SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Recall the example in Sec. 2 that using a lower level of Dirichlet abstraction by partitioning  $\mathcal{D}$  according to the classes allows us to correctly differentiate  $\mathbf{M}_i$  from its 'shifted' version  $\mathbf{M}_{i'}$ . Intuitively, partitioning  $\mathcal{D}$  according to the classes improves the similarity measure between  $\mathbf{M}_i$  and  $\mathbf{M}_{i'}$  (via some distributional distance) and we hypothesize that it can also improve the generalized symmetry, which exploits the similarity measure between two models (e.g., via  $d_{C}(\mathbb{Q}_{i},\mathbb{Q}_{i'})$ ). We verify this on MNIST and CIFAR-10 by partitioning the respective query sets  $\mathcal{D}$  (i.e., test set) according to  $C = 10$  classes s.t. query set  $\mathcal{D}_k$  is from class  $k$  and  $\gamma_k \coloneqq |\mathcal{D}_k|$ . We compute  $d_{C}(\mathbb{Q}_{i},\mathbb{Q}_{i'};\{\mathcal{D}_{k}\}_{k = 1,\dots,C}) \coloneqq \sum_{k = 1}^{C}\gamma_{k} d_{C}(\mathbb{Q}_{i,\mathcal{D}_{k}},\mathbb{Q}_{i'},\mathcal{D}_{k})$  and apply (P3) to compute  $\phi_i(\{\mathcal{D}_k\}_{k = 1,\dots,C}) = \sum_{k = 1}^{C}\gamma_k\phi_i(\mathcal{D}_k)$ , as shown in the right two plots of Fig. 6. Note that we use the Chernoff distance (instead of the Hellinger distance) due to its numerical stability, and will elaborate later on this point.
The setting for this experiment is as follows, we utilize the MNIST and CIFAR-10 datasets respectively. For each dataset, we train  $N = 150$  models with 3 different model types (50 of each). Specifically, for MNIST, we train 50 of LR, MLP and CNN while for CIFAR-10, we train 50 of ResNet-18, SqueezeNet and DenseNet-121. As in Sec. 4.1, we evaluate the performance via the Pearson correlation coefficient between  $d_{\mathrm{C}}(\mathbb{Q}_i,\mathbb{Q}_{i'};\{\mathcal{D}_k\}_{k = 1,\dots ,C})$  and  $|\phi_i(\{\mathcal{D}_k\}_{k = 1,\dots ,C}) - \phi_{i'}(\{\mathcal{D}_k\}_{k = 1,\dots ,C})|$  in Table 5.
Table 5: Comparison of Pearson coefficients between using single query set  $\mathcal{D}$  vs. partitioned query sets  $\{\mathcal{D}_k\}$  for MNIST (top two rows) and CIFAR-10 (bottom two rows).
Pearson \N	60	90	120	150
φi(D)	0.9493	0.9378	0.9146	0.9011
φi({Dk})	0.9440	0.9362	0.9326	0.9274
φi(D)	0.9958	0.9955	0.9934	0.9898
φi({Dk})	0.9961	0.9949	0.9936	0.9924
Results in Table 5 confirm our hypothesis that a lower level of Dirichlet abstraction can lead to better observation of the symmetry result. This is because a lower level of Dirichlet abstraction provides a more refined representation of each model (i.e., w.r.t. individual classes instead of overall). In particular, we note the performance improvement for a larger  $N$  is important so that model Shapley is learnable to be adopted in a large-scale marketplace.
Numerical overflow prevents the Hellinger distance from performing well. Due to the numerical overflow issue of  $d_{\mathrm{H}}$ : If the argument to  $\Gamma(\cdot)$  (e.g., the denominator in  $B(\alpha)$  in Definition 1) approaches 171.614479 in double precision floating point numbers, from [63]. Recall Definition 3 fuses models in a coalition  $\mathcal{C} \subseteq [N]$  to obtain  $\mathbb{Q}_{\mathcal{C}} = \mathrm{Dir}(\sum_{i=1}^{n} \alpha_{i,1}, \ldots, \sum_{i=1}^{n} \alpha_{i,\mathcal{C}})$  where  $|\mathcal{C}| = n$ . The summation  $\sum_{i=1}^{n} \alpha_{i,1}$  can lead to overflow when  $n$  (i.e.,  $|\mathcal{C}|$ ) is large, which tends to happen when  $N$  (the total number of models is large) because the calculation of MSV  $\phi_i$  requires the enumeration of  $\mathcal{C} \subseteq [N] \setminus \{i\}$  to obtain the respective  $\mathbb{Q}_{\mathcal{C}}$ . For instance  $N = 150$ , then the largest  $\mathcal{C} \subseteq [N] \setminus \{i\}$  contains 149 models and if their respective parameter  $\alpha_{i',k}$  for the class  $k$  is on average larger than 171.614479/149 ≈ 1.15, then it leads to numerical overflow for  $\Gamma(\cdot)$  (and thus Hellinger distance). Note that the Chernoff distance avoids this by combining the logarithmic operation with the  $\Gamma(\cdot)$ : in implementation  $\ln \Gamma(\cdot)$  can be computed directly instead of first computing  $\Gamma(\cdot)$  and then the logarithmic operation.
C.7 Same Model Types Produce Distributionally Close Dirichlet Abstractions
We perform clustering (using  $d_{\mathrm{C}}$  as the distance measure instead of  $d_{\mathrm{H}}$  because it shows a better visual illustration and the clustering results) to illustrate the effects of model types on the similarity between the resulting Dirichlet abstractions using the default test set as the query set without performing partitioning according to classes. Specifically, for the  $N = 150$  models trained on MNIST (50 CNNs, 50 MLPs and 50 LRs), we apply the density-based spatial clustering of applications with noise (DBSCAN) [17], because it is suitable for data which contain clusters of similar density (in our case, 3 clusters each of size 50). Fig. 11 shows good separation of different model types, supported by additional quantitative indices: homogeneity (1.000), completeness (0.971) and V-measure (0.985). Homogeneity measures the degree to which a single cluster contains only members of a single class. Completeness measures the degree to which the all the members of a single class are correctly classified into the same cluster. V-measure is a harmonic mean of both homogeneity and completeness. These high values suggest the Dirichlet abstractions contain sufficient model information to clearly distinguish between these model types. As a result, the clustering performance is quite good via the adjusted random index (ARI) (0.990 and optimal is 1.0), which measures the correctness of the given clustering via its match with the ground truth. We consider ARI because it is suitable for large equal-sized clusters [64], which is this case here: 3 equal-sized clusters of size 50. This result confirms the effectiveness of our approach to "convert" the heterogeneous models into homogeneous Dirichlet abstractions which live in the metric space (with metric  $d_{\mathrm{H}}$ , though we highlight the experiments are performed w.r.t.  $d_{\mathrm{C}}$  due to practical concerns). In particular, the numerical stability issue of  $d_{\mathrm{H}}$  [63] prevented us from obtaining meaningful clustering results using  $d_{\mathrm{H}}$ . Moreover, this result motivates our grouping paradigm used in our experiments to verify the generalized symmetry.
Figure 11: DBSCAN clustering of  $N = 150$  models trained on MNIST based on the Chernoff distance  $d_{\mathrm{C}}$  Definition 4.