SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In this section, we will outline a proof strategy for the results reported earlier in this section. Full technical details are reported in App. 8.4.
From its Taylor expansion, the  $(ij)$ -th element of  $(\mathbf{I}_N - \mathbf{U})^{-2}$  is nothing other than a sum over all possible walks between the nodes  $i$  and  $j$ , weighted by their lengths and the respective products of edge weights. GRFs use a Monte Carlo scheme to approximate this sum by sampling such walks at random – concretely, by first sampling separate random walks out of nodes  $i$  and  $j$  and summing contributions wherever they intersect. In order to sample the space of walks between nodes  $i$  and  $j$  more efficiently, it follows that we should make our ensemble of walks out of each node more diverse. q-GRFs achieve this by inducing negative correlations such that they are different lengths.
In more detail, it is clear that the variance of the kernel estimator will depend upon the expectation of the square of  $\phi (i)^{\top}\phi (j)$ . Each term in the resulting sum will take the form
$$
\sum_ {x, y \in \mathcal {N}} \sum_ {\omega_ {1} \in \Omega_ {i x}} \sum_ {\omega_ {2} \in \Omega_ {j x}} \sum_ {\omega_ {3} \in \Omega_ {i y}} \sum_ {\omega_ {4} \in \Omega_ {j y}} \frac {\widetilde {\omega} (\omega_ {1})}{p (\omega_ {1})} \frac {\widetilde {\omega} (\omega_ {2})}{p (\omega_ {2})} \frac {\widetilde {\omega} (\omega_ {3})}{p (\omega_ {3})} \frac {\widetilde {\omega} (\omega_ {4})}{p (\omega_ {4})} \tag {16}
$$
$$
\cdot p \left(\omega_ {1} \in \bar {\Omega} \left(k _ {1}, i\right), \omega_ {3} \in \bar {\Omega} \left(k _ {2}, i\right)\right) p \left(\omega_ {2} \in \bar {\Omega} \left(l _ {1}, j\right), \omega_ {4} \in \bar {\Omega} \left(l _ {2}, j\right)\right),
$$
where we direct the reader to Sec. 2.2 for the symbol definitions. Supposing that all edge weights are equal (an assumption we relax later), the summand is a function of the length of each of the walks  $\omega_{1,2,3,4}$ . It is then natural to write the sum over all walks between nodes  $i$  and  $x$  as a sum over walk lengths  $m$ , with each term weighted by a combinatorial factor that counts the number of walks of said length. This factor is  $(\mathbf{A}^m)_{ix}$ , with  $\mathbf{A}$  the unweighted adjacency matrix. That is,
$$
\sum_ {\omega_ {1} \in \Omega_ {i x}} (\cdot) = \sum_ {m = 0} ^ {\infty} \left(\mathbf {A} ^ {m}\right) _ {i x} (\cdot). \tag {17}
$$
Figure 2: Relative Frobenius norm error of estimator of the 2-regularised Laplacian kernel with GRFs (red circle) and q-GRFs (green cross). Lower is better. We also include q-RRW-GRFs [Choromanski, 2023] (blue triangle), instantiated with  $f = \exp$ , as a benchmark. Our novel q-GRFs perform the best on every graph considered.  $N$  is the number of nodes and, for the Erdős-Rényi (ER) graphs,  $p$  is the edge-generation probability. One standard deviation is shaded but it is too small to easily see.
$(\mathbf{A}^m)_{ix}$  is readily written as its eigendecomposition,  $\sum_{p=1}^{N} \lambda_p^m k_{pi} k_{px}$ , with  $\lambda_p$  the  $p$ th eigenvalue and  $k_{pi}$  the  $i$ -th coordinate of the  $p$ -th eigenvector  $\mathbf{k}_p$ . We put these into Eq. 16 and perform the sums over each walk length  $m_{1,2,3,4}$  from 1 to  $\infty$ , arriving at
$$
\sum_ {x, y \in \mathcal {N}} \sum_ {k _ {1}, k _ {2}, k _ {3}, k _ {4}} f \left(\lambda_ {1}, \lambda_ {2}, \lambda_ {3}, \lambda_ {4}\right) k _ {1 i} k _ {1 x} k _ {2 j} k _ {2 x} k _ {3 i} k _ {3 y} k _ {4 j} k _ {4 y} \tag {18}
$$
where  $f(\lambda_1, \lambda_2, \lambda_3, \lambda_4)$  is a function of the eigenvalues of  $\mathbf{A}$  that depends on whether we correlate the terminations of the walkers. Since the eigenvectors are orthogonal we have that  $\sum_{x \in \mathcal{N}} k_{1x} k_{2x} = \delta_{12}$ , so this reduces to
$$
\sum_ {k _ {1}, k _ {3}} f \left(\lambda_ {1}, \lambda_ {1}, \lambda_ {3}, \lambda_ {3}\right) k _ {1 i} k _ {1 j} k _ {3 i} k _ {4 j}. \tag {19}
$$
Our task becomes to prove that this expression becomes smaller when we induce antithetic termination. We achieve this by showing that a particular matrix (with elements given by the difference in  $f$  between the schemes) is negative definite: a task we achieve by appealing to Weyl's perturbation inequality [Bai et al., 2000].
In this section we report on empirical evaluations of q-GRFs. We confirm that they give lower kernel estimator variance than regular GRFs and show that this often leads to substantially better performance in downstream tasks, including simulation of graph diffusion,  $k$ -means node clustering and kernel regression for node attribute prediction. We use ensembles of antithetic pairs (Def. 3.1).
4.1 Estimation of the 2-regularised Laplacian kernel
We begin with the simplest of tasks: estimation of the 2-regularised Laplacian kernel,
$$
\mathbf {K} _ {\text {l a p}} ^ {(2)} = \left(\mathbf {I} _ {N} + \sigma^ {2} \widetilde {\mathbf {L}}\right) ^ {- 2}, \tag {20}
$$
where  $\widetilde{\mathbf{L}}\in \mathbb{R}^{N\times N}$  is the symmetrically normalised Laplacian.  $0 < \sigma < 1$  is a regulariser. We use both GRFs and q-GRFs to generate unbiased estimates  $\widetilde{\mathbf{K}}_{\mathrm{lap}}^{(2)}$  (see App. 8.2), then compute the relative Frobenius norm  $\| \mathbf{K}_{\mathrm{lap}}^{(2)} - \widetilde{\mathbf{K}}_{\mathrm{lap}}^{(2)}\|_{\mathrm{F}}^2 /\| \mathbf{K}_{\mathrm{lap}}^{(2)}\|_{\mathrm{F}}^2$  between the true and approximated kernel matrices. This enables us to compare the quality of the estimators. As a benchmark, we also include
an implementation of the high-level reinforced random walk QMC mechanism suggested (but not tested) by Choromanski [2023]. We choose the exponential mapping as the reinforcement function  $f$  (used to downweight the probability of traversing previously-visited edges), although the optimal choice remains an open problem. We refer to this mechanism as  $q$ -RRW-GRFs to disambiguate from our instantiation of q-GRFs (which uses antithetic termination).
Fig. 2 presents the results for a broad class of graphs: small Erdős-Rényi, larger Erdős-Rényi, a binary rooted tree, a ladder, and four real-world examples from Ivashkin [2023] (karate, dolphins, football and eurosis). We consider 2, 4, 8 and 16 walks, taking 100 repeats for the variance of the approximation error. We use the regulariser  $\sigma = 0.1$  and the termination probability  $p = 0.5$ .
The quality of kernel approximation naturally improves with the number of walkers. Inducing antibiotic termination consistently reduces estimator variance, with our q-GRF mechanism outperforming regular GRFs in every case. The exact size of the gain depends on the particular graph (according to the closed forms derived in App. 8.4), but improvement is always present. It is intriguing that tree-like and planar graphs tend to enjoy a bigger gap; we defer a rigorous theoretical analysis to future work. Meanwhile, the q-RRW-GRF variant with exponential  $f$  is often worse than the regular mechanism and is substantially more expensive. We do not include it in later experiments.
4.2 Scalable and accurate simulation of graph diffusion
In Sec. 2.1, we noted that the Laplacian  $\mathbf{L}$  (or  $\widetilde{\mathbf{L}}$ ) is the natural operator to describe diffusion on discrete spaces and that the 1-regularised Laplacian kernel constitutes the corresponding discrete time-evolution operator. Here we will show how, by leveraging q-GRFs to provide a lower-variance decomposition of  $\mathbf{K}_{\mathrm{lap}}^{(2)}$ , we can simulate graph diffusion in a scalable and accurate way.
Choosing a finite (even) number of discretisation timesteps  $N_{t}$ , we can approximate the final state by
$$
\mathbf {u} _ {t} \simeq \widetilde {\mathbf {u}} _ {t} := \left[ \left(\mathbf {I} _ {N} + \frac {t}{N _ {t}} \widetilde {\mathbf {L}}\right) ^ {- 2} \right] ^ {\frac {N _ {t}}{2}} \mathbf {u} _ {0} = \mathbf {K} _ {\mathrm {l a p}} ^ {(2) \frac {N _ {t}}{2}} \mathbf {u} _ {0} \tag {21}
$$
This is equivalent to approximating the action of the diffusion kernel  $\mathbf{K}_{\mathrm{diff}}$  (see App. 8.2). We can efficiently compute this using our GRF or q-GRF decomposition of  $\mathbf{K}_{\mathrm{lap}}^{(2)}$  and compare the accuracy of reconstruction of  $\mathbf{u}_t$ . In particular, we take an initial one-hot state  $\mathbf{u}_0 = (1,0,0,\dots,0)^\top$  and simulate diffusion for  $t = 1$  divided into  $N_{t} = 1000$  timesteps, using  $m = 10$  walkers and a termination probability  $p = 0.5$ . Fig. 3 gives a schematic. We average the MSE of  $\widetilde{\mathbf{u}}_t$  over 1000 trials. Table 1 reports the results; q-GRFs approximate the time evolution operator more accurately so consistently give a lower simulation error. Since the time-evolution operator is applied repeatedly, even modest improvements in its approximation can lead to a substantially better reconstruction of the final state. In extreme cases the simulation error is halved.
Table 1: MSE of the approximation of the final state  $\mathbf{u}_t$  by repeated application of an approximation of the discrete time-evolution operator, constructed with (q-)GRFs. Brackets give one standard deviation.
Graph	N	Sim error, MSE(ut)
GRFs	q-GRFs
Small ER	20	0.0210(5)	0.0160(3)
Larger ER	100	0.0179(9)	0.0085(3)
Binary tree	127	0.0161(6)	0.0106(3)
Ladder	100	0.0190(8)	0.0105(3)
karate	34	0.066(2)	0.054(1)
dolphins	62	0.0165(4)	0.0139(3)
football	115	0.0170(3)	0.0160(2)
eurosis	1272	0.089(1)	0.084(1)
Figure 3: Schematic of diffusion on a small graph, with heat initially localised on a single node spreading under the action of the Laplace operator. We use (q-)GRFs to estimate the quantity on the right.
2To be asymptotically unbiased, we should construct  $N_{t} / 2$  independent estimates of  $\mathbf{K}_{\mathrm{lap}}^{(2)}$  (see App. 8.2). However, here we observe that the error is small even when we apply the same approximation repeatedly, i.e. estimate  $\mathbf{K}_{\mathrm{lap}}^{(2)}$  once and use it to simulate diffusion at every timestep, despite this remaining biased as  $N_{t}\rightarrow \infty$
4.3 Kernelised  $k$ -means clustering for graph nodes
Next we evaluate the performance of GRFs and q-GRFs on the task of assigning nodes to clusters using the graph kernel, following the approach described by Dhillon et al. [2004]. We first run the algorithm to obtain  $N_{c} = 2$  clusters with the exact 2-regularised Laplacian kernel, then compare the results when we use its approximation via GRFs and q-GRFs. In each case, we report the clustering error, defined by
$$
E _ {c} := \frac {\text {n o . w r o n g p a i r s}}{N (N - 1) / 2}. \tag {22}
$$
This is simply the number of misclassified pairs (in the sense of being assigned to the same cluster when the converse is true or vice versa) di
vided by the total number of pairs. The results are less straightforward because curiously the clustering error does not generally vary monotonically with the variance on the kernel estimator, but nonetheless in six out of eight cases q-GRFs provide equally good or better results.
Table 2: Errors in kernelised  $k$ -means clustering when approximating the Gram matrix with q-GRFs and GRFs. Lower is better.
Graph	N	Clustering error, Ec
GRFs	q-GRFs
karate	34	0.11	0.05
dolphins	62	0.40	0.38
polbooks	105	0.28	0.28
football	115	0.10	0.09
eu-core	1005	0.04	0.05
databases	1046	0.17	0.11
eurosis	1285	0.15	0.19
citeseer	3300	0.02	0.01
4.4 Kernel regression for node attribute prediction
Lastly, we consider the problem of kernel regression on a triangular mesh graph. Each node is associated with a normal vector  $\pmb{v}^{(i)}$ , equal to the mean of the normal vectors of its surrounding faces. The task is to predict the directions of missing vectors (a random  $5\%$  split) from the remainder. We consider five meshes of different sizes, available from Dawson-Haggerty [2023]. Our prediction for the direction of the normal vector corresponding to the  $i$ th node is
$$
\widetilde {\boldsymbol {v}} ^ {(i)} := \frac {\sum_ {j} \mathbf {K} _ {\text {l a p}} ^ {(2)} (i , j) \boldsymbol {v} ^ {(j)}}{\sum_ {j} \mathbf {K} _ {\text {l a p}} ^ {(2)} (i , j)}, \tag {23}
$$
Table 3:  $1 - \cos \theta$  between true and predicted node vectors when approximating the Gram matrix with q-GRFs and GRFs. Lower is better. Brackets give one standard deviation.
Graph	N	Pred error, 1 - cos θ
GRFs	q-GRFs
cylinder	210	0.104(1)	0.101(1)
teapot	480	0.0531(5)	0.0493(5)
idler-riser	782	0.0881(6)	0.0852(5)
busted	1941	0.00690(4)	0.00661(4)
torus	4350	0.00131(1)	0.00120(1)
where  $j$  sums over the vertices with known vectors (a 95% split). This is simply a linear combination of all the training node normal vectors, weighted by their respective kernel evaluations. We compute the average angular error  $1 - \cos \theta$  between the prediction  $\widetilde{\pmb{v}}^{(i)}$  and groundtruth  $\pmb{v}^{(i)}$  across the missing vectors, comparing the result when  $\mathbf{K}_{\mathrm{lap}}^{(2)}$  is approximated with GRFs and q-GRFs with  $m = 6$  random walks at a termination probability  $p = 0.5$ . The regulariser is  $\sigma = 0.1$ . q-GRFs enjoy lower kernel estimator variance so consistently give better predictions for the missing vectors.
We have proposed a novel class of quasi-Monte Carlo graph random features (q-GRFs) for unbiased and efficient estimation of kernels defined on the nodes of a graph. We have proved that our new algorithm, which induces negative statistical correlations between the lengths of graph random walks via antithetic termination, enjoys better convergence properties than its regular predecessor (GRFs). This very often permits better performance in empirical tasks and for some applications the improvement is substantial. Our work ushers in further research in this new domain of quasi-Monte Carlo methods for kernels on combinatorial objects. It may be of broader interest including for algorithms that sample random walks.
6 Broader impacts and limitations
We envisage several possible impacts of our novel algorithm. First, graphs provide a natural way to describe systems characterised by complex biological interactions such as the proteins at the proteome scale or drugs in the body [Ingraham et al., 2019]. Our novel QMC algorithm might be of translational impact in this bioinformatics setting. Antithetic termination is at its heart a procedure to improve the sampling efficiency of random walks, so it could also help mitigate the notoriously high energy and carbon cost of large models [Strubell et al., 2019]. Lastly, we believe our results are of intrinsic interest as the first (to our knowledge) rigorously studied QMC scheme defined on a combinatorial object. They might spur further research in this new domain. Our work is foundational with no immediate direct negative societal impacts that we can see. However, it is important to note that increases in scalability afforded by GRF and q-GRF algorithms could amplify risks of graph-based machine learning, either from bad actors or as unintended consequences.
The work also has some limitations and directions for future research. First, although we have derived closed-form expressions for the kernel estimator variance with GRFs and q-GRFs in App. 8.4, they are still complicated functions of the spectra of the respective graph Laplacians. Understanding what characterises graphs that particularly benefit from antithetic termination (empirically, tree-like and planar graphs) is an important future direction. Moreover, our scheme only correlates walk lengths. A more sophisticated mechanism that couples walk directions might do even better. Lastly, further work is needed to fully understand the applicability of antithetic termination beyond the GRF setting.
7 Relative contributions and acknowledgements
IR devised the antithetic termination scheme, proved all theoretical results and ran the experiments in Secs 4.1, 4.2 and 4.4. KC provided crucial support throughout, particularly: running the clustering experiment in Sec. 4.3, showing how  $d$ -regularised Laplacian kernels can be used to approximate the graph diffusion process, and proposing to apply q-GRFs to the problems in Secs 4.2 and 4.4. AW gave important guidance and feedback on the manuscript.
IR acknowledges support from a Trinity College External Studentship. AW acknowledges support from a Turing AI fellowship under grant EP/V025279/1 and the Leverhulme Trust via CFI.
We thank Austin Tripp and Kenza Tazi for their thoughtful feedback on earlier versions of the text, and Michael Ren for his excellent suggestion to treat the negative definite property perturbatively. We are also grateful to the anonymous reviewers.
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8 Appendices
Notation: to reduce the burden of summation indices, in places we use Dirac's bra-ket notation from quantum mechanics.  $|y\rangle$  can be interpreted as the vector  $\mathbf{y}$  and  $\langle y|$  as  $\pmb{y}^{\top}$ .
8.1 Antithetic termination for PageRank
Here we demonstrate the broader applicability of antithetic termination beyond the GRF setting by showing how it can be used to improve the convergence of numerical estimators of the PageRank vector. The treatment is similar to App. 8.4; the reader may benefit from reviewing this section first or in parallel.
The PageRank vector is a measure of the relative importance of the nodes in a graph, proposed by Page et al. [1998] to rank websites in search engine results. It is the steady state of a Markov chain with a  $N \times N$  transition matrix
$$
\mathbf {P} := (1 - p) \mathbf {T} + \frac {p}{N} \mathbf {E}, \tag {24}
$$
where  $p\in (0,1)$  is a scalar and  $N = |\mathcal{N}|$  is the number of graph nodes. The matrix  $\mathbf{E}\coloneqq [1]_{i,j\in \mathcal{N}}$  has entries that are all equal to 1, and the matrix  $\mathbf{T}$  is given by
$$
\mathbf {T} _ {i j} = \left\{ \begin{array}{l l} \frac {1}{d _ {i}} & \text {i f} (i, j) \in \mathcal {E} \\ 0 & \text {o t h e r w i s e} \end{array} \right. \tag {25}
$$
with  $d_{i} = \sum_{j\in \mathcal{N}}\mathbb{I}((i,j)\in \mathcal{E})$ , the degree (number of neighbours) of the  $i$ th node. The state space of the Markov chain is the nodes  $\mathcal{N}$ .  $\mathbf{T}$  encodes the behaviour of a simple random walk.
Formally, we are solving for the vector  $\pi \in \mathbb{R}^N$  such that
$$
\boldsymbol {\pi} ^ {\top} \mathbf {P} = \boldsymbol {\pi} ^ {\top}, \quad \boldsymbol {\pi} ^ {\top} \mathbf {1} = 1 \tag {26}
$$
where  $\mathbf{1} \coloneqq (1, 1, \dots, 1)$ . Physically,  $\pi_j$  is the stationary probability that a surfer is located at node  $j$ , supposing that at every timestep it either i) moves to one of its neighbouring nodes (with probability  $1 - p$ ) or ii) teleports to a random node (with probability  $p$ ).
For large graphs,  $\pi$  becomes very expensive to compute exactly, motivating a host of research dedicated to approximating it. Solving Eq. 26 and Taylor expanding  $(1 - (1 - p)\mathbf{T})^{-1}$ , the solution is given by
$$
\boldsymbol {\pi} _ {i} = \frac {p}{N} \sum_ {j \in \mathcal {N}} \sum_ {k = 0} ^ {\infty} (1 - p) ^ {k} \mathbf {T} _ {j i} ^ {k}. \tag {27}
$$
Note that the term
$$
p \sum_ {k = 0} ^ {\infty} (1 - p) ^ {k} \mathbf {T} _ {j i} ^ {k} \tag {28}
$$
is nothing other than a sum over all walks between nodes  $j$  and  $i$ , weighted by  $p(1 - p)^k \prod_{i=1}^k \frac{1}{d_i}$  with  $k$  the walk length and  $d_i$  the degree of the  $i$ th node visited on the walk. Supposing a walker terminates with probability  $p$  at each timestep and picks a neighbour to visit at random, this weight is equal to the probability of sampling a particular walk between nodes  $j$  and  $i$ , allowing us to write
$$
\pi_ {i} = \frac {1}{N} \sum_ {j \in \mathcal {N}} p (\text {w a l k e r s t a r t i n g a t} j \text {t e r m i n a t e s a t} i). \tag {29}
$$
This invites the Monte Carlo estimate
$$
\widehat {\pi} _ {i} = \frac {1}{N m} \sum_ {j \in \mathcal {N}} \sum_ {l = 1} ^ {m} \mathbb {I} (\text {w a l k e r s t a r t i n g a t j t e r m i n a t e s a t i}) \tag {30}
$$
with  $m$  the number of walks sampled out of each node. We have arrived at the algorithm proposed by Fogaras et al. [2005]: simulate  $m$  walks out of each node  $j$  of the graph, and compute the estimator  $\widehat{\pi}_i$  as the proportion of walkers terminating at node  $i$ .
Since we are sampling random walks that terminate with probability  $p$  at every timestep, this is a natural setting to try antithetic termination. The following is true.
Theorem 8.1 (Superiority of antithetic termination for PageRank estimation). Supposing an ensemble of random walkers exhibits either i) i.i.d. or ii) antithetic termination and  $p \leq \frac{1}{2}$ ,
$$
\operatorname {V a r} _ {\text {a n t i t h e t i c}} \left(\widehat {\boldsymbol {\pi}} _ {i}\right) \leq \operatorname {V a r} _ {i. i. d.} \left(\widehat {\boldsymbol {\pi}} _ {i}\right). \tag {31}
$$
Proof: It is sufficient to consider the behaviour of a single pair of walkers out of node  $j$  that are either i.i.d. or exhibit antithetic termination. Since in both cases the marginal probability of walkers terminating at node  $i$  is identical, the estimators are unbiased. It is sufficient to determine whether the expectation value
$\mathbb{E}\left(\mathbb{I}[\Omega_1 \text{ terminates at node } i] \mathbb{I}[\Omega_2 \text{ terminates at node } i]\right) = p(\text{both terminate at } i)$  (32) is suppressed by antithetic termination, where  $\Omega_{1,2}$  denotes two walks simulated out of  $j$ .