Will Ma

Ali Aouad, Will Ma

The design of online algorithms for matching markets and revenue management settings is usually bound by the assumption that the demand process is formed by a fixed-length sequence of queries with unknown types, each drawn independently. This notion of serial independence implies that the demand of each type, i.e., the number of queries of a given type, has low variance and is approximately Poisson-distributed. This paper proposes a nonparametric framework for modeling arrival sequences in online stochastic matching that departs from the serial independent assumption. We propose two models, Indep and Correl, that capture different forms of serial correlations by combining a nonparametric distribution for the demand with standard assumptions on the arrival patterns -- adversarial or random order. The Indep model can capture arbitrary serial correlations within each customer type but assumes cross-sectional independence across types, whereas the Correl model captures common shocks across customer types. We demonstrate that fluid relaxations, which rely solely on demand expectations, have arbitrarily bad performance guarantees. In contrast, we develop new algorithms that achieve optimal (constant-factor) performance guarantees in each model. Our mathematical analysis includes tighter linear programming (LP) relaxations that leverage distribution knowledge, and a new lossless randomized LP rounding scheme for Indep. We test our new LP relaxations and rounding scheme in simulations on real and synthetic data, and find that they consistently outperform well-established matching algorithms, especially on real data sequences that exhibit greater demand variance.

Brian Brubach, Nathaniel Grammel, Will Ma et al.

Matching is one of the most fundamental and broadly applicable problems across many domains. In these diverse real-world applications, there is often a degree of uncertainty in the input which has led to the study of stochastic matching models. Here, each edge in the graph has a known, independent probability of existing derived from some prediction. Algorithms must probe edges to determine existence and match them irrevocably if they exist. Further, each vertex may have a patience constraint denoting how many of its neighboring edges can be probed. We present new ordered contention resolution schemes yielding improved approximation guarantees for some of the foundational problems studied in this area. For stochastic matching with patience constraints in general graphs, we provide a 0.382-approximate algorithm assuming each vertex has patience at least $2$. Under this assumption, we improve upon the previous best 0.31-approximation of Baveja et al. (2018). When the vertices do not have patience constraints, we describe a 0.432-approximate random order probing algorithm with several corollaries such as an improved guarantee for the Prophet Secretary problem under Edge Arrivals. Finally, for the special case of bipartite graphs with unit patience constraints on one of the partitions, we show a 0.632-approximate algorithm that improves on the recent $1/3$-guarantee of Hikima et al. (2021).

Jiashuo Jiang, Will Ma, Jiawei Zhang

We study the classical Network Revenue Management (NRM) problem with accept/reject decisions and $T$ IID arrivals. We consider a distributional form where each arrival must fall under a finite number of possible categories, each with a deterministic resource consumption vector, but a random value distributed continuously over an interval. We develop an online algorithm that achieves $O(\log^2 T)$ regret under this model, with the only (necessary) assumption being that the probability densities are bounded away from 0. We derive a second result that achieves $O(\log T)$ regret under an additional assumption of second-order growth. To our knowledge, these are the first results achieving logarithmic-level regret in an NRM model with continuous values that do not require any kind of "non-degeneracy" assumptions. Our results are achieved via new techniques including a new method of bounding myopic regret, a "semi-fluid" relaxation of the offline allocation, and an improved bound on the "dual convergence".

Omar Besbes, Will Ma, Omar Mouchtaki

How should one leverage historical data when past observations are not perfectly indicative of the future, e.g., due to the presence of unobserved confounders which one cannot "correct" for? Motivated by this question, we study a data-driven decision-making framework in which historical samples are generated from unknown and different distributions assumed to lie in a heterogeneity ball with known radius and centered around the (also) unknown future (out-of-sample) distribution on which the performance of a decision will be evaluated. This work aims at analyzing the performance of central data-driven policies but also near-optimal ones in these heterogeneous environments and understanding key drivers of performance. We establish a first result which allows to upper bound the asymptotic worst-case regret of a broad class of policies. Leveraging this result, for any integral probability metric, we provide a general analysis of the performance achieved by Sample Average Approximation (SAA) as a function of the radius of the heterogeneity ball. This analysis is centered around the approximation parameter, a notion of complexity we introduce to capture how the interplay between the heterogeneity and the problem structure impacts the performance of SAA. In turn, we illustrate through several widely-studied problems -- e.g., newsvendor, pricing -- how this methodology can be applied and find that the performance of SAA varies considerably depending on the combinations of problem classes and heterogeneity. The failure of SAA for certain instances motivates the design of alternative policies to achieve rate-optimality. We derive problem-dependent policies achieving strong guarantees for the illustrative problems described above and provide initial results towards a principled approach for the design and analysis of general rate-optimal algorithms.

Omar Besbes, Will Ma, Omar Mouchtaki

In this work, we study how the relevance/quality and quantity of past data influence performance by analyzing a contextual Newsvendor problem, in which a decision-maker trades off between underage and overage costs under uncertain demand. We consider a setting in which past demands observed under ``close by'' contexts come from close by distributions and analyze the performance of data-driven algorithms through a notion of context-dependent worst-case expected regret. We analyze the broad class of Weighted Empirical Risk Minimization (WERM) policies which weigh past data according to their similarity in the contextual space. This class includes classical policies such as ERM, k-Nearest Neighbors and kernel-based policies. Our main methodological contribution is to characterize exactly the worst-case regret of any WERM policy on any given configuration of contexts. To the best of our knowledge, this provides the first understanding of tight performance guarantees in any contextual decision-making problem, with past literature focusing on upper bounds via concentration inequalities. We instead take an optimization approach, and isolate a structure in the Newsvendor loss function that allows to reduce the infinite-dimensional optimization problem over worst-case distributions to a simple line search. This in turn allows us to unveil fundamental insights that were obfuscated by previous general-purpose bounds. We characterize actual guaranteed performance as a function of the contexts, as well as granular insights on the learning curve of algorithms.

Yaqi Xie, Xinru Hao, Jiaxi Liu et al.

Deep Reinforcement Learning (DRL) provides a general-purpose methodology for training inventory policies that can leverage big data and compute. However, off-the-shelf implementations of DRL have seen mixed success, often plagued by high sensitivity to the hyperparameters used during training. In this paper, we show that by imposing policy regularizations, grounded in classical inventory concepts such as "Base Stock", we can significantly accelerate hyperparameter tuning and improve the final performance of several DRL methods. We report details from a 100% deployment of DRL with policy regularizations on Alibaba's e-commerce platform, Tmall. We also include extensive synthetic experiments, which show that policy regularizations reshape the narrative on what is the best DRL method for inventory management.

Jackie Baek, Yunhan Chen, Ziyu Chi et al.

LLMs can generate a wealth of data, ranging from simulated personas imitating human valuations and preferences, to demand forecasts based on world knowledge. But how well do such LLM-generated distributions support downstream decision-making? For example, when pricing a new product, a firm could prompt an LLM to simulate how much consumers are willing to pay based on a product description, but how useful is the resulting distribution for optimizing the price? We refer to this approach as LLM-SAA, in which an LLM is used to construct an estimated distribution and the decision is then optimized under that distribution. In this paper, we study metrics to evaluate the quality of these LLM-generated distributions, based on the decisions they induce. Taking three canonical decision-making problems (assortment optimization, pricing, and newsvendor) as examples, we find that LLM-generated distributions are practically useful, especially in low-data regimes. We also show that decision-agnostic metrics such as Wasserstein distance can be misleading when evaluating these distributions for decision-making.

Jackie Baek, Yaopeng Fu, Will Ma et al.

Inventory control is a fundamental operations problem in which ordering decisions are traditionally guided by theoretically grounded operations research (OR) algorithms. However, such algorithms often rely on rigid modeling assumptions and can perform poorly when demand distributions shift or relevant contextual information is unavailable. Recent advances in large language models (LLMs) have generated interest in AI agents that can reason flexibly and incorporate rich contextual signals, but it remains unclear how best to incorporate LLM-based methods into traditional decision-making pipelines. We study how OR algorithms, LLMs, and humans can interact and complement each other in a multi-period inventory control setting. We construct InventoryBench, a benchmark of over 1,000 inventory instances spanning both synthetic and real-world demand data, designed to stress-test decision rules under demand shifts, seasonality, and uncertain lead times. Through this benchmark, we find that OR-augmented LLM methods outperform either method in isolation, suggesting that these methods are complementary rather than substitutes. We further investigate the role of humans through a controlled classroom experiment that embeds LLM recommendations into a human-in-the-loop decision pipeline. Contrary to prior findings that human-AI collaboration can degrade performance, we show that, on average, human-AI teams achieve higher profits than either humans or AI agents operating alone. Beyond this population-level finding, we formalize an individual-level complementarity effect and derive a distribution-free lower bound on the fraction of individuals who benefit from AI collaboration; empirically, we find this fraction to be substantial.

Zhuoxin Chen, Will Ma

In the Newsvendor problem, the goal is to guess the number that will be drawn from some distribution, with asymmetric consequences for guessing too high vs. too low. In the data-driven version, the distribution is unknown, and one must work with samples from the distribution. Data-driven Newsvendor has been studied under many variants: additive vs. multiplicative regret, high probability vs. expectation bounds, and different distribution classes. This paper studies all combinations of these variants, filling in many gaps in the literature and simplifying many proofs. In particular, we provide a unified analysis based on the notion of clustered distributions, which in conjunction with our new lower bounds, shows that the entire spectrum of regrets between $1/\sqrt{n}$ and $1/n$ can be possible. Simulations on commonly-used distributions demonstrate that our notion is the "correct" predictor of empirical regret across varying data sizes.

Yaqi Xie, Will Ma, Linwei Xin

Advances in computational power and AI have increased interest in reinforcement learning approaches to inventory management. This paper provides a theoretical foundation for these approaches and investigates the benefits of restricting to policy structures that are well-established by inventory theory. In particular, we prove generalization guarantees for learning several well-known classes of inventory policies, including base-stock and (s, S) policies, by leveraging the celebrated Vapnik-Chervonenkis (VC) theory. We apply the Pseudo-dimension and Fat-shattering dimension from VC theory to determine the generalization error of inventory policies, that is, the difference between an inventory policy's performance on training data and its expected performance on unseen data. We focus on a classical setting without contexts, but allow for an arbitrary distribution over demand sequences and do not make any assumptions such as independence over time. We corroborate our supervised learning results using numerical simulations. Managerially, our theory and simulations translate to the following insights. First, there is a principle of ``learning less is more'' in inventory management: depending on the amount of data available, it may be beneficial to restrict oneself to a simpler, albeit suboptimal, class of inventory policies to minimize overfitting errors. Second, the number of parameters in a policy class may not be the correct measure of overfitting error: in fact, the class of policies defined by T time-varying base-stock levels exhibits a generalization error an order of magnitude lower than that of the two-parameter (s, S) policy class. Finally, our research suggests situations in which it could be beneficial to incorporate the concepts of base-stock and inventory position into black-box learning machines, instead of having these machines directly learn the order quantity actions.

Eric Balkanski, Will Ma, Andreas Maggiori

Algorithms with predictions is a recent framework for decision-making under uncertainty that leverages the power of machine-learned predictions without making any assumption about their quality. The goal in this framework is for algorithms to achieve an improved performance when the predictions are accurate while maintaining acceptable guarantees when the predictions are erroneous. A serious concern with algorithms that use predictions is that these predictions can be biased and, as a result, cause the algorithm to make decisions that are deemed unfair. We show that this concern manifests itself in the classical secretary problem in the learning-augmented setting -- the state-of-the-art algorithm can have zero probability of accepting the best candidate, which we deem unfair, despite promising to accept a candidate whose expected value is at least $\max\{Ω(1) , 1 - O(ε)\}$ times the optimal value, where $ε$ is the prediction error. We show how to preserve this promise while also guaranteeing to accept the best candidate with probability $Ω(1)$. Our algorithm and analysis are based on a new "pegging" idea that diverges from existing works and simplifies/unifies some of their results. Finally, we extend to the $k$-secretary problem and complement our theoretical analysis with experiments.

Jingkai Huang, Will Ma, Zhengyuan Zhou

A simple strategy for improving LLM accuracy, especially in math and reasoning problems, is to sample multiple responses and submit the answer most consistently reached. In this paper we leverage Bayesian prior information to save on sampling costs, stopping once sufficient consistency is reached. Although the exact posterior is computationally intractable, we further introduce an efficient "L-aggregated" stopping policy that tracks only the L-1 most frequent answer counts. Theoretically, we prove that L=3 is all you need: this coarse approximation is sufficient to achieve asymptotic optimality, and strictly dominates prior-free baselines, while having a fast posterior computation. Empirically, this identifies the most consistent (i.e., mode) LLM answer using fewer samples, and can achieve similar answer accuracy while cutting the number of LLM calls (i.e., saving on LLM inference costs) by up to 50%.

Will Ma, Pan Xu, Yifan Xu

Matching markets involve heterogeneous agents (typically from two parties) who are paired for mutual benefit. During the last decade, matching markets have emerged and grown rapidly through the medium of the Internet. They have evolved into a new format, called Online Matching Markets (OMMs), with examples ranging from crowdsourcing to online recommendations to ridesharing. There are two features distinguishing OMMs from traditional matching markets. One is the dynamic arrival of one side of the market: we refer to these as online agents while the rest are offline agents. Examples of online and offline agents include keywords (online) and sponsors (offline) in Google Advertising; workers (online) and tasks (offline) in Amazon Mechanical Turk (AMT); riders (online) and drivers (offline when restricted to a short time window) in ridesharing. The second distinguishing feature of OMMs is the real-time decision-making element. However, studies have shown that the algorithms making decisions in these OMMs leave disparities in the match rates of offline agents. For example, tasks in neighborhoods of low socioeconomic status rarely get matched to gig workers, and drivers of certain races/genders get discriminated against in matchmaking. In this paper, we propose online matching algorithms which optimize for either individual or group-level fairness among offline agents in OMMs. We present two linear-programming (LP) based sampling algorithms, which achieve online competitive ratios at least 0.725 for individual fairness maximization (IFM) and 0.719 for group fairness maximization (GFM), respectively. We conduct extensive numerical experiments and results show that our boosted version of sampling algorithms are not only conceptually easy to implement but also highly effective in practical instances of fairness-maximization-related models.

Christian Tjandraatmadja, Ross Anderson, Joey Huchette et al.

We improve the effectiveness of propagation- and linear-optimization-based neural network verification algorithms with a new tightened convex relaxation for ReLU neurons. Unlike previous single-neuron relaxations which focus only on the univariate input space of the ReLU, our method considers the multivariate input space of the affine pre-activation function preceding the ReLU. Using results from submodularity and convex geometry, we derive an explicit description of the tightest possible convex relaxation when this multivariate input is over a box domain. We show that our convex relaxation is significantly stronger than the commonly used univariate-input relaxation which has been proposed as a natural convex relaxation barrier for verification. While our description of the relaxation may require an exponential number of inequalities, we show that they can be separated in linear time and hence can be efficiently incorporated into optimization algorithms on an as-needed basis. Based on this novel relaxation, we design two polynomial-time algorithms for neural network verification: a linear-programming-based algorithm that leverages the full power of our relaxation, and a fast propagation algorithm that generalizes existing approaches. In both cases, we show that for a modest increase in computational effort, our strengthened relaxation enables us to verify a significantly larger number of instances compared to similar algorithms.

Wang Chi Cheung, Will Ma, David Simchi-Levi et al.

We study a general problem of allocating limited resources to heterogeneous customers over time under model uncertainty. Each type of customer can be serviced using different actions, each of which stochastically consumes some combination of resources, and returns different rewards for the resources consumed. We consider a general model where the resource consumption distribution associated with each (customer type, action)-combination is not known, but is consistent and can be learned over time. In addition, the sequence of customer types to arrive over time is arbitrary and completely unknown. We overcome both the challenges of model uncertainty and customer heterogeneity by judiciously synthesizing two algorithmic frameworks from the literature: inventory balancing, which "reserves" a portion of each resource for high-reward customer types which could later arrive, and online learning, which shows how to "explore" the resource consumption distributions of each customer type under different actions. We define an auxiliary problem, which allows for existing competitive ratio and regret bounds to be seamlessly integrated. Furthermore, we show that the performance guarantee generated by our framework is tight, that is, we provide an information-theoretic lower bound which shows that both the loss from competitive ratio and the loss for regret are relevant in the combined problem. Finally, we demonstrate the efficacy of our algorithms on a publicly available hotel data set. Our framework is highly practical in that it requires no historical data (no fitted customer choice models, nor forecasting of customer arrival patterns) and can be used to initialize allocation strategies in fast-changing environments.