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Low-Rank Online Dynamic Assortment with Dual Contextual Information

Seong Jin Lee, Will Wei Sun, Yufeng Liu

TL;DR

This work addresses dynamic assortment optimization with dual user/item contexts by introducing a low-rank bilinear utility model $v_{it}=p_i^\top \Phi^* q_t$ and proposing the LOAD framework. The two-stage ELSA-UCB algorithm first estimates a low-dimensional subspace via rank-constrained MLE (Burer-Monteiro) and truncation-vectorization, then applies a UCB policy in the reduced space to select assortments. Theoretical guarantees include a non-asymptotic regret bound $\mathcal{R}_T = \tilde{O}(r(d_1+d_2)\sqrt{T})$, significantly improving over full-vectorization methods when the rank is small. Empirical results on synthetic data and the Expedia hotel dataset demonstrate rank-selection consistency via GIC and practical gains from learning in a reduced subspace, highlighting the method's scalability and robustness in dual-context online recommendation settings.

Abstract

As e-commerce expands, delivering real-time personalized recommendations from vast catalogs poses a critical challenge for retail platforms. Maximizing revenue requires careful consideration of both individual customer characteristics and available item features to continuously optimize assortments over time. In this paper, we consider the dynamic assortment problem with dual contexts -- user and item features. In high-dimensional scenarios, the quadratic growth of dimensions complicates computation and estimation. To tackle this challenge, we introduce a new low-rank dynamic assortment model to transform this problem into a manageable scale. Then we propose an efficient algorithm that estimates the intrinsic subspaces and utilizes the upper confidence bound approach to address the exploration-exploitation trade-off in online decision making. Theoretically, we establish a regret bound of $\tilde{O}((d_1+d_2)r\sqrt{T})$, where $d_1, d_2$ represent the dimensions of the user and item features respectively, $r$ is the rank of the parameter matrix, and $T$ denotes the time horizon. This bound represents a substantial improvement over prior literature, achieved by leveraging the low-rank structure. Extensive simulations and an application to the Expedia hotel recommendation dataset further demonstrate the advantages of our proposed method.

Low-Rank Online Dynamic Assortment with Dual Contextual Information

TL;DR

This work addresses dynamic assortment optimization with dual user/item contexts by introducing a low-rank bilinear utility model and proposing the LOAD framework. The two-stage ELSA-UCB algorithm first estimates a low-dimensional subspace via rank-constrained MLE (Burer-Monteiro) and truncation-vectorization, then applies a UCB policy in the reduced space to select assortments. Theoretical guarantees include a non-asymptotic regret bound , significantly improving over full-vectorization methods when the rank is small. Empirical results on synthetic data and the Expedia hotel dataset demonstrate rank-selection consistency via GIC and practical gains from learning in a reduced subspace, highlighting the method's scalability and robustness in dual-context online recommendation settings.

Abstract

As e-commerce expands, delivering real-time personalized recommendations from vast catalogs poses a critical challenge for retail platforms. Maximizing revenue requires careful consideration of both individual customer characteristics and available item features to continuously optimize assortments over time. In this paper, we consider the dynamic assortment problem with dual contexts -- user and item features. In high-dimensional scenarios, the quadratic growth of dimensions complicates computation and estimation. To tackle this challenge, we introduce a new low-rank dynamic assortment model to transform this problem into a manageable scale. Then we propose an efficient algorithm that estimates the intrinsic subspaces and utilizes the upper confidence bound approach to address the exploration-exploitation trade-off in online decision making. Theoretically, we establish a regret bound of , where represent the dimensions of the user and item features respectively, is the rank of the parameter matrix, and denotes the time horizon. This bound represents a substantial improvement over prior literature, achieved by leveraging the low-rank structure. Extensive simulations and an application to the Expedia hotel recommendation dataset further demonstrate the advantages of our proposed method.
Paper Structure (43 sections, 23 theorems, 134 equations, 11 figures, 2 tables, 6 algorithms)

This paper contains 43 sections, 23 theorems, 134 equations, 11 figures, 2 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Notations
  4. Problem Formulation
  5. Algorithm
  6. Estimation of Subspace
  7. UCB-based Stage
  8. Theory
  9. Assumptions
  10. Main Results
  11. Numerical Studies
  12. Simulation
  13. Real Data Application
  14. Data Pre-processing
  15. Analysis of the Expedia Dataset
  16. ...and 28 more sections

Key Result

Proposition 1

Suppose Assumptions assm:user_feature, assm:item_feature and assm:kappa hold. Further assume we have an initial estimate $\Phi_0\in \mathop{\mathrm{\mathbb{R}}}\limits^{d_2\times d_1}$ such that $\|\Phi_0 - \Phi^*\|_F \le c_2 \sqrt{\sigma_r}$ for some constant $c_2$, where $\sigma_r$ is $r$-th large for some constants $C_1, C_2$ and any $\delta \in (0,1)$, $B>0$, where $\Sigma = \Sigma_q \otimes \

Figures (11)

  • Figure 1: Illustration of the contextual dynamic assortment problem.
  • Figure 2: Stacked utility formulation of item and user features.
  • Figure 3: Vectorized utility formulation of interactions between user and item features.
  • Figure 4: Illustration of low-rank user-item interaction effect.
  • Figure 5: Accuracy of estimated rank using GIC with increasing $T$.
  • ...and 6 more figures

Theorems & Definitions (31)

  • Remark 1: Cold-start and Long-tail Issues
  • Proposition 1: Accuracy of Low-Rank Estimation
  • Remark 2: Initial Estimate $\Phi_0$
  • Lemma 1: Confidence bound on reduced space
  • Theorem 1: Regret Bound of ELSA-UCB
  • Corollary 1: High Probability Bound of Regret
  • Proposition 2: Consistency of Rank Estimation
  • Proposition 3: Bounds for Subspace Estimation
  • Lemma 2: Bound on Optimistic Utility
  • Lemma 3: Lemma 5 of oh2021multinomial
  • ...and 21 more