$$
\begin{array}{l} (\Theta - \Theta_ {c}) ^ {T} (g _ {i} - h _ {i}) \\ = \left(\Theta - \Theta_ {c}\right) ^ {T} \left(\frac {1}{\left| \mathcal {D} _ {i} \cup \mathcal {D} _ {i} ^ {\prime} \right|} \sum_ {\mathbf {z} ^ {\prime} \in \mathcal {D} _ {i} \cup \mathcal {D} _ {i} ^ {\prime}} \nabla_ {\Theta} \ell \left(\mathbf {z} ^ {\prime}; \Theta\right) - \frac {1}{\left| \mathcal {D} _ {i} \right|} \sum_ {\mathbf {z} \in \mathcal {D} _ {i}} \nabla_ {\Theta_ {c}} \ell \left(\mathbf {z}; \Theta_ {c}\right)\right) \quad \triangleright \text {d e f i n i t i o n o f} g _ {i} \text {a n d} h _ {i} (14) \\ = (\Theta - \Theta_ {c}) ^ {T} \left(\frac {1}{(1 + r _ {i} ^ {t}) | \mathcal {D} _ {i} |} \sum_ {\mathbf {z} ^ {\prime} \in \mathcal {D} _ {i} \cup \mathcal {D} _ {i} ^ {\prime}} \nabla_ {\Theta} \ell (\mathbf {z} ^ {\prime}; \Theta) - \frac {1}{| \mathcal {D} _ {i} |} \sum_ {\mathbf {z} \in \mathcal {D} _ {i}} \nabla_ {\Theta_ {c}} \ell (\mathbf {z}; \Theta_ {c})\right) \quad \triangleright r _ {i} ^ {t} = \frac {| \mathcal {D} _ {i} ^ {\prime} |}{| \mathcal {D} _ {i} |} (1 5) \\ = \frac {1}{\left| \mathcal {D} _ {i} \right| \left(1 + r _ {i} ^ {t}\right)} \left(\Theta - \Theta_ {c}\right) ^ {T} \left(\sum_ {\mathbf {z} ^ {\prime} \in \mathcal {D} _ {i} \cup \mathcal {D} _ {i} ^ {\prime}} \nabla_ {\Theta} \ell \left(\mathbf {z} ^ {\prime}; \Theta\right) - \left(1 + r _ {i} ^ {t}\right) \sum_ {\mathbf {z} \in \mathcal {D} _ {i}} \nabla_ {\Theta_ {c}} \ell \left(\mathbf {z}; \Theta_ {c}\right)\right) (16) \\ = \frac {1}{\left| \mathcal {D} _ {i} \right| \left(1 + r _ {i} ^ {t}\right)} \left(\Theta - \Theta_ {c}\right) ^ {T} \left(\sum_ {\mathbf {z} ^ {\prime} \in \mathcal {D} _ {i}} \nabla_ {\Theta} \ell \left(\mathbf {z} ^ {\prime}; \Theta\right) - \sum_ {\mathbf {z} \in \mathcal {D} _ {i}} \nabla_ {\Theta_ {c}} \ell (\mathbf {z}; \Theta_ {c}) \right. \\ + \sum_ {\mathbf {z} ^ {\prime} \in \mathcal {D} _ {i} ^ {\prime}} \nabla_ {\Theta} \ell \left(\mathbf {z} ^ {\prime}; \Theta\right) - r _ {i} ^ {t} \sum_ {\mathbf {z} \in \mathcal {D} _ {i}} \nabla_ {\Theta_ {c}} \ell \left(\mathbf {z}; \Theta_ {c}\right)) (17) \\ = \frac {1}{| \mathcal {D} _ {i} | (1 + r _ {i} ^ {t})} (\sum_ {\mathbf {z} \in \mathcal {D} _ {i}} (\Theta - \Theta_ {c}) ^ {T} (\nabla_ {\Theta} \ell (\mathbf {z}; \Theta) - \nabla_ {\Theta_ {c}} \ell (\mathbf {z}; \Theta_ {c})) \\ + \left(\Theta - \Theta_ {c}\right) ^ {T} \left(\sum_ {\mathbf {z} ^ {\prime} \in \mathcal {D} _ {i} ^ {\prime}} \nabla_ {\Theta} \ell \left(\mathbf {z} ^ {\prime}; \Theta\right) - r _ {i} ^ {t} \sum_ {\mathbf {z} \in \mathcal {D} _ {i}} \nabla_ {\Theta_ {c}} \ell \left(\mathbf {z}; \Theta_ {c}\right)\right) (18) \\ \geq \frac {1}{| \mathcal {D} _ {i} | (1 + r _ {i} ^ {t})} (\sum_ {\mathbf {z} \in \mathcal {D} _ {i}} (\Theta - \Theta_ {c}) ^ {T} (\nabla_ {\Theta} \ell (\mathbf {z}; \Theta) - \nabla_ {\Theta_ {c}} \ell (\mathbf {z}; \Theta_ {c})) \\ - \left\| \left(\Theta - \Theta_ {c}\right) ^ {T} \left(\sum_ {\mathbf {z} ^ {\prime} \in \mathcal {D} _ {i} ^ {\prime}} \nabla_ {\Theta} \ell \left(\mathbf {z} ^ {\prime}; \Theta\right) - r _ {i} ^ {t} \sum_ {\mathbf {z} \in \mathcal {D} _ {i}} \nabla_ {\Theta_ {c}} \ell \left(\mathbf {z}; \Theta_ {c}\right)\right) \| _ {1}\right) \quad \triangleright \forall x, x \geq - \| x \| _ {1} (19) \\ \end{array}
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