Library companion to the paper Efficient Shapley Performance Attribution for Least-Squares Regression by Logan Bell, Nikhil Devanathan, and Stephen Boyd.

The results provided in the reference paper were generated using a more performant, but harder to use implementation of the same algorithm. This benchmark code and the numerical experiments from the reference paper can be found at cvxgrp/ls-spa-benchmark. We recommend caution in trying to use the benchmark code.

To install this package, execute

pip install ls_spa

Import ls_spa by adding

from ls_spa import ls_spa

to the top of your Python file.

ls_spa has the following dependencies:

• numpy
• scipy
• pandas
• joblib

Optional dependencies are

• marimo for using the demo notebook
• matplotlib for plotting in the demo notebook

We assume that you have imported ls_spa and you have a $N\times p$ matrix of training data X_train, a $M\times p$ matrix of testing data X_test, a $N$ vector of training labels y_train, and a $M$ vector of testing labels y_test for positive integers $p, N, M$ with $N,M\geq p$. In this case, you can find the Shapley attribution of the out-of-sample $R^2$ on your data by executing

attrs = ls_spa(X_train, X_test, y_train, y_test).attribution

attrs will be a NumPy array containing the Shapley values of your features. The ls_spa function computes Shapley values for the given data using the LS-SPA method described in the companion paper. It takes arguments:

• X_train: Training feature matrix (NumPy array or pandas DataFrame).
• X_test: Testing feature matrix (NumPy array or pandas DataFrame).
• y_train: Training response vector (NumPy array or pandas Series).
• y_test: Testing response vector (NumPy array or pandas Series).

We present a complete Python script that utilizes LS-SPA to compute the Shapley attribution on the data from the toy example described in the companion paper.

# Imports
import numpy as np
from ls_spa import ls_spa

# Data loading
X_train, X_test, y_train, y_test = [np.load("./data/toy_data.npz")[key] for key in ["X_train","X_test","y_train","y_test"]]

# Compute Shapley attribution with LS-SPA
results = ls_spa(X_train, X_test, y_train, y_test)

# Print attribution
print(results)

This example uses data from the data directory of this repository.

The line print(results) prints a dashboard of information generated while computing the Shapley attribution such as the attribution, the $R^2$ of the model fitted with all of the features, the feature cofficients of the fitted model, and an error estimate on the attribution (since LS-SPA is a method of estimation).

To extract just the vector of Shapley values, use results.attribution. For more info, see optional arguments.

In this demo, we walk through the process of computing Shapley values on the data for the toy example in the companion paper. We then use ls_spa to compute the Shapley attribution on the same data.

ls_spa takes the optional arguments:

• reg: Ridge regularization parameter (Default 0.0).
• max_samples: Maximum number of feature permutations to sample (Default 8192).
• batch_size: Number of permutations to process per batch (Default 256).
• tolerance: Stopping criterion for estimation error (Default 0.01).
• seed: Seed for random number generation (Default 42).
• perms: Permutation sampling method (Default None). Options include:
  • None: Auto-select "exact" for p < 9 features, otherwise "random"
  • "exact": Enumerate all permutations (only feasible for p < 9)
  • "random": Uniformly random permutations
  • "argsort": Quasi-Monte Carlo permutations using argsort
  • "permutohedron": Quasi-Monte Carlo permutations from permutohedron lattice
  • Custom array or tuple of permutations
• antithetical: Use antithetical (paired) sampling for variance reduction (Default True).
• return_attribution_history: Return convergence history of attributions (Default False).
• n_jobs: Number of parallel jobs; use -1 for all CPU cores (Default 1).

ls_spa returns a ShapleyResults object. The ShapleyResults object has the fields:

• attribution: Array of Shapley values for each feature.
• theta: Array of regression coefficients with all features.
• r_squared: Out-of-sample R² with all features.
• overall_error: Estimated error (95th percentile L2 norm) in Shapley attribution vector.
• attribution_errors: Array of estimated errors for each feature's attribution.
• error_history: Array of error estimates after each batch. None if using exact computation.
• attribution_history: Array of attribution estimates over time. None if return_attribution_history=False.

If you use this code for research, please cite the associated paper.

@article{Bell2024,
  title = {Efficient Shapley performance attribution for least-squares regression},
  volume = {34},
  ISSN = {1573-1375},
  url = {http://dx.doi.org/10.1007/s11222-024-10459-9},
  DOI = {10.1007/s11222-024-10459-9},
  number = {5},
  journal = {Statistics and Computing},
  publisher = {Springer Science and Business Media LLC},
  author = {Bell,  Logan and Devanathan,  Nikhil and Boyd,  Stephen},
  year = {2024},
  month = jul
}