^d, \|v\|_2 =1} \lrn{Gv}_2.
        $.

Finally, we define a few useful constants which will be used throughout the paper: $$\begin{aligned}
            &\LN := \frac{4\beta L_\xi}{\cm}, \ \  \aq:=\frac{\LR+L_N^2}{2\cm^2},  \\ &\Rq:=\max\lrbb{R,{\frac{16\beta^2 L_N}{m\cdot \cm}}}  \\
            &\lambda :=\min\lrbb{\frac{m}{2}, \frac{2\cm^2}{32\Rq^2}}\exp\lrp{-\frac{7}{3}\aq\Rq^2}.
            \numberthis \label{d:constants}
            \end{aligned}$$

$\LN$ is the smoothness parameter of the matrix $N(x)$, and we show in Lemma \[l:N\_is\_regular\] that $\tr\lrp{\lrp{N(x) - N(y)}^2} \leq \LN^2\lrn{x-y}_2^2$. The constants $\aq$ and $\Rq$ are used to define a Lyapunov function $q$ in Appendix \[ss:defining-q\]. A key step in our proof uses the fact that, under the dynamics , $q$ contracts at a rate of $e^{-\lambda}$, plus discretization error.

[Main Results]{}\[s:main\_results\] In this section, we present our main convergence results beginning with convergence under Gaussian noise and proceeding to the non-Gaussian case.

\[t:main\_gaussian\] Let $x_t$ and $y_t$ have dynamics as defined in and respectively, and suppose that the initial conditions satisfy $\E{\lrn{x_0}_2^2}\leq R^2 + \beta^2/m$ and $\E{\lrn{y_0}_2^2}\leq R^2 + \beta^2/m$. Let $\hat{\epsilon}$ be a target accuracy satisfying $\hat{\epsilon} \leq  \lrp{\frac{16\lrp{L + \LN^2}}{\lambda}} \cdot \exp\lrp{7\aq\Rq/3} \cdot \frac{\Rq}{\aq\Rq^2 + 1}$. Let $\delta$ be a step size satisfying $$\begin{aligned}
        \delta \leq \min\twocase{\frac{\lambda^2 \hat{\epsilon}^2}{512 \beta^2\lrp{L^2 + \LN^4}\exp\lrp{\frac{14\aq\Rq^2}{3}}}}{\frac{2\lambda \hat{\epsilon}}{(L^2+\LN^4)\exp\lrp{\frac{7\aq\Rq^2}{3}}\sqrt{R^2 + \beta^2/m}}}.
        \end{aligned}$$

If we assume that $x_0 = y_0$, then there exists a coupling between $x_t$ and $y_t$ such that for any $k$, $$\begin{aligned}
        \E{\lrn{x_{k\delta} - y_{k\delta}}_2} \leq \hat{\epsilon}
        \end{aligned}$$

Alternatively, if we assume $n \geq \frac{ 3\aq\Rq^2}{\delta} \log \frac{R^2 + \beta^2/m}{