SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proof. Without loss of generality, we assume that  $d \geq \eta_k$  and  $\widetilde{f}_{\eta_k}(z_0) \geq 0$ . Following the arguments in Lemma 2.6 in Venkatraman (1992), it suffices to consider two cases: (i)  $\eta_{k+1} > e$  and (ii)  $\eta_{k+1} \leq e$ . Case (i). Note that
$$
\widetilde {f} _ {\eta_ {k}} ^ {(s, e ]} (z _ {0}) = \sqrt {\frac {(e - \eta_ {k}) (\eta_ {k} - s)}{e - s}} \left\{f _ {\eta_ {k}} (z _ {0}) - f _ {\eta_ {k + 1}} (z _ {0}) \right\}
$$
$$
\widetilde {f} _ {d} ^ {(s, e ]} \left(z _ {0}\right) = \left(\eta_ {k} - s\right) \sqrt {\frac {e - d}{(e - s) (d - s)}} \left\{f _ {\eta_ {k}} \left(z _ {0}\right) - f _ {\eta_ {k + 1}} \left(z _ {0}\right) \right\}.
$$
Therefore, it follows from (56) that
$$
\widetilde {f} _ {\eta_ {k}} ^ {(s, e ]} \left(z _ {0}\right) - \widetilde {f} _ {d} ^ {(s, e ]} \left(z _ {0}\right) = \left(1 - \sqrt {\frac {(e - d) (\eta_ {k} - s)}{(d - s) (e - \eta_ {k})}}\right) \widetilde {f} _ {\eta_ {k}} ^ {(s, e ]} \left(z _ {0}\right) \geq c \Delta \left| d - \eta_ {k} \right| (e - s) ^ {- 2} \widetilde {f} _ {\eta_ {k}} ^ {(s, e ]} \left(z _ {0}\right). \tag {61}
$$
The inequality follows from the following arguments. Let  $u = \eta_k - s, v = e - \eta_k$  and  $w = d - \eta_k$ . Then
$$
\begin{array}{l} 1 - \sqrt {\frac {(e - d) (\eta_ {k} - s)}{(d - s) (e - \eta_ {k})}} - c \Delta \left| d - \eta_ {k} \right| (e - s) ^ {2} \\ = 1 - \sqrt {\frac {(v - w) u}{(u + w) v} - c \frac {\Delta w}{(u + v) ^ {2}}} \\ = \frac {w (u + v)}{\sqrt {(u + w) v} (\sqrt {(v - w) u} + \sqrt {(u + w) v})} - c \frac {\Delta w}{(u + v) ^ {2}}. \\ \end{array}
$$
The numerator of the above equals
$$
\begin{array}{l} w (u + v) ^ {3} - c \Delta w (u + w) v - c \Delta w \sqrt {u v (u + w) (v - w)} \\ \geq 2 c _ {1} \Delta w \Bigl \{(u + v) ^ {2} - \frac {c (u + w) v}{2 c _ {1}} - \frac {c \sqrt {u v (u + w) (v - w)}}{2 c _ {1}} \Bigr \} \\ \geq 2 c _ {1} \Delta w \left\{\left(1 - c / \left(2 c _ {1}\right)\right) (u + v) ^ {2} - 2 ^ {- 1 / 2} c / c _ {1} u v \right\} > 0 \\ \end{array}
$$
$$
c <   \frac {\sqrt {2} c _ {1}}{4 + 1 / \left(\sqrt {2} c _ {1}\right)}.
$$
Case (ii). Let  $g = c_{1}\Delta / 16$ . We can write
$$
\widetilde {f} _ {\eta_ {k}} ^ {(s, e ]} (z _ {0}) = a \sqrt {\frac {e - s}{(\eta_ {k} - s) (e - \eta_ {k})}}, \quad \widetilde {f} _ {\eta_ {k} + g} ^ {(s, e ]} (z _ {0}) = (a + g \theta) \sqrt {\frac {e - s}{(e - \eta_ {k} - g) (\eta_ {k} + g - s)}},
$$
$$
\begin{array}{l} a = \sum_ {j = s + 1} ^ {\eta_ {k}} \left\{f _ {j} (z _ {0}) - \frac {1}{e - s} \sum_ {j = s + 1} ^ {e} f _ {j} (z _ {0}) \right\} \\ \theta = \frac {a \sqrt {\left(\eta_ {k} + g - s\right) \left(e - \eta_ {k} - g\right)}}{g} \Big \{\frac {1}{\sqrt {\left(\eta_ {k} - s\right) \left(e - \eta_ {k}\right)}} - \frac {1}{\left(\eta_ {k} + g - s\right) \left(e - \eta_ {k} - g\right)} + \frac {b}{a \sqrt {e - s}} \Big \}, \\ \end{array}
$$
and  $b = \widetilde{f}_{\eta_k + g}^{(s,e]}\Big(z_0\Big) - \widetilde{f}_{\eta_k}^{(s,e]}\Big(z_0\Big)$ . To ease notation, let  $d - \eta_{k} = l \leq g / 2$ ,  $N_{1} = \eta_{k} - s$  and  $N_{2} = e - \eta_{k} - g$ . We have
$$
E _ {l} = \widetilde {f} _ {\eta_ {k}} ^ {(s, e ]} \left(z _ {0}\right) - \widetilde {f} _ {d} ^ {(s, e ]} \left(z _ {0}\right) = E _ {1 l} \left(1 + E _ {2 l}\right) + E _ {3 l}, \tag {62}
$$
$$
\begin{array}{l} E _ {1 l} = \frac {a l (g - l) \sqrt {e - s}}{\sqrt {N _ {1} (N _ {2} + g)} \sqrt {(N _ {1} + l) (g + N _ {2} - l)} (\sqrt {(N _ {1} + l) (g + N _ {2} - l)} + \sqrt {N _ {1} (g + N _ {2})})}, \\ E _ {2 l} = \frac {\left(N _ {2} - N _ {1}\right) \left(N _ {2} - N _ {1} - l\right)}{\left(\sqrt {\left(N _ {1} + l\right) \left(g + N _ {2} - l\right)} + \sqrt {\left(N _ {1} + g\right) N _ {2}}\right) \left(\sqrt {N _ {1} \left(g + N _ {2}\right)} + \sqrt {\left(N _ {1} + g\right) N _ {2}}\right)}, \\ \end{array}
$$
$$
E _ {3 l} = - \frac {b l}{g} \sqrt {\frac {\left(N _ {1} + g\right) N _ {2}}{\left(N _ {1} + l\right) \left(g + N _ {2} - l\right)}}.
$$
Next, we notice that  $g - l \geq c_{1}\Delta / 32$ . It holds that
$$
\left. E _ {1 l} \geq c _ {1 l} \right| d - \eta_ {k} \left| \Delta \widetilde {f} _ {\eta_ {k}} ^ {(s, e ]} \left(z _ {0}\right) (e - s) ^ {- 2}, \right. \tag {63}
$$
where  $c_{1l} > 0$  is a sufficiently small constant depending on  $c_{1}$ . As for  $E_{2l}$ , due to (59), we have
$$
E _ {2 l} \geq - 1 / 2. \tag {64}
$$
As for  $E_{3l}$ , we have
$$
\begin{array}{l} E _ {3 l} \geq - c _ {3 l, 1} b \left| d - \eta_ {k} \right| (e - s) \Delta^ {- 2} \geq - c _ {3 l, 2} b \left| d - \eta_ {k} \right| \Delta^ {- 3} (e - s) ^ {3 / 2} \widetilde {f} _ {\eta_ {k}} ^ {(s, e ]} \left(z _ {0}\right) \kappa^ {- 1} (65) \\ \geq - c _ {1 l} / 2 \left| d - \eta_ {k} \right| \Delta \widetilde {f} _ {\eta_ {k}} ^ {(s, e ]} \left(z _ {0}\right) (e - s) ^ {- 2}, (66) \\ \end{array}
$$
where the second inequality follows from (57) and the third inequality follows from (58),  $c_{3l,1}, c_{3l,2} > 0$  are sufficiently small constants, depending on all the other absolute constants. Combining (62), (63), (64) and (65), we have
$$
\left. \widetilde {f} _ {\eta_ {k}} ^ {(s, e ]} \left(z _ {0}\right) - \widetilde {f} _ {d} ^ {(s, e ]} \left(z _ {0}\right) \geq c \mid d - \eta_ {k} \right| \Delta \widetilde {f} _ {\eta_ {k}} ^ {(s, e ]} \left(z _ {0}\right) (e - s) ^ {- 2}, \tag {67}
$$
where  $c > 0$  is a sufficiently small constant. In view of (61) and (67), the proof is complete.
Consider the following events
$$
\begin{array}{l} \mathcal{A}((s,e],\rho ,\gamma) = \Bigg\{\max_{t = s + \rho +1,\ldots ,e - \rho}\sup_{z\in \mathbb{R}^{p}}|\widetilde{F}_{t,h}^{s,e}(z) - \widetilde{f}_{t}^{s,e}(z)|\leq \gamma \Bigg\}, \\ \mathcal {B} (r, \rho , \gamma) = \left\{\max  _ {N = \rho , \dots , T - r} \sup  _ {z \in \mathbb {R} ^ {p}} \left| \frac {1}{\sqrt {N}} \sum_ {t = r + 1} ^ {r + N} \left(F _ {t, h} - f _ {t}\right) \right| \leq \gamma \right\} \\ \bigcup \Bigl\{\max_{N = \rho ,\ldots ,r}\left|\frac{1}{\sqrt{N}}\sum_{t = r - N + 1}^{r}\sup_{z\in \mathbb{R}^{p}}\bigl(F_{t,h}(z) - f_{t}(z)\bigr)\right|\leq \gamma \Bigr \} . \\ \end{array}
$$
Lemma 12. Suppose Assumption 5 holds. Let  $[s,e]$  be an subinterval of  $[1,T]$  with  $e - s\leq C_R\Delta$  and contain at least one change point  $\eta_r$  with  $\min \{\eta_r - s,e - \eta_r\} \geq cT$  for some constant  $c > 0$ . Let  $\kappa_{\max}^{s,e} = \max \{\kappa_p:\min \{\eta_p - s,e - \eta_p\} \geq cT\}$ . Let
$$
b \in \arg \max  _ {t = s + \rho , \dots , e - \rho} | | \widetilde {F} _ {t, h} ^ {s, e} | | _ {L _ {2}}.
$$
For some  $c_{1} > 0$ ,  $\lambda > 0$  and  $\delta > 0$ , suppose that the following events hold
$$
\mathcal {A} ((s, e ], \rho , \gamma), \tag {68}
$$
$$
\mathcal {B} (s, \rho , \gamma) \cup \mathcal {B} (e, \rho , \gamma) \cup \bigcup_ {\eta \in \left\{\eta_ {k} \right\} _ {k = 1} ^ {K}} \mathcal {B} (\eta , \rho , \gamma) \tag {69}
$$
$$
\max  _ {t = s + \rho , \dots , e - \rho} | | \widetilde {F} _ {t, h} ^ {s, e} | | _ {L _ {2}} = | | \widetilde {F} _ {b, h} ^ {s, e} | | _ {L _ {2}} \geq c _ {1} \kappa_ {\max } ^ {s, e} \sqrt {T} \tag {70}
$$
If there exists a sufficiently small  $c_2 > 0$  such that
$$
\gamma \leq c _ {2} \kappa_ {\max } ^ {s, e} \sqrt {T} \quad \text {a n d t h a t} \quad \rho \leq c _ {2} T, \tag {71}
$$
then there exists a change point  $\eta_{k}\in (s,e)$  such that
$$
\min  \left\{e - \eta_ {k}, \eta_ {k} - s \right\} > c _ {3} T \quad a n d \quad | \eta_ {k} - b | \leq C _ {3} \max  \left\{\gamma^ {2} \kappa_ {k} ^ {- 2}, \rho \right\},
$$
where  $c_3$  is some sufficiently small constant independent of  $T$ .
Proof. Let  $z_1 \in \arg \max_{z \in \mathbb{R}^p} \left| \widetilde{f}_b^{(s,e]}(z) \right|$ . Without loss of generality, assume that  $\widetilde{f}_b^{(s,e]}\left(z_1\right) > 0$  and that  $\widetilde{f}_b^{(s,e]}\left(z_1\right)$  as a function of  $t$  is locally decreasing at  $b$ . Observe that there has to be a change point  $\eta_k \in (s,b)$ , or otherwise  $\widetilde{f}_b^{(s,e]}\left(z_1\right) > 0$  implies that  $\widetilde{f}_t^{(s,e]}\left(z_1\right)$  is decreasing, as a consequence of Lemma 10. Thus, there exists a change point  $\eta_k \in (s,b)$  satisfying that
$$
\sup  _ {z \in \mathbb {R} ^ {p}} \left| \widetilde {f} _ {\eta_ {k}} ^ {(s, e ]} (z) \right| \geq \left| \widetilde {f} _ {\eta_ {k}} ^ {(s, e ]} \left(z _ {1}\right) \right| > \left| \widetilde {f} _ {b} ^ {(s, e ]} \left(z _ {1}\right) \right| \geq \sup  _ {z \in \mathbb {R} ^ {p}} \left| \widetilde {F} _ {b} ^ {(s, e ]} (z) \right| - \gamma \geq c \kappa_ {k} \sqrt {\Delta} \tag {72}
$$
where the second inequality follows from Lemma 10, the third because of the good event  $\mathcal{A}$ , and fourth inequalities by (70) and Assumption 1, and  $c > 0$  is an absolute constant. Observe that  $(s,e)$  has to contain at least one change point or otherwise  $\sup_{z\in \mathbb{R}}\left|\widetilde{f}_{\eta_k}^{(s,e]}(z)\right| = 0$  which contradicts (72).
Step 1. In this step, we are to show that
$$
\min  \left\{\eta_ {k} - s, e - \eta_ {k} \right\} \geq \min  \left\{1, c _ {1} ^ {2} \right\} \Delta / 1 6 \tag {73}
$$
Suppose that  $\eta_{k}$  is the only change point in  $(s,e)$ . Then (73) must hold or otherwise it follows from (14) that
$$
\sup  _ {z \in \mathbb {R} ^ {p}} \left| \widetilde {f} _ {\eta_ {k}} ^ {s, e} (z) \right| \leq \kappa_ {k} \frac {c _ {1} \sqrt {\Delta}}{4},
$$
which contradicts (72).
Suppose  $(s,e)$  contains at least two change points. Then arguing by contradiction, if  $\eta_k - s < \min \left\{1,c_1^2\right\} \Delta /16$ , it must be the cast that  $\eta_k$  is the left most change point in  $(s,e)$ . Therefore
$$
\begin{array}{l} \sup  _ {z \in \mathbb {R} ^ {p}} \left| \widetilde {f} _ {\eta_ {k}} ^ {s, e} (z) \right| \leq c _ {1} / 4 \sup  _ {z \in \mathbb {R} ^ {p}} \left| \widetilde {f} _ {\eta_ {k + 1}} ^ {s, e} (z) \right| + 2 \kappa_ {k} \sqrt {\eta_ {k} - s} (74) \\ <   c _ {1} / 4 \max  _ {s + \rho <   t <   e - \rho} \sup  _ {z \in \mathbb {R} ^ {p}} \left| \widetilde {f} _ {t} ^ {s, e} (z) \right| + 2 \sqrt {\Delta} \kappa_ {k} (75) \\ \leq c _ {1} / 4 \max  _ {s + \rho <   t <   e - \rho} \sup  _ {z \in \mathbb {R} ^ {p}} \left| \widetilde {F} _ {t} ^ {s, e} (z) \right| + c _ {1} / 4 \gamma + 2 \sqrt {\Delta} \kappa_ {k} (76) \\ \leq \sup  _ {z \in \mathbb {R} ^ {p}} \left| \widetilde {F} _ {b} ^ {s, e} (z) \right| - \gamma (77) \\ \end{array}
$$
where the first inequality follows from Lemma 9, the second follows from the assumption of  $\eta_k - s$ , the third from the definition of the event  $\mathcal{A}$  and the last from (70) and Assumption 1. The last display contradicts (72), thus (73) must hold.
Step 2. Let
$$
z_{0}\in \operatorname *{arg  max}_{z\in \mathbb{R}^{p}}\bigg|\widetilde{f}_{\eta_{k}}^{s,e}(z)\bigg|.
$$
It follows from Lemma 11 that there exists  $d \in \left(\eta_k, \eta_k + c_1\Delta / 32\right)$  such that
$$
\left. \widetilde {f} _ {\eta_ {k}} ^ {s, e} \left(z _ {0}\right) - \widetilde {f} _ {d} ^ {s, e} \left(z _ {0}\right) \geq 2 \gamma . \right. \tag {78}
$$
We claim that  $b \in (\eta_k, d) \subset (\eta_k, \eta_k + c_1 \Delta / 16)$ . By contradiction, suppose that  $b \geq d$ . Then
$$
\widetilde {f} _ {b} ^ {s, e} \left(z _ {0}\right) \leq \widetilde {f} _ {d} ^ {s, e} \left(z _ {0}\right) \leq \max  _ {s <   t <   e} \sup  _ {z \in \mathbb {R} ^ {p}} \left| \widetilde {f} _ {t} ^ {s, e} (z) \right| - 2 \gamma \leq \sup  _ {z \in \mathbb {R} ^ {p}} \left| \widetilde {F} _ {b} ^ {s, e} (z) \right| - \gamma , \tag {79}
$$
where the first inequality follows from Lemma 10, the second follows from (78) and the third follows from the definition of the event  $\mathcal{A}$ . Note that (79) is a contradiction to the bound in (72), therefore we have  $b \in \left(\eta_k, \eta_k + c_1\Delta / 32\right)$ .
Step 3. Let
$$
j^{*}\in \operatorname *{arg  max}_{j = 1,\ldots ,T}\bigg| \widetilde{F}_{b}^{s,e}(X(j))\bigg|,\quad f^{s,e} = \Big(f_{s + 1}\Big(X\Big(j^{*}\Big)\Big),\ldots ,f_{e}\Big(X\Big(j^{*}\Big)\Big)\Big)^{\top}\in \mathbb{R}^{(e - s)}
$$
$$
F ^ {s, e} = \Big (\frac {1}{h ^ {p}} k \Big (\frac {X (j ^ {*}) - X (s)}{h} \Big), \ldots , \frac {1}{h ^ {p}} k \Big (\frac {X (j ^ {*}) - X (e)}{h} \Big) \Big) \in \mathbb {R} ^ {(e - s)}.
$$
By the definition of  $b$ , it holds that
$$
\left\| F ^ {s, e} - \mathcal {P} _ {b} ^ {s, e} \left(F ^ {s, e}\right) \right\| ^ {2} \leq \left\| F ^ {s, e} - \mathcal {P} _ {\eta_ {k}} ^ {s, e} \left(F ^ {s, e}\right) \right\| ^ {2} \leq \left\| F ^ {s, e} - \mathcal {P} _ {\eta_ {k}} ^ {s, e} \left(f ^ {s, e}\right) \right\| ^ {2}
$$
where the operator  $\mathcal{P}^{s,e}(\cdot)$  is defined in Lemma 21 in Wang et al. (2020). For the sake of contradiction, throughout the rest of this argument suppose that, for some sufficiently large constant  $C_3 > 0$  to be specified,
$$
\eta_ {k} + C _ {3} \lambda_ {\mathcal {A}} ^ {2} \kappa_ {k} ^ {- 2} <   b. \tag {80}
$$
We will show that this leads to the bound
$$
\left\| F ^ {s, e} - \mathcal {P} _ {b} ^ {s, e} \left(F ^ {s, e}\right) \right\| ^ {2} > \left\| F ^ {s, e} - \mathcal {P} _ {\eta_ {k}} ^ {s, e} \left(f ^ {s, e}\right) \right\| ^ {2}, \tag {81}
$$
which is a contradiction. If we can show that
$$
2 \left\langle F ^ {s, e} - f ^ {s, e}, \mathcal {P} _ {b} ^ {s, e} \left(F ^ {s, e}\right) - \mathcal {P} _ {\eta_ {k}} ^ {s, e} \left(f ^ {s, e}\right) \right\rangle <   \left\| f ^ {s, e} - \mathcal {P} _ {b} ^ {s, e} \left(f ^ {s, e}\right) \right\| ^ {2} - \left\| f ^ {s, e} - \mathcal {P} _ {\eta_ {k}} ^ {s, e} \left(f ^ {s, e}\right) \right\| ^ {2}, \tag {82}
$$
then (81) holds. To derive (82) from (80), we first note that  $\min \left\{e - \eta_k, \eta_k - s\right\} \geq \min \left\{1, c_1^2\right\} \Delta / 16$  and that  $|b - \eta_k| \leq c_1 \Delta / 32$  implies that
$$
\min  \{e - b, b - s \} \geq \min  \left\{1, c _ {1} ^ {2} \right\} \Delta / 1 6 - c _ {1} \Delta / 3 2 \geq \min  \left\{1, c _ {1} ^ {2} \right\} \Delta / 3 2 \tag {83}
$$
As for the right-hand side of (82), we have
$$
\begin{array}{l} \left\| f ^ {s, e} - \mathcal {P} _ {b} ^ {s, e} \left(f ^ {s, e}\right) \right\| ^ {2} - \left\| f ^ {s, e} - \mathcal {P} _ {\eta_ {k}} ^ {s, e} \left(f ^ {s, e}\right) \right\| ^ {2} = \left(\widetilde {f} _ {\eta_ {k}} ^ {s, e} \left(X \left(j ^ {*}\right)\right)\right) ^ {2} - \left(\widetilde {f} _ {b} ^ {s, e} \left(X \left(j ^ {*}\right)\right)\right) ^ {2} (84) \\ \geq \left(\widetilde {f} _ {\eta_ {k}} ^ {s, e} \left(X \left(j ^ {*}\right)\right) - \widetilde {f} _ {b} ^ {s, e} \left(X \left(j ^ {*}\right)\right)\right) \left| \widetilde {f} _ {\eta_ {k}} ^ {s, e} \left(X \left(j ^ {*}\right)\right) \right| (85) \\ \end{array}
$$
On the event  $\mathcal{A} \cap \mathcal{B}$ , we are to use Lemma 11. Note that (57) holds due to the fact that here we have
$$
\left| \widetilde {f} _ {\eta_ {k}} ^ {s, e} (X (j ^ {*})) \right| \geq \left| \widetilde {f} _ {b} ^ {s, e} (X (j ^ {*})) \right| \geq \left| \widetilde {F} _ {b} ^ {s, e} (X (j ^ {*})) \right| - \gamma \geq c _ {1} \kappa_ {k} \sqrt {\Delta} - \gamma \geq (c _ {1}) / 2 \kappa_ {k} \sqrt {\Delta}, \tag {86}
$$
where the first inequality follows from the fact that  $\eta_{k}$  is a true change point, the second inequality holds due to the event  $\mathcal{A}$ , the third inequality follows from (70), and the final inequality follows from (71). Towards this end, it follows from Lemma 11 that
$$
\left. \mid \widetilde {f} _ {\eta_ {k}} ^ {s, e} \left(X \left(j ^ {*}\right)\right) \right| - \left. \mid \widetilde {f} _ {b} ^ {s, e} \left(X \left(j ^ {*}\right)\right) \right| | > c | b - \eta_ {k} | \Delta | \widetilde {f} _ {\eta_ {k}} ^ {s, e} \left(X \left(j ^ {*}\right)\right) \bigg) | (e - s) ^ {- 2}. \tag {87}
$$
Combining (84), (86) and (87), we have
$$
\left\| f ^ {s, e} - \mathcal {P} _ {b} ^ {s, e} \left(f ^ {s, e}\right) \right\| ^ {2} - \left\| f ^ {s, e} - \mathcal {P} _ {\eta_ {k}} ^ {s, e} \left(f ^ {s, e}\right) \right\| ^ {2} \geq \frac {c c _ {1} ^ {2}}{4} \Delta^ {2} \kappa_ {k} \mathcal {A} ^ {2} (e - s) ^ {- 2} | b - \eta_ {k} |. \tag {88}
$$
The left-hand side of (82) can be decomposed as follows.
$$
\begin{array}{l} 2 \left\langle F ^ {s, e} - f ^ {s, e}, \mathcal {P} _ {b} ^ {s, e} \left(F ^ {s, e}\right) - \mathcal {P} _ {\eta_ {k}} ^ {s, e} \left(f ^ {s, e}\right) \right\rangle (89) \\ = 2 \left\langle F ^ {s, e} - f ^ {s, e}, \mathcal {P} _ {b} ^ {s, e} \left(F ^ {s, e}\right) - \mathcal {P} _ {b} ^ {s, e} \left(f ^ {s, e}\right) \right\rangle + 2 \left\langle Y ^ {s, e} - f ^ {s, e}, \mathcal {P} _ {b} ^ {s, e} \left(f ^ {s, e}\right) - \mathcal {P} _ {\eta_ {k}} ^ {s, e} \left(f ^ {s, e}\right) \right\rangle (90) \\ = (I) + 2 \left(\sum_ {i = 1} ^ {\eta_ {k} - s} + \sum_ {i = \eta_ {k} - s + 1} ^ {b - s} + \sum_ {i = b - s + 1} ^ {e - s}\right) \left(F ^ {s, e} - f ^ {s, e}\right) _ {i} \left(\mathcal {P} _ {b} ^ {s, e} \left(f ^ {s, e}\right) - \mathcal {P} _ {\eta_ {k}} ^ {s, e} \left(f ^ {s, e}\right)\right) _ {i} (91) \\ = (I) + (I I. 1) + (I I. 2) + (I I. 3). (92) \\ \end{array}
$$
As for the term (I), we have
$$
(I) \leq 2 \gamma^ {2}. \tag {93}
$$
As for the term (II.1), we have
$$
(I I. 1) = 2 \sqrt {\eta_ {k} - s} \Big \{\frac {1}{\sqrt {\eta_ {k} - s}} \sum_ {i = 1} ^ {\eta_ {k} - s} \left(F ^ {s, e} - f ^ {s, e}\right) _ {i} \Big \} \Big \{\frac {1}{b - s} \sum_ {i = 1} ^ {b - s} \left(f ^ {s, e}\right) _ {i} - \frac {1}{\eta_ {k} - s} \sum_ {i = 1} ^ {\eta_ {k} - s} \left(f ^ {s, e}\right) _ {i} \Big \}.
$$
In addition, it holds that
$$
\begin{array}{l} \left| \frac {1}{b - s} \sum_ {i = 1} ^ {b - s} \left(f ^ {s, e}\right) _ {i} - \frac {1}{\eta_ {k} - s} \sum_ {i = 1} ^ {\eta_ {k} - s} \left(f ^ {s, e}\right) _ {i} \right| = \frac {b - \eta_ {k}}{b - s} \Big | - \frac {1}{\eta_ {k} - s} \sum_ {i = 1} ^ {\eta_ {k} - s} f _ {i} \left(X \left(j ^ {*}\right)\right) + f _ {\eta_ {k + 1}} \left(X \left(j ^ {*}\right)\right) \Big | \\ \leq \frac {b - \eta_ {k}}{b - s} \left(C _ {R} + 1\right) \kappa_ {s _ {0}, e _ {0}} ^ {\max }, \\ \end{array}
$$
where the inequality is followed by Lemma 8. Combining with the good events,
$$
\begin{array}{l} \left. (I I. 1) \leq 2 \sqrt {\eta_ {k} - s} \frac {b - \eta_ {k}}{b - s} \left(C _ {R} + 1\right) \kappa_ {s _ {0}, e _ {0}} ^ {\max } \gamma \right. (94) \\ \leq 2 \frac {4}{\min  \left\{1 , c _ {1} ^ {2} \right\}} \Delta^ {- 1 / 2} \gamma | b - \eta_ {k} | (C _ {R} + 1) \kappa_ {s _ {0}, e _ {0}} ^ {\max } (95) \\ \end{array}
$$
As for the term (II.2), it holds that
$$
(I I. 2) \leq 2 \sqrt {\left| b - \eta_ {k} \right|} \gamma (2 C _ {R} + 3) \kappa_ {s _ {0}, e _ {0}} ^ {\max } \tag {96}
$$
As for the term (II.3), it holds that
$$
(I I. 3) \leq 2 \frac {4}{\min  \left\{1 , c _ {1} ^ {2} \right\}} \Delta^ {- 1 / 2} \gamma \left| b - \eta_ {k} \right| \left(C _ {R} + 1\right) \kappa_ {s _ {0}, e _ {0}} ^ {\max } \tag {97}
$$
Therefore, combining (94), (96), (97), (88), (89) and (93), we have that (82) holds if
$$
\left. \Delta^ {2} \kappa_ {k} ^ {2} (e - s) ^ {- 2} \right| b - \eta_ {k} | \gtrsim \max  \left\{\gamma^ {2}, \Delta^ {- 1 / 2} \gamma \left| b - \eta_ {k} \right| \kappa_ {k}, \sqrt {\left| b - \eta_ {k} \right|} \gamma \kappa_ {k} \right\} \tag {98}
$$
The second inequality holds due to Assumption 3, the third inequality holds due to (79) and the first inequality is a consequence of the third inequality and Assumption 3.