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Optimal Sequential Recommendations: Exploiting User and Item Structure

Mina Karzand, Guy Bresler

TL;DR

The paper tackles online sequential recommendations where each user receives one item per time step and provides binary feedback; users and items are each partitioned into latent types, with a random preference matrix Xi capturing interactions. It demonstrates that exploiting both item and user structure is essential for near-optimal regret, delivering a novel algorithm that achieves regret bounds matching information-theoretic lower bounds up to log factors across five operating regimes. The work provides new lower bounds that jointly account for item and user structure, contrasts with prior work that considered only one dimension or offline settings, and offers regime-dependent insights for practical design, including cold-start and exploration costs. Overall, the results yield principled guidelines for balancing exploration and exploitation in collaborative filtering under dual-type structure, with implications for the design of near-optimal, horizon-aware recommender systems. The analysis integrates clustering-based information gain, partial learning of a latent preference matrix, and a carefully constructed information-theoretic framework to reveal when and how to leverage item versus user structure in online recommendations.

Abstract

We consider an online model for recommendation systems, with each user being recommended an item at each time-step and providing 'like' or 'dislike' feedback. A latent variable model specifies the user preferences: both users and items are clustered into types. The model captures structure in both the item and user spaces, as used by item-item and user-user collaborative filtering algorithms. We study the situation in which the type preference matrix has i.i.d. entries. Our main contribution is an algorithm that simultaneously uses both item and user structures, proved to be near-optimal via corresponding information-theoretic lower bounds. In particular, our analysis highlights the sub-optimality of using only one of item or user structure (as is done in most collaborative filtering algorithms).

Optimal Sequential Recommendations: Exploiting User and Item Structure

TL;DR

The paper tackles online sequential recommendations where each user receives one item per time step and provides binary feedback; users and items are each partitioned into latent types, with a random preference matrix Xi capturing interactions. It demonstrates that exploiting both item and user structure is essential for near-optimal regret, delivering a novel algorithm that achieves regret bounds matching information-theoretic lower bounds up to log factors across five operating regimes. The work provides new lower bounds that jointly account for item and user structure, contrasts with prior work that considered only one dimension or offline settings, and offers regime-dependent insights for practical design, including cold-start and exploration costs. Overall, the results yield principled guidelines for balancing exploration and exploitation in collaborative filtering under dual-type structure, with implications for the design of near-optimal, horizon-aware recommender systems. The analysis integrates clustering-based information gain, partial learning of a latent preference matrix, and a carefully constructed information-theoretic framework to reveal when and how to leverage item versus user structure in online recommendations.

Abstract

We consider an online model for recommendation systems, with each user being recommended an item at each time-step and providing 'like' or 'dislike' feedback. A latent variable model specifies the user preferences: both users and items are clustered into types. The model captures structure in both the item and user spaces, as used by item-item and user-user collaborative filtering algorithms. We study the situation in which the type preference matrix has i.i.d. entries. Our main contribution is an algorithm that simultaneously uses both item and user structures, proved to be near-optimal via corresponding information-theoretic lower bounds. In particular, our analysis highlights the sub-optimality of using only one of item or user structure (as is done in most collaborative filtering algorithms).
Paper Structure (80 sections, 48 theorems, 290 equations, 6 figures, 6 algorithms)

This paper contains 80 sections, 48 theorems, 290 equations, 6 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Our Recommendation System Model
  3. Our Contributions
  4. Comparison to Related Work
  5. Notation
  6. Model
  7. Problem Setup and Performance Metrics
  8. User Preferences
  9. Main result
  10. Conceptual Preliminaries
  11. Comparison of items and users
  12. Clustering of Items and Users
  13. Partial Learning of Preference Matrix
  14. Theorem Statement
  15. Discussion of Main Result
  16. ...and 65 more sections

Key Result

Theorem 3.1

Under the modeling assumptions in Figure f:assumptions, there are universal constants $c$ and $C$ such that the following holds. For any time horizon $T$, algorithm RecommendationSystem achieves $N\mathtt{regret}(T) \leq C\,N\mathtt{R}(T) \log^{3/2} (N\mathtt{R}(T))$, where $\mathtt{R}(T)$ is define The function $\mathtt{R}(T)$ is defined in a piece-wise manner in the table below, with the support

Figures (6)

  • Figure 1: Notation for the System Model
  • Figure 2: Model Assumptions
  • Figure 3: Two possible regret curves $\mathtt{R}(T)$ and their various operation regimes. The piece-wise curves are scaled by the appropriate constant factor so that regret is continuous.
  • Figure 4: Schematics of functions $f_1(\gamma), f_2(\gamma)$ and $f_3(\gamma)$.
  • Figure 5: Various regimes of lower bound for regret corresponding to three different cases (a) $8\texttt{q}_I\log\texttt{q}_I\leq N\mathsf{s}_{U}$, (b) $\sqrt{\texttt{q}_I}\leq N\mathsf{s}_{U}< 8\texttt{q}_I\log\texttt{q}_I$; and (c) $N\mathsf{s}_{U}<\sqrt{\texttt{q}_I}$.
  • ...and 1 more figures

Theorems & Definitions (82)

  • Theorem 3.1: Main Result
  • Corollary 3.2: Cold-Start
  • Remark 4.1
  • Proposition 4.2: Bad recommendations are uncertain
  • Corollary 4.3
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • proof
  • Lemma 4.6
  • ...and 72 more