\section{General Notations and Results}
This section presents some general notations and results for the logistic function that would be used throughout the Appendix. For a matrix $\bm{A}$, let $\eigmax{\bm{A}}$ and $\eigmin{\bm{A}}$ denote the maximum and minimum eigenvalue of $\bm{A}$ respectively. Similarly, we define $\singmax{\bm{A}}$ and $\singmin{\bm{A}}$ to be the maximum and minimum singular values respectively. 
We also define the following functions, borrowed from \cite{Faury2022}:

\begin{enumerate}
    \item $\gamma_t(\delta) = O(S^2Nd\log(t/\delta))$
    \item $\beta_t(\delta) = O(S^6 Nd \log(t/\delta))$
    \item $\eta_t(\delta) = O(S^2Nd\log(t/\delta))$
\end{enumerate}

\begin{claim}
    Let $\mu:\R\rightarrow\R$ be the logistic function, i.e., $\mu(x) = 1/(1+\exp(-x))$ and $\dot{\mu}, \ddot{\mu}$ be the first and second derivative of $\mu$. The following are true.
    \begin{enumerate}
        \item $|\ddot{\mu}(x)| \leq \dot{\mu}(x)$, $\forall x\in \R$
        \item $\dot{\mu}(x) \leq \dot{\mu}(y) \exp(|x-y|)$, $\forall x,y\in \R$
    \end{enumerate}
\end{claim}

\begin{definition}
    Let $\dot{\mu}$ be the derivative of the logistic function. Define functions $\alpha:\R\times \R\rightarrow \R$ and $\tilde\alpha:\R\times \R\rightarrow\R$ as follows.
    \begin{enumerate}
        \item $\alpha(x,y) = \int\limits_{0}^1 \dot{\mu}(x+v(y-x)) \diff v$
        \item $\tilde\alpha(x,y) = \int\limits_0^1 (1-v)\dot{\mu}(x+v(y-x)) \diff v$ 
    \end{enumerate}
\end{definition}

\begin{definition}
    (Exact Taylor Expansion for the Logistic Function) The logistic function $\mu(x)$ can be expanded using an Exact Taylor Expansion as follows:
    $$\mu(x) = \mu(y) + \dot{\mu}(y) (x-y) +  \int\limits_0^1 (1-v)\ddot{\mu}(x+v(y-x)) \diff v (x-y)^2$$
    \label{def:Exact_Taylor}
\end{definition}

\begin{definition}
    (Mean Value Theorem for the Logistic Function) The logistic function $\mu$ can be expanded using the Mean Value theorem as follows:
    \[\mu(x) = \mu(y) + \alpha(x,y) (x-y)\]
\end{definition}

Recall the following notations from Section \ref{section:main-algo}:

\begin{enumerate}
    \item $\bm{W}_t = \bm I + \sum\limits_{s=1}^{t-1} \dot{\mu}(\bm{x}_s^\top\bm\theta_{s+1}) \bm{x}_s \bm{x}_s^\top$
    \item $\bm{W}_t^i = \bm{I} + \sum\limits_{s=1}^{t-1} \dot{\mu}(\bm{x}_s^\top\bm\theta_{s+1})\bm{x}_s^i {\bm{x}_s^i}^\top$
    \item $\bm{V}_t^{\mathcal{H}} = \gamma_t(\delta)\bm{I} + \sum\limits_{\bm{x}\in \mathcal{H}_t} \bm{x}\bm{x}^\top/\kappa$
\end{enumerate}

We define the following additional matrices.
\begin{enumerate}
    \item $\bm{U}_t = \textrm{\textrm{diag}}(\bm{W}_t^1, \ldots, \bm{W}_t^N)$
    \item $\bm{W}_t^{i,j} = \sum\limits_{s=1}^{t-1} \dot{\mu}(\bm{x}_s^\top\bm\theta_{s+1})\bm{x}_s^i {\bm{x}_s^j}^\top$
    \item $\bm{V}_t^{\mathcal{H},i} = \gamma_t(\delta)\bm{I} + \sum\limits_{\bm{x}\in \mathcal{H}_t} \bm{x}^i{\bm{x}^i}^\top/\kappa$
    \item $\bm{V}_t^{\mathcal{H},i,j} = \gamma_t(\delta)\bm{I} + \sum\limits_{\bm{x}\in \mathcal{H}_t} \bm{x}^i{\bm{x}^j}^\top/\kappa$ 
    \item $\bm{U}^\H_t = \textrm{diag}(\bm{V}_t^{\mathcal{H},1}, \ldots, \bm{V}_t^{\mathcal{H},N})$
\end{enumerate}

\input{Sections/appendix_sections/appendix_OFU}
\input{Sections/appendix_sections/appendix_TS}
\input{Sections/appendix_sections/appendix_general}

\section{Other Useful Results and Definitions}
\begin{definition}
    (Hermitian Dilation) The Hermitian matrix for a matrix $\bm{A}$ is defined as 
    \begin{align*}
        \H(\bm{A}) = \begin{bmatrix}
            \mathbf{0} & \bm{A}\\
            \bm{A}^\top & \mathbf{0}
        \end{bmatrix}
    \end{align*}
    \label{def:HermitianDilation}
\end{definition}

\begin{lemma}
    (\cite{Das_2024}, Lemma 16) Let $\bm{\H(\bm{Z}}) = \begin{bmatrix}
        \mathbf{0} & \bm{Z}\\
        \bm{Z}^\top & \mathbf{0}
    \end{bmatrix}$ where $\bm{Z}$ has positive singular values.
    Then, it holds almost surely, $\eigmax{\H(\bm{Z})} = -\eigmin{\H(\bm{Z})} = \singmax{\bm{Z}}$
    \label{lemma: max_min_eig_hermitian}
\end{lemma}

\begin{lemma}
    (Matrix Freedman Inequality \cite{Tropp2011b} Corollary 1.3) Define a matrix martingale $\bm{Z}_s \in \R^{d_1\times d_2}$ with respect to the filtration  $\filteration{s}$ and a martingale difference sequence $\bm{X}_s = \bm{Z}_s - \bm{Z}_{s-1}$. Assume that the difference sequence is almost surely uniformly bounded, i.e $\norm{\bm{X}_s} \leq R$. Define the quantities
    $$\bm{W}_{row,t} = \summation{s=1}{t}\E\sbrak{\bm{X}_s\bm{X}_s^\top | \filteration{s-1}}$$
    $$\bm{W}_{col,t} = \summation{s=1}{t}\E\sbrak{\bm{X}_s^\top\bm{X}_s | \filteration{s-1}}$$
Then, for all $u \geq 0$ and $\omega^2 > 0$, we have

$$\P\cbrak{\exists t \geq 0: \norm{\bm{Z}_t} \geq u \text{ and } \max\cbrak{\norm{\bm{W}_{row,t}} , \norm{\bm{W}_{row,t}}}\leq \omega^2} \leq (d_1 + d_2) \exp\pbrak{-\frac{u^2/2}{\omega^2 + Ru/3}}$$
\label{lemma: freedman inequality}
\end{lemma}

\begin{lemma}
    (Matrix Azuma Inequality, \cite{Tropp_2011a}, Theorem 7.1) Let $\cbrak{\bm{X}_s}_{s=1}^\infty$ be a matrix martingale difference sequence in $\R^{d_1\times d_2}$ and let $\H(\bm{X}_s)$ represent the Hermitian Dilation (see def. \ref{def:HermitianDilation}) of $\bm{X}_s$. Let $\cbrak{\bm{A}_s}_{s=1}^\infty$ be a sequence of matrices in $\R^{(d_1+d_2) \times(d_1 + d_2)}$ such that $\E\sbrak{\bm{X}_s|\filteration{s-1}} = \mathbf{0}$ and $\H(\bm{X}_s)^2 \mleq \bm{A}_s^2$. Let $\sigma_t^2 = \lambda_{max}{\summation{s=1}{t}\bm{A}_k^2}$ for $t\geq 1$. Then, for all $\epsilon \geq 0$:
    $$\P\cbrak{\exists t\geq1: \singmax{\summation{s=1}{t}\bm{X}_s} \geq \epsilon} \leq (d_1+d_2)\exp\pbrak{-\frac{\epsilon^2}{8\sigma_t^2}}$$
    \label{lemma: Matrix Azuma}
\end{lemma}

\begin{lemma}
    (Elliptical Potential Lemma, \cite{Yadkori2011}, Lemma 11) Let $\cbrak{\bm{x}_s}_{s=1}^\top$ represent a set of vectors in $\R^d$ and let $\twonorm{\bm{x}_s} \leq L$. Let $\bm{V}_s = \lambda\bm{I}_d + \summation{m=1}{s-1}\bm{x}_m\bm{x}_m^\top$. Then, for $\lambda \geq 1$
    $$\summation{s=1}{t}\matnorm{\bm{x}_s}{\bm{V}_s\inv}^2 \leq 2d\log\pbrak{1 + \frac{tL^2}{\lambda d}} \leq 4d\log({tL^2})$$
    \label{lemma: elliptical potential lemma}
\end{lemma}

\begin{lemma}
    (\cite{Abeille2021}, Proposition 7) Let $b , c \geq 0$ and $x^2 - bx - c \leq 0$. Then, $x^2 \leq 2b^2 + 2c$.
    \label{lemma: quadratic inequality}
\end{lemma}
\begin{proof} Since the coefficient of the quadratic term is 1, the quadratic expression can attain non-positive values only if it has two distinct or equal real roots. We denote the roots by $\alpha_1$ and $\alpha_2$. Without loss of generality, assume $\alpha_1 = \frac{b - \sqrt{b^2 + 4c}}{2}$ and $\alpha_2 = \frac{b + \sqrt{b^2 + 4c}}{2}$. Then, the set of $x$ for which $x^2-bx-c \leq 0$ is true is $x \in \sbrak{\alpha_1 , \alpha_2}$. Thus, we can say
$$x \leq 
\alpha_2 = \frac{b + \sqrt{b^2 + 4c}}{2} \leq b + \sqrt{c}$$
using the fact that $\sqrt{a+b}\leq \sqrt{a} + \sqrt{b}$ for $a,b \geq 0$.
Finally, 
$$x^2 \leq b^2 + c + 2b\sqrt{c} \leq 2b^2 + 2c$$
using the fact that $(b - \sqrt{c})^2 \geq 0 \implies 2b\sqrt{c} \leq b^2 + c$ 

\end{proof}


\begin{definition}
    (Multivariate distribution for Thompson Sampling, \cite{Abeille2017}, Definition 1) $\mathcal{D}^{TS}$ is a suitable multivariate distribution on $\R^d$ for Thompson Sampling if it is absolutely continuous with respect to the Lebesgue measure and satisfies the following properties:
    \begin{enumerate}
        \item Concentration: There exist constants $c$ and $c^{'}$ such that $\forall \delta \in (0,1)$
        $$\P_{\bm{\eta} \sim\mathcal{D}^{TS}}\cbrak{\twonorm{\bm{\eta}} \leq \sqrt{cd\log \frac{c^{'}d}{\delta}}} \geq 1 -\delta$$

    \item Anti-Concentration: There exists a strictly positive probability $p$ such that for any $\bm{u} \in \R^d$
    $$\P_{\bm{\eta} \sim \mathcal{D}^{TS}} \cbrak{\bm{u}^\top \bm{\eta} \geq \twonorm{\bm{u}}} \geq p$$
    \end{enumerate}
\label{def: D_TS}
\end{definition}

\input{Sections/appendix_sections/appendix_experiments}