Optimal Design for Multinomial Logit Model with Applications to Best Assortment Identification

Joongkyu Lee, Min-Hwan Oh
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:64899-64932, 2026.

Abstract

We study optimal experimental design for multinomial logit (MNL) bandits, where an agent repeatedly selects a subset of $K$ items from a ground set of size $N$ and observes single-choice feedback. Unlike linear or generalized linear bandits, MNL bandits have a combinatorial action space, which makes classical optimal design approaches and naive optimization over all subsets computationally intractable. We propose a computationally efficient optimal design framework for MNL models that achieves both statistical efficiency and scalability through two complementary approaches: (i) an exact or certified-approximate reformulation of the design oracle as a $0$-$1$ mixed-integer linear program (MILP) with solver-certified early stopping, and (ii) a fully polynomial-time lifted design that replaces the nonlinear objective with a tractable surrogate. Using the Kiefer-Wolfowitz equivalence theorem, we establish near G-optimality guarantees and characterize the induced statistical-computational trade-offs. As an application, we develop a best assortment identification algorithm for MNL bandits with linear utilities and non-uniform revenues, and prove an instance-dependent sample complexity of $\tilde{\mathcal{O}}\big(\frac{d \log N}{\Delta^2}\big)$, where $d$ is the feature dimension, $N$ is the number of arms, and $\Delta$ is the minimum revenue gap.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-lee26ba, title = {Optimal Design for Multinomial Logit Model with Applications to Best Assortment Identification}, author = {Lee, Joongkyu and Oh, Min-Hwan}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {64899--64932}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/lee26ba/lee26ba.pdf}, url = {https://proceedings.mlr.press/v306/lee26ba.html}, abstract = {We study optimal experimental design for multinomial logit (MNL) bandits, where an agent repeatedly selects a subset of $K$ items from a ground set of size $N$ and observes single-choice feedback. Unlike linear or generalized linear bandits, MNL bandits have a combinatorial action space, which makes classical optimal design approaches and naive optimization over all subsets computationally intractable. We propose a computationally efficient optimal design framework for MNL models that achieves both statistical efficiency and scalability through two complementary approaches: (i) an exact or certified-approximate reformulation of the design oracle as a $0$-$1$ mixed-integer linear program (MILP) with solver-certified early stopping, and (ii) a fully polynomial-time lifted design that replaces the nonlinear objective with a tractable surrogate. Using the Kiefer-Wolfowitz equivalence theorem, we establish near G-optimality guarantees and characterize the induced statistical-computational trade-offs. As an application, we develop a best assortment identification algorithm for MNL bandits with linear utilities and non-uniform revenues, and prove an instance-dependent sample complexity of $\tilde{\mathcal{O}}\big(\frac{d \log N}{\Delta^2}\big)$, where $d$ is the feature dimension, $N$ is the number of arms, and $\Delta$ is the minimum revenue gap.} }
Endnote
%0 Conference Paper %T Optimal Design for Multinomial Logit Model with Applications to Best Assortment Identification %A Joongkyu Lee %A Min-Hwan Oh %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-lee26ba %I PMLR %P 64899--64932 %U https://proceedings.mlr.press/v306/lee26ba.html %V 306 %X We study optimal experimental design for multinomial logit (MNL) bandits, where an agent repeatedly selects a subset of $K$ items from a ground set of size $N$ and observes single-choice feedback. Unlike linear or generalized linear bandits, MNL bandits have a combinatorial action space, which makes classical optimal design approaches and naive optimization over all subsets computationally intractable. We propose a computationally efficient optimal design framework for MNL models that achieves both statistical efficiency and scalability through two complementary approaches: (i) an exact or certified-approximate reformulation of the design oracle as a $0$-$1$ mixed-integer linear program (MILP) with solver-certified early stopping, and (ii) a fully polynomial-time lifted design that replaces the nonlinear objective with a tractable surrogate. Using the Kiefer-Wolfowitz equivalence theorem, we establish near G-optimality guarantees and characterize the induced statistical-computational trade-offs. As an application, we develop a best assortment identification algorithm for MNL bandits with linear utilities and non-uniform revenues, and prove an instance-dependent sample complexity of $\tilde{\mathcal{O}}\big(\frac{d \log N}{\Delta^2}\big)$, where $d$ is the feature dimension, $N$ is the number of arms, and $\Delta$ is the minimum revenue gap.
APA
Lee, J. & Oh, M.. (2026). Optimal Design for Multinomial Logit Model with Applications to Best Assortment Identification. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:64899-64932 Available from https://proceedings.mlr.press/v306/lee26ba.html.

Related Material