Othman El Balghiti, Adam N. Elmachtoub, Paul Grigas, Ambuj Tewari
The predict-then-optimize framework is fundamental in many practical settings: predict the unknown parameters of an optimization problem and then solve the problem using the predicted values of the parameters. A natural loss function in this environment is to consider the cost of the decisions induced by the predicted parameters in contrast to the prediction error of the parameters. This loss function is referred to as the smart predict-then-optimize (SPO) loss. In this work, we seek to provide bounds on how well the performance of a prediction model fit on training data generalizes out of sample in the context of the SPO loss. Because the SPO loss is nonconvex and non-Lipschitz, standard results for deriving generalization bounds do not apply. We first derive bounds based on the Natarajan dimension that, in the case of a polyhedral feasible region, scale at most logarithmically in the number of extreme points but, in the case of a general convex feasible region, have linear dependence on the decision dimension. By exploiting the structure of the SPO loss function and a key property of the feasible region, which we denote as the strength property, we can dramatically improve the dependence on the decision and feature dimensions. Our approach and analysis rely on placing a margin around problematic predictions that do not yield unique optimal solutions and then providing generalization bounds in the context of a modified margin SPO loss function that is Lipschitz continuous. Finally, we characterize the strength property and show that the modified SPO loss can be computed efficiently for both strongly convex bodies and polytopes with an explicit extreme point representation.Funding: O. El Balghiti thanks Rayens Capital for their support. A. N. Elmachtoub acknowledges the support of the National Science Foundation (NSF) [Grant CMMI-1763000]. P. Grigas acknowledges the support of NSF [Grants CCF-1755705 and CMMI-1762744]. A. Tewari acknowledges the support of the NSF [CAREER grant IIS-1452099] and a Sloan Research Fellowship.
预测-优化框架在许多实际环境中都非常重要：预测优化问题的未知参数，然后使用参数的预测值解决问题。在这种环境下，一个自然的损失函数就是考虑预测参数所引起的决策成本与参数预测误差的对比。这种损失函数被称为智能预测-优化（SPO）损失。在这项工作中，我们试图在 SPO 损失的背景下，为预测模型在训练数据上的拟合性能在样本外的泛化程度提供约束。由于 SPO 损失是非凸和非 Lipschitz 的，因此推导泛化边界的标准结果并不适用。我们首先推导出基于 Natarajan 维度的边界，在多面体可行区域的情况下，边界最多与极值点的数量成对数关系，但在一般凸形可行区域的情况下，边界与决策维度成线性关系。通过利用 SPO 损失函数的结构和可行区域的一个关键属性（我们称之为强度属性），我们可以显著改善对决策维度和特征维度的依赖性。我们的方法和分析依赖于在不产生唯一最优解的问题预测周围设置一个边际，然后在修改边际 SPO 损失函数的背景下提供泛化边界，该函数是立普齐兹连续的。最后，我们描述了强度特性，并证明对于强凸体和具有明确极值点表示的多边形，都能有效计算修正的 SPO 损失：O. El Balghiti 感谢 Rayens Capital 的支持。A. N. Elmachtoub 感谢美国国家科学基金会 (NSF) [CMMI-1763000] 的支持。P. Grigas 感谢美国国家科学基金会 [CCF-1755705 和 CMMI-1762744] 的支持。A. Tewari 感谢美国国家科学基金会 [CAREER grant IIS-1452099] 和斯隆研究奖学金的资助。
{"title":"Generalization Bounds in the Predict-Then-Optimize Framework","authors":"Othman El Balghiti, Adam N. Elmachtoub, Paul Grigas, Ambuj Tewari","doi":"10.1287/moor.2022.1330","DOIUrl":"https://doi.org/10.1287/moor.2022.1330","url":null,"abstract":"The predict-then-optimize framework is fundamental in many practical settings: predict the unknown parameters of an optimization problem and then solve the problem using the predicted values of the parameters. A natural loss function in this environment is to consider the cost of the decisions induced by the predicted parameters in contrast to the prediction error of the parameters. This loss function is referred to as the smart predict-then-optimize (SPO) loss. In this work, we seek to provide bounds on how well the performance of a prediction model fit on training data generalizes out of sample in the context of the SPO loss. Because the SPO loss is nonconvex and non-Lipschitz, standard results for deriving generalization bounds do not apply. We first derive bounds based on the Natarajan dimension that, in the case of a polyhedral feasible region, scale at most logarithmically in the number of extreme points but, in the case of a general convex feasible region, have linear dependence on the decision dimension. By exploiting the structure of the SPO loss function and a key property of the feasible region, which we denote as the strength property, we can dramatically improve the dependence on the decision and feature dimensions. Our approach and analysis rely on placing a margin around problematic predictions that do not yield unique optimal solutions and then providing generalization bounds in the context of a modified margin SPO loss function that is Lipschitz continuous. Finally, we characterize the strength property and show that the modified SPO loss can be computed efficiently for both strongly convex bodies and polytopes with an explicit extreme point representation.Funding: O. El Balghiti thanks Rayens Capital for their support. A. N. Elmachtoub acknowledges the support of the National Science Foundation (NSF) [Grant CMMI-1763000]. P. Grigas acknowledges the support of NSF [Grants CCF-1755705 and CMMI-1762744]. A. Tewari acknowledges the support of the NSF [CAREER grant IIS-1452099] and a Sloan Research Fellowship.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"75 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139648200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a game-theoretic variant of the maximum circulation problem. In a flow allocation game, we are given a directed flow network. Each node is a rational agent and can strategically allocate any incoming flow to the outgoing edges. Given the strategy choices of all agents, a maximal circulation that adheres to the chosen allocation strategies evolves in the network. Each agent wants to maximize the amount of flow through his or her node. Flow allocation games can be used to express strategic incentives of clearing in financial networks. We provide a cumulative set of results on the existence and computational complexity of pure Nash and strong equilibria as well as tight bounds on the (strong) prices of anarchy and stability. Our results show an interesting dichotomy. Ranking strategies over individual flow units allows us to obtain optimal strong equilibria for many objective functions. In contrast, more intuitive ranking strategies over edges can give rise to unfavorable incentive properties.Funding: This work was supported by Deutsche Forschungsgemeinschaft Research Group ADYN [411362735].
{"title":"Flow Allocation Games","authors":"Nils Bertschinger, Martin Hoefer, Daniel Schmand","doi":"10.1287/moor.2022.0355","DOIUrl":"https://doi.org/10.1287/moor.2022.0355","url":null,"abstract":"We study a game-theoretic variant of the maximum circulation problem. In a flow allocation game, we are given a directed flow network. Each node is a rational agent and can strategically allocate any incoming flow to the outgoing edges. Given the strategy choices of all agents, a maximal circulation that adheres to the chosen allocation strategies evolves in the network. Each agent wants to maximize the amount of flow through his or her node. Flow allocation games can be used to express strategic incentives of clearing in financial networks. We provide a cumulative set of results on the existence and computational complexity of pure Nash and strong equilibria as well as tight bounds on the (strong) prices of anarchy and stability. Our results show an interesting dichotomy. Ranking strategies over individual flow units allows us to obtain optimal strong equilibria for many objective functions. In contrast, more intuitive ranking strategies over edges can give rise to unfavorable incentive properties.Funding: This work was supported by Deutsche Forschungsgemeinschaft Research Group ADYN [411362735].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"11 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139584217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study problem-dependent rates, that is, generalization errors that scale near-optimally with the variance, effective loss, or gradient norms evaluated at the “best hypothesis.” We introduce a principled framework dubbed “uniform localized convergence” and characterize sharp problem-dependent rates for central statistical learning problems. From a methodological viewpoint, our framework resolves several fundamental limitations of existing uniform convergence and localization analysis approaches. It also provides improvements and some level of unification in the study of localized complexities, one-sided uniform inequalities, and sample-based iterative algorithms. In the so-called “slow rate” regime, we provide the first (moment-penalized) estimator that achieves the optimal variance-dependent rate for general “rich” classes; we also establish an improved loss-dependent rate for standard empirical risk minimization. In the “fast rate” regime, we establish finite-sample, problem-dependent bounds that are comparable to precise asymptotics. In addition, we show that iterative algorithms such as gradient descent and first order expectation maximization can achieve optimal generalization error in several representative problems across the areas of nonconvex learning, stochastic optimization, and learning with missing data.Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2021.0076 .
{"title":"Towards Optimal Problem Dependent Generalization Error Bounds in Statistical Learning Theory","authors":"Yunbei Xu, Assaf Zeevi","doi":"10.1287/moor.2021.0076","DOIUrl":"https://doi.org/10.1287/moor.2021.0076","url":null,"abstract":"We study problem-dependent rates, that is, generalization errors that scale near-optimally with the variance, effective loss, or gradient norms evaluated at the “best hypothesis.” We introduce a principled framework dubbed “uniform localized convergence” and characterize sharp problem-dependent rates for central statistical learning problems. From a methodological viewpoint, our framework resolves several fundamental limitations of existing uniform convergence and localization analysis approaches. It also provides improvements and some level of unification in the study of localized complexities, one-sided uniform inequalities, and sample-based iterative algorithms. In the so-called “slow rate” regime, we provide the first (moment-penalized) estimator that achieves the optimal variance-dependent rate for general “rich” classes; we also establish an improved loss-dependent rate for standard empirical risk minimization. In the “fast rate” regime, we establish finite-sample, problem-dependent bounds that are comparable to precise asymptotics. In addition, we show that iterative algorithms such as gradient descent and first order expectation maximization can achieve optimal generalization error in several representative problems across the areas of nonconvex learning, stochastic optimization, and learning with missing data.Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2021.0076 .","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"164 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139517379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the online saddle point problem, an online learning problem where at each iteration, a pair of actions needs to be chosen without knowledge of the current and future (convex-concave) payoff functions. The objective is to minimize the gap between the cumulative payoffs and the saddle point value of the aggregate payoff function, which we measure using a metric called saddle point regret (SP-Regret). The problem generalizes the online convex optimization framework, but here, we must ensure that both players incur cumulative payoffs close to that of the Nash equilibrium of the sum of the games. We propose an algorithm that achieves SP-Regret proportional to [Formula: see text] in the general case, and [Formula: see text] SP-Regret for the strongly convex-concave case. We also consider the special case where the payoff functions are bilinear and the decision sets are the probability simplex. In this setting, we are able to design algorithms that reduce the bounds on SP-Regret from a linear dependence in the dimension of the problem to a logarithmic one. We also study the problem under bandit feedback and provide an algorithm that achieves sublinear SP-Regret. We then consider an online convex optimization with knapsacks problem motivated by a wide variety of applications, such as dynamic pricing, auctions, and crowdsourcing. We relate this problem to the online saddle point problem and establish [Formula: see text] regret using a primal-dual algorithm.
{"title":"The Online Saddle Point Problem and Online Convex Optimization with Knapsacks","authors":"Adrian Rivera Cardoso, He Wang, Huan Xu","doi":"10.1287/moor.2018.0330","DOIUrl":"https://doi.org/10.1287/moor.2018.0330","url":null,"abstract":"We study the online saddle point problem, an online learning problem where at each iteration, a pair of actions needs to be chosen without knowledge of the current and future (convex-concave) payoff functions. The objective is to minimize the gap between the cumulative payoffs and the saddle point value of the aggregate payoff function, which we measure using a metric called saddle point regret (SP-Regret). The problem generalizes the online convex optimization framework, but here, we must ensure that both players incur cumulative payoffs close to that of the Nash equilibrium of the sum of the games. We propose an algorithm that achieves SP-Regret proportional to [Formula: see text] in the general case, and [Formula: see text] SP-Regret for the strongly convex-concave case. We also consider the special case where the payoff functions are bilinear and the decision sets are the probability simplex. In this setting, we are able to design algorithms that reduce the bounds on SP-Regret from a linear dependence in the dimension of the problem to a logarithmic one. We also study the problem under bandit feedback and provide an algorithm that achieves sublinear SP-Regret. We then consider an online convex optimization with knapsacks problem motivated by a wide variety of applications, such as dynamic pricing, auctions, and crowdsourcing. We relate this problem to the online saddle point problem and establish [Formula: see text] regret using a primal-dual algorithm.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"78 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139460720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider direct mechanisms to sell heterogeneous objects when buyers have private additive valuations and nonunit demand. We completely characterize the class of strategy-proof and agent sovereign mechanisms that satisfy a local side-flatness condition. Further, we introduce a notion of “continuity up to utility” and show that any such mechanism allocating all objects at all profiles is continuous and anonymous only if it is efficient. We find that the only mechanism satisfying these properties is equivalent to operating simultaneous second-price auctions for each object—as was done by the New Zealand government in allocating license rights to the use of radio spectrum in 1990. Finally, we present a complete characterization of simultaneous second-price auctions with object-specific reserve prices in terms of these properties and a weak nonbossiness restriction.
{"title":"Strategy-Proof Multidimensional Mechanism Design","authors":"Ranojoy Basu, Conan Mukherjee","doi":"10.1287/moor.2022.0324","DOIUrl":"https://doi.org/10.1287/moor.2022.0324","url":null,"abstract":"We consider direct mechanisms to sell heterogeneous objects when buyers have private additive valuations and nonunit demand. We completely characterize the class of strategy-proof and agent sovereign mechanisms that satisfy a local side-flatness condition. Further, we introduce a notion of “continuity up to utility” and show that any such mechanism allocating all objects at all profiles is continuous and anonymous only if it is efficient. We find that the only mechanism satisfying these properties is equivalent to operating simultaneous second-price auctions for each object—as was done by the New Zealand government in allocating license rights to the use of radio spectrum in 1990. Finally, we present a complete characterization of simultaneous second-price auctions with object-specific reserve prices in terms of these properties and a weak nonbossiness restriction.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"46 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139421383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Guocheng Liao, Yu Su, Juba Ziani, Adam Wierman, Jianwei Huang
Whereas users claim to be concerned about privacy, often they do little to protect their privacy in their online actions. One prominent explanation for this privacy paradox is that, when an individual shares data, it is not just the individual’s privacy that is compromised; the privacy of other individuals with correlated data is also compromised. This information leakage encourages oversharing of data and significantly impacts the incentives of individuals in online platforms. In this paper, we study the design of mechanisms for data acquisition in settings with information leakage and verifiable data. We design an incentive-compatible mechanism that optimizes the worst case trade-off between bias and variance of the estimation subject to a budget constraint, with which the worst case is over the unknown correlation between costs and data. Additionally, we characterize the structure of the optimal mechanism in closed form and study monotonicity and nonmonotonicity properties of the marketplace.Funding: This work is supported by the National Natural Science Foundation of China [Grants 62202512 and 62271434], Shenzhen Science and Technology Program [Grant JCYJ20210324120011032], Guangdong Basic and Applied Basic Research Foundation [Grant 2021B1515120008], Shenzhen Key Laboratory of Crowd Intelligence Empowered Low-Carbon Energy Network [Grant ZDSYS20220606100601002], and the Shenzhen Institute of Artificial Intelligence and Robotics for Society. This work is also supported by the National Science Foundation [Grants CNS-2146814, CPS-2136197, CNS-2106403, and NGSDI-2105648].Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2023.0022 .
{"title":"The Privacy Paradox and Optimal Bias–Variance Trade-offs in Data Acquisition","authors":"Guocheng Liao, Yu Su, Juba Ziani, Adam Wierman, Jianwei Huang","doi":"10.1287/moor.2023.0022","DOIUrl":"https://doi.org/10.1287/moor.2023.0022","url":null,"abstract":"Whereas users claim to be concerned about privacy, often they do little to protect their privacy in their online actions. One prominent explanation for this privacy paradox is that, when an individual shares data, it is not just the individual’s privacy that is compromised; the privacy of other individuals with correlated data is also compromised. This information leakage encourages oversharing of data and significantly impacts the incentives of individuals in online platforms. In this paper, we study the design of mechanisms for data acquisition in settings with information leakage and verifiable data. We design an incentive-compatible mechanism that optimizes the worst case trade-off between bias and variance of the estimation subject to a budget constraint, with which the worst case is over the unknown correlation between costs and data. Additionally, we characterize the structure of the optimal mechanism in closed form and study monotonicity and nonmonotonicity properties of the marketplace.Funding: This work is supported by the National Natural Science Foundation of China [Grants 62202512 and 62271434], Shenzhen Science and Technology Program [Grant JCYJ20210324120011032], Guangdong Basic and Applied Basic Research Foundation [Grant 2021B1515120008], Shenzhen Key Laboratory of Crowd Intelligence Empowered Low-Carbon Energy Network [Grant ZDSYS20220606100601002], and the Shenzhen Institute of Artificial Intelligence and Robotics for Society. This work is also supported by the National Science Foundation [Grants CNS-2146814, CPS-2136197, CNS-2106403, and NGSDI-2105648].Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2023.0022 .","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"83 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139410295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The multi-objective optimization is to optimize several objective functions over a common feasible set. Because the objectives usually do not share a common optimizer, people often consider (weakly) Pareto points. This paper studies multi-objective optimization problems that are given by polynomial functions. First, we study the geometry for (weakly) Pareto values and represent Pareto front as the boundary of a convex set. Linear scalarization problems (LSPs) and Chebyshev scalarization problems (CSPs) are typical approaches for getting (weakly) Pareto points. For LSPs, we show how to use tight relaxations to solve them and how to detect existence or nonexistence of proper weights. For CSPs, we show how to solve them by moment relaxations. Moreover, we show how to check whether a given point is a (weakly) Pareto point or not and how to detect existence or nonexistence of (weakly) Pareto points. We also study how to detect unboundedness of polynomial optimization, which is used to detect nonexistence of proper weights or (weakly) Pareto points.Funding: J. Nie is partially supported by the National Science Foundation [Grant DMS-2110780].
多目标优化是在一个共同的可行集合上优化多个目标函数。由于这些目标通常不共享一个共同的优化器，人们通常会考虑（弱）帕累托点。本文研究多项式函数给出的多目标优化问题。首先，我们研究了（弱）帕累托值的几何形状，并将帕累托前沿表示为凸集的边界。线性标量化问题（LSPs）和切比雪夫标量化问题（CSPs）是获得（弱）帕累托点的典型方法。对于线性标度化问题，我们展示了如何使用严格松弛来解决它们，以及如何检测适当权重的存在与否。对于 CSP，我们展示了如何通过矩松弛来求解。此外，我们还展示了如何检查给定点是否为（弱）帕累托点，以及如何检测（弱）帕累托点是否存在。我们还研究了如何检测多项式优化的无界性，它可用于检测适当权重或（弱）帕累托点的不存在：J. Nie 由美国国家科学基金会 [Grant DMS-2110780] 部分资助。
{"title":"The Multi-Objective Polynomial Optimization","authors":"Jiawang Nie, Zi Yang","doi":"10.1287/moor.2023.0200","DOIUrl":"https://doi.org/10.1287/moor.2023.0200","url":null,"abstract":"The multi-objective optimization is to optimize several objective functions over a common feasible set. Because the objectives usually do not share a common optimizer, people often consider (weakly) Pareto points. This paper studies multi-objective optimization problems that are given by polynomial functions. First, we study the geometry for (weakly) Pareto values and represent Pareto front as the boundary of a convex set. Linear scalarization problems (LSPs) and Chebyshev scalarization problems (CSPs) are typical approaches for getting (weakly) Pareto points. For LSPs, we show how to use tight relaxations to solve them and how to detect existence or nonexistence of proper weights. For CSPs, we show how to solve them by moment relaxations. Moreover, we show how to check whether a given point is a (weakly) Pareto point or not and how to detect existence or nonexistence of (weakly) Pareto points. We also study how to detect unboundedness of polynomial optimization, which is used to detect nonexistence of proper weights or (weakly) Pareto points.Funding: J. Nie is partially supported by the National Science Foundation [Grant DMS-2110780].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"16 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139374433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a many-server queue in which each server can serve multiple customers in parallel. Such multitasking phenomena occur in various applications areas (e.g., in hospitals and contact centers), although the impact of the number of customers who are simultaneously served on system efficiency may vary. We establish diffusion limits of the queueing process under the quality-and-efficiency-driven scaling and for different policies of assigning customers to servers depending on the number of customers they serve. We show that for a broad class of routing policies, including routing to the least busy server, the same one-dimensional diffusion process is obtained in the heavy-traffic limit. In case of assignment to the most busy server, there is no state-space collapse, and the diffusion limit involves a custom regulator mapping. Moreover, we also show that assigning customers to the least (most) busy server is optimal when the cumulative service rate per server is concave (convex), motivating the routing policies considered. Finally, we also derive diffusion limits in the nonheavy-traffic scaling regime and in the heavy-traffic scaling regime where customers can be reassigned during service.Funding: The research of J. Storm is partly funded by the Netherlands Organization for Scientific Research (NWO) Gravitation project Networks [Grant 024.002.003].
{"title":"Diffusion-Based Staffing for Multitasking Service Systems with Many Servers","authors":"Jaap Storm, Wouter Berkelmans, René Bekker","doi":"10.1287/moor.2021.0051","DOIUrl":"https://doi.org/10.1287/moor.2021.0051","url":null,"abstract":"We consider a many-server queue in which each server can serve multiple customers in parallel. Such multitasking phenomena occur in various applications areas (e.g., in hospitals and contact centers), although the impact of the number of customers who are simultaneously served on system efficiency may vary. We establish diffusion limits of the queueing process under the quality-and-efficiency-driven scaling and for different policies of assigning customers to servers depending on the number of customers they serve. We show that for a broad class of routing policies, including routing to the least busy server, the same one-dimensional diffusion process is obtained in the heavy-traffic limit. In case of assignment to the most busy server, there is no state-space collapse, and the diffusion limit involves a custom regulator mapping. Moreover, we also show that assigning customers to the least (most) busy server is optimal when the cumulative service rate per server is concave (convex), motivating the routing policies considered. Finally, we also derive diffusion limits in the nonheavy-traffic scaling regime and in the heavy-traffic scaling regime where customers can be reassigned during service.Funding: The research of J. Storm is partly funded by the Netherlands Organization for Scientific Research (NWO) Gravitation project Networks [Grant 024.002.003].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"24 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139067240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Priyank Agrawal, Eric Balkanski, Vasilis Gkatzelis, Tingting Ou, Xizhi Tan
In this work, we introduce an alternative model for the design and analysis of strategyproof mechanisms that is motivated by the recent surge of work in “learning-augmented algorithms.” Aiming to complement the traditional worst-case analysis approach in computer science, this line of work has focused on the design and analysis of algorithms that are enhanced with machine-learned predictions. The algorithms can use the predictions as a guide to inform their decisions, aiming to achieve much stronger performance guarantees when these predictions are accurate (consistency), while also maintaining near-optimal worst-case guarantees, even if these predictions are inaccurate (robustness). We initiate the design and analysis of strategyproof mechanisms that are augmented with predictions regarding the private information of the participating agents. To exhibit the important benefits of this approach, we revisit the canonical problem of facility location with strategic agents in the two-dimensional Euclidean space. We study both the egalitarian and utilitarian social cost functions, and we propose new strategyproof mechanisms that leverage predictions to guarantee an optimal trade-off between consistency and robustness. Furthermore, we also prove parameterized approximation results as a function of the prediction error, showing that our mechanisms perform well, even when the predictions are not fully accurate.Funding: The work of E. Balkanski was supported in part by the National Science Foundation [Grants CCF-2210501 and IIS-2147361]. The work of V. Gkatzelis and X. Tan was supported in part by the National Science Foundation [Grant CCF-2210502] and [CAREER Award CCF-2047907].Supplemental Material: The e-companion is available at https://doi.org/10.1287/moor.2022.0225 .
{"title":"Learning-Augmented Mechanism Design: Leveraging Predictions for Facility Location","authors":"Priyank Agrawal, Eric Balkanski, Vasilis Gkatzelis, Tingting Ou, Xizhi Tan","doi":"10.1287/moor.2022.0225","DOIUrl":"https://doi.org/10.1287/moor.2022.0225","url":null,"abstract":"In this work, we introduce an alternative model for the design and analysis of strategyproof mechanisms that is motivated by the recent surge of work in “learning-augmented algorithms.” Aiming to complement the traditional worst-case analysis approach in computer science, this line of work has focused on the design and analysis of algorithms that are enhanced with machine-learned predictions. The algorithms can use the predictions as a guide to inform their decisions, aiming to achieve much stronger performance guarantees when these predictions are accurate (consistency), while also maintaining near-optimal worst-case guarantees, even if these predictions are inaccurate (robustness). We initiate the design and analysis of strategyproof mechanisms that are augmented with predictions regarding the private information of the participating agents. To exhibit the important benefits of this approach, we revisit the canonical problem of facility location with strategic agents in the two-dimensional Euclidean space. We study both the egalitarian and utilitarian social cost functions, and we propose new strategyproof mechanisms that leverage predictions to guarantee an optimal trade-off between consistency and robustness. Furthermore, we also prove parameterized approximation results as a function of the prediction error, showing that our mechanisms perform well, even when the predictions are not fully accurate.Funding: The work of E. Balkanski was supported in part by the National Science Foundation [Grants CCF-2210501 and IIS-2147361]. The work of V. Gkatzelis and X. Tan was supported in part by the National Science Foundation [Grant CCF-2210502] and [CAREER Award CCF-2047907].Supplemental Material: The e-companion is available at https://doi.org/10.1287/moor.2022.0225 .","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"26 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139053488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a general tractable model for default contagion and systemic risk in a heterogeneous financial network subjected to an exogenous macroeconomic shock. We show that under certain regularity assumptions, the default cascade model can be transformed into a death process problem represented by a balls-and-bins model. We state various limit theorems regarding the final size of default cascades. Under appropriate assumptions on the degree and threshold distributions, we prove that the final sizes of default cascades have asymptotically Gaussian fluctuations. We next state limit theorems for different system-wide wealth aggregation functions, which enable us to provide systemic risk measures in relation to the structure and heterogeneity of the financial network. Lastly, we demonstrate how these results can be utilized by a social planner to optimally target interventions during a financial crisis given a budget constraint and under partial information of the financial network.
{"title":"Limit Theorems for Default Contagion and Systemic Risk","authors":"Hamed Amini, Zhongyuan Cao, Agnès Sulem","doi":"10.1287/moor.2021.0283","DOIUrl":"https://doi.org/10.1287/moor.2021.0283","url":null,"abstract":"We consider a general tractable model for default contagion and systemic risk in a heterogeneous financial network subjected to an exogenous macroeconomic shock. We show that under certain regularity assumptions, the default cascade model can be transformed into a death process problem represented by a balls-and-bins model. We state various limit theorems regarding the final size of default cascades. Under appropriate assumptions on the degree and threshold distributions, we prove that the final sizes of default cascades have asymptotically Gaussian fluctuations. We next state limit theorems for different system-wide wealth aggregation functions, which enable us to provide systemic risk measures in relation to the structure and heterogeneity of the financial network. Lastly, we demonstrate how these results can be utilized by a social planner to optimally target interventions during a financial crisis given a budget constraint and under partial information of the financial network.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"30 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139053715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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