$$
\begin{array}{l} R _ {\text {s i n g l e}} (K) = \mathbb {E} _ {\mathcal {T} \sim \mathcal {P} (\mathcal {T}), \boldsymbol {x} \sim \mathcal {D} _ {\mathcal {T}}} \left[ \mathcal {L} \big (\boldsymbol {x}; \mathcal {I} (\mathcal {T}, \boldsymbol {\Theta} ^ {(K)}) \big) \right] - \mathbb {E} _ {\mathcal {T} \sim \mathcal {P} (\mathcal {T}), \boldsymbol {x} \sim \mathcal {D} _ {\mathcal {T}}} \left[ \mathcal {L} \big (\boldsymbol {x}; \mathcal {I} (\mathcal {T}, \boldsymbol {\Theta} ^ {(K), *}) \big) \right] \\ = \mathbb {E} _ {\mathcal {T} \sim \mathcal {P} (\mathcal {T}), \boldsymbol {x} \sim \mathcal {D} _ {\mathcal {T}}} \left[ \mathcal {L} \big (\boldsymbol {x}; \mathcal {I} (\mathcal {T}, \boldsymbol {\Theta} ^ {(K - 1)} [ \mathcal {T} _ {K} ]) \big) \right] - \mathbb {E} _ {\mathcal {T} \sim \mathcal {P} (\mathcal {T}), \boldsymbol {x} \sim \mathcal {D} _ {\mathcal {T}}} \left[ \mathcal {L} \big (\boldsymbol {x}; \mathcal {I} (\mathcal {T}, \boldsymbol {\Theta} ^ {(K - 1), *} [ \mathcal {T} _ {K} ^ {*} ]) \big) \right] \\ = h \left(\boldsymbol {\Theta} ^ {(K - 1), *} \left[ \mathcal {T} _ {K} ^ {*} \right]\right) - h \left(\boldsymbol {\Theta} ^ {(K - 1)} \left[ \mathcal {T} _ {K} \right]\right) \\ = h \left(\boldsymbol {\Theta} ^ {(K - 1), *} \left[ \mathcal {T} _ {K} ^ {*} \right]\right) - f \left(\chi_ {K} ^ {*}; \tilde {\boldsymbol {\theta}} ^ {(K - 1)}\right) + f \left(\chi_ {K} ^ {*}; \tilde {\boldsymbol {\theta}} ^ {(K - 1)}\right) - f \left(\chi_ {K}; \boldsymbol {\theta} ^ {(K - 1)}\right) \\ + f \left(\boldsymbol {\chi} _ {K}; \boldsymbol {\theta} ^ {(K - 1)}\right) - h \left(\boldsymbol {\Theta} ^ {(K - 1)} \left[ \mathcal {T} _ {K} \right]\right) \\ \leq h \left(\boldsymbol {\Theta} ^ {(K - 1), *} [ \mathcal {T} _ {K} ^ {*} ]\right) - f \left(\boldsymbol {\chi} _ {K} ^ {*}; \tilde {\boldsymbol {\theta}} ^ {(K - 1)}\right) + f \left(\boldsymbol {\chi} _ {K} ^ {*}; \tilde {\boldsymbol {\theta}} ^ {(K - 1)}\right) - f \left(\boldsymbol {\Theta} ^ {(K - 1)} [ \mathcal {T} _ {K} ^ {*} ]; \boldsymbol {\theta} ^ {(K - 1)}\right) \\ + f \left(\boldsymbol {\chi} _ {K}; \boldsymbol {\theta} ^ {(K - 1)}\right) - h \left(\boldsymbol {\Theta} ^ {(K - 1)} \left[ \mathcal {T} _ {K} \right]\right) \\ = h \left(\boldsymbol {\Theta} ^ {(K - 1), *} \left[ \mathcal {T} _ {K} ^ {*} \right]\right) - f \left(\boldsymbol {\Theta} ^ {(K - 1), *} \left[ \mathcal {T} _ {K} ^ {*} \right]; \tilde {\boldsymbol {\theta}} ^ {(K - 1)}\right) + f \left(\boldsymbol {\Theta} ^ {(K - 1), *} \left[ \mathcal {T} _ {K} ^ {*} \right]; \tilde {\boldsymbol {\theta}} ^ {(K - 1)}\right) - f \left(\boldsymbol {\Theta} ^ {(K - 1)} \left[ \mathcal {T} _ {K} ^ {*} \right]; \boldsymbol {\theta} ^ {(K - 1)}\right) \\ + f \left(\boldsymbol {\Theta} ^ {(K - 1)} \left[ \mathcal {T} _ {K} \right]; \boldsymbol {\theta} ^ {(K - 1)}\right) - h \left(\boldsymbol {\Theta} ^ {(K - 1)} \left[ \mathcal {T} _ {K} \right]\right) \\ \leq | h \left(\Theta^ {(K - 1), *} \left[ \mathcal {T} _ {K} ^ {*} \right]\right) - f \left(\Theta^ {(K - 1), *} \left[ \mathcal {T} _ {K} ^ {*} \right]; \tilde {\boldsymbol {\theta}} ^ {(K - 1)}\right) | + \underbrace {| f \left(\Theta^ {(K - 1) , *} \left[ \mathcal {T} _ {K} ^ {*} \right] ; \tilde {\boldsymbol {\theta}} ^ {(K - 1)}\right) - f \left(\Theta^ {(K - 1)} \left[ \mathcal {T} _ {K} ^ {*} \right] ; \boldsymbol {\theta} ^ {(K - 1)}\right) |} _ {I _ {0}} \\ + \left| f \left(\boldsymbol {\Theta} ^ {(K - 1)} \left[ \mathcal {T} _ {K} \right]; \boldsymbol {\theta} ^ {(K - 1)}\right) - h \left(\boldsymbol {\Theta} ^ {(K - 1)} \left[ \mathcal {T} _ {K} \right]\right) \right| \\ \end{array}
$$