1 Introduction
Identifying and avoiding overfitting are one of the most important factors in training a successful machine learning (ML) model. Failure to do so might result in an illusion of high accuracy, yet the model performs poorly in deployment. One basic prevention method is using a part of the data to train the model and evaluate it on the remaining, unseen test^{1}^{1}1We abuse the term test dataset to refer to both test and validation datasets, as we have the same problem with the validation dataset or even if we use the validation dataset. data, thereby measuring the overfit on the training data (Bishop, 2007).
However, there are several limitations to the existing approaches using the test dataset. First, recent studies show that we might not have adequate test dataset for this. Recht et al. (2018, 2019) claim that the datasets can be too biased overall, and the classifier might overfit unseen test data. For example, if both training and test datasets for image classification contain only green frogs, overfitting to the color feature might result in an illusional high accuracy. Another challenge is a dataset shift having training and test data from slightly different distributions, which is often the case for a real world deployment scenario (Muandet et al., 2013)
. That is, training a highly accurate classifier on ImageNet might not result in the best classifier for a smartphone camera as the pictures might have very different characteristics such as overall brightness, colors, resolution, and noise pattern.
Moreover, all existing approaches output a single global number, whereas overfitting can actually be a local phenomenon: The model can overfit some portion of data but not others, making overfitting less visible overall. Finally, existing approaches require a large amount of labeled test data, which is not always available and hard to create while avoiding all the previously mentioned pitfalls.
Consequently, recent efforts have been made to measure overfitting (Werpachowski et al., 2019) and memorization (Achille et al., 2019). However, these procedures are computationally expensive, or require labeled test dataset. Thus, we propose a novel measure, ^{2}^{2}2 is named in honor of Doctor Who, who first recognized wibbly wobbly timy wimy properties., that examines a local decision surface around an unlabeled test data sample. Specifically, we compute, for a given data point , how stable the decision surface is around .
is a local adaptation of the biasvariance decomposition
(Geman et al., 1992; Bishop, 2007) which was recently revisited for deep models (Ba et al., 2020; Yang et al., 2020). However, instead of computing a scalar value, we consider the distribution of measurements that can not only improve the understanding of overfitting, but also enable other applications such as detecting backdoors. Furthermore, unlike the biasvariance decomposition, our measure leverages only small portion of unlabeled data.Our empirical evaluation shows that covers many cases of overfitting: it reflects the better fit of seen training data as opposed to unseen test data. It reacts to different configurations that influence overfitting, like adversarial training (Miyato et al., 2016) or adversarial initialization (Liu et al., 2019a). Our experiments also confirm that overfitting can be local phenomenon, as the shows significant differences among classes. These results also confirm that our measure requires only a small number of unlabeled data: is sufficiently stable with as little as unlabeled data samples.
Moreover, we evaluate a hypothesis from adversarial machine learning, stating that backdoors are enabled by overfitting
(Wang et al., 2019). A backdoor for a ML model is a security threat where the attacker hides a particular pattern in the training data linked to a particular class. This pattern can later be used to evade the trained classifier at test time (Liu et al., 2018; Zhu et al., 2019). We show that backdoors are underfit rather than overfit: Backdoor features are, when present, reliably classified as one class. Furthermore, we find that neural networks in fact contain such stable trigger patterns for any class, supporting the pattern matching classification hypothesis
(Krotov and Hopfield, 2016).2 Related work
There are many ways to measure overfitting beyond the typical comparison of test and training loss (Bishop, 2007). Achille et al. (2019) derive a measure for learning complexity, which can be used to compare different datasets and can be further used to measure overfitting. Werpachowski et al. (2019) propose an overfitting measure based on a large batch of adversarial examples and a statistical test. Unlike the existing approaches, our method does not require costly computations, and requires only a small number of unlabeled data points. In general, it could be fruitful to combine our measure with other methods and run the costly, but fullfledged test when indicates overfitting.
Kohavi et al. (1996) decomposed the expected generalization loss of a classifier into variance, bias, and an irreducible error term, called biasvariancetradeoff. The underlying idea is to compare a classifier across different datasets, also called random design. Ba et al. (2020) assume instead one dataset, but that the data is noised. This is called fixed design, and although we noise the input data, not the labels, this is the setting our motivation stems from. Formerly, the biasvariancetradeoff implied that bias decreases with model complexity, whereas variance increases. Mei and Montanari (2019) and Yang et al. (2020) challenge this view: They find evidence that the variance term is instead ushaped, and decreases in regimes of higher model complexity. Our measure provides an orthogonal view that we can measure overfitting locally.
Finally we investigate backdoors in the context of overfitting. Defending backdoors is an open research problem (Tran et al., 2018; Chen et al., 2019; Wang et al., 2019; Liu et al., 2019b), which has shown to lead to an armsrace (Tan and Shokri, 2020). Our detection mechanism wrongly classifies all candidates generated by Wang et al. (2019) as backdoors. However, this wrong attribution implies the existence of benign backdoors in any network, increasing the difficulty of defending such attacks.
3 Local Wobbliness Measure
We first give a high level intuition of the measure, and then formally introduce the measure.
High level intuition. In Figure 1, we depict data points of different classes (in different blue shades) with their sampled areas (gray circles). On the left hand side, we show a wobbly, or overfitted classifier. On the right hand side, we observe a good, nonoverfitted classifier. The picture illustrates : at an appropriately chosen radius, overfitting becomes evident as the sampled ball around a test point is not consistently classified. We thus quantify overfitting in the input space by measuring the output space in the area around each test point. Note that this measurement is local, and it can vary across the input space.
Definition of the . An ML classifier trained on i.i.d. data is given. We denote training points as and the corresponding label as . After learning the parameters on pairs, the loss should be minimal. We finally define as a randomly drawn point around ; .
Given this formalization, we use the biasvariance decomposition to divide the loss into bias and variance as introduced by Kohavi et al. (1996)
. We start our analysis with the local measurement of the crossentropy loss function
, the most common loss function for DNN classification, around written(1) 
where and indicate the respective ith dimension element, and
is the mean over data points from the Gaussian distribution
. We decompose the loss in bias and variance by adapting (James, 2003) on cross entropy, yielding

(2) 
The part we are interested in is . This measures how consistent the prediction is around as we discussed in the high level idea section.
Finally, we define our measure ( for crossentropy loss) as the approximation to :
(3) 
where
, the mean of onehot vectors to use only the top1 class, and
is a small constant () to avoid computing the logarithm of zero. Using the top1 class makes it easier to interpret in terms of causing inputs, and enable us to compute this measure even for a blackbox model returning only top1 class. To obtain the measurements, instead of using just one , we use several data points and compute the measure for each, hence .Note again that we have a few critical differences to Kohavi et al. (1996) and James (2003). We compute the expectation around
by drawing random variable
from to obtain local measurements. This is important for two reasons. First, overfitting can be local, and the existing global measures can overlook regional overfitting. Second, global variance can be perceived much higher as it utilizes only a single mean value. Oftentimes, due to the diversity of the input points, using the mean of all points might not be suitable, as it can be far from every data point. Also, we measure many such points to build a distribution where we can apply diverse aggregation or statistical tests.In other ML models using mean squared error loss, we have . This version shows mostly similar results to , so we will focus on due to the page limit, and refer interested readers to the Appendix. We also want to briefly remark that Yang et al. (2020) decompose the cross entropy function into a sum of KLdivergences instead of crossentropies. James (2003) states that there can be multiple decompositions, especially for an asymmetric loss. Also, the KLdivergence can have similar trend since KLdivergence is crossentropy minus entropy.
4 Empirical Evaluation
Unfortunately, comes with many parameter choices. This includes the amount of noised points , the variance of these points, , and the number of test points. A detailed ablation study for all parameters can be found in the Appendix. However, as we see in this section, reasonable values can be found for each. In this section, we vary the amount of noised points between and , merely to show that there is little effect of this number on as long as it is decently large (). The choice of and the number of test points used to compute the measure are more important. The first, , is varied between , as low as . Small values are appropriate for finegrained task (distinguish test and training data), larger for more roughgrained (monitor progress during training). Finally, the amount of test points depends on the setting as well. We show that starting with testpoints, the measure is fairly stable.
We first describe the overall setting, and then show that is able to distinguish training and test data. Afterwards, we investigate how the measure captures small differences in the training setting, and conclude the section with a detailed study on the measure on individual classes.
Setting. To test , we deploy small networks on Fashion MNIST (Xiao et al., 2017), which achieve an accuracy of around %. These networks contain a convolution layer with
filters, a maxpooling layer of
, another convolution layer with filter, a dense layer with neurons, and a softmax layer with neurons. We further experiment on CIFAR10 (Krizhevsky and Hinton, 2009), where we train a ResNet18 (He et al., 2016) epochs to achieve % accuracy.Plots. We plot the distribution of over
test points using boxplots. These plots depict the mean (orange line), the quartiles (blue boxes, whiskers) and outliers (dots). We follow the standard definition for outliers in
Frigge et al. (1989): An outlier is defined as a point further away than 1.5 the interquartile range from the quartiles. More concretely, is the first quartile and is the third quartile (and is the median). Value is an outlier iff(4) 
in other words if is more than times the interquartile range away from either quartile or quartile .
4.1 Measuring Overfitting: Training vs Test Data.
An overfitted network will have adjusted better to the training data than to the unseen test data. Since captures overfitting, it should be possible to tell apart training and test data given measurement outputs. In the plots, we compare the distribution of our measure on test (blue) and training (purple) points from the same dataset. On each point, we sample noise points and compute with low to capture small differences. The results are plotted in Figure 2.
Results. In Figure 1(a), training and test data differ for . More specifically, the training data (purple) shows less spread than the test data. This translates to less entropy around training data, or more stable classification, as expected. To illustrate that the results are not cherry picked, we show the results for three different random draws on CIFAR in Figure 1(b). At , the results become very obvious and the difference of the spread is clearly visible. In contrast to Fashion MNIST, however, differences in the measure at smaller can be spotted, but are less pronounced.
Implications.
The question whether or not a data point formed part of the training data is critical knowledge for many applications such as medical sciences, social networks, public databases, or image data (e.g., facial recognition). The corresponding attack is called membership inference
(Shokri et al., 2017; Yeom et al., 2018; Salem et al., 2019). However, so far, even attacks with weak attackers (Salem et al., 2019)rely on output probabilities, and a straightforward defense is to only output the onehot labels. Our technique allows to circumvent these defenses, as the
can approximate a distribution that contains membership information based only on the onehot output.4.2 Measuring Overfitting during Training
As overfitting unfolds during training, we study how develops during training. We connect this with a study of different training variations that either increase or decrease overfitting.
Setting. We train networks on the Fashion MNIST dataset. In each iteration, we sample noised points. As we expect the surface to vary drastically during training, we choose a large around a given test point and compute the measure over the whole test set for stability. We choose five settings, where the first two cases are used to investigate the effect of the number of sampled points when computing the measure. We also investigate adversarial training, where one adds adversarial examples with the correct labels during training. This training makes the network more robust, and has been shown to reduce overfitting (Miyato et al., 2016). Further, we examine a backdoor scenario. Here, the attacker introduced a pattern in the training data which is always classified as the same class (Liu et al., 2018; Zhu et al., 2019). The last setting is adversarial initialization (Liu et al., 2019a). Such an initialization is generated by fitting random labels before training on the correct labels. The authors showed that adversarial initialization increases overfitting. More concretely, even though the training error of such networks is low, the test error tends to be high. We do not compute the during the pretraining with random labels.
Plots. Figure 3 depicts all five scenarios. The lines show the mean, the error bars the variance over the different runs. The blue line (‘2000 samples’) denotes the baseline networks, light blue (‘5000 samples’) the same baselines, where we sampled 5,000 noised points. The adversarial training is visualized as a red, backdoor as a yellow, and adversarial initialization setting as a green line.
Results. The left plot shows clean test accuracy. There is no a specific point at which the test error decreases. Hence, we also depict the difference between the training and test accuracy (middle plot). The adversarial training case (red line) is the lowest, corresponding to the least overfit networks. On the other hand, the adversarially initialized networks are the highest. The normal and backdoored models appear in the middle. This corresponds to our expectations about adversarial training and adversarial initialization. This order is preserved for in the right plot.
4.3 Classwise Differences
One might be tempted to think that a measure for overfitting should be classindependent. Yet, as accuracy for different classes may differ, so may overfitting properties. We study on individual classes, connecting it with experiments to test stability depending on the number of test points used to compute . In the previous experiments, measurements were done on points or the whole test data. We now compare different draws of a very low size, , and the initial size to confirm that the latter is sufficiently stable. To reduce the amount of plotted data, we randomly chose classes for the plots in Figure 4.
Results. On Fashion MNIST, varies when using points to compute the measure (Figure 3(a)). For example, the mean of the measurement changes more than 0.4 when test points from all classes are used. An extreme case is class 9 (ankle boot), where one measurement has half the spread of the other measurement. The measure is already fairly stable around test points (Figure 3(b)). Here, the largest change of mean is around . However, stability could be improved further if necessary by using more test points. On CIFAR, the results are analogous as visible in Figure 3(c) and Figure 3(d).
4.4 Conclusion of Empirical Study
We showed that indeed captures overfitting in different settings. After training, both training and test data exhibit differences in the measure. also reflects training, and in particular configurations during training that affect overfitting like adversarial training or adversarial initialization.
Resulting Complexity. We observe that points without labels are sufficient, where we sample points around each. Hence, the time complexity of computing the measure is or . The sample complexity is even lower with only unlabeled points.
5 Overfitting and Backdoors
To further show the practical usefulness of our measure, we investigate a common hypothesis from adversarial machine learning. This hypothesis by Wang et al. (2019) states that backdoors are enabled by overfitting. A backdoor is a particular pattern hidden in the training data, which can later be used to evade the trained classifier at test time (Liu et al., 2018; Zhu et al., 2019). Examples for such patterns on visual recognition or classification tasks are given in Figure 5. Most of these particular backdoors were also used by Chen et al. (2019) and Gu et al. (2019). In this section, we first present that allows to distinguish data points with and without a functional trigger. We then investigate under which circumstances detection given an functional trigger is possible.
Attacker and Hypothesis. In some cases, the attacker can only control the victim’s data. As the victim might inspect this data later, the amount of injected poisons is traditionally very small (roughly 250 for 50,000 training points) (Tran et al., 2018; Chen et al., 2019). In other cases, the attacker trains the model, and the trained model is then handed over to the victim. In this setting, the victim is not able to inspect the training data, and, thus, the attacker can poison a larger fraction of the training data to achieve better results (1020% in for example (Wang et al., 2019; Chen et al., 2019). We investigate both settings and use 250 points for the former case and fix the percentage to 15% for the latter case in our experiments. Previous work (Wang et al., 2019) has indicated that backdoors are related to overfitting, equating the backdoor pattern to a shortcut between the target and victim classes. In this setting, the model has been poisoned such that any input plus the trigger will result in the desired target class.
General Setting. We train a slightly larger network on Fashion MNIST to allow the network to fit the backdoors well: a convolution layer ( filters), a maxpooling layer (), another convolution layer ( filters), two dense layers with neurons are followed by a softmax layer with neurons. On CIFAR10, we use a smaller ResNet18. Due to the amount of networks trained, we train few epochs and the networks achieve only % accuracy on benign data. Both networks (Fashion MNIST and CIFAR) show a very high accuracy, %, on the backdoors.
and Backdoors. If backdoors indeed rely on overfitting, we expect to be high around backdoors, as the backdoors are close to the decision boundary. We thus pick as few as 25 test points and compute once on these clean points (blue). We are using the same 25 points with the leftmost backdoor pattern from Figure 5 added (yellow). As we are investigating the difference in behavior for those 25 points, such a low number is indeed sufficient. As a sanity check, we also evaluate an unseen/random pattern (right plot in Figure 6), the fourth trigger from Figure 5. The model does not overfit but rather underfits backdoors: is consistently low for the functional trigger, in particular at large . The classification output remains consistent regardless of the added noise, and the backdoors are consistent and stable. This behavior is the attacker’s goal: as soon as the backdoor is present, other features become irrelevant.
A Detection Mechanism. The differences in above are quite pronounced and should be detectable. Statistical tests with the hypothesis that both populations have equal variance are the Levene (Olkin, 1960) and the Fligner test (Fligner and Killeen, 1976)
. The latter is nonparametric, the former assumes a normal distribution but is robust if the actual distribution deviates. Alternatively, the KolmogorovSmirnov (KS) test can be used with the meanbased test statistic proposed by
Hodges (1958). The KS test evaluates the hypothesis that both samples are from the same distribution. To evaluate the performance of the tests with , we train three networks on clean data and nine networks with different implanted backdoors. Of the latter backdoored networks, three are trained with backdoors one, three and four each from Figure 5. To obtain a valid false positive rate, we test for all five backdoors of Figure 5 on all networks. We then compute a ROC curve and the AUC value given the pvalues and the ground truth of each test. In some cases, the test performs better when we remove outliers (defined as above or in Frigge et al. (1989)). In case that all points have the same value, we do not remove any point.Performance of Detection. Our results are depicted in Figure 7. In general, the performance of the tests is very good. The Levene test performs consistently best. The worst AUC () occurs using the CIFAR10 dataset, where 15% of the data is poisoned. The performance of the Fligner () and Levene test () is higher. On Fashion MNIST, the worst AUC is , although only 250 points in training are poisoned. The other two tests show perfect performance at AUC .
False negatives? To validate the above results, we repeat the previous experiments with universal examples, instead of backdoors, added to the test data. Universal adversarial examples are perturbations that can be added to a range of data points which are then missclassified (MoosaviDezfooli et al., 2017). They are hence somewhat similar to backdoors, yet they are not necessarily classified as one class consistently. We depict the results in Figure 8. Here, a true positive corresponds a universal perturbation classified as backdoor. Hence, we expect the performance to be low: The perturbation is not a backdoor, and should not be confirmed. The tests are now close to a random guess, with the exception of the Levene test with an AUC of on CIFAR. The other two tests are close to a random guess with AUC of (KS) and (Levene). On Fashion MNIST, the tests are also close to random guess, with KS (0.39) being furthest away, followed by Levene (0.456) and Fligner (0.47). We conclude that the test does not confirm universal adversarial examples as backdoors.
False negatives! We now test a method to generate backdoor candidates given a network by Wang et al. (2019). The idea behind their algorithm is to generate a perturbation with a minimal mask that, when applied to any image (in a given input set), will cause all images to return the same fixed class. In the original work, the size of the perturbation indicates if the trigger is functional or not. We generate a trigger candidate for each class on clean and backdoored models. As before, we add the candidates to the test points, compute the , and feed the measure into the test. In contrast to our expectations, however, the test detects all of them as backdoors with the same, very low pvalue. As a sanity check, we add the patterns found to the training data (which has not been used to generate them) and find that the accuracy on the targeted class is 100% in all cases.
Conclusion. Our measure, when combined with a test, reliably detects implanted backdoors. We further confirm that perturbations computed at test time (universal adversarial examples) are not detected. Yet, crafted backdoor candidates are found that manage to fool our detection. Hence, even benign networks contain variants of backdoors that emerge during training.
6 Conclusion
In this paper, we showed that overfitting can be measured by looking solely at the local decision boundary around comparatively few unlabeled data points. Our measure, , shows differences between seen training and unseen test data. Furthermore, we can show how the network adapts during training, and reflects techniques applied to increase or decrease overfitting. Finally, exhibits differences between the different classes the network learned, possibly linking it to an empirical measure for difficulty of a task as well.
Orthogonally, but also in strong relation to overfitting, we investigate a recent hypothesis from adversarial machine learning that backdooring in deep learning relies on overfitting. We find the reverse to be true: backdoors underfit rather than overfit. Applying in this area showed us, however, that any benign deep network contains benign backdoors classification relies on. Hence, we are confident that will provide many more insights when applied to other problems and open questions.
Acknowledgments
This work was supported by the German Federal Ministry of Education and Research (BMBF) through funding for the Center for ITSecurity, Privacy and Accountability (CISPA) (FKZ: 16KIS0753).
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Appendix A Mean Squared Error Loss and Study on and
We use the same networks as in the main paper, and study the measure based on the mean squared error loss and the influence of both and .
Variance of Noise Distribution. The plots evaluating different values of are depicted in Figure 9. As expected, when decreases, one class prevails around the given point eventually, and the output varies less. Indeed, when for both variance and entropy of the output decrease. Analogous to the values, the variance of both and decreases as decreases. We also plot the results on CIFAR. In particular for , the distributions do not show such a clear trend for large ( and ), possibly due to the higher dimensionality. With decreasing , however, the results become as expected and the trend can be found on CIFAR10 as well.
Number of Points Sampled. To determine the importance of the number of noise points sampled around each of the test points, we repeat the previous experiment with fixed and now vary , the number of noised points sampled. The results are plotted in Figure 10. Surprisingly, we find that the number of points does not have a strong influence on . In particular, for , the measure does not show significant differences. For , on the other hand, the spread continuously decreases, and even might be beneficial.
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