Title: AN EMPIRICAL AND COMPARATIVE ANALYSIS OF DATA VALUATION WITH SCALABLE ALGORITHMS Anonymous authors Paper under double-blind review

Abstract: This paper focuses on valuating training data for supervised learning tasks and studies the Shapley value, a data value notion originated in cooperative game theory. The Shapley value defines a unique value distribution scheme that satisfies a set of appealing properties desired by a data value notion. However, the Shapley value requires exponential complexity to calculate exactly. Existing approximation algorithms, although achieving great improvement over the exact algorithm, relies on retraining models for multiple times, thus remaining limited when applied to larger-scale learning tasks and real-world datasets. In this work, we develop a simple and efficient heuristic for data valuation based on the Shapley value with complexity independent with the model size. The key idea is to approximate the model via a K-nearest neighbor (KNN) classifier, which has a locality structure that can lead to efficient Shapley value calculation. We evaluate the utility of the values produced by the KNN proxies in various settings, including label noise correction, watermark detection, data summarization, active data acquisition, and domain adaption. Extensive experiments demonstrate that our algorithm achieves at least comparable utility to the values produced by existing algorithms while significant efficiency improvement. Moreover, we theoretically analyze the Shapley value and justify its advantage over the leave-one-out error as a data value measure.

1. INTRODUCTION:

Data valuation addresses the question of how to decide the worth of data. Data analysis based on machine learning (ML) has enabled various applications, such as targeted advertisement, autonomous driving, and healthcare, and creates tremendous business values; at the same time, there lacks a principled way to attribute these values to different data sources. Thus, recently, the problem of data valuation has attracted increasing attention in the research community. In this work, we focus on valuing data in relative terms when the data is used for supervised learning.

The Shapley value has been proposed to value data in recent works (Jia et al., 2019b; a; Ghorbani & Zou, 2019) . The Shapley value originates from cooperative game theory and is considered a classic way of distributing total gains generated by the coalition of a set of players. One can formulate supervised learning as a cooperative game between different training points and thus apply the Shapley value to data valuation. An important reason for employing the Shapley value is that it uniquely possesses a set of appealing properties desired by a data value notion, such as fairness and additivity of values in multiple data uses.

Algorithmically, the Shapley value inspects the marginal contribution of a point to every possible subset of training data and averages the marginal contributions over all subsets. Hence, computing the Shapley value is very expensive and the exact calculation has exponential complexity. The existing works on the Shapley value-based data valuation have been focusing on how to scale up the Shapley value calculation to large training data size. The-state-of-art methods to estimate the Shapley value for general models are based on Monte Carlo approximation (Jia et al., 2019b; Ghorbani & Zou, 2019) . However, these methods require to re-train the ML model for a large amount of times; thus, they are only applicable to simple models and small training data size. The first question we ask in the paper is: How can we value data when it comes to large models, such as neural networks, and massive datasets?

In this paper, we propose a simple and efficient heuristic for data valuation based on the Shapley value for any given classification models. Our algorithm obviates the need to retrain the model for multiple times. The complexity of the algorithm scales quasi-linearly with the training data size, linearly with the validation data size, and independent of the model size. The key idea underlying our algorithm is to approximate the model by a K-nearest neighbor classifier, whose locality structure can reduce the number of subsets examined for computing the Shapley value, thus enabling tremendous efficiency improvement.

Moreover, the existing works argue the use of the Shapley value mainly based on interpreting its properties (e.g., fairness, additivity) in the data valuation context. However, can we move beyond these known properties and reason about the "performance" of the Shapley value as a data value measure? In this paper, we formalize two performance metrics specific to data values: one focuses on the predictive power of a data value measure, studying whether it is indicative of a training point's contribution to some random set; the other focuses on the ability of a data value to discriminate "good" training points from "bad" ones for privacy-preserving models. We consider leave-one-out error as a simple baseline data value measure and investigate the advantage of the Shapley value in terms of the two above performance metrics.

Finally, we show that our algorithm is a versatile and scalable tool that can be applied to a wide range of tasks: correcting label noise, detecting watermarks used for claiming the ownership of a data source, summarizing a large dataset, guiding the acquisition of new data, and domain adaptation. On small datasets and models where the complexity of the existing algorithms is acceptable, we demonstrate that our approximation can achieve at least comparable performance on these tasks. We also experiment on large datasets and models in which case prior algorithms all fail to value data in a reasonable amount of time and highlight the scablability of our algorithm.

2. GENERAL FRAMEWORKS FOR DATA VALUATION:

We present two frameworks to interpret the value of each training point for supervised learning and discuss their computational challenges.

We first set up the notations to characterize the main ingredients of a supervised learning problem, including the training and testing data, the learning algorithm and the performance measure. Let D = {z i } N i=1 be the training set, where z i is a feature-label pair (x i , y i ), and D val be the testing data. Let A be the learning algorithm which maps a training dataset to a model. Let U be a performance measure which takes as input training data, any learning algorithm, and validation data and returns a score. We write U (S, A, D val ) to denote the performance score of the model trained on a subset S of training data using the learning algorithm A when testing on D val . When the learning algorithm and validation data are self-evident, we will suppress the dependence of U on them and just use U (S) for short. Our goal is to assign a score to each training point z i , denoted by ν(z i , D, A, D val , U ), indicating its value for the supervised learning problem specified by D, A, D val , U . We will often write it as ν(z i ) or ν(z i , U ) to simplify notation.

2.1. LEAVE-ONE-OUT METHOD:

One simple way of appraising each training point is to measure its contribution to the rest of the training data:

ν loo (z i ) = U (D) -U (D \ {z i }) (1)

This value measure is referred to as the Leave-one-out (LOO) value. The exact evaluation of the LOO values for N training points requires to re-train the model for N times and the associated computational cost is prohibitive for large training datasets and large models. For deep neural networks, Koh & Liang (2017) proposed to estimate the model performance change due to the removal of each training point via influence functions. However, in order to obtain the influence functions, one will need to evaluate the inverse of the Hessian for the loss function. With N training points and p model parameters, it requires O(N ×p 2 +p 3 ) operations. Koh & Liang (2017) introduced a method to approximate the influence function with O(N p) complexity, which is still expensive for large networks. In contrast, our method, which will be discussed in Section 3, has complexity independent of model size, thus preferable for models like deep neural networks.

2.2. SHAPLEY VALUE-BASED METHOD:

The Shapley value is a classic concept in cooperative game theory to distribute the total gains generated by the coalition of all players. One can think of a supervised learning problem as a cooperative game among training data instances and apply the Shapley value to value the contribution of each training point.

Given a performance measure U , the Shapley value for training data z i is defined as the average marginal contribution of z i to all possible subsets of D formed by other training points:

EQUATION

The reason why the Shapley value is appealing for data valuation is that it uniquely satisfies the following properties.

i Group rationality: The utility of the machine learning model is completely distributed among all training points, i.e., U (D) = zi∈D ν(z i ). This is a natural rationality requirement because any rational group of data contributors would expect to distribute the full yield of their coalition. ii Fairness: (1) Two data points which have identical contributions to the model utility should have the same value. That is, if for data z i and z j and any subset S ⊆ D \ {z i , z j }, we have

U (S ∪ {i}) = U (S ∪ {j}), then ν(z i ) = ν(z j ).

(2) Data points with zero marginal contributions to all subsets of the training set should be given zero value, i.e., ν

(z i ) = 0 if U (S ∪ {z i }) = 0 for all S ⊆ D \ {z i }.

iii Additivity: When the overall performance measure is the sum of separate performance measures, the overall value of a datum should be the sum of its value under each performance measure

ν(z i , U 1 ) + ν(z i , U 2 ) = ν(z i , U 1 + U 2 ) for z i ∈ D.

In machine learning applications, the model performance measure is often evaluated by summing up the individual loss of validation points. We expect the value of a data point in predicting multiple validation points to be the sum of the values in predicting each validation point.

Despite the desirable properties of the Shapley value, calculating the Shapley value is expensive.

Evaluating the exact Shapley value involves computing the marginal contribution of each training point to all possible subsets, which is O(2 N ). Such complexity is clearly impractical for valuating a large number of training points. Even worse, for ML tasks, evaluating the utility function per se (e.g., testing accuracy) is computationally expensive as it requires to re-train an ML model. Ghorbani & Zou (2019) introduced two approaches to approximating the Shapley value based on Monte Carlo approximation. The central idea behind these approaches is to treat the Shapley value of a training point as its expected contribution to a random subset and use sample average to approximate the expectation. By the definition of the Shapley value, the random set has size 0 to N -1 with equal probability (corresponding to the 1/N factor) and is also equally likely to be any subset of a given size (corresponding to the N -1 |S| factor). In practice, one can implement an equivalent sampler by drawing a random permutation of the training set. Then, the Shapley value can be estimated by computing the marginal contribution of a point to the points preceding it and averaging the marginal contributions across different permutations. However, these Monte Carlo-based approaches cannot circumvent the need to re-train models and therefore are not viable for large models. In our experiments, we found that the approaches in Ghorbani & Zou (2019) can manage data size up to one thousand for simple models such as logistic regression and shallow neural networks, while failing to estimate the Shapley value for larger data sizes and deep nets in a reasonable amount of time. We will evaluate runtime in more details in Section 5.

3. SCALABLE DATA VALUATION VIA KNN PROXIES:

In this section, we explicate our proposed method to achieve efficient data valuation for large training data size and large models like deep nets. The key idea is to approximate the model with a KNN, which enjoys efficient algorithms for computing both the LOO and the Shapley value due to its unique locality structure.

3.1 KNN SHAPLEY VALUE Given a single validation point x val with the label y val , the simplest, unweighted version of a KNN classifier first finds the top-K training points (x α1 , • • • , x α K ) that are most similar to x val and outputs the probability of x val taking the label y val as P

[x val → y val ] = 1 K K i=1 1[y αi = y val ].

We assume that the confidence of predicting the right label is used as the performance measure, i.e.,

EQUATION

where α k (S) represents the index of the training feature that is kth closest to x val among the training examples in S. Particularly, α k (D) is abbreviated to α k . Under this performance measure, the Shapley value can be calculated exactly using the following theorem.

Theorem 1 (Jia et al. (2019a) ). Consider the model performance measure in (3). Then, the Shapley value of each training point can be calculated recursively as follows:

EQUATION

Theorem 1 can be readily extended to the case of multiple validation points, wherein the utility function is defined as the average of the utility function with respect to each validation point. By the additivity property, the Shapley value with respect to multiple validation points is the average across the Shapley value with respect to every single validation point. We will call the values obtained from ( 4) and ( 5) the KNN Shapley value hereinafter. For each validation point, computing the KNN Shapley value requires only O(N log N ) time, which circumvents the exponentially large number of utility evaluations entailed by the Shapley value definition. The intuition for achieveing such exponential improvement is that for KNN, the marginal contribution U (S ∪ z i ) -U (S) only depends on the relative distance of z i and K-nearest neighbors in S to the validation point. When calculating the Shapley value, instead of considering all S ⊆ D \ {z i }, we only need to focus on the subsets that result in distinctive K-nearest neighbors. 2019b)), the proposed algorithm does not require to retrain models. It is well-suited for approximating values for large models as its complexity is independent of model size.

3.2. AN EFFICIENT ALGORITHM ENABLED:

Note that when applied to deep nets, this algorithm neglects the contribution of data points for feature learning as the feature extractor is fixed therein. In other words, the data values produced by this algorithm attempt to distribute the total yield of a cooperative game between deep features, rather than raw data. However, as we will later show in the experiment, these values can still reflect data usefulness in various applications, while being much more efficient than the existing works.

4. THEORETICAL COMPARISON BETWEEN LOO AND THE SHAPLEY VALUE:

One may ask how we choose between the LOO method and the Shapley value for valuing training data in machine learning tasks. We have seen that the Shapley value uniquely satisfies several appealing properties. Moreover, prior work (Ghorbani & Zou, 2019) has demonstrated empirical evidence that the Shapley value is more effective than the LOO value for assessing the quality of training data instances. Nevertheless, can we theoretically justify the "valuation performance" of the two value measures?

4.1. PREDICTIVE POWER OF THE VALUE MEASURES:

To justify that the data values produced by a valuation technique can reflect the data usefulness in practice, existing valuation techniques are often examined in terms of their performance to work as a pre-processing step to filter out low-quality data, such as mislabeled or noisy data, in a given dataset. Then, one may train a model based only on the remaining "good" data instances or their combination with additional data. Note that both the LOO and the Shapley value only measure the worth of a data point relative to other points in the given dataset. Since it is still uncertain what data will be used in tandem with the point being valued after data valuation is performed, we hope that the value measures of a point are indicative of the expected performance boost when combining the point with a random set of data points.

In particular, we consider two points that have different values under a given value measure and study whether the expected model performance improvements due to the addition of these two points will have the same order as the estimated values. With the same order, we can confidently select the higher-value point in favor of another when performing ML tasks. We formalize this desirable property in the following definition. Definition 1. We say a value measure ν to be order-preserving at a pair of training points z i , z j that have different values if

ν(z i , U ) -ν(z j , U ) × E U (T ∪ {z i }) -U (T ∪ {z j }) > 0 (6)

where T is an arbitrary random set drawn from some distribution.

For general model performance measures U , it is difficult to analyze the order-preservingness of the corresponding value measures. However, for KNN, we can precisely characterize this property for both the LOO and the Shapley value. The formula for the KNN Shapley value is given in Theorem 1 and we present the expression for the KNN LOO value in the following lemma.

Lemma 1 (KNN LOO Value). Consider the model performance measure in (3). Then, the KNN LOO value of each training point can be calculated by

ν loo (z αi ) = 1 K 1[y αi = y val ] -1[y α K+1 = y val ] if i ≤ K and 0 otherwise.

Now, we are ready to state the theorem that exhibits the order-preservingness of the KNN LOO value and the KNN Shapley value. Theorem 2. For any given D = {z 1 , . . . , z N }, where z i = (x i , y i ), and any given validation point z val = (x val , y val ), assume that z 1 , . . . , z N are sorted according to their similarity to x val . Let d(•, •) be the feature distance metric according to which D is sorted. Suppose that P (X,Y )∈D (d(X, x val ) ≥ d(x i , x val )) > δ for all i = 1, . . . , N and some δ > 0. Then, ν shap-knn is order-preserving for all pairs of points in I; ν LOO-knn is order-preserving only for (z i , z j ) such that max i, j ≤ K.

Due to the space limit, we will omit all proofs to the appendix. The assumption that P (X,Y )∈D (d(X, x val ) ≥ d(x i , x val )) > δ in Theorem 2 intuitively means that it is possible to sample points that are further away from x val than the points in D. This assumption can easily hold for reasonable data distributions in continuous space.

Theorem 2 indicates that the KNN Shapley value has more predictive power than the KNN LOO value-the KNN Shapley value can predict the relative utility of any two points in D, while the KNN LOO value is only able to correctly predict the relative utility of the K-nearest neighbors of x val . In Theorem 2, the relative data utility of two points is measured in terms of the model performance difference when using them in combination with a random dataset.

Theorem 2 can be generalized to the setting of multiple validation points using the additivity property. Specifically, for any two training points, the KNN Shapley value with respect to multiple validation points is order-preserving when the order remains the same on each validation point, while the KNN LOO value with respect to multiple validation points is order-preserving when the two points are within the K-nearest neighbors of all validation points and the order remains the same on each validation point. We can see that similar to the single-validation-point setting, the condition for the KNN LOO value with respect to multiple validation points to be order-preserving is more stringent than that for the KNN Shapley value.

Moreover, we would like to highlight that the definition of order-preservingness is proposed as a property for data value measures; nevertheless, it can also be regarded as a property of a data value measure estimator. A data value estimator will be order-preserving if the estimation error is much smaller than the minimum gap between the data value measures of any two points in the training set. Since the estimation error of a consistent estimator can be made arbitrarily small as long with enough samples, a consistent estimator (if exists) for an order-preserving data value measure is also order-preserving when the sample size is large. An example for such estimator is the sample average of the marginal contribution of a point to the ones preceding it in multiple random permutations.

4.2. USABILITY FOR DIFFERENTIALLY PRIVATE ALGORITHMS:

Since the datasets used for machine learning tasks often contain sensitive information (e.g., medical records), it has been increasingly prevalent to develop privacy-preserving learning algorithms. Hence, it is also interesting to study how to value data when the learning algorithm preserves some notion of privacy. Differential privacy (DP) has emerged as a strong privacy guarantee for algorithms on aggregate datasets. The idea of DP is to carefully randomize the algorithm so that the output does not depend too much on any individuals' data. Definition 2 (Differential privacy). A :

D N → H is ( , δ)-differentially private if for all R ⊆ H

and for all D, D ∈ D N such that D and D differ only in one data instance:

P [A(D) ∈ R] ≤ e P [A(D ) ∈ R] + δ.

By definition, differential private learning algorithms will hide the influence of one training point on the model performance. Thus, intuitively, it will be more difficult to differentiate "good" data from "bad" ones for differentially private models. We will show that the Shapley value could have more disciminative power than the LOO value when the learning algorithms satisfy DP.

The following theorem states that for differentially private learning algorithms, the values of training data are gradually indistinguishable from dummy points set as the training size grows larger using both the LOO and the Shapley value measures; nonetheless, the value differences vanish faster for the LOO value than the Shapley value. Theorem 3. For a learning algorithm A(•) that achieves ( (N ), δ(N ))-DP when training on N data points. Let the performance measure be

EQUATION

For typical differentially private learning algorithms, such as adding random noise to stochastic gradient descent, the privacy guarantees will be weaker if we reduce the size of training set (e.g., see Theorem 1 in Abadi et al. (2016) ). In other words, (n) and δ(n) are monotonically decreasing functions of n, and so is (n). Therefore, it holds that (N ) < 1 N N i=1 (i). The implications of Theorem 3 are three-fold. Firstly, the fact that the maximum difference between all training points' values and the dummy point's value is directly upper bounded by signifies that stronger privacy guarantees will naturally lead to the increased difficulty to distinguish between data values. Secondly, the monotonic dependence of on N indicates that both the LOO and the Shapley value are converging to zero when the training size is very large. Thirdly, by comparing the upper bound for the LOO and the Shapley value, we can see that the convergence rate of the Shapley value is slower and therefore it could have a better chance to differentiate between "good" data from the bad than the LOO value.

Note that our results are extendable to general stable learning algorithms, which are insensitive to the removal of an arbitrary point in the training dataset (Bousquet & Elisseeff, 2002) . Stable learning algorithms are appealing as they enjoy provable generalization error bounds. Indeed, differentially private algorithms are subsumed by the class of stable algorithms (Wang et al., 2015) . A broad variety of other learning algorithms are also stable, including all learning algorithms with Tikhonov regularization. We will leave the corresponding theorem for stable algorithms to Appendix C.

5. EXPERIMENTS:

In this section, we first compare the runtime of our algorithm with the existing works on various dataset. Then, we compare the usefulness of the data values produced by different algorithms based on various applications, including mislabeled data detection, watermark removal, data summarization, active data acquisition, and domain adaptation. We will leave the detailed experimental setting, such as model architecture and hyperparamters of training processes, to the appendix.

5.1. BASELINES:

We will compare our algorithm, termed KNN-Shapley, with the following baselines.

Truncated Monte Carlo Shapley (TMC-Shapley:

). This is a Monte Carlo-based approximation algorithm proposed by Ghorbani & Zou (2019) . Monte Carlo-based methods regard the Shapley value as the expectation of a training instance's marginal contribution to a random set and then use the sample mean to approximate it. Evaluating the marginal contribution to a different set requires to retrain the model, which bottlenecks the efficiency of Monte Carlo-based methods. TMC-Shapley combined the Monte Carlo method with a heuristic that ignores the random sets of large size since the contribution of a data point to those sets will be small.

Gradient Shapley (G-Shapley). This is another Monte Carlo-based method proposed by Ghorbani & Zou (2019) and employs a different heuristic to accelerate the algorithm. G-Shapley approximates the model performance change due to the addition of one training point by taking a gradient descent step using that point and calculating the performance difference. This method is applicable only to the models trained with gradient methods; hence, the method will be included as a baseline in our experimental results when the underlying models are trained using gradient methods. Random The random baseline does not differentiate between different data's value and just randomly selects data points from training set to perform a given task.

5.2. RUNTIME COMPARISON:

We compare their runtime on different datasets and models and exhibit the result for a logistic regression trained on MNIST and a larger model ResNet-18 on CIFAR-10 in Figure 1 . We can see that KNN Shapley outperforms the rest of baselines by several orders of magnitude for large training data size and large model size.

5.3. APPLICATIONS:

Most of the following applications are discussed in recent work on data valuation (Ghorbani & Zou, 2019) . In this paper, we hope to understand: Can a simple, scalable heuristic to approximate the Shapley value using a KNN surrogate outperforms these, often more computationally expensive, previous approaches on the same set of applications? As a result, our goal is not to outperform state-of-the-art methods for each application; instead, we hope to put our work in the context of current efforts in understanding the relationships between different data valuation techniques and their performance on these tasks.

Detecting Noisy Labels Labels in the real world are often noisy due to automatic labeling, nonexpert labeling, or label corruption by data poisoning adversaries. Even if a human expert can identify incorrectly labeled examples, it is impossible to manually verify all labels for large training data. We show that the notion of data value can help human experts prioritize the verification process, allowing them to review only the examples that are most likely to be contaminated. The key idea is to rank the data points according to their data values and prioritize the points with the lowest values. Following (Ghorbani & Zou, 2019) For the pattern-based watermark removal experiment, we consider three settings: two convolutional networks trained on 1000 images from fashion MNIST and 10000 images from MNIST, resepctively, and a ResNet18 model trained on 1000 images from a face recognition dataset, Pubfig-83. The watermark ratio is 10% for all three settings. The details about watermark patterns are provided in the appendix. Since for the last two settings, either due to large data size or model size, TMC-Shapley, G-Shapley, and Leave-one-out all fail to produce value estimates in 3 hours, we compare our algorithm only with the rest of baselines. The results are shown in Figure 3 . We can see that KNN-Shapley achieves similar performance to TMC-Shapley when the time complexity of TMC-Shapley is acceptable and outperforms all other baselines. For the instance-based watermark removal experiment, we consider the following settings: a logistic regression model trained on 10000 images from MNIST, a convolution network trained on 3000 images from CIFAR-10, and ResNet18 trained on 3000 images from SVHN. The watermark ratio is 10%, 3%, and 3%, respectively. The results of our experiment are shown in Figure 4 . For this experiment, we found that both watermarks and benign data tend to have low values on some validation points; therefore, watermarks and benign data are not quite separable in terms of the average value across the validation set. We propose to compute the max value across the validation set for each training point, which we call max-KNN-Shapley, and remove the points with lowest max-KNN-Shapley values. The intuition is that out-of-distribution samples are inessential to predict all normal validation points and thus the maximum of their Shapley values with respect to different validation points should be low. The results show that the max-KNN-Shapley is more effective measure to detect instance-based watermarks than all other baselines. Data Summarization Data summarization aims to find a small subset that well represents a massive dataset. The utility of the succinct summaries should be comparable to that of the whole dataset. This is a natural application of data values, since we can just summarize the dataset by removing low-value points. We consider two settings for this experiment: a single hidden layer neural network trained on UCI Adult Census dataset and a ResNet-18 trained on Tiny ImageNet. For the first setting, we randomly pick 1000 individuals as a training set, where half of them have income exceeding $50, 000 per year. We use another a balanced dataset of size 500 to calculate the values of training data and 1000 data points to evaluate the model performance. As Figure 5 Active Data Acquisition Annotated data is often hard and expensive to obtain, particularly for specialized domains where only experts can provide reliable labels. Active data acquisition aims to ease the data collection process by automatically deciding which instances an annotator should label to train a model as efficiently and effectively as possible. We assume that we start with a small training set and compute the data values. Then, we train a random forest to predict the value for new data based on their features and select the points with highest values. For this experiment, we consider two setups. For the first setup, we synthesize a dataset with disparate data qualities by add noise to partial MNIST. For the second, we use Tiny ImageNet, which has realistic variation of data quality, for evaluation. In the first setup, we choose 100 images from MNIST and add Gaussian white noise into half of them. We use another 100 images to calculate the training data values and a held-out dataset of size 1000 to evaluate the performance. In the second setup, we separate the training set into two parts with 5000 training points and 95000 new points. We calculate values of 1 , which shows that KNN-Shapley performs better than TMC-Shapley. For the transfer between SVHN and MNIST, we pick 2000 training data from SVHN, train a ResNet-18 model (He et al., 2016) , and evaluate the performance on the whole test set of MNIST. KNN-Shapley is able to implement on the data of this scale efficiently while TMC-Shapley algorithm cannot finish in 48 hours.

Table 1 : Domain adapation. We calculate the value of data in source domain based on a validation set from the target domain, and then pick the points with positive values to train the model for performing prediction tasks in the target domain. 

6. CONCLUSION:

In this paper, we propose an efficient algorithm to approximate the Shapley value based on KNN proxies, which for the first time, enables the data valuation for large-scale dataset and large model size. We demonstrate the utility of the approximate Shapley values produced by our algorithm on a variety of applications, from noisy label detection, watermark removal, data summarization, active data acquisition, to domain adaption. We also compare with the existing Shapley value approximation algorithms, and show that our values can achieve comparable performance while the computation is much more efficient and scalable. We characterize the advantage of the Shapley value over the LOO value from a theoretical perspective and show that it is preferable in terms of predictive power and discriminative power under differentially private learning algorithms.

To analyze the sign of the KNN LOO value difference, we first write out the expression for the KNN LOO value difference:

EQUATION

Therefore, ν loo-knn (z i ) -ν loo-knn (z i+l ) has the same sign as 1[y i = y val ] -1[y i+l = y val ] and

E[U (T ∪ {z i }) -U (T ∪ {z i+1 })] only when i + l ≤ K.

B PROOF OF THEOREM 3

We will need the following lemmas on group differential privacy for the proof of Theorem 3.

Lemma 2. If A is ( , δ)-differentially private with respect to one change in the database, then A is (c , ce c δ)-differentially private with respect to c changes in the database.

Lemma 3 (Jia et al. (2019b)) . For any z i , z j ∈ D, the difference in Shapley values between z i and z j is 

ν shap (z i ) -ν shap (z j ) = 1 N -1 T ⊆D\{zi,zj } U (T ∪ {z i }) -U (T ∪ {z j })

EQUATION

Proof. Let S be the set with one element in S replaced by a different value. Let the probability density/mass defined by A(S ) and A(S) be p(h) and p (h), respectively. Using Lemma 2, for any z val we have E h∼A(S) l(h, z val ) = We perform experiments to compare the ground truth Shapley value of raw data and the value estimates produced by different heuristics. The ground truth Shapley value is computed using the group testing algorithm in (Jia et al., 2019b) , which can approximate the Shapley value with provable error bounds. We use a fully-connected neural network with three hidden layers as the target model. Following the setting in (Jia et al., 2019b) , we construct a size-1000 training set using MNIST, which contains both benign and adversarial examples, as well as a size-100 validation set

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Figures and Tables:

BY THE KNN SHAPLEY VALUE By leveraging the KNN Shapley value as a proxy, we propose the following algorithm to value the training data importance for general models that do not enjoy efficient Shapley value calculation methods. For deep nets, the algorithm proceeds as follows: (1) Given the training set, we train the network and obtain the deep features (i.e., the input to the last softmax layer);(2) We train a KNN classifier on the deep features and corresponding labels and further calibrate K such that the resulting KNN mimics the performance of the original deep net. (3) With a proper choice of K obtained from the last step, we employ Theorem 1 to compute the Shapley value of the deep features. For other models, the algorithm directly computes the KNN Shapley value on the raw data as a surrogate for the true Shapley value.The complexity of the above algorithm is O(N d + N log N ) where d is the dimension of deep feature representation. As opposed to Monte Carlo-based methods (e.g.,Ghorbani & Zou (2019);Jia et al. (

Figure 1: Comparison of running time of different data valuation algorithms. (a) is implemented on a machine with 2.6GHz CPU and 32GB memory and (b) is implement on a machine with 1.80GHz and 32GB memory

Figure 2: Detecting noisy labels. We examine the label for the training points with lowest values and plot the fraction of mislabeled data detected changes with the fraction of training data checked.

Figure 4: Instance-based watermark removal. We examine the label for the training points with lowest values and plot the faction of watermarks detected with the fraction of training data checked. The max-KNN-Shapley in the plot calculates the maximum of the KNN Shapley values across all validation points.

(a) shows, our method remains a high performance even reducing 50% of the whole training set. The data selected by the Shapley value-based data values are more helpful to boost model performance than the LOO-based value measures. TMC-Shapley and G-Shapley can achieve slightly better performance than KNN-Shapley. In the second setting, we use 95000 points as the training set, 5000 points to calculate the values, and 10000 points as the held-out set. The result in Figure 5 (b) shows that KNN-Shapley is able to maintain model performance even after removing 40% of the whole training set. However, TMC-Shapley, G-Shapley, and LOO cannot finish in 24 hours and hence are omitted from the figure.

Figure 5: (a-b) Data summarization. We remove low-value points from the ground training set and evaluate the accuracy of the model trained with remaining data. (c-d) Active data acquisition. We compute the data values for a small starting set, train a Random forest to predict the values of new data, and then add the new points with highest values into the starting set. We plot the model accuracy change when combining the starting set with more and more new data added according to different value measures. 2500 data points in the training set based on the other 2500 points. Both Figure 5 (c) and (d) show that new data selected based on KNN-Shapley value can improve model accuracy faster than the rest of baselines.Domain Adaptation Machine learning models are know to have limited capablity of generalizing learned knowledge to new datasets or environments. In practice, there is a need to transfer the model from a source domain where sufficient training data is available to a target domain where few labeled data are available. Domain adaptation aims to better leverage the dataset from one domain for the prediction tasks in another domain. We will show that data values will be useful for this task. Specifically, we can compute the values of data in the source domain with respect to a held-out set from the target domain. In this way, the data values will reflect how useful different training points are for the task in the target domain. We then train the model based only on positive-value points in the source domain and evaluate the model performance in the target one. We perform experiments on three datasets, namely, MNIST, USPS, and SVHN. For the transfer between USPS and MNIST, we use the same experiment setting as(Ghorbani & Zou, 2019). We firstly train a multinomial logistic regression classifier. We randomly sample 1000 images from the source domain as training set, calculate the values for the training data based on 1000 instances from the target domain, and evaluate the performance of the model on another 1000 target domain data. The results are summarized in Table1, which shows that KNN-Shapley performs better than TMC-Shapley. For the transfer between SVHN and MNIST, we pick 2000 training data from SVHN, train a ResNet-18 model(He et al., 2016), and evaluate the performance on the whole test set of MNIST. KNN-Shapley is able to implement on the data of this scale efficiently while TMC-Shapley algorithm cannot finish in 48 hours. Table1: Domain adapation. We calculate the value of data in source domain based on a validation set from the target domain, and then pick the points with positive values to train the model for performing prediction tasks in the target domain.

For a learning algorithm A(•) that achieves ( (N ), δ(N ))-DP when training on N data points. Let the performance measure be U (S) = -1 M M i=1 E h∼A(S) l(h, z val,i ) for S ⊆ D. Let (N ) = e c (N ) -1 + ce c (N ) δ(N ). Then, it holds that max z i ∈D ν loo (zi) ≤ (N -1) max

h∼A(S) [l(h, z val ) > t]dt (P h∼A(S ) [l(h, z val ) > t] + ce c δ)dt (19) = e c E h∼A(S ) [l(h, z val )] + ce c δ(20)It follows thatE h∼A(S) l(h, z val ) -E h∼A(S ) [l(h, z val )] ≤ (e c -1)E h∼A(S ) [l(h, z val )] + ce c δ (21) ≤ e c -1 + ce c δ (22)By symmetry, it also holds thatE h∼A(S ) l(h, z val ) -E h∼A(S) [l(h, z val )] ≤ (e c -1)E h∼A(S) [l(h, z val )] + ce c δ (23) ≤ e c -1 + ce c δ (24)Thus, we have the following bound:|E h∼A(S) l(h, z val ) -E h∼A(S ) [l(h, z val )]| ≤ e c -1 + ce c δ ν shap (z i ) -ν shap (z j ) ≤ C stab (1 + log(N -1)) N -1As for the LOO value, we haveν loo (z i ) -ν loo (z j ) = U (D \ {z j }) -U (D \ {z i }DATA POINTS OF HIGH VALUEWe remove training points from most valuable to least valuable and evaluate the accuracy of the model trained with remaining data. The experimental setting, including datasets and models, is the same as that for the data summarization experiment. Figure 6 compares the power of different data value measures to detect high-utility data. We can see that removing high-value points based on KNN-Shapley, G-Shapley, and TMC-Shapley can all significantly reduce the model performance. It indicates that these three heuristics are effective in detecting most valuable training points. On UCI Census, TMC-Shapley achieves the best performance and KNN-Shapley performs similarly to G-Shapley. On Tiny ImageNet, both TMC-Shapley and G-Shapley cannot finish in 24 hours and are therefore omitted from comparison. Compared with the random baseline, KNN-Shapley can lead to a much faster performance drop when removing high-value points.

Figure 6: (a-b) We remove high-value points from the ground training set and evaluate the accuracy of the model trained with remaining data.

Peer review (iclr2020):
In this paper, the authors have developed an algorithm to estimate Shapley value with complexity independent of the model size, based on the KNN classifier. Although the paper is interesting in general, and the experiment results are strong, I still feel that the current version of the paper has not quite met the (very high) standard of ICLR, for the following reasons: 1) The authors need to better motivate the advantages of using Shapley value as a data valuation metric. It is not completely clear to me why Shapley value is a good data valuation metric, compared with other options. The authors argue that it is both fair and decomposable (linear in U). However, based on Section 2.2, it is only fair under two extreme cases (identical points and zero marginal contributions). Also, it seems that a lot of other metrics will also satisfy the decomposability condition. Please explain! 2) Section 3 and Section 4.1 focus on U defined in equation (3), in which the testing set is a singleton. It seems to be a major limitation of the paper and it is not clear to me whether or not it is easy to generalize the results in these two sections to the general testing set with multiple points. Please explain! 3) In Definition 3, the definition for the dummy point. This definition requires that U(S union {z_i}) = U(S) for any S subseteq D, and in particular it should hold for S=emptyset. Does U({z_i}) = U(emptyset) make sense in most practical problems? A typo: in Definition 2, n and N should be the same.