SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \text {s . t .} \operatorname {C o v} (v _ {n}) = \boldsymbol {\Sigma} ^ {[ n ]}. \end{array}
$$
Theorem 1. Under Assumption 1, and further assume  $\sigma_{12} < \min(\sigma_{11}, \sigma_{22})$ , the Dirac delta distribution  $\delta_{w^*}$  with
$$
w ^ {*} = \frac {\sigma_ {1 1} - \sigma_ {1 2}}{\sigma_ {1 1} + \sigma_ {2 2} - 2 \sigma_ {1 2}} \tag {8}
$$
is optimal to the problem (7) for every  $n \geq 1$ . In addition, the optimal solution is unique if  $n > 3$ .
Remark 2. Theorem 1 suggests that 0.5-weighted Banzhaf value (i.e., the Banzhaf value) is the most robust provided that  $\sigma_{11} = \sigma_{22}$ , which coincides with (Wang and Jia 2023, Theorem 4.6). Specifically,
they simulated some isotropic Gaussian noises added to deterministic utility functions to verify that the Banzhaf value is the most robust in ranking data, which is also supported by Theorem 1. However, the first column of Figure 3 shows that the Banzhaf value is not always the most robust if the added noises follow Kronecker noises with  $\sigma_{11} \neq \sigma_{22}$ . In practice, Table 2 also refutes that the Banzhaf value is universally the most robust.
Our experiments contain three aspects: 1) approximation efficiency of the MSR estimator for weighted Banzhaf values; 2) verifying Theorem 1 using synthetic and authentic noises; 3) empirical study to show that the most robust/effective semi-values tend to be some weighted Banzhaf values.
All datasets used are from open sources, and are classification tasks. Except for MNIST and FMNIST, each  $\mathcal{D}_{tr}$  or  $\mathcal{D}_{val}$  is balanced between different classes. Without explicitly stated, we set  $|\mathcal{D}_{val}| = 200$ . All utility functions are set to be the accuracy reported on  $\mathcal{D}_{val}$  with logistic regression models being trained on  $\mathcal{D}_{tr}$ , except that we implement LeNet (LeCun et al. 1998) for MNIST and FMNIST. To have the merit of efficiency, we adopt one-epoch one-mini-batch learning for training models in all types of experiments (Ghorbani and Zou 2019). More experiment details and results are included in Appendices D and E.
Figure 2: Comparison of three estimators for 0.8-Banzhaf value on 2dplanes dataset. The shaded region represents the standard deviation over 10 different random seeds.
4.1 Approximation Efficiency
The MSR estimator is compared with reweighted sampling lift and sampling lift estimators. To calculate the exact values, we set  $|\mathcal{D}_{tr}| = 16$ . Besides, the learning rate is set to be 1.0. To have fair comparison, we plot relative difference  $\| \hat{\phi} - \phi \|_2 / \| \phi \|_2$  along the number of average utility evaluations, i.e.  $\frac{\#\text{utility evaluations}}{n}$ . Figure 2 is plotted with 10 different random seeds used by estimators. It demonstrates that the MSR estimator is significantly the most efficient.
4.2 Verification for the Theorem 1
Experiments based on synthetic and authentic noises are designed to verify Theorem 1. We set  $|\mathcal{D}_{tr}| = 10$  so that the exact values can be cheaply computed.
For synthetic experiments, we simulate Kronecker noises with each  $\epsilon_{i}$  independently and identically following a Gaussian distribution  $\mathcal{N}(0,\Sigma)$ , which are employed as added noises to deterministic utility functions. Besides, the learning rate is set to be 0.05. Specifically, we fix  $\sigma_{11}$  and  $\sigma_{22}$  while varying  $\sigma_{12}$  along the horizontal axis. For each  $\sigma_{12}$ , the corresponding added noises are simulated using 10 different seeds. The Spearman's rank correlation coefficient is used to measure the similarity between the rankings of the ground-truth value and any noisy one. The results are reported in the first column of Figure 3, where the one tagged as "robust semi-value" refers to the corresponding weighted Banzhaf values predicted by Eq. (8). It is clear that Theorem 1 predicts the most robust semi-value well, and that the Banzhaf value is not universally the most robust under these synthetic noises.
Next, we examine Theorem 1 by fitting a Kronecker noise to the stochasticity inherent in the given utility function, and then compute the predicted most robust semi-value using Eq. (8). To produce simple utility functions, we set  $|\mathcal{D}_{val}| = 10$ . Besides, the learning rate for utility functions is set to be 1.0. The underlying covariance matrix is empirically approximated by  $\hat{\mathbf{D}}$ , and then we fit a Kronecker
Figure 3: (a) Synthetic noises: Each point represents the Spearman's rank correlation coefficient averaged over 10 random seeds. For each  $\sigma_{12}$ , the robust semi-value is the weighted Banzhaf value parameterized by  $(\sigma_{11} - \sigma_{12}) / (\sigma_{11} + \sigma_{22} - 2\sigma_{12})$ , as predicted by Theorem 1. For the first row, we set  $\sigma_{11} = 1.5$  and  $\sigma_{22} = 0.5$ , while it is  $\sigma_{11} = 0.5$  and  $\sigma_{22} = 1.5$  for the other. (b) Authentic noises: Ranking consistency is measured by the mean and standard deviation of the Spearman's rank correlation coefficients between all pairs of 10 random seeds. The variance refers to  $\sum_{i=1}^{10} \|\hat{\phi}^i - \mathbf{m}\|_2^2 / 10$  where  $\mathbf{m} = \frac{1}{10} \sum_{i=1}^{10} \hat{\phi}^i$ . The computed "robust" parameters are 0.5255 and 0.5341 for iris and phoneme datasets, respectively.
noise by optimizing the Kullback-Leibler divergence between Gaussian distributions, which is
$$
\operatorname *{argmin}_{\boldsymbol {\Sigma}\in \mathbb{R}^{2\times 2}}KL\Bigl(\mathcal{N}(0,\hat{\boldsymbol{D}})\mid \mathcal{N}(0,\overbrace{\boldsymbol{\Sigma}\otimes\boldsymbol{\Sigma}\otimes\cdots\otimes\boldsymbol{\Sigma}}^{10\text{repetitions}})\Bigr).
$$
The final results averaged over 10 random seeds are presented in the last two columns of Figure 3. Each parameter in  $\{0.00, 0.05, 0.10, \dots, 0.95, 1.00\}$  defines a specific weighted Banzhaf value, while  $\{(16,1), (4,1), (1,4), (1,16)\}$  are parameters for the Beta Shapley values. The one tagged as "robust" is generated from the approximate Kronecker noise using Eq. (8). As demonstrated, the predicted "robust" semi-values do achieve the best in both ranking consistency and variance.
4.3 Empirical Study
Following the previous study (Wang and Jia 2023), we implement an exploratory study on 23 datasets, including two types of experiment: 1) noisy label detection and 2) ranking consistency.
We fix  $|\mathcal{D}_{tr}| = 1,000$  for all datasets except that it is  $|\mathcal{D}_{tr}| = 2,000$  for MNIST and FMNIST, which means the induced stochasticity inherent in utility functions is more complicated than Assumption 1. For each dataset, the chosen model is individually fine-tuned for the learning rate. The randomness in each utility function stems from the underlying random seed that determines the order of feeding training data as well as the initialization of models. All results are reported using mean and standard deviation over 10 random seeds. For each estimator, the total number of utility evaluations is set to be 400,000. Permutation estimator (Castro et al. 2009) is used for Shapley value, reweighted sampling lift estimator for Beta Shapley, and MSR estimator for weighted Banzhaf values. Specifically, 0-weighted Banzhaf value and 1-weighted Banzhaf value require only  $n + 1$  utility evaluations to compute exactly, and thus we do not use any estimators for them.
Table 2: Ranking consistency. For weighted Banzhaf values, we only report the best one from the parameter list  $\{0.00, 0.05, 0.10, \ldots, 0.95, 1.00\}$ . The first value in the column of "Weighted Banzhaf" is the parameter that achieves the highest ranking consistency. Boldface numbers mark the best. The number following each dataset refers to the number of classes used in the experiments.
Dataset	Weighted Banzhaf	Banzhaf	Beta(16, 1)	Beta(4, 1)	Beta(1, 4)	Beta(1, 16)	Shapley
MNIST (10)	0.20	0.680 ± 0.120	0.581 ± 0.075	0.073 ± 0.082	0.020 ± 0.043	-0.004 ± 0.039	-0.009 ± 0.037	0.011 ± 0.019
FMNIST (10)	0.30	0.670 ± 0.042	0.562 ± 0.040	0.017 ± 0.092	0.007 ± 0.046	-0.001 ± 0.035	0.009 ± 0.029	0.002 ± 0.021
2dplanes (2)	0.80	0.950 ± 0.012	0.839 ± 0.091	0.424 ± 0.155	0.358 ± 0.133	0.046 ± 0.087	0.001 ± 0.130	0.256 ± 0.083
bank-marketing (2)	0.75	0.895 ± 0.028	0.814 ± 0.092	0.599 ± 0.096	0.445 ± 0.093	0.015 ± 0.094	0.041 ± 0.128	0.226 ± 0.033
bioresponse (2)	0.00	0.997 ± 0.000	0.948 ± 0.003	0.193 ± 0.032	0.169 ± 0.030	0.016 ± 0.037	0.004 ± 0.034	0.122 ± 0.021
covertype (7)	0.35	0.840 ± 0.033	0.832 ± 0.032	0.618 ± 0.066	0.490 ± 0.051	0.035 ± 0.134	-0.028 ± 0.227	0.234 ± 0.026
cpu (2)	0.20	0.926 ± 0.011	0.874 ± 0.008	0.549 ± 0.096	0.298 ± 0.084	0.003 ± 0.132	-0.004 ± 0.158	0.142 ± 0.028
default (2)	0.30	0.953 ± 0.011	0.941 ± 0.007	0.542 ± 0.078	0.369 ± 0.059	0.022 ± 0.090	0.004 ± 0.131	0.170 ± 0.034
gas (6)	0.05	0.959 ± 0.009	0.729 ± 0.019	0.204 ± 0.076	0.105 ± 0.057	0.003 ± 0.068	-0.017 ± 0.094	0.063 ± 0.032
har (6)	0.05	0.956 ± 0.003	0.741 ± 0.012	0.133 ± 0.063	0.074 ± 0.057	-0.004 ± 0.050	-0.001 ± 0.058	0.051 ± 0.031
letter (26)	0.50	0.966 ± 0.004	0.966 ± 0.004	0.530 ± 0.048	0.346 ± 0.065	0.021 ± 0.110	-0.015 ± 0.148	0.197 ± 0.031
optdigits (10)	0.25	0.923 ± 0.010	0.898 ± 0.009	0.687 ± 0.039	0.485 ± 0.049	0.024 ± 0.076	0.004 ± 0.145	0.278 ± 0.030
pendigits (10)	0.05	0.947 ± 0.014	0.806 ± 0.010	0.384 ± 0.100	0.259 ± 0.085	0.031 ± 0.096	0.024 ± 0.142	0.113 ± 0.024
phoneme (2)	0.85	0.783 ± 0.159	0.588 ± 0.309	0.601 ± 0.141	0.499 ± 0.148	0.028 ± 0.125	0.008 ± 0.153	0.325 ± 0.105
pol (2)	0.25	0.970 ± 0.003	0.938 ± 0.003	0.479 ± 0.108	0.320 ± 0.105	0.066 ± 0.144	0.033 ± 0.158	0.186 ± 0.026
satimage (6)	0.30	0.863 ± 0.024	0.814 ± 0.028	0.341 ± 0.135	0.189 ± 0.087	0.004 ± 0.086	0.006 ± 0.087	0.100 ± 0.033
segment (7)	0.25	0.941 ± 0.012	0.883 ± 0.006	0.361 ± 0.118	0.213 ± 0.069	0.001 ± 0.102	0.005 ± 0.149	0.113 ± 0.030
spambase (2)	0.20	0.939 ± 0.007	0.876 ± 0.007	0.466 ± 0.073	0.336 ± 0.070	-0.001 ± 0.094	0.004 ± 0.148	0.194 ± 0.030
texture (11)	0.05	0.961 ± 0.011	0.790 ± 0.009	0.217 ± 0.073	0.109 ± 0.050	0.003 ± 0.072	-0.020 ± 0.080	0.049 ± 0.032
wind (2)	0.00	0.793 ± 0.111	0.720 ± 0.036	0.398 ± 0.091	0.215 ± 0.078	0.004 ± 0.079	0.012 ± 0.114	0.116 ± 0.028
Table 3: Noisy label detection. For weighted Banzhaf values, we only report the best one from the parameter list  $\{0.00, 0.05, 0.10, \ldots, 0.95, 1.00\}$ . The first value in the column of "Weighted Banzhaf" is the parameter that achieves the highest F1-score. Boldface numbers mark the best. The number following each dataset refers to the number of classes used in the experiments.
Dataset	Weighted Banzhaf	Banzhaf	Beta(16, 1)	Beta(4, 1)	Beta(1, 16)	Beta(1, 4)	Shapley
MNIST (10)	0.50	0.625 ± 0.040	0.625 ± 0.040	0.228 ± 0.023	0.226 ± 0.017	0.210 ± 0.013	0.207 ± 0.013	0.229 ± 0.020
FMNIST (10)	0.25	0.496 ± 0.058	0.448 ± 0.027	0.217 ± 0.019	0.223 ± 0.017	0.208 ± 0.015	0.209 ± 0.016	0.212 ± 0.016
2dplanes (2)	0.05	0.609 ± 0.059	0.346 ± 0.014	0.572 ± 0.052	0.532 ± 0.055	0.219 ± 0.035	0.210 ± 0.030	0.412 ± 0.053
bank-marketing (2)	0.05	0.528 ± 0.015	0.460 ± 0.014	0.487 ± 0.025	0.452 ± 0.026	0.247 ± 0.024	0.226 ± 0.020	0.368 ± 0.025
bioresponse (2)	0.05	0.502 ± 0.012	0.481 ± 0.011	0.323 ± 0.031	0.316 ± 0.030	0.221 ± 0.022	0.204 ± 0.027	0.293 ± 0.025
covertype (7)	0.05	0.518 ± 0.026	0.318 ± 0.025	0.481 ± 0.025	0.398 ± 0.026	0.217 ± 0.014	0.212 ± 0.023	0.318 ± 0.032
cpu (2)	0.05	0.772 ± 0.017	0.648 ± 0.015	0.682 ± 0.031	0.508 ± 0.045	0.242 ± 0.028	0.212 ± 0.031	0.390 ± 0.027
default (2)	0.05	0.398 ± 0.010	0.346 ± 0.011	0.355 ± 0.019	0.316 ± 0.019	0.203 ± 0.022	0.211 ± 0.023	0.286 ± 0.029
gas (6)	0.10	0.767 ± 0.018	0.731 ± 0.021	0.402 ± 0.030	0.358 ± 0.020	0.220 ± 0.018	0.209 ± 0.025	0.321 ± 0.026
har (6)	0.10	0.869 ± 0.008	0.809 ± 0.013	0.388 ± 0.028	0.355 ± 0.025	0.222 ± 0.023	0.210 ± 0.022	0.328 ± 0.021
letter (26)	0.05	0.552 ± 0.022	0.467 ± 0.009	0.444 ± 0.038	0.384 ± 0.022	0.238 ± 0.030	0.226 ± 0.030	0.336 ± 0.022
optdigits (10)	0.05	0.852 ± 0.011	0.701 ± 0.016	0.739 ± 0.025	0.652 ± 0.023	0.248 ± 0.030	0.218 ± 0.033	0.513 ± 0.033
pendigits (10)	0.35	0.822 ± 0.006	0.814 ± 0.008	0.590 ± 0.056	0.489 ± 0.037	0.240 ± 0.027	0.224 ± 0.026	0.390 ± 0.027
phoneme (2)	0.05	0.492 ± 0.021	0.382 ± 0.023	0.482 ± 0.030	0.450 ± 0.034	0.242 ± 0.048	0.225 ± 0.033	0.376 ± 0.047
pol (2)	0.05	0.646 ± 0.011	0.554 ± 0.009	0.498 ± 0.035	0.476 ± 0.037	0.275 ± 0.046	0.248 ± 0.045	0.388 ± 0.030
satimage (6)	0.05	0.604 ± 0.082	0.476 ± 0.037	0.472 ± 0.055	0.398 ± 0.055	0.225 ± 0.021	0.209 ± 0.022	0.342 ± 0.031
segment (7)	0.15	0.763 ± 0.015	0.712 ± 0.014	0.502 ± 0.042	0.452 ± 0.042	0.249 ± 0.027	0.228 ± 0.031	0.351 ± 0.021
spambase (2)	0.05	0.757 ± 0.012	0.717 ± 0.008	0.586 ± 0.057	0.511 ± 0.033	0.243 ± 0.036	0.223 ± 0.038	0.412 ± 0.027
texture (11)	0.15	0.834 ± 0.009	0.762 ± 0.013	0.458 ± 0.041	0.388 ± 0.036	0.210 ± 0.018	0.194 ± 0.034	0.315 ± 0.028
wind (2)	0.05	0.686 ± 0.016	0.618 ± 0.019	0.520 ± 0.026	0.428 ± 0.044	0.227 ± 0.025	0.222 ± 0.031	0.359 ± 0.020
Ranking Consistency In data valuation, it could be the ranking of data that matters the most, but the randomness of utility functions may produce inconsistent rankings. For each semi-value, the averaged Spearman's rank correlation coefficients over all possible pairs, which is  $\binom{10}{2} = 45$  in total, are calculated to measure ranking consistency. The results are summarized in Table 2, while figures with more details are included in Appendix E. As presented, it is mostly some member of weighted Banzhaf values, not always the Banzhaf value, that achieves the highest ranking consistency compared with the baselines.
Noisy Label Detection Notice that a more robust semi-value does not necessarily produce a higher quality of data valuation. This experiment is to present how well weighted Banzhaf values would achieve in noisy label detection. For each dataset, we randomly flip the labels of 20 percent of data in  $\mathcal{D}_{tr}$  to be any of the rest in a uniform manner. Those flipped data are treated as having noisy/incorrect labels. Particularly, such a flipped  $\mathcal{D}_{tr}$  is also used for reporting ranking consistency in Table 2. Then, each estimated semi-value is used to detect noisy labels by marking data assigned with values of importance less than or equal to 20 percent percentile as mislabeled. The results are summarized in Table 3, and figures that contain more details are included in Appendix E. As observed, the best performance is mostly achieved by weighted Banzhaf values, and most of them are not the Banzhaf value.
In both types of experiments, it manifests itself that the semi-value that achieves the best in ranking consistency does not coincide with the one that achieves the highest F1-score on each dataset. Nevertheless, the detailed figures in Appendix E show that weighted Banzhaf values tend to be more robust than the Beta Shapley values, which include the Shapley value. Moreover, we can conclude that there is no single universally the best semi-value across various experiment settings in terms of either ranking consistency or noisy label detection. For each utility function, it would be useful if the best semi-value can be effectively determined in advance.
We introduce a noise-structure-specific framework to theoretically assert that it is always a member of weighted Banzhaf values that is deemed the most robust in terms of minimizing the worst-case entropy. Theorem 1 shows that the most robust semi-value would vary across different experiment settings. Our extensive empirical study demonstrates that weighted Banzhaf values are promising when dealing with inevitable noises inherent in utility functions. Particularly, all experiments suggest that there is no universally the best or the most robust semi-value for all experiment settings. Therefore, future investigation is to set up principled approaches that can pin down the most robust or the most effective value in advance.
The concept of Kronecker noise is still ideal for fitting the stochasticity in practice. The verification of Theorem 1 using fitted Kronecker noises can only be done based on simple utility functions, as one may expect. Nevertheless, the theoretical and empirical evidence is sufficient to argue that there is no single universally the most robust or the most effective semi-value across various experiment settings, and that weighted Banzhaf values are promising in practice.
We thank the reviewers and the area chair for thoughtful comments that have improved our final draft. YY gratefully acknowledges NSERC and CIFAR for funding support.
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A The Axiom of Equal Treatment 12
B Proofs for the Propositions 13
C Prerequisites and Proof for Theorem 1 15
C.1 Extend Semi-values to Be Semi-indices 15
C.2 Interaction Transform 17
C.3 Matrix Representation for Semi-indices 18
C.4 A Theorem of Robustness for Semi-indices 19
C.5 Proof for Theorem 1 22
C.6 Lemmas 23
DExperiment Settings 26
D.1 Fitting Kronecker Noises 26
D.2 Sampling Lift 26
E More Experiment Results 27
A The Axiom of Equal Treatment
In the existing references, the axiom of symmetry is sometimes used to refer to the axiom of equal treatment, but they are not always interchangeable.
Equal Treatment:  $\phi_i(v_n) = \phi_j(v_n)$  if  $v_{n}(S\cup i) = v_{n}(S\cup j)$  for every  $S\subseteq [n]\backslash \{i,j\}$ .
The axiom of equal treatment is not always equal to the axiom of symmetry. As a counterexample, pick  $n = 2$ , and consider a value  $\phi$  defined by  $\phi_1(v) = \frac{1}{2} v(1) + \frac{1}{2} v(2)$  and  $\phi_2(v) = \frac{1}{3} v(1) + \frac{2}{3} v(2)$ . The reader can verify that it satisfies the axiom of equal treatment but violates the axiom of symmetry (e.g., take  $v(1) = 3$  and  $v(2) = 6$ ).
Nevertheless, they are equivalent in some contexts. Recall that a weaker version of the axiom of dummy is
Null:  $\phi_i(v_n) = 0$  if  $v_{n}(S\cup i) = v_{n}(S)$  for every  $S\subseteq [n]\backslash i$
Proposition 4. Suppose a value  $\phi$  verifies the axioms of linearity and null, then  $\phi$  verifies the axiom of symmetry if and only if it verifies the axiom of equal treatment.
Proof. The only if part is trivial as the axiom of symmetry always implies the axiom of equal treatment, and thus we only prove the if part. Let  $n \geq 1$  be fixed, since  $\phi$  verifies the axioms of linearity and null, by (Weber 1977, Theorem 2),<sup>6</sup> we have
$$
\phi_ {i} (v _ {n}) = \sum_ {S \subseteq [ n ] \backslash i} p _ {S} ^ {i} \left(v _ {n} (S \cup i) - v _ {n} (S)\right) \text {f o r e v e r y} v _ {n} \in \mathcal {G} _ {n} \text {a n d} i \in [ n ].
$$
For every  $S \subseteq N$ , we define  $u_{S} \in \mathcal{G}_{n}$  by letting  $u_{S}(T) = 1$  if  $T = S$  and 0 otherwise. For every  $i \neq j$  and  $S \subseteq [n] \backslash ij$ , by the axiom of equal treatment, there is
$$
p _ {S} ^ {i} = \phi_ {i} \left(u _ {S \cup i} + u _ {S \cup j} + u _ {S \cup i j}\right) = \phi_ {j} \left(u _ {S \cup i} + u _ {S \cup j} + u _ {S \cup i j}\right) = p _ {S} ^ {j}. \tag {9}
$$
On the other hand, for every  $S$  with  $s > 1$ , and every  $i,j \in S$ , there is
$$
p _ {S \backslash i} ^ {i} = \phi_ {i} \left(u _ {S}\right) = \phi_ {j} \left(u _ {S}\right) = p _ {S \backslash j} ^ {j}. \tag {10}
$$
<sup>6</sup>By using the axiom of null instead of the axiom of dummy, the only difference is that there is no  $\sum_{S \subseteq [n] \setminus i} p_S^i = 1$ .
Combining Eqs. (9) and (10), we can get  $p_S^i = p_s$  for every  $i \in [n]$  and every  $S \subseteq [n] \backslash i$ . To demonstrate, take  $p_{\{2,3\}}^1$  and  $p_{\{5,6\}}^4$  as example, there is
$$
p _ {\{2, 3 \}} ^ {1} = p _ {\{2, 3 \}} ^ {4} = p _ {\{4, 3 \}} ^ {2} = p _ {\{4, 3 \}} ^ {5} = p _ {\{4, 5 \}} ^ {3} = p _ {\{4, 5 \}} ^ {6} = p _ {\{5, 6 \}} ^ {4}.
$$
Therefore, the equality  $p_S^i = p_s$  for every  $i \in [n]$  and every  $S \subseteq [n] \backslash i$  implies the axiom of symmetry.
B Proofs for the Propositions