SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \stackrel {(\Delta_ {2})} {\leq} \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} - 2 \eta^ {t} \gamma \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} + C _ {1} (\eta^ {t}) ^ {2} + 2 \eta^ {t} \ell_ {r} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2} \cdot \hat {k} \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} \\ \stackrel {(\Delta_ {3})} {\leq} \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} - 2 \eta^ {t} \gamma \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} + C _ {1} (\eta^ {t}) ^ {2} + \eta^ {t} \ell_ {r} \hat {k} \left[ \frac {\gamma}{\ell_ {r} \hat {k}} \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} + \frac {\ell_ {r} \hat {k}}{\gamma} \left\| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} \right] \\ \stackrel {(\Delta_ {4})} {\leq} \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} - \eta^ {t} \gamma \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} + C _ {1} (\eta^ {t}) ^ {2} + \eta^ {t} k \left\| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2}. \tag {58} \\ \end{array}
$$
In step  $(\Delta_1)$ ,  $\ell_r$  is the Lipschitz constant defined in Eq. (100) and the constant  $C_1$  is defined in Eq. (54). In step  $(\Delta_2)$ , we utilize Lemma F.3 and the following inequality
$$
\begin{array}{l} \sum_ {i \in \{H, L \}} \left(b _ {i} + c _ {i}\right) \left| p _ {i} ^ {\star \star} - p _ {i} ^ {t} \right| \leq \max  _ {i \in \{H, L \}} \left\{b _ {i} + c _ {i} \right\} \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {1} \\ \leq \sqrt {2} \max  _ {i \in \{H, L \}} \left\{b _ {i} + c _ {i} \right\} \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} \tag {59} \\ = \hat {k} \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2}, \\ \end{array}
$$
where we define  $\hat{k} \coloneqq \sqrt{2}\max_{i\in \{H,L\}}\{b_i + c_i\}$ . Step  $(\Delta_3)$  in Eq. (58) follows from the inequality of arithmetic and geometric means, i.e.,  $2xy \leq Ax^2 + (1 / A)y^2$  for any constant  $A > 0$ . The value of constant  $k$  in  $(\Delta_4)$  is given by  $k \coloneqq (\ell_r\hat{k})^2 /\gamma = 2\big(\ell_r\max_{i\in \{H,L\}}\{b_i + c_i\}\big)^2 /\gamma$ .
To upper-bound the right-hand side of Eq. (58), we first focus on the term  $\left\| \mathbf{r}^t - \mathbf{p}^t \right\|_2^2$  and inductively show that
$$
\left\| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} = \mathcal {O} \left(\frac {1}{t ^ {2}}\right), \quad \forall t \geq T _ {\lambda} := \frac {\sqrt {\lambda + 1}}{\sqrt {\lambda + 1} - \sqrt {2 \lambda}}, \tag {60}
$$
where  $\lambda := (1 + \alpha^2) / 2 < 1$ . We use the notation  $\mathbf{D}^t := \left(D_H^t, D_L^t\right) = \left((b_H + c_H)G_H(\mathbf{p}^t, \mathbf{r}^t), (b_L + c_L)G_L(\mathbf{p}^t, \mathbf{r}^t)\right)$ , where we recall that  $D_i^t$  is the partial derivative specified in Eq. (9) and function  $G_i(\cdot, \cdot)$  is defined in Eq. (25). Then, the term  $\left\| \mathbf{r}^t - \mathbf{p}^t \right\|_2^2$  can be upper-bounded as follows
$$
\begin{array}{l} \left\| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} \\ = \left\| \alpha \mathbf {r} ^ {t - 1} + (1 - \alpha) \mathbf {p} ^ {t - 1} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} \\ = \left\| \alpha \left(\mathbf {r} ^ {t - 1} - \mathbf {p} ^ {t - 1}\right) + \left(\mathbf {p} ^ {t - 1} - \mathbf {p} ^ {t}\right) \right\| _ {2} ^ {2} \\ = \alpha^ {2} \left\| \mathbf {r} ^ {t - 1} - \mathbf {p} ^ {t - 1} \right\| _ {2} ^ {2} + \left\| \mathbf {p} ^ {t - 1} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} + 2 \alpha \left(\mathbf {r} ^ {t - 1} - \mathbf {p} ^ {t - 1}\right) ^ {\top} \left(\mathbf {p} ^ {t - 1} - \mathbf {p} ^ {t}\right) \\ \end{array}
$$
$$
\begin{array}{l} \stackrel {(\Delta_ {1})} {\leq} \alpha^ {2} \| \mathbf {r} ^ {t - 1} - \mathbf {p} ^ {t - 1} \| _ {2} ^ {2} + \| \eta^ {t - 1} \mathbf {D} ^ {t - 1} \| _ {2} ^ {2} + 2 \alpha \| \mathbf {r} ^ {t - 1} - \mathbf {p} ^ {t - 1} \| _ {2} \| \eta^ {t - 1} \mathbf {D} ^ {t - 1} \| _ {2} \\ \stackrel {(\Delta_ {2})} {\leq} \alpha^ {2} \left\| \mathbf {r} ^ {t - 1} - \mathbf {p} ^ {t - 1} \right\| _ {2} ^ {2} + \left\| \eta^ {t - 1} \mathbf {D} ^ {t - 1} \right\| _ {2} ^ {2} + \frac {1 - \alpha^ {2}}{2} \left\| \mathbf {r} ^ {t - 1} - \mathbf {p} ^ {t - 1} \right\| _ {2} ^ {2} + \frac {2 \alpha^ {2}}{1 - \alpha^ {2}} \left\| \eta^ {t - 1} \mathbf {D} ^ {t - 1} \right\| _ {2} ^ {2} \\ = \frac {1 + \alpha^ {2}}{2} \left\| \mathbf {r} ^ {t - 1} - \mathbf {p} ^ {t - 1} \right\| _ {2} ^ {2} + \frac {1 + \alpha^ {2}}{1 - \alpha^ {2}} \left\| \eta^ {t - 1} \mathbf {D} ^ {t - 1} \right\| _ {2} ^ {2} \\ = \lambda \| \mathbf {r} ^ {t - 1} - \mathbf {p} ^ {t - 1} \| _ {2} ^ {2} + \frac {1 + \alpha^ {2}}{1 - \alpha^ {2}} \cdot \left(\sum_ {i \in \{H, L \}} (b _ {i} + c _ {i}) ^ {2} [ G _ {i} (\mathbf {p} ^ {t - 1}, \mathbf {r} ^ {t - 1}) ] ^ {2}\right) \cdot (\eta^ {t - 1}) ^ {2} \\ \end{array}
$$
$$
\stackrel {(\Delta_ {3})} {\leq} \lambda \left\| \mathbf {r} ^ {t - 1} - \mathbf {p} ^ {t - 1} \right\| _ {2} ^ {2} + \underbrace {\left(\frac {1 + \alpha^ {2}}{1 - \alpha^ {2}} \cdot \left(M _ {G}\right) ^ {2} \sum_ {i \in \{H , L \}} \left(b _ {i} + c _ {i}\right) ^ {2}\right)} _ {\lambda_ {0}} \cdot (\eta^ {t - 1}) ^ {2}, \tag {61}
$$
where  $(\Delta_1)$  holds due to the Cauchy-Schwarz inequality and the property of the projection operator (see Line 5 in Algorithm 1). Step  $(\Delta_2)$  is derived from the inequality of arithmetic and geometric means. Lastly, step  $(\Delta_3)$  applies the upper bound on function  $G_{i}(\cdot ,\cdot)$  in Lemma F.5 and the definition of  $C_1$  in Eq. (54). For the simplicity of notation, we denote the coefficient of  $(\eta^{t - 1})^2$  in the last line as  $\lambda_0$ .
Let the step-size be  $\eta^t = d_\eta / (t + 1)$  for  $t \geq 0$ , where  $d_\eta$  is some constant that will be determined later. Suppose that there exists a constant  $d_{rp}$  such that
$$
\left\| \mathbf {r} ^ {t - 1} - \mathbf {p} ^ {t - 1} \right\| _ {2} ^ {2} \leq \frac {d _ {r p}}{(t - 1) ^ {2}}, \tag {62}
$$
for some  $t \geq T_{\lambda} + 1$ . Then, together with Eq. (61), we have that
$$
\begin{array}{l} \left\| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} \leq \lambda \left\| \mathbf {r} ^ {t - 1} - \mathbf {p} ^ {t - 1} \right\| _ {2} ^ {2} + \lambda_ {0} (\eta^ {t - 1}) ^ {2} \\ \leq \frac {\lambda d _ {r p}}{(t - 1) ^ {2}} + \frac {\lambda_ {0} (d _ {\eta}) ^ {2}}{t ^ {2}} \\ = \frac {\lambda d _ {r p}}{t ^ {2}} \cdot \frac {t ^ {2}}{(t - 1) ^ {2}} + \frac {\lambda_ {0} (d _ {\eta}) ^ {2}}{t ^ {2}} \tag {63} \\ \stackrel {(\Delta)} {\leq} \frac {\lambda d _ {r p}}{t ^ {2}} \cdot \frac {\lambda + 1}{2 \lambda} + \frac {\lambda_ {0} (d _ {\eta}) ^ {2}}{t ^ {2}} \\ = \frac {0 . 5 (\lambda + 1) \cdot d _ {r p} + \lambda_ {0} (d _ {\eta}) ^ {2}}{t ^ {2}}, \\ \end{array}
$$
where the inequality  $(\Delta)$  results from the choice of  $T_{\lambda}$ . Hence, the induction follows if  $0.5(\lambda + 1)d_{rp} + \lambda_0(d_\eta)^2 \leq d_{rp}$ , which is further equivalent to
$$
d _ {r p} \geq \frac {2 \lambda_ {0} \left(d _ {\eta}\right) ^ {2}}{1 - \lambda}. \tag {64}
$$
Lastly, the base case of the induction requires that  $\left\| \mathbf{r}^{T_{\lambda}} - \mathbf{p}^{T_{\lambda}}\right\|_2^2 \leq \frac{d_{rp}}{(T_{\lambda})^2}$ . Therefore, by Eq. (64) and the definition of feasible price range  $\mathcal{P} = [\underline{p}, \overline{p}]$ , it suffices to choose
$$
d _ {r p} = \max  \left\{\frac {2 \lambda_ {0} (d _ {\eta}) ^ {2}}{1 - \lambda}, 2 \left(T _ {\lambda}\right) ^ {2} (\bar {p} - \underline {{p}}) ^ {2} \right\}, \tag {65}
$$
where the constants  $\lambda_0$  and  $T_{\lambda}$  are respectively defined in Eq. (61) and Eq. (60). Note that, under this choice of constant  $d_{rp}$ , it also holds that
$$
\left\| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} \leq 2 (\bar {p} - \underline {{p}}) ^ {2} <   \frac {2 (T _ {\lambda}) ^ {2} (\bar {p} - \underline {{p}}) ^ {2}}{t ^ {2}} \leq \frac {d _ {r p}}{t ^ {2}}, \quad \forall 1 \leq t <   T _ {\lambda}. \tag {66}
$$
Together with Eq. (60), we derive that
$$
\left\| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} \leq \frac {d _ {r p}}{t ^ {2}}, \quad \forall t \geq 1. \tag {67}
$$
For every  $t \geq T_{\epsilon_0}$ , we can further upper-bound the right-hand side of Eq. (58) by exploiting the upper bound of  $\left\| \mathbf{r}^t - \mathbf{p}^t \right\|_2^2$  in Eq. (67) and the choice of  $\eta_t = d_\eta / (t + 1)$ :
$$
\begin{array}{l} \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t + 1} \right\| _ {2} ^ {2} \leq \left(1 - \frac {\gamma d _ {\eta}}{t + 1}\right) \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} + \frac {C _ {1} (d _ {\eta}) ^ {2}}{(t + 1) ^ {2}} + \frac {d _ {\eta} k \cdot d _ {r p}}{(t + 1) t ^ {2}} \\ \leq \left(1 - \frac {\gamma d _ {\eta}}{t + 1}\right) \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} + \frac {1}{t (t + 1)} \underbrace {\left(C _ {1} \left(d _ {\eta}\right) ^ {2} + \frac {d _ {\eta} k d _ {r p}}{T _ {\epsilon_ {0}}}\right)} _ {C _ {3}}. \tag {68} \\ \end{array}
$$
Now, we inductively show that
$$
\left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} = \mathcal {O} \left(\frac {1}{t}\right), \quad \forall t \geq T _ {\epsilon_ {0}}. \tag {69}
$$
Suppose there exists a constant  $d_p$  such that for a fixed period  $t \geq T_{\epsilon_0}$
$$
\left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} \leq \frac {d _ {p}}{t}. \tag {70}
$$
To establish the induction, it suffices for the following inequality to hold
$$
\left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t + 1} \right\| _ {2} ^ {2} \leq \left(1 - \frac {\gamma d _ {\eta}}{t + 1}\right) \frac {d _ {p}}{t} + \frac {C _ {3}}{t (t + 1)} \leq \frac {d _ {p}}{t + 1}, \tag {71}
$$
which is further equivalent to
$$
\left(\gamma d _ {\eta} - 1\right) d _ {p} \geq C _ {3}. \tag {72}
$$
To satisfy the base case of the induction, we can select  $d_p$  such that  $d_p \geq T_{\epsilon_0} \cdot \left\| \mathbf{p}^{\star \star} - \mathbf{p}^{T_{\epsilon_0}} \right\|_2^2$ . In summary, one possible set of constants  $(d_\eta, d_{rp}, d_p)$  that satisfies all the requirements can be
$$
d _ {\eta} = \frac {2}{\gamma}, \quad d _ {r p} = \max  \left\{\frac {2 \lambda_ {0} \left(d _ {\eta}\right) ^ {2}}{1 - \lambda}, 2 \left(T _ {\lambda}\right) ^ {2} (\bar {p} - \underline {{p}}) ^ {2} \right\}, \quad d _ {p} = \max  \left\{C _ {3}, 2 T _ {\epsilon_ {0}} (\bar {p} - \underline {{p}}) ^ {2} \right\}, \tag {73}
$$
where the constants  $\lambda_0, T_\lambda$ , and  $C_3$  are respectively defined in Eq. (61), Eq. (60), and Eq. (68). Now, for any period  $1 \leq t < T_{\epsilon_0}$ , it follows that
$$
\left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} \leq 2 (\bar {p} - \underline {{p}}) ^ {2} <   \frac {2 T _ {\epsilon_ {0}} (\bar {p} - \underline {{p}}) ^ {2}}{t} \leq \frac {d _ {p}}{t}. \tag {74}
$$
Together with Eq. (69), this proves the convergence rate for the price path, i.e.,
$$
\left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} \leq \frac {d _ {p}}{t}, \quad \forall t \geq 1. \tag {75}
$$
Finally, the convergence rate of the reference price path can be deduced from the following triangular inequality:
$$
\begin{array}{l} \left\| \mathbf {p} ^ {\star \star} - \mathbf {r} ^ {t} \right\| _ {2} ^ {2} = \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} + \mathbf {p} ^ {t} - \mathbf {r} ^ {t} \right\| _ {2} ^ {2} \\ \leq 2 \left\| \mathbf {p} ^ {\star \star} - \mathbf {p} ^ {t} \right\| _ {2} ^ {2} + 2 \left\| \mathbf {p} ^ {t} - \mathbf {r} ^ {t} \right\| _ {2} ^ {2} \\ \stackrel {(\Delta)} {\leq} \frac {2 d _ {p}}{t} + \frac {2 d _ {r p}}{t ^ {2}} \tag {76} \\ \leq \frac {2 d _ {p} + 2 d _ {r p}}{t}, \quad \forall t \geq 1, \\ \end{array}
$$
where  $(\Delta)$  follows from Eq. (67) and Eq. (75). Therefore, it suffices to choose  $d_r \coloneqq 2d_p + 2d_{rp}$ , and this completes the proof.
Appendix E Proof of Theorem 6.1
Proof. Recall the definition of  $G_{i}(\mathbf{p},\mathbf{r})$  from Eq. (25). We can write the inexact derivative as  $D_{i}^{t} = (b_{i} + c_{i})G_{i}(\mathbf{p}^{t},\mathbf{r}^{t}) + n_{i}^{t}$ , where  $n_i^t$  represents the error satisfying  $|n_i^t| < \delta, \forall i \in \{H,L\}$  and  $\forall t \geq 0$ .
We also recall the two-part proof for Theorem 5.1: in Part 1, we show that the price path and reference price paths converge towards the SNE and visit the neighborhood  $N_{\epsilon}^{1}$  infinitely many times, for any  $\epsilon > 0$ . In Part 2, we establish that, given  $\epsilon$  is below a certain threshold  $\epsilon_0$ , the price vector would remain in the neighborhood  $N_{\epsilon}^{2}$  after entering it with a sufficiently small step-size. Since the relative scale of  $\delta$  and  $\epsilon_0$  is unsure, our proof below is mainly based on Part 1.
In Part 1, the original proof assuming the exact gradient oracle employs a contradiction-based argument. Suppose the price path does not converge to the SNE, the proof consists of the following major steps (different from Appendix C, we use  $\epsilon$  in lieu of  $\epsilon_0$  to avoid confusion in the following proof):
- In Eq. (29), it is demonstrated that the price path steadily approaches the SNE if it stays in the same quadrant defined in Eq. (28). The main technique we use is  $\mathcal{G}(\mathbf{p}) > M_{\epsilon}$  when  $\varepsilon (\mathbf{p}) > \epsilon$ . The definition of  $\mathcal{G}(\mathbf{p})$  and the validation for the technique are stated in Lemma F.2.
- In Eq. (34), we show that the price path only oscillates between adjacent quadrants provided the step-sizes are sufficiently small.
- From Eqs (37) to (41), we prove that even when the price path does not stay in the same quadrant, it will still converge toward the SNE if it is at the boundary regions between two quadrants. The pivotal inequality employed here is again  $\mathcal{G}((p_H^{\star \star}, p_L^{T_{1}^{\epsilon_1}})) \geq M_{\epsilon - \epsilon_1}$ , provided  $\varepsilon((p_H^{\star \star}, p_L^{T_{1}^{\epsilon_1}})) > \epsilon - \epsilon_1$  (see the end of Eqs. (38) and (40)).
- Finally, in Eq. (44) and subsequent equations, we provide supplementary justification for the above bullet that the price path remains adjacent to the boundaries given that the step-size is small. A crucial consideration here is opting for a much smaller  $\epsilon_{1}$  relative to  $\epsilon$  when defining the boundary region. In addition, the inequality  $\mathcal{G}((p_H^{\tau^{\epsilon_1}}, p_L^{\star \star})) \geq M_{\widehat{\epsilon}_1}$  is also utilized in Eq. (50).
To summarize, the key to the proof is Lemma F.2., i.e.,  $\mathcal{G}(\mathbf{p}) > M_{\epsilon} > 0$  as long as  $\varepsilon (\mathbf{p}) > \epsilon$ . By definition,  $\mathcal{G}(\mathbf{p}^t)$  approximately characterizes the difference  $\varepsilon (\mathbf{p}^t) - \varepsilon (\mathbf{p}^{t + 1})$  as seen in Eq. (29). As a result, the effect of the lower bound  $M_{\epsilon}$  is that, if other terms can be upper-bounded by  $M_{\epsilon}$ , we can show that the updated price is closer to the SNE. For example, in the right-hand side of Eq. (29), once the term  $2\ell_r\| \mathbf{r}^t -\mathbf{p}^t\| _2$  is below  $M_{\epsilon}$ , it suggests that the price vector is heading towards the SNE.
With inexact first-order oracles, the immediate consequence is that there will always be an error term accompanying  $\mathcal{G}(\mathbf{p})$ . For instance, given  $D_{i}^{t} = (b_{i} + c_{i})G_{i}(\mathbf{p}^{t},\mathbf{r}^{t}) + n_{i}^{t}$ , we observe that Eq. (29) evolves to
$$
\varepsilon \left(\mathbf {p} ^ {t + 1}\right) \leq \varepsilon \left(\mathbf {p} ^ {t}\right) - \eta^ {t} \left(M _ {\epsilon} - 2 \ell_ {r} \| \mathbf {r} ^ {t} - \mathbf {p} ^ {t} \| _ {2} - \sum_ {i \in \{H, L \}} \frac {n _ {i} ^ {t}}{\left(b _ {i} + c _ {i}\right)}\right). \tag {77}
$$
This observation is consistent across the proof. Thus, if the error  $n_i^t$  is substantially smaller than  $M_{\epsilon}$ , the proof of Theorem 5.1 remains valid, implying that the price vector converges towards the SNE. The subtlety arises when the size of  $n_i^t$  is comparable with  $M_{\epsilon}$ . Under such circumstances, the analysis in both Eq. (29) and Eqs. (37) to (41) becomes invalid, as the errors are substantial enough to negate any assurance that the price path strictly approaches the SNE.
However, if the error  $n_i^t$  is similar in magnitude to  $\mathcal{G}(\mathbf{p})$ , we can show that the price vector  $\mathbf{p}$  is already close to the SNE  $\mathbf{p}^{\star \star}$ . More precisely, since the errors are bounded by  $\delta$ , it is equivalent to show that  $\mathcal{G}(\mathbf{p}) = \mathcal{O}(\delta)$  also implies  $\| \mathbf{p} - \mathbf{p}^{\star \star}\|_2 = \mathcal{O}(\delta)$ . This can be proved by a refined version of Lemma F.2.
Lemma E.1 (Refined Lemma F.2) Let  $\mathcal{G}(\mathbf{p})$  be the function defined in Eq. (86). Then, it holds that
$$
\mathcal {G} (\mathbf {p}) \geq \sum_ {i} \frac {\left| p _ {i} - p _ {i} ^ {\star \star} \right|}{\left(b _ {i} + c _ {i}\right) p _ {i} ^ {\star \star} \bar {p}}, \tag {78}
$$
i.e.,  $\mathcal{G}(\mathbf{p})$  is lower-bounded by a weighted  $\ell_1$  distance between  $\mathbf{p}$  and  $\mathbf{p}^{\star \star}$ . By the equivalence of norms in the Euclidean space, there exists some constant  $C$  such that  $\mathcal{G}(\mathbf{p}) \geq C \cdot \| \mathbf{p} - \mathbf{p}^{\star \star} \|_2$ .
Proof. Similar to the proof of the original Lemma F.2, we separately consider the four possible scenarios where  $\mathbf{p}$  belongs to one of the quadrant defined in Eq. (28).
1. Suppose  $p_H > p_H^{\star \star}$  and  $p_L\geq p_L^{\star \star}$ , i.e.,  $\mathbf{p}\in N_1$ . Since  $G_{i}(\mathbf{p}^{\star \star}) = 0$ , we have that
$$
\mathcal {G} (\mathbf {p}) = \mathcal {G} (\mathbf {p}) - \mathcal {G} \left(\mathbf {p} ^ {\star \star}\right) = \sum_ {i} \frac {1}{b _ {i} + c _ {i}} \left(\frac {1}{p _ {i} ^ {\star \star}} - \frac {1}{p _ {i}}\right) + d _ {0} (\mathbf {p}, \mathbf {p}) - d _ {0} \left(\mathbf {p} ^ {\star \star}, \mathbf {p} ^ {\star \star}\right). \tag {79}
$$
By definition of the non-purchase probability, we observe that  $d_0(\mathbf{p},\mathbf{p}) - d_0(\mathbf{p}^{\star \star},\mathbf{p}^{\star \star}) > 0$ . Hence, the above equation implies that
$$
\mathcal {G} (\mathbf {p}) > \sum_ {i} \frac {1}{b _ {i} + c _ {i}} \cdot \frac {p _ {i} - p _ {i} ^ {\star \star}}{p _ {i} ^ {\star \star} p _ {i}} \geq \sum_ {i} \frac {1}{b _ {i} + c _ {i}} \cdot \frac {p _ {i} - p _ {i} ^ {\star \star}}{p _ {i} ^ {\star \star} \bar {p}} = \sum_ {i} \frac {| p _ {i} - p _ {i} ^ {\star \star} |}{(b _ {i} + c _ {i}) p _ {i} ^ {\star \star} \bar {p}}, \tag {80}
$$
where we use the fact that  $p_i \in \mathcal{P} = [\underline{p}, \overline{p}]$  and the presumption  $\mathbf{p} \in N_1$ .
2. Suppose  $p_H \leq p_H^{\star \star}$  and  $p_L > p_L^{\star \star}$ , i.e.,  $\mathbf{p} \in N_2$ . Again, using the fact that  $G_i(\mathbf{p}^{\star \star}) = 0$ , we derive that
$$
\begin{array}{l} \mathcal {G} (\mathbf {p}) = \frac {1}{b _ {H} + c _ {H}} \left(\frac {1}{p _ {H}} - \frac {1}{p _ {H} ^ {\star \star}}\right) + \frac {1}{b _ {L} + c _ {L}} \left(\frac {1}{p _ {L} ^ {\star \star}} - \frac {1}{p _ {L}}\right) \tag {81} \\ + \left[ d _ {H} (\mathbf {p}, \mathbf {p}) - d _ {H} \left(\mathbf {p} ^ {\star \star}, \mathbf {p} ^ {\star \star}\right) \right] + \left[ d _ {L} \left(\mathbf {p} ^ {\star \star}, \mathbf {p} ^ {\star \star}\right) - d _ {L} (\mathbf {p}, \mathbf {p}) \right]. \\ \end{array}
$$
Since  $\mathbf{p} \in N_2$ , we have that  $d_H(\mathbf{p}, \mathbf{p}) - d_H(\mathbf{p}^{\star \star}, \mathbf{p}^{\star \star}) > 0$  and  $d_L(\mathbf{p}^{\star \star}, \mathbf{p}^{\star \star}) - d_L(\mathbf{p}, \mathbf{p}) > 0$ . Hence, similar to the first case, it follows that
$$
\mathcal {G} (\mathbf {p}) > \frac {1}{b _ {H} + c _ {H}} \left(\frac {1}{p _ {H}} - \frac {1}{p _ {H} ^ {\star \star}}\right) + \frac {1}{b _ {L} + c _ {L}} \left(\frac {1}{p _ {L} ^ {\star \star}} - \frac {1}{p _ {L}}\right) \geq \sum_ {i} \frac {\left| p _ {i} - p _ {i} ^ {\star \star} \right|}{\left(b _ {i} + c _ {i}\right) p _ {i} ^ {\star \star} \bar {p}}. \tag {82}
$$
The same conclusion can be drawn when  $\mathbf{p} \in N_3 \cup N_4$ , and the proof is intrinsically the same. Hence, we conclude the proof of this lemma.
Finally, due to Lemma E.1, we have that  $\| \mathbf{p} - \mathbf{p}^{\star \star}\| _2\leq 1 / C\cdot \mathcal{G}(\mathbf{p})$ . Therefore, when  $\mathcal{G}(\mathbf{p}) = \mathcal{O}(\delta)$ , we also have  $\| \mathbf{p} - \mathbf{p}^{\star \star}\| _2 = \mathcal{O}(\delta)$ , i.e., the price vector is already in a  $\mathcal{O}(\delta)$  neighborhood of the SNE. This completes the proof of Theorem 6.1.
Appendix F Supporting lemmas
Lemma F.1 (Convergence of price to reference price) Let  $\{\mathbf{p}^t\}_{t\geq 0}$  and  $\{\mathbf{r}^t\}_{t\geq 0}$  be the price path and reference path generated by Algorithm 1 with non-increasing step-sizes  $\{\eta^t\}_{t\geq 0}$  such that  $\lim_{t\to \infty}\eta^t = 0$ . Then, their difference  $\{\mathbf{r}^t -\mathbf{p}^t\}_{t\geq 0}$  converges to 0 as  $t$  goes to infinity.
Proof. First, we recall that  $D_{i}^{t} = (b_{i} + c_{i}) \cdot G_{i}(\mathbf{p}^{t},\mathbf{r}^{t})$ , where  $G_{i}(\mathbf{p},\mathbf{r})$  is the scaled partial derivative defined in (25). Thus, it follows from Lemma F.5 that  $|D_i^t |\leq (b_i + c_i)M_G$ . Since  $\{\eta^t\}_{t\geq 0}$  is a non-increasing sequence with  $\lim_{t\to \infty}\eta^t = 0$ , for any constant  $\eta >0$ , there exists  $T_{\eta}\in \mathbb{N}$  such that  $|\eta^t D_i^t |\leq \eta$  for every  $t\geq T_{\eta}$  and for every  $i\in \{H,L\}$ . Therefore, it holds that
$$
\left| p _ {i} ^ {t + 1} - p _ {i} ^ {t} \right| = \left| \operatorname {P r o j} _ {\mathcal {P}} \left(p _ {i} ^ {t} + \eta^ {t} D _ {i} ^ {t}\right) - p _ {i} ^ {t} \right| \leq \left| \eta^ {t} D _ {i} ^ {t} \right| \leq \eta , \quad \forall t \geq T _ {\eta}, \tag {83}
$$
where the first inequality is due to the property of the projection operator. Then, by the reference price update, we have for  $t \geq T_{\eta}$  and for  $i \in \{H, L\}$  that
$$
\begin{array}{l} \left| r _ {i} ^ {t + 1} - p _ {i} ^ {t + 1} \right| = \left| \alpha r _ {i} ^ {t} + (1 - \alpha) p _ {i} ^ {t} - p _ {i} ^ {t + 1} \right| \\ = \left| \alpha \left(r _ {i} ^ {t} - p _ {i} ^ {t}\right) + \left(p _ {i} ^ {t + 1} - p _ {i} ^ {t}\right) \right| (84) \\ \leq \alpha \left| r _ {i} ^ {t} - p _ {i} ^ {t} \right| + \left| p _ {i} ^ {t + 1} - p _ {i} ^ {t} \right| (84) \\ \leq \alpha \left| r _ {i} ^ {t} - p _ {i} ^ {t} \right| + \eta , \\ \end{array}
$$
where the last line follows from the upper bound in Eq. (83). Applying Eq. (84) recursively from  $t$  to  $T_{\eta}$ , we further derive that
$$
\begin{array}{l} \left| r _ {i} ^ {t + 1} - p _ {i} ^ {t + 1} \right| \leq \alpha^ {t + 1 - T _ {\eta}} \cdot \left| r _ {i} ^ {T _ {\eta}} - p _ {i} ^ {T _ {\eta}} \right| + \eta \sum_ {\tau = T _ {\eta}} ^ {t} \alpha^ {\tau - T _ {\eta}} \tag {85} \\ \leq \alpha^ {t + 1 - T _ {\eta}} \cdot (\bar {p} - \underline {{p}}) + \frac {\eta}{1 - \alpha}, \quad \forall i \in \{H, L \}. \\ \end{array}
$$
Since  $\eta$  can be arbitrarily close to 0, we have that  $|r_i^t - p_i^t| \to 0$  as  $t \to \infty$ , which completes the proof of the convergence.
Lemma F.2 Define the function  $\mathcal{G}(\mathbf{p})$  as
$$
\mathcal {G} (\mathbf {p}) := \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H}\right) \cdot G _ {H} (\mathbf {p}, \mathbf {p}) + \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L}\right) \cdot G _ {L} (\mathbf {p}, \mathbf {p}), \tag {86}
$$