$$
\begin{array}{l} | \bar {R} _ {\mathrm {C D}} (g) - \widehat {R} _ {\mathrm {C D}} (g) | \\ = \frac {1}{2 n} \left| \sum_ {i = 1} ^ {n} \left(\left(\bar {\pi} _ {+} - \pi_ {+} + c _ {i} - \bar {c} _ {i}\right) \ell \left(g \left(\boldsymbol {x} _ {i}\right), + 1\right) + \left(\bar {\pi} _ {-} - \pi_ {-} + c _ {i} - \bar {c} _ {i}\right) \ell \left(g \left(\boldsymbol {x} _ {i} ^ {\prime}\right), - 1\right) \right. \right. \\ + \left(\bar {\pi} _ {+} - \pi_ {+} + \bar {c} _ {i} - c _ {i}\right) \ell \left(g \left(\boldsymbol {x} _ {i} ^ {\prime}\right), + 1\right) + \left(\bar {\pi} _ {-} - \pi_ {-} + \bar {c} _ {i} - c _ {i}\right) \ell \left(g \left(\boldsymbol {x} _ {i}\right), - 1\right) | \\ \leq \frac {1}{2 n} \sum_ {i = 1} ^ {n} \left(\left| \left(\bar {\pi} _ {+} - \pi_ {+} + c _ {i} - \bar {c} _ {i}\right) \ell (g (\boldsymbol {x} _ {i}), + 1) \right| + \left| \left(\bar {\pi} _ {-} - \pi_ {-} + c _ {i} - \bar {c} _ {i}\right) \ell (g (\boldsymbol {x} _ {i} ^ {\prime}), - 1) \right| \right. \\ + \left| \left(\bar {\pi} _ {+} - \pi_ {+} + \bar {c} _ {i} - c _ {i}\right) \ell \left(g \left(\boldsymbol {x} _ {i} ^ {\prime}\right), + 1\right) \right| + \left| \left(\bar {\pi} _ {-} - \pi_ {-} + \bar {c} _ {i} - c _ {i}\right) \ell \left(g \left(\boldsymbol {x} _ {i}\right), - 1\right) \right| \\ = \frac {1}{2 n} \sum_ {i = 1} ^ {n} \left(\left| \bar {\pi} _ {+} - \pi_ {+} + c _ {i} - \bar {c} _ {i} \right| \ell \left(g \left(\boldsymbol {x} _ {i}\right), + 1\right) + \left| \bar {\pi} _ {-} - \pi_ {-} + c _ {i} - \bar {c} _ {i} \right| \ell \left(g \left(\boldsymbol {x} _ {i} ^ {\prime}\right), - 1\right) \right. \\ + \left| \bar {\pi} _ {+} - \pi_ {+} + \bar {c} _ {i} - c _ {i} \right| \ell \left(g \left(\boldsymbol {x} _ {i} ^ {\prime}\right), + 1\right) + \left| \bar {\pi} _ {-} - \pi_ {-} + \bar {c} _ {i} - c _ {i} \right| \ell \left(g \left(\boldsymbol {x} _ {i}\right), - 1\right)) \\ \leq \frac {1}{2 n} \sum_ {i = 1} ^ {n} \left(\left| \bar {\pi} _ {+} - \pi_ {+} \right| + \left| c _ {i} - \bar {c} _ {i} \right|\right) \ell \left(g \left(\boldsymbol {x} _ {i}\right), + 1\right) + \left(\left| \bar {\pi} _ {-} - \pi_ {-} \right| + \left| c _ {i} - \bar {c} _ {i} \right|\right) \ell \left(g \left(\boldsymbol {x} _ {i} ^ {\prime}\right), - 1\right) \\ + \left(\left| \bar {\pi} _ {+} - \pi_ {+} \right| + \left| \bar {c} _ {i} - c _ {i} \right|\right) \ell \left(g \left(\boldsymbol {x} _ {i} ^ {\prime}\right), + 1\right) + \left(\left| \bar {\pi} _ {-} - \pi_ {-} \right| + \left| \bar {c} _ {i} - c _ {i} \right|\right) \ell \left(g \left(\boldsymbol {x} _ {i}\right), - 1\right)) \\ = \frac {1}{2 n} \sum_ {i = 1} ^ {n} \left(\left| \bar {\pi} _ {+} - \pi_ {+} \right| + \left| c _ {i} - \bar {c} _ {i} \right|\right) \ell \left(g \left(\boldsymbol {x} _ {i}\right), + 1\right) + \left(\left| \pi_ {+} - \bar {\pi} _ {+} \right| + \left| c _ {i} - \bar {c} _ {i} \right|\right) \ell \left(g \left(\boldsymbol {x} _ {i} ^ {\prime}\right), - 1\right) \\ + \left(\left| \bar {\pi} _ {+} - \pi_ {+} \right| + \left| \bar {c} _ {i} - c _ {i} \right|\right) \ell \left(g \left(\boldsymbol {x} _ {i} ^ {\prime}\right), + 1\right) + \left(\left| \pi_ {+} - \bar {\pi} _ {+} \right| + \left| \bar {c} _ {i} - c _ {i} \right|\right) \ell \left(g \left(\boldsymbol {x} _ {i}\right), - 1\right)) \\ \leq \frac {2 C _ {\ell} \sum_ {i = 1} ^ {n} | \bar {c} _ {i} - c _ {i} |}{n} + 2 C _ {\ell} | \bar {\pi} _ {+} - \pi_ {+} |. \\ \end{array}
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