### D.3 Proof of lemma D.2

In this proof, we heavily rely on the total covariance decomposition [Weiss, 2005]. Also, refer to [Hong et al., 2022b, Section 5.2] for a brief introduction to this decomposition. Now, from Eq. (17), we have that

$$ 
\begin{aligned}
\operatorname{cov}[\theta_{*,i} | H_t, \psi_{*,1}] &= \hat{\Sigma}_{t,i} = (\hat{G}_{t,i} + \Sigma_1^{-1})^{-1}, \\
\mathbb{E}[\theta_{*,i} | H_t, \psi_{*,1}] &= \hat{\mu}_{t,i} = \hat{\Sigma}_{t,i} (\hat{G}_{t,i} \hat{B}_{t,i} + \Sigma_1^{-1} W_1 \psi_{*,1}).
\end{aligned}
$$

First, given $H_t$, $\operatorname{cov}[\theta_{*,i} | H_t, \psi_{*,1}] = (\hat{G}_{t,i} + \Sigma_1^{-1})^{-1}$ is constant. Thus

$$ \mathbb{E}[\operatorname{cov}[\theta_{*,i} | H_t, \psi_{*,1}] | H_t] = \operatorname{cov}[\theta_{*,i} | H_t, \psi_{*,1}] = (\hat{G}_{t,i} + \Sigma_1^{-1})^{-1} = \hat{\Sigma}_{t,i}. $$

In addition, given $H_t$, $\hat{\Sigma}_{t,i}$, $\hat{G}_{t,i}$ and $\hat{B}_{t,i}$ are constant. Thus

$$ 
\begin{aligned}
\operatorname{cov}[\mathbb{E}[\theta_{*,i} | H_t, \psi_{*,1}] | H_t] &= \operatorname{cov}[\hat{\Sigma}_{t,i} (\hat{G}_{t,i} \hat{B}_{t,i} + \Sigma_1^{-1} W_1 \psi_{*,1}) | H_t], \\
&= \operatorname{cov}[\hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1 \psi_{*,1} | H_t], \\
&= \hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1 \operatorname{cov}[\psi_{*,1} | H_t] W_1^\top \Sigma_1^{-1} \hat{\Sigma}_{t,i}, \\
&= \hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1 \bar{\Sigma}_{t,1} W_1^\top \Sigma_1^{-1} \hat{\Sigma}_{t,i},
\end{aligned}
$$

where $\bar{\Sigma}_{t,1} = \operatorname{cov}[\psi_{*,1} | H_t]$ is the marginal posterior covariance of $\psi_{*,1}$. Finally, the total covariance decomposition [Weiss, 2005, Hong et al., 2022b] yields that

$$ 
\begin{aligned}
\hat{\Sigma}_{t,i} &= \operatorname{cov}[\theta_{*,i} | H_t] = \mathbb{E}[\operatorname{cov}[\theta_{*,i} | H_t, \psi_{*,1}] | H_t] + \operatorname{cov}[\mathbb{E}[\theta_{*,i} | H_t, \psi_{*,1}] | H_t], \\
&= \hat{\Sigma}_{t,i} + \hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1 \bar{\Sigma}_{t,1} W_1^\top \Sigma_1^{-1} \hat{\Sigma}_{t,i},
\end{aligned}
\quad (31) $$

However, $\bar{\Sigma}_{t,1} = \operatorname{cov}[\psi_{*,1} | H_t]$ is different from $\bar{\Sigma}_{t,1} = \operatorname{cov}[\psi_{*,1} | H_t, \psi_{*,2}]$ that we already derived in Eq. (18). Thus we do not know the expression of $\bar{\Sigma}_{t,1}$. But we can use the same total covariance decomposition trick to find it. Precisely, let $\bar{\Sigma}_{t,\ell} = \operatorname{cov}[\psi_{*,\ell} | H_t]$ for any $\ell \in [L]$. Then we have that

$$ 
\begin{aligned}
\bar{\Sigma}_{t,1} &= \operatorname{cov}[\psi_{*,1} | H_t, \psi_{*,2}] = (\Sigma_2^{-1} + \bar{G}_{t,1})^{-1}, \\
\bar{\mu}_{t,1} &= \mathbb{E}[\psi_{*,1} | H_t, \psi_{*,2}] = \bar{\Sigma}_{t,1} (\Sigma_2^{-1} W_2 \psi_{*,2} + \bar{B}_{t,1}).
\end{aligned}
$$

First, given $H_t$, $\operatorname{cov}[\psi_{*,1} | H_t, \psi_{*,2}] = (\Sigma_2^{-1} + \bar{G}_{t,1})^{-1}$ is constant. Thus

$$ \mathbb{E}[\operatorname{cov}[\psi_{*,1} | H_t, \psi_{*,2}] | H_t] = \operatorname{cov}[\psi_{*,1} | H_t, \psi_{*,2}] = \bar{\Sigma}_{t,1}. $$

In addition, given $H_t$, $\bar{\Sigma}_{t,1}$, $\tilde{\Sigma}_{t,1}$ and $\bar{B}_{t,1}$ are constant. Thus

$$ 
\begin{aligned}
\operatorname{cov}[\mathbb{E}[\psi_{*,1} | H_t, \psi_{*,2}] | H_t] &= \operatorname{cov}[\bar{\Sigma}_{t,1} (\Sigma_2^{-1} W_2 \psi_{*,2} + \bar{B}_{t,1}) | H_t], \\
&= \operatorname{cov}[\bar{\Sigma}_{t,1} \Sigma_2^{-1} W_2 \psi_{*,2} | H_t], \\
&= \bar{\Sigma}_{t,1} \Sigma_2^{-1} W_2 \operatorname{cov}[\psi_{*,2} | H_t] W_2^\top \Sigma_2^{-1} \bar{\Sigma}_{t,1}, \\
&= \bar{\Sigma}_{t,1} \Sigma_2^{-1} W_2 \bar{\Sigma}_{t,2} W_2^\top \Sigma_2^{-1} \bar{\Sigma}_{t,1}.
\end{aligned}
$$

Finally, total covariance decomposition [Weiss, 2005, Hong et al., 2022b] leads to

$$ 
\begin{aligned}
\bar{\Sigma}_{t,1} &= \operatorname{cov}[\psi_{*,1} | H_t] = \mathbb{E}[\operatorname{cov}[\psi_{*,1} | H_t, \psi_{*,2}] | H_t] + \operatorname{cov}[\mathbb{E}[\psi_{*,1} | H_t, \psi_{*,2}] | H_t], \\
&= \bar{\Sigma}_{t,1} + \bar{\Sigma}_{t,1} \Sigma_2^{-1} W_2 \bar{\Sigma}_{t,2} W_2^\top \Sigma_2^{-1} \bar{\Sigma}_{t,1}.
\end{aligned}
$$

Now using the techniques, this can be generalized using the same technique as above to

$$ \bar{\Sigma}_{t,\ell} = \bar{\Sigma}_{t,\ell} + \bar{\Sigma}_{t,\ell} \Sigma_{\ell+1}^{-1} W_{\ell+1} \bar{\Sigma}_{t,\ell+1} W_{\ell+1}^\top \Sigma_{\ell+1}^{-1} \bar{\Sigma}_{t,\ell}, \quad \forall \ell \in [L-1]. $$
Then, by induction, we get that

$$ \bar{\Sigma}_{t,1} = \sum_{\ell \in [L]} \bar{P}_{\ell} \bar{\Sigma}_{t,\ell} \bar{P}_{\ell}^{\top}, \quad \forall \ell \in [L-1], $$

where we use that by definition $\bar{\Sigma}_{t,L} = \text{cov}[\psi_{*,L} | H_t] = \bar{\Sigma}_{t,L}$ and set $\bar{P}_1 = I_d$ and $\bar{P}_\ell = \prod_{k=1}^{\ell-1} \bar{\Sigma}_{t,k} \Sigma_{k+1}^{-1} W_{k+1}$ for any $\ell \in [L]/\{1\}$. Plugging this in Eq. (31) leads to

$$ \begin{aligned} \hat{\Sigma}_{t,i} &= \hat{\Sigma}_{t,i} + \sum_{\ell \in [L]} \hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1 \bar{P}_{\ell} \bar{\Sigma}_{t,\ell} \bar{P}_{\ell}^{\top} W_1^{\top} \Sigma_1^{-1} \hat{\Sigma}_{t,i}, \\ &= \hat{\Sigma}_{t,i} + \sum_{\ell \in [L]} \hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1 \bar{P}_{\ell} \bar{\Sigma}_{t,\ell} (\hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1)^{\top}, \\ &= \hat{\Sigma}_{t,i} + \sum_{\ell \in [L]} P_{i,\ell} \bar{\Sigma}_{t,\ell} P_{i,\ell}^{\top}, \end{aligned} $$

where $P_{i,\ell} = \hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1 \bar{P}_{\ell} = \hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1 \prod_{k=1}^{\ell-1} \bar{\Sigma}_{t,k} \Sigma_{k+1}^{-1} W_{k+1}$.

### D.4 Proof of lemma D.3

We prove this result by induction. We start with the base case when $\ell = 1$.

(I) Base case. Let $u = \sigma^{-1} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} X_t$ From the expression of $\bar{\Sigma}_{t,1}$ in Eq. (18), we have that

$$ \begin{aligned} \bar{\Sigma}_{t+1,1}^{-1} - \bar{\Sigma}_{t,1}^{-1} &= W_1^{\top} (\Sigma_1^{-1} - \Sigma_1^{-1}(\hat{\Sigma}_{t,A_t}^{-1} + \sigma^{-2} X_t X_t^{\top})^{-1} \Sigma_1^{-1} - (\Sigma_1^{-1} - \Sigma_1^{-1} \hat{\Sigma}_{t,A_t} \Sigma_1^{-1})) W_1, \\ &= W_1^{\top} (\Sigma_1^{-1}(\hat{\Sigma}_{t,A_t} - (\hat{\Sigma}_{t,A_t}^{-1} + \sigma^{-2} X_t X_t^{\top})^{-1}) \Sigma_1^{-1}) W_1, \\ &= W_1^{\top} (\Sigma_1^{-1} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} (I_d - (I_d + \sigma^{-2} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} X_t X_t^{\top} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}})^{-1}) \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} \Sigma_1^{-1}) W_1, \\ &= W_1^{\top} (\Sigma_1^{-1} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} (I_d - (I_d + u u^{\top})^{-1}) \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} \Sigma_1^{-1}) W_1, \\ &\stackrel{(i)}{=} W_1^{\top} (\Sigma_1^{-1} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} \frac{u u^{\top}}{1+u^{\top} u} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} \Sigma_1^{-1}) W_1, \\ &\stackrel{(ii)}{=} \sigma^{-2} W_1^{\top} \Sigma_1^{-1} \hat{\Sigma}_{t,A_t} \frac{X_t X_t^{\top}}{1+u^{\top} u} \hat{\Sigma}_{t,A_t} \Sigma_1^{-1} W_1. \end{aligned} \quad (32) $$

In (i) we use the Sherman-Morrison formula. Note that (ii) says that $\bar{\Sigma}_{t+1,1}^{-1} - \bar{\Sigma}_{t,1}^{-1}$ is one-rank which we will also need in induction step. Now, we have that $\|X_t\|^2 = 1$. Therefore,

$$ 1 + u^{\top} u = 1 + \sigma^{-2} X_t^{\top} \hat{\Sigma}_{t,A_t} X_t \le 1 + \sigma^{-2} \lambda_1(\Sigma_1) \|X_t\|^2 = 1 + \sigma^{-2} \sigma_1^2 \le \sigma_{\text{MAX}}^2, $$

where we use that by definition of $\sigma_{\text{MAX}}^2$ in Lemma D.3, we have that $\sigma_{\text{MAX}}^2 \ge 1 + \sigma^{-2}\sigma_1^2$. Therefore, by taking the inverse, we get that $\frac{1}{1+u^{\top}u} \ge \sigma_{\text{MAX}}^{-2}$. Combining this with Eq. (32) leads to

$$ \bar{\Sigma}_{t+1,1}^{-1} - \bar{\Sigma}_{t,1}^{-1} \succeq \sigma^{-2} \sigma_{\text{MAX}}^{-2} W_1^{\top} \Sigma_1^{-1} \hat{\Sigma}_{t,A_t} X_t X_t^{\top} \hat{\Sigma}_{t,A_t} \Sigma_1^{-1} W_1 $$

Noticing that $P_{A_t,1} = \hat{\Sigma}_{t,A_t} \Sigma_1^{-1} W_1$ concludes the proof of the base case when $\ell = 1$.

(II) Induction step. Let $\ell \in [L]/\{1\}$ and suppose that $\bar{\Sigma}_{t+1,\ell-1}^{-1} - \bar{\Sigma}_{t,\ell-1}^{-1}$ is one-rank and that it holds for $\ell - 1$ that

$$ \bar{\Sigma}_{t+1,\ell-1}^{-1} - \bar{\Sigma}_{t,\ell-1}^{-1} \succeq \sigma^{-2} \sigma_{\text{MAX}}^{-2(\ell-1)} P_{A_t,\ell-1}^{\top} X_t X_t^{\top} P_{A_t,\ell-1}, \quad \text{where } \sigma_{\text{MAX}}^{-2} = \max_{\ell \in [L]} \{1 + \sigma^{-2}\sigma_{\ell}^2\}. $$

Then, we want to show that $\bar{\Sigma}_{t+1,\ell}^{-1} - \bar{\Sigma}_{t,\ell}^{-1}$ is also one-rank and that it holds that

$$ \bar{\Sigma}_{t+1,\ell}^{-1} - \bar{\Sigma}_{t,\ell}^{-1} \succeq \sigma^{-2} \sigma_{\text{MAX}}^{-2\ell} P_{A_t,\ell}^{\top} X_t X_t^{\top} P_{A_t,\ell}, \quad \text{where } \sigma_{\text{MAX}}^{-2} = \max_{\ell \in [L]} \{1 + \sigma^{-2}\sigma_{\ell}^2\}. $$
Then, by induction, we get that

$$ \bar{\Sigma}_{t,1} = \sum_{\ell \in [L]} \bar{P}_{\ell} \bar{\Sigma}_{t,\ell} \bar{P}_{\ell}^{\top}, \quad \forall \ell \in [L-1], $$

where we use that by definition $\bar{\Sigma}_{t,L} = \text{cov}[\psi_{*,L} | H_t] = \bar{\Sigma}_{t,L}$ and set $\bar{P}_1 = I_d$ and $\bar{P}_\ell = \prod_{k=1}^{\ell-1} \bar{\Sigma}_{t,k} \Sigma_{k+1}^{-1} W_{k+1}$ for any $\ell \in [L]/\{1\}$. Plugging this in Eq. (31) leads to

$$ \begin{aligned} \hat{\Sigma}_{t,i} &= \hat{\Sigma}_{t,i} + \sum_{\ell \in [L]} \hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1 \bar{P}_{\ell} \bar{\Sigma}_{t,\ell} \bar{P}_{\ell}^{\top} W_1^{\top} \Sigma_1^{-1} \hat{\Sigma}_{t,i}, \\ &= \hat{\Sigma}_{t,i} + \sum_{\ell \in [L]} \hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1 \bar{P}_{\ell} \bar{\Sigma}_{t,\ell} (\hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1)^{\top}, \\ &= \hat{\Sigma}_{t,i} + \sum_{\ell \in [L]} P_{i,\ell} \bar{\Sigma}_{t,\ell} P_{i,\ell}^{\top}, \end{aligned} $$

where $P_{i,\ell} = \hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1 \bar{P}_{\ell} = \hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1 \prod_{k=1}^{\ell-1} \bar{\Sigma}_{t,k} \Sigma_{k+1}^{-1} W_{k+1}$.

### D.4 Proof of lemma D.3

We prove this result by induction. We start with the base case when $\ell = 1$.

(I) Base case. Let $u = \sigma^{-1} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} X_t$ From the expression of $\bar{\Sigma}_{t,1}$ in Eq. (18), we have that

$$ \begin{aligned} \bar{\Sigma}_{t+1,1}^{-1} - \bar{\Sigma}_{t,1}^{-1} &= W_1^{\top} (\Sigma_1^{-1} - \Sigma_1^{-1}(\hat{\Sigma}_{t,A_t}^{-1} + \sigma^{-2} X_t X_t^{\top})^{-1} \Sigma_1^{-1} - (\Sigma_1^{-1} - \Sigma_1^{-1} \hat{\Sigma}_{t,A_t} \Sigma_1^{-1})) W_1, \\ &= W_1^{\top} (\Sigma_1^{-1}(\hat{\Sigma}_{t,A_t} - (\hat{\Sigma}_{t,A_t}^{-1} + \sigma^{-2} X_t X_t^{\top})^{-1}) \Sigma_1^{-1}) W_1, \\ &= W_1^{\top} (\Sigma_1^{-1} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} (I_d - (I_d + \sigma^{-2} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} X_t X_t^{\top} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}})^{-1}) \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} \Sigma_1^{-1}) W_1, \\ &= W_1^{\top} (\Sigma_1^{-1} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} (I_d - (I_d + u u^{\top})^{-1}) \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} \Sigma_1^{-1}) W_1, \\ &\stackrel{(i)}{=} W_1^{\top} (\Sigma_1^{-1} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} \frac{u u^{\top}}{1+u^{\top} u} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} \Sigma_1^{-1}) W_1, \\ &\stackrel{(ii)}{=} \sigma^{-2} W_1^{\top} \Sigma_1^{-1} \hat{\Sigma}_{t,A_t} \frac{X_t X_t^{\top}}{1+u^{\top} u} \hat{\Sigma}_{t,A_t} \Sigma_1^{-1} W_1. \end{aligned} \quad (32) $$

In (i) we use the Sherman-Morrison formula. Note that (ii) says that $\bar{\Sigma}_{t+1,1}^{-1} - \bar{\Sigma}_{t,1}^{-1}$ is one-rank which we will also need in induction step. Now, we have that $\|X_t\|^2 = 1$. Therefore,

$$ 1 + u^{\top} u = 1 + \sigma^{-2} X_t^{\top} \hat{\Sigma}_{t,A_t} X_t \le 1 + \sigma^{-2} \lambda_1(\Sigma_1) \|X_t\|^2 = 1 + \sigma^{-2} \sigma_1^2 \le \sigma_{\text{MAX}}^2, $$

where we use that by definition of $\sigma_{\text{MAX}}^2$ in Lemma D.3, we have that $\sigma_{\text{MAX}}^2 \ge 1 + \sigma^{-2} \sigma_1^2$. Therefore, by taking the inverse, we get that $\frac{1}{1+u^{\top} u} \ge \sigma_{\text{MAX}}^{-2}$. Combining this with Eq. (32) leads to

$$ \bar{\Sigma}_{t+1,1}^{-1} - \bar{\Sigma}_{t,1}^{-1} \succeq \sigma^{-2} \sigma_{\text{MAX}}^{-2} W_1^{\top} \Sigma_1^{-1} \hat{\Sigma}_{t,A_t} X_t X_t^{\top} \hat{\Sigma}_{t,A_t} \Sigma_1^{-1} W_1 $$

Noticing that $P_{A_t,1} = \hat{\Sigma}_{t,A_t} \Sigma_1^{-1} W_1$ concludes the proof of the base case when $\ell = 1$.

(II) Induction step. Let $\ell \in [L]/\{1\}$ and suppose that $\bar{\Sigma}_{t+1,\ell-1}^{-1} - \bar{\Sigma}_{t,\ell-1}^{-1}$ is one-rank and that it holds for $\ell - 1$ that

$$ \bar{\Sigma}_{t+1,\ell-1}^{-1} - \bar{\Sigma}_{t,\ell-1}^{-1} \succeq \sigma^{-2} \sigma_{\text{MAX}}^{-2(\ell-1)} P_{A_t,\ell-1}^{\top} X_t X_t^{\top} P_{A_t,\ell-1}, \quad \text{where } \sigma_{\text{MAX}}^{-2} = \max_{\ell \in [L]} 1 + \sigma^{-2} \sigma_{\ell}^2. $$

Then, we want to show that $\bar{\Sigma}_{t+1,\ell}^{-1} - \bar{\Sigma}_{t,\ell}^{-1}$ is also one-rank and that it holds that

$$ \bar{\Sigma}_{t+1,\ell}^{-1} - \bar{\Sigma}_{t,\ell}^{-1} \succeq \sigma^{-2} \sigma_{\text{MAX}}^{-2\ell} P_{A_t,\ell}^{\top} X_t X_t^{\top} P_{A_t,\ell}, \quad \text{where } \sigma_{\text{MAX}}^{-2} = \max_{\ell \in [L]} 1 + \sigma^{-2} \sigma_{\ell}^2. $$
This is achieved as follows. First, we notice that by the induction hypothesis, we have that $\tilde{\Sigma}_{t+1,\ell-1}^{-1} - \tilde{G}_{t,\ell-1} = \tilde{\Sigma}_{t+1,\ell-1}^{-1} - \tilde{\Sigma}_{t,\ell-1}^{-1}$ is one-rank. In addition, the matrix is positive semi-definite. Thus we can write it as $\tilde{\Sigma}_{t+1,\ell-1}^{-1} - \tilde{G}_{t,\ell-1} = uu^\top$ where $u \in \mathbb{R}^d$. Then, similarly to the base case, we have

$$ 
\begin{aligned}
\tilde{\Sigma}_{t+1,\ell}^{-1} - \tilde{\Sigma}_{t,\ell}^{-1} &= \tilde{\Sigma}_{t+1,\ell}^{-1} - \tilde{\Sigma}_{t,\ell}^{-1}, \\
&= W_{\ell}^{\top} (\tilde{\Sigma}_{\ell} + \tilde{\Sigma}_{t+1,\ell-1})^{-1} W_{\ell} - W_{\ell}^{\top} (\tilde{\Sigma}_{\ell} + \tilde{\Sigma}_{t,\ell-1})^{-1} W_{\ell}, \\
&= W_{\ell}^{\top} [(\tilde{\Sigma}_{\ell} + \tilde{\Sigma}_{t+1,\ell-1})^{-1} - (\tilde{\Sigma}_{\ell} + \tilde{\Sigma}_{t,\ell-1})^{-1}] W_{\ell}, \\
&= W_{\ell}^{\top} \tilde{\Sigma}_{\ell}^{-1} [(\tilde{\Sigma}_{\ell}^{-1} + \tilde{G}_{t,\ell-1})^{-1} - (\tilde{\Sigma}_{\ell}^{-1} + \tilde{\Sigma}_{t+1,\ell-1}^{-1})^{-1}] \tilde{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&= W_{\ell}^{\top} \tilde{\Sigma}_{\ell}^{-1} [(\tilde{\Sigma}_{\ell}^{-1} + \tilde{G}_{t,\ell-1})^{-1} - (\tilde{\Sigma}_{\ell}^{-1} + \tilde{G}_{t,\ell-1} + \tilde{\Sigma}_{t+1,\ell-1}^{-1} - \tilde{G}_{t,\ell-1})^{-1}] \tilde{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&= W_{\ell}^{\top} \tilde{\Sigma}_{\ell}^{-1} [(\tilde{\Sigma}_{\ell}^{-1} + \tilde{G}_{t,\ell-1})^{-1} - (\tilde{\Sigma}_{\ell}^{-1} + \tilde{G}_{t,\ell-1} + uu^{\top})^{-1}] \tilde{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&= W_{\ell}^{\top} \tilde{\Sigma}_{\ell}^{-1} [\tilde{\Sigma}_{t,\ell-1} - (\tilde{\Sigma}_{t,\ell-1}^{-1} + uu^{\top})^{-1}] \tilde{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&= W_{\ell}^{\top} \tilde{\Sigma}_{\ell}^{-1} [\tilde{\Sigma}_{t,\ell-1} \frac{uu^{\top}}{1+u^{\top}\tilde{\Sigma}_{t,\ell-1}u} \tilde{\Sigma}_{t,\ell-1}] \tilde{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&= W_{\ell}^{\top} \tilde{\Sigma}_{\ell}^{-1} \tilde{\Sigma}_{t,\ell-1} \frac{uu^{\top}}{1+u^{\top}\tilde{\Sigma}_{t,\ell-1}u} \tilde{\Sigma}_{t,\ell-1} \tilde{\Sigma}_{\ell}^{-1} W_{\ell}
\end{aligned}
$$

However, we it follows from the induction hypothesis that $uu^\top = \tilde{\Sigma}_{t+1,\ell-1}^{-1} - \tilde{G}_{t,\ell-1} = \tilde{\Sigma}_{t+1,\ell-1}^{-1} - \tilde{\Sigma}_{t,\ell-1}^{-1} \ge \sigma^{-2}\sigma_{\text{MAX}}^{-2(\ell-1)}P_{A_t,\ell-1}^\top X_t X_t^\top P_{A_t,\ell-1}$. Therefore,

$$ 
\begin{aligned}
\tilde{\Sigma}_{t+1,\ell}^{-1} - \tilde{\Sigma}_{t,\ell}^{-1} &= W_{\ell}^{\top} \tilde{\Sigma}_{\ell}^{-1} \tilde{\Sigma}_{t,\ell-1} \frac{uu^{\top}}{1+u^{\top}\tilde{\Sigma}_{t,\ell-1}u} \tilde{\Sigma}_{t,\ell-1} \tilde{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&\ge W_{\ell}^{\top} \tilde{\Sigma}_{\ell}^{-1} \tilde{\Sigma}_{t,\ell-1} \frac{\sigma^{-2}\sigma_{\text{MAX}}^{-2(\ell-1)}P_{A_t,\ell-1}^{\top} X_t X_t^{\top} P_{A_t,\ell-1}}{1+u^{\top}\tilde{\Sigma}_{t,\ell-1}u} \tilde{\Sigma}_{t,\ell-1} \tilde{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&= \frac{\sigma^{-2}\sigma_{\text{MAX}}^{-2(\ell-1)}}{1+u^{\top}\tilde{\Sigma}_{t,\ell-1}u} W_{\ell}^{\top} \tilde{\Sigma}_{\ell}^{-1} \tilde{\Sigma}_{t,\ell-1} P_{A_t,\ell-1}^{\top} X_t X_t^{\top} P_{A_t,\ell-1} \tilde{\Sigma}_{t,\ell-1} \tilde{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&= \frac{\sigma^{-2}\sigma_{\text{MAX}}^{-2(\ell-1)}}{1+u^{\top}\tilde{\Sigma}_{t,\ell-1}u} P_{A_t,\ell}^{\top} X_t X_t^{\top} P_{A_t,\ell}.
\end{aligned}
$$

Finally, we use that $1 + u^\top \tilde{\Sigma}_{t,\ell-1} u \le 1 + \|u\|_2 \lambda_1(\tilde{\Sigma}_{t,\ell-1}) \le 1 + \sigma^{-2}\sigma_{\ell}^2$. Here we use that $\|u\|_2 \le \sigma^{-2}$, which can also be proven by induction, and that $\lambda_1(\tilde{\Sigma}_{t,\ell-1}) \le \sigma_{\ell}^2$, which follows from the expression of $\tilde{\Sigma}_{t,\ell-1}$ in Section 3.1. Therefore, we have that

$$ 
\begin{aligned}
\tilde{\Sigma}_{t+1,\ell}^{-1} - \tilde{\Sigma}_{t,\ell}^{-1} &\ge \frac{\sigma^{-2}\sigma_{\text{MAX}}^{-2(\ell-1)}}{1+u^{\top}\tilde{\Sigma}_{t,\ell-1}u} P_{A_t,\ell}^{\top} X_t X_t^{\top} P_{A_t,\ell}, \\
&\ge \frac{\sigma^{-2}\sigma_{\text{MAX}}^{-2(\ell-1)}}{1+\sigma^{-2}\sigma_{\ell}^2} P_{A_t,\ell}^{\top} X_t X_t^{\top} P_{A_t,\ell}, \\
&\ge \sigma^{-2}\sigma_{\text{MAX}}^{-2\ell} P_{A_t,\ell}^{\top} X_t X_t^{\top} P_{A_t,\ell},
\end{aligned}
$$

where the last inequality follows from the definition of $\sigma_{\text{MAX}}^2 = \max_{\ell \in [L]} 1 + \sigma^{-2}\sigma_{\ell}^2$. This concludes the proof.

## D.5 Proof of theorem 4.1

We start with the following standard result which we borrow from [Hong et al., 2022a, Aouali et al., 2023b],

$$ \mathcal{BR}(n) \le \sqrt{2n \log(1/\delta)} \sqrt{\mathbb{E} \left[ \sum_{t=1}^{n} \|X_t\|_{\tilde{\Sigma}_{t,A_t}}^2 \right]} + cn\delta, \quad \text{where } c > 0 \text{ is a constant.} \quad (33) $$
This is achieved as follows. First, we notice that by the induction hypothesis, we have that $\tilde{\Sigma}_{t+1,\ell-1}^{-1} - \tilde{G}_{t,\ell-1} = \tilde{\Sigma}_{t+1,\ell-1}^{-1} - \tilde{\Sigma}_{t,\ell-1}^{-1}$ is one-rank. In addition, the matrix is positive semi-definite. Thus we can write it as $\tilde{\Sigma}_{t+1,\ell-1}^{-1} - \tilde{G}_{t,\ell-1} = uu^\top$ where $u \in \mathbb{R}^d$. Then, similarly to the base case, we have

$$
\begin{align*}
\tilde{\Sigma}_{t+1,\ell}^{-1} - \tilde{\Sigma}_{t,\ell}^{-1} &= \tilde{\Sigma}_{t+1,\ell}^{-1} - \tilde{\Sigma}_{t,\ell}^{-1}, \\
&= W_{\ell}^{\top} (\tilde{\Sigma}_{\ell} + \tilde{\Sigma}_{t+1,\ell-1})^{-1} W_{\ell} - W_{\ell}^{\top} (\tilde{\Sigma}_{\ell} + \tilde{\Sigma}_{t,\ell-1})^{-1} W_{\ell}, \\
&= W_{\ell}^{\top} [ (\tilde{\Sigma}_{\ell} + \tilde{\Sigma}_{t+1,\ell-1})^{-1} - (\tilde{\Sigma}_{\ell} + \tilde{\Sigma}_{t,\ell-1})^{-1} ] W_{\ell}, \\
&= W_{\ell}^{\top} \tilde{\Sigma}_{\ell}^{-1} [ (\tilde{\Sigma}_{\ell}^{-1} + \tilde{G}_{t,\ell-1})^{-1} - (\tilde{\Sigma}_{\ell}^{-1} + \tilde{\Sigma}_{t+1,\ell-1}^{-1})^{-1} ] \tilde{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&= W_{\ell}^{\top} \tilde{\Sigma}_{\ell}^{-1} [ (\tilde{\Sigma}_{\ell}^{-1} + \tilde{G}_{t,\ell-1})^{-1} - (\tilde{\Sigma}_{\ell}^{-1} + \tilde{G}_{t,\ell-1} + \tilde{\Sigma}_{t+1,\ell-1}^{-1} - \tilde{G}_{t,\ell-1})^{-1} ] \tilde{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&= W_{\ell}^{\top} \tilde{\Sigma}_{\ell}^{-1} [ (\tilde{\Sigma}_{\ell}^{-1} + \tilde{G}_{t,\ell-1})^{-1} - (\tilde{\Sigma}_{\ell}^{-1} + \tilde{G}_{t,\ell-1} + uu^{\top})^{-1} ] \tilde{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&= W_{\ell}^{\top} \tilde{\Sigma}_{\ell}^{-1} [ \tilde{\Sigma}_{t,\ell-1} - (\tilde{\Sigma}_{t,\ell-1}^{-1} + uu^{\top})^{-1} ] \tilde{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&= W_{\ell}^{\top} \tilde{\Sigma}_{\ell}^{-1} [ \tilde{\Sigma}_{t,\ell-1} \frac{uu^{\top}}{1+u^{\top}\tilde{\Sigma}_{t,\ell-1}u} \tilde{\Sigma}_{t,\ell-1} ] \tilde{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&= W_{\ell}^{\top} \tilde{\Sigma}_{\ell}^{-1} \tilde{\Sigma}_{t,\ell-1} \frac{uu^{\top}}{1+u^{\top}\tilde{\Sigma}_{t,\ell-1}u} \tilde{\Sigma}_{t,\ell-1} \tilde{\Sigma}_{\ell}^{-1} W_{\ell}
\end{align*}
$$

However, we it follows from the induction hypothesis that $uu^\top = \tilde{\Sigma}_{t+1,\ell-1}^{-1} - \tilde{G}_{t,\ell-1} = \tilde{\Sigma}_{t+1,\ell-1}^{-1} - \tilde{\Sigma}_{t,\ell-1}^{-1} \ge \sigma^{-2}\sigma_{\text{MAX}}^{-2(\ell-1)}P_{A_t,\ell-1}^\top X_t X_t^\top P_{A_t,\ell-1}$. Therefore,

$$
\begin{align*}
\bar{\Sigma}_{t+1,\ell}^{-1} - \bar{\Sigma}_{t,\ell}^{-1} &= W_{\ell}^{\top} \bar{\Sigma}_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} \frac{uu^{\top}}{1+u^{\top} \bar{\Sigma}_{t,\ell-1} u} \bar{\Sigma}_{t,\ell-1} \bar{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&\ge W_{\ell}^{\top} \bar{\Sigma}_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} \frac{\sigma^{-2} \sigma_{\text{MAX}}^{-2(\ell-1)} P_{A_t,\ell-1}^{\top} X_t X_t^{\top} P_{A_t,\ell-1}}{1+u^{\top} \bar{\Sigma}_{t,\ell-1} u} \bar{\Sigma}_{t,\ell-1} \bar{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&= \frac{\sigma^{-2} \sigma_{\text{MAX}}^{-2(\ell-1)}}{1+u^{\top} \bar{\Sigma}_{t,\ell-1} u} W_{\ell}^{\top} \bar{\Sigma}_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} P_{A_t,\ell-1}^{\top} X_t X_t^{\top} P_{A_t,\ell-1} \bar{\Sigma}_{t,\ell-1} \bar{\Sigma}_{\ell}^{-1} W_{\ell}, \\
&= \frac{\sigma^{-2} \sigma_{\text{MAX}}^{-2(\ell-1)}}{1+u^{\top} \bar{\Sigma}_{t,\ell-1} u} P_{A_t,\ell}^{\top} X_t X_t^{\top} P_{A_t,\ell}.
\end{align*}
$$

Finally, we use that $1 + u^\top \bar{\Sigma}_{t,\ell-1} u \le 1 + \|u\|_2 \lambda_1(\bar{\Sigma}_{t,\ell-1}) \le 1 + \sigma^{-2}\sigma_\ell^2$. Here we use that $\|u\|_2 \le \sigma^{-2}$, which can also be proven by induction, and that $\lambda_1(\bar{\Sigma}_{t,\ell-1}) \le \sigma_\ell^2$, which follows from the expression of $\bar{\Sigma}_{t,\ell-1}$ in Section 3.1. Therefore, we have that

$$
\begin{align*}
\bar{\Sigma}_{t+1,\ell}^{-1} - \bar{\Sigma}_{t,\ell}^{-1} &\ge \frac{\sigma^{-2}\sigma_{\text{MAX}}^{-2(\ell-1)}}{1+u^{\top}\bar{\Sigma}_{t,\ell-1}u} P_{A_t,\ell}^{\top} X_t X_t^{\top} P_{A_t,\ell}, \\
&\ge \frac{\sigma^{-2}\sigma_{\text{MAX}}^{-2(\ell-1)}}{1+\sigma^{-2}\sigma_{\ell}^2} P_{A_t,\ell}^{\top} X_t X_t^{\top} P_{A_t,\ell}, \\
&\ge \sigma^{-2}\sigma_{\text{MAX}}^{-2\ell} P_{A_t,\ell}^{\top} X_t X_t^{\top} P_{A_t,\ell},
\end{align*}
$$

where the last inequality follows from the definition of $\sigma_{\text{MAX}}^2 = \max_{\ell \in [L]} (1 + \sigma^{-2}\sigma_\ell^2)$. This concludes the proof.

### D.5 Proof of theorem 4.1

We start with the following standard result which we borrow from [Hong et al., 2022a, Aouali et al., 2023b],

$$ \mathcal{BR}(n) \le \sqrt{2n \log(1/\delta)} \sqrt{\mathbb{E} \left[ \sum_{t=1}^{n} \|X_t\|_{\bar{\Sigma}_{t,A_t}}^2 \right]} + cn\delta, \quad \text{where } c > 0 \text{ is a constant.} \quad (33) $$
Then we use Lemma D.2 and express the marginal covariance $\tilde{\Sigma}_{t,A_t}$ as

$$ \tilde{\Sigma}_{t,i} = \hat{\Sigma}_{t,i} + \sum_{\ell \in [L]} P_{i,\ell} \bar{\Sigma}_{t,\ell} P_{i,\ell}^{\top}, \quad \text{where } P_{i,\ell} = \hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1 \prod_{k=1}^{\ell-1} \bar{\Sigma}_{t,k} \Sigma_{k+1}^{-1} W_{k+1}. \quad (34) $$

Therefore, we can decompose $\|X_t\|_{\tilde{\Sigma}_{t,A_t}}^2$ as

$$ \begin{aligned} \|X_t\|_{\tilde{\Sigma}_{t,A_t}}^2 &= \sigma^2 \frac{X_t^\top \tilde{\Sigma}_{t,A_t} X_t}{\sigma^2} \stackrel{(i)}{=} \sigma^2 (\sigma^{-2} X_t^\top \hat{\Sigma}_{t,A_t} X_t + \sigma^{-2} \sum_{\ell \in [L]} X_t^\top P_{A_t,\ell} \bar{\Sigma}_{t,\ell} P_{A_t,\ell}^\top X_t), \\ &\le c_0 \log(1 + \sigma^{-2} X_t^\top \hat{\Sigma}_{t,A_t} X_t) + \sum_{\ell \in [L]} c_\ell \log(1 + \sigma^{-2} X_t^\top P_{A_t,\ell} \bar{\Sigma}_{t,\ell} P_{A_t,\ell}^\top X_t), \quad (35) \end{aligned} $$

where (i) follows from Eq. (34), and we use the following inequality in (ii)

$$ x = \frac{x}{\log(1+x)} \log(1+x) \le \left( \max_{x \in [0, u]} \frac{x}{\log(1+x)} \right) \log(1+x) = \frac{u}{\log(1+u)} \log(1+x), $$

which holds for any $x \in [0, u]$, where constants $c_0$ and $c_\ell$ are derived as

$$ c_0 = \frac{\sigma_1^2}{\log(1 + \frac{\sigma_1^2}{\sigma^2})}, \quad c_\ell = \frac{\sigma_{\ell+1}^2}{\log(1 + \frac{\sigma_{\ell+1}^2}{\sigma^2})}, \quad \text{with the convention that } \sigma_{L+1} = 1. $$

The derivation of $c_0$ uses that

$$ X_t^\top \hat{\Sigma}_{t,A_t} X_t \le \lambda_1(\hat{\Sigma}_{t,A_t}) \|X_t\|^2 \le \lambda_d^{-1}(\Sigma_1^{-1} + G_{t,A_t}) \le \lambda_d^{-1}(\Sigma_1^{-1}) = \lambda_1(\Sigma_1) = \sigma_1^2. $$

The derivation of $c_\ell$ follows from

$$ X_t^\top P_{A_t,\ell} \bar{\Sigma}_{t,\ell} P_{A_t,\ell}^\top X_t \le \lambda_1(P_{A_t,\ell} P_{A_t,\ell}^\top) \lambda_1(\bar{\Sigma}_{t,\ell}) \|X_t\|^2 \le \sigma_{\ell+1}^2. $$

Therefore, from Eq. (35) and Eq. (33), we get that

$$ \mathcal{BR}(n) \le \sqrt{2n \log(1/\delta)} \left( \mathbb{E} \left[ c_0 \sum_{t=1}^{n} \log(1 + \sigma^{-2} X_t^\top \hat{\Sigma}_{t,A_t} X_t) + \sum_{\ell \in [L]} c_\ell \sum_{t=1}^{n} \log(1 + \sigma^{-2} X_t^\top P_{A_t,\ell} \bar{\Sigma}_{t,\ell} P_{A_t,\ell}^\top X_t) \right] \right)^{\frac{1}{2}} + cn\delta \quad (36) $$

Now we focus on bounding the logarithmic terms in Eq. (36).

(I) First term in Eq. (36) We first rewrite this term as

$$ \begin{aligned} \log(1 + \sigma^{-2} X_t^\top \hat{\Sigma}_{t,A_t} X_t) &\stackrel{(i)}{=} \log \det(I_d + \sigma^{-2} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} X_t X_t^\top \hat{\Sigma}_{t,A_t}^{\frac{1}{2}}), \\ &= \log \det(\hat{\Sigma}_{t,A_t}^{-1} + \sigma^{-2} X_t X_t^\top) - \log \det(\hat{\Sigma}_{t,A_t}^{-1}) = \log \det(\hat{\Sigma}_{t+1,A_t}^{-1}) - \log \det(\hat{\Sigma}_{t,A_t}^{-1}), \end{aligned} $$

where (i) follows from the Weinstein-Aronszajn identity. Then we sum over all rounds $t \in [n]$, and get a telescoping

$$ \begin{aligned} & \sum_{t=1}^{n} \log \det(I_d + \sigma^{-2} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} X_t X_t^\top \hat{\Sigma}_{t,A_t}^{\frac{1}{2}}) = \sum_{t=1}^{n} \log \det(\hat{\Sigma}_{t+1,A_t}^{-1}) - \log \det(\hat{\Sigma}_{t,A_t}^{-1}), \\ &= \sum_{t=1}^{n} \sum_{i=1}^{K} \log \det(\hat{\Sigma}_{t+1,i}^{-1}) - \log \det(\hat{\Sigma}_{t,i}^{-1}) = \sum_{t=1}^{K} \sum_{i=1}^{n} \log \det(\hat{\Sigma}_{t+1,i}^{-1}) - \log \det(\hat{\Sigma}_{t,i}^{-1}), \\ &= \sum_{t=1}^{K} \log \det(\hat{\Sigma}_{n+1,i}^{-1}) - \log \det(\hat{\Sigma}_{1,i}^{-1}) \stackrel{(i)}{=} \sum_{t=1}^{K} \log \det(\Sigma_1^{\frac{1}{2}} \hat{\Sigma}_{n+1,i}^{-1} \Sigma_1^{\frac{1}{2}}), \end{aligned} $$
Then we use Lemma D.2 and express the marginal covariance $\tilde{\Sigma}_{t,A_t}$ as

$$ \tilde{\Sigma}_{t,i} = \hat{\Sigma}_{t,i} + \sum_{\ell \in [L]} P_{i,\ell} \bar{\Sigma}_{t,\ell} P_{i,\ell}^{\top}, \quad \text{where } P_{i,\ell} = \hat{\Sigma}_{t,i} \Sigma_1^{-1} W_1 \prod_{k=1}^{\ell-1} \bar{\Sigma}_{t,k} \Sigma_{k+1}^{-1} W_{k+1}. \quad (34) $$

Therefore, we can decompose $\|X_t\|_{\tilde{\Sigma}_{t,A_t}}^2$ as

$$ \begin{aligned} \|X_t\|_{\tilde{\Sigma}_{t,A_t}}^2 &= \sigma^2 \frac{X_t^\top \tilde{\Sigma}_{t,A_t} X_t}{\sigma^2} \stackrel{(i)}{=} \sigma^2 (\sigma^{-2} X_t^\top \hat{\Sigma}_{t,A_t} X_t + \sigma^{-2} \sum_{\ell \in [L]} X_t^\top P_{A_t,\ell} \bar{\Sigma}_{t,\ell} P_{A_t,\ell}^\top X_t), \\ &\le c_0 \log(1 + \sigma^{-2} X_t^\top \hat{\Sigma}_{t,A_t} X_t) + \sum_{\ell \in [L]} c_\ell \log(1 + \sigma^{-2} X_t^\top P_{A_t,\ell} \bar{\Sigma}_{t,\ell} P_{A_t,\ell}^\top X_t), \quad (35) \end{aligned} $$

where (i) follows from Eq. (34), and we use the following inequality in (ii)

$$ x = \frac{x}{\log(1+x)} \log(1+x) \le \left( \max_{x \in [0, u]} \frac{x}{\log(1+x)} \right) \log(1+x) = \frac{u}{\log(1+u)} \log(1+x), $$

which holds for any $x \in [0, u]$, where constants $c_0$ and $c_\ell$ are derived as

$$ c_0 = \frac{\sigma_1^2}{\log(1 + \frac{\sigma_1^2}{\sigma^2})}, \quad c_\ell = \frac{\sigma_{\ell+1}^2}{\log(1 + \frac{\sigma_{\ell+1}^2}{\sigma^2})}, \quad \text{with the convention that } \sigma_{L+1} = 1. $$

The derivation of $c_0$ uses that

$$ X_t^\top \hat{\Sigma}_{t,A_t} X_t \le \lambda_1(\hat{\Sigma}_{t,A_t}) \|X_t\|^2 \le \lambda_d^{-1}(\Sigma_1^{-1} + G_{t,A_t}) \le \lambda_d^{-1}(\Sigma_1^{-1}) = \lambda_1(\Sigma_1) = \sigma_1^2. $$

The derivation of $c_\ell$ follows from

$$ X_t^\top P_{A_t,\ell} \bar{\Sigma}_{t,\ell} P_{A_t,\ell}^\top X_t \le \lambda_1(P_{A_t,\ell} P_{A_t,\ell}^\top) \lambda_1(\bar{\Sigma}_{t,\ell}) \|X_t\|^2 \le \sigma_{\ell+1}^2. $$

Therefore, from Eq. (35) and Eq. (33), we get that

$$ \mathcal{BR}(n) \le \sqrt{2n \log(1/\delta)} \left( \mathbb{E} \left[ c_0 \sum_{t=1}^n \log(1 + \sigma^{-2} X_t^\top \hat{\Sigma}_{t,A_t} X_t) + \sum_{\ell \in [L]} c_\ell \sum_{t=1}^n \log(1 + \sigma^{-2} X_t^\top P_{A_t,\ell} \bar{\Sigma}_{t,\ell} P_{A_t,\ell}^\top X_t) \right] \right)^{\frac{1}{2}} + cn\delta \quad (36) $$

Now we focus on bounding the logarithmic terms in Eq. (36).

(I) First term in Eq. (36) We first rewrite this term as

$$ \begin{aligned} \log(1 + \sigma^{-2} X_t^\top \hat{\Sigma}_{t,A_t} X_t) &\stackrel{(i)}{=} \log \det(I_d + \sigma^{-2} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} X_t X_t^\top \hat{\Sigma}_{t,A_t}^{\frac{1}{2}}), \\ &= \log \det(\hat{\Sigma}_{t,A_t}^{-1} + \sigma^{-2} X_t X_t^\top) - \log \det(\hat{\Sigma}_{t,A_t}^{-1}) = \log \det(\hat{\Sigma}_{t+1,A_t}^{-1}) - \log \det(\hat{\Sigma}_{t,A_t}^{-1}), \end{aligned} $$

where (i) follows from the Weinstein-Aronszajn identity. Then we sum over all rounds $t \in [n]$, and get a telescoping

$$ \begin{aligned} & \sum_{t=1}^{n} \log \det(I_d + \sigma^{-2} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} X_t X_t^\top \hat{\Sigma}_{t,A_t}^{\frac{1}{2}}) = \sum_{t=1}^{n} \log \det(\hat{\Sigma}_{t+1,A_t}^{-1}) - \log \det(\hat{\Sigma}_{t,A_t}^{-1}), \\ &= \sum_{t=1}^{n} \sum_{i=1}^{K} \log \det(\hat{\Sigma}_{t+1,i}^{-1}) - \log \det(\hat{\Sigma}_{t,i}^{-1}) = \sum_{t=1}^{K} \sum_{i=1}^{n} \log \det(\hat{\Sigma}_{t+1,i}^{-1}) - \log \det(\hat{\Sigma}_{t,i}^{-1}), \\ &= \sum_{t=1}^{K} \log \det(\hat{\Sigma}_{n+1,i}^{-1}) - \log \det(\hat{\Sigma}_{1,i}^{-1}) \stackrel{(i)}{=} \sum_{t=1}^{K} \log \det(\Sigma_1^{\frac{1}{2}} \hat{\Sigma}_{n+1,i}^{-1} \Sigma_1^{\frac{1}{2}}), \end{aligned} $$
where (i) follows from the fact that $\hat{\Sigma}_{1,i} = \Sigma_1$. Now we use the inequality of arithmetic and geometric means and get

$$
\begin{align}
\sum_{t=1}^{n} \log \det(I_d + \sigma^{-2} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} X_t X_t^{\top} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}}) &= \sum_{i=1}^{K} \log \det(\Sigma_1^{\frac{1}{2}} \hat{\Sigma}_{n+1,i}^{-1} \Sigma_1^{\frac{1}{2}}), \nonumber \\
&\le \sum_{i=1}^{K} d \log \left( \frac{1}{d} \operatorname{Tr}(\Sigma_1^{\frac{1}{2}} \hat{\Sigma}_{n+1,i}^{-1} \Sigma_1^{\frac{1}{2}}) \right), \tag{37} \\
&\le \sum_{i=1}^{K} d \log \left( 1 + \frac{n}{d} \frac{\sigma_1^2}{\sigma^2} \right) = K d \log \left( 1 + \frac{n}{d} \frac{\sigma_1^2}{\sigma^2} \right). \nonumber
\end{align}
$$

(II) Remaining terms in Eq. (36) Let $\ell \in [L]$. Then we have that

$$
\begin{align*}
\log(1 + \sigma^{-2} X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^\top X_t) &= \sigma_{\text{MAX}}^{2\ell} \sigma_{\text{MAX}}^{-2\ell} \log(1 + \sigma^{-2} X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^\top X_t), \\
&\le \sigma_{\text{MAX}}^{2\ell} \log(1 + \sigma^{-2} \sigma_{\text{MAX}}^{-2\ell} X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^\top X_t), \\
&\stackrel{(i)}{=} \sigma_{\text{MAX}}^{2\ell} \log \det(I_d + \sigma^{-2} \sigma_{\text{MAX}}^{-2\ell} \bar{\Sigma}_{t, \ell}^{\frac{1}{2}} P_{A_t, \ell}^\top X_t X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell}^{\frac{1}{2}}), \\
&= \sigma_{\text{MAX}}^{2\ell} \left( \log \det(\bar{\Sigma}_{t, \ell}^{-1} + \sigma^{-2} \sigma_{\text{MAX}}^{-2\ell} P_{A_t, \ell}^\top X_t X_t^\top P_{A_t, \ell}) - \log \det(\bar{\Sigma}_{t, \ell}^{-1}) \right),
\end{align*}
$$

where we use the Weinstein-Aronszajn identity in (i). Now we know from Lemma D.3 that the following inequality holds $\sigma^{-2}\sigma_{\text{MAX}}^{-2\ell}P_{A_t,\ell}^\top X_t X_t^\top P_{A_t,\ell} \le \bar{\Sigma}_{t+1,\ell}^{-1} - \bar{\Sigma}_{t,\ell}^{-1}$. As a result, we get that $\bar{\Sigma}_{t,\ell}^{-1} + \sigma^{-2}\sigma_{\text{MAX}}^{-2\ell}P_{A_t,\ell}^\top X_t X_t^\top P_{A_t,\ell} \le \bar{\Sigma}_{t+1,\ell}^{-1}$. Thus,

$$
\log(1 + \sigma^{-2} X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^\top X_t) \le \sigma_{\text{MAX}}^{2\ell} \left( \log \det(\bar{\Sigma}_{t+1, \ell}^{-1}) - \log \det(\bar{\Sigma}_{t, \ell}^{-1}) \right),
$$

Then we sum over all rounds $t \in [n]$, and get a telescoping

$$
\begin{align*}
\sum_{t=1}^{n} \log(1 + \sigma^{-2} X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^\top X_t) &\le \sigma_{\text{MAX}}^{2\ell} \sum_{t=1}^{n} \log \det(\bar{\Sigma}_{t+1, \ell}^{-1}) - \log \det(\bar{\Sigma}_{t, \ell}^{-1}), \\
&= \sigma_{\text{MAX}}^{2\ell} \left( \log \det(\bar{\Sigma}_{n+1, \ell}^{-1}) - \log \det(\bar{\Sigma}_{1, \ell}^{-1}) \right), \\
&\stackrel{(i)}{=} \sigma_{\text{MAX}}^{2\ell} \left( \log \det(\bar{\Sigma}_{n+1, \ell}^{-1}) - \log \det(\bar{\Sigma}_{\ell+1}^{-1}) \right), \\
&= \sigma_{\text{MAX}}^{2\ell} \left( \log \det(\Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1, \ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}}) \right),
\end{align*}
$$

where we use that $\bar{\Sigma}_{1,\ell} = \Sigma_{\ell+1}$ in (i). Finally, we use the inequality of arithmetic and geometric means and get that

$$
\begin{align}
\sum_{t=1}^{n} \log(1 + \sigma^{-2} X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^\top X_t) &\le \sigma_{\text{MAX}}^{2\ell} \left( \log \det(\Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1, \ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}}) \right), \nonumber \\
&\le d \sigma_{\text{MAX}}^{2\ell} \log \left( \frac{1}{d} \operatorname{Tr}(\Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1, \ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}}) \right), \tag{38} \\
&\le d \sigma_{\text{MAX}}^{2\ell} \log \left( 1 + \frac{\sigma_{\ell+1}^2}{\sigma_{\ell}^2} \right), \nonumber
\end{align}
$$

The last inequality follows from the expression of $\bar{\Sigma}_{n+1,\ell}^{-1}$ in Eq. (18) that leads to

$$
\begin{align}
\Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1, \ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}} &= I_d + \Sigma_{\ell+1}^{\frac{1}{2}} \bar{G}_{t, \ell} \Sigma_{\ell+1}^{\frac{1}{2}}, \nonumber \\
&= I_d + \Sigma_{\ell+1}^{\frac{1}{2}} W_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t, \ell-1} \Sigma_{\ell}^{-1}) W_{\ell} \Sigma_{\ell+1}^{\frac{1}{2}}, \tag{39}
\end{align}
$$
where (i) follows from the fact that $\hat{\Sigma}_{1,i} = \Sigma_1$. Now we use the inequality of arithmetic and geometric means and get

$$ 
\begin{aligned}
\sum_{t=1}^{n} \log \det(I_d + \sigma^{-2} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}} X_t X_t^{\top} \hat{\Sigma}_{t,A_t}^{\frac{1}{2}}) &= \sum_{i=1}^{K} \log \det(\Sigma_1^{\frac{1}{2}} \hat{\Sigma}_{n+1,i}^{-1} \Sigma_1^{\frac{1}{2}}), \\
&\leq \sum_{i=1}^{K} d \log \left( \frac{1}{d} \operatorname{Tr}(\Sigma_1^{\frac{1}{2}} \hat{\Sigma}_{n+1,i}^{-1} \Sigma_1^{\frac{1}{2}}) \right), \quad (37) \\
&\leq \sum_{i=1}^{K} d \log \left( 1 + \frac{n}{d} \frac{\sigma_1^2}{\sigma^2} \right) = K d \log \left( 1 + \frac{n}{d} \frac{\sigma_1^2}{\sigma^2} \right).
\end{aligned}
$$

(II) Remaining terms in Eq. (36) Let $\ell \in [L]$. Then we have that

$$ 
\begin{aligned}
\log(1 + \sigma^{-2} X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^\top X_t) &= \sigma_{\max}^{2\ell} \sigma_{\max}^{-2\ell} \log(1 + \sigma^{-2} X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^\top X_t), \\
&\leq \sigma_{\max}^{2\ell} \log(1 + \sigma^{-2} \sigma_{\max}^{-2\ell} X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^\top X_t), \\
&\stackrel{(i)}{=} \sigma_{\max}^{2\ell} \log \det(I_d + \sigma^{-2} \sigma_{\max}^{-2\ell} \bar{\Sigma}_{t, \ell}^{\frac{1}{2}} P_{A_t, \ell}^\top X_t X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell}^{\frac{1}{2}}), \\
&= \sigma_{\max}^{2\ell} \left( \log \det(\bar{\Sigma}_{t, \ell}^{-1} + \sigma^{-2} \sigma_{\max}^{-2\ell} P_{A_t, \ell}^\top X_t X_t^\top P_{A_t, \ell}) - \log \det(\bar{\Sigma}_{t, \ell}^{-1}) \right),
\end{aligned}
$$

where we use the Weinstein-Aronszajn identity in (i). Now we know from Lemma D.3 that the following inequality holds $\sigma^{-2} \sigma_{\max}^{-2\ell} P_{A_t, \ell}^\top X_t X_t^\top P_{A_t, \ell} \leq \bar{\Sigma}_{t+1, \ell}^{-1} - \bar{\Sigma}_{t, \ell}^{-1}$. As a result, we get that $\bar{\Sigma}_{t, \ell}^{-1} + \sigma^{-2} \sigma_{\max}^{-2\ell} P_{A_t, \ell}^\top X_t X_t^\top P_{A_t, \ell} \leq \bar{\Sigma}_{t+1, \ell}^{-1}$. Thus,

$$ \log(1 + \sigma^{-2} X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^\top X_t) \leq \sigma_{\max}^{2\ell} \left( \log \det(\bar{\Sigma}_{t+1, \ell}^{-1}) - \log \det(\bar{\Sigma}_{t, \ell}^{-1}) \right), $$

Then we sum over all rounds $t \in [n]$, and get a telescoping

$$ 
\begin{aligned}
\sum_{t=1}^{n} \log(1 + \sigma^{-2} X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^\top X_t) &\leq \sigma_{\max}^{2\ell} \sum_{t=1}^{n} \log \det(\bar{\Sigma}_{t+1, \ell}^{-1}) - \log \det(\bar{\Sigma}_{t, \ell}^{-1}), \\
&= \sigma_{\max}^{2\ell} \left( \log \det(\bar{\Sigma}_{n+1, \ell}^{-1}) - \log \det(\bar{\Sigma}_{1, \ell}^{-1}) \right), \\
&\stackrel{(i)}{=} \sigma_{\max}^{2\ell} \left( \log \det(\bar{\Sigma}_{n+1, \ell}^{-1}) - \log \det(\bar{\Sigma}_{\ell+1}^{-1}) \right), \\
&= \sigma_{\max}^{2\ell} \left( \log \det(\bar{\Sigma}_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1, \ell}^{-1} \bar{\Sigma}_{\ell+1}^{\frac{1}{2}}) \right),
\end{aligned}
$$

where we use that $\bar{\Sigma}_{1, \ell} = \Sigma_{\ell+1}$ in (i). Finally, we use the inequality of arithmetic and geometric means and get that

$$ 
\begin{aligned}
\sum_{t=1}^{n} \log(1 + \sigma^{-2} X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^\top X_t) &\leq \sigma_{\max}^{2\ell} \left( \log \det(\Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1, \ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}}) \right), \\
&\leq d \sigma_{\max}^{2\ell} \log \left( \frac{1}{d} \operatorname{Tr}(\Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1, \ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}}) \right), \quad (38) \\
&\leq d \sigma_{\max}^{2\ell} \log \left( 1 + \frac{\sigma_{\ell+1}^2}{\sigma_{\ell}^2} \right),
\end{aligned}
$$

The last inequality follows from the expression of $\bar{\Sigma}_{n+1, \ell}^{-1}$ in Eq. (18) that leads to

$$ 
\begin{aligned}
\Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1, \ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}} &= I_d + \Sigma_{\ell+1}^{\frac{1}{2}} \bar{G}_{t, \ell} \Sigma_{\ell+1}^{\frac{1}{2}}, \\
&= I_d + \Sigma_{\ell+1}^{\frac{1}{2}} W_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t, \ell-1} \Sigma_{\ell}^{-1}) W_{\ell} \Sigma_{\ell+1}^{\frac{1}{2}},
\end{aligned}
\quad (39) $$
since $\bar{G}_{t,\ell} = W_{\ell}^{\top}(\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1}\bar{\Sigma}_{t,\ell-1}\Sigma_{\ell}^{-1})W_{\ell}$. This allows us to bound $\frac{1}{d}\text{Tr}(\Sigma_{\ell+1}^{\frac{1}{2}}\bar{\Sigma}_{n+1,\ell}^{-1}\Sigma_{\ell+1}^{\frac{1}{2}})$ as

$$ 
\begin{aligned}
\frac{1}{d} \text{Tr}(\Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1,\ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}}) &= \frac{1}{d} \text{Tr}(I_d + \Sigma_{\ell+1}^{\frac{1}{2}} W_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} \Sigma_{\ell}^{-1}) W_{\ell} \Sigma_{\ell+1}^{\frac{1}{2}}), \\
&= \frac{1}{d} (d + \text{Tr}(\Sigma_{\ell+1}^{\frac{1}{2}} W_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} \Sigma_{\ell}^{-1}) W_{\ell} \Sigma_{\ell+1}^{\frac{1}{2}}), \\
&\le 1 + \frac{1}{d} \sum_{k=1}^{d} \lambda_1 (\Sigma_{\ell+1}^{\frac{1}{2}} W_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} \Sigma_{\ell}^{-1}) W_{\ell} \Sigma_{\ell+1}^{\frac{1}{2}}, \\
&\le 1 + \frac{1}{d} \sum_{k=1}^{d} \lambda_1 (\Sigma_{\ell+1}) \lambda_1 (W_{\ell}^{\top} W_{\ell}) \lambda_1 (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} \Sigma_{\ell}^{-1}), \\
&\le 1 + \frac{1}{d} \sum_{k=1}^{d} \lambda_1 (\Sigma_{\ell+1}) \lambda_1 (W_{\ell}^{\top} W_{\ell}) \lambda_1 (\Sigma_{\ell}^{-1}), \\
&\le 1 + \frac{1}{d} \sum_{k=1}^{d} \frac{\sigma_{\ell+1}^2}{\sigma_{\ell}^2} = 1 + \frac{\sigma_{\ell+1}^2}{\sigma_{\ell}^2},
\end{aligned}
\quad (40) $$

where we use the assumption that $\lambda_1(W_{\ell}^{\top}W_{\ell}) = 1$ (A2) and that $\lambda_1(\Sigma_{\ell+1}) = \sigma_{\ell+1}^2$ and $\lambda_1(\Sigma_{\ell}^{-1}) = 1/\sigma_{\ell}^2$. This is because $\Sigma_{\ell} = \sigma_{\ell}^2 I_d$ for any $\ell \in [L+1]$. Finally, plugging Eqs. (37) and (38) in Eq. (36) concludes the proof.

## D.6 Proof of proposition 4.2

We use exactly the same proof in Appendix D.5, with one change to account for the sparsity assumption (A3). The change corresponds to Eq. (38). First, recall that Eq. (38) writes

$$ \sum_{t=1}^{n} \log(1 + \sigma^{-2} X_t^{\top} P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^{\top} X_t) \leq \sigma_{\text{MAX}}^{2\ell} \left( \log \det \left( \Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1, \ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}} \right) \right), $$

where

$$ 
\begin{aligned}
\Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1,\ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}} &= I_d + \Sigma_{\ell+1}^{\frac{1}{2}} W_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} \Sigma_{\ell}^{-1}) W_{\ell} \Sigma_{\ell+1}^{\frac{1}{2}}, \\
&= I_d + \sigma_{\ell+1}^2 W_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} \Sigma_{\ell}^{-1}) W_{\ell},
\end{aligned}
\quad (41) $$

where the second equality follows from the assumption that $\Sigma_{\ell+1} = \sigma_{\ell+1}^2 I_d$. But notice that in our assumption, (A3), we assume that $W_{\ell} = (\bar{W}_{\ell}, 0_{d,d-d_{\ell}})$, where $\bar{W}_{\ell} \in \mathbb{R}^{d \times d_{\ell}}$ for any $\ell \in [L]$. Therefore, we have that for any $d \times d$ matrix $B \in \mathbb{R}^{d \times d}$, the following holds, $W_{\ell}^{\top} BW_{\ell} = \begin{pmatrix} \bar{W}_{\ell}^{\top} B \bar{W}_{\ell} & 0_{d_{\ell},d-d_{\ell}} \\ 0_{d-d_{\ell},d_{\ell}} & 0_{d-d_{\ell},d-d_{\ell}} \end{pmatrix}$. In particular, we have that

$$ W_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t, \ell-1} \Sigma_{\ell}^{-1}) W_{\ell} = \begin{pmatrix} \bar{W}_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t, \ell-1} \Sigma_{\ell}^{-1}) \bar{W}_{\ell} & 0_{d_{\ell}, d-d_{\ell}} \\ 0_{d-d_{\ell}, d_{\ell}} & 0_{d-d_{\ell}, d-d_{\ell}} \end{pmatrix}. \quad (42) $$

Therefore, plugging this in Eq. (41) yields that

$$ \Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1, \ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}} = \begin{pmatrix} I_{d_{\ell}} + \sigma_{\ell+1}^2 \bar{W}_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t, \ell-1} \Sigma_{\ell}^{-1}) \bar{W}_{\ell} & 0_{d_{\ell}, d-d_{\ell}} \\ 0_{d-d_{\ell}, d_{\ell}} & I_{d-d_{\ell}} \end{pmatrix}. \quad (43) $$

As a result, $\det(\Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1,\ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}}) = \det(I_{d_{\ell}} + \sigma_{\ell+1}^2 \bar{W}_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} \Sigma_{\ell}^{-1}) \bar{W}_{\ell})$. This allows us to move the problem from a $d$-dimensional one to a $d_{\ell}$-dimensional one. Then we use the inequality
since $\bar{G}_{t,l} = W_{\ell}^{\top}(\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1}\bar{\Sigma}_{t,l-1}\Sigma_{\ell}^{-1})W_{\ell}$. This allows us to bound $\frac{1}{d}\text{Tr}(\Sigma_{\ell+1}^{\frac{1}{2}}\bar{\Sigma}_{n+1,l}^{-1}\Sigma_{\ell+1}^{\frac{1}{2}})$ as

$$
\begin{align}
\frac{1}{d} \operatorname{Tr}(\Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1,\ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}}) &= \frac{1}{d} \operatorname{Tr}(I_d + \Sigma_{\ell+1}^{\frac{1}{2}} W_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} \Sigma_{\ell}^{-1}) W_{\ell} \Sigma_{\ell+1}^{\frac{1}{2}}), \nonumber \\
&= \frac{1}{d} (d + \operatorname{Tr}(\Sigma_{\ell+1}^{\frac{1}{2}} W_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} \Sigma_{\ell}^{-1}) W_{\ell} \Sigma_{\ell+1}^{\frac{1}{2}}), \nonumber \\
&\le 1 + \frac{1}{d} \sum_{k=1}^{d} \lambda_1 (\Sigma_{\ell+1}^{\frac{1}{2}} W_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} \Sigma_{\ell}^{-1}) W_{\ell} \Sigma_{\ell+1}^{\frac{1}{2}}, \nonumber \\
&\le 1 + \frac{1}{d} \sum_{k=1}^{d} \lambda_1 (\Sigma_{\ell+1}) \lambda_1 (W_{\ell}^{\top} W_{\ell}) \lambda_1 (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} \Sigma_{\ell}^{-1}), \nonumber \\
&\le 1 + \frac{1}{d} \sum_{k=1}^{d} \lambda_1 (\Sigma_{\ell+1}) \lambda_1 (W_{\ell}^{\top} W_{\ell}) \lambda_1 (\Sigma_{\ell}^{-1}), \nonumber \\
&\le 1 + \frac{1}{d} \sum_{k=1}^{d} \frac{\sigma_{\ell+1}^2}{\sigma_{\ell}^2} = 1 + \frac{\sigma_{\ell+1}^2}{\sigma_{\ell}^2}, \tag{40}
\end{align}
$$

where we use the assumption that $\lambda_1(W_\ell^\top W_\ell) = 1$ (A2) and that $\lambda_1(\Sigma_{\ell+1}) = \sigma_{\ell+1}^2$ and $\lambda_1(\Sigma_\ell^{-1}) = 1/\sigma_\ell^2$. This is because $\Sigma_\ell = \sigma_\ell^2 I_d$ for any $\ell \in [L+1]$. Finally, plugging Eqs. (37) and (38) in Eq. (36) concludes the proof.

## D.6 Proof of proposition 4.2

We use exactly the same proof in Appendix D.5, with one change to account for the sparsity assumption (A3). The change corresponds to Eq. (38). First, recall that Eq. (38) writes

$$ \sum_{t=1}^{n} \log(1 + \sigma^{-2} X_t^{\top} P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^{\top} X_t) \leq \sigma_{\text{MAX}}^{2\ell} \left( \log \det \left( \Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1, \ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}} \right) \right), $$

where

$$
\begin{align}
\Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1,\ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}} &= I_d + \Sigma_{\ell+1}^{\frac{1}{2}} W_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} \Sigma_{\ell}^{-1}) W_{\ell} \Sigma_{\ell+1}^{\frac{1}{2}}, \nonumber \\
&= I_d + \sigma_{\ell+1}^2 W_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t,\ell-1} \Sigma_{\ell}^{-1}) W_{\ell}, \tag{41}
\end{align}
$$

where the second equality follows from the assumption that $\Sigma_{\ell+1} = \sigma_{\ell+1}^2 I_d$. But notice that in our assumption, (A3), we assume that $W_\ell = (\bar{W}_\ell, 0_{d,d-d_\ell})$, where $\bar{W}_\ell \in \mathbb{R}^{d \times d_\ell}$ for any $\ell \in [L]$. Therefore, we have that for any $d \times d$ matrix $B \in \mathbb{R}^{d \times d}$, the following holds, $W_\ell^\top B W_\ell = \begin{pmatrix} \bar{W}_\ell^\top B \bar{W}_\ell & 0_{d_\ell, d-d_\ell} \\ 0_{d-d_\ell, d_\ell} & 0_{d-d_\ell, d-d_\ell} \end{pmatrix}$. In particular, we have that

$$ W_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t, \ell-1} \Sigma_{\ell}^{-1}) W_{\ell} = \begin{pmatrix} \bar{W}_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t, \ell-1} \Sigma_{\ell}^{-1}) \bar{W}_{\ell} & 0_{d_{\ell}, d-d_{\ell}} \\ 0_{d-d_{\ell}, d_{\ell}} & 0_{d-d_{\ell}, d-d_{\ell}} \end{pmatrix}. \quad (42) $$

Therefore, plugging this in Eq. (41) yields that

$$ \Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1, \ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}} = \begin{pmatrix} I_{d_{\ell}} + \sigma_{\ell+1}^2 \bar{W}_{\ell}^{\top} (\Sigma_{\ell}^{-1} - \Sigma_{\ell}^{-1} \bar{\Sigma}_{t, \ell-1} \Sigma_{\ell}^{-1}) \bar{W}_{\ell} & 0_{d_{\ell}, d-d_{\ell}} \\ 0_{d-d_{\ell}, d_{\ell}} & I_{d-d_{\ell}} \end{pmatrix}. \quad (43) $$

As a result, $\det(\Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1,\ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}}) = \det(I_{d_\ell} + \sigma_{\ell+1}^2 \bar{W}_\ell^\top (\Sigma_\ell^{-1} - \Sigma_\ell^{-1} \bar{\Sigma}_{t,\ell-1} \Sigma_\ell^{-1}) \bar{W}_\ell)$. This allows us to move the problem from a $d$-dimensional one to a $d_\ell$-dimensional one. Then we use the inequality
of arithmetic and geometric means and get that

$$
\begin{align}
\sum_{t=1}^{n} \log(1 + \sigma^{-2} X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^\top X_t) &\le \sigma_{\text{MAX}}^{2\ell} \left( \log \det \left( \Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1, \ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}} \right) \right), \nonumber \\
&= \sigma_{\text{MAX}}^{2\ell} \log \det (I_{d_\ell} + \sigma_{\ell+1}^2 \bar{W}_\ell^\top (\Sigma_\ell^{-1} - \Sigma_\ell^{-1} \bar{\Sigma}_{t, \ell-1} \Sigma_\ell^{-1}) \bar{W}_\ell), \nonumber \\
&\le d_\ell \sigma_{\text{MAX}}^{2\ell} \log \left( \frac{1}{d_\ell} \operatorname{Tr} (I_{d_\ell} + \sigma_{\ell+1}^2 \bar{W}_\ell^\top (\Sigma_\ell^{-1} - \Sigma_\ell^{-1} \bar{\Sigma}_{t, \ell-1} \Sigma_\ell^{-1}) \bar{W}_\ell) \right), \nonumber \\
&\le d_\ell \sigma_{\text{MAX}}^{2\ell} \log \left( 1 + \frac{\sigma_{\ell+1}^2}{\sigma_\ell^2} \right). \tag{44}
\end{align}
$$

To get the last inequality, we use derivations similar to the ones we used in Eq. (40). Finally, the desired result in obtained by replacing Eq. (38) by Eq. (44) in the previous proof in Appendix D.5.

D.7 Additional discussion: link to two-level hierarchies

The linear diffusion (15) can be marginalized into a 2-level hierarchy using two different strategies.
The first one yields,

$$
\begin{align}
\psi_{*,L} &\sim \mathcal{N}(0, \sigma_{L+1}^2 B_L B_L^\top), \tag{45} \\
\theta_{*,i} | \psi_{*,L} &\sim \mathcal{N}(\psi_{*,L}, \Omega_1), \quad \forall i \in [K], \nonumber
\end{align}
$$

with $\Omega_1 = \sigma_1^2 I_d + \sum_{\ell=1}^{L-1} \sigma_{\ell+1}^2 B_\ell B_\ell^\top$ and $B_\ell = \prod_{k=1}^\ell W_k$. The second strategy yields,

$$
\begin{align}
\psi_{*,1} &\sim \mathcal{N}(0, \Omega_2), \tag{46} \\
\theta_{*,i} | \psi_{*,1} &\sim \mathcal{N}(\psi_{*,1}, \sigma_1^2 I_d), \quad \forall i \in [K], \nonumber
\end{align}
$$

where $\Omega_2 = \sum_{\ell=1}^L \sigma_{\ell+1}^2 B_\ell B_\ell^\top$. Recently, HierTS [Hong et al., 2022b] was developed for such two-level graphical models, and we call HierTS under (45) by HierTS-1 and HierTS under (46) by HierTS-2. Then, we start by highlighting the differences between these two variants of HierTS. First, their regret bounds scale as

$$
\text{HierTS-1: } \tilde{O}\left(\sqrt{nd\left(K \sum_{\ell=1}^{L} \sigma_{\ell}^{2}+L \sigma_{L+1}^{2}\right)}\right), \quad \text{HierTS-2: } \tilde{O}\left(\sqrt{nd\left(K \sigma_{1}^{2}+\sum_{\ell=1}^{L} \sigma_{\ell+1}^{2}\right)}\right).
$$

When $K \approx L$, the regret bounds of HierTS-1 and HierTS-2 are similar. However, when $K > L$, HierTS-2 outperforms HierTS-1. This is because HierTS-2 puts more uncertainty on a single $d$-dimensional latent parameter $\psi_{*,1}$, rather than $K$ individual $d$-dimensional action parameters $\theta_{*,j}$. More importantly, HierTS-1 implicitly assumes that action parameters $\theta_{*,j}$ are conditionally independent given $\psi_{*,L}$, which is not true. Consequently, HierTS-2 outperforms HierTS-1. Note that, under the linear diffusion model (15), dTS and HierTS-2 have roughly similar regret bounds. Specifically, their regret bounds dependency on $K$ is identical, where both methods involve multiplying $K$ by $\sigma_1^2$, and both enjoy improved performance compared to HierTS-1. That said, note that Theorem 4.1 and Proposition 4.2 provide an understanding of how dTS’s regret scales under linear score functions $f_\ell$, and do not say that using dTS is better than using HierTS when the score functions $f_\ell$ are linear since the latter can be obtained by a proper marginalization of latent parameters (i.e., HierTS-2 instead of HierTS-1). While such a comparison is not the goal of this work, we still provide it for completeness next.

When the mixing matrices $W_\ell$ are dense (i.e., assumption (A3) is not applicable), dTS and HierTS-2 have comparable regret bounds and computational efficiency. However, under the sparsity assumption (A3) and with mixing matrices that allow for conditional independence of $\psi_{*,1}$ coordinates given $\psi_{*,2}$, dTS enjoys a computational advantage over HierTS-2. This advantage explains why works focusing on multi-level hierarchies typically benchmark their algorithms against two-level structures akin to HierTS-1, rather than the more competitive HierTS-2. This is also consistent with prior works in Bayesian bandits using multi-level hierarchies, such as Tree-based priors [Hong et al., 2022a], which compared their method to HierTS-1. In line with this, we also compared dTS with HierTS-1 in our experiments. But this is only given for completeness as this is not the aim of Theorem 4.1 and Proposition 4.2. More importantly, HierTS is inapplicable in the general case in (1) with non-linear score functions since the latent parameters cannot be analytically marginalized.
of arithmetic and geometric means and get that

$$
\begin{align}
\sum_{t=1}^{n} \log(1 + \sigma^{-2} X_t^\top P_{A_t, \ell} \bar{\Sigma}_{t, \ell} P_{A_t, \ell}^\top X_t) &\le \sigma_{\text{MAX}}^{2\ell} \left( \log \det \left( \Sigma_{\ell+1}^{\frac{1}{2}} \bar{\Sigma}_{n+1, \ell}^{-1} \Sigma_{\ell+1}^{\frac{1}{2}} \right) \right), \nonumber \\
&= \sigma_{\text{MAX}}^{2\ell} \log \det (I_{d_\ell} + \sigma_{\ell+1}^2 \bar{W}_\ell^\top (\Sigma_\ell^{-1} - \Sigma_\ell^{-1} \bar{\Sigma}_{t, \ell-1} \Sigma_\ell^{-1}) \bar{W}_\ell), \nonumber \\
&\le d_\ell \sigma_{\text{MAX}}^{2\ell} \log \left( \frac{1}{d_\ell} \operatorname{Tr} (I_{d_\ell} + \sigma_{\ell+1}^2 \bar{W}_\ell^\top (\Sigma_\ell^{-1} - \Sigma_\ell^{-1} \bar{\Sigma}_{t, \ell-1} \Sigma_\ell^{-1}) \bar{W}_\ell) \right), \nonumber \\
&\le d_\ell \sigma_{\text{MAX}}^{2\ell} \log \left( 1 + \frac{\sigma_{\ell+1}^2}{\sigma_\ell^2} \right). \tag{44}
\end{align}
$$

To get the last inequality, we use derivations similar to the ones we used in Eq. (40). Finally, the desired result in obtained by replacing Eq. (38) by Eq. (44) in the previous proof in Appendix D.5.

D.7 Additional discussion: link to two-level hierarchies

The linear diffusion (15) can be marginalized into a 2-level hierarchy using two different strategies.
The first one yields,

$$
\begin{align}
\psi_{*,L} &\sim \mathcal{N}(0, \sigma_{L+1}^2 B_L B_L^\top), \tag{45} \\
\theta_{*,i} | \psi_{*,L} &\sim \mathcal{N}(\psi_{*,L}, \Omega_1), \quad \forall i \in [K], \nonumber
\end{align}
$$

with $\Omega_1 = \sigma_1^2 I_d + \sum_{\ell=1}^{L-1} \sigma_{\ell+1}^2 B_\ell B_\ell^\top$ and $B_\ell = \prod_{k=1}^\ell W_k$. The second strategy yields,

$$
\begin{align}
\psi_{*,1} &\sim \mathcal{N}(0, \Omega_2), \tag{46} \\
\theta_{*,i} | \psi_{*,1} &\sim \mathcal{N}(\psi_{*,1}, \sigma_1^2 I_d), \quad \forall i \in [K], \nonumber
\end{align}
$$

where $\Omega_2 = \sum_{\ell=1}^L \sigma_{\ell+1}^2 B_\ell B_\ell^\top$. Recently, HierTS [Hong et al., 2022b] was developed for such two-level graphical models, and we call HierTS under (45) by HierTS-1 and HierTS under (46) by HierTS-2. Then, we start by highlighting the differences between these two variants of HierTS. First, their regret bounds scale as

$$
\text{HierTS-1: } \tilde{\mathcal{O}}\left(\sqrt{nd\left(K \sum_{\ell=1}^{L} \sigma_{\ell}^{2}+L \sigma_{L+1}^{2}\right)}\right), \quad \text{HierTS-2: } \tilde{\mathcal{O}}\left(\sqrt{nd\left(K \sigma_{1}^{2}+\sum_{\ell=1}^{L} \sigma_{\ell+1}^{2}\right)}\right).
$$

When $K \approx L$, the regret bounds of HierTS-1 and HierTS-2 are similar. However, when $K > L$, HierTS-2 outperforms HierTS-1. This is because HierTS-2 puts more uncertainty on a single $d$-dimensional latent parameter $\psi_{*,1}$, rather than $K$ individual $d$-dimensional action parameters $\theta_{*,j}$. More importantly, HierTS-1 implicitly assumes that action parameters $\theta_{*,j}$ are conditionally independent given $\psi_{*,L}$, which is not true. Consequently, HierTS-2 outperforms HierTS-1. Note that, under the linear diffusion model (15), dTS and HierTS-2 have roughly similar regret bounds. Specifically, their regret bounds dependency on $K$ is identical, where both methods involve multiplying $K$ by $\sigma_1^2$, and both enjoy improved performance compared to HierTS-1. That said, note that Theorem 4.1 and Proposition 4.2 provide an understanding of how dTS’s regret scales under linear score functions $f_\ell$, and do not say that using dTS is better than using HierTS when the score functions $f_\ell$ are linear since the latter can be obtained by a proper marginalization of latent parameters (i.e., HierTS-2 instead of HierTS-1). While such a comparison is not the goal of this work, we still provide it for completeness next.

When the mixing matrices $W_\ell$ are dense (i.e., assumption (A3) is not applicable), dTS and HierTS-2 have comparable regret bounds and computational efficiency. However, under the sparsity assumption (A3) and with mixing matrices that allow for conditional independence of $\psi_{*,1}$ coordinates given $\psi_{*,2}$, dTS enjoys a computational advantage over HierTS-2. This advantage explains why works focusing on multi-level hierarchies typically benchmark their algorithms against two-level structures akin to HierTS-1, rather than the more competitive HierTS-2. This is also consistent with prior works in Bayesian bandits using multi-level hierarchies, such as Tree-based priors [Hong et al., 2022a], which compared their method to HierTS-1. In line with this, we also compared dTS with HierTS-1 in our experiments. But this is only given for completeness as this is not the aim of Theorem 4.1 and Proposition 4.2. More importantly, HierTS is inapplicable in the general case in (1) with non-linear score functions since the latent parameters cannot be analytically marginalized.
E Broader impact

This work contributes to the development and analysis of practical algorithms for online 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 domain. Nevertheless, we acknowledge the crucial need to consider potential biases that may be present in pre-trained diffusion models, given that our method relies on them.

F Limitations

Our work investigated contextual bandits, laying the groundwork for future exploration into reinforce-
ment learning. This exploration can be done from both practical (empirical) and theoretical angles.
While our method, which approximates rewards using a Gaussian distribution, worked well for linear
rewards and those following a generalized linear model, its effectiveness in real-world, complex
scenarios needs further testing. Another interesting direction for future research is pre-training the
diffusion model prior. Hsieh et al. [2023] proposed a method for this in multi-armed bandits, but its
application to contextual bandits remains unexplored.

G Amount of computation required

Our experiments were conducted on internal machines with 30 CPUs and thus they required a moder-
ate amount of computation. These experiments are also reproducible with minimal computational
resources.
E Broader impact

This work contributes to the development and analysis of practical algorithms for online 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 domain. Nevertheless, we acknowledge the crucial need to consider potential biases that may be present in pre-trained diffusion models, given that our method relies on them.

F Limitations

Our work investigated contextual bandits, laying the groundwork for future exploration into reinforce-
ment learning. This exploration can be done from both practical (empirical) and theoretical angles.
While our method, which approximates rewards using a Gaussian distribution, worked well for linear
rewards and those following a generalized linear model, its effectiveness in real-world, complex
scenarios needs further testing. Another interesting direction for future research is pre-training the
diffusion model prior. Hsieh et al. [2023] proposed a method for this in multi-armed bandits, but its
application to contextual bandits remains unexplored.

G Amount of computation required

Our experiments were conducted on internal machines with 30 CPUs and thus they required a moder-
ate amount of computation. These experiments are also reproducible with minimal computational
resources.
# NeurIPS Paper Checklist

## 1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: All claims are supported by the theory in Section 4 (with proofs provided in the appendix) and experiments in Section 5.

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* The answer NA means that the abstract and introduction do not include the claims made in the paper.

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* It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

## 2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

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* The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

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* While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

## 3. Theory Assumptions and Proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]
# NeurIPS Paper Checklist

## 1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: All claims are supported by the theory in Section 4 (with proofs provided in the appendix) and experiments in Section 5.

Guidelines:

* The answer NA means that the abstract and introduction do not include the claims made in the paper.

* The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

* The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

* It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

## 2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: Limitations were discussed in Section 6 and Appendix F.

Guidelines:

* The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

* The authors are encouraged to create a separate "Limitations" section in their paper.

* The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

* The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

* The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

* The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

* If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

* While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

## 3. Theory Assumptions and Proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]
Justification: Assumptions are mentioned in the main text. Complete proofs are provided in the appendix.

Guidelines:

* The answer NA means that the paper does not include theoretical results.

* All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.

* All assumptions should be clearly stated or referenced in the statement of any theorems.

* The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

* Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

* Theorems and Lemmas that the proof relies upon should be properly referenced.

4. **Experimental Result Reproducibility**

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

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Justification: Information needed to reproduce the main experimental results of the paper is described in Section 5.

Guidelines:

* The answer NA means that the paper does not include experiments.

* If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

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* Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

* While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. **Open access to data and code**
Justification: Assumptions are mentioned in the main text. Complete proofs are provided in the appendix.

Guidelines:

* The answer NA means that the paper does not include theoretical results.

* All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.

* All assumptions should be clearly stated or referenced in the statement of any theorems.

* The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

* Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

* Theorems and Lemmas that the proof relies upon should be properly referenced.

4. **Experimental Result Reproducibility**

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: Information needed to reproduce the main experimental results of the paper is described in Section 5.

Guidelines:

* The answer NA means that the paper does not include experiments.

* If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

* If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

* Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

* While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. **Open access to data and code**
Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: The code for the main experiments is shared in the supplementary material.

Guidelines:

* The answer NA means that the paper does not include experiments requiring code.

* Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.

* While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

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* The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

* The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

* At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

* Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

## 6. Experimental Setting/Details

Question: Does the paper specify all the training and test details (e.g., data splits, hyper-parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: All experimental details are described in Section 5.

Guidelines:

* The answer NA means that the paper does not include experiments.

* The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

* The full details can be provided either with the code, in appendix, or as supplemental material.

## 7. Experiment Statistical Significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: Standard error bars are included in the figures.

Guidelines:

* The answer NA means that the paper does not include experiments.

* The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

* The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

* The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: The code for the main experiments is shared in the supplementary material.

Guidelines:

* The answer NA means that the paper does not include experiments requiring code.

* Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.

* While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

* The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.

* The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

* The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

* At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

* Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

## 6. Experimental Setting/Details

Question: Does the paper specify all the training and test details (e.g., data splits, hyper-parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: All experimental details are described in Section 5.

Guidelines:

* The answer NA means that the paper does not include experiments.

* The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

* The full details can be provided either with the code, in appendix, or as supplemental material.

## 7. Experiment Statistical Significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: Standard error bars are included in the figures.

Guidelines:

* The answer NA means that the paper does not include experiments.

* The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

* The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

* The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
* The assumptions made should be given (e.g., Normally distributed errors).

* It should be clear whether the error bar is the standard deviation or the standard error of the mean.

* It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

* For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

* If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

## 8. Experiments Compute Resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: As mentioned in Appendix G, 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.

Guidelines:

* The answer NA means that the paper does not include experiments.

* The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

* The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

* The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

## 9. Code Of Ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?

Answer: [Yes]

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

Guidelines:

* 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 E.

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.
* The assumptions made should be given (e.g., Normally distributed errors).

* It should be clear whether the error bar is the standard deviation or the standard error of the mean.

* It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

* For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

* If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

## 8. Experiments Compute Resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: As mentioned in Appendix G, 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.

Guidelines:

* The answer NA means that the paper does not include experiments.

* The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

* The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

* The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

## 9. Code Of Ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?

Answer: [Yes]

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

Guidelines:

* 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 E.

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 simulated. 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 simulated. 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 5.

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 5.

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.

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

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