$$
\begin{array}{l} \left\| \widetilde {\mathbf {h}} _ {i} - \widetilde {\mathbf {h}} _ {i} ^ {\prime} \right\| _ {2} = \left\| \sum_ {m = 1} ^ {M} \frac {1}{N} \sum_ {j = 1} ^ {N} [ \sigma (\langle \mathbf {Q} _ {m} \mathbf {h} _ {i}, \mathbf {K} _ {m} \mathbf {h} _ {j} \rangle) \mathbf {V} _ {m} \mathbf {h} _ {j} - \sigma (\langle \mathbf {Q} _ {m} ^ {\prime} \mathbf {h} _ {i}, \mathbf {K} _ {m} ^ {\prime} \mathbf {h} _ {j} \rangle) \mathbf {V} _ {m} ^ {\prime} \mathbf {h} _ {j} ] \right\| _ {2} \\ \leq \sum_ {m = 1} ^ {M} \frac {1}{N} \sum_ {j = 1} ^ {N} \| \sigma (\langle \mathbf {Q} _ {m} \mathbf {h} _ {i}, \mathbf {K} _ {m} \mathbf {h} _ {j} \rangle) \mathbf {V} _ {m} - \sigma (\langle \mathbf {Q} _ {m} ^ {\prime} \mathbf {h} _ {i}, \mathbf {K} _ {m} ^ {\prime} \mathbf {h} _ {j} \rangle) \mathbf {V} _ {m} ^ {\prime} \| _ {\mathrm {o p}} \| \mathbf {h} _ {j} \| _ {2} \\ \leq \sum_ {m = 1} ^ {M} \frac {1}{N} \sum_ {j = 1} ^ {N} \| \mathbf {h} _ {j} \| _ {2} \left\{\left| \sigma \left(\left\langle \mathbf {Q} _ {m} \mathbf {h} _ {i}, \mathbf {K} _ {m} \mathbf {h} _ {j} \right\rangle\right) \right| \cdot \| \mathbf {V} _ {m} - \mathbf {V} _ {m} ^ {\prime} \| _ {\mathrm {o p}} \right. \\ + \left| \sigma (\langle \mathbf {Q} _ {m} \mathbf {h} _ {i}, \mathbf {K} _ {m} \mathbf {h} _ {j} \rangle) - \sigma (\langle \mathbf {Q} _ {m} ^ {\prime} \mathbf {h} _ {i}, \mathbf {K} _ {m} \mathbf {h} _ {j} \rangle) \right| \cdot \| \mathbf {V} _ {m} ^ {\prime} \| _ {\mathrm {o p}} \\ \left. + \left| \sigma (\langle \mathbf {Q} _ {m} ^ {\prime} \mathbf {h} _ {i}, \mathbf {K} _ {m} \mathbf {h} _ {j} \rangle) - \sigma (\langle \mathbf {Q} _ {m} ^ {\prime} \mathbf {h} _ {i}, \mathbf {K} _ {m} ^ {\prime} \mathbf {h} _ {j} \rangle) \right| \cdot \| \mathbf {V} _ {m} ^ {\prime} \| _ {\mathrm {o p}} \right\} \\ \leq \sum_ {m = 1} ^ {M} \frac {1}{N} \sum_ {j = 1} ^ {N} \mathsf {R} \Bigl \{B ^ {2} \mathsf {R} ^ {2} \cdot \| \mathbf {V} _ {m} - \mathbf {V} _ {m} ^ {\prime} \| _ {\mathrm {o p}} + \| \mathbf {Q} _ {m} \mathbf {h} _ {i} - \mathbf {Q} _ {m} ^ {\prime} \mathbf {h} _ {i} \| _ {2} \cdot \| \mathbf {K} _ {m} \mathbf {h} _ {j} \| _ {2} \cdot \| \mathbf {V} _ {m} ^ {\prime} \| _ {\mathrm {o p}} \\ \left. + \left\| \mathbf {Q} _ {m} ^ {\prime} \mathbf {h} _ {i} \right\| _ {2} \cdot \left\| \mathbf {K} _ {m} \mathbf {h} _ {j} - \mathbf {K} _ {m} ^ {\prime} \mathbf {h} _ {j} \right\| _ {2} \cdot \left\| \mathbf {V} _ {m} ^ {\prime} \right\| _ {\mathrm {o p}} \right\} \\ \leq \sum_ {m = 1} ^ {M} \mathsf {R} \left\{B ^ {2} \mathsf {R} ^ {2} \left\| \mathbf {V} _ {m} - \mathbf {V} _ {m} ^ {\prime} \right\| _ {\mathrm {o p}} + B \mathsf {R} ^ {2} \left\| \mathbf {Q} _ {m} - \mathbf {Q} _ {m} ^ {\prime} \right\| _ {\mathrm {o p}} \cdot \left\| \mathbf {V} _ {m} ^ {\prime} \right\| _ {\mathrm {o p}} + B \mathsf {R} ^ {2} \left\| \mathbf {K} _ {m} - \mathbf {K} _ {m} ^ {\prime} \right\| _ {\mathrm {o p}} \cdot \left\| \mathbf {V} _ {m} ^ {\prime} \right\| _ {\mathrm {o p}} \right\} \\ \leq B ^ {2} \mathsf {R} ^ {3} \Big \{\sum_ {m = 1} ^ {M} \| \mathbf {V} _ {m} - \mathbf {V} _ {m} ^ {\prime} \| _ {\mathrm {o p}} + \max _ {m} \| \mathbf {Q} _ {m} - \mathbf {Q} _ {m} ^ {\prime} \| _ {\mathrm {o p}} + \max _ {m} \| \mathbf {K} _ {m} - \mathbf {K} _ {m} ^ {\prime} \| _ {\mathrm {o p}} \Big \} \\ = B ^ {2} \mathsf {R} ^ {3} \| \boldsymbol {\theta} _ {\mathrm {a t t n}} - \boldsymbol {\theta} _ {\mathrm {a t t n}} ^ {\prime} \|, \\ \end{array}
$$