Assortment Optimization

At the start of every session, LLM agents load a fixed context file, such as $\texttt{AGENTS.md}$. Each loaded token in the file is charged again in every later round of the session, and these files can degrade performance as they grow in size. However, in practice, human or automated curators usually grow these files by appending. We formulate context curation as a capacitated assortment problem. Instructions consume tokens under a finite attention capacity; adding an instruction never raises the compliance of the others, while retained instructions incur a per-session setup cost. We prove an upper bound on the optimal file size, regardless of the number of available candidate instructions, and that appending every instruction with positive standalone value can be arbitrarily worse in net value than selecting an optimal subset. A token budget also limits the loss when the token price is underestimated. We then examine what can be learned from past sessions and how this information can guide decisions to add or remove instructions. Feedback is inherently censored: the benefits and harms of loaded instructions are observable, whereas missing instructions generate feedback only when their absence causes harm. In this setting, we show that deleting instructions ignored by agents can inevitably remove helpful ones. We characterize how much evidence should be collected before adding an instruction. Besides, we bound regret when human reviewers can inspect only a limited number of edits per period. Empirical experiments further show that irrelevant rules drawn from real context files reduce language-model compliance.

Assortment optimization is a fundamental problem in revenue management, typically addressed using parametric choice models such as the multinomial logit (MNL) and its variants. While these models enable tractable formulations, their performance is sensitive to model misspecification and often struggles to capture complex customer behavior. In this paper, we propose a model-agnostic framework for assortment optimization based on guided discrete diffusion. We represent assortments as binary vectors and perform stochastic search via a learned reverse diffusion process, avoiding explicit combinatorial enumeration. To incorporate decision objectives, we introduce a reward-guided mechanism that biases local transitions using estimates of expected revenue. This allows the method to effectively balance exploration and exploitation during generation. Empirically, we show that the proposed approach consistently identifies high-quality assortments and remains robust under model misspecification, often recovering near-optimal solutions in high-dimensional settings. Moreover, the generative nature of diffusion enables the production of diverse high-performing assortments, offering flexibility beyond a single deterministic solution. These results highlight the potential of generative modeling as a scalable and robust paradigm for combinatorial optimization in data-driven decision-making.

Existing contextual multinomial logit (MNL) bandits model relevance-driven choice but ignore the potential benefits of within-assortment diversity, while submodular/combinatorial bandits encode diversity in rewards but lack structured choice probabilities. We bridge this gap with the $\textit{diversified multinomial logit}$ (DMNL) contextual bandit, which augments MNL choice probabilities with a generally submodular diversity function, thereby formalizing the relevance--diversity trade-off within a single model. Incorporating diversity renders exact MNL assortment optimization intractable. We propose a $\textit{white-box}$ UCB-based algorithm, $\texttt{OFU-DMNL}$, that constructs assortments item-wise by maximizing optimistic marginal gains, avoids black-box optimization oracles. We show that $\texttt{OFU-DMNL}$ achieves at least a $(1-\frac{1}{e+1})$-$\textit{approximate}$ regret bound $\tilde{O}\left(d \sqrt{T/K}\right)$, where $d$ is the context dimension, $K$ the maximum assortment size, and $T$ the horizon, and attains an improved approximation factor over standard submodular baselines. Experiments demonstrate consistent gains and, relative to exhaustive enumeration, comparable regret with substantially lower runtime. Overall, DMNL bandits provide a practical foundation for diversity-aware assortment optimization under uncertainty, and $\texttt{OFU-DMNL}$ offers a statistically and computationally efficient solution.

We propose a framework for the Markov chain (MC) choice model with panel data, including parameter estimation, personalized choice prediction, and personalized assortment optimization. In contrast to the traditional setting, which assumes that each transaction is independently drawn from a random utility model, our framework accounts for dependencies among transactions for the same customer in historical data, captured by partial-ordering preference information. To the best of our knowledge, our framework initiates the study of choice modeling with panel data under MC. As our primary result, we propose novel expectation-maximization (EM) algorithms for MC parameter estimation by incorporating partial-ordering-based customer preference information. On synthetic datasets and the sushi dataset, our EM algorithms outperform the traditional EM algorithm of Simsek and Topaloglu (Operations Research, 66, 2018) and multinomial-logit-based partial-order benchmarks adapted from Jagabathula and Vulcano (Management Science, 64, 2018). As our secondary contribution, we present hardness and computational results for conditional choice prediction and assortment optimization problems. These results complement our estimation framework and clarify the computational landscape of conditional choice and assortment optimization, which may be of independent interest.

We study a dynamic assortment problem on a two-sided service platform with incomplete information and heterogeneous customers in a discrete-time setting. In each period, a customer arrives seeking service, and the platform chooses an assortment of sellers to display. The customer then proposes a transaction to at most one seller in the assortment according to a multinomial logit choice model. After a fixed number of periods, sellers review the proposals they have received and each chooses at most one customer according to another multinomial logit choice model, after which the cycle repeats. A key challenge is that the platform does not know the choice-model parameters of either customers or sellers in advance. To our knowledge, this is the first study of a dynamic assortment problem in which both sides' choice parameters are unknown. We develop a data-driven algorithm that learns these parameters while optimizing the platform's objective over time. We evaluate performance using regret, which measures revenue loss relative to a clairvoyant benchmark that knows all parameters and customer arrivals in advance. We show that the algorithm's worst-case regret grows polylogarithmically over time, and we derive a matching lower bound, establishing its rate optimality.

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 $\tilde{O}\big(\frac{d \log N}{Δ^2}\big)$, where $d$ is the feature dimension, $N$ is the number of arms, and $Δ$ is the minimum revenue gap.

We study the dynamic joint assortment selection and positioning problem, where the attraction of each product depends on both its intrinsic appeal and its display position under a Multinomial Logit (MNL) choice framework. Our study ranges from the multiplicative position effects model, in which each product's attraction is scaled by a position-specific factor, to a general position effects model assigning independent attraction parameters to every product--position pair to capture heterogeneous synergies. For both models, we design round-based learning algorithms that update decisions after every single feedback, and establish the first regret-optimal characterization. Besides, our round-based algorithms provide the prompt operations needed by modern platforms. For the multiplicative model, we develop a cross-position pairwise maximum likelihood estimator with a clipping mechanism, and prove that our algorithm P2MLE-UCB attains a regret of $\tilde{O}(\sqrt{NT})$, matching the lower bound and closing the $\sqrt{K}$ gap left by prior epoch-based analyses. For the general model, we establish a minimax lower bound and propose GP2-UCB with a matching upper bound. Moreover, we design an efficient subroutine for the per-round joint assortment and positioning optimization based on Dinkelbach's method and maximum-weight bipartite matching. Numerical experiments on synthetic data and the Expedia dataset show that our algorithms consistently outperform state-of-the-art benchmarks.

We study online learning for new products on a platform that makes capacity-constrained assortment decisions on which products to offer. For a newly listed product, its quality is initially unknown, and quality information propagates through social learning: when a customer purchases a new product and leaves a review, its quality is revealed to both the platform and future customers. Since reviews require purchases, the platform must feature new products in the assortment ("explore") to generate reviews to learn about new products. Such exploration is costly because customer demand for new products is lower than for incumbent products. We characterize the optimal assortments for exploration to minimize regret, addressing two questions. (1) Should the platform offer a new product alone or alongside incumbent products? The former maximizes the purchase probability of the new product but yields lower short-term revenue. Despite the lower purchase probability, we show it is always optimal to pair the new product with the top incumbent products. (2) With multiple new products, should the platform explore them simultaneously or one at a time? We show that the optimal number of new products to explore simultaneously has a simple threshold structure: it increases with the "potential" of the new products and, surprisingly, does not depend on their individual purchase probabilities. We also show that two canonical bandit algorithms, UCB and Thompson Sampling, both fail in this setting for opposite reasons: UCB over-explores while Thompson Sampling under-explores. Our results provide structural insights on how platforms should learn about new products through assortment decisions.

In the rapidly evolving landscape of retail, assortment planning plays a crucial role in determining the success of a business. With the rise of sponsored products and their increasing prominence in online marketplaces, retailers face new challenges in effectively managing their product assortment in the presence of sponsored products. Remarkably, previous research in assortment planning largely overlooks the existence of sponsored products and their potential impact on overall recommendation effectiveness. Instead, they commonly make the simplifying assumption that all products are either organic or non-sponsored. This research gap underscores the necessity for a more thorough investigation of the assortment planning challenge when sponsored products are in play. We formulate the assortment planning problem in the presence of sponsored products as a combinatorial optimization task. The ultimate objective is to compute an assortment plan that optimizes expected revenue while considering the specific requirements of placing sponsored products strategically.

Assortment optimization has received active explorations in the past few decades due to its practical importance. Despite the extensive literature dealing with optimization algorithms and latent score estimation, uncertainty quantification for the optimal assortment still needs to be explored and is of great practical significance. Instead of estimating and recovering the complete optimal offer set, decision-makers may only be interested in testing whether a given property holds true for the optimal assortment, such as whether they should include several products of interest in the optimal set, or how many categories of products the optimal set should include. This paper proposes a novel inferential framework for testing such properties. We consider the widely adopted multinomial logit (MNL) model, where we assume that each customer will purchase an item within the offered products with a probability proportional to the underlying preference score associated with the product. We reduce inferring a general optimal assortment property to quantifying the uncertainty associated with the sign change point detection of the marginal revenue gaps. We show the asymptotic normality of the marginal revenue gap estimator, and construct a maximum statistic via the gap estimators to detect the sign change point. By approximating the distribution of the maximum statistic with multiplier bootstrap techniques, we propose a valid testing procedure. We also conduct numerical experiments to assess the performance of our method.