SlowGuess/NeurIPS_2023_OCR huggingface dataset:
such that these modifications cancel out when we evaluate Eq. 53.
Case 3 (Sec. 66) is also straightforward. The prefactor in square brackets is equal to 85 and is again positive for any valid sampling strategy  $p(\omega_1 \in \bar{\Omega}(k_1, i))$  and  $\mathbf{J}$  does not change, so our arguments still hold and these variance contributions are reduced by antithetic coupling.
We note that these arguments will generalise straightforwardly to any weight-dependent sampling strategy and are not particular to the linear case.  $\widetilde{\omega} /\gamma$  can be replaced by some more complicated variant that defines a valid probability distribution  $p(\omega_1\in \bar{\Omega} (k_1,i))$  and antithetic termination will still prove effective.
8.5.4 Summary
In Sec. 8.5, our theoretical results for antithetic termination have proved robust to generalisations such as relaxing  $d$ -regularity and changing the walk sampling strategy. A qualitative explanation for this is as follows: upon making the changes, the ratio of the joint to marginal probabilities
$$
\frac {p \left(\omega_ {1} , \omega_ {3}\right)}{p \left(\omega_ {1}\right) p \left(\omega_ {3}\right)} \tag {91}
$$
is unmodified. This is because we know how we are modifying the probability over walks and construct the estimator to compensate for it. Meanwhile, the correlations between walk lengths are insensitive to the walk directions, so in every case they continue to suppress the kernel estimator variance. The only kink is the terms described in Sec. 8.4.3 which require a little more work, but the mathematics conspires that our arguments are again essentially unmodified, though perhaps without such an intuitive explanation.
8.6 Beyond antithetic coupling (proof of Theorem 3.4)
Our final theoretical contribution is to consider random walk behaviour when TRVs are offset by less than  $p$ ,  $\Delta < p$ . Unlike antithetic coupling, it permits simultaneous termination. Eqs 13 become
$$
p \left(s _ {1}\right) = p \left(s _ {2}\right) = p, \quad p \left(\bar {s} _ {1}\right) = p \left(\bar {s} _ {2}\right) = 1 - p, \quad p \left(s _ {2} \mid s _ {1}\right) = \frac {p - \Delta}{p}, \tag {92}
$$
$$
p (\bar {s} _ {2} | s _ {1}) = \frac {\Delta}{p}, \quad p (s _ {2} | \bar {s} _ {1}) = \frac {\Delta}{1 - p}, \quad p (\bar {s} _ {2} | \bar {s} _ {1}) = \frac {1 - p - \Delta}{1 - p}.
$$
The probability of two antithetic walks  $\bar{\Omega} (1,i)$  and  $\bar{\Omega} (3,i)$  containing subwalks  $\omega_{1}$  and  $\omega_{3}$  becomes
$$
p \left(\omega_ {3} \in \bar {\Omega} (3, i), \omega_ {1} \in \bar {\Omega} (1, i)\right) = \left\{ \begin{array}{l l} \frac {1}{d ^ {m + n}} \left(\frac {1 - p - \Delta}{1 - p}\right) ^ {n} (1 - p) ^ {m} & \text {i f} n <   m \\ \frac {1}{d ^ {2 m}} (1 - p - \Delta) ^ {m} & \text {i f} n = m \\ \frac {1}{d ^ {m + n}} \left(\frac {1 - p - \Delta}{1 - p}\right) ^ {m} (1 - p) ^ {n} & \text {i f} n > m, \end{array} \right. \tag {93}
$$
which the reader might compare to Eq. 57. In analogy to Eq. 63, this induces the matrix
$$
D _ {k _ {1}, k _ {3}} ^ {\Delta} := \frac {1}{1 - w ^ {2} \lambda_ {k _ {1}} \lambda_ {k _ {3}} \frac {1 - p - \Delta}{(1 - p) ^ {2}}} \left(\frac {1 - w ^ {2} \lambda_ {k _ {1}} \lambda_ {k _ {3}}}{(1 - w \lambda_ {k _ {1}}) (1 - w \lambda_ {k _ {3}})}\right). \tag {94}
$$
We can immediately observe that this is exactly equal to  $C_{k_1,k_3}$  when  $\Delta = p(1 - p)$ , so for a pair of walkers with this TRV offset the variance will be identical to the i.i.d. result. Replacing  $D$  by  $D^{\Delta}$  in  $E_{pq}$  and  $F_{pq}$  and  $J_{pq}$  and reasoning about negative definiteness via their respective Taylor expansions (as well as the new possible cross-term  $D_{k_1,k_3}D_{k_1,k_3}^{\Delta}$ ), it is straightforward to conclude that variance is suppressed compared to the i.i.d. case provided  $\Delta > p(1 - p)$  and  $\rho(\mathbf{U})$  or  $p$  is sufficiently small. The  $p \to 0$  limit demands a slightly more careful treatment: in order to stay in the regime  $p(1 - p) < \Delta < p$  we need to simultaneously take  $\Delta \to 0$ , e.g. by defining  $\Delta(p) := p(1 - p) + ap^2$  with the constant  $0 < a < 1$ .
This result was reported in Theorem 3.4 of the main text.
8.7 What about diagonal terms?
The alert reader might remark that all derivations in Sec. 8.4 have taken  $i \neq j$ , considering estimators of the off-diagonal elements of the matrix  $(\mathbf{I}_N - \mathbf{U})^{-2}$ . In fact, estimators of the diagonal elements  $\phi(i)^\top \phi(i)$  will be biased for both GRFs and q-GRFs if  $\phi(i)$  is constructed using the same ensemble of walkers because each walker is manifestly correlated with, rather than independent of, itself. This is rectified by taking two ensembles of walkers out of each node, each of which may exhibit antithetic correlations among itself, then taking the estimator  $\phi_1(i)^\top \phi_2(i)$ . It is straightforward to convince oneself that, in this setup, the estimator is unbiased and q-GRFs will outperform GRFs. In practice, this technicality has essentially no effect on (q-)GRF performance and doubles runtime so we omit further discussion.
8.8 Further experimental details: compute, datasets and uncertainties
The experiments in Secs. 4.1, 4.2 and 4.4 were carried out on an Intel® Core™ i5-7640X CPU @ 4.00GHz × 4. Each required  $\sim$  1 CPU hour. The experiments in Sec. 4.3 were carried out on a 2-core Xeon 2.2GHz with 13GB RAM and 33GB HDD. The computations for the largest considered graphs took  $\sim$  1 CPU hour.
The real-world graphs and meshes were accessed from Ivashkin [2023] and Dawson-Haggerty [2023], with further information about the datasets available therein. Where we were able to locate them, the original papers presenting the graphs are: Zachary [1977], Lusseau et al. [2003], Newman [2006], Bollacker et al. [1998], Leskovec et al. [2007].
All our experiments report standard deviations on the means, apart from the clustering task in Sec. 4.3 because running kernelised  $k$ -means on large graphs is expensive.