$$
\begin{array}{l} \hat {\mathcal {R}} _ {\mathcal {S}} \left(\mathcal {G} _ {T}\right) \leq \hat {\mathcal {R}} _ {\mathcal {S}} \left(\mathcal {G} _ {T} ^ {U}\right) \\ = \frac {1}{n} \mathbb {E} _ {\boldsymbol {\sigma}} \left[ \sup  _ {g \in \mathcal {G} _ {T} ^ {U}} \sum_ {i = 1} ^ {n} \sigma_ {i} g \left(\boldsymbol {z} _ {i}\right) \right] \\ = \frac {1}{n} \mathbb {E} _ {\boldsymbol {\sigma}} \left[ \sup  _ {\mathsf {K} (\cdot , \cdot ; \mathcal {S} ^ {\prime}) \in \mathcal {K} _ {T}} \sum_ {i = 1} ^ {n} \sigma_ {i} \left(\langle \boldsymbol {\beta}, \Phi_ {\mathcal {S} ^ {\prime}} (\boldsymbol {z} _ {i}) \rangle + \ell (\boldsymbol {w} _ {0}, \boldsymbol {z} _ {i})\right) \right] \\ = \frac {1}{n} \underset {\boldsymbol {\sigma}} {\mathbb {E}} \left[ \sup  _ {K (\cdot , \cdot ; \mathcal {S} ^ {\prime}) \in \mathcal {K} _ {T}} \sum_ {i = 1} ^ {n} \sigma_ {i} \left\langle \boldsymbol {\beta}, \Phi_ {\mathcal {S} ^ {\prime}} (\boldsymbol {z} _ {i}) \right\rangle \right] + \frac {1}{n} \underset {\boldsymbol {\sigma}} {\mathbb {E}} \left[ \sup  _ {K (\cdot , \cdot ; \mathcal {S} ^ {\prime}) \in \mathcal {K} _ {T}} \sum_ {i = 1} ^ {n} \sigma_ {i} \ell (\boldsymbol {w} _ {0}, \boldsymbol {z} _ {i}) \right] \\ = \frac {1}{n} \mathbb {E} _ {\boldsymbol {\sigma}} \left[ \sup  _ {\mathsf {K} (\cdot , \cdot ; \mathcal {S} ^ {\prime}) \in \mathcal {K} _ {T}} \left\langle \boldsymbol {\beta}, \sum_ {i = 1} ^ {n} \sigma_ {i} \Phi_ {\mathcal {S} ^ {\prime}} (\boldsymbol {z} _ {i}) \right\rangle \right] \\ = \frac {B}{n} \mathbb {E} _ {\boldsymbol {\sigma}} \left[ \sup  _ {K (\cdot , \cdot ; S ^ {\prime}) \in \mathcal {K} _ {T}} \left\| \sum_ {i = 1} ^ {n} \sigma_ {i} \Phi_ {S ^ {\prime}} \left(\boldsymbol {z} _ {i}\right) \right\| \right] \\ = \frac {B}{n} \underset {\boldsymbol {\sigma}} {\mathbb {E}} \left[ \sup  _ {\mathsf {K} (\cdot , \cdot ; \mathcal {S} ^ {\prime}) \in \mathcal {K} _ {T}} \left(\sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} \sigma_ {i} \sigma_ {j} \mathsf {K} (\boldsymbol {z} _ {i}, \boldsymbol {z} _ {j}; \mathcal {S} ^ {\prime})\right) ^ {\frac {1}{2}} \right] \\ = \frac {B}{n} \underset {\boldsymbol {\sigma}} {\mathbb {E}} \left[ \right.\left.\left(\sup  _ {\mathsf {K} (\cdot , \cdot ; \mathcal {S} ^ {\prime}) \in \mathcal {K} _ {T}} \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} \sigma_ {i} \sigma_ {j} \mathsf {K} (\boldsymbol {z} _ {i}, \boldsymbol {z} _ {j}; \mathcal {S} ^ {\prime})\right) ^ {\frac {1}{2}} \right] \\ \leq \frac {B}{n} \left(\mathbb {E} _ {\boldsymbol {\sigma}} \left[ \sup  _ {K (\cdot , \cdot ; \mathcal {S} ^ {\prime}) \in \mathcal {K} _ {T}} \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} \sigma_ {i} \sigma_ {j} K \left(\boldsymbol {z} _ {i}, \boldsymbol {z} _ {j}; \mathcal {S} ^ {\prime}\right) \right]\right) ^ {\frac {1}{2}} \quad (\text {J e s s e n ' s i n e q u a l i t y}) \\ = \frac {B}{n} \left(\mathbb {E} _ {\boldsymbol {\sigma}} \left[ \sup  _ {K (\cdot , \cdot ; \mathcal {S} ^ {\prime}) \in \mathcal {K} _ {T}} \left(\sum_ {i = 1} ^ {n} K (z _ {i}, z _ {i}) + \sum_ {i \neq j} \sigma_ {i} \sigma_ {j} K (z _ {i}, z _ {j}; \mathcal {S} ^ {\prime})\right) \right]\right) ^ {\frac {1}{2}} \\ \leq \frac {B}{n} \left(\mathbb {E} _ {\boldsymbol {\sigma}} \left[ \sup  _ {K (\cdot , \cdot ; \mathcal {S} ^ {\prime}) \in \mathcal {K} _ {T}} \sum_ {i = 1} ^ {n} K (z _ {i}, z _ {i}) + \sup  _ {K (\cdot , \cdot ; \mathcal {S} ^ {\prime}) \in \mathcal {K} _ {T}} \sum_ {i \neq j} \sigma_ {i} \sigma_ {j} K (z _ {i}, z _ {j}; \mathcal {S} ^ {\prime}) \right]\right) ^ {\frac {1}{2}} \\ = \frac {B}{n} \left(\sup  _ {\mathsf {K} (\cdot , \cdot ; \mathcal {S} ^ {\prime}) \in \mathcal {K} _ {T}} \operatorname {T r} (\mathsf {K} (\mathbf {Z}, \mathbf {Z}; \mathcal {S} ^ {\prime})) + \underset {\boldsymbol {\sigma}} {\mathbb {E}} \left[ \sup  _ {\mathsf {K} (\cdot , \cdot ; \mathcal {S} ^ {\prime}) \in \mathcal {K} _ {T}} \sum_ {i \neq j} \sigma_ {i} \sigma_ {j} \mathsf {K} (\boldsymbol {z} _ {i}, \boldsymbol {z} _ {j}; \mathcal {S} ^ {\prime}) \right]\right) ^ {\frac {1}{2}}. \\ \end{array}
$$