SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\left(s _ {1}, s _ {2}, s _ {3}, s _ {4}\right) = (0. 0 5, 0. 1, 0. 1, 0. 3).
$$
Instead of using the location-scale sequences of Figure 1, we instead considered tempered sequences of the form
$$
\log \pi_ {n} (x) = \epsilon_ {n} \log \pi_ {0} (x) + (1 - \epsilon_ {n}) \log u (x).
$$
For the converging sequences considered we set  $u$  to be the target (either  $u = p_1$  or  $u = p_2$ ) and set  $u = \mathcal{N}(x; 0, 0.4^2)$  for each of the non-converging sequences. The different sequences vary in choice of  $\pi_0$  and tempering sequence  $(\epsilon_n)_{n \in \mathbb{N}}$ . These choices are displayed in Figure S6 and are taken as the same for both of the targets considered.
The specification of our choices of  $q$  is the following:
- Prior: For  $p_1$ , we took  $q \sim \mathcal{N}(0, 0.5^2)$ . For  $p_2$ , we took  $q \sim \text{Student-T}(10, 0, 0.5)$ .
- Laplace: For  $p_1$ , the Laplace approximation computed was  $q \sim \mathcal{N}(0, 0.051^2)$ . For  $p_2$ , the Laplace approximation computed was  $q \sim \mathcal{N}(0, 0.125^2)$ .
- GMM: For both targets, the Gaussian mixture model was computed using 100 samples from the target. In both cases, the number of components used was 3, since this value minimised the Bayes information criterion [Schwarz, 1978].
- KDE: For both targets, the kernel density estimate was computed using 100 samples from the target. In both cases, we utilised a Gaussian kernel  $k(x,y) = \exp (-(x - y)^2 /\ell^2)$  with the lengthscale or bandwidth parameter  $\ell$  determined by Silverman's rule of thumb [Silverman, 1986].
Results for  $p_1$  are displayed in Figure S7 and results for  $p_2$  are displayed in Figure S8. It can be seen that for both target distributions and the different sequences considered, GF-KSD correctly detects convergence in each case. For both targets and for  $q = \mathrm{Laplace}$ , it can be seen that GF-KSD exhibits the same behaviour of Failure Mode (b), displayed in Figure 2b.
The values of GF-KSD reported in Figure S7 and Figure S8 were computed using a quasi Monte Carlo approximation to the integral (6), utilising a length 300 low-discrepancy sequence. Due to the lack of an easily computable inverse CDF, we performed an importance sampling estimate of GF-KSD as follows
$$
\begin{array}{l} \mathrm {D} _ {p, q} (\pi) = \iint \left(\mathcal {S} _ {p, q} \otimes \mathcal {S} _ {p, q}\right) k (x, y) \mathrm {d} \pi (x) \mathrm {d} \pi (y) \\ = \iint (\mathcal {S} _ {p, q} \otimes \mathcal {S} _ {p, q}) k (x, y) \frac {\pi (x) \pi (y)}{w (x) w (y)} \mathrm {d} w (x) \mathrm {d} w (y), \\ \end{array}
$$
where  $w$  is the proposal distribution. For each element of a sequence  $\pi_n$ , we used a Gaussian proposal  $w_n$  of the form:
$$
\log w _ {n} (x) = \epsilon_ {n} \log \pi_ {0} (x) + (1 - \epsilon_ {n}) \log \mathcal {N} (x; 0, 0. 4 ^ {2}).
$$
Since  $\pi_0$  is Gaussian for each sequence, this construction ensures that each  $w_{n}$  is both Gaussian and a good proposal distribution for  $\pi_{n}$ . The low-discrepancy sequences were then obtained by first specifying a uniform grid over [0, 1] and the performing an inverse CDF transformation using  $w_{n}$ .
C.5.2 Exploring the Effect of  $\sigma$  and  $\beta$
In this section we investigate the effect on convergence detection that results from changing the parameters  $\sigma$  and  $\beta$  in the inverse multi-quadric kernel (2). Utilising the same test sequences and choices of  $q$  used in Figure 1, we plot the values of GF-KSD in Figure S9. It can be seen that the convergence detection is robust to changing values of  $\sigma$  and  $\beta$ .
C.5.3 GF-KSD vs. KSD Importance Sampling
In this section we investigate the performance of gradient-free Stein importance sampling, standard Stein importance sampling, and self-normalised importance sampling, as the distribution  $q$  varies in quality as an approximation to  $p$ . We consider two different regimes:
1.  $p = \mathcal{N}(0,I)$  and  $q = \mathcal{N}(0,\lambda I)$  for  $0.7\leq \lambda \leq 1.3$
2.  $p = \mathcal{N}(0,I)$  and  $q = \mathcal{N}(c\mathbf{1},I)$  for  $-0.6\leq c\leq 0.6$  , where  $\mathbf{1} = (1,\dots ,1)^{\top}$
In both cases, we consider the performance of each approach for varying dimension  $d$  and number of samples  $n$ . Results are reported in Figure S10 and Figure S11 for each regime respectively. The quadratic programme defining the optimal weights of gradient-free Stein importance sampling and Stein importance sampling (refer to Theorem 3) was solved using the splitting conic solver of O'Donoghue et al. [2016].
Figure S7: Additional empirical assessment of gradient-free kernel Stein discrepancy using the target  $p_1$  defined in Appendix C.5.1. (a) Test sequences  $(\pi_n)_{n \in \mathbb{N}}$ , defined in Appendix C.5.1. The first column displays sequences (solid) that converge to the distributional target  $p$  (black), while the second column displays sequences (dashed) which converge instead to a fixed Gaussian target. (b) Performance of gradient-free kernel Stein discrepancy, when different approaches to selecting  $q$  are employed. The colour and style of each curve in (b) indicates which of the sequences in (a) is being considered. [Here we fixed the kernel parameters  $\sigma = 1$  and  $\beta = 1/2$ .]
Figure S8: Additional empirical assessment of gradient-free kernel Stein discrepancy using the target  $p_2$  defined in Appendix C.5.1. (a) Test sequences  $(\pi_n)_{n \in \mathbb{N}}$ , defined in Appendix C.5.1. The first column displays sequences (solid) that converge to the distributional target  $p$  (black), while the second column displays sequences (dashed) which converge instead to a fixed Gaussian target. (b) Performance of gradient-free kernel Stein discrepancy, when different approaches to selecting  $q$  are employed. The colour and style of each curve in (b) indicates which of the sequences in (a) is being considered. [Here we fixed the kernel parameters  $\sigma = 1$  and  $\beta = 1/2$ .]
Figure S9: Comparison of different values of  $\sigma$  and  $\beta$  in the inverse multiquadric kernel. Here the vertical axis displays the logarithm of the gradient free kernel Stein discrepancy. The colour and style of each of the curves indicates which of the sequences in Figure 1 is being considered.
Figure S10: Comparison of the performance of importance sampling methodologies in varying dimension  $d$  and number of sample points considered  $n$  under the regime  $q = \mathcal{N}(0, \lambda I)$ . The approximation quality is quantified as the logarithm of the Energy Distance (ED).
Figure S11: Comparison of the performance of importance sampling methodologies in varying dimension  $d$  and number of sample points considered  $n$  under the regime  $q = \mathcal{N}(c, I)$ . The approximation quality is quantified as the logarithm of the Energy Distance (ED).