Robust Assortment Optimization from Observational Data

TL;DR

This work addresses robust data-driven assortment optimization under potential shifts in customer preferences by introducing a distributionally robust framework around a nominal choice model. It develops a dual-formulation-based approach and two practical algorithms, PR^2B-C and PR^2B-V, that combine rank-breaking estimation with a double-pessimism principle to balance robustness and data efficiency. Theoretical contributions include finite-sample suboptimality bounds, minimax lower bounds, and the concept of robust item-wise coverage, establishing near-optimal data requirements under offline data. Empirically, the methods demonstrate strong sample efficiency and robustness against distributional shifts, with clear advantages over non-robust baselines across varying cardinality constraints. The results provide a principled, scalable toolkit for reliable assortment optimization in imperfect data settings with uncertainty in demand patterns.

Abstract

Assortment optimization is a fundamental challenge in modern retail and recommendation systems, where the goal is to select a subset of products that maximizes expected revenue under complex customer choice behaviors. While recent advances in data-driven methods have leveraged historical data to learn and optimize assortments, these approaches typically rely on strong assumptions -- namely, the stability of customer preferences and the correctness of the underlying choice models. However, such assumptions frequently break in real-world scenarios due to preference shifts and model misspecification, leading to poor generalization and revenue loss. Motivated by this limitation, we propose a robust framework for data-driven assortment optimization that accounts for potential distributional shifts in customer choice behavior. Our approach models potential preference shift from a nominal choice model that generates data and seeks to maximize worst-case expected revenue. We first establish the computational tractability of robust assortment planning when the nominal model is known, then advance to the data-driven setting, where we design statistically optimal algorithms that minimize the data requirements while maintaining robustness. Our theoretical analysis provides both upper bounds and matching lower bounds on the sample complexity, offering theoretical guarantees for robust generalization. Notably, we uncover and identify the notion of ``robust item-wise coverage'' as the minimal data requirement to enable sample-efficient robust assortment learning. Our work bridges the gap between robustness and statistical efficiency in assortment learning, contributing new insights and tools for reliable assortment optimization under uncertainty.

Figures (9)

• Figure 1: Left two figures: Illustration of degeneration of expected revenue value under customer choice pattern shifts. We consider a simple MNL model with $6$ items and a randomly generated nominal choice distribution $\mathbb{P}$. The non-robust assortment (left) is solved for the MNL model under $\mathbb{P}$, while the robust assortment (right) is solved for the robust assortment framework we consider in this paper (by Example \ref{['exp: new']}). The two axis's represent two parameters $\alpha_1$, $\alpha_2\in[0,1]$ that control the shift of the choice distribution $\mathbb{P}_{\alpha_1,\alpha_2}$ from the nominal model $\mathbb{P}$ to two adversarial choice models. The KL divergence between the shifted model and the nominal model is within $0.1$. Right two figures: The suboptimality gap in terms of robust expected revenue of different algorithms under two concrete examples of the framework we consider the paper. See Section \ref{['sec: experiments']}. Our algorithms P$\mathrm{R}^2$B-(C/V) enjoy superior sample efficiency.
• Figure 2: Comparing $|S^{\star}|$, avged over $5$ instances. Here $\rho$ refers to $\rho_0$ in Example \ref{['exp: new']}.
• Figure 3: Suboptimality gap of P$\mathrm{R}^2$B-C (Algorithm \ref{['alg: jin']}) compared to the single-pessimistic counterpart \ref{['eq: vanilla 1']}. All the parameter configurations are averaged over $25$ independent runs.
• Figure 4: Suboptimality gap of P$\mathrm{R}^2$B-V (Algorithm \ref{['alg: new']}) compared to single-pessimism counterpart \ref{['eq: vanilla 2']}. All the parameter configurations are averaged over $25$ independent runs.
• Figure 5: Results for experiments on robustness of the learned assortment. The first column is the distribution for the absolute improvement $\Delta$. The second column is the distribution of the relative improvement $\widetilde{\Delta}$, and the last column is the distribution of the optimal robust parameter. The first two rows are for P$\mathrm{R}^2$B-C, and the last two rows are for P$\mathrm{R}^2$B-V.

Theorems & Definitions (76)

• Proposition 2.1: Dual representation of robust expected revenue
• proof : Proof of Proposition \ref{['prop:_dual']}
• Example 2.1: Constant robust set size jin2022distributionally
• Example 2.2: Varying robust set size
• Proposition 3.1: Optimal robust assortment set without size constraints
• proof : Proof of Proposition \ref{['prop:_optimal_set_unconstrained']}
• Proposition 3.2: Optimal robust assortment set under uniform reward
• proof : Proof of Proposition \ref{['prop:_optimal_set_uniform']}
• Proposition 3.3: Solving optimal robust assortment: general case
• proof : Proof of Proposition \ref{['prop:_optimal_set_general']}