tr><tr><td>MAMujoco (Peng et al., 2021)</td><td>1 × N</td><td>Continuous</td><td>Partial</td><td>+++</td><td>—</td></tr><tr><td>NeuralMMO (Suarez et al., 2021)</td><td>M × N</td><td>Discrete</td><td>Partial</td><td>+</td><td>+++</td></tr><tr><td>GoBigger (Ours)</td><td>M × N</td><td>Hybrid</td><td>Partial</td><td>+++</td><td>+++</td></tr></table>
+
+environment with well-designed action space and observation space, giving partial view for all agents in the game. A reproducible benchmark including several state-of-the-art algorithms under different scenarios is accessible for users to quickly get started with GoBigger and demonstrate the potential of the environment. GoBigger also features game systems configurable for users to research on individual aspects of intelligence and on combinations thereof, with given rule-based built-in bots and visualization tools to make it easier for users to evaluate their agents. The overview of the research workflow for users is shown in Figure 1.
+
+# 2 RELATED WORK
+
+Most multi-agent reinforcement learning environments are mainly designed for cooperation or competition between no more than two teams of agents. SMAC (Samvelyan et al., 2019) consists of a set of carefully designed micro scenarios and necessitates learning one or more micromanagement techniques to defeat the enemy. But the SOTA algorithms can achieve high scores even without cooperation between agents. Google Research Football (Kurach et al., 2019) is a novel open-source reinforcement learning environment provides different modes for both single-agent and multi-agent settings, while there is no baseline indicating multi-agent cooperation and competition in the environment. Furthermore, both SMAC and Google Research Football can not handle situations where multiple (more than two) teams in a game, making them unable to be used for the research on swarm behaviors. Agar (staghuntrpg) implemented with python is also available and is evaluated on recent works (Tang et al., 2021). However, it focuses on cooperation and competition between two players but not multiple teams, and its scenarios are not complex. Neural MMO (Suarez et al., 2021) is a massively multi-agent environment for AI research, which support spans 1 to 1024 agents and minute- to hours-long time horizons. It also allows  $M \times N$  game mode in its scenarios, and agents need to cooperate with teammates to fight against other teams in the map. According to its given baselines, we find that agents in Neural MMO focus more on competition with other agents but less on cooperation up to now. And open-sourced agents in the challenge are more likely to play on their own instead of cooperating with teammates.
+
+# 3 DESIGN PRINCIPLES AND SYSTEM OVERVIEW
+
+GoBigger is inspired by the popular online multiplayer game Agar (SA). In a regular full game of Agar with OFA(One For All) mode, players cooperate with their teammates and fight against other teams for the final champion. On this basis, many modifications have been made to allow more
+
+cooperation and competition in GoBigger. Each player, which is represented by one or more balls (dubbed clone ball), increases its size by colliding and merging with other balls within a bounded rectangular area in a limited time. The larger the size of clone balls, the higher the player's score. At the beginning of the game, a single controllable clone ball (one player) is randomly spawned on the map. In addition to the clone balls, there are three uncontrollable neutral balls in the game map (dubbed as food ball, spore ball, and thorn ball), as shown in Figure 1.
+
+# 3.1 BASIC UNITS
+
+Food Ball: Food balls are neutral resources and they are always fixed on the map. Clone balls can eat food balls, and then absorb their size. Food balls will be randomly and continuously generated until the number of available food balls reaches the maximum.
+
+Spore Ball: Spore balls are ejected by clone balls. They will stay on the map and can be eaten by any other clone balls.
+
+Thorn Ball: Thorn balls are another kind of neutral resource in the game. Different from food balls, they have a larger size and less quantity on the map. In addition, they can eat the spore balls ejected from clone balls and absorb their size. When a clone ball eats a thorn ball, it will evenly split into several small balls. It means a player can push a thorn ball to an opponent's clone ball, force it to split, and eat its small balls.
+
+**Clone Ball:** A player consists of several clone balls and they are the only controllable balls. In addition, a clone ball can eat other smaller balls including food balls, thorn balls, spore balls, and clone balls (from teammates or opponents) by covering their center. Clone balls of an agent can be merged after a certain time. Clone balls will decay in size to ensure they will not grow too large. If all clone balls of a player are eaten, this player will respawn on the map randomly in the next frame. A player can control clone balls by the following three actions in total:
+
+- Move: Moving can help a clone ball eat other balls or escape from danger. The bigger the clone ball, the slower it moves.  
+- Eject: Ejecting a spore ball can help a clone ball drop some size and move faster (Figure 1). When a clone ball ejects, a new spore ball will appear on the clone ball's moving direction at high speed but quickly slow down and stop.  
+- Split: Splitting helps a clone ball to evenly split into two clone balls (Figure 1). Besides, a player can have at most 16 clone balls. To move faster, a player can split and turn the clone balls into smaller ones and get a higher speed.
+
+# 3.2 GAME SYSTEMS
+
+The base game representation is a limited map comprising the units declared in Section 3.1. Note that the map is not a grid map, which means that the continuous state is a complex challenge for agents. At the same time, GoBigger makes the operating efficiency of the game acceptable for users. The whole game can be split into the following systems: resources, cooperation, and competition.
+
+Resources: Resource system is designed for navigation and multi-task reasoning. In GoBigger, food balls and thorn balls can help the agents to grow larger. Eating food balls is safe but slow while eating thorn balls is dangerous but quick. There are food balls and thorn balls randomly distributed on the map. The agent needs to decide which ball to eat according to its current size and the state of other agents around it to obtain faster development. The initial state and the regeneration rates of resources are all configurable.
+
+Cooperation: Cooperation system is designed for direct cooperation among agents in a team. For a single agent, the maximum number of clone balls is 16 (configurable), which causes an agent cannot move fast by splitting indefinitely on its own. At the same time, each clone ball of the agent has a 20 seconds cooldown period, which is reset after each split. When the clone ball is on its cooldown period, it cannot merge with other clone balls that belong to the same agent. This means that splitting is dangerous because failing to merge quickly can easily lead to attacks by other agents. The above two constraints make multiple agents in the same team need to cooperate and transfer size to each other by splitting and ejecting to move faster and ensure safety. In addition, the final ranking of
+
+![](images/50848b3e6e996932086cf369c04b34b976487737068233d842a65d7c36d4a2e4.jpg)  
+(a) The full view of the game
+
+![](images/d2b9ae1375ad28161da4613ee23b45acab30a21446a3d5df19e6c0b42f73134a.jpg)  
+(b) The partial view of players  
+Figure 2: Screenshots for the different observation methods of GoBigger. (a) The full view of the game with 4 teams and 3 players in each team. (b) The players in (a) have their observations, and the three figures in each column represent the three players' view from each team.
+
+GoBigger is based on the size of the whole team. Therefore, frequent and reasonable cooperation within the team is an important condition for victory.
+
+**Competition:** Competition system is designed for direct competition among different teams. In GoBigger, the agent loses some size after each timestep. This is more obvious when the agent is very large. Therefore, as the resources on the map decrease as the game progress, the agent needs to eat other agents to ensure its size. Agents can attack other agents by moving and splitting to gain their size. In addition, gathering the size of the team to fight against other agents can often have an advantage in the competition. GoBigger encourages competition among teams.
+
+# 4 ENVIRONMENT
+
+GoBigger provides a set of standard game environment configurations. To facilitate access to the current observation and the efficient control of the balls, GoBigger follows the design of gym (Brockman et al., 2016) and develops easy-to-use interfaces of observation and action.
+
+# 4.1 OBSERVATION
+
+As mentioned in Section 3, the observation space of GoBigger is quite complex. It consists of GlobalState and PlayerState:
+
+GlobalState is shared by all players. It includes all essential global information, such as map size, total frames of the game, current running frame in the game, and the leaderboard which consists of the current scores and rankings of all teams.
+
+PlayerState is player-wise. It is the information that can be observed within the view of each player, making GoBigger a partially observable MDP (POMDP) (Drake, 1962; Sondik, 1971). Specifically, the player's view depends on the range of the positions of its clone balls. Figure 2 shows the view of different players in a game. When the clone balls of a player are dispersed enough, a larger view will be applied to this player. Given a view, PlayerState contains the positions and radius of all balls in the view, including food balls, thorn balls, spore balls, and clone balls, as well as their owner.
+
+More details can be found in Appendix A.1.
+
+# 4.2 ACTION SPACE
+
+The action space of the clone ball consists of move, eject and split could be shown as:
+
+$$
+a := (x, y, t). \tag {1}
+$$
+
+Table 2: GoBigger challenges. Agents Size denotes the number of agents on the map as Table 1. Map Size denotes the length and width of the map. Food denotes the range of food balls. Thorn denotes the range of thorn balls. Init Size denotes the size of clone balls at their first birth. Limited Frame denotes the number of frames in a game.  
+
+<table><tr><td>Name</td><td>Agents Size</td><td>Map Size</td><td>Food</td><td>Thorn</td><td>Init Size</td><td>Limited Frame</td></tr><tr><td colspan="7">Small maps</td></tr><tr><td>st_t1p1</td><td>1 × 1</td><td>32 × 32</td><td>[65, 75]</td><td>[1, 2]</td><td>1000</td><td>3600 (3min)</td></tr><tr><td>st_t1p2</td><td>1 × 2</td><td>48 × 48</td><td>[130, 150]</td><td>[2, 3]</td><td>1000</td><td>3600 (3min)</td></tr><tr><td>st_t2p1</td><td>2 × 1</td><td>48 × 48</td><td>[130, 150]</td><td>[2, 3]</td><td>1000</td><td>3600 (3min)</td></tr><tr><td>st_t2p2</td><td>2 × 2</td><td>64 × 64</td><td>[260, 300]</td><td>[3, 4]</td><td>13000</td><td>3600 (3min)</td></tr><tr><td>st_t3p2</td><td>3 × 2</td><td>88 × 88</td><td>[500, 560]</td><td>[5, 6]</td><td>13000</td><td>3600 (3min)</td></tr><tr><td colspan="7">Large maps</td></tr><tr><td>st_t4p3</td><td>4 × 3</td><td>128 × 128</td><td>[800, 900]</td><td>[9, 12]</td><td>1000</td><td>14400 (12min)</td></tr><tr><td>st_t5p3</td><td>5 × 3</td><td>128 × 128</td><td>[900, 1000]</td><td>[10, 12]</td><td>1000</td><td>14400 (12min)</td></tr><tr><td>st_t5p4</td><td>5 × 4</td><td>144 × 144</td><td>[900, 1000]</td><td>[10, 12]</td><td>1000</td><td>14400 (12min)</td></tr><tr><td>st_t6p4</td><td>6 × 4</td><td>144 × 144</td><td>[1000, 1100]</td><td>[11, 13]</td><td>1000</td><td>14400 (12min)</td></tr></table>
+
+Here  $a$  denotes the action,  $(x,y)$  is a unit vector for the direction of movement, and  $t$  is the action type for the clone ball to be executed in the current frame. When a player applies a direction  $(x,y)$ , enforcement in this direction will be applied on all this player's clone balls and smoothly alters their movement direction to this direction in several frames. The action types consist of moving, ejecting, and splitting. Agents can complete development and attack according to the combination of different action types. For more details, please refer to Appendix A.2.
+
+# 4.3 REWARD
+
+In Gobigger, the goal of all teams is to become bigger and bigger and finally become the biggest. Thus a natural reward for a single agent is to consider its score change of two adjacent steps:
+
+$$
+\text {R e w a r d} _ {t} := \text {S c o r e} _ {t + 1} - \text {S c o r e} _ {t}. \tag {2}
+$$
+
+The above reward can directly reflects the players' growth and induce agents to grow bigger. But it can't encourage agents to cooperate and confront each other very well. Therefore, GoBigger provides some other kinds of reward functions as follows:
+
+Example 1: Rapid development must come from eating a large number of thorn balls, so the reward function can be designed as the number of thorn balls eaten by the agent between two adjacent steps.
+
+Example 2: The cooperation among teammates consists of ejecting spore balls to each other and eating teammates' clone balls, so the reward function can be designed as the number of spores and the number of clone balls eaten by teammates between two adjacent steps.
+
+Example 3: To improve the offensive performance of the agent, the reward function can be designed as the number of clone balls that the agent eats from other teams between two adjacent steps.
+
+With the help of different kinds of rewards and constraints, an agent with strong cooperation and aggression will be obtained. For more details of reward functions, please refer to Section 5.
+
+# 4.4 SCENARIOS
+
+GoBigger consists of a set of scenarios that aim to evaluate the learning cooperation and competition of different agents in different game settings. The scenarios are carefully designed with different settings including agents size, map size, food number, horn number, and so on. Most of the scenarios are a confrontation among over two teams of players. In different scenarios, agents need to learn to acquire resources on the map and fight against other teams through cooperation. Teams are only different in their locations of birth. When the specified time limit is reached, the team with the
+
+![](images/a79c0aed8d23ee76ffffad08b33324f7cf5c1007c5efa76bfe61cd74163d22b8.jpg)  
+Figure 3: Agents trained on  $st\_t2p2$ ,  $st\_t3p2$ , and  $st\_t4p3$  with DQN (the first row) and PPO (the second row). Each column (left to right) denotes the score against built-in bots, the average number of eaten food balls, the average number of eaten horn balls, and the average number of eaten clone balls from teammates.
+
+higher score wins. In the evaluation phase, agents have to battle against other teams of agents in the scenarios. Built-in bots with different levels could be also chosen as competitors. The rule-based bots can increase their sizes quickly by eating the food balls and thorn balls and avoid being eaten by other players through the given rules. The complete list of challenges is presented in Table 2. The Init Size of  $st\_t2p2$  and  $st\_t3p2$  is 13000 to reduce farming time and make the agent more focused on cooperation and competition. According to the map size and the number of agents on the map, the scenarios are simply divided into two parts including small maps and large maps. The large maps may be more difficult as its larger observation while playing against more teams of agents.
+
+# 5 EXPERIMENTS
+
+In this section, we will present some experiments results of GoBigger, which are based on the most representative scenarios:  $st\_t2p2$ ,  $st\_t3p2$  and  $st\_t4p3$ . Details of observation encoding and neural network can be found in Appendix B.1. The agents learned by different RL algorithms are evaluated by fighting against the built-in bots of level 3 A.5 several times. The average score will be one of the most important metrics as it stands for the game level of the agents, showing cooperation and competition skills.
+
+# 5.1 SINGLE AGENT ALGORITHMS
+
+We apply DQN (Mnih et al., 2015) and PPO (Schulman et al., 2017) on GoBigger, where each agent only use its own observation and learns a value function independently. Training details are presented in Appendix B.3. The experiment results in different scenarios can be found in Figure 3. The score represents the game level of the agents. Agents can get high scores by eating more food balls and thorn balls in a limited time. It also shows that the popular single-agent algorithms can quickly converge in the given scenarios, inferring that GoBigger is a feasible and challenging environment for multi-agent reinforcement learning research.
+
+# 5.2 MULTI AGENT ALGORITHMS
+
+We also apply several state-of-the-art multi-agent algorithms including QMIX (Rashid et al., 2018), MAPPO (Yu et al., 2021), COMA (Foerster et al., 2018), and VMIX (Su et al., 2021) on GoBigger. The training details are presented in Appendix B.3. Figure 4 shows the performance of different multi-agent algorithms in GoBigger. Overall MAPPO achieves the highest scores when fighting against the built-in bots and is the best performer in all three scenarios with the fewest environment steps. Additionally, with a fixed number of steps, COMA has the worst performance of all algorithms, demonstrating the on-policy policy gradient methods can not have an advantage over different algorithms. Figure 4 also shows that the number of food balls and thorn balls grows rapidly in the training phase. This is in line with the expectations of the environments, in which the fastest way for an agent to farm is to eat food balls and thorn balls quickly. Besides, after enough steps, the agents
+
+![](images/8826ebeea49790aa71721faebb49864e4ecccfe1bd3a6cbf0ed619d3cf749162.jpg)  
+Figure 4: Agents trained on  $st\_t2p2$ ,  $st\_t3p2$ , and  $st\_t4p3$  with QMIX (the first row), VMIX (the second row), MAPPO (the third row), and COMA (the fourth row). Each column (left to right) denotes the score against built-in bots, the average number of eaten food balls, eaten thorn balls, and eaten clone balls from teammates.
+
+learn to cooperate with teammates, which is shown from the number of `clone-team`. The growth of `clone-team` means that the agents are trying to merge with teammates.
+
+The results in different scenarios are also shown in Figure 4.  $st\_t2p2$  is a small map with only two teams and four agents, while  $st\_t4p3$  is a large map containing four teams and each consists of three agents. QMIX plays well on small maps including  $st\_t2p2$  and  $st\_t3p2$  but plays bad on large map  $st\_t4p3$ . We infer that the map of  $st\_t4p3$  is too large and too difficult for agents to find their teammates, which causes a low level of cooperation.
+
+# 6 ABLATION STUDY
+
+GoBigger provides small and large maps for training on different scales. Here we explore some factors that have impacts on the performance of an agent, including the necessity of the observations from teammates, the frequency of acting in the game, and the ladder system.
+
+# 6.1 DROPPING OBSERVATIONS FROM TEAMMATES
+
+To explore the importance of cooperation in GoBigger, we drop the observations from teammates, making the agents decide their actions only based on their observations. Figure 5 shows that the agents with teammates' observations outperform the agents without observations from the whole team. Agents can not easily find their teammates without the observations sharing especially on a large map, i.e.,  $st\_t4p3$ , resulting that they could not cooperate with their teammates to fight against other teams, which is consistent with our original intention of designing the GoBigger. Appendix C.1 shows more details.
+
+![](images/d3b5a1c5282295952387e5ee5e09019c5247298340d702ba6d1e2aeb93c46971.jpg)  
+Figure 5: Comparing agents w/ or w/o observation from teammates. Each picture (left to right) denotes the performance of agents on  $st\_t2p2$ ,  $st\_t3p2$ , and  $st\_t4p3$ .
+
+![](images/67483769d0224e2334cc40fc8857526b54cf28909ae7511dbdb1543f6b2f750a.jpg)  
+Figure 6: Comparing agents with different frequencies of actions on  $st\_t2p2$ ,  $st\_t3p2$ , and  $st\_t4p3$  (left to right). Each line denotes the score against bots with a certain frequency of actions.
+
+# 6.2 FREQUENCY OF ACTIONS
+
+In most reinforcement learning environments, agents will execute actions immediately after environments step(Jia et al., 2021). However, it is not necessary in a real-time game. For example, AlphaStar(Vinyals et al., 2019) uses a delay head in their network to predict delay of the next action. To save computing resources, we explore the impact of the frequency of actions on an agent's performance. Figure 6 shows that the agents with frequency  $= 8$  outperform other settings. We infer this is because higher frequency will bring more redundant information, which is a little more difficult for agents to understand and distinguish.
+
+# 6.3 LADDER SYSTEM AND LEADERBOARD
+
+In multiplayer games, reward versus some specific opponents is not enough to fully measure agents skills in different situations. So we introduce a ladder system based on TrueSkill (Herbrich et al., 2006) to evaluate different kinds of agents objectively. This ladder system will make agents continuously play against each other, and then use their final ranks in each game to update their trueskill scores. Figure 11 shows the ladder system including all of our trained agents as well as built-in bot. More details of the ladder system could be found in Appendix C.2.
+
+# 7 CONCLUSION AND FUTURE WORK
+
+In this paper, GoBigger is presented as a scalable platform for multi-agent interactive simulation. GoBigger allows  $M \times N$  game mode that focuses on intra-team cooperation and inter-team competition. We offer a diverse set of challenge scenarios in GoBigger for best practices in benchmarking. A reproducible benchmark including several state-of-the-art algorithms under different scenarios is accessible. GoBigger also features a game system configurable with given rule-based built-in bots and visualization tools to make it easier for users to evaluate their agents.
+
+In the near future, we aim to explore more based on GoBigger. Considering the scenarios in GoBigger now are monotonous, we plan to develop more interesting and complex scenarios that require a higher level of coordination amongst agents. We also plan to expand GoBigger as a massive environment to host thousands of agents. With harder multi-agent coordination and competition problems, we aim to motivate further research in this domain, particularly in areas such as multi-agent exploration and coordination.
+
+Acknowledgement. This work was supported by the National Key RD Program of China under Grant STI 2030—Major Projects 2021ZD0201300.
+
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+Ruonan Jia, Qingming Li, Wenzhen Huang, Junge Zhang, and Xiu Li. Consistency regularization for ensemble model based reinforcement learning. In 18th Pacific Rim International Conference on Artificial Intelligence, Proceedings, Part III 18, pp. 3-16. Springer, 2021.  
+Matthew Johnson, Katja Hofmann, Tim Hutton, and David Bignell. The malmo platform for artificial intelligence experimentation. In *Ijcai*, pp. 4246-4247. CiteSeer, 2016.  
+Venkatesan Kanagaraj and Martin Treiber. Self-driven particle model for mixed traffic and other disordered flows. Physica A: Statistical Mechanics and its Applications, 509:1-11, 2018.  
+Karol Kurach, Anton Raichuk, Piotr Stanczyk, Michal Zajac, Olivier Bachem, Lasse Espeholt, Carlos Riquelme, Damien Vincent, Marcin Michalski, Olivier Bousquet, et al. Google research football: A novel reinforcement learning environment. arXiv preprint arXiv:1907.11180, 2019.
+
+Trevor McInroe. Repo for the multi-agent pressureplate environment. https://github.com/uoe-agents/pressureplate, 2022.  
+Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. nature, 518(7540):529-533, 2015.  
+Sharada Mohanty, Erik Nygren, Florian Laurent, Manuel Schneider, Christian Scheller, Nilabha Bhattacharya, Jeremy Watson, Adrian Egli, Christian Eichenberger, Christian Baumberger, Gereon Vienken, Irene Sturm, Guillaume Sartoretti, and Giacomo Spigler. Flatland-rl: Multi-agent reinforcement learning on trains, 2020.  
+Igor Mordatch and Pieter Abbeel. Emergence of grounded compositional language in multi-agent populations. arXiv preprint arXiv:1703.04908, 2017.  
+John Fredrik Wilhelm Norén. Derk gym environment. https://gym.derkgame.com, 2020.  
+Georgios Papoudakis, Filippos Christianos, Lukas Schäfer, and Stefano V Albrecht. Comparative evaluation of multi-agent deep reinforcement learning algorithms. arXiv preprint arXiv:2006.07869, 2020.  
+Georgios Papoudakis, Filippos Christianos, Lukas Schäfer, and Stefano V. Albrecht. Benchmarking multi-agent deep reinforcement learning algorithms in cooperative tasks. In Proceedings of the Neural Information Processing Systems Track on Datasets and Benchmarks (NeurIPS), 2021. URL: http://arxiv.org/abs/2006.07869.  
+Bei Peng, Tabish Rashid, Christian Schroeder de Witt, Pierre-Alexandre Kamienny, Philip Torr, Wendelin Böhmer, and Shimon Whiteson. Facmac: Factored multi-agent centralised policy gradients. Advances in Neural Information Processing Systems, 34, 2021.  
+Tabish Rashid, Mikayel Samvelyan, Christian Schroeder, Gregory Farquhar, Jakob Foerster, and Shimon Whiteson. Qmix: Monotonic value function factorisation for deep multi-agent reinforcement learning. In International conference on machine learning, pp. 4295-4304. PMLR, 2018.  
+Cinjon Resnick, Wes Eldridge, David Ha, Denny Britz, Jakob Foerster, Julian Togelius, Kyunghyun Cho, and Joan Bruna. Pommerman: A multi-agent playground. CoRR, abs/1809.07124, 2018. URL http://arxiv.org/abs/1809.07124.  
+Miniclip SA. Agar.io. https://agar.io/.  
+Mikayel Samvelyan, Tabish Rashid, Christian Schroeder De Witt, Gregory Farquhar, Nantas Nardelli, Tim GJ Rudner, Chia-Man Hung, Philip HS Torr, Jakob Foerster, and Shimon Whiteson. The starcraft multi-agent challenge. arXiv preprint arXiv:1902.04043, 2019.  
+John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017.  
+Edward Jay Sondik. The optimal control of partially observable Markov processes. Stanford University, 1971.  
+staghuntrpg. Python implement for agar. https://github.com/staghuntrpg/agar.  
+Jianyu Su, Stephen Adams, and Peter Beling. Value-decomposition multi-agent actor-critics. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp. 11352-11360, 2021.  
+Joseph Suarez, Yilun Du, Clare Zhu, Igor Mordatch, and Phillip Isola. The neural mmo platform for massively multiagent research. arXiv preprint arXiv:2110.07594, 2021.  
+Zhenggang Tang, Chao Yu, Boyuan Chen, Huazhe Xu, Xiaolong Wang, Fei Fang, Simon Du, Yu Wang, and Yi Wu. Discovering diverse multi-agent strategic behavior via reward randomization. arXiv preprint arXiv:2103.04564, 2021.  
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017.
+
+Oriol Vinyals, Igor Babuschkin, Wojciech M Czarnecki, Michael Mathieu, Andrew Dudzik, Junyoung Chung, David H Choi, Richard Powell, Timo Ewalds, Petko Georgiev, et al. Grandmaster level in starcraft ii using multi-agent reinforcement learning. Nature, 575(7782):350-354, 2019.  
+Ping Xuan, Victor Lesser, and Shlomo Zilberstein. Communication decisions in multi-agent cooperation: Model and experiments. In Proceedings of the fifth international conference on Autonomous agents, pp. 616-623, 2001.  
+Chao Yu, Akash Velu, Eugene Vinitsky, Yu Wang, Alexandre Bayen, and Yi Wu. The surprising effectiveness of ppo in cooperative, multi-agent games. arXiv preprint arXiv:2103.01955, 2021.
+
+# A GOBIGGER
+
+# A.1 OBSERVATION
+
+The observation in GoBigger is divided into two parts, including Global State and Player States. After each step, GoBigger will return observation for all agents in the environment. Here are the details for Global State. border denotes the map size of the game. total_frame denotes the number of total frames in the game. last_frame_count denotes the number of frames that have passed. leaderboard denotes the ranks and scores of all agents.
+
+```txt
+global_state = {  
+2 'border': [map_width, map_height],  
+3 'total_frame': total_frame,  
+4 'last_frame_count': last_frame_count,  
+5 'leaderboard': { team_name: team_size }  
+6 }
+```
+
+The Player States declare the specific states for each player (or agent) in each team. player_id denotes the identity of the player in the game. rectangle denotes the position of partial vision. overlap denotes all kinds of balls in the vision of the player, including the position, radius, score, velocity, direction, player_id, and team_id. team_name denotes the team's name of the player. score denotes the score of the player. can_eject denotes if the player can eject at this frame. can_split denotes if the player can split at this frame.
+
+```python
+1 player_states = {  
+2 player_id: {  
+3 'rectangle': [left_top_x, left_top_y, right_bottom_x, right_bottom_y],  
+4 'overlap': {  
+5 'food': [[position.x, position.y, radius, score], ...],  
+6 'thorns': [[position.x, position.y, radius, score, vel.x, vel.y], ...],  
+7 'spore': [[position.x, position.y, radius, score, vel.x, vel.y, owner], ...],  
+8 'clone': [[position.x, position.y, radius, score, vel.x, vel.y, direction.x, direction.y, player_id, team_id], ...],  
+9 },  
+10 'team_name': team_name,  
+11 'score': player_score,  
+12 'can_eject': bool,  
+13 'can_split': bool,  
+14 },  
+15 ...  
+16 }
+```
+
+The original structural observations may be difficult for new users in GoBigger. That's why we modified the observations into a simpler tensor array in Section B.1. New users can directly use the given observation encoder, while advanced users can build their own observation encoders which are more suitable for their algorithms.
+
+# A.2 ACTION
+
+In Section 4.2, we offers a simple action space for the control of agents. And the format of the given action could be shown as:
+
+$$
+a := (x, y, t). \tag {3}
+$$
+
+The value of  $t$  is in  $0,1,2$ , separately representing move, eject and split. Especially, when  $t$  is 1 or 2, the clone balls will eject or split in the given direction decided by  $(x,y)$ . At most time, a player has more than one clone ball. Under this circumstance, the action will be applied to all the clone balls of
+
+![](images/be2e0513e0335c6e7cb1cb28d1d734376648383494289a7450cf02523da28cfa.jpg)  
+(a)
+
+![](images/a2e3114b0542fb885faaa7e818a4466b323d105fd145947a4e91783bd5097a22.jpg)  
+(b)  
+Figure 7: Screenshots for one of the action combinations named middle-ejecting. This action combination will make agents gather their size into one big clone ball. (a) The clone ball is ready to eat the thorn ball. (b) After eating the thorn ball, the clone ball is forced to split into several pieces. (c) The agent ejects to the center. (d) The clone balls at the center eat the spore and merges their size together.
+
+![](images/39038fb384d38fb7b0b5b5494c7a913e1886540e06973a1c6b97b8b6a84620ae.jpg)  
+(c)
+
+![](images/c34e0ee06a30c3aae388320016d23608123f09f756af86ae32db19f97b524fa6.jpg)  
+(d)
+
+the player, i.e., all clone balls with sizes exceeding the split threshold will split when  $t$  is 1, and all clone balls with sizes exceeding the eject threshold will eject spores when  $t$  is 2.
+
+# A.3 ACTION COMBINATIONS
+
+Though the action space in GoBigger is simple, the combinations of different actions can lead to quick growth, attack, and escape. Figure 7 shows one of the action combinations named middle-ejecting, which is the combination of moving and ejecting. It is really useful to gather most of the size in one clone ball after eating the thorn ball. Besides, agents could use this action combination when they are split into too many pieces, which may help them to avoid being eaten by other agents. In the experiments in Section 5 we find that the agents can perform these action combinations.
+
+# A.4 COLLISION DETECTION
+
+The core problem of GoBigger is how to detect the collision of the balls in each frame and help the game update the state of the balls. The problem of collision detection could be simplified into a scenario where we need to get the collided balls from the given gallery balls for each ball in the given query balls. GoBigger has designed four collision detection algorithms as follows:
+
+Exhaustive For each ball in the query, we enumerate each ball in the gallery to determine whether there is a collision. This is an exhaustive solution that will take a long time if the number of balls in the gallery is large. The complexity of the algorithm could be written in:
+
+$$
+O (n * m) \tag {4}
+$$
+
+where  $m$  denotes the number of balls in the query, and  $n$  denotes the number of balls in the gallery.
+
+Precision Precision approximation algorithm divides the map into several rows according to the accuracy that has been set, dynamically maintain the row information in each frame and search by row. First, we need to sort the balls in each row according to their ordinate. For the balls in query, we abstract the boundary of the ball into a rectangle, then traverse each row in the rectangle, and find the first ball covered by the query through dichotomy in each row, and then Enumerate the balls in sequence until the ordinate exceeds the boundary of the query rectangle. The complexity of the algorithm could be written in:
+
+$$
+O (n * \log (n) + \Sigma r * \log (n) + p) \tag {5}
+$$
+
+where  $m$  denotes the number of balls in query,  $n$  denotes the number of balls in the gallery,  $k$  denotes the precision we set,  $r$  denotes the number of balls whose position status has changed and  $p$  denotes the number of balls that collide with other balls.
+
+RebuildQuadTree We build a Quadtree on a two-dimensional plane in every frame based on the positions of balls and search the collisions that happen among the balls in query and gallery according to the Quadtree. With the Quadtree, GoBigger can quickly find the nearby balls of the given query
+
+ball, which helps to search the collisions in a relatively small search area. The complexity of the algorithm could be written in:
+
+$$
+O (n * \log (n) + m * \log (n) + p) \tag {6}
+$$
+
+where  $m$  denotes the number of balls in query,  $n$  denotes the number of balls in the gallery, and  $p$  denotes the number of balls that collide with other balls.
+
+RemoveQuadTree This algorithm is based on the RebuildQuadTree A.4, and we add the operations of deleting nodes in the tree with dynamically maintaining the tree. The complexity of the algorithm could be written in:
+
+$$
+O (r * \log (n) + m * \log (n) + p) \tag {7}
+$$
+
+where  $m$  denotes the number of balls in query,  $n$  denotes the number of balls in the gallery,  $r$  denotes the number of balls whose position status has changed and  $p$  denotes the number of balls that collide with other balls.
+
+To test the efficiency of the above algorithms, we modify the parameters including the number of balls in query and gallery, the number of changed balls, and the iteration rounds to get a more fair and intuitive result. The data in Table 3 comes from the most representative scenarios. We finally choose Precision A.4 as our default algorithm for collision detection.
+
+Table 3: Comparison of the different algorithms of collision detection on different settings.  $T$  denotes the number of balls in the gallery.  $Q$  denotes the number of balls in the query.  $C$  denotes the number of changing balls, which means the number of collisions.  
+
+<table><tr><td></td><td>T=3000</td><td>T=3000</td><td>T=10000</td><td>T=10000</td><td>T=30000</td></tr><tr><td></td><td>Q=300</td><td>Q=300</td><td>Q=1000</td><td>Q=2000</td><td>Q=600</td></tr><tr><td></td><td>C=600</td><td>C=1500</td><td>C=2000</td><td>C=5000</td><td>C=3000</td></tr><tr><td>Exhaustive</td><td>688ms</td><td>1067ms</td><td>8384ms</td><td>12426ms</td><td>127000ms</td></tr><tr><td>Precision</td><td>14ms</td><td>16ms</td><td>61ms</td><td>86ms</td><td>403ms</td></tr><tr><td>Rebuild QuadTree</td><td>47ms</td><td>50ms</td><td>339ms</td><td>586ms</td><td>5691ms</td></tr><tr><td>Remove QuadTree</td><td>48ms</td><td>178ms</td><td>497ms</td><td>2460ms</td><td>8419ms</td></tr></table>
+
+# A.5 BUILT-IN BOTS
+
+Built-in bots of different levels can help users to get started with the environment and perform a standard evaluation for algorithm development. Level 1 The bot of level 1 aims to collect neutral resources on the map for quick development in size. It can only move and eat the closest food balls in its view. Level 2 We add the exploration of thorn balls in view for the bot of level 2. The bot prefers to eat thorn ball in its view. Level 3 The bot of level 3 can avoid being eaten by other larger players as they will move far away from the larger player in their view.
+
+# B EXPERIMENTS
+
+# B.1 OBSERVATION TRANSFORM
+
+It is challenging to encode thousands of entities compared to other multi-agent environments. In order to better model the observation information, we transform the original information into Scalar info, Spatial info, Team info, Ball info. As shown in (Figure 8), Scalar Info models the size of the agent's local vision, the current ranking and score, and the action type at the last moment. Spatial info models the location of different types of balls in vision. Team info models information of teammates in vision. Ball info models the properties of different balls, including relative position, size, and speed. Here are more details of the information after transformation.
+
+![](images/e91eeb9a1878fa0a2903c7a84597c030d93eb676849e5685b03f20719a3bcb64.jpg)  
+Figure 8: The overview of feature encoder.
+
+# Scalar info
+
+- (view_x, view_y): The positions of the center of the observation.  
+- view_width: The width of the observation.  
+- score: Score of the agent.  
+- team_score: Team score of the agent.  
+time: Remaining game time.  
+rank: Rank of the agent.  
+- last_action_type: Action type in the previous step.
+
+# Ball info
+
+- alliance: The numbering for different types in observation.  
+- score: Scores for different types of balls.  
+- radius: Radius for different types of balls.  
+- rank: Rank for different types of balls.  
+- x,y: position for different types of balls.  
+- next_x, next_y: The predicted position in the next frame for different types of balls.  
+- ball_num: The number of the balls(clone and thorn) in observation.
+
+# Team info
+
+- alliance: The teammate's identity information.  
+- view_x,view_y: The teammate's position information.  
+- player_num: The number of the teammates.
+
+# Spatial info
+
+- food_x, food_y: The position of food balls.  
+- spore_x, spore_y: The position of spore balls.  
+- ball_x, ball_y: The position of clone balls and thorn balls.  
+- food_num: The number of food balls.  
+- spore_num: The number of spore balls.
+
+![](images/919ec6de51abcf0b8cf6cea8982341d0ae6f80fd7366e782bbc82f6c316026f7.jpg)  
+Figure 9: Neural network in experiments.
+
+# B.2 NEURAL NETWORK
+
+The neural network in our experiments consists of MLP (Haykin, 1994), Transformer (Vaswani et al., 2017), and ResNet (He et al., 2016). The specific structure is shown in Figure 9. Finally, we concatenate all the output embeddings into a feature.
+
+# B.3 EXPERIMENTAL SETUP
+
+DQN The batch_size is set as 512. learning_rate is 1e-4, replay_buffer_size is 2e4.
+
+PPO The batch_size is set as 256. clip_range is 0.2, gae_lambda is 0.95, and grad Clip is 0.5.
+
+QMIX The batch_size is set as 512. learning_rate is 1e-4, replay_buffer_size is 2e4.
+
+MAPPO The batch_size is set as 512. clip_range is 0.2, gae_lambda is 0.95, and grad Clip is 0.5.
+
+VMIX The batch_size is set as 2048, and the data in every step contains all the observations and actions of all the agents. td_lambda is 0.95, and grad Clip is 0.5.
+
+COMA The batch_size is set as 2048, and the data in every step contains all the observations and actions of all the agents. td_lambda is 0.95, target_update_interval is 20, and grad Clip is 0.5.
+
+# C ABLATION STUDY
+
+# C.1 DROPPING OBSERVATIONS FROM TEAMMATES
+
+Figure 10 shows the food, thorn, spore, clone_self, clone-team, and clone_other of the experiments in Section 6.1. With the observations from teammates, agents are more likely to eat thorn balls instead of food balls. We can find that the number of eaten thorn balls is more than that without the observations, as eating thorn balls can provide more scores. A single agent can not eat too many thorn balls, as it will split into several pieces after eating a thorn ball. With the cooldown period, an agent can not merge quickly to gather enough size to eat the next thorn ball. But when an agent learns to cooperate with teammates, they can gather size quickly and eat the next thorn ball ignoring the cooldown period, which could be found by the number of clone_team in the figure. Eating more clone balls from teammates means that the agents can better cooperate with other agents in the same team.
+
+# C.2 LADDER SYSTEM AND LEADERBOARD
+
+Our ladder system is based on TrueSkill (Herbrich et al., 2006), which is a skill-based ranking system developed by Microsoft for use with video game matchmaking on Xbox Live. TrueSkill is designed to support games with more than two players. In our ladder system, we assume that each contestant
+
+![](images/fdee715c5168e71ecdf21d832b1e3e623573af3efda47df5ea54440f9ae56ce0.jpg)
+
+![](images/355cb8f35162a91c47e708dc3fed4342565791629c7b6b412453cef0eccee805.jpg)
+
+![](images/ecb1b6456dc772f9fdc84606a2f6f3fa81c066ff4c2dc54da9c15e022e75b8d8.jpg)
+
+![](images/72f2a00e078c12396972d9d313920e47e2702d6f397b5f85343ab70bfb819279.jpg)
+
+![](images/a2af20d5a98cbc6879e2612c6425f73380f2172ee54fc9ca803475bc27e66665.jpg)
+
+![](images/eb5d53eafa88c2c19150d1cb7a9ef6af36295041372263250746a9ab2bff98be.jpg)
+
+![](images/281bcb30cde6556a65e981eea34c24eb8224322b222cba96b7a293ca29632d2f.jpg)
+
+![](images/ef64585ae5ac02663c5288dfba6acd04716831d5a5e1600bfce80c4347b729fd.jpg)  
+(a)  $st\_t2p2$
+
+![](images/199f28aafde519021b27e1abf381e0c9e91ed2facbc8f51427d4fc24db8c3ea1.jpg)
+
+![](images/c12508ee91437588a18190097eae409c17bef47875013e9d428740fa25b2f5ff.jpg)
+
+![](images/a890918b214f231b99e6421e2aabc7dafac490ff3b3365a200e801a53f93a06d.jpg)
+
+![](images/1843ef6341cc0e76dc681c6e86012f4640644f88c7f280618a76d2c892a962b0.jpg)
+
+![](images/f8b1987fe65a2294932880561c8c23fa089fc6ac6bd32a6ea61c87b278782477.jpg)
+
+![](images/86541bfb2fc347ea480f005a8652750ef792e5bad82b48c56bc9d498ed516fa4.jpg)  
+(b)  $st\_ t3p2$
+
+![](images/47e885ff617b1d471bd4e95a2c24a955e64d1a131c4cbebb95b2f49863e1e727.jpg)
+
+![](images/2aaffdc9b04eacf3eab67fa607d167ea7fde8bdc5306babf190d0430668b2a7d.jpg)  
+Figure 10: Comparing agents w/ or w/o observation from teammates.
+
+![](images/f00101bf5bd605024987bfa715a0b854b71e009b64b354f35848fe645a81c8e2.jpg)  
+(c)  $st\_t4p3$
+
+![](images/eaabb879b4d39c9b930b7a0be60624298a43c6349385957f068d4f3e4f4211d4.jpg)
+
+controls a whole team. The contestant continuously plays against other contestant and update their scores based on the ranking at the end of the game. The initial score for each contestant is 1000. To ensure the balance of contestants' matches, we select the contestant with the least number of matches each time. Eq 8 shows the quality we use as the indicator for selecting opponents.
+
+$$
+q u a l i t y _ {d r a w} \left(\beta^ {2}, \mu_ {i}, \mu_ {j}, \sigma_ {i}, \sigma_ {j}\right) = \sqrt {\frac {2 \beta^ {2}}{2 \beta^ {2} + \sigma_ {i} ^ {2} + \sigma_ {j} ^ {2}}} \cdot e x p \left(- \frac {\left(\mu_ {i} - \mu_ {j}\right) ^ {2}}{2 \left(2 \beta^ {2} + \sigma_ {i} ^ {2} + \sigma_ {j} ^ {2}\right)}\right) \tag {8}
+$$
+
+![](images/44b0f7c55078741babf60fd4c17e937f95f4e38ab220dfed2a8562f928e7d3e5.jpg)
+
+![](images/e103be8fb1c93952f545d73b56b4db68ea40c3fb750e6154172d6a2cab0226ed.jpg)  
+Figure 11: Comparing agents on the ladder system. The scenario is  $st\_t2p2$ .  
+(a)
+
+![](images/319436eef472bcbe44a61a9750f1afffff52e99b8f4073918bbed52f604e857a.jpg)  
+(b)
+
+![](images/4c8272f74795c585b684f63f3fdef341050b3912f1d152214c3a2428664f0093.jpg)  
+(c)
+
+![](images/ddb633dc0f16a8cb39eccaa19a75551de54ce779e60ec8c389fcf03fc2f8bc16.jpg)  
+(d)
+
+![](images/8e286312b8f852dab814b02dac2972b6201d9bff66eeae4aea1a9940d693e9e2.jpg)  
+(e)
+
+![](images/ad8b1825393ba9221902781448b75b6dcddf9ef9454826896a7e55f868719312.jpg)  
+Figure 12: Screenshots of cooperating to eat the thorn ball. (a) The clone ball A and B get closed to each other. (b) The clone ball B eject to the A. (c) The clone ball A eat the spore and grow bigger. (d) The clone ball A eat several clone balls from teammates and grow big enough to eat the thorn ball. (e) The clone ball A eat the thorn ball.  
+(a)  
+Figure 13: Screenshots of competition between different agents. (a) The green clone balls find the purple clone balls nearby. (b) The green clone balls split to eat the purple ones. (c) The purple clone balls gather together to avoid being eaten by the green clone balls.
+
+![](images/566bf3b4d8a4dddd88e9b85f82a045bd29279dbeacba55547cdc107d78acd58e.jpg)  
+(b)
+
+![](images/672e89dafea86703ad2fb00073c25b1c1f15c039e9d0b9ed45204a518a5d5002.jpg)  
+(c)
+
+where contestant  $i$  is assumed to exhibit a performance  $p_i \sim \mathbb{N}(p_i; \mu_i, \sigma_i^2)$ , contestant  $j$  is assumed to exhibit a performance  $p_j \sim \mathbb{N}(p_j; \mu_j, \sigma_j^2)$ , and  $\beta^2 = \left(\frac{\sigma}{2}\right)^2$
+
+Figure 11 shows the ladder system including all the trained agents in Section 5. Most of the contestants have played over 1500 games with other contestants. The ladder system ensures that each contestant can match opponents with similar game levels as much as possible while retaining the possibility of matching opponents with higher or lower game levels.
+
+# D PERFORMANCE
+
+In this section, we post some screenshots to show the actual performance of the agent in the game, including the cooperation with teammates and the competition with opponents.
+
+![](images/5a1e95f448daa739e43551390b8b55a00bfc58727db9c199e5a17823edf2f30b.jpg)  
+(a)
+
+![](images/f15169ac1f3403a59e43d3b7d8c173a7da1616c1afda228557365e6f3f661489.jpg)  
+(b)  
+Figure 14: Screenshots of cooperation between different agents. (a) The clone ball C is ready to eat the thorn ball. (b) The clone balls split into many pieces. (c) The clone balls eject to the center of the agents. (d) The clone balls finally merge into a big one.
+
+![](images/e7508cc3a66c39a1d353e28b6a34dedd0f482a077d1812a071a57b40770003ec.jpg)  
+(c)
+
+![](images/a9758410235e2868ab1df91c99f89cca846001e576b332e0617e594d763d09e5.jpg)  
+(d)
+
+![](images/42ad6554bfb7b511617f811642d2f2b3166eaf1ec8b94b20e05f8180dabab3c3.jpg)  
+(a)  
+Figure 15: Screenshots of competition between different agents. (a) The bule and green agents are competing for the thorn ball. (b) The green clone balls split to eat the thorn ball. (c) The blue clone balls split and eat the green clone balls and win more score.
+
+![](images/a8ee1864638c9be75be415ee6ffa0e60b5b6b1a77644e012eed51f8189b4c9b0.jpg)  
+(b)
+
+![](images/c549fb9d12d4fcf87e9668e835aae91dcc7eeb79cde7fa8b8c2b96b334785c96.jpg)  
+(c)

2023/GoBigger_ A Scalable Platform for Cooperative-Competitive Multi-Agent Interactive Simulation/images.zip

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2023/Gradient Gating for Deep Multi-Rate Learning on Graphs/full.md

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+# GRADIENT GATING FOR DEEP MULTI-RATE LEARNING ON GRAPHS
+
+T. Konstantin Rusch
+
+ETH Zürich, ICSI and UC Berkeley
+
+konstantin.rusch@sam.math.ethz.ch
+
+Benjamin P. Chamberlain
+
+Charm Therapeutics
+
+Michael W. Mahoney
+
+ICSI, LBNL, and UC Berkeley
+
+Michael M. Bronstein
+
+University of Oxford
+
+Siddhartha Mishra
+
+ETH Zürich
+
+# ABSTRACT
+
+We present Gradient Gating  $(\mathbf{G}^2)$ , a novel framework for improving the performance of Graph Neural Networks (GNNs). Our framework is based on gating the output of GNN layers with a mechanism for multi-rate flow of message passing information across nodes of the underlying graph. Local gradients are harnessed to further modulate message passing updates. Our framework flexibly allows one to use any basic GNN layer as a wrapper around which the multi-rate gradient gating mechanism is built. We rigorously prove that  $\mathbf{G}^2$  alleviates the oversmoothing problem and allows the design of deep GNNs. Empirical results are presented to demonstrate that the proposed framework achieves state-of-the-art performance on a variety of graph learning tasks, including on large-scale heterophilic graphs.
+
+# 1 INTRODUCTION
+
+Learning tasks involving graph structured data arise in a wide variety of problems in science and engineering. Graph Neural Networks (GNNs) (Sperduti, 1994; Goller & Kuchler, 1996; Sperduti & Starita, 1997; Frasconi et al., 1998; Gori et al., 2005; Scarselli et al., 2008; Bruna et al., 2014; Defferrard et al., 2016; Kipf & Welling, 2017; Monti et al., 2017; Gilmer et al., 2017) are a popular deep learning architecture for graph-structured and relational data. GNNs have been successfully applied in domains including computer vision and graphics (Monti et al., 2017), recommender systems (Ying et al., 2018), transportation (Derrow-Pinion et al., 2021), computational chemistry (Gilmer et al., 2017), drug discovery (Gaudelet et al., 2021), particle physics (Shlomi et al., 2020) and social networks. See Zhou et al. (2019); Bronstein et al. (2021) for extensive reviews.
+
+Despite the widespread success of GNNs and a plethora of different architectures, several fundamental problems still impede their efficiency on realistic learning tasks. These include the bottleneck (Alon & Yahav, 2021), oversquashing (Topping et al., 2021), and oversmoothing (Nt & Maehara, 2019; Oono & Suzuki, 2020) phenomena. Oversmoothing refers to the observation that all node features in a deep (multi-layer) GNN converge to the same constant value as the number of layers is increased. Thus, and in contrast to standard machine learning frameworks, oversmoothing inhibits the use of very deep GNNs for learning tasks. These phenomena are likely responsible for the unsatisfactory empirical performance of traditional GNN architectures in heterophilic datasets, where the features or labels of a node tend to be different from those of its neighbors (Zhu et al., 2020).
+
+Given this context, our main goal is to present a novel framework that alleviates the oversmoothing problem and allows one to implement very deep multi-layer GNNs that can significantly improve performance in the setting of heterophilic graphs. Our starting point is the observation that in standard Message-Passing GNN architectures (MPNNs), such as GCN (Kipf & Welling, 2017) or GAT (Velickovic et al., 2018), each node gets updated at exactly the same rate within every hidden layer. Yet, realistic learning tasks might benefit from having different rates of propagation (flow) of information on the underlying graph. This insight leads to a novel multi-rate message passing scheme capable of learning these underlying rates. Moreover, we also propose a novel procedure that harnesses graph gradients to ameliorate the oversmoothing problem. Combining these elements leads to a new architecture described in this paper, which we term Gradient Gating  $(\mathbf{G}^2)$ .
+
+Main Contributions. We will demonstrate the following advantages of the proposed approach:
+
+-  $\mathbf{G}^2$  is a flexible framework wherein any standard message-passing layer (such as GAT, GCN, GIN, or GraphSAGE) can be used as the coupling function. Thus, it should be thought of as a framework into which one can plug existing GNN components. The use of multiple rates and gradient gating facilitates the implementation of deep GNNs and generally improves performance.  
+-  $\mathbf{G}^2$  can be interpreted as a discretization of a dynamical system governed by nonlinear differential equations. By investigating the stability of zero-Dirichlet energy steady states of this system, we rigorously prove that our gradient gating mechanism prevents oversmoothing. To complement this, we also prove a partial converse, that the lack of gradient gating can lead to oversmoothing.  
+- We provide extensive empirical evidence demonstrating that  $\mathbf{G}^2$  achieves state-of-the-art performance on a variety of graph learning tasks, including on large heterophilic graph datasets.
+
+# 2 GRADIENT GATING
+
+Let  $\mathcal{G} = (\mathcal{V},\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V})$  be an undirected graph with  $|\mathcal{V}| = v$  nodes and  $|\mathcal{E}| = e$  edges (unordered pairs of nodes  $\{i,j\}$  denoted  $i\sim j$ ). The 1-neighborhood of a node  $i$  is denoted  $\mathcal{N}_i = \{j\in \mathcal{V}:i\sim j\}$ . Furthermore, each node  $i$  is endowed with an  $m$ -dimensional feature vector  $\mathbf{X}_i$ ; the node features are arranged into a  $v\times m$  matrix  $\mathbf{X} = (\mathbf{X}_{ik})$  with  $i = 1,\dots ,v$  and  $k = 1,\ldots ,m$ .
+
+A typical residual Message-Passing GNN (MPNN) updates the node features by performing several iterations of the form,
+
+$$
+\mathbf {X} ^ {n} = \mathbf {X} ^ {n - 1} + \sigma \left(\mathbf {F} _ {\theta} \left(\mathbf {X} ^ {n - 1}, \mathcal {G}\right)\right), \tag {1}
+$$
+
+where  $\mathbf{F}_{\theta}$  is a learnable function with parameters  $\theta$ , and  $\sigma$  is an element-wise non-linear activation function. Here  $n \geq 1$  denotes the  $n$ -th hidden layer with  $n = 0$  being the input.
+
+One can interpret (1) as a discrete dynamical system in which  $\mathbf{F}$  plays the role of a coupling function determining the interaction between different nodes of the graph. In particular, we consider local (1-neighborhood) coupling of the form  $\mathbf{Y}_i = (\mathbf{F}(\mathbf{X},\mathcal{G}))_i = \mathbf{F}(\mathbf{X}_i,\{\{\mathbf{X}_{j\in \mathcal{N}_i}\})$  operating on the multiset of 1-neighbors of each node. Examples of such functions used in the graph machine learning literature (Bronstein et al., 2021) are graph convolutions  $\mathbf{Y}_i = \sum_{j\in \mathcal{N}_i}c_{ij}\mathbf{X}_j$  (GCN, (Kipf & Welling, 2017)) and graph attention  $\mathbf{Y}_i = \sum_{j\in \mathcal{N}_i}a(\mathbf{X}_i,\mathbf{X}_j)\mathbf{X}_j$  (GAT, (Velickovic et al., 2018)).
+
+We observe that in (1), at each hidden layer, every node and every feature channel gets updated with exactly the same rate. However, it is reasonable to expect that in realistic graph learning tasks one can encounter multiple rates for the flow of information (node updates) on the graph. Based on this observation, we propose a multi-rate (MR) generalization of (1), allowing updates to each node of the graph and feature channel with different rates,
+
+$$
+\mathbf {X} ^ {n} = \left(1 - \boldsymbol {\tau} ^ {n}\right) \odot \mathbf {X} ^ {n - 1} + \boldsymbol {\tau} ^ {n} \odot \sigma \left(\mathbf {F} _ {\theta} \left(\mathbf {X} ^ {n - 1}, \mathcal {G}\right)\right), \tag {2}
+$$
+
+where  $\pmb{\tau}$  denotes a  $v\times m$  matrix of rates with elements  $\tau_{ik}\in [0,1]$ . Rather than fixing  $\pmb{\tau}$  prior to training, we aim to learn the different update rates based on the node data  $\mathbf{X}$  and the local structure of the underlying graph  $\mathcal{G}$ , as follows
+
+$$
+\boldsymbol {\tau} ^ {n} (\mathbf {X} ^ {n - 1}, \mathcal {G}) = \bar {\sigma} (\hat {\mathbf {F}} _ {\hat {\theta}} (\mathbf {X} ^ {n - 1}, \mathcal {G})), \tag {3}
+$$
+
+where  $\hat{\mathbf{F}}_{\hat{\theta}}$  is another learnable 1-neighborhood coupling function, and  $\bar{\sigma}$  is a sigmoidal logistic activation function to constrain the rates to lie within [0, 1]. Since the multi-rate message-passing scheme (2) using (3) does not necessarily prevent oversmoothing (for any choice of the coupling function), we need to further constrain the rate matrix  $\tau^n$ . To this end, we note that the graph gradient of scalar node features  $\mathbf{y}$  on the underlying graph  $\mathcal{G}$  is defined as  $(\nabla \mathbf{y})_{ij} = \mathbf{y}_j - \mathbf{y}_i$  at the edge  $i \sim j$  (Lim, 2015). Next, we will use graph gradients to obtain the proposed Gradient Gating  $(\mathbf{G}^2)$  framework given by
+
+$$
+\hat {\boldsymbol {\tau}} ^ {n} = \sigma (\hat {\mathbf {F}} _ {\theta} (\mathbf {X} ^ {n - 1}, \mathcal {G})),
+$$
+
+$$
+\boldsymbol {\tau} _ {i k} ^ {n} = \tanh  \left(\sum_ {j \in \mathcal {N} _ {i}} | \hat {\boldsymbol {\tau}} _ {j k} ^ {n} - \hat {\boldsymbol {\tau}} _ {i k} ^ {n} | ^ {p}\right), \tag {4}
+$$
+
+$$
+\mathbf {X} ^ {n} = \left(1 - \boldsymbol {\tau} ^ {n}\right) \odot \mathbf {X} ^ {n - 1} + \boldsymbol {\tau} ^ {n} \odot \sigma (\mathbf {F} _ {\theta} (\mathbf {X} ^ {n - 1}, \mathcal {G})),
+$$
+
+where  $\hat{\pmb{\tau}}_{jk}^n -\hat{\pmb{\tau}}_{ik}^n = (\nabla \hat{\pmb{\tau}}_{*k}^n)_{ij}$  denotes the graph-gradient and  $\hat{\pmb{\tau}}_{*k}^{n}$  is the  $k$  -th column of the rate matrix  $\hat{\pmb{\tau}}^n$  and  $p\geq 0$  . Since  $\sum_{j\in \mathcal{N}_i}|\hat{\pmb{\tau}}_{jk}^n -\hat{\pmb{\tau}}_{ik}^n |^p\geq 0$  for all  $i\in \mathcal{V}$  , it follows that  $\pmb {\tau}^n\in [0,1]^{v\times m}$  for all  $n$  , retaining its interpretation as a matrix of rates. The sum over the neighborhood  $\mathcal{N}_i$  in (4) can be replaced by any permutation-invariant aggregation function (e.g., mean or max). Moreover, any standard message-passing procedure can be used to define the coupling functions  $\mathbf{F}$  and  $\hat{\mathbf{F}}$  (and, in particular, one can set  $\hat{\mathbf{F}} = \mathbf{F}$  ). As an illustration, Fig. 1 shows a schematic diagram of the layer-wise update of the proposed  $\mathbf{G}^2$  architecture.
+
+![](images/d875dd67d6235e24fb15e32ad38327c2306dd6176fbd23625a6a408a551eb4b0.jpg)  
+Figure 1: Schematic diagram of  $\mathbf{G}^2$  (4) showing the layerwise update of the latent node features  $\mathbf{X}$  (at layer  $n$ ). The norm of the graph-gradient (i.e., sum in second equation in (4)) is denoted as  $\|\nabla\|_p^p$ .
+
+The intuitive idea behind gradient gating in (4) is the following: If for any node  $i \in \mathcal{V}$  local oversmoothing occurs, i.e.,  $\lim_{n \to \infty} \sum_{j \in \mathcal{N}_i} \| \mathbf{X}_i^n - \mathbf{X}_j^n \| = 0$ , then  $\mathbf{G}^2$  ensures that the corresponding rate  $\tau_i^n$  goes to zero (at a faster rate), such that the underlying hidden node feature  $\mathbf{X}_i$  is no longer updated. This prevents oversmoothing by early-stopping of the message passing procedure.
+
+# 3 PROPERTIES OF  $\mathbf{G}^2$ -GNN
+
+$\mathbf{G}^2$  is a flexible framework. An important aspect of  $\mathbf{G}^2$  (4) is that it can be considered as a "wrapper" around any specific MPNN architecture. In particular, the hidden layer update for any form of message passing (e.g., GCN (Kipf & Welling, 2017), GAT (Velickovic et al., 2018), GIN (Xu et al., 2018) or GraphSAGE (Hamilton et al., 2017)) can be used as the coupling functions  $\mathbf{F}$ ,  $\hat{\mathbf{F}}$  in (4). By setting  $\boldsymbol{\tau} \equiv \mathbf{I}$ , (4) reduces to
+
+$$
+\mathbf {X} ^ {n} = \sigma \left(\mathbf {F} _ {\theta} \left(\mathbf {X} ^ {n - 1}, \mathcal {G}\right)\right), \tag {5}
+$$
+
+a standard (non-residual) MPNN. As we will show in the following, the use of a non-trivial gradient-gated learnable rate matrix  $\tau$  allows implementing very deep architectures that avoid oversmoothing.
+
+Maximum Principle for node features. Node features produced by  $\mathbf{G}^2$  satisfy the following Maximum Principle.
+
+Proposition 3.1. Let  $\mathbf{X}^n$  be the node feature matrix generated by iteration formula (4). Then, the features are bounded as follows:
+
+$$
+\min  \left(- 1, \underline {{\sigma}}\right) \leq \mathbf {X} _ {i k} ^ {n} \leq \max  \left(1, \bar {\sigma}\right), \quad \forall 1 \leq n, \tag {6}
+$$
+
+where the scalar activation function is bounded by  $\underline{\sigma} \leq \sigma(z) \leq \overline{\sigma}$  for all  $z \in \mathbb{R}$ .
+
+The proof follows readily from writing (4) component-wise and using the fact that  $0 \leq \tau_{ik}^{n} \leq 1$ , for all  $1 \leq i \leq v, 1 \leq k \leq m$  and  $1 \leq n$ .
+
+Continuous limit of  $\mathbf{G}^2$ . It has recently been shown (see Avelar et al. (2019); Poli et al. (2019); Zhuang et al. (2020); Xhonneux et al. (2020); Chamberlain et al. (2021a); Eliasof et al. (2021); Chamberlain et al. (2021b); Topping et al. (2021); Rusch et al. (2022a) and references therein)
+
+that interesting properties of GNNs (with residual connections) can be understood by taking the continuous (infinite-depth) limit and analyzing the resulting differential equations.
+
+In this context, we can derive a continuous version of (4) by introducing a small-scale  $0 < \Delta t < 1$  and rescaling the rate matrix  $\tau^n$  to  $\Delta t\tau^n$  leading to
+
+$$
+\mathbf {X} ^ {n} = \left(1 - \Delta t \boldsymbol {\tau} ^ {n}\right) \odot \mathbf {X} ^ {n - 1} + \Delta t \boldsymbol {\tau} ^ {n} \odot \sigma \left(\mathbf {F} _ {\theta} \left(\mathbf {X} ^ {n - 1}, \mathcal {G}\right)\right). \tag {7}
+$$
+
+Rearranging the terms in (7), we obtain
+
+$$
+\frac {\mathbf {X} ^ {n} - \mathbf {X} ^ {n - 1}}{\Delta t} = \boldsymbol {\tau} ^ {n} \odot \left(\sigma \left(\mathbf {F} _ {\theta} \left(\mathbf {X} ^ {n - 1}, \mathcal {G}\right)\right) - \mathbf {X} ^ {n - 1}\right). \tag {8}
+$$
+
+Interpreting  $\mathbf{X}^n\approx \mathbf{X}(n\Delta t) = \mathbf{X}(t^n)$ , i.e., marching in time, corresponds to increasing the number of hidden layers. Letting  $\Delta t\to 0$ , one obtains the following system of graph-coupled ordinary differential equations (ODEs):
+
+$$
+\frac {d \mathbf {X} (t)}{d t} = \boldsymbol {\tau} (t) \odot (\sigma (\mathbf {F} _ {\theta} (\mathbf {X} (t), \mathcal {G})) - \mathbf {X} (t)), \quad \forall t \geq 0,
+$$
+
+$$
+\boldsymbol {\tau} _ {i k} (t) = \tanh  \left(\sum_ {j \in \mathcal {N} _ {i}} | \hat {\boldsymbol {\tau}} _ {i k} (t) - \hat {\boldsymbol {\tau}} _ {j k} (t) | ^ {p}\right), \tag {9}
+$$
+
+$$
+\hat {\pmb {\tau}} (t) = \hat {\sigma} (\hat {\mathbf {F}} _ {\hat {\boldsymbol {\theta}}} (\mathbf {X} ^ {n - 1}, \mathcal {G})).
+$$
+
+We observe that the iteration formula (4) acts as a forward Euler discretization of the ODE system (9). Hence, one can follow Chamberlain et al. (2021a) and design more general (e.g., higher-order, adaptive, or implicit) discretizations of the ODE system (9). All these can be considered as design extensions of (4).
+
+Oversmoothing. Using the interpretation of (4) as a discretization of the ODE system (9), we can adapt the mathematical framework recently proposed in Rusch et al. (2022a) to study the oversmoothing problem. In order to formally define oversmoothing, we introduce the Dirichlet energy defined on the node features  $\mathbf{X}$  of an undirected graph  $\mathcal{G}$  as
+
+$$
+\mathcal {E} (\mathbf {X}) = \frac {1}{v} \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \| \mathbf {X} _ {i} - \mathbf {X} _ {j} \| ^ {2}. \tag {10}
+$$
+
+Following Rusch et al. (2022a), we say that the scheme (9) oversmoothes if the Dirichlet energy decays exponentially fast,
+
+$$
+\mathcal {E} (\mathbf {X} (t)) \leq C _ {1} e ^ {- C _ {2} t}, \quad \forall t > 0, \tag {11}
+$$
+
+for some  $C_{1,2} > 0$ . In particular, the discrete version of (11) implies that oversmoothing happens when the Dirichlet energy, decays exponentially fast as the number of hidden layers increases ((Rusch et al., 2022a) Definition 3.2).
+
+Next, one can prove the following proposition further characterizing oversmoothing with the standard terminology of dynamical systems (Wiggins, 2003).
+
+Proposition 3.2. The oversmoothing problem occurs for the ODEs (9) iff the hidden states  $\mathbf{X}_i^* = \mathbf{c}$ , for all  $i \in \mathcal{V}$  are exponentially stable steady states (fixed points) of the ODE (9), for some  $\mathbf{c} \in \mathbb{R}^m$ .
+
+In other words, for the oversmoothing problem to occur for this system, all the trajectories of the ODE (9) that start within the corresponding basin of attraction have to converge exponentially fast in time (according to (11)) to the corresponding steady state  $\mathbf{c}$ . Note that the basins of attraction will be different for different values of  $\mathbf{c}$ . The proof of this Proposition is a straightforward adaptation of the proof of Proposition 3.3 of Rusch et al. (2022a).
+
+Given this precise formulation of oversmoothing, we will investigate whether and how gradient gating in (9) can prevent oversmoothing. For simplicity, we set  $m = 1$  to consider only scalar node features (extension to vector node features is straightforward). Moreover, we assume coupling functions of the form  $\mathbf{F}(\mathbf{X}) = \mathbf{A}(\mathbf{X})\mathbf{X}$ , expressed element-wise as (see also Chamberlain et al. (2021a); Rusch et al. (2022a)),
+
+$$
+(\mathbf {F} (\mathbf {X})) _ {i} = \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} \left(\mathbf {X} _ {i}, \mathbf {X} _ {j}\right) \mathbf {X} _ {j}. \tag {12}
+$$
+
+Here,  $\mathbf{A}(\mathbf{X})$  is a matrix-valued function whose form covers many commonly used coupling functions stemming from the graph attention (GAT, where  $\mathbf{A}_{ij} = \mathbf{A}(\mathbf{X}_i,\mathbf{X}_j)$  is learnable) or convolution operators (GCN, where  $\mathbf{A}_{ij}$  is fixed). Furthermore, the matrices are right stochastic, i.e., the entries satisfy
+
+$$
+0 \leq \mathbf {A} _ {i j} \leq 1, \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} = 1. \tag {13}
+$$
+
+Finally, as the multi-rate feature of (9) has no direct bearing on the oversmoothing problem, we focus on the contribution of the gradient feedback term. To this end, we deactivate the multi-rate aspects and assume that  $\hat{\pmb{\tau}}_i = \mathbf{X}_i$  for all  $i\in \mathcal{V}$ , leading to the following form of (9):
+
+$$
+\begin{array}{l} \frac {d \mathbf {X} _ {i} (t)}{d t} = \boldsymbol {\tau} _ {i} (t) \left(\sigma \left(\sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} \mathbf {X} _ {j} (t)\right) - \mathbf {X} _ {i} (t)\right), \quad \forall t \geq 0, \tag {14} \\ \boldsymbol {\tau} _ {i} (t) = \tanh  \left(\sum_ {j \in \mathcal {N} _ {i}} \| \mathbf {X} _ {j} (t) - \mathbf {X} _ {i} (t) \| ^ {p}\right). \\ \end{array}
+$$
+
+Lack of  $\mathbf{G}^2$  can lead to oversmoothing. We first consider the case where the Gradient Gating is switched off by setting  $p = 0$  in (14). This yields a standard GNN in which node features are evolved through message passing between neighboring nodes, without any explicit information about graph gradients. We further assume that the activation function is ReLU i.e.,  $\sigma(x) = \max(x, 0)$ . Given this setting, we have the following proposition on oversmoothing:
+
+Proposition 3.3. Assume the underlying graph  $\mathcal{G}$  is connected. For any  $c\geq 0$ , let  $\mathbf{X}_i^*\equiv c$ , for all  $i\in \mathcal{V}$  be a (zero-Dirichlet energy) steady state of the ODEs (14). Moreover, assume no Gradient Gating  $(p = 0$  in (14)) and
+
+$$
+\mathbf {A} _ {i j} (c, c) = \mathbf {A} _ {j i} (c, c), \text {a n d} \mathbf {A} _ {i j} (c, c) \geq \underline {{a}}, \quad 1 \leq i, j \leq v, \tag {15}
+$$
+
+with  $0 < \underline{a} \leq 1$  and that there exists at least one node denoted w.l.o.g. with index 1 such that  $\mathbf{X}_1(t) \equiv c$ , for all  $t \geq 0$ . Then, the steady state  $\mathbf{X}_i^* = \mathbf{c}$ , for all  $i \in \mathcal{V}$ , of (14) is exponentially stable.
+
+Proposition 3.2 implies that without gradient gating  $(\mathbf{G}^2)$ , (9) can lead to oversmoothing. The proof, presented in SM C.1 relies on analyzing the time-evolution of small perturbations around the steady state  $\mathbf{c}$  and showing that these perturbations decay exponentially fast in time (see (20)).
+
+$\mathbf{G}^2$  prevents oversmoothing. We next investigate the effect of Gradient Gating in the same setting of Proposition 3.3. The following Proposition shows that gradient gating prevents oversmoothing:
+
+Proposition 3.4. Assume the underlying graph  $\mathcal{G}$  is connected. For any  $c\geq 0$  and for all  $i\in \mathcal{V}$ , let  $\mathbf{X}_i^* \equiv c$  be a (zero-Dirichlet energy) steady state of the ODEs (14). Moreover, assume Gradient Gating  $(p > 0)$  and that the matrix  $\mathbf{A}$  in (14) satisfies (15) and that there exists at least one node denoted w.l.o.g. with index 1 such that  $\mathbf{X}_1(t)\equiv c,$  for all  $t\geq 0$ . Then, the steady state  $\mathbf{X}_i^* = \mathbf{c},$  for all  $i\in \mathcal{V}$  is not exponentially stable.
+
+The proof, presented in SM C.2 clearly elucidates the role of gradient gating by showing that the energy associated with the quasi-linearized evolution equations (SM Eqn. (21)) is balanced by two terms (SM Eqn. (23)), both resulting from the introduction of gradient gating by setting  $p > 0$  in (14). One of them is of indefinite sign and can even cause growth of perturbations around a steady state  $\mathbf{c}$ . The other decays initial perturbations. However, the rate of this decay is at most polynomial (SM Eqn. (28)). For instance, the decay is merely linear for  $p = 2$  and slower for higher values of  $p$ . Thus, the steady state  $\mathbf{c}$  cannot be exponentially stable and oversmoothing is prevented. This justifies the intuition behind gradient gating, namely, if oversmoothing occurs around a node  $i$ , i.e.,  $\lim_{n\to \infty}\sum_{j\in \mathcal{N}_i}\| \mathbf{X}_i^n -\mathbf{X}_j^n\| = 0$ , then the corresponding rate  $\tau_i^n$  goes to zero (at a faster rate), such that the underlying hidden node feature  $\mathbf{X}_i$  stops getting updated.
+
+# 4 EXPERIMENTAL RESULTS
+
+In this section, we present an experimental study of  $\mathbf{G}^2$  on both synthetic and real datasets. We use  $\mathbf{G}^2$  with three different coupling functions: GCN (Kipf & Welling, 2017), GAT (Velickovic et al., 2018) and GraphSAGE (Hamilton et al., 2017).
+
+Effect of  $\mathbf{G}^2$  on Dirichlet energy. Given that oversmoothing relates to the decay of Dirichlet energy (11), we follow the experimental setup proposed by Rusch et al. (2022a) to probe the dynamics of the Dirichlet energy of Gradient-Gated GNNs, defined on a 2-dimensional  $10\times 10$  regular grid with 4-neighbor connectivity. The node features  $\mathbf{X}$  are randomly sampled from  $\mathcal{U}([0,1])$  and then propagated through a 1000-layer GNN with random weights. We compare GAT, GCN and their gradient-gated versions ( $\mathbf{G}^2$ -GAT and  $\mathbf{G}^2$ -GCN) in this experiment. Fig. 2 depicts on log-log scale the Dirichlet energy of each layer's output with respect to the layer number. We clearly observe that GAT and GCN oversmooth as the underlying Dirichlet energy converges exponentially fast to zero, resulting in the node features becoming indistinguishable. In practice, the Dirichlet energy for these architectures is  $\approx 0$  after just ten hidden layers. On the other hand, and as suggested by the theoretical results of the previous section, adding  $\mathbf{G}^2$  decisively prevents this behavior and the Dirichlet energy remains (near) constant, even for very deep architectures (up to 1000 layers).
+
+![](images/c7e715d2b860951c816f41f6a68a9e7e2c2effaa61dce9c4ba1ee94974752e89.jpg)  
+Figure 2: Dirichlet energy  $\mathcal{E}(\mathbf{X}^n)$  of layer-wise node features  $\mathbf{X}^n$  propagated through a GAT, GCN and their gradient gated versions (G $^2$ -GAT, G $^2$ -GCN).
+
+![](images/fc54f5a11dc9115158320f468d311e05cdb707550c308b1bee5a10dfdf8e982e.jpg)  
+Figure 3: Test accuracies of GCN with gradient gating  $(\mathbf{G}^2$ -GCN) as well as plain GCN and GCN combined with other methods on the Cora dataset for increasing number of layers.
+
+$\mathbf{G}^2$  for very deep GNNs. Oversmoothing inhibits the use of large number of GNN layers. As  $\mathbf{G}^2$  is designed to alleviate oversmoothing, it should allow very deep architectures. To test this assumption, we reproduce the experiment considered in Chamberlain et al. (2021a): a node-level classification task on the Cora dataset using increasingly deeper GCN architectures. In addition to  $\mathbf{G}^2$ , we also compare with two recently proposed mechanisms to alleviate oversmoothing, DropEdge (Rong et al., 2020) and GraphCON (Rusch et al., 2022a). The results are presented in Fig. 3, where we plot the test accuracy for all the models with the number of layers ranging from 2 to 128. While a plain GCN seems to suffer the most from oversmoothing (with the performance rapidly deteriorating after 8 layers), GCN+DropEdge as well as GCN+GraphCON are able to mitigate this behavior to some extent, although the performance eventually starts dropping (after 16 and 64 layers, respectively). In contrast,  $\mathbf{G}^2$ -GCN exhibits a small but noticeable increase in performance for increasing number of layers, reaching its peak performance for 128 layers. This experiment suggests that  $\mathbf{G}^2$  can indeed be used in conjunction with deep GNNs, potentially allowing performance gains due to depth.
+
+$\mathbf{G}^2$  for multi-scale node-level regression. We test the multi-rate nature of  $\mathbf{G}^2$  on node-level regression tasks, where the target node values exhibit multiple scales. Due to a lack of widely available node-level regression tasks, we propose regression experiments based on the Wikipedia article networks Chameleon and Squirrel, (Rozemberczki et al., 2021). While Chameleon and Squirrel are already widely used as heterophilic node-level classification tasks, the original datasets consist of continuous node targets (average monthly web-page traffic). We normalize the provided webpage traffic values for every node between 0 and 1 and note that the resulting node values exhibit values on a wide range of different scales ranging between  $10^{-5}$  and 1 (see Fig. 4). Table 1 shows the test normalized
+
+Table 1: Normalized test MSE on multiscale node-level regression tasks.  
+
+<table><tr><td></td><td>Chameleon</td><td>Squirrel</td></tr><tr><td>#Nodes</td><td>2,277</td><td>5,201</td></tr><tr><td>#Edges</td><td>31,421</td><td>198,493</td></tr><tr><td>GCNII</td><td>0.170 ± 0.034</td><td>0.093 ± 0.031</td></tr><tr><td>PairNorm</td><td>0.207 ± 0.038</td><td>0.140 ± 0.040</td></tr><tr><td>GCN</td><td>0.207 ± 0.039</td><td>0.143 ± 0.039</td></tr><tr><td>GAT</td><td>0.207 ± 0.038</td><td>0.143 ± 0.039</td></tr><tr><td>G2-GCN</td><td>0.137 ± 0.033</td><td>0.070 ± 0.028</td></tr><tr><td>G2-GAT</td><td>0.136 ± 0.029</td><td>0.069 ± 0.029</td></tr></table>
+
+mean-square error (mean and standard deviation based on the ten pre-defined splits in Pei et al. (2020)) for two standard GNN architectures (GCN and GAT) with and without  $\mathbf{G}^2$ . We observe from
+
+Table 1 that adding  $\mathbf{G}^2$  to the baselines significantly reduces the error, demonstrating the advantage of using multiple update rates.
+
+![](images/c77041985217400db80ef5ae2c4ce29e61e9f07f4990a39e2be88ba1b188676d.jpg)  
+Figure 4: Histogram of the target node values of the Chameleon and Squirrel node-level regression tasks.
+
+![](images/a79e1b2228c5e0c56fa9f498ea231c2309424f246a41614a998ca5fe6a88bfb2.jpg)  
+Figure 5: Test accuracy of GCN and GAT with / without gradient gating  $(\mathbf{G}^2)$  on synthetic Cora with a varying level of true label homophily.
+
+$\mathbf{G}^2$  for varying homophily (Synthetic Cora). We test  $\mathbf{G}^2$  on a node-level classification task with varying levels of homophily on the synthetic Cora dataset Zhu et al. (2020). Standard GNN models are known to perform poorly in heterophilic settings. This can be seen in Fig. 5, where we present the classification accuracy of GCN and GAT on the synthetic-Cora dataset with a level of homophily varying between 0 and 0.99. While these models succeed in the homophilic case (reaching nearly perfect accuracy), their performance drops to  $\approx 20\%$  when the level of homophily approaches 0. Adding  $\mathbf{G}^2$  to GCN or GAT mitigates this phenomenon: the resulting models reach a test accuracy of over  $80\%$ , even in the most heterophilic setting, thus leading to a four-fold increase in the accuracy of the underlying GCN or GAT models. Furthermore, we notice an increase in performance even in the homophilic setting. Moreover, we compare with a state-of-the-art model GGCN (Yan et al., 2021), which has been recently proposed to explicitly deal with heterophilic graphs. From Fig. 5 we observe that  $\mathbf{G}^2$  performs on par and slightly better than GGCN in strongly heterophilic settings.
+
+Table 2: Results on heterophilic graphs. The three best performing methods are highlighted in red (First), blue (Second), and violet (Third).  
+
+<table><tr><td></td><td>Texas</td><td>Wisconsin</td><td>Film</td><td>Squirrel</td><td>Chameleon</td><td>Cornell</td></tr><tr><td>Hom level</td><td>0.11</td><td>0.21</td><td>0.22</td><td>0.22</td><td>0.23</td><td>0.30</td></tr><tr><td>#Nodes</td><td>183</td><td>251</td><td>7,600</td><td>5,201</td><td>2,277</td><td>183</td></tr><tr><td>#Edges</td><td>295</td><td>466</td><td>26,752</td><td>198,493</td><td>31,421</td><td>280</td></tr><tr><td>#Classes</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td></tr><tr><td>GGCN</td><td>84.86 ± 4.55</td><td>86.86 ± 3.29</td><td>37.54 ± 1.56</td><td>55.17 ± 1.58</td><td>71.14 ± 1.84</td><td>85.68 ± 6.63</td></tr><tr><td>GPRGNN</td><td>78.38 ± 4.36</td><td>82.94 ± 4.21</td><td>34.63 ± 1.22</td><td>31.61 ± 1.24</td><td>46.58 ± 1.71</td><td>80.27 ± 8.11</td></tr><tr><td>H2GCN</td><td>84.86 ± 7.23</td><td>87.65 ± 4.98</td><td>35.70 ± 1.00</td><td>36.48 ± 1.86</td><td>60.11 ± 2.15</td><td>82.70 ± 5.28</td></tr><tr><td>FAGCN</td><td>82.43 ± 6.89</td><td>82.94 ± 7.95</td><td>34.87 ± 1.25</td><td>42.59 ± 0.79</td><td>55.22 ± 3.19</td><td>79.19 ± 9.79</td></tr><tr><td>F2GAT</td><td>82.70 ± 5.95</td><td>87.06 ± 4.13</td><td>36.65 ± 1.13</td><td>47.32 ± 2.43</td><td>67.81 ± 2.05</td><td>83.51 ± 6.70</td></tr><tr><td>MixHop</td><td>77.84 ± 7.73</td><td>75.88 ± 4.90</td><td>32.22 ± 2.34</td><td>43.80 ± 1.48</td><td>60.50 ± 2.53</td><td>73.51 ± 6.34</td></tr><tr><td>GCNII</td><td>77.57 ± 3.83</td><td>80.39 ± 3.40</td><td>37.44 ± 1.30</td><td>38.47 ± 1.58</td><td>63.86 ± 3.04</td><td>77.86 ± 3.79</td></tr><tr><td>Geom-GCN</td><td>66.76 ± 2.72</td><td>64.51 ± 3.66</td><td>31.59 ± 1.15</td><td>38.15 ± 0.92</td><td>60.00 ± 2.81</td><td>60.54 ± 3.67</td></tr><tr><td>PairNorm</td><td>60.27 ± 4.34</td><td>48.43 ± 6.14</td><td>27.40 ± 1.24</td><td>50.44 ± 2.04</td><td>62.74 ± 2.82</td><td>58.92 ± 3.15</td></tr><tr><td>LINKX</td><td>74.60 ± 8.37</td><td>75.49 ± 5.72</td><td>36.10 ± 1.55</td><td>61.81 ± 1.80</td><td>68.42 ± 1.38</td><td>77.84 ± 5.81</td></tr><tr><td>GloGNN</td><td>84.32 ± 4.15</td><td>87.06 ± 3.53</td><td>37.35 ± 1.30</td><td>57.54 ± 1.39</td><td>69.78 ± 2.42</td><td>83.51 ± 4.26</td></tr><tr><td>GraphSAGE</td><td>82.43 ± 6.14</td><td>81.18 ± 5.56</td><td>34.23 ± 0.99</td><td>41.61 ± 0.74</td><td>58.73 ± 1.68</td><td>75.95 ± 5.01</td></tr><tr><td>ResGatedGCN</td><td>80.00 ± 5.57</td><td>81.57 ± 5.35</td><td>36.02 ± 1.19</td><td>37.60 ± 1.80</td><td>49.82 ± 2.71</td><td>73.51 ± 4.95</td></tr><tr><td>GCN</td><td>55.14 ± 5.16</td><td>51.76 ± 3.06</td><td>27.32 ± 1.10</td><td>31.52 ± 0.71</td><td>38.44 ± 1.92</td><td>60.54 ± 5.30</td></tr><tr><td>GAT</td><td>52.16 ± 6.63</td><td>49.41 ± 4.09</td><td>27.44 ± 0.89</td><td>36.77 ± 1.68</td><td>48.36 ± 1.58</td><td>61.89 ± 5.05</td></tr><tr><td>MLP</td><td>80.81 ± 4.75</td><td>85.29 ± 3.31</td><td>36.53 ± 0.70</td><td>28.77 ± 1.56</td><td>46.21 ± 2.99</td><td>81.89 ± 6.40</td></tr><tr><td>G2-GAT</td><td>84.59 ± 5.55</td><td>87.65 ± 4.64</td><td>37.30 ± 0.87</td><td>46.48 ± 1.41</td><td>64.12 ± 1.96</td><td>87.30 ± 4.84</td></tr><tr><td>G2-GCN</td><td>84.86 ± 3.24</td><td>87.06 ± 3.19</td><td>37.09 ± 1.16</td><td>39.62 ± 2.91</td><td>55.83 ± 2.88</td><td>86.49 ± 5.27</td></tr><tr><td>G2-GraphSAGE</td><td>87.57 ± 3.86</td><td>87.84 ± 3.49</td><td>37.14 ± 1.01</td><td>64.26 ± 2.38</td><td>71.40 ± 2.38</td><td>86.22 ± 4.90</td></tr></table>
+
+Heterophilic datasets. In Table 2, we test the proposed framework on several real-world heterophilic graphs (with a homophily level of  $\leq 0.30$ ) (Pei et al., 2020; Rozemberczki et al., 2021) and benchmark it against baseline models GraphSAGE (Hamilton et al., 2017), GCN (Kipf & Welling,
+
+2017), GAT (Velickovic et al., 2018) and MLP (Goodfellow et al., 2016), as well as recent state-of-the-art models on heterophilic graph datasets, i.e., GGCN (Yan et al., 2021), GPRGNN (Chien et al., 2020), H2GCN (Zhu et al., 2020), FAGCN (Bo et al., 2021),  $\mathbf{F}^2$ GAT (Wei et al., 2022), MixHop (Abu-El-Haija et al., 2019), GCNII (Chen et al., 2020b), Geom-GCN (Pei et al., 2020), PairNorm (Zhao & Akoglu, 2019). We can observe that  $\mathbf{G}^2$  added to GCN, GAT or GraphSAGE outperforms all other methods (in particular recent methods such as GGCN, GPRGNN, H2GCN that were explicitly designed to solve heterophilic tasks). Moreover, adding  $\mathbf{G}^2$  to the underlying base GNN model improves the results on average by  $45.75\%$  for GAT,  $45.4\%$  for GCN and  $18.6\%$  for GraphSAGE.
+
+Large-scale graphs. Given the exceptional performance of  $\mathbf{G}^2$ -GraphSAGE on small and medium sized heterophilic graphs, we test the proposed  $\mathbf{G}^2$  (applied to GraphSAGE, i.e.,  $\mathbf{G}^2$ -GraphSAGE) on large-scale datasets. To this end, we consider three different experiments based on large graphs from Lim et al. (2021), which range from highly heterophilic (homophily level of 0.07) to fairly homophilic (homophily level of 0.61). The sizes range from large graphs with  $\sim 170\mathrm{K}$  nodes and  $\sim 1\mathrm{M}$  edges to a very large graph with  $\sim 3\mathrm{M}$  nodes and  $\sim 14\mathrm{M}$  edges.
+
+<table><tr><td colspan="4">Table 3: Results on large-scale datasets.</td></tr><tr><td></td><td>snap-patents</td><td>arXiv-year</td><td>genius</td></tr><tr><td>Hom level</td><td>0.07</td><td>0.22</td><td>0.61</td></tr><tr><td>#Nodes</td><td>2,923,922</td><td>169,343</td><td>421,961</td></tr><tr><td>#Edges</td><td>13,975,788</td><td>1,166,243</td><td>984,979</td></tr><tr><td>#Classes</td><td>5</td><td>5</td><td>2</td></tr><tr><td>MLP</td><td>31.34 ± 0.05</td><td>36.70 ± 0.21</td><td>86.68 ± 0.09</td></tr><tr><td>GCN</td><td>45.65 ± 0.04</td><td>46.02 ± 0.26</td><td>87.42 ± 0.37</td></tr><tr><td>GAT</td><td>45.37 ± 0.44</td><td>46.05 ± 0.51</td><td>55.80 ± 0.87</td></tr><tr><td>MixHop</td><td>52.16 ± 0.09</td><td>51.81 ± 0.17</td><td>90.58 ± 0.16</td></tr><tr><td>LINKX</td><td>61.95 ± 0.12</td><td>56.00 ± 1.34</td><td>90.77 ± 0.27</td></tr><tr><td>LINK</td><td>60.39 ± 0.07</td><td>53.97 ± 0.18</td><td>73.56 ± 0.14</td></tr><tr><td>GCNII</td><td>37.88 ± 0.69</td><td>47.21 ± 0.28</td><td>90.24 ± 0.09</td></tr><tr><td>APPNP</td><td>32.19 ± 0.07</td><td>38.15 ± 0.26</td><td>85.36 ± 0.62</td></tr><tr><td>GloGNN</td><td>62.09 ± 0.27</td><td>54.68 ± 0.34</td><td>90.66 ± 0.11</td></tr><tr><td>GPR-GNN</td><td>40.19 ± 0.03</td><td>45.07 ± 0.21</td><td>90.05 ± 0.31</td></tr><tr><td>ACM-GCN</td><td>55.14 ± 0.16</td><td>47.37 ± 0.59</td><td>80.33 ± 3.91</td></tr><tr><td>G2-GraphSAGE</td><td>69.50 ± 0.39</td><td>63.30 ± 1.84</td><td>90.85 ± 0.64</td></tr></table>
+
+Table 3 shows the results of  $\mathbf{G}^2$ -
+
+GraphSAGE together with other standard GNNs, as well as recent state-of-the-art models, i.e., MLP(Goodfellow et al., 2016), GCN (Kipf & Welling, 2017), GAT (Velickovic et al., 2018), MixHop (Abu-El-Haija et al., 2019), LINK(X) (Lim et al., 2021), GCNII (Chen et al., 2020b), APPNP (Klicpera et al., 2018), GloGNN (Li et al., 2022), GPR-GNN (Chien et al., 2020) and ACM-GCN (Luan et al., 2021). We can see that  $\mathbf{G}^2$ -GraphSAGE significantly outperforms current state-of-the-art (by up to  $13\%$ ) on the two heterophilic graphs (i.e., snap-patents and arXiv-year). Moreover,  $\mathbf{G}^2$ -GraphSAGE is on-par with the current state-of-the-art on the homophilic graph dataset genius.
+
+We conclude that the proposed gradient gating method can successfully be scaled up to large graphs, reaching state-of-the-art performance, in particular on heterophilic graph datasets.
+
+# 5 RELATED WORK
+
+Gating. Gating is a key component of our proposed framework. The use of gating (i.e., the modulation between 0 and 1) of hidden layer outputs has a long pedigree in neural networks and sequence modeling. In particular, classical recurrent neural network (RNN) architectures such as LSTM (Hochreiter & Schmidhuber, 1997) and GRU (Cho et al., 2014) rely on gates to modulate information propagation in the RNN. Given the connections between RNNs and early versions of GNNs (Zhou et al., 2019), it is not surprising that the idea of gating has been used in designing GNNs Bresson & Laurent (2017); Li et al. (2016); Zhang et al. (2018). However, to the best of our knowledge, the use of local graph-gradients to further modulate gating in order to alleviate the oversmoothing problem is novel, and so is its theoretical analysis.
+
+Multi-scale methods. The multi-rate gating procedure used in  $\mathbf{G}^2$  is a particular example of multi-scale mechanisms. The use of multi-scale neural network architectures has a long history. An early example is Hinton & Plaut (1987), who proposed a neural network with each connection having a fast changing weight for temporary memory and a slow changing weight for long-term learning. The classical convolutional neural networks (CNNs, LeCun et al. (1989)) can be viewed as multi-scale architectures for processing multiple spatial scales in images (Bai et al., 2020). Moreover, there is a close connection between our multi-rate mechanism (4) and the use of multiple time scales in recently proposed sequence models such as UnICORNN (Rusch & Mishra, 2021) and long expressive memory (LEM) (Rusch et al., 2022b).
+
+Neural differential equations. Ordinary and partial differential equations (ODEs and PDEs) are playing an increasingly important role in designing, interpreting, and analyzing novel graph machine learning architectures Avelar et al. (2019); Poli et al. (2019); Zhuang et al. (2020); Xhonneux et al. (2020). Chamberlain et al. (2021a) designed attentional GNNs by discretizing parabolic diffusion-type PDEs. Di Giovanni et al. (2022) interpreted GCNs as gradient flows minimizing a generalized version of the Dirichlet energy. Chamberlain et al. (2021b) applied a non-Euclidean diffusion equation ("Beltrami flow") yielding a scheme with adaptive spatial derivatives ("graph rewiring"), and Topping et al. (2021) studied a discrete geometric PDE similar to Ricci flow to improve information propagation in GNNs. Eliasof et al. (2021) proposed a GNN framework arising from a mixture of parabolic (diffusion) and hyperbolic (wave) PDEs on graphs with convolutional coupling operators, which describe dissipative wave propagation. Finally, Rusch et al. (2022a) used systems of nonlinear oscillators coupled through the associated graph structure to rigorously overcome the oversmoothing problem. In line with these works, one contribution of our paper is a continuous version of  $\mathbf{G}^2$  (9), which we used for a rigorous analysis of the oversmoothing problem. Understanding whether this system of ODEs has an interpretation as a known physical model is a topic for future research.
+
+# 6 DISCUSSION
+
+We have proposed a novel framework, termed  $\mathbf{G}^2$ , for efficient learning on graphs.  $\mathbf{G}^2$  builds on standard MPNNs, but seeks to overcome their limitations. In particular, we focus on the fact that for standard MPNNs such as GCN or GAT, each node (in every hidden layer) is updated at the same rate. This might inhibit efficient learning of tasks where different node features would need to be updated at different rates. Hence, we equip a standard MPNN with gates that amount to a multi-rate modulation for the hidden layer output in (4). This enables multiple rates (or scales) of flow of information across a graph. Moreover, we leverage local (graph) gradients to further constrain the gates. This is done to alleviate oversmoothing where node features become indistinguishable as the number of layers is increased.
+
+By combining these ingredients, we present a very flexible framework (dubbed  $\mathbf{G}^2$ ) for graph machine learning wherein any existing MPNN hidden layer can be employed as the coupling function and the multi-rate gradient gating mechanism can be built on top of it. Moreover, we also show that  $\mathbf{G}^2$  corresponds to a time-discretization of a system of ODEs (9). By studying the (in)-stability of the corresponding zero-Dirichlet energy steady states we rigorously prove that gradient gating can mitigate the oversmoothing problem, paving the way for the use of very deep GNNs within the  $\mathbf{G}^2$  framework. In contrast, the lack of gradient gating is shown to lead to oversmoothing.
+
+We also present an extensive empirical evaluation to illustrate different aspects of the proposed  $\mathbf{G}^2$  framework. Starting with synthetic, small-scale experiments, we demonstrate that i)  $\mathbf{G}^2$  can prevent oversmoothing by keeping the Dirichlet energy constant, even for a very large number of hidden layers, ii) this feature allows us to deploy very deep architectures and to observe that the accuracy of classification tasks can increase with increasing number of hidden layers, iii) the multi-rate mechanism significantly improves performance on node regression tasks when the node features are distributed over a range of scales, and iv)  $\mathbf{G}^2$  is very accurate at classification on heterophilic datasets, witnessing an increasing gain in performance with increasing heterophily.
+
+This last feature was more extensively investigated, and we observed that  $\mathbf{G}^2$  can significantly outperform baselines as well as recently proposed methods on both benchmark medium-scale and large-scale heterophilic datasets, achieving state-of-the-art performance. Thus, by a combination of theory and experiments, we demonstrate that the  $\mathbf{G}^2$ -framework is a promising approach for learning on graphs.
+
+Future work. As future work, we would like to better understand the continuous limit of  $\mathbf{G}^2$ , i.e., the ODEs (9), especially in the zero spatial-resolution limit and investigate if the resulting continuous equations have interesting geometric and analytical properties. Moreover, we would like to use  $\mathbf{G}^2$  for solving scientific problems, such as in computational chemistry or the numerical solutions of PDEs. Finally, the promising results for  $\mathbf{G}^2$  on large-scale graphs encourage us to use it for even larger industrial-scale applications.
+
+# ACKNOWLEDGEMENTS
+
+The research of TKR and SM was performed under a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 770880). MWM would like to acknowledge the IARPA (contract W911NF20C0035), NSF, and ONR for providing partial support of this work. MB is supported in part by ERC Grant No. 724228 (LEMAN). The authors thank Emmanuel de Bézenac (ETH) for his constructive suggestions.
+
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+
+# Supplementary Material for:
+
+Gradient Gating for Deep Multi-Rate Learning on Graphs
+
+# A ADDITIONAL EXPERIMENTS
+
+In this section, we describe additional empirical results to complement those in the main text.
+
+On the multi-rate effect of  $\mathbf{G}^2$ . Here, we analyze the performance of  $\mathbf{G}^2$  on the multi-scale node-level regression task of the main text. As we see in the main text,  $\mathbf{G}^2$  applied to GCN or GAT outperforms their plain counterparts (GCN and GAT) on the multi-scale node-level regression task by more than  $50\%$  on Chameleon and more than  $100\%$  on Squirrel. The question therefore arises whether this better performance can be explained by the multi-rate nature of gradient gating.
+
+To empirically analyse this, we begin by adding a control parameter  $\alpha$  to  $\mathbf{G}^2$  (4) as follows,
+
+$$
+\mathbf {X} ^ {n} = \left(1 - \left(\boldsymbol {\tau} ^ {n}\right) ^ {\alpha}\right) \odot \mathbf {X} ^ {n - 1} + \left(\boldsymbol {\tau} ^ {n}\right) ^ {\alpha} \odot \sigma \left(\mathbf {F} _ {\theta} \left(\mathbf {X} ^ {n - 1}, \mathcal {G}\right)\right).
+$$
+
+Clearly, setting  $\alpha = 1$  recovers the original gradient gating message-passing update,
+
+$$
+\mathbf {X} ^ {n} = \left(1 - \tau^ {n}\right) \odot \mathbf {X} ^ {n - 1} + \tau^ {n} \odot \sigma (\mathbf {F} _ {\theta} (\mathbf {X} ^ {n - 1}, \mathcal {G})),
+$$
+
+while setting  $\alpha = 0$  disables any explicit multi-rate behavior and a plain message-passing scheme is recovered,
+
+$$
+\mathbf {X} ^ {n} = \sigma (\mathbf {F} _ {\theta} (\mathbf {X} ^ {n - 1}, \mathcal {G})).
+$$
+
+Note that by continuously changing  $\alpha$  from 0 to 1 controls the level of multi-rate behavior in the proposed gradient gating method.
+
+In Fig. 6 we plot the test NMSE of the best performing  $\mathbf{G}^2$ -GCN and  $\mathbf{G}^2$ -GAT on the Chameleon multi-scale node-level regression task for increasing values of  $\alpha \in [10^{-3}, 1]$  in log-scale. We can see that the test NMSE monotonically decreases (lower error means better performance) for both  $\mathbf{G}^2$ -GCN and  $\mathbf{G}^2$ -GAT for increasing values of  $\alpha$ , i.e., increasing level of multi-rate behavior. We can conclude that the multi-rate behavior of  $\mathbf{G}^2$  is instrumental in successfully learning multi-scale regression tasks.
+
+![](images/9d3ea21a44223070e2caec229b673b3471cbc383173137598e788d3d971c1eda.jpg)  
+Figure 6: Test NMSE on the multi-scale chameleon node-level regression task of  $\mathbf{G}^2$ -GCN and  $\mathbf{G}^2$ -GAT for continuously decreasing level of multi-rate behavior.
+
+![](images/1f04484ef9ce50d576e6efc91f21051310bdf601ae74bbc5b80eff3f7d072654.jpg)  
+Figure 7: Test accuracies of  $\mathbf{G}^2$ -GraphSAGE on Texas, Squirrel, Film, Wisconsin and Chameleon graph datasets for varying values of  $p$  in (4).
+
+On the sensitivity of performance of  $\mathbf{G}^2$  to the hyperparameter  $p$ . The proposed gradient gating model implicitly depends on the hyperparameter  $p$ , which defines the multiple rates  $\tau$ , i.e.,
+
+$$
+\hat {\boldsymbol {\tau}} ^ {n} = \hat {\sigma} (\hat {\mathbf {F}} _ {\theta} (\mathbf {X} ^ {n - 1}, \mathcal {G})),
+$$
+
+$$
+\boldsymbol {\tau} _ {i k} ^ {n} = \tanh  \left(\sum_ {j \in \mathcal {N} _ {i}} | \hat {\tau} _ {i k} ^ {n} - \hat {\tau} _ {j k} ^ {n} | ^ {p}\right).
+$$
+
+While any value  $p > 0$  can be used in practice, a standard hyperparameter tuning procedure (see B for the training details) on  $p$  has been applied in every experiment included in this paper. Thus,
+
+it is natural to ask how sensitive the performance of  $\mathbf{G}^2$  is with respect to different values of the hyperparameter  $p$ .
+
+To answer this question, we trained different  $\mathbf{G}^2$ -GraphSAGE models on a variety of different graph datasets (i.e., Texas, Squirrel, Film, Wisconsin and Chameleon) for different values of  $p \in [1,5]$ . Fig. 7 shows the resulting performance of  $\mathbf{G}^2$ -GraphSAGE. We can see that different values of  $p$  do not significantly change the performance of the model. However, including the hyperparameter  $p$  to the hyperparameter fine-tuning procedure will further improve the overall performance of  $\mathbf{G}^2$ .
+
+On the sensitivity of performance of  $\mathbf{G}^2$  to the number of parameters. All results of  $\mathbf{G}^2$  provided in the main paper are obtained using standard hyperparameter tuning (i.e., random search). Those hyperparameters include the number of hidden channels for each hidden node of the graph, which directly correlates with the total number of parameters used in  $\mathbf{G}^2$ . It is thus natural to ask how  $\mathbf{G}^2$  performs compared to its plain counter-version (e.g.  $\mathbf{G}^2$ -GCN to GCN) for the exact same number of total parameters of the underlying model. To this end, Fig. 8 shows the test accuracies of  $\mathbf{G}^2$ -GCN and GCN for increasing number of total parameters in its corresponding model. We can see that first, using more parameters has only a slight effect on the overall performance of both models. Second, and most importantly,  $\mathbf{G}^2$ -GCN constantly reaches significantly higher test accuracies for the exact same number of total parameters. We can thus rule out that the better performance of  $\mathbf{G}^2$  compared to its plain counter-versions is explained by the usage of more parameters.
+
+![](images/cac4db8de33334c405392ba7a0b078e94ac391bd025beeed6f0828ce5a8f560f.jpg)  
+Figure 8: Test accuracies of plain GCN and  $\mathbf{G}^2$ -GCN on Texas for varying number of total parameters in the GNN.
+
+Ablation of  $\hat{\mathbf{F}}_{\theta}$  in  $\mathbf{G}^2$ . In its most general form  $\mathbf{G}^2$  (4) uses an additional GNN  $\hat{\mathbf{F}}_{\theta}$  to construct the multiple rates  $\tau^n$ . Is this additional GNN needed? To answer this question, Table 4 shows in which of the provided experiments (using  $\mathbf{G}^2$ -GraphSAGE) we actually used an additional GNN  $\hat{\mathbf{F}}_{\theta}$  (as suggested by our hyperparameter tuning protocol). We can see that on small-scale experiments having an additional GNN is not needed. However, on the considered medium and large-scale graph datasets it is beneficial to use it. Motivated by this, Table 5 shows the results for  $\mathbf{G}^2$ -GraphSAGE on the three medium-scale graph datasets (Film, Squirrel and Chameleon) without using an additional GNN in (4) (i.e.,  $\mathbf{F}_{\theta} = \hat{\mathbf{F}}_{\theta}$ ) as well as with using an additional GNN (i.e., the results provided in the main paper). We can see that while  $\mathbf{G}^2$ -GraphSAGE without an additional GNN (i.e., w/o  $\hat{\mathbf{F}}_{\theta}$ ) yields competitive results, using an additional GNN is needed in order to obtain state-of-the-art results on these three datasets.
+
+Table 4: Usage of  $\hat{\mathbf{F}}_{\theta}$  in  $\mathbf{G}^2$  (4) for each result with  $\mathbf{G}^2$ -GraphSAGE presented in the main paper (YES indicates the usage of  $\hat{\mathbf{F}}_{\theta}$ , while NO indicates that no additional GNN is used to construct the multiple rates, i.e.,  $\mathbf{F}_{\theta} = \hat{\mathbf{F}}_{\theta}$ )  
+
+<table><tr><td>Texas</td><td>Wisconsin</td><td>Film</td><td>Squirrel</td><td>Chameleon</td><td>Cornell</td><td>snap-patents</td><td>arXiv-year</td><td>genius</td></tr><tr><td>NO</td><td>NO</td><td>YES</td><td>YES</td><td>YES</td><td>NO</td><td>YES</td><td>YES</td><td>YES</td></tr></table>
+
+Table 5: Test accuracies of  $\mathbf{G}^2$ -GraphSAGE with and without additional GNN (i.e., w/  $\hat{\mathbf{F}}_{\theta}$  and w/o  $\hat{\mathbf{F}}_{\theta}$  in (4)) on Film, Squirrel and Chameleon graph dataset.  
+
+<table><tr><td></td><td>Film</td><td>Squirrel</td><td>Chameleon</td></tr><tr><td>G2-GraphSAGE w/ˆFθ</td><td>37.14 ± 1.01</td><td>64.26 ± 2.38</td><td>71.40 ± 2.38</td></tr><tr><td>G2-GraphSAGE w/oˆFθ</td><td>36.83 ± 1.26</td><td>55.78 ± 1.61</td><td>65.04 ± 2.27</td></tr></table>
+
+Ablation of multi-rate channels in  $\mathbf{G}^2$ . The corner stone of the proposed  $\mathbf{G}^2$  is the multi-rate matrix  $\tau^n$  in (4), which automatically solves the oversmoothing issue for any given GNN (Proposition 3.3). This multi-rate matrix learns different rates for every node but also for every channel of every node. It is thus natural to ask if the multi-rate property for the channels is necessary, or if having multiple rates for the different nodes is sufficient, i.e., having a multi-rate vector  $\tau^n \in \mathbb{R}^v$ . A direct construction of such multi-rate vector (derived from our proposed  $\mathbf{G}^2$ ) is:
+
+$$
+\hat {\boldsymbol {\tau}} ^ {n} = \sigma (\hat {\mathbf {F}} _ {\theta} (\mathbf {X} ^ {n - 1}, \mathcal {G})),
+$$
+
+$$
+\boldsymbol {\tau} _ {i} ^ {n} = \tanh  \left(\sum_ {j \in \mathcal {N} _ {i}} \| \hat {\boldsymbol {\tau}} _ {j} ^ {n} - \hat {\boldsymbol {\tau}} _ {i} ^ {n} \| _ {p}\right), \tag {16}
+$$
+
+$$
+\mathbf {X} ^ {n} = \left(1 - \tau^ {n}\right) \odot \mathbf {X} ^ {n - 1} + \tau^ {n} \odot \sigma \left(\mathbf {F} _ {\theta} \left(\mathbf {X} ^ {n - 1}, \mathcal {G}\right)\right).
+$$
+
+Note that the only difference to our proposed  $\mathbf{G}^2$  is in the second equation of (16), where we sum over the node-wise  $p$ -norms of the differences of adjacent nodes. This way, we compute a single scalar  $\tau_i^n$  for every node  $i \in \mathcal{V}$ .
+
+Table 6 shows the results of our proposed  $\mathbf{G}^2$ -GraphSAGE as well as the single-rate channels ablation of  $\mathbf{G}^2$  (eq. (16)) on the Film, Squirrel and Chameleon graph datasets. As a baseline, we also include the results of a plain GraphSAGE. We can see that while  $\mathbf{G}^2$  with single-scale channels outperforms the base GraphSAGE model, our proposed  $\mathbf{G}^2$  with multi-rate channels vastly outperforms the single-rate channels version of  $\mathbf{G}^2$ .
+
+Table 6: Test accuracies of plain GraphSAGE,  $\mathbf{G}^2$ -GraphSAGE with multi-rate channels for each node (i.e., standard  $\mathbf{G}^2$  (4)) as well as with only a single rate for every channel on Film, Squirrel and Chameleon.  
+
+<table><tr><td></td><td>Film</td><td>Squirrel</td><td>Chameleon</td></tr><tr><td>GraphSAGE</td><td>34.23 ± 0.99</td><td>41.61 ± 0.74</td><td>58.73 ± 1.68</td></tr><tr><td>G2-GraphSAGE w/ multi-rate channels G2 (4)</td><td>37.14 ± 1.01</td><td>64.26 ± 2.38</td><td>71.40 ± 2.38</td></tr><tr><td>G2-GraphSAGE w/ single-rate channels G2 (16)</td><td>36.67 ± 0.56</td><td>44.03 ± 1.01</td><td>60.29 ± 3.45</td></tr></table>
+
+Alternative measures of oversmoothing. The proof of Proposition 3.2 and Proposition 3.3 as well as the first experiment in the main paper is based on the definition of oversmoothing using the Dirichlet energy, which was proposed in Rusch et al. (2022a). However, there exist alternative measures to describe the oversmoothing phenomenon in deep GNNs. One such measure is the mean average distance (MAD), which was proposed in Chen et al. (2020a). In order to check if our proposed  $\mathbf{G}^2$  mitigates oversmoothing defined through the MAD measure we repeat the first experiment in the main paper and plot the MAD instead of the Dirichlet energy for increasing number of layers in Fig. 9. We can see that while the MAD of a plain GCN and GAT converges exponentially with increasing number of layers, it remains constant for  $\mathbf{G}^2$ -GCN and  $\mathbf{G}^2$ -GAT. We can thus conclude that  $\mathbf{G}^2$  mitigates oversmoothing defined through the MAD measure.
+
+![](images/66e82d98fd5362955b0775912bb46110b35812eb01e314749d21fd0cd299f0c6.jpg)  
+Figure 9: Mean average distance (MAD) of layerwise node features  $\mathbf{X}^n$  propagated through a GAT, GCN and their gradient gated versions ( $\mathbf{G}^2$ -GAT,  $\mathbf{G}^2$ -GCN).
+
+# B TRAINING DETAILS
+
+All small and medium-scale experiments have been run on NVIDIA GeForce RTX 2080 Ti, GeForce RTX 3090, Titan RTX and Quadro RTX 6000 GPUs. The large-scale experiments have been run on Nvidia Tesla A100 (40GiB) GPUs.
+
+All hyperparameters were tuned using random search. Table 7 shows the ranges of each hyperparameter as well as the random distribution used to randomly sample from it. Moreover, Table 8 shows the rounded hyperparameter  $p$  in  $\mathbf{G}^2$  (4) of each best performing network.
+
+Table 7: Hyperparameter ranges.  
+
+<table><tr><td></td><td>range</td><td>rand. distribution</td></tr><tr><td>learning rate</td><td>[10-4, 10-2]</td><td>log uniform</td></tr><tr><td>hidden size m</td><td>{32, 64, 128, 256, 512}</td><td>disc. uniform</td></tr><tr><td>dropout input</td><td>[0, 0.9]</td><td>uniform</td></tr><tr><td>dropout output</td><td>[0, 0.9]</td><td>uniform</td></tr><tr><td>weight decay</td><td>[10-8, 10-2]</td><td>log uniform</td></tr><tr><td>G2-exponent p</td><td>[1, 5]</td><td>uniform</td></tr><tr><td>Usage of Fθ in (4)</td><td>{YES, NO}</td><td>disc. uniform</td></tr></table>
+
+Table 8: Rounded hyperparameter  $p$  in  ${\mathbf{G}}^{2}$  of each best performing network.  
+
+<table><tr><td></td><td>Texas</td><td>Wisconsin</td><td>Film</td><td>Squirrel</td><td>Chameleon</td><td>Cornell</td><td>snap-patents</td><td>arXiv-year</td><td>genius</td></tr><tr><td>G2-GAT</td><td>3.06</td><td>1.68</td><td>1.23</td><td>3.48</td><td>3.54</td><td>3.54</td><td>/</td><td>/</td><td>/</td></tr><tr><td>G2-GCN</td><td>3.93</td><td>2.92</td><td>3.79</td><td>1.99</td><td>1.08</td><td>3.87</td><td>/</td><td>/</td><td>/</td></tr><tr><td>G2-GraphSAGE</td><td>4.47</td><td>1.14</td><td>2.89</td><td>3.04</td><td>2.00</td><td>3.27</td><td>1.60</td><td>3.40</td><td>4.40</td></tr></table>
+
+# C MATHEMATICAL DETAILS
+
+In this section, we provide proofs for Propositions 3.3 and 3.4 in the main text. We start with the following technical result which is necessary in the subsequent proofs.
+
+A Poincare Inequality on Connected Graphs. Poincare inequalities for functions (Evans, 2010) bound function values in terms of their gradients. Similar bounds on node values in terms of graph-gradients can be derived and a particular instance is given below,
+
+Proposition C.1. Let  $\mathcal{G} = (\mathcal{V},\mathcal{E})$  be a connected graph and the corresponding (scalar) node features are denoted by  $\mathbf{y}_i\in \mathbb{R}$ , for all  $i\in \mathcal{V}$ . Let  $\mathbf{y}_1 = 0$ . Then, the following bound holds,
+
+$$
+\sum_ {i \in \mathcal {V}} \mathbf {y} _ {i} ^ {2} \leq \bar {d} \Delta_ {1} \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} | \mathbf {y} _ {j} - \mathbf {y} _ {i} | ^ {2}, \tag {17}
+$$
+
+where  $\overline{d} = \max_{i\in \mathcal{V}}\deg (i)$  and  $\Delta_1$  is the eccentricity of the node 1.
+
+Proof. Fix a node  $i \in \mathcal{V}$ . By assumption, the graph  $\mathcal{G}$  is connected. Hence, there exists a path connecting  $i$  and the node 1. Denote the shortest path as  $\mathcal{P}(i,1)$ . This path can be expressed in terms of the nodes  $\ell_{i,1}$  with  $0 \leq \ell \leq \delta$ , where  $0_{i,1} = 1$  and  $\delta_{i,1} = i$ . For any  $\ell$ , we require  $\ell_{i,1} \sim (\ell + 1)_{i,1}$ . Moreover,  $\delta_{i,1}$  is the graph distance between the nodes  $i$  and 1 and  $\Delta_1 = \max_{i \in \mathcal{V}} \delta_{i,1}$  is the eccentricity of the node 1.
+
+Given the node feature  $\mathbf{y}_i$ , we can rewrite it as,
+
+$$
+\mathbf {y} _ {i} = \mathbf {y} _ {1} + \sum_ {\ell = 0} ^ {\delta - 1} \mathbf {y} _ {(\ell + 1) _ {i, 1}} - \mathbf {y} _ {\ell_ {i, 1}} = \sum_ {\ell = 0} ^ {\delta - 1} \mathbf {y} _ {(\ell + 1) _ {i, 1}} - \mathbf {y} _ {\ell_ {i, 1}},
+$$
+
+as by assumption  $\mathbf{y}_1 = 0$
+
+Using Cauchy-Schwartz inequality on the previous identity yields,
+
+$$
+\mathbf {y} _ {i} ^ {2} \leq \Delta_ {1} \sum_ {\ell = 0} ^ {\delta - 1} \left(\mathbf {y} _ {(\ell + 1) _ {i, 1}} - \mathbf {y} _ {\ell_ {i, 1}}\right) ^ {2}.
+$$
+
+Summing the above inequality over  $i \in \mathcal{V}$  and using the fact that  $\ell_{i,1} \sim (\ell + 1)_{i,1}$ , we obtain the desired Poincaré inequality (17).
+
+![](images/2fa7aaedde9e7f73399412ee7270c91ce04a1f0f14a1decbfede04bc665f3c5a.jpg)
+
+# C.1 PROOF OF PROPOSITION 3.3 OF MAIN TEXT
+
+Proof. By the definition of exponential stability, we consider a small perturbation around the steady state  $\mathbf{c}$  and study whether this perturbation grows or decays in time. To this end, define the perturbation as,
+
+$$
+\hat {\mathbf {X}} _ {i} = \mathbf {X} _ {i} - c, \quad 1 \leq i \leq v. \tag {18}
+$$
+
+A tedious but straightforward calculation shows that these perturbations evolve by the following linearized system of ODEs,
+
+$$
+\frac {d \hat {\mathbf {X}} _ {i} (t)}{d t} = \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) \left(\hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i}\right), \quad \forall t, \forall i \in \mathcal {V}. \tag {19}
+$$
+
+Multiplying  $\hat{\mathbf{x}}_i$  to both sides of (19) yields,
+
+$$
+\begin{array}{l} \hat {\mathbf {X}} _ {i} \frac {d \hat {\mathbf {X}} _ {i} (t)}{d t} = \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) \hat {\mathbf {X}} _ {i} (\hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i}), \\ \Rightarrow \frac {d \hat {\mathbf {X}} _ {i} ^ {2} (t)}{d t} = \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) (\hat {\mathbf {X}} _ {j} ^ {2} - \hat {\mathbf {X}} _ {i} ^ {2}) - \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) (\hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i}) ^ {2}. \\ \end{array}
+$$
+
+Summing the above identity over all nodes  $i\in \mathcal{V}$  yields,
+
+$$
+\begin{array}{l} \frac {d}{d t} \sum_ {i \in \mathcal {V}} \hat {\mathbf {X}} _ {i} ^ {2} (t) = \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) (\hat {\mathbf {X}} _ {j} ^ {2} - \hat {\mathbf {X}} _ {i} ^ {2}) - \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) (\hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i}) ^ {2} \\ = \frac {1}{2} \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \underbrace {\left(\mathbf {A} _ {i j} (c , c) - \mathbf {A} _ {j , i} (c , c)\right)} _ {= 0 (1 5)} \left(\hat {\mathbf {X}} _ {j} ^ {2} - \hat {\mathbf {X}} _ {i} ^ {2}\right) - \frac {1}{2} \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \underbrace {\left(\mathbf {A} _ {i j} (c , c) + \mathbf {A} _ {j , i} (c , c)\right)} _ {= 2 \mathbf {A} _ {i j} (1 5)} \left(\hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i}\right) ^ {2}, \\ = - \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) \left(\hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i}\right) ^ {2}, \\ \leq - \underline {{a}} \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \left(\hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i}\right) ^ {2}, \quad (\mathrm {b y (1 5)}), \\ \leq - \frac {a}{\bar {d} \Delta_ {1}} \sum_ {i \in \mathcal {V}} \hat {\mathbf {X}} _ {i} ^ {2}. \\ \end{array}
+$$
+
+Here, the last inequality comes from applying the Poincaré inequality (17) for the perturbations  $\hat{\mathbf{X}}$  and from the fact that by assumption  $\hat{\mathbf{X}}_1 = 0$ .
+
+Applying Gronwall's inequality yields,
+
+$$
+\sum_ {i \in \mathcal {V}} \hat {\mathbf {X}} _ {i} ^ {2} (t) \leq \sum_ {i \in \mathcal {V}} \hat {\mathbf {X}} _ {i} ^ {2} (0) e ^ {- \frac {a}{d \Delta_ {1}} t}. \tag {20}
+$$
+
+Thus, the initial perturbations around the steady state  $\mathbf{c}$  are damped down exponentially fast and the steady state  $\mathbf{c}$  is exponentially stable implying that this architecture will lead to oversmoothing.
+
+# C.2 PROOF OF PROPOSITION 3.4 OF MAIN TEXT
+
+Proof. As in the proof of Proposition 3.3, we consider small perturbations of form (18) of the steady state  $\mathbf{c}$  and investigate how these perturbations evolve in time. Assuming that the initial perturbations are small, i.e., that there exists an  $0 < \epsilon < \epsilon$  such that  $\max_{i\in \mathcal{V}}|\hat{\mathbf{x}}_i(0)|\leq \epsilon$ , we perform a straightforward calculation to obtain that the perturbations (for a short time) evolve with the following quasi-linearized system of ODEs,
+
+$$
+\frac {d \hat {\mathbf {X}} _ {i} (t)}{d t} = \bar {\boldsymbol {\tau}} _ {i} (t) \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) \left(\hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i}\right), \forall i \in \mathcal {V}, \tag {21}
+$$
+
+$$
+\overline {{\boldsymbol {\tau}}} _ {i} (t) = \sum_ {j \in \mathcal {N} _ {i}} | \hat {\mathbf {X}} _ {j} (t) - \hat {\mathbf {X}} _ {i} (t) | ^ {p}, \forall i \in \mathcal {V}.
+$$
+
+Note that we have used the fact that  $\sigma'(x) = 1$  and  $\tanh'(0) = 1$  in obtaining (21) from (14).
+
+Next, we multiply  $\hat{\mathbf{x}}_i$  to both sides of (21) to obtain,
+
+$$
+\begin{array}{l} \hat {\mathbf {X}} _ {i} \frac {d \hat {\mathbf {X}} _ {i} (t)}{d t} = \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) \overline {{\boldsymbol {\tau}}} _ {i} \hat {\mathbf {X}} _ {i} \left(\hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i}\right), \\ \Rightarrow \quad \frac {d \hat {\mathbf {X}} _ {i} ^ {2} (t)}{d t} = \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) \bar {\boldsymbol {\tau}} _ {i} \left(\hat {\mathbf {X}} _ {j} ^ {2} - \hat {\mathbf {X}} _ {i} ^ {2}\right) - \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) \bar {\boldsymbol {\tau}} _ {i} \left(\hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i}\right) ^ {2} \tag {22} \\ \end{array}
+$$
+
+Trivially,
+
+$$
+\left| \hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i} \right| ^ {p} \leq \bar {\boldsymbol {\tau}} _ {i}, \forall j \in \mathcal {N} _ {i}, \quad \forall i.
+$$
+
+Applying this inequality to the last line of the identity (22), we obtain,
+
+$$
+\frac {d \hat {\mathbf {X}} _ {i} ^ {2} (t)}{d t} \leq \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) \overline {{\boldsymbol {\tau}}} _ {i} \left(\hat {\mathbf {X}} _ {j} ^ {2} - \hat {\mathbf {X}} _ {i} ^ {2}\right) - \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) \left| \hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i} \right| ^ {p + 2}.
+$$
+
+Summing the above inequality over  $i \in \mathcal{V}$  leads to,
+
+$$
+\begin{array}{l} \frac {d}{d t} \sum_ {i \in \mathcal {V}} \hat {\mathbf {X}} _ {i} ^ {2} (t) \leq \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) \overline {{\boldsymbol {\tau}}} _ {i} \left(\hat {\mathbf {X}} _ {j} ^ {2} - \hat {\mathbf {X}} _ {i} ^ {2}\right) - \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) \left| \hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i} \right| ^ {p + 2} \\ \leq \frac {1}{2} \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) (\overline {{\boldsymbol {\tau}}} _ {i} - \overline {{\boldsymbol {\tau}}} _ {j}) (\hat {\mathbf {X}} _ {j} ^ {2} - \hat {\mathbf {X}} _ {i} ^ {2}) (\mathbf {A} _ {i j} = \mathbf {A} _ {j, i}) \\ - \underline {{a}} \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \left| \hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i} \right| ^ {p + 2} \quad (\text {f r o m (1 5)}). \\ \end{array}
+$$
+
+Therefore, we have the following inequality,
+
+$$
+\frac {d}{d t} \sum_ {i \in \mathcal {V}} \hat {\mathbf {X}} _ {i} ^ {2} (t) \leq T _ {1} - T _ {2},
+$$
+
+$$
+T _ {1} = \frac {1}{2} \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \mathbf {A} _ {i j} (c, c) (\bar {\boldsymbol {\tau}} _ {i} - \bar {\boldsymbol {\tau}} _ {j}) (\hat {\mathbf {X}} _ {j} ^ {2} - \hat {\mathbf {X}} _ {i} ^ {2}) \tag {23}
+$$
+
+$$
+T _ {2} = \underline {{a}} \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \left| \hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i} \right| ^ {p + 2}.
+$$
+
+We analyze the differential inequality (23) by starting with the term  $T_{1}$  in (23). We observe that this term does not have a definite sign and can be either positive or negative. However, we can upper bound this term in the following manner. Given that the right-hand side of the ODE system (21) is Lipschitz continuous, the well-known Cauchy-Lipschitz theorem states that the solutions  $\hat{\mathbf{x}}$  depend continuously on the initial data. Given that  $\max_{i\in \mathcal{V}}|\hat{\mathbf{X}}_i(0)|\leq \epsilon <  < 1$  and the bounds on the hidden states (1), there exists a time  $\bar{t} >0$  such that
+
+$$
+\max _ {i \in \mathcal {V}} | \hat {\mathbf {X}} _ {i} (t) | \leq 1, \forall t \in [ 0, \overline {{t}} ].
+$$
+
+Using the definitions of  $\overline{\tau}$  and the right stochasticity of the matrix  $\mathbf{A}$ , we easily obtain the following bound,
+
+$$
+\left| T _ {1} \right| \leq 2 ^ {p + 1} \bar {d} ^ {2} v, \tag {24}
+$$
+
+where  $\overline{d} = \max_{i\in \mathcal{V}}\deg (i)$
+
+On the other hand, the term  $T_{2}$  in (23) is clearly positive. Hence, the solutions of resulting ODE,
+
+$$
+\frac {d}{d t} \sum_ {i \in \mathcal {V}} \hat {\mathbf {X}} _ {i} ^ {2} (t) \leq - T _ {2}, \tag {25}
+$$
+
+will clearly decay in time. The key question is whether or not the decay is exponentially fast. We answer this question below.
+
+To this end, we have the following calculation using the Hölder's inequality,
+
+$$
+\begin{array}{l} \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \left| \hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i} \right| ^ {2} \leq (\bar {d} v) ^ {\frac {p}{p + 2}} \left(\sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \left| \hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i} \right| ^ {p + 2}\right) ^ {\frac {2}{p + 2}}, \\ \Rightarrow \quad \frac {1}{(\bar {d} v) ^ {\frac {p}{2}}} \left(\sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \left| \hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i} \right| ^ {2}\right) ^ {\frac {p + 2}{2}} \leq \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \left| \hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i} \right| ^ {p + 2}. \\ \end{array}
+$$
+
+Observing that  $\tilde{\mathbf{X}}_1 = 0$  by assumption, we can applying the Poincaré inequality (17) in the above inequality to further obtain,
+
+$$
+\frac {1}{\bar {d} ^ {p + 1} v ^ {\frac {p}{2}} \Delta_ {1} ^ {\frac {p + 2}{2}}} \left(\sum_ {i \in \mathcal {V}} | \hat {\mathbf {X}} _ {i} | ^ {2}\right) ^ {\frac {p + 2}{2}} \leq \sum_ {i \in \mathcal {V}} \sum_ {j \in \mathcal {N} _ {i}} \left| \hat {\mathbf {X}} _ {j} - \hat {\mathbf {X}} _ {i} \right| ^ {p + 2}.
+$$
+
+Hence, from the definition of  $T_{2}$  (23), we have,
+
+$$
+T _ {2} \geq \frac {a}{\bar {d} ^ {p + 1} v ^ {\frac {p}{2}} \Delta_ {1} ^ {\frac {p + 2}{2}}} \left(\sum_ {i \in \mathcal {V}} | \hat {\mathbf {X}} _ {i} | ^ {2}\right) ^ {\frac {p + 2}{2}}. \tag {26}
+$$
+
+Therefore, the differential inequality (25) now reduces to,
+
+$$
+\frac {d}{d t} \sum_ {i \in \mathcal {V}} \hat {\mathbf {X}} _ {i} ^ {2} (t) \leq - \frac {a}{\bar {d} ^ {p + 1} v ^ {\frac {p}{2}} \Delta_ {1} ^ {\frac {p + 2}{2}}} \left(\sum_ {i \in \mathcal {V}} | \hat {\mathbf {X}} _ {i} | ^ {2}\right) ^ {\frac {p + 2}{2}}. \tag {27}
+$$
+
+The differential inequality (27) can be explicitly solved to obtain,
+
+$$
+\sum_ {i \in \mathcal {V}} \hat {\mathbf {X}} _ {i} ^ {2} (t) \leq \left(2 + p t \frac {a}{\bar {d} ^ {p + 1} v ^ {\frac {p}{2}} \Delta_ {1} ^ {\frac {p + 2}{2}}} \sum_ {i \in \mathcal {V}} \hat {\mathbf {X}} _ {i} ^ {2} (0) ^ {\frac {p}{2}}\right) ^ {- \frac {2}{p}} \sum_ {i \in \mathcal {V}} \hat {\mathbf {X}} _ {i} ^ {2} (0). \tag {28}
+$$
+
+From (28), we see that the initial perturbations decay but only algebraically at a rate of  $t^{-\frac{2}{p}}$  in time. For instance, the decay is only linear in time for  $p = 2$  and even slower for higher value of  $p$ .
+
+Combining the analysis of the terms  $T_{1,2}$  in the differential inequality (23), we see that the one of the terms can lead to a growth in the initial perturbations whereas the second term only leads to polynomial decay. Even if the contribution of the term  $T_{1} \equiv 0$ , the decay of initial perturbations is only polynomial. Thus, the steady state  $\mathbf{c}$  is not exponentially stable.
+
+![](images/cc57e77f0a8c5b7a52d5250e7c243339e9b1a097650a6738fee2b81df5e64f5e.jpg)
+
+Remark C.2. We note that the Proposition 3.4 assumes a certain structure of the matrix  $\mathbf{A}$  in (14). A careful perusal of the proof presented above reveal that this assumptions can be further relaxed. To start with, if the matrix  $\mathbf{A}(c, c)$  is not symmetric, then there will be an additional term in the inequality (23), which would be proportional to  $\mathbf{A}_{ij} - \mathbf{A}_{ji}$ . This term will be of indefinite sign and can cause further growth in the perturbations of the steady state  $c$ . In any case, it can only further destabilize the quasi-linearized system. The assumption that the entries of  $\mathbf{A}$  are uniformly positive amounts to assuming positivity of the weights of the underlying GNN layer. This can be replaced by requiring that the corresponding eigenvalues are uniformly positive. If some eigenvalues are negative, this will cause further instability and only strengthen the conclusion of lack of (exponential) stability. Finally, the assumption that one node is not perturbed during the quasi-linearization is required for the Poincaré inequality (17). If this is not true, an additional term, of indefinite sign, is added to the inequality (23). This term can cause further growth of the perturbations and will only add instability to the system. Hence, all the assumptions in Proposition 3.4 can be relaxed and the conclusion of lack of exponential stability of the zero-Dirichlet energy steady state still holds.

2023/Gradient Gating for Deep Multi-Rate Learning on Graphs/images.zip

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2023/Gradient-Guided Importance Sampling for Learning Binary Energy-Based Models/20b76a9d-9de7-4097-b3cb-317945cb6e10_origin.pdf

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2023/Gradient-Guided Importance Sampling for Learning Binary Energy-Based Models/full.md

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+# GRADIENT-GUIDED IMPORTANCE SAMPLING FOR LEARNING BINARY ENERGY-BASED MODELS
+
+Meng Liu, Haoran Liu, Shuiwang Ji
+
+Department of Computer Science & Engineering
+
+Texas A&M University
+
+College Station, TX 77843, USA
+
+{mengliu, liuhr99,sji}@tamu.edu
+
+# ABSTRACT
+
+Learning energy-based models (EBMs) is known to be difficult especially on discrete data where gradient-based learning strategies cannot be applied directly. Although ratio matching is a sound method to learn discrete EBMs, it suffers from expensive computation and excessive memory requirements, thereby resulting in difficulties in learning EBMs on high-dimensional data. Motivated by these limitations, in this study, we propose ratio matching with gradient-guided importance sampling (RMwGGIS). Particularly, we use the gradient of the energy function w.r.t. the discrete data space to approximately construct the provably optimal proposal distribution, which is subsequently used by importance sampling to efficiently estimate the original ratio matching objective. We perform experiments on density modeling over synthetic discrete data, graph generation, and training Ising models to evaluate our proposed method. The experimental results demonstrate that our method can significantly alleviate the limitations of ratio matching, perform more effectively in practice, and scale to high-dimensional problems. Our implementation is available at https://github.com/divelab/RmwGGIS.
+
+# 1 INTRODUCTION
+
+Energy-Based models (EBMs), also known as unnormalized probabilistic models, model distributions by associating unnormalized probability densities. Such methods have been developed for decades (Hopfield, 1982; Ackley et al., 1985; Cipra, 1987; Dayan et al., 1995; Zhu et al., 1998; Hinton, 2012) and are unified as energy-based models (EBMs) (LeCun et al., 2006) in the machine learning community. EBMs have great simplicity and flexibility since energy functions are not required to integrate or sum to one, thus enabling the usage of various energy functions. In practice, given different data types, we can parameterize the energy function with different neural networks as needed, such as multi-layer perceptrons (MLPs), convolutional neural networks (CNNs) (LeCun et al., 1998), and graph neural networks (GNNs) (Gori et al., 2005; Scarselli et al., 2008). Recently, EBMs have been drawing increasing attention and are demonstrated to be effective in various domains, including images (Ngiam et al., 2011; Xie et al., 2016; Du & Mordatch, 2019), videos (Xie et al., 2017), texts (Deng et al., 2020), 3D objects (Xie et al., 2018), molecules (Liu et al., 2021; Hataya et al., 2021), and proteins (Du et al., 2020b).
+
+Nonetheless, learning (a.k.a., training) EBMs is known to be challenging since we cannot compute the exact likelihood due to the intractable normalization constant. As reviewed in Section 4, many approaches have been proposed to learn EBMs, such as maximum likelihood training with MCMC sampling (Hinton, 2002) and score matching (Hyvarinen & Dayan, 2005). However, most recent advanced methods cannot be applied to discrete data directly since they usually leverage gradients over the continuous data space. For example, for many methods based on maximum likelihood training with MCMC sampling, they use the gradient w.r.t. the data space to update samples in each MCMC step. However, if we update discrete samples using such gradient, the resulting samples are usually invalid in the discrete space. Therefore, learning EBMs on discrete data remains challenging.
+
+Ratio matching (Hyvarinen, 2007) is a method to learn discrete EBMs on binary data by matching ratios of probabilities between the data distribution and the model distribution, as detailed in Sec-
+
+tion 2.2. However, as analyzed in Section 3.1, it requires expensive computations and excessive memory usages, which is infeasible if the data is high-dimensional.
+
+In this work, we propose to use the gradient of the energy function w.r.t. the discrete data space to guide the importance sampling for estimating the original ratio matching objective. More specifically, we use such gradient to approximately construct the provably optimal proposal distribution for importance sampling. Thus, the proposed approach is termed as ratio matching with gradient-guided importance sampling (RMwGGIS). Our RMwGGIS can significantly overcome the limitations of ratio matching. In addition, it is demonstrated to be more effective than the original ratio matching in practice. We perform extensive analysis for this improvement by connecting it with hard negative mining, and further propose an advanced version of RMwGGIS accordingly by reconsidering the importance weights. Experimental results on synthetic discrete data, graph generation, and Ising model training demonstrate that our RMwGGIS significantly alleviates the limitations of ratio matching, achieves better performance with obvious margins, and has the ability of scaling to high-dimensional relevant problems.
+
+# 2 PRELIMINARIES
+
+# 2.1 ENERGY-BASED MODELS
+
+Let  $\pmb{x}$  be a data point and  $E_{\pmb{\theta}}(\pmb{x}) \in \mathbb{R}$  be the corresponding energy, where  $\pmb{\theta}$  represents the learnable parameters of the parameterized energy function  $E_{\pmb{\theta}}(\cdot)$ . The probability density function of the model distribution is given as  $p_{\theta}(\pmb{x}) = \frac{e^{-E_{\theta}(\pmb{x})}}{Z_{\theta}} \propto e^{-E_{\theta}(\pmb{x})}$ , where  $Z_{\theta} \in \mathbb{R}$  is the normalization constant (a.k.a., partition function). To be specific,  $Z_{\theta} = \int e^{-E_{\theta}(\pmb{x})} d\pmb{x}$  if  $\pmb{x}$  is in the continuous space and  $Z_{\theta} = \sum e^{-E_{\theta}(\pmb{x})}$  for discrete data. Hence, computing  $Z_{\theta}$  is usually infeasible due to the intractable integral or summation. Note that  $Z_{\theta}$  is a variable depending on  $\pmb{\theta}$  but a constant w.r.t.  $\pmb{x}$ .
+
+# 2.2 RATIO MATCHING
+
+Ratio matching (Hyvärinen, 2007) is developed for learning EBMs on binary discrete data by matching ratios of probabilities between the data distribution and the model distribution. Note that we focus on  $d$ -dimensional binary discrete data  $\pmb{x} \in \{0,1\}^d$  in this work.
+
+Specifically, ratio matching considers the ratio of  $p(\pmb{x})$  and  $p(\pmb{x}_{-i})$ , where  $\pmb{x}_{-i} = (x_1, x_2, \dots, \bar{x}_i, \dots, x_d)$  denotes a point in the data space obtained by flipping the  $i$ -th dimension of  $\pmb{x}$ . The key idea is to force the ratios  $\frac{p_{\theta}(\pmb{x})}{p_{\theta}(\pmb{x}_{-i})}$  defined by the model distribution  $p_{\theta}$  to be as close as possible to the ratios  $\frac{p_{\mathcal{D}}(\pmb{x})}{p_{\mathcal{D}}(\pmb{x}_{-i})}$  given by the data distribution  $p_{\mathcal{D}}$ . The benefit of considering ratios of probabilities is that they do not involve the intractable normalization constant  $Z_{\theta}$  since  $\frac{p_{\theta}(\pmb{x})}{p_{\theta}(\pmb{x}_{-i})} = \frac{e^{-E_{\theta}(\pmb{x})}}{Z_{\theta}} \cdot \frac{Z_{\theta}}{e^{-E_{\theta}(\pmb{x}_{-i})}} = e^{E_{\theta}(\pmb{x}_{-i}) - E_{\theta}(\pmb{x})}$ . To achieve the match between ratios, Hyvarinen (2007) proposes to minimize the objective function
+
+$$
+\mathcal {J} _ {R M} (\boldsymbol {\theta}) = \mathbb {E} _ {\boldsymbol {x} \sim p _ {\mathcal {D}} (\boldsymbol {x})} \sum_ {i = 1} ^ {d} \left[ g \left(\frac {p _ {\mathcal {D}} (\boldsymbol {x})}{p _ {\mathcal {D}} (\boldsymbol {x} _ {- i})}\right) - g \left(\frac {p _ {\boldsymbol {\theta}} (\boldsymbol {x})}{p _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {- i})}\right) \right] ^ {2} + \left[ g \left(\frac {p _ {\mathcal {D}} (\boldsymbol {x} _ {- i})}{p _ {\mathcal {D}} (\boldsymbol {x})}\right) - g \left(\frac {p _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {- i})}{p _ {\boldsymbol {\theta}} (\boldsymbol {x})}\right) \right] ^ {2}. \tag {1}
+$$
+
+The sum of two square distances with the role of  $\pmb{x}$  and  $\pmb{x}_{-i}$  switched is specifically designed since it is essential for the following simplification. In addition, the function  $g(u) = \frac{1}{1 + u}$  is also carefully chosen in order to obtain the subsequent simplification. To compute the objective defined in Eq. (1), it is known that the expectation over data distribution (i.e.,  $\mathbb{E}_{\pmb{x} \sim p_{\mathcal{D}}(\pmb{x})}$ ) can be unbiasedly estimated by the empirical mean of samples  $\pmb{x} \sim p_{\mathcal{D}}(\pmb{x})$ . However, to obtain the ratios between  $p_{\mathcal{D}}(\pmb{x})$  and  $p_{\mathcal{D}}(\pmb{x}_{-i})$  in Eq. (1), the exact data distribution is required to be known, which is usually impossible.
+
+Fortunately, thanks to the above carefully designed objective, Hyvarinen (2007) demonstrates that the objective function in Eq. (1) is equivalent to the following simplified version
+
+$$
+\mathcal {J} _ {R M} (\boldsymbol {\theta}) = \mathbb {E} _ {\boldsymbol {x} \sim p _ {\mathcal {D}} (\boldsymbol {x})} \sum_ {i = 1} ^ {d} \left[ g \left(\frac {p _ {\boldsymbol {\theta}} (\boldsymbol {x})}{p _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {- i})}\right) \right] ^ {2} \tag {2}
+$$
+
+which does not require the data distribution to be known and can be easily computed by evaluating the energy of  $\pmb{x}$  and  $\pmb{x}_{-i}$ , according to  $\frac{p_{\theta}(\pmb{x})}{p_{\theta}(\pmb{x}_{-i})} = e^{E_{\theta}(\pmb{x}_{-i}) - E_{\theta}(\pmb{x})}$ . Further, Lyu (2009) shows that the above objective function of ratio matching agree with the following objective
+
+$$
+\mathcal {J} _ {R M} (\pmb {\theta}) = \mathbb {E} _ {\pmb {x} \sim p _ {\mathcal {D}} (\pmb {x})} \sum_ {i = 1} ^ {d} \left[ \frac {p _ {\pmb {\theta}} (\pmb {x} _ {- i})}{p _ {\pmb {\theta}} (\pmb {x})} \right] ^ {2} = \mathbb {E} _ {\pmb {x} \sim p _ {\mathcal {D}} (\pmb {x})} \sum_ {i = 1} ^ {d} \left[ e ^ {E _ {\pmb {\theta}} (\pmb {x}) - E _ {\pmb {\theta}} (\pmb {x} _ {- i})} \right] ^ {2}. \qquad (3)
+$$
+
+They agree with each other since function  $g(\cdot)$  decreases monotonically in  $[0, +\infty)$ , which aligns with the value range of probability ratios  $\frac{p_{\theta}(\boldsymbol{x})}{p_{\theta}(\boldsymbol{x}_{-i})}$ . Note that Eq. (3) is originally named as the discrete extension of generalized score matching in Lyu (2009). Since it agrees with Eq. (2) and includes ratios of probabilities, we also treat it as an objective function of ratio matching in our context. In the rest of this paper, we provide our analysis and develop our method based on Eq. (3) for clarity. Nonetheless, our proposed method below can be naturally performed on Eq. (2) as well.
+
+Intuitively, the objective function of ratio matching, as formulated in Eq. (3) or Eq. (2), can push down the energy of the training sample  $\pmb{x}$  and push up the energies of other data points obtained by flipping one dimension of  $\pmb{x}$ . Thus, this objective faithfully expect that each training sample  $\pmb{x}$  has higher probability than its local neighboring points that are hamming distance 1 from  $\pmb{x}$ .
+
+It is worth mentioning that ratio matching is not suitable for binary image data. This is because it faithfully treats  $\pmb{x}$  as a positive sample and  $x_{-i}$  for  $i = 1,2,\dots ,d$  as negative samples. However, this assumption does not hold for binary image data. Specifically, if we flip one dimension/pixel of a binary image such as a digit image from MNIST, the resulting image is almost unchanged and is still a positive sample in nature. Thus, it is not reasonable to use ratio matching to train EBMs on binary images. Similarly, our RMwGGIS presented below also has this limitation. Having said this, our RMwGGIS is still an effective and efficient method for a broad of binary discrete data. We leave extending its applicability to general discrete EBMs for future work.
+
+# 3 THE PROPOSED METHOD
+
+We first analyze the limitations of the ratio matching method from the perspective of computational time and memory usage. Then, we describe our proposed method, ratio matching with gradient-guided importance sampling (RMwGGIS), which uses the gradient of the energy function w.r.t. the discrete input  $\pmb{x}$  to guide the importance sampling for estimating the original ratio matching objective. Our approach can alleviate the limitations significantly and is shown to be more effective in practice.
+
+# 3.1 ANALYSIS OF RATIO MATCHING
+
+Time-intensive computations. Based on Eq. (3), for a sample  $\pmb{x}$ , we have to compute the energies for all  $x_{-i}$ , where  $i = 1, \dots, d$ . This needs  $\mathcal{O}(d)$  evaluations of the energy function for each training sample. This is computationally intensive, especially when the data dimension  $d$  is large.
+
+Excessive memory usages. The memory usage of ratio matching is another limitation that cannot be ignored, especially when we learn the energy function using modern GPUs with limited memory. As shown in Eq. (3), the objective function consists of  $d$  terms for each training sample. When we do backpropagation, computing the gradient of the objective function w.r.t. the learnable parameters of the energy function is required. Therefore, in order to compute such gradient, we have to store the whole computational graph and the intermediate tensors for all of the  $d$  terms, thereby leading to excessive memory usages especially if the data dimension  $d$  is large.
+
+# 3.2 RATIO MATCHING WITH GRADIENT-GUIDED IMPORTANCE SAMPLING
+
+The key idea is to use the well-known importance sampling technique to reduce the variance of estimating  $\mathcal{I}_{RM}(\pmb{\theta})$  with fewer than  $d$  terms. The most critical and challenging part of using the importance sampling technique is choosing a good proposal distribution. In this work, we propose to use the gradient of the energy function w.r.t. the discrete input  $\pmb{x}$  to approximately construct the optimal proposal distribution for importance sampling. We describe the details of our method below.
+
+The objective for each sample  $\pmb{x}$ , defined by Eq. (3), can be reformulated as
+
+$$
+\mathcal {J} _ {R M} (\boldsymbol {\theta}, \boldsymbol {x}) = d \sum_ {i = 1} ^ {d} \frac {1}{d} \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {- i})} \right] ^ {2} = d \mathbb {E} _ {\boldsymbol {x} _ {- i} \sim m (\boldsymbol {x} _ {- i})} \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {- i})} \right] ^ {2}, \tag {4}
+$$
+
+where  $m(\pmb{x}_{-i}) = \frac{1}{d}$  for  $i = 1, \dots, d$  is a discrete distribution. Thus, the objective of ratio matching for each sample  $\pmb{x}$  can be viewed as the expectation of  $\left[e^{E_{\theta}(\pmb{x}) - E_{\theta}(\pmb{x}_{-i})}\right]^2$  over the discrete distribution  $m(\pmb{x}_{-i})$ . In the original ratio matching method, as described in Section 2.2, we compute such expectation exactly by considering all possible  $\pmb{x}_{-i}$ , leading to expensive computations and excessive memory usages. Naturally, we can estimate the desired expectation with Monte Carlo method by considering fewer terms sampled based on  $m(\pmb{x}_{-i})$ . However, such estimation usually has a high variance, and is empirically verified to be ineffective by our experiments in Section 5.
+
+Further, we can apply the importance sampling method to reduce the variance of Monte Carlo estimation. Intuitively, certain values have more impact on the expectation than others. Hence, the estimator variance can be reduced if such important values are sampled more frequently than others. To be specific, instead of sampling based on the distribution  $m(\boldsymbol{x}_{-i})$ , importance sampling aims to sample from another distribution  $n(\boldsymbol{x}_{-i})$ , namely, proposal distribution. Formally,
+
+$$
+\mathcal {J} _ {R M} (\boldsymbol {\theta}, \boldsymbol {x}) = d \mathbb {E} _ {\boldsymbol {x} _ {- i} \sim m \left(\boldsymbol {x} _ {- i}\right)} \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {- i}\right)} \right] ^ {2} = d \mathbb {E} _ {\boldsymbol {x} _ {- i} \sim n \left(\boldsymbol {x} _ {- i}\right)} \frac {m \left(\boldsymbol {x} _ {- i}\right) \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {- i}\right)} \right] ^ {2}}{n \left(\boldsymbol {x} _ {- i}\right)}. \tag {5}
+$$
+
+A short derivation of Eq. (5) is in Appendix A. Afterwards, we can apply Monte Carlo estimation based on the proposal distribution  $n(\pmb{x}_{-i})$ . Specifically, we sample  $s$  terms, denoted as  $\pmb{x}_{-i}^{(1)}, \dots, \pmb{x}_{-i}^{(s)}$ , according to the proposal distribution  $n(\pmb{x}_{-i})$ . Note that  $s$  is usually chosen to be much smaller than  $d$ . Then the estimation for  $\mathcal{J}_{RM}(\pmb{\theta}, \pmb{x})$  is computed based on these  $s$  terms. Formally,
+
+$$
+\widehat {\mathcal {J} _ {R M} (\boldsymbol {\theta} , \boldsymbol {x})} _ {n} = d \frac {1}{s} \sum_ {t = 1} ^ {s} \frac {m \left(\boldsymbol {x} _ {- i} ^ {(t)}\right) \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {- i} ^ {(t)}\right)} \right] ^ {2}}{n \left(\boldsymbol {x} _ {- i} ^ {(t)}\right)}, \quad \boldsymbol {x} _ {- i} ^ {(t)} \sim n \left(\boldsymbol {x} _ {- i}\right). \tag {6}
+$$
+
+To make it clear, we stop the gradient flowing through the importance weights during backpropagation. It is known that the estimator obtained by Monte Carlo estimation with importance sampling is an unbiased estimator, as the conventional Monte Carlo estimator. The key point of importance sampling is to choose an appropriate proposal distribution  $n(\boldsymbol{x}_{-i})$ , which determines the variance of the corresponding estimator. Based on Robert et al. (1999), the optimal proposal distribution  $n^{*}(\boldsymbol{x}_{-i})$ , which yields the minimum variance, is given by the following fact.
+
+Fact 1. Let  $n^*(\pmb{x}_{-i}) = \frac{\left[e^{E_{\pmb{\theta}}(\pmb{x}) - E_{\pmb{\theta}}(\pmb{x}_{-i})}\right]^2}{\sum_{k=1}^d \left[e^{E_{\pmb{\theta}}(\pmb{x}) - E_{\pmb{\theta}}(\pmb{x}_{-k})}\right]^2}$  be a discrete distribution on  $\pmb{x}_{-i}$ , where  $i = 1, \dots, d$ . Then for any discrete distribution  $n(\pmb{x}_{-i})$  on  $\pmb{x}_{-i}$ , where  $i = 1, \dots, d$ , we have  $Var\left(\widehat{\mathcal{J}_{RM}(\pmb{\theta}, \pmb{x})}_{n^*}\right) \leq Var\left(\widehat{\mathcal{J}_{RM}(\pmb{\theta}, \pmb{x})}_n\right)$ .
+
+Proof. The proof is included in Appendix B.
+
+To construct the exact optimal proposal distribution  $n^{*}(\pmb{x}_{-i})$ , we still have to evaluate the energies of all  $\pmb{x}_{-i}$ , where  $i = 1, \dots, d$ . To avoid such complexity, we propose to leverage the gradient of the energy function w.r.t. the discrete input  $\pmb{x}$  to approximately construct the optimal proposal distribution. Only  $\mathcal{O}(1)$  evaluations of the energy function is needed to construct the proposal distribution.
+
+It is observed by Grathwohl et al. (2021) that many discrete distributions are implemented as continuous and differentiable functions, although they are evaluated only in discrete domains. Grathwohl et al. (2021) further proposes a scalable sampling method for discrete distributions by using the gradients of the underlying continuous functions w.r.t. the discrete input. In this study, we extend this idea to improve ratio matching. More specifically, in our case, even though our input  $x$  is discrete, our parameterized energy function  $E_{\theta}(\cdot)$ , such as a neural network, is usually continuous and differentiable. Hence, we can use such gradient information to efficiently and approximately construct the optimal proposal distribution given by Fact 1.
+
+Algorithm 1 Ratio Matching with Gradient-Guided Importance Sampling (RMwGGIS)  
+1: Input: Observed dataset  $\mathcal{D} = \{\pmb{x}^{(m)}\}_{m=1}^{|D|}$ , parameterized energy function  $E_{\theta}(\cdot)$ , number of samples  $s$  for Monte Carlo estimation with importance sampling  
+2: for  $\pmb{x} \sim \mathcal{D}$  do ▷ Batch training is applied in practice  
+3: Compute  $E_{\theta}(\pmb{x})$   
+4:Compute  $\nabla_{\pmb{x}} E_{\theta}(\pmb{x})$   
+5:Compute the proposal distribution  $\widetilde{n}^{*}(\pmb{x}_{-i})$  ▷ Eq. (10)  
+6:Sample  $s$  terms, denoted as  $\pmb{x}_{-i}^{(1)}, \dots, \pmb{x}_{-i}^{(s)}$ , according to  $\widetilde{n}^{*}(\pmb{x}_{-i})$   
+7:Compute  $\widehat{\mathcal{J}_{RM}(\pmb{\theta}, \pmb{x})}_{\widetilde{n}^{*}}$  ▷ Eq. (6) (or Eq. (11))  
+8:Update  $\pmb{\theta}$  based on  $\nabla_{\pmb{\theta}} \widehat{\mathcal{J}_{RM}(\pmb{\theta}, \pmb{x})}_{\widetilde{n}^{*}}$   
+9:end for
+
+The basic idea is that we can approximate  $E_{\theta}(\pmb{x}_{-i})$  based on the Taylor series of  $E_{\theta}(\cdot)$  at  $\pmb{x}$ , given that  $\pmb{x}_{-i}$  is close to  $\pmb{x}$  in the data space because they only have differences in one dimension<sup>1</sup>. Formally,
+
+$$
+E _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {- i}\right) \approx E _ {\boldsymbol {\theta}} (\boldsymbol {x}) + \left(\boldsymbol {x} _ {- i} - \boldsymbol {x}\right) ^ {T} \nabla_ {\boldsymbol {x}} E _ {\boldsymbol {\theta}} (\boldsymbol {x}). \tag {7}
+$$
+
+Thus, we can approximately obtain the desired term  $E_{\theta}(\pmb{x}) - E_{\theta}(\pmb{x}_{-i})$  in Fact 1 using Eq. (7). Note that  $\nabla_{\pmb{x}}E_{\theta}(\pmb{x}) \in \mathbb{R}^{d}$  contains the information for approximating all  $E_{\theta}(\pmb{x}) - E_{\theta}(\pmb{x}_{-i})$ , where  $i = 1,\dots ,d$ . Hence, we can consider the following  $d$ -dimensional vector
+
+$$
+(2 \boldsymbol {x} - 1) \odot \nabla_ {\boldsymbol {x}} E _ {\boldsymbol {\theta}} (\boldsymbol {x}) \quad \in \mathbb {R} ^ {d}, \tag {8}
+$$
+
+where  $\odot$  denotes the element-wise multiplication. Note that we have  $x_{i} - \bar{x}_{i} = -1$  if  $x_{i} = 0$  and  $x_{i} - \bar{x}_{i} = 1$  if  $x_{i} = 1$ , which can be unified as  $x_{i} - \bar{x}_{i} = 2x_{i} - 1$ . Therefore, we have
+
+$$
+E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {- i}\right) \approx \left[ (2 \boldsymbol {x} - 1) \odot \nabla_ {\boldsymbol {x}} E _ {\boldsymbol {\theta}} (\boldsymbol {x}) \right] _ {i}, i = 1, \dots , d. \tag {9}
+$$
+
+Afterwards, we can provide a proposal distribution  $\widetilde{n}^{*}(\pmb{x}_{-i})$  as an approximation of the optimal proposal distribution  $n^{*}(\pmb{x}_{-i})$  given by Fact 1. Formally,
+
+$$
+\widetilde {n} ^ {*} \left(\boldsymbol {x} _ {- i}\right) = \frac {\left[ e ^ {2 (2 \boldsymbol {x} - 1) \odot \nabla_ {\boldsymbol {x}} E _ {\boldsymbol {\theta}} (\boldsymbol {x})} \right] _ {i}}{\sum_ {k = 1} ^ {d} \left[ e ^ {2 (2 \boldsymbol {x} - 1) \odot \nabla_ {\boldsymbol {x}} E _ {\boldsymbol {\theta}} (\boldsymbol {x})} \right] _ {k}}, i = 1, \dots , d. \tag {10}
+$$
+
+Then  $\widetilde{n}^{*}(\pmb{x}_{-i})$  is used as the proposal distribution for Monte Carlo estimation with importance sampling, as in Eq. (6). The overall process of our RMwGGIS method is summarized in Algorithm 1.
+
+# 3.3 COMPARISON BETWEEN RATIO MATCHING AND RMWGGIS
+
+Time and memory. Since only  $s$  ( $s < d$ ) terms are considered in the objective function of our RMwGGIS, as shown in Eq. (6), we have better computational efficiency and less memory requirement compared to the original ratio matching method. To be specific, our RMwGGIS only needs  $\mathcal{O}(s)$  evaluations of the energy function compared with  $\mathcal{O}(d)$  in ratio matching, leading to a linear speedup, which is significant especially when the data is high-dimensional. The improvement in terms of memory usage is similar. In Section 5.1, we compare the real running time and memory usage between ratio matching and our proposed RMwGGIS on datasets with different data dimensions.
+
+Better optimization? In Section 3.2, we propose our RMwGGIS based on the motivation to approximate the objective of ratio matching with fewer terms. Although our RMwGGIS can approximate the original ratio matching objective numerically, our objective only includes  $s$  terms compared to  $d$  terms in the original ratio matching objective. In other words, the objective of ratio matching, as shown in Eq. (3), intuitively pushes up the energies of all  $x_{-i}$  for  $i = 1,\dots ,d$ , while our RMwGGIS only considers pushing up energies of  $s$  terms among them, as formulated by Eq. (6). Thus, one may wonder why our objective is effective for learning EBMs without pushing up the energies of all  $d$  terms? In practice, we even observe that RMwGGIS achieves better density modeling performance
+
+than ratio matching. We conjecture that this is because our RMwGGIS can lead to better optimization of Eq. (3) in practice for the following two properties, which are empirically verified in Section 5.
+
+(1) RMwGGIS introduces stochasticity. Without involving all  $d$  terms in the objective function, our method can introduce stochasticity, which could lead to better optimization in practice. This has the same philosophy as the comparison between mini-batch gradient descent and vanilla gradient descent. The gradient is obtained based on each batch in mini-batch gradient descent, while it is computed over the entire dataset in vanilla gradient descent. It is known that the mini-batch gradient descent usually performs better in practice since the stochasticity introduced by mini-batch training could help escape from the saddle points in non-convex optimization (Ge et al., 2015). Therefore, the stochasticity introduced by sampling only  $s$  terms in RMwGGIS might help the optimization especially when  $d$  is large.
+
+(2) RMwGGIS focuses on neighbors with low energies. Even though only energies of  $s$  terms are pushed up in our method, these  $s$  terms correspond to the neighboring points that have low energies. According to  $n^*(\boldsymbol{x}_{-i})$  given by Fact 1, a neighbor of  $\boldsymbol{x}$ , denoted as  $\boldsymbol{x}_{-i}$ , is more likely to be sampled if its corresponding energy value is lower. Hence, we choose to push up the energies of  $s$  neighbors according to their current energies. The lower the energy, the more likely it is to be selected. This is intuitively sound because the terms that have low energies are the most offending terms, which should have the higher priorities to be pushed up. In other words, neighbors with lower energies contribute more to the objective function according to Eq. (3). That is, the loss values for these neighbors are larger. Thus, RMwGGIS has the same philosophy as hard negative mining, which pays more attention to hard negative samples during training. More detailed explanation about this connection is provided in Appendix C.
+
+Following the hard negative mining perspective, we observe that the coefficients used in Eq. (6) provide smaller weights for terms with lower energies. In other words, among its selected offending terms (i.e., hard negative samples), it pay least attention to the most offending terms, which is intuitively less effective. Therefore, we further propose the following advanced version as an objective function by removing the coefficients in Eq. (6). Formally,
+
+$$
+\overbrace {\mathcal {J} _ {R M} (\boldsymbol {\theta} , \boldsymbol {x}) _ {\widetilde {n} ^ {*}}} ^ {a d v} = \sum_ {t = 1} ^ {s} \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {- i} ^ {(t)}\right)} \right] ^ {2}, \quad \boldsymbol {x} _ {- i} ^ {(t)} \sim \widetilde {n} ^ {*} \left(\boldsymbol {x} _ {- i}\right). \tag {11}
+$$
+
+This advanced version is essentially a heuristic extension of the basic version in Eq. (6). It is demonstrated to be more effective in practice. The explanation about this is discussed in Appendix D.
+
+# 4 RELATED WORKS
+
+Learning EBMs has been drawing increasing attention recently. Maximum likelihood training with MCMC sampling, also known as contrastive divergence (Hinton, 2002), is the most representative method. It contrasts samples from training set and samples from the model distribution. To draw samples from the model distribution, we can employ MCMC sampling approaches, such as Langevin dynamics (Welling & Teh, 2011) and Hamiltonian dynamics (Neal et al., 2011). Such methods are further improved and shown to be effective by recent studies (Xie et al., 2016; Gao et al., 2018; Du & Mordatch, 2019; Nijkamp et al., 2019; Grathwohl et al., 2019; Jacob et al., 2020; Qiu et al., 2019; Du et al., 2020a). These methods, however, require the gradient w.r.t. the data space to update samples in each MCMC step. Thus, they cannot be applied to discrete data directly. To enable maximum likelihood training with MCMC sampling on discrete data, we can naturally use discrete sampling methods, such as Gibbs sampling and Metropolis-Hastings algorithm (Zanella, 2020), to replace the above gradient-based sampling algorithms. Unfortunately, sampling from a discrete distribution is extremely time-consuming and not scalable. Recently, Dai et al. (2020) develops a learnable sampler parameterized as a local discrete search algorithm to propose negative samples for contrasting. Grathwohl et al. (2021) proposes a scalable sampling method for discrete distributions by surprisingly using the gradient w.r.t. the data space, which inspires our work a lot.
+
+An alternative method for learning EBMs is score matching (Hyvärinen & Dayan, 2005; Vincent, 2011; Song et al., 2020; Song & Ermon, 2019), where the scores, i.e., the gradients of the logarithmic
+
+Table 1: Results on 32-dimensional synthetic discrete data in terms of MMD. The lower the better. The top two results on each dataset are highlighted as 1st and 2nd.  
+
+<table><tr><td>Method</td><td>2spirals</td><td>8gaussians</td><td>circles</td><td>moons</td><td>pinwheel</td><td>swissroll</td><td>checkerboard</td></tr><tr><td>Ratio Matching</td><td>0.01514</td><td>0.10270</td><td>0.11856</td><td>0.02901</td><td>0.31353</td><td>0.05820</td><td>0.00059</td></tr><tr><td>RMwGGIS (basic)</td><td>0.01099</td><td>0.09763</td><td>0.11017</td><td>0.03111</td><td>0.27885</td><td>0.05176</td><td>0.00050</td></tr><tr><td>RMwGGIS (advanced)</td><td>0.00876</td><td>0.08414</td><td>0.10230</td><td>0.02787</td><td>0.26188</td><td>0.04477</td><td>0.00026</td></tr></table>
+
+probability distribution w.r.t. the data space, of the energy function are forced to match the scores of the training data. Ratio matching (Hyvarinen, 2007; Lyu, 2009) is obtained by extending the idea of score matching to discrete data. Our work is motivated by the limitations of ratio matching, as analyzed in Section 3.1. Stochastic ratio matching (Dauphin & Bengio, 2013) also aims to make ratio matching more efficient by considering the sparsity of input data. Stochastic ratio matching is limited to sparse data, while our method is effective for general EBMs
+
+There are some other methods for learning EBMs, such as noise contrastive estimation (Gutmann & Hyvarinen, 2010; Bose et al., 2018; Ceylan & Gutmann, 2018; Gao et al., 2020) and learning the stein discrepancy (Grathwohl et al., 2020). We recommend readers to refer to Song & Kingma (2021) for a comprehensive introduction on learning EBMs. We note that several works (Elvira et al., 2015; Schuster, 2015) use the gradient information of the target distribution to iteratively optimize the proposal distributions for adaptive importance sampling. However, compared to our method, they can only applied to continuous distributions and require expensive iterative processes.
+
+# 5 EXPERIMENTS
+
+# 5.1 DENSITY MODELING ON SYNTHETIC DISCRETE DATA
+
+Setup. For both quantitative results and qualitative visualization, we follow the experimental setting of Dai et al. (2020) for density modeling on synthetic discrete data. We firstly draw 2D data points from 2D continuous space according to some unknown distribution  $\hat{p}$ , which can be naturally visualized. Then, we convert each 2D data point  $\hat{x} \in \mathbb{R}^2$  to a discrete data point  $x \in \{0,1\}^d$ , where  $d$  is the desired number of data dimensions. To be specific, we transform each dimension of  $\hat{x}$ , which is a floating-point number, into a  $\frac{d}{2}$ -bit Gray code and concatenate the results to obtain a  $d$ -bit vector  $x$ . Thus, the unknown distribution in discrete space is  $p(x) = \hat{p}\left(\left[\mathrm{GrayToFloat}(x_{1:\frac{d}{2}}), \mathrm{GrayToFloat}(x_{\frac{d}{2} + 1:d})\right]\right)$ . This density modeling task is challenging since the transformation from  $\hat{x}$  to  $x$  is non-linear.
+
+To quantitatively evaluate the performance of density modeling, we adopt the maximum mean discrepancy (MMD) (Gretton et al., 2012) with a linear kernel corresponding to  $(d$ -HammingDistance) to compare distributions. In our case, particularly, the MMD metric is computed based on 4000 samples, drawn from the learned energy function via Gibbs sampling, and the same number of samples from the training set. Lower MMD indicates that the distribution defined by the learned energy function is closer to the unknown data distribution. In addition, in order to qualitatively visualize the learned energy function, we firstly uniformly obtain  $10k$  data points from 2D continuous space. Afterwards, they are converted into bit vectors and evaluated by the learned energy function. Subsequently, we can visualize the obtained corresponding energies in 2D space.
+
+The energy function is parameterized by an MLP with the Swish (Ramachandran et al., 2017) activation and 256 hidden dimensions. The number of samples  $s$ , involved in the objective functions of our RMwGGIS method, is set to be 10. In the following, we compare our basic and advanced RMwGGIS methods, as formulated in Eq. (6) and Eq. (11) respectively, with the original ratio matching method (Hyvärinen, 2007; Lyu, 2009).
+
+Quantitative and qualitative results. The quantitative results on 32-dimensional datasets are shown in Table 1. Our RMwGGIS, including the basic and the advanced versions, consistently outperforms the original ratio matching by large margins, which demonstrates that it is effective for our proposed gradient-guided importance sampling to stochastically push up neighbors with low energies. This
+
+verifies our analysis in Section 3.3. In Figure 1, we qualitatively visualize the learned energy functions of our proposed RMwGGIS. It is observed that EBMs learned by our method can fit the data distribution accurately. Note that we choose  $d = 32$  for quantitative evaluation because Gibbs sampling cannot obtain appropriate samples from the learned energy function with an affordable time budget if the data dimension is too high, thus leading to invalid MMD results. We will compare the results on higher-dimensional data in the following by observing the qualitative visualization. To further demonstrate that the performance improvement of RMwGGIS over ratio matching is brought by better optimization, we show that energy functions learned with our methods actually lead to lower value for the objective function defined by Eq. (3). The details are included in Appendix E.
+
+Observations on higher-dimensional data. As analyzed in Section 3.3, the advantages of our approach can be greater on higher-dimensional data. To evaluate this, we conduct experiments on the 256-dimensional 2spirals dataset, and visualize the learned energy functions corresponding to different learning iterations. We construct a ablation method, named as RMwRAND, which estimates the original ratio matching objective by randomly sampling  $s = 10$  terms.
+
+![](images/626f005a93db307acd82510a5241ff4d9effedb955891598dcd891a8ee4df6fd.jpg)  
+Figure 1: Visualization of learned energy functions on 32-dimensional synthetic discrete datasets.
+
+The only difference between our RMwGGIS method and RMwRAND is that we focus more on the terms corresponding to low energies, based on our proposed gradient-guided importance sampling.
+
+As shown in Figure 4, Appendix F, our RMwGGIS accurately captures the data distribution, while the original ratio matching method cannot. This further verifies that RMwGGIS leads to better optimization than ratio matching especially when the data dimension is high, as analyzed in Section 3.3. In addition, although RMwRAND can also introduce stochasticity as RMwGGIS by randomly sampling, it fails to capture the data distribution. This observation is intuitively reasonable since randomly pushing up  $s = 10$  terms among  $d = 256$  terms leads to large variance. Instead, our RMwGGIS performs well since we focus on pushing up terms with low energies, which are the most offending terms and should be pushed up first. Overall, these experiments can show the superiority of RMwGGIS endowed by our proposed gradient-guided importance sampling on high-dimensional data.
+
+Selection of  $s$ . Note that we did not perform extensive tuning for  $s$ , although tuning it might bring performance improvement. To further show the influence of  $s$  and the effectiveness of RMwGGIS (advanced) over RMwRAND. We perform an experiment to train both models on 256-dimensional 2spirals with  $s = 5, 10, 50, 100$ . The visualization of the learned energy functions is shown in Figure 2. Our RMwGGIS is observed to achieve good density modeling performance that is robust to the selection of  $s$ . In contrast, RMwRAND fails
+
+![](images/2a283fb9cd5c0445495294bac1b68eb21a1b3942997c02826c5dc94e3588ea96.jpg)  
+Figure 2: Comparison of RMwG-GIS(advanced) and RMwRAND with several configurations of  $s$ .
+
+to capture the data distribution even with  $s = 100$ . This further demonstrates the effectiveness and robustness of our proposed gradient-guided importance sampling strategy.
+
+Running time and memory usage. To verify RMwGGIS has better efficiency than ratio matching, we compare the real running time and memory usage on datasets of various dimensions. Specifically, we construct several 2spirals datasets with different data dimensions and train parameterized energy functions using ratio matching and our RMwGGIS, respectively. We choose batch size to be 256. The reported time corresponds to the average training time per batch. For RMwGGIS, we report the results of the advanced version.
+
+As summarized in Table 5, Appendix G, our RMwGGIS is much more efficient in terms of running time and memory usage, compared to ratio matching. In addition, the improvement is more obvious with the increasing of data dimension. Specifically, compared with ratio matching, our RMwGGIS can achieve 6.2 times speedup and save  $90.1\%$  memory usage on the 2048-dimensional dataset.
+
+# 5.2 GRAPH GENERATION
+
+Setup. We further evaluate our RMwGGIS on graph generation using the Ego-small dataset (You et al., 2018). It is a set of one-hop ego graphs, where the number of nodes  $4 \leq |V| \leq 18$ , obtained
+
+from the Citeseer network (Sen et al., 2008). Following the experimental setting of You et al. (2018) and Liu et al. (2019),  $80\%$  of the graphs are used for training and the rest for testing. New graphs can be generated via Gibbs sampling on the learned energy function. To evaluate the graph generation performance based on the generated graphs and the test graphs, we calculate MMD over three statistics, i.e., degrees, clustering coefficients, and orbit counts, as proposed in You et al. (2018).
+
+We parameterize the energy function by a 5-layer RGCN (Schlichtkrull et al., 2018) model with the Swish activation and 32 hidden dimensions, whose input is the upper triangle of the graph adjacency matrix. The number of samples  $s$  used in our RMwGGIS objective is 50. We apply our advanced version to learn the energy function since it is more effective in Section 5.1. Besides ratio matching, we consider the recent proposed method EBM (GWG) (Grathwohl et al., 2021) as a baseline. We also consider the recent works developed for graph generation as baselines, including GraphVAE (Simonovsky & Komodakis, 2018), DeepGMG (Li et al., 2018), GraphRNN (You et al., 2018),
+
+Table 2: Graph generation results in terms of MMD. Avg. denotes the average over three MMD results.  
+
+<table><tr><td>Method</td><td>Degree</td><td>Cluster</td><td>Orbit</td><td>Avg.</td></tr><tr><td>GraphVAE</td><td>0.130</td><td>0.170</td><td>0.050</td><td>0.117</td></tr><tr><td>DeepGMG</td><td>0.040</td><td>0.100</td><td>0.020</td><td>0.053</td></tr><tr><td>GraphRNN</td><td>0.090</td><td>0.220</td><td>0.003</td><td>0.104</td></tr><tr><td>GNF</td><td>0.030</td><td>0.100</td><td>0.001</td><td>0.044</td></tr><tr><td>EDP-GNN</td><td>0.052</td><td>0.093</td><td>0.007</td><td>0.050</td></tr><tr><td>GraphAF</td><td>0.030</td><td>0.110</td><td>0.001</td><td>0.047</td></tr><tr><td>GraphDF</td><td>0.040</td><td>0.130</td><td>0.010</td><td>0.060</td></tr><tr><td>EBM (GWG)</td><td>0.093</td><td>0.027</td><td>0.053</td><td>0.058</td></tr><tr><td>Ratio Matching</td><td>0.062</td><td>0.066</td><td>0.008</td><td>0.045</td></tr><tr><td>RMwGGIS</td><td>0.044</td><td>0.059</td><td>0.013</td><td>0.039</td></tr></table>
+
+GNF (Liu et al., 2019), EDP-GNN (Niu et al., 2020), GraphAF (Shi et al., 2019), and GraphDF (Luo et al., 2021). The detailed setup is provided in Appendix H
+
+Quantitative and qualitative results. As summarized in Table 2, our RMwGGIS outperforms baselines in terms of the average over three MMD results. This shows that our method can learn EBMs to generate graphs that align with various characteristics of the training graphs. The generated samples are visualized in Figure 5, Appendix I. It can be observed that the generated samples are realistic one-hop ego graphs that have similar characteristics as the training samples.
+
+# 5.3 TRAINING ISING MODELS
+
+To further demonstrate the scaling ability of our method and compare with recent baselines more thoroughly, we use our
+
+Table 3: Comparison of training Ising models.  
+
+<table><tr><td colspan="2">MCMC #Steps</td><td>5</td><td>10</td><td>25</td><td>50</td><td>100</td><td>Ratio Matching</td><td>RMwGGIS</td></tr><tr><td rowspan="2">log(RMSE)</td><td>EBM (Gibbs)</td><td>-1.60</td><td>-1.90</td><td>-2.50</td><td>-3.00</td><td>-3.60</td><td rowspan="2">-3.99</td><td rowspan="2">-4.06</td></tr><tr><td>EBM (GWG)</td><td>-4.02</td><td>-4.49</td><td>-4.87</td><td>-4.94</td><td>-5.05</td></tr><tr><td rowspan="2">Time/iter</td><td>EBM (Gibbs)</td><td>263.5ms</td><td>437.3ms</td><td>1113.2ms</td><td>2524.9ms</td><td>4670.1ms</td><td rowspan="2">450.7ms</td><td rowspan="2">13.9ms</td></tr><tr><td>EBM (GWG)</td><td>37.5ms</td><td>63.4ms</td><td>100.5ms</td><td>222.6ms</td><td>395.7ms</td></tr></table>
+
+RMwGGIS to train the Ising model with a 2D cyclic lattice structure, following Grathwohl et al. (2021). We consider a  $25 \times 25$  lattice, thus leading to a 625-dimension problem, which can be used to evaluate the ability of scaling to high-dimensional problems. We compare methods in terms of the RMSE between the inferred connectivity matrix  $\hat{J}$  and the true  $J$  and the running time per iteration. The experimental details are included in Appendix J. As shown in Table 3, our RMwGGIS is more effective than ratio matching and EBM (Gibbs) with various sample steps. The recently proposed EBM (GWG) (Grathwohl et al., 2021) achieves better RMSE than ours. In terms of running time, our method is much more efficient than baselines since we avoid the expensive MCMC sampling during training and do not have to consider flipping all dimensions as ratio matching. According to this experiment, one future direction could be further improving the effectiveness of RMwGGIS while preserving the efficiency advantage.
+
+# 6 CONCLUSION
+
+We propose ratio matching with gradient-guided importance sampling (RMwGGIS) for learning EBMs on binary discrete data. In particular, we use the gradient of the energy function w.r.t. the discrete input space to guide the importance sampling for estimating the original ratio matching objective. We further connect our method with hard negative mining, and obtain the advanced version of RMwGGIS. Compared to ratio matching, our RMwGGIS methods, including the basic and advanced versions, are more efficient in terms of computation and memory usage, and are shown to be more effective for density modeling. Extensive experiments on synthetic data density modeling, graph generation, and Ising model training demonstrate that our RMwGGIS achieves significant improvements over previous methods in terms of both effectiveness and efficiency.
+
+# ACKNOWLEDGMENTS
+
+This work was supported in part by National Science Foundation grant IIS-1908220.
+
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+
+# A THE DETAILED DERIVATION OF EQ. (5)
+
+The detailed derivation of Eq. (5) is as follows.
+
+$$
+\begin{array}{l} \mathcal {J} _ {R M} (\pmb {\theta}, \pmb {x}) = d \mathbb {E} _ {\pmb {x} _ {- i} \sim m (\pmb {x} _ {- i})} \left[ e ^ {E _ {\pmb {\theta}} (\pmb {x}) - E _ {\pmb {\theta}} (\pmb {x} _ {- i})} \right] ^ {2} \\ = d \sum_ {i = 1} ^ {d} m (\boldsymbol {x} _ {- i}) \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {- i})} \right] ^ {2} \\ = d \sum_ {i = 1} ^ {d} \frac {m \left(\boldsymbol {x} _ {- i}\right) \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {- i}\right)} \right] ^ {2}}{n \left(\boldsymbol {x} _ {- i}\right)} n \left(\boldsymbol {x} _ {- i}\right) \tag {12} \\ = d \mathbb {E} _ {\pmb {x} _ {- i} \sim n (\pmb {x} _ {- i})} \frac {m (\pmb {x} _ {- i}) \left[ e ^ {E _ {\pmb {\theta}} (\pmb {x}) - E _ {\pmb {\theta}} (\pmb {x} _ {- i})} \right] ^ {2}}{n (\pmb {x} _ {- i})}. \\ \end{array}
+$$
+
+# B PROOF OF FACT 1
+
+Proof. According to Eq. (6), we have
+
+$$
+\operatorname {V a r} \left(\widehat {\mathcal {J} _ {R M} (\boldsymbol {\theta} , \boldsymbol {x})} _ {n}\right) = \frac {d ^ {2}}{s} \operatorname {V a r} \left(\frac {m \left(\boldsymbol {x} _ {- i}\right) \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {- i}\right)} \right] ^ {2}}{n \left(\boldsymbol {x} _ {- i}\right)}\right). \tag {13}
+$$
+
+Then we can compare the variance of the estimator based on  $n^{*}(\pmb{x}_{-i})$  and  $n(\pmb{x}_{-i})$ . Formally,
+
+$$
+\begin{array}{l} \operatorname {V a r} \left(\widehat {\mathcal {J} _ {R M} (\boldsymbol {\theta} , \boldsymbol {x})} _ {n ^ {*}}\right) (14) \\ = \frac {d ^ {2}}{s} \operatorname {V a r} \left(\frac {m \left(\boldsymbol {x} _ {- i}\right) \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {- i}\right)} \right] ^ {2}}{n ^ {*} \left(\boldsymbol {x} _ {- i}\right)}\right) (15) \\ = \frac {d ^ {2}}{s} \left\{\mathbb {E} _ {\boldsymbol {x} _ {- i} \sim n ^ {*} (\boldsymbol {x} _ {- i})} \left[ \frac {m (\boldsymbol {x} _ {- i}) \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {- i})} \right] ^ {2}}{n ^ {*} (\boldsymbol {x} _ {- i})} \right] ^ {2} - \left[ \mathbb {E} _ {\boldsymbol {x} _ {- i} \sim n ^ {*} (\boldsymbol {x} _ {- i})} \frac {m (\boldsymbol {x} _ {- i}) \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {- i})} \right] ^ {2}}{n ^ {*} (\boldsymbol {x} _ {- i})} \right] ^ {2} \right\} (16) \\ = \frac {d ^ {2}}{s} \left\{\mathbb {E} _ {\boldsymbol {x} _ {- i} \sim n ^ {*} (\boldsymbol {x} _ {- i})} \left[ \frac {m (\boldsymbol {x} _ {- i}) \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {- i})} \right] ^ {2}}{n ^ {*} (\boldsymbol {x} _ {- i})} \right] ^ {2} - \left[ \frac {\mathcal {I} _ {R M} (\boldsymbol {\theta} , \boldsymbol {x})}{d} \right] ^ {2} \right\} (17) \\ = \frac {d ^ {2}}{s} \left\{\frac {1}{d} \sum_ {i = 1} ^ {d} n ^ {*} (\boldsymbol {x} _ {- i}) \left[ \frac {m (\boldsymbol {x} _ {- i}) \left[ e ^ {E _ {\theta} (\boldsymbol {x}) - E _ {\theta} (\boldsymbol {x} _ {- i})} \right] ^ {2}}{n ^ {*} (\boldsymbol {x} _ {- i})} \right] ^ {2} - \left[ \frac {\mathcal {J} _ {R M} (\boldsymbol {\theta} , \boldsymbol {x})}{d} \right] ^ {2} \right\} (18) \\ = \frac {d ^ {2}}{s} \left\{\frac {1}{d} \sum_ {i = 1} ^ {d} m ^ {2} (\boldsymbol {x} _ {- i}) \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {- i})} \right] ^ {2} \sum_ {k = 1} ^ {d} \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {- k})} \right] ^ {2} - \left[ \frac {\mathcal {J} _ {R M} (\boldsymbol {\theta} , \boldsymbol {x})}{d} \right] ^ {2} \right\} (19) \\ = \frac {d ^ {2}}{s} \left\{\frac {1}{d} \left[ \sum_ {i = 1} ^ {d} m \left(\boldsymbol {x} _ {- i}\right) \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {- i}\right)} \right] ^ {2} \right] ^ {2} - \left[ \frac {\mathcal {J} _ {R M} (\boldsymbol {\theta} , \boldsymbol {x})}{d} \right] ^ {2} \right\} (20) \\ = \frac {d ^ {2}}{s} \left\{\frac {1}{d} \left[ \sum_ {i = 1} ^ {d} \frac {m \left(\boldsymbol {x} _ {- i}\right) \left[ e ^ {E _ {\theta} (\boldsymbol {x}) - E _ {\theta} \left(\boldsymbol {x} _ {- i}\right)} \right] ^ {2}}{n \left(\boldsymbol {x} _ {- i}\right)} \sqrt {n \left(\boldsymbol {x} _ {- i}\right)} \sqrt {n \left(\boldsymbol {x} _ {- i}\right)} \right] ^ {2} - \left[ \frac {\mathcal {J} _ {R M} (\boldsymbol {\theta} , \boldsymbol {x})}{d} \right] ^ {2} \right\} (21) \\ \leq \frac {d ^ {2}}{s} \left\{\frac {1}{d} \sum_ {i = 1} ^ {d} \left[ \frac {m \left(\boldsymbol {x} _ {- i}\right) \left[ e ^ {E _ {\theta} (\boldsymbol {x}) - E _ {\theta} \left(\boldsymbol {x} _ {- i}\right)} \right] ^ {2}}{n \left(\boldsymbol {x} _ {- i}\right)} \sqrt {n \left(\boldsymbol {x} _ {- i}\right)} \right] ^ {2} \sum_ {i = 1} ^ {d} n \left(\boldsymbol {x} _ {- i}\right) - \left[ \frac {\mathcal {J} _ {R M} (\boldsymbol {\theta} , \boldsymbol {x})}{d} \right] ^ {2} \right\} (22) \\ \end{array}
+$$
+
+$$
+\begin{array}{l} = \frac {d ^ {2}}{s} \left\{\frac {1}{d} \sum_ {i = 1} ^ {d} n \left(\boldsymbol {x} _ {- i}\right) \left[ \frac {m \left(\boldsymbol {x} _ {- i}\right) \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {- i}\right)} \right] ^ {2}}{n \left(\boldsymbol {x} _ {- i}\right)} \right] ^ {2} - \left[ \frac {\mathcal {J} _ {R M} (\boldsymbol {\theta} , \boldsymbol {x})}{d} \right] ^ {2} \right\} (23) \\ = \frac {d ^ {2}}{s} \left\{\mathbb {E} _ {\boldsymbol {x} _ {- i} \sim n (\boldsymbol {x} _ {- i})} \left[ \frac {m (\boldsymbol {x} _ {- i}) \left[ e ^ {E _ {\boldsymbol {\theta}} (\boldsymbol {x}) - E _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {- i})} \right] ^ {2}}{n (\boldsymbol {x} _ {- i})} \right] ^ {2} - \left[ \frac {\mathcal {I} _ {R M} (\boldsymbol {\theta} , \boldsymbol {x})}{d} \right] ^ {2} \right\} (24) \\ = \operatorname {V a r} \left(\widehat {\mathcal {J} _ {R M} (\boldsymbol {\theta} , \boldsymbol {x})} _ {n}\right). (25) \\ \end{array}
+$$
+
+Eq. (17) can be derived because the estimator is unbiased no matter what proposal distribution is applied. Eq. (19) is obtained by choosing  $n^{*}(\pmb{x}_{-i}) = \frac{\left[e^{E_{\theta}(\pmb{x}) - E_{\theta}(\pmb{x}_{-i})}\right]^{2}}{\sum_{k=1}^{d}\left[e^{E_{\theta}(\pmb{x}) - E_{\theta}(\pmb{x}_{-k})}\right]^{2}}$ . Eq. (20) holds since  $m(\pmb{x}_{-i}) = \frac{1}{d}$  for  $i = 1, \dots, d$ . To derive Eq. (22), we apply the Cauchy-Schwarz inequality  $\left(\sum_{i=1}^{d} a_{i} b_{i}\right)^{2} \leq \left(\sum_{i=1}^{d} a_{i}^{2}\right)\left(\sum_{i=1}^{d} b_{i}^{2}\right)$ . This completes the proof of Fact 1.
+
+# C CONNECTION WITH HARD NEGATIVE MINING
+
+Here, we provide an insight to understand why the second property, described in Section 3.3, can lead to better optimization, by connecting it with hard sample mining.
+
+Hard sample mining (Felzenszwalb et al., 2009; Rowley et al., 1998) has been widely applied to train deep neural networks (Shrivastava et al., 2016). Our RMwGGIS is particularly highly related to hard negative training strategies. The basic idea for hard negative mining is to pay more attention to hard negative samples during training, which can usually achieve better performance since it can reduce false positives. In our setting of discrete EBMs, each training sample  $\pmb{x}$  is a positive sample, and its energy should be pushed down. For each positive sample  $\pmb{x}$ , all  $x_{-i}$  for  $i = 1, 2, \dots, d$  are negative samples, and their energy should be pushed up. Our RMwGGIS with the specific proposal distribution shown in Eq. (10) can approximately choose the  $x_{-i}$ 's that currently have low energies with larger probabilities. This has the same philosophy as hard negative mining. Specifically, in our case,  $x_{-i}$ 's with low energies are hard negative samples since they are the most offending terms, which are close to the positive sample  $\pmb{x}$  and have low energies.
+
+Since our proposal distribution defined in Eq. (10) approximates the provable optimal proposal distribution given in Theorem 1, our proposal distribution thus approximately performs "hard negative mining". The natural follow-up question is how accurate is the approximation and how does it affect the learning process? We answer this question by analyzing the following two stages during learning, which can intuitively show that our RMwGGIS is technically sound.
+
+![](images/c3d9922a33e21022c2ff920e681cfc306a819d1c2bfa1a75bf1dee4471eb1bde.jpg)  
+(a)
+
+![](images/98df5d79d96deb5b24f035738ed1779ee898fddb1ab7a16ebf812841c52f7cf6.jpg)  
+(b)  
+Figure 3: An intuitive illustration of the approximation: (a) Stage I and (b) Stage II. The black curves denote the energy functions. Green nodes and brown nodes represent the true energy values and approximated values of neighbors, respectively. Note that the approximated values are obtained based on Taylor series of  $E_{\theta}(\cdot)$  at  $x$ , as shown in Eq. (7). For clarity, we only show two neighbors of  $x$  in this figure, but this illustration can also be extended to include all neighbors.
+
+Stage I. As shown in Figure 3 (a), at the early stage of learning, the energy function is not learned well, thus the energy  $E_{\theta}(\pmb{x})$  of positive sample  $\pmb{x}$  is not smaller than its all neighbors. In this case,
+
+Table 4: Comparison of resulting objective values. The top two lowest values on each dataset are highlighted as 1st and 2nd.  
+
+<table><tr><td>Method</td><td>2spirals</td><td>8gaussians</td><td>circles</td><td>moons</td><td>pinwheel</td><td>swissroll</td><td>checkerboard</td></tr><tr><td>Ratio Matching</td><td>46.02</td><td>39.26</td><td>31.82</td><td>28.57</td><td>28.50</td><td>37.52</td><td>26.05</td></tr><tr><td>RMwGGIS (basic)</td><td>26.84</td><td>30.15</td><td>29.89</td><td>27.80</td><td>28.08</td><td>32.20</td><td>26.09</td></tr><tr><td>RMwGGIS (advanced)</td><td>27.15</td><td>29.31</td><td>29.54</td><td>27.75</td><td>27.30</td><td>29.45</td><td>26.06</td></tr></table>
+
+there are some neighbors of  $\pmb{x}$  which have lower energies than  $E_{\theta}(\pmb{x})$ , such as  $\pmb{x}_{-1}$  in Figure 3 (a). Therefore, "hard negative mining" is in demand in this stage. Under this situation, our approximated energies of neighbors could help to perform "hard negative mining". To be specific, the estimated energies of neighbors are close to the true energies, and the estimated energy of  $\pmb{x}_{-1}$  is much lower than the estimated energy of  $\pmb{x}_{-2}$ . Thus, our proposal distribution will sample  $\pmb{x}_{-1}$  with a higher probability. This actually works as "hard negative mining" since  $\pmb{x}_{-1}$  is the current most offending term, i.e., the so-called hard negative sample.
+
+Stage II. After learning for a while, we can obtain a relatively good energy function, where the positive sample  $\pmb{x}$  locates in the low energy area compared to its local neighbors. In this case our approximation is less accurate. Fortunately, in this case, "hard negative mining" is not that necessary since there do not exist many offending terms. Specifically, as shown in Figure 3 (b), the energies of  $\pmb{x}_{-1}$  and  $\pmb{x}_{-2}$  are safely higher than  $E_{\theta}(\pmb{x})$ .
+
+Even though the above analysis is based on a simplified example, we believe it can serve as a good intuitive understanding of why our RMwGGIS performs better than ratio matching.
+
+# D WHY DOES THE ADVANCED VERSION PERFORM BETTER?
+
+Here, we provide an intuitive explanation on why our advanced version usually performs better than the basic version.
+
+Following our analysis of the connection between our method and hard negative mining, as described in Appendix C, it is obvious that both basic version (i.e., Eq. (6)) and advanced version (i.e., Eq. (11)) perform "hard negative mining". The difference lies in the coefficients for different terms. Specifically, the advanced version gives the same weights to all sampled terms. In contrast, the basic version provides a weight  $\frac{m(\boldsymbol{x}_{-i}^{(t)})}{n(\boldsymbol{x}_{-i}^{(t)})}$  to each sampled term  $\boldsymbol{x}_{-i}^{(t)}$ , as shown in Eq. (6). Note that  $m(\boldsymbol{x}_{-i}^{(t)}) = \frac{1}{d}$  for all terms and  $n(\boldsymbol{x}_{-i}^{(t)})$  would be larger if  $\boldsymbol{x}_{-i}^{(t)}$  has lower energy. Hence, the basic version provides smaller weights for terms with lower energies. In other words, among its selected offending terms (i.e., hard negative samples), it pay least attention to the most offending terms, which could be less effective than the advanced version with equal weights. This could explain why our advanced RMwGGIS usually performs better than the basic version.
+
+# E COMPARISON OF ACHIEVED OBJECTIVE VALUES
+
+Specifically, for all the learned energy functions in Table 1, we sample 4000 data points on each dataset and evaluate the resulting objective value defined by Eq. (3). The results are summarized in Table 4. We can observe that our basic and advanced RMwGGIS indeed achieve lower objective values, which further demonstrates that our proposed RMwGGIS can lead to better optimization. This is quite natural and straightforward since our method focuses on neighbors with low energies. Specifically, according to the objective function of ratio matching (i.e., Eq. (3)), neighbors with lower energies contribute more to the objective function. In other words, the loss values for these neighbors are larger. This explains why our methods, focusing on reducing the loss values (i.e., pushing up energies) of these neighbors, indeed lead to lower value for the objective function defined by Eq. (3).
+
+![](images/5c532ffbcfee4b0c1689d010b70a4fe5d35e5408382dffb7f1a94f16bae61537.jpg)  
+Figure 4: Visualization of learned energy functions w.r.t. number of learning iterations on the 256-dimensional 2spiralts dataset.
+
+Table 5: Comparison between ratio matching and our RMwGGIS on 2spirals datasets with different dimensions in terms of running time and memory usage.  
+
+<table><tr><td></td><td># Data Dimensions</td><td>32</td><td>64</td><td>128</td><td>256</td><td>512</td><td>1024</td><td>2048</td></tr><tr><td rowspan="3">Time</td><td>Ratio Matching</td><td>63.9ms</td><td>106.5ms</td><td>185.6ms</td><td>372.9ms</td><td>735.2ms</td><td>1390.1ms</td><td>2684.1ms</td></tr><tr><td>RMwGGIS</td><td>41.2ms</td><td>47.2ms</td><td>58.8ms</td><td>86.9ms</td><td>137.7ms</td><td>244.9ms</td><td>434.1ms</td></tr><tr><td>Speedup</td><td>1.6×</td><td>2.3×</td><td>3.2×</td><td>4.3×</td><td>5.3×</td><td>5.7×</td><td>6.2×</td></tr><tr><td rowspan="3">Memory</td><td>Ratio Matching</td><td>957MB</td><td>1031MB</td><td>1189MB</td><td>1545MB</td><td>2315MB</td><td>4237MB</td><td>9633MB</td></tr><tr><td>RMwGGIS</td><td>891MB</td><td>893MB</td><td>893MB</td><td>915MB</td><td>919MB</td><td>931MB</td><td>951MB</td></tr><tr><td>Memory Saving</td><td>6.9%</td><td>13.4%</td><td>24.9%</td><td>40.8%</td><td>60.3%</td><td>78.0%</td><td>90.1%</td></tr></table>
+
+# F VISUALIZATION OF LEARNED ENERGY FUNCTIONS ON HIGHER-DIMENSIONAL DATA
+
+The learned energy functions w.r.t. number of learning iterations on the 256-dimensional 2spirals dataset are qualitatively visualized in Figure 4
+
+# G COMPARISON OF RUNNING TIME AND MEMORY USAGE
+
+The comparison of running time and memory usage are summarized in Table 5.
+
+# H DETAILED SETTINGS OF GRAPH GENERATION
+
+The results of baselines in Table 2 are reported from You et al. (2018), Liu et al. (2019), Niu et al. (2020), Shi et al. (2019), and Luo et al. (2021). We obtain the result of EBM (GWG) by using their official implementation, and the detailed settings are provided as follows.
+
+We use the official open-sourced implementation<sup>4</sup> of EBM (GWG) to perform its graph generation experiment. We train models with persistent contrastive divergence (Tieleman, 2008) with a buffer size of 200 samples. We use the Adam optimizer (Kingma & Ba, 2015) with a learning rate of  $1e-4$  and a batch size of 200. The following hyperparameters are tuned and the finally chosen ones are underlined: buffer initialization rate  $\in \{0, 0.2, \underline{0.4}, 0.6\}$  and MCMC steps  $\in \{100, 200, \underline{500}, 1000, 2000\}$ .
+
+# I VISUALIZATION OF GENERATED GRAPHS
+
+Generated graph samples are shown in Figure 5.
+
+![](images/17ebd07574d835227ecda22dc32eef0fc90603b3f3d083e13d846ba705bab929.jpg)  
+Training data  
+Figure 5: Visualization of training data and samples drawn from the energy function learned by our RMwGGIS for graph generation.
+
+![](images/62bf5d56821ba415dec387fe7b63588783c834bbef8f49b70ae2bd930f504a7c.jpg)  
+RMwGGIS samples
+
+# J DETAILS OF TRAINING ISING MODELS
+
+Lattice Ising Models. Ising models (Cipra, 1987) is firstly developed to model the spin magnetic particles (Ising, 1925). For Ising models, our energy function can be naturally defined as
+
+$$
+E (\boldsymbol {x}) = - \boldsymbol {x} ^ {T} \boldsymbol {J} \boldsymbol {x} - \boldsymbol {b} ^ {T} \boldsymbol {x}, \tag {26}
+$$
+
+where  $J$  and  $b$  are the parameters.  $J$  is the connectivity matrix which indicates the correlation across dimensions in  $x$ . We follow one specific setting in Grathwohl et al. (2021), where all of the non-zero entries of  $J$  are identical (denoted as  $\sigma$ ) and  $J$  is the adjacency matrix of a cyclic 2D lattice structure. Therefore,
+
+$$
+E (\boldsymbol {x}) = - \sigma \boldsymbol {x} ^ {T} \boldsymbol {J} \boldsymbol {x} - \boldsymbol {b} ^ {T} \boldsymbol {x}. \tag {27}
+$$
+
+Setup. We follow Grathwohl et al. (2021) for our experimental setting. To be specific, we create a model using a  $25 \times 25$  lattice and  $\sigma = 0.25$ , thus leading to a 625 dimensional distribution. For training the model, 2000 examples are generated via 1,000,000 steps of Gibbs sampling. We apply our proposed RMwGGIS method to train the model. The number of samples  $s$  used in our RMwGGIS objective is set to 10. We use Adam optimizer (Kingma & Ba, 2015) with a learning rate of  $1e-4$  and a batch size of 100.  $\ell 1$  penalty with strength 0.01 is used to encourage sparsity. In terms of baselines, in addition to ratio matching, we further consider the approaches which train discrete EBMs with persistent contrastive divergence (Tieleman, 2008). The number of steps for MCMC per training iteration is  $\in \{5,10,25,50,100\}$ . The samplers are Gibbs and Gibbs-With-Gradient (GWG) (Grathwohl et al., 2021). Results of EBM (GWG) are obtained by running the official implementation from Grathwohl et al. (2021). Results of EBM (Gibbs) are obtained by reading from Figure 6 in Grathwohl et al. (2021).
+
+Evaluation. We evaluate the performance by computing the root-mean-squared-error (RMSE) between the learned connectivity matrix  $\hat{J}$  and the true matrix  $J$ . In addition, we compare the efficiency by reporting the running time for each iteration. To be specific, for comparing efficiency, we use the same batch size 100 for our method and baselines. The report time is the average over 100 iterations.
+
+![](images/93f561f762172d1af4434ef7091f4e1026de2e46bb693a29be061249c5740d27.jpg)

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2023/Graph Contrastive Learning for Skeleton-based Action Recognition/full.md

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+# GRAPH CONTRASTIVE LEARNING FOR SKELETON-BASED ACTION RECOGNITION
+
+Xiaohu Huang  $^{1,2*}$  Hao Zhou  $^{2\dagger}$  Jian Wang  $^{2}$  Haocheng Feng  $^{2}$  Junyu Han  $^{2}$  Errui Ding  $^{2}$  Jingdong Wang  $^{2}$  Xinggang Wang  $^{1}$  Wenyu Liu  $^{1}$  Bin Feng  $^{1\dagger}$
+
+<sup>1</sup> School of EIC, Huazhong University of Science & Technology  
+$^{2}$  Department of Computer Vision Technology (VIS), Baidu Inc., China  
+{huangxiaohu, xgwang, liuwy, fengbin}@hust.edu.cn  
+{zhouhao14, wangjian33, fenghaocheng, hanjunyu}@baidu.com  
+{dingerrui, wangjingdong}@baidu.com
+
+# ABSTRACT
+
+In the field of skeleton-based action recognition, current top-performing graph convolutional networks (GCNs) exploit intra-sequence context to construct adaptive graphs for feature aggregation. However, we argue that such context is still local since the rich cross-sequence relations have not been explicitly investigated. In this paper, we propose a graph contrastive learning framework for skeleton-based action recognition (SkeletonGCL) to explore the global context across all sequences. In specific, SkeletonGCL associates graph learning across sequences by enforcing graphs to be class-discriminative, i.e., intra-class compact and inter-class dispersed, which improves the GCN capacity to distinguish various action patterns. Besides, two memory banks are designed to enrich cross-sequence context from two complementary levels, i.e., instance and semantic levels, enabling graph contrastive learning in multiple context scales. Consequently, SkeletonGCL establishes a new training paradigm, and it can be seamlessly incorporated into current GCNs. Without loss of generality, we combine SkeletonGCL with three GCNs (2S-ACGN, CTR-GCN, and InfoGCN), and achieve consistent improvements on NTU60, NTU120, and NW-UCLA benchmarks. The source code will be available at https://github.com/OliverHxh/SkeletonGCL.
+
+# 1 INTRODUCTION
+
+Graph convolutional networks (GCNs) have been widely applied in skeleton-based action recognition since they can naturally process non-grid skeleton sequences. For GCN-based methods, how to effectively learn the graphs remains a core and challenging problem. In particular, ST-GCN (Yan et al., 2018) is a milestone work, using pre-defined graphs to extract the action patterns. However, the pre-defined graphs only enable each joint to access the fixed neighboring joints but are hard to capture long-range dependency adaptively. Therefore, a mainstream of subsequent works (Li et al., 2019; Shi et al., 2019; Zhang et al., 2020b;a; Ye et al., 2020; Chen et al., 2021b; Chi et al., 2022) take efforts to solve this issue by generating adaptive graphs. The adaptive graphs can dynamically aggregate the features within each sequence and thus show significant advantages in performance comparison.
+
+Generally, adaptive graphs are constructed by using intra-sequence context. However, such context will still be "local" when viewing the cross-sequence information as an available context. Therefore, we wonder: Is it possible to involve the cross-sequence context in graph learning? To find out the answer, in Fig. 1, we visualize the adaptive graphs learned from sequences of two easily confused classes ("point to something" and "take a selfie"). The graphs are learned by a strong GCN, i.e.,
+
+![](images/8b5be2228b3e57190cf2e57410c5224ab00d7392159e11a391a157b09e64f38d.jpg)  
+Figure 1: Graph visualization of sequences from two easily confused classes ("point to something" and "take a selfie"). The graphs are learned by CTR-GCN (Chen et al., 2021b). We take the tip of the hand that does the action as the anchor. The size of the red circles and the width of the blue lines both denote the strengths of connections between joints. For simplicity, only representative frames are visualized. (a) Three sequences from class "point to something" are correctly classified, where the graphs contain connections to the body joints. (b) Three sequences from class "take a selfie" are correctly classified, where the graphs highly emphasize the connections to the hands, while the connections to the body are suppressed. (c) A sequence from class "point to something" is misclassified as "take a selfie", whose graph resembles the graphs in (b), but is dissimilar from graphs in (a). Hence, we realize that the class-ambiguous graph representations would make negative impacts on recognition performance.
+
+![](images/eecb22f77c4bb4caa031ced4303ec47603c185a3157b54afa34aff83adad9c45.jpg)
+
+![](images/466dc3250a279410033799d394e18bb03143d26d3567062fb7dd880a9a4ca684.jpg)
+
+CTR-GCN (Chen et al., 2021b). From the visualization, we find that (1) For sequences that are correctly classified in Fig. 1 (a) and Fig. 1 (b), the learned graphs in the same class look similar, while graphs in different classes have distinct differences. (2) For a misclassified sequence in Fig. 1 (c), the learned graph resembles the graphs from the misclassified class more than those from the ground truth class. These observations remind us that graph learning in current adaptive GCNs can implicitly learn class-specific graph representations to some extent. But without explicit constraints, it leads to class-ambiguous representations in some cases, thereby affecting the GCN capacity to discriminate classes (in Tab. 9 of Sec. 4.4, we provide quantitative results to further support our hypothesis). Therefore, we speculate that if the cross-sequence semantic relations are incorporated as guidance in graph learning, the class-ambiguity issue will be alleviated and the graph representations will better express individual characteristics of actions.
+
+In recent years, contrastive learning has achieved great success in self-supervised representation learning (He et al., 2020; Chen et al., 2020; 2021a). It studies cross-sample relations in the dataset. The essence of contrastive learning is "comparing", which pulls together the feature embedding from positive pairs and pushes away the feature embedding from negative pairs.
+
+Based on the analysis above and the advances in contrastive learning, we propose a graph contrastive learning framework for skeleton-based action recognition in the fully-supervised setting, dubbed SkeletonGCL. Instead of just using the local information within each sequence, SkeletonGCL explores the cross-sequence global context to improve graph learning. The core idea is to pull together the learned graphs from the same class while pushing away the learned graphs from different classes. Since graphs can reveal the action patterns of sequences, enforcing graph consistency in the same class and inconsistency among different classes helps the model understand various motion modes. In addition, to enrich the cross-sequence context, we build memory banks to store the graphs from historical sequences. In specific, an instance-level memory bank stores the sequence-wise graphs, which hold the individual properties of each sequence. In contrast, a semantic-level memory bank stores the aggregation of graphs from each class, which contains the class-level representation. The two banks are complementary to each other, enabling us to leverage more samples. SkeletonGCL can be seamlessly combined with existing GCNs. Eventually, we combine SkeletonGCL with three GCNs (2S-AGCN (Shi et al., 2019), CTR-GCN (Chen et al., 2021b), and InfoGCN (Chi et al., 2022)), and conduct experiments on three popular datasets (NTU60 (Shahroudy et al., 2016), NTU120 (Liu et al., 2019) and NW-UCLA (Wang et al., 2014)). SkeletonGCL achieves consistent improvements with these models using different testing protocols (single-modal or multi-modal) on all three datasets, which widely demonstrates the effectiveness of our design. Notably, SkeletonGCL only introduces a small amount of training consumption but has no impact at the test stage.
+
+Though there exist some works that apply contrastive learning in skeleton-based action recognition (Li et al., 2021; Guo et al., 2022; Mao et al., 2022), our method differs from them as follows: (1) The previous methods took pooled feature vectors to conduct contrastive learning as in (He et al., 2020; Chen et al., 2020), where the structural properties in skeletons are lost. In contrast, SkeletonGCL uses graphs to contrast, which maintains the structure details of skeletons and offers high-order connection information between joints. (2) The previous methods used memory banks to store instance-level representations only. Differently, our memory banks store both instance-level and semantic-level representations, allowing us to leverage context from individual sequences and class-specific aggregations, which are complementary to each other. (3) The previous methods were used in the pre-training stage, while SkeletonGCL is incorporated into the fully-supervised setting without extra pre-training cost.
+
+Summarily, the contribution of this paper can be concluded as follows:
+
+- We present a new perspective for graph learning of GCN models in skeleton-based action recognition. In specific, we propose to make use of the cross-sequence context to guide graph learning, whose goal is to enforce graphs to be intra-class compact and inter-class dispersed.  
+- Motivated by the advances in contrastive learning, we smoothly combine the ideas of contrastive learning and cross-sequence graph learning together, then propose a new training paradigm for skeleton-based action recognition, called SkletonGCL. SkeletonGCL incorporates an instance-level and a semantic-level memory bank to enrich the cross-sequence context comprehensively. Besides, it can be seamlessly incorporated into current GCNs.  
+- SkeletonGCL achieves consistent improvements combined with three GCNs (2S-AGCN, CTR-GCN, and InfoGCN) on three popular benchmarks (NTU60, NTU120, NW-UCLA) using both single-modal and multi-modal testing protocols. In addition, SkeletonGCL is training-efficient and has no impact at the test stage.
+
+# 2 RELATED WORKS
+
+# 2.1 SKELETON-BASED ACTION RECOGNITION
+
+Skeleton-based action recognition is to classify actions from sequences of estimated key points. The early deep-learning methods applied convolution neural networks (CNNs) (Chéron et al., 2015; Liu et al., 2017b) or recurrent neural networks (RNNs) (Du et al., 2015; Lev et al., 2016; Wang & Wang, 2017; Liu et al., 2017a) to model the skeletons, but they could not explicitly explore the topological structure of skeletons, thus the performances were limited. Recently, PoseC3D (Duan et al., 2022) revisited the CNN-based method by stacking the heatmaps as 3D volumes, which maintained the spatial-temporal properties of skeletons and obtained marginal performance improvements. In the past few years, the mainstream works in skeleton-based action recognition were GCN models. ST-GCN (Yan et al., 2018) was the first work that adopted GCN as the feature extractor, which heuristically designed fixed graphs to model the skeletons. The follow-up methods proposed spatial-temporal graphs (Liu et al., 2020), multi-scale graph convolutions (Chen et al., 2021c), channel-decoupled graphs (Chen et al., 2021b; Cheng et al., 2020a) and adaptive graphs (Li et al., 2019; Shi et al., 2019; Ye et al., 2020; Zhang et al., 2020b; Chen et al., 2021b; Chi et al., 2022) to improve the capacity of GCNs. Tracking the development of GCN-based methods, we find that graph learning has always been a core problem and now the adaptive GCNs are leading since they can model the intrinsic topology of skeletons.
+
+However, current adaptive GCNs generated the graphs based on the local context within each sequence, where the cross-sequence relations have been neglected. In contrast, we propose to explore the cross-sequence global context to shape graph representations. In this way, the learned graphs can not only describe the individual characteristics within each sequence but also emphasize the similarity and dissimilarity of motion patterns across sequences.
+
+# 2.2 CONTRASTIVE LEARNING
+
+In recent years, numerous representation learning methods (Wu et al., 2018; Oord et al., 2018; He et al., 2020; Chen et al., 2020; Wang et al., 2021) with contrastive learning have emerged, especially
+
+in self-supervised representation learning. The key idea is to pull together the positive pairs and push away the negative pairs in the feature space. Generally, the features are vectors obtained from feature extractors followed by a pooling layer. A standard approach to obtaining the positive pairs is augmenting an original sample into two different views. The negative samples are selected randomly or using hard mining strategies (Khosla et al., 2020; Robinson et al., 2020; Kalantidis et al., 2020). To increase the capacity of negative samples, the memory bank mechanism was devised in (He et al., 2020; Misra & Maaten, 2020) to store more negative instances. By contrasting positive pairs against negative pairs, the model can learn to focus on semantic representations.
+
+In the field of skeleton-based action recognition, prior works (Li et al., 2021; Mao et al., 2022; Guo et al., 2022) proposed to apply contrastive learning in the pre-training stage by roughly following the frameworks mentioned above. CrossCLR (Li et al., 2021) mined positive pairs in the data space and explored the cross-modal distribution relationships. Further, CMD (Mao et al., 2022) transferred the cross-modal knowledge in a distillation manner. And AimCLR (Guo et al., 2022) used extreme augmentations to improve the representation universality.
+
+Compared with the above methods, we use graph representations to contrast instead of using pooled feature vectors. Meanwhile, we establish two different memory banks at complementary levels, i.e., instance and semantic levels, to enrich the context scales. Besides, the proposed method is used with GCNs under the fully-supervised setting, which requires no pre-training procedure.
+
+# 3 METHOD
+
+# 3.1 PRELIMINARY
+
+We denote a human skeleton as a vertex set  $\mathcal{V} = \{v_{1}, v_{2}, \dots, v_{N}\}$ , where  $N$  denotes the number of vertices. For each vertex  $v_{i}$ , the feature dimension is set as  $C$ . Hence, a skeleton sequence with  $T$  frames can be denoted as  $\mathbf{X} \in \mathbb{R}^{T \times N \times C}$ . Graph topology is used to represent the correlations between joints, formulated as  $\mathbf{g}$ .
+
+GCNs in Skeleton-Based Action Recognition. Generally, GCN models alternatively apply graph convolutions and temporal convolutions to extract the spatial configuration and motion pattern of skeletons, respectively. The graph  $\mathbf{g}$  is vital for graph convolutions since it determines the message passing among joints. In current adaptive GCNs,  $\mathbf{g}$  is learned within each sequence and has different sizes, e.g.,  $\mathbf{g} \in \mathbb{R}^{K_S \times N \times N}$  in 2S-AGCN (Shi et al., 2019) and  $\mathbf{g} \in \mathbb{R}^{K_S \times C \times N \times N}$  in CTR-GCN (Chen et al., 2021b). The  $K_S$  denotes the number of sub-graphs, normally set as 3. In general, the graph convolution is defined as:
+
+$$
+\mathbf {X} _ {\mathbf {S}} = \sum_ {k = 1} ^ {K _ {S}} \mathbf {g} ^ {k} \mathbf {X} \mathbf {W} _ {\mathbf {S}} ^ {\mathbf {k}}, \tag {1}
+$$
+
+where  $\mathbf{X}_{\mathbf{S}} \in \mathbb{R}^{T \times N \times C'}$  denotes the spatial extracted feature with  $C'$  channels, and  $\mathbf{W}_{\mathbf{S}} \in \mathbb{R}^{K_S \times C \times C'}$  denotes the spatial feature transformation filters. Next, temporal convolutions are applied on  $\mathbf{X}_{\mathbf{S}}$ , producing motion extracted feature  $\mathbf{X}_{\mathbf{T}} \in \mathbb{R}^{T \times N \times C'}$ . After stacking layers of graph convolutions and temporal convolutions, a global average pooling (GAP) layer summarizes the global features, then a classification head (fully-connected layer) followed by a Softmax activation function is applied to obtain the class prediction  $\hat{\mathbf{y}} \in \mathbb{R}^{C_k}$ , where  $C_k$  denotes the number of classes. Finally, a cross-entropy loss  $\mathcal{L}_{\mathrm{CE}}$  supervises the class prediction with the ground truth label  $\mathbf{y}$  as follows:
+
+$$
+\mathcal {L} _ {\mathrm {C E}} = - \sum_ {i} \mathbf {y} _ {\mathrm {i}} \log \hat {\mathbf {y}} _ {\mathrm {i}} \tag {2}
+$$
+
+Self-Supervised Contrastive Learning. In the context of self-supervised contrastive learning, each input sample is processed by data augmentations to produce a positive pair:  $I$  and  $I^{+}$ . Through a feature extraction network,  $I$  and  $I^{+}$  are transformed into feature vectors  $\mathbf{f} \in \mathbb{R}^{D}$  and  $\mathbf{f}^{+} \in \mathbb{R}^{D}$ . As for the negative samples, they are selected from the dataset excluding  $I$  and  $I^{+}$ , represented as a negative set  $\mathcal{N}^{-}$ . Each negative in  $\mathcal{N}^{-}$  is denoted as  $\mathbf{f}^{-} \in \mathbb{R}^{D}$ . The similarity between two feature vectors is calculated as  $sim(\mathbf{f}^{+}, \mathbf{f}^{-}) = \frac{\mathbf{f}^{+}\mathbf{f}^{-}}{\|\mathbf{f}^{+}\|_{\|\mathbf{f}^{-}\|}}$ . InfoNCE (Gutmann & Hyvarinen, 2010; Oord
+
+![](images/7cd03dd7401716da5192f2bac84fcf763791d21f732a146855dd7507beee56fb.jpg)  
+Figure 2: Overview of SkeletonGCL. An input skeleton sequence  $I$  is fed into a GCN encoder, producing a feature vector  $\mathbf{f}$  for classification and a learned graph  $\mathbf{g}$  for graph contrastive learning. The graph  $\mathbf{g}$  is embedded into a vector by a projection head. And two memory banks are built to store the embedded graphs. From the memory banks, we sample the positives and negatives according to the labels, then perform contrastive loss. The memory banks are only used in the training stage but will be removed during the testing stage.
+
+et al., 2018) is widely adopted for contrastive learning, which is formulated as:
+
+$$
+\mathcal {L} _ {\mathrm {N C E}} = - \log \frac {\operatorname {s i m} \left(\mathbf {f} , \mathbf {f} ^ {+}\right) / \tau}{\operatorname {s i m} \left(\mathbf {f} , \mathbf {f} ^ {+}\right) / \tau + \sum_ {\mathbf {f} ^ {-} \in \mathcal {N} ^ {-}} \operatorname {s i m} \left(\mathbf {f} , \mathbf {f} ^ {-}\right) / \tau}, \tag {3}
+$$
+
+where temperature  $\tau > 0$  is a hyper-parameter.
+
+# 3.2 GRAPH CONTRASTIVE LEARNING
+
+The proposed SkeletonGCL is illustrated in Fig. 2. The framework consists of two branches, where the classification branch is parallel to the graph contrast branch. Taking a skeleton sequence  $I$  as input, the GCN encoder outputs a feature vector  $\mathbf{f}$  for classification and a graph  $\mathbf{g}$  for graph contrast.
+
+Graph Projection Head. In order to contrast the graphs in a common feature space, we embed the graphs into vectors by a graph projection head. The projection heads for different GCNs are similar (see App. 6.1 for details). In Fig. 2, taking the graph  $\mathbf{g} \in \mathbb{R}^{K_S \times C \times N \times N}$  learned in CTR-GCN (Chen et al., 2021b) as an example, we first squeeze  $\mathbf{g}$  along the channel dimension by an average pooling layer into  $\overline{\mathbf{g}} \in \mathbb{R}^{K_S \times N \times N}$ . Then, we flatten graph  $\overline{\mathbf{g}}$  into a 1D vector as  $\mathbf{g}' \in \mathbb{R}^{K_S N^2}$  and project  $\mathbf{g}'$  into a vector  $\mathbf{v} \in \mathbb{R}^{C_g}$  by an FC layer  $\mathbf{W_G} \in \mathbb{R}^{K_S N^2 \times C_g}$ . Since different channels in  $\mathbf{W_G}$  are specific to different vertices in the graph, the graph projection is vertex-aware and thus can encode the structures of skeletons. Afterward, we update two memory banks with  $\mathbf{v}$ . The memory banks are illustrated in Fig. 2, and detailed next.
+
+Memory Bank. To enrich the cross-sequence context, we build memory banks to store the cross-batch graphs. In specific, two memory banks are constructed, i.e., an instance-level memory bank  $\mathcal{M}_{\mathrm{Ins}} \in \mathbb{R}^{C_k \times P \times C_g}$  and a semantic-level memory bank  $\mathcal{M}_{\mathrm{Sem}} \in \mathbb{R}^{C_k \times C_g}$ .  $P$  denotes the number of instances stored for each class in  $\mathcal{M}_{\mathrm{Ins}}$ . Particularly, each element in  $\mathcal{M}_{\mathrm{Ins}}$  denotes a graph instance from a class. In contrast, each element in  $\mathcal{M}_{\mathrm{Sem}}$  denotes the graph aggregation of a class. Therefore, the two memory banks are on complementary levels, where the instance-level memory bank emphasizes the instance discrimination of each sequence, while the semantic-level memory bank covers the class properties across sequences.
+
+We update  $\mathcal{M}_{\mathrm{Ins}}$  in a first-in-first-out manner, which maintains the number of instances for each class as  $P$ . As for  $\mathcal{M}_{\mathrm{Sem}}$ , we use a momentum update strategy, which integrates the graphs of the same class from the current timestamp and all previous timestamps, regarded as a long-term representation. The momentum update is defined as follows:
+
+$$
+\mathbf {m} _ {\text {s e m}} ^ {c _ {*}} \leftarrow \alpha \mathbf {m} _ {\text {s e m}} ^ {c _ {*}} + (1 - \alpha) \mathbf {v}, \tag {4}
+$$
+
+where  $\mathbf{m}_{\mathrm{sem}}^{c_*}$  is the representation for class  $c_*, c_*$  is the class label for the input  $I$  and  $0 < \alpha < 1$  is a hyper-parameter.
+
+Loss. To achieve the graph contrast, we sample positives and negatives from the memory banks  $\mathcal{M}_{\mathrm{Ins}}$  and  $\mathcal{M}_{\mathrm{Sem}}$ . For  $\mathcal{M}_{\mathrm{Ins}}$ , vector  $\mathbf{v}$  is set as the anchor, hence samples in the positive set  $\mathcal{N}_{\mathrm{Ins}}^{+}$  are with label  $c_{*}$ , and samples in the negative set  $\mathcal{N}_{\mathrm{Ins}}^{-}$  are with different labels. Consequently, the InfoNCE loss in Eq. 3 can be rewritten as:
+
+$$
+\mathcal {L} _ {\mathrm {N C E}} ^ {\text {I n s}} = - \sum_ {\mathbf {v} ^ {+} \in \mathcal {N} _ {\text {I n s}} ^ {+}} \log \frac {\operatorname {s i m} (\mathbf {v} , \mathbf {v} ^ {+}) / \tau}{\operatorname {s i m} (\mathbf {v} , \mathbf {v} ^ {+}) / \tau + \sum_ {\mathbf {v} ^ {-} \in \mathcal {N} _ {\text {I n s}} ^ {-}} \operatorname {s i m} (\mathbf {v} , \mathbf {v} ^ {-}) / \tau}, \tag {5}
+$$
+
+$$
+\mathcal {L} _ {\mathrm {N C E}} ^ {\text {S e m}} = - \sum_ {\mathbf {v} ^ {+} \in \mathcal {N} _ {\text {S e m}} ^ {+}} \log \frac {\operatorname {s i m} \left(\mathbf {v} , \mathbf {v} ^ {+}\right) / \tau}{\operatorname {s i m} \left(\mathbf {v} , \mathbf {v} ^ {+}\right) / \tau + \sum_ {\mathbf {v} ^ {-} \in \mathcal {N} _ {\text {S e m}} ^ {-}} \operatorname {s i m} \left(\mathbf {v} , \mathbf {v} ^ {-}\right) / \tau}. \tag {6}
+$$
+
+$\mathcal{L}_{\mathrm{NCE}}^{\mathrm{Ins}}$  leverages multiple positives compared with Eq. 3 by using label information, which mines more semantic-related samples. Similarly, we can define the InfoNCE loss  $\mathcal{L}_{\mathrm{NCE}}^{\mathrm{Sem}}$ , which is specific for the memory bank  $\mathcal{M}_{\mathrm{Sem}}$ . Summarily, the overall contrastive loss is written as follows:
+
+$$
+\mathcal {L} _ {\mathrm {N C E}} = \mathcal {L} _ {\mathrm {N C E}} ^ {\text {I n s}} + \mathcal {L} _ {\mathrm {N C E}} ^ {\text {S e m}}. \tag {7}
+$$
+
+And the overall loss function is defined as follows:
+
+$$
+\mathcal {L} = \mathcal {L} _ {\mathrm {N C E}} + \mathcal {L} _ {\mathrm {C E}}. \tag {8}
+$$
+
+**Hard Sampling.** As the training continues, most samples become too easy, which contribute less to the training. Therefore, methods in (Tabassum et al., 2022; Robinson et al., 2020; Kalantidis et al., 2020; Wang et al., 2021) are proposed to use hard mining strategies to focus on informative samples. In this paper, considering the massive number of instances in  $\mathcal{M}_{\mathrm{Ins}}$ , contrasting with all these instances naturally leads to redundancy and hinders the training. To alleviate this issue, we propose to mine hard examples in  $\mathcal{M}_{\mathrm{Ins}}$ . Specifically, we take the similarity calculation  $sim(\mathbf{v}, \mathbf{v}^{\prime})$  as a criterion to evaluate hardness. Harder positives are with lower similarities, and harder negatives are with higher similarities. In total, for  $\mathcal{M}_{\mathrm{Ins}}$ , we select  $K_{\mathrm{H}}^{+}$ hardest positive examples,  $K_{\mathrm{H}}^{-}$ hardest negative examples, and  $K_{\mathrm{R}}^{-}$ random negative examples.
+
+# 4 EXPERIMENTS
+
+# 4.1 DATASETS
+
+NTU RGB+D. NTU RGB+D (NTU60) (Shahroudy et al., 2016) is a large-scale skeleton-based action recognition dataset, which contains 60 action classes and 56,880 sequences. Each sequence is annotated as skeletons with 25 joints. All the sequences are performed by 40 subjects and filmed by 3 Kinect cameras from three different views. Generally, two protocols are used to evaluate the performances: (1) cross-subject (X-Sub): train data are performed by 20 subjects, and test data are performed by other 20 subjects. (2) cross-view (X-View): train data from cameras 2 and 3, and test data from camera 1.
+
+NTU RGB+D 120. NTU RGB+D 120 (NTU120) (Liu et al., 2019) is an extension of NTU RGB+D dataset, which newly includes 57,367 skeletons of 60 extra classes. All the sequences are performed by 106 subjects and filmed by three cameras from three different views. In addition, NTU RGB+D 120 has 32 setups, where each denotes a unique location. Generally, two protocols are used to evaluate the performances: (1) cross-subject (X-Sub): train data are performed by 53 subjects, and test data are performed by other 53 subjects. (2) cross-setup (X-Set): train data are samples with even setup IDs, and test data are samples with odd setup IDs.
+
+Northwestern-UCLA. Northwestern-UCLA (NW-UCLA) dataset (Wang et al., 2014) contains 1494 sequences from 10 action classes. Each sequence is annotated as skeletons with 20 joints. All sequences are performed by 10 subjects and filmed by three Kinect cameras from different views. We follow the official evaluation protocol: train data are captured by the first two cameras, and test data are captured by the third camera.
+
+# 4.2 IMPLEMENTATION DETAILS
+
+To thoroughly validate SkeletonGCL, we take three GCNs (2S-AGCN (Shi et al., 2019), CTR-GCN (Chen et al., 2021b), and InfoGCN (Chi et al., 2022)) as baseline models. For CTR-GCN and
+
+Table 1: Top-1 accuracy comparison (\%) with the state-of-the-art methods on NTU 60 and NTU120 datasets. The numbers in gray indicate the results reported in their papers. * indicates that we retrain the models using their officially released code. Particularly, 2S-AGCN is retrained using a stronger train recipe from CTR-GCN.  
+
+<table><tr><td>Dataset</td><td colspan="8">NTU 60</td><td colspan="8">NTU 120</td></tr><tr><td>Setting</td><td colspan="4">X-Sub</td><td colspan="4">X-View</td><td colspan="4">X-Sub</td><td colspan="4">X-Set</td></tr><tr><td>Method/Modality</td><td>J</td><td>B</td><td>J+B</td><td>4S</td><td>J</td><td>B</td><td>J+B</td><td>4S</td><td>J</td><td>B</td><td>J+B</td><td>4S</td><td>J</td><td>B</td><td>J+B</td><td>4S</td></tr><tr><td>SGCN (Zhang et al., 2020a)</td><td>-</td><td>-</td><td>89.0</td><td>-</td><td>-</td><td>-</td><td>94.5</td><td>-</td><td>-</td><td>-</td><td>79.2</td><td>-</td><td>-</td><td>-</td><td>81.5</td><td>-</td></tr><tr><td>ST-TR-GCN (Plizzari et al., 2021)</td><td>89.2</td><td>-</td><td>90.3</td><td>-</td><td>95.8</td><td>-</td><td>96.3</td><td>-</td><td>82.7</td><td>-</td><td>85.1</td><td>-</td><td>85.0</td><td>-</td><td>87.1</td><td>-</td></tr><tr><td>Shift-GCN (Cheng et al., 2020b)</td><td>87.8</td><td>-</td><td>89.7</td><td>90.7</td><td>95.1</td><td>-</td><td>96.0</td><td>96.5</td><td>80.9</td><td>-</td><td>85.3</td><td>85.9</td><td>83.2</td><td>-</td><td>86.6</td><td>87.6</td></tr><tr><td>DC-GCN+ADG (Cheng et al., 2020a)</td><td>-</td><td>-</td><td>90.8</td><td>-</td><td>-</td><td>-</td><td>96.6</td><td>-</td><td>-</td><td>-</td><td>86.5</td><td>-</td><td>-</td><td>-</td><td>88.1</td><td>-</td></tr><tr><td>Dynamic GCN (Ye et al., 2020)</td><td>-</td><td>-</td><td>-</td><td>91.5</td><td>-</td><td>-</td><td>-</td><td>96.0</td><td>-</td><td>-</td><td>-</td><td>87.3</td><td>-</td><td>-</td><td>-</td><td>88.6</td></tr><tr><td>MS-G3D (Liu et al., 2020)</td><td>89.4</td><td>90.1</td><td>91.5</td><td>-</td><td>95.0</td><td>95.3</td><td>96.2</td><td>-</td><td>-</td><td>-</td><td>86.9</td><td>-</td><td>-</td><td>-</td><td>88.4</td><td>-</td></tr><tr><td>MST-GCN (Chen et al., 2021c)</td><td>89.0</td><td>89.5</td><td>91.1</td><td>91.5</td><td>95.1</td><td>95.2</td><td>96.4</td><td>96.6</td><td>82.8</td><td>84.8</td><td>87.0</td><td>87.5</td><td>84.5</td><td>86.3</td><td>88.3</td><td>88.8</td></tr><tr><td>2S-AGCN (Shi et al., 2019)</td><td>-</td><td>-</td><td>88.5</td><td>-</td><td>93.7</td><td>93.2</td><td>95.1</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>2S-AGCN* (Shi et al., 2019)</td><td>88.9</td><td>89.2</td><td>91.0</td><td>91.5</td><td>94.5</td><td>94.1</td><td>95.7</td><td>95.9</td><td>84.0</td><td>85.1</td><td>87.8</td><td>88.2</td><td>85.3</td><td>86.3</td><td>89.0</td><td>89.6</td></tr><tr><td>2S-AGCN* w/SkeletonGCL</td><td>89.9</td><td>90.0</td><td>91.6</td><td>92.2</td><td>95.0</td><td>94.4</td><td>96.1</td><td>96.4</td><td>84.7</td><td>86.0</td><td>88.4</td><td>88.7</td><td>86.1</td><td>86.8</td><td>89.7</td><td>90.2</td></tr><tr><td>CTR-GCN (Chen et al., 2021b)</td><td>-</td><td>-</td><td>-</td><td>92.4</td><td>-</td><td>-</td><td>-</td><td>96.8</td><td>-</td><td>85.7</td><td>88.7</td><td>88.9</td><td>-</td><td>87.5</td><td>90.1</td><td>90.6</td></tr><tr><td>CTR-GCN* (Chen et al., 2021b)</td><td>89.8</td><td>90.2</td><td>92.0</td><td>92.4</td><td>94.8</td><td>94.8</td><td>96.3</td><td>96.8</td><td>84.9</td><td>85.7</td><td>88.7</td><td>88.9</td><td>86.7</td><td>87.5</td><td>90.1</td><td>90.5</td></tr><tr><td>CTR-GCN* w/SkeletonGCL</td><td>90.8</td><td>91.1</td><td>92.6</td><td>93.1</td><td>95.3</td><td>95.4</td><td>96.6</td><td>97.0</td><td>85.6</td><td>86.9</td><td>89.2</td><td>89.5</td><td>87.3</td><td>88.2</td><td>90.5</td><td>91.0</td></tr><tr><td>InfoGCN (Chi et al., 2022)</td><td>89.8</td><td>90.6</td><td>91.6</td><td>92.7</td><td>95.2</td><td>95.5</td><td>96.5</td><td>96.9</td><td>85.1</td><td>87.3</td><td>88.5</td><td>89.4</td><td>86.3</td><td>88.5</td><td>89.7</td><td>90.7</td></tr><tr><td>InfoGCN* (Chi et al., 2022)</td><td>89.4</td><td>90.6</td><td>91.3</td><td>92.3</td><td>95.2</td><td>95.4</td><td>96.2</td><td>96.7</td><td>84.2</td><td>86.9</td><td>88.2</td><td>89.2</td><td>86.3</td><td>88.5</td><td>89.4</td><td>90.7</td></tr><tr><td>InfoGCN* w/SkeletonGCL</td><td>90.1</td><td>91.0</td><td>91.9</td><td>92.8</td><td>95.5</td><td>95.7</td><td>96.6</td><td>97.1</td><td>85.2</td><td>87.4</td><td>88.8</td><td>89.8</td><td>87.2</td><td>88.7</td><td>90.0</td><td>91.2</td></tr></table>
+
+InfoGCN, we follow their training recipes. Particularly, for 2S-AGCN, since its training recipe is out of date, we borrow the training recipe from CTR-GCN, which effectively improves its baseline performance.  $P$ , the number of stored instances for each class in  $\mathcal{M}_{\mathrm{Ins}}$ , is set as 684 on NTU60 and NTU120, and 342 on NW-UCLA. The dimension of graph vector  $C_g$  is set to 256. For all datasets, the number of sampling examples  $K_{\mathrm{H}}^{+}$ ,  $K_{\mathrm{H}}^{-}$ , and  $K_{\mathrm{R}}^{-}$  are set as 128, 512, and 512, respectively. For different models used in different modalities, we experiment with temperature  $\tau$  of 0.5, 0.8, 1.0, and 1.5, and choose the best one. The hyper-parameter  $\alpha$  for momentum updating is set as 0.85. Besides, we fix the random seed to ensure experiment reproducibility. All experiments are conducted using a single NVIDIA V100 GPU.
+
+# 4.3 COMPARED WITH THE STATE-OF-THE-ART
+
+In this section, we combine our method with three GCNs, and compare them with the state-of-the-art (SoTA) methods. In Tab. 1 and Tab. 2, we list current SoTA methods in skeleton-based action recognition except PoseC3D (Duan et al., 2022). PoseC3D is a promising CNN-based method, but it uses non-official skeleton data and applies a multi-crop test protocol (GCN methods typically use one crop), which are unfair for comparison here. In evaluation, four modalities are used: "joint stream"  $(J)$  denotes the joint coordinates, "bone stream"  $(B)$  denotes the coordinate difference between spatially connected joints, "joint motion"  $(J - M)$  denotes the coordinate difference between temporally adjacent frames, and "bone motion"  $(B - M)$  denotes the bone difference between temporally adjacent frames. The 4-stream ensemble  $(4S)$  denotes using the four modalities together. Following the widely-adopted protocol, we evaluate models using  $J$ ,  $B$ ,  $J + B$ , and  $4S$  modalities.
+
+NTU60 and NTU120. Tab. 1 lists the results on NTU60 and NTU120. From the results, we find that: (1) Combined with SkeletonGCL, all three baseline models achieve solid improvements on these two benchmarks over different settings and modalities. Taking the  $J$  modality on NTU60 X-Sub as an example, 2S-AGCN improves by  $1.0\%$  ( $88.9\%$  to  $89.9\%$ ), CTR-GCN improves by  $1.0\%$  ( $89.8\%$  to  $90.8\%$ ), and InfoGCN improves by  $0.7\%$  ( $89.4\%$  to  $90.1\%$ ). Considering NTU60 is an extensively-benchmarked dataset, such improvements are quite hard. (2) With SkeletonGCL, CTR-GCN and InfoGCN can set new SoTA performance.
+
+NW-UCLA. Tab. 2 lists the results on NW-UCLA. SkeletonGCL can still achieve consistent improvements based on the three models. And new state-of-the-art performances are achieved by combining SkeletonGCL with CTR-GCN and InfoGCN.
+
+Table 2: Top-1 accuracy Comparison (%) with the state-of-the-art methods on the NW-UCLA dataset. Numbers in gray denote the results reported in their papers. * indicates that we retrain the models using their officially released codes. Particularly, 2S-AGCN is retrained using a stronger train recipe from CTR-GCN.  
+
+<table><tr><td>Dataset</td><td colspan="4">NW-UCLA</td></tr><tr><td>Method/Modality</td><td>J</td><td>B</td><td>J+B</td><td>4S</td></tr><tr><td>AGC-LSTM (Si et al., 2019)</td><td>93.3</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DC-GCN+ADG (Cheng et al., 2020a)</td><td>-</td><td>-</td><td>95.3</td><td>-</td></tr><tr><td>Shift-GCN (Cheng et al., 2020b)</td><td>92.5</td><td>-</td><td>94.2</td><td>94.6</td></tr><tr><td>2S-AGCN (Shi et al., 2019)</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>2S-AGCN* (Shi et al., 2019)</td><td>92.0</td><td>92.2</td><td>95.0</td><td>95.5</td></tr><tr><td>2S-AGCN* w/SkeletonGCL</td><td>92.6</td><td>93.0</td><td>95.7</td><td>96.3</td></tr><tr><td>CTR-GCN (Chen et al., 2021b)</td><td>-</td><td>-</td><td>-</td><td>96.5</td></tr><tr><td>CTR-GCN* (Chen et al., 2021b)</td><td>94.6</td><td>91.8</td><td>94.2</td><td>96.5</td></tr><tr><td>CTR-GCN* w/SkeletonGCL</td><td>95.1</td><td>95.0</td><td>95.9</td><td>96.8</td></tr><tr><td>InfoGCN (Chi et al., 2022)</td><td>94.0</td><td>95.3</td><td>96.3</td><td>96.6</td></tr><tr><td>InfoGCN* (Chi et al., 2022)</td><td>93.8</td><td>94.2</td><td>95.5</td><td>96.1</td></tr><tr><td>InfoGCN* w/SkeletonGCL</td><td>94.8</td><td>94.6</td><td>96.1</td><td>96.8</td></tr></table>
+
+# 4.4 DIAGNOSTIC EXPERIMENTS
+
+In this section, we conduct diagnostic experiments to verify the design of SkeletonGCL. Otherwise stated, we use CTR-GCN as the GCN encoder to perform the experiments on the NTU60 dataset under the X-Sub setting using the joint modality  $(J)$ . See App. 6.2 for more diagnostic experiments.
+
+Table 3: Comparison of intrabatch and inter-batch contrast.  
+
+<table><tr><td>Model (CTR-GCN)</td><td>Acc (%)</td></tr><tr><td>Baseline (w/o contrast)</td><td>89.8</td></tr><tr><td>Intra-batch Contrast (No Bank)</td><td>90.2</td></tr><tr><td>Inter-batch Contrast</td><td>90.8</td></tr></table>
+
+Table 6: Impact of sampling strategies. R: Random; H: Hard.  
+
+<table><tr><td colspan="2">Sampling</td><td rowspan="2">Acc(%)</td></tr><tr><td>Positive</td><td>Negative</td></tr><tr><td rowspan="3">R</td><td>R</td><td>90.1</td></tr><tr><td>H</td><td>90.2</td></tr><tr><td>R+H</td><td>90.5</td></tr><tr><td rowspan="3">H</td><td>R</td><td>90.4</td></tr><tr><td>H</td><td>90.6</td></tr><tr><td>R+H</td><td>90.8</td></tr><tr><td>R + H</td><td>R + H</td><td>90.3</td></tr></table>
+
+Table 9: Graph Distance (dis.) comparison using Euclidean distance  $(10^{-5})$  
+Table 4: Comparison of feature and graph contrast.  
+
+<table><tr><td colspan="2">Model (CTR-GCN) | Acc (%)</td></tr><tr><td>Baseline</td><td>89.8</td></tr><tr><td>Feature Contrast</td><td>90.2</td></tr><tr><td>Graph contrast</td><td>90.8</td></tr></table>
+
+Table 7: Comparison with using triplet loss.  
+
+<table><tr><td colspan="2">Model (CTR-GCN) | Acc (%)</td></tr><tr><td>Baseline</td><td>89.8</td></tr><tr><td>Triplet loss</td><td>90.7</td></tr><tr><td>InfoNCE loss</td><td>90.8</td></tr></table>
+
+Table 5: Impact of memory banks.  
+
+<table><tr><td>Model (CTR-GCN)</td><td>Acc (%)</td></tr><tr><td>Baseline</td><td>89.8</td></tr><tr><td>Instance Memory</td><td>90.3</td></tr><tr><td>Semantic Memory</td><td>90.2</td></tr><tr><td>Instance + Semantic</td><td>90.8</td></tr></table>
+
+Table 8: Training consumption on NTU60.  
+
+<table><tr><td>Model</td><td>Time (hours)</td></tr><tr><td>2S-AGCN</td><td>3.4</td></tr><tr><td>2S-AGCN w/ours</td><td>3.5 ↑ 2.9%</td></tr><tr><td>CTR-GCN</td><td>11.4</td></tr><tr><td>CTR-GCN w/ours</td><td>11.7 ↑ 2.6%</td></tr><tr><td>InfoGCN</td><td>4.3</td></tr><tr><td>InfoGCN w/ours</td><td>4.6 ↑ 7.0%</td></tr></table>
+
+Table 10: Performance  $(\%)$  of samples with different graph distance ranks to the correct class.  
+
+<table><tr><td>Sample</td><td>Average dis. to all classes</td><td>Dis. to correct class</td><td>Dis. to misclassified class</td></tr><tr><td>Incorrectly-Classified Samples</td><td>1.70</td><td>0.74</td><td>0.68</td></tr><tr><td>Correctly-Classified Samples</td><td>2.20</td><td>0.47</td><td>-</td></tr></table>
+
+<table><tr><td></td><td>CTR</td><td>CTR w/Ours</td><td>+/-</td></tr><tr><td>rank1</td><td>97.2</td><td>97.4</td><td>+0.2</td></tr><tr><td>rank2-5</td><td>92.1</td><td>92.9</td><td>+0.8</td></tr><tr><td>rank6-10</td><td>88.3</td><td>89.3</td><td>+1.0</td></tr><tr><td>rank11-20</td><td>84.9</td><td>86.3</td><td>+1.4</td></tr><tr><td>rank21-40</td><td>83.2</td><td>84.8</td><td>+1.6</td></tr><tr><td>rank41-60</td><td>80.8</td><td>82.9</td><td>+2.1</td></tr></table>
+
+Intra-batch vs. Inter-batch Graph Contrast. In Tab. 3, the effectiveness of introducing cross-sequence context is investigated. We find that only contrasting the graphs within one batch can bring improvement with  $0.4\%$  (89.8% to 90.2%), which owes to the cross-sequence relation mining. And further exploring the inter-batch relations can bring more improvements to  $1.0\%$  (89.8% to 90.8%), which explains that different batches provide richer context than a single batch.
+
+Graph Contrast vs. Feature Contrast. In Tab. 4, the comparison of using features  $f$  to contrast and using graphs  $g$  to contrast is investigated. We find that feature contrast can improve the performance on the baseline with  $0.4\%$  (89.8% to 90.2%). But graph contrast can obviously outperform it by  $0.6\%$  (90.2% to 90.8%). The results suggest that, due to the high-order structural information in graphs, graph contrast can better benefit graph convolution learning in GCNs.
+
+Memory Banks. In Tab. 5, the effectiveness of instance-level and semantic-level memory banks is investigated. We find that both memory banks benefit the recognition, and using them together achieves much higher performance, which proves their complementary properties.
+
+Sampling Strategy. In Tab. 6, we compare different sampling strategies for SkeletonGCL. We find that selecting hard positive/negative examples can generally improve recognition. And also random negative samples are meaningful to recognition, which allows the contrastive loss to involve more negative samples.
+
+InfoNCE Loss vs. Triplet Loss. In Tab. 7, we compare the performance of using another popular metric learning loss, i.e., triplet loss (Schroff et al., 2015). We find that using triplet loss can achieve similar performance compared to InfoNCE loss. The results indicate the generality of our idea that it does not depend on a certain loss but can boost the performance using different losses.
+
+Traning Consumption. In Tab. 8, we report the training consumption on NTU60. With our method, the training time only slightly increases with different baseline models, ranging from  $2.6\%$  to  $7.0\%$ , which proves the efficiency of the design.
+
+Quantitative Results about Graph Similarities. As shown in Tab. 9, we statistically calculate the graph distances between each sample and all classes (detailed in App. 6.4). For incorrectly-classified samples, we find that: (1) The graph distance to the misclassified class (0.68) is much lower than the average distance (1.70) to all classes. (2) The graph distance to the misclassified class (0.68) is indeed slightly lower than the distance to the correct class (0.74), which explains that not learning class-specific graphs could truly degrade recognition performance. In addition, for correctly-classified samples, we notice that: (1) The average graph distance (2.20) is higher than that for the misclassified ones (1.70), which indicates that the inter-class graph representations are more dispersed for the correctly-classified samples. (2) The distance to the correct class (0.47) is lower than that for the misclassified ones (0.74), which reveals that the intra-class representations are more compact for the correctly classified samples. To sum up, these quantitative results illustrate the strong correlation between recognition performance and class-specific graph representation.
+
+Performance vs. Graph Quality. In Tab. 10, we first calculate the graph distances between each sample and all classes (detailed in App. 6.4) for CTR-GCN. Then, we rank the distances from low to high. In Tab. 10, we report the recognition accuracies of samples according to their distance ranks to the correct class. Here, higher ranks indicate that graphs are of higher quality (intra-class compact and inter-class dispersed), while lower ranks indicate that graphs are of lower quality (intra-class dispersed and inter-class aliasing). We note that: (1) Considering samples from lower ranks to higher ranks, performances improve monotonically, revealing the significant correlations between graph quality and recognition performance. (2) Combined with the proposed method, we improve performances in all cases, where larger improvements are obtained on the samples with lower-quality graphs. These results prove that our method can alleviate the problem caused by learning low-quality graphs.
+
+# 5 CONCLUSION
+
+In this paper, we establish a new training paradigm for skeleton-based action recognition, called SkeletonGCL, which explicitly explores the rich semantic context across sequences. Concretely, SkeletonGCL contrasts the learned graphs among sequences, guiding the graph representations to be class-associated, hence improving GCN capacity to recognize different actions. We improve the current methods significantly to achieve SoTA on three benchmarks.
+
+Limitation. In this paper, we push away the negative pairs from different classes in the same way without considering their intrinsic relations. Therefore, a comprehensive contrasting manner may be more suitable by delicately involving cross-class relations. We leave this for future work.
+
+# ACKNOWLEDGEMENTS
+
+This research is supported by the NSFC (grants No. 61773176 and No. 61733007).
+
+# REFERENCES
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+
+# 6 APPENDIX
+
+# 6.1 IMPLEMENTATIONS OF GRAPH PROJECTION HEADS FOR GCNS.
+
+![](images/4de5c48d857c13ab9c46fd49629f8c2ae1a533fad3264072a7265914de86f692.jpg)  
+Figure 3: Illustration of graph projection heads for GCNs.
+
+In Fig. 3, we illustrate the implementation details of graph projection heads for different GCNs. Particularly, for CTR-GCN and InfoGCN, we first apply an average pooling layer to summarize the information along the channel and temporal dimensions, respectively. Then, the same as 2S-AGCN, we flatten the graphs and embed them with an FC layer.
+
+# 6.2 MORE DIAGNOSTIC EXPERIMENTS.
+
+Table 11: Comparison of using cross-entropy loss to supervise graph learning.  
+
+<table><tr><td>Model (CTR-GCN)</td><td>Acc (%)</td></tr><tr><td>Baseline</td><td>89.8</td></tr><tr><td>w/Cross-entropy loss</td><td>89.7</td></tr><tr><td>w/SkeletonGCL</td><td>90.8</td></tr></table>
+
+Table 12: Impact of the FC layer in graph projection head.  
+
+<table><tr><td>Model (CTR-GCN)</td><td>Acc (%)</td></tr><tr><td>Baseline</td><td>89.8</td></tr><tr><td>Graph Projection w/FC</td><td>90.1</td></tr><tr><td>Graph Projection w/FC</td><td>90.8</td></tr><tr><td>Graph Projection w/MLP</td><td>90.6</td></tr></table>
+
+Table 13: Apply graph contrast on different layers.  
+Table 14: Impact of the size of Table 15: Performance comparison with different  $C_g$  
+
+<table><tr><td colspan="2">Model (CTR-GCN) | Acc (%)</td></tr><tr><td>Baseline</td><td>89.8</td></tr><tr><td>7th layer</td><td>89.8</td></tr><tr><td>8th layer</td><td>90.0</td></tr><tr><td>9th layer</td><td>90.4</td></tr><tr><td>10th layer</td><td>90.8</td></tr></table>
+
+Table 16: Impact of  $\tau$  .  
+
+<table><tr><td>τ</td><td>Acc (%)</td></tr><tr><td>0.5</td><td>90.4</td></tr><tr><td>0.8</td><td>90.6</td></tr><tr><td>1.0</td><td>90.8</td></tr><tr><td>1.5</td><td>90.5</td></tr></table>
+
+<table><tr><td>P</td><td>Acc (%)</td></tr><tr><td>128</td><td>90.3</td></tr><tr><td>342</td><td>90.5</td></tr><tr><td>684</td><td>90.8</td></tr><tr><td>1368</td><td>90.4</td></tr><tr><td>2736</td><td>90.0</td></tr></table>
+
+Table 17: Impact of  $\alpha$  .  
+
+<table><tr><td>α</td><td>Acc (%)</td></tr><tr><td>0.75</td><td>90.6</td></tr><tr><td>0.80</td><td>90.7</td></tr><tr><td>0.85</td><td>90.8</td></tr><tr><td>0.9</td><td>90.6</td></tr></table>
+
+<table><tr><td>Cg</td><td>Acc (%)</td></tr><tr><td>64</td><td>90.4</td></tr><tr><td>128</td><td>90.6</td></tr><tr><td>256</td><td>90.8</td></tr><tr><td>512</td><td>90.5</td></tr></table>
+
+Table 18: Performance comparison with different  $K_H^+$  
+
+<table><tr><td>KH+</td><td>Acc (%)</td></tr><tr><td>64</td><td>90.6</td></tr><tr><td>128</td><td>90.8</td></tr><tr><td>256</td><td>90.6</td></tr><tr><td>512</td><td>90.5</td></tr></table>
+
+Table 19: Performance comparison with different  $K_H^-$  
+Table 20: Performance comparison with different  $K_{R}^{-}$ .  
+
+<table><tr><td>KH</td><td>Acc (%)</td></tr><tr><td>128</td><td>90.5</td></tr><tr><td>256</td><td>90.7</td></tr><tr><td>512</td><td>90.8</td></tr><tr><td>1024</td><td>90.5</td></tr></table>
+
+<table><tr><td>KR</td><td>Acc (%)</td></tr><tr><td>128</td><td>90.6</td></tr><tr><td>256</td><td>90.7</td></tr><tr><td>512</td><td>90.8</td></tr><tr><td>1024</td><td>90.6</td></tr></table>
+
+Comparison of using cross-entropy loss. Since cross-entropy loss is a widely used classification loss in learning class-discriminative representations, in Tab. 11, we investigate its performance to supervise graph learning. We find that directly using cross-entropy loss for graph learning has negligible effects on the performance (89.8% to 89.7%), which indicates that it is impractical to learn favorable class-discriminative graphs by naively using a classification loss. In this paper, we find a practical way to achieve this goal by introducing the cross-sequence context for guiding graph learning.
+
+Impact of FC in Projection Head. In Tab. 12, the effectiveness of transformation layer (FC layer) in the graph projection head is investigated. We find that the model achieves obvious improvement  $(90.1\%)$  to  $90.8\%$  equipped with the FC layer, which proves the importance of vertex-aware graph encoding. In addition, we find that using an MLP achieves a similar but lower accuracy, hence we use a simple FC in the framework.
+
+Which Layer to Contrast Graphs? In Tab. 13, we apply graph contrast on different layers. We find that contrasting graphs on deeper layers outperform on shallower layers. One possible explanation is that deeper layers can provide higher-level semantics that is relevant to recognition.
+
+Impact of the size of  $\mathcal{M}_{\mathrm{Ins}}$ . In Tab. 14, the impact of the size of  $\mathcal{M}_{\mathrm{Ins}}$  is investigated, where we use different values of  $P$  to control the size. We find that appropriately increasing the size can effectively expand the cross-sequence context, and improve recognition performance. However, an over large memory bank stores old samples from a few batches ago, which hinders representation learning.
+
+Impact of dimension  $C_g$ . In Tab. 15, the influences of the dimension of graph vector  $C_g$  are investigated. For pursuing the best performance, we set  $C_g$  as 256.
+
+Impact of temperature  $\tau$ . In Tab. 16, the influences of the temperature  $\tau$  are investigated. For pursuing the best performance, we set  $\tau$  as 1.0.
+
+Impact of  $\alpha$ . In Tab. 17, the influences of the momentum updating hyper-parameter  $\alpha$  are investigated. For pursuing the best performance, we set  $\alpha$  as 0.85.
+
+Impact of the number of sampling examples. In Tab. 18, the impact of selecting  $K_H^+$  hardest positive examples is investigated. In Tab. 19, the impact of selecting  $K_H^-$  hardest negative examples is investigated. In Tab. 20, the impact of selecting  $K_R^-$  random negative examples is investigated. For pursuing the best performance, we set  $K_H^+, K_H^-$ , and  $K_R^-$  to 128, 512 and 512, respectively.
+
+The quantitative analysis of accuracy improvement. In Tab. 21, the recognition accuracies of the top-10 hardest classes for CTR-GCN on NTU-60 are presented. The improvements in four classes (i.e., "reading", "typing on a keyboard", "headache" and "point to something") are over  $4\%$ . Though performances in three classes decrease, they are relatively small  $(-1.5\%$  on "writing",  $-1.0\%$  on "take off a shoe", and  $-2.5\%$  on "sneeze/cough") vs. others' increase. Overall, we obtain an average improvement in the 10 classes of  $2.7\%$ .
+
+In Tab. 22, the recognition accuracies of top-10 improved classes for CTR-GCN on NTU-60 are presented. The accuracy of the above 10 classes shows an average gain of  $4.6\%$ .
+
+Table 21: Performance  $(\%)$  on top 10 hardest classes for CTR-GCN.  
+
+<table><tr><td>classes</td><td>CTR-GCN</td><td>CTR-GCN w/GCL</td><td>+/-</td></tr><tr><td>reading</td><td>58.2</td><td>67.8</td><td>+9.6</td></tr><tr><td>type on a keyboard</td><td>66.5</td><td>70.9</td><td>+4.4</td></tr><tr><td>writing</td><td>67.3</td><td>65.8</td><td>-1.5</td></tr><tr><td>play with phone/tablet</td><td>68.0</td><td>70.9</td><td>+2.9</td></tr><tr><td>eat meal</td><td>71.6</td><td>75.3</td><td>+3.7</td></tr><tr><td>take off a shoe</td><td>75.5</td><td>74.5</td><td>-1.0</td></tr><tr><td>headache</td><td>78.3</td><td>83.0</td><td>+4.7</td></tr><tr><td>point to something</td><td>81.9</td><td>85.9</td><td>+4.0</td></tr><tr><td>clapping</td><td>82.1</td><td>85.7</td><td>+3.6</td></tr><tr><td>sneeze/cough</td><td>82.2</td><td>79.7</td><td>-2.5</td></tr><tr><td>Average</td><td>73.2</td><td>75.9</td><td>+2.7</td></tr></table>
+
+Table 22: Performance (%) on top 10 improved classes for CTR-GCN.  
+
+<table><tr><td>classes</td><td>CTR-GCN</td><td>CTR-GCN w/GCL</td><td>+/-</td></tr><tr><td>reading</td><td>58.2</td><td>67.8</td><td>+9.6</td></tr><tr><td>headache</td><td>78.3</td><td>83.0</td><td>+4.7</td></tr><tr><td>rub two hands</td><td>87.3</td><td>92.0</td><td>+4.7</td></tr><tr><td>punch/slap</td><td>89.4</td><td>93.8</td><td>+4.4</td></tr><tr><td>type on a keyboard</td><td>66.6</td><td>70.9</td><td>+4.3</td></tr><tr><td>point to something</td><td>81.9</td><td>85.9</td><td>+4.0</td></tr><tr><td>clapping</td><td>82.1</td><td>85.7</td><td>+3.6</td></tr><tr><td>reach into pocket</td><td>82.5</td><td>86.1</td><td>+3.6</td></tr><tr><td>eat meal</td><td>71.6</td><td>75.2</td><td>+3.6</td></tr><tr><td>neck pain</td><td>88.0</td><td>91.3</td><td>+3.3</td></tr><tr><td>Average</td><td>78.6</td><td>83.2</td><td>+4.6</td></tr></table>
+
+# 6.3 QUALITATIVE RESULTS
+
+In Fig. 4, we visualize the t-SNE distribution of graph and feature representations of sequences from six classes, illustrating the impact of SkeletonGCL. As shown in Fig. 4(a), SkeletonGCL can shape the graph representation structure, where the graphs from the same class get together and graphs from different classes spread out. Consequently, in Fig. 4(b), with SkeletonGCL, the features from different classes become more distinguishable, which indicates that graph contrast indeed improves the feature extraction capacity.
+
+![](images/a54096b693e88efa6699a86aeee8bf920d188f12387c3fb8888e3bd89e1badb3.jpg)  
+CTR-GCN
+
+![](images/2e9bb985c1ca104c6cd222f0a9f8dbbbda467d4f6bf54131b882fe5ab70bafcb.jpg)  
+CTR-GCN w/SkeletonGCL  
+(a) t-SNE visualization of graph representation.
+
+![](images/116fdbd452c2ca1b184f951c578d86fe8427fd3eb613af65adb2c70dc009dd71.jpg)  
+Feature Representation t-SNE  
+CTR-GCN
+
+![](images/b9e6235605dc66bc5d9a007c888a50a0fd0218ad9876b41866e978ef24125aef.jpg)  
+CTR-GCN w/ours  
+(b) t-SNE visualization of feature representation.  
+Figure 4: t-SNE visualization. t-SNE (Van der Maaten & Hinton, 2008) visualization of graph and feature representations from sequences in the test set of NTU 60. Each color denotes a certain class. Best viewed with zoom in.
+
+# 6.4 THE CALCULATION DETAILS OF GRAPH DISTANCE
+
+In Tab. 9, the statistics of graph distance for all samples are investigated. In CTR-GCN, the graph  $\mathbf{g} \in \mathbb{R}^{K_S \times C \times N \times N}$  is learned for graph convolution. For the convenience of calculation, we use an average pooling to squeeze  $\mathbf{g}$  and reshape it into  $\mathbf{g} = \overline{\mathbf{g}} \in \mathbb{R}^{1 \times N^2}$  as the graph embedding to conduct distance measurement.
+
+To acquire the graph embedding for each class  $c_{*}$ , we first calculate the centroid vector  $s$  as follows,
+
+$$
+\boldsymbol {s} _ {c _ {*}} = \frac {1}{N _ {c _ {*}}} \sum_ {i = 1} ^ {N _ {c _ {*}}} \boldsymbol {g} _ {i} \tag {9}
+$$
+
+where  $N_{c*}$  denotes the number of samples which belongs to class  $c_*$ . To effectively reveal the relation between graph quality and classification accuracy, we introduce three types of graph distance, i.e.,  $d_{\mathrm{all}}$ ,  $d_{\mathrm{cor}}$  and  $d_{\mathrm{mis}}$ .  $d_{\mathrm{all}}$  measures the average distance to all classes.  $d_{\mathrm{cor}}$  measures the distance to the correct class  $c_*$ .  $d_{\mathrm{mis}}$  measures the distance to the misclassified class  $c_{\mathrm{mis}}$ .
+
+$$
+d _ {\text {a l l}} = \frac {1}{K} \sum_ {k = 1} ^ {K} \left\| \boldsymbol {g} - \boldsymbol {s} _ {c _ {k}} \right\| ^ {2} \tag {10}
+$$
+
+$$
+d _ {\mathrm {c o r}} = \left\| \boldsymbol {g} - \boldsymbol {s} _ {c _ {*}} \right\| ^ {2}, \tag {11}
+$$
+
+$$
+d _ {\text {m i s}} = \left\| \boldsymbol {g} - \boldsymbol {s} _ {c _ {\text {m i s}}} \right\| ^ {2}, \tag {12}
+$$
+
+where  $K$  denotes the number of classes in the dataset.

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2023/Graph Neural Network-Inspired Kernels for Gaussian Processes in Semi-Supervised Learning/full.md

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+# GRAPH NEURAL NETWORK-INSPIRED Kernels FOR GAUSSIAN PROCESSES IN SEMI-SUPERVISED LEARNING
+
+Zehao Niu $^{1}$ , Mihai Anitescu $^{1,2}$
+
+<sup>1</sup>University of Chicago, <sup>2</sup>Argonne National Laboratory  
+niuzehao@uchicago.edu  
+anitescu@mcs.anl.gov
+
+Jie Chen*
+
+MIT-IBM Watson AI Lab IBM Research chenjie@us.ibm.com
+
+# ABSTRACT
+
+Gaussian processes (GPs) are an attractive class of machine learning models because of their simplicity and flexibility as building blocks of more complex Bayesian models. Meanwhile, graph neural networks (GNNs) emerged recently as a promising class of models for graph-structured data in semi-supervised learning and beyond. Their competitive performance is often attributed to a proper capturing of the graph inductive bias. In this work, we introduce this inductive bias into GPs to improve their predictive performance for graph-structured data. We show that a prominent example of GNNs, the graph convolutional network, is equivalent to some GP when its layers are infinitely wide; and we analyze the kernel universality and the limiting behavior in depth. We further present a programmable procedure to compose covariance kernels inspired by this equivalence and derive example kernels corresponding to several interesting members of the GNN family. We also propose a computationally efficient approximation of the covariance matrix for scalable posterior inference with large-scale data. We demonstrate that these graph-based kernels lead to competitive classification and regression performance, as well as advantages in computation time, compared with the respective GNNs.
+
+# 1 INTRODUCTION
+
+Gaussian processes (GPs) (Rasmussen & Williams, 2006) are widely used in machine learning, uncertainty quantification, and global optimization. In the Bayesian setting, a GP serves as a prior probability distribution over functions, characterized by a mean (often treated as zero for simplicity) and a covariance. Conditioned on observed data with a Gaussian likelihood, the random function admits a posterior distribution that is also Gaussian, whose mean is used for prediction and the variance serves as an uncertainty measure. The closed-form posterior allows for exact Bayesian inference, resulting in great attractiveness and wide usage of GPs.
+
+The success of GPs in practice depends on two factors: the observations (training data) and the covariance kernel. We are interested in semi-supervised learning, where only a small amount of data is labeled while a large amount of unlabeled data can be used together for training (Zhu, 2008). In recent years, graph neural networks (GNNs) (Zhou et al., 2020; Wu et al., 2021) emerged as a promising class of models for this problem, when the labeled and unlabeled data are connected by a graph. The graph structure becomes an important inductive bias that leads to the success of GNNs. This inductive bias inspires us to design a GP model under limited observations, by building the graph structure into the covariance kernel.
+
+An intimate relationship between neural networks and GPs is known: a neural network with fully connected layers, equipped with a prior probability distribution on the weights and biases, converges to a GP when each of its layers is infinitely wide (Lee et al., 2018; de G. Matthews et al., 2018). Such a result is owing to the central limit theorem (Neal, 1994; Williams, 1996) and the GP covariance
+
+can be recursively computed if the weights (and biases) in each layer are iid Gaussian. Similar results for other architectures, such as convolution layers and residual connections, were subsequently established in the literature (Novak et al., 2019; Garriga-Alonso et al., 2019).
+
+One focus of this work is to establish a similar relationship between GNNs and the limiting GPs. We will derive the covariance kernel that incorporates the graph inductive bias as GNNs do. We start with one of the most widely studied GNNs, the graph convolutional network (GCN) (Kipf & Welling, 2017), and analyze the kernel universality as well as the limiting behavior when the depth also tends to infinity. We then derive covariance kernels from other GNNs by using a programmable procedure that corresponds every building block of a neural network to a kernel operation.
+
+Meanwhile, we design efficient computational procedures for posterior inference (i.e., regression and classification). GPs are notoriously difficult to scale because of the cubic complexity with respect to the number of training data. Benchmark graph datasets used by the GNN literature may contain thousands or even millions of labeled nodes (Hu et al., 2020b). The semi-supervised setting worsens the scenario, as the covariance matrix needs to be (recursively) evaluated in full because of the graph convolution operation. We propose a Nyström-like scheme to perform low-rank approximations and apply the approximation recursively on each layer, to yield a low-rank kernel matrix. Such a matrix can be computed scalably. We demonstrate through numerical experiments that the GP posterior inference is much faster than training a GNN and subsequently performing predictions on the test set.
+
+We summarize the contributions of this work as follows:
+
+1. We derive the GP as a limit of the GCN when the layer widths tend to infinity and study the kernel universality and the limiting behavior in depth.  
+2. We propose a computational procedure to compute a low-rank approximation of the covariance matrix for practical and scalable posterior inference.  
+3. We present a programmable procedure to compose covariance kernels and their approximations and show examples corresponding to several interesting members of the GNN family.  
+4. We conduct comprehensive experiments to demonstrate that the GP model performs favorably compared to GNNs in prediction accuracy while being significantly faster in computation.
+
+# 2 RELATED WORK
+
+It has long been observed that GPs are limits of standard neural networks with one hidden layer when the layer width tends to infinity (Neal, 1994; Williams, 1996). Recently, renewed interests in the equivalence between GPs and neural networks were extended to deep neural networks (Lee et al., 2018; de G. Matthews et al., 2018) as well as modern neural network architectures, such as convolution layers (Novak et al., 2019), recurrent networks (Yang, 2019), and residual connections (Garriga-Alonso et al., 2019). The term NNGP (neural network Gaussian process) henceforth emerged under the context of Bayesian deep learning. Besides the fact that an infinite neural network defines a kernel, the training of a neural network by using gradient descent also defines a kernel—the neural tangent kernel (NTK)—that describes the evolution of the network (Jacot et al., 2018; Lee et al., 2019). Library supports in Python were developed to automatically construct the NNGP and NTK kernels based on programming the corresponding neural networks (Novak et al., 2020).
+
+GNNs are neural networks that handle graph-structured data (Zhou et al., 2020; Wu et al., 2021). They are a promising class of models for semi-supervised learning. Many GNNs use the message-passing scheme (Gilmer et al., 2017), where neighborhood information is aggregated to update the representation of the center node. Representative examples include GCN (Kipf & Welling, 2017), GraphSAGE (Hamilton et al., 2017), GAT (Velicković et al., 2018), and GIN (Xu et al., 2019). It is found that the performance of GNNs degrades as they become deep; one approach to mitigating the problem is to insert residual/skip connections, as done by JumpingKnowledge (Xu et al., 2018), APPNP (Gasteiger et al., 2019), and GCNII (Chen et al., 2020).
+
+GP inference is too costly, because it requires the inverse of the  $N \times N$  dense kernel matrix. Scalable approaches include low-rank methods, such as Nyström approximation (Drineas & Mahoney, 2005), random features (Rahimi & Recht, 2007), and KISS-GP (Wilson & Nickisch, 2015); as well as multi-resolution (Katzfuss, 2017) and hierarchical methods (Chen et al., 2017; Chen & Stein, 2021).
+
+Prior efforts on integrating graphs into GPs exist. Ng et al. (2018) define a GP kernel by combing a base kernel with the adjacency matrix; it is related to a special case of our kernels where the network has only one layer and the output admits a robust-max likelihood for classification. Hu et al. (2020a) explore a similar route to us, by taking the limit of a GCN, but its exploration is less comprehensive because it does not generalize to other GNNs and does not tackle the scalability challenge.
+
+# 3 GRAPH CONVOLUTIONAL NETWORK AS A GAUSSIAN PROCESS
+
+We start with a few notations used throughout this paper. Let an undirected graph be denoted by  $\mathcal{G} = (\mathcal{V},\mathcal{E})$  with  $N = |\mathcal{V}|$  nodes and  $M = |\mathcal{E}|$  edges. For notational simplicity, we use  $A\in \mathbb{R}^{N\times N}$  to denote the original graph adjacency matrix or any modified/normalized version of it. Using  $d_{l}$  to denote the width of the  $l$ -th layer, the layer architecture of GCN reads
+
+$$
+X ^ {(l)} = \phi \left(A X ^ {(l - 1)} W ^ {(l)} + b ^ {(l)}\right), \tag {1}
+$$
+
+where  $X^{(l-1)} \in \mathbb{R}^{N \times d_{l-1}}$  and  $X^{(l)} \in \mathbb{R}^{N \times d_l}$  are layer inputs and outputs, respectively;  $W^{(l)} \in \mathbb{R}^{d_{l-1} \times d_l}$  and  $b^{(l)} \in \mathbb{R}^{1 \times d_l}$  are the weights and the biases, respectively; and  $\phi$  is the ReLU activation function. The graph convolutional operator  $A$  is a symmetric normalization of the graph adjacency matrix with self-loops added (Kipf & Welling, 2017).
+
+For ease of exposition, it will be useful to rewrite the matrix notation (1) in element-sums and products. To this end, for a node  $x$ , let  $z_{i}^{(l)}(x)$  and  $x_{i}^{(l)}(x)$  denote the pre- and post-activation value at the  $i$ -th coordinate in the  $l$ -th layer, respectively. Particularly, in an  $L$ -layer GCN,  $x^{(0)}(x)$  is the input feature vector and  $z^{(L)}(x)$  is the output vector. The layer architecture of GCN reads
+
+$$
+y _ {i} ^ {(l)} (x) = \sum_ {j = 1} ^ {d _ {l - 1}} W _ {j i} ^ {(l)} x _ {j} ^ {(l - 1)} (x), \quad z _ {i} ^ {(l)} (x) = b _ {i} ^ {(l)} + \sum_ {v \in \mathcal {V}} A _ {x v} y _ {i} ^ {(l)} (v), \quad x _ {i} ^ {(l)} (x) = \phi \left(z _ {i} ^ {(l)} (x)\right). \tag {2}
+$$
+
+# 3.1 LIMIT IN THE WIDTH
+
+The following theorem states that when the weights and biases in each layer are iid zero-mean Gaussians, in the limit on the layer width, the GCN output  $z^{(L)}(x)$  is a multi-output GP over the index  $x$ .
+
+Theorem 1. Assume  $d_{1}, \ldots, d_{L-1}$  to be infinite in succession and let the bias and weight terms be independent with distributions
+
+$$
+b _ {i} ^ {(l)} \sim \mathcal {N} (0, \sigma_ {b} ^ {2}), \quad W _ {i j} ^ {(l)} \sim \mathcal {N} (0, \sigma_ {w} ^ {2} / d _ {l - 1}), \quad l = 1, \ldots , L.
+$$
+
+Then, for each  $i$ , the collection  $\{z_i^{(l)}(x)\}$  over all graph nodes  $x$  follows the normal distribution  $\mathcal{N}(0,K^{(l)})$ , where the covariance matrix  $K^{(l)}$  can be computed recursively by
+
+$$
+C ^ {(l)} = \operatorname {E} _ {z _ {i} ^ {(l)} \sim \mathcal {N} (0, K ^ {(l)})} [ \phi (z _ {i} ^ {(l)}) \phi (z _ {i} ^ {(l)}) ^ {T} ], \quad l = 1, \dots , L, \tag {3}
+$$
+
+$$
+K ^ {(l + 1)} = \sigma_ {b} ^ {2} \mathbf {1} _ {N \times N} + \sigma_ {w} ^ {2} A C ^ {(l)} A ^ {T}, \quad l = 0, \dots , L - 1. \tag {4}
+$$
+
+All proofs of this paper are given in the appendix. Note that different from a usual GP, which is a random function defined over a connected region of the Euclidean space, here  $z^{(L)}$  is defined over a discrete set of graph nodes. In the usual use of a graph in machine learning, this set is finite, such that the function distribution degenerates to a multivariate distribution. In semi-supervised learning, the dimension of the distribution,  $N$ , is fixed when one conducts transductive learning; but it will vary in the inductive setting because the graph will have new nodes and edges. One special care of a graph-based GP over a usual GP is that the covariance matrix will need to be recomputed from scratch whenever the graph alters.
+
+Theorem 1 leaves out the base definition  $C^{(0)}$ , whose entry denotes the covariance between two input nodes. The traditional literature uses the inner product  $C^{(0)}(x, x') = \frac{x \cdot x'}{d_0}$  (Lee et al., 2018),
+
+but nothing prevents us using any positive-definite kernel alternatively. $^{1}$  For example, we could use the squared exponential kernel  $C^{(0)}(x,x^{\prime}) = \exp \left(-\frac{1}{2}\sum_{j = 1}^{d_0}\left(\frac{x_j - x_j'}{\ell_j}\right)^2\right)$ . Such flexibility in essence performs an implicit feature transformation as preprocessing.
+
+# 3.2 UNIVERSALITY
+
+A covariance kernel is positive definite; hence, the Moore-Aronszajn theorem (Aronszajn, 1950) suggests that it defines a unique Hilbert space for which it is a reproducing kernel. If this space is dense, then the kernel is called universal. One can verify universality by checking if the kernel matrix is positive definite for any set of distinct points. For the case of graphs, it suffices to verify if the covariance matrix for all nodes is positive definite.
+
+We do this job for the ReLU activation function. It is known that the kernel  $\mathrm{E}_{w\sim \mathcal{N}(0,I_d)}[\phi (w\cdot x)\phi (w\cdot x')]$  admits a closed-form expression as a function of the angle between  $x$  and  $x'$ , hence named the arc-cosine kernel (Cho & Saul, 2009). We first establish the following lemma that states that the kernel is universal over a half-space.
+
+Lemma 2. The arc-cosine kernel is universal on the upper-hemisphere  $S = \{x \in \mathbb{R}^d : \| x \|_2 = 1, x_1 > 0\}$  for all  $d \geq 2$ .
+
+It is also known that the expectation in (3) is proportional to the arc-cosine kernel up to a factor  $\sqrt{K^{(l)}(x,x)K^{(l)}(x',x')}$  (Lee et al., 2018). Therefore, we iteratively work on the post-activation covariance (3) and the pre-activation covariance (4) and show that the covariance kernel resulting from the limiting GCN is universal, for any GCN with three or more layers.
+
+Theorem 3. Assume  $A$  is irreducible and non-negative and  $C^{(0)}$  does not contain two linearly dependent rows. Then,  $K^{(l)}$  is positive definite for all  $l \geq 3$ .
+
+# 3.3 LIMIT IN THE DEPTH
+
+The depth of a neural network exhibits interesting behaviors. Deep learning tends to favor deep networks because of their empirically outstanding performance, exemplified by generations of convolutional networks for the ImageNet classification (Krizhevsky et al., 2012; Wortsman et al., 2022); while graph neural networks are instrumental to be shallow because of the over-smoothing and oversquashing properties (Li et al., 2018; Topping et al., 2022). For multi-layer perceptrons (networks with fully connected layers), several previous works have noted that the recurrence relation of the covariance kernel across layers leads to convergence to a fixed-point kernel, when the depth  $L \to \infty$  (see, e.g., Lee et al. (2018); in Appendix B.5, we elaborate this limit). In what follows, we offer the parallel analysis for GCN.
+
+Theorem 4. Assume  $A$  is symmetric, irreducible, aperiodic, and non-negative with Perron-Frobenius eigenvalue  $\lambda > 0$ . The following results hold as  $l \to \infty$ .
+
+1. When  $\sigma_b^2 = 0$ ,  $\rho_{\min}(K^{(l)}) \nearrow 1$ , where  $\rho_{\min}$  denotes the minimum correlation between any two nodes  $x$  and  $x'$ .  
+2. When  $\sigma_w^2 < 2 / \lambda^2$ , a subsequence of  $K^{(l)}$  converges to some matrix.  
+3. When  $\sigma_w^2 > 2 / \lambda^2$ , let  $c_l = (\sigma_w^2 \lambda^2 / 2)^l$ ; then,  $K^{(l)} / c_l \to vv^T$  where  $v$  is an eigenvector corresponding to  $\lambda$ .
+
+A few remarks follow. The first case implies that the correlation matrix converges monotonously to a matrix of all ones. As a consequence, up to some scaling  $c_{l}^{\prime}$  that may depend on  $l$ , the scaled covariance matrix  $K^{(l)} / c_{l}^{\prime}$  converges to a rank-1 matrix. The third case shares a similar result, with the limit explicitly spelled out, but note that the eigenvector  $v$  may not be normalized. The second case is challenging to analyze. According to empirical verification, we speculate a stronger result—convergence of  $K^{(l)}$  to a unique fixed point—may hold.
+
+Table 1: Computational costs.  $M$  : number of edges;  $N$  : number of nodes;  ${N}_{b}$  : number of training nodes;  ${N}_{ * }$  : number of prediction nodes;  ${N}_{a}$  : number of landmark nodes;  $L$  : number of layers. Assume  ${N}_{b} \gg  {N}_{a}$  . For posterior variance,assume only the diagonal is needed.  
+
+<table><tr><td></td><td>Time O(·)</td><td>Storage O(·)</td></tr><tr><td>Computation of Q(L)</td><td>LMNa + LNN2a + LNa3</td><td>NNa</td></tr><tr><td>Posterior mean (6)</td><td>NaNa + NbNa2 + Na3</td><td>(Nb + N*)Na</td></tr><tr><td>Posterior variance (7)</td><td>NaNa + NbNa2 + Na3</td><td>(Nb + N*)Na</td></tr></table>
+
+# 4 SCALABLE COMPUTATION THROUGH LOW-RANK APPROXIMATION
+
+The computation of the covariance matrix  $K^{(L)}$  through recursion (3)-(4) is the main computational bottleneck for GP posterior inference. We start the exposition with the mean prediction. We compute the posterior mean  $\widehat{y}_{*} = K_{*b}^{(L)}(K_{bb}^{(L)} + \epsilon I)^{-1}y_{b}$ , where the subscripts  $b$  and  $*$  denote the training set and the prediction set, respectively; and  $\epsilon$ , called the nugget, is the noise variance of the training data. Let there be  $N_{b}$  training nodes and  $N_{*}$  prediction nodes. It is tempting to compute only the  $(N_{b} + N_{*}) \times N_{b}$  submatrix of  $K^{(L)}$  for the task, but the recursion (4) requires the full  $C^{(L-1)}$  at the presence of  $A$ , and hence all the full  $C^{(l)}$ 's and  $K^{(l)}$ 's.
+
+To reduce the computational costs, we resort to a low-rank approximation of  $C^{(l)}$ , from which we easily see that  $K^{(l + 1)}$  is also low-rank. Before deriving the approximation recursion, we note (again) that for the ReLU activation  $\phi$ ,  $C^{(l)}$  in (3) is the arc-cosine kernel with a closed-form expression:
+
+$$
+C _ {x x ^ {\prime}} ^ {(l)} = \frac {1}{2 \pi} \sqrt {K _ {x x} ^ {(l)} K _ {x ^ {\prime} x ^ {\prime}} ^ {(l)}} \left(\sin \theta_ {x x ^ {\prime}} ^ {(l)} + \left(\pi - \theta_ {x x ^ {\prime}} ^ {(l)}\right) \cos \theta_ {x x ^ {\prime}} ^ {(l)}\right) \quad \text {w h e r e} \quad \theta_ {x x ^ {\prime}} ^ {(l)} = \arccos  \left(\frac {K _ {x x ^ {\prime}} ^ {(l)}}{\sqrt {K _ {x x} ^ {(l)} K _ {x ^ {\prime} x ^ {\prime}} ^ {(l)}}}\right). \tag {5}
+$$
+
+Hence, the main idea is that starting with a low-rank approximation of  $K^{(l)}$ , compute an approximation of  $C^{(l)}$  by using (5), and then obtain an approximation of  $K^{(l + 1)}$  based on (4); then, repeat.
+
+To derive the approximation, we use the subscript  $a$  to denote a set of landmark nodes with cardinality  $N_{a}$ . The Nystrom approximation (Drineas & Mahoney, 2005) of  $K^{(0)}$  is  $K_{:a}^{(0)}(K_{aa}^{(0)})^{-1}K_{a:}^{(0)}$ , where the subscript : denotes retaining all rows/columns. We rewrite this approximation in the Cholesky style as  $Q^{(0)}Q^{(0)^T}$ , where  $Q^{(0)} = K_{:a}^{(0)}(K_{aa}^{(0)})^{-\frac{1}{2}}$  has size  $N\times N_{a}$ . Proceed with induction. Let  $K^{(l)}$  be approximated by  $\widehat{K}^{(l)} = Q^{(l)}Q^{(l)^T}$ , where  $Q^{(l)}$  has size  $N\times (N_a + 1)$ . We apply (5) to compute an approximation to  $C_{:a}^{(l)}$ , namely  $\widehat{C}_{:a}^{(l)}$ , by using  $\widehat{K}_{:a}^{(l)}$ . Then, (4) leads to a Cholesky style approximation of  $K^{(l + 1)}$ :
+
+$$
+\widehat {K} ^ {(l + 1)} = \sigma_ {b} ^ {2} \mathbf {1} _ {N \times N} + \sigma_ {w} ^ {2} A \widehat {C} ^ {(l)} A ^ {T} \equiv Q ^ {(l + 1)} Q ^ {(l + 1) ^ {T}},
+$$
+
+where  $Q^{(l + 1)} = \left[ \sigma_w A \widehat{C}_{:a}^{(l)} (\widehat{C}_{aa}^{(l)})^{-\frac{1}{2}} \sigma_b \mathbf{1}_{N \times 1} \right]$ . Clearly,  $Q^{(l + 1)}$  has size  $N \times (N_a + 1)$ , completing the induction.
+
+In summary,  $K^{(L)}$  is approximated by a rank-  $(N_{a} + 1)$  matrix  $\widehat{K}^{(L)} = Q^{(L)}Q^{(L)^T}$ . The computation of  $Q^{(L)}$  is summarized in Algorithm 1. Once it is formed, the posterior mean is computed as
+
+$$
+\widehat {y} _ {*} \approx \widehat {K} _ {* b} ^ {(L)} \left(\widehat {K} _ {b b} ^ {(L)} + \epsilon I\right) ^ {- 1} y _ {b} = Q _ {*:} ^ {(L)} \left(Q _ {b:} ^ {(L) ^ {T}} Q _ {b:} ^ {(L)} + \epsilon I\right) ^ {- 1} Q _ {b:} ^ {(L) ^ {T}} y _ {b}, \tag {6}
+$$
+
+where note that the matrix to be inverted has size  $(N_{a} + 1)\times (N_{a} + 1)$ , which is assumed to be significantly smaller than  $N_{b}\times N_{b}$ . Similarly, the posterior variance is
+
+$$
+\widehat {K} _ {* *} ^ {(L)} - \widehat {K} _ {* b} ^ {(L)} \left(\widehat {K} _ {b b} ^ {(L)} + \epsilon I\right) ^ {- 1} \widehat {K} _ {b *} ^ {(L)} = \epsilon Q _ {*:} ^ {(L)} \left(Q _ {b:} ^ {(L) ^ {T}} Q _ {b:} ^ {(L)} + \epsilon I\right) ^ {- 1} Q _ {*:} ^ {(L) ^ {T}}. \tag {7}
+$$
+
+The computational costs of  $Q^{(L)}$  and the posterior inference (6)-(7) are summarized in Table 1.
+
+Algorithm 1 Computing  $K^{(L)} \approx \widehat{K}^{(L)} = Q^{(L)}Q^{(L)^T}$  
+Require:  $Q^{(0)}$  such that  $K^{(0)}\approx Q^{(0)}Q^{(0)T}$    
+1: for  $l = 0,\dots ,L - 1$  do   
+2: Compute  $\widehat{K}_{:a}^{(l)} = Q_{:a}^{(0)}Q_{a:}^{(0)^T}$    
+3: Compute  $\widehat{C}_{:a}^{(l)}$  by (5), where  $C^{(l)}$  (resp.  $K^{(l)})$  entries are replaced by  $\widehat{C}^{(l)}$  (resp.  $\widehat{K}^{(l)})$  entries   
+4: Compute  $Q^{(l + 1)} = \left[\sigma_wA\widehat{C}_{:a}^{(l)}(\widehat{C}_{aa}^{(l)})^{-\frac{1}{2}}\sigma_b\mathbf{1}_{N\times 1}\right]$    
+5: end for
+
+Table 2: Neural network building blocks, kernel operations, and the low-rank counterpart.  
+
+<table><tr><td>Building block</td><td>Neural network</td><td>Kernel operation</td><td>Low-rank operation</td></tr><tr><td>Input</td><td>X← X(0)</td><td>K← C(0)</td><td>Q← Chol(C(0))</td></tr><tr><td>Bias term</td><td>X← X + b</td><td>K← K + σb21N×N</td><td>Q← [Q σb1N×1]</td></tr><tr><td>Weight term</td><td>X← XW</td><td>K← σw2K</td><td>Q← σwQ</td></tr><tr><td>Mixed weight term</td><td>X← X(αI + βW)</td><td>K← (α2 + β2σw2)K</td><td>Q← √α2 + β2σw2Q</td></tr><tr><td>Graph convolution</td><td>X← AX</td><td>K← AKAT</td><td>Q← AQ</td></tr><tr><td>Activation</td><td>X← φ(X)</td><td>K← g(K)</td><td>Q← Chol(g(QQT))</td></tr><tr><td>Independent addition</td><td>X← X1 + X2</td><td>K← K1 + K2</td><td>Q← [Q1 Q2]</td></tr></table>
+
+# 5 COMPOSING GRAPH NEURAL NETWORK-INSPIRED KERNELS
+
+Theorem 1, together with its proof, suggests that the covariance matrix of the limiting GP can be computed in a composable manner. Moreover, the derivation of Algorithm 1 indicates that the low-rank approximation of the covariance matrix can be similarly composed. Altogether, such a nice property allows one to easily derive the corresponding covariance matrix and its approximation for a new GNN architecture, like writing a program and obtaining a transformation of it automatically through operator overloading (Novak et al., 2020): the covariance matrix is a transformation of the GNN and the composition of the former is in exactly the same manner and order as that of the latter. We call the covariance matrices programmable.
+
+For example, we write a GCN layer as  $X \gets A\phi(X)W + b$ , where for notational simplicity,  $X$  denotes pre-activation rather than post-activation as in the earlier sections. The activation  $\phi$  on  $X$  results in a transformation of the kernel matrix  $K$  into  $g(K)$ , defined as:
+
+$$
+g (K) := C = \operatorname {E} _ {z \sim \mathcal {N} (0, K)} [ \phi (z) \phi (z) ^ {T} ], \tag {8}
+$$
+
+due to (3). Moreover, if  $K$  admits a low-rank approximation  $QQ^{T}$ , then  $g(K)$  admits a low-rank approximation  $PPT$  where  $P = \mathrm{Chol}(g(K))$  with
+
+$$
+\operatorname {C h o l} (C) := C _ {: a} C _ {a a} ^ {- \frac {1}{2}}.
+$$
+
+The next operation—graph convolution—multiplies  $A$  to the left of the post-activation. Correspondingly, the covariance matrix  $K$  is transformed to  $A K A^T$  and the low-rank approximation factor  $Q$  is transformed to  $A Q$ . Then, the operation—multiplying the weight matrix  $W$  to the right—will transform  $K$  to  $\sigma_w^2 K$  and  $Q$  to  $\sigma_w Q$ . Finally, adding the bias  $b$  will transform  $K$  to  $K + \sigma_b^2 \mathbf{1}_{N \times N}$  and  $Q$  to  $[Q - \sigma_b \mathbf{1}_{N \times 1}]$ . Altogether, we have obtained the following updates per layer:
+
+$$
+\operatorname {G C N}: \quad X \leftarrow A \phi (X) W + b
+$$
+
+$$
+K \leftarrow \sigma_ {w} ^ {2} A g (K) A ^ {T} + \sigma_ {b} ^ {2} \mathbf {1} _ {N \times N}
+$$
+
+$$
+Q \leftarrow \left[ \sigma_ {w} A \operatorname {C h o l} (g (Q Q ^ {T})) \quad \sigma_ {b} \mathbf {1} _ {N \times 1} \right].
+$$
+
+One may verify the  $K$  update against (3)-(4) and the  $Q$  update against Algorithm 1. Both updates can be automatically derived based on the update of  $X$ .
+
+We summarize the building blocks of a GNN and the corresponding kernel/low-rank operations in Table 2. The independent-addition building block is applicable to skip/residual connections. For
+
+example, here is the composition for the GCNII layer (Chen et al., 2020) without a bias term, where a skip connection with  $X^{(0)}$  occurs:
+
+$$
+\operatorname {G C N I I}: \quad X \leftarrow \left((1 - \alpha) A \phi (X) + \alpha X ^ {(0)}\right) ((1 - \beta) I + \beta W)
+$$
+
+$$
+K \leftarrow \left((1 - \alpha) ^ {2} A g (K) A ^ {T} + \alpha^ {2} K ^ {(0)}\right) ((1 - \beta) ^ {2} + \beta^ {2} \sigma_ {w} ^ {2})
+$$
+
+$$
+Q \leftarrow \left[ (1 - \alpha) A \operatorname {C h o l} (g (Q Q ^ {T})) \quad \alpha Q ^ {(0)} \right] \sqrt {(1 - \beta) ^ {2} + \beta^ {2} \sigma_ {w} ^ {2}}.
+$$
+
+For another example of the composability, we consider the popular GIN layer (Xu et al., 2019), which we assume uses a 2-layer MLP after the neighborhood aggregation:
+
+$$
+\operatorname {G I N}: \quad X \leftarrow \phi (A \phi (X) W + b) W ^ {\prime} + b ^ {\prime}
+$$
+
+$$
+K \leftarrow \sigma_ {w} ^ {2} g (B) + \sigma_ {b} ^ {2} \mathbf {1} _ {N \times N} \quad \text {w h e r e} \quad B = \sigma_ {w} ^ {2} A g (K) A ^ {T} + \sigma_ {b} ^ {2} \mathbf {1} _ {N \times N}
+$$
+
+$$
+Q \leftarrow \left[ \sigma_ {w} \operatorname {C h o l} (g (P P ^ {T})) \quad \sigma_ {b} \mathbf {1} _ {N \times 1} \right] \quad \text {w h e r e} \quad P = \left[ \sigma_ {w} A \operatorname {C h o l} (g (Q Q ^ {T})) \quad \sigma_ {b} \mathbf {1} _ {N \times 1} \right].
+$$
+
+Additionally, the updates for a GraphSAGE layer (Hamilton et al., 2017) are given in Appendix C.
+
+# 6 EXPERIMENTS
+
+In this section, we conduct a comprehensive set of experiments to evaluate the performance of the GP kernels derived by taking limits on the layer width of GCN and other GNNs. We demonstrate that these GPs are comparable with GNNs in prediction performance, while being significantly faster to compute. We also show that the low-rank version scales favorably, suitable for practical use.
+
+Datasets. The experiments are conducted on several benchmark datasets of varying sizes, covering both classification and regression. They include predicting the topic of scientific papers organized in a citation network (Cora, Citeseer, PubMed, and ArXiv); predicting the community of online posts based on user comments (Reddit), and predicting the average daily traffic of Wikipedia pages using hyperlinks among them (Chameleon, Squirrel, and Crocodile). Details of the datasets (including sources and preprocessing) are given in Appendix D.
+
+Experiment environment, training details, and hyperparameters are given in Appendix D.
+
+Prediction Performance: GCN-based comparison. We first conduct the semi-supervised learning tasks on all datasets by using GCN and GPs with different kernels. These kernels include the one equivalent to the limiting GCN (GCNGP), a usual squared-exponential kernel (RBF), and the GGP kernel proposed by Ng et al. (2018). Each of these kernels has a low-rank version (simplified with -X). RBF-X and GGP-X<sup>4</sup> use the Nyström approximation, consistent with GCNGP-X.
+
+GPs are by nature suitable for regression. For classification tasks, we use the one-hot representation of labels to set up a multi-output regression. Then, we take the coordinate with the largest output as the class prediction. Such an ad hoc treatment is widely used in the literature, as other more principled approaches (such as using the Laplace approximation on the non-Gaussian posterior) are too time-consuming for large datasets, meanwhile producing no noticeable gain in accuracy.
+
+Table 3 summarizes the accuracy for classification and the coefficient of determination,  $R^2$ , for regression. Whenever randomness is involved, the performance is reported as an average over five runs. The results of the two tasks show different patterns. For classification, GCNGP(-X) is slightly better than GCN and GGP(-X), while RBF(-X) is significantly worse than all others; moreover, the low-rank version is outperformed by using the full kernel matrix. On the other hand, for regression, GCNGP(-X) significantly outperforms GCN, RBF(-X), and GGP(-X); and the low-rank version becomes better. The less competitive performance of RBF(-X) is expected, as it does not leverage the graph inductive bias. It is attractive that GCNGP(-X) is competitive with GCN.
+
+Prediction Performance: Comparison with other GNNs. In addition to GCN, we conduct experiments with several popularly used GNN architectures (GCNII, GIN, and GraphSAGE) and GPs
+
+Table 3: Performance of GCNGP, in comparison with GCN and typical GP kernels. The Micro-F1 score is reported for classification tasks and  $R^2$  is reported for regression tasks.  
+
+<table><tr><td></td><td>GCN</td><td>GCNGP</td><td>GCNGP-X</td><td>RBF</td><td>RBF-X</td><td>GGP</td><td>GGP-X</td></tr><tr><td>Cora</td><td>0.8183±0.0055</td><td>0.8280</td><td>0.7980</td><td>0.5860</td><td>0.5850</td><td>0.7850</td><td>0.7410</td></tr><tr><td>Citeseer</td><td>0.6941±0.0079</td><td>0.7090</td><td>0.7080</td><td>0.6120</td><td>0.6090</td><td>0.7060</td><td>0.6470</td></tr><tr><td>PubMed</td><td>0.7649±0.0058</td><td>0.7960</td><td>0.7810</td><td>0.7360</td><td>0.7340</td><td>0.7820</td><td>0.7380</td></tr><tr><td>ArXiv</td><td>0.6990±0.0014</td><td>OOM</td><td>0.7011±0.0011</td><td>OOM</td><td>0.5382</td><td>OOM</td><td>0.6527</td></tr><tr><td>Reddit</td><td>0.9330±0.0006</td><td>OOM</td><td>0.9465±0.0003</td><td>OOM</td><td>0.5920</td><td>OOM</td><td>0.9058</td></tr><tr><td>Chameleon</td><td>0.5690±0.0376</td><td>0.6720</td><td>0.6852</td><td>0.5554</td><td>0.5613</td><td>0.5280</td><td>0.5311</td></tr><tr><td>Squirrel</td><td>0.4243±0.0393</td><td>0.4926</td><td>0.4998</td><td>0.3187</td><td>0.3185</td><td>0.2440</td><td>0.2251</td></tr><tr><td>Crocodile</td><td>0.6976±0.0323</td><td>0.8002</td><td>0.8013</td><td>0.6643</td><td>0.6710</td><td>0.6952</td><td>0.6810</td></tr></table>
+
+Table 4: Performance comparison (Micro-F1) between GNNs and the corresponding GP kernels.  
+
+<table><tr><td rowspan="2">architecture</td><td colspan="2">PubMed</td><td colspan="2">ArXiv</td><td colspan="2">Reddit</td></tr><tr><td>GNN</td><td>GNNGP</td><td>GNN</td><td>GNNGP-X</td><td>GNN</td><td>GNNGP-X</td></tr><tr><td>GCN</td><td>0.7649±0.0058</td><td>0.7960</td><td>0.6989±0.0016</td><td>0.7011±0.0011</td><td>0.9330±0.0006</td><td>0.9465±0.0003</td></tr><tr><td>GCNII</td><td>0.7558±0.0096</td><td>0.7840</td><td>0.7008±0.0021</td><td>0.6955±0.0011</td><td>0.9482±0.0007</td><td>0.9500±0.0003</td></tr><tr><td>GIN</td><td>0.7406±0.0112</td><td>0.7690</td><td>0.6340±0.0056</td><td>0.6652±0.0012</td><td>0.9398±0.0016</td><td>0.9428±0.0005</td></tr><tr><td>GraphSAGE</td><td>0.7535±0.0047</td><td>0.7900</td><td>0.6984±0.0021</td><td>0.6962±0.0007</td><td>0.9628±0.0007</td><td>0.9539±0.0003</td></tr></table>
+
+with the corresponding kernels. We test with the three largest datasets: PubMed, ArXiv, and Reddit, for the latter two of which a low-rank version of the GPs is used for computational feasibility.
+
+Table 4 summarizes the results. The observations on other GNNs extend similarly those on the GCN. In particular, on PubMed the GPs noticeably improve over the corresponding GNNs, while on ArXiv and Reddit the two families perform rather similarly. An exception is GIN for ArXiv, which significantly underperforms the GP counterpart, as well as other GNNs. It may improve with an extensive hyperparameter tuning.
+
+Running time. We compare the running time of the methods covered by Table 3. Different from usual neural networks, the training and inference of GNNs do not decouple in full-batch training. Moreover, there is not a universally agreed split between the training and the inference steps in GPs. Hence, we compare the total time for each method.
+
+Figure 1 plots the timing results, normalized against the GCN time for ease of comparison. It suggests that GCNGP(-X) is generally faster than GCN. Note that the vertical axis is in the logarithmic scale. Hence, for some of the datasets, the speedup is even one to two orders of magnitude.
+
+Scalability. For graphs, especially under the semi-supervised learning setting, the computational cost of a GP is much more complex than that of a usual one (which can be simply described as "cubic in the training set size"). One sees in Table 1 the many factors that determine the cost of our graph-based low-rank kernels. To explore the practicality of the proposed method, we use the timings gathered for Figure 1 to obtain an empirical scaling with respect to the graph size,  $M + N$ .
+
+Figure 2 fits the running times, plotted in the log-log scale, by using a straight line. We see that for neither GCN nor GCNGP(-X), the actual running time closely follows a polynomial complexity. However, interestingly, the least-squares fittings all lead to a slope approximately 1, which agrees with a linear cost. Theoretically, only GCNGP-X and GCN are approximately linear with respect to  $M + N$ , while GCNGP is cubic.
+
+Analysis on the depth. The performance of GCN deteriorates with more layers, known as the oversmoothing phenomenon. Adding residual/skip connections mitigates the problem, such as in GCNII. A natural question asks if the corresponding GP kernels behave similarly.
+
+Figure 3 shows that the trends of GCN and GCNII are indeed as expected. Interestingly, their GP counterparts both remain stable for depth  $L$  as large as 12. Our depth analysis (Theorem 4) suggests that in the limit, the GPs may perform less well because the kernel matrix may degenerate to rank 1. This empirical result indicates that the drop in performance may have not started yet.
+
+![](images/0cfd00d6875dc9992e33764d26642617a112ed776af4e10ac7c9d416dcc5b874.jpg)  
+Figure 1: Timing comparison. For each dataset, the times are normalized against that of GCN.
+
+![](images/9944e45d8d5ff6c10af71ce02557b76db79b3dc8c17bdd4b65464dec62f3a8a8.jpg)  
+Figure 2: Scaling of the running time with respect to the graph size,  $M + N$ .
+
+![](images/cf44c96ff8ab15d7914b7547344286260ff55ff9d65b615241310b6f7621e130.jpg)  
+Figure 3: Performance of GNNs and GNNGPs as depth  $L$  increases. Dataset: Pubmed.
+
+![](images/9923eebc5b2965bf3b35cfaef4a79ffa962ae64766a9069505b86ace7973fd2b.jpg)  
+Figure 4: Performance of GCNGP-X as number of landmark nodes increases. Dataset: Reddit.
+
+Analysis on the landmark set. The number of landmark nodes,  $N_{a}$ , controls the approximation quality of the low-rank kernels and hence the prediciton accuracy. On the other hand, the computational costs summarized in Table 1 indicate a dependency on  $N_{a}$  as high as the third power. It is crucial to develop an empirical understanding of the accuracy-time trade-off it incurs.
+
+Figure 4 clearly shows that as  $N_{a}$  becomes larger, the running time increase is not linear, while the increase of accuracy diminishes as the landmark set approaches the training set. It is remarkable that using only 1/800 of the training set as landmark nodes already achieves an accuracy surpassing that of GCN, by using time that is only a tiny fraction of the time otherwise needed to gain an additional 1% increase in the accuracy.
+
+# 7 CONCLUSIONS
+
+We have presented a GP approach for semi-supervised learning on graph-structured data, where the covariance kernel incorporates a graph inductive bias by exploiting the relationship between a GP and a GNN with infinitely wide layers. Similar to other neural networks priorly investigated, one can work out the equivalent GP (in particular, the covariance kernel) for GCN; and inspired by this equivalence, we formulate a procedure to compose covariance kernels corresponding to many other members of the GNN family. Moreover, every building block in the procedure has a low-rank counterpart, which allows for building a low-rank approximation of the covariance matrix that facilitates scalable posterior inference. We demonstrate the effectiveness of the derived kernels used for semi-supervised learning and show their advantages in computation time over GNNs.
+
+# ACKNOWLEDGMENTS
+
+Mihai Anitescu was supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research (ASCR) under Contract DE-AC02-06CH11347. Jie Chen acknowledges supports from the MIT-IBM Watson AI Lab.
+
+# REFERENCES
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+
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+
+# A CODE
+
+Code is available at https://github.com/niuzehao/gnn-gp.
+
+# B PROOFS AND ADDITIONAL THEOREMS
+
+# B.1 PROOF OF THEOREM 1
+
+We use mathematical induction. First, write the matrix form of (2) as follows:
+
+$$
+\boldsymbol {Y} _ {N \times d _ {l}} ^ {(l)} = \boldsymbol {X} _ {N \times d _ {l - 1}} ^ {(l - 1)} \boldsymbol {W} _ {d _ {l - 1} \times d _ {l}} ^ {(l)}, \quad \boldsymbol {Z} _ {N \times d _ {l}} ^ {(l)} = \mathbf {1} _ {N \times 1} \boldsymbol {b} _ {1 \times d _ {l}} ^ {(l)} + \boldsymbol {A} _ {N \times N} \boldsymbol {Y} _ {N \times d _ {l}} ^ {(l)}, \quad \boldsymbol {X} _ {N \times d _ {l}} ^ {(l)} = \phi (\boldsymbol {Z} _ {N \times d _ {l}} ^ {(l)}).
+$$
+
+Assume each column of  $Z^{(l-1)}$  is iid, then  $X^{(l-1)}$  also has iid columns. Hence,  $y_i^{(l)}(x)$  is a sum of iid terms and thus normal. Also,  $(y_i^{(l)}(x_1), \dots, y_i^{(l)}(x_N))$  are jointly multivariate normal, and identically distributed for different  $i$ ; so they form a GP.
+
+Moreover, terms like  $y_{i}^{(l)}(x)$  and  $y_{i'}^{(l)}(x')$  with  $i \neq i'$  are jointly Gaussian, with zero covariance:
+
+$$
+\operatorname {C o v} \left(y _ {i} ^ {(l)} (x), y _ {i ^ {\prime}} ^ {(l)} \left(x ^ {\prime}\right)\right) = \sum_ {j = 1} ^ {d _ {l - 1}} \sum_ {j ^ {\prime} = 1} ^ {d _ {l - 1}} \operatorname {C o v} \left(W _ {j i} ^ {(l)} x _ {j} ^ {(l - 1)} (x), W _ {j ^ {\prime} i ^ {\prime}} ^ {(l)} x _ {j ^ {\prime}} ^ {(l - 1)} \left(x ^ {\prime}\right)\right) = 0.
+$$
+
+Thus, they are independent, despite the fact that they may share the same  $X^{(l-1)}$  terms. In conclusion, each column of  $Y^{(l)}$  is an iid GP. Hence, each column of  $Z^{(l)}$  is also an iid GP  $\mathcal{N}(0, K^{(l)})$ .
+
+The covariance  $K^{(l)}$  can be computed recursively. Similar to the case of a fully connected network, the covariance of  $Y^{(l)}$  is
+
+$$
+\operatorname {E} [ y _ {i} ^ {(l)} (x) y _ {i} ^ {(l)} (x ^ {\prime}) ] = \sigma_ {w} ^ {2} \operatorname {E} [ x _ {j} ^ {(l - 1)} (x) x _ {j} ^ {(l - 1)} (x ^ {\prime}) ] = \sigma_ {w} ^ {2} C ^ {(l - 1)} (x, x ^ {\prime}),
+$$
+
+where
+
+$$
+C ^ {(l - 1)} (x, x ^ {\prime}) = \operatorname {E} _ {z \sim \mathcal {N} (0, K ^ {(l - 1)})} [ \phi (z _ {j} (x)) \phi (z _ {j} (x ^ {\prime})) ] \quad \text {f o r a n y} j.
+$$
+
+In the matrix form, this is to say  $\mathrm{Cov}(Y_j^{(l)}) = \sigma_w^2\operatorname {E}[X_j^{(l - 1)}X_j^{(l - 1)^T}] = \sigma_w^2 C^{(l - 1)}$ . Then, because  $Z_{i}^{(l)} = A Y_{i}^{(l)} + \mathbf{1}_{N\times 1}b_{i}^{(l)}$ , we obtain
+
+$$
+K ^ {(l)} = \sigma_ {b} ^ {2} \mathbf {1} _ {N \times N} + A \operatorname {C o v} \left(Y _ {i} ^ {(l)}\right) A ^ {T} = \sigma_ {b} ^ {2} \mathbf {1} _ {N \times N} + \sigma_ {w} ^ {2} A C ^ {(l - 1)} A ^ {T},
+$$
+
+which concludes the proof.
+
+# B.2 PROOF OF LEMMA 2
+
+We know that  $\phi (w\cdot x)$  is the feature mapping of the arc-cosine kernel. It suffices to show for different  $x_{1},\ldots ,x_{m}\in S$  , the  $\phi (w,x_i)$  's are linearly independent.
+
+We prove this by contradiction. Assume there exist  $c_{1},\ldots ,c_{m}$  not simultaneously zero, such that
+
+$$
+\sum_ {i = 1} ^ {m} c _ {i} \phi (w \cdot x _ {i}) \equiv 0. \tag {9}
+$$
+
+WLOG, we further assume that  $m$  is the smallest integer that satisfies (9). Let  $e_1 = (1,0,\dots ,0)$  then  $x\cdot e_1 > 0,\forall x\in S$ . Assume  $x_{m}\cdot e_{1} = \min_{1\leq i\leq m}\{x_{i}\cdot e_{1}\} >0$ . Then, for  $1\leq i\leq m - 1$
+
+$$
+\frac {x _ {i} \cdot x _ {m}}{x _ {i} \cdot e _ {1}} <   \frac {1}{x _ {m} \cdot e _ {1}} = \frac {x _ {m} \cdot x _ {m}}{x _ {m} \cdot e _ {1}}.
+$$
+
+Therefore, there exists  $t$  such that
+
+$$
+\frac {x _ {i} \cdot x _ {m}}{x _ {i} \cdot e _ {1}} <   t <   \frac {x _ {m} \cdot x _ {m}}{x _ {m} \cdot e _ {1}}, \quad 1 \leq i \leq m - 1.
+$$
+
+Let  $w = x_{m} - te_{1}$ ; then
+
+$$
+w \cdot x _ {m} > 0, \quad w \cdot x _ {i} <   0, \quad 1 \leq i \leq m - 1.
+$$
+
+Using (9) we know  $c_{m}(w\cdot x_{m}) = 0$ ; so  $c_{m} = 0$ . Thus,  $c_{1}, \dots, c_{m-1}$  and  $\phi(w\cdot x_{1}), \dots, \phi(w\cdot x_{m-1})$  also satisfy (9), with a smaller number  $m$ , which forms a contradiction.
+
+# B.3 PROOF OF THEOREM 3
+
+From the recursive relation (3)-(4), we know  $C^{(0)}$  is element-wise non-negative. Hence,  $K^{(1)}$  is element-wise non-negative,  $C^{(1)}$  is element-wise positive, and  $K^{(2)}$  is element-wise positive.
+
+Assume the Cholesky decomposition  $K^{(2)} = LL^T$  and denote  $L = (l_1, \dots, l_n)^T$ . Let  $r_i = \| l_i \|_2 > 0$  and  $x_i = \frac{l_i}{r_i}$ . Then,  $x_1 = \pm e_1$ , and WLOG we assume  $x_1 = e_1$ . Then for each  $x_i$ , we know
+
+$$
+x _ {i} \cdot e _ {1} = \frac {l _ {i} \cdot l _ {1}}{r _ {i} r _ {1}} = \frac {K ^ {(2)} (i , 1)}{r _ {i} r _ {1}} > 0
+$$
+
+and each  $\| x_{i}\|_{2} = 1$ . Therefore,  $x_{i}\in S$  is defined in Lemma 2.
+
+By the assumption that  $C^{(0)}$  does not contain two linearly dependent rows, we know so are  $K^{(1)}$  and  $K^{(2)}$ . Thus, no two  $l_i$ 's are linearly dependent and hence  $x_i$ 's are pairwise distinct. Using the universality of the arc-cosine kernel (Lemma 2), we know  $C^{(2)} = g(K^{(2)})$  is positive definite. Hence,  $K^{(3)}$  is positive definite. An induction argument on the layer index completes the proof.
+
+# B.4 PROOF OF THEOREM 4
+
+For a covariance matrix  $K$ , denote its standard deviation and correlation by
+
+$$
+\sigma_ {K} (x) = \sqrt {K (x , x)},
+$$
+
+$$
+\rho_ {K} (x, x ^ {\prime}) = \frac {K (x , x ^ {\prime})}{\sqrt {K (x , x) K (x ^ {\prime} , x ^ {\prime})}}.
+$$
+
+From the closed-form formula (5) for the arc-cosine kernel, we have
+
+$$
+\rho_ {C ^ {(l)}} (x, x ^ {\prime}) = \frac {1}{\pi} \left(\sin \theta_ {x, x ^ {\prime}} ^ {(l)} + (\pi - \theta_ {x, x ^ {\prime}} ^ {(l)}) \cos \theta_ {x, x ^ {\prime}} ^ {(l)}\right),
+$$
+
+$$
+\rho_ {K ^ {(l)}} (x, x ^ {\prime}) = \cos \theta_ {x, x ^ {\prime}} ^ {(l)}.
+$$
+
+We define the correlation mapping from  $\rho_{K^{(l)}}$  to  $\rho_{C^{(l)}}$ :
+
+$$
+f: \cos \theta \mapsto \frac {1}{\pi} (\sin \theta + (\pi - \theta) \cos \theta), \quad \theta \in [ 0, \pi ].
+$$
+
+We first establish a few properties of  $f$ . A pictorial illustration of  $f$  is given in Figure 5.
+
+![](images/7f5ffd22b78b377384438049f1aeab78d1f012af6bbc813a142ca745ffd0aa26.jpg)  
+Figure 5: Correlation mapping  $f$ .
+
+Lemma 5. The following facts hold.
+
+1.  $f$  is increasing and it is a contraction mapping on  $[-1, 1]$ .
+
+2.  $f(\rho)\geq \rho$  and the equality holds only when  $\rho = 1$
+
+Proof. For part 1, denote  $\rho = \cos \theta \in [-1,1]$ . Then,
+
+$$
+f ^ {\prime} (\rho) = \frac {\partial f (\rho)}{\partial \theta} \frac {\partial \theta}{\partial \rho} = \frac {- 1}{\pi} (\pi - \theta) \sin \theta \cdot \frac {- 1}{\sin \theta} = \frac {\pi - \theta}{\pi} \in [ 0, 1 ].
+$$
+
+Thus,  $f$  is increasing and it is a contraction mapping on  $[-1, 1]$ .
+
+For part 2, we know that  $f(1) = 1$ . Using the contraction mapping, we know  $f(\rho) \geq \rho$  for  $\rho \in [-1,1]$ . Clearly, the equality holds only when  $\rho = 1$ .
+
+After the correlation mapping  $f$ , we further consider the covariance mapping  $g: K^{(l)} \mapsto C^{(l)}$  defined in (3) (and equivalently in (8)). We establish a few properties of  $g$  below.
+
+Lemma 6. For positive semi-definite matrices  $B$ , the following facts hold.
+
+1.  $g(B) = \frac{1}{2} D^{1/2} f(D^{-1/2} BD^{-1/2}) D^{1/2}$ , where  $D = \mathrm{diag}\{B\} \in \mathbb{R}^{N \times N}$  and  $f$  is elementwise.  
+2.  $g(B) \succ \frac{1}{2} B$  and  $\operatorname{diag}\{g(B)\} = \frac{1}{2} \operatorname{diag}\{B\}$ , where  $\succ$  means elementwise greater.  
+3.  $\operatorname{tr}\left(A g(B) A^T\right) \leq \frac{1}{2} \|A\|_2^2 \operatorname{tr}(B)$ , where  $\|A\|_2$  is the spectral norm of matrix  $A$ .
+
+Proof. For part 1, it is obvious according to (5).
+
+For part 2, using the properties of  $f$  we straightforwardly obtain
+
+$$
+g (B) = \frac {1}{2} D ^ {1 / 2} f (D ^ {- 1 / 2} B D ^ {- 1 / 2}) D ^ {1 / 2} \succ \frac {1}{2} D ^ {1 / 2} (D ^ {- 1 / 2} B D ^ {- 1 / 2}) D ^ {1 / 2} = \frac {1}{2} B.
+$$
+
+For part 3, note that  $g(B)$  is positive semi-definite. Using von Neumann's trace inequality,
+
+$$
+\operatorname {t r} \left(A g (B) A ^ {T}\right) = \operatorname {t r} \left(A ^ {T} A g (B)\right) \leq \sum_ {i = 1} ^ {N} \lambda_ {i} \left(A ^ {T} A\right) \lambda_ {i} (g (B)) \leq \| A ^ {T} A \| _ {2} \sum_ {i = 1} ^ {N} \lambda_ {i} (g (B)),
+$$
+
+where the last term is equal to  $\| A\| _2^2\mathrm{tr}\left(g(B)\right) = \frac{1}{2}\| A\| _2^2\mathrm{tr}\left(B\right)$ .
+
+Now, we are ready to prove Theorem 4.
+
+Proof of Theorem 4, part 1. We focus on uniform convergence over all pairwise correlations. Denote
+
+$$
+\rho_ {\mathrm {m i n}} (K) = \min _ {x, x ^ {\prime}} \rho_ {K} (x, x ^ {\prime}) \in [ - 1, 1 ].
+$$
+
+To simplify notation, we temporarily let  $C = C^{(l)}$ . Then,
+
+$$
+\begin{array}{l} K ^ {(l + 1)} (x, x ^ {\prime}) = \sigma_ {w} ^ {2} \sum_ {v, v ^ {\prime}} A _ {x v} A _ {x ^ {\prime} v ^ {\prime}} C (v, v ^ {\prime}) \\ = \sigma_ {w} ^ {2} \sum_ {v, v ^ {\prime}} A _ {x v} A _ {x ^ {\prime} v ^ {\prime}} \sigma_ {C} (v) \sigma_ {C} (v ^ {\prime}) \rho_ {C} (v, v ^ {\prime}) \\ \geq \rho_ {\mathrm {m i n}} (C) \cdot \sigma_ {w} ^ {2} \sum_ {v, v ^ {\prime}} A _ {x v} A _ {x ^ {\prime} v ^ {\prime}} \sigma_ {C} (v) \sigma_ {C} (v ^ {\prime}) \\ = \rho_ {\mathrm {m i n}} (C) \cdot \sigma_ {w} ^ {2} \left(\sum_ {v} A _ {x v} \sigma_ {C} (v)\right) \left(\sum_ {v ^ {\prime}} A _ {x ^ {\prime} v ^ {\prime}} \sigma_ {C} (v ^ {\prime})\right). \\ \end{array}
+$$
+
+Meanwhile,
+
+$$
+\begin{array}{l} K ^ {(l + 1)} (x, x) = \sigma_ {w} ^ {2} \sum_ {v, v ^ {\prime}} A _ {x v} A _ {x v ^ {\prime}} C (v, v ^ {\prime}) \\ \leq \sigma_ {w} ^ {2} \sum_ {v, v ^ {\prime}} A _ {x v} A _ {x v ^ {\prime}} \sigma_ {C} (v) \sigma_ {C} (v ^ {\prime}) \\ = \sigma_ {w} ^ {2} \left(\sum_ {v} A _ {x v} \sigma_ {C} (v)\right) ^ {2} \\ \end{array}
+$$
+
+and similarly
+
+$$
+K ^ {(l + 1)} \left(x ^ {\prime}, x ^ {\prime}\right) \leq \sigma_ {w} ^ {2} \left(\sum_ {v ^ {\prime}} A _ {x ^ {\prime} v ^ {\prime}} \sigma_ {C} \left(v ^ {\prime}\right)\right) ^ {2}.
+$$
+
+Combining the above inequalities, we obtain
+
+$$
+\rho (K ^ {(l + 1)}) (x, x ^ {\prime}) = \frac {K ^ {(l + 1)} (x , x ^ {\prime})}{\sqrt {K ^ {(l + 1)} (x , x) K ^ {(l + 1)} (x ^ {\prime} , x ^ {\prime})}} \geq \rho_ {\min } (C ^ {(l)}), \quad \forall x, x ^ {\prime},
+$$
+
+and thus
+
+$$
+\rho_ {\min } (K ^ {(l + 1)}) \geq \rho_ {\min } (C ^ {(l)}).
+$$
+
+From the correlation mapping, we know
+
+$$
+\rho_ {\min } (C ^ {(l)}) = f \left(\rho_ {\min } \left(K ^ {(l)}\right)\right).
+$$
+
+Therefore, by the properties of  $f$ ,
+
+$$
+\rho_ {\min } (K ^ {(l + 1)}) \geq f (\rho_ {\min } (K ^ {(l)})) \geq \rho_ {\min } (K ^ {(l)}).
+$$
+
+Thus,  $\rho_{\min}(K^{(l)}) \nearrow$  some  $\rho_{\infty} \in [-1, 1]$ . Because  $\rho_{\infty} = f(\rho_{\infty})$ , we conclude that  $\rho_{\infty} = 1$ .
+
+![](images/749a0b533d7ac4bad7b7b6395808f0dc7d4000260e6f86c19cd6711a95daa89c.jpg)
+
+Proof of Theorem 4, part 2. Consider each  $h: K^{(l)} \mapsto K^{(l+1)}$ :
+
+$$
+K ^ {(l + 1)} = \sigma_ {b} ^ {2} \mathbf {1} _ {N \times N} + \sigma_ {w} ^ {2} A C ^ {(l)} A ^ {T} = \sigma_ {b} ^ {2} + \sigma_ {w} ^ {2} A g (K ^ {(l)}) A ^ {T} := h (K ^ {(l)}).
+$$
+
+Let  $\delta = \frac{1}{2}\sigma_w^2\lambda^2 < 1$ . Using part 3 of Lemma 6,
+
+$$
+\begin{array}{l} \operatorname {t r} \left(K ^ {(l + 1)}\right) = \operatorname {t r} \left(\sigma_ {b} ^ {2} \mathbf {1} _ {N \times N}\right) + \operatorname {t r} \left(\sigma_ {w} ^ {2} A g (K ^ {(l)}) A ^ {T}\right) \\ = N \sigma_ {b} ^ {2} + \sigma_ {w} ^ {2} \operatorname {t r} \left(A g (K ^ {(l)}) A ^ {T}\right) \\ \leq N \sigma_ {b} ^ {2} + \frac {1}{2} \sigma_ {w} ^ {2} \lambda^ {2} \operatorname {t r} \left(K ^ {(l)}\right) \\ = N \sigma_ {b} ^ {2} + \delta \cdot \operatorname {t r} \left(K ^ {(l)}\right). \\ \end{array}
+$$
+
+Therefore, for sufficiently large  $l$ ,
+
+$$
+\operatorname {t r} \left(K ^ {(l)}\right) \leq \frac {N \sigma_ {b} ^ {2}}{1 - \delta} + 1.
+$$
+
+Because each  $K^{(l)}$  is positive semi-definite, we know  $K^{(l)}$  is bounded. Thus, a subsequence of  $K^{(l)} \to$  some matrix.
+
+Proof of Theorem 4, part 3. Define  $\kappa^{(l)}\coloneqq K^{(l)} / c_l$ . We have the following recursive relationship between  $\kappa^{(l)}$  and  $\kappa^{(l + 1)}$ :
+
+$$
+\kappa^ {(l + 1)} = \frac {1}{c _ {l + 1}} (\sigma_ {b} ^ {2} \mathbf {1} _ {N \times N} + \sigma_ {w} ^ {2} A g (K ^ {(l)}) A ^ {T}) = \frac {\sigma_ {b} ^ {2}}{c _ {l + 1}} \mathbf {1} _ {N \times N} + \frac {2}{\lambda^ {2}} A g (\kappa^ {(l)}) A ^ {T}.
+$$
+
+Then,
+
+$$
+\operatorname {t r} \left(\kappa^ {(l + 1)}\right) = \operatorname {t r} \left(\frac {\sigma_ {b} ^ {2}}{c _ {l + 1}} \mathbf {1} _ {N \times N}\right) + \frac {2}{\lambda^ {2}} \operatorname {t r} \left(A g (\kappa^ {(l)}) A ^ {T}\right) \leq \frac {N \sigma_ {b} ^ {2}}{c _ {l + 1}} + \operatorname {t r} \left(\kappa^ {(l)}\right),
+$$
+
+where the inequality results from part 3 of Lemma 8. Therefore,  $\kappa^{(l)}$  is bounded.
+
+Using the Perron-Frobenius theorem, we know the eigenspace  $V$  corresponding to  $\lambda$  is one-dimensional, and there exists  $v\in V$  such that  $\| v\| _2 = 1$  and  $v\succ 0$ . Then,
+
+$$
+v ^ {T} \kappa^ {(l + 1)} v = \frac {\sigma_ {b} ^ {2}}{c _ {l + 1}} v ^ {T} \mathbf {1} _ {N \times N} v + \frac {2}{\lambda^ {2}} v ^ {T} A g (\kappa^ {(l)}) A ^ {T} v \geq 2 v ^ {T} g (\kappa^ {(l)}) v \geq v ^ {T} \kappa^ {(l)} v.
+$$
+
+Therefore,  $v^T \kappa^{(l)} v$  has a limit  $c > 0$ .
+
+Let  $P = \operatorname{Proj}_{V^{\perp}} = I - \operatorname{Proj}_V = I - vv^T$ . Then  $PA = AP$ ,  $\| PA \|_2 = \lambda' < \lambda$ , and
+
+$$
+\operatorname {t r} \left(P g (\kappa^ {(l)}) P ^ {T}\right) = \operatorname {t r} \left(g (\kappa^ {(l)})\right) - v ^ {T} g (\kappa^ {(l)}) v \leq \frac {1}{2} \operatorname {t r} \left(\kappa^ {(l)}\right) - \frac {1}{2} v ^ {T} \kappa^ {(l)} v = \frac {1}{2} \operatorname {t r} \left(P \kappa^ {(l)} P ^ {T}\right).
+$$
+
+Therefore,
+
+$$
+\begin{array}{l} \operatorname {t r} \left(P \kappa^ {(l + 1)} P ^ {T}\right) = \operatorname {t r} \left(\frac {\sigma_ {b} ^ {2}}{c _ {l + 1}} P \mathbf {1} _ {N \times N} P ^ {T}\right) + \frac {2}{\lambda^ {2}} \operatorname {t r} \left(P A g (\kappa^ {(l)}) A ^ {T} P ^ {T}\right) \\ \leq \frac {N \sigma^ {2}}{c _ {l + 1}} + \frac {2}{\lambda^ {2}} \operatorname {t r} \left(P A P g (\kappa^ {(l)}) P ^ {T} A ^ {T} P ^ {T}\right) \\ \leq \frac {N \sigma^ {2}}{c _ {l + 1}} + \frac {2}{\lambda^ {2}} \| P A \| _ {2} ^ {2} \operatorname {t r} \left(P g (\kappa^ {(l)}) P ^ {T}\right) \\ \leq \frac {N \sigma^ {2}}{c _ {l + 1}} + (\frac {\lambda^ {\prime}}{\lambda}) ^ {2} \operatorname {t r} \left(P \kappa^ {(l)} P ^ {T}\right). \\ \end{array}
+$$
+
+Thus,  $P\kappa^{(l)}P^T\to 0$
+
+Combining the results, we have  $\kappa^{(l)}\to cvv^T$
+
+![](images/b4ac9806db67683df89ac927a16b71f08fdfff27bc588391ec80fec9e62af136.jpg)
+
+# B.5 RESULTS FOR MULTI-LAYER PERCEPTRONS
+
+Theorem 4 has parallel results for fully connected layers.
+
+Theorem 7. Let  $A$  be the identity matrix. The following results hold as  $l \to \infty$ .
+
+1. When  $\sigma_w^2 < 2$  and  $\sigma_b^2 > 0$ , then  $K^{(l)} \to q\mathbf{1}_{N \times N}$  where  $q = \frac{\sigma_b^2}{1 - \sigma_w^2 / 2}$ .  
+2. When  $\sigma_w^2 > 2$ , let  $c_l = (\sigma_w^2 / 2)^l$ ; then  $K^{(l)} / c_l \to vv^T$  where  $v_x = \left[\frac{\sigma_b^2}{\sigma_w^2 / 2 - 1} + K^{(0)}(x, x)\right]^{\frac{1}{2}}$ .
+
+Proof of Theorem 7, part 1. Starting with the diagonal elements, we have
+
+$$
+K ^ {(l + 1)} (x, x) = \sigma_ {b} ^ {2} + \frac {\sigma_ {w} ^ {2}}{2} K ^ {(l)} (x, x).
+$$
+
+Therefore,  $K^{(l)}(x,x) \to q = \frac{\sigma_b^2}{1 - \sigma_w^2 / 2}$ .
+
+For the off-diagonal elements, we have
+
+$$
+\begin{array}{l} K ^ {(l + 1)} (x, x ^ {\prime}) = \sigma_ {b} ^ {2} + \frac {\sigma_ {w} ^ {2}}{2} \sigma_ {K ^ {(l)}} (x) \sigma_ {K ^ {(l)}} (x ^ {\prime}) f (\rho_ {K ^ {(l)}} (x, x ^ {\prime})) \\ \geq \sigma_ {b} ^ {2} + \frac {\sigma_ {w} ^ {2}}{2} \sigma_ {K ^ {(l)}} (x) \sigma_ {K ^ {(l)}} (x ^ {\prime}) \rho_ {K ^ {(l)}} (x, x ^ {\prime}) \\ = \sigma_ {b} ^ {2} + \frac {\sigma_ {w} ^ {2}}{2} K ^ {(l)} (x, x ^ {\prime}). \\ \end{array}
+$$
+
+Therefore,
+
+$$
+\lim _ {l} \inf  K ^ {(l + 1)} (x, x ^ {\prime}) \geq \sigma_ {b} ^ {2} + \frac {\sigma_ {w} ^ {2}}{2} \lim _ {l} \inf  K ^ {(l)} (x, x ^ {\prime}) \Rightarrow \lim _ {l} \inf  K ^ {(l)} (x, x ^ {\prime}) \geq q.
+$$
+
+Meanwhile,
+
+$$
+\lim _ {l} \sup K ^ {(l)} (x, x ^ {\prime}) \leq \lim _ {l} \sup \sqrt {K ^ {(l)} (x , x) K ^ {(l)} (x ^ {\prime} , x ^ {\prime})} = q.
+$$
+
+Thus,  $K^{(l)}(x,x') \to q$ . In conclusion,  $K^{(l)} \to q\mathbf{1}_{N \times N}$ .
+
+Proof of Theorem 7, part 2. Again, we start with diagonal elements. Define  $\kappa^{(l)}\coloneqq K^{(l)} / c_l$ . When  $\sigma_w^2 >2$ , we have
+
+$$
+\kappa^ {(l + 1)} (x, x) = \frac {\sigma_ {b} ^ {2}}{(\sigma_ {w} ^ {2} / 2) ^ {l + 1}} + \kappa^ {(l)} (x, x).
+$$
+
+Therefore,
+
+$$
+\lim _ {l \to \infty} \kappa^ {(l)} (x, x) = \sum_ {l = 0} ^ {\infty} \frac {\sigma_ {b} ^ {2}}{(\sigma_ {w} ^ {2} / 2) ^ {l + 1}} + \kappa^ {(0)} (x, x) = \frac {\sigma_ {b} ^ {2}}{\sigma_ {w} ^ {2} / 2 - 1} + K ^ {(0)} (x, x) = v _ {x} ^ {2}.
+$$
+
+For the off-diagonal elements, we have
+
+$$
+\begin{array}{l} \kappa^ {(l + 1)} (x, x ^ {\prime}) = \frac {\sigma_ {b} ^ {2}}{c _ {l + 1}} + \sigma_ {\kappa^ {(l)}} (x) \sigma_ {\kappa^ {(l)}} (x ^ {\prime}) f (\rho_ {\kappa^ {(l)}} (x, x ^ {\prime})) \\ \geq \sigma_ {\kappa^ {(l)}} (x) \sigma_ {\kappa^ {(l)}} \left(x ^ {\prime}\right) \rho_ {\kappa^ {(l)}} (x, x ^ {\prime}) \\ = \kappa^ {(l)} (x, x ^ {\prime}). \\ \end{array}
+$$
+
+Thus,  $\kappa^{(l)}(x,x^{\prime})$  has a limit  $c$ . Taking limits on both sides, we have
+
+$$
+c = v _ {x} v _ {x ^ {\prime}} \cdot f \left(c / \left(v _ {x} v _ {x ^ {\prime}}\right)\right),
+$$
+
+which suggests that  $c = v_{x}v_{x^{\prime}}$ . In conclusion,  $\kappa^{(l)}\to vv^T$
+
+![](images/adf52f2790c79c9db3173765858b27aae545d9bba17a47a1e5173e1df4edcb6a.jpg)
+
+# C COMPOSING GRAPHSAGE
+
+As before,  $X$  denotes pre-activation rather post activation. A GraphSAGE layer has two weight matrices,  $W_{1}$  and  $W_{2}$ . Hence, we use two variance parameters  $\sigma_{w_1}$  and  $\sigma_{w_2}$  for them, respectively. For simplicity, we focus on the mean-aggregation version of GraphSAGE, where  $A$  is effectively the random-walk matrix. The updates are:
+
+GraphSAGE:  $X \gets \phi(X)W_{1} + A\phi(X)W_{2}$
+
+$$
+K \gets \sigma_ {w _ {1}} ^ {2} g (K) + \sigma_ {w _ {2}} ^ {2} A g (K) A ^ {T}
+$$
+
+$$
+Q \leftarrow \left[ \sigma_ {w _ {1}} P \quad \sigma_ {w _ {2}} A P \right] \quad \text {w h e r e} \quad P = \operatorname {C h o l} (g (Q Q ^ {T})).
+$$
+
+Note that we do not use a bias term. If it is desired, one can easily add it to the updates following the example of GCN.
+
+# D EXPERIMENT DETAILS
+
+Datasets. A summary is given in Table 5. The datasets Cora/CiteSeer/PubMed/Reddit, with predefined training/validation/test splits, are downloaded from the PyTorch Geometric library (Fey & Lenssen, 2019) and used as is. The dataset ArXiv comes from the Open Graph Benchmark (Hu et al., 2020b). The datasets Chameleon/Squirrel/Crocodile come from MUSAE (Rozemberczki et al., 2021). The training/validation/test splits of the former two sets of datasets come from GeomGCN (Pei et al., 2020), in accordance with the PyTorch Geometric library. The split for Crocodile
+
+Table 5: Dataset summary. The edge column lists the number of directed edges.  
+
+<table><tr><td>Dataset</td><td>Task</td><td>Nodes</td><td>Edges</td><td>Features</td><td>Train/Val/Test Ratio</td></tr><tr><td>Cora</td><td>Classification</td><td>2,708</td><td>5,429</td><td>1,433</td><td>0.05/0.18/0.37</td></tr><tr><td>Citeseer</td><td>Classification</td><td>3,327</td><td>4,732</td><td>3,703</td><td>0.04/0.15/0.30</td></tr><tr><td>PubMed</td><td>Classification</td><td>19,717</td><td>44,338</td><td>500</td><td>0.003/0.025/0.051</td></tr><tr><td>ArXiv</td><td>Classification</td><td>169,343</td><td>1,166,243</td><td>128</td><td>0.54/0.18/0.28</td></tr><tr><td>Reddit</td><td>Classification</td><td>232,965</td><td>11,606,919</td><td>602</td><td>0.66/0.10/0.24</td></tr><tr><td>Chameleon</td><td>Regression</td><td>2,277</td><td>36,101</td><td>3,132</td><td>0.48/0.32/0.20</td></tr><tr><td>Squirrel</td><td>Regression</td><td>5,201</td><td>217,073</td><td>3,148</td><td>0.48/0.32/0.20</td></tr><tr><td>Crocodile</td><td>Regression</td><td>11,631</td><td>180,020</td><td>13,183</td><td>0.48/0.32/0.20</td></tr></table>
+
+is not available, so we conduct a random split with the same 0.48/0.32/0.20 proportion as that used for Chameleon and Squirrel (Rozemberczki et al., 2021).
+
+Dataset preprocessing. For each graph, we treat the edges as undirected and construct a binary, symmetric adjacent matrix  $\underline{A}$ . Then, we apply the normalization proposed by Kipf & Welling (2017) to define  $A$  used in (1). Specifically,  $A = (I_N + D)^{-1/2}(I + \underline{A})(I_N + D)^{-1/2}$ , where  $D = \mathrm{diag}\{\sum_{j} \underline{A}_{ij}\}$ . For GraphSAGE and GGP, we instead use the row-normalized version  $A = (I_N + D)^{-1}(I_N + \underline{A})$ . No feature preprocessing is done except running the GPs on ArXiv, for which we apply a centering on the input features.
+
+Environment. All experiments are conducted on a Nvidia Quadro GV100 GPU with 32GB of HBM2 memory. The code is written in Python 3.10.4 as distributed with Ubuntu 22.04 LTS. We use PyTorch 1.11.0 and PyTorch Geometric 2.1.0 with CUDA 11.3.
+
+Training. For all datasets except Reddit, we train GCN using the Adam optimizer in full batch for 100 epochs. Reddit is too large for GPU computation and hence we conduct mini-batch training (with a batch size 10240) for 10 epochs.
+
+Hyperparameters. For classification tasks in Table 3, the hyperparameters are set to  $\sigma_{b} = 0.0$ ,  $\sigma_{w} = 1.0$ ,  $L = 2$ , hidden = 256, and dropout = 0.5. GCN is trained with learning rate 0.01. For regression tasks, they are set to  $\sigma_{b} = \sqrt{0.1}$ ,  $\sigma_{w} = 1.0$ ,  $L = 2$ , hidden = 256, and dropout = 0.5. GCN is trained with learning rate  $\sqrt{0.1}$
+
+We choose  $C^{(0)}$  to be the inner product kernel. The exception is when working with the low-rank versions, we apply PCA on the input features to ensure the number of features is smaller than the number of landmark nodes.
+
+For Table 4: GCNII uses a 2-layer architecture and sets  $\alpha = 0.1$ ,  $\lambda = 0.5$ ,  $\beta_{l} = \log \left(\frac{\lambda}{l} + 1\right)$ . GIN uses  $L = 2$  and  $\epsilon = 0.0$ . GraphSAGE uses a 2-layer architecture and sets  $\sigma_{w_1} = \sqrt{0.1}$  for Reddit and  $\sigma_{w_1} = 0.0$  for other datasets, and  $\sigma_{w_2} = 1.0$ .
+
+The nugget  $\epsilon$  is chosen using a grid search over  $[10^{-3}, 10^{1}]$ .
+
+For GCNGP-X, GNNGP-X and RBF-X, the landmark points are chosen to be the training set for small datasets, while for large datasets (ArXiv and Reddit) the landmark points are a random 1/50 of the training set.
+
+Kernels. For the RBF kernel,
+
+$$
+K (x, x ^ {\prime}) = \exp (- \gamma \| x - x ^ {\prime} \| _ {2} ^ {2}),
+$$
+
+where  $\gamma$  is chosen using a grid search over  $[10^{-2}, 10^{2}]$ . For the GGP kernel,
+
+$$
+K = A K _ {0} A ^ {T}, \quad K _ {0} (x, x ^ {\prime}) = (x ^ {T} x ^ {\prime} + c) ^ {d},
+$$
+
+where  $c = 5.0, d = 3.0$ . Note that we use only the kernel but not the robust-max likelihood nor the ELBO training proposed by Ng et al. (2018), because their code written in Python 2 is outdated and cannot be used. Note also that GGP-X in Ng et al. (2018) has a different meaning—training GGP by using additionally the 500 nodes in the validation set—while we use GGP-X to denote the Nyström approximation of GGP.

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