Curriculum Graph Machine Learning: A Survey

1 Introduction

Graph structured data is ubiquitous in the real world, which has been widely used to model the complex relationships and dependencies among various entities. In the past decade, graph machine learning approaches, especially graph neural networks (GNNs) [Kipf and Welling, 2017; Veliˇckovi´c et al., 2018; Hamilton et al., 2017], have drawn ever-increasing attention in both academia and industry, which make great progress in a variety of applications across wide-ranging domains, ranging from physics [Sanchez-Gonzalez et al., 2020] to chemistry [Gilmer et al., 2017], and from neuroscience [de Vico Fallani et al., 2014] to social science [Zhang and Tong, 2016]. Other areas, such as recommender systems [Wu et al., 2022], knowledge graphs [Wang et al., 2017], molecular prediction [Hu et al., 2020a], medical detection [Horry et al.,

2020] etc., also provide an increasing demand for the applications of graph machine learning.

Despite the popularity of graph machine learning approaches, the existing literature generally trains graph models by feeding the data samples in a random order during the training process. For example, when training GNNs, the widely adopted mini-batch stochastic gradient descent optimization strategy as well as its variants select the data samples in each mini-batch randomly. However, such training strategies largely ignore the importance of different graph data samples and how their orders can affect the optimization status, which may result in suboptimal performance of the graph learning models [Wei et al., 2022; Wang et al., 2021b]. Typically, humans tend to learn much better when the data examples are organized in a meaningful order rather than randomly presented, e.g., learning from basic easy concepts to advanced hard concepts resembling the "curriculum" taught in schools [Elman, 1993; Rohde and Plaut, 1999]. To this end, curriculum learning (CL) [Bengio et al., 2009; Wang et al., 2021a; Soviany et al., 2022] is proposed to mimic human's learning process, and has been proved to be effective in boosting the model performances as well as improving the generalization capacity and convergence of learning models in various scenarios including computer vision [Guo et al., 2018; Jiang et al., 2014], natural language processing [Platanios et al., 2019; Tay et al., 2019] etc.

Curriculum graph machine learning (Graph CL), which combines the strength of graph machine learning and curriculum learning, has become a promising research direction and attracted an increasing number of interests from the community, spanning over a variety of graph learning methodologies and applications recently. Facing opportunities as well as challenges, we believe it is the right time to review and promote the studies of Graph CL approaches. In this survey, we provide a comprehensive and systematic review of Graph CL. Specifically, we first analyze the key challenges of Graph CL and introduce the basic formulations. Then, we summarize the existing methodologies into three categories based on the granularity of graph tasks, i.e., node-level, link-level, and graph-level tasks, and elaborate representative approaches in each category. Last but not least, we discuss potential future research topics, which could shed light on the development of this promising area. We deeply hope this survey may provide insights to promote the Graph CL research in the community.

2 Challenges and ProblemFormulation

Curriculum graph machine learning (Graph CL), as an emerging research topic in the machine learning community, which non-trivially combines the strength of graph machine learning and curriculum learning, faces the following challenges.

• Uniqueness of curriculum graph learning problem. Unlike image or text, graph data lies in a non-Euclidean space. Besides, there exist complex relationships and dependencies between entities in graphs. Therefore, the nonEuclidean nature and complexity of graph-structured data bring unique challenges in tackling Graph CL problem.
• Complexity of curriculum graph learning method. Although general CL algorithms [Wang et al., 2021a; Soviany et al., 2022] have been extensively studied, it is non-trivial to combine the advantages of graph machine learning and curriculum learning into one unified framework, especially when customizing the design of the key CL components such as the difficulty measurer and the training scheduler that are compatible with graph models.
• Diversity of curriculum graph learning task. Curriculum graph learning tasks range from node-level and link-level to graph-level problems with different settings, objectives, constraints, and domain knowledge. Therefore, it is important yet difficult to develop graph CL approaches tailored for different graph tasks or applications.

The methodologies reviewed in this paper target on dealing with at least one of these three key challenges. Before introducing the problem formulation of curriculum graph machine learning (Graph CL), we briefly describe the background of graph machine learning and curriculum learning.

2.1 GraphMachine Learning

Existing graph machine learning models can be generally divided into the following two groups: network embedding (NE) [Cui et al., 2018] and graph neural networks (GNNs) [Zhang et al., 2020]. Specifically, GNNs as the current state-of-the-art in graph machine learning, have been widely adopted to serve as the backbone of a graph curriculum learning method.

Let G = (V, E) be a graph with the node set V and the edge (link) set E ⊆ V × V. GNNs first learn node representations by the following message-passing [Gilmer et al., 2017; Zhang et al., 2022a] function:

where h(l) v is the node representation of node v at the lth layer, which is usually initialized as node feature at the first layer. Nv denotes the neighbors of node v. AGG(·) and COM(·) denote the aggregation and combination function of GNNs [Wu et al., 2020a]. After deriving the node representations, the link representations can be obtained based on the representations of the two connected nodes. Furthermore, graph-level representations can be computed by the readout (or pooling) [Xu et al., 2019] operation on all nodes in this graph: h(l) G = READOUT {h(l) v | v ∈ V }. Finally, the representations of nodes, links, or graphs can be applied to different levels of graph tasks.

2.2 CurriculumLearning

Curriculum learning (CL), which mimics the human's learning process of learning data samples in a meaningful order, aims to enhance the machine learning models by using designed training curriculum, typically following an easy-tohard pattern. As a general and flexible plug-in, the CL strategy has demonstrated its power in improving the model performance, generalization, robustness, and even convergence in a wide range of scenarios [Liu et al., 2023a]. In general, the framework of curriculum learning consists of a difficulty measurer and a training scheduler [Wang et al., 2021a]. The difficulty measurer is adopted to calculate the difficulty score for each data sample and the training scheduler aims to arrange data samples in a meaningful order as the curriculum for training, based on the judgment of the difficulty measurer.

The existing CL methods fall into two groups: (1) predefined CL that manually designs heuristic-based policies to decide the training order, and (2) automatic CL that relies on computable metrics (e.g., the training loss) to automatically design the curriculum for model training. Predefined CL can utilize expert knowledge but ignores the model's feedback, and automatic CL is domain-agnostic and more general as well as able to consider the model's feedback. That is to say, automatic CL is more data-driven or model-driven instead of human-driven, and more dynamically adaptive to the current training status, than predefined CL. In addition, the training schedulers can be divided into discrete and continuous schedulers. The discrete schedulers adjust training after every fixed number (> 1) of epochs or convergence on the current data samples, while the continuous schedulers adjust the training at every epoch.

2.3 CurriculumGraphMachine Learning

Let C = ⟨Q1, . . . , Qt, . . . , QT ⟩ denote a curriculum for training the graph model, which consists of a sequence of training data subsets over T training steps. Each data subset Qt contains current training samples to be fed into the learning model at time step t. The order of all data subsets is determined by the difficulty measurer and the training scheduler. Therefore, the problem of curriculum graph machine learning (Graph CL) can be formulated in the following:

Problem 1. (Curriculum Graph Machine Learning). Given the training set of data instances (i.e., nodes, links, or graphs) D = {(Xi, Yi)}|D| i=1 , where Xi is the input instance and Yi denotes the label, the goal is to optimize an optimal graph learning model guided by the curriculum C, so as to achieve the best performance on testing graph data.

3 Categorization

To tackle the non-trivial challenges introduced in Section 2, considerable efforts have been made in the literature, which integrate the strengths of graph machine learning and curriculum learning, and further propose tailored methods. Next, we will comprehensively review the existing methodologies by categorizing them into three groups based on the granularity of graph tasks, i.e., node-level, link-level, and graph-level tasks, followed by elaborations on representative approaches.

Furthermore, for each category, we divide the methodologies into predefined graph CL and automatic graph CL, based on the type of curriculum learning strategy, i.e., whether adopting manually designed heuristic-based policies or automatically computable metrics to derive the curriculum for training. The categorization and characteristics of surveyed graph CL methods are summarized in Table 1.

4 Node-level Graph CL

Node is the fundamental unit for graph formation. Typically, the key to tackle node-level tasks is to learn node representations by training graph models (e.g., GNNs). Therefore, several studies on node-level graph CL are proposed to train graph models by starting with easy nodes and gradually include harder nodes during the training process, including both predefined and automatic node-level graph CL approaches.

4.1 Predefined Node-level Graph CL

Some methods define heuristic metrics to measure the difficulty of nodes in advance of the training process, e.g., by considering nodes' properties in terms of the topology, the feature, or the label.

CLNode (Curriculum Learning for Node Classification) [Wei et al., 2022] is a curriculum learning framework for node-level representation learning of GNNs. It boosts the performance of backbone GNNs via incrementally introducing nodes into the training process, starting with easy nodes and progressing to harder nodes. The multi-perspective difficulty measurer is proposed to measure the difficulty of training nodes based on the label information. Specifically, the local perspective's difficulty measurer calculates local label distribution to recognize inter-class difficult nodes whose neighbors have diverse labels. And the global perspective's difficulty measurer recognizes mislabeled difficult nodes in terms of node feature. The continuous training scheduler is introduced to select appropriate training nodes in each epoch to mitigate the detrimental effect of difficult nodes. CLNode can be compatible with most existing GNNs and boost their performances without increasing the time complexity just by feeding training nodes in order from easy to hard.

GNN-CL (Graph Neural Network with Curriculum Learning) [Li et al., 2022c] introduces curriculum learning into the imbalanced node classification task by controlling the training procedure from easy to hard, which consists of two modules. The first one is an adaptive graph oversampling module, which interpolates the most significant samples related to the original structure, so as to dynamically adjust the data distribution in the graph from imbalance to balance. The second one is a neighbor-based metric learning module. The distances between nodes and the connected neighbors are regularized based on the pseudo labels, which can dynamically adjust the position of the embeddings of minority class nodes in feature space. GNN-CL balances the label classification loss and neighbor-based triplet loss [Wang et al., 2019] in the whole training process. The curriculum strategy consisting of two opposite learning curves is adopted. At the beginning of the training process, it pays more attention to optimizing feature propagation as well as reducing biased noises in the soft feature space. And it gradually focuses more on the average accuracy in each class, leading to strong accuracy on imbalanced node classification datasets.

SMMCL (Soft Multi-Modal Curriculum Learning) [Gong et al., 2019] proposes a graph CL method for the label propagation on graphs [Iscen et al., 2019]. The goal is to learn the labels for predictions of unlabeled samples on graphs. Specifically, the authors assume that different unlabeled samples have different difficulty levels for propagation, so it should follow an easy-to-hard sequence with updated curriculum for label propagation. They also claim that the real-world graph data is often in multiple modalities [Gao et al., 2017], where each modality should be equipped with a "teacher" that not only evaluates the difficulties of samples from its own viewpoint, but cooperates with other teachers to generate the overall simplest curriculum samples for propagation. They take the curriculum of the teachers as a whole, so that the common preference (i.e., commonality) of teachers on selecting samples can be captured. Finally, an accurate curriculum sequence is established and the propagation quality can thus be improved, leading to more accurate label prediction results.

In addition to the supervised methods above that heavily rely on labels for training, there exist some self-supervised contrastive node-level graph CL methods.

DiGCL (Directed Graph Contrastive Learning) [Tong et al., 2021] considers multi-task curriculum learning to progressively learn from multiple easy-to-hard contrastive views in directed graph contrastive learning. Specifically, to maximize the mutual information between the representations of different contrastive views [You et al., 2020] and produce informative node representations, it introduces a generalized dynamic-view contrastive objective. The multi-task curriculum learning strategy is proposed to divide multiple contrastive views into sub-tasks with various difficulties and progressively learn from easy-to-hard sub-tasks. The different contrastive view pairs generated from Laplacian perturbation on the input graph are scored by a difficulty measurer that is a predefined function in terms of the Laplacian perturbation [Tong et al., 2020]. And three common function families are considered as the training scheduler, including logarithmic, exponential, and linear functions. Note that this method is first to introduce curriculum learning in directed graph contrastive learning for node-level representation learning.

HSAN (Hard Sample Aware Network) [Liu et al., 2023b] also considers curriculum learning scheme in contrastive clustering on graphs. Specifically, it first introduces a similarity measure criterion between training pairs in graph contrastive learning, which measures the difficulty considering both attribute and structure information and improves the representativeness of the selected hard negative samples. Besides, to overcome the drawback of classical graph contrastive learning that the hard node pairs are treated equally, it proposes a dynamic weight modulating function to adjust the weights of sample pairs during training, which can upweight the hard node pairs and down-weight the negative ones. The focusing factor controls the down-weighting rate of easy sample pairs, as the training scheduler of this method. Thus, the discriminative capability of learned representations is enhanced, leading to better performances.

MTGNN (Multivariate Time Series GNN) [Wu et al., 2020b] is a graph CL method designed specifically for multivariate time series data [Box et al., 2015], which advocates a curriculum learning strategy to find a better local optimum of the GNN and splits multivariate time series into subgroups during training. Specifically, since directly optimizing the traditional objective enforces the model to focus too much on improving the accuracy of long-term predictions while ignoring short-term ones, this method proposes a curriculum learning strategy for the multi-step forecasting task. The training is scheduled by starting with solving the easiest problem, i.e., only predicting the next one-step, which helps the model to find a good starting point. As the training progresses, the prediction length of the model is gradually increased, so that the model can learn the hard task step by step. Overall, it is one effective trial combining GNN and curriculum learning in the application of multivariate time series forecasting.

4.2 Automatic Node-level Graph CL

We next describe automatic node-level graph CL methods, which consider the model's feedback during training to dynamically adapt to the optimization status.

MentorGNN [Zhou et al., 2022a] derives a curriculum for pre-training GNNs to learn informative node representations. In order to tailor complex graph signals to boost the generalization performances, it develops a curriculum learning paradigm that automatically reweights graph signals for good generalization of the pre-trained GNNs in the target domain. Specifically, a teacher model that is a graph signal reweighting scheme gradually generates a domain-adaptive curriculum to guide the pre-training process of the student model that is a GNN architecture, so that the generalization performance in the node classification tasks can be enhanced. The curriculum is a sequence of graph signals that are extracted from the given graph. And the learned sample weighting scheme specifies a curriculum under which the GNNs are pre-trained gradually from the easy samples to the hard samples. The difficulty of training samples is measured by the teacher model and the training process is scheduled by the introduced learning threshold controlling the sample selection, so it belongs to automatic graph CL. The accuracy of this method on node classification in the graph transfer setting [Hu et al., 2020b] is largely enhanced.

RCL (Relational Curriculum Learning) [Zhang et al., 2023] claims that existing GNNs learn suboptimal node representations since they usually consider every edge of the input graph equally. Also, most graph CL methods simply consider nodes as independent samples for training, and introduce curriculum learning for these independent samples, which largely ignore the fundamental and unique dependency information behind the graph topology structure, and thus can not well deal with the correlation between nodes. To tackle this problem, it proposes a graph CL method, which leverages the various underlying difficulties of data dependencies, to train better GNNs that can improve the quality of learned node representations. Specifically, it considers the relation between nodes gradually into training based on the relation's difficulty from easy to hard, where the degree of difficulty is measured by a self-supervised learning paradigm instead of a predefined heuristic-based metric. Then it develops an optimization model to iteratively increment the training structure according to the model training status and a theoretical guarantee of the convergence on the optimization algorithm is provided. Finally, it presents an edge reweighting strategy to smooth the transition of the training structure between iterations, and reduce the influence of edges that connect nodes with relatively low confidence embeddings.

DRL (Learn curricula with Deep Reinforcement Learning) [Qu et al., 2018] studies learning curriculum for node representations in heterogeneous star network [Sun et al., 2009] that has a center node type linked with multiple attribute node types through different types of edges, namely learning a sequence of edges of different types for node representation learning. A curriculum is defined as a sequence of edge types used for training, so the problem is formulated as a Markov decision process [Puterman, 1990]. It learns the optimal curriculum by estimating the Q value of each state-action pair, namely the expected cumulative reward after taking the action from the state, by a planning module and a learning module. Finally, the meaningful curriculum can be learned with high accuracy and low time costs for enhancing the performance of node classification.

GAUSS (Graph ArchitectUre Search at Scale) [Guan et al., 2022] is one large-scale node-level representation learning method by searching the GNN's architecture with curriculum learning. Since this method focuses on the large-scale graph whose efficiency issue becomes the main obstacle, it proposes a graph sampling-based single-path one-shot supernet [Pham et al., 2018] to reduce the computation burden. To address the consistency collapse issues, it explicitly considers the joint architecture-graph sampling via a GNN architecture curriculum learning mechanism on the sampled sub-graphs and an architecture importance sampling algorithm [Tokdar and Kass, 2010]. Specifically, it first forms a curriculum learning group for the GNN's architecture, and then makes the best learner as the teacher to decide a smoother learning objective for the group. The importance sampling is also utilized to reduce the variance during architecture sampling to form better learning group. Experiments of node classification on the large-scale graph show the effectiveness of this graph CL method.

CGCT (Curriculum Graph Co-Teaching) [Roy et al., 2021] presents a graph CL method for the multi-target domain adaptation based on feature aggregation and curriculum learning. The authors claim that learning robust and generalized representations in a unified space is just one prerequisite for tackling minimum risk across multiple target domains, where GNNs can play an important role in aggregating semantic information from neighbors across different domains on graphs represented as the source and target nodes. Then, the curriculum learning strategy is advocated in the proposed co-teaching framework to obtain pseudo-labels in an episodic fashion for mitigating information absence for the target nodes. Furthermore, an easy-to-hard curriculum learning strategy for domain selection is also proposed, where the feature alignment starts with the target domain that is closest to the source (easy one) and then gradually progresses towards the hard one, making the feature alignment process smoother. The experiments show that the performance on multi-target domain adaptation settings is improved by this curriculum graph co-teaching scheme.

5 Link-level Graph CL

Links explicitly interconnect nodes in a graph, representing the relations and dependencies between nodes. Compared with node-level graph CL, link-level graph CL measures the difficulty and schedules the training on links.

5.1 Predefined Link-level Graph CL

GCN-WSRS (GCN for Web-Scale Recommender Systems) [Ying et al., 2018] presents a curriculum learning for negative sampling in link prediction tasks, whose effectiveness is validated in recommendation datasets. Specifically, the GNN model for learning link representations is fed easyto-hard samples during the training process, resulting in performance gains. At the first training epoch, there are no hard negative links used for training, so the GNN model can quickly converge to find an area in the parameter space where the loss is small. Note that the hard samples are those related to the query samples, but not as related as the positive samples in link predictions. The training scheduler outputs the rank of items in a graph according to the Personalized PageRank scores [Gasteiger et al., 2018] with respect to the query sample. As training proceeds, the harder negative links are added to the training set in subsequent epochs, which encourages the GNN model turns to learn how to distinguish highly related samples from only slightly related ones. In particular, e - 1 hard negative samples are added to the training set at the epoch e.

TUNEUP [Hu et al., 2022] is a two-stage curriculum learning strategy for better training GNNs, showing competitive performance gains against traditional GNNs in predicting new links in a graph given existing links on citation networks, protein-protein networks, and recommendation benchmarks. The link scores are produced by the inner product of the two connected nodes. Therefore, TUNEUP first trains a GNN to perform well on relatively easy head nodes (nodes with large degrees) and then proceeds to finetune the GNN to also perform well on hard tail nodes (nodes with small degrees). The node degrees are used to measure the difficulty of training samples. Specifically, in the first stage, TUNEUP randomly presents training samples to train the GNN model for obtaining a strong base GNN model. This base GNN model is encouraged to learn better representations for the head nodes, but performs poorly on the tail nodes. Thus, to handle this problem, in the second training stage, it further finetunes the base GNN model with increased supervisions on tail nodes, which follow the two steps: synthesizing more tail node inputs and adding target supervisions on the synthetic tail nodes. In addition to improving the performance of link predictions, it also performs well in learning node representations.

CHEST (Curriculum pre-training based HEterogeneous Subgraph Transformer) [Wang et al., 2023] designs a curriculum graph pre-training strategy to gradually learn from both local and global contexts in the subgraph, which helps the GNN model to more effectively capture useful information for link prediction in recommendation. Compared with the supervised methods above, this method focuses on curriculum graph pre-training tailored to the link prediction. Specifically, the difficulty measurer is predefined on several pretraining tasks. Three easy pre-training curricula are related to node, edge and meta-path [Sun et al., 2011], focusing on local context information within the subgraph. And a hard pretraining curriculum is a subgraph contrastive learning task, focusing on global context information at subgraph level for user-item interaction. The training scheduler is hand-crafted that schedules the pre-training tasks from the curricula in an "easy-to-hard" order, which is necessary to model complex data relations. The learned GNN model after curriculum learning can produce representations that are aggregated into obvious clusters, by gradually extracting useful information for user-item interaction, to improve the link prediction tasks.

5.2 Automatic Link-level Graph CL

GTNN (Graph Text Neural Network) [Vakil and Amiri, 2022] is one representative automatic link-level graph CL method. It trains GNN with trend-aware curriculum learning, which improves the performance on relation extraction that is one type of link-level task. Inspired by the SuperLoss (SL) [Castells et al., 2020] which is a generic curriculum learning approach that dynamically learns a curriculum from model status, this method further designs Trend-SL curriculum learning approach, which belongs to self-paced curriculum learning [Wang et al., 2021a]. Specifically, SuperLoss ignores the trend of instantaneous losses at sample-level that can not only improve the difficulty estimations of the model by making them local, sample dependent and potentially more precise, but also enable the model to distinguish samples with similar losses based on their known loss trajectories. In contrast, Trend-SL takes into account the loss information from the local time window before each iteration for capturing a form of momentum of loss on rising or falling trends and producing individual sample weights. Trend-SL adopts trend dynamics to shift the difficulty boundaries and adjust global difficulty using local sample-level loss trends. With the help of the Trend-SL, the performance of relation extraction can be improved on several benchmarks.

6 Graph-level Graph CL

Compared with node-level and link-level graph CL, graphlevel graph CL focuses more on the global high-level properties of the whole graph to design the difficulty measurer and training scheduler for curriculum learning.

6.1 Predefined Graph-level Graph CL

CurGraph [Wang et al., 2021b] proposes a curriculum learning method for graph classification tasks via easy-to-hard curriculum. It first obtains graph-level embeddings via unsupervised GNN scheme InfoGraph [Sun et al., 2019] and further derives a neural density estimator to model embedding distributions. Then, to tackle the challenges of evaluating the difficulty of graphs induced by high irregular nature of graph data, it analyzes the graph difficulty in the high-level semantic feature space. The difficulty scores of graphs are calculated by a predefined difficulty measurer based on the intra-class and inter-class distributions of their embeddings. For the training scheduler, a smooth-step method is proposed to provide a soft transition from easy to hard graphs for GNNs. At each training step, the trained GNNs focus on the samples that are near the border of capability and neither too easy nor too hard, to expand the border gradually. Finally, the performances of graph classification are enhanced without extra inference cost by feeding the graphs in an easy-to-hard order for training.

CuCo [Chu et al., 2021] incorporates curriculum learning into self-supervised graph-level representation learning. Compared with the supervised graph CL methods, this method focuses more on designing curriculum for selecting and training negative samples effectively, which are important steps in learning self-supervised graph representations. Similarly, it follows the learning process of humans by starting with easy negative samples when learning a new model and then learning difficult negative samples gradually. The difficulty measurer evaluates the difficulty of negative samples in the training dataset, which is calculated based on the embedding's similarity between the negative and positive samples. In addition, the training scheduler is proposed to decide how the negative samples are introduced to the training procedure by utilizing common function families: logarithmic, linear, quadratic, and root [Wang et al., 2021a; Soviany et al., 2022]. This method is the first to study the impact of negative samples in graph-level contrastive representation learning by introducing curriculum learning.

HACL (Hyper-graph based Attention Curriculum Learning) [Ahmed et al., 2022] is a graph attention curriculum learning approach to learn hypergraph representations. Note that the hypergraphs are converted from the text written by the patients, which can be used to identify depressive symptoms of the patients. Based on the semantic vectors from an emotion-driven context extraction technique and the structural hypergraph, this method separates the important boundary elements from the unlabeled sample and then incorporates them into the curriculum learning mechanism for the training process. Finally, the performances of classifying the nine particular symptoms are highly boosted due to the tailored designs on graph CL.

Dual-GCN [Dong et al., 2021] is one GNN model encoding structure information from both local and global levels with curriculum learning as the training strategy, whose capability is improved on the image captioning task. Specifically, the local object-level GCN first converts one image into one graph, where the region to region relation in the image is modeled by the graph topology. Besides, it also introduces auxiliary information to take into account the image's similarity relation by the global image-level GCN. To train this DualGCN, curriculum learning is adopted as the training strategy, where a cross-review scheme is introduced to distinguish the difficulty of the training samples. The difficulty is measured by a predefined metric, which is used to sort the training samples and divide them into several subsets. After that, all the training samples are scheduled in the order from easy subsets to difficult subsets.

6.2 Automatic Graph-level Graph CL

CurrMG [Gu et al., 2022] designs a curriculum learning strategy to learn representations for molecular graphs. It arranges the easy-to-hard training curriculum in predefined, (automatic) transfer teacher, and hybrid types [Hacohen and Weinshall, 2019]. To design molecular graph CL with high robustness and applicability, six optional difficulty measurers, inspired by chemical domain knowledge and prior task information, are proposed, which can be divided into structurespecific, task-specific and hybrid types. For the training scheduler, it is infeasible to split the training data based on molecular difficulties explicitly, as the difference of difficulty distributions is calculated by different difficulty measurers. Therefore, a monotonically increasing smooth curve called competence function is introduced to control the data sampling space. The competence value is treated as a threshold for decreasing the data sampling space continuously, which is used for sampling current batch data during the training process. Finally, the training scheduler encourages the GNN model to pay attention to the easy graphs in the early training stage, and gradually broaden the learning scope by contacting those difficult graphs.

HAC-TSP [Zhang et al., 2022b] proposes to design hardness-adaptive curriculum for solving travelling salesman problem (TSP) [Applegate et al., 2011] from the perspective of graphs. Compared with classical graph prediction tasks, defining a quantitative hardness measurement is challenging since obtaining the ground-truth solution of a TSP instance is NP-hard [Hochba, 1997]. Therefore, the difficulty measurer calculates the hardness as greedy self-improving potentials by comparing the current cost with a surrogate model, avoiding the unbearable computational costs of calculating the ground-truth optimal solution for TSP. Then, a hardnessadaptive generator is designed to efficiently and continuously generate instances with different levels of hardness tailored for model training. Finally, it proposes a curriculum learner to fully utilize the hardness-adaptive TSP instances. By learning instance weights, this method can train the TSP solvers more efficiently through curriculum learning. This method moves an important step towards graph CL for solving NPhard problems.

7 Future Directions

Curriculum graph machine learning (Graph CL) is an emerging research topic. Although significant progresses have been made for Graph CL, there still remain plenty of research directions worthy of future explorations.

• Theoretical guarantees: Although various graph CL methods have been proposed and demonstrated effectiveness empirically, it remains to be further explored to derive more fundamental theoretical analysis on graph CL. A promising direction is to develop such theoretical analysis inspired by general curriculum learning from the perspective of optimization problem [Weinshall et al., 2018] or data distribution [Gong et al., 2016] for better understanding the mechanism and effectiveness of the graph CL methods.
• More principled Graph CL methods: Although the existing works studied how curriculum learning methods are extended to graphs, it is still worth investigating to develop more principled models for graph curriculum learning by considering more detailed graph assumptions (e.g., homophily, heterophily), more complex graph types (e.g., attributed graphs, heterogeneous graphs, signed graphs, multiplex graphs), or more specific graph properties (e.g., data dependencies), into the model design. These methods are expected to further boost the model capacity.
• Generalization and transferability: Currently, most existing graph CL methods are overly dependent on the graph labels, so that learned graph models tend to inherit a strong inductive bias for new testing tasks. However, for realworld graphs, there will inevitably be scenarios with distribution shifts between testing and training graph data [Li et al., 2022b; Li et al., 2022a], which can induce significant performance drop for most existing approaches lacking the ability of generalization and transferability. One interesting direction is to learn label-irrelevant, generalizable and transferable representations via self-supervised learning via unsupervised pretext tasks, alleviating excessively relying on labeled data [Chu et al., 2021]. The other feasible direction is to explicitly consider distribution shifts in the design of graph CL methods for learning better graph representations [Zhu et al., 2021].
• Comprehensive evaluation protocols: While the graph CL methods have made great progress in performance boost, few works have made efforts on evaluating them with general graph benchmarks. The adopted datasets and evaluation metrics of existing works mainly follow classical settings. It is essential to develop a unified benchmark with unified metrics to evaluate and compare different methods, which should carefully incorporate datasets with different hardness (e.g., different levels of sparsity, heterogeneity, noises), and different evaluation metrics (e.g., relative performance boost, convergence speedup, additional computational costs). Besides, publicly available graph CL libraries are also important to facilitate and advance the research, which can be developed upon the existing library [Zhou et al., 2022b].
• Broader applications: Although graph CL methods have been applied on several tasks as discussed in this work, it is worth further exploring the potential capacity in more diverse real-world applications, including recommendation [Ying et al., 2018], healthcare [Ahmed et al., 2022], biochemistry [Gu et al., 2022], etc., for more effective and satisfying predictions. One of the major challenges is how to incorporate proper domain knowledge as additional priors to guide the model design.

Acknowledgments

This work was supported by the National Key Research and Development Program of China No. 2020AAA0106300, National Natural Science Foundation of China (No. 62250008, 62222209, 62102222), BNRist Grant No. BNR2023RC01003 and Beijing Key Lab of Networked Multimedia.

Contribution Statement

Haoyang Li and Xin Wang contribute equally. Xin Wang and Wenwu Zhu are the corresponding authors.