SlowGuess/NeurIPS_2023_OCR huggingface dataset:
As explained in Section 3, one of the main themes in proving error bounds for sampling is the natural relation between sampling and optimization in the Wasserstein space. This point of view, when applied to strongly-convex potentials, has produced numerous non-asymptotic guarantees; see [14, 18] for a recent account and the references therein. Note that strong convexity is crucial for the analysis used in the aforementioned work. Moreover, the error bounds for biased and noisy discretizations do not decrease with the step-size or iteration count; see [18, Theorem 4, Eqn. (14)]. This means that while the bound is non-asymptotic, it does not automatically result in an asymptotic convergence. Finally, we stress that these approaches are orthogonal to our techniques: We view a sampling algorithm as a (noisy and biased) discretization of a dynamical system (and not necessarily a gradient flow), and use tools from dynamical system theory to provide asymptotic convergence results.
C Proofs for Section 4
C.1 Proof of Theorem 1
In this appendix, we bring the detailed proof of Theorem 1. Recall that we interpolate the iterates of the LRM scheme  $\{x_{k}\}$  as
$$
X _ {t} = x _ {k} + \left(t - \tau_ {k}\right) \left\{v \left(x _ {k}\right) + \mathbb {E} \left[ Z _ {k + 1} \mid \mathcal {F} _ {t} \right] \right\} + \sigma \left(x _ {k}\right) \left(B _ {t} - B _ {\tau_ {k}}\right). \tag {3}
$$
Moreover, for a fixed  $t > 0$ , we considered the Brownian motion  $B_{s}^{(t)} = B_{t + s} - B_{t}$ , and constructed two important processes: the Langevin flow defined via
$$
\mathrm {d} \Phi_ {s} ^ {(t)} = v \left(\Phi_ {s} ^ {(t)}\right) \mathrm {d} s + \sigma \left(\Phi_ {s} ^ {(t)}\right) \mathrm {d} B _ {s} ^ {(t)}, \quad \Phi_ {0} ^ {(t)} = X _ {t}, \tag {12}
$$
and the Picard process (6) constructed as
$$
Y _ {s} ^ {(t)} = X _ {t} + \int_ {0} ^ {s} v \left(X _ {t + u}\right) \mathrm {d} u + \int_ {0} ^ {s} \sigma \left(X _ {t + u}\right) \mathrm {d} B _ {u} ^ {(t)}. \tag {6}
$$
Let us fix  $T > 0$ , and for  $s \in [0, T]$  decompose the distance between the interpolation and the Langevin flow as
$$
\frac {1}{2} \left\| X _ {t + s} - \Phi_ {s} ^ {(t)} \right\| ^ {2} \leq \left\| Y _ {s} ^ {(t)} - \Phi_ {s} ^ {(t)} \right\| ^ {2} + \left\| X _ {t + s} - Y _ {s} ^ {(t)} \right\| ^ {2}, \tag {7}
$$
where we have used  $\| a + b\| ^2\leq 2\| a\| ^2 +2\| b\| ^2$ . We now bound each term of this decomposition. Notice that due to the synchronous coupling of the processes, the Brownian motion cancels out in the differences.
The first term controls how close the Picard process is to the Langevin flow, and is bounded in the following lemma.
Lemma 3. For fixed  $t, T > 0$  and  $0 \leq s \leq T$ , the distance of the Picard process and the Langevin flow is bounded as
$$
\| Y _ {s} ^ {(t)} - \Phi_ {s} ^ {(t)} \| ^ {2} \leq 2 (T + 1) L ^ {2} \int_ {0} ^ {s} \| \Phi_ {u} ^ {(t)} - X _ {t + u} \| ^ {2} \mathrm {d} u.
$$
Proof. By the auxiliary Lemma 4 below, Lipschitzness of  $v, \sigma$ , Itô isometry (see, e.g., [62]) and  $s \leq T$ , we have
$$
\begin{array}{l} \mathbb {E} \| Y _ {s} ^ {(t)} - \Phi_ {s} ^ {(t)} \| ^ {2} = \mathbb {E} \left\| \int_ {0} ^ {s} v (\Phi_ {u} ^ {(t)}) - v (X _ {t + u}) \mathrm {d} u + \int_ {0} ^ {s} \sigma (\Phi_ {u} ^ {(t)}) - \sigma (X _ {t + u}) \mathrm {d} B _ {u} ^ {(t)} \right\| ^ {2} \\ \leq 2 s \int_ {0} ^ {s} \mathbb {E} \left\| v \left(\Phi_ {u} ^ {(t)}\right) - v \left(X _ {t + u}\right) \right\| ^ {2} d u + 2 \mathbb {E} \int_ {0} ^ {s} \left\| \sigma \left(X _ {t + u}\right) - \sigma \left(\Phi_ {u} ^ {(t)}\right) \right\| _ {F} ^ {2} d u \\ \leq 2 (T + 1) L ^ {2} \int_ {0} ^ {s} \mathbb {E} \left\| \Phi_ {u} ^ {(t)} - X _ {t + u} \right\| ^ {2} d u. \\ \end{array}
$$
For the rest of the proof, we need to define the continuous-time piecewise-constant processes  $\overline{X}(\tau_k + s) = X_k$ ,  $\overline{\gamma}(\tau_k + s) = \gamma_{k+1}$ ,  $\overline{Z}(\tau_k + s) = Z_{k+1}$ , and  $Z(\tau_k + s) = \mathbb{E}[Z_{k+1} | \mathcal{F}_{\tau_k + s}]$ , for  $0 \leq s < \gamma_{k+1}$ . Also, let  $m(t) = \sup \{k \geq 0 : \tau_k \leq t\}$  so that  $\tau_{m(t)} \leq t < \tau_{m(t) + 1}$ .
To bound the second term in (7), we have seen that
$$
\begin{array}{l} X _ {t + s} - Y _ {s} ^ {(t)} = \int_ {t} ^ {t + s} v (\bar {X} (u)) d u - \int_ {0} ^ {s} v (X _ {t + u}) d u \\ + \int_ {t} ^ {t + s} \sigma (\bar {X} (u)) d B _ {u} - \int_ {0} ^ {s} \sigma (X _ {t + u}) d B _ {u} ^ {(t)} \\ + \Delta_ {Z} (t, s), \\ \end{array}
$$
where  $\Delta_Z(t,s)$  plays the role of accumulated noise and bias from time  $t$  to  $t + s$ , and is defined as
$$
\Delta_ {Z} (t, s) := \sum_ {i = n} ^ {k - 1} \gamma_ {i + 1} Z _ {i + 1} + (t + s - \tau_ {k}) \mathbb {E} \left[ Z _ {k + 1} \mid \mathcal {F} _ {t + s} \right] - (t - \tau_ {n}) \mathbb {E} \left[ Z _ {n + 1} \mid \mathcal {F} _ {t} \right], \tag {13}
$$
with  $k = m(t + s)$  and  $n = m(t)$ . We therefore have
$$
\begin{array}{l} \mathbb {E} \left\| X _ {t + s} - Y _ {s} ^ {(t)} \right\| ^ {2} \leq 3 \mathbb {E} \left\| \int_ {t} ^ {t + s} v (X _ {u}) - v (\overline {{X}} (u)) \mathrm {d} u \right\| ^ {2} \\ + 3 \mathbb {E} \left\| \int_ {t} ^ {t + s} \sigma \left(X _ {u}\right) - \sigma (\bar {X} (u)) \mathrm {d} B _ {u} \right\| ^ {2} + 3 \mathbb {E} \| \Delta_ {Z} (t, s) \| ^ {2} \\ \leq 3 s \int_ {t} ^ {t + s} \mathbb {E} \left\| v \left(X _ {u}\right) - v (\bar {X} (u)) \right\| ^ {2} d u \\ + 3 \mathbb {E} \int_ {t} ^ {t + s} \left\| \sigma (X _ {u}) - \sigma (\overline {{X}} (u)) \right\| _ {F} ^ {2} d u + 3 \mathbb {E} \| \Delta_ {Z} (t, s) \| ^ {2} \\ \leq 3 (s + 1) L ^ {2} \int_ {t} ^ {t + s} \mathbb {E} \| X _ {u} - \bar {X} (u) \| ^ {2} \mathrm {d} u + 3 \mathbb {E} \| \Delta_ {Z} (t, s) \| ^ {2}. \tag {14} \\ \end{array}
$$
For bounding the term inside the integral, we have
$$
\begin{array}{l} \mathbb {E} \| X _ {u} - \bar {X} (u) \| ^ {2} = \mathbb {E} \| (u - \tau_ {m (u)}) \{v (\bar {X} (u)) + Z (u) \} + \sigma (\bar {X} (u)) \left(B _ {u} - B _ {\tau_ {m (u)}}\right) \| ^ {2} \\ \leq 4 \bar {\gamma} (u) ^ {2} \left(\mathbb {E} \| v (\bar {X} (u)) \| ^ {2} + \mathbb {E} \| Z (u) \| ^ {2}\right) + 2 \bar {\gamma} (u) \mathbb {E} \operatorname {t r} \left(\sigma (\bar {X} (u)) ^ {\top} \sigma (\bar {X} (u))\right). \\ \end{array}
$$
We have used the fact that
$$
\begin{array}{l} \mathbb {E} \| \sigma (\bar {X} (u)) (B _ {u} - B _ {\tau_ {m (u)}}) \| ^ {2} = \mathbb {E} \left((B _ {u} - B _ {\tau_ {m (u)}}) ^ {\top} \sigma (\bar {X} (u)) ^ {\top} \sigma (\bar {X} (u)) (B _ {u} - B _ {\tau_ {m (u)}})\right) \\ = \mathbb {E} \operatorname {t r} \left(\sigma (\bar {X} (u)) ^ {\top} \sigma (\bar {X} (u)) \left(B _ {u} - B _ {\tau_ {m (u)}}\right) \left(B _ {u} - B _ {\tau_ {m (u)}}\right) ^ {\top}\right) \\ = \mathbb {E} \left[ \mathbb {E} \left[ \operatorname {t r} \left(\sigma (\bar {X} (u)) ^ {\top} \sigma (\bar {X} (u)) \left(B _ {u} - B _ {\tau_ {m (u)}}\right) \left(B _ {u} - B _ {\tau_ {m (u)}}\right) ^ {\top}\right) \mid \mathscr {F} _ {\tau_ {m (u)}} \right] \right] \\ = (u - \tau_ {m (u)}) \mathbb {E} \left[ \operatorname {t r} \left(\sigma (\bar {X} (u)) ^ {\top} \sigma (\bar {X} (u))\right) \right] \\ \end{array}
$$
Notice that since conditional expectation is a projection in  $L^2$ , we have  $\mathbb{E}\| Z(u)\|^2 \leq \mathbb{E}\|\overline{Z}(u)\|^2$ . Using this fact, along with boundedness of  $\sigma(\cdot)$  by  $C_\sigma$ , and Lemma 2 we get
$$
\begin{array}{l} \mathbb {E} \left[ \| X _ {u} - \bar {X} (u) \| ^ {2} \right] \leq 4 \bar {\gamma} (u) ^ {2} \left(\mathbb {E} \| v (\bar {X} (u)) \| ^ {2} + \mathbb {E} \| \bar {Z} (u) \| ^ {2}\right) + 2 \bar {\gamma} (u) \mathbb {E} \operatorname {t r} \left(\sigma (\bar {X} (u)) ^ {\top} \sigma (\bar {X} (u))\right) \\ \leq 4 \bar {\gamma} (u) ^ {2} \mathbb {E} \| v (\bar {X} (u)) \| ^ {2} + 8 \bar {\gamma} (u) ^ {2} \sigma^ {2} + 4 \bar {\gamma} (u) ^ {2} \mathcal {O} (\bar {\gamma} (u)) + 2 C _ {\sigma} \bar {\gamma} (u) \leq C \bar {\gamma} (u), \\ \end{array}
$$
for some constant  $C > 0$ . Plugging this estimate into (14) after taking expectation yields
$$
\begin{array}{l} \mathbb {E} \left[ \left\| X _ {t + s} - Y _ {s} ^ {(t)} \right\| ^ {2} \right] \leq 3 (s + 1) L ^ {2} C \int_ {t} ^ {t + s} \bar {\gamma} (u) d u + 3 \mathbb {E} \| \Delta_ {Z} (t, s) \| ^ {2} \\ \leq 3(s + 1)sL^{2}C\sup_{u\in [t,t + s]}\overline{\gamma} (u) + 3\mathbb{E}\| \Delta_{Z}(t,s)\|^{2} \\ \leq 3 (T + 1) ^ {2} L ^ {2} C \sup  _ {u \in [ t, t + T ]} \overline {{\gamma}} (u) + 3 \sup  _ {u \in [ 0, T ]} \mathbb {E} \| \Delta_ {Z} (t, u) \| ^ {2} \\ \end{array}
$$
Taking supremum over  $s \in [0, T]$  and noticing that the right-hand-side is independent of  $s$  and  $\gamma_k \to 0$ , together with Lemma 1 yields
$$
\begin{array}{l} A _ {t} := \sup  _ {0 \leq s \leq T} \mathbb {E} \left[ \| X _ {t + s} - Y _ {s} ^ {(t)} \| ^ {2} \right] \tag {15} \\ \leq 3 (T + 1) ^ {2} L ^ {2} C \sup  _ {t \leq u \leq t + T} \overline {{\gamma}} (u) + 3 \sup  _ {0 \leq u \leq T} \mathbb {E} \left[ \| \Delta_ {Z} (t, u) \| ^ {2} \right] \\ \rightarrow 0 \quad \text {a s} t \rightarrow \infty , \\ \end{array}
$$
showing that the Picard process gets arbitrary close to the original interpolation, as  $t \to \infty$ .
Let us return to the decomposition (7). By taking expectation and using (8) and (15) we obtain
$$
\begin{array}{l} \mathbb {E} \left[ \| X _ {t + s} - \Phi_ {s} ^ {(t)} \| ^ {2} \right] \leq 2 (T + 1) L ^ {2} \int_ {0} ^ {s} \mathbb {E} \left[ \| X _ {t + u} - \Phi_ {u} ^ {(t)} \| ^ {2} \right] \mathrm {d} u + 2 A _ {t} \\ \leq 2 A _ {t} \exp (s (T + 1) L ^ {2}) \\ \leq 2 A _ {t} \exp \left(\left(T + 1\right) ^ {2} L ^ {2}\right), \\ \end{array}
$$
where in the last line we have used the Gronwall lemma. Thus,
Recall that the Wasserstein distance between  $X_{t+s}$  and  $\Phi_s^{(t)}$  is the infimum over all possible couplings between them, having the correct marginals. As  $\Phi_s^{(t)}$  has the same marginal as the Langevin diffusion started from  $X_t$  at time  $s$ , and the synchronous coupling of the interpolation and the Langevin flow produces a specific coupling between them, we directly get
$$
W _ {2} \left(X _ {t + s}, \Phi_ {s} ^ {(t)}\right) \leq \mathbb {E} \left[ \left\| X _ {t + s} - \Phi_ {s} ^ {(t)} \right\| ^ {2} \right] ^ {\frac {1}{2}},
$$
$$
\lim_{t\to \infty}\sup_{s\in [0,T]}W_{2}(X_{t + s},\Phi_{s}^{(t)}) = 0,
$$
C.2 Auxiliary Lemmas
Proof. Define  $\Delta_b$  and  $\Delta_U$  the same way as in (13). By Cauchy-Schwarz we have
$$
\begin{array}{l} \left\| \Delta_ {b} (t, s) \right\| ^ {2} \\ \leq \left(\sum_ {i = n} ^ {k - 1} \gamma_ {i + 1} \| b _ {i + 1} \| + (t + s - \tau_ {k}) \| \mathbb {E} [ b _ {k + 1} | \mathcal {F} _ {t + s} ] \| + (t - \tau_ {n}) \| \mathbb {E} [ b _ {n + 1} | \mathcal {F} _ {t} ] \|\right) ^ {2} \\ \leq \left(2 \gamma_ {n + 1} + s\right) \left(\sum_ {i = n} ^ {k - 1} \gamma_ {i + 1} \| b _ {i + 1} \| ^ {2} + (t + s - \tau_ {k}) \| \mathbb {E} \left[ b _ {k + 1} \mid \mathcal {F} _ {t + s} \right] \| ^ {2} + (t - \tau_ {n}) \| \mathbb {E} \left[ b _ {n + 1} \mid \mathcal {F} _ {t} \right] \| ^ {2}\right), \\ \end{array}
$$
where the last inequality comes from  $\sum_{i=n}^{k-1} \gamma_{i+1} \leq s, t+s-\tau_k \leq \gamma_{k+1}, t-\tau_n \leq \gamma_{n+1}$ , and  $\gamma_{k+1} \leq \gamma_{n+1}$ .
Noticing that conditional expectation is a contraction in  $L^2$  and letting  $k' = m(t + T)$ , we get
$$
\sup  _ {0 \leq s \leq T} \mathbb {E} \left[ \| \Delta_ {b} (t, s) \| ^ {2} \right] \leq (2 + T) \left(\sum_ {i = n} ^ {k ^ {\prime} - 1} \gamma_ {i + 1} \mathbb {E} \| b _ {i + 1} \| ^ {2} + \sup  _ {n \leq j \leq k ^ {\prime} + 1} \gamma_ {j + 1} \mathbb {E} \| b _ {j + 1} \| ^ {2} + \gamma_ {n + 1} \mathbb {E} \| b _ {n + 1} \| ^ {2}\right)
$$
Now, invoking Lemma 2 yields
$$
\begin{array}{l} \sup  _ {0 \leq s \leq T} \mathbb {E} \left[ \| \Delta_ {b} (t, s) \| ^ {2} \right] \leq C (2 + T) \left(\sum_ {i = n} ^ {k ^ {\prime} - 1} \gamma_ {i + 1} ^ {2} + \sup  _ {n \leq j \leq k ^ {\prime} + 1} \gamma_ {j + 1} ^ {2} + \gamma_ {n + 1} ^ {2}\right) \\ \leq C (2 + T) \left(\sum_ {i = n} ^ {k ^ {\prime} - 1} \gamma_ {i + 1} ^ {2} + 2 \gamma_ {n + 1} ^ {2}\right) \\ \leq C(2 + T)(T + 2\gamma_{n + 1})\sup_{0\leq s\leq T}\overline{\gamma} (t + s). \\ \end{array}
$$
As  $t\to \infty$  , the last quantity vanishes, since  $\gamma_{n}\rightarrow 0$
For the noise we have
$$
\begin{array}{l} \| \Delta_ {U} (t, s) \| ^ {2} \leq 2 \left\| \sum_ {i = n} ^ {k - 1} \gamma_ {i + 1} U _ {i + 1} \right\| ^ {2} + 4 \| (t + s - \tau_ {k}) \mathbb {E} [ U _ {k + 1} | \mathcal {F} _ {t + s} ] \| ^ {2} + 4 \| (t - \tau_ {n}) \mathbb {E} [ U _ {n + 1} | \mathcal {F} _ {t} ] \| ^ {2} \\ \leq 2 \left\| \sum_ {i = n} ^ {k - 1} \gamma_ {i + 1} U _ {i + 1} \right\| ^ {2} + 4 \gamma_ {k + 1} ^ {2} \| U _ {k + 1} \| ^ {2} + 4 \gamma_ {n + 1} ^ {2} \| U _ {n + 1} \| ^ {2}. \\ \end{array}
$$
Taking expectations and then sup, we get
$$
\sup  _ {0 \leq s \leq T} \mathbb {E} \left[ \| \Delta_ {U} (t, s) \| ^ {2} \right] \leq 2 \sup  _ {n + 1 \leq k \leq m (t + T)} \mathbb {E} \left\| \sum_ {i = n} ^ {k - 1} \gamma_ {i + 1} U _ {i + 1} \right\| ^ {2} + 4 \gamma_ {k + 1} ^ {2} \sigma^ {2} + 4 \gamma_ {n + 1} ^ {2} \sigma^ {2}.
$$
Since  $\{U_i\}$  is a martingale difference sequence, we have that  $\left\{\sum_{i=n}^{k-1} \gamma_{i+1} U_{i+1}\right\}_{k > n}$  is a martingale. Thus, by the boundedness of the second moments of  $U_i$ , we get
$$
\mathbb {E} \left\| \sum_ {i = n} ^ {k - 1} \gamma_ {i + 1} U _ {i + 1} \right\| ^ {2} = \sum_ {i = n} ^ {k - 1} \gamma_ {i + 1} ^ {2} \mathbb {E} \| U _ {i + 1} \| ^ {2} \leq \sigma^ {2} \sum_ {i = n} ^ {k - 1} \gamma_ {i + 1} ^ {2}.
$$
$$
\lim  _ {n \rightarrow \infty} \sup  \left\{\mathbb {E} \| \sum_ {i = n} ^ {k - 1} \gamma_ {i + 1} U _ {i + 1} \| ^ {2}: n <   k \leq m (\tau_ {n} + T) \right\} \leq \lim  _ {n \rightarrow \infty} \sigma^ {2} \sum_ {i = n} ^ {\infty} \gamma_ {i + 1} ^ {2} = 0.
$$
Proof. Without loss of generality, suppose  $v$  has a stationary point at 0. We repeatedly use the fact that  $\mathbb{E}\|v(x_k)\|^2 \leq L^2\mathbb{E}\|x_k\|^2$ . Moreover, by Assumption 1 we have  $\langle v(x), x \rangle \leq C_v(\|x\| + 1)$ , and  $\| \sigma(x)\|_F^2 \leq C_\sigma$ .
Define  $a_{k}\coloneqq \mathbb{E}\| x_{k}\|^{2}$  .We have
$$
\begin{array}{l} a _ {k + 1} - a _ {k} = \gamma_ {k + 1} ^ {2} \mathbb {E} \| v (x _ {k}) + Z _ {k + 1} \| ^ {2} + \gamma_ {k + 1} \mathbb {E} \| \sigma (x _ {k}) \xi_ {k + 1} \| ^ {2} + 2 \gamma_ {k + 1} \mathbb {E} \langle x _ {k}, v (x _ {k}) + Z _ {k + 1} \rangle \\ + 2 \gamma_ {k + 1} ^ {1 / 2} \mathbb {E} \langle x _ {k}, \sigma (x _ {k}) \xi_ {k + 1} \rangle + 2 \gamma_ {k + 1} ^ {3 / 2} \mathbb {E} \langle v (x _ {k}) + Z _ {k + 1}, \sigma (x _ {k}) \xi_ {k + 1} \rangle \\ \leq 2 L ^ {2} \gamma_ {k + 1} ^ {2} a _ {k} + 2 \gamma_ {k + 1} ^ {2} \mathbb {E} \| Z _ {k + 1} \| ^ {2} + \gamma_ {k + 1} C _ {\sigma} + 2 \gamma_ {k + 1} C _ {v} (\sqrt {a _ {k}} + 1) + 2 \gamma_ {k + 1} \sqrt {a _ {k}} \sqrt {\mathbb {E} \| Z _ {k + 1} \| ^ {2}} \\ + 2 \gamma_ {k + 1} ^ {3 / 2} \sqrt {C _ {\sigma}} \sqrt {\mathbb {E} \| Z _ {k + 1} \| ^ {2}} \tag {16} \\ \end{array}
$$
By Assumption 3, there is some  $C_b > 0$  such that  $\mathbb{E}\| b_{k + 1}\| ^2\leq C_b(\gamma_{k + 1}^2 a_k + \gamma_{k + 1})$ , and we have
$$
\mathbb {E} \| Z _ {k + 1} \| ^ {2} \leq 2 \mathbb {E} \| b _ {k + 1} \| ^ {2} + 2 \mathbb {E} \| U _ {k + 1} \| ^ {2} \leq 2 C _ {b} \left(\gamma_ {k + 1} ^ {2} a _ {k} + \gamma_ {k + 1}\right) + 2 \sigma^ {2}. \tag {17}
$$
Moreover, as  $\sqrt{p + q} \leq \sqrt{p} + \sqrt{q}$ , we have
$$
\sqrt {\mathbb {E} \left\| Z _ {k + 1} \right\| ^ {2}} \leq \sqrt {2 C _ {b}} \left(\gamma_ {k + 1} \sqrt {a _ {k}} + \sqrt {\gamma_ {k + 1}}\right) + \sqrt {2} \sigma . \tag {18}
$$
Plugging the bounds from (17) and (18) into (16) gives
$$
\begin{array}{l} a _ {k + 1} - a _ {k} \leq 2 L ^ {2} \gamma_ {k + 1} ^ {2} a _ {k} + 4 C _ {b} \gamma_ {k + 1} ^ {4} a _ {k} + 4 C _ {b} \gamma_ {k + 1} ^ {3} + 4 \gamma_ {k + 1} ^ {2} \sigma^ {2} \\ + \gamma_ {k + 1} C _ {\sigma} + 2 \gamma_ {k + 1} C _ {v} \sqrt {a _ {k}} + 2 \gamma_ {k + 1} C _ {v} \\ + 2 \sqrt {2 C _ {b}} \gamma_ {k + 1} ^ {2} a _ {k} + 2 \sqrt {2 C _ {b}} \gamma_ {k + 1} ^ {3 / 2} \sqrt {a _ {k}} + 2 \sqrt {2} \sigma \gamma_ {k + 1} \sqrt {a _ {k}} \tag {19} \\ + 2 \sqrt {2 C _ {b} C _ {\sigma}} \gamma_ {k + 1} ^ {5 / 2} \sqrt {a _ {k}} + 2 \sqrt {2 C _ {b} C _ {\sigma}} \gamma_ {k + 1} ^ {2} + 2 \gamma_ {k + 1} ^ {3 / 2} \sqrt {2 C _ {\sigma}} \sigma \\ =: P \gamma_ {k + 1} ^ {2} a _ {k} + Q \gamma_ {k + 1} \sqrt {a _ {k}} + R \gamma_ {k + 1}, \\ \end{array}
$$
$$
P = 2 L ^ {2} + 4 C _ {b} \gamma_ {k + 1} ^ {2} + 2 \sqrt {2 C _ {b}}
$$
$$
Q = 2 C _ {v} + 2 \sqrt {2 C _ {b}} \sqrt {\gamma_ {k + 1}} + 2 \sqrt {2} \sigma + 2 \sqrt {2 C _ {b}} \gamma_ {k + 1} + 2 \sqrt {2 C _ {b} C _ {\sigma}} \gamma_ {k + 1} ^ {3 / 2}
$$
$$
R = 4 C _ {b} \gamma_ {k + 1} ^ {2} + 4 \gamma_ {k + 1} \sigma^ {2} + C _ {\sigma} + 2 C _ {v} + 2 \sqrt {2 C _ {b} C _ {\sigma}} \gamma_ {k + 1} + 2 \gamma_ {k + 1} ^ {1 / 2} \sqrt {2 C _ {\sigma}} \sigma .
$$
The exact values of  $P$ ,  $Q$ , and  $R$  are irrelevant, and we only need upper bounds for them. Assuming that  $\gamma_{k+1} < 1$  for all  $k$ , we replace the three quantities by
$$
P = 2 L ^ {2} + 4 C _ {b} + 2 \sqrt {2 C _ {b}}
$$
$$
Q = 2 C _ {v} + 2 \sqrt {2 C _ {b}} + 2 \sqrt {2} \sigma + 2 \sqrt {2 C _ {b}} + 2 \sqrt {2 C _ {b} C _ {\sigma}} \tag {20}
$$
$$
R = 4 C _ {b} + 4 \sigma^ {2} + C _ {\sigma} + 2 C _ {v} + 2 \sqrt {2 C _ {b} C _ {\sigma}} + 2 \sqrt {2 C _ {\sigma}} \sigma .
$$
Now, define  $h_k = \gamma_{k+1}^2 a_k$ . The recursion (19) in terms of  $h_k$  becomes
$$
h _ {k + 1} \leq h _ {k} \left(1 + P \gamma_ {k + 1} ^ {2}\right) \frac {\gamma_ {k + 2} ^ {2}}{\gamma_ {k + 1} ^ {2}} + \sqrt {h _ {k}} Q \gamma_ {k + 2} ^ {2} + R \gamma_ {k + 1} \gamma_ {k + 2} ^ {2}.
$$
We now prove that there exists some  $M > 0$  so that  $h_k \leq M\gamma_{k+1}$  by induction. Suppose it is the case for  $k$ , and we prove it for  $k+1$ . Using the induction hypothesis we get
$$
\begin{array}{l} h _ {k + 1} \leq M \gamma_ {k + 1} (1 + P \gamma_ {k + 1} ^ {2}) \frac {\gamma_ {k + 2} ^ {2}}{\gamma_ {k + 1} ^ {2}} + \sqrt {M \gamma_ {k + 1}} Q \gamma_ {k + 2} ^ {2} + R \gamma_ {k + 1} \gamma_ {k + 2} ^ {2} \\ = M (1 + P \gamma_ {k + 1} ^ {2}) \frac {\gamma_ {k + 2} ^ {2}}{\gamma_ {k + 1}} + \sqrt {M} Q \sqrt {\gamma_ {k + 1}} \gamma_ {k + 2} ^ {2} + R \gamma_ {k + 1} \gamma_ {k + 2} ^ {2} \\ \end{array}
$$
For the last to be less than  $M\gamma_{k + 2}$ , we have to verify
$$
M \left(1 + P \gamma_ {k + 1} ^ {2}\right) \frac {\gamma_ {k + 2}}{\gamma_ {k + 1}} + \sqrt {M} Q \sqrt {\gamma_ {k + 1}} \gamma_ {k + 2} + R \gamma_ {k + 1} \gamma_ {k + 2} \leq M
$$
$$
M \left(\frac {\gamma_ {k + 2}}{\gamma_ {k + 1}} + P \gamma_ {k + 1} \gamma_ {k + 2} - 1\right) + \sqrt {M} Q \sqrt {\gamma_ {k + 1}} \gamma_ {k + 2} + R \gamma_ {k + 1} \gamma_ {k + 2} \leq 0.
$$
This is a quadratic equation in  $\sqrt{M}$ , and for this inequality to hold, we prove that the leading coefficient is negative, and the largest root is bounded above by some constant not depending on  $n$ .
Negativity of the leading coefficient is equivalent to
$$
\frac {\gamma_ {k + 2}}{\gamma_ {k + 1}} + P \gamma_ {k + 1} \gamma_ {k + 2} <   1,
$$
which is implied by our assumption on the step size.
The larger root of the equation is
$$
\begin{array}{l} \frac {\left(- 4 \gamma_ {k + 1} ^ {2} \gamma_ {k + 2} ^ {2} P R + \gamma_ {k + 1} \gamma_ {k + 2} \left(\gamma_ {k + 2} Q ^ {2} + 4 R\right) - 4 R \gamma_ {k + 2} ^ {2}\right) ^ {1 / 2} + \sqrt {\gamma_ {k + 1}} \gamma_ {k + 2} Q}{2 \left(1 - \gamma_ {k + 1} \gamma_ {k + 2} P - \gamma_ {k + 2} / \gamma_ {k + 1}\right)} \\ <   \frac {\sqrt {\gamma_ {k + 1}} \gamma_ {k + 2} Q + \sqrt {R \gamma_ {k + 1} \gamma_ {k + 2}}}{(1 - \gamma_ {k + 1} \gamma_ {k + 2} P - \gamma_ {k + 2} / \gamma_ {k + 1})} \\ \leq \frac {\sqrt {\gamma_ {k + 1}} \gamma_ {k + 1} Q + \sqrt {R} \gamma_ {k + 1}}{\left(1 - \gamma_ {k + 1} \gamma_ {k + 2} P - \gamma_ {k + 2} / \gamma_ {k + 1}\right)}. \\ \end{array}
$$
By our assumption on the step size that
$$
\frac {\gamma_ {k + 2}}{\gamma_ {k + 1}} + P \gamma_ {k + 1} \gamma_ {k + 2} <   1 - \gamma_ {k + 1},
$$
we get that the larger root is smaller than
$$
\frac {\sqrt {\gamma_ {k + 1}} \gamma_ {k + 1} Q + \sqrt {R} \gamma_ {k + 1}}{\gamma_ {k + 1}} = \sqrt {\gamma_ {k + 1}} Q + \sqrt {R} <   Q + \sqrt {R}.
$$
Letting  $M \coloneqq Q + \sqrt{R}$  gives the desired result.
The second argument of the lemma follows from Assumption 3 and the first result of the lemma.
Lemma 4. For a vector valued function  $g\in L^{2}(\mathbb{R};\mathbb{R}^{d})$  , one has
$$
\left\| \int_ {0} ^ {s} g (u) d u \right\| ^ {2} \leq \left(\int_ {0} ^ {s} \| g (u) \| d u\right) ^ {2} \leq s \int_ {0} ^ {s} \| g (u) \| ^ {2} d u.
$$
D Proofs for Section 5
D.1 Proof of Theorem 3
For brevity, let us write  $\mathcal{F}_k$  instead of  $\mathcal{F}_{\tau_k}$ . Opening up  $\| x_{k + 1}\| ^2 = \| x_k + \gamma_{k + 1}\{v(x_k) + Z_{k + 1}\} +$ $\sqrt{\gamma_{k + 1}}\sigma (x_k)\xi_{k + 1}\| ^2$  and ignoring every term that is zero-mean under  $\mathbb{E}[\cdot |\mathcal{F}_k]$ , we get
$$
\begin{array}{l} \mathbb {E} \left[ \| x _ {k + 1} \| ^ {2} \mid \mathcal {F} _ {k} \right] = \mathbb {E} \left[ \| x _ {k} \| ^ {2} + 2 \gamma_ {k + 1} \langle x _ {k}, v (x _ {k}) + Z _ {k + 1} \rangle \right. \\ + \gamma_ {k + 1} ^ {2} \| v (x _ {k}) + Z _ {k + 1} \| ^ {2} + \gamma_ {k + 1} \| \sigma (x _ {k}) \xi_ {k + 1} \| ^ {2} + 2 \gamma_ {k + 1} ^ {\frac {3}{2}} \langle \sigma (x _ {k}) \xi_ {k + 1}, b _ {k + 1} \rangle \left| \mathcal {F} _ {k} \right] \\ \leq \left\| x _ {k} \right\| ^ {2} + 2 \gamma_ {k + 1} \left(\left\langle x _ {k}, v (x _ {k}) \right\rangle + C _ {\sigma} / 2\right) + 2 \gamma_ {k + 1} ^ {2} \left\| v (x _ {k}) \right\| ^ {2} \\ + \mathbb {E} \left[ 2 \gamma_ {k + 1} ^ {2} \| Z _ {k + 1} \| ^ {2} + 2 \gamma_ {k + 1} \langle x _ {k}, Z _ {k + 1} \rangle + 2 \gamma_ {k + 1} ^ {\frac {3}{2}} \langle \sigma (x _ {k}) \xi_ {k + 1}, b _ {k + 1} \rangle \mid \mathcal {F} _ {k} \right] \\ \leq \left\| x _ {k} \right\| ^ {2} + 2 \gamma_ {k + 1} \left(\left\langle x _ {k}, v \left(x _ {k}\right) \right\rangle + C _ {\sigma} / 2 + \gamma_ {k + 1} ^ {\frac {1}{2}} C _ {\sigma} / 4\right) + 2 \gamma_ {k + 1} ^ {2} \| v \left(x _ {k}\right) \| ^ {2} \tag {21} \\ + \mathbb {E} \left[ 2 \gamma_ {k + 1} ^ {2} \| Z _ {k + 1} \| ^ {2} | \mathcal {F} _ {k} \right] + \gamma_ {k + 1} ^ {\frac {3}{2}} \mathbb {E} \left[ \| b _ {k + 1} \| ^ {2} | \mathcal {F} _ {k} \right] + 2 \mathbb {E} \left[ \gamma_ {k + 1} \langle x _ {k}, b _ {k + 1} \rangle | \mathcal {F} _ {k} \right]. \\ \end{array}
$$
Recalling (5) in Assumption 3, we have for some  $C > 0$
$$
\mathbb {E} \| Z _ {k + 1} \| ^ {2} \leq 2 \sigma^ {2} + 2 C \left(\gamma_ {k + 1} ^ {2} \mathbb {E} \| v (x _ {k}) \| ^ {2} + \gamma_ {k + 1}\right) \tag {22}
$$
Without loss of generality, assume  $\gamma_{k} \leq 1$  and  $\mathbb{E}\|x_k\|^2 \geq 1$  (so that  $\left(\mathbb{E}\|x_k\|^2\right)^2 \geq \mathbb{E}\|x_k\|^2$ ) for all  $k$ . Then,  $\|v(x_k)\|^2 \leq L^2\|x_k\|^2$ , together with Assumption 4 and the Cauchy-Schwartz inequality on the last term of (21), implies
$$
\begin{array}{l} \mathbb {E} \| x _ {k + 1} \| ^ {2} \leq \mathbb {E} \| x _ {k} \| ^ {2} - 2 \alpha \gamma_ {k + 1} \mathbb {E} \| x _ {k} \| ^ {2} + 2 \gamma_ {k + 1} \left(\beta + C _ {\sigma} + \frac {1}{2} \gamma_ {k + 1} ^ {\frac {1}{2}} C _ {\sigma}\right) + 2 L ^ {2} \gamma_ {k + 1} ^ {2} \mathbb {E} \| x _ {k} \| ^ {2} \\ + 2 \gamma_ {k + 1} ^ {2} \left[ 2 \sigma^ {2} + 2 C \left(L ^ {2} \gamma_ {k + 1} ^ {2} \mathbb {E} \| x _ {k} \| ^ {2} + \gamma_ {k + 1}\right) \right] \\ + \gamma_ {k + 1} ^ {\frac {3}{2}} C \left(L ^ {2} \gamma_ {k + 1} ^ {2} \mathbb {E} \| x _ {k} \| ^ {2} + \gamma_ {k + 1}\right) \\ + 2 \gamma_ {k + 1} \sqrt {C} \sqrt {L ^ {2} \gamma_ {k + 1} ^ {2} \left(\mathbb {E} \| x _ {k} \| ^ {2}\right) ^ {2} + \gamma_ {k + 1} \mathbb {E} \| x _ {k} \| ^ {2}} \\ \leq \mathbb {E} \| x _ {k} \| ^ {2} \left(1 - C _ {1} \gamma_ {k + 1} + C _ {2} \gamma_ {k + 1} ^ {\frac {3}{2}}\right) + C _ {3} \gamma_ {k + 1} \\ \end{array}
$$