SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Dependence on  $\| \lambda \| _2$  An apparent weakness of Theorem 3.2 is its reliance on the unknown value  $J_{\epsilon}^{\star}$ , which depends on the specifics of the kernel  $K$  and base measure  $\mu$ . Determining it is potentially highly nontrivial even for simple kernels. Slightly weakening Theorem 3.2 we obtain the following corollary, which shows that the dependence on  $\| \lambda \| _2$  is tight, at least for small  $\epsilon$ .
Corollary 3.5. Suppose  $J \geq 2$  is such that  $\sum_{j=2}^{J} \lambda_j^2 \geq c^2 \| \lambda \|_2^2$  for some  $c \leq 1$ . Then  $\| \lambda \|_{2J_\epsilon^*}$  can be replaced by  $c \| \lambda \|_2$  in Theorem 3.2 whenever  $\epsilon \leq c \| \lambda \|_2 / (2\sqrt{J - 1})$ .
Dependence on  $R$  and  $\alpha$  Due to the general nature of our lower bound constructions, it is difficult to capture the dependence on the misspecification parameter  $R$ . As for the probability of error  $\alpha$ , based on related work [16] we expect the gap of size  $\sqrt{\log(1 / \alpha)}$  to be a shortcoming of Theorem 3.1 and not the lower bound. Closing this gap may require a different approach, however, as tests based on empirical  $L^2$  distances are known to have a sub-optimal concentration [27].
Dependence on  $\delta$  The correct dependence on the signal rate  $\delta$  is the most important question left open by our theoretical results. Any method requiring  $n$  larger than a function of  $\delta$  irrespective of  $m$  (as in Theorem 3.1) is provably sub-optimal because taking  $m \gtrsim 1 / (\delta \epsilon)^2$  and  $n$  large enough to estimate both  $P_X, P_Y$  to within accuracy  $\epsilon / 10$  always suffices to reach a fixed level of total error.
3.4 Relation to Prior Results
In this section we discuss some connections of Theorem 3.1 to prior work. Specifically, we discuss how Theorem 3.1 recovers some known results in the literature [4; 18; 37] that are minimax optimal. Details omitted in this section are included in Appendix B.
Binary Hypothesis Testing Suppose the two distributions  $P_{X}, P_{Y}$  are known, we are given  $m$  i.i.d. observations  $Z_{1}, \ldots, Z_{m} \sim P_{Z}$  and our task is to decide between the hypotheses  $H_{0}: P_{X} = P_{Z}$  versus  $H_{1}: P_{Y} = P_{Z}$ . Then, we may take  $n = \infty, R = 0, \delta = 1$  in Theorem 3.1 to conclude that
$$
m \geq c \cdot \frac {C \| \lambda \| _ {\infty} \log (1 / \alpha)}{\epsilon^ {2}}
$$
observations suffice to perform the test at total error  $\alpha$ .
Two-Sample Testing Suppose we have two i.i.d. samples \( X \) and \( Y \), both of size \( n \), from unknown distributions \( P_{X}, P_{Y} \) respectively and our task is to decide between \( H_{0}: P_{X} = P_{Y} \) against \( H_{1}:
$\mathrm{MMD}(P_X, P_Y) \geq \epsilon$ . We split our  $Y$  sample in half resulting in  $Y^{(1)}$  and  $Y^{(2)}$  and form the statistic  $\Psi_{\mathrm{TS}} \triangleq \Psi_{1/2}(X, Y^{(1)}, Y^{(2)}) - \Psi_{1/2}(Y^{(1)}, X, Y^{(2)})$ , where  $\Psi_{1/2}$  is defined in Section 2.2. Then  $|\mathbb{E} \Psi_{\mathrm{TS}}|$  is equal to 0 under the null hypothesis and is at least  $1 - 2\alpha_1$  under the alternative, where  $\alpha_1$  is the target total error probability of  $\Psi_{1/2}$ . Taking  $\alpha_1 = 5\%$ , by repeated sample splitting and majority voting we may amplify the success probability to  $\alpha$  provided
$$
n \geq c ^ {\prime} \frac {C \| \lambda \| _ {2} \log (1 / \alpha)}{\epsilon^ {2}}, \tag {7}
$$
where  $c' > 0$  is universal (see Appendix for details). The upper bound (7) partly recovers [37, Theorem 3 and 5] where authors show that thresholding the MMD with Gaussian kernel  $G_{\sigma}(x,y) = \sigma^{-d}\exp (-\| x - y\| ^2 /\sigma^2)$  achieves the minimax optimal sample complexity  $n\asymp \epsilon^{-(2\beta +d / 2) / \beta}$  for the problem of two-sample testing over the class  $\mathcal{P}_{\beta ,d}$  of  $d$  -dimensional  $(\beta ,2)$  -Sobolev-smooth distributions (defined in Appendix B.3) under  $\epsilon -L^2$  -separation. For this, taking  $\sigma \asymp \epsilon^{1 / \beta}$  ensures that  $\| P - Q\|_{L^2}\lesssim \mathrm{MMD}(P,Q)$  over  $P,Q\in \mathcal{P}_{\beta ,d}$  . Taking e.g.  $\mathrm{d}\mu (x) = \exp (-\| x\| _2^2)\mathrm{d}x$  as the base measure, (7) recovers the claimed sample complexity since  $\| \lambda \| _2^2 = \int G_\sigma^2 (x,y)\mathrm{d}\mu (x)\mathrm{d}\mu (y) = \mathcal{O}(\sigma^{-d})$  hiding dimension dependent constants. Our result requires a bounded density with respect to a Gaussian.
Likelihood-Free Hypothesis Testing By taking  $\alpha \asymp R \asymp \delta = \Theta(1)$  in Theorem 3.1 we can recover many of the results in [18; 32; 33]. When  $\mathcal{X}$  is finite, we can take the kernel  $K(x,y) = \sum_{z\in \mathcal{X}}\mathbb{1}\{x = y = z\}$  in Theorem 3.1 to obtain the results for bounded discrete distributions (defined in Appendix B.1) which state that under  $\epsilon$ -TV-separation the minimax optimal sample complexity is given by  $m \gtrsim 1/\epsilon^2$ ;  $\min\{n,\sqrt{nm}\} \gtrsim \sqrt{|\mathcal{X}|}/\epsilon^2$ . A similar kernel recovers the optimal result for the class of  $\beta$ -Hölder smooth densities on the hypercube  $[0,1]^d$  (see Appendix B.2).
**Curse of Dimensionality** Using the Gaussian kernel  $G_{\sigma}$  as for two-sample testing above, one can conclude by Theorem 3.1 that the required number of samples for (mLFHT) over the class  $\mathcal{P}_{\beta,d}$  under  $\epsilon-L^2$ -separation grows at most like  $\Omega\left(\left(\frac{c}{\epsilon}\right)^{\Omega(d)} \frac{1}{\delta^2}\right)$  for some  $c > 1$ , instead of the expected  $\Omega\left(\left(\frac{c}{\epsilon\delta}\right)^{\Omega(d)}\right)$ . This may be interpreted as theoretical support for the success of LFI in practice where signal and background can be rather different (cf. [5, Figures 2-3]) and the difficulty of the problem stems from the rate of signal events being small (i.e.  $\epsilon \approx 1$  but  $\delta \ll 1$ ).
4 Learning Kernels from Data
Given a fixed kernel  $K$ , our Theorems 3.1 and 3.2 show that the sample complexity is heavily dependent on the separation  $\epsilon$  under the given MMD as well as the spectrum  $\lambda = \lambda(\mu, K)$  of the kernel. Thus, to have good test performance we need to use a kernel  $K$  that is well-adapted to the problem at hand. In practice, however, instead of using a fixed kernel it would be only natural to use part of the simulation sample to try to learn a good kernel.
In Sections 4 and 5 we report experimental results after training a kernel parameterized by a neural network on part of the simulation data. In particular, due to the dependence between the data and the kernel, Theorem 3.1 doesn't directly apply. Our main contribution here is showing the existence of an asymmetric simulation-experimentation trade-off (cf. Figure 1.2 and also Section 1.1) even in this realistic setting. Figure 1.2 plots the total number  $n$  of simulations used, including those used for training, so as to provide a realistic view of the amount of data used. The experiments also illustrate that the (trained-)kernel-based statistic of Section 2.2 achieves state-of-the-art performance.
4.1 Proposed Training Algorithm
Consider splitting the data into three parts:  $(X^{\mathrm{tr}}, Y^{\mathrm{tr}})$  is used for training (optimizing) the kernel;  $(X^{\mathrm{ev}}, Y^{\mathrm{ev}})$  is used to evaluate our test statistic at test time; and  $(X^{\mathrm{cal}}, Y^{\mathrm{cal}})$  is used for calibrating the distribution of the test statistic under the null hypothesis. We write  $n_s = |X^s| = |Y^s|$  for  $s \in \{\mathrm{tr}, \mathrm{ev}, \mathrm{cal}\}$ . Given the training data  $X^{\mathrm{tr}}, Y^{\mathrm{tr}}$  with empirical measures  $\widehat{P}_{X^{\mathrm{tr}}}, \widehat{P}_{Y^{\mathrm{tr}}}$ , we maximize the objective in
$$
\widehat {J} \left(X ^ {\mathrm {t r}}, Y ^ {\mathrm {t r}}; K\right) = \frac {\operatorname {M M D} _ {u} ^ {2} \left(\widehat {P} _ {X ^ {\mathrm {t r}}}, \widehat {P} _ {Y ^ {\mathrm {t r}}}; K\right)}{\widehat {\sigma} \left(X ^ {\mathrm {t r}} , Y ^ {\mathrm {t r}} ; K\right)}, \tag {8}
$$
which was introduced in [48]. Here  $\hat{\sigma}^2$  is an estimator of the variance of  $\mathrm{MMD}_u^2 (\widehat{P}_{X^{\mathrm{tr}}},\widehat{P}_{Y^{\mathrm{tr}}};K)$  and is defined in Appendix F.1.
Algorithm 1 mLFHT with a learned deep kernel
Input:  $(X^{\mathrm{tr}}, X^{\mathrm{ev}}, X^{\mathrm{cal}})$ ,  $(Y^{\mathrm{tr}}, Y^{\mathrm{ev}}, Y^{\mathrm{cal}})$ ; parametrized kernel  $K_{\omega}$ ; hyperparameters and initialization.
# Phase 1: Kernel training (optimization) on  $X^{\mathrm{tr}}$  and  $Y^{\mathrm{tr}}$ .
$\omega \leftarrow \arg \max_{\omega}^{\mathrm{Optimize}}\widehat{J} (X^{\mathrm{tr}},Y^{\mathrm{tr}};K_{\omega})$  # maximize objective as defined in (8)
# Phase 2: Distributional calibration of test statistic (under null hypothesis).
for  $r = 1,2,\dots ,k$  do
$Z^{\mathrm{cal},r}\gets$  sample  $m$  points without replacement from  $X^{\mathrm{cal}}$
$$
T _ {r} \gets \frac {1}{n _ {\mathrm {e v}} m} \sum_ {i, j} \Big (K _ {\omega} (Z _ {i} ^ {\mathrm {c a l}, r}, Y _ {j} ^ {\mathrm {e v}}) - K _ {\omega} (Z _ {i} ^ {\mathrm {c a l}, r}, X _ {j} ^ {\mathrm {e v}}) \Big);
$$
Phase 3: Inference with input  $Z$ .
$$
\widehat {T} \leftarrow \frac {1}{n _ {\mathrm {e v}} m} \sum_ {i, j} \left(K _ {\omega} \left(Z _ {i}, Y _ {j} ^ {\mathrm {e v}}\right) - K _ {\omega} \left(Z _ {i}, X _ {j} ^ {\mathrm {e v}}\right)\right);
$$
Output: Estimated  $p$ -value:  $\frac{1}{k}\sum_{i = 1}^{k}\mathbb{1}\{\widehat{T} < T_i\}$ .
Intuitively, the objective  $J$  aims to separate  $P_{X}$  from  $P_{Y}$  while keeping variance low. For a heuristic justification of its use for (mLFHT) see Appendix.
In Algorithm 1 we describe the training and testing procedure, which produces unbiased  $p$ -values for (mLFHT) when there is no misspecification ( $R = 0$  in Theorem 3.1). During training, we use the Adam optimizer [34] with stochastic batches.
Proposition 4.1. When there is no misspecification ( $R = 0$  in Theorem 3.1), Algorithm 1 outputs an unbiased estimate of the  $p$ -value that is consistent as  $\min \{n_{\mathrm{cal}}, k\} \to \infty$ .
Time complexity Algorithm 1 runs in three separate stages: training, calibration, and inference. The first two take  $O\left(\# \text{epochs} \cdot B^2 + k n_{\text{ev}} m\right)$  total time, where  $B$  is the batch size, whereas Phase 3 takes only  $O(n_{\text{ev}} m)$  time, which is generally much faster especially if  $n_{\text{ev}} \ll n_{\text{tr}}$ .
Sample usage Empirically, data splitting in Algorithm 1 can have non-trivial effects on performance. Instead of training the kernel on only a fraction of the data  $\{X^{\mathrm{tr}},Y^{\mathrm{tr}}\} \cap \{X^{\mathrm{ev}},Y^{\mathrm{ev}}\} = \emptyset$ , we discovered that taking  $\{X^{\mathrm{ev}},Y^{\mathrm{ev}}\} \subseteq \{X^{\mathrm{tr}},Y^{\mathrm{tr}}\}$  results in more efficient use of data. The stronger condition  $\{X^{\mathrm{ev}},Y^{\mathrm{ev}}\} = \{X^{\mathrm{tr}},Y^{\mathrm{tr}}\}$  can also be applied; we take  $\subseteq$  to reduce time complexity. We do, however, crucially require  $X^{\mathrm{cal}}$ ,  $Y^{\mathrm{cal}}$  in Phase 2 to be independently sampled ("held-out") for consistent  $p$ -value estimation. Finally, we remark also that splitting this way is only valid in the context of Algorithm 1. For the test (4) using the data-dependent threshold  $\gamma$ , one needs  $\{X^{\mathrm{tr}},Y^{\mathrm{tr}}\} \cap \{X^{\mathrm{ev}},Y^{\mathrm{ev}}\} = \emptyset$  to estimate  $\gamma$ .
4.2 Classifier-Based Tests and Other Benchmarks
Let  $\phi : \mathcal{X} \to [0,1]$  be a classifier, assigning small values to  $P_{X}$  and high values to  $P_{Y}$  by minimizing cross-entropy loss of a classifier net. There are two natural test statistics based on  $\phi$ :
Scheffé's Test. The first idea, attributed to Scheffé in folklore [15, Section 6], is to take the statistic  $T(Z) = \frac{1}{m}\sum_{i = 1}^{m}\mathbb{1}\{\phi (Z_i) > t\}$  where  $t$  is some (learn-able) threshold.
Approximate Neyman-Pearson / Logit Methods. If  $\phi$  is trained to perfection, then  $\phi(z) = \mathbb{P}(P_X|z)$  would be the likelihood and  $\phi(z)/(1 - \phi(z))$  would equal precisely the likelihood ratio between  $P_Y$  and  $P_X$  at  $z$ . This motivates the use of  $T(Z) = \frac{1}{m}\sum_{i=1}^{m}\log(\phi(Z_i)/(1 - \phi(Z_i)))$ . See also [10].
Let us list the testing procedures that we benchmark against each other in our experiments.
1. MMD-M: The MMD statistic (3) using  $K$  with the mixing architecture
$$
K (x, y) = \left[ (1 - \tau) G _ {\sigma} \left(\varphi_ {\omega} (x), \varphi_ {\omega} (y)\right) + \tau \right] \cdot G _ {\sigma_ {0}} \left(x + \varphi_ {\omega^ {\prime}} ^ {\prime} (x), y + \varphi_ {\omega^ {\prime}} ^ {\prime} (y)\right).
$$
Here  $G_{\sigma}$  is the Gaussian kernel with variance  $\sigma^2$ ;  $\varphi_{\omega}, \varphi_{\omega'}'$  are NN's (with parameters  $\omega, \omega'$ ), and  $\sigma, \sigma_0, \tau, \omega, \omega'$  are trained.
2. MMD-G: The MMD statistic (3) using the Gaussian kernel architecture  $K(x,y) = G_{\sigma}(\varphi_{\omega}(x),\varphi_{\omega}(y))$  where  $\varphi_{\omega}$  is the feature mapping parametrized by a trained network and  $\sigma$  is a trainable parameter.
3. MMD-O: The MMD statistic (3) using the Gaussian Kernel  $K(x,y) = G_{\sigma}(x,y)$  with optimized bandwidth  $\sigma$ . First proposed in [7; 39].
Figure 3: Empirical performance on (9) for the CIFAR detection problem when  $n_{\mathrm{tr}} = 1920$ . Plots from left to right are as follows. (a) rejection rate under the alternative if test rejects whenever the estimated  $p$ -value is smaller than  $5\%$ ; (b) expected  $p$ -value [45] under the alternative; (c) the average of type-I and II error probabilities when thresholded at 0 (different from (4), see Appendix); and (d) ROC curves for different  $m$  using MMD-M and Algorithm 1. Shaded area shows the standard deviation across 10 independent runs. Missing benchmarks (thresholded MMD, MMD-O, LBI, RFM) are weaker; see Appendix for full plot.
4. UME: An interpretable model comparison algorithm proposed by [31], which evaluates the kernel mean embedding on a chosen "witness set".
5. SCHE, LBI: Scheffé's test and Logit Based Inference methods [10], based on a binary classifier network  $\phi$  trained via cross-entropy, introduced above.
6. RFM: Recursive Feature Machines, a recently proposed kernel learning algorithm by [44].
4.3 Additive Statistics and the Thresholding Trick
Given a function  $f$  (usually obtained by training) and test data  $Z = (Z_{1},\dots,Z_{m})$ , we call a test additive if its output is obtained by thresholding  $T_{f}(Z)\triangleq \frac{1}{m}\sum_{i = 1}^{m}f(Z_{i})$ . We point out that all of MMD-M/G/O, SCHE, LBI, UME, RFM are of this form, see the Appendix for further details. Similarly to [39], we observe that any such statistic can be realized by our kernel-based approach.
Proposition 4.2. The kernel-based statistic defined in (3) with the kernel  $K(x,y) = f(x)f(y)$  is equal to  $T_{f}$  up to a multiplicative and additive constant independent of  $Z$ .
Motivated by the Scheffé's test, instead of directly thresholding the additive statistic  $T_{f}(Z)$ , we found empirically that replacing  $f$  by  $f_{t}(x) \triangleq \mathbb{1}\{f(x) > t\}$  can yield improved power. We set  $t$  by maximizing an estimate of the significance under the null using a normal approximation, i.e. by solving  $t_{\mathrm{opt}} \triangleq \arg \max_t \frac{T_{f_t}(Y^{\mathrm{opt}}) - T_{f_t}(X^{\mathrm{opt}})}{\sqrt{T_{f_t}(X^{\mathrm{opt}})(1 - T_{f_t}(X^{\mathrm{opt}}))}}$ , where  $X^{\mathrm{opt}}, Y^{\mathrm{opt}}$  satisfy  $\{X^{\mathrm{opt}}, Y^{\mathrm{opt}}\} \cap (\{X^{\mathrm{cal}}, Y^{\mathrm{cal}}\} \cup \{X^{\mathrm{ev}}, Y^{\mathrm{ev}}\}) = \emptyset$ . This trick improves the performance of our tester on the Higgs dataset in Section 5.2 but not for the image detection problem in Section 5.1.
Our code can be found at https://github.com/Sr-11/LFI.
5.1 Image Source Detection
Our first empirical study looks at the task of detecting whether images come from the CIFAR-10 [36] dataset or a SOTA generative model (DDPM) [25; 51]. While source detection is on its own interesting, it turns out that detecting whether a group of images comes from the generative model versus the real dataset can be too "easy" (see experiments in [31]). Therefore, we consider a mixed alternative, where the alternative hypothesis is not simply the generative model but CIFAR with planted DDPM images. Namely, our  $n$  labeled images come from the following distributions:
$$
P _ {X} = \text {C I F A R}, \quad \text {a n d} \quad P _ {Y} = \frac {1}{3} \cdot \text {D D P M} + \frac {2}{3} \cdot \text {C I F A R}. \tag {9}
$$
The goal is to test whether the  $m$  unlabeled observations  $Z$  have been corrupted with  $\rho$  or more fraction of DDPM images (versus uncorrupted CIFAR); this corresponds to (LFHT) (or equivalently (mLFHT) with  $\delta = 1$ ). Figure 3 shows the performance of our approach with this mixed alternative.
Network Architecture With a standard deep CNN, the difference is only at the final layer: for the kernel-based tests it is a feature output; for classifiers, we add an extra linear layer to logits.
We see from Figure 3 that our kernel-based test outperforms other benchmarks at a fixed training set size  $n_{tr}$ . One potential cause is that MMD has an "optimization" subroutine (which it solves in closed form) as it is an IPM. This additional layer of optimization may lead to better performance at small sample sizes. The thresholding trick does not seem to improve power empirically. We omit several benchmarks from this figure for graphic presentation and they do not exhibit good separating power; see the Appendix for the complete results. The bottom plot of Figure 1.2 shows  $m$  and  $n$  on log-scale against the total probability of error, exhibiting the simulation-experimentation trade-off.
5.2 Higgs-Boson Discovery
The 2012 announcement of the Higgs boson's discovery by the ATLAS and CMS experiments [1; 9] marked a significant milestone in physics. The statistical problem inherent in the experiment is well-modeled by (mLFHT), using a signal rate  $\delta$  predicted by theory and misspecification parameter  $R = 0$  (as was assumed in the original discovery). We consider our algorithm's power against past studies in the physics literature [5] as measured by the significance of discovery. We note an important distinction from Algorithm 1 in this application.
Estimating the Significance In physics, the threshold for claiming a "discovery" is usually at a significance of  $5\sigma$  , corresponding to a  $p$  value of  $2.87\times 10^{-7}$  .Approximately  $n_{\mathrm{cal}}\sim$ $(2.87)^{-1}\times 10^{7}$  samples would be necessary for Algorithm 1 to reach such a precision. Fortunately the distribution of the test statistic is approximated by a Gaussian customarily. We adopt this approach for our experiment hereby assuming that  $m$  is large enough for the CLT to
Figure 4: Expected significance of discovery on a mixture of 1000 backgrounds and 100 signals in the Higgs experiment. Shaded area shows the standard deviation over 10 independent runs. See Appendix for full plot including missing benchmarks.
apply. We use the "expected significance of discovery" as our metric [5] which, for the additive statistic  $T_{f} = \frac{1}{m}\sum_{i = 1}^{m}f(Z_{i})$ , is given by  $\frac{\delta(T_f(Y^{\mathrm{cal}}) - T_f(X^{\mathrm{cal}}))}{\sqrt{\widehat{\mathrm{var}}(f(X^{\mathrm{cal}})) / m}}$ . If the thresholding trick (Section 4.3) is applied we use the more precise Binomial tail, in which case the significance is estimated by  $-\Phi^{-1}(\mathbb{P}(\mathrm{Bin}(m,T_{f_{t_{\mathrm{opt}}}}(X^{\mathrm{cal}}))\geq T_{f_{t_{\mathrm{opt}}}}(Z))),$  where  $\Phi$  is the standard normal CDF.
Newtowk Architecture The architecture is a 6-layer feedforward net similar for all tests (kernel-based and classifiers) except for the last layer. We leave further details to the Appendix.
As can be seen in Figure 4, Scheffé's test and MMD-M with threshold  $t_{\mathrm{opt}}$  are the best methods, achieving similar performance as the algorithm of [5]; reaching the significance level of  $5\sigma$  on 2.6 million simulated datapoints and a test sample made up of a mixture of 1000 backgrounds and 100 signals. The top plot of Figure 1.2 shows  $m$  and  $n$  on log-scale against the total probability of error through performing the test (4), exhibiting the asymmetric simulation-experimentation trade-off.
In this paper, we introduced (mLFHT) as a theoretical model of real-world likelihood-free signal detection problems arising in science. We proposed a kernel-based test statistic and analyzed its minimax sample complexity, obtaining both upper (Theorem 3.1) and lower bounds (Theorem 3.2) in terms of multiple problem parameters, and discussed their tightness (Section 3.3) and connections to prior work (Section 3.4). On the empirical side, we described a method for training a parametrized kernel and proposed a consistent  $p$ -value estimate (Algorithm 1 and Proposition 4.1). We examined the performance of our method in two experiments and found that parametrized kernels achieve state-of-the-art performance compared to relevant benchmarks from the literature. Moreover, we confirmed experimentally the existence of the asymmetric simulation-experimentation trade-off (Figure 1.2) which is suggested by minimax analysis. We defer further special cases of Theorem 3.1, all relevant proofs and experimental details to the Appendix.
YP was supported in part by the National Science Foundation under Grant No CCF-2131115. Research was sponsored by the United States Air Force Research Laboratory and the Department of the Air Force Artificial Intelligence Accelerator and was accomplished under Cooperative Agreement Number FA8750-19-2-1000. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Department of the Air Force or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.