Now we start by proving that $\mathbb{E}[r(X, \hat{\pi}_G(X); \hat{\mu}) - r(X, \hat{\pi}_G(X); \theta_*)] = 0$. This is achieved as follows

$$
\begin{align*}
\mathbb{E}[r(X, \hat{\pi}_G(X); \hat{\mu}) - r(X, \hat{\pi}_G(X); \theta_*)] &= \mathbb{E}[\mathbb{E}[r(X, \hat{\pi}_G(X); \hat{\mu}) - r(X, \hat{\pi}_G(X); \theta_*) | X, S]], \\
&= \mathbb{E}[\mathbb{E}[\phi(X)^\top \hat{\mu}_{\hat{\pi}_G(X)} - \phi(X)^\top \theta_{*, \hat{\pi}_G(X)} | X, S]], \\
&\stackrel{(i)}{=} \mathbb{E}[\mathbb{E}[\Phi(X, \hat{\pi}_G(X))^\top \hat{\mu} - \Phi(X, \hat{\pi}_G(X))^\top \theta_* | X, S]], \\
&\stackrel{(ii)}{=} \mathbb{E}[\Phi(X, \hat{\pi}_G(X))^\top \mathbb{E}[\hat{\mu} - \theta_* | X, S]], \\
&\stackrel{(iii)}{=} \mathbb{E}[\Phi(X, \hat{\pi}_G(X))^\top (\hat{\mu} - \mathbb{E}[\theta_* | X, S])], \\
&\stackrel{(iv)}{=} 0.
\end{align*}
$$

In (i), we used that by definition of $\Phi(x, a) = e_a \otimes \phi(X) \in \mathbb{R}^{dK}$, we have $\phi(x)^\top \hat{\mu}_a = \Phi(x, a)^\top \hat{\mu}$ for any $(x, a)$, and the same holds for $\theta_*$. In (ii), we used that $\Phi(X, \hat{\pi}_G(X))$ is deterministic given $X$ and $S$. In (iii), we used that $\hat{\mu}$ is deterministic given $X$ and $S$. Finally, in (iv), we used that $\mathbb{E}[\theta_* | X, S] = \mathbb{E}[\theta_* | S] = \hat{\mu}$, which follows from the assumption that $\theta_*$ does not depend on $X$ and the assumption that $\theta_*$ is drawn from the prior, and hence when conditioned on $S$, it is drawn from the posterior whose mean is $\hat{\mu}$. Therefore, $\mathbb{E}[r(X, \hat{\pi}_G(X); \hat{\mu}) - r(X, \hat{\pi}_G(X); \theta_*)] = 0$ which leads to

$$
\mathrm{Bso}(\hat{\pi}_G) \leq \mathbb{E}[r(X, \pi_*(X); \theta_*) - r(X, \pi_*(X); \hat{\mu})].
$$

Now let $\delta \in (0, 1)$, we define the following high-probability events

$$
E_a = \left\{ \forall x \in \mathcal{X} : |r(x, a; \theta_*) - r(x, a; \hat{\mu})| \le \alpha(d, \delta) \|\phi(x)\|_{\hat{\Sigma}_a} \right\}, \quad \forall a \in \mathcal{A},
$$

where $\alpha(d, \delta) = \sqrt{d + 2\sqrt{d \log \frac{1}{\delta}} + 2 \log \frac{1}{\delta}}$. Then we decompose $\mathrm{Bso}(\hat{\pi}_G)$ as

$$
\begin{align*}
\mathrm{Bso}(\hat{\pi}_G) &\le \mathbb{E}[r(X, \pi_*(X); \theta_*) - r(X, \pi_*(X); \hat{\mu})], \\
&\le \mathbb{E}[|r(X, \pi_*(X); \theta_*) - r(X, \pi_*(X); \hat{\mu})|], \\
&\le \mathbb{E}[|r(X, \pi_*(X); \theta_*) - r(X, \pi_*(X); \hat{\mu})|1\{E_{\pi_*(X)}\}] + \mathbb{E}[|r(X, \pi_*(X); \theta_*) - r(X, \pi_*(X); \hat{\mu})|1\{\bar{E}_{\pi_*(X)}\}], \\
&\le \alpha(d, \delta)\mathbb{E}[\|\phi(X)\|_{\hat{\Sigma}_{\pi_*(X)}}] + \mathbb{E}[|r(X, \pi_*(X); \theta_*) - r(X, \pi_*(X); \hat{\mu})|1\{\bar{E}_{\pi_*(X)}\}].
\end{align*}
$$

Now we deal with the term $\mathbb{E}[|r(X, \pi_*(X); \theta_*) - r(X, \pi_*(X); \hat{\mu})|1\{\bar{E}_{\pi_*(X)}\}]$. Let $Z_a = r(X, a; \theta_*) - r(X, a; \hat{\mu})$, so that $Z_{\pi_*(X)} = r(X, \pi_*(X); \theta_*) - r(X, \pi_*(X); \hat{\mu})$. Then we have that

$$
\begin{align*}
\mathbb{E}[|r(X, \pi_*(X); \theta_*) - r(X, \pi_*(X); \hat{\mu})|1\{\bar{E}_{\pi_*(X)}\}] &= \mathbb{E}[|Z_{\pi_*(X)}|1\{|Z_{\pi_*(X)}| > \alpha(d, \delta)\|\phi(X)\|_{\hat{\Sigma}_{\pi_*(X)}}\}], \\
&= \mathbb{E}\left[\mathbb{E}\left[|Z_{\pi_*(X)}|1\{|Z_{\pi_*(X)}| > \alpha(d, \delta)\|\phi(X)\|_{\hat{\Sigma}_{\pi_*(X)}}\} \mid X, S\right]\right], \\
&\stackrel{(i)}{\le} \mathbb{E}\left[\frac{2}{\|\phi(X)\|_{\hat{\Sigma}_{\pi_*(X)}}\sqrt{2\pi}} \int_{u=\alpha(d, \delta)}^{\infty} u \exp\left[-\frac{u^2}{2\|\phi(X)\|_{\hat{\Sigma}_{\pi_*(X)}}^2}\right] du\right], \\
&\stackrel{(ii)}{\le} \mathbb{E}\left[\|\phi(X)\|_{\hat{\Sigma}_{\pi_*(X)}} \frac{2}{\sqrt{2\pi}} \int_{u=\alpha(d, \delta)}^{\infty} u \exp\left[-\frac{u^2}{2}\right] du\right], \\
&\stackrel{(iii)}{\le} \sqrt{\frac{2}{\pi}} \exp\left(-\frac{\alpha(d, \delta)^2}{2}\right) \mathbb{E}\left[\|\phi(X)\|_{\hat{\Sigma}_{\pi_*(X)}}\right],
\end{align*}
$$

where (i) follows from the facts that $Z_{\pi_*(X)} | X, S \sim \mathcal{N}(0, \|\phi(X)\|_{\hat{\Sigma}_{\pi_*(X)}})$. In (ii), we use the change of variables $u \leftarrow u/\|\phi(X)\|_{\hat{\Sigma}_{\pi_*(X)}}$. Finally, in (iii), we compute the integral. This leads to the following result

$$
\begin{align}
\mathrm{Bso}(\hat{\pi}_G) &\le \left( \sqrt{\frac{2}{\pi}} \exp\left(-\frac{\alpha(d, \delta)^2}{2}\right) + \alpha(d, \delta) \right) \mathbb{E} \left[ \|\phi(X)\|_{\hat{\Sigma}_{\pi_*(X)}} \right]. \tag{41} \\
&\le \sqrt{\frac{2}{\pi}} \exp\left(-\frac{\alpha(d, \delta)^2}{2}\right) \tag{42}
\end{align}
$$

This concludes the proof.
## E.4 Proof of Theorem 5.1

*Proof.* The proof is similar to that of Theorem 5.2. Precisely, we have that

$$
\begin{align*}
\mathrm{BMSE}(\hat{r}(x, a)) &= \mathbb{E}[(\hat{r}(x, a) - r(x, a; \theta_*))^2], \\
&= \mathbb{E}[\mathbb{E}[(\hat{r}(x, a) - r(x, a; \theta_*))^2 | S]], \\
&= \mathbb{E}[\mathbb{E}[(\phi(x)^\top \hat{\mu}_a - \phi(x)^\top \theta_{*,a})^2 | S]], \\
&= \mathbb{E}[\mathbb{E}[(\phi(x)^\top \theta_{*,a} - \phi(x)^\top \hat{\mu}_a)^2 | S]].
\end{align*}
$$

But we assumed that the true action parameters $\theta_{*,a}$ are random and their prior distribution matches our model in (10). Therefore, $\theta_{*,a} | S$ have the same density as our posterior $\theta_a | S$, and hence $\theta_{*,a} | S \sim \mathcal{N}(\hat{\mu}_a, \hat{\Sigma}_a)$. This means that $\phi(x)^\top \theta_{*,a} | S \sim \mathcal{N}(\phi(x)^\top \hat{\mu}_a, \|\phi(x)\|_{\hat{\Sigma}_a}^2)$. But $\mathbb{E}[(\phi(x)^\top \theta_{*,a} - \phi(x)^\top \hat{\mu}_a)^2 | S]$ is exactly the variance of $\phi(x)^\top \theta_{*,a} | S$ and thus it is equal to $\|\phi(x)\|_{\hat{\Sigma}_a}^2$. Taking the expectation concludes the proof. $\square$

## E.5 Optimality of greedy policies

Here, we show that Greedy policy $\hat{\pi}_G$ should be preferred to any other choice of policies when considering the BSO as our performance metric. This is because $\hat{\pi}_G$ minimizes the BSO. To see this, note that by definition the Greedy policy $\hat{\pi}_G$ is deterministic, that is for any context $x \in \mathcal{X}$, there exists $\hat{a}_G$, such that $\hat{\pi}_G(\hat{a}_G | x) = 1$. Thus, for any context $x \in \mathcal{X}$, we simplify the notation by letting $\hat{\pi}_G(x)$ denote the action that has a mass equal to 1. Then, we have that

$$ \mathbb{E}_{A \sim \hat{\pi}_G(\cdot | x)} [\mathbb{E}_{\theta_*} [r(x, A; \theta_*) | S]] = \mathbb{E}_{\theta_*} [r(x, \hat{\pi}_G(x); \theta_*) | S] \geq \mathbb{E} [r(x, a; \theta_*) | S] \quad \forall x, a \in \mathcal{X} \times \mathcal{A}. \quad (43) $$

where this follows from the definition of $\hat{r}(x, a) = \mathbb{E}[r(x, a; \theta) | S]$, the definition of $\hat{\pi}_G$ and the fact that $\theta_*$ is sampled from the prior, which leads to $\mathbb{E}[r(x, a; \theta) | S] = \mathbb{E}[r(x, a; \theta_*) | S]$. Now (43) holds for any $x \in \mathcal{X}$ and $a \in \mathcal{A}$, and hence it holds in expectation under $X \sim \nu$ and $A \sim \pi(\cdot | X)$ for any policy $\pi$. That is,

$$ \mathbb{E}_{X \sim \nu, A \sim \hat{\pi}_G(\cdot | X)} [\mathbb{E}_{\theta_*} [r(x, A; \theta_*) | S]] \geq \mathbb{E}_{X \sim \nu, A \sim \pi(\cdot | X)} [\mathbb{E}_{\theta_*} [r(x, A; \theta_*) | S]]. \quad (44) $$

Taking another expectation w.r.t. the sample set $S$ and using Fubini's theorem and the tower rule leads to $\mathbb{E}[V(\hat{\pi}_G; \theta_*)] \geq \mathbb{E}[V(\pi; \theta_*)]$ for any stationary policy $\pi$. Then, subtracting $\mathbb{E}[V(\pi_*; \theta_*)]$ from both sides of the previous inequality yields that the BSO is minimized by $\hat{\pi}_G$ compared to any stationary policy $\pi$, in particular, compared to the policy $\pi_P$ induced by pessimism.

## E.6 Discussing the main assumption

In this section, we discuss our main assumption of the existence of the latent parameter $\psi$. Let us take the Gaussian case, where we were assume that

$$
\begin{align}
\psi &\sim \mathcal{N}(\mu, \Sigma), \tag{45} \\
\theta_a | \psi &\sim \mathcal{N}(W_a\psi, \Sigma_a), \nonumber
\end{align}
$$

Now we discuss if (45) is as mild as assuming that the action parameters are jointly sampled as $\theta \sim \mathcal{N}(\mu_{\text{ACT}}, \Sigma_{\text{ACT}})$, where $\mu_{\text{ACT}} \in \mathbb{R}^{dK}$ and $\Sigma_{\text{ACT}} \in \mathbb{R}^{dK \times dK}$. To do so, assume that $\theta \sim \mathcal{N}(0, \Sigma_{\text{ACT}})$, where we set $\mu_{\text{ACT}} = 0$ to simplify notation. Then we want to prove that there exists $\psi \sim \mathcal{N}(0, I_{d'})$, with $\mu \in \mathbb{R}^{d'}$ for some $d' \ge 1$ and $W \in \mathbb{R}^{dK \times d'}$ such that $\theta | \psi \sim \mathcal{N}(W\psi, \tilde{\Sigma})$, where $\tilde{\Sigma}$ is diagonal by $d \times d$ blocks. First, note that $(\theta, \psi)$ is also Gaussian with a mean equal to 0 and covariance $\Omega = (\Sigma_{\text{ACT}} \quad \Omega_2)$, $\Omega_2 \in \mathbb{R}^{(dK+d') \times (dK+d')}$. Then it suffices to find $W$ such that $\theta - W\psi \perp \psi$. That is, $W$ such that $\text{cov}(\theta - W\psi, \psi) = \text{cov}(\theta, \psi) - W\text{cov}(\psi, \psi) = \Omega_2 - W = 0$, which leads to $W = \Omega_2$. Using this conditional independence and that $W = \Omega_2$, we get that the distribution of $\theta - W\psi | \psi$ is the same as the distribution of $\theta - W\psi$ and it writes $\mathcal{N}(0, \Sigma_{\text{ACT}} - WW^\top)$. Thus,
## E.4 Proof of Theorem 5.1

*Proof.* The proof is similar to that of Theorem 5.2. Precisely, we have that

$$
\begin{align*}
\mathrm{BMSE}(\hat{r}(x, a)) &= \mathbb{E}[(\hat{r}(x, a) - r(x, a; \theta_{*}))^2], \\
&= \mathbb{E}[\mathbb{E}[(\hat{r}(x, a) - r(x, a; \theta_{*}))^2 | S]], \\
&= \mathbb{E}[\mathbb{E}[(\phi(x)^{\top}\hat{\mu}_a - \phi(x)^{\top}\theta_{*,a})^2 | S]], \\
&= \mathbb{E}[\mathbb{E}[(\phi(x)^{\top}\theta_{*,a} - \phi(x)^{\top}\hat{\mu}_a)^2 | S]].
\end{align*}
$$

But we assumed that the true action parameters $\theta_{*,a}$ are random and their prior distribution matches our model in (10). Therefore, $\theta_{*,a} | S$ have the same density as our posterior $\theta_a | S$, and hence $\theta_{*,a} | S \sim \mathcal{N}(\hat{\mu}_a, \hat{\Sigma}_a)$. This means that $\phi(x)^{\top}\theta_{*,a} | S \sim \mathcal{N}(\phi(x)^{\top}\hat{\mu}_a, \|\phi(x)\|_{\hat{\Sigma}_a}^2)$. But $\mathbb{E}[(\phi(x)^{\top}\theta_{*,a} - \phi(x)^{\top}\hat{\mu}_a)^2 | S]$ is exactly the variance of $\phi(x)^{\top}\theta_{*,a} | S$ and thus it is equal to $\|\phi(x)\|_{\hat{\Sigma}_a}^2$. Taking the expectation concludes the proof. $\square$

## E.5 Optimality of greedy policies

Here, we show that Greedy policy $\hat{\pi}_G$ should be preferred to any other choice of policies when considering the BSO as our performance metric. This is because $\hat{\pi}_G$ minimizes the BSO. To see this, note that by definition the Greedy policy $\hat{\pi}_G$ is deterministic, that is for any context $x \in \mathcal{X}$, there exists $\hat{a}_G$, such that $\hat{\pi}_G(\hat{a}_G | x) = 1$. Thus, for any context $x \in \mathcal{X}$, we simplify the notation by letting $\hat{\pi}_G(x)$ denote the action that has a mass equal to 1. Then, we have that

$$ \mathbb{E}_{A \sim \hat{\pi}_G(\cdot | x)} [\mathbb{E}_{\theta_*} [r(x, A; \theta_*) | S]] = \mathbb{E}_{\theta_*} [r(x, \hat{\pi}_G(x); \theta_*) | S] \geq \mathbb{E} [r(x, a; \theta_*) | S] \quad \forall x, a \in \mathcal{X} \times \mathcal{A}. \quad (43) $$

where this follows from the definition of $\hat{r}(x, a) = \mathbb{E}[r(x, a; \theta) | S]$, the definition of $\hat{\pi}_G$ and the fact that $\theta_*$ is sampled from the prior, which leads to $\mathbb{E}[r(x, a; \theta) | S] = \mathbb{E}[r(x, a; \theta_*) | S]$. Now (43) holds for any $x \in \mathcal{X}$ and $a \in \mathcal{A}$, and hence it holds in expectation under $X \sim \nu$ and $A \sim \pi(\cdot | X)$ for any policy $\pi$. That is,

$$ \mathbb{E}_{X \sim \nu, A \sim \hat{\pi}_G(\cdot | X)} [\mathbb{E}_{\theta_*} [r(x, A; \theta_*) | S]] \geq \mathbb{E}_{X \sim \nu, A \sim \pi(\cdot | X)} [\mathbb{E}_{\theta_*} [r(x, A; \theta_*) | S]]. \quad (44) $$

Taking another expectation w.r.t. the sample set $S$ and using Fubini's theorem and the tower rule leads to $\mathbb{E}[V(\hat{\pi}_G; \theta_*)] \geq \mathbb{E}[V(\pi; \theta_*)]$ for any stationary policy $\pi$. Then, subtracting $\mathbb{E}[V(\pi_*; \theta_*)]$ from both sides of the previous inequality yields that the BSO is minimized by $\hat{\pi}_G$ compared to any stationary policy $\pi$, in particular, compared to the policy $\pi_P$ induced by pessimism.

## E.6 Discussing the main assumption

In this section, we discuss our main assumption of the existence of the latent parameter $\psi$. Let us take the Gaussian case, where we were assume that

$$
\begin{align}
\psi &\sim \mathcal{N}(\mu, \Sigma), \tag{45} \\
\theta_a | \psi &\sim \mathcal{N}(W_a\psi, \Sigma_a), \nonumber
\end{align}
$$

Now we discuss if (45) is as mild as assuming that the action parameters are jointly sampled as $\theta \sim \mathcal{N}(\mu_{\text{ACT}}, \Sigma_{\text{ACT}})$, where $\mu_{\text{ACT}} \in \mathbb{R}^{dK}$ and $\Sigma_{\text{ACT}} \in \mathbb{R}^{dK \times dK}$. To do so, assume that $\theta \sim \mathcal{N}(0, \Sigma_{\text{ACT}})$, where we set $\mu_{\text{ACT}} = 0$ to simplify notation. Then we want to prove that there exists $\psi \sim \mathcal{N}(0, I_{d'})$, with $\mu \in \mathbb{R}^{d'}$ for some $d' \geq 1$ and $W \in \mathbb{R}^{dK \times d'}$ such that $\theta | \psi \sim \mathcal{N}(W\psi, \tilde{\Sigma})$, where $\tilde{\Sigma}$ is diagonal by $d \times d$ blocks. First, note that $(\theta, \psi)$ is also Gaussian with a mean equal to 0 and covariance $\Omega = (\frac{\Sigma_{\text{ACT}}}{\Omega_2} \quad I_{d'}) \in \mathbb{R}^{(dK+d') \times (dK+d')}$. Then it suffices to find $W$ such that $\theta - W\psi \perp \psi$. That is, $W$ such that $\text{cov}(\theta - W\psi, \psi) = \text{cov}(\theta, \psi) - W\text{cov}(\psi, \psi) = \Omega_2 - W = 0$, which leads to $W = \Omega_2$. Using this conditional independence and that $W = \Omega_2$, we get that the distribution of $\theta - W\psi | \psi$ is the same as the distribution of $\theta - W\psi$ and it writes $\mathcal{N}(0, \Sigma_{\text{ACT}} - WW^\top)$. Thus,
$\theta | \psi \sim \mathcal{N}(W\psi, \Sigma_{\text{ACT}} - WW^{\top})$. But $\Sigma_{\text{ACT}}$ is positive-definite and hence there exists $\lambda > 0$ such that $\Sigma_{\text{ACT}} - \lambda I_{dK}$ is symmetric and positive semi-definite. Therefore, $\Sigma_{\text{ACT}} - \lambda I_{dK}$ can be written as $\Sigma_{\text{ACT}} - \lambda I_{dK} = BB^{\top}$ where $B \in \mathbb{R}^{dK \times d'}$ is such that $d' \ge \text{rank}(\Sigma_{\text{ACT}} - \lambda I_{dK})$ and setting $W = B$ concludes the proof.

# F Additional experiments

As mentioned in Appendix F, our experiments were conducted on internal machines with 30 CPUs and thus they required a moderate amount of computation. These experiments are also reproducible with minimal computational resources.

## F.1 OPE experiments

We consider the same synthetic setting as in Section 6, but now we fix a target policy $\pi$ and assess algorithm performance in evaluating the value function of that target policy using the mean squared error (MSE) averaged over 50 problem instances sampled from the prior. This can be seen as a proxy for our theoretical metric, Bayesian MSE (BMSE). The target policy is defined as $\epsilon$-greedy with $\epsilon = 0.5$; it chooses the best action with probability 0.5 and a random action with probability 0.5.

Results are shown in Fig. 6. While sDM maintains its advantage over baselines, particularly in low-data settings, an interesting observation emerged. IPS-based methods (snIPS and DR) outperformed standard direct methods (DM (Bayes) and DM (Freq)) in the OPE experiments, which was not the case in OPL experiments. The only direct method that outperformed them in OPE is sDM.

<figure><img src="image_9.png" /><figcaption>Figure 6: The average MSE of an $\epsilon$-greedy target policy on synthetic problems with varying $n$, $K$ and $d'$.</figcaption></figure>

## F.2 Different logging policy: $\epsilon$-greedy

Here we consider the setting in Section 6 and provide additional results with the $\epsilon$-greedy logging policy, where $\epsilon = 0.5$. We consider both synthetic and MovieLens datasets. The results are shown in Fig. 7. The conclusions are similar to those in Section 6, except that IPS outperforms the other variants DR and snIPS when the logging policy is performing well. Also, the results for the MovieLens problems are given in Fig. 8 and the conclusions are similar to those made in Section 6.
$\theta | \psi \sim \mathcal{N}(W\psi, \Sigma_{\text{ACT}} - WW^{\top})$. But $\Sigma_{\text{ACT}}$ is positive-definite and hence there exists $\lambda > 0$ such that $\Sigma_{\text{ACT}} - \lambda I_{dK}$ is symmetric and positive semi-definite. Therefore, $\Sigma_{\text{ACT}} - \lambda I_{dK}$ can be written as $\Sigma_{\text{ACT}} - \lambda I_{dK} = BB^{\top}$ where $B \in \mathbb{R}^{dK \times d'}$ is such that $d' \ge \text{rank}(\Sigma_{\text{ACT}} - \lambda I_{dK})$ and setting $W = B$ concludes the proof.

# F Additional experiments

As mentioned in Appendix F, our experiments were conducted on internal machines with 30 CPUs and thus they required a moderate amount of computation. These experiments are also reproducible with minimal computational resources.

## F.1 OPE experiments

We consider the same synthetic setting as in Section 6, but now we fix a target policy $\pi$ and assess algorithm performance in evaluating the value function of that target policy using the mean squared error (MSE) averaged over 50 problem instances sampled from the prior. This can be seen as a proxy for our theoretical metric, Bayesian MSE (BMSE). The target policy is defined as $\epsilon$-greedy with $\epsilon = 0.5$; it chooses the best action with probability 0.5 and a random action with probability 0.5.

Results are shown in Fig. 6. While sDM maintains its advantage over baselines, particularly in low-data settings, an interesting observation emerged. IPS-based methods (snIPS and DR) outperformed standard direct methods (DM (Bayes) and DM (Freq)) in the OPE experiments, which was not the case in OPL experiments. The only direct method that outperformed them in OPE is sDM.

<figure><img src="image_10.png" /><figcaption>Figure 6: The average MSE of an $\epsilon$-greedy target policy on synthetic problems with varying $n$, $K$ and $d'$.</figcaption></figure>

## F.2 Different logging policy: $\epsilon$-greedy

Here we consider the setting in Section 6 and provide additional results with the $\epsilon$-greedy logging policy, where $\epsilon = 0.5$. We consider both synthetic and MovieLens datasets. The results are shown in Fig. 7. The conclusions are similar to those in Section 6, except that IPS outperforms the other variants DR and snIPS when the logging policy is performing well. Also, the results for the MovieLens problems are given in Fig. 8 and the conclusions are similar to those made in Section 6.
<figure><img src="image_10.png" /><figcaption>Figure 7: The relative reward of the learned policy on synthetic problems with varying *n*, *K* and *d*′ and varying logging policies.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 8: The relative reward of the learned policy on MovieLens problems with varying *n* and *K* and varying logging policies.</figcaption></figure>

### F.3 Robustness to misspecification

We strengthened our evaluation by assessing sDM's robustness to prior/model misspecification. In these experiments, the true data-generating process (same as the synthetic experiments in Section 6) differed from sDM's assumptions in four different ways:

**Misspecified prior means (Fig. 9).** This was achieved by adding uniformly sampled noise from $[v, v+0.5]$ to the true prior mean parameters $\mu$ and $W_a$, with $v$ controlling the level of misspecification. We varied $v \in \{0.5, 1, 1.5\}$ and analyzed its impact on sDM's performance.

**Misspecified prior covariances (Fig. 10).** This is achieved by adding uniformly sampled noise from $[v, v+0.5]$ to the true prior covariance parameters $\Sigma$ and $\Sigma_a$, with $v$ controlling the level of misspecification. We varied $v \in \{0.5, 1, 1.5\}$ and analyzed its impact on sDM's performance.

**Misspecified prior means and covariances(Fig. 11).** This is achieved by adding uniformly sampled noise from $[v, v+0.5]$ to both the true prior mean and covariance parameters $\mu, \Sigma, W_a, \Sigma_a$, with $v$ controlling the level of misspecification. We varied $v \in \{0.5, 1, 1.5\}$ and analyzed its impact on sDM's performance. For comparison, we included the well-specified sDM and the most competitive baseline, DM (Bayes), while omitting other baselines to reduce clutter. In all figures (Figs. 9 to 11) sDM's performance decreases with increasing misspecification, yet sDM with misspecification still outperforms the most competitive baseline, especially when *K* is large. We also observe that the impact of prior covariance misspecification is less significant compared to prior mean misspecification.

**Misspecified likelihood (Fig. 12).** We also simulate the case when the true reward distribution differed from the likelihood assumed by sDM. For example, we simulated binary rewards using a
<figure><img src="image_11.png" /><figcaption>Figure 7: The relative reward of the learned policy on synthetic problems with varying *n*, *K* and *d*′ and varying logging policies.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 8: The relative reward of the learned policy on MovieLens problems with varying *n* and *K* and varying logging policies.</figcaption></figure>

### F.3 Robustness to misspecification

We strengthened our evaluation by assessing sDM's robustness to prior/model misspecification. In these experiments, the true data-generating process (same as the synthetic experiments in Section 6) differed from sDM's assumptions in four different ways:

**Misspecified prior means (Fig. 9).** This was achieved by adding uniformly sampled noise from $[v, v+0.5]$ to the true prior mean parameters $\mu$ and $W_a$, with $v$ controlling the level of misspecification. We varied $v \in \{0.5, 1, 1.5\}$ and analyzed its impact on sDM's performance.

**Misspecified prior covariances (Fig. 10).** This is achieved by adding uniformly sampled noise from $[v, v+0.5]$ to the true prior covariance parameters $\Sigma$ and $\Sigma_a$, with $v$ controlling the level of misspecification. We varied $v \in \{0.5, 1, 1.5\}$ and analyzed its impact on sDM's performance.

**Misspecified prior means and covariances(Fig. 11).** This is achieved by adding uniformly sampled noise from $[v, v+0.5]$ to both the true prior mean and covariance parameters $\mu, \Sigma, W_a, \Sigma_a$, with $v$ controlling the level of misspecification. We varied $v \in \{0.5, 1, 1.5\}$ and analyzed its impact on sDM's performance. For comparison, we included the well-specified sDM and the most competitive baseline, DM (Bayes), while omitting other baselines to reduce clutter. In all figures (Figs. 9 to 11) sDM's performance decreases with increasing misspecification, yet sDM with misspecification still outperforms the most competitive baseline, especially when *K* is large. We also observe that the impact of prior covariance misspecification is less significant compared to prior mean misspecification.

**Misspecified likelihood (Fig. 12).** We also simulate the case when the true reward distribution differed from the likelihood assumed by sDM. For example, we simulated binary rewards using a
Bernoulli-logistic model while sDM used a linear-Gaussian likelihood. Other DMs: DM (Bayes) and DM (Freq) also use a misspecified likelihood model and to emphasize this we add the suffix Lin to all DMs names. Overall, as known in the literature, IPS variants performed better than standard DMs when the likelihood was misspecified. However, this is not the case for sDM. While misspecification narrowed the performance gap between sDM and baselines, sDM still consistently outperformed them, particularly in low-data regimes or large action spaces. In a few cases, some baselines, e.g., snIPS, showed competitive performance with enough data.

<figure><img src="image_12.png" /><figcaption>Figure 9: Effect of prior mean misspecification: The relative reward of the learned policy on synthetic problems using both misspecified prior means and covariances with varying *n* and *K*.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Figure 10: Effect of prior covariance misspecification: The relative reward of the learned policy on synthetic problems using both misspecified prior means and covariances with varying *n* and *K*.</figcaption></figure>
Bernoulli-logistic model while sDM used a linear-Gaussian likelihood. Other DMs: DM (Bayes) and DM (Freq) also use a misspecified likelihood model and to emphasize this we add the suffix Lin to all DMs names. Overall, as known in the literature, IPS variants performed better than standard DMs when the likelihood was misspecified. However, this is not the case for sDM. While misspecification narrowed the performance gap between sDM and baselines, sDM still consistently outperformed them, particularly in low-data regimes or large action spaces. In a few cases, some baselines, e.g., snIPS, showed competitive performance with enough data.

<figure><img src="image_13.png" /><figcaption>Figure 9: Effect of prior mean misspecification: The relative reward of the learned policy on synthetic problems using both misspecified prior means and covariances with varying *n* and *K*.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 10: Effect of prior covariance misspecification: The relative reward of the learned policy on synthetic problems using both misspecified prior means and covariances with varying *n* and *K*.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Figure 11: Effect of prior mean and covariance misspecification: The relative reward of the learned policy on synthetic problems using both misspecified prior means and covariances with varying n and K.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Figure 12: Effect of likelihood misspecification: The relative reward of the learned policy on synthetic problems using misspecified likelihood with varying n and K.</figcaption></figure>

## F.4 Comparison to MIPS

MIPS was designed for OPE only and thus we consider the OPE setting in Appendix F.1. MIPS assumes access to embeddings $E_i$ in the sample set S and defines its estimator as follows

$$ \hat{V}^{\text{MIPS}}(\pi, S) = \frac{1}{n} \sum_{i=1}^{n} \frac{\pi(E_i | X_i)}{\pi_0(E_i | X_i)} R_i = \frac{1}{n} \sum_{i=1}^{n} w(X_i, E_i) R_i, $$

where the sample set $S = \{(X_i, A_i, E_i, R_i)\}_{i=1}^n$ now includes action embeddings $E_i$ for each data point $i \in [n]$, and the marginal importance weight is given by $w(x, e) = \frac{\pi(e|x)}{\pi_0(e|x)} = \frac{\sum_a p(e|x,a)\pi(a|x)}{\sum_a p(e|x,a)\pi_0(a|x)}$, where $p(\cdot | x, a)$ is the embedding distribution such that $E_i \sim p(\cdot | X_i, A_i)$. They assume that R is independent of A given X and E and hence we should define embeddings that have a direct effect on the reward. From our prior (10), these correspond to the true action parameters $\theta_*$. In fact, creating MIPS's structure from our prior in (10) is not straightforward, but we approach it as follows. We define the embedding distribution as $p(\cdot | x, a) = N(\cdot; \mu_a, \Sigma_a)$, where $\mu_a$ and $\Sigma_a$ are obtained by marginalizing out the latent parameters $\psi$ in (10). Additionally, we set $A_i = \theta_{*,A_i}$, having access to $\theta_*$ since these are synthetic problems. This is the best version of MIPS that we could create from our prior (10). However, we stress again that this comparison is only provided for completeness as the setting may be favorable to sDM. In this setting, sDM outperforms MIPS.
<figure><img src="image_15.png" /><figcaption>Figure 11: Effect of prior mean and covariance misspecification: The relative reward of the learned policy on synthetic problems using both misspecified prior means and covariances with varying n and K.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 12: Effect of likelihood misspecification: The relative reward of the learned policy on synthetic problems using misspecified likelihood with varying n and K.</figcaption></figure>

## F.4 Comparison to MIPS

MIPS was designed for OPE only and thus we consider the OPE setting in Appendix F.1. MIPS assumes access to embeddings $E_i$ in the sample set S and defines its estimator as follows

$$ \hat{V}^{\text{MIPS}}(\pi, S) = \frac{1}{n} \sum_{i=1}^{n} \frac{\pi(E_i | X_i)}{\pi_0(E_i | X_i)} R_i = \frac{1}{n} \sum_{i=1}^{n} w(X_i, E_i) R_i, $$

where the sample set $S = \{(X_i, A_i, E_i, R_i)\}_{i=1}^n$ now includes action embeddings $E_i$ for each data point $i \in [n]$, and the marginal importance weight is given by $w(x, e) = \frac{\pi(e|x)}{\pi_0(e|x)} = \frac{\sum_a p(e|x,a)\pi(a|x)}{\sum_a p(e|x,a)\pi_0(a|x)}$, where $p(\cdot | x, a)$ is the embedding distribution such that $E_i \sim p(\cdot | X_i, A_i)$. They assume that R is independent of A given X and E and hence we should define embeddings that have a direct effect on the reward. From our prior (10), these correspond to the true action parameters $\theta_*$. In fact, creating MIPS's structure from our prior in (10) is not straightforward, but we approach it as follows. We define the embedding distribution as $p(\cdot | x, a) = N(\cdot; \mu_a, \Sigma_a)$, where $\mu_a$ and $\Sigma_a$ are obtained by marginalizing out the latent parameters $\psi$ in (10). Additionally, we set $A_i = \theta_{*,A_i}$, having access to $\theta_*$ since these are synthetic problems. This is the best version of MIPS that we could create from our prior (10). However, we stress again that this comparison is only provided for completeness as the setting may be favorable to sDM. In this setting, sDM outperforms MIPS.
<figure><img src="image_16.png" /><figcaption>Figure 13: Comparison of sDM and MIPS in OPE: The average MSE of an $\epsilon$-greedy policy on synthetic problems with varying $n$, $K$ and $d'$.</figcaption></figure>

## F.5 Comparison of greedy and pessimistic policies

To validate our theory that a greedy policy should be preferred over the commonly adopted pessimistic policy in our Bayesian setting, we used a performance metric averaged over multiple bandit problems sampled from the prior. To verify this, we considered the same OPL synthetic setting as in Section 6 and compared sDM with a greedy policy to sDM with a pessimistic policy. Recall that a greedy policy with respect to our reward estimate writes

$$ \hat{\pi}_G(a | x) = \mathbb{1}\{a = \underset{b \in A}{\operatorname{argmax}} \hat{r}(x, b)\}, \qquad (46) $$

while a pessimistic one writes

$$ \hat{\pi}_P(a | x) = \mathbb{1}\{a = \underset{b \in A}{\operatorname{argmax}} \hat{r}(x, b) - u(x, a)\}, \qquad (47) $$

where $u(x, a) = \alpha(d, \delta)\|\phi(X)\|_{\hat{\Sigma}_a}$ with $\alpha(d, \delta) = \sqrt{d + 2\sqrt{d \log \frac{1}{\delta}} + 2\log \frac{1}{\delta}}$ and this is derived in (29) in Appendix E.2. As predicted by our theory, the results show that the greedy policy has better average performance over multiple bandit instances sampled from the prior.

<figure><img src="image_17.png" /><figcaption>Figure 14: Comparison of sDM with greedy policy and sDM with pessimistic policy in OPL: The average MSE of an $\epsilon$-greedy policy on synthetic problems with varying $n$, $K$, and $d'$.</figcaption></figure>

## G Broader impact

This work contributes to the development and analysis of practical algorithms for offline learning to act under uncertainty. While our generic setting and algorithms have broad potential applications, the specific downstream social impacts are inherently dependent on the chosen application.
<figure><img src="image_17.png" /><figcaption>Figure 13: Comparison of sDM and MIPS in OPE: The average MSE of an $\epsilon$-greedy policy on synthetic problems with varying $n$, $K$ and $d'$.</figcaption></figure>

## F.5 Comparison of greedy and pessimistic policies

To validate our theory that a greedy policy should be preferred over the commonly adopted pessimistic policy in our Bayesian setting, we used a performance metric averaged over multiple bandit problems sampled from the prior. To verify this, we considered the same OPL synthetic setting as in Section 6 and compared sDM with a greedy policy to sDM with a pessimistic policy. Recall that a greedy policy with respect to our reward estimate writes

$$ \hat{\pi}_G(a | x) = \mathbb{1}\{a = \underset{b \in A}{\operatorname{argmax}} \hat{r}(x, b)\}, \qquad (46) $$

while a pessimistic one writes

$$ \hat{\pi}_P(a | x) = \mathbb{1}\{a = \underset{b \in A}{\operatorname{argmax}} \hat{r}(x, b) - u(x, a)\}, \qquad (47) $$

where $u(x, a) = \alpha(d, \delta)\|\phi(X)\|_{\hat{\Sigma}_a}$ with $\alpha(d, \delta) = \sqrt{d + 2\sqrt{d \log \frac{1}{\delta}} + 2\log \frac{1}{\delta}}$ and this is derived in (29) in Appendix E.2. As predicted by our theory, the results show that the greedy policy has better average performance over multiple bandit instances sampled from the prior.

<figure><img src="image_18.png" /><figcaption>Figure 14: Comparison of sDM with greedy policy and sDM with pessimistic policy in OPL: The average MSE of an $\epsilon$-greedy policy on synthetic problems with varying $n$, $K$, and $d'$.</figcaption></figure>

## G Broader impact

This work contributes to the development and analysis of practical algorithms for offline learning to act under uncertainty. While our generic setting and algorithms have broad potential applications, the specific downstream social impacts are inherently dependent on the chosen application.
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# NeurIPS Paper Checklist

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## 3. Theory Assumptions and Proofs

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## 8. Experiments Compute Resources

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Justification: This work contributes to the development and theoretical analysis of offline learning to act under uncertainty and it adheres to the Neurips Code Of Ethics.

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* The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

* If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

* The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

## 10. Broader Impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: Broader Impacts are discussed in Appendix G.

Guidelines:

* The answer NA means that there is no societal impact of the work performed.

* If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.
* Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

* The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

* The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

* If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

## 11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: Our paper is mainly theoretical and the used data is either simulated or openly accessible. Thus, we believe that our work poses no such risks.

Guidelines:

* The answer NA means that the paper poses no such risks.

* Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

* Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

* We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

## 12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: To the best of our knowledge, all relevant and used papers were cited.

Guidelines:

* The answer NA means that the paper does not use existing assets.

* The authors should cite the original paper that produced the code package or dataset.

* The authors should state which version of the asset is used and, if possible, include a URL.

* The name of the license (e.g., CC-BY 4.0) should be included for each asset.

* For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
* Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

* The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

* The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

* If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

## 11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: Our paper is mainly theoretical and the used data is either simulated or openly accessible. Thus, we believe that our work poses no such risks.

Guidelines:

* The answer NA means that the paper poses no such risks.

* Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

* Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

* We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

## 12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: To the best of our knowledge, all relevant and used papers were cited.

Guidelines:

* The answer NA means that the paper does not use existing assets.

* The authors should cite the original paper that produced the code package or dataset.

* The authors should state which version of the asset is used and, if possible, include a URL.

* The name of the license (e.g., CC-BY 4.0) should be included for each asset.

* For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
* If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

* For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

* If this information is not available online, the authors are encouraged to reach out to the asset's creators.

## 13. New Assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [Yes]

Justification: We include our code as supplementary material, with all details needed for reproducibility given in Section 6 and Appendix F.

Guidelines:

* The answer NA means that the paper does not release new assets.

* Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

* The paper should discuss whether and how consent was obtained from people whose asset is used.

* At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

## 14. Crowdsourcing and Research with Human Subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: This work does not involve crowdsourcing nor research with human subjects.

Guidelines:

* The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

* Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

* According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

## 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: This work does not involve crowdsourcing nor research with human subjects.

Guidelines:

* The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

* Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
* If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

* For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

* If this information is not available online, the authors are encouraged to reach out to the asset's creators.

## 13. New Assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [Yes]

Justification: We include our code as supplementary material, with all details needed for reproducibility given in Section 6 and Appendix F.

Guidelines:

* The answer NA means that the paper does not release new assets.

* Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

* The paper should discuss whether and how consent was obtained from people whose asset is used.

* At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

## 14. Crowdsourcing and Research with Human Subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: This work does not involve crowdsourcing nor research with human subjects.

Guidelines:

* The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

* Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

* According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

## 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: This work does not involve crowdsourcing nor research with human subjects.

Guidelines:

* The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

* Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
* We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

* For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
* We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

* For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.