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Title: mixup: Beyond Empirical Risk Minimization

Paper content:
Large deep neural networks are powerful, but exhibit undesirable behaviors such as memorization and sensitivity to adversarial examples.
In this work, we propose mixup, a simple learning principle to alleviate these issues.
In essence, mixup trains a neural network on convex combinations of pairs of examples and their labels.
By doing so, mixup regularizes the neural network to favor simple linear behavior in-between training examples.
Our experiments on the ImageNet-2012, CIFAR-10, CIFAR-100, Google commands and UCI datasets show that mixup improves the generalization of state-of-the-art neural network architectures.
We also find that mixup reduces the memorization of corrupt labels, increases the robustness to adversarial examples, and stabilizes the training of generative adversarial networks.
Large deep neural networks have enabled breakthroughs in fields such as computer vision BID22 , speech recognition , and reinforcement learning BID28 .
In most successful applications, these neural networks share two commonalities.
First, they are trained as to minimize their average error over the training data, a learning rule also known as the Empirical Risk Minimization (ERM) principle BID35 .
Second, the size of these state-of-theart neural networks scales linearly with the number of training examples.
For instance, the network of BID31 used 10 6 parameters to model the 5 · 10 4 images in the CIFAR-10 dataset, the network of BID30 Strikingly, a classical result in learning theory BID36 tells us that the convergence of ERM is guaranteed as long as the size of the learning machine (e.g., the neural network) does not increase with the number of training data.
Here, the size of a learning machine is measured in terms of its number of parameters or, relatedly, its VC-complexity BID16 .This
contradiction challenges the suitability of ERM to train our current neural network models, as highlighted in recent research. On the
one hand, ERM allows large neural networks to memorize (instead of generalize from) the training data even in the presence of strong regularization, or in classification problems where the labels are assigned at random . On the
other hand, neural networks trained with ERM change their predictions drastically when evaluated on examples just outside the training distribution BID33 , also known as adversarial examples. This evidence
suggests that ERM is unable to explain or provide generalization on testing distributions that differ only slightly from the training data. However, what
is the alternative to ERM?The method of
choice to train on similar but different examples to the training data is known as data augmentation BID29 , formalized by the Vicinal Risk Minimization (VRM) principle BID3 . In VRM, human
knowledge is required to describe a vicinity or neighborhood around each example in the training data. Then, additional
virtual examples can be drawn from the vicinity distribution of the training examples to enlarge the support of the training distribution. For instance, when
performing image classification, it is common to define the vicinity of one image as the set of its horizontal reflections, slight rotations, and mild scalings. While data augmentation
consistently leads to improved generalization BID29 , the procedure is dataset-dependent, and thus requires the use of expert knowledge. Furthermore, data augmentation
assumes that the examples in the vicinity share the same class, and does not model the vicinity relation across examples of different classes.Contribution Motivated by these issues, we introduce a simple and data-agnostic data augmentation routine, termed mixup (Section 2). In a nutshell, mixup constructs
virtual training examples DISPLAYFORM0 where x i , x j are raw input vectors y = λy i + (1 − λ)y j , where y i , y j are one-hot label encodings (x i , y i ) and (x j , y j ) are two examples drawn at random from our training data, and λ ∈ [0, 1]. Therefore, mixup extends the training
distribution by incorporating the prior knowledge that linear interpolations of feature vectors should lead to linear interpolations of the associated targets. mixup can be implemented in a few lines
of code, and introduces minimal computation overhead.Despite its simplicity, mixup allows a new state-of-the-art performance in the CIFAR-10, CIFAR-100, and ImageNet-2012 image classification datasets (Sections 3.1 and 3.2). Furthermore, mixup increases the robustness
of neural networks when learning from corrupt labels (Section 3.4), or facing adversarial examples (Section 3.5). Finally, mixup improves generalization on speech
(Sections 3.3) and tabular (Section 3.6) data, and can be used to stabilize the training of GANs (Section 3.7). The source-code necessary to replicate our CIFAR-10
experiments is available at:https://github.com/facebookresearch/mixup-cifar10.To understand the effects of various design choices in mixup, we conduct a thorough set of ablation study experiments (Section 3.8). The results suggest that mixup performs significantly
better than related methods in previous work, and each of the design choices contributes to the final performance. We conclude by exploring the connections to prior work
(Section 4), as well as offering some points for discussion (Section 5).
We have proposed mixup, a data-agnostic and straightforward data augmentation principle.
We have shown that mixup is a form of vicinal risk minimization, which trains on virtual examples constructed as the linear interpolation of two random examples from the training set and their labels.
Incorporating mixup into existing training pipelines reduces to a few lines of code, and introduces little or no computational overhead.
Throughout an extensive evaluation, we have shown that mixup improves the generalization error of state-of-the-art models on ImageNet, CIFAR, speech, and tabular datasets.
Furthermore, mixup helps to combat memorization of corrupt labels, sensitivity to adversarial examples, and instability in adversarial training.In our experiments, the following trend is consistent: with increasingly large α, the training error on real data increases, while the generalization gap decreases.
This sustains our hypothesis that mixup implicitly controls model complexity.
However, we do not yet have a good theory for understanding the 'sweet spot' of this bias-variance trade-off.
For example, in CIFAR-10 classification we can get very low training error on real data even when α → ∞ (i.e., training only on averages of pairs of real examples), whereas in ImageNet classification, the training error on real data increases significantly with α → ∞.
Based on our ImageNet and Google commands experiments with different model architectures, we conjecture that increasing the model capacity would make training error less sensitive to large α, hence giving mixup a more significant advantage.mixup also opens up several possibilities for further exploration.
First, is it possible to make similar ideas work on other types of supervised learning problems, such as regression and structured prediction?
While generalizing mixup to regression problems is straightforward, its application to structured prediction problems such as image segmentation remains less obvious.
Second, can similar methods prove helpful beyond supervised learning?
The interpolation principle seems like a reasonable inductive bias which might also help in unsupervised, semi-supervised, and reinforcement learning.
Can we extend mixup to feature-label extrapolation to guarantee a robust model behavior far away from the training data?
Although our discussion of these directions is still speculative, we are excited about the possibilities mixup opens up, and hope that our observations will prove useful for future development.

assistant: text: Training on convex combinations between random training examples and their labels improves generalization in deep neural networks