SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Theorem 2. Under Assumption 2, suppose we run Algorithm 1 with budget  $B = C\varepsilon^{-2}\log n$  for large enough  $C > 0$ , to obtain  $\{\widehat{\ell}_i([AX| - \mathbf{y}]),\forall i\in [n]\}$ . Subsequently, if we run LEVERAGESCORESAMPLING(  $A,\{\widehat{\ell}_i([AX| - \mathbf{y})],\forall i\in [n]\}$  ), we obtain  $\tilde{A}\in \mathbb{R}^{O(d\varepsilon^{-2}\log n)\times n}$  and  $\tilde{\mathbf{y}}\in \mathbb{R}^{O(d\varepsilon^{-2}\log n)}$  such that REGRESSIONSOLVER  $(\tilde{A} X,\tilde{\mathbf{y}})$  provides  $\tilde{\mathbf{w}}$  satisfying
$$
\| \tilde {\mathbf {y}} - \tilde {A} X \tilde {\mathbf {w}} \| _ {2} ^ {2} \leq (1 + \varepsilon) \cdot \min  _ {\mathbf {w}} \| \mathbf {y} - A X \mathbf {w} \| _ {2} ^ {2}.
$$
Theorem 2 captures our main contribution. By only observing  $O(d\log n)$  rows of  $A$ , we can obtain a reduced adjacency matrix  $\tilde{A}$  (and corresponding label vector  $\tilde{\mathbf{y}}$ ). Running REGRESSIONSOLVER  $(\tilde{A} X, \tilde{\mathbf{y}})$  then yields a solution  $\tilde{\mathbf{w}}$  with an approximation guarantee. We refer to detailed descriptions of LEVERAGESCORESAMPLING  $(A, \{\hat{\ell}_i([AX| - \mathbf{y}]), \forall i \in [n]\})$  and REGRESSIONSOLVER  $(\tilde{A} X, \tilde{\mathbf{y}})$  in Appendix B.
Sample complexity and run-time: We note that the sample complexity is  $O(nd\varepsilon^{-2}\log n)$ . First, ESTIMATELEVERAGESCORES(A,X) exploits  $O(\varepsilon^{-2}\log n)$  columns of  $A$  and the same number of rows of  $X$ , thus accessing  $O(n\varepsilon^{-2}\log n)$  entries of  $A$  and  $O(d\varepsilon^{-2}\log n)$  of  $X$ . In addition, LEVERAGESCORESAMPLING(A,  $\{\widehat{\ell_i} ([AX| - \mathbf{y})],\forall i\in [n]\}$ ) exploits  $O(d\varepsilon^{-2}\log n)$  rows and thus uses  $O(nd\varepsilon^{-2}\log n)$  entries. Hence, the total number of entry observations is  $O(nd\varepsilon^{-2}\log n)$ .
For the time complexity, the run time of ESTIMATELEVERAGESCORES  $(A,X)$  is  $O(nd\varepsilon^{-2}\log n)$ ; that of LEVERAGESCORESAMPLING  $(A,\{\widehat{\ell_i} ([AX| - \mathbf{y}]),\forall i\in [n]\})$  is  $O(nd^{2}\varepsilon^{-2}\log n)$ . Finally, we note that the run time of REGRESSIONSOLVER  $(\tilde{A} X,\tilde{\mathbf{y}})$  is  $O(nd^2 +d^3)^2$ . Hence, the total run time is  $O(nd^{2}\varepsilon^{-2}\log n)$  given the setting that we consider has  $d < n$ .
2We refer to [24-28] that introduce a variety of acceleration algorithms for regression.
Extension to the more-than-two-layer case: Note that without non-linearities, considering a multi-layer network is equivalent to performing additional matrix multiplication by  $A$  to the output of the previous layer. Since our proposed scheme yields a spectral approximation guarantee for matrix multiplication, an analysis similar to that of Algorithm 1 should be possible for this extension to obtain a  $(1 + \varepsilon)^{L}$ -approximation guarantee, where  $L$  denotes the number of layers. Since the approximation guarantee comes with high probability, a union bound could be used to obtain a high probability guarantee for the multi-layer case (since the order or error for each approximate matrix multiplication is the same).
4 Variance Minimization via Data-dependent Random Sampling
As stated in the previous section, Algorithm 1 produces an unbiased estimate of  $AX$ . Yet the uniform sampling strategy used in Algorithm 1 does not take the variance of the estimator into consideration. This motivates us to develop an estimator of  $AX$  that further reduces the estimator variance while maintaining unbiasedness. Due to the unbiasedness of  $\widehat{AX}$ , the total variance of the estimator coincides with the mean square error (MSE):  $\operatorname{Var}(\widehat{AX}) = \mathbb{E}\| AX - \widehat{AX}\|_F^2 = \operatorname{MSE}(\widehat{AX})$ .
In order to reduce the estimator variance  $\operatorname{Var}(\widehat{AX})$ , we consider a data-dependent sampling probability  $p_j$  that depends on the  $j$ th column of  $A$  and the  $j$ th row of  $X$ . As we view  $A_{:j}X_{j}$ : as capturing the impact of node  $j$ 's features after a one-step message passing operation, one way to set the sampling probability for node  $j$  is to take the norm of  $A_{:j}X_{j}$  into account. Specifically, we set
$$
p _ {j} = \min  \left[ B \cdot \frac {\| A _ {: j} \| \| X _ {j :} \|}{\sum_ {j} \| A _ {: j} \| \| X _ {j :} \|}, 1 \right]. \tag {11}
$$
Algorithm 2 describes the detailed procedure, where the only distinction relative to Algorithm 1 is the choice of  $p_j$ . We claim that this sampling probability choice minimizes the total variance (as well as the MSE) of the estimator. First notice that the total variance can be expressed as
$$
\begin{array}{l} \operatorname {V a r} (\widehat {A X}) = \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {d} \operatorname {V a r} \left(\left[ \widehat {A X} \right] _ {i j}\right) \stackrel {(i)} {=} \sum_ {k = 1} ^ {n} \left(\frac {1}{p _ {k}} \left(\sum_ {i = 1} ^ {n} a _ {i k} ^ {2}\right) \left(\sum_ {j = 1} ^ {d} x _ {k j} ^ {2}\right)\right) - C _ {2} \left(C _ {2} := n d C _ {1}\right) \\ \stackrel {(i i)} {=} \left(\sum_ {k = 1} ^ {n} \frac {\| A _ {: k} \| ^ {2} \| X _ {k :} \| ^ {2}}{p _ {k}}\right) \cdot \left(\sum_ {k = 1} ^ {n} p _ {k}\right) - C _ {2} \\ \stackrel {(i i i)} {\geq} \left(\sum_ {k = 1} ^ {n} \| A _ {: k} \| \| X _ {k:} \|\right) ^ {2} - C _ {2}, \\ \end{array}
$$
where  $(i)$  follows from the fact that
$$
\begin{array}{l} \operatorname {V a r} \left(\widehat {\left[ A X \right]} _ {i j}\right) = \sum_ {k = 1} ^ {n} \operatorname {V a r} \left(\frac {I _ {k}}{p _ {k}} a _ {i k} x _ {k j}\right) = \sum_ {k = 1} ^ {n} \frac {1}{p _ {k} ^ {2}} a _ {i k} ^ {2} x _ {k j} ^ {2} \operatorname {V a r} \left(I _ {k}\right) = \sum_ {k = 1} ^ {n} \frac {1 - p _ {k}}{p _ {k}} a _ {i k} ^ {2} x _ {k j} ^ {2} \\ = \sum_ {k = 1} ^ {n} \frac {1}{p _ {k}} a _ {i k} ^ {2} x _ {k j} ^ {2} - \sum_ {k = 1} ^ {n} a _ {i k} ^ {2} x _ {k j} ^ {2} = \sum_ {k = 1} ^ {n} \frac {1}{p _ {k}} a _ {i k} ^ {2} x _ {k j} ^ {2} - C _ {1} (\text {f o r s o m e c o n s t a n t} C _ {1}) \\ \end{array}
$$
and (ii) follows from using the fact that  $\sum_{j}p_{j} = 1$ ; and (iii) follows from Cauchy-Schwarz inequality. Notice that, in order for Cauchy-Schwarz to hold with equality, we must have  $p_k$  proportional to  $\| A_{:k}\| \| X_{k:}\|$ . This implies that (11) is an optimal choice of the sampling probabilities.
As it turns out, the same theoretical guarantees obtained for Algorithm 1 hold in this case. See Appendix G for the detailed proof. While the theoretical performance guarantee is the same, in settings where the budget  $B$  is very small, the data-dependent sampling probabilities  $p_j$  lead to a better estimate  $\widehat{AX}$ , which improves the overall performance of the algorithm.
One important observation is that it is not obvious how to use the data-dependent sampling probabilities  $p_j$  while avoiding the an  $O(n^2)$  computation complexity. In particular, a naive computation of  $\| A_{j} \|$ ,  $j \in [n]$  would require observing all  $n^2$  entries of  $A$ . However, standard sampling techniques could be used in order to also estimate  $\| A_{j} \|$ ,  $j \in [n]$  fairly accurate, and use those estimates to compute the sampling probabilities  $p_j$ ,  $j \in [n]$ .
Algorithm 2 ESTIMATELEVERAGESCORES(A,X) via Data-dependent Sampling
Input: Adjacency matrix  $A\in \mathbb{R}^{n\times n}$  data matrix  $X\in \mathbb{R}^{n\times d}$  budget  $B$  , threshold  $\varepsilon >0$    
Output: Leverage score estimates  $\widehat{\ell_i} ([AX| - \mathbf{y})$ $\forall i\in [n]$    
1: Draw  $I_{j}\sim \mathrm{Bern}(p_{j})$ $\forall j\in [n]$  independently, where  $p_j = \min \left[B\cdot \frac{\|A_{:j}\|\|X_{j:\|}}{\sum_j\|A_{:j}\|\|X_{j:\|}},1\right]$    
2:  $\widehat{A_{:j}X_{j:}}\gets \frac{I_j}{p_j} A_{:j}X_{j:},\forall j\in [n]$    
3:  $\widehat{AX}\gets \sum_{j = 1}^{n}\widehat{A_{:j}X_{j:}}$ $\triangleright O(Bd)$    
4:  $S\gets$  LEVERAGESCORE([AX- y])  $\triangleright \tilde{O} (\mathtt{nnz}([\widehat{AX} -\mathbf{y}] + d^3)$    
5:  $\{\widehat{\ell}_i([AX - \mathbf{y}]),\forall i\in [n]\} \gets \frac{(1 + \varepsilon)^2}{(1 - \varepsilon)^2}\cdot S$    
6: return  $\widehat{\ell_i} ([AX - y]),\forall i\in [n]$
We support our theoretical findings via numerical experiments<sup>3</sup> to validate the performance of Algorithm 1 and Algorithm 2 in a non-asymptotic setting.
Dataset and evaluation methods: We consider benchmark datasets from the Open Graph Benchmark (OGB) [29], Stanford Network Analysis Project (SNAP) [30], and House dataset [31]. We defer detailed explanations of the datasets to Appendix H. For evaluation methods, we compute the mean squared error (MSE), wall-clock run-time, and peak memory usage of our two proposed schemes and five baselines: (1) REGRESSIONSOLVER  $(AX, y)$  with fully known  $A$  and  $X$ ; (2) REGRESSIONSOLVER  $(\tilde{A} X, y)$  with parts of rows of  $A$ , say  $\tilde{A}$ , obtained from sampling rows of  $A$  uniformly at random; (3) REGRESSIONSOLVER  $(\tilde{A} X, y)$  with the exact leverage score sampling for sampling rows of  $A$  to obtain partial  $\tilde{A}$ ; (4) REGRESSIONSOLVER  $(\tilde{A} X, \tilde{y})$  with Algorithm 1; (5) REGRESSIONSOLVER  $(\tilde{A} X, \tilde{y})$  with Algorithm 2; (6) REGRESSIONSOLVER  $(\tilde{A} X, \tilde{y})$  with a sampling scheme inspired by GraphSage [6] that selects the neighborhood of feature aggregation for each node; and (7) REGRESSIONSOLVER  $(\tilde{A} X, \tilde{y})$  with a sampling scheme motivated by Graph-Saint [32] that selects a subgraph of the original graph. We note that (6) and (7) are not the tools themselves [6, 32], but graph subsampling strategies inspired by these tools.
Results and analysis:
Mean squared error: Figure 2 plots the mean squared error with respect to the budget allocation. First, we observe that Algorithm 1 and Algorithm 2 have a similar performance to the one with exact leverage scores sampling, and the performance of the two proposed schemes tends to be similar to that of REGRESSIONSOLVER with full  $(AX,y)$ , as the budget increases. Particularly seen from Figure 2a, the error reduction on Algorithm 2 from Algorithm 1 is  $84\%$  when the budget is at  $3\%$ . Even for the sparse datasets, as seen from Figure 2b and Figure 2c, the corresponding improvements of  $86\%$  and  $67\%$  exist when the budget is at  $3\%$  and  $7\%$  respectively. We also observe that the error decrease for Algorithm 2 over Algorithm 1; for instance Figure 2b demonstrates the existence of  $91\%$  improvement at  $5\%$  observation budget. On the other hand, we observe that other subsampling approaches—including those employed in uniform sampling, GraphSage, and GraphSaint—do not perform the graph-weighted regression task as well, especially in the low-budget regime.
Extension to nonlinear GCN: While our theoretical results are for linear GCNs, we also assess the efficacy of our proposed algorithm on nonlinear GCNs. Using the ReLU activation function and one hidden layer, Figure 2d plots the mean squared error as a function of the observation budget, shown as a percentage of the number of observed nodes in the graph. Interestingly, we observe that running the non-linear GCN with uniformly sampled  $(AX, y)$  fails to perform the graph-weighted regression task, especially in the low-budget regime. On the other hand, we observe that running the non-linear GCN with our proposed sampling technique has a similar performance to the one with exact leverage scores sampling, and the performance of the two proposed schemes tends to be similar to that of Regression with exact  $(AX, y)$ , as the budget increases. Particularly, the error reduction on Algorithm 1 from Regression with uniformly sampled  $(AX, y)$  is  $60\%$  when the budget is at  $5\%$ .
3Our code is publicly available online at https://github.com/seiyun-shin/gnn_node_subsampling.
(a) Linear GCN on ogbl-ddi dataset [29] and generated according to the Cauchy distribution.
$X$  (b) Linear GCN on ego-Facebook dataset [30] and  $X$  generated by the Cauchy distribution.
(c) Linear GCN on house dataset [33].
(d) Non-linear GCN on house dataset [33].
Figure 2: MSE w.r.t. observation budget  $(\%)$ : (1) Regression with full  $AX$ ; (2) one with partially observed  $AX$  obtained by sampling rows of  $A$  uniformly at random; (3) one with partial  $AX$  based on the exact leverage score sampling; (4) and (5) one with Algorithm 1 and Algorithm 2; and (6) and (7) ones with GraphSage [6] inspired sampling and GraphSaint [32] inspired sampling algorithms.
Table 1: Wall-clock time comparison on end-to-end processes
Dataset	# Nodes	# Edges	# Features	Wall-clock time (sec)
use of full AX	use of Algorithm 1 and leverage score sampling
ogbl-ddi [29]	4.3K	1.3M	100	1.49	1.39
ogbn-arxiv [29]	169.3K	1.2M	128	299.48	7.40
Synthetic data (Gaussian)	50.0K	625.0M	500	27.28	5.77
Synthetic data (Gaussian)	100.0K	2.5B	500	107.10	8.97
Synthetic data (Gaussian)	150.0K	5.6B	500	247.70	9.96
Wall-clock time: In Table 1, we provide a run-time comparison for the end-to-end training process for a regression task. We compare the wall-clock time of performing a regression task with full  $AX$  computation and that with Algorithm 1 that uses partial observations of  $A$  and  $X$ . The results show that our proposed scheme requires orders of magnitude less wall-clock time for large-scale graphs than the regression with exact  $AX$  computation. In particular, for the ogbn-arxiv dataset [29], our algorithm runs about 40x faster than the regression with the exact computation of  $AX$ .
Peak memory usage: We also demonstrate the efficacy on the memory usage of our proposed algorithms. See Table 3 in Appendix H for details.
Motivated by the prohibitive computational/storage costs of running GNNs with large-scale graphs, we considered the problem of subsampling nodes' information for performing a regression task.
Our main contribution lies in providing an efficient sampling algorithm for learning linear GNNs with a large number of nodes with a theoretical guarantee. Specifically, for a two-layer linear GNN with adjacency matrix  $A \in \mathbb{R}^{n \times n}$  and data matrix  $X \in \mathbb{R}^{n \times d}$ , we show that it is possible to learn the model accurately while observing only  $O(nd\log n)$  entries of  $A$ . This in turn yields a run-time of  $O(nd^2\log n)$ , avoiding the computation time  $O(n^2d)$  of using the full matrix multiplication  $AX$ . While we view our main contribution to be of a theoretical flavor, on real-world benchmark datasets, we also demonstrate the run-time/memory improvements of our proposed schemes over other baselines via wall-clock time and peak-memory usage comparisons and MSE comparisons with respect to the observation budget.
Our result on learnability with the run-time complexity reduction is particularly significant when the number of nodes  $n$  is large and the feature dimension is much smaller compared to  $n$  (i.e.,  $d \ll n$ ). In addition, we note that the result holds even when the graph adjacency matrix and the data matrix are dense (e.g., weighted graphs where the  $a_{ij}$  entries of the adjacency matrix are non-negative real values, and not restricted to  $\{0,1\}$ ). Examples include pairwise similarity matrices and Erdos-Renyi graphs with constant edge probability  $p$ , which have a number of edges that scales as  $n^2$ . In such cases, sparse matrix multiplication would not help much, and our approach could be very useful.
Our proposed scheme is not only useful for speeding up the training of linear GCNs. It is also useful in partial observation settings where the algorithm only has access to a partial view of the graph, and we provide theoretical guarantees that this partial view is still sufficient to accurately perform the regression task. On the contrary, conventional speed-up methods such as sparse matrix multiplication are purely a technique for computational purposes and still require complete information on  $A$  and  $X$ . For instance, consider a social network setting where, in order to know friendships between users, a learner needs to use an API to query the friendships. Note that one needs fewer queries for using our method (since it requires partial  $A$ ) than for using the sparse matrix multiplication (which needs full knowledge of  $A$ ). If the graph is large, the query complexity gain becomes larger.
Several extensions of our results are possible. First, establishing theoretical guarantees for the nonlinear GCN framework, and for other target tasks, such as graph link prediction and node classification, would be of significant interest. Based on the empirical observation that our training scheme works well for a two-layer non-linear GCN, extending our theoretical results to the non-linear GCN under some assumptions (as in [34]) on non-linear activation functions seems to be promising. Note that our proposed sampling scheme (based on approximating the leverage score) and the theoretical guarantee we made are tailored for the node-level regression task in the training phase. The reason we focused on this specific setting was so that we could take advantage of the theoretical results on leverage score sampling and linear regression in order to obtain theoretical results for GNN training using a subsampled graph. In particular, we are helped by the fact that in the linear GCN setting, the optimal solution for the linear regression is a matrix-vector product which can be expressed as  $((AX)^{\top}(AX))^{\dagger}(AX)^{\top}\mathbf{y}$ , where  $A$ ,  $X$ , and  $\mathbf{y}$  denote the adjacency matrix, data matrix, and the labels respectively.
For graph link prediction and classification tasks, on the other hand, since the optimal solution cannot be expressed as a matrix-vector product (due to non-linearities and different cost functions), our proposed algorithm does not extend to the classification problems in a straightforward way, and similarly, our theoretical guarantee may not hold anymore. For future work, we want to study whether our same subsampling process can have any guarantee for a node classification task. For more general tasks, we believe that we need different sampling strategies other than the leverage score sampling and hence we may need other tools for developing a theoretical guarantee for sampling algorithms.
Another direction for future work is to establish generalization guarantees beyond the approximation errors [35]. Lastly, motivated by the fast acceleration of gradient descent algorithms [24-28] and iterative schemes that sample a subset of the entries of  $A$  [21, 36], developing algorithms that adaptively select a subset  $(A,X)$  to observe based on the information at each gradient descent step would be an interesting direction to explore.
We thank Ruosong Wang, Yun Yang for the helpful discussion and the anonymous reviewers for their constructive feedback. Han Zhao and Seiyun Shin were partly supported by the Defense Advanced Research Projects Agency (DARPA) under Cooperative Agreement Number: HR00112320012, an IBM-IL Discovery Accelerator Institute research award, and Amazon AWS Cloud Credit. The work of Ilan Shomorony was supported in part by the National Science Foundation (NSF) under grant CCF-2046991.
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Leverage score sampling for linear regression: The history of leverage scores traces back to linear regression problems in statistics [37, 38]. The statistical leverages of a matrix indicate the importance of rows in solving a least-squares problem. It has been well-known that leverage score sampling [39, 40] uses leverage scores to sample a subset of the rows of the matrix and solve the regression task with  $(1 + \varepsilon)$  accuracy using only  $O(d\varepsilon^{-2}\log d)$  subsampled rows of the data matrix  $X$  and in time  $O(nd^{2} + d^{3})$ , and there have been extensive works on reducing the time complexity as well as the sample complexity [41, 18-22]. However, we note that our setting is a graph-weighted linear regression, which involves a matrix multiplication of  $A$  and  $X$ , taking  $O(n^{2}d)$  time. To the best of our knowledge, our work is the first one that addresses the setting of graph-weighted linear regression and proposes an algorithm with subquadratic time complexity in  $n$ .
Sampling on Graph Convolutional Networks (GCNs): Most of the related works focus on reducing the receptive field size via subsampling neighbors of nodes in the input graph. The motivation is that the neighborhood size can grow exponentially with respect to the number of layers due to the message passing operation of each layer. In contrast, we focus on the effect of partial observation in one hidden layer and show that node subsampling can reduce the run time as well as the sample complexity while ensuring the desired performance guarantee. We also highlight that our work provides a theoretical guarantee, whereas many of the cited works below provide empirical results without a theoretical guarantee. For readers interested in works related to sampling on GCNs, we refer to the comprehensive survey in [42]. Here we provide a brief summary.