SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The work of [56] presents a number of different definitions of fairness in ML to cater to different situations. In general, these definitions each describe a particular way for the model to make classifications w.r.t. specific conditions on (the features of) the data in order to be fair. For simplicity, we illustrate with equal opportunity (EO) [56, Definition 2] which "means that the probability of a person in a positive class being assigned to a positive outcome should be equal for both protected and unprotected (female and male) group members." To relate this to CE, we can define two query sets:  $\mathcal{D}_{\mathrm{prot}}^{+}$  containing positive training examples under the protected group and  $\mathcal{D}_{\mathrm{unprot}}^{+}$  containing positive training examples under the unprotected group. To achieve equal opportunity, the average CE losses on both query sets should be (approximately) equal (i.e., both groups have equal true positive rates) or, equivalently, the difference in the CE losses should be small:
$$
\mathrm {o b j e c t i v e} _ {\mathrm {E O}} := \left| \ell_ {\mathrm {C E}} (\mathbf {M}; \mathcal {D} _ {\mathrm {p r o t}} ^ {+}) - \ell_ {\mathrm {C E}} (\mathbf {M}; \mathcal {D} _ {\mathrm {u n p r o t}} ^ {+}) \right|.
$$
Table 1 gives the specific definitions of query sets with the corresponding (possible) choices of the characteristic function  $\nu$  adapted from the above-mentioned minimization objectives by replacing  $\ell_{\mathrm{CE}}(\mathbf{M};\mathcal{D})$  with  $d(\mathbb{Q},\mathbb{Q}^*;\mathcal{D})$  and adding a negation since these are minimization objectives.
Firstly, we show the original implementations/formulations using the CE loss  $\ell_{\mathrm{CE}}$  [43, 67, 56] can be reformulated using the CE between Dirichlet abstractions. We provide the reformulation of objective $_{\mathrm{DRO}}$ ,
$$
\max  _ {g \in \mathcal {G}} \operatorname {C E} (\mathbb {Q}, \mathbb {Q} ^ {*}; \mathcal {D})  ,
$$
by replacing  $\ell_{\mathrm{CE}}$  with CE on the respective Dirichlet abstraction  $\mathbb{Q}$  (for  $\mathbf{M}$ ) and an expert  $\mathbb{Q}^*$  (from the test set) and omit the explicit derivations of the other two for brevity.
Such reformulation is enabled by the connection between CE and  $\ell_{\mathrm{CE}}$ : The CE loss  $\ell_{\mathrm{CE}}(\mathbf{M};\mathcal{D})$  implicitly assumes an expert who provides the correct labels to compute the loss on the query set  $\mathcal{D}$ .
On the other hand,  $\mathrm{CE}(\mathbb{Q},\mathbb{Q}^*;\mathcal{D})$  explicitly uses the expert and measures the cross-entropy between the two Dirichlet abstractions (i.e., one for the model  $\mathbf{M}$  and the other for the expert) on the query set  $\mathcal{D}$ . Therefore, if a model  $\mathbf{M}$  makes predictions similar to the expert on a fixed query set  $\mathcal{D}$ , then both  $\ell_{\mathrm{CE}}(\mathbf{M};\mathcal{D})$  and  $\mathrm{CE}(\mathbb{Q},\mathbb{Q}^*;\mathcal{D})$  are small (i.e., optimum is 0). Next, inspired by the relationship between  $d_{\mathrm{H}}$  and CE in Proposition 2, the expressions in CE are reformulated using  $d_{\mathrm{H}}$  in Table 1. Note that Proposition 2 provides us with the intuition and is not used exactly. Hence, our approach of decoupling the query set(s) from the model valuation makes it general enough to subsume these more sophisticated criteria (of a model's performance) for model valuation. A question one might ask is that: (How) can multiple such evaluation criteria be combined? The answer is yes, by leveraging (P3) to linearly combine selected evaluation criteria, as discussed next.
Combining multiple evaluation criteria. Specifically, for a user (e.g., potential buyer of the model) who knows the relative importance of several different criteria (formalized by the specific query sets such as in Table 1), then the user can specify the weights to achieve a desirable trade-off. This is because different users might have different preferences and there is no one-size-fits-all solution. To elaborate, suppose the user only cares about whether the model makes accurate predictions but not at all about adversarial robustness because the user intends to deploy it in a controlled and safe environment, then the task constructed for adversarial robustness is not very relevant to this user. In contrast, if the user does care about the adversarial robustness (which, is often at trade-off against pure predictive performance), then the user can set the weights between the two tasks according to their preferences.
On the other hand, if the trade-offs of the tasks are unknown, for instance the objectives are very complex, then uncovering the relationship between tasks (which are potentially trade-offs of each other) is useful. Specifically, the approach to obtain the connections in Table 1 is useful. For instance, predictive accuracy and adversarial robustness are trade-offs of each other since the objective of adversarial robustness "balances" between the clean and adversarial cross entropy (CE) losses. Upon identifying this theoretical connection, the user can then specify the weight between the two accordingly.
Remark 1. Note that our discussion on how to extend and combine multiple model evaluation criteria aims to provide the technical tools for doing so, instead of identifying how a buyer or a seller should use them. To elaborate, the buyer can use our method to combine several evaluation criteria based on known preferences of the relative importance of these criteria. Our discussion does not aim to guide the buyer in identifying such preferences or consider the potential asymmetry of information in a marketplace where only the buyer (or the seller) knows such preferences, how the other party should react. We believe these are further and interesting questions for future exploration.
Other useful technical results.
Lemma 3 (Precision-weighted fusion preserves Dirichlet). The precision-weighted fusion in Definition 3 follows a Dirichlet distribution:  $\mathbb{Q}_{\mathcal{C}} = \mathrm{Dir}\bigl ([\sum_{i = 1}^{n}\alpha_{i,1},\ldots ,\sum_{i = 1}^{n}\alpha_{i,C}]\bigr)$  [34, Theorem 2.1].
Lemma 4 (Bhattacharyya coefficient between Dirichlet distributions [63]). Let  $\alpha$  and  $\alpha'$  denote the  $C$ -dimensional parameters specifying the two Dirichlet distributions  $p$  and  $q$ , respectively. Then, the Bhattacharyya coefficient is
$$
\mathbf {B C} (p, q) = \frac {\prod_ {k = 1} ^ {C} \Gamma ((\alpha_ {k} + \alpha_ {k} ^ {\prime}) / 2)}{\Gamma (\sum_ {k = 1} ^ {C} (\alpha_ {k} + \alpha_ {k} ^ {\prime}) / 2)} \times \frac {\sqrt {\Gamma (| \boldsymbol {\alpha} | _ {1}) \Gamma (| \boldsymbol {\alpha} ^ {\prime} | _ {1})}}{\sqrt {\prod_ {k = 1} ^ {C} \Gamma (\alpha_ {k}) \Gamma (\alpha_ {k} ^ {\prime})}}.
$$
Lemma 5. The Hellinger distance between two Dirichlet distributions  $p$  and  $q$  parameterized by the respective  $\alpha$  and  $\alpha'$  is
$$
d _ {\mathrm {H}} (p, q) = \sqrt {2} \times \left(1 - \frac {\prod_ {k = 1} ^ {C} \Gamma ((\alpha_ {k} + \alpha_ {k} ^ {\prime}) / 2)}{\Gamma (\sum_ {k = 1} ^ {C} (\alpha_ {k} + \alpha_ {k} ^ {\prime}) / 2)} \times \frac {\sqrt {\Gamma (| \boldsymbol {\alpha} | _ {1}) \Gamma (| \boldsymbol {\alpha} ^ {\prime} | _ {1})}}{\sqrt {\prod_ {k = 1} ^ {C} \Gamma (\alpha_ {k}) \Gamma (\alpha_ {k} ^ {\prime})}}\right) ^ {1 / 2},
$$
which follows directly from an equivalent definition of  $d_{\mathrm{H}}(p,q) \coloneqq \sqrt{2} \times \sqrt{1 - \mathrm{BC}(p,q)}$ .
A.3 Lipschitz Continuity of Model Shapley
We provide the proof of Theorem 1 here and describe a sufficient condition for fusion (i.e., Definition 3) to increase similarity.
Further elaboration on Lipschitz continuity. We adopt the following general definition for Lipschitz continuity: For two metric spaces  $(\mathbb{X},d_{\mathbb{X}})$  and  $(\mathbb{Y},d_{\mathbb{Y}})$ , a function  $f:\mathbb{X}\mapsto \mathbb{Y}$  is  $L_{f}$ -Lipschitz continuous if there exists a constant  $L_{f} > 0$  s.t.  $\forall x_{1},x_{2}\in \mathbb{X}$
$$
d _ {\mathbb {Y}} \left(f \left(x _ {1}\right), f \left(x _ {2}\right)\right) \leq L _ {f} d _ {\mathbb {X}} \left(x _ {1}, x _ {2}\right).
$$
In our formulation, for the model Shapley function  $\Phi$ , the input space is the set  $[N]$  (or more precisely the set  $\{\mathbb{Q}_i : i \in [N]\}$  of Dirichlet abstractions) of the  $N$  models and the metric is the Hellinger distance  $d_{\mathrm{H}}$ , which is a proper metric for probability distributions (in this case Dirichlet distributions); the output space is  $\mathbb{R}$  (i.e., for model Shapley values) and the metric is the absolute difference (i.e.,  $|\phi_i - \phi_{i'}|$  for two inputs  $\mathbb{Q}_i, \mathbb{Q}_{i'}$ ).
In summary, recall that  $\phi_i \gets \Phi(i, \nu, \{\mathbf{M}_i\}_{i \in [N]})$  in Sec. 3, the Lipschitz continuity of  $\Phi$  is w.r.t. its first argument (i.e.,  $i$ ), when  $\nu$  is defined as in (3) and the set  $\{\mathbf{M}_i\}_{i \in [N]}$  of models is fixed (i.e., correspondingly the set  $\{\mathbb{Q}_i\}_{i \in [N]}$  is also fixed). For brevity, we suppress the notational dependence on the latter arguments and write as  $\phi_i \coloneqq \Phi(i; \cdot): [N] \mapsto \mathbb{R}$  where as mentioned above, the metric for the input space is defined as  $d(i, i') \coloneqq d_{\mathrm{H}}(\mathbb{Q}_i, \mathbb{Q}_{i'})$  for  $i, i' \in [N]$  and the metric for the outputs is the absolute difference  $|\phi_i - \phi_{i'}|$ .
Proof of Theorem 1. We follow an idea that the similarity between the Dirichlet abstractions  $\mathbb{Q}_i$  and  $\mathbb{Q}_{i'}$  will lead to a small difference in their expected marginal contributions  $\phi_i$  and  $\phi_{i'}$  when fused with a common  $\mathbb{Q}_C$  for any  $\mathcal{C} \subseteq [N] \setminus \{i, i'\}$ . Consequently, we can apply Lemma 6 in App. A.4.
Firstly, from  $\nu_{\mathrm{H}}(\mathcal{C}) = -d_{\mathrm{H}}(\mathbb{Q}_{\mathcal{C}},\mathbb{Q}^{*})$  as in (3),
$$
\nu_ {\mathrm {H}} (\mathcal {C} \cup \{i \}) - \nu_ {\mathrm {H}} (\mathcal {C} \cup \{i ^ {\prime} \}) = - d _ {\mathrm {H}} \left(\mathbb {Q} _ {\mathcal {C} \cup \{i \}}, \mathbb {Q} ^ {*}\right) + d _ {\mathrm {H}} \left(\mathbb {Q} _ {\mathcal {C} \cup \{i ^ {\prime} \}}, \mathbb {Q} ^ {*}\right).
$$
Then, using the property of triangle inequality of  $d_{\mathrm{H}}$ , it follows that
$$
\left| - d _ {\mathrm {H}} \left(\mathbb {Q} _ {\mathcal {C} \cup \{i \}}, \mathbb {Q} ^ {*}\right) + d _ {\mathrm {H}} \left(\mathbb {Q} _ {\mathcal {C} \cup \{i ^ {\prime} \}}, \mathbb {Q} ^ {*}\right) \right| \leq d _ {\mathrm {H}} \left(\mathbb {Q} _ {\mathcal {C} \cup \{i \}}, \mathbb {Q} _ {\mathcal {C} \cup \{i ^ {\prime} \}}\right) \leq d _ {\mathrm {H}} \left(\mathbb {Q} _ {i}, \mathbb {Q} _ {i ^ {\prime}}\right)
$$
where the last inequality is due to the fusion-increases-similarity condition stated in Theorem 1 (and examined below by Proposition 4). The final result can be obtained applying Lemma 6: Note that  $d(i,i^{\prime})\coloneqq d_{\mathrm{H}}(\mathbb{Q}_i,\mathbb{Q}_{i^{\prime}}) < 1$  (since the Hellinger distance is upper bounded by 1), so the condition in Lemma 6 is satisfied with  $L = 1$ . In other words, for a different distance (other than the Hellinger distance), a different value of  $L$  may be necessary. The constant  $Z$  is directly inherited to be the Lipschitz constant.
Proposition 4 (Fusion increases similarity). Suppose that  $\alpha_{i}$  and  $\alpha_{i'}$  parameterize  $\mathbb{Q}_{i}$  and  $\mathbb{Q}_{i'}$ , respectively. Then,
$$
\begin{array}{l} \left[ \forall k \in [ C ] \quad \psi \left(\frac {\alpha_ {i , k} + \alpha_ {i ^ {\prime} , k}}{2}\right) - \psi \left(\alpha_ {i, k}\right) \geq \psi \left(\frac {\left| \boldsymbol {\alpha} _ {i} \right| _ {1} + \left| \boldsymbol {\alpha} _ {i ^ {\prime}} \right| _ {1}}{2}\right) - \psi \left(\left| \boldsymbol {\alpha} _ {i} \right| _ {1}\right) \right. \\ \wedge \psi \left(\frac {\alpha_ {i , k} + \alpha_ {i ^ {\prime} , k}}{2}\right) - \psi (\alpha_ {i ^ {\prime}, k}) \geq \psi \left(\frac {| \boldsymbol {\alpha} _ {i} | _ {1} + | \boldsymbol {\alpha} _ {i ^ {\prime}} | _ {1}}{2}\right) - \psi (| \boldsymbol {\alpha} _ {i ^ {\prime}} | _ {1}) \Big ] \\ \Rightarrow \quad d _ {\mathrm {H}} \left(\mathbb {Q} _ {\mathcal {C} \cup \{i \}}, \mathbb {Q} _ {\mathcal {C} \cup \{i ^ {\prime} \}}\right) \leq d _ {\mathrm {H}} \left(\mathbb {Q} _ {i}, \mathbb {Q} _ {i ^ {\prime}}\right) \\ \end{array}
$$
where  $\psi$  is the digamma function.
Proof of Proposition 4. It can be observed from Lemma 5 that  $d_{\mathrm{H}}(p,q)$  increases iff  $\mathrm{BC}(p,q)$  (Lemma 4) decreases. Then, it is equivalent to show that
$$
\mathbf {B C} \left(\mathbb {Q} _ {\mathcal {C} \cup \{i \}}, \mathbb {Q} _ {\mathcal {C} \cup \{i ^ {\prime} \}}\right) \geq \mathbf {B C} \left(\mathbb {Q} _ {i}, \mathbb {Q} _ {i ^ {\prime}}\right).
$$
It can also be observed that BC can be viewed as a differentiable function taking in  $2C$  parameters (i.e.,  $\alpha_{i}$  and  $\alpha_{i'}$ ). Consider w.l.o.g. its partial derivative w.r.t.  $\alpha_{i,k}$  and w.r.t.  $\alpha_{i',k}$  for  $k = 1, \ldots, C$ :
$$
\frac {\partial \mathrm {B C}}{\partial \alpha_ {i , k}} = \operatorname {c o e f f} \times \left[ - \psi \left(\alpha_ {i, k}\right) + \psi \left(\frac {\alpha_ {i , k} + \alpha_ {i ^ {\prime} , k}}{2}\right) - \psi \left(\frac {\left| \boldsymbol {\alpha} _ {i} \right| _ {1} + \left| \boldsymbol {\alpha} _ {i ^ {\prime}} \right| _ {1}}{2}\right) + \psi \left(\left| \boldsymbol {\alpha} _ {i} \right| _ {1}\right) \right]
$$
$$
\frac {\partial \mathbf {B C}}{\partial \alpha_ {i ^ {\prime} , k}} = \operatorname {c o e f f} \times \left[ - \psi \left(\alpha_ {i ^ {\prime}, k}\right) + \psi \left(\frac {\alpha_ {i , k} + \alpha_ {i ^ {\prime} , k}}{2}\right) - \psi \left(\frac {| \boldsymbol {\alpha} _ {i} | _ {1} + | \boldsymbol {\alpha} _ {i ^ {\prime}} | _ {1}}{2}\right) + \psi \left(| \boldsymbol {\alpha} _ {i ^ {\prime}} | _ {1}\right) \right]
$$
$$
\operatorname {c o e f f} = \frac {\sqrt {\Gamma (| \boldsymbol {\alpha} _ {i} | _ {1}) \Gamma (| \boldsymbol {\alpha} _ {i ^ {\prime}} | _ {1})} \sum_ {k = 1} ^ {C} \Gamma ((\alpha_ {i , k} + \alpha_ {i ^ {\prime} , k}) / 2)}{2 \sqrt {\prod_ {k = 1} ^ {C} \Gamma (\alpha_ {i , k}) \Gamma (\alpha_ {i ^ {\prime} , k})} \Gamma (\sum_ {k = 1} ^ {C} (\alpha_ {i , k} + \alpha_ {i ^ {\prime} , k}) / 2)} > 0
$$
due to the positivity of  $\Gamma$  over the positive domain.11
Now, Lemma 3 implies that the fusion always increases  $\alpha_{i,k}$ : Since  $\alpha_{i,k}$  denotes the  $k$ -th component of  $\alpha_{i}$  and  $\alpha_{\mathcal{C} \cup \{i\}, k}$  denotes the  $k$ -th component of  $\alpha_{\mathcal{C} \cup \{i\}}, \alpha_{\mathcal{C} \cup \{i\}, k} \geq \alpha_{i,k}$ . So, if  $\partial \mathrm{BC} / \partial \alpha_{i,k} \geq 0$  and  $\partial \mathrm{BC} / \partial \alpha_{i', k} \geq 0$  for  $k = 1, \ldots, C$ , then the resulting BC increases (or, equivalently,  $d_{\mathrm{H}}$  decreases) after fusion.
Since  $\text{coeff} > 0$ ,
$$
\begin{array}{l} - \psi \left(\alpha_ {i, k}\right) + \psi \left(\frac {\alpha_ {i , k} + \alpha_ {i ^ {\prime} , k}}{2}\right) - \psi \left(\frac {\left| \boldsymbol {\alpha} _ {i} \right| _ {1} + \left| \boldsymbol {\alpha} _ {i ^ {\prime}} \right| _ {1}}{2}\right) + \psi \left(\left| \boldsymbol {\alpha} _ {i} \right| _ {1}\right) \geq 0 \Rightarrow \frac {\partial \mathbf {B C}}{\partial \alpha_ {i , k}} \geq 0 \\ - \psi \left(\alpha_ {i ^ {\prime}, k}\right) + \psi \left(\frac {\alpha_ {i , k} + \alpha_ {i ^ {\prime} , k}}{2}\right) - \psi \left(\frac {| \boldsymbol {\alpha} _ {i} | _ {1} + | \boldsymbol {\alpha} _ {i ^ {\prime}} | _ {1}}{2}\right) + \psi \left(| \boldsymbol {\alpha} _ {i ^ {\prime}} | _ {1}\right) \geq 0 \Rightarrow \frac {\partial \mathbf {B C}}{\partial \alpha_ {i ^ {\prime} , k}} \geq 0 \\ \end{array}
$$
for  $k = 1,\dots ,C$  . So, the final result follows.
Let  $D_{\mathrm{sum}} \coloneqq \sum_{k=1}^{C} \psi((\alpha_{i,k} + \alpha_{i',k})/2) - (\psi(\alpha_{i,k}) + \psi(\alpha_{i',k}))/2$  and  $D_{|\cdot|} \coloneqq C[\psi((|\alpha_i|_1 + |\alpha_{i'}|_1)/2) - (\psi(|\alpha_i|_1) + \psi(|\alpha_{i'}|_1))/2]$ . The sufficient condition for Proposition 4 implies that  $D_{\mathrm{sum}} \geq D_{|\cdot|}$ . Intuitively,  $D_{\mathrm{sum}}$  is a sum of dimension-/class-wise differences between  $\mathbb{Q}_i$  and  $\mathbb{Q}_{i'}$  and it is large if every pair of  $\alpha_{i,k}$  and  $\alpha_{i',k}$  are different and relatively small (i.e., low concentration for dimension/class  $k$ ).  $D_{\mathrm{sum}}$  is likely large if the dimension  $C$  is large as there are more pairs of  $\alpha_{i,k}$  and  $\alpha_{i',k}$  whose difference contributes towards  $D_{\mathrm{sum}}$ . On the other hand,  $D_{|\cdot|}$  is a measure of the difference in the overall precisions of  $\mathbb{Q}_i$  and  $\mathbb{Q}_{i'}$ .  $D_{|\cdot|}$  is large if  $|\alpha_i|_1$  and  $|\alpha_{i'}|_1$  are different and have small values.  $D_{|\cdot|}$  is likely small if  $C$  is large. As  $C$  increases, the precisions  $|\alpha_i|_1$  and  $|\alpha_{i'}|_1$  will increase, which will cause  $\psi((|\alpha_i|_1 + |\alpha_{i'}|_1)/2)$  and  $(\psi(|\alpha_i|_1) + \psi(|\alpha_{i'}|_1))/2$  to be very close due to the converging behavior of  $\psi$ .
The condition  $D_{\mathrm{sum}} \geq D_{|\cdot|}$  says that if the class-wise difference between  $\mathbb{Q}_i$  and  $\mathbb{Q}_{i'}$  outweighs the difference in their precisions, then fusion increases similarity. Intuitively, if the 'shapes' of  $\mathbb{Q}_i$  and  $\mathbb{Q}_{i'}$  are very different, then fusing each with a common distribution  $\mathbb{Q}_C$  'evens out' the difference in their shapes and increases the similarity. In particular, if  $C$  is large (i.e., a high-dimensional classification task), then the condition is more likely to be satisfied.
Empirical verification of Theorem 1. Specifically, we verify whether a small  $d_{\mathrm{H}}(\mathbb{Q}_i,\mathbb{Q}_{i'})$  leads to a small  $|\phi_i - \phi_{i'}|$  via the Pearson coefficient between  $d_{\mathrm{H}}(\mathbb{Q}_i,\mathbb{Q}_{i'})$  and  $|\phi_i - \phi_{i'}|$  over all  $i,i'$ , and visualizing the corresponding heatmaps (Fig. 6). The setting is as follows. We investigate 5 real-world datasets (and various ML models), including MNIST, CIFAR-10 [41], two medical datasets: a drug reviews dataset that classifies the type of prescribed medicine based on the text reviews (DrugRe) [23] and a medical imaging dataset that classifies the medical department from the diagnostic scans (MedNIST) [58], and a cyber-threat detection dataset that classifies network intrusion based on input features such as IP addresses and network communication protocol (KDD99) [28]. Recall these are some of the highlighted application domains of model valuation (i.e., medicine and cyber-defense) in Sec. 1. Query set  $\mathcal{D}$  is the respective test set of each dataset without partitioning.
We adopt a grouping paradigm where the grouped models have the same model type (but undergo independent training with the same data) to ensure some similarity within each group. So, we can verify whether the models within the same group have similar MSVs. To see why models of the same type can produce similar Dirichlet abstractions, we provide a clustering result of  $\mathbb{Q}_i$  (Fig. 11 in App. C). Specifically, for MNIST, we implement 3 model types: logistic regression (LR), multilayer perceptron (MLP), and a 2-layer convolutional neural network (CNN). For CIFAR-10, we utilize 3 known model types with pre-trained weights: ResNet-18 [26], SqueezeNet [37], and DenseNet-121 [35]. For DrugRe, we use a CNN and a bi-directional long-short term memory (BiLSTM) network. More details on other datasets are in App. C.
<sup>11</sup>The code for verifying this partial derivative using an automatic differentiation package is included in the supplementary material.
The high Pearson coefficients in Table 3 provide some verification for Theorem 1. The matched color intensities in the heatmaps in Fig. 6 confirm that similar models have similar MSVs. Left (right) of Fig. 6 is w.r.t. general (class-specific) Dirichlet abstractions.
Figure 6: Left two plots are heatmaps for  $d_{\mathrm{H}}(\mathbb{Q}_i,\mathbb{Q}_{i'})$  and  $|\phi_i - \phi_{i'}|$ . Right two plots are heatmaps for  $d_{\mathrm{H}}(\mathbb{Q}_i,\mathbb{Q}_{i'};\{\mathcal{D}_k\})$  and  $|\phi_i(\{\mathcal{D}_k\}) - \phi_{i'}(\{\mathcal{D}_k\})|$ .
Table 3: Pearson coefficient between  $d_{\mathrm{H}}(\mathbb{Q}_i, \mathbb{Q}_{i'})$  and  $|\phi_i - \phi_{i'}|$  using the test set as a single query set. A high coefficient verifies Theorem 1.
MNIST	CIFAR-10	MedNIST	KDD	DrugRe
0.974	0.996	0.974	0.999	0.955
A.4 Model Shapley Value
Closed-form expression of  $\nu$  and the resulting computational complexity. Recall that an advantage of the combined choices of Definition 3 and the Hellinger distance  $d_{\mathrm{H}}$  (described in Sec. 3) is an available closed form evaluation of  $d_{\mathrm{H}}$  between two Dirichlet abstractions, which has a constant time computational cost (i.e.,  $\mathcal{O}(1)$ ). This is important as  $d_{\mathrm{H}}$  is used to define the characteristic function  $\nu$  (3) for model Shapley (2). Specifically, note that the definition of model Shapley (2) requires evaluating the characteristic function  $\nu$  for an exponential number of times due to the summation over all possible subsets  $\mathcal{C} \subseteq [N] \setminus \{i\}$ . In other words, even with a characteristic function  $\nu$  that can be evaluated in constant time (e.g.,  $d_{\mathrm{H}}$ ), the computational complexity of MSV is still at least exponential in  $N$ , which is almost intractable for large  $N$ ; if evaluating the characteristic function has a higher computational complexity, then the computational complexity of MSV would be even more intractable.
Similarity-bounded difference in SVs. The following lemma provides a general result for bounding the difference in two Shapley values (not necessarily MSVs) and is used in the proof of Theorem 1. It may be of independent interest.
Lemma 6 (Similarity-bounded difference in the Shapley values). For all  $i, i' \in [N]$ ,
$$
(\forall \mathcal {C} \subseteq [ N ] \backslash \{i, i ^ {\prime} \} | \nu (\mathcal {C} \cup \{i \}) - \nu (\mathcal {C} \cup \{i ^ {\prime} \}) | \leq L d (i, i ^ {\prime})) \Longrightarrow | \phi_ {i} - \phi_ {i ^ {\prime}} | \leq Z L d (i, i ^ {\prime})
$$
where  $L \geq 0$  is a constant and  $d(i, i')$  is some distance measure between  $i$  and  $i'$  and  $Z$  is the linear scaling as in (2).
Proof of Lemma 6. The difference  $|\phi_i - \phi_i'|$  can be bounded by enumerating the coalitions in a paired way as follows:
$$
|\phi_{i} - \phi_{i^{\prime}}|\leq \sum_{\mathcal{C}\subseteq [N]\setminus \{i,i^{\prime}\}}Z\omega_{\mathcal{C}}|\nu (\mathcal{C}\cup \{i\}) - \nu (\mathcal{C}\cup \{i^{\prime}\})| + \sum_{\substack{(\mathcal{C},\mathcal{C}^{\prime}),\\ \mathcal{C}\subseteq [N]\setminus \{i\} ;i^{\prime}\in \mathcal{C},\\ \mathcal{C}^{\prime}\subseteq [N]\setminus \{i^{\prime}\} ;i\in \mathcal{C}^{\prime},\\ \mathcal{C}\cup \{i\} = \mathcal{C}^{\prime}\cup \{i^{\prime}\}}}Z\omega_{\mathcal{C}}|\nu (\mathcal{C}^{\prime}) - \nu (\mathcal{C})|
$$
where  $\omega_{\mathcal{C}} := |\mathcal{C}|!(N - |\mathcal{C}| - 1)! / N!$  and we have multiplied the constant  $Z$  into the separate summations. Note that this enumeration (considering both summations) exhausts (w.l.o.g. from the viewpoint of  $i$ )  $\mathcal{C} \subseteq [N] \setminus \{i\}$  in the calculation of  $\phi_i$ . The first summation enumerates all  $\mathcal{C} \subseteq [N] \setminus \{i, i'\}$ , so the remaining  $\mathcal{C}$  to consider for  $i$ 's marginal contributions as in (2) are the ones that include  $i'$  but not  $i$  (considered in the second summation). The summands in the second summation are in fact also in the form of  $|\nu(\underline{\mathcal{C}} \cup \{i\}) - \nu(\underline{\mathcal{C}} \cup \{i'\})|$  for some  $\underline{\mathcal{C}} \subseteq [N] \setminus \{i, i'\}$ . Note that in the second summation,