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Modeling population dynamics is a fundamental problem with broad scientific applications. Motivated by real-world applications including biosystems with diverse populations, we consider a class of population dynamics modeling with two technical challenges: (i) dynamics to learn for individual particles are heterogeneous and (ii) available data to learn from are not time-series (i.e, each individual’s state trajectory over time) but cross-sectional (i.e, the whole population’s aggregated states without individuals matched over time). To address the challenges, we introduce a novel computational framework dubbed correlational Lagrangian Schrödinger bridge (CLSB) that builds on optimal transport to “bridge" cross-sectional data distributions. In contrast to prior methods regularizing all individuals’ transport “costs” and then applying them to the population homogeneously, CLSB directly regularizes population cost allowing for population heterogeneity and potentially improving model generalizability. Specifically our contributions include (1) a novel population perspective of the transport cost and a new class of population regularizers capturing the temporal variations in multivariate relations, with the tractable formulation derived, (2) three domain-informed instantiations of population regularizers on covariance, and (3) integration of population regularizers into data-driven generative models as constrained optimization and an approximate numerical solution, with further extension to conditional generative models. Empirically, we demonstrate the superiority of CLSB in single-cell sequencing data analyses (including cell differentiation and drug-conditioned cell responses) and opinion depolarization. 

This paper considers the problem of offline optimization, where the objective function is unknown except for a collection of “offline" data examples. While recent years have seen a flurry of work on applying various machine learning techniques to the offline optimization problem, the majority of these works focused on learning a surrogate of the unknown objective function and then applying existing optimization algorithms. While the idea of modeling the unknown objective function is intuitive and appealing, from the learning point of view it also makes it very difficult to tune the objective of the learner according to the objective of optimization. Instead of learning and then optimizing the unknown objective function, in this paper we take on a less intuitive but more direct view that optimization can be thought of as a process of sampling from a generative model. To learn an effective generative model from the offline data examples, we consider the standard technique of “re-weighting", and our main technical contribution is a probably approximately correct (PAC) lower bound on the natural optimization objective, which allows us to jointly learn a weight function and a score-based generative model from a surrogate loss function. The robustly competitive performance of the proposed approach is demonstrated via empirical studies using the standard offline optimization benchmarks.