Title: Model Shapley: Equitable Model Valuation with Black-box Access
Venue: NeurIPS 2023
Accepted: yes
Keywords: Model Valuation

Dirichlet Abstraction

Shapley Value
Abstract: Valuation methods of data and machine learning (ML) models are essential to the establishment of AI marketplaces. Importantly, certain practical considerations (e.g., operational constraints, legal restrictions) favor the use of model valuation over data valuation. Also, existing marketplaces that involve trading of pre-trained ML models call for an equitable model valuation method to price them. In particular, we investigate the black-box access setting which allows querying a model (to observe predictions) without disclosing model-specific information (e.g., architecture and parameters). By exploiting a Dirichlet abstraction of a model’s predictions, we propose a novel and equitable model valuation method called model Shapley. We also leverage a Lipschitz continuity of model Shapley to design a learning approach for predicting the model Shapley values (MSVs) of many vendors’ models (e.g., 150) in a large-scale marketplace. We perform extensive empirical validation on the effectiveness of model Shapley using various real-world datasets and heterogeneous model types.

Model Shapley: Equitable Model Valuation with Black-box Access

Abstract

Valuation methods of data and machine learning (ML) models are essential to the establishment of AI marketplaces. Importantly, certain practical considerations (e.g., operational constraints, legal restrictions) favor the use of model valuation over data valuation. Also, existing marketplaces that involve trading of pre-trained ML models call for an equitable model valuation method to price them. In particular, we investigate the black-box access setting which allows querying a model (to observe predictions) without disclosing model-specific information (e.g., architecture and parameters). By exploiting a Dirichlet abstraction of a model's predictions, we propose a novel and equitable model valuation method called model Shapley . We also leverage a Lipschitz continuity of model Shapley to design a learning approach for predicting the model Shapley values (MSVs) of many vendors' models (e.g., 150 ) in a large-scale marketplace. We perform extensive empirical validation on the effectiveness of model Shapley using various real-world datasets and heterogeneous model types.

1 Introduction

Data valuation methods have important roles in extensive applications such as dataset pricing in a data marketplace [1], evaluating the contributions of data from multiple collaborating parties [71], and identifying valuable data (or filtering out less valuable data) for training higher-quality machine learning (ML) models [20]. However, the following practical considerations favor the use of model valuation over data valuation: (A) Data valuation can be operationally infeasible due to the characteristics of training data, such as being massively distributed over millions of sources [73], enormous in size (e.g., 400 billion byte-pair-encoded tokens for a language model [7]), and/or transient (e.g., the data for online learning are not stored persistently [14]). In contrast, a trained model is not distributed by nature (and hence does not need to be aggregated from distributed sources), often much smaller in size than the training data (e.g., less than 1% ), 1 and usually stored persistently for inference. (B) Data privacy regulations (e.g., GDPR) can be a legal impediment to data valuation in designing fair payments for ML-based collaborations (e.g., in medicine [16] or cyber-defense [25]) because the centralization of (possibly) private data, which seems necessary for data valuation [39], is prohibited by such regulations. In contrast, the ML models trained on private data without centralization [40] can be available for valuation. (C) Existing marketplaces that involve trading of pre-trained ML models (e.g., AWS marketplace, Modzy) call for an equitable model valuation to price them. These practical considerations motivate the use of equitable model valuation over data valuation.

For model valuation, practitioners (e.g., model vendors or clinicians who use the trained models) would likely prefer their models to be examined via only black-box access , 2 i.e., by querying the model with input for the corresponding predictions without observing its internal mechanism [8]. It does not disclose the proprietary model information (for model vendors) nor the sensitive information contained in the model (for clinicians), and provides an added advantage of the model valuation being model-agnostic since no model-specific information is used. However, these make model valuation challenging by restricting the available model information to only a selected query set (e.g., a validation set) with the observed predictions. In contrast, with white-box access, there is more model information available (e.g., information-theoretic criteria for probabilistic models [71] or norms of the parameters for deep neural networks [22]) to assess model complexity or certain analytical properties (e.g., uniqueness of an optimal model in logistic regression).

Intuitively, the value of a model depends on its intended task: For example, a model trained to classify MNIST digits [45] is not very valuable to a clinician trying to classify diagnostic scans. This dependency is useful in practice for selecting the most 'valuable' model for the desired task (i.e., as a query set). We refer to task and query set interchangeably hereafter. While the predictive accuracy of a model on the task provides an intuitive value [66], it can be too reductive: Suppose that two models M i and M i ′ have identical predictive accuracies on the task but M i ( M i ′ ) makes highly (barely) certain predictions, i.e., M i ( M i ′ ) predicts the true class with over 90% (barely over 50% ) probability. Intuitively, the values for M i and M i ′ should not be the same, which cannot be achieved with accuracy alone, hence suggesting that additional model information beyond accuracy is required. (1) What then should be a suitable abstraction of a model w.r.t. a task, for model valuation?

Satisfying certain equitability properties in model valuation is imperative in the application of model pricing to ensuring a fair market. For instance, consistent valuation of identical models is important as it is unfair to price them differently otherwise. Furthermore, the market economy dictates that the value of a model depends on other available models (e.g., more available substitutes cause depreciation), which is important to guarantee a fair market by preventing price fixing (i.e., an exploitative pricing scheme often made illegal by anti-trust laws). (2) How then should model valuation be formulated to satisfy these equitability properties?

In particular, the Shapley value is shown to satisfy these equitability properties but raises another practical challenge that computing it exactly incurs O (2 N ) time where N is the number of models in a marketplace. So, for a given computational budget, there is a fundamental ceiling to the size of the marketplace such that including more models causes the marketplace to be unable to determine their values (within reasonable time). (3) How can the desirable equitability properties of the Shapley value still be exploited without imposing a significant restriction on the size of the marketplace?

This paper presents a novel model-agnostic valuation framework to tackle the above challenges. For (1) , we use an insight that the predictive pattern of a model (w.r.t. a task) is an suitable abstraction for valuation, especially since the black-box access rules out other model information. Specifically, we use a Dirichlet distribution to approximate a model's predictive pattern/distribution w.r.t. a task, which we call the Dirichlet abstraction of this model to encode both its predictive accuracy and certainty (Sec. 2). We describe how to adjust the level of abstraction to trade off the amount of model information (i.e., higher abstraction level) for the availability of a smaller query set. Then, for (2) , we observe that ensuring equitability requires a similarity measure of the models (e.g., to ensure identical models are valued consistently or to identify substitutes). We exploit the ability of the Dirichlet abstractions to preserve the similarity between models for proposing model Shapley as an equitable model valuation (Sec. 3). As an illustration, identical models produce identical Dirichlet abstractions which result in equal model Shapley values (MSVs). To address (3) , based on the Dirichlet abstraction, we leverage a Lipschitz continuity of model Shapley to justify and propose a learning approach of training a model appraiser (i.e., a regression learner) on a small subset of models (and their MSVs) for predicting other models' MSVs to validate model Shapley's practical feasibility in a large-scale marketplace (Sec. 4), as empirically verified on real-world datasets and up to 150 heterogeneous models. We empirically validate that better predictive accuracy (e.g., F 1 score) due to better training data, more suitable model types, and/or higher predictive certainty result in higher MSVs, and demonstrate a use case for identifying a valuable subset of models from the marketplace to construct a larger learner (e.g., random forests) (Sec. 5).

2 Dirichlet Abstraction of a Model

We give some preliminaries on the Dirichlet distribution and Hellinger distance below: 3

Definition 1 ( Dirichlet distribution [63]) . The probability density function of a C -dimensional Dirichlet random variables Z ∼ Dir ( α ) with parameters α = [ α 1 , . . . , α C ] ∈ (0 , ∞ ) C is p ( z ; α ) = ∏ C k =1 z α k -1 k /{[ ∏ C k =1 Γ( α k ) ] / Γ ( ∑ C k =1 α k )} where Γ is the gamma function.

Definition 2 ( Hellinger distance [27]) . The Hellinger distance between distributions (whose probability density functions are) p and q is d H ( p, q ) := [1 -∫ √ p ( x ) q ( x ) d x ] 1 / 2 .

2.1 Dirichlet Abstraction and MLE

Wewill now describe an abstraction of a model via a Dirichlet approximation to the model's predictive pattern over some task and a maximum likelihood estimation (MLE) for it. Each (learnt) C -way classification model M i : X ↦→ △ ( C ) is a mapping from the input space X to the ( C -1) -probability simplex △ ( C ) (i.e., space of C -dimensional probability vectors). Denote a random variable X ∼ P X whose support supp ( X ) = X is the input space, then its distribution P X induces a predictive distribution P M i ( X ) over △ ( C ) . Concretely, P X is represented by a task/query set D := { ( x j ∈ X , y j ∈ { 1 , . . . , C } ) } j =1 ,...,D where each x j is a realization of X ∼ P X , so that M i ( x j ) is a realization of P M i ( X ) .

Dirichlet abstraction. Note that the predictive distribution P M i ( X ) (induced by P X ) (i) has the exactly same support to that of a C -dimensional Dirichlet distribution, so P M i ( X ) can be mathematically modeled using a Dirichlet distribution; and (ii) can be statistically modeled using a Dirichlet distribution because the predictive pattern of a classification model can be (statistically) characterized well with a Dirichlet distribution [68]. Informally, the 'shape' of predictive distribution of a classification model is similar to that of a Dirichlet distribution. Hence, we let P M i ( X ) = Q i := Dir ( α i ) , whose parameters α i can be learnt using MLE (described later) and refer to α i or Q i as M i 's Dirichlet abstraction . In words, we abstract the predictive distribution P M i ( X ) of the model M i induced by P X into a Dirichlet distribution, hence the name Dirichlet abstraction. Note that the notational dependence on P X (or D ) is suppressed when the context is clear.

The proposed Dirichlet abstraction offers several important advantages (further elaborated in Sec. 3): (a) By design, this formulation replaces the heterogeneity in models with the homogeneity in their Dirichlet abstractions, and allows the subsequently proposed model valuation to be applicable to models of different types: The respective Dirichlet abstractions of a multi-layer perceptron and a logistic regression can be compared directly. (b) Q i encodes the predictive accuracy and certainty of M i through a theoretical connection between the Hellinger distance d H of two Dirichlet abstractions and the cross entropy of M i (formalized by Proposition 2). (c) Importantly for model valuation, an appealing analytic property of the Dirichlet distribution is that d H of two Dirichlet abstractions can be evaluated in closed-form and incurs O (1) computational complexity.

MLE. We adopt the MLE approximation of α i using the predictions of M i on D since the loglikelihood function is concave with a unique maximizer and can thus be efficiently optimized [57]. Since M i ( x j ) denotes a realized predictive probability vector of M i ( X ) ∼ Dir ( α i ) , we use the observed sufficient statistics log ¯ h i := D -1 ∑ D j =1 log M i ( x j ) (i.e., with an element-wise log operation) to derive the log-likelihood as follows [36]:

where G ( α ) := log Γ ( ∑ k α k ) -∑ k log Γ( α k ) . From (1), ¯ h i arises as an alternative (to α i ) abstraction of M i w.r.t. D . Hence, we compare α i and ¯ h i theoretically (Proposition 3 in App. A) and empirically (Sec. 4.2).

2.2 Class-specific Dirichlet Abstraction

Since the Dirichlet abstraction Q i of M i w.r.t. some task D does not explicitly account for the class information in D (i.e., the true classification label y j of each data x j ), models with certain (different) predictive patterns can be indistinguishable. This can be problematic because the Dirichlet abstractions of an optimal model and another model with zero predictive accuracy but a specific predictive pattern can identical, making it difficult to distinguish between these two models.

Suppose that there are C = 3 classes in a balanced query set D (i.e., equal data size for each class). For some M i (with optimal predictive accuracy), artificially construct M i ′ to have the same Dirichlet abstraction, but zero predictive accuracy in the following way: Since D is balanced, group the data into triplets { ( x j, 1 , x j, 2 , x j, 3 ) } j =1 ,...,D/ 3 for the input data from 3 classes. Next, define M i ′ ( x j,c ) := M i ( x j, ( c mod 3)+1 ) for c = 1 , 2 , 3 . Intuitively, for each prediction (i.e., a 3 -dimensional probability vector in a 2 -simplex) by M i , M i ′ makes an identical one but on an input data from a wrong class, i.e., we 'shift' M i 's predictions by one class to construct the predictions of M i ′ . Obviously, the predictive accuracy of M i ′ is zero, but its predictions (in aggregation) are identical to those of M i , which means ¯ h i = ¯ h i ′ in (1), hence resulting in Q i = Q i ′ . In Fig. 1: Plots 1 and 2 are the predictions of M i and M i ′ respectively, on D , which are (visually) indistinguishable. Plots 3 and 4 are samples from Q i and Q i ′ respectively, which are also (visually) indistinguishable. The implication is that, in this case (of M i and such specially constructed M i ′ ), the highest level of abstraction (i.e., using the entire query set D to construct Dirichlet abstractions Q i and Q i ′ ) is not effective to distinguish between i and i ′ . Hence, we adopt a lower level of abstraction, described next.

The remedy is the so-called class-specific Dirichlet abstraction : Partition the query set D = ∪ C k =1 D k where D k contains only data from the k -th class. The Dirichlet abstraction α i, D k of M i on D k is called the class-specific Dirichlet abstraction w.r.t. class k . Based on the example above, we verify using a small experiment. 4 Over the entire D , we obtain α i = α i ′ = [0 . 5040 , 0 . 5339 , 0 . 5306] . Restricting to query set D 1 from class 1 only, gives α i, D 1 = [21 . 8601 , 2 . 2005 , 2 . 0215] and α i ′ , D 1 = [1 . 5817 , 1 . 7576 , 19 . 4037] , which are clearly different. In Fig. 1: Plots 5 and 6 are the predictions of M i and M i ′ , respectively, on D 1 while plots 7 and 8 are samples from Q i, D 1 and Q i ′ , D 1 . We see that M i is clearly different from M i ′ , and Q i, D 1 is clearly different from Q i ′ , D 1 , demonstrating the effectiveness of a lower level of abstraction via the class-specific Dirichlet abstraction.

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Trade-off between level of Dirichlet abstraction and size of query set. Note that we need not stop at partitioning D at the class level. For instance, a more refined partition can be first according to the classes and then certain groups of input feature values. Doing so produces Dirichlet abstractions with a lower level of abstraction (i.e., less abstract and containing more model information), but also requires a larger query set D to begin with so that each smaller partitioned query set contains sufficient data (for predictions) to obtain an accurate MLE estimate. 5 For an extremely refined partition of D where each partitioned query set is very small (e.g., size ≤ 5 ), the obtained MLE estimates and the corresponding Dirichlet abstraction can be inaccurate and thus not useful. It is thus important to find a suitable trade-off between the abstraction level vs. query set size. In particular, in Sec. 5, we observe that partitioning D according to the classes provides such a suitable trade-off: E.g., it can distinguish models with almost equal predictive accuracy due to the high class imbalance

in the data 6 but different F 1 scores. Since the Dirichlet abstraction and the class-specific version are both Dirichlet distributions, and share the same theoretical properties (e.g., enabling a closed-form expression of the Hellinger distance), subsequently, we refer to Dirichlet abstractions generally (i.e., omitting the specific dependence on D or D k ), unless otherwise specified.

3 Equitable Valuation via Model Shapley

We discuss and formalize several equitability properties to derive a general formulation called model Shapley to satisfy them. Then, we will specify the characteristic function and a precision-weighted fusion to encode a model's predictive accuracy and certainty into its MSV.

3.1 Equitability Properties and Model Shapley

Consider N models in a marketplace and denote M i 's equitable value by ϕ i . (P1) If a model M i has not been queried at all, then its value is indeterminate and we set ϕ i = 0 by default. (P2) If two models M i and M i ′ give identical predictions (over the task), then their values are equal, i.e., ϕ i = ϕ i ′ . (P3) If some buyer is interested in multiple tasks simultaneously and a model (from some vendor) performs very well on only one of the tasks, then it is unfair for the vendor to set the value/price solely based on this performance. Instead, an equitable value should be based on the model's joint performance on these tasks (e.g., a linear combination according to the buyer's interests in these tasks). (P4) The existence of perfect substitutes depreciates the value of a model. (P1)-(P4) are useful for equitable valuation in ML model marketplaces [51, 66]. To formalize these properties, some notations are needed: Let ν : 2 [ N ] ↦→ R denote a characteristic function (specified later) s.t. ν ( C ) quantifies the value of a collection C ⊆ [ N ] := { 1 , . . . , N } of models to capture the dependence of M i 's value on other existing models M i ′ (e.g., substitutes). Denote M i 's value by ϕ i ← Φ( i, ν, { M i } i ∈ [ N ] ) which is fully specified (up to linear scaling) by the properties:

P1 Null player: ( ∀C ⊆ [ N ] \ { i } ν ( C ∪ { i } ) -ν ( C ) = 0) = ⇒ ϕ i = 0 .

P2 Symmetry: ( ∀C ⊆ [ N ] \ { i, i ′ } ν ( C ∪ { i } ) = ν ( C ∪ { i ′ } )) = ⇒ ϕ i = ϕ i ′ .

P3 Linearity: ∀ γ, γ ′ ∈ R ( ν D∪D ′ := γ ν D + γ ′ ν D ′ ) = ⇒ ϕ i ( D∪D ′ ) = γ ϕ i ( D ) + γ ′ ϕ i ( D ′ ) .

P4 Diminished marginal utility: Add a perfect substitute (i.e., duplicate/copy) M i c of M i to the pool of N models already containing M i and denote the new pool by [ N ′ ] := [ N ] ∪ { i c } . Denote the value of M i w.r.t. [ N ] by ϕ i and w.r.t. [ N ′ ] by ϕ ′ i . Then, ϕ ′ i ≤ ϕ i .

Proposition 1. Properties P1, P2, and P3 fully specify Φ( · ) up to a linear scaling Z :

Its proof follows directly from [20, Proposition 2.1]. Since (2) coincides with the Shapley value [69], we refer to Φ as model Shapley and ϕ i as the model Shapley value (MSV). Note that P3 requires two distinct tasks D and D ′ to distinguish between the MSVs of M i w.r.t. to these two tasks. P4 additionally requires ν to be conditionally redundant 7 (Proposition 5 in App. A): The benefit of a redundant copy M i c (conditioned on model M i already being added) is not more than the initial benefit of adding M i . Intuitively, as adding M i is already sufficient for the desired task, subsequently adding M i c does not yield (as much) extra benefit [24]. In contrast, [51] also adopts (2) but assumes ν already exists while we explicitly design ν to encode predictive accuracy and certainty:

where Q C is the precision-weighted fusion of the Dirichlet abstractions in C and d ( · , · ) is a distributional distance measure between two Dirichlet abstractions. Recall from Sec. 2 that the predictive distribution of M i is represented in its Dirichlet abstraction (visualized in Fig. 1). In particular, the more accurate M i is, the closer (distributionally) Q i is to the Dirichlet abstraction Q ∗ of an expert (i.e., optimal classifier). Hence, a (high) similarity between Q i and Q ∗ can suggest M i 's (high) predictive accuracy. Specifically, Q ∗ is implemented as follows: For an input data-label pair ( x j , y j ) , take the one-hot encoded vector e y j ∈ [0 , 1] C of y j , add an independent uniform noise ϵ j ∈ [0 , 0 . 01] C (to avoid degeneracy during MLE), normalize it to sum to 1 to yield e y j ,ϵ j as a

'prediction' of the expert, and solve (1) using these 'predictions' over D . The predictive certainty of M i is encoded in the precision | α i | 1 of its Dirichlet abstraction, which is then used as a weight to fuse the individual Dirichlet abstractions Q i in C to obtain Q C :

Definition 3 ( Precision-weighted Fusion). Let the random vector [ β 1 , . . . , β n ] ∼ Dir ([ | α 1 | 1 , . . . , | α n | 1 ]) be independent of M i ( x ) for all i ∈ C ⊆ [ N ] where n := |C| . Then, the precision-weighted fusion Q C is the distribution of ∑ n i =1 β i M i ( x ) .

In Definition 3, the weight β i on M i ( x ) is large if | α i | 1 is large, i.e., Q i has a high precision/ M i has a high predictive certainty. Definition 3 has an important implication: Q C is a fully specified Dirichlet, i.e., Q C = Dir ([ ∑ n i =1 α i, 1 , . . . , ∑ n i =1 α i,C ]) (Lemma 3 in App. A) with four advantages: (A) Q i and Q C are both Dirichlets so that a single d (e.g., d H) in (3) can be used for both singleton and non-singleton C 's. Interestingly, it gives a perspective that each M i lives as Q i in a metric space w.r.t. d H (Fig. 11 in App. C). (B) We can theoretically justify learning model Shapley (Theorem 1). (C) (3) using d H can be evaluated in closed form with O (1) time, which is important since (2) requires O (2 N ) evaluations of ν ( C ) for different C 's. (D) An alternative to specify ν ( C ) is a linear combination of the performance of M i , ∀ i ∈ C as ν ( C ) . However, it is unclear what the weights in this linear combination should be. In contrast, Definition 3 'automatically' resolves this issue by fusing each Q i according to its precision | α i | 1 into Q C .

Interpreting MSV. Note that (2) is an average (over all possible C ⊆ [ N ] \ { i } ) of how much M i (or, more precisely, Q i ) improves the distributional similarity between Q C and Q ∗ (i.e., the expert) after i joins C . Both the predictive accuracy and certainty of M i can affect (2). To see this, a high predictive accuracy of M i implies that Q i is (distributionally) close to Q ∗ ; a high predictive certainty of M i ensures its weight β i is large when fused into Q C , so the predictive certainty can amplify M i 's effect in bringing Q C close to Q ∗ (if Q i is close to Q ∗ ). Hence, a model M i with a high predictive accuracy and certainty is likely to have a high MSV ϕ i . When considering several models jointly [66], we can use ϕ i to indicate how well (on average) M i combines with other models (i.e., whether M i joining C leads to a performance improvement), as verified in Sec. 5. In contrast to the work of [66], which supports up to N = 32 simpler binary classifiers, our approach -more scalable and general- supports up to N = 150 C -way classifiers (Sec. 4.2).

3.2 Connection to Cross-entropy and Other Model Evaluation Criteria

Wefirst formalize the connection between cross-entropy (CE) and our approach that uses the Hellinger distance, and then use this connection to extend our approach to other evaluation criteria such as adversarial robustness [43], distributional robustness [67], and algorithmic fairness in ML [56].

As a distributional distance measure, CE can be used specify (3) because the CE loss is used to evaluate the performance of a model. Then, specifying (3) with CE or the Hellinger distance (as proposed in Sec. 4) can be connected in how they evaluate the model performance.

Proposition 2. Let ν CE ( C ) := -CE ( Q C , Q ∗ ) , 8 ν 2 H ( C ) := -d 2 H ( Q C , Q ∗ ) and H ( · ) be differential entropy. Then,

Proposition 2 shows that ν CE ( C ) has upper and lower bounds that are monotonic in ν 2 H , providing some justification for using ν H (instead of ν CE). Note that while Proposition 2 utilizes ν 2 H due to a key Lemma 2 (in App. A), we adopt ν H := -d H (Sec. 4) as d H satisfies the triangle inequality (for Theorem 1). Moreover, Proposition 2 confirms that ν H encodes the predictive accuracy and certainty of a model since ν CE encodes the predictive accuracy and certainty (see the example in App. A).

Interestingly, this connection to CE enables the extension to adversarial robustness, distributional robustness and algorithmic fairness, with different practical motivations. For instance, adversarial robustness (in model valuation) is important to application scenarios where the model can encounter adversarial attacks in deployment [43]. Formally, the respective objective functions objective adv [43], objective DRO [67, Equation 5] and objective EO [56, Definition 2] can be achieved from a suitable definition of ν using d H (precise definitions and full deviations are deferred to App. A), as summarized

in Table 1. We highlight that this illustrates the potential generality of MSVs using the Hellinger distance w.r.t. Dirichlet abstractions (i.e., using ν H ( C ) in (2)), and defer the formal treatment of such theoretical connections to future work. 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 in App. A.

4 Learning Model Shapley

To address challenge (3) in Sec. 1, we propose a learning approach to train a model appraiser (i.e., a regression learner) from the MSVs of a small subset of models for predicting MSVs of the remaining models (further elaborated in App. C). If we can learn a good appraiser with only 20% of all models (empirically verified), then the marketplace size can (theoretically) quintuple. To justify this learning approach, we derive a Lipschitz continuity of model Shapley, which is also empirically verified using 5 real-world datasets and various model types. Next, we implement this learning approach by training a Gaussian process regression (as the model appraiser) on a subset (from 5% to 50% in size) of 150 model-MSV pairs and examine its predictive performance on the rest. Our implementation is available at https://github.com/XinyiYS/ModelShapley .

4.1 Lipschitz Continuity of Model Shapley

We derive a Lipschitz continuity of the model Shapley function Φ : [ N ] ↦→ R : The difference between the MSVs of two models M i , M i ′ (i.e., inputs to Φ ) is bounded by the distance d H ( Q i , Q i ′ ) between them, multiplied by a constant factor.

Theorem 1 ( Lipschitz Continuity). Let d := d H in (3). Then, ∀ i, i ′ ∈ [ N ] ( ∀C ⊆ [ N ] \ { i, i ′ } d H ( Q C∪ i , Q C∪ i ′ ) ≤ d H ( Q i , Q i ′ )) = ⇒ | ϕ i -ϕ i ′ | ≤ Zd H ( Q i , Q i ′ ) .

Its proof is in App. A. Theorem 1 states that the difference in MSVs of two models is bounded by the Hellinger distance between their Dirichlet abstractions, and the constant Z from (2) is the Lipschitz constant. This is based on a simple fusion-increases-similarity condition: When Q i and Q i ′ are each fused with a common Q C , the resulting similarity is higher (i.e., smaller d H) since Q C∪ i and Q C∪ i ′ have Q C in common (see Proposition 4 in App. A). Moreover, Table 3 and Fig. 6 (in App. A) empirically verify Theorem 1. Then, Theorem 1 provides a theoretical justification for the learning approach because it guarantees that similar inputs (i.e., small d H ( Q i , Q i ′ ) ) imply similar outputs (i.e., small | ϕ i -ϕ i ′ | ). namely, the model Shapley function is well-behaved w.r.t. its inputs, and hence learnable. This reasoning is applied to justify learning the data Shapley value [21].

4.2 Empirical Learning Performance via Gaussian Process Regression (GPR)

To exploit the Lipschitz continuity (i.e., Theorem 1), we adopt the Gaussian process regression (GPR) due to a uniform error bound of GPR on Lipschitz continuous functions [47]. Our implementation trains a GPR (as the model appraiser) on the MSVs of a subset of N = 150 models and examines its predictive performance on the remaining ones.

Regression setting. We train N = 150 independent models on MNIST (CIFAR-10): 50 of logistic regression (LR), multi-layer perceptron (MLP), and convolutional neural network (CNN) each (ResNet-18, SqueezeNet, and DenseNet-121 each). For simplicity, we use the test set without

partitioning as the query set D . For each M i , we obtain ¯ h i via its predictions on D and solve (1) to obtain α i as input features for separate regressions. For the regression labels, as calculating ϕ i exactly incurs O (2 150 ) time, we use the ( ϵ = 0 . 1 , δ = 0 . 1) -approximation [53] ˆ ϕ i as the average of 3745 Monte-Carlo samples. This results in two sets { α i , ˆ ϕ i } and { ¯ h i , ˆ ϕ i } of model-MSV pairs of size 150 each. We train a GPR on a random subset of 150 model-MSV pairs to learn to predict the MSV on the remaining pairs. In GPR, we use the squared exponential kernel exp( -d ( i, i ′ ) / (2 σ 2 )) (the lengthscale σ is learnt) where d ( i, i ′ ) := d H ( Q i , Q i ′ ) for { α i , ˆ ϕ i } , and d ( i, i ′ ) := | ¯ h i ′ -¯ h i | 1 for { ¯ h i , ˆ ϕ i } .

High regression performance verifies learnability. We examine the test performance using two error metrics: mean-squared error (MSE) and maximum error (MaxE) w.r.t. varied training ratios from 5% to 50% , in Fig. 2. In particular, results for training ratio of 20% are in Table 2. We observe that even using only 20% of model-MSV pairs for training, the learning is effective (i.e., low test errors), which shows its feasibility in a large-scale model marketplace. This can be attributed to the learnability justified by Theorem 1 and the uniform error bound of GPR [47]. In addition, learning on α i is more effective than learning on ¯ h i (as Table 2 and Fig. 2 show higher errors for the latter), since the average operation to get ¯ h i loses some model information.

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0.010

S

hi

-0.02

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M

0.005

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0.0000

0.25

0.50

0.25

0.50

Training ratio

Training ratio

5 MSV vs. Common Evaluation Criteria

This work is motivated by the lack of a standardized model valuation, but there are different evaluation criteria useful in different scenarios (e.g., accuracy, F 1 score, predictive certainty). Interestingly, we find that MSV can produce consistent model values with these criteria. Additionally, we evaluate the utility of MSV directly in a use case where a buyer wishes to purchase multiple models with black-box access to build a larger learner (e.g., random forest or voting classifier) [33, 66] and show that MSV can be used to effectively identify the most 'valuable' models for this purpose.

MSV vs. predictive performance. Fig. 3 (left) compares the MSVs of different model types (independently trained on the same data) for MNIST. Here D consists of misclassified data (from the original test set) of all the models to highlight the difference in their predictive accuracies. Without

needing to partition D , we observe that CNNs significantly outperform both MLPs and LRs (in terms of accuracy) and have the highest MSVs. This is expected, since CNNs are more capable of performing well in image-based tasks, and hence the MSVs for CNNs are correspondingly higher. Then, we examine predictive certainty. We use the same CNN model type independently trained on the same MNIST data (i.e., their accuracies are essentially equal), but artificially increase the predictive certainty for some: We multiply the highest probability of M i ( x j ) by a factor of [1 , 5 , 10] and then normalize the resulting vector to sum to 1 without affecting the predicted class/accuracy. Fig. 3 (right) shows that models with higher predictive certainty have higher MSVs, confirmed by additional results on CIFAR-10, MedNIST, and DrugRe in App. C. These results confirm our intuition that a model with high predictive accuracy and certainty is likely to have a high MSV.

0.02

0.000

100

75

-0.005

0.01

MSV

50

ISV

-0.010

50

M

0.00

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-25

MSV

-0.01

-0.01

0

CNN MLP

LR

1

5

10

Next, we train 3 groups of 3 LR models independently with varying sizes of training data from 0 . 001 to 0 . 01 and 0 . 1 of the entire KDD99 dataset containing highly imbalanced data; the top 3 classes out of 23 constitute 98 . 22% of total data. Intuitively, the models trained on more data should perform better and thus be more valuable, but due to the class imbalance, Fig. 4 (left) shows difficulties in differentiating these models based on their accuracies (or MSVs w.r.t. the entire/unpartitioned query set). Intuitively, to differentiate them, we need a lower level (i.e., more refined) Dirichlet abstraction, namely the class-specific Dirichlet abstractions: Partition the entire query set according to the C = 23 classes with γ k := |D k | . Define d H ( Q i , Q i ′ ; {D k } k =1 ,...,C ) := ∑ C k =1 γ k d H ( Q i, D k , Q i ′ , D k ) to leverage P3 to compute ϕ i ( {D k } k =1 ,...,C ) = ∑ C k =1 γ k ϕ i ( D k ) (right of Fig. 4). Then we can see that MSVs are indeed consistent with F 1 score (a criterion especially suited for imbalanced data) without explicitly using F 1 score in the computation. In addition, for KDD, due to the high class imbalance, there are classes with extremely small data size (i.e., ≤ 5 ) and our calculation of d H ( Q i , Q i ′ ; {D k } ) naturally suppresses their effect (possibly inaccurate Q i ) via γ k := |D k | . However, in practice, it should be noted that partitioning the query set to obtain a lower level of Dirichlet abstraction should be considered w.r.t. the size of available query set (Sec. 2). In other words, to obtain a lower level of Dirichlet abstraction (and thus a more refined representation), it incurs a higher cost from collecting a larger query set. In our experiments, we find that the size of each partitioned query set D k should contain at least 10 2 samples (e.g., for KDD most classes have at least or close to 10 2 samples).

0.015

100

0.6

0.02

score

MSV

0.010

MSV

0.4

50

0.01

S

0.005

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MSV

0.2

1

F

F1S

score

0.000

0.00

0.0

0.0010.01 0.1

0.0010.01

0.1

Identifying valuable models to purchase. For a more end-to-end use case (instead of a single evaluation criterion), we evaluate the MSVs for up to 50 models and the performance of a larger

learner by including a subset of these models based on their MSVs in a highest/lowest-first sequence in Fig. 5. The larger learner is random forests (voting classifier) and models are decision trees (LeNets [46]) for Breast Cancer (CIFAR-10). As the test accuracy of highest MSV-first increases more quickly (orange line), it verifies our previous comment on models that perform well when combined with other models are likely to have high MSVs. This characteristic offers some practical utility. If a buyer is looking to purchase models from a marketplace [66], then following the highest MSV sequence, the buyer only needs to purchase a subset of 15 (left of Fig. 5) or 25 (right of Fig. 5) out all 50 available models, thus saving cost. More results on CIFAR-100 with ResNet-18 are in App. C.

Test Accuracy

Test Accuracy

0.95

0.44

Maxaccuracy

Maxaccuracy

0.42

+

0.90

Lowest first

Lowest first

Highest first

Highest first

0.40

0

20

40

0

20

40

No.models

No.models

6 Related Work

The work of [66] investigates the binary classification setting and is not suitable for empirical comparison as we consider problems with multiple classes. The works of [10, 11, 13, 51] approach the design of a model marketplace from an economics perspective by addressing issues like arbitrage (e.g., via horizontal/vertical pricing). In contrast, we formalize the value of a model via what it has learned w.r.t. a task. The black-box access setting is appealing in a model marketplace as it accommodates different model types. Some existing methods [8, 29, 30, 33, 44] focus on how to learn a fused model from several trained models (possibly with black-box access) instead of how to value these models. We highlight that we design the fusion (Definition 3) to leverage its analytic properties in Theorem 1. The approach of learning the Shapley value arises in data valuation problems but has not been considered in model valuation. Interestingly, we can draw parallels between Theorem 1 and [19, Theorem 2.8]. For brevity, we include a more extensive contrasting comparison with data valuation in App. B.

7 Discussion and Future Work

We exploit a Dirichlet abstraction of classification models with only black-box access for proposing a novel equitable model valuation called the model Shapley . We discuss that choosing a suitable level of the Dirichlet abstraction can improve how accurately MSV reflects a model's predictive performance and empirically show that using the partitioned query sets (according to the classes) can provide a suitable trade-off between the level of abstraction and the size of the available query set. MSV behaves consistently (in our experiments) with some common model evaluation criteria (i.e., predictive accuracy and certainty, F 1 score) and can be extended to more sophisticated criteria. This implies MSV can potentially help unify existing evaluation criteria to provide a simplified model valuation in practice, without needing to explicitly perform separate evaluations.

For future work, it is interesting to explore how model valuation can help address the practical considerations encountered in existing data valuation methods [70, 78, 80] and to apply this technique to existing collaborative (learning) frameworks which require a valuation of models, both non-parametric ones [2, 62, 72, 74, 81] and parameterized ones [18, 49, 79]. Moreover, a more detailed investigation into satisfying the equitability of Shapley value [59] and its trade-off with the computational cost [82] is of practical interest, such as by applying more sophisticated analyses [47] or methods [9, 31, 32, 52] for our proposed Gaussian process regression learning approach.

Acknowledgments and Disclosure of Funding

This research/project is supported by the National Research Foundation Singapore and DSO National Laboratories under the AI Singapore Programme (AISG Award No: AISG 2 -RP2020 -018 ). Xinyi Xu is supported by the Institute for Infocomm Research of Agency for Science, Technology and Research (A ∗ STAR).

References

A. Agarwal, M. Dahleh, and T. Sarkar. A marketplace for data: An algorithmic solution. In Proc. ACM-EC , 2019.

L. Agussurja, X. Xu, and B. K. H. Low. On the convergence of the shapley value in parametric Bayesian learning games. In Proc. ICML , pages 180-196, 2022.

S. Arora, R. Ge, Y. Halpern, D. Mimno, A. Moitra, D. Sontag, Y. Wu, and M. Zhu. A practical algorithm for topic modeling with provable guarantees. In Proc. ICML , pages 280-288, 2013.

M. Basseville. Distance measures for signal processing and pattern recognition. Signal Processing , 18(4):349-369, 1989.

J. A. Blackard and D. J. Dean. Comparative accuracies of artificial neural networks and discriminant analysis in predicting forest cover types from cartographic variables. Computers and Electronics in Agriculture , 24(3):131-151, 1999.

S. Boyd, S. P. Boyd, and L. Vandenberghe. Convex optimization . Cambridge Univ. Press, 2004.

T. Brown, B. Mann, N. Ryder, M. Subbiah, J. D. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, S. Agarwal, A. Herbert-Voss, G. Krueger, T. Henighan, R. Child, A. Ramesh, D. Ziegler, J. Wu, C. Winter, C. Hesse, M. Chen, E. Sigler, M. Litwin, S. Gray, B. Chess, J. Clark, C. Berner, S. McCandlish, A. Radford, I. Sutskever, and D. Amodei. Language models are few-shot learners. In Proc. NeurIPS , 2020.

H. Chang, V. Shejwalkar, R. Shokri, and A. Houmansadr. Cronus: Robust and heterogeneous collaborative learning with black-box knowledge transfer. In Proceedings of the 1st NeurIPS Workshop on New Frontiers in Federated Learning (NFFL 2021) , 2021.

J. Chen, N. Cao, K. H. Low, R. Ouyang, C. K.-Y. Tan, and P. Jaillet. Parallel gaussian process regression with low-rank covariance matrix approximations. In Proc. UAI , page 152-161, 2013.

L. Chen, P. Koutris, and A. Kumar. Model-based pricing: Do not pay for more than what you learn! In Proc. ACM SIGMOD Workshop on Data Management for End-to-end Machine Learning , 2017.

L. Chen, P. Koutris, and A. Kumar. Towards model-based pricing for machine learning in a data marketplace. In Proc. ACM SIGMOD , page 1535-1552, 2019.

H. Chernoff. A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations. The Annals of Mathematical Statistics , 23(4):493-507, 1952.

Z. Cong, X. Luo, J. Pei, F. Zhu, and Y. Zhang. Data pricing in machine learning pipelines. Knowledge and Information Systems , 64:1417-1455, 2022.

A. S. Das, M. Datar, A. Garg, and S. Rajaram. Google news personalization: Scalable online collaborative filtering. In Proc. WWW , 2007.

P. Devijver and J. Kittler. Pattern Recognition: A Statistical Approach . Prentice-Hall, London, 1982.

J. M. Drazen, S. Morrissey, D. Malina, M. B. Hamel, and E. W. Campion. The importance and the complexities - of data sharing. New England Journal of Medicine , 375(12):1182-1183, 2016.

M. Ester, H.-P. Kriegel, J. Sander, and X. Xu. A density-based algorithm for discovering clusters in large spatial databases with noise. In Proc. KDD , 1996.

F. X. Fan, Y. Ma, Z. Dai, W. Jing, C. Tan, and B. K. H. Low. Fault-tolerant federated reinforcement learning with theoretical guarantee. In Proc. NeurIPS , 2021.

A. Ghorbani, M. Kim, and J. Zou. A distributional framework for data valuation. In Proc. ICML , pages 3535-3544, 2020.

A. Ghorbani and J. Zou. Data Shapley: Equitable valuation of data for machine learning. In Proc. ICML , pages 2242-2251, 2019.

A. Ghorbani, J. Zou, and A. Esteva. Data Shapley valuation for efficient batch active learning, 2021.

I. Goodfellow, Y. Bengio, and A. Courville. Deep Learning . MIT Press, 2016.

F. Gr¨ aßer, S. Kallumadi, H. Malberg, and S. Zaunseder. Aspect-based sentiment analysis of drug reviews applying cross-domain and cross-data learning. In Proc. International Conference on Digital Health , page 121-125, 2018.

D. Han, M. Wooldridge, A. Rogers, S. Tople, O. Ohrimenko, and S. Tschiatschek. Replicationrobust payoff-allocation for machine learning data markets. Journal of IEEE Transactions on Artificial Intelligence , 2022.

M. F. Haque and R. Krishnan. Toward automated cyber defense with secure sharing of structured cyber threat intelligence. Information Systems Frontiers , 23(4):883-896, 2021.

K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In Proc. CVPR , 2016.

E. Hellinger. Neue begr¨ undung der theorie quadratischer formen von unendlichvielen ver¨ anderlichen. Journal f¨ ur die reine und angewandte Mathematik , page 210-271, 1909.

S. Hettich and S. D. Bay. KDD Cup 1999 data data set. https://archive.ics.uci.edu/ ml/datasets/kdd+cup+1999+data , 1999.

N. Hoang, T. Lam, B. K. H. Low, and P. Jaillet. Learning task-agnostic embedding of multiple black-box experts for multi-task model fusion. In Proc. ICML , pages 4282-4292, 2020.

Q. M. Hoang, T. N. Hoang, B. K. H. Low, and C. Kingsford. Collective model fusion for multiple black-box experts. In Proc. ICML , pages 4857-4867, 2019.

Q. M. Hoang, T. N. Hoang, and K. H. Low. A generalized stochastic variational Bayesian hyperparameter learning framework for sparse spectrum Gaussian process regression. In Proc. AAAI , volume 31, 2017.

T. N. Hoang, Q. M. Hoang, and B. K. H. Low. A unifying framework of anytime sparse gaussian process regression models with stochastic variational inference for big data. In Proc. ICML , volume 37, pages 569-578, 2015.

T. N. Hoang, S. Hong, C. Xiao, B. K. H. Low, and J. Sun. AID: Active distillation machine to leverage pre-trained black-box models in private data settings. In Proc. WWW , pages 3569-3581, 2021.

H. Homei. Randomly weighted averages: A multivariate case, 2016.

G. Huang, Z. Liu, L. van der Maaten, and K. Q. Weinberger. Densely connected convolutional networks. In Proc. CVPR , 2017.

J. Huang. Maximum likelihood estimation of dirichlet distribution parameters. CMU Technique report , 18, 2005.

F. N. Iandola, S. Han, M. W. Moskewicz, K. Ashraf, W. J. Dally, and K. Keutzer. Squeezenet: Alexnet-level accuracy with 50x fewer parameters and¡ 0.5 mb model size. arXiv preprint arXiv:1602.07360 , 2016.

H. Jeffreys. An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences , 186(1007):453-461, 1946.

R. Jia, D. Dao, B. Wang, F. A. Hubis, N. Hynes, N. M. G¨ urel, B. Li, C. Zhang, D. Song, and C. J. Spanos. Towards efficient data valuation based on the shapley value. In Proc. AISTATS , pages 1167-1176, 2019.

G. A. Kaissis, M. R. Makowski, D. R¨ uckert, and R. F. Braren. Secure, privacy-preserving and federated machine learning in medical imaging. Nature Machine Intelligence , 2:305-311, 2020.

A. Krizhevsky. Learning multiple layers of features from tiny images. Master's thesis, Department of Computer Science, University of Toronto, 2009.

S. Kullback and R. A. Leibler. On information and sufficiency. The Annals of Mathematical Statistics , 22(1):79 - 86, 1951.

A. Kurakin, I. J. Goodfellow, and S. Bengio. Adversarial machine learning at scale. In Proc. ICLR , 2017.

T. C. Lam, N. Hoang, B. K. H. Low, and P. Jaillet. Model fusion for personalized learning. In Proc. ICML , pages 5948-5958, 2021.

Y. LeCun, B. Boser, J. Denker, D. Henderson, R. Howard, W. Hubbard, and L. Jackel. Handwritten digit recognition with a back-propagation network. In Proc. NeurIPS , 1990.

Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE , 86(11):2278-2324, 1998.

A. Lederer, J. Umlauft, and S. Hirche. Uniform error bounds for Gaussian process regression with application to safe control. In Proc. NeurIPS , 2019.

J. Lin. On the Dirichlet distribution., 2016. Master's thesis, Department of Mathematics and Statistics, Queen's University.

X. Lin, X. Xu, S.-K. Ng, C.-S. Foo, and B. K. H. Low. Fair yet asymptotically equal collaborative learning. In Proc. ICML , pages 21223-21259, 2023.

T. Lissack and K.-S. Fu. Error estimation in pattern recognition via l α -distance between posterior density functions. IEEE Transactions on Information Theory , 22(1):34-45, 1976.

J. Liu, J. Lou, J. Liu, L. Xiong, J. Pei, and J. Sun. Dealer: An end-to-end model marketplace with differential privacy. Proc. VLDB Endow. , 14(6):957-969, 2021.

B. K. H. Low, J. Yu, J. Chen, and P. Jaillet. Parallel gaussian process regression for big data: Low-rank representation meets markov approximation. In Proc. AAAI , page 2821-2827, 2015.

S. Maleki, L. Tran-Thanh, G. Hines, T. Rahwan, and A. Rogers. Bounding the estimation error of sampling-based Shapley value approximation with/without stratifying, 2013.

A. Malinin. Uncertainty estimation in deep learning with application to spoken language assessment . PhD thesis, Department of Engineering, University of Cambridge, 2019.

A. Malinin and M. Gales. Predictive uncertainty estimation via prior networks. In Proc. NeurIPS , 2018.

N. Mehrabi, F. Morstatter, N. Saxena, K. Lerman, and A. Galstyan. A survey on bias and fairness in machine learning. ACM Computing Surveys , 54(6), 2021.

T. Minka. Estimating a dirichlet distribution, 2000.

MONAI Consortium. MONAI: Medical open network for AI. https://github.com/ Project-MONAI/MONAI , 2020. Version 0.6.0.

Q. P. Nguyen, B. K. H. Low, and P. Jaillet. Trade-off between payoff and model rewards in shapley-fair collaborative machine learning. In Proc. NeurIPS , pages 30542-30553, 2022.

E. Patrick and F. Fischer. Nonparametric feature selection. IEEE Transactions on Information Theory , 15(5):577-584, 1969.

Y. Polyanskiy. Lecture notes on f -divergences, 2019. Lecture notes, https://people.lids. mit.edu/yp/homepage/data/LN_fdiv.pdf .

R. Qiao, X. Xu, and B. K. H. Low. Collaborative causal inference with fair incentives. In Proc. ICML , pages 28300-28320, 2023.

T. W. Rauber, T. Braun, and K. Berns. Probabilistic distance measures of the Dirichlet and Beta distributions. Pattern Recognition , 41(2):637-645, 2008.

S. Romano, N. X. Vinh, J. Bailey, and K. Verspoor. Adjusting for chance clustering comparison measures. Journal of Machine Learning Research , 17(1):4635-4666, 2016.

G. Ronning. Maximum likelihood estimation of Dirichlet distributions. Journal of Statistical Computation and Simulation , 32(4):215-221, 1989.

B. Rozemberczki and R. Sarkar. The Shapley value of classifiers in ensemble games. In Proc. CIKM , pages 1558-1567, 2021.

S. Sagawa, P. W. Koh, T. B. Hashimoto, and P. Liang. Distributionally robust neural networks. In Proc. ICLR , 2020.

M. Sensoy, L. Kaplan, and M. Kandemir. Evidential deep learning to quantify classification uncertainty. In Proc. NeurIPS , 2018.

L. S. Shapley. A value for n -person games. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games , volume 2, pages 307-317. Princeton University Press, 1953.

R. H. L. Sim, X. Xu, and B. K. H. Low. Data valuation in machine learning: 'ingredients', strategies, and open challenges. In Proc. IJCAI-22 , pages 5607-5614, 2022. Survey Track.

R. H. L. Sim, Y. Zhang, M. C. Chan, and B. K. H. Low. Collaborative machine learning with incentive-aware model rewards. In Proc. ICML , 2020.

R. H. L. Sim, Y. Zhang, T. N. Hoang, X. Xu, B. K. H. Low, and P. Jaillet. Incentives in private collaborative machine learning. In Proc. NeurIPS , 2023.

K. Singhal, H. Sidahmed, Z. Garrett, S. Wu, J. K. Rush, and S. Prakash. Federated reconstruction: Partially local federated learning. In Proc. NeurIPS , 2021.

S. S. Tay, X. Xu, C. S. Foo, and B. K. H. Low. Incentivizing collaboration in machine learning via synthetic data rewards. In Proc. AAAI , volume 36, pages 9448-9456, 2022.

C. Tosh and S. Dasgupta. The relative complexity of maximum likelihood estimation, MAP estimation, and sampling. In Proc. COLT , pages 2993-3035, 2019.

T. Wang, Y. Yang, and R. Jia. Learnability of learning performance and its application to data valuation, 2021.

L. Watson, R. Andreeva, H.-T. Yang, and R. Sarkar. Differentially private Shapley values for data evaluation, 2022.

Z. Wu, Y. Shu, and B. K. H. Low. DAVINZ: Data valuation using deep neural networks at initialization. In Proc. ICML , pages 24150-24176, 2022.

X. Xu, L. Lyu, X. Ma, C. Miao, C. S. Foo, and B. K. H. Low. Gradient driven rewards to guarantee fairness in collaborative machine learning. In Proc. NeurIPS , 2021.

X. Xu, Z. Wu, C.-S. Foo, and B. K. H. Low. Validation free and replication robust volume-based data valuation. In Proc. NeurIPS , 2021.

X. Xu, Z. Wu, A. Verma, C. S. Foo, and B. K. H. Low. Fair: Fair collaborative active learning with individual rationality for scientific discovery. In Proc. AISTATS , pages 4033-4057, 2023.

Z. Zhou, X. Xu, R. H. L. Sim, and C.-S. F. B. K. H. Low. Probably approximate Shapley fairness with applications in machine learning. In Proc. AAAI , 2023.

A Theoretical Discussion

A.1 Results Related to Maximum Likelihood Estimation (MLE)

Log-likelihood function F (1) . For completeness, we provide the full log-likelihood function [36, 57, 65] using the predictions as follows: To avoid notational overload, let y j := M i ( x j ) denote the predictive probability vector of M i on x j and its dependence on i is suppressed. Similarly, we suppress the dependence of α on i since the context is clear. Then,

where log ¯ h i,k is the k -th component of log ¯ h i . Since the final expression of the above log-likelihood function F only depends on the log-predictions { log y j } j =1 ,...,D through the observed sufficient statistics log ¯ h i (i.e., with an element-wise log operation), we can directly use ¯ h i in (1) instead of { y j } j =1 ,...,D .

Similar predictions between models imply similar MLE approximations of their Dirichlet abstractions. Intuitively, two models produce similar predictions if and only if their Dirichlet abstractions are similar. The result below precisely formalizes this intuition by exploiting the analytic tractability of the log-likelihood function F (1) which is concave with a unique maximizer:

Proposition 3. Suppose that the observed sufficient statistics log ¯ h i ( log ¯ h i ′ ) based on the predictions of M i ( M i ′ ) on D and the MLE approximation of its Dirichlet abstraction α i ( α i ′ ) are given. Then,

Proof. From (1),

Note that the RHS expression of (4) to be maximized remains concave since F ( α , ¯ h i ′ ) is concave in α [36] and the D ( log ¯ h i -log ¯ h i ′ ) α ⊤ term is linear in α . If | ¯ h i -¯ h i ′ | 1 = 0 , then the D ( log ¯ h i -log ¯ h i ′ ) α ⊤ term in (4) becomes 0 . So, α i = argmax α F ( α , ¯ h i ) = argmax α F ( α , ¯ h i ′ ) = α i ′ since F is concave with a unique maximizer. Therefore, | ¯ h i -¯ h i ′ | 1 = 0 = ⇒ α i = α i ′ .

From Lemma 1 below, log ¯ h i,k = ψ ( α i,k ) -ψ ( | α i | 1 ) and log ¯ h i ′ ,k = ψ ( α i ′ ,k ) -ψ ( | α i ′ | 1 ) where α i,k ( α i ′ ,k ) is the k -th component of α i ( α i ′ ) and ψ is the digamma function. 9 It follows immediately that

Importantly, both abstractions are able to preserve the similarity between M i and M i ′ , via either the Hellinger distance d H ( α i , α i ′ ) (further discussed in Sec. 4.1) or the ℓ 1 distance | ¯ h i -¯ h i ′ | 1

(Proposition 3 above) where a lower ℓ 1 or d H distance is equivalent to a higher similarity. We also empirically compare α i , ¯ h i in Sec. 4.2 and find that in general α i is better.

Lemma 1. Suppose that the observed sufficient statistics log ¯ h i based on the predictions of M i on D and the MLE approximation of its Dirichlet abstraction α i are given. Then,

Proof. The partial derivative of F ( α , ¯ h i ) w.r.t. α k can be explicitly derived as follows:

For a concave optimization problem (i.e., maximizing the log-likelihood) subject to a non-negative orthant constraint (i.e., α ⪰ 0 ), the work of [6] has shown that the optimality conditions can be expressed as

The last condition is known as the complementarity . Focusing on the complementarity condition w.r.t. the k -th components of α i and ¯ h i gives

Since α i,k > 0 (Definition 1),

which is simplified to and thus,

A.2 Results Related to Dirichlet Distribution and Distributional Distance

Suitable measures of distance between Dirichlet distributions. We highlight the challenges in applying the distance measures beside the Hellinger distance in App. A. Specifically, the Hellinger distance provides unique and desirable theoretical properties (e.g., satisfying the triangle inequality) which we exploit (e.g., to derive Theorem 1)

In our context, we seek a distance measure that is well-defined between two Dirichlet distributions, can be evaluated in closed form, and has analytic properties. The work of [63] has painstakingly compared several probabilistic distance measures including the Kullback-Leibler (KL) and symmetric KLdivergences [38, 42], Patrick-Fischer distance [60], generalized Matusita distance [4], Lissack-Fu distance [50], Kolmogorov [15] distance, Chernoff distance [12], and Hellinger distance [27], and made the following observations: The KL and symmetric KL divergences and the Patrick-Fischer distance can encounter cases where the distance becomes undefined. The generalized Matusita distance, Lissack-Fu distance, and Kolmogorov distance all lack an anti-derivative. So, we consider and compare the remaining two options: Chernoff and Hellinger distances, which are connected as follows.

Chernoff vs. Hellinger distances. As mentioned earlier, the Chernoff distance d H is the other theoretically appealing choice for Dirichlet distributions [63], so we discuss its connection with the Hellinger distance as follows. We start by recalling the definition for Chernoff distance and derive an analytic connection between them.

Definition 4 ( Chernoff distance [12]) . The Chernoff distance between two distributions p and q is d C ( p, q ; λ ) := -ln ∫ p λ ( x ) q 1 -λ ( x ) d x where λ ∈ (0 , 1) . Note that d C ( p, q ; λ = 1 / 2) is symmetric in p and q .

For two continuous distributions p, q , their Hellinger distance d H ( p, q ) and their Chernoff distance (with λ = 1 / 2 ) d C ( p, q ; λ = 1 / 2) are connected via the Bhattacharyya coefficient BC ( p, q ) := ∫ √ p ( x ) q ( x ) dx ∈ [0 , 1] as follows,

From Equ.(6), d C is monotonic w.r.t. d H, but d C has a logarithmic dependence which results in it not being a proper metric (i.e., does not satisfy the triangle inequality) unlike d H. We specifically leverage the triangle inequality to prove Theorem 1, which seems difficult to do for the Chernoff distance.

Interestingly, this logarithmic dependence turns out to be a practical advantage where d H can run the risk of numerical overflow [63]. Our clustering experiment of the models (App. C) runs into the issue of numerical overflow of the Hellinger distance and we resort to the Chernoff distance. Our empirical results (App. C) also support this that using d C can obtain better results than using d H.

Proof of Proposition 2. Continuing from our discussion in Sec. 3, ν CE ( C ) ≤ ν 2 H ( C ) follows from Lemma 2 below by requiring a sufficient condition on the differential entropy of Q C , as stated in Proposition 2. 10 On the other hand, constructing a lower bound of ν CE ( C ) via ν 2 H ( C ) is less direct, as can be seen from the upper bound on CE using d 2 H in Lemma 2. Deriving such a lower bound (Proposition 2) involves exploiting a property specific to Dirichlet distributions that the differential entropy is non-positive and bounded from above.

Proof of Proposition 2. By substituting p = Q C and q = Q ∗ into the first inequality of Lemma 2, 2 d 2 H ( Q C , Q ∗ ) ≤ CE ( Q C , Q ∗ ) -H ( Q C ) . Using H ( Q C ) + d 2 H ( Q C , Q ∗ ) ≥ 0 , it follows that d 2 H ( Q C , Q ∗ ) ≤ CE ( Q C , Q ∗ ) , so ν CE ( C ) ≤ ν 2 H ( C ) by plugging in the definitions of ν CE ( C ) := -CE ( Q C , Q ∗ ) and ν 2 H ( C ) := -d 2 H ( Q C , Q ∗ ) .

Next, the differential entropy of a Dirichlet distribution p parameterized by α [48, Table 2.1] is

which is maximized at α = 1 C [55, 54], and its maximum value is therefore -log Γ( C ) . Substituting H ( p ) ≤ -log Γ( C ) into the second inequality of Lemma 2 gives

By substituting p = Q C and q = Q ∗ into the above and plugging in the definitions of ν CE ( C ) := -CE ( Q C , Q ∗ ) and ν 2 H ( C ) := -d 2 H ( Q C , Q ∗ ) ,

which can be rearranged to complete the proof.

Lemma 2. For any two continuous distributions p and q ,

where H ( p ) := -∫ p ( x ) log p ( x ) d x is the differential entropy of p , d H ( p, q ) ∈ [0 , 1] is the Hellinger distance, and const q := [log(1 /q min -1)] / (1 -2 q min ) with q min := min z q ( z ) .

by substituting the equalities and

As a result, or equivalently,

Proof. Firstly,

To prove (8),

where the inequality is due to -log z ≥ 1 -z for all z ≥ 0 by setting z = √ q ( x ) / √ p ( x ) ≥ 0 . Moreover, we have an upper bound of d KL ( p, q ) from [61, Equation 7.27]:

Lastly, substituting d KL ( p, q ) = CE ( p, q ) -H ( p ) from (7) into (8) and (9) completes the proof.

An example on how cross-entropy loss encodes the predictive accuracy and certainty. The CE loss is used to construct upper and lower bounds for our proposed method (i.e., ν in Eq. (3)). Hence, our proposed method also encodes the predictive accuracy and certainty, as exemplified below.

Recall that the CE loss of a C -dimensional predicted probability vector ˆ y w.r.t. the one-hot encoded true label y :

W.l.o.g., assume that y 1 = 1 (i.e., the correct class is the first class).

For two predictions [0 . 9 , 0 . 1 , 0 , . . . , 0] vs. [0 . 1 , 0 . 9 , 0 , . . . , 0] . The CE losses are 0 . 105 and 2 . 30 , respectively. Note that the first prediction is correct while the second in incorrect and that both predictions are 'equally certain'. Hence, a higher predictive accuracy implies a lower CE .

where d KL ( p, q ) := ∫ p ( x ) log( p ( x ) /q ( x )) d x is the Kullback-Leibler distance. Next, we derive a lower bound of d KL ( p, q ) in terms of d H ( p, q ) :

For two predictions [0 . 9 , 0 . 1 , 0 , . . . , 0] vs. [0 . 6 , 0 . 4 , 0 , . . . , 0] . The CE losses are 0 . 105 and 0 . 511 , respectively. Note that both predictions are correct while the first prediction is 'more certain'. Hence, a higher predictive certainty implies a lower CE , if the prediction is correct.

Extension to other model evaluation criteria. In addition to predictive accuracy and certainty, there are other possible desirable criteria for model valuation such as adversarial robustness [43], distributional robustness [67], and algorithmic fairness (i.e., by removing prediction bias) in ML [56]. Put differently, it is possible to devise other model valuations based on adversarial robustness, distributional robustness, or the algorithmic fairness of a model. Interestingly, since these more sophisticated criteria all utilize the same building blocks (i.e., the predictive accuracy and certainty of models on carefully selected query sets), our proposed approach can subsume these different criteria, as described below and summarized in Table 1. To be a little technical, the high-level idea is that, since these criteria leverage the CE loss ℓ CE in very specific ways, and we have derived the relationship between d H and CE above (Lemma 2), our approach can be extended to incorporate these criteria through careful choices of ν .

For instance, the work of [43] explicitly defines 'adversarial' training examples D adv to be distinguished from 'clean' training examples D clean and evaluates a model's performance (i.e., a linear combination of the CE losses w.r.t. the adversarial and clean training examples separately) as follows:

for some weight υ > 0 where we suppress the constants that linearly depend on the sizes of D adv and D clean for simplicity and ℓ CE ( M ; D ) is the common CE loss incurred by the predictions of M on query set D .

The work of [67, Equation 5] gives the group-adjusted distributionally robust optimization (DRO) estimator where each 'group' g ∈ G contains training examples from a possibly different data distribution. Effectively, the optimizer minimizes the maximum CE loss over different groups/query sets s.t. each query set is a dataset from a possibly different data distribution:

where, for simplicity, a group size-dependent constant and a model capacity-dependent constant are ignored.

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: D + prot containing positive training examples under the protected group and D + 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:

Table 1 gives the specific definitions of query sets with the corresponding (possible) choices of the characteristic function ν adapted from the above-mentioned minimization objectives by replacing ℓ CE ( M ; D ) with d ( Q , Q ∗ ; D ) and adding a negation since these are minimization objectives.

Firstly, we show the original implementations/formulations using the CE loss ℓ CE [43, 67, 56] can be reformulated using the CE between Dirichlet abstractions. We provide the reformulation of objective DRO ,

by replacing ℓ CE with CE on the respective Dirichlet abstraction Q (for M ) and an expert 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 ℓ CE: The CE loss ℓ CE ( M ; D ) implicitly assumes an expert who provides the correct labels to compute the loss on the query set D .

On the other hand, CE ( Q , Q ∗ ; D ) explicitly uses the expert and measures the cross-entropy between the two Dirichlet abstractions (i.e., one for the model M and the other for the expert) on the query set D . Therefore, if a model M makes predictions similar to the expert on a fixed query set D , then both ℓ CE ( M ; D ) and CE ( Q , Q ∗ ; D ) are small (i.e., optimum is 0 ). Next, inspired by the relationship between d H and CE in Proposition 2, the expressions in CE are reformulated using d 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: Q C = Dir ([ ∑ n i =1 α i, 1 , . . . , ∑ n i =1 α i,C ]) [34, Theorem 2.1]. Lemma 4 ( Bhattacharyya coefficient between Dirichlet distributions [63]) . Let α and α ′ denote the C -dimensional parameters specifying the two Dirichlet distributions p and q , respectively. Then, the Bhattacharyya coefficient is

Lemma 5. The Hellinger distance between two Dirichlet distributions p and q parameterized by the respective α and α ′ is

which follows directly from an equivalent definition of d H ( p, q ) := 2 × √ 1 -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 ( X , d X ) and ( Y , d Y ) , a function f : X ↦→ Y is L f -Lipschitz continuous if there exists a constant L f > 0 s.t. ∀ x 1 , x 2 ∈ X

In our formulation, for the model Shapley function Φ , the input space is the set [ N ] (or more precisely the set { Q i : i ∈ [ N ] } of Dirichlet abstractions) of the N models and the metric is the Hellinger distance d H, which is a proper metric for probability distributions (in this case Dirichlet distributions); the output space is R (i.e., for model Shapley values) and the metric is the absolute difference (i.e., | ϕ i -ϕ i ′ | for two inputs Q i , Q i ′ ).

In summary, recall that ϕ i ← Φ( i, ν, { M i } i ∈ [ N ] ) in Sec. 3, the Lipschitz continuity of Φ is w.r.t. its first argument (i.e., i ), when ν is defined as in (3) and the set { M i } i ∈ [ N ] of models is fixed (i.e., correspondingly the set { Q i } i ∈ [ N ] is also fixed). For brevity, we suppress the notational dependence on the latter arguments and write as ϕ i := Φ( i ; · ) : [ N ] ↦→ R where as mentioned above, the metric for the input space is defined as d ( i, i ′ ) := d H ( Q i , Q i ′ ) for i, i ′ ∈ [ N ] and the metric for the outputs is the absolute difference | ϕ i -ϕ i ′ | .

Proof of Theorem 1. We follow an idea that the similarity between the Dirichlet abstractions Q i and Q i ′ will lead to a small difference in their expected marginal contributions ϕ i and ϕ i ′ when fused with a common Q C for any C ⊆ [ N ] \ { i, i ′ } . Consequently, we can apply Lemma 6 in App. A.4.

Firstly, from ν H ( C ) = -d H ( Q C , Q ∗ ) as in (3),

Then, using the property of triangle inequality of d H, it follows that

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 ′ ) := d H ( Q i , Q i ′ ) < 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 α i and α i ′ parameterize Q i and Q i ′ , respectively. Then,

where ψ is the digamma function.

Proof of Proposition 4. It can be observed from Lemma 5 that d H ( p, q ) increases iff BC ( p, q ) (Lemma 4) decreases. Then, it is equivalent to show that

It can also be observed that BC can be viewed as a differentiable function taking in 2 C parameters (i.e., α i and α i ′ ). Consider w.l.o.g. its partial derivative w.r.t. α i,k and w.r.t. α i ′ ,k for k = 1 , . . . , C :

where

due to the positivity of Γ over the positive domain. 11

Now, Lemma 3 implies that the fusion always increases α i,k : Since α i,k denotes the k -th component of α i and α C∪{ i } ,k denotes the k -th component of α C∪{ i } , α C∪{ i } ,k ≥ α i,k . So, if ∂ BC /∂α i,k ≥ 0 and ∂ BC /∂α i ′ ,k ≥ 0 for k = 1 , . . . , C , then the resulting BC increases (or, equivalently, d H decreases) after fusion.

Since coeff > 0 ,

for k = 1 , . . . , C . So, the final result follows.

Let D sum := ∑ C k =1 ψ (( α i,k + α i ′ ,k ) / 2) -( ψ ( α i,k ) + ψ ( α i ′ ,k )) / 2 and D |·| := C [ ψ (( | α i | 1 + | α i ′ | 1 ) / 2) -( ψ ( | α i | 1 ) + ψ ( | α i ′ | 1 )) / 2] . The sufficient condition for Proposition 4 implies that D sum ≥ D |·| . Intuitively, D sum is a sum of dimension-/class-wise differences between Q i and Q i ′ and it is large if every pair of α i,k and α i ′ ,k are different and relatively small (i.e., low concentration for dimension/class k ). D sum is likely large if the dimension C is large as there are more pairs of α i,k and α i ′ ,k whose difference contributes towards D sum. On the other hand, D |·| is a measure of the difference in the overall precisions of Q i and Q i ′ . D |·| is large if | α i | 1 and | α i ′ | 1 are different and have small values. D |·| is likely small if C is large. As C increases, the precisions | α i | 1 and | α i ′ | 1 will increase, which will cause ψ (( | α i | 1 + | α i ′ | 1 ) / 2) and ( ψ ( | α i | 1 ) + ψ ( | α i ′ | 1 )) / 2 to be very close due to the converging behavior of ψ .

The condition D sum ≥ D |·| says that if the class-wise difference between Q i and Q i ′ outweighs the difference in their precisions, then fusion increases similarity. Intuitively, if the 'shapes' of Q i and Q i ′ are very different, then fusing each with a common distribution 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 H ( Q i , Q i ′ ) leads to a small | ϕ i -ϕ i ′ | via the Pearson coefficient between d H ( Q i , Q i ′ ) and | ϕ i -ϕ 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 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 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 DenseNet121 [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.

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.

0

0.1

0

0

8642

0.75

8642

3642

0.10

2

0.05

0.50

64

0.05

0.25

0.0

8

8

0.00

02468

02468

02468

02468

A.4 Model Shapley Value

Closed-form expression of ν and the resulting computational complexity. Recall that an advantage of the combined choices of Definition 3 and the Hellinger distance d H (described in Sec. 3) is an available closed form evaluation of d H between two Dirichlet abstractions, which has a constant time computational cost (i.e., O (1) ). This is important as d H is used to define the characteristic function ν (3) for model Shapley (2). Specifically, note that the definition of model Shapley (2) requires evaluating the characteristic function ν for an exponential number of times due to the summation over all possible subsets C ⊆ [ N ] \ { i } . In other words, even with a characteristic function ν that can be evaluated in constant time (e.g., d 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 ′ ∈ [ N ] ,

where L ≥ 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 | ϕ i -ϕ ′ i | can be bounded by enumerating the coalitions in a paired way as follows:

where ω C := |C| !( N - |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 ) C ⊆ [ N ] \ { i } in the calculation of ϕ i . The first summation enumerates all C ⊆ [ N ] \ { i, i ′ } , so the remaining 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 | ν ( C ∪ { i } ) -ν ( C ∪ { i ′ } ) | for some C ⊆ [ N ] \ { i, i ′ } . Note that in the second summation,

To complete the final step, first consider the simpler case of Z = 1 and the coefficient ω C is defined in a way such that ϕ i satisfies efficiency [69] (i.e., ∑ C⊆ [ N ] \{ i } ω C = 1 ), the condition ∀C ⊆ [ N ] \ { i, i ′ } | ν ( C ∪ { i } ) -ν ( C ∪ { i ′ } ) | ≤ Ld ( i, i ′ ) can be used to bound the overall sum of the RHS as

̸

More generally for Z = 1 , it can be directly multiplied to the RHS as follows

since every term is multiplied by the same constant in the above two summations.

Proposition 5 ( Diminishing Model Shapley Value due to Substitutes). According to the definitions as in (P4), ϕ i ( ϕ ′ i ) denotes the model Shapley value of M i w.r.t. [ N ] ( [ N ′ ] := [ N ] ∪ { i c } ) and M i c = M i is a perfect substitute/identical duplicate/copy. Then,

Proof of Proposition 5. The inequality ϕ i ≥ ϕ ′ i is shown by examining the pairwise difference over their respective summands in the summation of model Shapley.

For ϕ i , ϕ i := ∑ C⊆ [ N ] \{ i } s C where the summand s C := ω C MC i ( C ) . ω C is as in (2) and MC i ( C ) := ν ( C ∪ { i } ) -ν ( C ) is the marginal contribution of i w.r.t. C .

For ϕ ′ i , ϕ ′ i := ∑ C⊆ [ N ] \{ i } C ′ = C∪{ i c } s ′ C where the summand

ϕ ′ i is obtained by observing that adding i c to [ N ] means additionally enumerating all the C ⊆ [ N ] \{ i } but added with i c, as shown above.

where the inequality is due to the conditionally redundant property of ν . 12 Since ϕ i and ϕ ′ i enumerate the same summation and individual summands have s ′ C ≤ s C , it follows that ϕ ′ i ≤ ϕ i .

Remedy for duplication from a dishonest seller. While the marginal utility of each duplicate model decreases, the combined utility of all the duplicated models may be higher than if there is only one such model. Hence, a dishonest seller might exploit this by duplicating a model to receive a higher combined utility. As a hypothetical example to illustrate this: two models M i , M j from vendors i, j respectively, each have values 0 . 5 , then if the model seller i decides to fraudulently duplicate model M i to another M i c . Although the value for M i depreciates, such duplication can lead to a higher value for vendor i , namely the value of M i and M i c combined might be higher than if only M i is present.

We note that our proposed approach can be adapted to address this issue relatively easily, by substituting our proposed in Eq. (3) into the variant of the Shapley value [24, Theorem 4.5], which importantly continues to satisfy the properties (P1), (P2) and (P3) [24]. However, we highlight that (the robustness to) such duplication is beyond the scope of this work.

B Additional Literature Review

B.1 Relation to Data Valuation and its Design Approach

Data valuation, originating from the motivating application scenario of AI marketplaces [1, 13, 24], studies how to determine the intrinsic worth of data, often in the context of ML. Intuitively, as these marketplaces treat data as commodities for trading, a pricing mechanism (i.e., a valuation function) is necessary. There have been some works exploring data valuation [10, 20, 39, 80], by leveraging ML principles and assumptions. For instance, data are more valuable if training on the data produces an MLmodel with higher performance (i.e., more accurate). In addition to the application scenario of AI marketplaces, the value of data can also be/has been used in interpretable ML [20], data sharing [16] and collaborative ML [76, 79]. However, as motivated in Sec. 1, there are various practical scenarios where data valuation is difficult. Hence, we explore an alternative by shifting our focus onto the ML models in these scenarios to consider model valuation.

In terms of the design approach, although the existing data valuation works have different technical perspectives and thus different solutions, they often adopt a common first-principle approach, i.e., data which can produce more accurate models are more valuable . This approach provides an interpretation for the valuation function or the values assigned to the data. While it may seem counter-intuitive to market economics where the price/value can naturally arise from the demand and supply, this approach is sensible because the lack of effective demand (i.e., the willingness and ability of buyers to purchase goods/data/ML model at different prices). To elaborate, effective demand requires the buyers to have some intrinsic valuation function for the data/ML model which helps determine the quantity the buyers are willing to purchase at some fixed price. In contrast to the more conventional goods, the market for data/ML model is relatively niche in the sense that even the buyers themselves do not already have a good intrinsic valuation function for the data/ML model. As a result, the buyers are unable to specify the effective demand, which makes it difficult for the price/valuation to arise naturally from the demand and supply in a market. To this end, the (proposed) valuation methods (i.e., existing data valuation methods and the model valuation method in this paper) aim to fill in this gap by explicitly designing such a valuation function where the value is determined through the utility of the data/ML model in the context of ML (e.g., predictive performance). In this vein, the existing data valuation methods and our proposed model valuation method share a common perspective of designing the valuation function reflect the utility of the data/ML model in terms of some performance in the ML context. An added benefit of this design approach is the interpretability of the value. To see this, suppose in the marketplace (e.g., AWS marketplace), an auditor questions the basis for certain pricing of some ML model, our proposed valuation function can provide some insight to that question and can potentially be used by regulators to oversee the ML model marketplaces.

In terms of the practical setting, data valuation can be viewed as (mostly) white-box (i.e., the actual data are used as the input to the designed valuation function and thus completely observed). 13 Intuitively, in order to determine the value of some data, the valuation function must 'see' the data. In this regard, this setting for data valuation leads to a relatively straightforward formal representation of the data, which is the data. In contrast, model valuation can encounter additional practical difficulties which make the formal representation of ML much less straightforward. Similar to in data valuation, we might want to use the model itself as its formal representation (e.g., the parameters of the parametric models). However, the so-called black-box access which is particularly appealing in model valuation (Sec. 1), excludes the choice of using the model itself (e.g., the parameters). Then, it becomes unclear what the formal representation of a model to use under this black-box access setting. In other words, for model valuation, precisely what is the input to the valuation function?

To briefly summarize the comparison between data and model valuation: In light of the practical obstacles of applying data valuation, we explore the alternative of model valuation. We adopt a similar design approach to those adopted by existing data valuation works to explicitly design a model valuation function that reflects the utility of the ML model/data in terms of a performance in the ML context. To address the additional challenges due to the black-box access setting (which is not encountered in data valuation), we propose a novel formal representation of an ML model (for classification).

B.2 Model Evaluation Criteria and Model Valuation

Model evaluation criteria. The value of an ML model depends on its utility/performance in the ML context. But the performance is a multi-faceted concept because there are different evaluation criteria, motivated by and suitable in different scenarios. For instance, one of the most commonly used evaluation criteria is the predictive accuracy (i.e., the proportion of correct predictions of the model). Another useful criterion is the predictive certainty (i.e., the certainty with which the model makes the predictions), as illustrated in Sec. 1. Furthermore, there are other sophisticated and practically important evaluation criteria such as fairness in prediction [56], robustness to adversaries [43] and robustness to distributional shifts in data [67]. In this regard, one approach is to design a model valuation bespoke to each of these criteria separately. However, it is more scalable and appealing to leverage a common theoretical connection among these criteria to design a more general model valuation that can be specified to different criteria as needed.

Model valuation. The work of [66] investigates the binary classification setting and is not suitable for empirical comparison as we consider problems with multiple classes. The works of [10, 11, 13, 51] approach the design of a model marketplace from an economics perspective by addressing issues like arbitrage (e.g., via horizontal/vertical pricing). In contrast, we formalize the value of a model via what it has learned w.r.t. a task. In addition, the black-box access setting is appealing in a model marketplace as it accommodates different model types. Some existing methods [8, 30, 33] focus on how to learn a fused model from several trained models with black-box access instead of how to value these models.

C Additional Experiments

C.1 Additional Experiment Settings

Licenses of used datasets and computational resources. MNIST [45]: Creative Commons Attribution-Share Alike 3.0. CIFAR-10 [41]: The MIT License (MIT). MedNIST [58]: Apache License 2.0. DrugRe [23]: Apache License 2.0 KDD99 [28]: Apache License 2.0.

We perform our experiments on a server with Intel(R) Xeon(R) Gold 6226R CPU @ 2 . 90 GHz and four NVIDIA GeForce RTX 3080's. As in our experiments, we use the pre-trained weights for the models (where available) instead of training from scratch. This is because our method is w.r.t. trained models, instead of focusing on the training procedure. As a result, the usage of GPUs is moderate (mainly for performing inference on the trained models, typically within 1 -2 hours depending on the complexity of the models).

Multiple independent training for robustness of results. We train a particular setting multiple (i.e., 3 ) independent times and compare the results from different settings to ensure the robustness of results. For instance, Fig. 7 examines the effect of training data ratio on MSVs. For each particular training data ratio, we perform 3 independent training using the same model type and set of hyperparameters so that the plotted results are robust to randomness in the training (via stochastic gradient descent).

C.2 Additional Discussion and Experiment Details for Learning MSV

Learning approach. Conventionally, the model Shapley value (MSV) of each model M i is calculated (for sufficiently small N ) or approximated (for larger N , e.g., larger than 30 ). The computational complexity of exact calculation scales exponentially in the number of models (i.e., O (2 N ) ), and that of approximation scales polynomial in N [53], depending on the approximation requirement (i.e., a better approximation with smaller error would incur a higher computational cost). Furthermore, each of these computational complexities has to be multiplied by the number N of models, since the calculation or approximation is performed for each model individually. Our learning approach aims to reduce the number of models for which calculation or approximation is performed, following the steps (i) perform calculation or approximation (e.g., Monte Carlo) for a subset (of size Z ) of all N models individually; (ii) use the obtained Z model-MSV (or model-approximate MSV) pairs to fit a regression learner (e.g., Gaussian process regression); (iii) use the regression learner to predict the MSV for the remaining ( N -Z ) models.

We highlight that this approach does not aim to reduce the computational complexity of the MSV of a model. Instead, this approach aims to reduce the total computational complexity of obtaining the

MSVs of N models, by a factor of N/Z (e.g., if Z = 30 for N = 150 models such as in Table 2, the total computational complexity is reduced to 1 / 5 th). Importantly, our learning approach is parallel to the prior and existing efforts that aim to reduce the computational complexity of the MSV of a model. It means that if a more efficient approximation (than the oft-used Monte Carlo) to the MSV of a model is proposed (in the future), it can be directly integrated with our learning approach in step (i) described above, namely replacing Monte Carlo.

Why Gaussian process regression. There are several reasons that we adopt the Gaussian process regression (GPR) as the specific choice for the learning approach: (i) GPR is a kernel-based method, which can exploit a suitably defined distance function between inputs. The Hellinger distance between two Dirichlet abstractions, or the ℓ 1 distance between the observed sufficient statistics ¯ h i of two models are both such suitable distance functions between the inputs (i.e., models). Hence, we adopt GPR to exploit the squared exponential kernel exp( -d ( i, i ′ ) / (2 σ 2 )) on the similarity measure between models (where the lengthscale σ is learned). Specifically, for { α i , ˆ ϕ i } , d ( i, i ′ ) := d H ( Q i , Q i ′ ) , while for { ¯ h i , ˆ ϕ i } , d ( i, i ′ ) := | ¯ h i ′ -¯ h i | 1 . (ii) While the exact MSVs satisfy (P1)-(P4), the predicted MSVs are not guaranteed to satisfy these properties. Fortunately, if the predicted MSVs have a bounded error to the exact MSVs, then these properties can be approximately satisfied [82]. In particular, GPR has such an error guarantee [47]. The result [47, Theorem 3.1] requires the function to be learnt to be Lipschitz continuous (which we derive in Theorem 1), and also the kernel to be Lipschitz continuous, so we adopt the squared exponential kernel, which is Lipschitz continuous [47]. Moreover, note that the result [47, Theorem 3.1] is w.r.t. a continuous input space (i.e., a subset of R d for some d ), but the error guarantee depends on the Lipschitz continuity only through the metric between two inputs (i.e., ℓ 2 norm of the input vectors). This is to say, their result can be adapted to our setting (where the input space is not continuous, but discrete): In our formulation for Φ , the metric between two inputs is the Hellinger distance (i.e., d H ( Q i , Q i ′ ) . It is thus an appealing future direction provide a formal guarantee based on these two theoretical results.

C.3 Additional Results for MSV vs. Predictive Accuracy/Certainty

Varying size of training data. We use the same model type and vary the size of training data. For MNIST, we use a CNN, CIFAR-10, we use ResNet-18, for MedNIST we use a specific architecture called MedNet, 14 and for DrugRe we use a CNN for text. Fig. 7 shows more training data generally lead to (better trained models, and thus) higher MSVs. Although in some cases the MSVs are negative, it can be mitigated (if necessary) by exploiting the linearity property of MSV to linear translate all MSVs by a positive amount. For instance, if only ranking of the models is needed, then negative values are acceptable as long as the ordering is correct.

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Fig. 8 shows the following. For KDD99, MLP performs the worst (very negative MSVs) while LR performs the best. For CIFAR-10, DenseNet-121 outperforms the rest while SqueezeNet performs

the worst. For MedNIST, the bespoke MedNet performs the best while the tiny CNN performs the worst. For DrugRe, BiLSTM outperforms CNN for text.

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Fig. 9 shows increasing predictive certainty (while maintaining the predictive accuracy) improves MSVs.

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C.4 Additional Results for Identifying Valuable Models for a Larger Learner

We perform additional experiments on CovType [5], MNIST and CIFAR-100. For CovType, the models used are decision trees of depths at most 3 and the larger learner is a random forest. For MNIST (CIFAR-100) the models used are LeNet-5 [46] (Resnet-18) and the larger learner is a voting classifier. The total number of models is 50 for CovType and CIFAR-10, and 25 for CIFAR-100.

Fig. 10 shows for all three datasets, MSVs effectively identify the valuable subset of models since following the highest-first sequence increases the test accuracy more quickly. In particular, we identify overfitting for CovType and MNIST since the max attained accuracy occurs before all the models are included.

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C.5 Empirical Advantage of the Chernoff Distance over the Hellinger Distance

Additional performance metrics of learning MSV. While using d H produces high regression performance of learning MSV (i.e., low test errors), we find that the performance may be somewhat limiting when we apply other performance metrics such as coefficient of determinant ( R 2 ) and

explained variance (exVar). In particular, we find thet d C outperforms d H on these two metrics, shown in Table 4.

Notice in Table 4, the right two columns ( d C) outperform the left two columns ( d H). We hypothesize that this can to the logarithmic dependence in d C resulting a more 'linear' relationship between Q i and ϕ i . For Dirichlet distributions (i.e., Dirichlet abstractions) over a C -dimensional space ( C > 1 ), the product of the pdf (appears in both d C and d H) over the space may not be well-behaved, and the integral of this product makes it more intractable (possibly due to the curse of dimensionality). The logarithmic operation in d C can help mitigate this, resulting in the better regression performance using GPR, in terms of exVar and R 2 .

C.6 Additional Benefit of A Lower Level of Dirichlet Abstraction

Recall the example in Sec. 2 that using a lower level of Dirichlet abstraction by partitioning D according to the classes allows us to correctly differentiate M i from its 'shifted' version M i ′ . Intuitively, partitioning D according to the classes improves the similarity measure between M i and 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 ( Q i , Q i ′ ) ). We verify this on MNIST and CIFAR-10 by partitioning the respective query sets D (i.e., test set) according to C = 10 classes s.t. query set D k is from class k and γ k := |D k | . We compute d C ( Q i , Q i ′ ; {D k } k =1 ,...,C ) := ∑ C k =1 γ k d C ( Q i, D k , Q i ′ , D k ) and apply (P3) to compute ϕ i ( {D k } k =1 ,...,C ) = ∑ C k =1 γ k ϕ i ( 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 C ( Q i , Q i ′ ; {D k } k =1 ,...,C ) and | ϕ i ( {D k } k =1 ,...,C ) -ϕ i ′ ( {D k } k =1 ,...,C ) | in Table 5.

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 H: If the argument to Γ( · ) (e.g., the denominator in B ( α ) in Definition 1) approaches 171 . 614479 in double precision floating point numbers, from [63]. Recall Definition 3 fuses models in a coalition C ⊆ [ N ] to obtain Q C = Dir ( ∑ n i =1 α i, 1 , . . . , ∑ n i =1 α i,C ) where |C| = n . The summation ∑ n i =1 α i, 1 can lead to overflow when n (i.e., |C| ) is large, which tends to happen when N (the total number of models is large) because the calculation of MSV ϕ i requires the enumeration of C ⊆ [ N ] \ { i } to obtain the respective Q C . For instance N = 150 , then the largest C ⊆ [ N ] \ { i } contains 149 models and if their respective parameter α i ′ ,k for the class k is on average larger than 171 . 614479 / 149 ≈ 1 . 15 , then it leads to numerical overflow for Γ( · ) (and thus Hellinger distance). Note that the Chernoff distance avoids this by combining the logarithmic operation with the Γ( · ) : in implementation ln Γ( · ) can be computed directly instead of first computing Γ( · ) and then the logarithmic operation.

C.7 Same Model Types Produce Distributionally Close Dirichlet Abstractions

We perform clustering (using d C as the distance measure instead of d H because it shows a better visual illustration and the cluserting 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 perform 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 equalsized 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 H, though we highlight the experiments are performed w.r.t. d C due to practical concerns). In particular, the numerical stability issue of d H [63] prevented us from obtaining meaningful clustering results using d H. Moreover, this result motivates our grouping paradigm used in our experiments to verify the generalized symmetry.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies an abstract problem of model valuation.\nThey propose a notion of valuing models, called the *Model Shapley Value*, based on the classic notion of Shapley Value from the literature on cooperative games.\nAdditionally, the work proposes an abstraction for models, a *Dirichlet abstraction*, that is meant to enable efficient comparison of different models.\nThey propose a method of learning to valuate models, and evaluate their method empirically.\n\nSTRENGTHS:\nThe problem of valuing different predictive models is well-motivated.  This paper nicely motivates the study of valuation of models, rather than simply data, on a number of axes.  The reliance on Shapley value is standard in the ML literature, but also well-motivated, due to the strong theoretical guarantees and study of Shapley.\n\nWEAKNESSES:\nMajor concern:  I find the presentation of the paper to be a substantial barrier to understanding and evaluating the work.\n\nConcretely, the task at hand of model valuation is never actually described formally.\n- What is the task?  What is the input and output?\n- What is the trivial solution?\n- Why is the proposed solution better?\n\nUnfortunately, I do not understand the description of Dirichlet abstractions.\n- Again, what is the point of the abstraction?  We are representing a model as a Dirichlet probability distribution?  Why?\n- What is $\\mathbf{M}_i(x)$?  Specifically, what is $x$ here?  Is this the induced distribution on predictions given a randomly sampled input $x_j$?\n- Why can this be viewed as a Dirchlet distribution?  I see the citation, but it feels like a substantial enough point that it should be justified and motivated in the text of this paper.\n- \"$\\mathbb{Q}_i$ encodes the predictive accuracy and certainty of $\\mathbf{M}_i$ through a theoretical connection...\" << Is this a formal statement?  How does this work?  How is it justified?\n\n\nMy lack of understanding of the basic building blocks behind this paper makes it difficult for me to evaluate the theory or experiments appropriately.  I've spent a decent amount of time trying to understand, and it is still not clear to me what is happening.\n\nQUESTIONS:\nMany of my questions are listed above.\n- What is the task at hand?\n- What is the key goal of this paper?\n- What are the key contributions to achieving this goal?\n\nI believe that there is probably something interesting happening in this paper, but I am not able to evaluate due to my lack of clarity on the basic objects studied.\n\nLIMITATIONS:\nAdequately addressed.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors represent the model's predictive accuracy and certainty with Dritchlet abstractions and formalize model Shapley values, which measure the value of models to a given task defined by a query set. \nThey evaluate their work on MNIST, CIFA-10, DrugRe, and medNIST datasets.\n\nSTRENGTHS:\n- The authors lay out an interesting problem, with clear background and supporting statements. \n\n- The authors formulate a solid, well-thought-out solution to the problem.\n\n- The theoretical statements are sound, and the results of the empirical analyses are clearly explained. \n\n- The paper is well structured and written, thus facilitating a good read.\n\nWEAKNESSES:\nP4: Diminished marginal utility \n- While it might be the case that the marginal utility of duplicate models reduces, the model sellers might negatively affect the marketplace with untruthful model presentations. Say, for example, two models (A, B), each have values 0.5, then if one model seller decides to fraudulently duplicate model A, the value although depreciates, might assign a high value collectively to A (2/3).\n\nQuery set \n- The authors say they have black-box access to the models. This is somewhat okay with image data, where one doesn't need explicit knowledge of the key features. For example, if I'm interested in testing the value of cat/dog classification models, I will bring cat/do images in the query set. However, for things like loan worthiness, with black box access, how does one construct the query set?\n- What is the upper and lower bound on the query set size before one reverse engineer the model? \n\nP3\n- Multiple complementary tasks might be a fair way to measure value of models with p3 assumption. How about if these tasks are tradeoffs of each other?\n\nQUESTIONS:\nFor the rebuttal, the authors should respond to all the questions highlighted in the weaknesses subsection.\n\nLIMITATIONS:\nThe authors do not mention any limitations of their work.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-04"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel approach to model comparison and valuation, utilizing a method known as Dirichlet Abstraction. The fundamental idea is to abstract the predictive behavior of different models via a Dirichlet distribution. This abstraction allows the comparison of diverse models on an equal footing.\n\nThree main challenges in model valuation are identified in the paper: developing a suitable abstraction for model valuation relative to a task, satisfying equitability properties in model valuation, and exploiting the equitability properties of the Shapley value in a large marketplace.\n\nTo address these challenges, the authors introduce an innovative approach to model valuation, utilizing a Dirichlet distribution to approximate a model’s predictive pattern or distribution with respect to a task. This abstraction, termed the Dirichlet abstraction, incorporates both the model's predictive accuracy and certainty. The authors then propose using the model Shapley value as an equitable valuation method, leveraging the Dirichlet abstractions' ability to preserve similarity between models. To address the computational challenge of Shapley value in a large marketplace, the paper suggests a learning approach for training a model appraiser. This model appraiser, trained on a small subset of models and their model Shapley values (MSVs), can predict other models’ MSVs, thus validating model Shapley’s practical feasibility in a large-scale marketplace.\n\nThe paper's empirical validation, performed on real-world datasets and with up to 150 heterogeneous models, confirms that higher predictive accuracy, more suitable model types, and higher predictive certainty correlate with higher model Shapley values. The authors also provide a use case for identifying a valuable subset of models from the marketplace to construct a more complex learner.\n\nSTRENGTHS:\nThe paper presents a novel approach to model comparison and valuation using Dirichlet abstraction, offering a creative combination of existing statistical concepts applied in a new manner. The concept of utilizing Dirichlet distributions to abstract predictive behaviors of models is quite innovative. Furthermore, the introduction of class-specific Dirichlet abstraction is an original refinement to their method, preserving more detailed information about the model's behavior.\n\nThe quality of the research seems to be high, given the soundness of the mathematical framework used in the paper and the logical structure of the presented methodology. The authors provide a theoretical background to support their claims and supplement it with visual aids, offering a detailed explanation of how their method works. The decision to use Hellinger distance to measure the dissimilarity between two probability distributions shows a thoughtful choice in ensuring computational efficiency.\n\nThe ability to compare different types of machine learning models in a unified framework is an important advancement in the field. This has potential implications for model selection in various machine learning tasks, enabling a more flexible, efficient, and fair comparison of models. Moreover, the efficient computation of the Hellinger distance between Dirichlet distributions has implications in other fields where these statistical concepts are used. Overall, the paper presents a valuable addition to the machine learning and statistical literature.\n\nWEAKNESSES:\nWhile the paper does present a novel approach to model valuation, there are several areas where it could be improved.\n\nThe paper assumes the availability of a large query set for accurate MLE estimation. It should address situations where the available data is sparse or not balanced across classes. A detailed discussion or a possible solution for handling these scenarios would strengthen the paper.\n\nWhile the authors discuss the trade-off between the abstraction level vs. query set size, there is a lack of clear guidelines or a framework for determining the optimal trade-off. This could lead to difficulties in implementing the proposed method in real-world applications. Adding a more formal discussion or a proposed methodology to address this trade-off would be beneficial.\n\nThe approach assumes homogeneity across models after the Dirichlet abstraction, which might not always be the case in practical scenarios. Some models might have specific characteristics that cannot be captured by the Dirichlet distribution. More discussion on how such cases can be handled would improve the paper.\n\nThe paper lacks a comparison with other existing model valuation methods. It would be beneficial to include an evaluation of how the proposed method fares against existing methods in terms of accuracy, efficiency, and robustness.\n\nQUESTIONS:\nHow does the proposed technique compare with existing model valuation techniques, in terms of accuracy, computational efficiency, and other relevant metrics? It would be beneficial if the authors could provide a comparative study in this regard.\n\nHow does the method perform when the available data is sparse or unbalanced across classes? Could the authors elaborate on how their approach handles diverse model architectures, especially those with characteristics that cannot be adequately captured by a Dirichlet distribution?\n\nCould the authors elaborate on how to choose an optimal trade-off between the level of Dirichlet abstraction and the size of the query set? A concrete set of guidelines or a decision-making framework would be helpful for practitioners seeking to apply this method.\n\nSome of the assumptions in the paper (like model homogeneity post-abstraction) might not hold in all practical scenarios. Could the authors elaborate on the implications if these assumptions are violated and how such situations could be handled?\n\nLIMITATIONS:\nThe authors have not explicitly addressed the potential limitations and broader societal impacts of their work.\n\nThe authors could provide more discussion on the potential limitations of their work. For instance, they might discuss the sensitivity of the model's valuation to the choice of the task or query set, and the limitations of the Dirichlet abstraction in approximating a model's predictive pattern or distribution.\n\nIn terms of broader societal impacts, the authors might consider discussing how their proposed model valuation framework could potentially affect the machine learning marketplace. For instance, could this valuation approach inadvertently favor certain types of models or tasks over others, or influence the development and use of machine learning models in ways that could have unintended consequences? They could also consider the potential impacts on data privacy, given that model valuation, unlike data valuation, does not require centralization of potentially private data.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers the problem of assigning a value to a ML model (e.g., in a marketplace with multiple models). The proposed idea is to estimate (via MLE) the Dirichlet abstraction of a model (potentially conditioned on the class) and to compute the Shapley value of a game where the value function is the Hellinger distance between the Dirichlet abstraction of a model that fuses the models in the coalition and an (almost) optimal model. The paper shows that this proposal is well-behaved and allows for learning the Shapley value to reduce the complexity (the number of Shapley values to compute). The paper also shows a number of experiments on standard datasets (MNIST, CIFAR et al.) and compares to other more standard quality metrics.\n\nSTRENGTHS:\nThe problem of valuing a model with black-box access (i.e., we can access predictions but not the model itself) is interesting because of many considerations that restrict sharing data and model parameters. The paper proposes an original solution in considering the Dirichlet abstraction of the model to compute metrics. This indeed helps because then models fusion as well as Hellinger distance can be computed easily. In a sense, the framework itself is the main contribution more than the theoretical results (which is totally fine). \n\nOne of the advantages of the Dirichlet abstraction framework is that it handles easily and naturally multiple classes, as opposed to binary classification which is much more standard in that type of literature. \n\nI find the paper well written and easy to follow (up to the issue of Figures to understand numerical experiments, which I discuss later).\n\nWEAKNESSES:\nI had a doubt about the Shapley value: Shapley assumes an axiom of efficiency. Here, $Z$ seems free. So $\\phi_i$ will be the Shapley value only for a particular value of $Z$. Does it have any impact? \nMore generally, this should be clarified. For instance, Thm 1 clearly depends on $Z$ and I have not seen that said anywhere. In the proof, it looks like $Z$ is taken to be $1$, is that the case? Does it correspond to Shapley then?\n\nThe condition in Prop 4 (App A), which implies that fusion increases similarity, is not too explicit. Is there a nice intuitive interpretation of it? And does it hold (theoretically) for the models used in the experiments for instance?\n\nThe notation can be improved in some places. For instance, l. 208, $n$ is the size of a subset that depends on $\\mathcal{C}$ whereas $n$ is almost always the size of the full set $N$. Also the fact that $C$ is the dimension of the label whereas $\\mathcal{C}$ is a subset of $N$ is confusing. \n\nThe definition of $\\mathbb{Q}^*$ poses some questions. It is in a sense supposed to represent an optimal model, but then some noise is added with range $[0, 0.01]$ to avoid degeneracy in MLE. So it means that theoretically, there could be a better model with a non-zero distance and hence strictly speaking the Hellinger distance is not decreasing with the model quality. Is it correct? If so, isn't there a nicer way to define $\\mathbb{Q}^*$ theoretically?\n \nOn the learning of the Shapley value: the proposal is to compute the Shapley value for a subset of models (say, 20 out the 150 models) and to use that to infer the value of others because the Shapley value has some regularity wrt the Dirichlet abstraction. This is nice, but poses two questions: \na- is it obvious that one can just compute the Shapley values of a subset of models? Normally, computing the Shapley value requires computing the marginal increment to some subcoalitions (even if not all, using MC). Here, we cannot do it for coalitions that aren't in the subset of 20, so how do we do? This requires clarification. Perhaps the special form of the game allows that but it needs to be clarified. \nb- in the experiments (l. 305-312), my understanding is that the paper computes the Shapley value of all 150 models (used as ground-truth), and then uses only a subset of them to infer the others. That by-passes the issue I mentioned just above... but then it does not allow to get the reduced complexity in practice so the issue stands completely. I think a better experiment would be to compute the Shapley value of the 20 models using whatever solution the authors came up with to my point a- above (which needs to be clarified), and then using it to infer the SV of the others, comparing it to the case where one can indeed compute all 150 SVs.\n\nFigures are really too small. I understand the page limit but it is excessive. Some of them are literally unreadable (like the plot is hidden behind the legend, e.g., Fig 3 left, for MLP). Also in many of them, I could not understand what the x-axis is (mostly because it was not written).\n\n\nUPDATE AFTER REBUTTAL: I raise my score to 7, due to a better understanding, in particular of the experiments.\n\nQUESTIONS:\n[Please also see the weaknesses part where I put questions as well (probably the most important); I am putting here whatever I have not already written above.]\n\n- The Dirichlet abstraction allows handling multiple classes nicely, but it was not clear to me what would simplify or get more standard if we have binary classification. \n\n- I understand why it helps (l. 210-220) with the Shapley computation, but it would be good to better explain why doing the fusion as it is done makes sense from the prediction perspective. \n\n- For clarification: the Dirichlet abstraction (e.g., l. 100) is not allowed to depend on x?\n\n- Is it possible to extend the framework to assign different weights to different types of errors?\n\n- It may be good to add a bit more background on the Dirichlet distribution and Hellinger distance (in App), just enough to help follow the rest of the paper for those who are not familiar with these.\n\n- Fig 1: although I understand it doesn't matter for the point the paper makes there, I find that plots 1-2 aren't similar to plots 3-4. Is this normal?\n\nLIMITATIONS:\nSee above.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-13"}