ence\_vt\] Let $v_t$ be as defined in , initialized at $v_0$. Then for any $T=n\delta$, $$\begin{aligned}
        \E{\lrn{v_T - v_0}_2^2} \leq T^2 L^2 \E{\lrn{v_0}_2^2} + T\beta^2
        \end{aligned}$$ If we additionally assume that $\E{\lrn{v_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$ and $T \leq \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}$, then $$\begin{aligned}
        \E{\lrn{v_T - v_0}_2^2} \leq 2 T\beta^2
        \end{aligned}$$

From , $$\begin{aligned}
    v_T - v_0 = - T \nabla U(v_0) + \sqrt{\delta} \sum_{i=0}^{n-1} \xi(v_0, \eta_i)
    \end{aligned}$$ Conditioned on the randomness up to time $i$, $\E{\xi(v_0,\eta_{i+1})}=0$. Thus $$\begin{aligned}
    & \E{\lrn{v_T - v_0}_2^2}\\
    =& T^2\E{\lrn{\nabla U(v_0)}_2^2} + \delta \sum_{i=0}^{n-1} \E{\lrn{\xi(v_0,\eta_i)}_2^2}\\
    \leq& T^2 L^2 \E{\lrn{v_0}_2^2} + T \beta^2
    \end{aligned}$$ where the inequality is by item 1 of Assumption \[ass:U\_properties\] and item 2 of Assumption \[ass:xi\_properties\].

\[l:divergence\_wt\] Let $w_t$ be as defined in , initialized at $w_0$. Then for any $T=n\delta$ such that $T \leq \frac{1}{2L}$, $$\begin{aligned}
        \E{\lrn{w_T - w_0}_2^2} \leq 16 \lrp{T^2 L^2 \E{\lrn{w_0}_2^2} + T \beta^2}
        \end{aligned}$$ If we additionally assume that $\E{\lrn{w_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$ and $T \leq \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}$, then $$\begin{aligned}
        \E{\lrn{w_T - w_0}_2^2} \leq 32 T\beta^2
        \end{aligned}$$

$$\begin{aligned}
        & \E{\lrn{w_{(k+1)\delta} - w_0}_2^2}\\
        =& \E{\lrn{w_{k\delta} - \delta \nabla U(w_{k\delta}) + \sqrt{\delta} \xi\lrp{w_{k\delta}, \eta_k} - w_0}_2^2}\\
        =& \E{\lrn{w_{k\delta} - \delta \nabla U(w_{k\delta}) - w_0}_2^2}  + \delta \E{\lrn{\xi\lrp{w_{k\delta}, \eta_k}}_2^2} \numberthis \label{e:t:sfddd}
        \end{aligned}$$

We can bound $\delta \E{\lrn{\xi\lrp{w_{k\delta}, \eta_k}}_2^2} \leq \delta \beta^2$ by item 2 of Assumption \[ass:xi\_properties\]. $$\begin{aligned}
        & \E{\lrn{w_{k\delta} - \delta \nabla U(w_{k\delta})