$$
\begin{array}{l} \mathbb{I}\{\mathcal{E}\} \epsilon_{*}\sum_{t = T_{0} + 1}^{T}\sum_{\substack{s\in [t - 1]\\ i_{s}\in \overline{V}_{t}}}\left|\boldsymbol{x}_{a_{t}}^{\top}\overline{M}_{\overline{V}_{t},t - 1}^{-1}\boldsymbol{x}_{a_{s}}\right| = \mathbb{I}\{\mathcal{E}\} \epsilon_{*}\sum_{t = T_{0} + 1}^{T}\sum_{k = 1}^{m_{t}}\mathbb{I}\{i_{t}\in \tilde{V}_{t,k}^{\prime}\} \sum_{\substack{s\in [t - 1]\\ i_{s}\in \overline{V}_{t,k}^{\prime}}} \left|\boldsymbol{x}_{a_{t}}^{\top}\overline{M}_{\overline{V}_{t,k}^{\prime} ,t - 1}^{-1}\boldsymbol{x}_{a_{s}}\right| \\ \leq \mathbb {I} \left\{\mathcal {E} \right\} \epsilon_ {*} \sum_ {t = T _ {0} + 1} ^ {T} \sum_ {k = 1} ^ {m _ {t}} \mathbb {I} \left\{i _ {t} \in \tilde {V} _ {t, k} ^ {\prime} \right\} \sqrt {\sum_ {\substack {s \in [ t - 1 ] \\ i _ {s} \in \overline {{V}} _ {t , k} ^ {\prime}}} 1 \sum_ {\substack {s \in [ t - 1 ] \\ i _ {s} \in \overline {{V}} _ {t , k} ^ {\prime}}} \left| \boldsymbol {x} _ {a _ {t}} ^ {\top} \overline {{M}} _ {\overline {{V}} _ {t , k} ^ {\prime}, t - 1} ^ {- 1} \boldsymbol {x} _ {a _ {s}} \right| ^ {2}} (19) \\ \leq \mathbb {I} \left\{\mathcal {E} \right\} \epsilon_ {*} \sum_ {t = T _ {0} + 1} ^ {T} \sum_ {k = 1} ^ {m _ {t}} \mathbb {I} \left\{i _ {t} \in \tilde {V} _ {t, k} ^ {\prime} \right\} \sqrt {T _ {\bar {V} _ {t , k} ^ {\prime} , t - 1} \left\| \boldsymbol {x} _ {a _ {t}} \right\| _ {\bar {M} _ {V _ {t , k} ^ {\prime} , t - 1}} ^ {2}} (20) \\ \leq \mathbb {I} \{\mathcal {E} \} \epsilon_ {*} \sum_ {t = T _ {0} + 1} ^ {T} \sqrt {\sum_ {k = 1} ^ {m _ {t}} \mathbb {I} \{i _ {t} \in \tilde {V} _ {t , k} ^ {\prime} \} \sum_ {k = 1} ^ {m _ {t}} \mathbb {I} \{i _ {t} \in \tilde {V} _ {t , k} ^ {\prime} \} T _ {\overline {{V}} _ {t , k} ^ {\prime}, t - 1} \| \boldsymbol {x} _ {a _ {t}} \| _ {\overline {{M}} _ {\overline {{V}} _ {t , k} ^ {\prime}, t - 1}} ^ {2}} (21) \\ \leq \mathbb {I} \left\{\mathcal {E} \right\} \epsilon_ {*} \sqrt {T} \sum_ {t = T _ {0} + 1} ^ {T} \sqrt {\sum_ {k = 1} ^ {m _ {t}} \mathbb {I} \left\{i _ {t} \in \tilde {V} _ {t , k} ^ {\prime} \right\} \left\| \boldsymbol {x} _ {a _ {t}} \right\| _ {\overline {{M}} _ {\overline {{V}} _ {t , k} ^ {\prime}, t - 1}} ^ {2}} (22) \\ \leq \mathbb {I} \left\{\mathcal {E} \right\} \epsilon_ {*} \sqrt {T} \sqrt {\sum_ {t = T _ {0} + 1} ^ {T} 1 \sum_ {t = T _ {0} + 1} ^ {T} \sum_ {k = 1} ^ {m _ {t}} \mathbb {I} \left\{i _ {t} \in \tilde {V} _ {t , k} ^ {\prime} \right\} \left\| \boldsymbol {x} _ {a _ {t}} \right\| _ {\overline {{M}} _ {V _ {t, k} ^ {\prime}, t - 1}} ^ {2}} (23) \\ \leq \mathbb {I} \left\{\mathcal {E} \right\} \epsilon_ {*} \sqrt {T} \sqrt {T \sum_ {t = T _ {0} + 1} ^ {T} \sum_ {j = 1} ^ {m} \mathbb {I} \left\{i _ {t} \in V _ {j} \right\} \left\| \boldsymbol {x} _ {a _ {t}} \right\| _ {\overline {{M}} _ {V _ {j}, t - 1}} ^ {2}} (24) \\ = \mathbb {I} \{\mathcal {E} \} \epsilon_ {*} T \sqrt {\sum_ {j = 1} ^ {m} \sum_ {t = T _ {0} + 1} ^ {T} \mathbb {I} \{i _ {t} \in V _ {j} \} \| \boldsymbol {x} _ {a _ {t}} \| _ {\overline {{M}} _ {V _ {j} , t - 1}} ^ {2}} \\ \leq \mathbb {I} \left\{\mathcal {E} \right\} \epsilon_ {*} T \sqrt {2 m d \log \left(1 + \frac {T}{\lambda d}\right)}, (25) \\ \end{array}
$$