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Public goods are often either over-consumed in the absence of regulatory mechanisms, or remain completely unused, as in the Covid-19 pandemic, where social distance constraints are enforced to limit the number of people who can share public spaces. In this work, we plug this gap through market based mechanisms designed to efficiently allocate capacity constrained public goods. To design these mechanisms, we leverage the theory of Fisher markets, wherein each agent in the economy is endowed with an artificial currency budget that they can spend to avail public goods. While Fisher markets provide a strong methodological backbone to model resource allocation problems, their applicability is limited to settings involving two types of constraints - budgets of individual buyers and capacities of goods. Thus, we introduce a modified Fisher market, where each individual may have additional physical constraints, characterize its solution properties and establish the existence of a market equilibrium. Furthermore, to account for additional constraints we introduce a social convex optimization problem where we perturb the budgets of agents such that the KKT conditions of the perturbed social problem establishes equilibrium prices. Finally, to compute the budget perturbations we present a fixed point scheme and illustrate convergence guarantees through numerical experiments. Thus, our mechanism, both theoretically and computationally, overcomes a fundamental limitation of classical Fisher markets, which only consider capacity and budget constraints. 

Abstract MotivationProperties of molecules are indicative of their functions and thus are useful in many applications. With the advances of deep-learning methods, computational approaches for predicting molecular properties are gaining increasing momentum. However, there lacks customized and advanced methods and comprehensive tools for this task currently. ResultsHere, we develop a suite of comprehensive machine-learning methods and tools spanning different computational models, molecular representations and loss functions for molecular property prediction and drug discovery. Specifically, we represent molecules as both graphs and sequences. Built on these representations, we develop novel deep models for learning from molecular graphs and sequences. In order to learn effectively from highly imbalanced datasets, we develop advanced loss functions that optimize areas under precision–recall curves (PRCs) and receiver operating characteristic (ROC) curves. Altogether, our work not only serves as a comprehensive tool, but also contributes toward developing novel and advanced graph and sequence-learning methodologies. Results on both online and offline antibiotics discovery and molecular property prediction tasks show that our methods achieve consistent improvements over prior methods. In particular, our methods achieve #1 ranking in terms of both ROC-AUC (area under curve) and PRC-AUC on the AI Cures open challenge for drug discovery related to COVID-19. Availability and implementationOur source code is released as part of the MoleculeX library (https://github.com/divelab/MoleculeX) under AdvProp. Supplementary informationSupplementary data are available at Bioinformatics online.