Trivial or Impossible --- dichotomous data difficulty masks model differences (on ImageNet and beyond)

1 INTRODUCTION . Let ’ s play a game we call Find those tricky images ! In Figure 1 , we show pairs of images . One image is impossible for a CNN regardless of its architecture , optimiser , random seed etc.—it never gets the label correct . The other image always yields a correct classification—can you find the tricky images ? Done ? We will wait . You have probably never seen these images before , and neither have CNNs seen them during training . How exactly a decision maker—be it a neural network , or a biological brain—generalises to previously unseen images is influenced by the decision maker ’ s inductive bias ( Goyal and Bengio , 2020 ) —in fact , as already recognised in 1980 , “ the power of a generalisation system follows directly from its biases ” ( Mitchell , 1980 ) . Commonly , the inductive bias is defined as the set of assumptions and choices that determine which hypothesis space is available to the model , before the model is exposed to data . For instance , starting from the set of all possible hypotheses , the hypothesis space of linear models is a tiny subset ( linearity is one example of a strong inductive bias ) . After the “ choice ” of the inductive bias , the dataset then influences which particular decision boundary ( or concrete hypothesis ) is selected from the model ’ s hypothesis space . Finding the right inductive bias for a given problem is at the core of machine learning . Therefore it is only consequent that a tremendous amount of work is being invested in improved architectures ( Alzubaidi et al. , 2021 ) , optimisers ( Ruder , 2016 ) , learning rate schedules ( Loshchilov and Hutter , 2016 ) , etc.—surely we would expect these choices to make a difference on the resulting model ’ s decisions even if trained on the exact same dataset . However , in the present work , we have tested various factors related to the inductive bias—among other aspects , architecture , optimiser , learning rate , and initialisation—and yet , on ImageNet , all models agree in the sense that they all make largely similar errors . This is shown in Figure 2 : even radically different state-of-the-art ( SOTA ) models make surprisingly similar errors on the ImageNet validation set . To a certain degree , image difficulty appears dichotomous : nearly 60 % of all images are either “ trivial ” ( all models correct ) or “ impossible ” ( all models wrong ) . As we will demonstrate later , this dataset issue masks and overshadows hidden differences between models . 1.1 RELATED WORK . Metrics for CNN comparisons Given the scientific , practical and engineering implications of model inductive biases , it is perhaps not surprising that a number of studies investigated differences between neural networks . For this purpose , the standard metric is accuracy . Some studies also focus on learned features and decision boundaries ( e.g . Hermann and Lampinen , 2020 ; Nguyen et al. , 2020 ; Wang et al. , 2018 ; Hermann et al. , 2019 ; Shah et al. , 2020 ) , or internal representations ( Kriegeskorte et al. , 1The tricky ( =misclassified ) images are : 1. bottom , 2. bottom , 3. top , 4. top , 5. bottom , 6. top . This game is an homage to “ Name that dataset ” by Torralba and Efros ( 2011 ) . 2008 ; Kornblith et al. , 2019 ) . Using representational similarity analysis ( RSA ) and most similar to our work , Mehrer et al . ( 2020 ) and Akbarinia and Gegenfurtner ( 2019 ) investigated whether different CNNs yield correlated representations and found that many neural networks show differences on a representational level . How intermediate representations are related to classification behaviour largely remains unclear . In order to compare networks on a behavioural level directly , metrics such as error consistency can be used . Error consistency ( measured by κ ) assesses the degree of agreement between two decision-makers on an image-by-image basis , not just average performance ( Geirhos et al. , 2020a ; b ) . Consistent model errors Tramèr et al . ( 2017 ) observe that the decision boundaries of two models are highly similar , an issue that is related to the transferability of adversarial examples between models . Additionally , it has been shown that standard vanilla models systematically agree on their errors both on IID ( independent and identically distributed ) data ( Mania et al. , 2019 ) and OOD ( out-of-distribution ) data ( Geirhos et al. , 2020a ) . It is unclear whether , if at all , there is a connection between model inductive bias , dataset difficulty and consistent model errors . Problems of datasets The ImageNet dataset ( Russakovsky et al. , 2015 ) has numerous issues . Next to those affecting most datasets—such as dataset bias ( Torralba and Efros , 2011 ) —a number of problems have been identified . Very recently , Northcutt et al . ( 2021b ) showed that around 6 % of ImageNet validation images suffer from label errors . Additionally , many images simply require more than a single label since multiple objects are present , and the distinctions between classes seem rather arbitrary at times ( Tsipras et al. , 2020 ; Beyer et al. , 2020 ) . Even when trying to replicate the original ImageNet labeling procedure in order to create a new test set , models trained on ImageNet have an accuracy drop of 11–14 % on this new test set ( Recht et al. , 2019 ) . Finally , ImageNet labels are based on the WordNet hierarchy , which contains many problematic categories . For instance , many categories in the “ person ” subtree have labels ranging from outdated to outrageous and racist ( Crawford and Paglen ; Yang et al. , 2020 ) . Furthermore and similar to our work , authors already investigated image sampling strategies during training ( Jiang et al. , 2019 ; Katharopoulos and Fleuret , 2018 ) . However , these studies focused on accelerating the training and not how the ImageNet issues may obscure differences between models as we explore here . Example difficulty A number of papers have investigated what makes images easy or difficult—e.g . Agarwal et al . ( 2020 ) ; Baldock et al . ( 2021 ) ; Mangalam and Prabhu ( 2019 ) ; Paul et al . ( 2021 ) for MNIST/CIFAR , and e.g . Hacohen et al . ( 2020 ) for ImageNet . Additionally , it was previously known that models often make similar errors—and often learn examples in the same order , see e.g . Toneva et al . ( 2018 ) ; Kalimeris et al . ( 2019 ) . To summarise : it was clear that there are easier and harder images and that models often make similar errors . In contrast and most importantly , the relationship between these two findings was not recognised . We here show for the first time that underlying model differences are masked by dichotomous data difficulty . 2 METHODS . All details regarding our software , hardware and dataset can be found in the appendix ( section A.2 ) . Similarity measure For the investigation of network similarities , we mainly use the behavioural measure error consistency ( κ ) ( Geirhos et al. , 2020a ) based on Cohen ’ s work ( Cohen , 1960 ) . κ > 0 represents that two decision-makers systematically make errors on the same images ; κ = 0 indicates no more error overlap than what could be expected by chance alone . κ < 0 shows that two decisionmakers systematically disagree . Network variations In our experiments we investigated the systematic agreement between CNNs , varying not only architecture but carefully controlling for number of epochs , optimiser , batch size , random initialisation , learning rate , hardware randomness , data order , architecture , and disjoint data sampling . Unless stated otherwise , we only changed one of the above parameters at a time . Our main results are based on the ImageNet ILSVRC dataset ( Russakovsky et al. , 2015 ) . We first used systematic variations on ResNet-18 ( called ResNet-18 variants ) . Details can be found in section A.1 in the Appendix . In total , 30 networks were trained on each of the three data sets ( See below : ImageNet , CIFAR-100 , Gaussian ) presented in the main text , as well as 60 more networks for control experiments reported in the Appendix . We stored all network states and all responses for each epoch . This allows us to analyse the agreement on different training stages epoch by epoch ( and image by image ) . Later , we investigated different state-of-the-art network architectures . When we investigated these SOTA models , implementations provided by modelvshuman ( Geirhos et al. , 2021 ) were used ( which focuses on various out-of-distribution datasets but not on ImageNet as we do ) . Psychophysical experiment In order to test whether humans can infer which images are easy and hard for CNNs , we conducted a psychophysical two-alternative forced choice experiment ( Wichmann and Jäkel , 2018 ) . In the experiment , observers were instructed to indicate by button press which image of an image pair they believe to be more difficult for a network to classify correctly—Is the right or the left image easier to classify for a CNN ? An example trial can be found in the appendix , see Figure 22 . Images were chosen from the ImageNet validation set such that the image pairs consisted of one image which all networks with different inductive bias classified correctly and another image which all networks misclassified ( see also Figure 17 ) . Stimuli were non-normalized images of size 224× 224 px . Observers performed 149 self-paced trials . Overall , nine observers ( mean age = 34.6 yrs , 2 female , 7 male ) participated . Two observers were entirely naïve to CNN research , a further four were naïve to the purpose of the experiments , but knew about CNNs . Subjects received monetary compensation of 10 C per hour . The total duration of the experiment was 30 minutes per observer . 3 RESULTS . 3.1 MODEL ERRORS ARE ALIGNED DUE TO DICHOTOMOUS DATA DIFFICULTY ( DDD ) . Figure 3 shows the result of our controlled study of model differences on ImageNet . A positive error consistency score means that the networks agree beyond what is expected by independent models . Regardless of the parameter changed ( architecture , optimiser , etc . ) , we find very high error consistencies ( around 0.7 ) —thus all models agree which images are easy or difficult to classify irrespective of the model differences investigated.2 Strikingly , changes that we hypothesized would make a larger difference , e.g . different architecture , have basically the same error consistency as “ minor ” changes like enabling hardware randomness on the GPUs . All networks achieved similar top1 accuracies ( mean : 69.05 % after 90 epochs ; range : 65.87 % to 71.47 % ; standard deviation : 1.60 % , cf . Figure 12 in the Appendix ) . Another popular method for agreement analysis is RSA . All our results also hold here , see Figure 7 in the Appendix . Additionally , switching the base architecture to VGG-11 or DenseNet-121 does not make a difference either ( see Figures 14 and 15 in the Appendix ) . We go into a deeper analysis of why model differences play such an insignificant role in Figure 4 , which plots , for the base network , whether ImageNet validation images are classified correctly ( white ) or incorrectly ( blue ) across epochs . There are three take-aways from this visualization . ( 1 . ) , 2In addition to the mean across several runs , we also plot the consistencies of single runs in gray . However , these are non-visible since the variance within the conditions is very small ( except , perhaps , for the “ cuda nondeterministic condition ” ) . one immediately notices the influence of the standard learning rate steps after 30 and 60 epochs . However , after this step , some images ( bottom ) are also “ forgotten ” ( classified correctly before step but incorrectly afterwards ) , which contrasts with the usual expectation that a model gradually improves over time . ( 2 . ) , some images are learned immediately during the very first epoch and never forgotten later ( top right region ) , while some are never learned at all . We will later see that this is not an effect of label errors , see Figure 2a . ( 3 . ) , while accuracy usually only improves minimally from one epoch to the next ( e.g . 0.04 % from epoch 89 to epoch 90 , or 14 additionally correctly classified images out of 50,000 ) , on average 12.37 % of the models ’ image classification decisions swap every epoch , corresponding to 6,184 images ! ( See Figure 16 in the Appendix for a plot which shows the percentage of swapped labels from epoch to epoch ) . The key takeaways from Figure 4 are already known from previous works investigating model errors over training time ( Toneva et al. , 2018 ; Mangalam and Prabhu , 2019 ; Kalimeris et al. , 2019 ) —we do not intend to claim any conceptual novelty in this regard , Figure 4 simply intends to visualise these intriguing patterns clearly . This last finding becomes even more prominent in Figure 5 , where we overlay the previous figure for all of the 13 networks with different hyperparameters , architectures etc . ( explained in section 2 ) . A very light red entry indicates that all networks correctly classify the image , a very dark red entry that all networks misclassify the corresponding image ; shades of red indicate the cases in-between ( where , e.g. , some but not all networks make errors ) . The figure illustrates that the previous findings hold across very different inductive biases for ResNet-18 variants : We observe that 48.2 % images are learned by all models regardless of their inductive bias ; 14.3 % images are consistently misclassified by all models3 ; only roughly a third ( 37.5 % ) of images are responsible for the differences between two model ’ s decisions . We call this phenomenon dichotomous data difficulty ( DDD ) : While the inductive bias restricts the hyperparameter space for a given model , the nature of the dataset—and especially its highly non-uniform image difficulties—seems to be an important cause for the high similarity in the decisions of different networks . Model differences may play a bigger role for images of intermediate difficulty ( where there is substantial consistency variation across models ) , but only a minor role for easy and hard images . As the dataset primarily consists of images that all models either classify correctly or incorrectly , all models end up with similar classification behaviour . Let us consider two extreme cases in order to put these findings into context . On one end of the spectrum , if all images were equally difficult and if all networks were independent ( i.e . their different inductive biases would result in independent decision boundaries ) , then we could expect a binomial distribution of model errors : out of 13 investigated ResNet-18 models , very few images should be misclassified by all models and very few correctly classified by all models—instead , most images 3Of course , these numbers change if one uses an architecture with higher top-1 accuracy , see next section for SOTA models should be correctly classified by a handful of models . Figure 2a shows , in green , exactly this distribution expected for independent models and equally difficult data 4 . On the extreme end of the spectrum , if the inductive bias had no influence at all and the dataset only contained “ trivial ” and “ impossible ” images , we would expect a histogram with only two “ spikes ” : given ImageNet accuracies of 69.05 % on average , one spike at “ None ” ( 30.95 % for ImageNet ) and one at “ All ” ( 69.05 % for ImageNet ) . Clearly , the empirically obtained histogram ( blue ) much more resembles the latter , i.e . the scenario where the ( nearly ) dichotomous data difficulty dominates over inductive bias . We observe that DDD on ImageNet is amplified , but not caused , by label errors ( Northcutt et al. , 2021a ; Beyer et al. , 2020 ; Tsipras et al. , 2020 ) which only have a minor influence on the “ None-Bar ” from our histogram in Figure 2 . Hence : removing erroneous labels is beneficial and laudable , but it will not solve DDD . Is dichotomous data difficulty ( DDD ) only a problem for ImageNet ? This is not the case : DDD is also present in CIFAR-100 and in the synthetic Gaussian dataset we ( purposefully ) generated . As a first indication , for both of these datasets we find similarly high error consistencies between different models , just like we found for ImageNet ( see section A.5 in the appendix ) .

Summary: The authors investigate why networks with highly different architectures and objectives seem to produce similar decision boundaries. By analyzing the model behavior of each sample of the validation set during training, they find that over half of the samples of the ImageNet validation set are either trivial or impossible for almost all analyzed models, which they name "dichotomous data difficulty". They then show that the model agreement is almost only caused by these trivial and impossible samples, by measuring the agreement over all models with and without the trivial examples. Finally, they show through human trial that humans can with a relatively high accuracy (~81.36%) predict which samples are easier for neural networks to predict, showing that there is a level of difficulty to the image samples.