$$
\begin{array}{l} \mathcal {L} \left(\sum_ {k = 1} ^ {K} \left(\left(\mathbf {X} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {W}} _ {2 k}\right) _ {+} \hat {\mathbf {w}} _ {3 k}, \mathbf {y}\right) \\ = \mathcal {L} \left(\sum_ {k = 1} ^ {K} \sum_ {j _ {2} = 1} ^ {m _ {2}} \left[ \left(\left(\mathbf {X} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \hat {w} _ {3 k j _ {2}} - \left(\left(\hat {\mathbf {X}} _ {r} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \hat {w} _ {3 k j _ {2}} + \left(\left(\hat {\mathbf {X}} _ {r} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \hat {w} _ {3 k j _ {2}} \right], \mathbf {y}\right) \\ \stackrel {(1)} {\leq} \mathcal {L} \left(\sum_ {k = 1} ^ {K} \sum_ {j _ {2} = 1} ^ {m _ {2}} \left[ \left(\left(\mathbf {X} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \hat {w} _ {3 k j _ {2}} - \left(\left(\hat {\mathbf {X}} _ {r} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \hat {w} _ {3 k j _ {2}} \right], \mathbf {y}\right) + \mathcal {L} \left(\sum_ {k = 1} ^ {K} \sum_ {j _ {2} = 1} ^ {m _ {2}} \left(\left(\hat {\mathbf {X}} _ {r} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \hat {w} _ {3 k j _ {2}}, \mathbf {y}\right) \\ \stackrel {(2)} {\leq} R \left\| \sum_ {k = 1} ^ {K} \sum_ {j _ {2} = 1} ^ {m _ {2}} \left[ \left(\left(\mathbf {X} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \hat {w} _ {3 k j _ {2}} - \left(\left(\hat {\mathbf {X}} _ {r} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \hat {w} _ {3 k j _ {2}} \right] \right\| _ {2} + \mathcal {L} \left(\sum_ {k = 1} ^ {K} \sum_ {j _ {2} = 1} ^ {m _ {2}} \left(\left(\hat {\mathbf {X}} _ {r} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \hat {w} _ {3 k j _ {2}}, \mathbf {y}\right) \\ = R \left\| \sum_ {k = 1} ^ {K} \sum_ {j _ {2} = 1} ^ {m _ {2}} \left(\left(\left(\mathbf {X} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} - \left(\left(\hat {\mathbf {X}} _ {r} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+}\right) \hat {w} _ {3 k j _ {2}} \right\| _ {2} + \mathcal {L} \left(\sum_ {k = 1} ^ {K} \sum_ {j _ {2} = 1} ^ {m _ {2}} \left(\left(\hat {\mathbf {X}} _ {r} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \hat {w} _ {3 k j _ {2}}, \mathbf {y}\right) \\ \stackrel {(3)} {\leq} R \sum_ {k = 1} ^ {K} \sum_ {j _ {2} = 1} ^ {m _ {2}} \left\| \left(\left(\mathbf {X} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} - \left(\left(\hat {\mathbf {X}} _ {r} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \right\| _ {2} | \hat {w} _ {3 k j _ {2}} | + \mathcal {L} \left(\sum_ {k = 1} ^ {K} \sum_ {j _ {2} = 1} ^ {m _ {2}} \left(\left(\hat {\mathbf {X}} _ {r} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \hat {w} _ {3 k j _ {2}}, \mathbf {y}\right) \\ \leq R \max  _ {k \in [ K ], j _ {2} \in [ m _ {2} ]} \left\| \left(\left(\mathbf {X} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} - \left(\left(\hat {\mathbf {X}} _ {r} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \right\| _ {2} \sum_ {k = 1} ^ {K} \| \hat {\mathbf {w}} _ {3 k} \| _ {1} + \mathcal {L} \left(\sum_ {k = 1} ^ {K} \sum_ {j _ {2} = 1} ^ {m _ {2}} \left(\left(\hat {\mathbf {X}} _ {r} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \hat {w} _ {3 k j _ {2}}, \mathbf {y}\right) \\ \stackrel {(4)} {\leq} R \max  _ {k \in [ K ]} \left\| \mathbf {X} - \hat {\mathbf {X}} _ {r} \right\| _ {2} \sum_ {j _ {1} = 1} ^ {m _ {1}} \| \hat {\mathbf {w}} _ {1 k j _ {1}} \| _ {2} | \hat {w} _ {2 k j _ {1} j _ {2}} | \sum_ {k = 1} ^ {K} \| \hat {\mathbf {w}} _ {3 k} \| _ {1} + \mathcal {L} \left(\sum_ {k = 1} ^ {K} \sum_ {j _ {2} = 1} ^ {m _ {2}} \left(\left(\hat {\mathbf {X}} _ {r} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \hat {w} _ {3 k j _ {2}}, \mathbf {y}\right) \\ \stackrel {(5)} {\leq} \sqrt {m _ {1}} R \max  _ {k \in [ K ]} \left\| \mathbf {X} - \hat {\mathbf {X}} _ {r} \right\| _ {2} \sqrt {\sum_ {j _ {1} = 1} ^ {m _ {1}} \| \hat {\mathbf {w}} _ {1 k j _ {1}} \| _ {2} ^ {2} | \hat {w} _ {2 k j _ {1} j _ {2}} | ^ {2}} \sum_ {k = 1} ^ {K} \| \hat {\mathbf {w}} _ {3 k} \| _ {1} + \mathcal {L} \left(\sum_ {k = 1} ^ {K} \sum_ {j _ {2} = 1} ^ {m _ {2}} \left(\left(\hat {\mathbf {X}} _ {r} \hat {\mathbf {W}} _ {1 k}\right) _ {+} \hat {\mathbf {w}} _ {2 k j _ {2}}\right) _ {+} \hat {w} _ {3 k j _ {2}}, \mathbf {y}\right) \\ \end{array}
$$