User-Creator Feature Polarization in Recommender Systems with Dual Influence

Authors: tao  lin
kun  jin
  google
andrew  estornell
xiaoying  zhang
yiling  chen
yang  liu
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-14 22:08:54

Abstract: Recommender systems serve the dual purpose of presenting relevant content to users and helping content creators reach their target audience. The dual nature of these systems naturally influences both users and creators: users' preferences are affected by the items they are recommended, while creators may be incentivized to alter their content to attract more users. We define a model, called user-creator feature dynamics, to capture the dual influence of recommender systems. We prove that a recommender system with dual influence is guaranteed to polarize, causing diversity loss in the system. We then investigate, both theoretically and empirically, approaches for mitigating polarization and promoting diversity in recommender systems. Unexpectedly, we find that common diversity-promoting approaches do not work in the presence of dual influence, while relevancy-optimizing methods like top-$k$ truncation can prevent polarization and improve diversity of the system.

Main paper: 
 introduction: 
 From restaurant selection, video watching, to apartment renting, recommender systems play a pivotal role across a plethora of real-world domains. These systems match users with content they like, and help creators (those producing the content) identify their target audiences. Nevertheless, behind such success, concerns have emerged regarding possible harmful outcomes of recommender systems, in particular, filter bubbles [32, 5] and polarization [36] -outcomes with insufficient recommendation diversity and creation diversity. Recommendation diversity, meaning the diversity of the contents recommended to a user, is key to users' engagement and retention on the platform. Meanwhile, creation diversity, meaning the variety of content created on the platform, is a determinant of the platform's long-term health. In extreme cases, insufficient creation diversity can lead to consensus or polarization, where the latter can cause conflict and hatred, diminish people's mutual understanding, and cause societal crises. Therefore, from both business and social responsibility perspectives, championing and improving diversity in recommender systems is equally important as optimizing recommendation relevancy. and creators resembles a bipartite social network, and our conclusion that the system converges to polarization is conceptually similar to people reaching consensus on social networks [1, 10, 31, 46] . However, the technique we use to prove our conclusion (absorbing Markov chain) significantly differs from the main technique (stability of ODE) in the mentioned works.Performative effects of recommender systems The phenomenon that predictive systems like recommender systems can impact the individuals interacting with those systems (e.g., users and creators) is related to the literature of performative prediction [34, 18] . These impacts can be direct, such as individuals ostensibly modifying their features in order to obtain more desirable outcomes [27] . Prior works on the performative effects of recommender systems (e.g., [7, 24, 14, 41, 15, 42, 35, 20, 3, 43, 2, 23] ) only consider one-sided impact, either on users or on creators. Differing from them, our work studies two-sided impacts, i.e., on both users and creators. We provide a table to compare our work with previous works in Appendix A.2 Model: User-Creator Feature DynamicsWe define a dynamics model for user preferences and content/creator features in a recommender system. Let U t = [u t j ] m j=1 = [u t 1 , . . . , u t m ] ∈ R d×m be a population of m users andV t = [v t i ] n i=1 = [v t1 , . . . , v t n ] ∈ R d×n be a population of n creators at time t, where each vector u t j , v t i ∈ S d-1 represent the preference/feature vector of each user and creator respectively, assumed to be on the unit sphere S d-1 with ℓ 2 -norm. Then (U t , V t ) denotes the state of the dynamics at time t. The dynamics evolve as follows at each time step t ≥ 0:1) Recommendation: Each user j ∈ [m] is recommended a creator, where creator i ∈ [n] is chosen with a probabilityEQUATIONWhile we allow a wide array of different functions p t ij (•), a common example of such functions is the so-called softmax function:EQUATIONA larger β means that the recommendation is more sensitive to the relevance of a creator to a user, measured by ⟨u t j , v t i ⟩. 2) User update: After recommendation, each user j ∈ [m] updates their feature vector u t j , based on which creator, say i t j , was recommended to them:EQUATIONHere, η u ∈ [0, 1] is a parameter controlling the rate of update, f (v i , u j ) is a function that quantifies the impact of creator i's content on user j (discussed in detail later), and P(x) = x ∥x∥2 is the projection back onto the unit sphere. Our user update model generalizes [14] , which considers u t+1 j = P(u t j + η u ⟨v t i t j , u t j ⟩v t i t j ), by replacing the inner product with a general function f .3) Creator update: Creators also update their feature vectors based on which users are recommended their content. For each creator i ∈ [n], let J t i = {j : i t j = i} be the set of users being recommended creator i, then v t i is updated by:EQUATIONwhere η c ∈ [0, 1] is a parameter controlling the rate of update, and g(u j , v i ) is a function that quantifies the impact of user j on creator i.Impact functions f and g Our results apply to any impact functions f and g that satisfy the following natural assumptions. First, f (v i , u j ) and the inner product ⟨v i , u j ⟩ have the same sign:f (v i , u j ) is > 0 if ⟨v i , u j ⟩ > 0 < 0 if ⟨v i , u j ⟩ < 0 = 0 if ⟨v i , u j ⟩ = 0.This means that if a user likes the content (⟨v t i , u t j ⟩ > 0), then the user vector u t j will be updated towards the direction of the creator vector v t j . If the user dislikes the content (⟨v t i , u t j ⟩ < 0), then the user vector u t j will move away from v t j . Such "biased assimilation" user behavior is well documented in the literature [14] . Further, we assume upper and lower bounds on |f |:|f(v i , u j )| ≤ 1, |f (v i , u j )| ≥ L f > 0 whenever ⟨v i , u j ⟩ ̸ = 0.The lower bound |f (v i , u j )| ≥ L f means that the exposure to an item that a user likes or dislikes always has some non-negligible impact on the user's preference. For example, f (v i , u j ) = sign(⟨v i , u j ⟩)a + b⟨v i , u j ⟩ satisfies both assumptions when L f = a > 0 and b ≥ 0.For g, likewise assume that its sign is the same as ⟨u j , v i ⟩: g(u j , v i ) is> 0 if ⟨u j , v i ⟩ > 0 < 0 if ⟨u j , v i ⟩ < 0 = 0 if ⟨u j , v i ⟩ = 0.Intuitively, this captures the incentive of a creator who aims to maximize the average ratings from users who are recommended their items. On video platforms for example, if the creators are rewarded based on the average rating of their videos, they will try to reinforce their creation styles based on the users who give positive feedback (⟨u j , v i ⟩ > 0) so that their creations are more likely to be recommended to those users. Meanwhile, the creators will also change their creation styles based on negative feedback (⟨u j , v i ⟩ < 0), but in the opposite direction of the negative-feedback users' interests, so that their creations are less likely to be recommended to those users. Taking both scenarios into account, the creator moves towards the weighted average of user preferences j∈J t i g(u t j , v t i )u t j , which is captured by our update rule (4). A particular example of g is the sign function g(u j , v i ) = sign(⟨u j , v i ⟩) ∈ {-1, 0, 1}. We will only consider the sign function g in order to simplify the theoretical presentation. We believe that all our results can be generalized to other g functions satisfying similar conditions as f ; the details are left as future work.Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?Answer: [Yes]Justification:Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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limitations: 
 Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes]Justification: Limitations are discussed in Section 6 and Section H.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
unavoidable polarization: 
 Having defined the user-creator feature dynamics in a recommender system with dual influence, we now theoretically study how such dynamics evolve. Our main result is: if every creator can be recommended to every user with some non-zero probability, then the dynamics must eventually polarize. Definition 3.1 (consensus and bi-polarization). Let R > 0. The dynamics (U t , V t ) is said to reach:• R-consensus if there exists a vector c ∈ R d such that every feature vector is R-close to c: ∀u t j , ∥u t j -c∥ 2 ≤ R and ∀v t i , ∥v t i -c∥ 2 ≤ R. • R-bi-polarization if there exists a vector c ∈ R d such that every feature vector is R-close to +c or -c: ∀u t j , ∥u t j -c∥ 2 ≤ R or ∥u t j + c∥ 2 ≤ R, and ∀v t i , ∥v t i -c∥ 2 ≤ R or ∥v t i + c∥ 2 ≤ R. The dynamics is said to reach (R, c)-consensus (or (R, c)-bi-polarization) if the dynamics reaches R-consensus (or R-bi-polarization) with the vector c.Consensus is any state where all users and creators have similar feature vectors (with maximum difference R), implying that they have similar interests or preferences. Bi-polarization is any state where all users and creators are clustered into two groups with exactly opposite features (e.g., Republicans vs Democrats). Mathematically, consensus is a special case of bi-polarization. Proposition 3.2. Bi-polarization states are absorbing: once the dynamics reaches (R, c)-bipolarization with some R ∈ [0, 1] and c ∈ S d-1 , it will satisfy (R, c)-bi-polarization forever. The same holds for consensus.A natural property of a recommender system is that every creator can be recommended to every user with some non-zero probability: p t ij ≥ p 0 > 0 with some constant p 0 . This is satisfied by the softmax function, which is a rough model of real-world recommendation algorithms [13, 26] :p t ij = exp(β⟨u t j ,v t i ⟩) n i=1 exp(β⟨u t j ,v t i ⟩) ≥ exp(-β) n exp(β) = p 0 > 0.Moreover, many large-scale real-world recomendation systems (e.g., Yahoo! [28] and Kuaishou [16] ) intentionally insert small random traffic attempting to improve recommendation diversity or explore users' interests [25, 40] , which will cause all recommendation probabilities to be non-zero. We show in Theorem 3.3 that, however, a recommender system satisfying p t ij ≥ p 0 > 0 must converge to polarization, under some additional conditions on the users' and creators' update rates:Theorem 3.3. Suppose g(u j , v i ) = sign(⟨u j , v i ⟩), the update rates η c ≤ ηuL f 2and η u < 1 2 , and the recommendation probability p t ij ≥ p 0 > 0, ∀i, j, t. Then, from almost all initial states, the dynamics (U t , V t ) will eventually reach R-consensus or R-bi-polarization for any R > 0.In other words, if the users' and creators' updates are not too fast and all recommendation probabilities are non-zero, then all users and creators will eventually converge to at most two clusters (regardless of the feature dimension d). Since creators in one cluster produce similar contents, users in such a polarized system can never receive diverse recommendations. This means that the naïve attempt of imposing p t ij ≥ p 0 > 0 cannot improve the diversity of a recommender system with dual influence. The conditions on the update rates η u , η c are only assumed to simplify the proof of Theorem 3.3. Our experiments (in Section 5) will show that polarization still occurs even without those conditions. Theorem 3.3 does not characterize the rate of convergence of the user-creator feature dynamics to polarization, which we leave as an open question.The proof of Theorem 3.3 is an absorbing Markov chain argument. It uses the following lemma:Lemma 3.4. Suppose η c ≤ ηuL f 2and η u < 1 2 . For any R > 0, for almost every state (U t , V t ) in the state space, there exists a path(U t , V t ) → (U t+1 , V t+1 ) → • • • → (U t+T , V t+T ) of finite length that leads to an R-bi-polarization state (U t+T , V t+T ).The proof of this lemma (in Appendix F) is involved. It uses induction on the number of creators n. The base case of n = 1 is proved by a potential function argument. For n ≥ 2, we first construct a path that leads the subsystem of n -1 creators and all users to R-bi-polarization. Then, depending on where the remaining creator is, we construct a sequence of recommendations that leads the remaining creator to one of the two clusters formed by the n -1 creators and all users. Such recommendations will move some users out of the formed clusters, which requires extra care in the proof.Proof of Theorem 3.3. For any state (U t , V t ) in the state space, by Lemma 3.4 there exists a path(U t , V t ) → • • • → (U t+T , V t+T ) of length T that leads to R-bi-polarization.Because every creator can be recommended to a user with probability at least p 0 , each transition(U t ′ , V t ′ ) → (U t ′ +1 , V t ′ +1) happens with probability at least p m 0 . So, the path of length T has probability at least p mT 0 > 0, and the probability that the dynamics does not reach R-bi-polarization after KT steps is at most (1 -p mT 0 ) K , which → 0 as K → ∞. Therefore, with probability 1 the dynamics will reach R-bi-polarization eventually.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Answer: [Yes] Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. • The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
discussions on real-world designs: 
 Next, we discuss how 4 types of real-world recommender system designs affect the user-creator feature dynamics: top-k truncation, threshold truncation, diversity-boosting, and uniform traffic.(1) Top-k Truncation A prevalent practice in modern two-stage recommendation algorithms on large-scale platforms, such as YouTube [13] , is to first filter out items that are unlikely to be relevant to a user, then make recommendations from the remaining items. In particular, we consider the top-k truncation policy: for every user j, find the k most relevant creators, namely, the k creators whose inner products with the user ⟨v t i , u t j ⟩ are largest (equivalently, the k creators whose probabilities p t ij of being recommended to user j are highest), then recommend one of those k creators to user j with probability proportional to p t ij . The other creators will not be recommended. This practice significantly reduces the computation cost and improves the relevancy of recommendations. Interestingly, we show that such a practice also has the potential to improve the long-term diversity of a recommender system with dual influence. Definition 4.1 (clusters). We say a state (U t , V t ) forms q clusters if there exist c 1 , . . . , c q ∈ R d and a small number R > 0 such that every feature vector is in the ℓ 2 ball of some c i with radius R (denoted by B(c ℓ , R) = {x : ∥x -c ℓ ∥ 2 ≤ R}), andB(c ℓ , 2R) ∩ B(c ℓ ′ , 2R) = ∅ for ℓ ̸ = ℓ ′ .It is clear that consensus has a single cluster, and bi-polarization has two. Proposition 4.2. With top-k truncation, there exist states (U t , V t ) that form ⌊n/k⌋ clusters and are absorbing (i.e., once the system forms ⌊n/k⌋ clusters, it forms ⌊n/k⌋ clusters forever).This result is in contrast with Theorem 3.3 which shows that a recommender system where every creator can be recommended to every user (p t ij > 0) is doomed to polarize. With top-k truncation where some p t ij = 0, polarization can be avoided. Experiments in Section 5.3 support our prediction that top-k truncation can reduce polarization and improve diversity.(2) Threshold Truncation Besides top-k truncation, threshold truncation is another way to filter out irrelevant creators: set a threshold τ ∈ [-1, 1] such that any user-creator pair with inner product ⟨u i , v j ⟩ < τ is not recommended. A natural choice is τ = 0, meaning that users will not receive recommendations predicted to be "disliked" by them. Increasing τ is similar to increasing the β in the softmax function, which improves recommendation relevance. Proposition 4.3. In d-dimensional feature space, if user-creator pairs with ⟨u i , v j ⟩ < 0 are not recommended, then there exist stable states with d + 1 clusters.Although truncation at τ = 0 allows stable states with d + 1 clusters to exist, the dynamics does not necessarily converge to such states; it can still end up with stable states with fewer clusters. In fact, experiments (in Appendix B.2) show that truncation at τ = 0 is not good for diversity and causes severe polarization, while truncation at a large threshold like τ = 0.707 is better at reducing polarization.(3) Diversity Boosting Diversity boosting aims to explore users' interests and improve users' experience by diversifying recommendation. For example, when making recommendations, the model optimizes the objective:EQUATIONwhere h rel , h div rewards the recommendation relevance and diversity respectively and list i records the recent list of recommended items to user i. h div can take a simple form of j ′ ∈listi 1 -⟨v j ′ , v j ⟩, and ρ > 0 controls the strength of diversity-boosting. Despite being successful when users' preferences and items are fixed, this design alone cannot prevent bi-polarization in our dual-influence dynamics, since the conditions in Theorem 3.3 are still satisfied and the users' and creators' update rules remain the same. Experiments in Section 5.2 support our claim.(4) Uniform Traffic Adding a small fraction of uniform traffic to the personalized recommendations is another method proposed in previous works to improve recommendation diversity or to explore user preferences [25, 16, 9, 8, 30] . This method gives a non-zero lower bound on the probability of every creator being recommended to every user. So, as a corollary of our Theorem 3.3, it causes a recommender system with dual influence to polarize. Such an observation is striking as it demonstrates that optimizing for recommendation diversity in a static setting can ultimately lead to a huge loss of the system diversity in the long run. 
experiments: 
 We present experimental results on the behavior of user-creator feature dynamics on synthetic data and real-world (MovieLens 20M) data and the effect of top-k truncation and threshold truncation on the dynamics.Setup The dynamics is initialized by randomly generating user and creator features on the unit sphere in R d . We pick d = 10, number of creators n = 50, number of users m = 100. We use the softmax recommendation probability function (2) . We simulate the dynamics for T = 1000 steps, repeated 100 times each with a new initialization. We choose the sign impact function g(u j , v i ) = sign(⟨u j , v i ⟩) for creator updates. For user updates, we choose inner product f (v i , u j ) = ⟨v i , u j ⟩.The inner product function is studied in previous works on users' preference dynamics (but not creators') [14] . Note that the inner product does not satisfy the condition |f (v i , u j )| ≥ L f needed in Theorem 3.3. However, we still observe convergence to polarization in nearly all experiments. Thus, even when this condition does not hold, users and creators still tend towards polarization in practice.Three key parameters in our model are β (sensitivity of the softmax function), η c (creator update rate), and η u (user update rate). We set them to β = 1, η c = η u = 0.1, and change one parameter at a time to see its effect on the dynamics. We also test what happens when some dimensions of the user features are fixed features that are not updated.Measures To quantify the behavior of the dynamics, given user and creator feature vectors (U , V ) we compute the following measures, which cover diversity, relevancy, and polarization of the system:• Creator Diversity (CD): diversity of the creator features, measured by their average pairwise distance [47, 33] :CD(V ) = 1 n(n-1) n i=1 j̸ =i ∥v i -v j ∥.• Recommendation Diversity (RD): diversity of the contents recommended to a user, measured by the weighted variance of the contents:RD(U , V ; β) = 1 m m j=1 n i=1 p ij ∥v i -v j ∥ 2 , where v j = n i=1 p ij v i and p ij = exp(β⟨uj ,vi⟩) n i=1 exp(β⟨uj ,vi⟩) .• Recommendation Relevance (RR): relevance of the contents recommended to a user, measured by the weighted average of inner products:RR(U , V ; β) = 1 m m j=1 n i=1 p ij ⟨u j , v i ⟩.• Tendency to Polarization (TP): This is a novel measure we propose to quantify how close the system is to consensus or bi-polarization, measured by the average absolute inner products between the creators:TP(V ) = 1 n 2 n i=1 n k=1 |⟨v i , v k ⟩|.TP(V ) being closer to 1 means that the system is more polarized, because the term |⟨v i , v k ⟩| is 1 iff the two vectors v i , v k are equal or opposite to each other.It is worth noting that a high creator diversity is necessary for simultaneously achieving high recommendation relevance and high recommendation diversity. For example, they cannot be simultaneously achieved in a polarized state.Sensitivity Parameter β A larger β means that a user will be recommended more relevant content/creator with a higher probability. β = 0, on the other hand, means that the user receives uniform recommendations across all creators. Our main observation from the experiments is: a larger β leads to higher creator diversity and alleviated polarization in the long run.Figure 1 shows snapshots of the dynamics at different time steps under different β values. Here, we choose dimension d = 3 instead of 10 so the feature vectors can be visualized on a 3d sphere. We see that the system tends to form more clusters at time t = 200 as β increases.Figure 2 shows the changes of the 4 measures CD, RD, RR, TP over time under different β values. We see that a more diverse recommendation policy (a smaller β) leads to lower creator diversity and a higher level of polarization in the long run. In particular, while Creator Diversity reaches a similar level under different β in the end, it drops at a slower rate with a larger β (see β = 5, 6). Moreover, from the plot of Tendency to Polarization, we see that a larger β alleviates polarization, which means improvement in the diversity of the whole system.An explanation for our observation is the following: When β is smaller, each user receives more uniform recommendations across all creators. So, for different creators, the sets of users recommended to those creators have larger intersections. Since the creator updates are based on the sets of recommended users, different creators will be moving towards more similar directions. This leads to faster polarization. One can also predict this observation from Theorem 3.3: when β is large, the minimum recommendation probability p 0 of the softmax function tends to 0, so it might take a long time for the system to converge to polarization, while with a small β the system polarizes quickly.Update Rates η c and η u A larger η c means that creator features are updated faster, and intuitively should lead to faster polarization. This is validated in experiments: Figure 3 shows that a larger η c indeed causes more extreme polarization and lower diversity (both CD and RD). A larger η u means that user features are updated faster. It has a similar effect of exacerbating polarization as η c does, as shown in Figure 4 . We also consider the scenario where some dimensions of the user feature vectors are fixed features and thus not updated from round to round (e.g., age, gender), which is a realistic scenario. Detailed results are in Appendix B.1. The main observation is: as the number of fixed dimensions increases, the diversity of the system improves and the degree of polarization is reduced. This is similar to the effect of decreasing user update rate η u . The observation that fixed dimensions of user features help to improve diversity might be a reason why the recommender systems in practice are not as polarized as our theoretical prediction.In this part, we conduct experiments on the MovieLens 20M dataset [19] . We use a real-world twotower recommendation model with 16-dimensional tower tops as the user and creator embeddings (Figure 9 ). The model is initialized by fitting a two-tower model [21] on the existing MovieLens rating data and using the tower tops as the initial user and creator embeddings. Then we follow Algorithm 1 to simulate the dynamics.Figure 5 shows the effect of the recommendation sensitivity parameter β on the system. Similar to the synthetic data experiments, a smaller β (more diverse recommendation for the users in the short term) results in faster polarization. We note that the joint results on CD and TP are more informative than each one alone: despite β = 0 has a higher creator diversity than β = 2 at T = 500, the system reaches polarization more quickly under β = 0. The higher creator diversity under β = 0 is because the two clusters in the bi-polarized state are more balanced so the average pairwise distance between the creators is higher under β = 0 than under β = 2.Figure 6 shows the effect of using diversity-aware objective (Eq. 5) for diversity boosting. We see that myopically promoting the short-term recommendation diversity (using a larger ρ) results a higher creation diversity but also a higher tendency to polarization. A possible explanation for this phenomenon is, similar to the case with β, the system polarizes into two balanced clusters which actually have a large average pairwise distance. In this case, Tendency to Polarization is a better measure for diversity loss than Creator Diversity (average pairwise distance). We experimented with top-k truncation on the synthetic data (Table 1 ) and the MovieLens dataset (Appendix C). Our main observation is: a small k improves the diversity of the recommender system and reduces polarization. This is consistent with our theoretical prediction (Proposition 4.2). However, there is a tradeoff between the diversity of recommendations to users (RD) and the diversity of creations in the system (CD and TP). A top-k truncation policy with small k is "not diverse" for users because it exposes a user only to a small set of contents. However, such a policy can lead to a more diverse outcome in the whole system. This tradeoff is worth further studying. We also experimented with threshold truncation on synthetic data (Appendix B.2) and MovieLens data (Appendix C). The effect of a large truncation threshold τ is similar to the effect of a small k in top-k truncation. 
conclusion: 
 Our work defines a dynamics model to capture the dual influence of recommender systems on user preferences and content creation. Although our model is a theoretical abstraction, we believe that it captures the essence of a real-world recommender system, and our effort is an important initial endeavor to study diversity in recommender systems with dual influence. (See Appendix H for some additional discussions on real-world recommender systems.) We specifically point out different concepts of diversity in recommender systems (creation diversity, recommendation diversity, and tendency to polarization) and provide theoretical and empirical evidences to show that, due to dual influence, myopically optimizing recommendation diversity might hurt the long-term creation diversity and result in polarization of the system. We also explore popular design choices in recommender systems and show an interesting and somewhat counter-intuitive result that designs purely targeting efficiency improvement (e.g., top-k truncation) can alleviate polarization. We believe that the insights from our work are valuable to building healthy and sustainable recommender systems, and our results can inspire more sophisticated solutions for improving the long-term diversity of recommender systems to be developed.  
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 Appendix: In this part, we consider the case where some dimensions of the user feature vectors are fixed features and thus not updated from round to round (e.g., ages, genders). Formally, we fix the firstk ≤ d dimensions. The remaining d -k dimensions u t j [k + 1 : d] = (u t j [k + 1], . . . , u t j [d]) are updated according to the following rule:u t+1 j [k + 1 : d] = ∥u t j [k + 1 : d]∥ • P u t j [k + 1 : d] + η u f (v t i , u t j )v t i [k + 1 : d] .The multiplication by ∥u t j [k + 1 : d]∥ ensures unit norm ∥u t+1 j ∥ = 1. The effect of the number of fixed dimensions on the dynamics is shown in Figure 7 . We see that the diversity of the system improves as the number of fixed dimensions increases, and the degree of polarization is reduced. This is similar to the effect of decreasing user update rate η u in Figure 4 . The observation that fixed user features encourage diversity might be a reason why the recommender systems in practice are not as polarized as our theoretical prediction. 3 shows that the effect of different thresholds in threshold truncation on the long-term diversity of the system. We see that truncating at τ = 0, which corresponds to 90 • angle between u j and v i , is not good for diversity, resulting in the lowest creator diversity measure (CD) and highest tendency to polarization (TP). Truncating at a large threshold like 0.707 is good for diversity, instead.Figure 8 shows how the diversity measures change over time, under different truncation thresholds. -cos(60 • ) = -0.5 1.00 ± 0.03 0.00 ± 10 -3 1.00 ± 10 -3 0.99 ± 10 -3 -cos(72 • ) = -0.309 0.96 ± 0.06 0.01 ± 0.02 1.00 ± 0.02 0.92 ± 0.10 cos(90 • ) = 0 0.03 ± 0.16 0.00 ± 10 -3 1.00 ± 10 -3 0.99 ± 0.04 cos(72 • ) = 0.309 0.72 ± 0.30 0.00 ± 10 -3 1.00 ± 10 -3 0.81 ± 0.12 cos(60 • ) = 0.5 1.16 ± 0.11 0.00 ± 10 -3 1.00 ± 10 -3 0.47 ± 0.10 cos(45 • ) = 0.707 1.37 ± 0.02 0.00 ± 10 -3 1.00 ± 10 -3 0.33 ± 0.02 cos(30 • ) = 0.866 1.30 ± 0.03 0.00 ± 10 -3 1.00 ± 10 -3 0.55 ± 0.05 -cos(60 • ) = -0.5 0.98 ± 0.04 0.00 ± 0.02 1.00 ± 0.01 0.96 ± 0.04 -cos(72 • ) = -0.309 0.92 ± 0.08 0.00 ± 0.02 0.99 ± 0.02 0.87 ± 0.10 cos(90 • ) = 0 0.13 ± 0.31 0.00 ± 10 -3 1.00 ± 10 -3 0.97 ± 0.08 cos(72 • ) = 0.309 0.85 ± 0.16 0.00 ± 10 -3 1.00 ± 10 -3 0.76 ± 0.11 cos(60 • ) = 0.5 1.21 ± 0.07 0.00 ± 10 -3 1.00 ± 10 -3 0.43 ± 0.08 cos(45 • ) = 0.707 1.38 ± 0.01 0.00 ± 10 -3 1.00 ± 10 -3 0.30 ± 0.01 cos(30 • ) = 0.866 1.33 ± 0.02 0.00 ± 10 -3 1.00 ± 10 -3 0.47 ± 0.04 -cos(60 • ) = -0.5 0.91 ± 0.18 0.01 ± 0.02 1.00 ± 0.01 0.83 ± 0.10 -cos(72 • ) = -0.309 0.85 ± 0.23 0.00 ± 10 -3 1.00 ± 10 -3 0.78 ± 0.11 cos(90 • ) = 0 0.64 ± 0.33 0.00 ± 10 -3 1.00 ± 10 -3 0.81 ± 0.12 cos(72 • ) = 0.309 1.01 ± 0.14 0.00 ± 10 -3 1.00 ± 10 -3 0.64 ± 0.14 cos(60 • ) = 0.5 1.26 ± 0.05 0.00 ± 10 -3 1.00 ± 10 -3 0.38 ± 0.06 cos(45 • ) = 0.707 1.39 ± 0.01 0.00 ± 10 -3 1.00 ± 10 -3 0.28 ± 0.01 cos(30 • ) = 0.866 1.37 ± 0.01 0.00 ± 10 -3 1.00 ± 10 -3 0.34 ± 0.01 Input: t = 0, actual embedding U (0) , V (0) , true labelsEQUATIONj ), initial parameter θ (0) (which includes the predicted embeddingÛ (0) , V (0) ) repeat Let temporary parameter w (0) ← θ (t) Compute loss L(θ (t) , Y (t) ) for s = 1 to m -1 do w (s+1) ← θ (s) -η∇ w L(w (s) , Y (t) ) end for θ (t+1) ← w (m)Deliver recommendations based on Û (t+1) , V (t+1) Update U (t+1) , V (t+1) , and Y (t+1) t ← t + 1 until ∥θ (t) -θ Results for threshold truncation are in Table 5 and Figure 11 . Similar to synthetic data, we see that a large (but not too large) threshold like 0.707 is good for improving CD and TP. ) = -0.5 1.00 ± 10 -3 0.00 ± 10 -3 1.00 ± 10 -3 1.00 ± 10 -3 -cos(72 • ) = -0.309 1.00 ± 10 -3 0.00 ± 10 -3 1.00 ± 10 -3 1.00 ± 10 -3 cos(90 • ) = 0 0.01 ± 0.01 0.00 ± 10 -3 1.00 ± 10 -3 1.00 ± 10 -3 cos(72 • ) = 0.309 0.83 ± 0.08 0.00 ± 10 -3 1.00 ± 10 -3 0.72 ± 0.09 cos(60 • ) = 0.5 1.20 ± 0.05 0.00 ± 10 -3 1.00 ± 10 -3 0.46 ± 0.07 cos(45 • ) = 0.707 1.39 ± 10 -3 0.00 ± 10 -3 1.00 ± 10 -3 0.20 ± 10 -3 cos(30 • ) = 0.866 1.36 ± 10 -3 0.00 ± 10 -3 1.00 ± 10 -3 0.40 ± 0.01 1 -cos(60 • ) = -0.5 1.00 ± 10 -3 0.00 ± 10 -3 1.00 ± 10 -3 1.00 ± 10 -3 -cos(72 • ) = -0.309 0.96 ± 0.03 0.00 ± 10 -3 1.00 ± 10 -3 0.95 ± 0.03 cos(90 • ) = 0 0.02 ± 0.02 0.00 ± 10 -3 1.00 ± 10 -3 0.99 ± 10 -3 cos(72 • ) = 0.309 0.83 ± 0.07 0.00 ± 10 -3 1.00 ± 10 -3 0.66 ± 0.10 cos(60 • ) = 0. This section provides some lemmas that will be used in the proofs. They are mainly about some properties of the dynamics update rule. Claim D.1. For vectors x, y ∈ R d with unit norm ∥x∥ 2 = ∥y∥ 2 = 1, we have:• ∥x -y∥ 2 2 = 2(1 -⟨x, y⟩). • ⟨x, y⟩ = 1 -1 2 ∥x -y∥ 2 2. Lemma D.2 (Convex Cone Property). Let z 1 , . . . , z k ∈ R d be vectors with norm ∥z t i ∥ 2 = 1. Suppose ⟨z i , y⟩ > 0 for every i = 1, . . . , k for some y ∈ R d . Let x = P( k i=1 a i z i ) for some a 1 , . . . , a k ≥ 0 (namely, x is the normalization of some vector in the convex cone formed by z 1 , . . . , z k ). Then, we have⟨x, y⟩ ≥ k min i=1 ⟨z i , y⟩ > 0 and ∥x -y∥ 2 ≤ k max i=1 ∥z i -y∥ 2 > 0. Proof. ⟨x, y⟩ = k i=1 a i z i ∥ k i=1 a i z i ∥ 2 , y = 1 ∥ k i=1 a i z i ∥ 2 k i=1 a i ⟨z i , y⟩ ≥ 1 ∥ k i=1 a i z i ∥ 2 k i=1 a i k min i=1 ⟨z i , y⟩ = k min i=1 ⟨z i , y⟩ k i=1 a i ∥ k i=1 a i z i ∥ 2 ≥ k min i=1 ⟨z i , y⟩ k i=1 a i k i=1 a i = k min i=1 ⟨z i , y⟩.This proves the first inequality. To prove the second inequality, we use Claim D.1 and the first inequality:∥x -y∥ 2 = 2(1 -⟨x, y⟩) ≤ 2(1 -min i ⟨z i , y⟩) = max i 2(1 -⟨z i , y⟩) = k max i=1 ∥z i -y∥ 2 .Lemma D.3. Let x t , y, z t ∈ R d be vectors with norm ∥x t ∥ 2 = 1, ∥y∥ 2 ≥ 0, ∥z t ∥ 2 ≤ 1. Suppose ⟨x t , y⟩ ≥ 0, ⟨z t , y⟩ ≥ 0. After the update x t+1 = P(x t + ηz t ), we have⟨x t+1 -x t , y⟩ ≥ η 1 + η∥z t ∥ 2 ⟨z t , y⟩ -∥z t ∥ 2 ⟨x t , y⟩ .As a corollary, if y = z t and ∥z t ∥ 2 = 1, then⟨x t+1 -x t , z t ⟩ ≥ η 1 + η 1 -⟨x t , z t ⟩ .Proof. By definition,⟨x t+1 -x t , y⟩ = x t + ηz t ∥x t + ηz t ∥ 2 -x t , y = 1 ∥x t + ηz t ∥ 2 -1 • ⟨x t , y⟩ + η ∥x t + ηz t ∥ 2 • ⟨z t , y⟩ (because ∥x t + ηz t ∥2 ≤ 1 + η∥z t ∥2) ≥ 1 1 + η∥z t ∥ 2 -1 • ⟨x t , y⟩ + η 1 + η∥z t ∥ 2 • ⟨z t , y⟩ = η 1 + η∥z t ∥ 2 ⟨z t , y⟩ -∥z t ∥ 2 ⟨x t , y⟩ .Lemma D.4. Let x t , z t ∈ R d be vectors with norm ∥x t ∥ 2 = 1, ∥z t ∥ 2 ≤ 1. Suppose ⟨x t , z t ⟩ ≥ 0 and η > 0. Then the update x t+1 = P(x t + ηz t ) satisfies• ⟨x t+1 -x t , z t ⟩ ≥ 1 η ∥x t+1 -x t ∥ 2 2 . • ∥x t+1 -x t ∥ 2 ≤ η∥z t ∥ 2 .Proof. Let xt+1 = x t + ηz t , so x t = P( xt+1 ) and z t = 1 η ( xt+1 -x t ). Then we have⟨x t+1 -x t , z t ⟩ = 1 η ⟨x t+1 -x t , xt+1 -x t ⟩.Because ⟨x t , z t ⟩ ≥ 0, the vector xt+1 = x t + ηz t has length ≥ 1 and hence is outside (or on the surface) of the d-dimensional unit ball. Since x t = P( xt+1 ) is the projection of xt+1 onto the unit ball, and z t is another vector inside the unit ball, by the "Pythagorean property" (Proposition 2.2 in [6] ), we must have ⟨x tx t+1 , xt+1 -x t+1 ⟩ ≤ 0. This implies⟨x t+1 -x t , z t ⟩ ≥ 1 η ⟨x t+1 -x t , xt+1 -x t ⟩ + ⟨x t -x t+1 , xt+1 -x t+1 ⟩ = 1 η ⟨x t+1 -x t , x t+1 -x t ⟩ = 1 η ∥x t+1 -x t ∥ 2 2 ,which proves the first claim. To prove the second claim, we use Cauchy-Schwarz inequality:1 η ∥x t+1 -x t ∥ 2 2 ≤ ⟨x t+1 -x t , z t ⟩ ≤ ∥x t+1 -x t ∥ 2 ∥z t ∥ 2 .This implies ∥x t+1 -x t ∥ 2 ≤ η∥z t ∥ 2 .Lemma D.5. Consider a creator v t i and a user u t j . Suppose the user is always recommended creator i (so the user is updated byu t+1 j = P(u t j + η u f (v t i , u t j )v t i )), and creator i is updated byv t+1 i = P(v t i + η c α t i ) with ∥α t i ∥ 2 ≤1 and ⟨v t i , α t i ⟩ ≥ 0 at each time step. Assume:• The inner product ⟨u 0 j , v 0 i ⟩ > 0 initially. (Note that ⟨u 0 j , u 0 j ′ ⟩ needs not hold.)• There exists some constantL f > 0 such that f (v i , u j ) ≥ L f > 0 whenever ⟨u j , v i ⟩ > 0. • η c ≤ ηuL f 2 and 0 ≤ η u < 1 2 .Then, we have ⟨u t j , v t i ⟩ > 0 in all time steps.Proof. We prove by induction. Suppose ⟨u t j , v t i ⟩ > 0 already holds. We prove that ⟨u t+1 j , v t+1 i ⟩ > 0 will also hold. Take the difference between ⟨u t+1 j , v t+1 i ⟩ and ⟨u t j , v t i ⟩:⟨u t+1 j , v t+1 i ⟩ -⟨u t j , v t i ⟩ = ⟨u t+1 j , v t+1 i -v t i ⟩ + ⟨u t+1 j -u t j , v t i ⟩. For ⟨u t+1 j -u t j , v t i ⟩, using Lemma D.3 with x t = u t j , z t = v t i , and η = η u f (v t i , u t j ), we get ⟨u t+1 j -u t j , v t i ⟩ ≥ ηuf (v t i ,u t j ) 1+ηuf (v t i ,u t j ) 1 -⟨u t j , v t i ⟩ ≥ ηuL f 1+ηuL f 1 -⟨u t j , v t i ⟩ .For ⟨u t+1 j , v t+1 i -v t i ⟩, by Cauchy-Schwarz inequality and Lemma D.4,⟨u t+1 j , v t+1 i -v t i ⟩ ≥ -∥u t+1 j ∥ 2 • ∥v t+1 i -v t i ∥ 2 ≥ -1 • η c ∥α t i ∥ 2 ≥ -η c . • If 1 -⟨u t j , v t i ⟩ > 1 2 (1 + η u L f ), then we have ⟨u t+1 j , v t+1 i ⟩ -⟨u t j , v t i ⟩ > η u L f 1 2 -η c ≥ 0 by the assumption of η c ≤ ηuL f 2 . • If 1 -⟨u t j , v t i ⟩ ≤ 1 2 (1 + η u L f ), then we have ⟨u t+1 j , v t+1 i ⟩ -⟨u t j , v t i ⟩ ≥ 0 -η c =⇒ ⟨u t+1 j , v t+1 i ⟩ ≥ ⟨u t j , v t i ⟩ -η c ≥ 1 2 -1 2 η u L f -η c > 0 under the assumption of η c ≤ ηuL f 2 and η u < 1 2 .The above two cases together ensure ⟨u t+1 j , v t+1 i ⟩ > 0.Lemma D.6. Consider a system of one user and one creator that satisfies ⟨u 0 j , v 0 i ⟩ > 0 and ⟨u 0 j , y⟩ > ⟨v 0 i , y⟩ > 0 for some y ∈ R d with ∥y∥ ≤ 1 initially. The creator is always recommended to the user (so the updates are u t+1 j = P(ut j + η u f (v t i , u t j )v t i ) and v t+1 i = P(v t i + η c u t j )). Suppose η c ≤ ηuL f 2and 0 ≤ η u < 1 2 . Then, we have:• ⟨u t j , y⟩ > ⟨v t i , y⟩ > 0 for all t ≥ 1.• Suppose ⟨u 0 j , y⟩ -⟨v 0 i , y⟩ = D > 0. For any R < D, after T = 8 3ηuL f ln 2 R 2 steps, we have ⟨v T i , y⟩ -⟨v 0 i , y⟩ ≥ ηc ηu+ηc (D -R).Proof. We prove the first item by induction. Suppose ⟨u t j , y⟩ > ⟨v t i , y⟩ > 0 holds. Consider t + 1. First, by Lemma D.2, ⟨v t+1 i , y⟩ > 0 holds. Then, we prove ⟨u t+1 j , y⟩EQUATIONEQUATION1 ∥v t i + η c u t j ∥ 2 - η u f ∥u t j + η u f v t i ∥ 2 ⟨v t i , y⟩ = 1 + η u f ∥u t j + η u f v t i ∥ 2 - 1 + η c ∥v t i + η c u t j ∥ 2 ⟨v t i , y⟩ = 1 + η u f 1 + 2η u f ⟨u t j , v t i ⟩ + (η u f ) 2 - 1 + η c 1 + 2η c ⟨u t j , v t i ⟩ + (η c ) 2 ⟨v t i , y⟩.Let a = ⟨u t j , v t i ⟩ ≤ 1. We note that the functionh(η) = 1 + η 1 + 2ηa + η 2 = 1 + 2η + η 2 1 + 2ηa + η 2 = 1 + (2 -2a)η 1 + 2ηa + η 2 = 1 + 2(1 -a) 1 η + 2a + η is increasing in η ∈ [0, 1]. Under the assumption of η c ≤ ηuL f 2 ≤ ηuf 2 < η u f , we have h(η c ) ≤ h(η u f ) and hence ⟨u t+1 j , y⟩ -⟨v t+1 i , y⟩ > h(η u f ) -h(η c ) ⟨v t i , y⟩ ≥ 0.We then prove the second item. Using Lemma D.3 for v t+1 i = P(v t i + η c u t j ), we get⟨v t+1 i -v t i , y⟩ ≥ η c 1 + η c ⟨u t j , y⟩ -⟨v t i , y⟩ . Using Lemma D.3 for u t+1 j = P(u t j + η u f (v t i , u t j )v t i) and using the fact ⟨v t i , y⟩ -⟨u t j , y⟩ < 0 proved in item 1,⟨u t+1 j -u t j , y⟩ ≥ η u f (v t i , u t j ) 1 + η u f (v t i , u t j ) ⟨v t i , y⟩ -⟨u t j , y⟩ ≥ η u 1 + η u ⟨v t i , y⟩ -⟨u t j , y⟩ .Rearranging the above two inequalities:1 + η c η c ⟨v t+1 i , y⟩ -⟨v t i , y⟩ ≥ ⟨u t j , y⟩ -⟨v t i , y⟩; 1 + η u η u ⟨u t+1 j , y⟩ -⟨u t j , y⟩ ≥ ⟨v t i , y⟩ -⟨u t j , y⟩.Summing the above two inequalities over t = 0, 1, . . . , T -1:EQUATIONAccording to Lemma F.1, after at mostEQUATION) Multiplying ( 7) by 1+ηu ηu and adding to ( 6):1 + η c η c + 1 + η u η u ⟨v T i , y⟩ -⟨v 0 i , y⟩ ≥ 1 + η u η u (D -R).This implies⟨v T i , y⟩-⟨v 0 i , y⟩ ≥ 1+ηu ηu 1+ηc ηc + 1+ηu ηu (D-R) = η c (1 + η u ) η u (1 + η c ) + η c (1 + η u ) (D-R) ≥ η c η u + η c (D-R).given η c ≤ η u .The following lemma shows that, when we reflect some of the feature vectors in a system(U t , V t ) = ({u t j } j∈[m] , {v t i } i∈[n]), there is a correspondence between the behaviors of the system with the reflected vectors and the original system. Lemma D.7 (Reflection).Let (U t , V t ) = ({u t j } j∈[m] , {v t i } i∈[n]) be a system of m users and n creators with impact functions f, g. Let a i , b j ∈ {+1, -1}, ∀i ∈ [n], ∀i ∈ [m] be some binary constants. Define:ũt j = b j u t j = ±u t j , ṽt i = a i v t i = ±ṽ t i . and impact functions f (ṽ i , ũj ) = a i b j f (v i , u j ), g( ũj , ṽi ) = a i b j g(u j , v i ). Then:• There is a "correspondence" between the evolution of the system (U t , V t ) with impact functions f, g and the evolution of the system( Ũ t , Ṽ t ) = ({ ũt j } j∈[m] , {ṽ t i } i∈[n]) with impact functions f , g. Formally, suppose every user is recommended the same creator in the two systems, then the updated vectors in the two systems still satisfy the relations:ũt+1 j = b j u t+1 j , ṽt+1 i = a i v t+1 i . • If the system ( Ũ t , Ṽ t ) is in R-bi-polarization, then the original system (U t , V t ) is also in R-bi-polarization.Proof. Consider the first item. Suppose user i is recommended creator j at time step t in the two systems. Then by definition, the updated user vectors in the two systems satisfyũt+1 j = P ũt j + η u f (ṽ t i , ũt j )ṽ t i = P b j u t j + η u a i b j f (v t i , u t j )a i v t i = P b j u t j + η u b j f (v t i , u t j )v t i = b j P u t j + η u f (v t i , u t j )v t i = b j u t+1 jSuppose creator i is recommended to the set of users J at time step t in the two systems. Then,ṽt+1 i = P ṽt i + ηc |J| j∈J g( ũt j , ṽt i ) ũt j = P a i v t i + ηc |J| j∈J a i b j g(u t j , v t i )b j u t j = P a i v t i + ηc |J| j∈J a i g(u t j , v t i )u t j = a i P v t i + ηc |J| j∈J g(u t j , v t i )u t j = a i v t+1 i .This means that the evolution of the system ( Ũ t , Ṽ t ) has a correspondence to the evolution of the original system (U t , V t ).Consider the second item. Suppose ( Ũ t , Ṽ t ) is in R-bi-polarization, so ṽt i = ±v t i is R-close to ±c and ũt j = ±u t j is R-close to ±c with some vector c ∈ S d-1 . This implies that v t i is R-close to ±c and u t j is R-close to ±c. So, the system (U t , V t ) satisfies R-bi-polarization.Proof. Let (U t , V t ) be an (R, c)-bi-polarization state with R ∈ [0, 1] and c ∈ S d-1 , where all u t j and v t i are within distance R to +c or -c. We show that, after one step of update, u t+1 j and v t+1 i are still within distance R to +c or -c, so (U t+1 , V t+1 ) still satisfies (R, c)-bi-polarization.Consider u t j . Without loss of generality, suppose u t j is close to +c, so ∥u t j -c∥ 2 ≤ R. Suppose user j is recommended creator i at step t.Let ṽt i = v t i if ⟨v t i , u t j ⟩ ≥ 0 and ṽt i = -v t i if ⟨v t i , u t j ⟩ < 0.Then, the user update isu t+1 j = P u t j + η u f (v t i , u t j )v t i = P u t j + η u |f (v t i , u t j )|ṽ t i .Since ṽt i is close to +c or -c, ⟨ṽ t i , u t j ⟩ > 0, and u t j is close to +c, it must be that ṽt i is close to +c, so ∥ṽ t i -c∥ 2 ≤ R. Then, since u t+1 j is the normalization of a vector in the convex cone formed by u t j and ṽt i , by Lemma D.2, we have∥u t+1 j -c∥ 2 ≤ max ∥u t j -c∥ 2 , ∥ṽ t i -c∥ 2 ≤ R.Consider v t i . Suppose ∥v t i -c∥ 2 ≤ R. Let J be the set of users that are recommended creator i at step t. For each j ∈ J, let ũt j = u t j if ⟨u t j , v t i ⟩ ≥ 0 and ũt j = -u t j if ⟨u t j , v t i ⟩ < 0. Then, the creator update isv t+1 i = P v t i + η c |J| j∈J g(u t j , v t i )u t j = P v t i + η c |J| j∈J |g(u t j , v t i )| ũt j .We note that every ũt j satisfies ∥ ũt j -c∥ 2 ≤ R (by the same reasoning as above). Then, since v t+1 i is the normalization of a vector in the convex cone formed by v t i and { ũt j } j∈J , by Lemma D.2, we have∥v t+1 i -c∥ 2 ≤ min ∥v t i -c∥ 2 , min j∈J ∥ ũt j -c∥ 2 ≤ R.F Proof of Lemma 3.4Lemma 3.4 is proved by induction on the number n of creators. We first show that any system with 1 creator and multiple users must converge to R-bi-polarization in finite steps for any R > 0. Using the result for 1 creator, we then construct a finite length path that leads to R-bi-polarization for any system with n ≥ 2 creators.We prove some convergence results for the special case of only one creator. This will serve as the basis for the proof for n ≥ 2 creators. Recall that we have the following dynamics update rule:• User:u t+1 j = P(u t j + η u f (v t i , u t j )v t i )where v t i is the creator recommended to user j; f (v i , u j ) satisfies:EQUATION• Creator:v t+1 i = P(v t j + ηc |J| j∈J g(u t j , v t i )u t j )where J is the set of users being recommended creator i.Lemma F.1. Consider a system of 1 creator v t i and |J| users {u t j } j∈J , where the creator is recommended to all users at every time step. Assume:• Initially, ∀j ∈ J, ⟨u 0 j , v 0 i ⟩ > 0.• There exists some constantL f > 0 such that f (v i , u j ) ≥ L f > 0 whenever ⟨v i , u j ⟩ > 0. • g(u j , v i ) = 1 when ⟨u j , v i ⟩ > 0. • η c ≤ ηuL f 2 and 0 ≤ η u < 1 2 .Then, for any R > 0, after at most 8 3ηuL f ln 2|J| R 2 steps, j∈J ∥u t j -v t i ∥ 2 2 ≤ R 2 will hold forever. In particular, each user vector will satisfy ∥u t j -v t i ∥ 2 ≤ R.Proof. We first note that, by Lemma D.5, all user vectors satisfy ⟨u t j , v t i ⟩ > 0 in all time steps t > 0. Hence, the creator update is alwaysv t+1 i = P(v t i + ηc |J| j∈J g(u t j , v t i )u t j ) = P(v t i + η c 1 |J| j∈J u t j ). Let a t = 1/(1 - 3ηuL f 8) t . Define the following potential function:EQUATIONWe will show that Φ t is monotonically decreasing. Take the difference between Φ t+1 and Φ t :Φ t+1 -Φ t = a t+1 j∈J ⟨u t j , v t i ⟩ -⟨u t+1 j , v t+1 i ⟩ + (a t+1 -a t ) j∈J 1 -⟨u t j , v t i ⟩ = a t+1 j∈J ⟨v t i , u t j -u t+1 j ⟩ + j∈J ⟨u t j , v t i -v t+1 i ⟩ + j∈J ⟨u t+1 j -u t j , v t i -v t+1 i ⟩ + (a t+1 -a t ) j∈J 1 -⟨u t j , v t i ⟩ .Using Lemma D.3 with x t = u t j , z t = v t i , and η = η u f (v t i , u t j ), we get⟨v t i , u t j -u t+1 j ⟩ ≤ - η u f (v t i , u t j ) 1 + η u f (v t i , u t j ) 1 -⟨u t j , v t i ⟩ ≤ - η u L f 2 1 -⟨u t j , v t i ⟩ .Using Lemma D.4 with x t = u t j , z t = v t i , and η = η u f (v t i , u t j ), we get⟨v t i , u t j -u t+1 j ⟩ ≤ - 1 η u f (v t i , u t j ) ∥u t+1 j -u t j ∥ 2 2 ≤ - 1 η u ∥u t+1 j -u t j ∥ 2 2 .Using Lemma D.4 withx t = v t i , z t = 1|J| j∈J u t j , and η = η c , we getj∈J ⟨u t j , v t i -v t+1 i ⟩ = |J|⟨ 1 |J| j∈J u t j , v t i -v t+1 i ⟩ ≤ - |J| η c ∥v t+1 i -v t i ∥ 2 2 .Using the above three inequalities, we can upper bound Φ t+1 -Φ t :Φ t+1 -Φ t = a t+1 3 4 j∈J ⟨v t i , u t j -u t+1 j ⟩ + 1 4 j∈J ⟨v t i , u t j -u t+1 j ⟩ + j∈J ⟨u t j , v t i -v t+1 i ⟩ + j∈J ⟨u t+1 j -u t j , v t i -v t+1 i ⟩ + (a t+1 -a t ) j∈J 1 -⟨u t j , v t i ⟩ ≤ a t+1 - 3 4 j∈J η u L f 2 1 -⟨u t j , v t i ⟩ - 1 4 j∈J 1 η u ∥u t+1 j -u t j ∥ 2 2 - |J| η c ∥v t+1 i -v t i ∥ 2 2 + j∈J ∥u t+1 j -u t j ∥ 2 • ∥v t+1 i -v t i ∥ 2 + (a t+1 -a t ) j∈J 1 -⟨u t j , v t i ⟩ = a t+1 - 3η u L f 8 j∈J 1 -⟨u t j , v t i ⟩ - j∈J 1 4η u ∥u t+1 j -u t j ∥ 2 2 + 1 η c ∥v t+1 i -v t i ∥ 2 2 ≥2 1 4ηu ηc ∥u t+1 j -u t j ∥ 2 2 ∥v t+1 i -v t i ∥ 2 2 -∥u t+1 j -u t j ∥ 2 • ∥v t+1 i -v t i ∥ 2 + (a t+1 -a t ) j∈J 1 -⟨u t j , v t i ⟩ ≤ a t+1 - 3η u L f 8 j∈J 1 -⟨u t j , v t i ⟩ - j∈J 1 ηuηc -1 ≥0 ∥u t+1 j -u t j ∥ 2 • ∥v t+1 i -v t i ∥ 2 + (a t+1 -a t ) j∈J 1 -⟨u t j , v t i ⟩ ≤ a t+1 - 3η u L f 8 j∈J 1 -⟨u t j , v t i ⟩ + 0 + (a t+1 -a t ) j∈J 1 -⟨u t j , v t i ⟩ = 1 - 3η u L f 8 a t+1 -a t j∈J 1 -⟨u t j , v t i ⟩ = 0where the last step is because (1 -3ηuL f 8)a t+1 = a t .We have shown that Φ t is monotonically decreasing. Thus,1 2 j∈J ∥u T j -v T i ∥ 2 = Φ T a T ≤ Φ 0 a T ≤ j∈J 1 a T = 1 - 3ηuL f 8 T |J| ≤ e -3ηu L f 8 T |J| ≤ 1 2 R 2 whenever T ≥ 8 3ηuL f ln 2|J| R 2 .Corollary F.2 (of Lemma F.1). Consider a system of 1 creator v t i and |J| users {u t j } j∈J , where the creator is recommended to all users at every time step. Assume:• Initially, ⟨u 0 j , v 0 i ⟩ ̸ = 0 for every j ∈ J.• There exists some constantL f > 0 such that |f (v i , u j )| ≥ L f > 0 whenever ⟨v i , u j ⟩ ̸ = 0. • g(u j , v i ) = sign(⟨u j , v i ⟩). • η c ≤ ηuL f 2 and 0 ≤ η u < 1 2 .Then, for any R > 0, after at most 8 3ηuL f ln 2|J| R 2 steps, the system will reach R-bi-polarization.Repeat steps (2) and (3) for K times. Then, using (15) for K times,F T1+K(T2+T3) ≤ ηu ηu+ηc F T1+(K-1)(T2+T3) + ηc ηu+ηc R 2 1 2 + R 1 + 2R 1 ≤ ηu ηu+ηc ηu ηu+ηc F T1+(K-2)(T2+T3) + ηc ηu+ηc R 2 1 2 + R 1 + 2R 1 + ηc ηu+ηc R 2 1 2 + R 1 + 2R 1 . . . ≤ ηu ηu+ηc K F T1 + 1 + ηu ηt+ηc + • • • + ηu ηt+ηc K-1 ηc ηu+ηc R 2 1 2 + R 1 + 2R 1 ≤ ηu ηu+ηc K • 1 + 1 1 -ηu ηu+ηc ηc ηu+ηc R 2 1 2 + R 1 + 2R 1 = ηu ηu+ηc K + R 2 1 2 + R 1 + ηu+ηc ηc 2R 1 ≤ R 2 1 2 + R 2 1 2 + R 1 + ηu+ηc ηc 2R 1 ≤ ηu ηc + 2)2R 1 , by choosing K = ln 2 R 2 1 ln ηu+ηc ηu ≤ ηu+ηc ηc ln 2 R 2 1. This means that, after repeating steps ( 2) and ( 3) for K times, we must have∥v T1+K(T2+T3) n -c T1+K(T2+T3) ∥ = 2 1 -⟨v T1+K(T2+T3) n , c T1+K(T2+T3) ⟩ = 2F T1+K(T2+T3) ≤ 2 ηu ηc + 2)2R 1 = R.The above inequality, together with the fact that other creators i ̸ = n and all users in J already satisfy (R 1 ≤ R, c T1+K(T2+T3) )-consensus after step (3), implies that the whole system has reached (R, c T1+K(T2+T3) )-consensus.The length of the path constructed above is at most:T 1 + K(T 2 + T 3 ) ≤ L n-1,R1 + ηu+ηc ηc ln 2 R 2 1 8 3ηuL f ln 2|J| R 2 1 + L n-1,R1 = L n,R < + ∞,which is finite.Lemma F.4. Consider a subsystem of n creators {v t 1 , . . . , v t n } and |J| users {u t j } j∈J . Assume:• Initially, the first n -1 creators and all users are in R 0 -consensus:∥v 0 i -c∥ ≤ R 0 , ∥u 0 j -c∥ ≤ R 0 , with 0 < R 0 < ηc 5(ηc+ηu) .• ⟨v 0 n , u 0 j0 ⟩ > 0 for some j 0 ∈ J.• g(u j , v i ) = sign(⟨u j , v i ⟩).• Assumption of Lemma F.1.Then, for any R ∈ (0, 1), there exists a path of finite length that leads the initial state (U 0 , V 0 ) to R-consensus.Proof. First, we recommend creator n to user j 0 for T = 8 3ηuL f ln 2 R 2 0 steps, while recommending other creators to other users arbitrarily. Applying Lemma D.6 with y = u 0 j0 , we getEQUATIONOn the other hand, because the first n -1 creators and all users in J \ {j 0 } form an (R 0 , c)-consensus at time step 0, according to Observation 3.2, they still form an (R 0 , c)-consensus at time step T , so ∥v T i -c∥ ≤ R 0 for every i ∈ {1, . . . , n -1}. This implies, for every i ∈ {1, . . . , n -1},EQUATIONG Missing Proofs from Section 4G.1 Proof of Proposition 4.2Let R > 0 be any small number. Let c 1 , . . . , c ⌊n/k⌋ ∈ R d be ⌊n/k⌋ vectors that satisfy B(c ℓ , 2R) ∩ B(c ℓ ′ , 2R) = ∅ for ℓ ̸ = ℓ ′ , where B(c, R) is the ball centered at c with radius R: {x ∈ R d : ∥x -c∥ 2 ≤ R}. Consider user and creator features (U t , V t ) that satisfy: every ball B(c ℓ , R) (ℓ = 1, . . . , ⌊n/k⌋) contains k creator vectors, and every user vector u t j is in one of the balls B(c ℓ , R). By definition, (U t , V t ) form ⌊n/k⌋ clusters. We show that, after one step of update, the new state (U t+1 , V t+1 ) must still form ⌊n/k⌋ clusters. Consider any user j. Suppose u t j ∈ B(c ℓ , R), then the distance from u t j to any creator v t i ∈ B(c ℓ , R) is at most 2R:∥u t j -v t i ∥ ≤ 2R.The distance from u t j to any creator v t i ′ not in B(c ℓ , R) is greater than 2R:∥u t j -v t i ′ ∥ > 2R because v t i ′ is in some other ball B(c ℓ ′ , R) that satisfies B(c ℓ ′ , 2R) ∩ B(c ℓ , 2R) = ∅.This implies that the inner products between user j and the creators in ball B(c ℓ , R) are greater than that with the creators in other ball:∀v t i ∈ B(c ℓ , R), ⟨u t j , v t i ⟩ = 1 - 1 2 ∥u t j -v t i ∥ 2 2 ≥ 1 - 1 2 (2R) 2 > 1 - 1 2 ∥u t j -v t i ′ ∥ = ⟨u t j , v t i ⟩, ∀v t i ′ ∈ B(c ℓ ′ , R).Since B(c ℓ , R) contains k creators, these k creators are the k-most relevant ones to user j, so user j will only be recommended these creators. Then, by applying Observation 3.2 to each of the ⌊n/k⌋ balls separately, we see that each ball is a R-consensus and hence absorbing. So, the new state (U t+1 , V t+1 ) still forms ⌊n/k⌋ clusters with these ⌊n/k⌋ balls.The d-dimensional simplex centered at the original has d+1 vectors with negative inner products with each other. They form d+1 clusters. Since user-creator pairs with negative inner product ⟨u i , v j ⟩ < 0 are not recommended, recommendations only happen within each cluster. By Observation 3.2, each cluster is absorbing, so the whole system is stable, keep forming d + 1 clusters forever.Here we further discuss real-world recommender systems' properties and designs that are currently not covered in our main paper. We plan to generalize our model in the future to further capture these features and discuss insightful findings, but having them in the current paper may be a distraction to our main findings.In our current model, the users and creators will stay in the system from the start to the end. However, in real-world recommender systems, users and creators may leave the platform either permanently or for a certain period. Meanwhile, new users and creators will join the platform. Such join and leave dynamics are also influenced by the recommendations' relevance and diversity, which further complicate the problem. Moreover, users and creators have different activeness levels on the platform, e.g., some users may watch a lot more videos than others, and some creators may post a lot more creations, these effects will also be strongly correlated with the dual influence of the recommender system.Creation quality is a major factor influencing users' feedback in addition to the creation style, e.g., well-made cuisine videos could also be fun and liked by gamers and pet lovers, which we need more than a collaborative filtering type of modeling like our current model to capture such features. A potential solution to boost both long-term system diversity and single-shot recommendation diversity is to design mechanisms that can incentivize creators to create higher-quality videos instead of changing their creation styles.Cold Start is widely used in real-world recommender systems for newly published items. Due to the lack of user-item interactions on new items, the systems randomly recommend these new items to users and collect data for collaborative filtering. In our current model, if we consider the creators creating new items in each time step under their current time creation style, then cold start guarantees the conditions in Theorem 3.3. But if we consider the system to have good enough content understanding ability and can accurately predict the new creations' embeddings, the cold start is not necessary and our model and results in the top-k truncation and threshold truncation parts are valid. We also highlight a subtle difference between cold start and random traffic, if cold start is used on creators instead of items, then after the creator is exposed to users a certain number of times, the system will not guarantee to provide a non-zero probability of recommending this creator, and thus the conditions in Theorem 3.3 may not hold.NeurIPS Paper Checklist 


Decision: The paper studies the dual influence of recommender systems on users and creators, and provides theoretical proof on the convergence towards polarization under the model and certain conditions, and presented findings showing relevance focusing systems such as top-K recommendations rather than diversity promoting systems are more effective at preventing polarization.  Reviewers find the work solid and the result interesting. Some minor comments are missing references and clarification on assumptions were addressed during rebuttal.

Reviews: review: paper_summary: paper_summary: This paper studies how recommendations become polarized over the long run when user and creator features dynamically change over time. The authors theoretically prove that, under the assumption that every creator can be recommended to every user with some non-zero probability, recommender systems will eventually converge to polarization. They also simulate some real-world models, including top-k recommendation, truncation, diversity boosting, and lower-bounding probabilities in a long-term setting. The key observation is that top-k recommendation (i.e., only recommending top-k items to users) can reduce polarization to some extent, while existing diversity-boosting methods will worsen polarization when user/creator features dynamically change over time in the system.
strength_weakness: strength_weakness: 1. Despite the novelty and interestingness, I have concerns about the key assumptions of the theoretical and empirical analyses. The assumption that all items can be recommended to users is not realistic. In practice, almost all recommender systems rely on top-k recommendations for either effectiveness or resource constraints like screen size. For example, on platforms like Netflix or Amazon, customers can only see a certain number of items on the webpage (i.e., p=0 for items that users can't see). Even if they can scroll down and the system continually recommends new items, they cannot physically see all items in the system. Thus, I believe the top-k setting is the most realistic and natural for real-world scenarios, and this seems like a hole in the authors' analyses. In this sense, the measures for empirical analysis should also only consider top-k items, not all items.
2. For the real-world designs, it would be more extensive if users included trustworthiness-aware recommender systems that consider dynamic/continual settings. For example, [1] consider performance difference between two different user groups when the user/item features are continually updated over time in the systems.
3. For the analysis with Movielens, considering the interaction timestamp in the simulation would more accurately reflect real-world scenarios, for example, for determining the true labels.

[1] Yoo et al., Ensuring User-side Fairness in Dynamic Recommender Systems, WWW'24
questions: questions: 1. Please address the points I raised in Weaknesses.
2. (Minor) Are both consensus and bi-polarization conceptually polarization?
3. (Minor) How are the initial user/creator embeddings initialized in the Movielens experiment?
limitations: limitations: The authors adequately addressed the limitations.
paper_summary: paper_summary: The paper explores the dynamics between users and content creators in recommender systems, highlighting the dual influence where users’ preferences are shaped by recommendations and creators modify their content to align with what is more likely to be recommended. The study defines a model called user-creator feature dynamics to capture these interactions, demonstrating that such systems are prone to polarization, resulting in a loss of diversity. The paper then examines various approaches to mitigate polarization and improve diversity, finding that relevancy-optimizing methods, such as top-k recommendations, can prevent polarization more effectively than traditional diversity-promoting approaches.
strength_weakness: strength_weakness: 1. There are two lines of works focusing on modeling content creator dynamics and user preference evolving dynamics that are neglected by the authors. I listed several representative works and it would be great to include a comprehensive literature review regarding these works in the related work section.

2. One of your main observation (larger $\beta$ leads to higher creator diversity and alleviated polarization) is actually pointed out in [1] under a similar model, where content creators compete for a fixed user population (see section 3.2 in [1]). And another main observation in section 5.3 that smaller $k$ improves diversity does not echo the result in [2], which shows that larger $k$ improves the total creator utilities. It would be better to include some detailed discussions regarding these two works. 

3. The user/creator preference updating dynamics need more justifications and empirical evidence.

4. The dynamical model makes some sense to me, but it would be more interesting to understand whether the observations still hold in the presence of noise. If the noisy version is hard to analyze theoretically, additional simulation results could also be valuable.


[1]. Modeling Content Creator Incentives on Algorithm-Curated Platforms
[2]. How Bad is Top-K Recommendation under Competing Content Creators?
[3]. Online recommendations for agents with discounted adaptive preferences
[4]. Recommender systems as dynamical systems: Interactions with viewers and creators
[5]. Learning from a learning user for optimal recommendations
[6]. Supply-side equilibria in recommender systems
questions: questions: 1. In theorem 3.3, how does the convergence rate depending on the temperature parameter $\beta$? I ask this because when $\beta\rightarrow +\infty$, the softmax recommendation strategy is equivalent to the top-1 recommendation strategy. In this case, Proposition 4.2 predicts that the top-1 recommendation lead to $n$ clusters rather than bi-polarization, which seems to contradict theorem 3.3.
2. I do not fully get why the specific forms of function $f$ and $g$ do not affect the analysis. Is it because your main results only depend on the range of $f$ and $g$?
3. In the experiments, the range of $\beta$ is quite conservative. I'm curious about the result under a larger range of $\beta$.
limitations: limitations: yes
paper_summary: paper_summary: This paper models dynamics of both users and creators in a recommender system. The user features shift in the direction of the content recommended to them. The creator dynamics are strategically motivated i.e. they try to align content to attract their audience, to increase profit. 

The authors then provide sufficient conditions for this model of dual dynamics to converge to polarization under a natural assumption that each creator has some non zero probability of being recommended to every user.

The paper then discusses four real world recommendation designs, and whether they cause polarization or multiple clusters etc. They also provide results on synthetic and Movielens data complementing theory results and show that certain recommender designs do lead to polarization vs diverse clusters.
strength_weakness: strength_weakness: A criticism I had while reading the paper are gaps in literature for the discussion on dynamics in recommender systems. In addition to [Eilat & Rosenfeld] referenced in the introduction, [1,2,3,4,5,6] consider creator dynamics in recommender systems. These works assume static user features and provide results on content at equilibrium and user welfare. In the context of these works, it would be beneficial to highlight how your work is the first to consider both creator and user dynamics.

[1] A Game-Theoretic Approach to Recommendation Systems with Strategic Content Providers (Ben-Porat and Tennenholtz)

[2] Supply-side equilibria in recommender systems (Jagadeesan et al)

[3] How Bad is Top-k Recommendation under Competing Content Creators? (Yao et al)

[4] Modeling content creator incentives on algorithm-curated platforms (Hron et al)

[5] Producers Equilibria and Dynamics in Engagement-Driven Recommender Systems (Acharya et al)

[6] User Welfare Optimization in Recommender Systems with Competing Content Creators (Yao et al)
questions: questions: -  I understand the motivation for the form of user update in equation (3) . In each recommendation step an item is recommended and the user preference shifts in that direction, this is like in [Dean and Morgenstern]. Can you motivate the update in Eq (4), is this myopically optimal for the creator to do, and how does it generalize [Eilat & Rosenfeld]?
- Minor Typo? For Figure 6,  larger $\rho$ seems to lead to higher creator diversity (green curve).
limitations: limitations: The authors discuss limitations of their results in the paper.

Comments: title: Re: Rebuttal
Correct, recommendations are based on probability distribution.
Clarifications on "top-k recommendation", "non-zero probability of not shown items", and "random intervention"
Acknowledging rebuttal
comment: I thank the authors for their detailed response, which addressed most of my concerns. And I appreciate the summarized table which emphasizes the contribution. I decide to maintain my score, leaning towards acceptance.
Thank you!  We will surely include our rebuttal in the final version.
Thank you for the clarifications. Most of my concerns have been resolved, so I'll increase the credits. I hope the authors can include their rebuttal in the final version.
Yes, we are recommending creators based on their probability distribution.

Without top-k truncation, each creator $j$ (among the N total creators) is recommended with $p_{ij}$ probability.

With top-k truncation, the N creators with the $k$ highest relevance scores pass the filter/truncation stage, and these $k$ creators are then recommended with probability proportional to $p_{ij}$ (calculated and normalized using these top $k$ relevance scores). We are not deterministically selecting the top-$L$ ($L < k$) creators to show to the user.
Thank you for the reply. I agree the screen size L could be much smaller than K. 
Let me ask you something more just to ensure my understanding on this setting is correct. Whether we use top-K truncation or use all N creators, we are assuming that recommendations are based on their probability distribution, correct? Rather than just recommending the Top-X creators with the highest probabilities.
We would like to provide some clarification regarding item delivery. When referring to "top-$k$”, we actually mean “top-$k$ truncation". For a given user $u_i$, we compute probability $p_{ij}$ for each creator $v_j$. Let $p^{(k)}$ be the $k^{\text{th}}$ largest $p_{ij}$, then all $p_{ij} < p^{(k)}$ are set to 0. Creators are then recommended to $u_i$ based on the remaining nonzero probability $p_{ij}$. This filtering process corresponds to the (first) recall stage commonly found in large-scale recommendation systems with two (or more) stages (see, e.g., [[Youtube's DNN recommendation paper]](https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/45530.pdf)).

Top-$k$ truncation does not mean showing $k$ items to the users. Let's say that a user sees $L$ items in a short time frame. In real-world scenarios, the value of $L$ is usually smaller than $k$ and can depend on factors like screen size. Moreover, the set of $L$ items shown to the user is a random sample from a distribution of items. Reviewer EsTG wrote "(p=0 for items that users can't see)" and "Therefore, I am unclear on why there would be a non-zero probability when only K items are shown". That seems to be a misunderstanding. Even if an item $j$ is not shown to the user in one time step, its probability $p_{ij}$ can still be $>0$, and this item can be sampled and shown to the user in the next time step. 

The random traffic intervention (which has been used in large-scale recommendation systems as we discussed in Lines 225 - 227; see also [[KuaiRand, page 4, left column (ii)]](https://arxiv.org/pdf/2208.08696)) bypasses all the steps (scoring, filter, etc) in the multi-stage recommendation system and replaces the candidates returned by the multi-stage recommendation process with randomly chosen candidates at a low probability. The random traffic intervention does not cause efficiency loss for the system.  And with random traffic, a user might be recommended any possible creator (namely, all $p_{ij} > 0$), even if $L$ is small.
Thanks for the detailed response. However, I still feel that my question about top-k recommendation has not been adequately answered. Let me clarify the question further.

1. In the paper, lines 192-193 state, "In particular, we consider the top-k recommendation policy where each user is recommended only he k most relevant creators, so pt ij = 0 if i is not one of k creators i′ that maximize ⟨vt i′ , ut j ⟩.", meaning that p=0 in the case of top-K recommendation. My thought was that nearly all recommendation scenarios are essentially top-K recommendation practices, which means p=0 in these cases as well. Could you provide examples where this is not the case?
As mentioned in line 188-189, involves filtering out items unlikely to be relevant to a user and then recommending from the remaining items. This filtering process is typically based on relevance scores, where only the highest scores are retained. In this sense, it seems equivalent to "only K items are shown." Therefore, I am unclear on why there would be a non-zero probability when only K items are shown. What is the difference between top-K recommendation in Section 4-1 and these real-world scenarios?

2. Regarding random interventions, are you suggesting or is it possible that these interventions occur on top of top-k recommendations (i.e., so if there are n interventions, there would be K+n items in the list)? I believe this scenario could indeed have a non-zero probability even within the top-k recommendation practice. But without this scenario, I think most recommendation scenarios are basically top-k recommendations in Section 4-1. What are your thoughts on this?
Thanks for the addressing my questions, I stand with my assessment of accepting this paper.