User-Creator Feature Polarization in Recommender Systems with Dual Influence

Q1: Can you motivate the update in Eq (4)? Is this myopically optimal for the creator to do, and how does it generalize [Eilat & Rosenfeld]?

[Eilat & Rosenfeld] assumes that creators aim to maximize exposure (defined as the sum of inner products between user embeddings and the creator embedding) minus the content adjustment cost (see their Section 2.1 and equation (6)). They show that such creators will move towards the average of the user embeddings by some step size (their equation (11)). In our notations, their update is $v_i^{t+1} = \mathcal P(v_i^t + \frac{\eta_c}{|J_i^t|} \sum_{j\in J_i^t} u_j^t)$, which is the special case of our equation (4) where $g$ is the constant positive function $g(u_j, v_i) = 1$.

Nevertheless, as we wrote in Lines 122 - 129, we are motivated by a different type of creators: rating-maximizing creators. This means that the $g(u_j, v_i)$ function should have the same sign as $\langle u_j, v_i\rangle$. Intuitively, for all the users $J_i^t$ being recommended creator $i$, if the user likes the item, $\langle u_j^t, v_i^t\rangle > 0 \implies g(u_j^t, v_i^t) > 0$, then the creator is incentivized to move towards that user to receive more positive rating from that user. Otherwise, $\langle u_j^t, v_i^t\rangle < 0 \implies g(u_j^t, v_i^t) < 0$, the creator is incentivized to move away from the user in order to be recommended less often to that user (to avoid being negatively-rated by the user). Taking both scenarios into account, the creator moves toward the weighted average $\sum_{j\in J_i^t} g(u_j^t, v_i^t) u_j^t$, which gives equation (4). Under this particular assumption on $g$, our equation (4) does not capture [Eilat & Rosenfeld].

Q2: Minor Typo? For Figure 6, larger $\rho$ seems to lead to higher creator diversity (green curve).

Yes, this is a minor typo. The figure shows that a higher $\rho$ leads to higher creator diversity but also higher tendency to polarization at the same time. A possible explanation for this phenomenon is (similar to our reasoning in Lines 307 to 309): the system polarizes into two balanced clusters which actually have a large average pairwise distance. So in this case, Tendency to Polarization may be a better measure for diversity loss than the Creator Diversity measure (average pairwise distance).

Weakness 1: related works.

Thank you for listing those related works! We will discuss them in the revision. We also provide comparisons between some of those works and our work in a table in our global response.

Weakness 2: It would be better to include some detailed discussions regarding these two works [1] [2].

[1] Modeling Content Creator Incentives on Algorithm-Curated Platforms

[2] How Bad is Top-K Recommendation under Competing Content Creators?

[1] indeed also observes that a larger $\beta$ leads to higher creator diversity. We view our work as complementary to [1]. Our findings corroborate those of [1] despite two key differences between our setting and that of [1]: in [1], creators maximize exposure while we consider creators who maximize user engagement; second, [1]'s user population is fixed while ours is adaptive. This helps us understand that there may be some fundamental mitigation strategies (such as selecting $\beta$), which have similar effects regardless of changes to the underlying problem formulation.

[2] shows that a larger $k$ improves the social welfare, defined to be the total utility/relevance of the users (which is also the total utility of creators in their model), assuming a fixed user population. In contrast, we focus on the diversity/polarization of creators, and we have adaptive users. So the $k$ plays a different role in our setting: a larger $k$ leads to worse creator diversity.

Weakness 3: the user/creator preference updating dynamics need more justifications and empirical evidence.

Our user preference update model is a generalization of [Dean & Morgenstern, EC 22], which models user update as $u^{t+1}_j = \mathcal{P}(u^t_j + \eta \langle v_i^t, u_j^t\rangle v_i^t)$. Our work replaces the inner produce with a general function $f(v_i^t, u_j^t)$ (our Equation 3), with some constraints outlined on lines 113 - 121 of our paper. The motivation for this update rule, as outlined by [Dean & Morgenstern, EC 22], is the "biased assimilation" phenomenon, which is in turn inspired by the opinion polarization literature. (We mentioned this in Lines 107, 115, and 241.)

Our creator update model, as we wrote in Lines 122 - 129, is motivated by rating-maximizing creators. This means that the $g(u_j, v_i)$ function has the same sign as $\langle u_j, v_i\rangle$. Intuitively, for all the users $J_i^t$ being recommended creator $i$, if the user likes the item, $\langle u_j^t, v_i^t\rangle > 0 \implies g(u_j^t, v_i^t) > 0$, then the creator is incentivized to move towards that user to receive more positive rating from that user. Otherwise, $\langle u_j^t, v_i^t\rangle < 0 \implies g(u_j^t, v_i^t) < 0$, the creator is incentivized to move away from the user in order to be recommended less often to that user (to avoid being negatively-rated by the user). Taking both scenarios into account, the creator moves toward the weighted average $\sum_{j\in J_i^t} g(u_j^t, v_i^t) u_j^t$, which gives our update rule (4).

If any additional details would help improve the justification of the user/creator update rules, please let us know. This is an important aspect of our work and we would like to polish it as much as possible.

Weakness 4: it would be more interesting to understand whether the observations still hold in the presence of noise.

In the newly uploaded PDF, we provide simulation results where the user and creator updates include normally distributed unbiased noises inside the projection operator: $u_j^{t+1} = P (u_j^t + \eta_u f( v_{i_j^t}^t, u_j^t) v_{i_j^t}^t + \eta_u \epsilon_j^t)$ where $\epsilon_j^t \sim Normal(0, \sigma^2I)$ and $v_i^{t+1} = P ( v_i^t + \frac{\eta_c}{|J_i^t|} \sum_{j \in J_i^t} g(u_j^t, v_i^t) u_j^t + \eta_c \epsilon_i^t)$ where $\epsilon_i^t \sim Normal(0, \sigma^2 I)$.

We observe that a small noise still leads to near polarization, while a large noise (large $\sigma$) reduces the tendency to polarization. And the observation that top-k recommendation reduces polarization still holds. (see Figures R2 - R5 in the uploaded PDF.)

Q1: how does the convergence rate depend 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, which by Proposition 4.2 leads to $n$ clusters rather than bi-polarization, which seems to contradict theorem 3.3.

Finite $\beta$ and infinite $\beta$ are qualitatively different. Finite $\beta$ leads to bi-polarization, while $\beta = +\infty$ is equivalent to top-1 recommendation and does not lead to bi-polarization. Indeed, the rate of convergence to bi-polarization with a large but finite $\beta$ might be slow (see the "Tendency to Polarization" plot in Figure R1 in the PDF we uploaded during rebuttal). Our Theorem 3.3 is more of an asymptotic result. Analyzing the convergence rate would be an interesting yet challenging direction for future work.

Q2: I do not fully get why the specific forms of function do not affect the analysis.

For $f$, our results and analysis only require the assumptions in Lines 114 - 121 that $f(v_i, u_j)$ has the same sign as $\langle v_i, u_j\rangle$ and is two-sided bounded $L_f \le |f(v_i, u_j)| \le 1$.

For $g$, our current analysis assumes the specific form of $g(u_j, v_i) = sign(\langle u_j, v_i\rangle)$. We believe our analysis (in particular, Lemma E.1) can be generalized to other $g$ functions satisfying similar assumptions as $f$ (but as of now this generalization remains an open problem).

Q3: results under a large range of $\beta$.

In Figure R1 of the uploaded PDF, we provide experiment results for $\beta \in [0, 10]$ and $+\infty$. We note that $\beta = 10$, although not very large, has similar effects as $\beta=+\infty$ (top-1 recommendation) in 1000 time steps, because the softmax probability of non-max creators when $\beta =10$ is very close to 0.

Q1: Please address the points I raised in Weaknesses.

W1: The assumption that all items can be recommended to users is not realistic. ... Customers can only see a certain number of items on the webpage (i.e., p=0 for items that users can't see).

First, we note that customers not seeing some items due to screen size limit does not mean that those items are recommended with probability p=0. The set of items seen by a customer is randomly drawn from a distribution over items which can have positive probability on all items, even if only $k$ items are shown.

Second, we argue that non-zero probability of recommendation is realistic, even in large-scale real-world recommendation systems used by Yahoo [Marlin et al, 2009], [Li et al, 2010], and Kuaishou [Gao et al, 2022]. As noted in our Section 4.4, practical recommendation systems insert small random traffic (uniformly random recommendations, or missing-at-random MAR [Yang et al, 2018] data) to improve recommendation diversity [Section 2.2 of Moller et al, 2018] or to explore users' interests for unseen contents [Gao et al, 2022] or for debias purposes [Liu et al, 2023]. These interventions will cause all recommendation probabilities to be non-zero (although they may be very small).

... In this sense, the measures for empirical analysis should also only consider top-k items, not all items.

Note that our measures RD and RR include the recommendation probability $p_{ij}$ (where $p_{ij} = 0$ for non-top-k creators), so RD and RR do consider top-k items. Our other two measures, CD and TP, aim to measure the diversity of the entire creator pool, independent of the recommendation scheme, so they do not consider the recommendation probability for users.

W2: 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.

We thank the reviewer for the suggestion and we agree that fairness-aware recommendation systems are important in practice. However, we want to make the best use of the limited space in the main article to introduce the notion of dual influence and outline its relationship with polarization in recommendation systems. Whether a system polarizes its users or creators is by itself an interesting trustworthiness question, separated from fairness considerations. Moreover, [1] does not study strategic content creators in recommendation systems and has fundamental differences with our work. Applying this work in case of adaptive users and creators will require a careful re-designing of the method proposed in [1]. We see the consideration of fairness-aware systems as a deeply important area, and hope that our work on dual influence will inspire future research to consider both fairness-aware recommendation and dual influence (as we fully agree with you that most realistic settings would consider both aspects). We are happy to add further discussions on fairness-aware recommendation systems and how our work relate to them in the appendix in the final version.

W3: 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.

We agree with the reviewer that this will be an interesting simulation. However, we need to model the dual dynamics (that include both user updates and creator updates) in the system, such dynamics are not currently captured in any publicly available dataset that we are aware of. Since we need to model dual dynamics, solely relying on existing MovieLens data (or any other dataset) is not enough, we need to create pseudo-real-world data which simulates these dual dynamics.

Q2: (Minor) Are both consensus and bi-polarization conceptually polarization?

Yes.

Q3: (Minor) How are the initial user/creator embeddings initialized in the Movielens experiment?

The initialization is done by fitting a two-tower model [Huang et al.] on the existing MovieLens ratings data and we use the tower tops as the initial user and creator embeddings.

[Huang et al.] Learning Deep Structured Semantic Models for Web Search using Clickthrough Data. CIKM, 2013.

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?

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 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.