Towards Understanding Data Values: Empirical Results on Synthetic Data

1 INTRODUCTION . Machine learning algorithms stand and fall with the training data . Although the process of data collection and labeling is highly time consuming , the creation of quality training data sets is of paramount importance . However , it turns out that not all data points contribute equally to the quality of the trained models . Recently , several works have addressed the problem of measuring the value of individual data points for the performance of the algorithm such as data Shapley ( Ghorbani & Zou , 2019 ) , catastrophic forgetting ( Toneva et al. , 2018 ) , influence functions ( Koh & Liang , 2017 ) , and data valuation using reinforcement learning ( DVRL , ( Yoon et al. , 2019 ) ) . In this paper we address the question : what distinguishes a high value data point from a low value one according to the state of the art works listed above ? Accordingly , our goals are to support a better understanding of the models and a more efficient data collection . Our contributions can be summarized as follows : We propose a framework to describe data values as the resulting change of a model ’ s decision boundary with respect to added data points . Subsequently , we probe the above data evaluation methods using the introduced framework on a synthetic 2D data set and visualize the resulting data values . Finally , we discuss the implications of our findings and place them in the recent literature . 1.1 BACKGROUND . There are several directions of research that can be interpreted as being related to estimating the value of data . Active Learning ( Settles , 2010 ) and core-set estimation ( Mirzasoleiman et al. , 2020 ; Sener & Savarese , 2018 ) historically belong to the earlier approaches . Points chosen for labeling in active learning and core-sets should intuitively have high data values . On top of that , recent methods such as influence functions ( Koh & Liang , 2017 ) , DVRL ( Yoon et al. , 2019 ) and data Shapley ( Ghorbani & Zou , 2019 ) directly aim at estimating the value of data . Finally , several other works are related or can be applied to the estimation of data values . These works include forgetting events discovered by Toneva et al . ( 2018 ) and memorization effects described by Feldman ( 2020b ) and Jiang et al . ( 2020 ) . Another related work by Paul & Dziugaite ( 2021 ) deals with estimating important samples early in training . In this paper , we empirically compare four of these methods on a synthetic data set in order to develop an intuition of how they work and to understand what makes a point important . To measure the importance of data points we consider two definitions . An important point either causes a large absolute change in the model ’ s estimated decision boundary , or reduces the relative distance between the estimated decision boundary and the ground-truth decision boundary . The reason to consider both is that some points could result in a large change of the decision boundary while increasing the error on the test set ( e.g. , a miss-labelled point ) . However , access to the groundtruth decision boundary is rarely available in practice . As our results on noise-free data show , there is no big difference between these two definitions , provided that the train data set is free of label noise . 1.2 RELATED WORK . In this section , we briefly outline a few research fields related to data valuation and revisit the four different data valuation methods from our experiments more closely . We use the standard notation to refer to the data set D = { ( xi , yi ) } ni=0 and sometimes the shorter form D = { X , y } . Core-Set Estimation Historically , core-set estimation is one of the first approaches that indirectly address data values ( Feldman , 2020a ) . The goal in core-set estimation is the selection of a small core-set DC ∈ D resulting in the same or similar performance of the model as if it was trained on the entire training set D. Many core-set-selection methods are model dependent . A recent work by Sener & Savarese ( 2018 ) applies core-sets to active learning and convolutional neural networks . As mentioned before , the connection between core-sets and data values is that one would expect points in the core-set to be of high data value . Active Learning Another related field is active learning ( Settles , 2010 ) . Given a labeled data set { X , y } , and a large set of unlabeled points XU , the algorithm can query an oracle for the label of a number of unlabeled data points xUi ∈ XU up to some budget b . An easy baseline for active learning is to select data points predicted with low confidence score . We , therefore , evaluate some out-ofdistribution methods ( Lakshminarayanan et al. , 2017 ; Gal & Ghahramani , 2016 ) on the task of data valuation in Appendix A.2 . A link between active learning and data valuation can be established quite naturally : the points chosen by active learning methods should also have a high data value . Leave-One-Out Valuation ( LOO ) A straight forward way of estimating the impact of data points is cancelling individual data points iteratively and comparing model performances after training . Let fX refer to the model trained on the entire train set X and fX\xi on the train set without xi . The data value of xi is then directly given by the performance difference of fX and fX\xi on the test set . Influence Function Influence Functions were proposed by Koh & Liang ( 2017 ) as a way of measuring the influence a data point will have on the model without retraining . As Koh & Liang ( 2017 ) propose , influence functions should be an approximation for leave-one-out retraining . The influence is estimated by the change in the model parameters θ . That is , the influence of data point zi = ( xi , yi ) is given by θ̂−zi − θ̂ , where θ̂−zi refers to the model parameters if the point zi was not part of the training data . Since our synthetic data set is small , we can afford retraining and do not consider influence functions . Data Shapley Ghorbani & Zou ( 2019 ) criticize that leave-one-out valuation does not capture interdependence between points . They propose data Shapley as a way of tackling this . The value of a data point xi is measured by its average contribution to the model performance when trained on all subsets of the remaining points X \ xi . Since this means exponential complexity , Ghorbani & Zou ( 2019 ) propose a truncated Monte Carlo scheme to estimate the value . Sample Forgetting Toneva et al . ( 2018 ) found that data points frequently forgotten during training are also important for training performance . A point is considered as forgotten if it was correctly classified at some time step t during training but again miss-classified later . They also report that this scheme of finding important points works between different model architectures . Memorization is another related line of work . Feldman ( 2020b ) and Jiang et al . ( 2020 ) independently use it for two different approaches . A training instance xi is referred to as singleton or rare if removing xi from the train set reduces the probability for xi to be classified correctly . That is , xi is only memorized if it is part of the train data . Formally , P ( fX ( xi ) = yi ) P ( fX\xi ( xi ) = yi ) . Feldman uses it to support the long-tail theory in Feldman ( 2020b ) and provides empirical evidence for the theory in Feldman & Zhang ( 2020 ) . In short , long tail refers to the fact that most data sets contain a long tail with many atypical instances . According to the theory , these examples are important for the generalization of the algorithm . Jiang et al . ( 2020 ) use memorization as a measure to characterize the regularities of a data set . They find that removing examples with the highest irregularity can improve training performance . These examples typically are the miss-labeled ones . DVRL Yoon et al . ( 2019 ) propose reinforcement learning as a way of estimating the value of data points . They train a data valuator to output values for each instance ( xi , yi ) in the training set , and use the resulting data values to sample training batches . Each points ’ probability to be in the batch is proportional to this data value . A predictor model fB is trained on the selected batch B and the resulting performance of the predictor is evaluated on a separate target set . The performance of the predictor model serves in turn as reward for the reinforcement learning of the data valuator . The valuator thereby learns to estimate important samples in the training set for the target distribution represented by the target set { XV , yV } . The data valuator ( DVRL ) is a function of dvrl ( xi , yi , ( fX ( xi ) − yi ) ) → R. In some initial experiments we were interested in whether we can remove the true labels yi from the DVRL while retaining the benefits to some extend but were not successful . Nevertheless , we provide results without labels in our experiments . 2 CONTRIBUTIONS . In this paper we use a synthetic 2D data set to visualize the importance of data points on the model performance . We span a mesh grid over the relevant feature plane and empirically compare the estimated data values of some of the methods introduced above . A comprehensible metric is then implemented to measure the difference between two decision boundaries . In the following section we first introduce the data set , describe how we estimate data values and , finally , explain our experimental setup . 2.1 DATA SET . The synthetic data set D = { ( xi , yi ) } ni=0 is two dimensional ( xi ∈ R2 ) and binary labeled ( yi ∈ { 0 , 1 } ) . We randomly divided it into train and test split . It is sampled from a mixture of Gaussians for each label class and ground-truth labels are assigned by means of one-nearestneighbor classification . We will refer to this ground-truth decision boundary as g. The data is not centered around the origin in order to have a non-linear decision boundary learned from the model and , thus , to make the graphs more comprehensible . On data centered in the origin a multi-layer perceptron learns linear decision boundaries as visible in the appendix in Figure 14 . A detailed description of the data can be found in Appendix A.1 . Furthermore , the data set DM = { XM , yM } = { ( xMi , yMi ) } m 2 i=0 is a 2D mesh grid used for plotting and estimating data values . For values { a1 , ... , am } and { b1 , ... , bm } on the x and y axes , the mesh grid contains all pairs of values : XM = { ( a1 , b1 ) , ( a1 , b2 ) , ... , ( am , bm ) } with corresponding labels yM ∈ { 0 , 1 } m 2 . Labels are computed with the ground-truth decision boundary g. In all experiments , the mesh grid is evenly spaced . The model will be referred to as fT , where T = { ( xTi , yTi ) } nt i=0 is the data set it was trained on . Hence , fT = fT ( x , θ ) = arg minθ 1 nt ∑nt i=0 L ( f ( x T i ) , y T i ) , where L ( . ) is the loss function .

Summary: This paper uses a synthetic two-dimensional dataset to visualize the importance of different data points on machine learning model performance. In particular, they used a multi-layer perceptron as the model, and they used four different schemes by which to measure the importance of individual data points. Not surprisingly, they show that regions of data that are misclassified tend to have relatively high importance in determining model performance. However, there are notable differences among the different schemes in their performances. This paper discusses the nature of those differences.