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Title: Feature-Robustness, Flatness and Generalization Error for Deep Neural Networks

Paper content:
The performance of deep neural networks is often attributed to their automated, task-related feature construction.
It remains an open question, though, why this leads to solutions with good generalization, even in cases where the number of parameters is larger than the number of samples.
Back in the 90s, Hochreiter and Schmidhuber observed that flatness of the loss surface around a local minimum correlates with low generalization error.
For several flatness measures, this correlation has been empirically validated.
However, it has recently been shown that existing measures of flatness cannot theoretically be related to generalization: if a network uses ReLU activations, the network function can be reparameterized without changing its output in such a way that flatness is changed almost arbitrarily.
This paper proposes a natural modification of existing flatness measures that results in invariance to reparameterization.
The proposed measures imply a robustness of the network to changes in the input and the hidden layers.
Connecting this feature robustness to generalization leads to a generalized definition of the representativeness of data.
With this, the generalization error of a model trained on representative data can be bounded by its feature robustness which depends on our novel flatness measure.
Neural networks (NNs) have become the state of the art machine learning approach in many applications.
An explanation for their superior performance is attributed to their ability to automatically learn suitable features from data.
In supervised learning, these features are learned implicitly through minimizing the empirical error E emp (f, S) = 1 /|S| (x,y)∈S (f
(x),
y) for a training set S ⊂ X × Y drawn iid according to a target distribution D : X × Y → [0, 1], and a loss function : Y × Y → R + .
Here, f : X → Y denotes the function represented by a neural network.
It is an open question why minimizing the empirical error during deep neural network training leads to good generalization, even though in many cases the number of network parameters is higher than the number of training examples.
That is, why deep neural networks have a low generalization error
which is the difference between expected error on the target distribution D and the empirical error on a finite dataset S ⊂ X × Y.
It has been proposed that good generalization correlates with flat minima of the non-convex loss surface (Hochreiter & Schmidhuber, 1997; 1995) and this correlation has been empirically validated (Keskar et al., 2016; Novak et al., 2018; Wang et al., 2018) .
Thus, for deep neural networks trained with stochastic gradient descent (SGD), this could present a (partial) explanation for their generalization performance (Zhang et al., 2016) , since minibatch SGD tends to converge to flat local minima (Zhang et al., 2018; Jastrzębski et al., 2017) .
This idea was elaborated on by Chaudhari et al. (2016) who suggest a new training method that favors flat over sharp minima even at the cost of a slightly higher empirical error -indeed solutions found by this algorithm exhibit better generalization performance.
Similarly, Dziugaite & Roy (2017) augment the loss to improve generalization and find that this promotes flat minima.
However, as Dinh et al. (2017) remarked, current flatness measures-which are based only on the Hessian of the loss function-cannot theoretically be related to generalization: For deep neural networks with ReLU activation functions, there are layer-wise reparameterizations that leave the network function unchanged (hence, also the generalization performance), but change any measure derived only from the loss Hessian.
Another, more intuitive explanation for generalization is that the function generalizes well if the extracted features encode a semantic similarity of the input that is robust to small changes-both in the input and the features.
This allows to generalize from the training set to novel, sufficiently similar data.
Starting from such a concept of robustness with respect to changes of features, we derive a measure of flatness that is invariant under the mentioned reparameterizations and that reduces to the well-known ridge regression penalty in the special case of a linear regression.
This brings three seemingly related properties into our focus: flatness, robustness, and generalization.
The exact relationship, however, between flatness of the loss surface around local minima (measuring changes of the empirical error for perturbations in parameter space), robustness (measuring changes of the error for perturbations in either input or feature space), and generalization (performance on unseen data from the target distribution) is not well-understood.
This paper provides new insights into this relationship.
The notion of feature robustness proposed in this paper measures the robustness of a function f = ψ • φ (e.g., a neural network) toward local changes in a feature space.
That is, f can be split into a composition of functions f (x) = (ψ • φ)(x) for x ∈ X , φ : X → R m and ψ : R m → Y. The function φ is considered as a feature extraction, mapping the input X into a feature space R m , while the function ψ corresponds to the model (e.g., a classifier) with R m as its domain (see Figure 1 for illustration).
It is the feature space defined by φ where we measure robustness toward small perturbations.
For neural networks, the activation values of any but the output layer can be viewed as a feature space.
A function f is called -feature robust on a dataset S ⊂ X × Y if small changes in the feature space defined by φ do not change the empirical error by more than .
This differs from the notion of robustness defined by Xu & Mannor (2012) using a cover of the sample space, which has been theoretically connected to generalization.
Flatness of the loss surface, however, is a local property and we require a more local version of robustness to derive a connection between flatness and robustness.
Then, indeed, feature-robustness is upper bounded by the proposed flatness measure.
To finally connect the two local properties of robustness and flatness to generalization, we necessarily need a notion describing how representative the given samples are for the true distribution.
We define a suitable notion, leading to an upper bound for the generalization error given by feature robustness together with representativeness.
In summary, our contributions are as follows: (i) For models of the form f (x) = (ψ • φ)(x) (e.g. most (deep) neural networks) that split up into a feature extractor φ and a model ψ on the feature space defined by φ, we define a property of feature robustness that measures the change of the loss function under small perturbations of the features.
This property is strongly related to flatness of the loss surface at local minima.
(ii) We propose a novel flatness measure.
For neural networks with ReLU activation functions, it is invariant under layer-wise reparameterization, addressing a shortcoming of previous measures of flatness.
(iii) We define a suitable notion of representativeness of a dataset connecting feature robustness to the generalization error in form of an upper bound.
(iv) The proposed flatness measure is empirically shown to strongly correlate with good generalization performance.
Thereby, we recover Hessian based quantities as measures of flatness.
We established a theoretical connection between flatness, feature robustness and, under the assumption of representative data, the generalization error.
The relation between feature robustness and Hessianbased flatness measures has been established for κ l , which takes into account the maximum eigenvalue of the Hessian, and κ l T r , which uses the trace instead.
Empirically, the measure κ l T r based on the trace of the Hessian shows a stronger correlation with the generalization error.
This is not surprising, since it takes into account the whole spectrum of the Hessian and every eigenvalue corresponds to a feature selection matrix of feature robustness.
The tracial measure can be related to feature robustness by either bounding the maximum eigenvalue of the loss Hessian by its unnormalized trace or by averaging feature robustness over all orthogonal matrices A ∈ O m .
It is interesting to note that strong feature robustness does not exclude the possibility of adversarial examples, first observed by Szegedy et al. (2013) , since large changes of loss for individual samples (i.e. adversarial examples) may be hidden in the mean in the definition of feature robustness.
In Appendix C.2 we briefly discuss the freedom of perturbing individual points by suitable feature selection matrices A.
In contrast to existing measures of flatness, our proposed measure is invariant to layer-wise reparameterizations of ReLU networks.
However, we note that other reparameterizations are possible, e.g., we can use the positive homogeneity and multiply all incoming weights into a single neuron by a positive number λ > 0 and multiply all outgoing weights of the same neuron by 1 /λ.
While the Fisher-Rao norm suggested by Liang et al. (2019) is invariant to such reparameterizations, our proposed measures of flatness κ l and κ l T r are in general not.
In principle, variations of our flatness measures can be found that are invariant to such reparameterizations as well (see Appendix B) but their analysis, except for some empirical evaluations in Appendix E, is left for future work.
The second term in the generalization bound of Theorem 10 is given by our notion of representativeness.
In order to find specific bounds for the -representativeness of (S, A δ ), a distribution over matrices is required that induces a distribution which is similar to a localized kernel density estimation (KDE).
While our notion of representativeness is a generalization of classical representativeness, it remains open whether it is efficiently computable.
The more feature robust a model is, the more freedom there is to finding specific distributions over matrices that lead to bounds on the generalization error.
In Appendix D we give a computation of representativeness for a KDE with Gaussian kernels.
Taking things together, we proposed a novel and practically useful flatness measure that strongly correlates with the generalization error.
We theoretically investigated this connection by relating this measure to feature robustness.
This notion of robustness, together with a novel notion of representativeness provides a link to the generalization error.
To the best of our knowledge, this yields the first theoretical connection between a notion of robustness, flatness of the loss surface, and generalization error and can help to better understand the performance of deep neural networks.

assistant: text: We introduce a novel measure of flatness at local minima of the loss surface of deep neural networks which is invariant with respect to layer-wise reparameterizations and we connect flatness to feature robustness and generalization.