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SV is a particular case of a more general concept called semi-value, which is a generalization to different weighting schemes. A semi-value is any valuation function with the form:

\[ v_\text{semi}(i) = \sum_{i=1}^n w(k) \sum_{S \subset D_{-i}^{(k)}} [u(S_{+i}) - u(S)], \]

where the coefficients \(w(k)\) satisfy the property:

\[\sum_{k=1}^n w(k) = 1,\]

the set \(D_{-i}^{(k)}\) contains all subsets of \(D\) of size \(k\) that do not include sample \(x_i\), \(S_{+i}\) is the set \(S\) with \(x_i\) added, and \(u\) is the utility function.

Two instances of this are Banzhaf indices (Wang and Jia, 2023)1, and Beta Shapley (Kwon and Zou, 2022)2, with better numerical and rank stability in certain situations.


Shapley values are a particular case of semi-values and can therefore also be computed with the methods described here. However, as of version 0.8.1, we recommend using compute_shapley_values instead, in particular because it implements truncation policies for TMCS.

Beta Shapley

For some machine learning applications, where the utility is typically the performance when trained on a set \(S \subset D\), diminishing returns are often observed when computing the marginal utility of adding a new data point.

Beta Shapley is a weighting scheme that uses the Beta function to place more weight on subsets deemed to be more informative. The weights are defined as:

\[ w(k) := \frac{B(k+\beta, n-k+1+\alpha)}{B(\alpha, \beta)}, \]

where \(B\) is the Beta function, and \(\alpha\) and \(\beta\) are parameters that control the weighting of the subsets. Setting both to 1 recovers Shapley values, and setting \(\alpha = 1\), and \(\beta = 16\) is reported in (Kwon and Zou, 2022)2 to be a good choice for some applications. Beta Shapley values are available in pyDVL through compute_beta_shapley_semivalues:

from pydvl.value import *

utility = Utility(model, data)
values = compute_beta_shapley_semivalues(
    u=utility, done=AbsoluteStandardError(threshold=1e-4), alpha=1, beta=16

See however the Banzhaf indices section for an alternative choice of weights which is reported to work better.

Banzhaf indices

As noted in the section Problems of Data Values, the Shapley value can be very sensitive to variance in the utility function. For machine learning applications, where the utility is typically the performance when trained on a set \(S \subset D\), this variance is often largest for smaller subsets \(S\). It is therefore reasonable to try reducing the relative contribution of these subsets with adequate weights.

One such choice of weights is the Banzhaf index, which is defined as the constant:

\[w(k) := 2^{n-1},\]

for all set sizes \(k\). The intuition for picking a constant weight is that for any choice of weight function \(w\), one can always construct a utility with higher variance where \(w\) is greater. Therefore, in a worst-case sense, the best one can do is to pick a constant weight.

The authors of (Wang and Jia, 2023)1 show that Banzhaf indices are more robust to variance in the utility function than Shapley and Beta Shapley values. They are available in pyDVL through compute_banzhaf_semivalues:

from pydvl.value import *

utility = Utility(model, data)
values = compute_banzhaf_semivalues(
    u=utility, done=AbsoluteStandardError(threshold=1e-4), alpha=1, beta=16

Banzhaf semi-values with MSR sampling

Wang et. al. propose a more sample-efficient method for computing Banzhaf semivalues in their paper Data Banzhaf: A Robust Data Valuation Framework for Machine Learning (Wang and Jia, 2023)1. This method updates all semivalues per evaluation of the utility (i.e. per model trained) based on whether a specific data point was included in the data subset or not. The expression for computing the semivalues is

\[\hat{\phi}_{MSR}(i) = \frac{1}{|\mathbf{S}_{\ni i}|} \sum_{S \in \mathbf{S}_{\ni i}} U(S) - \frac{1}{|\mathbf{S}_{\not{\ni} i}|} \sum_{S \in \mathbf{S}_{\not{\ni} i}} U(S)\]

where \(\mathbf{S}_{\ni i}\) are the subsets that contain the index \(i\) and \(\mathbf{S}_{\not{\ni} i}\) are the subsets not containing the index \(i\).

The function implementing this method is compute_msr_banzhaf_semivalues.

from pydvl.value import compute_msr_banzhaf_semivalues, RankCorrelation, Utility

utility = Utility(model, data)
values = compute_msr_banzhaf_semivalues(
  u=utility, done=RankCorrelation(rtol=0.001),
For further details on how to use this method and a comparison of the sample efficiency, we suggest to take a look at the example notebook msr_banzhaf_spotify.

General semi-values

As explained above, both Beta Shapley and Banzhaf indices are special cases of semi-values. In pyDVL we provide a general method for computing these with any combination of the three ingredients that define a semi-value:

  • A utility function \(u\).
  • A sampling method
  • A weighting scheme \(w\).

You can construct any combination of these three ingredients with compute_generic_semivalues. The utility function is the same as for Shapley values, and the sampling method can be any of the types defined in the samplers module. For instance, the following snippet is equivalent to the above:

from pydvl.value import *

data = Dataset(...)
utility = Utility(model, data)
values = compute_generic_semivalues(
  coefficient=beta_coefficient(alpha=1, beta=16),

Allowing any coefficient can help when experimenting with models which are more sensitive to changes in training set size. However, Data Banzhaf indices are proven to be the most robust to variance in the utility function, in the sense of rank stability, across a range of models and datasets (Wang and Jia, 2023)1.

Careful with permutation sampling

This generic implementation of semi-values allowing for any combination of sampling and weighting schemes is very flexible and, in principle, it recovers the original Shapley value, so that compute_shapley_values is no longer necessary. However, it loses the optimization in permutation sampling that reuses the utility computation from the last iteration when iterating over a permutation. This doubles the computation requirements (and slightly increases variance) when using permutation sampling, unless the cache is enabled. In addition, as mentioned above, truncation policies are not supported by this generic implementation (as of v0.8.1). For these reasons it is preferable to use compute_shapley_values whenever not computing other semi-values.

  1. Wang, J.T., Jia, R., 2023. Data Banzhaf: A Robust Data Valuation Framework for Machine Learning, in: Proceedings of The 26th International Conference on Artificial Intelligence and Statistics. Presented at the International Conference on Artificial Intelligence and Statistics, PMLR, pp. 6388–6421. 

  2. Kwon, Y., Zou, J., 2022. Beta Shapley: A Unified and Noise-reduced Data Valuation Framework for Machine Learning, in: Proceedings of the 25th International Conference on Artificial Intelligence and Statistics (AISTATS) 2022,. Presented at the AISTATS 2022, PMLR.