An ODE Model for Dynamic Matching in Heterogeneous Networks

We study the problem of dynamic matching in heterogeneous networks, where agents are subject to compatibility restrictions and stochastic arrival and departure times. In particular, we consider networks with one type of easy-to-match agents and multiple types of hard-to-match agents, each subject to its own compatibility constraints. Such a setting arises in many real-world applications, including kidney exchange programs and carpooling platforms. We introduce a novel approach to modeling dynamic matching by establishing the ordinary differential equation (ODE) model, which offers a new perspective for evaluating various matching algorithms. We study two algorithms, namely the Greedy and Patient Algorithms, where both algorithms prioritize matching compatible hard-to-match agents over easy-to-match agents in heterogeneous networks. Our results demonstrate the trade-off between the conflicting goals of matching agents quickly and optimally, offering insights into the design of real-world dynamic matching systems. We provide simulations and a real-world case study using data from the Organ Procurement and Transplantation Network to validate theoretical predictions.

Continuous Time Analysis of Dynamic Matching in Heterogeneous Networks

This paper addresses the problem of dynamic matching in heterogeneous networks, where agents are subject to compatibility restrictions and stochastic arrival and departure times. In particular, we consider networks with one type of easy-to-match agents and multiple types of hard-to-match agents, each subject to its own set of compatibility constraints. Such a setting arises in many real-world applications, including kidney exchange programs and carpooling platforms, where some participants may have more stringent compatibility requirements than others. We introduce a novel approach to modeling dynamic matching by establishing ordinary differential equation (ODE) models, offering a new perspective for evaluating various matching algorithms. We study two algorithms, the Greedy Algorithm and the Patient Algorithm, which prioritize the matching of compatible hard-to-match agents over easy-to-match agents in heterogeneous networks. Our results show the trade-off between the conflicting goals of matching agents quickly and optimally, offering insights into the design of real-world dynamic matching systems. We present simulations and a real-world case study using data from the Organ Procurement and Transplantation Network to validate theoretical predictions.

Robust multi-item auction design using statistical learning: Overcoming uncertainty in bidders' types distributions

This paper presents a novel mechanism design for multi-item auction settings with uncertain bidders' type distributions. Our proposed approach utilizes nonparametric density estimation to accurately estimate bidders' types from historical bids, and is built upon the Vickrey-Clarke-Groves (VCG) mechanism, ensuring satisfaction of Bayesian incentive compatibility (BIC) and $\delta$-individual rationality (IR). To further enhance the efficiency of our mechanism, we introduce two novel strategies for query reduction: a filtering method that screens potential winners' value regions within the confidence intervals generated by our estimated distribution, and a classification strategy that designates the lower bound of an interval as the estimated type when the length is below a threshold value. Simulation experiments conducted on both small-scale and large-scale data demonstrate that our mechanism consistently outperforms existing methods in terms of revenue maximization and query reduction, particularly in large-scale scenarios. This makes our proposed mechanism a highly desirable and effective option for sellers in the realm of multi-item auctions.

Double Matching Under Complementary Preferences

In this paper, we propose a new algorithm for addressing the problem of matching markets with complementary preferences, where agents' preferences are unknown a priori and must be learned from data. The presence of complementary preferences can lead to instability in the matching process, making this problem challenging to solve. To overcome this challenge, we formulate the problem as a bandit learning framework and propose the Multi-agent Multi-type Thompson Sampling (MMTS) algorithm. The algorithm combines the strengths of Thompson Sampling for exploration with a double matching technique to achieve a stable matching outcome. Our theoretical analysis demonstrates the effectiveness of MMTS as it is able to achieve stability at every matching step, satisfies the incentive-compatibility property, and has a sublinear Bayesian regret over time. Our approach provides a useful method for addressing complementary preferences in real-world scenarios.

Incentive-Aware Recommender Systems in Two-Sided Markets

Online platforms in the Internet Economy commonly incorporate recommender systems that recommend arms (e.g., products) to agents (e.g., users). In such platforms, a myopic agent has a natural incentive to exploit, by choosing the best product given the current information rather than to explore various alternatives to collect information that will be used for other agents. We propose a novel recommender system that respects agents' incentives and enjoys asymptotically optimal performances expressed by the regret in repeated games. We model such an incentive-aware recommender system as a multi-agent bandit problem in a two-sided market which is equipped with an incentive constraint induced by agents' opportunity costs. If the opportunity costs are known to the principal, we show that there exists an incentive-compatible recommendation policy, which pools recommendations across a genuinely good arm and an unknown arm via a randomized and adaptive approach. On the other hand, if the opportunity costs are unknown to the principal, we propose a policy that randomly pools recommendations across all arms and uses each arm's cumulative loss as feedback for exploration. We show that both policies also satisfy an ex-post fairness criterion, which protects agents from over-exploitation.

On Large Batch Training and Sharp Minima: A Fokker-Planck Perspective

We study the statistical properties of the dynamic trajectory of stochastic gradient descent (SGD). We approximate the mini-batch SGD and the momentum SGD as stochastic differential equations (SDEs). We exploit the continuous formulation of SDE and the theory of Fokker-Planck equations to develop new results on the escaping phenomenon and the relationship with large batch and sharp minima. In particular, we find that the stochastic process solution tends to converge to flatter minima regardless of the batch size in the asymptotic regime. However, the convergence rate is rigorously proven to depend on the batch size. These results are validated empirically with various datasets and models.

Post-Regularization Confidence Bands for Ordinary Differential Equations

Ordinary differential equation (ODE) is an important tool to study the dynamics of a system of biological and physical processes. A central question in ODE modeling is to infer the significance of individual regulatory effect of one signal variable on another. However, building confidence band for ODE with unknown regulatory relations is challenging, and it remains largely an open question. In this article, we construct post-regularization confidence band for individual regulatory function in ODE with unknown functionals and noisy data observations. Our proposal is the first of its kind, and is built on two novel ingredients. The first is a new localized kernel learning approach that combines reproducing kernel learning with local Taylor approximation, and the second is a new de-biasing method that tackles infinite-dimensional functionals and additional measurement errors. We show that the constructed confidence band has the desired asymptotic coverage probability, and the recovered regulatory network approaches the truth with probability tending to one. We establish the theoretical properties when the number of variables in the system can be either smaller or larger than the number of sampling time points, and we study the regime-switching phenomenon. We demonstrate the efficacy of the proposed method through both simulations and illustrations with two data applications.

Orthogonal Statistical Inference for Multimodal Data Analysis

Multimodal imaging has transformed neuroscience research. While it presents unprecedented opportunities, it also imposes serious challenges. Particularly, it is difficult to combine the merits of interpretability attributed to a simple association model and flexibility achieved by a highly adaptive nonlinear model. In this article, we propose an orthogonal statistical inferential framework, built upon the Neyman orthogonality and a form of decomposition orthogonality, for multimodal data analysis. We target the setting that naturally arises in almost all multimodal studies, where there is a primary modality of interest, plus additional auxiliary modalities. We successfully establish the root-$N$-consistency and asymptotic normality of the estimated primary parameter, the semi-parametric estimation efficiency, and the asymptotic honesty of the confidence interval of the predicted primary modality effect. Our proposal enjoys, to a good extent, both model interpretability and model flexibility. It is also considerably different from the existing statistical methods for multimodal data integration, as well as the orthogonality-based methods for high-dimensional inferences. We demonstrate the efficacy of our method through both simulations and an application to a multimodal neuroimaging study of Alzheimer's disease.

Multi-Stage Decentralized Matching Markets: Uncertain Preferences and Strategic Behaviors

Matching markets are often organized in a multi-stage and decentralized manner. Moreover, participants in real-world matching markets often have uncertain preferences. This article develops a framework for learning optimal strategies in such settings, based on a nonparametric statistical approach and variational analysis. We propose an efficient algorithm, built upon concepts of "lower uncertainty bound" and "calibrated decentralized matching," for maximizing the participants' expected payoff. We show that there exists a welfare-versus-fairness trade-off that is characterized by the uncertainty level of acceptance. Participants will strategically act in favor of a low uncertainty level to reduce competition and increase expected payoff. We study signaling mechanisms that help to clear the congestion in such decentralized markets and find that the effects of signaling are heterogeneous, showing a dependence on the participants and matching stages. We prove that participants can be better off with multi-stage matching compared to single-stage matching. The deferred acceptance procedure assumes no limit on the number of stages and attains efficiency and fairness but may make some participants worse off than multi-stage matching. We demonstrate aspects of the theoretical predictions through simulations and an experiment using real data from college admissions.