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Influence of Recommender Systems on Users: A Dynamical Systems Analysis

Prabhat Lankireddy, Jayakrishnan Nair, D Manjunath

TL;DR

This work develops a formal model of how a contextual linear bandit recommender system interacts with users whose preferences evolve toward the recommendations. It employs the ODE method of stochastic approximation to derive a deterministic asymptotic dynamical system that captures the coupled evolution of algorithm state and user preferences, both in single-user and multi-user settings. The analysis reveals how exploration-exploitation tradeoffs shape long-term preferences, including filter bubbles and polarization under high exploitation, and it identifies conditions under which the RS can still learn true preferences despite model mismatch. The results highlight the potential for feedback loops in recommender environments and provide a rigorous framework for understanding and mitigating unintended consequences in both single- and multi-user contexts.

Abstract

We analyze the unintended effects that recommender systems have on the preferences of users that they are learning. We consider a contextual multi-armed bandit recommendation algorithm that learns optimal product recommendations based on user and product attributes. It is well known that the sequence of recommendations affects user preferences. However, typical learning algorithms treat the user attributes as static and disregard the impact of their recommendations on user preferences. Our interest is to analyze the effect of this mismatch between the model assumption of a static environment and the reality of an evolving environment affected by the recommendations. To perform this analysis, we introduce a model for the coupled evolution of a linear bandit recommendation system and its users, whose preferences are drawn towards the recommendations made by the algorithm. We describe a method, that is grounded in stochastic approximation theory, to come up with a dynamical system model that asymptotically approximates the mean behavior of the stochastic model. The resulting dynamical system captures the coupled evolution of the population preferences and the learning algorithm. Analyzing this dynamical system gives insight into the long-term properties of user preferences and the learning algorithm. Under certain conditions, we show that the RS is able to learn the population preferences in spite of the model mismatch. We discuss and characterize the relation between various parameters of the model and the long term preferences of users in this work. A key observation is that the exploration-exploitation tradeoff used by the recommendation algorithm significantly affects the long term preferences of users. Algorithms that exploit more can polarize user preferences, leading to the well-known filter bubble phenomenon.

Influence of Recommender Systems on Users: A Dynamical Systems Analysis

TL;DR

This work develops a formal model of how a contextual linear bandit recommender system interacts with users whose preferences evolve toward the recommendations. It employs the ODE method of stochastic approximation to derive a deterministic asymptotic dynamical system that captures the coupled evolution of algorithm state and user preferences, both in single-user and multi-user settings. The analysis reveals how exploration-exploitation tradeoffs shape long-term preferences, including filter bubbles and polarization under high exploitation, and it identifies conditions under which the RS can still learn true preferences despite model mismatch. The results highlight the potential for feedback loops in recommender environments and provide a rigorous framework for understanding and mitigating unintended consequences in both single- and multi-user contexts.

Abstract

We analyze the unintended effects that recommender systems have on the preferences of users that they are learning. We consider a contextual multi-armed bandit recommendation algorithm that learns optimal product recommendations based on user and product attributes. It is well known that the sequence of recommendations affects user preferences. However, typical learning algorithms treat the user attributes as static and disregard the impact of their recommendations on user preferences. Our interest is to analyze the effect of this mismatch between the model assumption of a static environment and the reality of an evolving environment affected by the recommendations. To perform this analysis, we introduce a model for the coupled evolution of a linear bandit recommendation system and its users, whose preferences are drawn towards the recommendations made by the algorithm. We describe a method, that is grounded in stochastic approximation theory, to come up with a dynamical system model that asymptotically approximates the mean behavior of the stochastic model. The resulting dynamical system captures the coupled evolution of the population preferences and the learning algorithm. Analyzing this dynamical system gives insight into the long-term properties of user preferences and the learning algorithm. Under certain conditions, we show that the RS is able to learn the population preferences in spite of the model mismatch. We discuss and characterize the relation between various parameters of the model and the long term preferences of users in this work. A key observation is that the exploration-exploitation tradeoff used by the recommendation algorithm significantly affects the long term preferences of users. Algorithms that exploit more can polarize user preferences, leading to the well-known filter bubble phenomenon.

Paper Structure

This paper contains 51 sections, 12 theorems, 123 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Research objectives
  4. Literature Review
  5. Interactions between RS and users
  6. Modeling the learning algorithm
  7. Algorithms that assume static preferences
  8. Algorithms that explicitly model user dynamics
  9. Algorithms that learn user preference dynamics from data
  10. Modeling the influence on the users
  11. Outline
  12. Other related work
  13. Empirical work
  14. Bias and causality
  15. Performative prediction
  16. ...and 36 more sections

Key Result

Lemma 2.1

Let $(x_n)$ satisfy the recurrence relation given by eq:ode-method-sa-recursion along with A1A5. Then, almost surely, the sequence $(x_n)$ converges to its limit set, and that limit set is a (possibly sample path-dependent) connected internally chain transitive invariant set of the ODE given by eq:o

Figures (7)

  • Figure 1: Numerical simulation of a trajectory of the asymptotic ODE
  • Figure 2: Effect of the value of $a$ on the set of equilibria.
  • Figure 3: Effect of $\rho$ on the trajectories of the asymptotic ODE.
  • Figure 4: Effect of $a$ on preferences of $N$ users.
  • Figure 5: Effect of user attributes $\{v_n\}$ and arrival probabilities $\{\lambda_n\}$ on user preferences.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Lemma 2.1: ODE method
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • proof
  • Lemma 2.5: La Salle's Invariance Principle
  • Theorem 2.6
  • proof
  • Theorem 3.1
  • proof
  • ...and 10 more