arXiv subjects

Mo Liu

Publications and source records attributed to Mo Liu.

12 recordsLinked to original sources

arXiv2026

Automated Design of Inventory Policy with Large Language Models: An Exploratory Study

Firms making inventory decisions have access to operational data, optimization tools, and large language models (LLMs). Typically, data characterize the operating environment, optimization selects parameters within a prespecified inventory policy class, and LLMs support coding and decision analysis. We develop an integrated framework that combines these resources to automate inventory policy design. Given demand data, the framework iteratively uses an LLM to generate parameterized policy classes and an external solver to optimize its parameters within each class. Across 30 lost-sales inventory instances, the mean cost reduction relative to optimized base-stock benchmarks increases from 17.5% after one generation to 30.0% after ten generations. Parameter optimization is central to this performance: an LLM-only variant performs substantially worse, whereas optimization-guided feedback improves policy quality, accelerates search, and directs the LLM toward better policy classes rather than merely better parameter values within a fixed class. The strongest discovered policies are also interpretable: they combine recognizable inventory-control motifs, including capped orders, discounted or weighted pipeline inventory, and threshold-based replenishment logic. The search thereby produces new policy-class functional forms that, to our knowledge, have not previously been studied in the lost-sales inventory literature. These functional forms are not specified ex ante but emerge from the search process. Moreover, after their parameters are re-optimized, three discovered policy classes achieve average cost reductions of 21.75% to 22.60% across 10,064 new inventory instances. Overall, the results show that data-driven parameter optimization can guide LLM-based search over a broad space of inventory policy classes and identify high-performing, interpretable, and transferable decision rules.

cs.AI↗

arXiv2026

Decision-Focused Learning: When and Why Traditional Prediction Models Fail

Plugging predictions of unknown parameters into downstream optimization problems, often referred to as the ``predict-then-optimize'' paradigm, has long been a standard approach in decision-making under uncertainty. However, improved predictive accuracy does not, in general, translate into improved decision quality. This disconnect has motivated growing interest in decision-focused learning (DFL) within the operations research community. This tutorial reviews recent developments in DFL and highlights key methodological insights, with a particular focus on stochastic linear programming as the downstream decision-making problem. We discuss why several widely used tools in traditional statistical learning are not directly suited to decision-focused settings and must be rethought, including (i) data collection strategies driven purely by predictive uncertainty and (ii) distributional distance measures such as the Wasserstein distance. We summarize properties of DFL that distinguish it from conventional predictive modeling and provide insights into the development of new decision-focused tools.

cs.LG↗

arXiv2026

A Solver-Free Training Method for Predict-then-Optimize

We propose a scalable method for training prediction (machine learning) models in the predict-then-optimize paradigm, where model outputs serve as coefficients for a subsequent linear optimization task. Directly minimizing the empirical decision regret is intractable for linear programming and combinatorial optimization since the decision mapping is piecewise constant, and the gradients are zero almost everywhere. While existing methods address this by smoothing the differentiation process, they suffer from scalability issues, since a computationally expensive solver call is required for every gradient evaluation. To address this, we propose a decision-focused learning pipeline based on a measure transformation principle, which yields a new surrogate loss that is completely optimization-solver-free during training. We establish theoretical guarantees, including Fisher consistency and excess risk bounds. Empirically, our method achieves decision quality competitive with state-of-the-art methods while reducing training time by orders of magnitude.

stat.ML↗

arXiv2026

Asymptotically Optimal Sequential Testing with Heterogeneous LLMs

We study a Bayesian binary sequential hypothesis testing problem with multiple large language models (LLMs). Each LLM $j$ has per-query cost $c_j>0$, random waiting time with mean $\mu_j>0$ and sub-Gaussian tails, and \emph{asymmetric} accuracies: the probability of returning the correct label depends on the true hypothesis $\theta\in\{A,B\}$ and needs not be the same under $A$ and $B$. This asymmetry induces two distinct information rates $(I_{j,A}, I_{j,B})$ per LLM, one under each hypothesis. The decision-maker chooses LLMs sequentially, observes their noisy binary answers, and stops when the posterior probability of one hypothesis exceeds $1-\alpha$. The objective is to minimize the sum of expected query cost and expected waiting cost, $\mathbb{E}[C_\pi] + \mathbb{E}[g(W_\pi)]$, where $C_\pi$ is the total query cost, $W_\pi$ is the total waiting time and $g$ is a polynomial function (e.g., $g(x)=x^\rho$ with $\rho\ge 1$). We prove that as the error tolerance $\alpha\to0$, the optimal policy is asymptotically equivalent to one that uses at most two LLMs. In this case, a single-LLM policy is \emph{not} generically optimal: optimality now requires exploiting a two-dimensional tradeoff between information under $A$ and information under $B$. Any admissible policy induces an expected information-allocation vector in $\mathbb{R}_+^2$, and we show that the optimal allocation lies at an extreme point of the associated convex set when $\alpha$ is relatively small, and hence uses at most two LLMs. We construct belief-dependent policies that first mix between two LLMs when the posterior is ambiguous, and then switch to a single "specialist" LLM when the posterior is sufficiently close to one of the hypotheses. These policies match the universal lower bound up to a $(1+o(1))$ factor as $\alpha\rightarrow 0$.

cs.DS↗

arXiv2026

Decision-Focused Sequential Experimental Design: A Directional Uncertainty-Guided Approach

We consider the sequential experimental design problem in the predict-then-optimize paradigm. In this paradigm, the outputs of the prediction model are used as coefficient vectors in a downstream linear optimization problem. Traditional sequential experimental design aims to control the input variables (features) so that the improvement in prediction accuracy from each experimental outcome (label) is maximized. However, in the predict-then-optimize setting, performance is ultimately evaluated based on the decision loss induced by the downstream optimization, rather than by prediction error. This mismatch between prediction accuracy and decision loss renders traditional decision-blind designs inefficient. To address this issue, we propose a directional-based metric to quantify predictive uncertainty. This metric does not require solving an optimization oracle and is therefore computationally tractable. We show that the resulting sequential design criterion enjoys strong consistency and convergence guarantees. Under a broad class of distributions, we demonstrate that our directional uncertainty-based design attains an earlier stopping time than decision-blind designs. This advantage is further supported by real-world experiments on an LLM job allocation problem.

stat.ML↗

arXiv2026

Decision-Focused Optimal Transport

We propose a fundamental metric for measuring the distance between two distributions. This metric, referred to as the decision-focused (DF) divergence, is tailored to stochastic linear optimization problems in which the objective coefficients are random and may follow two distinct distributions. Traditional metrics such as KL divergence and Wasserstein distance are not well-suited for quantifying the resulting cost discrepancy, because changes in the coefficient distribution do not necessarily change the optimizer of the underlying linear program. Instead, the impact on the objective value depends on how the two distributions are coupled (aligned). Motivated by optimal transport, we introduce decision-focused distances under several settings, including the optimistic DF distance, the robust DF distance, and their entropy-regularized variants. We establish connections between the proposed DF distance and classical distributional metrics. For the calculation of the DF distance, we develop efficient computational methods. We further derive sample complexity guarantees for estimating these distances and show that the DF distance estimation avoids the curse of dimensionality that arises in Wasserstein distance estimation. The proposed DF distance provides a foundation for a broad range of applications. As an illustrative example, we study the interpolation between two distributions. Numerical studies, including a toy newsvendor problem and a real-world medical testing dataset, demonstrate the practical value of the proposed DF distance.

math.ST↗

arXiv2025

Decision-Focused Bias Correction for Fluid Approximation

We revisit the multi-period newsvendor network problem, in which demands from multiple customers are correlated and jointly time-varying. Due to the curse of dimensionality associated with estimating the full joint demand distribution, we consider fluid approximation, a widely used approach for solving two-stage stochastic optimization problems such as large-scale service-system design. However, replacing the underlying random distribution (e.g., the demand distribution) with its mean (e.g., the time-varying average arrival rate) introduces bias in performance estimation and can lead to suboptimal decisions. In this paper, we investigate how to identify an alternative point statistic, not necessarily the mean, such that substituting this statistic into the two-stage newsvendor network problem yields an optimal decision. We refer to this statistic as the decision-corrected point estimate (a time-varying arrival rate). Although the critical fractile is well known to be the decision-corrected point forecast for the single-item newsvendor problem, counterexamples show that such a point statistic may not exist for newsvendor networks. We establish necessary and sufficient conditions for the existence of such a corrected point estimate and propose an algorithm for computing it. Numerical experiments on real data demonstrate that using the proposed decision-corrected point forecast in fluid approximation achieves substantially lower cost than traditional fluid approximation and sample average approximation benchmarks.

math.OC↗

arXiv2025

Bizard: A Community-Driven Platform for Accelerating and Enhancing Biomedical Data Visualization

Biomedical research increasingly relies on heterogeneous, high-dimensional datasets, yet effective visualization remains hindered by fragmented code resources, steep programming barriers, and limited domain-specific guidance. Bizard is an open-source visualization code repository engineered to streamline data analysis in biomedical research. It aggregates a diverse array of executable visualization scripts, empowering researchers to select and tailor optimal graphical methods for their specific investigative demands. The platform features an intuitive interface equipped with sophisticated browsing and filtering capabilities, exhaustive tutorials, and interactive discussion forums that foster knowledge dissemination. Through its community-driven paradigm, Bizard promotes continual refinement and functional expansion, establishing itself as an essential resource for elevating biomedical data visualization and analytical standards. By harnessing Bizard's infrastructure, researchers can augment their visualization proficiency, propel methodological progress, and enhance interpretive rigor, ultimately accelerating precision medicine and personalized therapeutics. Bizard is freely accessible at https://openbiox.github.io/Bizard/.

q-bio.GN↗

arXiv2024

A Re-solving Heuristic for Dynamic Assortment Optimization with Knapsack Constraints

In this paper, we consider a multi-stage dynamic assortment optimization problem with multi-nomial choice modeling (MNL) under resource knapsack constraints. Given the current resource inventory levels, the retailer makes an assortment decision at each period, and the goal of the retailer is to maximize the total profit from purchases. With the exact optimal dynamic assortment solution being computationally intractable, a practical strategy is to adopt the re-solving technique that periodically re-optimizes deterministic linear programs (LP) arising from fluid approximation. However, the fractional structure of MNL makes the fluid approximation in assortment optimization non-linear, which brings new technical challenges. To address this challenge, we propose a new epoch-based re-solving algorithm that effectively transforms the denominator of the objective into the constraint, so that the re-solving technique is applied to a linear program with additional slack variables amenable to practical computations and theoretical analysis. Theoretically, we prove that the regret (i.e., the gap between the resolving policy and the optimal objective of the fluid approximation) scales logarithmically with the length of time horizon and resource capacities.

math.OC↗

arXiv2023

Active Learning For Contextual Linear Optimization: A Margin-Based Approach

We develop the first active learning method for contextual linear optimization. Specifically, we introduce a label acquisition algorithm that sequentially decides whether to request the ``labels'' of feature samples from an unlabeled data stream, where the labels correspond to the coefficients of the objective in the linear optimization. Our method is the first to be directly informed by the decision loss induced by the predicted coefficients, referred to as the Smart Predict-then-Optimize (SPO) loss. Motivated by the structure of the SPO loss, our algorithm adopts a margin-based criterion utilizing the concept of distance to degeneracy. In particular, we design an efficient active learning algorithm with theoretical excess risk (i.e., generalization) guarantees. We derive upper bounds on the label complexity, defined as the number of samples whose labels are acquired to achieve a desired small level of SPO risk. These bounds show that our algorithm has a much smaller label complexity than the naive supervised learning approach that labels all samples, particularly when the SPO loss is minimized directly on the collected data. To address the discontinuity and nonconvexity of the SPO loss, we derive label complexity bounds under tractable surrogate loss functions. Under natural margin conditions, these bounds also outperform naive supervised learning. Using the SPO+ loss, a specialized surrogate of the SPO loss, we establish even tighter bounds under separability conditions. Finally, we present numerical evidence showing the practical value of our algorithms in settings such as personalized pricing and the shortest path problem.

cs.LG↗

arXiv2022

Are Bundles Good Deals for FOML?

Bundled products are often offered as good deals to customers. When we bundle quantifiers and modalities together (as in $\exists x \Box$, $\Diamond \forall x$ etc.) in first-order modal logic (FOML), we get new logical operators whose combinations produce interesting fragments of FOML without any restriction on the arity of predicates, the number of variables, or the modal scope. It is well-known that finding decidable fragments of FOML is hard, so we may ask: do bundled fragments that exploit the distinct expressivity of FOML constitute good deals in balancing the expressivity and complexity? There are a few positive earlier results on some particular fragments. In this paper, we try to fully map the terrain of bundled fragments of FOML in (un)decidability, and in the cases without a definite answer yet, we show that they lack the finite model property. Moreover, whether the logics are interpreted over constant domains (across states/worlds) or increasing domains presents another layer of complexity. We also present the \textit{loosely bundled fragment}, which generalizes the bundles and yet retain decidability (over increasing domain models).

cs.LO↗

arXiv2022

Decision problem of some bundled FOML fragments

Over increasing domain interpretations, \exists\Box and \forall\Box bundled fragments are decidable and over constant domain interpretations, \exists\Box bundled fragment is decidable while \forall\Box bundled fragment is undecidable. Based on the existing results,we show that over increasing domain interpretations, \Box\exists and \Box\forall bundled fragments are decidable as well. On the other hand, over constant domain interpretations, \Box\forall bundled fragment is undecidable and \Box\exists^2 bundled fragment, an extension of \Box\exists bundled fragment, is undecidable neither.

cs.LO↗