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Title: I Am Going MAD: Maximum Discrepancy Competition for Comparing Classifiers Adaptively

Paper content:
The learning of hierarchical representations for image classification has experienced an impressive series of successes due in part to the availability of large-scale labeled data for training.
On the other hand, the trained classifiers have traditionally been evaluated on a handful of test images, which are deemed to be extremely sparsely distributed in the space of all natural images.
It is thus questionable whether recent performance improvements on the excessively re-used test sets generalize to real-world natural images with much richer content variations.
In addition, studies on adversarial learning show that it is effortless to construct adversarial examples that fool nearly all image classifiers, adding more complications to relative performance comparison of existing models.
This work presents an efficient framework for comparing image classifiers, which we name the MAximum Discrepancy (MAD) competition.
Rather than comparing image classifiers on fixed test sets, we adaptively sample a test set from an arbitrarily large corpus of unlabeled images so as to maximize the discrepancies between the classifiers, measured by the distance over WordNet hierarchy.
Human labeling on the resulting small and model-dependent image sets reveals the relative performance of the competing classifiers and provides useful insights on potential ways to improve them.
We report the MAD competition results of eleven ImageNet classifiers while noting that the framework is readily extensible and cost-effective to add future classifiers into the competition.
Large-scale human-labeled image datasets such as ImageNet (Deng et al., 2009 ) have greatly contributed to the rapid progress of research in image classification.
In recent years, considerable effort has been put to designing novel network architectures (He et al., 2016; Hu et al., 2018) and advanced optimization algorithms (Kingma & Ba, 2015) to improve the training of image classifiers based on deep neural networks (DNNs), while little attention has been paid to comprehensive and fair evaluation/comparison of their model performance.
Conventional model evaluation methodology for image classification generally follows a three-step approach (Burnham & Anderson, 2003) .
First, pre-select a number of images from the space of all possible natural images (i.e., natural image manifold) to form the test set.
Second, collect the human label for each image in the test set to identify its ground-truth category.
Third, rank the competing classifiers according to their goodness of fit (e.g., accuracy) on the test set; the one with the best goodness of fit is declared the winner.
A significant problem with this methodology is the apparent contradiction between the enormous size and high dimensionality of natural image manifold and the limited scale of affordable testing (i.e., human labeling, or verifying predicted labels, which is expensive and time consuming).
As a result, a typical "large-scale" test set for image classification allows for only tens of thousands of natural images to be examined, which are deemed to be extremely sparsely distributed in natural image manifold.
Model comparison based on a limited number of samples assume that they are sufficiently representative of the whole population, an assumption that has been proven to be doubtful in image classification.
Specifically, Recht et al. (2019) found that a minute natural distribution shift leads to a large drop in accuracy for a broad range of image classifiers on both CIFAR-10 ( Krizhevsky, 2009) and ImageNet (Deng et al., 2009) , suggesting that the current test sets may far less suffice to represent hard natural images encountered in the real world.
Another problem with the conventional model comparison methodology is that the test sets are pre-selected and therefore fixed.
This leaves open the door of adapting classifiers to the test images, deliberately or unintentionally, via extensive hyperparameter tuning, raising the risk of overfitting.
As a result, it is never guaranteed that image classifiers with highly competitive performance on such small and fixed test sets can generalize to real-world natural images with much richer content variations.
In addition, recent studies on adversarial learning (Goodfellow et al., 2015; Madry et al., 2018) indicate that adversarial images produced to mislead a specific image classifier have strong transferability to fool other classifiers even if their design philosophies differ substantially, which further complicates the relative performance comparison of existing classifiers.
In order to reliably measure the progress in image classification and to fairly test the generalizability of existing classifiers in a natural setting, we believe a much larger test set in the order of millions or even billions must be used.
Apparently, the main challenge here is how to exploit such a large-scale test set under the constraint of the very limited budget for human labeling, knowing that collecting ground-truth labels for all images is extremely difficult, if not impossible.
In this work, we propose an efficient and practical methodology, namely the MAximum Discrepancy (MAD) competition, to meet this challenge.
Instead of trying to prove an image classifier to be correct using a small and fixed test set, MAD starts with a large-scale unlabeled image set and attempts to falsify a classifier by finding a set of images, whose predictions are in strong disagreements with the rest competing classifiers (See Figure 1) .
A classifier that is harder to be falsified is considered better.
The initial image set for MAD to explore can be made arbitrarily large provided that the cost of computational prediction for all competing classifiers is cheap.
To quantify the discrepancy between two classifiers on one image, we propose a weighted distance over WordNet hierarchy (Miller, 1998) , which is more semantically aligned with human cognition compared with traditional binary judgment (agree vs. disagree).
The set of model-dependent images selected by MAD are the most informative in discriminating the competing classifiers.
Subjective experiments on the MAD test set reveal the relative strengths and weaknesses among the classifiers and identify the training techniques and architecture choices that improve the generalizability to natural image manifold.
Moreover, careful inspection of the selected images may suggest potential ways to improve a classifier or to combine aspects of multiple classifiers.
We apply the MAD competition to compare eleven ImageNet classifiers and find that MAD verifies the relative improvements achieved by recent DNN-based methods, with a minimal subjective testing budget.
MAD is readily extensible, allowing future classifiers to be added into competition with little additional cost.
Moreover, the application scope of MAD is far beyond image classification.
It can be extended to many other research fields that expect discrete-valued outputs, and is especially useful when the sample space is enormous and the ground-truth measurement is expensive.
The general problem of model comparison in image classification may be formulated as follows.
We work with the natural image manifold X , upon which we define a class label f (x) ∈ Y for all natural images x ∈ X , where Y = {1, 2, . . .
, c} indicates which of c categories the object in x belongs to.
We assume a subjective assessment environment, in which a human subject can identify the category membership for any natural image x among all possible categories.
A group of image classifiers F = {f i } m i=1 are also assumed, each of which takes a natural image x as input and makes a prediction of f (x), collectively denoted by
.
The goal is to compare the relative performance of m classifiers under very limited resource for subjective testing.
A conventional model comparison method for image classification first samples a natural image set D = {x k } n k=1 ⊂ X .
For each image x k ∈ D, we ask human annotators to provide the ground-truth label f (x k ) ∈ Y. Since human labeling is an expensive and time consuming process and DNNbased classifiers are hungry for labeled data in the training stage (Krizhevsky et al., 2012) , n is typically small in the order of tens of thousands (Russakovsky et al., 2015) .
The predictions of the classifiers are compared with the human labels by computing the empirical classification accuracy
The classifier f i with a higher classification accuracy is said to outperform f j with a lower accuracy.
As an alternative, the proposed MAD competition methodology aims to falsify a classifier in the most efficient way with the help of other competing classifiers.
A classifier that is more likely to be falsified is considered worse.
The MAD competition methodology starts by sampling an image set D = {x k } n k=1 from the natural image manifold X .
Since the number of images selected by MAD for subjective testing is independent of the size of D , we may make n arbitrarily large such that D provides dense coverage of (i.e., sufficiently represents) X .
MAD relies on a distance measure to quantify the degree of discrepancy between the predictions of any two classifiers.
The most straightforward measure is the 0-1 loss:
Unfortunately, it ignores the semantic relations between class labels, which may be crucial in distinguishing two classifiers, especially when they share similar design philosophies (e.g., using DNNs as backbones) and is trained on the same image set (e.g., ImageNet).
For example, misclassifying a "chihuahua" as a dog of other species is clearly more acceptable compared to misclassifying it as a "watermelon".
We propose to leverage the semantic hierarchies in WordNet (Miller, 1998) to measure the distance between two (predicted) class labels.
Specifically, we model WordNet as a weighted undirected graph G(V, E) 1 .
Each edge e = (u, v) ∈ E connects a parent node u of a more general level (e.g., canine) to its child node v of a more specific level (e.g., dog), for u, v ∈ V .
A nonnegative weight w(e) is assigned to each edge e = (u, v) to encode the semantic similarity between u and v. A larger w(e) indicates that u and v are semantically more dissimilar.
We measure the distance between two labels as the sum of the weights assigned to the edges along the shortest path P connecting them Figure 2 : Comparison of weighted and unweighted distances.
In the sub-tree of WordNet, we highlight the shortest paths from "fountain" to "church" and from "drake" to "American coot" in green and red, respectively.
The semantic distance between the two aquatic birds is much shorter than that between the two constructions of completely different functionalities (verified by our internal subjective testing).
The proposed weighted distance is well aligned with human cognition by assigning a much smaller distance to the red path (0.0037) compared to the green one (0.0859).
Eq. (3) reduces to the standard graph hop distance between two vertices by setting w(e) = 1.
Here we design w(e) to be inversely proportional to the tree depth level l of the parent node (e.g., w(e) ∝ 2 −l ).
In other words, we prefer the shortest paths to traverse the root node (or nodes with smaller l) as a way of encouraging f i (x) and f j (x) to differ in a more general level (e.g., vehicle rather than watercraft).
Figure 2 shows the semantic advantages of our choice of weights compared to the equal weight.
With the distance measure at hand, the optimal image in terms of discriminating f i and f j can be obtained by maximizing the discrepancies between the two classifiers on D
The queried image label f (x ) leads to three possible outcomes (see Figure 1 ):
• Case I. Both classifiers make correct predictions.
Although theoretically impossible based on the general problem formulation, it is not uncommon in practice that a natural image may contain multiple distinct objects (e.g., guacamole and mortar).
In this case, f i and f j successfully recognize different objects in x , indicating that both classifiers tend to perform at a high level and cannot be differentiated.
By restricting D to contain only natural images with a single salient object, we may reduce the possibility of this outcome.
• Case II.
f i (or f j ) makes correct prediction, while f j (or f i ) makes incorrect prediction.
In this case, MAD automatically identifies a strong failure case to falsify one classifier, not the other; a clear winner is obtained.
The selected image x provides the strongest evidence in ranking the relative performance between the two classifiers.
• Case III.
Both classifiers make incorrect predictions in a multiclass image classification problem (i.e., c ≥ 3).
Although both classifiers make mistakes, they differ substantially during inference, which in turn provides a strong indication of their respective weaknesses and suggests potential ways to combine them into a single better classifier 2 .
However, in this case, x contributes little to performance comparison between the two classifiers.
To obtain a reliable performance comparison between f i and f j in practice, we choose top-k images in D with k largest distances computed by Eq. (3) to form the test subset S {i,j} .
MAD runs this game among all m 2 distinct pairs of classifiers, resulting in the final MAD test set S = S {i,j} .
The number of natural images in S is at most m(m − 1)k/2, which is independent of the size n of D. As a result, applying MAD to a larger image set has no impact on the cost of human labeling.
In scenarios where the cost of computational prediction can be ignored, MAD encourages to expand D to cover as many "free" natural images as possible.
We now describe our subjective assessment environment for collecting human labels.
Given an image x ∈ S, which is associated with two classifiers f i and f j , we pick two binary questions for human annotators: "Does x contain an f i (x)?" and "Does x contain an f j (x)?".
When both answers are no (corresponding to Case III), we cease to source the ground-truth label of x because f (x) is uninformative in distinguishing f i and f j .
Nevertheless, it is difficult for humans to select one among c classes, especially when c is large and the ontology of classes is complex.
Select top-k images with k largest distances to form S {i,j}
2 Compute the predictions of the new image classifier {f m+1 (x), x ∈ D} 3 for i ← 1 to m do 4
Compute the distances using Eq. (3)
Select top-k images with k largest distances to form S {i,m+1} 6 S = S S {i,m+1} 7 end 8 Source human labels for S 9 for i ← 1 to m do 10 a i,m+1 = Acc(f i ; S {i,m+1} ) a m+1,i = Acc(f m+1 ; S {i,m+1} ) 11 end 12 Update the pairwise dominance matrix B with
13 Compute the global ranking vector r ∈ R m+1 using Eq. (5) After subjective testing, we first compare the classifiers in pairs and aggregate the pairwise statistics into a global ranking.
Specifically, we compute the empirical classification accuracies of f i and f j on S {i,j} using Eq. (1), denoted by a ij and a ji , respectively.
When k is small, Laplace smoothing is employed to smooth the estimation.
Note that a ij + a ji may be greater than one because of Case I.
The pairwise accuracy statistics of all classifiers form a matrix A, from which we compute another matrix B with b ij = a ij /a ji indicating the pairwise dominance of f i over f j .
We aggregate pairwise comparison results into a global ranking r ∈ R m using Perron rank (Saaty & Vargas, 1984) :
where 1 is an m-dimensional vector of all ones.
The limit of Eq. (5) is the normalized principal eigenvector of B corresponding to the largest eigenvalue, where r i > 0 for i = 1, . . .
m and i r i = 1.
The larger r i is, the better f i performs in the MAD competition.
Other ranking aggregation methods such as HodgeRank (Jiang et al., 2011) may also be applied.
We summarize the workflow of the MAD competition in Algorithm 1.
Finally, it is straightforward and cost-effective to add the m + 1-th classifier into the current MAD competition.
No change is necessary for the sampled S and the associated subjective testing.
The only additional work is to select a total of mk new images from D for human labeling.
We then enlarge B by one along its row and column, and insert the pairwise comparison statistics between f m+1 and the previous m classifiers.
An updated global ranking vector r ∈ R m+1 can be computed using Eq. (5).
We summarize the procedure of adding a new classifier in Algorithm 2.
In this section, we apply the proposed MAD competition methodology to comparing ImageNet classifiers.
We focus on ImageNet (Deng et al., 2009 ) not only because it is one of the first large-scale datasets in image classification, but also because the improvements on ImageNet seem to plateau, which sets up an ideal platform for MAD to distinguish the newly proposed image classifiers finer.
Constructing D Inspired by , we focus on the same 200 out of 1, 000 classes to avoid rare and abstract classes, and classes that have changed much since 2012.
For each class, we crawl a large number of images from Flickr, resulting in a total of n = 168, 000 natural images.
Although MAD allows us to arbitrarily increase n with essentially no cost, we choose the size of D to be only approximately three times larger than the ImageNet validation set to provide a relatively easy environment for probing the generalizability of the classifiers.
As will be cleared in Section 3.2, the current setting of n is sufficient to discriminate the competing classifiers.
To guarantee the content independence between ImageNet and our test set, we collect images that have been uploaded after 2013.
It is worth noting that no data cleaning (e.g., inappropriate content and near-duplicate removal) is necessary at this stage since we only need to ensure the selected subset S for human labeling are eligible. (Tan & Le, 2019) , and WSL-ResNeXt101-32×48 (Mahajan et al., 2018) .
Since VGG16BN and ResNet34 have nearly identical accuracies (both top-1 and top-5) on ImageNet validation set, it is particularly interesting to see which method generalizes better in natural image manifold.
We compare ResNet34 with ResNet101 to see the influence of DNN depth on generalizability.
WRN101-2, ResNeXt101-32×4, SE-ResNet-101 are different improved versions over ResNet-101.
We also include two state-ofthe-art classifiers on ImageNet validation set: WSL-ResNeXt101-32×48 and EfficientNet-B7.
The former leverages the power of weakly supervised pre-training on Instagram data, while the latter makes use of compound scaling method.
We use publicly available code repositories for all DNNbased models, whose top-1 accuracies on ImageNet validation set are listed in Table 1 for reference.
Constructing S When constructing S using the maximum discrepancy principle, we add another constraint based on prediction confidence.
Specifically, a candidate image x associated with f i and f j is filtered out if it does not satisfy min(p i (x), p j (x)) ≥ T , where p i (x) is the confidence score (i.e., probability produced by the last softmax layer) of f i (x) and T is a predefined threshold set to 0.8.
We include the confidence constraint for two main reasons.
First, if f i misclassifies x with low confidence, it is highly likely that x is near the decision boundary and thus contains less information on improving the decision rules of f i .
Second, some images in D do not necessarily fall into the 1, 000 classes in ImageNet, which are bound to be misclassified (a problem closely related to outof-distribution detection).
If they are misclassified by f i with high confidence, we consider them as hard counterexamples of f i .
To encourage class diversity in S, we remain a maximum of three images with the same predicted label by f i .
In addition, we exclude images that are non-natural.
Figure 3 visually compares representative "manhole cover" images in S and ImageNet validation set (see more in Figure 6 ).
Collecting Human Labels As described in 2.1, given an image x ∈ S, human annotators need to answer two binary questions.
In our subjective experiment, we choose k = 30 and invite five volunteer graduate students, who are experts in computer vision to label a total of 11 × 10 × 30/2 = 1, 650 images.
If more than three of them find difficulty in labeling x (associated with f i and f j ), it is discarded and replaced by x ∈ D with the k + 1-th largest distance d w (f i (x ), f j (x )).
Majority vote is adopted to decide the final label when disagreement occurs.
After subjective testing, we find that 53.5% of annotated images belong to Case II, which form the cornerstone of the subsequent data analysis.
Besides, 32.9% and 13.6% images pertain to Case I and Case III, respectively.
Figure 4 shows the pairwise accuracy matrix A in the current MAD competition, where a larger value of an entry (a brighter color) indicates a higher accuracy in the subset selected together by the corresponding row and column models.
An interesting phenomenon All "manhole cover" images in ImageNet validation set.
The MAD-selected images are visually much harder, which contain diverse and non-trivial distortions, e.g., occlusion, shading and unbalanced lightening, complex background, rare colors, and untypical viewing points.
In contrast, ImageNet images mainly include a single center-positioned object with relatively clean background, whose shape, color and viewing point are common.
we find is that when two classifiers f i and f j perform at a similar level on S {i,j} (i.e., |Acc(
is also small.
That is, more images on which they both make incorrect but different predictions (Case III) have been selected compared to images falling into Case I. Taking a closer look at images in S {i,j} , we may reveal the respective model biases of f i and f j .
For example, we find that WSL-ResNeXt101-32×48 tend to focus on foreground objects while EfficientNet-B7 attends more to background objects (See Figure 8) .
We also observe several common failure modes of the competing classifiers through pairwise comparison, e.g., excessive reliance on relation inference (see Figure 9 ), bias towards low-level visual features (see Figure 10) , and difficulty in recognizing rare instantiations of objects (see Figures 3 and 6 ).
We present the global ranking results by MAD in Table 1 , where we find that MAD tracks the steady progress in image classification, as verified by a reasonable Spearman rank-order correlation coefficient (SRCC) of 0.89 between the accuracy rank on ImageNet validation set and the MAD rank on our test set D. Moreover, by looking at the differences between the two rankings, we obtain a number of interesting findings.
First, VGG16BN outperforms not only ResNet34 but also ResNet101, suggesting that under similar computation budgets, VGG-like networks may exhibit better generalizability to hard samples than networks with residual connections.
Second, both networks equipped with the squeezeand-extraction mechanism, i.e. SE-ResNet-101 and SENet154, move up by two places in the MAD ranking.
This indicates that explicitly modeling dependencies between channel-wise feature maps seem quite beneficial to image classification.
Third, for the two models that Models ImageNet top-1 Acc Acc Rank MAD Rank ∆ Rank WSL-ResNeXt101-32×48 (Mahajan et al., 2018) 85.44 1 2 -1 EfficientNet-B7 (Tan & Le, 2019) 84.48 2 1 1 PNASNet-5-Large (Liu et al., 2018) 82.74 3 7 -4 NASNet-A-Large (Zoph et al., 2018) 82.51 4 4 0 SENet154 (Hu et al., 2018) 81.30 5 3 2 WRN101-2 (Zagoruyko & Komodakis, 2016) 78.85 6 6 0 SE-ResNet-101 (Hu et al., 2018) 78.40 7 5 2 ResNeXt101-32×4 (Xie et al., 2017) 78.19 8 8 0 ResNet101 (He et al., 2016) 77.37 9 10 -1 VGG16BN (Simonyan & Zisserman, 2014) 73.36 10 9 1 ResNet34 (He et al., 2016) 73.31 11 11 0 exploit neural architecture search, NASNet-A-Large is still ranked high by MAD; interestingly, the rank of PNASNet-5-Large drops a lot.
That implies MAD may prefer the global search strategy used in NASNet-A-Large to the progressive cell-wise search strategy adopted in PNASNet-5-Large, although the former is slighter inferior in ImageNet top-1 accuracy.
Last but not least, the top-2 performers, WSL-ResNeXt101-32×48 and EfficientNet-B7, are still the best in MAD competition (irrespective of their relative rankings), verifying the effectiveness of large-scale hashtag data pretraining and compound scaling method in the context of image classification.
Ablation Study We analyze the key hyperparameter k in MAD, i.e., the number of images in S {i,j} selected for subjective testing.
We calculate the SRCC values between the top-30 ranking (as reference) and other top-k rankings with k = {1, 2, . . .
, 29}. As shown in Figure 5 , the ranking results are fairly stable (SRCC > 0.90) when k > 15.
This supports our choice of k = 30 since the final global ranking already seems to enter a stable plateau.
We have presented a new methodology for comparing image classification models.
MAD effectively mitigates the conflict between the prohibitively large natural image manifold that we have to evaluate against and the expensive human labeling effort that we aim to minimize.
Much of our endeavor has been dedicated to selecting natural images that are optimal in term of distinguishing or falsifying classifiers.
MAD requires explicit specification of image classifiers to be compared, and provides an effective means of exposing the respective flaws of competing classifiers.
It also directly contributes to model interpretability and helps us analyze the models' focus and bias when making predictions.
We have demonstrated the effectiveness of MAD competition on ImageNet classifiers, and concluded a number of interesting observations, which were not apparently drawn from the (often quite close) accuracy numbers on the small and fixed ImageNet validation set.
MAD is widely applicable to computational models that produce discrete-valued outputs, and is particularly useful when the sample space is large and the ground-truth label being predicted is expensive to measure.
Examples include medical and hyperspectral image classification (Filipovych & Davatzikos, 2011; Wang et al., 2014) , where signification domain expertise is crucial to obtain correct labels.
MAD can also be applied towards spotting rare but fatal failures in high-cost and failure-sensitive applications, e.g., comparing perception systems of autonomous cars (Chen et al., 2015) in unconstrained real-world weathers, lighting conditions, and road scenes.
In addition, by restricting the test set to some domain of interest, MAD allows comparison of classifiers in more specific applications, e.g., fine-grained image recognition (Xiao et al., 2015) .
We also feel it important to note the current limitations of MAD.
First, MAD cannot prove a model's predictions to be correct and therefore it should be viewed as complementary to, rather than a replacement for, the conventional accuracy comparison for image classification.
Second, although the distance in Eq. (3) is sufficient to distinguish multiple classifiers in the current experimental setting, it does not yet fully reflect human cognition of image label semantics.
Third, the computation of the confidence used to select images is not perfectly grounded.
How to marry the MAD competition with Bayesian probability theory to model uncertainties during image selection is an interesting direction for future research.
Our method arises as a natural combination of concepts drawn from two separate lines of research.
The first explores the idea of model falsification as a model comparison.
Wang & Simoncelli (2008) introduced the maximum differentiation competition for comparing computational models of continuous perceptual quantities, which was further extended by (Ma et al., 2019) .
Berardino et al. (2017) developed a computational method for comparing hierarchical image representations in terms of their ability to explain perceptual sensitivity in humans.
MAD, on the other hand, is tailored to applications with discrete model responses and relies on a semantic distance measure to compute model discrepancy.
The second endeavour arises from machine learning literature on generating adversarial examples (Szegedy et al., 2013; Goodfellow et al., 2015; Madry et al., 2018) and evaluating image classifiers on new test sets (Geirhos et al., 2019; Recht et al., 2019; Hendrycks & Dietterich, 2019; .
The images selected by MAD can be seen as a form of natural adversarial examples as each of them is able to fool at least one classifier (when Case I is eliminated).
Unlike adversarial images with inherent transferability to mislead most classifiers, MAD-selected images emphasize on their discriminability of the competing models.
Different from recently created test sets, the MAD-selected test set is adapted to the competing classifiers with the goal of minimizing human labeling effort.
Disk brake is correlated to freewheel and spokes.
Similar with how humans recognize objects, it would be reasonable for DNN-based classifiers to make predictions by inferring useful information from object relationships, only when their prediction confidence is low.
However, this is not the case in our experiments, which show that classifiers may make high-confidence predictions by leveraging object relations only without really "seeing" the predicted object.
Figure 10 : Examples of network bias to low-level visual features, such as color, shape and texture, while overlooking conflicting semantic cues.
An ideal classifier is expected to utilize both low-level (appearance) and high-level (semantic) features when making predictions.

assistant: text: We present an efficient and adaptive framework for comparing image classifiers to maximize the discrepancies between the classifiers, in place of comparing on fixed test sets.