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Stop Relying on No-Choice and Do not Repeat the Moves: Optimal, Efficient and Practical Algorithms for Assortment Optimization

Aadirupa Saha, Pierre Gaillard

TL;DR

This work tackles Active Optimal Assortment (AOA) under Plackett-Luce preferences, addressing the lack of practical, regret-optimal solutions that do not rely on a strongest No-Choice (NC) item or repetitive posting of the same assortment. The authors introduce Rank-Breaking (RB) to obtain tight concentration guarantees for PL parameter estimation and develop AOA-RB_PL, a practical algorithm that optimistically selects assortments using UCB-based PL parameters without repeating actions. They prove regret bounds for Top-$m$ and Weighted Top-$m$ objectives, and further improve dependence on $\theta_{ ext{max}}$ via Adaptive Pivot selection, achieving nearly optimal rates with reduced sensitivity to the NC parameter. Empirical results on PL datasets corroborate theoretical findings and demonstrate superior performance over baselines like MNL-UCB, particularly in regimes where NC is weak. These contributions yield scalable, realistic assortment optimization methods for applications in ads, online retail, and recommender systems.

Abstract

We address the problem of active online assortment optimization problem with preference feedback, which is a framework for modeling user choices and subsetwise utility maximization. The framework is useful in various real-world applications including ad placement, online retail, recommender systems, fine-tuning language models, amongst many. The problem, although has been studied in the past, lacks an intuitive and practical solution approach with simultaneously efficient algorithm and optimal regret guarantee. E.g., popularly used assortment selection algorithms often require the presence of a `strong reference' which is always included in the choice sets, further they are also designed to offer the same assortments repeatedly until the reference item gets selected -- all such requirements are quite unrealistic for practical applications. In this paper, we designed efficient algorithms for the problem of regret minimization in assortment selection with \emph{Plackett Luce} (PL) based user choices. We designed a novel concentration guarantee for estimating the score parameters of the PL model using `\emph{Pairwise Rank-Breaking}', which builds the foundation of our proposed algorithms. Moreover, our methods are practical, provably optimal, and devoid of the aforementioned limitations of the existing methods. Empirical evaluations corroborate our findings and outperform the existing baselines.

Stop Relying on No-Choice and Do not Repeat the Moves: Optimal, Efficient and Practical Algorithms for Assortment Optimization

TL;DR

This work tackles Active Optimal Assortment (AOA) under Plackett-Luce preferences, addressing the lack of practical, regret-optimal solutions that do not rely on a strongest No-Choice (NC) item or repetitive posting of the same assortment. The authors introduce Rank-Breaking (RB) to obtain tight concentration guarantees for PL parameter estimation and develop AOA-RB_PL, a practical algorithm that optimistically selects assortments using UCB-based PL parameters without repeating actions. They prove regret bounds for Top- and Weighted Top- objectives, and further improve dependence on via Adaptive Pivot selection, achieving nearly optimal rates with reduced sensitivity to the NC parameter. Empirical results on PL datasets corroborate theoretical findings and demonstrate superior performance over baselines like MNL-UCB, particularly in regimes where NC is weak. These contributions yield scalable, realistic assortment optimization methods for applications in ads, online retail, and recommender systems.

Abstract

We address the problem of active online assortment optimization problem with preference feedback, which is a framework for modeling user choices and subsetwise utility maximization. The framework is useful in various real-world applications including ad placement, online retail, recommender systems, fine-tuning language models, amongst many. The problem, although has been studied in the past, lacks an intuitive and practical solution approach with simultaneously efficient algorithm and optimal regret guarantee. E.g., popularly used assortment selection algorithms often require the presence of a `strong reference' which is always included in the choice sets, further they are also designed to offer the same assortments repeatedly until the reference item gets selected -- all such requirements are quite unrealistic for practical applications. In this paper, we designed efficient algorithms for the problem of regret minimization in assortment selection with \emph{Plackett Luce} (PL) based user choices. We designed a novel concentration guarantee for estimating the score parameters of the PL model using `\emph{Pairwise Rank-Breaking}', which builds the foundation of our proposed algorithms. Moreover, our methods are practical, provably optimal, and devoid of the aforementioned limitations of the existing methods. Empirical evaluations corroborate our findings and outperform the existing baselines.
Paper Structure (24 sections, 11 theorems, 76 equations, 3 figures, 1 algorithm)

This paper contains 24 sections, 11 theorems, 76 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Formulation
  4. Feedback model
  5. Performance Objectives:
  6. AOA with PL: A Practical and Efficient Algorithm for $\textit{Reg}^{\texttt{top}}$ and $\textit{Reg}^{\texttt{wtd}}$ Objective
  7. Algorithm Design
  8. Analysis: Concentration Lemmas
  9. Analysis: Top-$m$ Objective:
  10. Analysis: Wtd-Top-$m$ Objective
  11. Generalization: Improved dependance on $\theta_{\max}$ with Adaptive Pivot Selection
  12. Experiments
  13. Conclusion
  14. Preliminaries: Some Useful Concepts for PL choice models
  15. Plackett-Luce (PL): A Discrete Choice Model
  16. ...and 9 more sections

Key Result

Lemma 1

Let $(i,j) \in [K]\times[K]$. Let $T\geq 1$ and $\delta >0$. Then, with probability at least $1- 3 Te^{-x}$, simultaneously for all $t \in [T]$.

Figures (3)

  • Figure 1: Averaged Regret for $m = 5$, $\theta_0 = 0.01$
  • Figure 2: Comparative performances of the algorithms for varying strength of NC $(\theta_0)$ for $m = 5$
  • Figure 3: Tradofff: Averaged Regret vs length of the $k$ rank-ordered feedback for $k = 1,2,4,8$, $m = 30$

Theorems & Definitions (24)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 4: : Regret Analysis for Objective
  • proof : Proof Sketch of \ref{['thm:topm_wf']}
  • Lemma 5: Lemma A.3 of assort-mnl
  • Theorem 6: : Regret Analysis for Objective
  • Remark 2
  • proof : Proof sketch of Thm. \ref{['thm:utilm_wf']}
  • ...and 14 more