Paper:
Title
Deep neural networks (DNNs) are poorly-calibrated when trained in conventional ways. To improve confidence calibration of DNNs, we propose a novel training method, distance-based learning from errors (DBLE). DBLE bases its confidence estimation on distances in the representation space. We first adapt prototypical learning for training of a classification model for DBLE. It yields a representation space where the distance from a test sample to its ground-truth class center can calibrate the model performance. At inference, however, these distances are not available due to the lack of ground-truth labels. To circumvent this by approximately inferring the distance for every test sample, we propose to train a confidence model jointly with the classification model by merely learning from mis-classified training samples, which we show to be highly beneficial for effective learning. On multiple datasets and DNN architectures, we demonstrate that DBLE outperforms alternative single-modal confidence calibration approaches. DBLE also achieves comparable performance with computationally-expensive ensemble approaches with lower computational cost and lower number of parameters.

Review:
Summary:
The paper proposes a method to do confidence calibration for deep neural networks. It uses standard
episodic training for prototypical networks, and first shows empirically that the distances of the
embedded test point to its ground truth class center embedding (*not* the predicted class embedding)
are indicative of the confidence of the prediction. Further it proposes to exploit this by training
an auxiliary confidence prediction MLP carefully. To do so they demonstrate that the training needs
to be done of erroneously predicted training examples cf. all the traning examples. They show
results with MLP+MNIST, VGG11+CIFAR10, ResNet50+CIFAR100 and ResNet50+TinyImageNet.
Detailed comments:
The paper is interesting but largely empirical. It shows empirically that:
1. when prototypical networks (and episodic training) is used the distance of test example to true
class center reflects the confidence.
2. this does not hold when `vanilla' training is used
3. an auxiliary MLP can be used to learn to predict this while training only with erroneously 
classified training examples cf. all training examples
                                      
The results are reported on three networks (MLP, VGG11 and ResNet50) on different benchmarks of 
image classification. The confidence prediction improvements wrt baselines are non trivial, while 
keeping the accuracy similar, and the computation cost lower than competing methods. Ablations 
studies are also convincing.                    
            
I would have two broad critical comments on the paper:
1. Would this generalize to other image classification tasks and datasets. Generally distance
(embedding) based networks perform less than softmax based networks on bigger datasets, so an
immediate disadvantage if that happens, is that you would be trading off accuracy cf. vanilla
networks, for better confidence prediction using the required distance based network here.
2. The vanilla training is never formally detailed. I am assuming it was softmax + cross entropy loss 
with gradient descent. Would some other loss be helpful? Specially the metric learning based losses like 
contrastive or triplet losses come to mind, since they are also distance based. Does the method work
with prototypical networks only or it generalizes to other distance based methods as well?
                                                                                    
Minor comments:
The notations are a bit confusing sometimes, and require going back and forth a bit. Eg. \mu is used
for representation of feature (Eq4) while it usually denotes a mean of some sort (so the reader's expectation
could be that it represents class center). Similarly, boldface p is used for class centers, which is
again a bit confusing as being a probability of some sort. In general the notations are different
from the original prototypical networks paper (which I needed to revise); keeping them similar would
help the reader. The contribution of the present paper is more than that anyway.
The erroneously classified training examples are called errors (eg. just before eq8). By errors one
could think that it is a difference between some sort of prediction and the ground truth. Explicitly
calling them erroneously classified training examples would help the reader as well.
The notation \odot is not explained (before eq.9).