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

arXiv: https://arxiv.org/abs/2309.08923 · HTML: https://mlsanity.com/paper/2309.08923/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 36 references that are in the map.
- First posted on arXiv 16 Sep 2023.

## Builds on (most cited first, 6 of 6)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [The many Shapley values for model explanation](https://mlsanity.com/paper/1908.08474/) (2019)
- [Consistent Individualized Feature Attribution for Tree Ensembles](https://mlsanity.com/paper/1802.03888/) (2018)
- [Explaining black box decisions by Shapley cohort refinement](https://mlsanity.com/paper/1911.00467/) (2019)
- [A simple algorithm for global sensitivity analysis with Shapley effects](https://mlsanity.com/paper/2009.00874/) (2020)
- [The Design of Order-of-Addition Experiments](https://mlsanity.com/paper/1701.02786/) (2017)

## Abstract (arXiv)

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 2d 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.