Improving KernelSHAP: Practical Shapley Value Estimation via Linear Regression

The Shapley value solution concept from cooperative game theory has become popular for interpreting ML models, but efficiently estimating Shapley values remains challenging, particularly in the model-agnostic setting. We revisit the idea of estimating Shapley values via linear regression to understand and improve upon this approach. By analyzing KernelSHAP alongside a newly proposed unbiased estimator, we develop techniques to detect its convergence and calculate uncertainty estimates. We also find that that the original version incurs a negligible increase in bias in exchange for a significant reduction in variance, and we propose a variance reduction technique that further accelerates the convergence of both estimators. Finally, we develop a version of KernelSHAP for stochastic cooperative games that yields fast new estimators for two global explanation methods.

[1]  Scott Lundberg,et al.  A Unified Approach to Interpreting Model Predictions , 2017, NIPS.

[2]  Carlos Guestrin,et al.  "Why Should I Trust You?": Explaining the Predictions of Any Classifier , 2016, ArXiv.

[3]  Looking Deeper into Tabular LIME , 2020, 2008.11092.

[4]  B. Welford Note on a Method for Calculating Corrected Sums of Squares and Products , 1962 .

[5]  Clement Adebamowo,et al.  A Comprehensive Pan-Cancer Molecular Study of Gynecologic and Breast Cancers. , 2018, Cancer cell.

[6]  Erik Strumbelj,et al.  Explaining prediction models and individual predictions with feature contributions , 2014, Knowledge and Information Systems.

[7]  D. Monderer,et al.  Variations on the shapley value , 2002 .

[8]  L. Shapley A Value for n-person Games , 1988 .

[9]  Anh Nguyen,et al.  SAM: The Sensitivity of Attribution Methods to Hyperparameters , 2020, 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).

[10]  Yair Zick,et al.  Algorithmic Transparency via Quantitative Input Influence: Theory and Experiments with Learning Systems , 2016, 2016 IEEE Symposium on Security and Privacy (SP).

[11]  R. Tibshirani Regression Shrinkage and Selection via the Lasso , 1996 .

[12]  Art B. Owen,et al.  Sobol' Indices and Shapley Value , 2014, SIAM/ASA J. Uncertain. Quantification.

[13]  Paulo Cortez,et al.  A data-driven approach to predict the success of bank telemarketing , 2014, Decis. Support Syst..

[14]  Claudio Borio,et al.  Risk Attribution Using the Shapley Value: Methodology and Policy Applications , 2016 .

[15]  Anh Nguyen,et al.  SAM: The Sensitivity of Attribution Methods to Hyperparameters , 2020, 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).

[16]  Michel Grabisch,et al.  Equivalent Representations of Set Functions , 2000, Math. Oper. Res..

[17]  Le Song,et al.  L-Shapley and C-Shapley: Efficient Model Interpretation for Structured Data , 2018, ICLR.

[18]  Abraham Charnes,et al.  Prior Solutions: Extensions of Convex Nucleus Solutions to Chance-Constrained Games. , 1973 .

[19]  Scott Lundberg,et al.  Explaining by Removing Explaining by Removing: A Unified Framework for Model Explanation , 2020 .

[20]  Scott Lundberg,et al.  Understanding Global Feature Contributions With Additive Importance Measures , 2020, NeurIPS.

[21]  Daniel Gómez,et al.  Polynomial calculation of the Shapley value based on sampling , 2009, Comput. Oper. Res..

[22]  Damien Garreau,et al.  An Analysis of LIME for Text Data , 2020, AISTATS.

[23]  A. Charnes,et al.  Extremal Principle Solutions of Games in Characteristic Function Form: Core, Chebychev and Shapley Value Generalizations , 1988 .

[24]  Abraham Charnes,et al.  Coalitional and Chance-Constrained Solutions to N-Person Games. I. The Prior Satisficing Nucleolus. , 1976 .

[25]  Tie-Yan Liu,et al.  LightGBM: A Highly Efficient Gradient Boosting Decision Tree , 2017, NIPS.

[26]  James Zou,et al.  Neuron Shapley: Discovering the Responsible Neurons , 2020, NeurIPS.

[27]  Jean-Luc Marichal,et al.  Weighted Banzhaf power and interaction indexes through weighted approximations of games , 2010, Eur. J. Oper. Res..

[28]  Markus H. Gross,et al.  Explaining Deep Neural Networks with a Polynomial Time Algorithm for Shapley Values Approximation , 2019, ICML.

[29]  James Y. Zou,et al.  Data Shapley: Equitable Valuation of Data for Machine Learning , 2019, ICML.

[30]  Ankur Taly,et al.  The Explanation Game: Explaining Machine Learning Models Using Shapley Values , 2020, CD-MAKE.

[31]  R. Aumann Economic Applications of the Shapley Value , 1994 .

[32]  Anna Veronika Dorogush,et al.  CatBoost: unbiased boosting with categorical features , 2017, NeurIPS.

[33]  Hugh Chen,et al.  From local explanations to global understanding with explainable AI for trees , 2020, Nature Machine Intelligence.

[34]  Jianhua Chen,et al.  Transforms of pseudo-Boolean random variables , 2010, Discret. Appl. Math..

[35]  Talal Rahwan,et al.  Bounding the Estimation Error of Sampling-based Shapley Value Approximation With/Without Stratifying , 2013, ArXiv.

[36]  G. Zaccour,et al.  Time-consistent Shapley value allocation of pollution cost reduction , 1999 .

[37]  Peter L. Hammer,et al.  Approximations of pseudo-Boolean functions; applications to game theory , 1992, ZOR Methods Model. Oper. Res..

[38]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[39]  Ulrike von Luxburg,et al.  Looking deeper into LIME , 2020, ArXiv.

[40]  Jianhua Chen,et al.  Formulas for approximating pseudo-Boolean random variables , 2008, Discret. Appl. Math..

[41]  Barry L. Nelson,et al.  Shapley Effects for Global Sensitivity Analysis: Theory and Computation , 2016, SIAM/ASA J. Uncertain. Quantification.

[42]  S. Lipovetsky,et al.  Analysis of regression in game theory approach , 2001 .