Approximating the Shapley Value via Multi-Issue Decomposition

The Shapley value provides a fair method for the division of value in coalitional games. Motivated by the application of crowdsourcing for the collection of suitable labels and features for regression and classification tasks, we develop a method to approximate the Shapley value by identifying a suitable decomposition into multiple issues, with the decomposition computed by applying a graph partitioning to a pairwise similarity graph induced by the coalitional value function. The method is significantly faster and more accurate than existing random-sampling based methods on both synthetic data and data representing user contributions in a real world application of crowdsourcing to elicit labels and features for classification.Engineering and Applied Science

Approximating the Shapley Value without Marginal Contributions

The Shapley value is arguably the most popular approach for assigning a meaningful contribution value to players in a cooperative game, which has recently been used intensively in explainable artificial intelligence. The meaningfulness is due to axiomatic properties that only the Shapley value satisfies, which, however, comes at the expense of an exact computation growing exponentially with the number of agents. Accordingly, a number of works are devoted to the efficient approximation of the Shapley values, most of them revolve around the notion of an agent's marginal contribution. In this paper, we propose with SVARM and Stratified SVARM two parameter-free and domain-independent approximation algorithms based on a representation of the Shapley value detached from the notion of marginal contributions. We prove unmatched theoretical guarantees regarding their approximation quality and provide empirical results including synthetic games as well as common explainability use cases comparing ourselves with state-of-the-art methods

Explaining Data-Driven Decisions made by AI Systems: The Counterfactual Approach

We examine counterfactual explanations for explaining the decisions made by model-based AI systems. The counterfactual approach we consider defines an explanation as a set of the system's data inputs that causally drives the decision (i.e., changing the inputs in the set changes the decision) and is irreducible (i.e., changing any subset of the inputs does not change the decision). We (1) demonstrate how this framework may be used to provide explanations for decisions made by general, data-driven AI systems that may incorporate features with arbitrary data types and multiple predictive models, and (2) propose a heuristic procedure to find the most useful explanations depending on the context. We then contrast counterfactual explanations with methods that explain model predictions by weighting features according to their importance (e.g., SHAP, LIME) and present two fundamental reasons why we should carefully consider whether importance-weight explanations are well-suited to explain system decisions. Specifically, we show that (i) features that have a large importance weight for a model prediction may not affect the corresponding decision, and (ii) importance weights are insufficient to communicate whether and how features influence decisions. We demonstrate this with several concise examples and three detailed case studies that compare the counterfactual approach with SHAP to illustrate various conditions under which counterfactual explanations explain data-driven decisions better than importance weights

Fast Approximation of the Shapley Values Based on Order-of-Addition Experimental Designs

Shapley value is originally a concept in econometrics to fairly distribute both gains and costs to players in a coalition game. In the recent decades, its application has been extended to other areas such as marketing, engineering and machine learning. For example, it produces reasonable solutions for problems in sensitivity analysis, local model explanation towards the interpretable machine learning, node importance in social network, attribution models, etc. However, its heavy computational burden has been long recognized but rarely investigated. Specifically, in a $d$-player coalition game, calculating a Shapley value requires the evaluation of $d!$ or $2^d$ marginal contribution values, depending on whether we are taking the permutation or combination formulation of the Shapley value. Hence it becomes infeasible to calculate the Shapley value when $d$ is reasonably large. A common remedy is to take a random sample of the permutations to surrogate for the complete list of permutations. We find an advanced sampling scheme can be designed to yield much more accurate estimation of the Shapley value than the simple random sampling (SRS). Our sampling scheme is based on combinatorial structures in the field of design of experiments (DOE), particularly the order-of-addition experimental designs for the study of how the orderings of components would affect the output. We show that the obtained estimates are unbiased, and can sometimes deterministically recover the original Shapley value. Both theoretical and simulations results show that our DOE-based sampling scheme outperforms SRS in terms of estimation accuracy. Surprisingly, it is also slightly faster than SRS. Lastly, real data analysis is conducted for the C. elegans nervous system and the 9/11 terrorist network

Beyond Oaxaca-Blinder: Accounting for Differences in Household Income Distributions Across Countries

This paper develops a micro-econometric method to account for differences across distributions of household income. Going beyond the determination of earnings in labor markets, we also estimate statistical models for occupational choice and for the conditional distributions of education, fertility and non-labor incomes. We import combinations of estimated parameters from these models to simulate counterfactual income distributions. This allows us to decompose differences between functionals of two income distributions (such as inequality or poverty measures) into shares due to differences in the structure of labor market returns (price effects); differences in the occupational structure; and differences in the underlying distribution of assets (endowment effects). We apply the method to the differences between the Brazilian income distribution and those of the United States and Mexico, and find that most of Brazil's excess income inequality is due to underlying inequalities in the distribution of two key endowments: access to education and to sources of non-labor income, mainly pensions.http://deepblue.lib.umich.edu/bitstream/2027.42/39863/3/wp478.pd

Beyond Oaxaca-Blinder: Accounting for Differences in Household Income Distributions Across Countries

This paper develops a micro-econometric method to account for differences across distributions of household income. Going beyond the determination of earnings in labor markets, we also estimate statistical models for occupational choice and for the conditional distributions of education, fertility and non-labor incomes. We import combinations of estimated parameters from these models to simulate counterfactual income distributions. This allows us to decompose differences between functionals of two income distributions (such as inequality or poverty measures) into shares due to differences in the structure of labor market returns (price effects); differences in the occupational structure; and differences in the underlying distribution of assets (endowment effects). We apply the method to the differences between the Brazilian income distribution and those of the United States and Mexico, and find that most of Brazil's excess income inequality is due to underlying inequalities in the distribution of two key endowments: access to education and to sources of non-labor income, mainly pensions.Inequality, Distribution, Micro-simulations

Beyond Oaxaca-Blinder: accounting for differences in household income distributions across countries

This paper develops a micro-econometric method to account for differences across distributions of household income. Going beyond the determination of earnings in labor markets, we also estimate statistical models for occupational choice and for the conditional distributions of education, fertility and non-labor incomes. We import combinations of estimated parameters from these models to simulate counterfactual income distributions. This allows us to decompose differences between functionals of two income distributions (such as inequality or poverty measures) into shares due to differences in the structure of labor market returns (price effects); differences in the occupational structure; and differences in the underlying distribution of assets (endowment effects). We apply the method to the differences between the Brazilian income distribution and those of the United States and Mexico, and find that most of Brazil's excess income inequality is due to underlying inequalities in the distribution of two key endowments: access to education and to sources of non-labor income, mainly pensions.

On Sparse Discretization for Graphical Games

This short paper concerns discretization schemes for representing and computing approximate Nash equilibria, with emphasis on graphical games, but briefly touching on normal-form and poly-matrix games. The main technical contribution is a representation theorem that informally states that to account for every exact Nash equilibrium using a nearby approximate Nash equilibrium on a grid over mixed strategies, a uniform discretization size linear on the inverse of the approximation quality and natural game-representation parameters suffices. For graphical games, under natural conditions, the discretization is logarithmic in the game-representation size, a substantial improvement over the linear dependency previously required. The paper has five other objectives: (1) given the venue, to highlight the important, but often ignored, role that work on constraint networks in AI has in simplifying the derivation and analysis of algorithms for computing approximate Nash equilibria; (2) to summarize the state-of-the-art on computing approximate Nash equilibria, with emphasis on relevance to graphical games; (3) to help clarify the distinction between sparse-discretization and sparse-support techniques; (4) to illustrate and advocate for the deliberate mathematical simplicity of the formal proof of the representation theorem; and (5) to list and discuss important open problems, emphasizing graphical-game generalizations, which the AI community is most suitable to solve.Comment: 30 pages. Original research note drafted in Dec. 2002 and posted online Spring'03 (http://www.cis.upenn. edu/~mkearns/teaching/cgt/revised_approx_bnd.pdf) as part of a course on computational game theory taught by Prof. Michael Kearns at the University of Pennsylvania; First major revision sent to WINE'10; Current version sent to JAIR on April 25, 201

Utilizing Machine Learning to Greatly Expand the Range and Accuracy of Bottom-Up Coarse-Grained Models Through Virtual Particles

Coarse-grained (CG) models parameterized using atomistic reference data, i.e., 'bottom up' CG models, have proven useful in the study of biomolecules and other soft matter. However, the construction of highly accurate, low resolution CG models of biomolecules remains challenging. We demonstrate in this work how virtual particles, CG sites with no atomistic correspondence, can be incorporated into CG models within the context of relative entropy minimization (REM) as latent variables. The methodology presented, variational derivative relative entropy minimization (VD-REM), enables optimization of virtual particle interactions through a gradient descent algorithm aided by machine learning. We apply this methodology to the challenging case of a solvent-free CG model of a 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) lipid bilayer and demonstrate that introduction of virtual particles captures solvent-mediated behavior and higher-order correlations which REM alone cannot capture in a more standard CG model based only on the mapping of collections of atoms to the CG sites.Comment: 35 pages, 9 figure