[Paper Review] A Nonparametric Framework for Online Stochastic Matching with Correlated Arrivals

Ali Aouad, Will Ma|arXiv (Cornell University)|Aug 3, 2022

Optimization and Search Problems4 citations

TL;DR

This paper introduces a nonparametric framework for online stochastic matching that relaxes the standard assumption of serial independence in demand arrivals, allowing for arbitrary, correlated, and high-variance demand distributions. By proposing two models—Indep (independent type demands) and Correl (correlated type distribution given total demand)—the authors show that fluid LP relaxations based on expectations alone can be arbitrarily poor benchmarks, and instead develop tightened LP relaxations that achieve constant-factor competitive ratios under adversarial and random-order arrivals.

ABSTRACT

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.

Motivation & Objective

• To address the limitation of existing online stochastic matching models that assume serial independence and low-variance demand, which restricts applicability to real-world data with high variability.
• To develop a data-driven, nonparametric framework that captures arbitrary demand distributions and correlations across query types without relying on parametric assumptions like Poisson or i.i.d. arrivals.
• To demonstrate that fluid LP relaxations—commonly used in prior work—are arbitrarily poor benchmarks under high-variance, correlated demand, and thus require tightening.
• To design competitive online matching algorithms with provable constant-factor performance guarantees under adversarial and random-order arrival sequences in the new models.
• To provide a general modeling framework that combines empirical demand distributions with standard online arrival models, enabling richer, more realistic stochastic matching formulations.

Proposed method

• Proposes two new models: Indep, where each type's demand is drawn independently from an arbitrary distribution; and Correl, where total demand is drawn first, followed by i.i.d. type assignment conditional on total demand.
• Introduces tightened linear programming (LP) relaxations that incorporate the full distribution of demand, rather than just expectations, to better capture variance and correlation.
• Applies prophet inequality techniques to bound the performance of resource-specific threshold policies, showing that each resource's contribution is at least half the value of an optimal threshold policy.
• Uses a rounding strategy based on the optimal solution of the tightened LP to assign queries to resources, ensuring feasibility and competitive performance.
• Establishes a lower bound on the expected reward of the online policy by relating it to the optimal threshold policy’s payoff via stochastic dominance and expectation arguments.
• Employs a decomposition approach where the total reward is analyzed per resource, leveraging structural properties of the LP to show that constraints are nearly binding, enabling tight performance analysis.

Figure 1: Illustration of TypeRound in stage $t=3$ for an example with $L=5$ , $\bar{p}_{\ell}=\frac{1}{2^{\ell-1}}$ for all $\ell\in[L]$ , and ${\bf x}_{j}=(\frac{1}{8},\frac{3}{8},\frac{7}{8},\frac{1}{4},0)$ . The $y$ -axis visualizes the probabilistic space, including the disjointedness of events

Experimental results

Research questions

• RQ1 Can online matching policies maintain constant-factor performance guarantees when demand distributions are non-Poisson and exhibit high variance, violating the standard serial independence assumption?
• RQ2 To what extent does the fluid LP relaxation—based solely on expected demands—fail as a benchmark under correlated or heavy-tailed demand distributions?
• RQ3 How can LP relaxations be tightened to incorporate full distributional information of demand, especially in settings with arbitrary or correlated type arrivals?
• RQ4 Can constant-factor competitive algorithms be designed for online matching under adversarial or random-order arrival sequences when demand variance is not bounded by the mean?
• RQ5 What is the role of nonparametric demand modeling in improving the robustness and accuracy of online matching policies in data-driven operational settings?

Key findings

• The fluid LP relaxation, which uses only expected demands, can be arbitrarily poor as a benchmark in settings with high-variance or correlated demand, invalidating its use as a performance proxy.
• The proposed Indep and Correl models allow for arbitrary marginal or total demand distributions, respectively, and generalize standard models like Poisson or i.i.d. arrivals.
• For both models, the authors design online matching algorithms that achieve a 1/2-competitive ratio under adversarial and random-order arrivals, respectively.
• The performance guarantee is derived by showing that each resource’s contribution is at least half the payoff of an optimal threshold policy, leveraging prophet inequality results.
• The tightened LP relaxations, which incorporate the full distribution of demand, are essential for achieving constant-factor performance and outperform standard fluid relaxations in high-variance settings.
• The framework enables data-driven modeling by allowing empirical estimation of demand distributions from real data, which can then be combined with standard arrival order models for algorithmic design.

Figure 2: Each of the 1600 dots represents a (SKU, location)-combination $j$ . The $x$ -coordinate is the empirical mean weekly demand for that SKU from that location, and the $y$ -coordinate is the natural logarithm of the ratio of empirical variance to empirical mean. The property $\mathrm{Var}(D_