SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We choose the EMCM method [6] for three main reasons. First, it is a performance-based approach that directly assesses the expected impact on the model if a sample is selected for a query, aligning perfectly with the goal of AL. Second, it offers lower computational costs compared to other performance-based methods like EER. Third, it is ideally supported by our proposed Bayesian probabilistic learning framework which provides an efficient way to estimate the expected model change without re-training. In the sequential active learning setting, our objective is to select the most informative node  $k^{*} = \arg \max_{k\in \mathcal{V}_{u}}\mathcal{A}(k)$  with a designed acquisition function  $\mathcal{A}(\cdot)$ . Similar to other performance-based methods, EMCM utilizes a look-ahead model [31] with the modified objective as  $\mathcal{L}^{k,y_k^+} = \mathcal{L} + \ell (\sigma (u_k(\Theta)),y_k^+)$  with  $\mathcal{L}$  from Eq. (5), where we add the unlabeled node  $k\in \mathcal{V}_u$  with the hypothetical label  $y_{k}^{+}\in \{0,1\}$  due to the unavailability of the true label  $y_{k}$ . We select the node that could maximally change the node embeddings if it is added to the query set, via
$$
k ^ {*} = \underset {k \in \mathcal {V} _ {u}} {\arg \max } \mathcal {A} (k) = \underset {k \in \mathcal {V} _ {u}} {\arg \max } \sum_ {y _ {k} ^ {+} \in \{0, 1 \}} \mathbb {P} \left(y _ {k} ^ {+} \mid k\right) \| \mathbf {u} \left(\tilde {\boldsymbol {\Theta}} ^ {*}\right) - \mathbf {u} \left(\boldsymbol {\Theta} ^ {*}\right) \| _ {2}. \tag {7}
$$
Here,  $\tilde{\Theta}^{*} = \arg \min \mathcal{L}^{k,y_{k}^{+}}$ ,  $\Theta^{*} = \arg \min \mathcal{L}$  and  $\mathbb{P}(y_k^+ \mid k)$  is the predicted probability of label  $y_{k}^{+}$  for node  $k$  estimated by the current model. The high-level motivation of Eq. (7) is that the generalization error changes only when the current model is updated. Nodes that do not update the node embeddings are useless for AL. Nodes causing significant model changes in terms of node embeddings are expected to lead to faster convergence to the optimal model. More detailed illustrations of EMCM can be found in Appendices B.3 and C.2.1.
Re-training the look-ahead model for the fixed node  $k$  in Eq. (7) to obtain the exact solution is computationally infeasible due to the large number of unlabeled nodes. However, the proposed equivalent Bayesian learning framework Eq. (6) offers an efficient alternative. By utilizing rank-one updates of the current model's posterior mean and covariance (Eq. (5)), the posterior mean and covariance of the look-ahead model can be computed efficiently without the need for re-training. It significantly reduces the computational cost involved in applying the EMCM method on graphs.
3.2.2 Truncated Spectral Projection and Laplace Approximation for Efficiency
The computation of  $\mathbf{u}(\boldsymbol{\Theta}^{*})$  or  $\mathbf{u}(\tilde{\boldsymbol{\Theta}}^{*})$  through the Bayesian learning method in Eq. (6) still requires storing a large covariance matrix  $\tilde{\mathbf{L}}_{\lambda}^{-1}\in \mathbb{R}^{n\times n}$ . Motivated by spectral clustering [48], we utilize the first  $m$  ( $m\ll n$ ) smallest eigenvalues of  $\tilde{\mathbf{L}}$  and their corresponding eigenvectors that contain important geometric information of the graph. Namely, we introduce  $\Lambda_{\lambda} = \mathrm{diag}(\lambda_1 + \lambda ,\dots ,\lambda_m + \lambda)$  and  $\mathbf{V} = [\mathbf{v}^1,\dots ,\mathbf{v}^m ]\in \mathbb{R}^{n\times m}$ , where  $\mathbf{v}^i$  is the eigenvector corresponding to the  $i$ -th eigenvalue  $\lambda_{i}$ . We can now project the node embedding  $\mathbf{u}(\boldsymbol {\Theta})$  onto the space spanned by these  $m$  eigenvectors by  $\alpha = \mathbf{V}^{\intercal}\mathbf{u}(\boldsymbol {\Theta})$ . Based on the orthogonality of  $\mathbf{V}$ $(\mathbf{V}^{\intercal}\mathbf{V} = \mathbf{I})$ , we can easily convert Eq. (5) as,
$$
\min  _ {\boldsymbol {\alpha}} \tilde {\mathcal {L}} = \sum_ {i \in \mathcal {V} _ {l}} \ell \left(\sigma \left(\mathbf {e} _ {i} ^ {\intercal} \mathbf {V} \boldsymbol {\alpha}\right), y _ {i}\right) + \operatorname {T r} \left(\boldsymbol {\alpha} ^ {\intercal} \boldsymbol {\Lambda} _ {\lambda} \boldsymbol {\alpha}\right). \tag {8}
$$
Eq. (8) now restricts the model latent space from  $\mathbb{R}^n$  to  $\mathbb{R}^m$ , speeding up the model training and reducing the spatial complexity. A similar Bayesian interpretation of Eq. (8) regarding  $\alpha \in \mathbb{R}^m$  can be trivially extended from Theorem 3.2. We term this technique as truncated spectral projection.
To further reduce the computational overhead, we apply Laplace approximation on the posterior distribution of  $\mathbb{P}(\boldsymbol{\alpha} \mid \mathbf{y})$  (instead of  $\mathbb{P}(\mathbf{u}(\boldsymbol{\Theta}) \mid \mathbf{y}))$  corresponding to Eq. (6). The key idea is to approximate the non-Gaussian posterior via suitable Gaussian distributions here so as to obtain the analytical closed-form of the look-ahead model's posterior mean and covariance later [47].
Theorem 3.3. The Laplace approximation of the posterior distribution regarding  $\alpha = \mathbf{V}^{\top}\mathbf{u}(\Theta) \in \mathbb{R}^{m}$  according to Eq. (8) and Eq. (6) is given as follows.
$$
\boldsymbol {\alpha} \mid \mathbf {y} \sim \mathcal {N} (\hat {\boldsymbol {\alpha}}, \hat {\mathbf {C}} _ {\hat {\boldsymbol {\alpha}}}), \hat {\boldsymbol {\alpha}} = \underset {\boldsymbol {\alpha}} {\arg \min } \tilde {\mathcal {L}}, \hat {\mathbf {C}} _ {\hat {\boldsymbol {\alpha}}} = \left(\boldsymbol {\Lambda} _ {\lambda} + \mathbf {V} ^ {\intercal} \left(\sum_ {i \in \mathcal {V} _ {l}} F ^ {\prime} \left(\sigma \left(\mathbf {e} _ {i} ^ {\intercal} \mathbf {V} \hat {\boldsymbol {\alpha}}\right), y _ {i}\right) \mathbf {e} _ {i} \mathbf {e} _ {i} ^ {\intercal}\right) \mathbf {V}\right) ^ {- 1}. \tag {9}
$$
Here, we define  $F'(x,y) \coloneqq \frac{\partial^2 \ell}{\partial x^2}(x,y)$ .  $\mathbf{e}_i$  represents the  $i$ -th standard basis vector.
Theorem 3.3 gives a closed-form approximated Gaussian posterior distribution of  $\alpha$ . Note that  $\hat{\alpha} = \mathbf{V}^{\top}\mathbf{u}(\Theta^{*})$  denotes the current model's  $\tilde{\mathcal{L}}$  MAP estimator in Eq. (8) based on Theorem 3.2.
3.2.3 Expected Model Change Acquisition Function
Following the definition of  $\mathcal{L}^{k,y_k^+}$  over  $\Theta$ , the look-ahead model now optimizes  $\tilde{\mathcal{L}}^{k,y_k^+} = \tilde{\mathcal{L}} + \ell(\sigma(\mathbf{e}_i^\top \mathbf{V}\boldsymbol{\alpha}), y_k^+)$  with  $\tilde{\mathcal{L}}$  in Eq. (8). We use  $\hat{\alpha}^{k,y_k^+}$  to denote the look-ahead model's  $\tilde{\mathcal{L}}^{k,y_k^+}$  MAP estimator. The exact value of  $\hat{\alpha}^{k,y_k^+}$  can be obtained by re-training the look-ahead model. We instead compute an approximation of  $\hat{\alpha}^{k,y_k^+}$  by one single step of Newton's method on the objective  $\tilde{\mathcal{L}}^{k,y_k^+}$  without re-training, given by Theorem 3.4. Since the look-ahead model is built upon the current model, using a single step of Newton's Method can quickly approximate the solution for  $\tilde{\mathcal{L}}^{k,y_k^+}$  based on the local information (gradient and curvature) around the current model's MAP estimator [47].
Theorem 3.4. The look-ahead MAP estimator  $\hat{\alpha}^{k,y_k^+}$  can be obtained by performing one step of Newton's method on  $\tilde{\mathcal{L}}^{k,y_k^+}$  from the current MAP estimator  $\hat{\alpha}$  as
$$
\hat {\boldsymbol {\alpha}} ^ {k, y _ {k} ^ {+}} = \hat {\boldsymbol {\alpha}} - \frac {F (\sigma (\mathbf {e} _ {k} ^ {\intercal} \mathbf {V} \hat {\boldsymbol {\alpha}}) , y _ {k} ^ {+})}{1 + F ^ {\prime} (\sigma (\mathbf {e} _ {k} ^ {\intercal} \mathbf {V} \hat {\boldsymbol {\alpha}}) , y _ {k} ^ {+}) (\mathbf {e} _ {k} ^ {\intercal} \mathbf {V}) \hat {\mathbf {C}} _ {\hat {\boldsymbol {\alpha}}} (\mathbf {e} _ {k} ^ {\intercal} \mathbf {V}) ^ {\intercal}} \hat {\mathbf {C}} _ {\hat {\boldsymbol {\alpha}}} (\mathbf {e} _ {k} ^ {\intercal} \mathbf {V}) ^ {\intercal} \tag {10}
$$
We define  $F(x,y) \coloneqq \frac{\partial\ell}{\partial x} (x,y)$ ,  $F^{\prime}(x,y) \coloneqq \frac{\partial^{2}\ell}{\partial x^{2}} (x,y)$ .  $\mathbf{e}_i$  represents the  $i$ -th standard basis vector.
Now, we finally come back to the original acquisition function for applying EMCM on graphs in Eq. (7). The predicted probability of label  $y_{k}^{+}$  can be set as the soft label prediction for node  $k$  by the current model  $\mathbb{P}(y_{k}^{+} = 1 \mid k) = \sigma(\mathbf{e}_{k}^{\top}\mathbf{u}(\boldsymbol{\Theta}))$  ( $\mathbb{P}(y_{k}^{+} = 0 \mid k) = 1 - \sigma(\mathbf{e}_{k}^{\top}\mathbf{u}(\boldsymbol{\Theta}))$ ). Thanks to the truncated spectral projection  $(\hat{\alpha} = \mathbf{V}^{\top}\mathbf{u}(\boldsymbol{\Theta}^{*}), \hat{\alpha}^{k,y_{k}^{+}} = \mathbf{V}^{\top}\mathbf{u}(\tilde{\boldsymbol{\Theta}}^{*}))$  and Laplace approximation, we can now measure the change of  $\| \mathbf{u}(\tilde{\boldsymbol{\Theta}}^{*}) - \mathbf{u}(\boldsymbol{\Theta}^{*}) \|_{2}$  via  $\| \hat{\alpha}^{k,y_{k}^{+}} - \hat{\alpha} \|_{2}$  due to the orthogonal invariance of  $l_{2}$  norm. Based on Theorem 3.4 and Eq. (7), the final designed acquisition function is
$$
\mathcal {A} (k) = \sum_ {y _ {k} ^ {+} \in \{0, 1 \}} \mathbb {P} \left(y _ {k} ^ {+} = 1 | k\right) ^ {y _ {k} ^ {+}} \mathbb {P} \left(y _ {k} ^ {+} = 0 | k\right) ^ {1 - y _ {k} ^ {+}} \left| \frac {F \left(\sigma \left(\mathbf {v} _ {k} ^ {\mathsf {T}} \hat {\boldsymbol {\alpha}}\right) , y _ {k} ^ {+}\right)}{1 + F ^ {\prime} \left(\sigma \left(\mathbf {v} _ {k} ^ {\mathsf {T}} \hat {\boldsymbol {\alpha}}\right) , y _ {k} ^ {+}\right) \mathbf {v} _ {k} ^ {\mathsf {T}} \hat {\mathbf {C}} _ {\hat {\boldsymbol {\alpha}}} \mathbf {v} _ {k}} \right| \| \hat {\mathbf {C}} _ {\hat {\boldsymbol {\alpha}}} \mathbf {v} _ {k} \| _ {2}. \tag {11}
$$
Here,  $\mathbb{P}(y_k^+ = 1|k) = \sigma (\mathbf{e}_k^\top \mathbf{u}(\Theta))$  and define  $\mathbf{v}_k\coloneqq (\mathbf{e}_k^\top \mathbf{V})^\top$  as the  $k$ -th row vector of  $\mathbf{V}$ . The proposed active learning algorithm expecteD mOdel Change maximizaTion On gRaphs (DOCTOR) in the sequential setting is presented in Algorithm 2 in Appendix D.2. For the batch active learning setting and the multi-class classification setting, we refer to Appendices D.3 and D.4.
3.3 Theoretical Insights of the Proposed Method DOCTOR
We analyze the profound connection between the proposed EMCM method and the Expected Error Minimization (EEM) or EER method for graph active learning. We will show that the designed acquisition function Eq. (11) will be reduced to the one for EEM under some assumptions and thus it has a direct connection to the ultimate goal for graph AL in Eq. (1). For clarity, we focus on the fundamental regression task within the sequential AL setting. We choose the regression for theoretical analysis to get rid of the non-linear activation functions in the Bayesian learning framework  $(\sigma(\cdot)$  in Eq. (5)) from the beginning. We also fix the loss function  $\ell(\cdot, \cdot)$  as the common squared error loss.
Theorem 3.5. We assume  $\ell(x, y) = \frac{1}{2}(x - y)^2$  is used for the regression task in the sequential active learning setting ( $b = 1$ ). In the  $t$ -th iteration step, the proposed DOCTOR algorithm aims to solve
$$
\max  _ {k \in \mathcal {V} _ {u} ^ {(t)}} \mathcal {A} (k) = \max  _ {k \in \mathcal {V} _ {u} ^ {(t)}} \frac {1}{1 + \mathbf {v} _ {k} ^ {\intercal} \hat {\mathbf {C}} _ {\hat {\alpha}}} \| \hat {\mathbf {C}} _ {\hat {\alpha}} \mathbf {v} _ {k} \| _ {2} ^ {2}. \tag {12}
$$
And the Expected Error Minimization on graphs aims to solve  $\min_{k\in \mathcal{V}_u^{(t)}}\mathbb{E}_{i\in \mathcal{V}_u^{(t)}\setminus \{k\}}[\ell (\hat{y}_i,y_i)]$ . Then,
$$
\underset {k \in \mathcal {V} _ {u} ^ {(t)}} {\arg \min } \mathbb {E} _ {i \in \mathcal {V} _ {u} ^ {(t)} \backslash \{k \}} [ \ell (\hat {y} _ {i}, y _ {i}) ] = \underset {k \in \mathcal {V} _ {u} ^ {(t)}} {\arg \max } \frac {1}{1 + \mathbf {v} _ {k} ^ {\intercal} \hat {\mathbf {C}} _ {\hat {\alpha}}} \| \hat {\mathbf {C}} _ {\hat {\alpha}} \mathbf {v} _ {k} \| _ {2} ^ {2}. \tag {13}
$$
Theorem 3.5 demonstrates the equivalence between the proposed DOCTOR algorithm and the EEM method on graphs, given certain basic assumptions. Consequently, from a theoretical standpoint, DOCTOR can implicitly select the node that minimizes the expected prediction error if added into the training set during the next iteration, as directly correlated with the initial objective in Eq. (1). Moreover, DOCTOR offers significantly reduced computational overhead through truncated spectral projection and Laplace approximation, as compared to EEM-based approaches like GEEM [51].
We now verify the effectiveness of DOCTOR on five real-world graphs. We aim to evaluate DOCTOR in terms of prediction accuracy, efficiency, generalization ability, and interpretability (Appendix G).
Datasets We evaluate DOCTOR for the pool-based active learning setting on five datasets, including three citation networks (i.e., Citeseer, Cora and PubMed) [32], one social network (Reddit) [22], and one large-scale OGB dataset (ogbn-arxiv) [26]. Details can be found in Appendix F.1.
Baselines We compare DOCTOR with six state-of-the-art baseline methods for active learning on graphs. Random selects the nodes to query randomly. AGE [4] Combines different query strategies linearly with time-sensitive parameters. ALG [88] select nodes that maximize the effective reception field in GCN. RIM [89] further considers the influence quality during node selection. IGP [90] choose nodes that maximize information gain propagation. GraphPart [41] selects representative nodes within each graph partition to query. Implementation details are referred to in Appendix F.2.
Accuracy Comparison We compare DOCTOR with baselines for multi-class node classification in the batch AL setting (Appendix D.4). We choose a small set of two randomly sampled labeled nodes for each method as the initial pool. The query node size is fixed as  $b = 5$  in each iteration and the labeling budget is fixed as 20 labels per class ( $B = 20c$ ). The backbone is set as the SGC model. Table 1 shows the classification accuracy on five datasets with the same labeling budget. We repeat each method 10 times and report the mean and variance regarding the accuracy. Remarkably, the proposed DOCTOR achieves the best in all the cases and improves the second-best baseline by nearly  $1\%$  on some datasets. The improvement on the Reddit dataset is rather marginal since it is designed for the inductive setting and DOCTOR is originally proposed for the transductive setting.
To show the influence of the labeling budget, we display the accuracy of different AL methods under different labeling budgets on three citation datasets in Figure 2. We range the labeling budgets  $B$  from  $2c$  to  $20c$  with  $c$  as the number of classes. The results in Figure 2 demonstrate that with the increase in labeling budget, the accuracy of DOCTOR grows as well, outperforming many of the baselines with a greater margin.
Efficiency Comparison To evaluate the efficiency of DOCTOR, we conducted an analysis of its running time per AL iteration alongside other baselines using the Cora dataset. We introduce another performance-based graph AL method called GEEM [51], which extends the EEM principle
for graphs. However, GEEM's scalability is extremely limited on large-scale datasets. We consider the training time of GEEM as the baseline and measure the relative running time of each method. Figure 1 illustrates these results, where the numbers atop each bar represent the multiplier improvement in running time compared to GEEM. Our method, DOCTOR, exhibits significant efficiency improvements, surpassing GEEM dozens of times while achieving comparable results to ALG. The more detailed time complexity of the proposed DOCTOR method can be found in Appendix G.1.
Generalization to Other GNNs Note that the acquisition function in the proposed DOCTOR algorithm is specifically designed for the SGC method [68]. Nonetheless, we can still substitute the backbone training model, SGC, with other GNNs in each iteration of the AL loop (line 5 in Algorithm 2) without any difficulties. The resulting acquisition function will be computed based on the new node embeddings generated by other GNNs without any other changes. We select some common GNNs, including GCN [32], APPNP [33] and GraphSAGE [22], to verify the generalization ability of the proposed acquisition function. We change the backbone model of other baselines accordingly as well. We fix the labeling budget as 20 labels per class. All experiments are conducted in the transductive AL setting on the Citeseer dataset and the results are summarized in Table 2. Like other baselines, the proposed acquisition function in the proposed DOCTOR method generalizes well to other GNN backbone models without any significant changes regarding prediction accuracy, even though it is originally designed for SGC. The underlying reason is that apart from SGC, most other GNNs can also be incorporated into the proposed Bayesian learning framework with more complex forms of prior and the currently derived form of the acquisition function can approximate the exact solution when other GNNs are used as backbones.
Interpretability We perform a case study on a single iteration step of DOCTOR in the sequential AL setting using the Cora dataset. We choose one query node from a pool of 10 randomly sampled unlabeled nodes for visualization purposes. We focus on one iteration step in the exact middle of the entire AL process. We compare the approximate EMC, computed based on Eq. (11) in the DOCTOR method, with the exact EMC, computed based on Eq. (7) by adding the node's label for re-training. We use the maximum value of the exact EMC among these 10 nodes as the baseline, and we report all other EMC values in a relative manner in Figure 3. The results show that our efficient computation of the designed acquisition function via truncated spectral projection and Laplace approximation approximates the exact solution quite well. Additionally, we sort the candidate nodes in descending
order based on the exact EMC value and obtain the respective test error by re-training the model with the node's label. Figure 3 illustrates that by choosing the node that has the maximum approximate EMC value without re-training via the proposed acquisition function Eq. (11), we can minimize the test error on the remaining unlabeled nodes by adding this node to the labeled node set for the next round of training. This key observation empirically validates the insights of Theorem 3.5.
In this work, we extend the EMCM principle for GNNs based on a provided Bayesian learning framework. This allows us to efficiently compute a closed-form acquisition function that can be used to select the most informative nodes to label. The proposed DOCTOR method establishes theoretical connections with the ultimate goal of minimizing expected prediction error in AL. This makes it a promising tool for training GNNs in real-world applications with limited labeled data.
The acquisition function in DOCTOR is derived from the SGC model but has good generalizability to other GNNs empirically. We leave the investigation of the closed-form MAP estimator of other GNNs as future work. The acquisition function in DOCTOR is originally designed for the sequential active learning setting, and we only use a simple sampling method to extend it for the batch active learning setting. We also leave the investigation of a more advanced batch AL extension of DOCTOR as future work. The theoretical connection between our method with entropy minimization in terms of strictly proper scoring rules can be further investigated in detail [64]. Other potential applications of our method can be explored in the hyperbolic space [76, 79, 77, 80, 78, 75, 81] or in the natural language processing domain [34, 36, 35, 20, 19, 63, 44, 99, 100, 98].
The work described in this paper was partially supported by the National Key Research and Development Program of China (No. 2018AAA0100204), RGC Research Impact Fund (RIF), R5034-18 (CUHK 2410021), and RGC General Research Funding Scheme (GRF) 14222922 (CUHK 2151185).
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