SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Algorithm 7 (Riemannian stochastic extra-gradient). Inspired by the original work of Korpelevich [38], the Riemannian stochastic extra-gradient (RSEG) method [51, 67] proceeds as
$$
X _ {n} ^ {+} = \exp_ {X _ {n}} \left(\gamma_ {n} V \left(X _ {n}; \theta_ {n}\right)\right),
$$
$$
X _ {n + 1} = \exp_ {X _ {n}} \left(\Gamma_ {X _ {n} ^ {+} \rightarrow X _ {n}} \left(\gamma_ {n} V \left(X _ {n} ^ {+}; \theta_ {n} ^ {+}\right)\right)\right) \tag {RSEG}
$$
where  $\theta_{n}$  and  $\theta_{n}^{+}$  are independent seeds for (SFO). Thus, to cast (RSEG) in the RRM framework, it suffices to take  $U_{n} = \Gamma_{X_{n}^{+}\to X_{n}}(\mathrm{err}(X_{n}^{+};\theta_{n}^{+}))$  and  $b_{n} = \Gamma_{X_{n}^{+}\to X_{n}}(v(X_{n}^{+})) - v(X_{n})$
Under the assumptions in Proposition 1, one can show that Algorithms 6-7 also avoid strict saddle points of  $f$ ; we provide the relevant details in Appendix B.3.
B Missing Proofs of Section 4
B.1. Proof of Theorem 2. We begin by proving Theorem 2, which will play a crucial role in our proof of Theorem 1. For the reader's convenience, we restate the result below:
Proof. To begin, let  $\{e_i(n)\}_{i=1}^d$  be an arbitrary sequence of orthonormal bases for  $\mathcal{T}_{X_n}\mathcal{M}$ , and let  $U_n^{\parallel}$  be the (Euclidean) noise vector composed of components of the noise  $U_n$  in the basis  $\{e_i(n)\}_{i=1\dots d}$ , viz.
$$
U _ {i, n} ^ {\parallel} := \left\langle U _ {n}, e _ {i} (n) \right\rangle_ {X _ {n}}. \tag {B.1}
$$
It is then easy to see that  $\mathbb{E}[U_n^{\parallel}|\mathcal{F}_n] = 0$ , and, moreover
$$
\left\| U _ {n} ^ {\parallel} \right\| = \| U _ {n} \| _ {X _ {n}} \leq \sigma \tag {B.2}
$$
by Assumption 3. Then, following Benaim [10], consider the "continuous-to-discrete" counter
$$
M (t) = \sup  \{n \geq 1: t \geq \tau_ {n} \} \tag {B.3}
$$
which measures the number of iterations required for the effective time  $\tau_{n} = \sum_{k=1}^{n-1} \gamma_{k}$  to reach a given timestamp  $t \geq 0$ . We further denote the piecewise-constant interpolation of the noise sequence as
$$
\bar {U} ^ {\parallel} (t) = U _ {n} ^ {\parallel} \quad \text {f o r a l l} t \in \left[ \tau_ {n}, \tau_ {n + 1}\right), n \geq 1 \tag {B.4}
$$
and we let
$$
\Delta (t; T) := \sup  _ {0 \leq h \leq T} \left\| \int_ {t} ^ {t + h} \bar {U} ^ {\parallel} (s) d s \right\|. \tag {B.5}
$$
Moving forward, since  $X_{n}\to \mathcal{S}$  by assumption, we will also have
$$
\sup  _ {n} \operatorname {d i s t} \left(X _ {n}, x\right) =: R <   \infty \quad \text {f o r a l l} x \in \mathcal {S} \tag {B.6}
$$
for some (possibly random)  $R \geq 0$ . Moreover, since  $f$  is assumed to be  $C^2$ , (B.6) implies that
$$
\sup  _ {n} \| v \left(X _ {n}\right) \| _ {X _ {n}} =: G <   \infty \tag {B.7}
$$
for some (possibly random) non-negative constant  $G \geq 0$ . Moreover, since the manifold  $\mathcal{M}$  is assumed to be smooth, the sectional curvatures at each  $X_{n}$ ,  $n = 1,2,\ldots$ , must be likewise bounded by some constant  $K_{\mathrm{max}}$ . Then, by the analysis of [34, Eq. 53], there exists a constant  $C \equiv C_{L,G,K_{\mathrm{max}},R}$  depending only on  $L,G,K_{\mathrm{max}}$  and  $R$  such that
$$
\sup  _ {0 \leq h \leq T} \operatorname {d i s t} (X (t + h), \Phi_ {h} (X (t))) \leq C _ {L, G, K _ {\max }, R} \cdot \left[ \sup  _ {n \geq M (t)} \left(\left\| b _ {n} \right\| _ {x _ {n}} + \gamma_ {n}\right) + \Delta (t - 1; T + 1) \right] \tag {B.8}
$$
By Assumption 3, we have  $\|b_n\|_{x_n} = \mathcal{O}(\gamma_n)$ . Since  $\gamma_n \to 0$ , it suffices to show that  $\Delta(t;T) \to 0$  with probability 1 under Assumptions 2 and 3. This is equivalent to showing that, for any  $\varepsilon > 0$ , we have
$$
\lim  _ {t \rightarrow \infty} \Delta (t; T) \leq \varepsilon \quad \text {w i t h p r o b a b i l i t y} 1. \tag {B.9}
$$
To this end, let  $n = M(t)$  and recall that, by (B.2), we have
$$
\mathbb {E} \left[ \exp \left(\langle w, \bar {U} _ {n} ^ {\parallel} \rangle\right) \mid \mathcal {F} _ {n} \right] \leq \exp \left(\frac {\sigma^ {2}}{2} \| w \| ^ {2}\right) \tag {B.10}
$$
for all  $w \in \mathbb{R}^d$  where  $d$  is the dimension of  $\mathcal{M}$ . Therefore, for each  $w \in \mathbb{R}^d$ , the sequence  $Y_{n}(w)$  defined by
$$
Y _ {n} (w) := \exp \left(\sum_ {k = 1} ^ {n} \langle w, \gamma_ {k} \bar {U} _ {k} ^ {\parallel} \rangle - \frac {\sigma^ {2} \| w \| ^ {2}}{2} \sum_ {k = 1} ^ {n} \gamma_ {k} ^ {2}\right) \tag {B.11}
$$
is a supermartingale. Since  $Y_{n}$  is a supermartingale, we have
$$
\begin{array}{l} \mathbb {P} \left(\sup  _ {n <   k \leq M (\tau_ {n} + T)} \sum_ {i = n} ^ {k - 1} \langle w, \gamma_ {i} \bar {U} _ {i} ^ {\parallel} \rangle \geq \delta\right) \\ \leq \mathbb {P} \left(\sup  _ {n <   k \leq M (\tau_ {n} + T)} Y _ {k} (w) \geq Y _ {n} (w) \exp \left(\delta - \frac {\sigma^ {2} \| w \| ^ {2}}{2} \sum_ {i = n} ^ {M (\tau_ {n} + T) - 1} \gamma_ {i} ^ {2}\right)\right) \\ \leq \exp \left(\frac {\sigma^ {2} \| w \| ^ {2}}{2} \sum_ {i = n} ^ {M (\tau_ {n} + T) - 1} \gamma_ {i} ^ {2} - \delta\right) \tag {B.12} \\ \end{array}
$$
for any  $\delta > 0$ . Now, let  $e_i$  be the  $i$ -th basis vector of  $\mathbb{R}^d$ . Then, by (B.12), we have
$$
\begin{array}{l} \mathbb {P} \left(\sup  _ {n <   k \leq M (\tau_ {n} + T)} \sum_ {i = n} ^ {k - 1} \langle \pm d e _ {i}, \gamma_ {k} \bar {U} _ {k} ^ {\parallel} \rangle \geq \varepsilon\right) = \mathbb {P} \left(\sup  _ {n <   k \leq M (\tau_ {n} + T)} \sum_ {i = n} ^ {k - 1} \langle \pm \varepsilon^ {- 1} \delta d e _ {i}, \gamma_ {k} \bar {U} _ {k} ^ {\parallel} \rangle \geq \delta\right) \\ \leq \exp \left(\frac {\sigma^ {2} \delta^ {2} d ^ {2}}{2 \varepsilon^ {2}} \sum_ {i = n} ^ {M (\tau_ {n} + T) - 1} \gamma_ {i} ^ {2} - \delta\right). \tag {B.13} \\ \end{array}
$$
Optimizing (B.13) over  $\delta$ , we get
$$
\mathbb {P} \left(\sup  _ {n <   k \leq M (\tau_ {n} + T)} \sum_ {i = n} ^ {k - 1} \langle \pm d e _ {i}, \gamma_ {k} \bar {U} _ {k} ^ {\mathrm {I I}} \rangle \geq \varepsilon\right) \leq \exp \left(- \frac {\varepsilon^ {2}}{2 \sigma^ {2} d ^ {2} \sum_ {i = n} ^ {M (\tau_ {n} + T) - 1} \gamma_ {i} ^ {2}}\right). \tag {B.14}
$$
Since  $\gamma_{n}\to 0$ , with loss of generality we may assume that  $\gamma_{n}\leq 1$ , and hence
$$
\begin{array}{l} \mathbb {P} \left(\sup  _ {n <   k \leq M (\tau_ {n} + T)} \sum_ {i = n} ^ {k - 1} \langle \pm d e _ {i}, \gamma_ {k} \bar {U} _ {k} ^ {\parallel} \rangle \geq \varepsilon\right) \leq \exp \left(- \frac {\varepsilon^ {2}}{2 \sigma^ {2} d ^ {2} \sum_ {i = n} ^ {M (\tau_ {n} + T) - 1} \gamma_ {i}}\right) \\ \leq \exp \left(- \frac {\varepsilon^ {2}}{2 \sigma^ {2} d ^ {2} \int_ {t} ^ {t + T} \bar {\gamma} (t)}\right) \tag {B.15} \\ \end{array}
$$
where, analogously to (B.4), we have defined the piece-wise constant interpolated step-size sequence
$$
\bar {\gamma} (t) = \gamma_ {n} \quad \text {f o r a l l} t \in [ \tau_ {n}, \tau_ {n + 1}), n \geq 1. \tag {B.16}
$$
$$
\left\| \sum_ {i = n} ^ {k - 1} \gamma_ {i} \bar {U} _ {i} ^ {\parallel} \right\| \geq \varepsilon \quad \Rightarrow \quad \exists i \text {s u c h t h a t} \sum_ {i = n} ^ {k - 1} \langle \pm d e _ {i}, \gamma_ {i} \bar {U} _ {i} ^ {\parallel} \rangle \geq \varepsilon , \tag {B.17}
$$
by the union bound, we have
$$
\mathbb {P} (\Delta (t, T) \geq \varepsilon) \leq 2 d \cdot \exp \left(- \frac {\varepsilon^ {2}}{2 \sigma^ {2} d ^ {2} \int_ {t} ^ {t + T} \bar {\gamma} (t)}\right) \leq 2 d \cdot \exp \left(- \frac {\varepsilon^ {2}}{2 \sigma^ {2} d ^ {2} T \bar {\gamma} (s)}\right) \tag {B.18}
$$
for some  $t \leq s \leq t + T$ . Therefore, by setting  $\lambda \coloneqq \exp \left(-\frac{\varepsilon^2}{2\sigma^2d^2T}\right) < 1$ , we have
$$
\sum_ {k} \mathbb {P} (\Delta (k T, T) \geq \varepsilon) \leq 2 d \sum_ {k} \lambda^ {1 / \gamma_ {k}} <   \infty \tag {B.19}
$$
by Assumption 2. The Borel-Cantelli Lemma then implies that the following event happens almost surely:
$$
\lim  _ {k \rightarrow \infty} \Delta (k T; T) \leq \varepsilon . \tag {B.20}
$$
The proof is finished by noting that, for  $kT \leq t < (k + 1)T$ ,
$$
\Delta (t; T) \leq 2 \Delta (k T; T) + \Delta (k T + T; T) \tag {B.21}
$$
by triangle inequality.
B.2. Proof of Theorem 1. We are now in a position to present our proof of Theorem 1, which we restate below for convenience:
Proof. Assume that  $X_{n} \to S$ . We will show that this event has zero probability in a series of steps which we outline below.
Step 1: Isometrically embedded Robbins-Monro iterates. Since  $\mathcal{M}$  is assumed to be smooth, the second Nash embedding theorem [37] implies there exists a smooth and isometric embedding  $\iota : \mathcal{M} \to \mathbb{R}^M$  such that, for all  $x \in \mathcal{M}$  and all  $z, w \in \mathcal{T}_x\mathcal{M}$ , we have
$$
\langle z, w \rangle_ {x} = \left\langle \mathrm {D} \iota_ {x} (z), \mathrm {D} \iota_ {x} (w) \right\rangle . \tag {B.22}
$$
Since  $\iota$  is an embedding, it is surjective. Since it is isometric, it preserves distance and hence must be one-to-one. Therefore,  $\iota$  is an diffeomorphism since it is also smooth. We can therefore define the pushforward of the vector field  $v$  on  $\mathcal{M}$  to a vector field on the image  $\mathcal{M}^E\subset \mathbb{R}^M$  in the usual way as
$$
v _ {0} ^ {E} \left(x ^ {E}\right) := \mathrm {D} \iota_ {x} v (x) \quad \text {f o r a l l} x ^ {E} = \iota (x) \in \mathbb {R} ^ {M}. \tag {B.23}
$$
We also set  $\mathcal{S}^E\coloneqq \iota (\mathcal{S})$
By the Tietze extension theorem and the smooth manifold extension lemma [44],  $v_0^E(x^E)$  can be extended to a Lipschitz continuous vector field on all of  $\mathbb{R}^M$ , which we still denote by  $v_0^E(x^E)$ . To avoid trivialities, we will also need to ensure that  $v_0^E(x^E)$  is not 0 in a neighborhood of  $S^E$ : If this is the case, then we set our target field  $v^E(x^E) \coloneqq v_0^E(x^E)$ ; otherwise, let 1 denote the vector of 1's in all coordinates, and define a new vector field  $v^E$  on  $\mathbb{R}^M$  as
$$
v ^ {E} \left(x ^ {E}\right) := v _ {0} ^ {E} \left(x ^ {E}\right) + \operatorname {d i s t} ^ {E} \left(x ^ {E}, \mathcal {S} ^ {E}\right) ^ {2} \cdot \mathbf {1} \tag {B.24}
$$
where  $\mathrm{dist}^E (x^E,\mathcal{S}^E)\coloneqq \inf_{y^E\in \mathcal{S}^E}\| x^E -y^E\|$ . Obviously, this new vector field agrees with  $v_0^E (x^E)$  on  $\mathcal{M}^E$  and therefore is still the pushforward of  $v$  under  $\iota$ . Moreover, it is not uniformly 0 in a neighborhood of  $\mathcal{S}^E$ . [It is worth noting that the so-defined vector field  $v^{E}$  is in general not the (Euclidean) gradient of any function, a fact which presents significant difficulty to our analysis.]
Step 2:  $\mathcal{S}^E$  is an unstable invariant set. Our next goal is to show that there exists an unstable neighborhood  $\mathcal{U}^E$  around  $\mathcal{S}^E$  in the following sense: First, for each  $\hat{x} \in S$ , consider its image  $\hat{x}^E = \iota(\hat{x}) \in \mathcal{S}^E$ . Since  $v^E$  agrees with the pushforward of  $v \equiv \operatorname{grad} f$  under  $\iota$ , and since  $\iota$  is an isometry, we have the following relation for all tangent vector  $z \in T_{\hat{x}}\mathcal{M}$ :
$$
\langle \mathrm {D} v _ {x E} ^ {E} \mathrm {D} \iota_ {\hat {x}} (z), \mathrm {D} \iota_ {\hat {x}} (z) \rangle = \left\langle \operatorname {H e s s} f (\hat {x}) z, z \right\rangle_ {\hat {x}}. \tag {B.25}
$$
Since  $\hat{x}$  is a strict saddle point, (B.25) implies that  $\lambda_{\min} \left( \mathrm{D} v_{\hat{x}^E}^E \right) < -c_+ < 0$  for all  $\hat{x}^E \in S^E$ . By an established series of arguments [10, 12, 46], using the stable manifold theorem for a strict saddle [63] and the transversality of the strict saddle manifold [1], there exists a  $(M-m)$ -dimensional embedded submanifold  $\mathcal{Q}^E$  in  $\mathbb{R}^M$  that contains  $S^E$ . [Here,  $1 \leq m \leq M$ , and  $M-m$  represents the dimension of the unstable manifold of  $v^E$ .] Moreover, writing  $\Phi^E$  for the flow generated by  $v^E$ , it follows that  $\mathcal{Q}^E$  is locally invariant under  $\Phi^E$ . Hence, there exists a neighborhood  $\mathcal{N}^E$  of  $S^E$  in  $\mathbb{R}^M$  and a positive time  $t_0$  such that for all  $|t| \leq t_0$ , the following inclusion holds:
$$
\Phi_ {t} ^ {E} \left(\mathcal {N} ^ {E} \cap \mathcal {Q} ^ {E}\right) \subset \mathcal {Q} ^ {E}. \tag {B.26}
$$
To proceed, note that  $\mathbb{R}^M$  can be decomposed further as the direct sum of the tangent space to  $\mathcal{Q}^E$  at  $x^E$ , denoted by  $\mathcal{T}_{x^E}\mathcal{Q}^E$ , and an additional complementary subspace denoted by  $\mathcal{E}_{x^E}^u$ :
$$
\mathbb {R} ^ {M} = \mathcal {T} _ {x ^ {E}} \mathcal {Q} ^ {E} \oplus \mathcal {E} _ {x ^ {E}} ^ {u}. \tag {B.27}
$$
The mapping  $x^{E} \to \mathcal{E}_{x^{E}}^{u}$  is continuous, where  $x^{E}$  varies over  $S^{E}$  and  $\mathcal{E}_{x^{E}}^{u}$  belongs to the Grassmanian manifold  $G(m,M)$ . It is important to note that  $\mathcal{E}_{x^{E}}^{u}$  contains at least one direction in  $\mathcal{T}_{x^{E}}\mathcal{M}^{E}$  due to (B.25). Then, for all  $t \in \mathbb{R}$  and  $x \in S^{E}$ , the Jacobian of  $\Phi_t^E$  evaluated at  $x^{E}$  maps  $\mathcal{E}_{x^{E}}^{u}$  to  $\mathcal{E}_{\Phi_t^E (x^E)}^u$ , i.e.,
$$
\mathrm {D} \Phi_ {t} ^ {E} \left(x ^ {E}\right) \mathcal {E} _ {x ^ {E}} ^ {u} = \mathcal {E} _ {\Phi_ {t} ^ {E} \left(x ^ {E}\right)} ^ {u}. \tag {B.28}
$$
Finally, and most importantly, we have the following characterization that formalizes the idea that all directions in the unstable manifold should diverge under  $\Phi_t^E$ : There exist positive constants  $c$  and  $C$  such that for all  $x^E \in S^E$ ,  $w^E \in \mathcal{E}_{x^E}^u$ , and  $t \geq 0$ , the following inequality holds:
$$
\left\| \mathrm {D} \Phi_ {t} ^ {E} \left(x ^ {E}\right) w ^ {E} \right\| \geq C e ^ {c t} \| w ^ {E} \|. \tag {B.29}
$$
The above verifies all the conditions for a unstable invariant set for  $S^E$  in the sense of Benaim [10]. A deep result by Benaim & Hirsch [11] then asserts the existence of a local Lyapunov function near a neighborhood of  $S^E$ , whose construction we outline below.
Step 3: Local Lyapunov function  $\eta^E$ . For a right-differentiable function  $\eta^E \colon \mathbb{R}^M \to \mathbb{R}$  we define its right derivative  $\mathrm{D}\eta^E$  applied to a vector  $h^E \in \mathbb{R}^M$  by
$$
\mathrm {D} \eta^ {E} (x ^ {E}) h ^ {E} = \lim  _ {t \rightarrow 0 ^ {+}} \frac {\eta^ {E} \left(x ^ {E} + t h ^ {E}\right) - \eta^ {E} \left(x ^ {E}\right)}{t}. \tag {B.30}
$$
If  $\eta^E$  is differentiable, then (B.30) is simply  $\langle \nabla \eta^{E}(x^{E}),h^{E}\rangle$ . In view of all this, Benaim [10] provides the following crucial result:
Proposition B.1 (Benaïm, 1999, Prop. 9.5). There exists a compact neighborhood  $\mathcal{U}^E(S^E)$  of  $S^E$ , positive numbers  $l, \beta > 0$ , and a map  $\eta^E \colon \mathcal{U}^E(S^E) \to \mathbb{R}^+$  such that  $\eta^E(x^E) = 0$  if and only if  $x^E \in \mathcal{Q}^E$ , and the following holds:
(i)  $\eta^E$  is  $C^2$  on  $\mathcal{U}^E (\mathcal{S}^E)\setminus \mathcal{Q}^E$
(ii) For all  $x^{E} \in \mathcal{U}^{E}(\mathcal{S}^{E}) \cap \mathcal{Q}^{E}$ ,  $\eta^{E}$  admits a right derivative  $\mathrm{D}\eta^{E}(x^{E}) \colon \mathbb{R}^{M} \to \mathbb{R}^{M}$  which is Lipschitz, convex and positively homogeneous.
(iii) There exists  $k > 0$  and a neighborhood  $\mathcal{W}^E \subset \mathbb{R}^M$  of 0 such that for all  $x^E \in \mathcal{U}^E(\mathcal{S}^E)$  and  $z^E \in \mathcal{W}^E$ ,
$$
\eta^ {E} \left(x ^ {E} + z ^ {E}\right) \geq \eta^ {E} \left(x ^ {E}\right) + D \eta^ {E} \left(x ^ {E}\right) z ^ {E} - k \| z ^ {E} \| ^ {2}. \tag {B.31}
$$
(iv) There exists  $c_{1} > 0$  such that for all  $x^{E}\in \mathcal{U}^{E}(\mathcal{S}^{E})\setminus \mathcal{Q}^{E}$
$$
\left\| \Pi_ {\mathcal {M} ^ {E}} \left(\mathrm {D} \eta^ {E} \left(x ^ {E}\right)\right) \right\| \geq c _ {1} \tag {B.32}
$$
where  $\Pi_{\mathcal{M}^E}$  is the projection on  $\mathcal{M}^E$ . In addition, for all  $x^{E}\in \mathcal{U}^{E}(\mathcal{S}^{E})\cap \mathcal{Q}^{E}$  and  $z^{E}\in \mathbb{R}^{M}$
$$
\langle \mathrm {D} \eta^ {E} \left(x ^ {E}\right), z ^ {E} \rangle \geq c _ {1} \| z ^ {E} - \mathrm {D} \Pi \left(x ^ {E}\right) z ^ {E} \| \tag {B.33}
$$
where  $\Pi$  is the projection of a neighborhood of  $S^E$  onto  $\mathcal{Q}^E$ .
(v) For all  $x^{E}\in \mathcal{U}^{E}(\mathcal{S}^{E})\cap \mathcal{Q}^{E}$ ,  $w^{E}\in \mathcal{T}_{x^{E}}\mathcal{Q}^{E}$  and  $z^{E}\in \mathbb{R}^{M}$ ,
$$
D \eta^ {E} \left(x ^ {E}\right) \left(w ^ {E} + z ^ {E}\right) = D \eta^ {E} \left(x ^ {E}\right) z ^ {E}. \tag {B.34}
$$
(vi) For all  $x^{E}\in \mathcal{U}^{E}(\mathcal{S}^{E})$  we have
$$
D \eta^ {E} (x) v ^ {E} \left(x ^ {E}\right) \geq \beta \eta^ {E} \left(x ^ {E}\right). \tag {B.35}
$$
The function  $\eta^E$  will serve as a local "energy function" that plays an instrumental role in our analysis; the well-posedness of  $\Pi$  is guaranteed by [27, Chap 4].
Step 4: Geodesic offset. Consider the image of an RRM scheme  $X_{n}^{E}\coloneqq \iota (X_{n})$ . If  $X_{n}^{E}\notin \mathcal{U}^{E}(\mathcal{S}^{E})$  for all  $n$ , then there is nothing to prove. Otherwise, without loss of generality we may assume that  $X_{1}^{E}\in \mathcal{U}^{E}(\mathcal{S}^{E})$ . Accordingly, define the first exit time  $T$  from  $\mathcal{U}^E (\mathcal{S}^E)$  as
$$
T := \inf  \left\{k \geq 1: X _ {n} ^ {E} \notin \mathcal {U} ^ {E} \left(\mathcal {S} ^ {E}\right) \right\}. \tag {B.36}
$$
Evidently,  $T$  is a stopping time adaptive to  $\mathcal{F}_n$ , so it suffices to show that
$$
\mathbb {P} (T = \infty) = 0. \tag {B.37}
$$
To this end, a notion that plays a central role in our analysis is the geodesic offset, defined as follows. Define the pushforward of the respective noise and bias vectors in the RRM scheme  $X_{n}$  by
$$
U _ {n} ^ {E} := \mathrm {D} \iota_ {x _ {n}} U _ {n}, \quad b _ {n} ^ {E} := \mathrm {D} \iota_ {x _ {n}} b _ {n}. \tag {B.38}
$$
It is important to remember that  $U_{n}^{E}\in \mathcal{T}_{X_{n}^{E}}\mathcal{M}^{E}$ , a fact that we will use freely in the sequel.
We now formally define the geodesic offset  $\Delta (x;z)\in \mathbb{R}^{M}$  as, for any  $x\in \mathcal{M}$  and  $z\in T_x\mathcal{M}$
$$
\Delta (x; z) := \iota \left(\exp_ {x} (z)\right) - \iota (x) - D \iota_ {x} (z). \tag {B.39}
$$
By Assumption 4, there exists  $\varrho >0$  such that, for all  $\| z\| _x < \varrho$ , the exponential mapping is the unique minimizing geodesic. Furthermore, for all such  $z$ 's, define the curve  $\gamma^E (t)\coloneqq \iota (x) + t\mathrm{D}\iota_x(z)$  then
$$
\gamma^ {E} (0) = \iota (x), \quad \dot {\gamma} ^ {E} (0) = D \iota_ {x} (z) \tag {B.40}
$$
so that  $\gamma^{E}(t)$  agrees with the image of the geodesics  $\iota(\exp_{x}(tz))$ . As a result, for any  $\|z\|_x < \varrho$ , we have  $\Delta(x;z) = \mathcal{O}(\|z\|_x^2)$ . Now, setting  $x \gets X_n$  and  $z \gets \gamma_n(v(X_n) + U_n + b_n)$ , we have
$$
X _ {n + 1} ^ {E} = X _ {n} ^ {E} + \gamma_ {n} \left(v ^ {E} \left(X _ {n} ^ {E}\right) + U _ {n} ^ {E} + b _ {n} ^ {E}\right) + \epsilon_ {n} ^ {E} \tag {B.41}
$$
where  $\epsilon_n^E \coloneqq \Delta(X_n; \gamma_n(v(X_n) + U_n + b_n))$ . By Assumption 3, we know that  $U_n^E + b_n^E = \mathcal{O}(1)$  almost surely. Moreover, since  $v$  is smooth, on the event  $T = \infty$ ,  $X_n^E \in \mathcal{U}^E(S^E)$  for all  $n$  and therefore  $\sup_{n \geq 1} \| v^E(X_n^E) \| < \infty$ . Since  $\gamma_n \to 0$ , for any  $n$  large enough, we get
$$
\epsilon_ {n} ^ {E} = \mathcal {O} \left(\gamma_ {n} ^ {2}\right). \tag {B.42}
$$
Now, define two sequences of random variables  $\{Y_{n}\}_{n\geq 1}$  and  $\{S_n\}_{n\geq 1}$  as
$$
Y _ {n + 1} = \left(\eta^ {E} \left(X _ {n + 1} ^ {E}\right) - \eta^ {E} \left(X _ {n} ^ {E}\right)\right) \mathbb {1} _ {\{n \leq T \}} + \gamma_ {n} \mathbb {1} _ {\{n > T \}}, \tag {B.43a}
$$
$$
S _ {0} = \eta^ {E} \left(X _ {0} ^ {E}\right), \quad S _ {n} = S _ {0} + \sum_ {k = 1} ^ {n} Y _ {k}. \tag {B.43b}
$$
The importance of these sequences is that the event  $\{T = \infty\}$  is contained in the event  $\{S_n \to 0\}$ . To see this, assume  $T = \infty$ . Then we have  $Y_{n+1} = \eta^E(X_{n+1}^E) - \eta^E(X_n^E)$  and  $S_n = \eta^E(X_n)$  by Eqs. (B.43a) and (B.43b). In addition, since  $\{X_n^E\}$  remains in  $\mathcal{U}^E(\mathcal{S}^E)$  by definition of the stopping time  $T$ , Theorem 2 combined with [10, Theorem 5.7] asserts that the limit set  $L(\{X_n^E\})$  of  $\{X_n^E\}$  is a nonempty compact invariant subset of  $\mathcal{U}^E(\mathcal{S}^E)$ , so that for all  $y^E \in L(\{X_n^E\})$  and  $t \in \mathbb{R}$ ,  $\Phi_t^E(y^E) \in \mathcal{U}^E(\mathcal{S}^E)$ . But then Proposition B.1(vi) implies that  $\eta^E(\Phi_t^E(y^E)) \geq e^{\beta t} \eta^E(y^E)$  for all  $t > 0$ , forcing  $\eta^E(y^E)$  to be zero. Since  $\eta^E(x^E) = 0$  if and only if  $x^E \in \mathcal{Q}^E$ , we have  $L(\{X_n^E\}) \subset \mathcal{Q}^E$ , which implies  $S_n = \eta^E(X_n^E) \to 0$ .
Therefore, the rest of the proof is devoted to showing that  $\mathbb{P}(\lim_{n\to \infty}S_n = 0) = 0$
2The original statement in [10] is  $\left\| \mathrm{D}\eta^{E}(x^{E})\right\| \geq c_{1}$ . The inequality in (B.32) is obtained via the same proof and noting that  $\mathcal{E}_{x^E}^u$  contains at least one direction in  $\mathcal{T}_{x^E}\mathcal{M}^E$ .
Theorem 1 follows from (B.37) because the event  $\{\mathrm{dist}(X_n,\mathcal{S})\to 0\}$  is contained in  $\{T = \infty \}$ .
Step 5: Probabilistic estimates. To this end, we will need two technical lemmas, originally due to Pemantle [55], and extended to their current form by Benaim & Hirsch [12].
Lemma B.1. Let  $S_{n}$  be a nonnegative stochastic process,  $S_{n} = S_{0} + \sum_{k=1}^{n} Y_{k}$  where  $Y_{n}$  is  $\mathcal{F}_{n}$ -measurable. Let  $\alpha_{n} \coloneqq \sum_{k=n}^{\infty} \gamma_{k}^{2}$ . Assume there exist a sequence  $0 \leq \varepsilon_{n} = o(\sqrt{\alpha_{n}})$ , constants  $a_{1}, a_{2} > 0$  and an integer  $N_{0}$  such that for all  $n \geq N_{0}$ ,
(i)  $|Y_{n}| = o(\sqrt{\alpha_{n}})$ .
(ii)  $\mathbb{1}_{\{S_n > \varepsilon_n\}}\mathbb{E}[Y_{n + 1}|\mathcal{F}_n]\geq 0.$
(iii)  $\mathbb{E}[S_{n + 1}^2 - S_n^2 | \mathcal{F}_n] \geq a_1 \gamma_n^2$ .
(iv)  $\mathbb{E}\bigl [Y_{n + 1}^2\bigr |\mathcal{F}_n\bigr ]\leq a_2\gamma_n^2.$
Then  $\mathbb{P}(\lim_{n\to \infty}S_n = 0) = 0$
Lemma B.2. Let  $S_{n}$  be a nonnegative stochastic process,  $S_{n} = S_{0} + \sum_{k=1}^{n} Y_{k}$  where  $Y_{n}$  is  $\mathcal{F}_{n}$ -measurable and  $|Y_{n}| \leq C$  almost surely for some constant  $C$ . Assume that  $\sum_{n} \gamma_{n}^{2} = \infty$ , and there exists  $c > 0, N' \in \mathbb{N}$  such that for all  $n \geq N'$ ,
$$
\mathbb {E} \left[ S _ {n + 1} ^ {2} - S _ {n} ^ {2} \mid \mathcal {F} _ {n} \right] \geq c \gamma_ {n} ^ {2}. \tag {B.44}
$$
$$
\mathbb {P} \left(\lim  _ {n \rightarrow \infty} S _ {n} = 0\right) = 0. \tag {B.45}
$$
Our proof will be concluded by verifying all the premises of Lemmas B.1-B.2. To that end, note first that the sequence  $S_{n}$  defined in (B.43b) is nonnegative by construction. We will then separate the analysis into two cases:
Case 1. Square-summable step-sizes, i.e.,  $\sum_{n}\gamma_{n}^{2} < \infty$ . In this case, we have
$$
\lim  _ {n \rightarrow \infty} \frac {\gamma_ {n}}{\sqrt {\sum_ {k = n} ^ {\infty} \gamma_ {k} ^ {2}}} = 0 \tag {B.46}
$$
so  $\gamma_{n} = o\left(\sqrt{\sum_{k=n}^{\infty}\gamma_{k}^{2}}\right)$ . This fact will be used in the proof when we invoke Lemma B.1 below with  $\varepsilon_{n} = \mathcal{O}(\gamma_{n})$  and  $\alpha_{n} = \sum_{k=n}^{\infty}\gamma_{k}^{2}$  therein. The verification process then proceeds as follows:
- Verifying Lemma B.1(i) and (iv): By the Lipschitz continuity of  $\eta^E$ , we know that
$$
\begin{array}{l} \left\| \eta^ {E} X _ {n} ^ {E} - \eta^ {E} X _ {n + 1} ^ {E} \right\| \leq L ^ {\prime} \| X _ {n} ^ {E} - X _ {n + 1} ^ {E} \| \\ = \gamma_ {n} \| v ^ {E} \left(X _ {n} ^ {E}\right) + U _ {n} ^ {E} + b _ {n} ^ {E} \| \tag {B.47} \\ \end{array}
$$