SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u}, p _ {1} \neq v _ {m}, \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {\tilde {v} _ {t} ^ {u}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]} <   \beta \right] \\ \leq \mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u}, \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {\tilde {v} _ {t} ^ {u}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]} <   \beta \right] \\ \leq \mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u}, \eta_ {\tilde {v} _ {t} ^ {u}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \geq \eta_ {p _ {2}} (\tilde {\mathbf {x}} _ {t} ^ {u}), \sum_ {h = 1} ^ {H} w _ {h} \left[ \tilde {\eta} _ {\tilde {v} _ {t} ^ {u}} (\tilde {\mathbf {x}} _ {t} ^ {u}) - \epsilon \right] <   \beta \sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right] \right] \\ = \mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u}, \eta_ {\tilde {v} _ {t} ^ {u}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \geq \eta_ {p _ {2}} (\tilde {\mathbf {x}} _ {t} ^ {u}), \tilde {\eta} _ {\tilde {v} _ {t} ^ {u}} (\tilde {\mathbf {x}} _ {t} ^ {u}) <   \beta \sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right] + \epsilon \right] \\ = \mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u}, \eta_ {\tilde {v} _ {t} ^ {u}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \geq \eta_ {p _ {2}} (\tilde {\mathbf {x}} _ {t} ^ {u}), \sum_ {v _ {j} \in \mathcal {V}} \tau_ {v _ {j} \tilde {v} _ {t} ^ {u}} \eta_ {v _ {j}} (\tilde {\mathbf {x}} _ {t} ^ {u}) <   \beta \sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right] + \epsilon \right] \\ \leq \mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u}, \eta_ {p _ {2}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \leq \eta_ {\tilde {v} _ {t} ^ {u}} (\tilde {\mathbf {x}} _ {t} ^ {u}) <   \frac {\beta \sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right] + \epsilon - \sum_ {v _ {j} \neq \tilde {v} _ {t} ^ {u}} \left[ \tau_ {v _ {j} \tilde {v} _ {t} ^ {u}} \eta_ {v _ {j}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right]}{\tau_ {\tilde {v} _ {t} ^ {u} \tilde {v} _ {t} ^ {u}} (\tilde {\mathbf {x}} _ {t} ^ {u})} \right] \\ \leq \mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u}, \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \leq \eta_ {\tilde {v} _ {t} ^ {u}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) <   \frac {\left(\beta_ {1} + \xi_ {1}\right) \sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right] + \epsilon - \sum_ {v _ {j} \neq \tilde {v} _ {t} ^ {u}} \left[ \tau_ {v _ {j} \tilde {v} _ {t} ^ {u}} \eta_ {v _ {j}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\tau_ {\tilde {v} _ {t} ^ {u} \tilde {v} _ {t} ^ {u}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right)} \right]. \tag {9} \\ \end{array}
$$
We substitute  $\beta_{1}$  with  $\frac{\tau_{\tilde{v}_t^u\tilde{v}_t^u}(\tilde{\mathbf{x}}_t^u)\eta_{p_2}(\tilde{\mathbf{x}}_t^u) + \sum_{v_j\neq \tilde{v}_t^u}\tau_{v_j\tilde{v}_t^u}\eta_{v_j}(\tilde{\mathbf{x}}_t^u)}{\sum_{h = 1}^{H}[w_hf_{vm}^h(\tilde{\mathbf{x}}_t^u)]}$  and obtain:
$$
\begin{array}{l} \mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u}, p _ {1} \neq v _ {m}, \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {\tilde {v} _ {t} ^ {u}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]} <   \beta \right] \\ \leq \mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u}, \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \leq \eta_ {\tilde {v} _ {t} ^ {u}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) <   \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) + \frac {\epsilon}{\tau_ {\tilde {v} _ {t} ^ {u} \tilde {v} _ {t} ^ {u}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right)} + \frac {\xi_ {1} \sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\tau_ {\tilde {v} _ {t} ^ {u} \tilde {v} _ {t} ^ {u}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right)} \right] \tag {10} \\ \leq \mathbb {P} \left[ \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \leq \eta_ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) <   \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) + \frac {\epsilon + \xi_ {1}}{\tau_ {\tilde {v} _ {t} ^ {u}} \tilde {v} _ {t} ^ {u} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right)} \right]. \\ \end{array}
$$
Recall that  $\epsilon \leq \alpha_0\tau_{\tilde{v}_t^u\tilde{v}_t^u}(\tilde{\mathbf{x}}_t^u) - \xi_1$ , which implies  $\frac{\epsilon + \xi_1}{\tau_{\tilde{v}_t^u\tilde{v}_t^u}(\tilde{\mathbf{x}}_t^u)} \leq \alpha_0$ . Hence, the Multiclass Tsybakov Condition holds and the probability of Case-2 is bounded by  $C\left(\frac{\epsilon + \xi_1}{\tau_{\tilde{v}_t^u\tilde{v}_t^u}(\tilde{\mathbf{x}}_t^u)}\right)^{\lambda}$ , namely,  $C\big(\mathcal{O}(\epsilon +\xi_1)\big)^{\lambda}$ .
For Case-2 of  $E_{\mathrm{DRUT}}$ , we have:
We replace  $\beta_{2}$  with  $\frac{\sum_{h = 1}^{H}[w_{h}f_{\tilde{v}_{t}^{u}}^{h}(\tilde{\mathbf{x}}_{t}^{u})]}{\tau_{vmvm}(\tilde{\mathbf{x}}_{t}^{u})\eta_{P2}(\tilde{\mathbf{x}}_{t}^{u}) + \sum_{v_{j}\neq v_{m}}\tau_{v_{j}vm}(\tilde{\mathbf{x}}_{t}^{u})\eta_{v_{j}}(\tilde{\mathbf{x}}_{t}^{u})}$  and continue the calculation:
$$
\begin{array}{l} \mathbb {P} \left[ p _ {1} \neq \tilde {v} _ {t} ^ {u}, p _ {1} = v _ {m}, \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {\tilde {v} _ {t}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]} \geq \beta \right] \\ \leq \mathbb {P} \left[ p _ {1} = v _ {m}, \eta_ {p 2} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \leq \eta_ {m} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \leq \eta_ {p 2} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) + \frac {\epsilon}{\tau_ {v _ {m} v _ {m}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right)} + \frac {\xi_ {2} \sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\beta_ {2} \left(\beta_ {2} - \xi_ {2}\right) \tau_ {v _ {m} v _ {m}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right)} \right] \tag {12} \\ \leq \mathbb {P} \left[ \eta_ {p _ {2}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \leq \eta_ {p _ {1}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \leq \eta_ {p _ {2}} (\tilde {\mathbf {x}} _ {t} ^ {u}) + \frac {\epsilon}{\tau_ {v _ {m} v _ {m}} (\tilde {\mathbf {x}} _ {t} ^ {u})} + \frac {\xi_ {2}}{\beta_ {2} (\beta_ {2} - \xi_ {2}) \tau_ {v _ {m} v _ {m}} (\tilde {\mathbf {x}} _ {t} ^ {u})} \right]. \\ \end{array}
$$
Recall that  $\epsilon \leq \frac{\alpha_0\tau_{vm}v_m(\tilde{\mathbf{x}}_t^u)\beta_2(\beta_2 - \xi_2) - \xi_2}{\beta_2(\beta_2 - \xi_2)}$ , which implies  $\frac{\epsilon}{\tau_{v_m v_m}(\tilde{\mathbf{x}}_t^u)} + \frac{\xi_2}{\beta_2(\beta_2 - \xi_2)\tau_{v_m v_m}(\tilde{\mathbf{x}}_t^u)} \leq \alpha_0$ . Hence, the relaxed Multiclass Tsybakov Condition holds and the probability of Case-2 is bounded by  $C\left(\frac{\epsilon\beta_2(\beta_2 - \xi_2) + \xi_2}{\beta_2(\beta_2 - \xi_2)\tau_{v_m v_m}(\tilde{\mathbf{x}}_t^u)}\right)^{\lambda}$ , namely,  $C\big(\mathcal{O}(\epsilon +\xi_2)\big)^{\lambda}$ .
Finally, for Case-3 of  $E_{\mathrm{DRUT}}$ , we have:
Recall that  $\epsilon \leq \frac{1}{2}\Big[\big[\tau_{p_1p_1}(\tilde{\mathbf{x}}_t^u) - \tau_{p_1v_m}(\tilde{\mathbf{x}}_t^u)\big]\big[\alpha_0 + \eta_{p_2}(\tilde{\mathbf{x}}_t^u)\big] - \sum_{v_j\neq p_1}\big[\tau_{v_jv_m}(\tilde{\mathbf{x}}_t^u) - \tau_{v_jp_1}(\tilde{\mathbf{x}}_t^u)\big]\eta_{v_j}(\tilde{\mathbf{x}}_t^u)\Big]$ , which implies  $\frac{\sum_{v_j\neq p_1}\left[\tau_{v_jv_m}(\tilde{\mathbf{x}}_t^u) - \tau_{v_jp_1}(\tilde{\mathbf{x}}_t^u)\right]\eta_{v_j}(\tilde{\mathbf{x}}_t^u) + 2\epsilon -\left[\tau_{p_1p_1}(\tilde{\mathbf{x}}_t^u) - \tau_{p_1v_m}(\tilde{\mathbf{x}}_t^u)\right]\eta_{p_2}(\tilde{\mathbf{x}}_t^u)}{\left[\tau_{p_1p_1}(\tilde{\mathbf{x}}_t^u) - \tau_{p_1v_m}(\tilde{\mathbf{x}}_t^u)\right]}\leq \alpha_0$ . Hence, the relaxed Multiclass Tsybakov Condition holds and the probability of Case-3 is bounded by  $C\Bigg(\frac{\sum_{v_j\neq p_1}\left[\tau_{v_jv_m}(\tilde{\mathbf{x}}_t^u) - \tau_{v_jp_1}(\tilde{\mathbf{x}}_t^u)\right]\eta_{v_j}(\tilde{\mathbf{x}}_t^u) + 2\epsilon -\left[\tau_{p_1p_1}(\tilbar{\mathbf{x}}_t^u) - \tau_{p_1v_m}(\tilde{\mathbf{x}}_t^u)\right]\eta_{p_2}(\tilde{\mathbf{x}}_t^u)}{\left[\tau_{p_1p_1}(\tilde{\mathbf{x}}_t^u) - \tau_{p_1v_m}(\tilde{\mathbf{x}}_t^{u})\right]}\Bigg)^{\lambda}$ , namely,  $C\big(\mathcal{O}(\epsilon)\big)^{\lambda}$ .
B.4 The Proof of Theorem 3
$$
\boldsymbol {E} _ {\mathrm {D D U I}} = \mathbb {P} \left[ p _ {1} ^ {\prime} = \tilde {v} _ {t - l} ^ {u}, \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} \bar {f} _ {\bar {v} _ {t - l}} ^ {h} \left(\bar {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\sum_ {h = 1} ^ {H} \left[ w _ {h} \bar {f} _ {\bar {v} _ {m}} ^ {h} \left(\bar {\mathbf {x}} _ {t} ^ {u}\right) \right]} <   \beta^ {\prime} \right] + \mathbb {P} \left[ p _ {1} ^ {\prime} \neq \tilde {v} _ {t - l} ^ {u}, \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} \bar {f} _ {\bar {v} _ {t - l}} ^ {h} \left(\bar {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\sum_ {h = 1} ^ {H} \left[ W _ {h} \bar {f} _ {\bar {v} _ {m}} ^ {h} \left(\bar {\mathbf {x}} _ {t} ^ {u}\right) \right]} \geq \beta^ {\prime} \right]. \tag {14}
$$
Theorem 3 (The Upper Bound of  $E_{\mathrm{DDUI}}$ ). Given Assumption 1 and the set of weights  $\{w_h \mid 1 \leq h \leq H, 0 \leq w_h \leq 1, \sum_{h=1}^{H} w_h = 1\}$ ,  $\forall \langle \tilde{\mathbf{x}}_t^u, \tilde{v}_{t-l}^u \rangle$ , let  $\beta_1' = \left[ \frac{\tau_{\tilde{v}_{t-l}^u \tilde{v}_{t-l}^u}^{\prime} (\tilde{\mathbf{x}}_t^u) \eta_{p_2'}^{\prime} (\tilde{\mathbf{x}}_t^u) + \sum_{v_j \neq \tilde{v}_{t-l}^u} \tau_{v_j \tilde{v}_{t-l}^u}^{\prime} \eta_{v_j}' (\tilde{\mathbf{x}}_t^u)}{\sum_{h=1}^{H} [w_h \overline{f}_vh_m(\tilde{\mathbf{x}}_t^u)]} \right]$ ,
$$
\beta_ {2} ^ {\prime} = \left[ \frac {\sum_ {h = 1} ^ {H} [ w _ {h} \bar {f} _ {\tilde {v} _ {t} - l} ^ {h} (\bar {\mathbf {x}} _ {t} ^ {u}) ]}{\tau_ {\tilde {v} _ {m} \tilde {v} _ {m}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) \eta_ {p _ {2} ^ {\prime}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) + \sum_ {v _ {j} \neq \tilde {v} _ {m}} \tau_ {v _ {j} \tilde {v} _ {m}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) \eta_ {v _ {j}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u})} \right], \xi_ {1} ^ {\prime} = | \beta^ {\prime} - \beta_ {1} ^ {\prime} |, \xi_ {2} ^ {\prime} = | \beta^ {\prime} - \beta_ {2} ^ {\prime} |. A s s u m e \xi_ {2} ^ {\prime} <   \beta_ {2} ^ {\prime},
$$
$$
\epsilon^ {\prime} \leq \min \left[ \alpha_ {0} \tau_ {\tilde {v} _ {t - l} ^ {u} \tilde {v} _ {t - l} ^ {u}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) - \xi_ {1} ^ {\prime}, \frac {\alpha_ {0} \tau_ {\bar {v} _ {m} \bar {v} _ {m}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) \beta_ {2} ^ {\prime} (\beta_ {2} ^ {\prime} - \xi_ {2} ^ {\prime}) - \xi_ {2} ^ {\prime}}{\beta_ {2} ^ {\prime} (\beta_ {2} ^ {\prime} - \xi_ {2} ^ {\prime})}, \frac {1}{2} \left[ [ \tau_ {p _ {1} ^ {\prime} p _ {1} ^ {\prime}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) - \tau_ {p _ {1} ^ {\prime} \bar {v} _ {m}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) ] [ \alpha_ {0} + \eta_ {p _ {2} ^ {\prime}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) ] - \right. \right.
$$
$$
\left. \sum_ {v _ {j} \neq p _ {1} ^ {\prime}} [ \tau_ {v _ {j} \bar {v} _ {m}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) - \tau_ {v _ {j} p _ {1} ^ {\prime}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) ] \eta_ {v _ {j}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) ] \right]. W e h a v e: E _ {\mathrm {D D U I}} \leq C \big (\mathcal {O} (\epsilon^ {\prime} + \xi_ {1} ^ {\prime}) \big) ^ {\lambda} + C \big (\mathcal {O} (\epsilon^ {\prime} + \xi_ {2} ^ {\prime}) \big) ^ {\lambda} + C \big (\mathcal {O} (\epsilon^ {\prime}) \big) ^ {\lambda}.
$$
Proof. For the first term of  $E_{\mathrm{DDUI}}$ , we have:
We substitute  $\beta_{1}^{\prime}$  with  $\frac{\tau_{\tilde{v}_{t - l}^{u}\tilde{v}_{t - l}^{u}}^{\prime}(\tilde{\mathbf{x}}_{t}^{u})\eta_{p_{2}^{\prime}}^{\prime}(\tilde{\mathbf{x}}_{t}^{u}) + \sum_{v_{j}\neq \tilde{v}_{t - l}^{u}}\tau_{v_{j}\tilde{v}_{t - l}^{u}}^{\prime}\eta_{v_{j}}^{\prime}(\tilde{\mathbf{x}}_{t}^{u})}{\sum_{h = 1}^{H}[w_{h}\tilde{f}_{\tilde{v}_{m}}^{h}(\tilde{\mathbf{x}}_{t}^{u})]}$  and obtain:
$$
\begin{array}{l} \mathbb {P} \left[ p _ {1} ^ {\prime} = \tilde {v} _ {t - l} ^ {u}, \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} \tilde {f} _ {\tilde {v} _ {t - l}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\sum_ {h = 1} ^ {H} \left[ w _ {h} \tilde {f} _ {\tilde {v} _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]} <   \beta^ {\prime} \right] \\ \leq \mathbb {P} \left[ p _ {1} ^ {\prime} = \tilde {v} _ {t - l} ^ {u}, \eta_ {p _ {2} ^ {\prime}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) \leq \eta_ {\tilde {v} _ {t - l} ^ {u}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) <   \eta_ {p _ {2} ^ {\prime}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) + \frac {\epsilon^ {\prime}}{\tau_ {\tilde {v} _ {t - l} ^ {u} \tilde {v} _ {t - l} ^ {u}} (\bar {\mathbf {x}} _ {t} ^ {u})} + \frac {\xi_ {1} ^ {\prime} \sum_ {h = 1} ^ {H} \left[ w _ {h} \tilde {f} _ {v _ {m}} ^ {h} (\bar {\mathbf {x}} _ {t} ^ {u}) \right]}{\tau_ {\tilde {v} _ {t - l} ^ {u} \tilde {v} _ {t - l} ^ {u}} (\bar {\mathbf {x}} _ {t} ^ {u})} \right] \\ \leq \mathbb {P} \left[ \eta_ {p _ {2} ^ {\prime}} ^ {\prime} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \leq \eta_ {\tilde {v} _ {t - l} ^ {u}} ^ {\prime} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) <   \eta_ {p _ {2} ^ {\prime}} ^ {\prime} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) + \frac {\epsilon^ {\prime} + \xi_ {1} ^ {\prime}}{\tau_ {\tilde {v} _ {t - l} ^ {u} \tilde {v} _ {t - l} ^ {u}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right)} \right]. \tag {16} \\ \end{array}
$$
Recall that  $\epsilon' \leq \alpha_0 \tau_{\tilde{v}_{t-l}^u \tilde{v}_{t-l}^u}^{\prime} (\tilde{\mathbf{x}}_t^u) - \xi_1'$ , which implies  $\frac{\epsilon' + \xi_1'}{\tau_{\tilde{v}_{t-l}^u \tilde{v}_{t-l}^u}^{\prime} (\tilde{\mathbf{x}}_t^u)} \leq \alpha_0$ . Hence, the relaxed Multiclass Tsybakov Condition holds, and the probability of the first term of  $E_{\text{DDUI}}$  (Eq. 14) is bounded by  $C \left( \frac{\epsilon' + \xi_1'}{\tau_{\tilde{v}_{t-l}^u \tilde{v}_{t-l}^u}^{\prime} (\tilde{\mathbf{x}}_t^u)} \right)^{\lambda}$ , namely,  $C \left( \mathcal{O}(\epsilon' + \xi_1') \right)^{\lambda}$ .
For the second term of  $E_{\mathrm{DDUI}}$ , we have:
$$
\begin{array}{l} \mathbb {P} \left[ p _ {1} ^ {\prime} \neq \tilde {v} _ {t - l} ^ {u}, \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} \tilde {f} _ {\tilde {v} _ {t - l}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\sum_ {h = 1} ^ {H} \left[ w _ {h} \tilde {f} _ {\tilde {v} _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]} \geq \beta^ {\prime} \right] \\ = \mathbb {P} \left[ p _ {1} ^ {\prime} \neq \tilde {v} _ {t - l} ^ {u}, p _ {1} ^ {\prime} = \bar {v} _ {m}, \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} \bar {f} _ {\bar {v} _ {t - l}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\sum_ {h = 1} ^ {H} \left[ w _ {h} \bar {f} _ {\bar {v} _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]} \geq \beta^ {\prime} \right] \tag {17} \\ + \mathbb {P} \left[ p _ {1} ^ {\prime} \neq \tilde {v} _ {t - l} ^ {u}, p _ {1} ^ {\prime} \neq \bar {v} _ {m}, \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} \bar {f} _ {\tilde {v} _ {t - l} ^ {u}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right]}{\sum_ {h = 1} ^ {H} \left[ w _ {h} \bar {f} _ {\tilde {v} _ {m}} ^ {h} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right]} \geq \beta^ {\prime} \right] \\ \end{array}
$$
For the first term of Eq. 17, we have:
We replace  $\beta_{2}^{\prime}$  with  $\frac{\sum_{h = 1}^{H}[w_{h}\vec{f}_{\vec{v}_{t}^{u}}^{h}(\vec{\mathbf{x}}_{t}^{u})]}{\tau_{\vec{v}_{m}\vec{v}_{m}}^{\prime}(\vec{\mathbf{x}}_{t}^{u})\eta_{p_{2}}^{\prime}(\vec{\mathbf{x}}_{t}^{u}) + \sum_{v_{j}\neq v_{m}}\tau_{v_{j}v_{m}}^{\prime}(\vec{\mathbf{x}}_{t}^{u})\eta_{v_{j}}^{\prime}(\vec{\mathbf{x}}_{t}^{u})}$  and continue the calculation:
$$
\begin{array}{l} \mathbb {P} \left[ p _ {1} ^ {\prime} \neq \tilde {v} _ {t} ^ {u}, p _ {1} ^ {\prime} = v _ {m}, \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} \bar {f} _ {\tilde {v} _ {t} ^ {u}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\sum_ {h = 1} ^ {H} \left[ w _ {h} \bar {f} _ {\tilde {v} _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]} \geq \beta \right] \\ \leq \mathbb {P} \left[ p _ {1} ^ {\prime} = \bar {v} _ {m}, \eta_ {p _ {2} ^ {\prime}} ^ {\prime} \left(\bar {\mathbf {x}} _ {t} ^ {u}\right) \leq \eta_ {\bar {v} _ {m}} ^ {\prime} \left(\bar {\mathbf {x}} _ {t} ^ {u}\right) \leq \eta_ {p _ {2} ^ {\prime}} ^ {\prime} \left(\bar {\mathbf {x}} _ {t} ^ {u}\right) + \frac {\epsilon^ {\prime}}{\tau_ {\bar {v} _ {m} \bar {v} _ {m}} ^ {\prime} \left(\bar {\mathbf {x}} _ {t} ^ {u}\right)} + \frac {\xi_ {2} ^ {\prime} \sum_ {h = 1} ^ {H} \left[ w _ {h} \bar {f} _ {\bar {v} _ {m}} ^ {h} \left(\bar {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\beta_ {2} ^ {\prime} \left(\beta_ {2} ^ {\prime} - \xi_ {2} ^ {\prime}\right) \tau_ {\bar {v} _ {m} \bar {v} _ {m}} ^ {\prime} \left(\bar {\mathbf {x}} _ {t} ^ {u}\right)} \right] \tag {19} \\ \leq \mathbb {P} \left[ \eta_ {p _ {2} ^ {\prime}} ^ {\prime} (\tilde {\mathbf {x}} _ {t} ^ {u}) \leq \eta_ {\tilde {v} _ {m}} ^ {\prime} (\tilde {\mathbf {x}} _ {t} ^ {u}) \leq \eta_ {p _ {2} ^ {\prime}} ^ {\prime} (\tilde {\mathbf {x}} _ {t} ^ {u}) + \frac {\epsilon^ {\prime}}{\tau_ {\tilde {v} _ {m} \tilde {v} _ {m}} ^ {\prime} (\tilde {\mathbf {x}} _ {t} ^ {u})} + \frac {\xi_ {2} ^ {\prime}}{\beta_ {2} ^ {\prime} (\beta_ {2} ^ {\prime} - \xi_ {2} ^ {\prime}) \tau_ {\tilde {v} _ {m} \tilde {v} _ {m}} ^ {\prime} (\tilde {\mathbf {x}} _ {t} ^ {u})} \right]. \\ \end{array}
$$
Recall that  $\epsilon' \leq \frac{\alpha_0 \tau_{\bar{v}_m \bar{v}_m}^{\prime} (\bar{\mathbf{x}}_t^u) \beta_2^{\prime} (\beta_2^{\prime} - \xi_2^{\prime}) - \xi_2^{\prime}}{\beta_2^{\prime} (\beta_2^{\prime} - \xi_2^{\prime})}$ , which implies  $\frac{\epsilon'}{\tau_{\bar{v}_m \bar{v}_m}^{\prime} (\bar{\mathbf{x}}_t^u)} + \frac{\xi_2^{\prime}}{\beta_2^{\prime} (\beta_2^{\prime} - \xi_2^{\prime}) \tau_{\bar{v}_m \bar{v}_m}^{\prime} (\bar{\mathbf{x}}_t^u)} \leq \alpha_0$ . Hence, the relaxed Multiclass Tsybakov Condition holds and the probability of the first term of Eq. 17 is bounded by  $C \left( \frac{\epsilon' \beta_2' (\beta_2' - \xi_2') + \xi_2'}{\beta_2' (\beta_2' - \xi_2') \tau_{\bar{v}_m \bar{v}_m}^{\prime} (\bar{\mathbf{x}}_t^u)} \right)^{\lambda}$ , namely,  $C \left( \mathcal{O}(\epsilon' + \xi_2') \right)^{\lambda}$ .
Finally, for the second term of Eq. 17, we have:
$$
\begin{array}{l} \left. \tau_ {p _ {1} ^ {\prime} \bar {v} _ {m}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) \eta_ {p _ {1} ^ {\prime}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) + \sum_ {v _ {j} \neq p _ {1} ^ {\prime}} \tau_ {\bar {v} _ {m}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) \eta_ {v _ {j}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) + 2 \epsilon^ {\prime} \right] \\ = \mathbb {P} \left[ \eta_ {p _ {1} ^ {\prime}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) \geq \eta_ {p _ {2} ^ {\prime}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}), \eta_ {p _ {1} ^ {\prime}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) \big [ \tau_ {p _ {1} ^ {\prime} p _ {1} ^ {\prime}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) - \tau_ {p _ {1} ^ {\prime} \bar {v} _ {m}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) \big ] <   \right. \\ \left. \sum_ {v _ {j} \neq p _ {1} ^ {\prime}} \left[ \tau_ {\bar {v} _ {m}} ^ {\prime} \left(\bar {\mathbf {x}} _ {t} ^ {u}\right) - \tau_ {v _ {j} p _ {1} ^ {\prime}} ^ {\prime} \left(\bar {\mathbf {x}} _ {t} ^ {u}\right) \right] \eta_ {v _ {j}} ^ {\prime} \left(\bar {\mathbf {x}} _ {t} ^ {u}\right) + 2 \epsilon^ {\prime} \right] \\ = \mathbb {P} \left[ \eta_ {p _ {2} ^ {\prime}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) \leq \eta_ {p _ {1} ^ {\prime}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) <   \eta_ {p _ {2} ^ {\prime}} ^ {\prime} (\bar {\mathbf {x}} _ {t} ^ {u}) + \right. \\ \frac {\sum_ {v _ {j} \neq p _ {1} ^ {\prime}} \left[ \tau_ {\bar {v} _ {m}} ^ {\prime} (\tilde {\mathbf {x}} _ {t} ^ {u}) - \tau_ {v _ {j} p _ {1} ^ {\prime}} ^ {\prime} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right] \eta_ {v _ {j}} ^ {\prime} (\tilde {\mathbf {x}} _ {t} ^ {u}) + 2 \epsilon^ {\prime} - \left[ \tau_ {p _ {1} ^ {\prime} p _ {1} ^ {\prime}} ^ {\prime} (\tilde {\mathbf {x}} _ {t} ^ {u}) - \tau_ {p _ {1} ^ {\prime} \bar {v} _ {m}} ^ {\prime} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right] \eta_ {p _ {2} ^ {\prime}} ^ {\prime} (\tilde {\mathbf {x}} _ {t} ^ {u})}{\left[ \tau_ {p _ {1} ^ {\prime} p _ {1} ^ {\prime}} ^ {\prime} (\tilde {\mathbf {x}} _ {t} ^ {u}) - \tau_ {p _ {1} ^ {\prime} \bar {v} _ {m}} ^ {\prime} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right]} \\ \end{array}
$$
Recall that  $\epsilon' \leq \frac{1}{2} \left[ [\tau_{p_1' p_1'}' (\tilde{\mathbf{x}}_t^u) - \tau_{p_1' \bar{v}_m}' (\tilde{\mathbf{x}}_t^u)] [\alpha_0 + \eta_{p_2'}' (\tilde{\mathbf{x}}_t^u)] - \sum_{v_j \neq p_1'} [\tau_{\bar{v}_m}' (\tilde{\mathbf{x}}_t^u) - \tau_{v_j p_1'}' (\tilde{\mathbf{x}}_t^u)] \eta_{v_j}' (\tilde{\mathbf{x}}_t^u) + 2\epsilon' - [\tau_{p_1' p_1'}' (\tilde{\mathbf{x}}_t^u) - \tau_{p_1' \bar{v}_m}' (\tilde{\mathbf{x}}_t^u)] \eta_{p_2'}' (\tilde{\mathbf{x}}_t^u) \right]$ , which implies  $\frac{\sum_{v_j \neq p_1'} [\tau_{\bar{v}_m}' (\tilde{\mathbf{x}}_t^u) - \tau_{v_j p_1'}' (\tilde{\mathbf{x}}_t^u)] \eta_{v_j}' (\tilde{\mathbf{x}}_t^u) + 2\epsilon' - [\tau_{p_1' p_1'}' (\tilde {\mathbf{x}}_t^u) - \tau_{p_1' \bar{v}_m}' (\tilde {\mathbf{x}}_t^u)] \eta_{p_2'}' (\tilde {\mathbf{x}}_t^u)}{[\tau_{p_1' p_1'}' (\tilde{\mathbf{x}}_t^u) - \tau_{p_1' \bar{v}_m}' (\tilde{\mathbf{x}}_t^u)]} \leq \alpha_0$ . Hence, the relaxed Multiclass Tsybakov Condition holds and the probability of Case-3 is bounded by  $C\left(\frac{\sum_{v_j \neq p_1'} [\tau_{\bar{v}_m}' (\tilde{\mathbf{x}}_t^u) - \tau_{v_j p_1'}' (\tilde{\mathbf{x}}_t^u)] \eta_{v_j}' (\tilde{\mathbf{x}}_t^u) + 2\epsilon' - [\tau_{p_1' p_1'}'(\tilde{\mathbf{x}}_t^u) - \tau_{p_1' \bar{v}_m}'(\tilde{\mathbf{x}}_t^u)] \eta_{p_2'}'(\tilde{\mathbf{x}}_t^u)}{[\tau_{p_1' p_1'}'(\tilde{\mathbf{x}}_t^u) - \tau_{p_1' \bar{v}_m}'(\tilde{\mathbf{x}}_t^u)]}\right)^{\lambda}$ , namely,  $C\big(\mathcal{O}(\epsilon')\big)^{\lambda}$ .
B.5 The Proof of Theorem 4
Theorem 4. Let  $\mathcal{R}$  be a list of  $K$  items randomly sampled from  $\nu$  with replacement,  $\zeta \in (0,1)$ ,  $r_H(\mathbf{x}_t^u,v_i) = \sum_{v_j\in \mathcal{V}}\mathbb{I}\big[\sum_{h = 1}^H [w_hf_{v_j}^h (\tilde{\mathbf{x}}_t^u)] > \sum_{h = 1}^H [w_hf_{v_i}^h (\tilde{\mathbf{x}}_t^u)]\big]$  be the rank of item  $v_{i}$  over the entire item set at the  $H$ -th epoch, and  $\hat{r}_H(\mathbf{x}_t^u,v_i) = \sum_{v_j\in \mathcal{R}}\mathbb{I}\big[\sum_{h = 1}^H [w_hf_{v_j}^h (\tilde{\mathbf{x}}_t^u)] > \sum_{h = 1}^H [w_hf_{v_i}^h (\tilde{\mathbf{x}}_t^u)]\big]$  be the rank of  $v_{i}$  over  $\mathcal{R}$  at the  $H$ -th epoch. We have:  $\mathbb{P}\big[|\frac{\hat{r}_H(\mathbf{x}_t^u,v_i)}{K} -\frac{r_H(\mathbf{x}_t^u,v_i)}{|\mathcal{V}|} |\geq \zeta \big]\leq \exp (-2K\zeta^2)$ .
Proof.  $\hat{r}_H(\mathbf{x}_t^u,v_i)$  can be deemed as a random variable that counts the number of items ranked higher than  $v_{i}$  in  $\mathcal{R}$ . Since the items in  $\mathcal{R}$  are randomly sampled from  $\nu$  with replacement,  $\hat{r}_H(\mathbf{x}_t^u,v_i)$  follows a Binomial distribution Binomial  $(K,\frac{r_H(\mathbf{x}_t^u,v_i)}{|\mathcal{V}|})$ . Thus, we have: