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Title: Predicting the accuracy of neural networks from final and intermediate layer outputs

Paper content:
We show that information about whether a neural network's output will be correct or incorrect is present in the outputs of the network's intermediate layers.
To demonstrate this effect, we train a new "meta" network to predict from either the final output of the underlying "base" network or the output of one of the base network's intermediate layers whether the base network will be correct or incorrect for a particular input.
We find that, over a wide range of tasks and base networks, the meta network can achieve accuracies ranging from 65% - 85% in making this determination.
What do neural networks know and where do they know it?
At what stage of a network's processing does a "decision" get made and are there reliable markers of a correct or incorrect decision either in the output or during a network's operation at one of its intermediate layers?
To begin this investigation, we ask where in a neural network's operation it becomes possible to determine whether the network might be correct or incorrect in its output for a particular input.
We feed a second, "meta" network the outputs of either an intermediate or final layer of the first, "base", network and train the meta network to predict whether the base network will be correct for an individual input.
We call the second network a meta or metacognitive network because humans and other animals are known to make so-called metacognitive judgments to assess their confidence in the correctness of their beliefs or actions BID4 .We
find that the meta network is able to predict whether a base network will be correct or incorrect on previously unseen inputs with up to 69% accuracy for base networks 1 Computer Science Department, Columbia University, New York, New York, USA 2 Mechanical Engineering Department and Data Science Institute, Columbia University, New York, New York, USA. Correspondence
to: Chad DeChant <chad.dechant@columbia.edu>.Identifying and
Understanding Deep Learning Phenomena Workshop at the International Conference on Machine Learning 2019 FIG0 . Meta network pipeline
: the Meta network receives as input the output of one of the base network's layers for a particular input and predicts whether the base network will be correct.classifying ImageNet images and 85% accuracy for a base network classifying CIFAR 10 images. As these two examples
suggest, the accuracy of the meta network is higher for simpler underlying tasks in our experiments.The usefulness of the layers' outputs for predicting the accuracy of the network is lowest at the earliest layers in the network and increases to be highest either at the last hidden layer or, in most cases, the final output. Meta networks trained
on different layers' outputs have significant but not complete overlap in which examples they are able to correctly predict will go on to be accurately or inaccurately classified, suggesting that there is slightly different information at each level which can be used to make assessments of accuracy.
It is clear that the meta networks are able to learn something about the intermediate and final outputs which are indicative of the networks' accuracy.
Just what that is and whether it can be useful in improving or interpreting the networks is as yet unclear.It is difficult to estimate the accuracy of a neural network at runtime.
On tasks that involve a choice between discrete options, the value of the highest output after it is put through a softmax is often considered to represent the network's confidence or estimate of the probability of the corresponding class's being correct.
However, it is not clear that this interpretation is warranted.
Recent work has shown that these outputs are not reliable BID2 .
It is interesting, then, to consider whether when a meta network is trained on the final outputs it learns to simply classify those outputs in which the predicted class has very high values as correct and those with relatively low values as incorrect.
This would correspond to the general intuition that high values for predicted classes indicate meaningfully high confidence.Figure 2 graphically illustrates the outputs of a ResNet18 network trained on ImageNet, with sample outputs of the highest confidence class arrayed along the x axis (a similar chart for outputs of the BiDAF model is found in the Appendix).
It shows that while there is certainly a correlation between a base network's accuracy and the value of the output corresponding to the highest predicted class, it is not a simple or completely reliable one.
On average, the base network indeed tends to be more confident in its correct answers than its wrong answers, and the set of examples the meta network is correct on shows this pattern clearly while the examples the meta network gets wrong show less distinct base "confidence" numbers.
However, it is apparent that the base network is often very "confident" of a wrong answer and not confident of a correct answer.
From inspecting the plots it is clear that the meta network is not judging the net- FIG1 .
Examples of maximum values (arrayed along the x axis) output by a Resnet18 network on ImageNet after the softmax function.
The meta network is correct in both cases in the top row and incorrect in the bottom row; the Resnet base classifier is correct on the left and incorrect on the right in both rows.
The mean value in each category is given.
This shows that the meta network does not learn to simply classify the output based on the value of the class prediction, which is often interpreted as the network's 'confidence'.work
's output simply by learning a threshold "confidence" level above which it predicts it will be correct and below which it predicts it will be incorrect. This
is evident by the large number of incorrect high "confidence" outputs of the base network which the meta network accurately marks as incorrect, as well as the correct low "confidence" outputs which the meta networks finds correct. Further
study will be required to better understand what features the meta network has learned to look for to measure accuracy.Neural networks designed for a classification-type task are generally trained to give an answer, not to also indicate whether they are likely to be right or wrong. While there
has has certainly been work to address this, notably that involving Bayesian networks BID0 , the present work and its future extensions may point in other fruitful directions for characterizing a network's likely accuracy at runtime. There may also
be interesting connections to work studying neural networks from an information theoretic perspective BID9 . We train meta
networks to judge whether a base network is correct or incorrect on particular inputs by feeding the meta network outputs, final or intermediate, from the base network. The blue arrows
show which outputs of the base Bi-Directional Attention Flow model the meta network examines when classifying the base network's output as accurate or inaccurate. Image adapted from
BID8

assistant: text: Information about whether a neural network's output will be correct or incorrect is somewhat present in the outputs of the network's intermediate layers.