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Title: Manifold Mixup: Learning Better Representations by Interpolating Hidden States

Paper content:
Deep networks often perform well on the data distribution on which they are trained, yet give incorrect (and often very confident) answers when evaluated on points from off of the training distribution.
This is exemplified by the adversarial examples phenomenon but can also be seen in terms of model generalization and domain shift.
Ideally, a model would assign lower confidence to points unlike those from the training distribution.
We propose a regularizer which addresses this issue by training with interpolated hidden states and encouraging the classifier to be less confident at these points.
Because the hidden states are learned, this has an important effect of encouraging the hidden states for a class to be concentrated in such a way so that interpolations within the same class or between two different classes do not intersect with the real data points from other classes.
This has a major advantage in that it avoids the underfitting which can result from interpolating in the input space.
We prove that the exact condition for this problem of underfitting to be avoided by Manifold Mixup is that the dimensionality of the hidden states exceeds the number of classes, which is often the case in practice.
Additionally, this concentration can be seen as making the features in earlier layers more discriminative.
We show that despite requiring no significant additional computation, Manifold Mixup achieves large improvements over strong baselines in supervised learning, robustness to single-step adversarial attacks, semi-supervised learning, and Negative Log-Likelihood on held out samples.
Machine learning systems have been enormously successful in domains such as vision, speech, and language and are now widely used both in research and industry.
Modern machine learning systems typically only perform well when evaluated on the same distribution that they were trained on.
However machine learning systems are increasingly being deployed in settings where the environment is noisy, subject to domain shifts, or even adversarial attacks.
In many cases, deep neural networks which perform extremely well when evaluated on points on the data manifold give incorrect answers when evaluated on points off the training distribution, and with strikingly high confidence.This manifests itself in several failure cases for deep learning.
One is the problem of adversarial examples (Szegedy et al., 2014) , in which deep neural networks with nearly perfect test accuracy can produce incorrect classifications with very high confidence when evaluated on data points with small (imperceptible to human vision) adversarial perturbations.
These adversarial examples could present serious security risks for machine learning systems.
Another failure case involves the training and testing distributions differing significantly.
With deep neural networks, this can often result in dramatically reduced performance.To address these problems, our Manifold Mixup approach builds on following assumptions and motivations: (1) we adopt the manifold hypothesis, that is, data is concentrated near a lower-dimensional non-linear manifold (this is the only required assumption on the data generating distribution for Manifold Mixup to work); (2) a neural net can learn to transform the data non-linearly so that the transformed data distribution now lies on a nearly flat manifold; (3) as a consequence, linear interpolations between examples in the hidden space also correspond to valid data points, thus providing novel training examples.
(a,b,c) shows the decision boundary on the 2d spirals dataset trained with a baseline model (a fully connected neural network with nine layers where middle layer is a 2D bottleneck layer), Input Mixup with α = 1.0, and Manifold Mixup applied only to the 2D bottleneck layer.
As seen in
(b), Input Mixup can suffer from underfitting since the interpolations between two samples may intersect with a real sample.
Whereas Manifold Mixup
(c), fits the training data perfectly (more intuitive example of how Manifold Mixup avoids underfitting is given in Appendix H).
The bottom row (d,e,f) shows the hidden states for the baseline, Input Mixup, and manifold mixup respectively.
Manifold Mixup concentrates the labeled points from each class to a very tight region, as predicted by our theory (Section 3) and assigns lower confidence classifications to broad regions in the hidden space.
The black points in the bottom row are the hidden states of the points sampled uniformly in x-space and it can be seen that manifold mixup does a better job of giving low confidence to these points.
Additional results in Figure 6 of Appendix B show that the way Manifold Mixup changes the representations is not accomplished by other well-studied regularizers (weight decay, dropout, batch normalization, and adding noise to the hidden states).Manifold
Mixup performs training on the convex combinations of the hidden state representations of data samples. Previous
work, including the study of analogies through word embeddings (e.g. king -man + woman ≈ queen), has shown that such linear interpolation between hidden states is an effective way of combining factors (Mikolov et al., 2013) . Combining
such factors in the higher level representations has the advantage that it is typically lower dimensional, so a simple procedure like linear interpolation between pairs of data points explores more of the space and with more of the points having meaningful semantics. When we combine
the hidden representations of training examples, we also perform the same linear interpolation in the labels (seen as one-hot vectors or categorical distributions), producing new soft targets for the mixed examples.In practice, deep networks often learn representations such that there are few strong constraints on how the states can be distributed in the hidden space, because of which the states can be widely distributed through the space, (as seen in FIG0 ). As well as, nearly
all points in hidden space correspond to high confidence classifications even if they correspond to off-the-training distribution samples (seen as black points in FIG0 ). In contrast, the consequence
of our Manifold Mixup approach is that the hidden states from real examples of a particular class are concentrated in local regions and the majority of the hidden space corresponds to lower confidence classifications. This concentration of the hidden
states of the examples of a particular class into a local regions enables learning more discriminative features. A low-dimensional example of this
can be seen in FIG0 and a more detailed analytical discussion for what "concentrating into local regions" means is in Section 3.Our method provides the following contributions:• The introduction of a novel regularizer which outperforms competitive alternatives such as Cutout BID4 , Mixup (Zhang et al., 2018) , AdaMix BID10 Dropout (Hinton et al., 2012) . On CIFAR-10, this includes a 50%
reduction in test Negative Log-Likelihood (NLL) from 0.1945 to 0.0957.• Manifold Mixup achieves significant robustness to single step adversarial attacks.• A new method for semi-supervised
learning which uses a Manifold Mixup based consistency loss. This method reduces error relative
to Virtual Adversarial Training (VAT) (Miyato et al., 2018a) by 21.86% on CIFAR-10, and unlike VAT does not involve any additional significant computation.• An analysis of Manifold Mixup and
exact sufficient conditions for Manifold Mixup to achieve consistent interpolations. Unlike Input Mixup, this doesn't require
strong assumptions about the data distribution (see the failure case of Input Mixup in FIG0 ): only that the number of hidden units exceeds the number of classes, which is easily satisfied in many applications.
Deep neural networks often give incorrect yet extremely confident predictions on data points which are unlike those seen during training.
This problem is one of the most central challenges in deep learning both in theory and in practice.
We have investigated this from the perspective of the representations learned by deep networks.
In general, deep neural networks can learn representations such that real data points are widely distributed through the space and most of the area corresponds to high confidence classifications.
This has major downsides in that it may be too easy for the network to provide high confidence classification on points which are off of the data manifold and also that it may not provide enough incentive for the network to learn highly discriminative representations.
We have presented Manifold Mixup, a new algorithm which aims to improve the representations learned by deep networks by encouraging most of the hidden space to correspond to low confidence classifications while concentrating the hidden states for real examples onto a lower dimensional subspace.
We applied Manifold Mixup to several tasks and demonstrated improved test accuracy and dramatically improved test likelihood on classification, better robustness to adversarial examples from FGSM attack, and improved semi-supervised learning.
Manifold Mixup incurs virtually no additional computational cost, making it appealing for practitioners.

assistant: text: A method for learning better representations, that acts as a regularizer and despite its no significant additional computation cost , achieves improvements over strong baselines on Supervised and Semi-supervised Learning tasks.