SlowGuess/NeurIPS_2023_OCR huggingface dataset:
1. Suppose that  $\mathbf{p}^{T_2^{\epsilon_1}} \in N_1 \cup N_3$ . According to the definition of set  $\mathcal{T}_{\epsilon_1}$  in Eq. (36), we have that  $\mathbf{p}^{T_1^{\epsilon_1}} \in N_1 \cup N_3$  since  $T_1^{\epsilon_1} \in \mathcal{T}_{\epsilon_1}$  is a jumping period. Together with the established fact that  $\mathrm{sign}\left(p_L^{\star \star} - p_L^{T_2^{\epsilon_1}}\right) = \mathrm{sign}\left(p_L^{\star \star} - p_L^{T_1^{\epsilon_1}}\right)$  (see the argument below Eq. (43)), we conclude that  $\mathbf{p}^{T_2^{\epsilon_1}}$  and  $\mathbf{p}^{T_1^{\epsilon_1}}$  are in the same quadrant, and  $\mathrm{sign}\left(p_H^{\star \star} - p_H^{T_2^{\epsilon_1}}\right) = \mathrm{sign}\left(p_H^{\star \star} - p_H^{T_1^{\epsilon_1}}\right)$ . Then, we can write that
$$
\begin{array}{l} \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \cdot G _ {H} \left(\mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}}, \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \\ \geq \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \cdot G _ {H} \left(\mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}}, \mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) - \ell_ {r} \| \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}} \| _ {2} \\ \geq \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \cdot G _ {H} \left(\left(p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right), \left(p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right) - \ell_ {r} \| \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}} \| _ {2} \tag {46} \\ = \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot G _ {H} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) - \ell_ {r} \| \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}} \| _ {2} \\ + \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot \left(G _ {H} \left(\left(p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right), \left(p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right) - G _ {H} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right), \\ \end{array}
$$
where we apply Lemma F.4 in  $(\Delta)$  by utilizing the relation  $\left|p_L^{\star \star} - p_L^{T_2^{\epsilon 1}}\right|\leq \left|p_L^{\star \star} - p_L^{T_1^{\epsilon 1}}\right|$  proved in Eq. (43). Next, as  $T_{1}^{\epsilon_{1}}$  is a jumping period, it follows from the price update rule in Eq. (26) that  $\left|p_H^{\star \star} - p_H^{T_1^{\epsilon_1}}\right| / (b_H + c_H)\leq \left|\eta^{T_1^{\epsilon_1}}G_H\bigl (\mathbf{p}^{T_1^{\epsilon_1}},\mathbf{p}^{T_1^{\epsilon_1}}\bigr)\right|\leq \eta^{T_1^{\epsilon_1}}M_G$ . Thus, when the chosen jumping period  $T_{1}^{\epsilon_{1}}$  is sufficiently large so that the step-size  $\eta^{T_1^{\epsilon_1}}$  is considerably smaller than  $\epsilon_{1}$ , we further derive that
$$
\left| p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}} \right| \geq \left(b _ {H} + c _ {H}\right) \cdot \left(\epsilon_ {1} - \eta^ {T _ {1} ^ {\epsilon_ {1}}} M _ {G}\right) \geq \left(b _ {H} + c _ {H}\right) \cdot \eta^ {T _ {1} ^ {\epsilon_ {1}}} M _ {G} \geq \left| p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {1} ^ {\epsilon_ {1}}} \right|. \tag {47}
$$
With Eq. (47), we use Lemma F.4 again to upper-bound the last term on the right-hand side of Eq. (46):
$$
\begin{array}{l} \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot \left(G _ {H} \left(\left(p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right), \left(p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right) - G _ {H} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right) \\ = \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \cdot G _ {H} \left(\left(p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right), \left(p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right) - \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot G _ {H} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \\ \end{array}
$$
$$
\begin{array}{l} \stackrel {(\Delta_ {1})} {=} \left| \left(\frac {1}{(b _ {H} + c _ {H}) p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}} + d _ {H} \big ((p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}), (p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}, p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}) \big) - 1\right) \right. \\ \left. - \left(\frac {1}{\left(b _ {H} + c _ {H}\right) p _ {H} ^ {T _ {1} ^ {\epsilon_ {1}}}} + d _ {H} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) - 1\right) \right| \\ = \left| \underbrace {\frac {1}{b _ {H} + c _ {H}} \left(\frac {1}{p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}} - \frac {1}{p _ {H} ^ {T _ {1} ^ {\epsilon_ {1}}}}\right)} _ {\text {d i f f} _ {1}} + \underbrace {\left[ d _ {H} \left(\left(p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}} , p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) , \left(p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}} , p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)\right) - d _ {H} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \right]} _ {\text {d i f f} _ {2}} \right| \\ \end{array}
$$
$$
\geq \frac {1}{b _ {H} + c _ {H}} \left| \frac {1}{p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}} - \frac {1}{p _ {H} ^ {T _ {1} ^ {\epsilon_ {1}}}} \right|, \tag {48}
$$
where  $(\Delta_{1})$  holds since the difference in the previous step is non-negative by Lemma F.4. Furthermore, it is straightforward to observe that the terms  $\mathrm{diff}_1$  and  $\mathrm{diff}_2$  have the same sign, which results in the final inequality  $(\Delta_2)$ . We substitute Eq. (48) back into Eq. (46):
$$
\begin{array}{l} \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \cdot G _ {H} \left(\mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}}, \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \\ \geq \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot G _ {H} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) - \ell_ {r} \| \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}} \| _ {2} + \frac {1}{b _ {H} + c _ {H}} \left| \frac {1}{p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}} - \frac {1}{p _ {H} ^ {T _ {1} ^ {\epsilon_ {1}}}} \right| \\ \geq \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) \cdot G _ {H} \left(\mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}}, \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}}\right) - \ell_ {r} \left(\| \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}} \| _ {2} + \| \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}} \| _ {2}\right) \\ + \frac {1}{b _ {H} + c _ {H}} \left| \frac {1}{p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}} - \frac {1}{p _ {H} ^ {T _ {1} ^ {\epsilon_ {1}}}} \right| \\ \end{array}
$$
$$
\begin{array}{l} \stackrel {(\Delta_ {1})} {\geq} - \ell_ {r} \left(\| \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}} \| _ {2} + \| \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}} \| _ {2}\right) + \frac {\left| p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}} - p _ {H} ^ {T _ {1} ^ {\epsilon_ {1}}} \right|}{\left(b _ {H} + c _ {H}\right) \cdot \left(p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}} \cdot p _ {H} ^ {T _ {1} ^ {\epsilon_ {1}}}\right)} \\ \stackrel {(\Delta_ {2})} {\geq} - \ell_ {r} \left(\| \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}} \| _ {2} + \| \mathbf {r} ^ {T _ {1} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {1} ^ {\epsilon_ {1}}} \| _ {2}\right) + \frac {\left| \left(\epsilon_ {1} - \eta^ {T _ {1} ^ {\epsilon_ {1}}} M _ {G}\right) - \eta^ {T _ {1} ^ {\epsilon_ {1}}} M _ {G} \right|}{\left(b _ {H} + c _ {H}\right) \cdot \bar {p} ^ {2}}. \tag {49} \\ \end{array}
$$
In inequality  $(\Delta_1)$ , we use the fact that  $\mathrm{sign}\left(p_H^{\star \star} - p_H^{T_1^{\epsilon_1}}\right) \cdot G_H\left(\mathbf{p}^{T_1^{\epsilon_1}}, \mathbf{r}^{T_1^{\epsilon_1}}\right) > 0$  since  $T_1^{\epsilon_1}$  is a jumping period in  $\mathcal{T}_{\epsilon_1}$ . Then, inequality  $(\Delta_2)$  is implied by the properties that  $\mathrm{sign}\left(p_H^{\star \star} - p_H^{T_2^{\epsilon_1}}\right) = \mathrm{sign}\left(p_H^{\star \star} - p_H^{T_1^{\epsilon_1}}\right), \left|p_H^{\star \star} - p_H^{T_2^{\epsilon_1}}\right| / (b_H + c_H) \geq \epsilon_1 - \eta^{T_1^{\epsilon_1}}M_G$ , and  $\left|p_H^{\star \star} - p_H^{T_1^{\epsilon_1}}\right| / (b_H + c_H) \leq \eta^{T_1^{\epsilon_1}}M_G$ . Since both the step-size  $\eta^t$  and the difference  $\|\mathbf{r}^t - \mathbf{p}^t\|_2$  decrease to 0 as  $t \to \infty$ , by choosing a sufficiently large jumping period  $T_1^{\epsilon_1}$  from  $\mathcal{T}_{\epsilon_1}$ , the right-hand side of Eq. (46) is guaranteed to be non-negative, i.e.,  $\mathrm{sign}\left(p_H^{\star \star} - p_H^{T_2^{\epsilon_1}}\right) \cdot G_H\left(\mathbf{p}^{T_2^{\epsilon_1}}, \mathbf{r}^{T_2^{\epsilon_1}}\right) \geq 0$ . This completes the proof of scenario 1.
2. Suppose  $\mathbf{p}^{T_2^{\epsilon_1}} \in N_2 \cup N_4$ . We deduce that
$$
\begin{array}{l} \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \cdot G _ {H} \left(\mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}}, \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \\ \geq \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \cdot G _ {H} \left(\mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}}, \mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) - \ell_ {r} \| \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}} \| _ {2} \\ \geq \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}\right) \cdot G _ {H} \left(\left(p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}, p _ {L} ^ {\star \star}\right), \left(p _ {H} ^ {T _ {2} ^ {\epsilon_ {1}}}, p _ {L} ^ {\star \star}\right)\right) - \ell_ {r} \| \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}} \| _ {2} \tag {50} \\ \stackrel {(\Delta_ {2})} {=} \mathcal {G} \left(\left(p _ {H} ^ {T _ {2} ^ {\epsilon 1}}, p _ {L} ^ {\star \star}\right)\right) - \ell_ {r} \| \mathbf {r} ^ {T _ {2} ^ {\epsilon 1}} - \mathbf {p} ^ {T _ {2} ^ {\epsilon 1}} \| _ {2} \\ \stackrel {(\Delta_ {3})} {\geq} M _ {\widehat {\epsilon} _ {1}} - \ell_ {r} \| \mathbf {r} ^ {T _ {2} ^ {\epsilon_ {1}}} - \mathbf {p} ^ {T _ {2} ^ {\epsilon_ {1}}} \| _ {2}, \quad w h e r e \widehat {\epsilon} _ {1} := \epsilon_ {1} - \eta^ {T _ {2} ^ {\epsilon_ {1}}} M _ {G}. \\ \end{array}
$$
The inequality  $(\Delta_1)$  follows from Lemma F.4, where  $\mathrm{sign}\left(p_H^{\star \star} - p_H\right) \cdot G_H(\mathbf{p}, \mathbf{p})$  is deceasing when  $|p_L^{\star \star} - p_L|$  decreases. The equality  $(\Delta_2)$  is due to the definition of  $\mathcal{G}(\mathbf{p})$  in Eq. (31) and the fact that  $\mathrm{sign}(p_L^{\star \star} - p_L^{\star \star}) = 0$ . Finally, in the last line  $(\Delta_3)$ , we leverage the initial assumption of Case 2, which implies that  $\varepsilon\left(\left(p_H^{T_2^{\epsilon_1}}, p_L^{\star \star}\right)\right) = \left|p_H^{\star \star} - p_H^{T_2^{\epsilon_1}}\right| / (b_H + c_H) \in \left[\epsilon_1 - \eta^{T_1^{\epsilon_1}}M_G,\epsilon_1\right)$ . Then, by Lemma F.2, there must exist  $M_{\widehat{\epsilon}_1} > 0$  with  $\widehat{\epsilon}_1 := \epsilon_1 - \eta^{T_2^{\epsilon_1}}M_G$  such that  $\mathcal{G}\left(\left(p_H^{T_2^{\epsilon_1}}, p_L^{\star \star}\right)\right) \geq M_{\widehat{\epsilon}_1}$ . As  $|\mathcal{T}_{\epsilon_1}| = \infty$ , we can pick a sufficiently large  $T_1^{\epsilon_1} \in T_{\epsilon_1}$  to guarantee that  $\ell_r\left\|\mathbf{r}^t - \mathbf{p}^t\right\|_2 \leq M_{\widehat{\epsilon}_1}$  for  $t \geq T_1^{\epsilon_1}$  by Lemma F.1. Consequently, we obtain the desired conclusion that  $\mathrm{sign}\left(p_H^{\star \star} - p_H^{T_2^{\epsilon_1}}\right) \cdot G_H\left(\mathbf{p}^{T_2^{\epsilon_1}}, \mathbf{r}^{T_2^{\epsilon_1}}\right) > 0$  when  $\mathbf{p}^{T_2^{\epsilon_1}} \in N_2 \cup N_4$ , which completes the proof of scenario 2.
Combining Case 1 and Case 2, we have shown that  $\left|p_H^{\star \star} - p_H^{T_2^{\epsilon_1} + 1}\right| / (b_H + c_H) < \epsilon_1$ , which completes the strong induction, i.e.,  $\left|p_H^{\star \star} - p_H^t\right| / (b_H + c_H) < \epsilon_1$  for all  $t \geq T_1^{\epsilon_1}$ . Under the assumption that  $\{\mathbf{p}^t\}_{t \geq 0}$  never visits  $N_{\epsilon_0}^1$  if  $t \geq T^{\epsilon_0}$ , we accordingly have that  $\left|p_L^{\star \star} - p_L^t\right| / (b_L + c_L) > (\epsilon_0 - \epsilon_1)$  for all  $t \geq T_1^{\epsilon_1}$ . Therefore, following the derivations from Eq. (37) to Eq. (41), we deduce that Eq. (42) holds for all  $t \geq T_1^{\epsilon_1}$ . Using a telescoping sum, it holds for any period  $T > T_1^{\epsilon_1}$  that
$$
\frac {\left| p _ {L} ^ {\star \star} - p _ {L} ^ {T} \right|}{b _ {L} + c _ {L}} \leq \frac {\left| p _ {L} ^ {\star \star} - p _ {L} ^ {T _ {1} ^ {\epsilon_ {1}}} \right|}{b _ {L} + c _ {L}} - \frac {M _ {\epsilon_ {0} - \epsilon_ {1}}}{2} \sum_ {t = T _ {1} ^ {\epsilon_ {1}}} ^ {T - 1} \eta^ {t}. \tag {51}
$$
Since  $M_{\epsilon_0 - \epsilon_1} > 0$  is a constant and  $\lim_{T\to \infty}\sum_{t = T_1^{\epsilon_1}}^{T - 1}\eta^t = \infty$ , we arrive at the contradiction that  $\left|p_L^{\star \star} - p_L^T\right| / (b_L + c_L)\rightarrow -\infty$  as  $T\rightarrow \infty$ . Consequently, our initial assumption is incorrect and the price path  $\{\mathbf{p}^t\}_{t\geq 0}$  would visit the  $\ell_1$ -neighborhood  $N_{\epsilon_0}^1$  infinitely many times for any  $\epsilon_0 > 0$ , which completes the proof of Part 1.
C.2 Proof of Part 2
In particular, we show that when  $\mathbf{p}^t \in N_{\epsilon_0}^2$  for some sufficiently large  $t$ , then it also holds  $\mathbf{p}^{t+1} \in N_{\epsilon_0}^2$ . The value of  $\epsilon_0$  will be specified later in the proof.
By the update rule of Algorithm 1, it holds that
$$
\begin{array}{l} \left| p _ {i} ^ {\star \star} - p _ {i} ^ {t + 1} \right| ^ {2} = \left| p _ {i} ^ {\star \star} - \operatorname {P r o j} _ {\mathcal {P}} \left(p _ {i} ^ {t} + \eta^ {t} D _ {i} ^ {t}\right) \right| ^ {2} \\ \leq \left| p _ {i} ^ {\star \star} - \left(p _ {i} ^ {t} + \eta^ {t} D _ {i} ^ {t}\right) \right| ^ {2} \\ = \left| \left(p _ {i} ^ {\star \star} - p _ {i} ^ {t}\right) - \eta^ {t} \left(b _ {i} + c _ {i}\right) \cdot G _ {i} \left(\mathbf {p} ^ {t}, \mathbf {r} ^ {t}\right) \right| ^ {2} \\ = \left| p _ {i} ^ {\star \star} - p _ {i} ^ {t} \right| ^ {2} - 2 \eta^ {t} \left(b _ {i} + c _ {i}\right) \cdot G _ {i} \left(\mathbf {p} ^ {t}, \mathbf {r} ^ {t}\right) \cdot \left(p _ {i} ^ {\star \star} - p _ {i} ^ {t}\right) + \left(\eta^ {t} \left(b _ {i} + c _ {i}\right) \cdot G _ {i} \left(\mathbf {p} ^ {t}, \mathbf {r} ^ {t}\right)\right) ^ {2} \\ = \left| p _ {i} ^ {\star \star} - p _ {i} ^ {t} \right| ^ {2} - 2 \eta^ {t} \left(b _ {i} + c _ {i}\right) \cdot G _ {i} \left(\mathbf {p} ^ {t}, \mathbf {p} ^ {t}\right) \cdot \left(p _ {i} ^ {\star \star} - p _ {i} ^ {t}\right) + \left(\eta^ {t} \left(b _ {i} + c _ {i}\right) \cdot G _ {i} \left(\mathbf {p} ^ {t}, \mathbf {r} ^ {t}\right)\right) ^ {2} \\ + 2 \eta^ {t} \left(b _ {i} + c _ {i}\right) \cdot \left(G _ {i} \left(\mathbf {p} ^ {t}, \mathbf {p} ^ {t}\right) - G _ {i} \left(\mathbf {p} ^ {t}, \mathbf {r} ^ {t}\right)\right) \cdot \left(p _ {i} ^ {\star \star} - p _ {i} ^ {t}\right) \\ \leq \left| p _ {i} ^ {\star \star} - p _ {i} ^ {t} \right| ^ {2} - 2 \eta^ {t} \left(b _ {i} + c _ {i}\right) \cdot G _ {i} \left(\mathbf {p} ^ {t}, \mathbf {p} ^ {t}\right) \cdot \left(p _ {i} ^ {\star \star} - p _ {i} ^ {t}\right) + \left(\eta^ {t} M _ {G} \left(b _ {i} + c _ {i}\right)\right) ^ {2} \\ + 2 \eta^ {t} \left(b _ {i} + c _ {i}\right) \cdot \left| p _ {i} ^ {\star \star} - p _ {i} ^ {t} \right| \cdot \ell_ {r} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2}, \tag {52} \\ \end{array}
$$
where we use  $|G_{i}(\mathbf{p},\mathbf{r})| \leq M_{G}$  and the mean value theorem in the last inequality (see Lemma F.5). Let  $\mathcal{H}(\mathbf{p})$  be a function defined as
$$
\mathcal {H} (\mathbf {p}) := \left(b _ {H} + c _ {H}\right) \cdot G _ {H} (\mathbf {p}, \mathbf {p}) \cdot \left(p _ {H} ^ {\star \star} - p _ {H}\right) + \left(b _ {L} + c _ {L}\right) \cdot G _ {L} (\mathbf {p}, \mathbf {p}) \cdot \left(p _ {L} ^ {\star \star} - p _ {L}\right). \tag {53}
$$
Then, by summing Eq. (52) over both products  $i \in \{H, L\}$ , we have that
$$
\begin{array}{l} \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t + 1} \right\| _ {2} ^ {2} \leq \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} - 2 \eta^ {t} \sum_ {i \in \{H, L \}} (b _ {i} + c _ {i}) \cdot G _ {i} (\mathbf {p} ^ {t}, \mathbf {p} ^ {t}) \cdot (p _ {i} ^ {\star \star} - p _ {i} ^ {t}) \\ + \left(\eta^ {t} M _ {G}\right) ^ {2} \sum_ {i \in \{H, L \}} \left(b _ {i} + c _ {i}\right) ^ {2} + 2 \eta^ {t} \ell_ {r} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2} \sum_ {i \in \{H, L \}} \left(b _ {i} + c _ {i}\right) \cdot \left| p _ {i} ^ {\star \star} - p _ {i} ^ {t} \right| \\ = \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} - 2 \eta^ {t} \mathcal {H} (\mathbf {p} ^ {t}) + \left(\eta^ {t} M _ {G}\right) ^ {2} \sum_ {i \in \{H, L \}} \left(b _ {i} + c _ {i}\right) ^ {2} \\ + 2 \eta^ {t} \ell_ {r} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2} \sum_ {i \in \{H, L \}} (b _ {i} + c _ {i}) \cdot | p _ {i} ^ {\star \star} - p _ {i} ^ {t} | \\ \leq \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} - \eta^ {t} \left(2 \mathcal {H} (\mathbf {p} ^ {t}) - \eta^ {t} C _ {1} - C _ {2} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2}\right) \tag {54} \\ \end{array}
$$
where we denote  $C_1 \coloneqq (M_G)^2 \cdot \sum_{i \in \{H,L\}} (b_i + c_i)^2$  and  $C_2 = 2\ell_r|\bar{p} - \underline{p}| \cdot \sum_{i \in \{H,L\}} (b_i + c_i)$ .
By Lemma F.3, there exist  $\gamma > 0$  and a open set  $U_{\gamma} \ni \mathbf{p}^{\star \star}$  such that  $\mathcal{H}(\mathbf{p}) \geq \gamma \cdot \| \mathbf{p} - \mathbf{p}^{\star \star} \|_2^2$ ,  $\forall \mathbf{p} \in U_{\gamma}$ . Consider  $\epsilon_0 > 0$  such that the  $\ell_2$ -neighborhood  $N_{\epsilon_0}^2 = \left\{\mathbf{p} \in \mathcal{P}^2 \mid \| \mathbf{p} - \mathbf{p}^{\star \star} \|_2 < \epsilon_0\right\} \subset U_{\gamma}$ . Furthermore, let  $T_{\gamma}$  be some period such that
$$
\eta^ {t} \left(\eta^ {t} C _ {1} + \sqrt {2} C _ {2} (\bar {p} - \underline {{p}})\right) \leq \frac {\epsilon_ {0} ^ {2}}{4}, \text {a n d} \eta^ {t} C _ {1} + C _ {2} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2} \leq \frac {\gamma \left(\epsilon_ {0}\right) ^ {2}}{2}, \quad \forall t > T _ {\gamma}. \tag {55}
$$
The existence of such a number  $T_{\gamma}$  follows from the fact that  $\lim_{t\to \infty}\eta^t = 0$  and  $\lim_{t\to \infty}\| \mathbf{r}^t - \mathbf{p}^t\| _2 = 0$  (see Lemma F.1). Below, we discuss two cases depending on the location of  $\mathbf{p}^t$  in  $N_{\epsilon_0}^2$ .
Case 1:  $\mathbf{p}^t\in N_{\epsilon_0 / 2}^2\subset N_{\epsilon_0}^2$  , i.e.,  $\left\| \mathbf{p}^{\star \star} - \mathbf{p}^t\right\| _2 <   \epsilon_0 / 2.$
Since  $\mathcal{H}(\mathbf{p})\geq 0,\forall \mathbf{p}\in U_{\gamma}$  by Lemma F.3, it follows from Eq. (54) and Eq. (55) that
$$
\begin{array}{l} \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t + 1} \right\| _ {2} ^ {2} \leq \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} + \eta^ {t} \left(\eta^ {t} C _ {1} + C _ {2} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2}\right) \\ \stackrel {(\Delta)} {\leq} \frac {\left(\epsilon_ {0}\right) ^ {2}}{4} + \eta^ {t} \left(\eta^ {t} C _ {1} + \sqrt {2} C _ {2} (\bar {p} - \underline {{p}})\right) \tag {56} \\ \leq \frac {(\epsilon_ {0}) ^ {2}}{4} + \frac {(\epsilon_ {0}) ^ {2}}{4} \\ <   (\epsilon_ {0}) ^ {2}, \\ \end{array}
$$
where inequality  $(\Delta)$  is due to  $\| \mathbf{r}^t - \mathbf{p}^t \|_2 \leq \sqrt{2} (\bar{p} - \underline{p})$ . Eq. (56) implies that  $\mathbf{p}^{t+1} \in N_{\epsilon_0}^2$ .
Case 2:  $\mathbf{p}^t\in N_{\epsilon_0}^2\backslash N_{\epsilon_0 / 2}^2$  , i.e.,  $\left\| \mathbf{p}^{\star \star} - \mathbf{p}^t\right\| _2\in [\epsilon_0 / 2,\epsilon_0)$
By Lemma F.3, we have that  $\mathcal{H}(\mathbf{p}^t) \geq \gamma \left\| \mathbf{p}^{\star \star} - \mathbf{p}^t \right\|_2^2 \geq \gamma (\epsilon_0)^2 / 4$ . Thus, again by Eq. (54) and Eq. (55), we have that
$$
\begin{array}{l} \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t + 1} \right\| _ {2} ^ {2} \leq \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} - \eta^ {t} \left(2 \mathcal {H} (\mathbf {p} ^ {t}) - \eta^ {t} C _ {1} - C _ {2} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2}\right) \\ \leq \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} - \eta^ {t} \left(\frac {\gamma \left(\epsilon_ {0}\right) ^ {2}}{2} - \eta^ {t} C _ {1} - C _ {2} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2}\right) \tag {57} \\ \leq \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} \\ \leq (\epsilon_ {0}) ^ {2}, \\ \end{array}
$$
which implies  $\mathbf{p}^{t + 1}\in N_{\epsilon_0}^2$ . Therefore, we conclude by induction that the price path will stay in the  $\ell_2$ -neighborhood  $N_{\epsilon_0}^2$ . This completes the proof of Theorem 5.1.
Appendix D Proof of Theorem 5.2
Proof. Recall the function  $\mathcal{H}(\mathbf{p})$  defined in Eq. (53). By Lemma F.3, there exist  $\gamma > 0$  and a open set  $U_{\gamma} \ni \mathbf{p}^{\star \star}$  such that  $\mathcal{H}(\mathbf{p}) \geq \gamma \cdot \| \mathbf{p} - \mathbf{p}^{\star \star} \|_2^2$ ,  $\forall \mathbf{p} \in U_{\gamma}$ . Consider  $\epsilon_0 > 0$  such that the  $\ell_2$ -neighborhood  $N_{\epsilon_0}^2 = \left\{\mathbf{p} \in \mathcal{P}^2 \mid \| \mathbf{p} - \mathbf{p}^{\star \star} \|_2 < \epsilon_0\right\} \subset U_{\gamma}$ . Below, we first show that the price path  $\{\mathbf{p}^t\}_{t \geq 0}$  enjoys the sublinear convergence rate in  $N_{\epsilon_0}^2$  when  $t$  is greater than some constant  $T_{\epsilon_0}$ . Then, we will show that this convergence rate also holds for any  $t \leq T_{\epsilon_0}$ .
By Part 2 in the proof of Theorem 5.1, there exists  $T_{\epsilon_0} > 0$  such that  $\mathbf{p}^t \in N_{\epsilon_0}^2$  for every  $t \geq T_{\epsilon_0}$ . Following a similar argument as Eq. (52) and Eq. (54), we have that
$$
\begin{array}{l} \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t + 1} \right\| _ {2} ^ {2} \\ \stackrel {(\Delta_ {1})} {\leq} \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} - 2 \eta^ {t} \mathcal {H} (\mathbf {p} ^ {t}) + C _ {1} (\eta^ {t}) ^ {2} + 2 \eta^ {t} \ell_ {r} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2} \sum_ {i \in \{H, L \}} (b _ {i} + c _ {i}) \cdot | p _ {i} ^ {\star \star} - p _ {i} ^ {t} | \\ \end{array}
$$