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Title: The Nonlinearity Coefficient - Predicting Generalization in Deep Neural Networks

Paper content:
For a long time, designing neural architectures that exhibit high performance was considered a dark art that required expert hand-tuning.
One of the few well-known guidelines for architecture design is the avoidance of exploding or vanishing gradients.
However, even this guideline has remained relatively vague and circumstantial, because there exists no well-defined, gradient-based metric that can be computed {\it before} training begins and can robustly predict the performance of the network {\it after} training is complete.
We introduce what is, to the best of our knowledge, the first such metric: the nonlinearity coefficient (NLC).
Via an extensive empirical study, we show that the NLC, computed in the network's randomly initialized state, is a powerful predictor of test error and that attaining a right-sized NLC is essential for attaining an optimal test error, at least in fully-connected feedforward networks.
The NLC is also conceptually simple, cheap to compute, and is robust to a range of confounders and architectural design choices that comparable metrics are not necessarily robust to.
Hence, we argue the NLC is an important tool for architecture search and design, as it can robustly predict poor training outcomes before training even begins.
Designing neural architectures that perform well can be a difficult process.
In particular, the exploding / vanishing gradient problem has been a major challenge for building very deep neural networks at least since the advent of gradient-based parameter learning (Hochreiter, 1991; Hochreiter & Schmidhuber, 1997; BID4 .
However, there is still no consensus about which metric should be used for determining the presence of pathological exploding or vanishing gradients.
Should we care about the length of the gradient vector (He et al., 2015) , or about the size of individual components of the gradient vector (Schoenholz et al., 2017; Yang & Schoenholz, 2017; Glorot & Bengio, 2010) , or about the eigenvalues of the Jacobian (Saxe et al., 2014; Pascanu et al., 2013; Pennington et al., 2017) ?
Depending on the metric used, different strategies arise for combating exploding and vanishing gradients.
For example, manipulating the width of layers as suggested by e.g. Yang & Schoenholz (2018) ; Han et al. (2017) can greatly impact the size of gradient vector components but tends to leave the length of the entire gradient vector relatively unchanged.
The popular He initialization for ReLU networks (He et al., 2015) is designed to stabilize gradient vector length, wheareas the popular Xavier initialization for tanh networks (Glorot & Bengio, 2010) is designed to stabilize the size of gradient vector components.
While the papers cited above provide much evidence that gradient explosion / vanishing when defined according to some metrics is associated with poor performance when certain architectures are paired with certain optimization algorithms, it is often unclear how general those results are.We make the following core contributions.1.
We introduce the nonlinearity coefficient (NLC), a gradient-based measurement of the degree of nonlinearity of a neural network (section 3).2.
We show that the NLC, computed in the networks randomly initialized state, is a powerful predictor of test error and that attaining a right-sized NLC is essential for achieving an optimal test error, at least in fully-connected feedforward networks (section 4).The
NLC (defined at the top of page 3) combines the Frobenius norm of the Jacobian of a neural network with the global variability of the input data and the global variability of the network's outputs into a single metric. Despite
its simplicity, it is tied to many important properties of the network. It is a
remarkably accurate predictor of the network's nonlinearity as measured by the relative diameter of the regions in input space that can be well-approximated by a linear function (section 3 and figure 1). It is closely
related to the nonlinearity of the individual activation functions used in the network and the degree to which they can be approximated by a linear function (section 5). It is tied to
the susceptibility of the network's output to small random input perturbations.
We introduced the nonlinearity coefficient, a measure of neural network nonlinearity that is closely tied to the relative diameter of linearly approximable regions in the input space of the network, to the sensitivity of the network output with respect to small input changes, as well as to the linear approximability of activation functions used in the network.
Because of this conceptual grounding, because its value in the randomly initialized state is highly predictive of test error while also remaining somewhat stable throughout training, because it is robust to simple network changes that confound other metrics such as raw gradient size or correlation information, because it is cheap to compute and conceptually simple, we argue that the NLC is the best standalone metric for predicting test error in fully-connected feedforward networks.
It has clear applications to neural architecture search and design as it allows sub-optimal architectures to be discarded before training.
In addition to a right-sized NLC, we also found that avoiding excessive output bias and using skip connections play important independent roles in performance.
This paper makes important contributions to several long-standing debates.
We clearly show that neural networks are capable of overfitting when the output is too sensitive to small changes in the input.
In fact, our random architecture sampling scheme shows that such architectures are not unlikely to arise.
However, overfitting seems to be tied not to depth or the number of parameters, but rather to nonlinearity.
In contrast to Schoenholz et al. (2017); Xiao et al. (2018) , we find that a very high output sensitivity does not harm trainability, but only generalization.
This difference is likely caused by our very extensive learning rate search and 64 bit precision training.While the popular guidance for architecture designers is to avoid exploding and vanishing gradients, we argue that achieving an ideal nonlinearity level is the more important criterion.
While the raw gradient is susceptible to confounders and cannot be directly linked to meaningful network properties, the NLC captures what appears to be a deep and robust property.
It turns out that architectures that were specifically designed to attain a stable gradient, such as He-initialized ReLU networks, in fact display a divergent NLC at great depth.It has been argued that the strength of deep networks lies in their exponential expressivity (e.g. Raghu et al. (2017); Telgarsky (2015) ).
While we show that the NLC indeed exhibits exponential behavior, we find this property to be largely harmful, not helpful, as did e.g. Schoenholz et al. (2017) .
While very large datasets likely benefit from greater expressivity, in our study such expressivity only leads to lack of generalization rather than improved trainability.
In fact, at least in fully-connected feedforward networks, we conjecture that great depth does not confer significant practical benefit.In future work, we plan to study whether the ideal range of NLC values we discovered for our three datasets (1 N LC 3) holds also for larger datasets and if not, how we might predict this ideal range a priori.
We plan to investigate additional causes for why certain architectures perform badly despite a right-sized NLC, as well as extend our study to convolutional and densely-connected networks.
We are interested in studying the connection of the NLC to e.g. adversarial robustness, quantizability, sample complexity, training time and training noise.
Finally, unfortunately, we found the empirical measurement of the NLC to be too noisy to conclusively detect an underfitting regime.
We plan to study this regime in future.
The goal of this section is to provide an intuitive, graphical explanation of the NLC in addition to the mathematical derivation and analysis in section 3 for readers interested in developing a better intuition of this concept.

assistant: text: We introduce the NLC, a metric that is cheap to compute in the networks randomly initialized state and is highly predictive of generalization, at least in fully-connected networks.