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PASTA: A Unified Framework for Offline Assortment Learning

Juncheng Dong, Weibin Mo, Zhengling Qi, Cong Shi, Ethan X. Fang, Vahid Tarokh

TL;DR

PASTA addresses offline assortment optimization with an unknown choice model and data with incomplete coverage. It introduces a pessimistic framework that constructs a likelihood-ratio–based uncertainty set and solves a max–min revenue problem to robustly identify an optimal assortment. The authors prove finite-sample regret bounds for MNL and NL families and establish a minimax lower bound under MNL, confirming the method's optimality in sample and model complexity. Empirical results on simulated data demonstrate substantial improvements over standard as-if optimization, especially when the optimal assortment is sparsely represented in the offline data.

Abstract

We study a broad class of assortment optimization problems in an offline and data-driven setting. In such problems, a firm lacks prior knowledge of the underlying choice model, and aims to determine an optimal assortment based on historical customer choice data. The combinatorial nature of assortment optimization often results in insufficient data coverage, posing a significant challenge in designing provably effective solutions. To address this, we introduce a novel Pessimistic Assortment Optimization (PASTA) framework that leverages the principle of pessimism to achieve optimal expected revenue under general choice models. Notably, PASTA requires only that the offline data distribution contains an optimal assortment, rather than providing the full coverage of all feasible assortments. Theoretically, we establish the first finite-sample regret bounds for offline assortment optimization across several widely used choice models, including the multinomial logit and nested logit models. Additionally, we derive a minimax regret lower bound, proving that PASTA is minimax optimal in terms of sample and model complexity. Numerical experiments further demonstrate that our method outperforms existing baseline approaches.

PASTA: A Unified Framework for Offline Assortment Learning

TL;DR

PASTA addresses offline assortment optimization with an unknown choice model and data with incomplete coverage. It introduces a pessimistic framework that constructs a likelihood-ratio–based uncertainty set and solves a max–min revenue problem to robustly identify an optimal assortment. The authors prove finite-sample regret bounds for MNL and NL families and establish a minimax lower bound under MNL, confirming the method's optimality in sample and model complexity. Empirical results on simulated data demonstrate substantial improvements over standard as-if optimization, especially when the optimal assortment is sparsely represented in the offline data.

Abstract

We study a broad class of assortment optimization problems in an offline and data-driven setting. In such problems, a firm lacks prior knowledge of the underlying choice model, and aims to determine an optimal assortment based on historical customer choice data. The combinatorial nature of assortment optimization often results in insufficient data coverage, posing a significant challenge in designing provably effective solutions. To address this, we introduce a novel Pessimistic Assortment Optimization (PASTA) framework that leverages the principle of pessimism to achieve optimal expected revenue under general choice models. Notably, PASTA requires only that the offline data distribution contains an optimal assortment, rather than providing the full coverage of all feasible assortments. Theoretically, we establish the first finite-sample regret bounds for offline assortment optimization across several widely used choice models, including the multinomial logit and nested logit models. Additionally, we derive a minimax regret lower bound, proving that PASTA is minimax optimal in terms of sample and model complexity. Numerical experiments further demonstrate that our method outperforms existing baseline approaches.

Paper Structure

This paper contains 41 sections, 25 theorems, 94 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Offline Assortment Optimization under Insufficient Data Coverage
  4. Offline Assortment Optimization
  5. An Example of Insufficient Data Coverage
  6. Pessimistic Assortment Learning
  7. PASTA Framework
  8. General Regret Guarantees
  9. Notations
  10. Validity of Uncertainty Set
  11. Applications
  12. Multinomial Logit (MNL) Model
  13. Latent Class Logit (LCL) Model
  14. Nested Logit (NL) Model
  15. Minimax Optimality
  16. ...and 26 more sections

Key Result

Lemma 4.1

If $\pi_{S}(s^\star) > 0$ for an optimal assortment $s^\star$, and the uncertainty set eqn:uncertainty-set includes the true choice model, i.e., $p_{0} \in \Omega_{n}$, then where $\mathcal{C} = r_{s^\star}\sqrt{1/\pi_S(s^\star)}$ is a constant independent of the sample size $n$, and $r_{s^{\star}} = \max_{j \in s^\star}r(s^\star,j)$ is the maximal revenue among the items in $s^\star$.

Figures (2)

  • Figure 1: Performance comparison between PASTA and the baseline method on the MNL model. The left panel shows average regret (lower is better), while the right panel presents assortment accuracy (higher is better).Our proposed method PASTA consistently outperforms the baseline method in both metrics. The performance gain remains consistent across varying probabilities of observing the optimal assortment (middle row) and varying model complexity, measured by the dimension of the model parameter (bottom row).
  • Figure 2: PASTA also consistently outperforms the baseline method when the underlying choice model is an NL model. The performance gain in terms of the regret difference increases with increasing model complexity (bottom row).

Theorems & Definitions (52)

  • Lemma 4.1
  • proof
  • Theorem 4.1
  • Definition 4.1: Bracketing Number
  • Lemma 4.2
  • Theorem 4.2
  • Corollary 4.1
  • Theorem 5.1
  • Theorem 5.2
  • Theorem 5.3
  • ...and 42 more