$$
\begin{array}{l} \Delta_ {t} = r _ {t, a _ {t} ^ {*}} - r _ {t, a _ {t}} \\ = \left\langle \left[ \begin{array}{c} \boldsymbol {x} _ {t} \\ \phi^ {\star} (\boldsymbol {x} _ {t}) \end{array} \right], \left[ \begin{array}{c} \boldsymbol {\theta} _ {a _ {t} ^ {\star}} - \boldsymbol {\theta} _ {a _ {t}} ^ {\star} \\ \boldsymbol {\beta} _ {a _ {t} ^ {\star}} - \boldsymbol {\beta} _ {a _ {t}} ^ {\star} \end{array} \right] \right\rangle \\ \stackrel {\mathrm {(a)}} {\leq} \left\langle \left[ \begin{array}{c} \boldsymbol {x} _ {t} \\ \widetilde {\phi} _ {t} (\boldsymbol {x} _ {t}) \end{array} \right], \left[ \begin{array}{c} \widetilde {\boldsymbol {\theta}} _ {a _ {t}} - \boldsymbol {\theta} _ {a _ {t}} ^ {\star} \\ \widetilde {\boldsymbol {\beta}} _ {a _ {t}} - \boldsymbol {\beta} _ {a _ {t}} ^ {\star} \end{array} \right] \right\rangle + \left\langle \widetilde {\phi} _ {t} (\boldsymbol {x} _ {t}) - \phi^ {\star} (\boldsymbol {x} _ {t}), \boldsymbol {\beta} _ {a _ {t}} ^ {\star} \right\rangle \\ = \left\langle \left[ \begin{array}{c} \mathbf {0} \\ \widetilde {\phi_ {t}} (\boldsymbol {x} _ {t}) - \phi^ {\star} (\boldsymbol {x} _ {t}) \end{array} \right] + \left[ \begin{array}{c} \boldsymbol {x} _ {t} \\ \phi^ {\star} (\boldsymbol {x} _ {t}) \end{array} \right], \left[ \begin{array}{c} \widetilde {\boldsymbol {\theta}} _ {a _ {t}} - \boldsymbol {\theta} _ {a _ {t}} ^ {\star} \\ \widetilde {\boldsymbol {\beta}} _ {a _ {t}} - \boldsymbol {\beta} _ {a _ {t}} ^ {\star} \end{array} \right] \right\rangle + \left\langle \widetilde {\phi_ {t}} (\boldsymbol {x} _ {t}) - \phi^ {\star} (\boldsymbol {x} _ {t}), \boldsymbol {\beta} _ {a _ {t}} ^ {\star} \right\rangle \\ = \left\langle \widetilde {\phi} _ {t} (\boldsymbol {x} _ {t}) - \phi^ {\star} (\boldsymbol {x} _ {t}), \widetilde {\boldsymbol {\beta}} _ {a _ {t}} \right\rangle + \left\langle \left[ \begin{array}{c} \boldsymbol {x} _ {t} \\ \phi^ {\star} (\boldsymbol {x} _ {t}) \end{array} \right], \left[ \begin{array}{c} \widetilde {\boldsymbol {\theta}} _ {a _ {t}} - \boldsymbol {\theta} _ {a _ {t}} ^ {\star} \\ \widetilde {\boldsymbol {\beta}} _ {a _ {t}} - \boldsymbol {\beta} _ {a _ {t}} ^ {\star} \end{array} \right] \right\rangle \\ \stackrel {(b)} {\leq} \left\| \widetilde {\phi} _ {t} (\boldsymbol {x} _ {t}) - \phi^ {\star} (\boldsymbol {x} _ {t}) \right\| \cdot \left\| \widetilde {\boldsymbol {\beta}} _ {a _ {t}} \right\| + \left\langle \left[ \begin{array}{c} \boldsymbol {x} _ {t} \\ \phi^ {\star} (\boldsymbol {x} _ {t}) \end{array} \right], \left[ \begin{array}{c} \widetilde {\boldsymbol {\theta}} _ {a _ {t}} - \boldsymbol {\theta} _ {a _ {t}} ^ {\star} \\ \widetilde {\boldsymbol {\beta}} _ {a _ {t}} - \boldsymbol {\beta} _ {a _ {t}} ^ {\star} \end{array} \right] \right\rangle \\ \stackrel {\mathrm {(c)}} {\leq} \left\| \widetilde {\phi} _ {t} (\boldsymbol {x} _ {t}) - \boldsymbol {\phi} ^ {\star} (\boldsymbol {x} _ {t}) \right\| \cdot \left\| \widetilde {\boldsymbol {\beta}} _ {a _ {t}} \right\| + \left\| \left[ \begin{array}{c} \boldsymbol {x} _ {t} \\ \boldsymbol {\phi} ^ {\star} (\boldsymbol {x} _ {t}) \end{array} \right] \right\| _ {\boldsymbol {A} _ {t - 1, a _ {t}} ^ {- 1}} \left\| \left[ \begin{array}{c} \widetilde {\boldsymbol {\theta}} _ {a _ {t}} - \boldsymbol {\theta} _ {a _ {t}} ^ {\star} \\ \widetilde {\boldsymbol {\beta}} _ {a _ {t}} - \boldsymbol {\beta} _ {a _ {t}} ^ {\star} \end{array} \right] \right\| _ {\boldsymbol {A} _ {t - 1, a _ {t}}}. \\ \end{array}
$$