SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Yatong Sun $^{1,2}$ , Xiaochun Yang $^{1*}$ , Zhu Sun $^{3*}$ , Bin Wang $^{1}$ , Yan Wang $^{2}$
$^{1}$ School of Computer Science and Engineering, Northeastern University, China
$^{2}$ School of Computing, Macquarie University, Australia
<sup>3</sup>Center for Frontier AI Research, Institute of High Performance Computing, A*STAR, Singapore
yatong@stumail.neu.edu.cn, yangxc@mail.neu.edu.cn, sunzhunta@gmail.com,
binwang@mail.neu.edu.cn, yan.wang@mq.edu.au
Sequential recommender systems (SRSs) are typically trained to predict the next item as the target given its preceding (and succeeding) items as the input. Such a paradigm assumes that every input-target pair is reliable for training. However, users can be induced to click on items that are inconsistent with their true preferences, resulting in unreliable instances, i.e., mismatched input-target pairs. Current studies on mitigating this issue suffer from two limitations: (i) they discriminate instance reliability according to models trained with unreliable data, yet without theoretical guarantees that such a seemingly contradictory solution can be effective; and (ii) most methods can only tackle either unreliable input or targets but fail to handle both simultaneously. To fill the gap, we theoretically unveil the relationship between SRS predictions and instance reliability, whereby two error-bounded strategies are proposed to rectify unreliable targets and input, respectively. On this basis, we devise a model-agnostic Bidirectional Data Rectification (BirDRec) framework, which can be flexibly implemented with most existing SRSs for robust training against unreliable data. Additionally, a rectification sampling strategy is devised and a self-ensemble mechanism is adopted to reduce the (time and space) complexity of BirDRec. Extensive experiments on four real-world datasets verify the generality, effectiveness, and efficiency of our proposed BirDRec.
Recently, the study on sequential recommender systems (SRSs) [1, 2, 3, 4, 5] has garnered much attention as users' preferences are inherently dynamic and evolving in real-world scenarios. The goal of SRSs is learning to predict the next item a user interacts with given the preceding (and succeeding) items. Therefore, a training instance for SRSs is typically composed of an input item sequence and its next item as the target. However, distractions in daily lives (e.g. recommendations from friends, account sharing, and accidental clicks) can induce users to click on items that are inconsistent with their true preferences, resulting in unreliable training instances with mismatched input-target pairs. The mismatch can be categorized into Complete Mismatch and Partial Mismatch when the item caused by distractions acts as an unreliable target and unreliable input of an instance, respectively. To illustrate, the romantic film 'La La Land', in the first instance of Figure 1, serves as an unreliable target, which is recommended by friends and completely mismatched with the previous superhero movies. By contrast, in the second instance, 'La La Land' acts as an unreliable input item which renders the input sequence partially mismatched with the target superhero film. Both
* denotes the corresponding authors
Figure 1: Examples of two types of mismatch caused by external distractions.
types of mismatch would cause unreliable training instances that mislead SRSs to learn sequential relationships between irrelevant items and eventually undermine the recommendation accuracy.
Although there are a number of studies aiming to combat such unreliable data for more robust SRSs, they suffer from two core limitations. (i) They discriminate instance reliability based on the intermediate or final output of a model (either the SRS itself [6, 7, 8, 9] or an additional corrector [10]) that is trained with unreliable data. However, there is no theoretical guarantee that such seemingly contradictory solutions can be trustworthy for detecting and correcting unreliable instances. (ii) Most prior studies only focus on tackling either unreliable input [6, 7, 11, 12, 13] or targets [8], but fail to handle both simultaneously. Only one recently proposed method [10] attempts to address this issue, but it relies on a corrector trained with unreliable data, yet without any theoretical guarantees.
As such, we, for the first time, theoretically unveil the relationship between the SRS predictions and instance reliability, proving that a target with consistently low prediction scores is unlikely to be reliable, assuming that the randomness of user behavior is limited. It then inspires us to devise two error-bounded rectification strategies to (1) detect consistently low-scored targets (i.e., unreliable targets) and replace them with steadily high-scored items and (2) detect and delete consistently low-scored items within the input (i.e., unreliable input), where the score is estimated by a backward SRS. Note that the unreliable input items, as the interruptions in the input, are not replaced but directly removed to bridge the preceding and succeeding items. Based on these strategies, we propose a model-agnostic Bidirectional Data Rectification (BirDRec) framework which contains two SRSs in opposite directions for tackling both unreliable targets and input. In addition, to reduce the time complexity, a rectification sampling strategy is devised to efficiently obtain consistently high-scored items; to reduce space complexity, a self-ensemble mechanism [14] is adopted to approximate the weighted average prediction scores across different training epochs.
Contributions. (1) We are the first to provide theoretically guaranteed data rectification strategies based on SRS predictions to tackle both unreliable input and targets for more robust SRSs. (2) We devise a model-agnostic bidirectional data rectification framework that can be flexibly implemented with most existing SRSs for robust training against unreliable data. (3) We devise a rectification sampling strategy and adopt a self-ensemble mechanism to ensure better scalability of BirDRec. (4) Extensive experiments with SRSs based on representative backbones and datasets across various domains validate the generality, effectiveness, and efficiency of BirDRec.
Early SRSs [15, 16] adopt Markov Chains to encode users' interaction sequences, assuming users' latest interactions affect the future behavior linearly and independently. Later, powerful deep learning backbones such as recurrent neural networks (RNNs) [17, 18, 19], convolution neural networks (CNNs) [20, 21, 22], graph neural networks (GNNs) [23, 24, 25, 26, 27], and Transformers [28, 29] are employed to extract complex non-linear patterns within users' sequences [30]. They posit each training instance is a definitely matched input-target pair and thus cannot handle unreliable data.
To resist unreliable data, existing robust SRSs can be categorized into three types. The first type focuses on handling the complete mismatch by identifying and eliminating instances with unreliable targets. For example, BERD [8] empirically finds that instances with high training loss and low uncertainty tend to have unreliable targets. This idea is relevant to studies on clean label selection [31, 32, 33, 34, 35] and label correction [36, 37, 38]. The second type concentrates on addressing partial mismatch by reducing the importance of unreliable input when formulating users' dynamic preference representations. Accordingly, various advanced mechanisms are integrated into SRSs, such as memory networks [7], gating networks [39, 40, 41], autoencoders [12], reinforcement learning [6],
uncertainty modelling [11, 42], and Fast Fourier Transform [9]. To the best of our knowledge, there is only one recently proposed method STEAM [10] falling into the third type which attempts to tackle both unreliable targets and input with an additional corrector producing reliable data. Nonetheless, existing robust SRSs all rely on models (either SRSs or additional correctors) trained with unreliable data, yet without theoretical proof that such seemingly contradictory solutions can be effective.
3 Theoretical Guarantees for Rectifying Unreliable Data
This section presents theoretical guarantees for rectifying unreliable data via SRS predictions for more robust SRSs. In particular, Section 3.1 introduces important preliminaries. Subsequently, Section 3.2 unveils the relationship between the prediction score of SRSs and the reliability of a target, inspiring us to propose an error-bounded strategy for handling unreliable targets; and Section 3.3 provides the error-bounded strategy for dealing with unreliable input.
Problem Statement of SRSs. Let  $\mathcal{U}$  and  $\mathcal{V}$  be the sets of users and items, respectively. Each user  $u\in \mathcal{U}$  chronologically interacts with a sequence of items  $\mathbf{s}^u = [\tilde{v}_1^u,\tilde{v}_2^u,\dots,\tilde{v}_{|\mathbf{s}^u|}^u]$ , where  $\tilde{v}_t^u\in \mathcal{V}$  is the  $t$ -th item user  $u$  interacts with and  $|\mathbf{s}^u|$  is the length of sequence  $\mathbf{s}^u$ . The goal of SRSs is to predict the target item  $\tilde{v}_t^u$  given the input  $\tilde{\mathbf{x}}_t^u = \{u,[\tilde{v}_{t - L}^u,\dots,\tilde{v}_{t - 2}^u,\tilde{v}_{t - 1}^u]\}$ , where  $L$  is the length of  $\tilde{\mathbf{x}}_t^u$ . Thus, the training instance of SRSs can be represented as an input-target pair  $\langle \tilde{\mathbf{x}}_t^u,\tilde{v}_t^u\rangle$ . Note that we use  $\sim$  to denote the observed data that may be unreliable due to external distractions.
Core Assumptions. Ideally, each user-item interaction should be drawn from users' true preference distribution  $\eta$  without any distractions, where  $\eta_{v_i}(\mathbf{x}_t^u) = \mathbb{P}(v_t^u = v_i|\mathbf{x}_t^u)$ . We define  $p_1$  to be the true item for recommendation, i.e., the top-1 item according to  $\eta$ ,
$$
\sum_ {v _ {i} \in \mathcal {V}} \mathbb {I} \left[ \eta_ {p _ {1}} \left(\mathbf {x} _ {t} ^ {u}\right) \geq \eta_ {v _ {i}} \left(\mathbf {x} _ {t} ^ {u}\right) \right] = | \mathcal {V} |, \tag {1}
$$
where  $\mathbb{I}[\cdot ]$  is an indicator function that equals 1 if the condition is true; otherwise 0. Meanwhile, we define  $p_2$  to be the middle-ranked item (ranked  $\lfloor |\mathcal{V}| / 2\rfloor$  -th) according to  $\eta$ , namely,
$$
\sum_ {v _ {i} \in \mathcal {V}} \mathbb {I} \left[ \eta_ {p _ {2}} \left(\mathbf {x} _ {t} ^ {u}\right) \geq \eta_ {v _ {i}} \left(\mathbf {x} _ {t} ^ {u}\right) \right] = \lfloor | \mathcal {V} | / 2 \rfloor . \tag {2}
$$
In general, SRSs are built upon the hypothesis that users usually select items with a tendency rather than randomly. In other words, the randomness of users' true preferences is restricted, i.e., the probability gap between the top-1 and middle-ranked items regarding  $\eta$  is unlikely to be small. This assumption can be formally defined as follows.
Assumption 1. The users' true preference distribution  $\eta$  fulfills the relaxed Multiclass Tsybakov Condition [43] with constants  $C > 0$ ,  $\lambda > 0$ , and  $\alpha_0 \in (0,1]$ , such that for all  $\alpha \in (0,\alpha_0]$
$$
\mathbb {P} \left[ \eta_ {p _ {1}} \left(\mathbf {x} _ {t} ^ {u}\right) - \eta_ {p _ {2}} \left(\mathbf {x} _ {t} ^ {u}\right) \leq \alpha \right] \leq C \alpha^ {\lambda}. \tag {3}
$$
The feasibility of Assumption 1 relies on small  $C$  and large  $\lambda$ , which are satisfied on public (observed) datasets based on our empirical analysis in the Appendix with  $C \in (0.55, 0.70)$  and  $\lambda \in (1.37, 4.01)$ .
Connecting  $\eta$  with SRS Predictions. Obviously,  $\eta$  is the ideal corrector to rectify unreliable data, however, due to its unavailability, many existing methods [8, 10] leverage SRS predictions as the substitution with no theoretical guarantees. This urges us to explore the connection between  $\eta$  and SRS predictions. To achieve this goal, we first investigate the relationship between  $\eta$  and users' observed preference distribution  $\tilde{\eta}$ , since SRSs are trained with the observed data that may be distorted by external distractions. Formally,  $\tilde{\eta}_{v_i}(\tilde{\mathbf{x}}_t^u) = \mathbb{P}(\tilde{v}_t^u = v_i|\tilde{\mathbf{x}}_t^u)$ , where  $\tilde{v}_t^u$  is the observed target that may be unreliable. We then define a transition probability  $\tau_{v_jv_i}(\tilde{\mathbf{x}}_t^u) = \mathbb{P}(\tilde{v}_t^u = v_i|v_t^u = v_j,\tilde{\mathbf{x}}_t^u)$  as the chance that a true target  $v_{t}^{u}$  is flipped from item  $v_{j}$  to item  $v_{i}$  owing to external distractions. Thus, for any pair  $(v_{i},v_{j})\in \mathcal{V}$ , there is a linear relationship between  $\eta$  and  $\tilde{\eta}$ :
$$
\tilde {\eta} _ {v _ {i}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) = \sum_ {v _ {j} \in \mathcal {V}} \mathbb {P} \left(\tilde {v} _ {t} ^ {u} = v _ {i} \mid v _ {t} ^ {u} = v _ {j}, \tilde {\mathbf {x}} _ {t} ^ {u}\right) \mathbb {P} \left(v _ {t} ^ {u} = v _ {j} \mid \tilde {\mathbf {x}} _ {t} ^ {u}\right) = \sum_ {v _ {j} \in \mathcal {V}} \tau_ {v _ {j} v _ {i}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \eta_ {v _ {j}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right). \tag {4}
$$
To bridge  $\eta$  and SRS predictions via  $\tilde{\eta}$ , we then study the relationship between  $\tilde{\eta}$  and SRS predictions. Let  $f^h$  be an SRS at the  $h$ -th training epoch, and the prediction of  $f^h$  be  $\epsilon$ -close to  $\tilde{\eta}$ ,
$$
\epsilon = \max  _ {\tilde {\mathbf {x}} _ {t} ^ {u}, v _ {i}} \left| \tilde {\eta} _ {v _ {i}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) - f _ {v _ {i}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right|, \tag {5}
$$
where  $f_{v_i}^h (\tilde{\mathbf{x}}_t^u)$  is the predicted probability (score) of the target being  $v_{i}$  given the input  $\tilde{\mathbf{x}}_t^u$  at the  $h$ -th training epoch. Eqs. (4-5) indicate that there is indeed a connection between  $\eta$  and SRS predictions, laying the foundation for the proposed strategies to rectify unreliable data as what follows.
3.2 Theorems for Rectifying Unreliable Targets
We now explore how to properly use SRS predictions for rectifying unreliable targets. Prior works empirically find that unreliable targets tend to possess consistently low prediction scores at different epochs [8, 44]. Yet, there is no guarantee that such prediction scores given by an SRS trained with unreliable data can be trustworthy for detecting unreliable targets. This prompts us to theoretically unveil the relationship between target reliability and SRS predictions at different epochs. Specifically, Theorem 1 proves that a reliable target is unlikely to keep low prediction scores during training.
Theorem 1. Given Assumption 1, let  $\{w_h\mid 1\leq h\leq H,0\leq w_h\leq 1,\sum_{h = 1}^H w_h = 1\}$  be the weights for averaging prediction scores of different epochs.  $\forall \langle \tilde{\mathbf{x}}_t^u,\tilde{\mathbf{v}}_t^u\rangle$  assume  $\epsilon \leq \alpha_0\tau_{\tilde{v}_t^u\tilde{v}_t^u}(\tilde{\mathbf{x}}_t^u)$ . Let  $\gamma = \tau_{\tilde{v}_t^u\tilde{v}_t^u}(\tilde{\mathbf{x}}_t^u)\eta_{p_2}(\tilde{\mathbf{x}}_t^u) + \sum_{v_j\neq \tilde{v}_t^u}\tau_{v_j\tilde{v}_t^u}(\tilde{\mathbf{x}}_t^u)\eta_{v_j}(\tilde{\mathbf{x}}_t^u)$ . We have:  $\mathbb{P}\Big[p_1 = \tilde{v}_t^u,\sum_{h = 1}^H\big[w_hf_{\tilde{v}_t^u}^h (\tilde{\mathbf{x}}_t^u)\big]\leq \gamma \Big]\leq C(\mathcal{O}(\epsilon))^{\lambda}$ . Proof.
$$
\begin{array}{l} \mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u}, \sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {\tilde {v} _ {t} ^ {u}} ^ {h} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right] \leq \gamma \right] \\ \leq \mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u}, \sum_ {h = 1} ^ {H} w _ {h} \left[ \tilde {\eta} _ {\tilde {v} _ {t} ^ {u}} (\tilde {\mathbf {x}} _ {t} ^ {u}) - \epsilon \right] \leq \gamma \right] \\ = \mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u}, \eta_ {\tilde {v} _ {t} ^ {u}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \geq \eta_ {P 2} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right), \sum_ {h = 1} ^ {H} w _ {h} \left[ \sum_ {v _ {j} \in \nu} \tau_ {v _ {j} \tilde {v} _ {t} ^ {u}} \eta_ {v _ {j}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) - \epsilon \right] \leq \gamma \right] \tag {6} \\ \leq \mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u}, \eta_ {\tilde {v} _ {t} ^ {u}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \geq \eta_ {p _ {2}} (\tilde {\mathbf {x}} _ {t} ^ {u}), \tau_ {\tilde {v} _ {t} ^ {u} \tilde {v} _ {t} ^ {u}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \eta_ {\tilde {v} _ {t} ^ {u}} (\tilde {\mathbf {x}} _ {t} ^ {u}) + \sum_ {v _ {j} \neq \bar {v} _ {t} ^ {u}} \tau_ {v _ {j} \tilde {v} _ {t} ^ {u}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \eta_ {v _ {j}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \leq \gamma + \epsilon \right] \\ = \mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u}, \eta_ {p 2} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \leq \eta_ {\tilde {v} _ {t} ^ {u}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \leq \frac {\gamma + \epsilon - \sum_ {v _ {j} \neq \tilde {v} _ {t} ^ {u}} \tau_ {v _ {j} \tilde {v} _ {t} ^ {u}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \eta_ {v _ {j}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right)}{\tau_ {v _ {t} ^ {u} \tilde {v} _ {t} ^ {u}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right)} \right]. \\ \end{array}
$$
By replacing  $\gamma$  with  $\tau_{\tilde{v}_t^u\tilde{v}_t^u}(\tilde{\mathbf{x}}_t^u)\eta_{p_2}(\tilde{\mathbf{x}}_t^u) + \sum_{v_j\neq \tilde{v}_t^u}\tau_{v_j}\tilde{v}_t^u (\tilde{\mathbf{x}}_t^u)\eta_{v_j}(\tilde{\mathbf{x}}_t^u)$ , we obtain:
$$
\mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u}, \sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {\tilde {v} _ {t} ^ {u}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right] \leq \gamma \right] \leq \mathbb {P} \left[ \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \leq \eta_ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \leq \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) + \frac {\epsilon}{\tau_ {\tilde {v} _ {t} ^ {u}} v _ {t} ^ {u} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right)} \right]. \tag {7}
$$
Recall that  $\epsilon \leq \alpha_0\tau_{\tilde{v}_t^u}\tilde{v}_t^u (\tilde{\mathbf{x}}_t^u)$ , which indicates  $\frac{\epsilon}{\tau_{\tilde{v}_t^u}\tilde{v}_t^u(\tilde{\mathbf{x}}_t^u)}\leq \alpha_0$ . Hence, the relaxed Multiclass Tsybakov Condition holds and the probability is bounded by  $C\big(\frac{\epsilon}{\tau_{\tilde{v}_t^u}\tilde{v}_t^u(\tilde{\mathbf{x}}_t^u)}\big)^\lambda$ , namely,  $C\big(\mathcal{O}(\epsilon)\big)^{\lambda}$ .
Note that a small  $\epsilon$  relies on large observed data and a powerful  $f$ , so that (i) the observed data can accurately approximate  $\tilde{\eta}$ , and (ii)  $f$  can closely fit the observed data. Both requirements can be satisfied thanks to the large-scale datasets and deep learning advancements in recommendation.
Theorem 1 indicates that the probability of a reliable target  $(p_1 = \tilde{v}_t^u)$  keeping low prediction scores across different epochs  $\left(\sum_{h=1}^{H}\left[w_h f_{\tilde{v}_t^u}^h(\tilde{\mathbf{x}}_t^u)\right] \leq \gamma\right)$  is bounded to be low, i.e., no more than  $C\big(\mathcal{O}(\epsilon)\big)^{\lambda}$ . This inspires us to rectify unreliable targets by replacing consistently low-scored targets with steadily high-scored items as the following strategy.
DRUT: Detecting and Replacing Unreliable Targets. Given an SRS  $f$  that is  $\epsilon$ -close to  $\tilde{\eta}$ , an instance  $\langle \tilde{\mathbf{x}}_t^u, \tilde{v}_t^u \rangle$ , the consistently high-scored item  $v_m = \arg \max_{v_i \neq \tilde{v}_t^u} \sum_{h=1}^{H} [w_h f_{v_i}^h(\tilde{\mathbf{x}}_t^u)]$ , and  $\beta \in (0, 1]$ , we stipulate that if  $\sum_{h=1}^{H} [w_h f_{\tilde{v}_t^u}^h(\tilde{\mathbf{x}}_t^u)] / \sum_{h=1}^{H} [w_h f_{v_m}^h(\tilde{\mathbf{x}}_t^u)] < \beta$ , i.e., the target  $\tilde{v}_t^u$  is consistently lower-scored than  $v_m$  to some extent, then  $\tilde{v}_t^u$  should be replaced by  $v_m$  in the  $H$ -th epoch. We denote the output instance of DRUT as  $\langle \tilde{\mathbf{x}}_t^u, \hat{v}_t^u \rangle$ .
Different from existing methods, DRUT is theoretically error-bounded. Specifically, the error of DRUT, denoted as  $\pmb{E}_{\mathrm{DRUT}}$ , comes from three cases: (Case-1) the true target  $p_1$  is  $\tilde{v}_t^u$  but is replaced
<sup>1</sup>The detailed setting of the weights is related to the self-ensemble mechanism introduced in Section 4.2.
by  $v_{m}$ . (Case-2) the true target  $p_1$  is  $v_{m}$  but  $\tilde{v}_t^u$  is kept. (Case-3) the true target is neither  $\tilde{v}_t^u$  nor  $v_{m}$ . Correspondingly,  $E_{\mathrm{DRUT}}$  can be formulated as below:
$$
\boldsymbol {E} _ {\mathrm {D R U T}} = \underbrace {\mathbb {P} \left[ p _ {1} = \tilde {v} _ {t} ^ {u} , p _ {1} \neq v _ {m} , \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {\tilde {v} _ {t} ^ {h}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]} <   \beta \right]} _ {\text {C a s e - 1}} + \underbrace {\mathbb {P} \left[ p _ {1} \neq \tilde {v} _ {t} ^ {u} , p _ {1} = v _ {m} , \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {\tilde {v} _ {t} ^ {h}} ^ {u} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]} \geq \beta \right]} _ {\text {C a s e - 2}} + \underbrace {\mathbb {P} \left[ p _ {1} \neq \tilde {v} _ {t} ^ {u} , p _ {1} \neq v _ {m} \right]} _ {\text {C a s e - 3}}
$$
Subsequently, Lemma 1 first provides the thresholds  $\beta_{1}$  and  $\beta_{2}$  that can respectively guarantee bounded probabilities for Case-1 and Case-2 of  $E_{\mathrm{DRUT}}$ . Then in Theorem 2, we prove that even if the chosen  $\beta$  deviates from  $\beta_{1}$  and  $\beta_{2}$ , DRUT is still error-bounded<sup>2</sup>.
Lemma 1. Given Assumption 1 and the set of weights  $\{w_h \mid 1 \leq h \leq H, 0 \leq w_h \leq 1, \sum_{h=1}^{H} w_h = 1\}$ ,  $\forall \langle \tilde{\mathbf{x}}_t^u, \tilde{v}_t^u \rangle$ , assume  $\epsilon \leq \min \left[ \alpha_0 \tau_{\tilde{v}_t^u \tilde{v}_t^u} (\tilde{\mathbf{x}}_t^u), \alpha_0 \tau_{v_m v_m} (\tilde{\mathbf{x}}_t^u) \right]$ . Let  $\beta_1 = \left[ \frac{\tau_{\tilde{v}_t^u \tilde{v}_t^u} (\tilde{\mathbf{x}}_t^u) \eta_{p_2} (\tilde{\mathbf{x}}_t^u) + \sum_{v_j \neq \tilde{v}_t^u} \tau_{v_j \tilde{v}_t^u} \eta_{v_j} (\tilde{\mathbf{x}}_t^u)}{\sum_{h=1}^{H} [w_h f_{v_m}^h (\tilde{\mathbf{x}}_t^u)]} \right]$  and  $\beta_2 = \left[ \frac{\sum_{h=1}^{H} [w_h f_{v_t}^h (\tilde{\mathbf{x}}_t^u)]}{\tau_{v_m v_m} (\tilde{\mathbf{x}}_t^u) \eta_{p_2} (\tilde{\mathbf{x}}_t^u) + \sum_{v_j \neq v_m} \tau_{v_j v_m} (\tilde{\mathbf{x}}_t^u) \eta_{v_j} (\tilde{\mathbf{x}}_t^u)} \right]$ . We have:  $\beta \leq \beta_1$  guarantees the probability of Case-1 in  $E_{\mathrm{DRUT}}$  is bounded by  $C(\mathcal{O}(\epsilon))^{\lambda}$ , and  $\beta \geq \beta_2$  guarantees the probability of Case-2 in  $E_{\mathrm{DRUT}}$  is bounded by  $C(\mathcal{O}(\epsilon))^{\lambda}$ .