SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Table 1: Remaining challenges of invariant graph learning: no existing works can handle both cases.
H(S|Y) < H(C|Y)	H(S|Y) > H(C|Y)
DisC	✓	✗
CIGA	✗	✓
GALA (Ours)	✓	✓
propose CIGA to maximize the intra-class mutual information of the estimated invariant subgraphs to tackle the case when  $H(C|Y) < H(S|Y)$ . While for the case when  $H(S|Y) < H(C|Y)$ , Fan et al. [17] propose DisC that adopts GCE loss [40] to extract the spurious subgraph with a larger learning step size such that the left subgraph is invariant. However, both of them can fail when there is no prior knowledge about the relations between  $H(C|Y)$  and  $H(S|Y)$ . We verify the failures of DisC and CIGA in Table. 2. The failure thus raises a challenging question:
Given the established minimal assumptions, is there a unified framework that tackles both cases when  $H(C|Y) < H(S|Y)$  and  $H(C|Y) > H(S|Y)$ ?
4 Learning Invariant Graph Representations with Environment Assistant
We give an affirmative answer by proposing a new framework, GALA: Graph invArient Learning Assistant, which adopts an assistant model to provide proxy information about the environments.
4.1 Learning with An Environment Assistant
Intuitively, a straightforward approach to tackle the aforementioned challenge is to extend the framework of either DisC [17] or CIGA [8] to resolve the other case. As DisC always destroys the first learned features and tends to be more difficult to extend (which is empirically verified in Sec. 5), we are motivated to extend the framework of CIGA to resolve the case when  $H(S|Y) < H(C|Y)$ .
Understanding the success and failure of CIGA. The principle of CIGA lies in maximizing the intra-class mutual information of the estimated invariant subgraphs, i.e.,
$$
\max  _ {f _ {c}, g} I \left(\widehat {G} _ {c}; Y\right), \text {s . t .} \widehat {G} _ {c} \in \underset {\widehat {G} _ {c} = g (G), | \widehat {G} _ {c} | \leq s _ {c}} {\arg \max } I \left(\widehat {G} _ {c}; \widehat {G} _ {c} ^ {8} \mid Y\right), \tag {3}
$$
where  $\widehat{G}_c^s = g(G^s)$  and  $G^{s}\sim \mathbb{P}(G|Y)$ , i.e.,  $\widehat{G}$  is sampled from training graphs that share the same label  $Y$  as  $\widehat{G}$ . The key reason for the success of Eq. 3 is that, given the data generation process as in Fig. 2 and the same  $C$ , the underlying invariant subgraph  $G_{c}$  maximizes the mutual information of subgraphs from any two environments, i.e.,  $\forall e_1,e_2\in \mathcal{E}_{\mathrm{all}}$ ,
$$
G _ {c} ^ {e _ {1}} \in \arg \max  _ {\widehat {G} _ {c} ^ {e _ {1}}} I \left(\widehat {G} _ {c} ^ {e _ {1}}; \widehat {G} _ {c} ^ {e _ {2}} | C\right), \tag {4}
$$
where  $\widehat{G}_c^{e_1}$  and  $\widehat{G}_c^{e_2}$  are the estimated invariant subgraphs corresponding to the same latent causal variable  $C = c$  under the environments  $e_1, e_2$ , respectively. Since  $C$  is not observable, CIGA adopts  $Y$  as a proxy for  $C$ , as when  $H(S|Y) > H(C|Y)$ ,  $G_c$  maximizes  $I(\widehat{G}_c^{e_1}; \widehat{G}_c^{e_2}|Y)$  and thus  $I(\widehat{G}_c; \widehat{G}_c^s|Y)$ . However, when  $H(S|Y) < H(C|Y)$ , the proxy no longer holds. Given the absence of  $E$ , simply maximizing intra-class mutual information favors the spurious subgraph  $G_s$  instead, i.e.,
$$
G _ {s} \in \arg \max  _ {\widehat {G} _ {c}} I (\widehat {G} _ {c}; \widehat {G} _ {c} ^ {s} | Y). \tag {5}
$$
Invalidating spuriousness dominance. To mitigate the issue, we are motivated to find a new proxy that samples  $\widehat{G}_c$  for Eq. 5, while preserving only the  $G_{c}$  as the solution under both cases.
To begin with, we consider the case of  $H(S|Y) < H(C|Y)$ . Although the correlation between  $G_{s}$  and  $Y$  dominates the intra-class mutual information, Assumption 3.3 implies that there exists a subset of training data where  $P(Y|G_s)$  varies, while  $P(Y|G_c)$  remains invariant. Therefore, the dominance of spurious correlations no longer holds for samples from the subset. Incorporating samples from the subset into Eq. 3 as  $\widehat{G}_c^s$  invalidates the dominance of  $G_{s}$ . Denote the subset as  $\{\widehat{G}_c^n\}$ , then
$$
G _ {c} \in \arg \max  _ {\widehat {G} _ {c} ^ {p}} I \left(\widehat {G} _ {c} ^ {p}; \widehat {G} _ {c} ^ {n} | Y\right), \tag {6}
$$
where  $\widehat{G}_c^p\in \{\widehat{G}_c^p\}$  is sampled from the subset  $\{\widehat{G}_c^p\}$  dominated by spurious correlations, while  $\widehat{G}_c^n\in \{\widehat{G}_c^n\}$  is sampled from the subset  $\{\widehat{G}_c^n\}$  where spurious correlation no long dominates, or is dominated by invariant correlations. We prove the effectiveness of Eq. 6 in Theorem 4.1.
Environment assistant model  $A$ . To find the desired subsets  $\{\widehat{G}_c^p\}$  and  $\{\widehat{G}_c^n\}$ , inspired by the success in tackling spuriousness-dominated OOD generalization via learning from a biased predictors [40, 44, 58, 85], we propose to incorporate an assistant model  $A$  that is prone to spurious correlations. Simply training  $A$  with ERM using the spuriousness-dominated data enables  $A$  to learn spurious correlations, and hence identifies the subsets where the spurious correlations hold or shift, according to whether the predictions of  $A$  are correct or not, respectively. Let  $A = \arg \max_{\widehat{A}} I(\widehat{A}(G); Y)$ , we have
$$
\left\{\widehat {G} _ {c} ^ {p} \right\} = \left\{g \left(G _ {i} ^ {p}\right) \mid A \left(G _ {i} ^ {p}\right) = Y _ {i} \right\}, \left\{\widehat {G} _ {c} ^ {n} \right\} = \left\{g \left(G _ {i} ^ {n}\right) \mid A \left(G _ {i} ^ {n}\right) \neq Y _ {i} \right\}. \tag {7}
$$
Reducing to invariance dominance case. After showing that Eq. 6 resolves the spuriousness dominance case, we still need to show that Eq. 6 preserves  $\widehat{G}_c$  as the only solution when  $H(S|Y) > H(C|Y)$ . Considering training  $A$  with ERM using the invariance-dominated data,  $A$  will learn both invariant correlations and spurious correlations [10, 17]. Therefore,  $\{\widehat{G}_c^n\}$  switches to the subset that is dominated by spurious correlations, while  $\{\widehat{G}_c^p\}$  switches to the subset dominated by invariant correlations. Then, Eq. 6 establishes a lower bound for the intra-class mutual information, i.e.,
$$
I \left(\widehat {G} _ {c} ^ {p}; \widehat {G} _ {c} ^ {n} | Y\right) \leq I \left(\widehat {G} _ {c}; \widehat {G} _ {c} ^ {s} | Y\right), \tag {8}
$$
where  $\widehat{G}_c^p\in \{\widehat{G}_c^p\}$ $\widehat{G}_c^n\in \{\widehat{G}_c^n\}$  , and  $\widehat{G}_c,\widehat{G}_c^s$  are the same as in Eq. 3. The inequality in Eq. 8 holds as any subgraph maximizes the left hand side can also be incorporated in right hand side, while the sampling space of  $\widehat{G}_c$  and  $\widehat{G}_c^s$  in the right hand side (i.e., both  $\widehat{G}_c$  and  $\widehat{G}_c^s$  are sampled from the whole train set) is larger than that of the left hand side. The equality is achieved by taking the ground truth  $G_{c}$  as the solution for the featurizer  $g$  We verify the correctness of Eq. 6 and Eq. 8 in Fig. 3(a).
4.2 Practical implementations.
The detailed algorithm description of
GALA is shown as in Algorithm 1. In practice, the environment assistant can have multiple implementation choices so long as it is prone to distribution shifts. As discussed in Sec. 4.1, ERM trained
Algorithm 1 GALA: Graph invAriant Learning Assistant
1: Input: Training data  $\mathcal{D}_{\mathrm{tr}}$ ; environment assistant  $A$ ; featurizer GNN  $g$ ; classifier GNN  $f_{c}$ ; length of maximum training epochs  $e$ ; batch size  $b$ ;
2: Initialize environment assistant  $A$ ;
3: for  $p \in [1, \dots, e]$  do
4: Sample a batch of data  $\{G_i, Y_i\}_{i=1}^b$  from  $\mathcal{D}_{\mathrm{tr}}$ ;
5: Obtain Environment Assistant predictions  $\{\hat{y}_i^e\}_{i = 1}^b$
6: for each sample  $G_{i},y_{i}\in \{G_{i},Y_{i}\}_{i = 1}^{b}$  do
7: Find positive graphs with same  $y_{i}$  and different  $\hat{y}_i^e$ ;
8: Find negative graphs with different  $y_{i}$  but same environment assistant prediction  $\hat{y}_i^e$ ;
9: Calculate GALA risk via Eq. 9;
10: Update  $f_{c}, g$  via gradients from GALA risk;
11: end for
12: end for
13: return final model  $f_{c} \circ g$
model can serve as a reliable environment assistant, since ERM tends to learn the dominant features no matter whether the features are invariant or spurious. For example, when  $H(S|Y) < H(C|Y)$ , ERM will first learn to use spurious subgraphs  $G_{s}$  to make predictions. Therefore, we can obtain  $\{G^{p}\}$  by finding samples where ERM correctly predicts the labels, and  $\{G^{n}\}$  for samples where ERM predicts incorrect labels. In addition to label predictions, the clustering predictions of the hidden representations yielded by environment assistant models can also be used for sampling  $\{G^{p}\}$  and  $\{G^{n}\}$  [85]. Besides, we can also incorporate models that are easier to overfit to the first dominant features to better differentiate  $\{G^{p}\}$  from  $\{G^{n}\}$ . When the number of positive or negative samples is imbalanced, we can upsample the minor group to avoid trivial solutions. In addition, the final GALA objective is given in Eq. 9 and implemented as in Eq. 18. We provide more discussions about the implementation options in Appendix F.
In the following theorem, we show that the GALA objective derived in Sec. 4.1 can identify the underlying invariant subgraph and yields an invariant GNN defined in Sec. 2.
Theorem 4.1. Given  $i$ ) the same data generation process as in Fig. 2; ii)  $\mathcal{D}_{\mathrm{tr}}$  that satisfies variation sufficiency (Assumption 3.3) and variation consistency (Assumption 3.5); iii)  $\{G^p\}$  and  $\{G^n\}$  are distinct subsets of  $\mathcal{D}_{\mathrm{tr}}$  such that  $I(G_s^p; G_s^n | Y) = 0$ ,  $\forall G_s^p = \arg \max_{\widehat{G}_s^p} I(\widehat{G}_s^p; Y)$  under  $\{G^p\}$ , and  $\forall G_s^n = \arg \max_{\widehat{G}_s^n} I(\widehat{G}_s^n; Y)$  under  $\{G^n\}$ ; suppose  $|G_c| = s_c$ ,  $\forall G_c$ , resolving the following GALA objective elicits an invariant GNN defined via Eq. 1,
$$
\max  _ {f _ {c}, g} I \left(\widehat {G} _ {c}; Y\right), \text {s . t .} g \in \underset {\hat {g}, | \widehat {G} _ {c} ^ {p} | \leq s _ {c}} {\arg \max } I \left(\widehat {G} _ {c} ^ {p}; \widehat {G} _ {c} ^ {n} | Y\right), \tag {9}
$$
where  $\widehat{G}_c^p\in \{\widehat{G}_c^p = g(G^p)\}$  and  $\widehat{G}_c^n\in \{\widehat{G}_c^n = g(G^n)\}$  are the estimated invariant subgraphs via  $g$  from  $\{G^p\}$  and  $\{G^n\}$ , respectively.
The proof is given in Appendix E.5. Essentially, assumption iii) in Theorem 4.1 is an implication of the variation sufficiency (Assumption 3.3). When given the distinct subsets  $\{G^p\}$  and  $\{G^n\}$  with different relations of  $H(C|Y)$  and  $H(S|Y)$ , since  $H(C|Y)$  remains invariant across different subsets, the variation happens mostly to the spurious correlations between  $S$  and  $Y$ . By differentiating spurious correlations into distinct subsets, maximizing the intra-class mutual information helps identify the true invariance. The fundamental rationale for why GALA resolves two seemingly conversed cases essentially relies on the commutative law of mutual information.
We evaluated GALA with both synthetic and realistic graph distribution shifts. Specifically, we are interested in the following two questions: (a) Can GALA improve over the state-of-the-art invariant graph learning methods when the spurious subgraph has a stronger correlation with the labels? (b) Will GALA affect the performance when the invariant correlations are stronger?
5.1 Datasets and experiment setup
We prepare both synthetic and realistic graph datasets containing various distribution shifts to evaluate GALA. We will briefly introduce each dataset and leave more details in Appendix G.1.
Two-piece graph datasets. We adopt BA-2motifs [50] to implement 4 variants of 3-class two-piece graph (Def. 3.1) datasets. The datasets contain different relationships of  $H(C|Y)$  and  $H(S|Y)$  by controlling the  $\alpha$  and  $\beta$  in the mixed environment, respectively. We consider 4 cases of  $\alpha - \beta$ , ranging from  $\{+0.2, +0.1, -0.1, -0.2\}$ , to verify our discussion in Sec. 4.3.
Realistic datasets. We also adopt datasets containing various realistic graph distribution shifts to comprehensively evaluate the OOD performance of GALA. We adopt 6 datasets from DrugOOD benchmark [28], which focuses on the challenging real-world task of AI-aided drug affinity prediction. The DrugOOD datasets include splits using Assay, Scaffold, and Size from the EC50 category (denoted as EC50-*) and the Ki category (denoted as Ki-*). We also adopt graphs converted from the ColoredMNIST dataset [3] using the algorithm from Knyazev et al. [36], which contains distribution
shifts in node attributes (denoted as CMNIST-sp). In addition, we adopt Graph-SST2 [84], where we split graphs with a larger average degree in the training set while smaller in the test set.
Experiment setup. We adopt the state-of-the-art OOD methods from the Euclidean regime, including IRMv1 [3], VREx [38], EIIL [13] and IB-IRM [2], and from the graph regime, including GREA [45], GSAT [55], CAL [64], MoleOOD [77], GIL [41], DisC [17] and CIGA [8]. We exclude DIR [73] and GIB [81] as GREA and GSAT are their sophisticated variants. In addition to the ERM baseline that trained a vanilla GNN with ERM objective, in two-piece motif datasets, we also include XGNN to demonstrate the failures of previous approaches, which is an interpretable GNN trained with ERM. We also exclude CIGAv2 [8] as GALA focuses on improving the contrastive sampling via environment assistant for the objective in CIGAv1. All methods use the same GIN backbone [75], the same interpretable GNN architecture as in [55], and optimization protocol for fair comparisons. We tune the hyperparameters following the common practice. Details are given in Appendix G.2.
5.2 Experimental results and analysis
Proof-of-concept study. The results in two-piece graph datasets are reported in Table 2. It can be found that the previous environment augmentation approaches fail either in datasets where the invariant correlations dominate or where the spurious correlations dominate, aligned with our discussions in Sec. 3. In particular, GREA, CIGA and GIL achieve high performance when the invariant correlation dominates, but suffer great performance decrease when the spurious correlations are stronger. Although DisC is expected to succeed when spurious correlations dominate, DisC fails to outperform others because of its excessive destruction of the learned information. Mo1e00D also yields degraded performance, which
Table 2: OOD generalization performance under various invariant and spurious correlation degrees in the two-piece graph datasets. Each dataset is generated from a variation of two-piece graph model, denoted as  $\{a,b\}$ , where  $a$  refers to the invariant correlation strength and  $b$  refers to the spurious correlation strength. The blue shadowed entries are the results with the mean-1*std larger than the mean of the corresponding second best results.
Datasets	{0.8, 0.6}	{0.8, 0.7}	{0.8, 0.9}	{0.7, 0.9}	Avg.
ERM	77.33±0.47	75.65±1.62	51.37±1.20	42.73±3.82	61.77
IRM	78.32±0.70	75.13±0.77	50.76±2.56	41.32±2.50	61.38
V-Rex	77.69±0.38	74.96±1.40	49.47±3.36	41.65±2.78	60.94
IB-IRM	78.00±0.68	73.93±0.79	50.93±1.87	42.05±0.79	61.23
EIIL	76.98±1.24	74.25±1.74	51.45±4.92	39.71±2.64	60.60
XGNN	83.84±0.59	83.05±0.20	53.37±1.32	38.28±1.71	64.63
GREA	82.86±0.50	82.72±0.50	50.34±1.74	39.01±1.21	63.72
GSAT	80.54±0.88	78.11±1.23	48.63±2.18	36.62±0.87	63.32
CAL	76.98±6.03	62.95±8.58	51.57±6.33	46.23±3.93	59.43
MoleOOD	49.93±2.25	49.85±7.31	38.49±4.25	34.81±1.65	43.27
GIL	83.51±0.41	82.67±1.18	51.76±4.32	40.07±2.61	64.50
DisC	60.47±17.9	54.29±15.0	45.06±7.82	39.42±8.59	50.81
CIGA	84.03±0.53	83.21±0.30	57.87±3.38	43.62±3.20	67.18
GALA	84.27±0.34	83.65±0.44	76.42±3.53	72.50±1.06	79.21
Oracle	84.73±0.36	85.42±0.25	84.28±0.15	78.38±0.19
could be caused by the failures to infer reliable environment labels. In contrast, GALA achieves consistently high performance under both cases and improves CIGA up to  $30\%$  under  $\{0.7, 0.9\}$  and  $13\%$  in average, which validates our theoretical results in Sec. 4.3.
Table 3: OOD generalization performance under realistic graph distribution shifts. The blue shadowed entries are the results with the mean-1*std larger than the mean of the respective second best results.
Datasets	EC50-Assay	EC50-Sca	EC50-Size	Ki-Assay	Ki-Sca	Ki-Size	CMNIST-sp	Graph-SST2	Avg.(Rank)†
ERM	76.42±1.59	64.56±1.25	61.61±1.52	74.61±2.28	69.38±1.65	76.63±1.34	21.56±5.38	81.54±1.13	65.79 (6.50)
IRM	77.14±2.55	64.32±0.42	62.33±0.86	75.10±3.38	69.32±1.84	76.25±0.73	20.25±3.12	82.52±0.79	65.91 (6.13)
V-Rex	75.57±2.17	64.73±0.53	62.80±0.89	74.16±1.46	71.40±2.77	76.68±1.35	30.71±11.8	81.11±1.37	67.15 (5.25)
IB-IRM	64.70±2.50	62.62±2.05	58.28±0.99	71.98±3.26	69.55±1.66	70.71±1.95	23.58±7.96	81.56±0.82	62.87 (10.6)
EIIL	64.20±5.40	62.88±2.75	59.58±0.96	74.24±2.48	69.63±1.46	76.56±1.37	23.55±7.68	82.46±1.48	64.14 (8.00)
XGNN	72.99±2.56	63.62±1.35	62.55±0.81	72.40±3.05	72.01±1.34	73.15±2.83	20.96±8.00	82.55±0.65	65.03 (7.13)
GREA	66.87±7.53	63.14±2.19	59.20±1.42	73.17±1.80	67.82±4.67	73.52±2.75	12.77±1.71	82.40±1.98	62.36 (10.1)
GSAT	76.07±1.95	63.58±1.36	61.12±0.66	72.26±1.76	70.16±0.80	75.78±2.60	15.24±3.72	80.57±0.88	64.35 (8.63)
CAL	75.10±2.71	64.79±1.58	63.38±0.88	75.22±1.73	71.08±4.83	72.93±1.71	23.68±4.68	82.38±1.01	66.07 (5.38)
DisC	61.94±7.76	54.10±5.69	57.64±1.57	54.12±8.53	55.35±10.5	50.83±9.30	50.26±0.40	76.51±2.17	56.59 (12.4)
MoleOOD	61.49±2.19	62.12±1.91	58.74±1.73	75.10±0.73	60.35±11.3	73.69±2.29	21.04±3.36	81.56±0.35	61.76 (10.0)
GIL	70.56±4.46	61.59±3.16	60.46±1.91	75.25±1.14	70.07±4.31	75.76±2.23	12.55±1.26	83.31±0.50	63.69 (8.00)
CIGA	75.03±2.47	65.41±1.16	64.10±1.08	73.95±2.50	71.87±3.32	74.46±2.32	15.83±2.56	82.93±0.63	65.45 (5.88)
GALA	77.56±2.88	66.28±0.45	64.25±1.21	77.92±2.48	73.17±0.88	77.40±2.04	68.94±0.56	83.60±0.66	73.64 (1.00)
Oracle	84.77±0.58	82.66±1.19	84.53±0.60	91.08±1.43	88.58±0.64	92.50±0.53	67.76±0.60	91.40±0.26
${}^{ \dagger  }$  Averaged rank is also reported in the parentheses because of dataset heterogeneity. A lower rank is better.
OD generalization in realistic graphs. The results in realistic datasets are reported in Table 3. Aligned with our previous discussion, existing environment augmentation approaches sometimes yield better performance than ERM, such as CAL in EC50-Size, Mo1e00D in Ki-Assay, GIL in Graph-SST2, or CIGA in EC50-Size, however, inevitably fail to bring consistent improvements than ERM, due to the existence of failure cases. DisC is suspected to work only for graph distribution shifts on node features and bring impressive improvements in CMNIST-sp, but can destroy the learned information under more challenging settings. In contrast, GALA consistently outperform ERM by a non-trivial margin in all datasets. Notably, GALA achieves near oracle performance in CMNIST-sp and improves CIGA by  $53\%$ . The consistent improvements of GALA confirm the effectiveness of GALA.
(a) Correlation strengths
(b) CIGAv2 compatibility
(c) Hyperparameter sensitivity
Figure 3: Ablation studies.
Correlation strengths of  $\{G^p\}$  and  $\{G^n\}$ . We conduct experiments with the two-piece graph datasets evaluated in Table 2 to verify the correctness of Eq. 6 and Eq. 8. Eq. 6 and Eq. 8 imply that the underlying invariant subgraph will be the subgraph that maximizes the mutual information among subgraphs from  $\{G^p\}$  and  $\{G^n\}$ , no matter whether the dominant correlation is spurious or not. We measure the invariant and spurious correlation strengths in terms of co-occur probability of the invariant and spurious subgraphs with the labels. The results are shown in Fig. 3(a). It can be found that, under both cases, the underlying invariant subgraph maintains the predictivity with the label in an invariant manner. Hence, maximizing the intra-class subgraph mutual information between  $\{G^p\}$  and  $\{G^n\}$  in GALA succeeds in identifying the underlying invariant subgraph.
CIGAv2 compatibility. Although GALA focuses on the contrastive term in CIGA, both GALA and CIGA are compatible with the additional CIGAv2 term that facilitates constraining the graph sizes. To verify, we compare the OOD performances of CIGA, CIGAv2, GALA, and GALA +CIGAv2 using two challenging datasets, Ki-Scaffold and CMNIST-sp. The results are given in Fig. 3(b). It can be found that, despite incorporating the additional CIGAv2 constraint, CIGA can not outperform GALA, while GALA can bring more improvements with the additional CIGAv2 constraint. In CMNIST-sp, since GALA already achieve the upper bound, incorporating CIGAv2 can only achieve a similar result.
Hyperparameter sensitivity. We also test the hyperparameter sensitivity of GALA to the contrastive penalty weights as well as the upsampling times that are introduced to mitigate the imbalance of positive and negative graphs. We conduct the experiments with two-piece graph dataset  $\{0.7, 0.9\}$ . As shown in Fig. 3(c), it can be found that GALA is generically robust to different hyperparameter choices. In addition, when the penalty weight or the upsampling times turn to 0, the performance will decrease a lot, which serves as strong evidence for the effectiveness of GALA.
Computational analysis. We also conduct computational analysis of GALA and other methods, and defer the results to Table. 6 in Appendix G.4, due to space constraints. The results show that GALA costs only a competitive training time as environment generation based methods, while achieving much better OOD generalization performance.
We conducted a retrospective study on the faithfulness of the augmented environment information for OOD generalization on graphs. By showing hardness cases and impossibility results of the existing approaches, we developed a set of minimal assumptions for feasible invariant graph learning. Built upon the assumptions, we proposed GALA to learn the invariant graph representations guided by an environment assistant model. Extensive experiments with 12 datasets verified the superiority of GALA.
We thank the reviewers for their valuable comments. This work was supported by CUHK direct grant 4055146. BH was supported by the NSFC Young Scientists Fund No. 62006202, NSFC General Program No. 62376235, Guangdong Basic and Applied Basic Research Foundation No. 2022A1515011652, HKBU Faculty Niche Research Areas No. RC-FNRA-IG/22-23/SCI/04, and Tencent AI Lab Rhino-Bird Gift Fund.