SlowGuess/NeurIPS_2023_OCR huggingface dataset:
[1] Deepak Agarwal and Bee-Chung Chen. Regression-based latent factor models. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 19-28, 2009.
[2] Keping Bi, Qingyao Ai, Yongfeng Zhang, and W Bruce Croft. Conversational product search based on negative feedback. In Proceedings of the 28th acm international conference on information and knowledge management, pages 359-368, 2019.
[3] Haibin Chen, Qianli Ma, Zhenxi Lin, and Jiangyue Yan. Hierarchy-aware label semantics matching network for hierarchical text classification. In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers), pages 4370-4379, 2021.
[4] Heng-Tze Cheng, Levent Koc, Jeremiah Harmsen, Tal Shaked, Tushar Chandra, Hrishi Aradhye, Glen Anderson, Greg Corrado, Wei Chai, Mustafa Ispir, et al. Wide & deep learning for recommender systems. In Proceedings of the 1st workshop on deep learning for recommender systems, pages 7-10, 2016.
[5] Konstantina Christakopoulou, Alex Beutel, Rui Li, Sagar Jain, and Ed H Chi. Q&R: A two-stage approach toward interactive recommendation. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pages 139-148, 2018.
[6] Konstantina Christakopoulou, Filip Radlinski, and Katja Hofmann. Towards conversational recommender systems. In Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining, pages 815-824, 2016.
[7] Alipay Dataset. https://tianchi.aliyun.com/dataset/datadetail?dataid=53.
[8] Amazon Dataset. https://jmcauley.ucsd.edu/data/amazon/.
[9] LastFM Dataset. https://grouplens.org/datasets/hetrec-2011/.
[10] Taobao Dataset. https://tianchi.aliyun.com/dataset/datadetail?dataid=649.
[11] Tmall Dataset. https://tianchi.aliyun.com/dataset/datadetail?dataid=42.
[12] Yelp Dataset. https://www.yelp.com/dataset/.
[13] Yang Deng, Yaliang Li, Fei Sun, Bolin Ding, and Wai Lam. Unified conversational recommendation policy learning via graph-based reinforcement learning. In Proceedings of the 44th International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 1431-1441, 2021.
[14] Mehdi Elahi, Francesco Ricci, and Neil Rubens. A survey of active learning in collaborative filtering recommender systems. Computer Science Review, 20:29-50, 2016.
[15] Chongming Gao, Wenqiang Lei, Xiangnan He, Maarten de Rijke, and Tat-Seng Chua. Advances and challenges in conversational recommender systems: A survey. AI Open, 2:100-126, 2021.
[16] Nadav Golbandi, Yehuda Koren, and Ronny Lempel. On bootstrapping recommender systems. In Proceedings of the 19th ACM international conference on Information and knowledge management, pages 1805-1808, 2010.
[17] Nadav Golbandi, Yehuda Koren, and Ronny Lempel. Adaptive bootstrapping of recommender systems using decision trees. In Proceedings of the fourth ACM international conference on Web search and data mining, pages 595-604, 2011.
[52] Guorui Zhou, Xiaogiang Zhu, Chenru Song, Ying Fan, Han Zhu, Xiao Ma, Yanghui Yan, Junqi Jin, Han Li, and Kun Gai. Deep interest network for click-through rate prediction. In Proceedings of the 24th ACM SIGKDD international conference on knowledge discovery & data mining, pages 1059-1068, 2018.
[53] Feng Zhu, Chaochao Chen, Yan Wang, Guanfeng Liu, and Xiaolin Zheng. Dtcdr: A framework for dual-target cross-domain recommendation. In Proceedings of the 28th ACM International Conference on Information and Knowledge Management, pages 1533-1542, 2019.
A1 Conversational Agent Design on Uncertainty Minimization Framework
A1.1 Detailed Deviations
This paper introduces a conversational agent built upon the recommender system to interact with a human user. We begin by summarizing the interaction principles, taking Figure 1 as an example.
Definition A1 (Conversational Agent and Human User Interactions). Our conversational agent is designed to act in the following four ways.
(i) Query an item  $v \in \mathcal{V}_{k-1}$ , where  $\mathcal{V}_{k-1}$  is the set of unchecked items after  $k-1$  interactions (e.g., recommend Hotel A to the user).
(ii) Query an attribute  $x \in \mathcal{X}_{k-1}$ , where  $\mathcal{X}_{k-1}$  is the set of unchecked attributes after  $k-1$  interactions (e.g., query what Hotel Level does the user want).
(iii) Query whether the user's preference on an attribute  $x \in \mathcal{X}_{k-1}$  is equal to a specific attribute value  $w_x \in \mathcal{W}_x$  where  $\mathcal{W}_x$  is the set of values of attribute  $x$  (e.g., query whether the user likes a hotel with Hotel Level=5).
(iv) Query whether the user's preference on an attribute is not smaller than a specific value  $w_{x} \in \mathbb{R}$  (e.g., query whether the user likes a hotel with Hotel Level  $\geq 3.5$ ).
The human user is supposed to respond in the following ways.
(i) For queried item  $v$ , the user should answer Yes (if  $v$  satisfies the user) or No (otherwise) (e.g., answer Yes, if the user likes Hotel A).
(ii) For queried attribute  $x$ , the user should answer her preferred attribute value, denoted as  $w_{x}^{*} \in \mathcal{W}_{x}$  (e.g., answer 3 for queried attribute Hotel Level), or answer Not Care to represent that any attribute value works.
(iii) For queried attribute value  $w_x$ , the user should answer Yes (if  $w_x$  matches the user preference) or No (otherwise) (e.g., answer Yes, if the user wants a hotel with Hotel Level=5), or answer Not Care to represent that any attribute value works.
(iv) For queried attribute value  $w_{x}$ , the user should answer Yes (if the user wants an item whose value of attribute  $x$  is not smaller than  $w_{x}$ ) or No (otherwise) (e.g., answer Yes, if the user wants a hotel with Hotel Level=5), or answer Not Care to represent that any attribute value works.
Then, we separately describe the key concepts, including uncertainty, certainty gain, and expected certainty gain, introduced in this paper.
Definition A2 (Uncertainty). For the  $k$ -th turn, we define uncertainty, denoted as  $\mathrm{U}_k$ , to measure how many estimated relevance scores are still unchecked, which can be formulated as:
$$
\mathrm {U} _ {k} := \operatorname {S U M} \left(\left\{\Psi_ {\mathrm {R E}} \left(v _ {m}\right) \mid v _ {m} \in \mathcal {V} _ {k} \right\}\right), \tag {14}
$$
where  $\Psi_{\mathrm{RE}}(v_m)$  outputs the estimated relevance score for item  $v_m$ . The above equation tells us that the uncertainty of each turn is decided by the unchecked items.
It is straightforward to derive the certainty gain, as the uncertainty reduction at each turn.
Definition A3 (Certainty Gain). For the  $k$ -th turn, we define certainty gain of  $k$ -th interaction as:
$$
\Delta \mathrm {U} _ {k} := \mathrm {U} _ {k - 1} - \mathrm {U} _ {k} = \operatorname {S U M} \left(\left\{\Psi_ {\mathrm {R E}} \left(v _ {m}\right) \mid v _ {m} \in \Delta \mathcal {V} _ {k} \right\}\right), \tag {15}
$$
where  $\Delta \mathcal{V}_k = \mathcal{V}_{k - 1}\setminus \mathcal{V}_k$ . For simplicity, we use a to denote the  $k$ -th action of the conversational agent. According to the Human-AI interactions introduced in Definition A1, we can derive:
$$
\Delta \mathcal {V} _ {k} = \left\{ \begin{array}{c c} \mathcal {V} _ {k}, & a \in \mathcal {V} _ {k - 1} \text {a n d t h e a n s w e r t o q u e r y i n g (i) i s Y e s ,} \\ \{a \}, & a \in \mathcal {V} _ {k - 1} \text {a n d t h e a n s w e r t o q u e r y i n g (i) i s N o ,} \\ \mathcal {V} _ {a _ {\text {v a l u e}} \neq w _ {a} ^ {*}} \cap \mathcal {V} _ {k - 1}, & a \in \mathcal {X} _ {k - 1} \text {a n d t h e a n s w e r t o q u e r y i n g (i i)} \text {i s} w _ {a} ^ {*}, \\ \mathcal {V} _ {x _ {\text {v a l u e}} \neq w _ {x}} \cap \mathcal {V} _ {k - 1}, & a \in \mathcal {W} _ {x} \text {w h e r e x} \in \mathcal {X} _ {k - 1} \text {a n d t h e a n s w e r t o q u e r y i n g (i i i)} \text {i s Y e s ,} \\ \mathcal {V} _ {x _ {\text {v a l u e}} = w _ {x}} \cap \mathcal {V} _ {k - 1}, & a \in \mathcal {W} _ {x} \text {w h e r e x} \in \mathcal {X} _ {k - 1} \text {a n d t h e a n s w e r t o q u e r y i n g (i i i)} \text {i s N o ,} \\ \mathcal {V} _ {x _ {\text {v a l u e}} <   w _ {x}} \cap \mathcal {V} _ {k - 1}, & a \in \mathbb {R}, x \in \mathcal {X} _ {k - 1} \text {a n d t h e a n s w e r t o q u e r y i n g (i v)} \text {i s Y e s ,} \\ \mathcal {V} _ {x _ {\text {v a l u e}} \geq w _ {x}} \cap \mathcal {V} _ {k - 1}, & a \in \mathbb {R}, x \in \mathcal {X} _ {k - 1} \text {a n d t h e a n s w e r t o q u e r y i n g (i v)} \text {i s N o .} \\ \emptyset , & \text {t h e a n s w e r t o q u e r y i n g e i t h e r (i i) , (i i i) o r (i v) i s N o t C a r e ,} \end{array} \right. \tag {16}
$$
where  $\mathcal{V}_{a_{\text{value}}} \neq w_{a}^{*}$  is the set of unchecked items whose value of attribute  $a$  is not equal to the user answer  $w_{a}^{*}$ ,  $\mathcal{V}_{x_{\text{value}}} \neq w_{x}$  is the set of unchecked items whose value of attribute  $x$  is not equal to the queried attribute value  $w_{x}$ ,  $\mathcal{V}_{x_{\text{value}}} = w_{x}$ , a subset of  $\mathcal{V}_{k-1}$ , is the set of unchecked items whose value of attribute  $x$  is equal to the queried attribute value  $w_{x}$ ,  $\mathcal{V}_{x_{\text{value}} < w_{x}}$  is the set of unchecked items whose value of attribute  $x$  is smaller than the queried attribute value  $w_{x}$ ,  $\mathcal{V}_{x_{\text{value}} \geq w_{x}}$  is the set of unchecked items whose value of attribute  $x$  is not smaller than the queried attribute value  $w_{x}$ .
To estimate the certainty gain from taking each possible action, we introduce the expected certainty gain as follows.
Definition A4 (Expected Certainty Gain). For the  $k$ -th turn, we define expected certainty gain to estimate  $\Delta U_k$  on  $\Psi_{\mathbb{C}\mathbb{O}}(\cdot)$  taking a different action.
$$
\Psi_ {\mathrm {C G}} (\cdot) = \left\{ \begin{array}{l l} E q. (4), & a \in \mathcal {V} _ {k - 1}, i. e., q u e r y i n g (i), \\ E q. (6), & a \in \mathcal {X} _ {k - 1}, i. e., q u e r y i n g (i i), \\ E q. (9), & a \in \mathcal {W} _ {x}, i. e., q u e r y i n g (i i i), \\ E q. (1 1), & a \in \mathbb {R}, x \in \mathcal {X} _ {k - 1}, i. e., q u e r y i n g (i v). \end{array} \right. \tag {17}
$$
Then, at each turn, we can compute the candidate action, getting the maximum expected certainty gain, as the action to take, denoted as  $a_{\text{query}}$ . In practice, as shown in Proposition 1, for each attribute, querying attribute IDs, i.e., (ii), and querying attribute values, i.e., (iii), is not compatible. And, (iv) is particularly designed for a large discrete or continuous value space, which can be regarded as a specific attribute value generation engineering for (iii) (i.e., using Eq. (12) to directly compute the queried value for each attribute), and thus, we treat (iv) as a part of (iii). Therefore, we organize two querying strategies. One is querying (i) and (ii), whose objective can be formulated as Eq. (3). The other one is querying (i) and (iii), and the objective can be written as Eq. (10).
Besides  $\mathcal{V}_k$ , we further summarize the update of  $\mathcal{X}_k$  as follows. Similarly, we can define  $\Delta \mathcal{X}_k \coloneqq \mathcal{X}_{k-1} \setminus \mathcal{X}_k$ , then  $\Delta \mathcal{X}_k$  can be written as:
$$
\Delta \mathcal {X} _ {k} = \left\{ \begin{array}{c c} \{a \}, & \text {q u e r y i n g (i i) ,} \\ \{x \}, & \text {q u e r y i n g e i t h e r (i i i) o r (i v) , a n d t h e r e i s n o u n c h e c k e d a t t r i b u t e v a l u e i n x ,} \\ \emptyset , & \text {q u e r y i n g e i t h e r (i) o r (i v) .} \end{array} \right. \tag {18}
$$
Based on the above, CORE runs as Algorithm 1 shows.
Remark. One of the advantages of querying attribute values, compared to querying attributes, is that the user's answer to queried attribute would be out of the candidate attribute values (i.e.,  $\mathcal{W}_x$  for queried attribute  $x$ ). We are also aware that one possible solution is that the conversational agent would list all the candidate attribute values in the query. However, we argue that this approach would work only when the number of candidate values is small (namely,  $|\mathcal{W}_x|$  is small) such as attributes Color and Hotel Level, but can not work when there are many candidate values, e.g., attribute Brand, since listing all of them would significantly reduce the user satisfaction.
A1.2 Proofs
Proposition A1. For any attribute  $x \in \mathcal{X}_{k-1}$ ,  $\Psi_{\mathbb{CG}}(\mathbf{query}(x)) \geq \Psi_{\mathbb{CG}}(\mathbf{query}(x) = w_x)$  holds for all the possible value  $w_x \in \mathcal{W}_x$ .
Proof. For consistency, we re-formulate Eq. (6) as:
$$
\Psi_ {\mathbb {C G}} (\operatorname {q u e r y} (x)) = \sum_ {w _ {x} \in \mathcal {W} _ {x}} \left(\Psi_ {\mathbb {C G}} \left(w _ {x} = w _ {x} ^ {*}\right) \cdot \Pr \left(w _ {x} = w _ {x} ^ {*}\right)\right), \tag {19}
$$
where  $x$  is the queried attribute, and  $w_{x}^{*}$  represents the user preference on  $x$  (corresponding to the notations  $a$  and  $w_{a}^{*}$  respectively). We can also re-write  $\Psi_{\mathbb{CG}}(w_x = w_x^*)$  as:
$$
\begin{array}{l} \Psi_ {\mathbf {C G}} \left(w _ {x} ^ {\prime} = w _ {x} ^ {*}\right) = \operatorname {S U M} \left(\left\{\Psi_ {\mathrm {R E}} \left(v _ {m}\right) \mid v _ {m} \in \mathcal {V} _ {k - 1} \cap \mathcal {V} _ {x _ {\text {v a l u e}} \neq w _ {x} ^ {\prime}} \right\}\right) \\ = \sum_ {w _ {x} ^ {\prime \prime} \in \mathcal {W} _ {x} \backslash \left\{w _ {x} ^ {\prime} \right\}} \left(\operatorname {S U M} \left(\left\{\Psi_ {\mathrm {R E}} \left(v _ {m}\right) \mid v _ {m} \in \mathcal {V} _ {k - 1} \cap \mathcal {V} _ {x _ {\text {v a l u e}} = w _ {x} ^ {\prime \prime}} \right\}\right)\right) \tag {20} \\ = \sum_ {w _ {x} ^ {\prime \prime} \in \mathcal {W} _ {x} \backslash \{w _ {x} ^ {\prime} \}} \Psi_ {\mathbb {C G}} (w _ {x} ^ {\prime \prime} \neq w _ {x} ^ {*}) \geq \Psi_ {\mathbb {C G}} (w _ {x} \neq w _ {x} ^ {*}), \\ \end{array}
$$
where  $w_{x}$  is an arbitrary attribute value in  $\mathcal{W}_x \backslash \{w_x'\}$ . The above equation is built upon the simple fact that after an attribute  $x$ , the answer of the user preferring  $w_{x}'$  is equivalent to the answer of the user not preferring all the other  $w_{x}''$ s, which can remove all the unchecked items whose value is equal to any  $w_{x}''$ . Thus, the expected certainty gain of knowing the user preferring  $w_{x}'$  is not smaller than knowing the user not preferring any one  $w_{x} \in \mathcal{W}_{x} \backslash \{w_{x}'\}$ , and the equality holds only in the case where  $\mathcal{W}_x = \{w_x, w_x'\}$ , namely there are only two candidate attribute values.
Based on the above equations, we can derive:
$$
\begin{array}{l} \Psi_ {\mathbb {C G}} (\text {q u e r y} (x)) = \sum_ {w _ {x} \in \mathcal {W} _ {x}} \left(\Psi_ {\mathbb {C G}} \left(w _ {x} = w _ {x} ^ {*}\right) \cdot \Pr \left(w _ {x} ^ {\prime} = w _ {x} ^ {*}\right)\right) \\ = \Psi_ {\mathbb {C G}} (w _ {x} = w _ {x} ^ {*}) \cdot \Pr (w _ {x} = w _ {x} ^ {*}) + \sum_ {w _ {x} ^ {\prime} \in \mathcal {W} _ {x} \backslash \{w _ {x} \}} \left(\Psi_ {\mathbb {C G}} (w _ {x} ^ {\prime} = w _ {x} ^ {*}) \cdot \Pr (w _ {x} ^ {\prime} = w _ {x} ^ {*})\right) \\ \geq \Psi_ {\mathrm {C G}} \left(w _ {x} = w _ {x} ^ {*}\right) \cdot \Pr \left(w _ {x} = w _ {x} ^ {*}\right) + \sum_ {w _ {x} ^ {\prime} \in \mathcal {W} _ {x} \backslash \left\{w _ {x} \right\}} \left(\Psi_ {\mathrm {C G}} \left(w _ {x} \neq w _ {x} ^ {*}\right) \cdot \Pr \left(w _ {x} ^ {\prime} = w _ {x} ^ {*}\right)\right) \tag {21} \\ \geq \Psi_ {\mathbb {C G}} (w _ {x} = w _ {x} ^ {*}) \cdot \Pr (w _ {x} = w _ {x} ^ {*}) + \Psi_ {\mathbb {C G}} (w _ {x} \neq w _ {x} ^ {*}) \cdot \sum_ {w _ {x} ^ {\prime} \in \mathcal {W} _ {x} \backslash \{w _ {x} \}} \Big (\Pr (w _ {x} ^ {\prime} = w _ {x} ^ {*}) \Big) \\ \geq \Psi_ {\mathrm {C G}} \left(w _ {x} = w _ {x} ^ {*}\right) \cdot \Pr \left(w _ {x} = w _ {x} ^ {*}\right) + \Psi_ {\mathrm {C G}} \left(w _ {x} \neq w _ {x} ^ {*}\right) \cdot \Pr \left(w _ {x} \neq w _ {x} ^ {*}\right) \\ \geq \Psi_ {\mathrm {C G}} (\text {q u e r y} (x) = w _ {x}). \\ \end{array}
$$
Since we put no constraint on  $x \in \mathcal{X}_{k-1}$ , thus it proves the proposition.
Proposition A2. In the context of querying attribute values, an ideal choice is always the one that can partition all the unchecked relevance scores into two equal parts (i.e., the ideal  $w_{x} \in \mathcal{W}_{x}, x \in \mathcal{X}_{k-1}$  is the one that makes  $\Psi_{\mathbb{CG}}(w_{x} = w_{x}^{*}) = \mathrm{SUM}(\{\Psi_{\mathrm{RE}}(v_{m}) | v_{m} \in \mathcal{V}_{k-1}\}) / 2$  hold), if it is achievable. And the certainty gain in this case is  $\Psi_{\mathbb{CG}}(\text{query}(x) = w_{x}) = \mathrm{SUM}(\{\Psi_{\mathrm{RE}}(v_{m}) | v_{m} \in \mathcal{V}_{k-1}\}) / 2$ .
Proof. Without loss of generalizability, in the context of querying attribute values, we recap the formulation of  $\Psi_{\mathrm{CG}}(\mathbf{query}(x) = w_x)$ , shown in Eq. (9) as:
$$
\begin{array}{l} \Psi_ {\mathbb {C G}} (\text {q u e r y} (x) = w _ {x}) = \Psi_ {\mathbb {C G}} \left(w _ {x} = w _ {x} ^ {*}\right) \cdot \Pr \left(w _ {x} = w _ {x} ^ {*}\right) + \Psi_ {\mathbb {C G}} \left(w _ {x} \neq w _ {x} ^ {*}\right) \cdot \Pr \left(w _ {x} \neq w _ {x} ^ {*}\right) \\ = \operatorname {S U M} \left(\left\{\Psi_ {\mathrm {R E}} \left(v _ {m}\right) \mid v _ {m} \in \mathcal {V} _ {k - 1} \cap \mathcal {V} _ {a \text {v a l u e} \neq w _ {a}} \right\}\right) \cdot \frac {\operatorname {S U M} \left(\left\{\Psi_ {\mathrm {R E}} \left(v _ {m}\right) \mid v _ {m} \in \mathcal {V} _ {k - 1} \cap \mathcal {V} _ {a \text {v a l u e} = w _ {a}} \right\}\right)}{\operatorname {S U M} \left(\left\{\Psi_ {\mathrm {R E}} \left(v _ {m}\right) \mid v _ {m} \in \mathcal {V} _ {k - 1} \right\}\right)} \\ + \operatorname {S U M} \left(\left\{\Psi_ {\mathrm {R E}} \left(v _ {m}\right) \mid v _ {m} \in \mathcal {V} _ {k - 1} \cap \mathcal {V} _ {a _ {\text {v a l u e}} = w _ {a}} \right\}\right) \cdot \frac {\operatorname {S U M} \left(\left\{\Psi_ {\mathrm {R E}} \left(v _ {m}\right) \mid v _ {m} \in \mathcal {V} _ {k - 1} \cap \mathcal {V} _ {a _ {\text {v a l u e}} \neq w _ {a}} \right\}\right)}{\operatorname {S U M} \left(\left\{\Psi_ {\mathrm {R E}} \left(v _ {m}\right) \mid v _ {m} \in \mathcal {V} _ {k - 1} \right\}\right)} \tag {22} \\ = R _ {\text {Y E S}} \cdot \frac {R - R _ {\text {Y E S}}}{R} + (R - R _ {\text {Y E S}}) \cdot \frac {R _ {\text {Y E S}}}{R}, \\ \end{array}
$$
where we use  $R_{\mathrm{YES}}$  to denote  $\mathrm{SUM}(\{\Psi_{\mathrm{RE}}(v_m)|v_m \in \mathcal{V}_{k-1} \cap \mathcal{V}_{a_{\mathrm{value}} \neq w_a}\})$ , the expected certainty gain of the event  $w_x = w_x^*$  happening (i.e., the user answers Yes to querying  $w_x$ ), and use  $R$  to denote the summation of relevance scores of all the unchecked items, i.e.,  $\mathrm{SUM}(\{\Psi_{\mathrm{RE}}(v_m)|v_m \in \mathcal{V}_{k-1}\})$ . For convenience, we use  $\Psi$  to denote  $\Psi_{\mathrm{CG}}(\text{query}(x) = w_x)$ . Then,  $\Psi$  can be regarded as a function of  $R_{\mathrm{YES}}$ . Namely,  $R_{\mathrm{YES}}$  is the independent variable and  $\Psi$  is the dependent variable.
To maximize  $\Psi$ , we have:
$$
\frac {\partial \Psi}{\partial R _ {\mathrm {Y E S}}} = \frac {2}{R} \cdot (R - 2 \cdot R _ {\mathrm {Y E S}}) = 0. \tag {23}
$$
Therefore, we have  $R_{\mathrm{Yes}} = R / 2$ , and in this case,  $\Psi = R / 2$ . Then, we can reach the conclusion that the ideal partition is the one dividing all the unchecked relevance scores, i.e.,  $R$ , into two equal parts; and in this case,  $\Psi_{\mathrm{CG}}(\mathbf{query}(x) = w_x) = R / 2 = \mathrm{SUM}(\{\Psi_{\mathrm{RE}}(v_m)|v_m\in \mathcal{V}_{k - 1}\}) / 2$ , which indicates that querying  $w_{x}$  can check half of the relevance scores in expectation.
Lemma A1. In the context of querying attribute values, suppose that  $\Psi_{\mathrm{RE}}(v_m) = 1 / M$  holds for any  $v_m \in \mathcal{V}$ , then the expected number of turns (denoted as  $\widehat{K}$ ) is bounded by  $\log_2^{M + 1} \leq \widehat{K} \leq (M + 1) / 2$ .
Proof. We begin by considering the best case. According to Proposition 2, if we can find an attribute value  $w_{x}$ , where querying  $w_{x}$  can partition the unchecked relevance scores into two equal parts, then
we can build a binary tree, where we can check  $M / 2^k$  at the  $k$ -th turn. Therefore, we have:
$$
1 + 2 + \dots + 2 ^ {\widehat {K} - 1} = M, \tag {24}
$$
which derives  $\widehat{K} = \log_2^{M + 1}$ . In the worst case, we only can query one item during one query. Then, the expected number of turns is:
$$
\widehat {K} = 1 \cdot \frac {1}{M} + 2 \cdot \left(1 - \frac {1}{M}\right) \cdot \frac {1}{M - 1} + \dots + M \cdot \prod_ {i = 0} ^ {M - 1} \left(1 - \frac {1}{M - i}\right) \cdot 1 = \frac {M + 1}{2}. \tag {25}
$$
Combining Eqs. (24) and (25) together, we can draw  $\log_2^{M + 1}\leq \widehat{K}\leq (M + 1) / 2$
A2 Plugging the Conversational Agent into Recommender Systems
A2.1  $\Psi_{\mathrm{CG}}(\cdot)$  for Querying Attributes in Large Discrete or Continuous Space
The main idea of our conversational agent is to recursively query the user to reduce uncertainty. The core challenge is that there exist some cases where querying any attribute values or items can not effectively reduce uncertainty. Most of these cases occur when some key attributes have a large discrete space or a continuous space, leading to a broad decision tree. Formally, for a key attribute  $x \in \mathcal{X}_{k-1}$ , a "small" discrete space usually means  $|\mathcal{W}_x| \ll |\mathcal{V}_{k-1}|$ . For example, for attribute Hotel Price, then querying  $x$ , the user would not respond with an accurate value, and querying  $x$  = one possible value could be ineffective.
To address this issue, we propose to find a  $w_{x} \in \mathbb{R}$  instead of  $w_{x} \in \mathcal{W}_{x}$ , and then we can query whether the user's preference is not smaller than it or not, i.e.,  $\mathrm{query}(x) \geq w_{x}$  instead of whether the user's preference is equal to  $w_{x}$  or not, i.e.,  $\mathrm{query}(x) = w_{x}$ . Then, the expected certainty gain, in this case, can be written as:
$$
\Psi_ {\mathbb {C G}} (\text {q u e r y} (x) \geq w _ {x}) = \Psi_ {\mathbb {C G}} \left(w _ {x} \geq w _ {x} ^ {*}\right) \cdot \Pr \left(w _ {x} \geq w _ {x} ^ {*}\right) + \Psi_ {\mathbb {C G}} \left(w _ {x} <   w _ {x} ^ {*}\right) \cdot \Pr \left(w _ {x} <   w _ {x} ^ {*}\right), \tag {26}
$$
$$
\Psi_ {\mathrm {C G}} \left(w _ {x} \geq w _ {x} ^ {*}\right) = \operatorname {S U M} \left(\left\{\Psi_ {\mathrm {R E}} \left(v _ {m}\right) \mid v _ {m} \in \mathcal {V} _ {x <   w _ {x}} \cap \mathcal {V} _ {k - 1} \right\}\right),
$$
$$
\Psi_ {\mathrm {C G}} \left(w _ {x} <   w _ {x} ^ {*}\right) = \operatorname {S U M} \left(\left\{\Psi_ {\mathrm {R E}} \left(v _ {m}\right) \mid v _ {m} \in \mathcal {V} _ {x \geq w _ {x}} \cap \mathcal {V} _ {k - 1} \right\}\right), \tag {27}
$$
where  $\mathcal{V}_{x\geq w_x}$  is the set of items whose value of attribute  $x$  is not smaller than  $w_{x}$ , and  $\mathcal{V}_{x < w_x}$  is the set of the rest items, namely  $\mathcal{V}_{x\geq w_x} \cup \mathcal{V}_{x < w_x} = \mathcal{V}_{k - 1}$ ; and
$$
\mathsf {P r} (w _ {x} \geq w _ {x} ^ {*}) = \frac {\mathsf {S U M} (\{\Psi_ {\mathrm {R E}} (v _ {m}) | v _ {m} \in \mathcal {V} _ {x \geq w _ {x}} \cap \mathcal {V} _ {k - 1} \})}{\mathsf {S U M} (\{\Psi_ {\mathrm {R E}} (v _ {m ^ {\prime}}) | v _ {m ^ {\prime}} \in \mathcal {V} _ {k - 1} \})},
$$
$$
\Pr (w _ {x} <   w _ {x} ^ {*}) = \frac {\operatorname {S U M} (\{\Psi_ {\mathrm {R E}} (v _ {m}) | v _ {m} \in \mathcal {V} _ {x <   w _ {x}} \cap \mathcal {V} _ {k - 1} \})}{\operatorname {S U M} (\{\Psi_ {\mathrm {R E}} (v _ {m ^ {\prime}}) | v _ {m ^ {\prime}} \in \mathcal {V} _ {k - 1} \})}.
$$