Feature-Robustness, Flatness and Generalization Error for Deep Neural Networks
Authors: Henning Petzka
Linara Adilova
Michael Kamp
Cristian Sminchisescu
Venue:  (2020)
Keywords: robustness
flatness
generalization error
loss surface
deep neural networks
feature space

Abstract: 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.

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FEATURE-ROBUSTNESS. FLATNESS AND GENERAL- IZATION ERROR FOR DEEP NEURAL NETWORKS
Anonymous authors Paper under double-blind review
ABSTRACT
The performance of deep neural networks is often attributed to their automated, task- related feature construction. remains an open question, though, why this leads 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 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 network uses ReLU activations, the network function output in such a way that flatness is changed almost arbitrarily. This paper proposes natural modification of existing flatness measures that results in invariance reparameterization. The proposed measures imply a robustness of the network to changes the input and the hidden layers. Connecting this feature robustness 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.
be reparameterized without changing its
INTRODUCTION
Neural networks (NNs) have become the state of the art machine learning approach in many appli- cations. An explanation for their superior performance is attributed their ability automatically learn suitable features from data. In supervised learning, these features are learned implicitly through minimizing the empirical error Eemp (f. S) 1/1s)E(n ((f()) y) for a training
C X x y drawn iid according to a target distribution D x y + [0, 1]. and a loss function e: ) x R., Here, X y denotes the function represented by neural network.
is an open question why minimizing the empirical error during deep neural network training leads good generalization, the number of training examples. That is, why deep neural networks have a low generalization error
though in many cases the number of network parameters is higher than
Egen (e.))D [(f()),
{((xx,v) (H.D)ES
Σ
which is the difference between expected error on the target distribution D and the empirical error on finite dataset S C X x
It has been proposed that good generalization correlates with flat minima of the non-convex loss surface (Hochreiter Schmidhuber, [197] 1995) and this correlation has been empirically vali- dated (Keskar et al. 6016; Novak et | 2018] Wang et al. 2018). Thus, for deep neural networks trained with stochastic gradient descent (SGD), this could present (partial) explanation for their generalization performance (Zhang et al.) 2016), since minibatch SGD tends to converge to flat local minima (Zhang et al. Jastrzebbk et al. 2017). This idea was elaborated on by [Caudhari] et al. (2016) who suggest new training method that favors flat over sharp minima even at the cost of slightly higher empirical error indeed solutions found by this algorithm exhibit bet- ter generalization performance. Similarly, [Diiugaite & Roy (211)) augment the loss to improve generalization and find that this promotes tlat minima. However, as Dinh et al. (2017) remarked, current flatness which are based only on the Hessian of the loss function-ccanee cannot

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theoretically be related to generalization: For deep neural networks with ReLU activation func- tions, there are wiser-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 semantic similarity of the input that is robust to small changes-both in the input and the features. This al- lows 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 reparameteri- zations and that reduces to the well-known ridge regression penalty in the special case of a linear regression.
input layer
hidden
output
layers
layer
This brings three seemingly related properties into our fo- cus: flatness, robustness, and generalization. The exact relationship, however, between flatness of the loss surface around local minima (measuring changes of the empiri- cal 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) well-understood. This paper provides new insights into this relationship.
Figure I: Illustration of the decomposi- tion of f + Φ.
The notion of feature robustness proposed in this paper measures the robustness of a function
v e (e.g.. a neural network) toward local changes in a feature space. That is, can be split into a composition of functions f(x) (vb (I) for x E X, Ø X' + I=' and ei R" + y. The function á considered while the function z corresponds to the model (e.g., a classifier) with Ihm as its domain (see Figure 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 e-feature robust on a dataset S X x ) small changes in the feature space defined by & do not change the empirical error by more than e. 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 local property and we require a more local version of robustness to derive a connection between flatness and robustness. Then, indeed, febutre-rouustnsss 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 suitable notion, leading to an upper bound for the generalization error given by feature robustness together with representativeness.
feature extraction, mapping the input X' into a feature space ET
In summary, our contributions are as follows: (i) For models of the form f(z) = (vb o (z) (c.g. most (deep) neural networks) that split up into a feature extractor & and model v on the feature space defined by é, we define 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 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 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 strongly correlate with good generalization performance. Thereby, we recover Hessian based quantities as measures of flatness.
FEATURE ROBUSTNESS
We will define a notion of robustness in feature space RT for the model f = (v )) V, which depends small number 8 0, training set S, and feature selection defined by matrix A e mm of operator norm ||||| < 1. In the case of neural networks split into a composition according to Figure i traditionally, the activation values Φy (xr) of neurons considered as feature
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values. The feature value defined by the j-th neuron in the feature space ((x) R can be written as (x) = (o(z), ez). where e, denotes the j-th unit vector and (.-)) )) the scalar product in R" However, was shown by Szegedy et al. (2013) that, for any other direction
|/"l = the values (Ø(x). u) = proj. ((x:) obtained from the projection ((xx) onto v, can be likewise semantically interpreted feature. We single out the feature defined by from (xx)) by multiplication with the projection matrix E, = vu"" Similarly, multiplication of ((x) with matrix A corresponds to weighted selection of (A)-mayy features in parallel (e.g., projection matrices on d-dimensional subspaces correspond to the selection of d many features). This justifies our terminology considering a matrix as feature selection. The same way that, for sample imput z, non-activated neurons (x) = 0 are considered as non-expressed features, we call selection of features defined by matrix as non-expressed whenever AΦ(x) 0.
We define our notion of feature robustness. In words, feature robustness measures the mean change in loss over a dataset under small changes of features in the feature space. Hereby, matrix A determines which features shall be perturbed. For each sample, the perturbation is linear in the expression of the feature. Thereby, we only perturb features that are relevant for the output for a given sample and leave feature values unchanged that are not expressed (in the sense explained above). With
S( == SS|
Σ (x.N)ES
[e(b(ó(z)) + 5A0(z)). v) 1 v))(
(2)
the precise definition is given follows:
Definition 1. Let É y x y R+ denote loss function, ô and two strictly positive (small) real numbers, {
-= I......, .......N, N S X x y a set, and A E Rmxm a matrix such that ||A|| < 1. model f(x) (1b (z) which is composition of functions 6 X - R. and ; : Rm + +Y, is called ((6, S, ef-future e) robust, Ff(F('', S, S for all [8'| S 6.
More generally, if A C mm denotes probability space over matrices such that ||A|] I for all A E A, then we call the model
robust on average over A, if
EA-A [1F(8',
S efor all I S 8.
We will bound feature robustness local minima for a dataset S uniformly over all feature selections A and dependent on 8. With our interpretation, this corresponds to an upper bound of the change in loss when perturbing features in feature space IR. In Appendix C.T we note how feature robustness related to noise injection in the layer of consideration, which is known to be related to generalization
(An 1996 Bishop 1995).
3 FEATURE ROBUSTNESS IS CONNECTED TO FLATNESS OF THE LOss CURVE
Consider a function f(z w) (www, o(x)) g(wo(z)), where & is the composition of twice differentiable function g Rd + y and a matrix product with a matrix w E Rdxmm As before, : X' Rm be considered as feature extractor. We fix loss function E y x y R+ for supervised learning and let w. denote choice of parameters for which the empirical error Cemp(w; (w, S) 2/1s1 ((.y)es (((r,, w), v). considered as a function on w, is at local minimum on the training set = { 34)
,,..., J.. In the following, we write 2 = p(x).
For any matrix A E R we have that
n(w, + 5A2)= = g(w(= 8A2)) s((w + (ww))) (o Aw.,
(3)
Therefore.
S,), (emp(w,S)
Σ
+
(x.))ES
(4)
S
(3.D)ES
The latter is the empirical error Eamp(w + 8w S) of the model f on the dataset S at parameters dw A. If 5 is sufficiently small, then by Taylor expansion of Eomp (w, (w,S) S) with respect to
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parameters around the critical point w., we have that
Eemp(w. + A, = EEmpp (w., S) + (ów. A, V Vempp(... (w,;
(öw.A, A, HEemp( (W.. (Ów. (68)) Al)))
(5)
82 Eempl w.: 5) + 2 (W.A, H.Emp(We.) S) (w+ A)) - ( |-
with Hempp( (w., denoting the Hessian of the empirical error with respect to w, (:5-) the scalar product with vectorized versions of the parameters and ||w]| the Frobenius norm of
Subtracting Cemp(w.,, S) from (5) maximizing over matrices || S 1 and using (4), we get that, for any feature selection A, the function 2 defining feature robustness is bounded by
8²
max F(6, 5,4) ||||S
X (w.) + O
where X (w.) denotes the largest eigenvalue of the Hessian H& (w., 5) of the empirical error at w.. Here we used the identity that |ax||=]|==1 x² Mx = Mazz for any symmetrie matrix M, and that for matrices of norm ||A|| S we have |w S |/wllF. We show details of the proof of (6) in the appendix. We summarize the connection between feature robustness and flatness in terms of the Hessian in the following theorem.
Theorem 2. Let e: y Y -> R denote loss function, 5 strictly positive (small) real number, A E mmmm a matrix with || S 1, and let far w) = g((((()) be a model with an arbitrary twice differentiable function on a matrix product of parameters w and the image of x under a (feature) function 6h Let w. denote a local minimum of the empirical error on a dataset S. Then the model f(w.) is ((), (,), e robust for 82 //l A (w.) + (
4 MEASURES OF FLATNESS OF THE LOSS CURVE
Motivated by the relation of feature robustness with the Hessian H, we define novel measure of flatness. Note that the Hessian is computed with respect to those parameters w that are applied linearly on the feature space o(X)) I
Definition 3. Let E: y x ) + R., denote a loss function and f(x, w) g(w((()) be a model with y an arbitrary twice differentiable function on matrix product of parameters w and the
g: @.R.. Th image of x under a (feature) function & X - Rm Then xe (w) shall denote a flatness measure of the loss surface defined by
(w) == || 12 - (w).
Note that small values of (w) indicate flatness and high values indicate sharpness.
Linear regression with squared loss In the case of linear regression, f(x, w) = wa E R (X' = Rd, g = id and Φ id), for any loss function f, we compute second derivatives with respect the parameters w E RRd as
a2e
028
Ow.Ow; ØJ((z, w)) Tiz)
(8)
If L is the squared loss function É(G, y) = (g - y)². then d't/oy² = 2 and the Hessian is independent of the parameters w. In this case, nd = c- |/wl| with a constant c 2Amaz (Ezes zx²) and the
measure rd reduces to (a constant multiple of) the well-known Tikhonov (ridge) regression penalty.
Layers of Neural Networks We consider neural network functions
f(x) = Wwa(. Wwoa(...(Waed (W2d (w,x x + by) + bz) + bz.
of a neural network of L layers with nonlinear activation function a. We hide a possible non- coneaiie at the output by integrating it in loss function & chosen for neural network training. By letting o' (z) = o (w,-110(... ......(( oo (Wo (W x + b)) b2)...) + by-1) denote the output of the composition of the first 1- 1 layers and gf (=) (. (( (z=-b) ... b) the composition of the activation function
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of the 1-th layer together with the rest of layers, write for each layer I, f(x, wi) = (wio' (x)) Using (7) we obtain for each layer of the neural network a measure of flatness at parameter values w:
n² (w) == ||w/l2 H. (wi)
(10)
with X A (wi) the largest eigenvalue of the Hessian of the empirical error with respect to the parameters of the 1-th layer. By Theorem 2 x² is related to small changes of feature values in layer I. Corollary 4. Let f denote neural network function of an L-layer fully connected neural network. For each layer 1, 1,1 S S L of size ny let A E xe with || S correspond to feature selections of features in the 1-th layer of the neural network. Let Wie denote weights of the l-th layer at local minimum of the empirical error.
Then the neural network is ((6, A), feature robust in layer at w. for = x (w.)) + +0(0))
For everywhere well-defined Hessian of the loss function, we assumed network function to be twice differentiable. With the usual adjustments (equations only hold almost everywhere in parameter space), we can also consider neural networks with ReLU activation functions. In this case, Dinh et al. (2017) noted that a linear reparameterization of one layer, wi + Aw; for X > 0, can lead to the same network function by simultaneously multiplying another layer by the inverse of X, W& + 1/AWa, k + I. Representing the same function, the generalization performance remains unchanged. However, this linear reparameterization changes all common measures of the Hessian of the loss. This constitutes an issue in relating flatness of the loss curve to generalization. We counteract this behavior by the multiplication with ||w:||?
Theorem 5. Let f f(W1, W2..... WL) denote neural network function parameterized by weights Wt of the 1-th layer. Suppose there are positive numbers Xi....; d such that f(W1, W2,...,WW. (>/W,,,2222... ..WW....A..... for all wi. Then, with w (W1; ....... Wt.) X2w2.....
and w (21W1;A2W2.. AI.W.... we have
nt (w) = xx))=<'(w k'(w") all SISL.
(11)
We provide a proof in Appendix A.2
An Averaging Alternative Experimental work (Ghorbani et al. 2019) suggests that the spectrum of the Hessian has a lot of small values and only a few large outliers. In this case, our flatness measure serving
upper bound for feature robustness is governed by the outlier. However, feature robustness for different feature selections is governed by different eigenvalues of the Hessian, according to (5). We therefore consider the trace an average of the spectrum. We will show that this tracial averaging corresponds to feature robustness on average over all orthogonal feature selection matrices. The following theorem specifies this connection between feature robustness and the unnormalized trace Tr(HE (w. (w.)) of the empirical error at a local minimum w., The details and the proof can be found in Appendix A.3
Theorem 6. Let e: y x y + TR,, denote a loss function, 5 a strictly positive (small) real number, and let (Ix w) = ((w((x)) be a model with g an arbitrary twice differentiable function on matrix product of parameters w E Rdxx and the image of x under a (feature) ) function 6. Let w. denote a local minimum of the empirical error on dataset S and Om Rmxx denote the set of orthogonal matrices. Then, for each feature selection matrix ||A||| I 1 the model f(w.) is ((5, S, A), c--eaturee robust for - - 1l..11? Tr(HE (W.)) + O (83), and (ii) the model (w.) is ((5, Om); )-feature robust on average over O. for 2m 8" |w. ww..l? Tr (r(Eeem((..) + ((89).
We therefore consider the unnormalized trace as suitable and efficiently computable measure of flatness and define for each layer l of neural network
k4, (w) : ||w/lfe- Tr((FE (w1, S)).
(12)
The same arguments those used to prove Theorem alsso show the measure nly, to be independent with respect to the same layer-wise reparameterizations. The analogue of Corollary iis as follows. Corollary 7. Let denote neural network function of an I--layer fully connected neural network. For each layer 1, < 1. S I, of size nu, let On, C RRxM denote the set of orthogonal feature selections in the 1-th layer of the neural network. Let Wis &Iee+1 xn; denote weights of the 1-th layer at local minimum of the empirical error. Then the neural network is ((), On), e)- feature robust in layer I average over On at w. for
82 2n
(w.) O
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Remark 8. Other Hessian-basede measures have been proposed that are invariant under the given reparamnterizaations. Liang al. 2019) consider the Fisher-Rat Rao norm defined by l|w" H (w)w] where H (w) (Vwl)(V w)² denotes the Gauss-Newton cpproximation of the loss Hessian. Therefore. the Fisher-Rac norm considers the second partial derivative only into the direction defined by the given weight values w. Our measure considers all directions of moving away from a local minimum. In particular. in contrast to these measures, we take the full spectrum of the Hessian into account, which results a natural measure of flatness around a local minimum. We also came across preprints by Tsuzuku et al. (2019) and Rangamani et al. (209)) which propose a similar measure of flatness. While the first one derives the flatness measure from PAC Baysian approach, the latter considers the Riemannian metric on the quotient manifold obtained from the equivalence relation given by the refactorization of layers as above.
FEATURE ROBUSTNESS AND GENERALIZATION
In this section we aim to study the relation between flatness, feature robustness and the generalization error (defined in (I). The connection of flat local minima with generalization in PAC Baysian bounds has been considered in several works (Neyshabur et al. 2017 Tsuzuuu et al. 2019, Dziugaite & Roy 2017). Arora et (2018) relates noise injection to generalization under the same setting. The work of McAllester 1998 1999) and Langford& & Caruana (200T) initiated the PAC-Baysian approach to generalization, which measures the generalization error (usually for the 0-1 loss) of stochastic classifiers. This leads to bounds on the expected true error over distribution of models Q in terms of the expectation of the empirical error over Q and the Kullback-Leiblee divergence between Q and some prior distribution P. For example, Neyshabur et al. (2017) use work by McAAlester (2003) to derive an inequality relating the generalization error for the 0-1 loss & and expected sharpness To := Ep [Eemp(w ( + v, V, -Eemp (w, over distribution With probability (1 - c) over datasets S of size N, the expected eттоr Ew+z == E, (za))) [e(f(z, + v), y)]] over the "posterior"
distribution (w + v) is bounded by Eemp(w, S) + %o + 4 2/ (KL(W + w| P) In2 Here, P denotes a "prior" distribution which is fixed before seeing any data and KL denotes the Kullback- Leibler divergence. robustness and the Taylor expansion relating feature robustness to flatness) with data-independent prior P, the KI. term goes to infinity as the distribution v becomes increasingly localized. Since feature robustness connect the local properties of flatness and feature robustness to generalization of a specific model by following different approach. Our approach will be independent of the loss function and work in the sample space instead of averages over models in parameter space.
we aim to use distributions with local support (as considered in feature
local property is related generalization (Morcos et al., 2018), aim to
Since feature robustness is local property in neighborhoods around the points (xx y) E S, connect feature robustness to generalization we necessarily need an assumption of representativeness of the given data samples S. A simple computation shows that
Egen(J.)) = EAMA (F(5,S,
(13)
+
E ROOWDD [(f((x)))) v)l
Σ (xi:B)es
EAJ ((((((()))++)))))))))))))))))) + (AM(1;))
SS|
The first term is exactly feature robustness on average over probability distribution of feature matrices. For the second term, we accordingly define a notion datasets S that describes how well the loss on the true distribution can be approximated by certain probability distributions. The distributions consider are composed of a dataset and (local) probability distributions around its points suitably restricted to local distributions X; and 1/4 centered around the origin 0.
Definition 9. Let ib RT + y be a model, E strictly positive (small) real number, and S = (A1,2)) sisn denote a family of pairs of probability distributions on RT x y, where each Xa and vi have support contained in a neighborhood of the origin O. (i) The pair (S, A) is called
R. denote a loss function, e a N) C X x y set. Let

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e-representative for +p (with respect the loss l and distribution D) Df Rep(s < where R))(S,A) ==
Σ
((GxxS))-(Axxn))) [e(xb(z; + Ex), + Ey)l
[||
(E..8)ES
(14)
(ū) With St a collection of families A as above and H hypothesis space, we say that S is (e, 9 representative for H for all vo H there is Ag. S2 such that Av) is e-representative for th.
Interestingly, we naturally derived a definition of representativeness which is a generalization of classical e-representativeness (see e.g. Definition 4.1 in (Shalev-Shwartz & Ben-David, (2014), justifying the terminology. Indeed, let Ao denote the family of probability distributions where each X óg and V = 6, have full weight on the origin. Then S is (e, rrpresseeeeeettaavvee exactly when S is e-representative in the classical sense. Further, if S' is e-representative and S is (c', Q) represeentteee for some Q containing Ag, then é' <.
In our setting of a model f(x) (z), which is split up into feature extractor é and a model v. we consider (o()), reprrssentttt for model v and specific choices for A AAS.A- Here, As.A is a family of probability distributions induced by a distribution A on feature matrices A such that
||A|| S as follows: We assume that a Borel measure μd is defined by a probability distribution A
on matrices Rmxmm We then define Borel measures μ4; on RT by [c(C)) M Aœ(x;) C})
for Borel sets C C R". Then X; is the probability distribution defined by (4s. We fix the distributions i/ du and denote the set containing all families of distributions (Aç, that can be generated this
way by 2g.. The following result is a direct consequence of Equation [3 and our Definition 9
Theorem 10. Let f(x)
(e) (z) be model composed of functions Φ X - Rm and 3 : Rd Y. If is is (16, S, A), featuue robust for all ||A|| S and Ø(S) is (c', 24.) represaareee
for some H containing v. then the generalization error of bounded by Egen(J,5) Seté'.
Hence, for generalization we need a model that is Teature-robuse and training data that is sampled densely enough. In the trivial case with .A 6, the distribution with full weight on the O-matrix, we can choose 6 = O to obtain e = O and Egen S e'. The more feature robust model is, the larger á we the flexibility of choosing nontrivial A to lower the bound on representativeness and therefore the generalization error. We hope that in future work it will be possible to find suitable distributions A that lead to computable generalization bounds.
can consider
EMPIRICAL EVALUATION
In this section we empirically validate the practical usefulness of the proposed flatness measure. A correlation between generalization and cusssian-based flatness measures at local minima has been observed previously, but the results Dinh et al. (2017) questioned the usefulness of these measures. We show that our measure does not only overcome the theoretical issues, but also preserves the strong correlation with the generalization error. Previous works mostly use accuracy of the trained model on the testing dataset (Rangamani et al. 2019 Keskar et al. (0066) for evaluating the generalization properties of the achieved minimum. Nevertheless this does not directly correspond the theoretical definition of the generalization eиror [I For measuring the generalization error, we employ Monte Carlo approximation of the target distribution defined by the testing dataset and measure the difference between loss value on this approximation and empirical error. In order to track the correlation of the flatness measure to the generalization error, sufficiently different minima should be achieved by training. The most popular technique is to train the model with small and large batch size (Rangamani jet al. 2019 Keskar et al., 2016 Novak et al., 2018 Wang et al. 2018), which we also employed.
A neural network (LeNets (LeCun et al.)) is trained on CIFAR10 10 multiple times until convergence with various training setups. This way, we obtain network configurations in multiple local minima. In particular four different initialization schemes were considered (Xavier normal, Kaiming uniform, uniform in (-0.1, 0.11,, normal with μ = 0 and o² 0.1), with four different mini-batch sizes (4, 32, 64, 512) and corresponding learning rates keep the ration between them equal (0.001, 0.008, 0.02, 0.1) for the standard SGD optimizer. Each of the setups was run for 9 times with different random initializations.
Here the generalization error is the difference between summed error values on test samples multiplied by (since the size of the training set is 5 times larger) and summed error values the training
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90000
90000
00000
50000
00000
50000
10000
12000
90000
V1000
20000
20000
weight
weight
LeNet5 characteristics after training on CIFARIO. Each color corresponds to different setup of training, characterized by initialization strategy, mini batch size and learning rate. The setups are ordered in ascending order by the mini batch size, with the largest corresponding to the brightest color of the displayed points.
Figure 2:
0.05
-0.34
-0.03
-0,02
-0.01
0.00
0.01
0.02
90000
With weight norm
0.91
Platnees x 8//2
90000
Without weight nom
Rebustness
60000
50000
30000
20000
10000
p- 0.64
rExperiment
weight
Figure 3: LeNet5 configurations trained on CI- FARIO with random reparameterizatioos. The cor- relation stays the same for the proposed measure, while it breaks for classic Hessian-based measure.
Figure 4: Robustness and flatness for LeNet5 con- figurations trained on CIFARIO. Results ordered by flatness, showing that robustness is bound by our flatness measure.
examples. Figure[]shooe the approximated generalization error with respect to the flatness measure (for both se and KTr with / corresponding the last hidden layer) for all network configurations. The correlation is significant for both measures, and it is stronger (with p = 0.91) for K.,,. This indicates that taking into account the full spectrum of the Hessian is beneficial. To investigate the invariance of the proposed measure to reparameterization, we apply the reparameterizztion discussed in Sec. 4 all networks using random factors in the interval 25). The impact of the reparameterization on the proposed flatness measure based on the trace in comparison to the traditional one is shown in Figure 3 While the proposed flatness measure is not affected, the one purely based on the Hessian has very weak correlation with the generalization eπror after the modifications. To verify the relation described by Equation 6 we also compared feature robustness with 6 = 0.001 and feature matrices A that have only non-zero value 1 the diagonal. Figure 4 shows that up to outliers the robustness is bound by the flatness measure. Additional experiments conducted on MNIST dataset described in Appendix E where obtain correlation factors between the generalization error and tracial flatness nly, of 0.73, 0.70, 0.72, 0.71 for the network's 's hidden layers 1 1,2, 3, respectively.
7 DISCUSSION AND CONCLUSION
We established theoretical connection between flatness, feature robustness and, under the assumption of representative data, the generalization error. The relation between feature robustness and Hessian- based flatness measures has been established for w', which takes into account the maximum eigenvalue of the Hessian, and w'y,, which uses the trace instead. Empirically, the measure siy, based on the trace of the Hessian shows stronger correlation with the generalization error. This surprising,
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since it takes into account the whole spectrum of the Hessian and every eigenvalue corresponds to 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 E Om- It is interesting to note that strong feature robustness does not exclude the possibility of adversarial examples, first observed by |Szegedy ct al. (2013). since large changes of loss for individual samples (i.e. adversarial examples) may be hidden in the mean in the definition feature robustness. Appendix C.2 we briefly discuss the freedom of perturbing individual points by suitable feature selection matrices A.
In contrast existing measures of flatness, our proposed measure is invariant layer-wise repa- rameterizations of ReLU networks. However, we note that other reparameterizations are possible, e.g., we can use the positive homogeneity and multiply incoming weights into a single neuron by positive number À 0 and multiply all outgoing weights of the same neuron by 1/a. While the Fisher-Raa Rao norm suggested by [Liang et al. (2019) invariant to such reparameterizations, our proposed measures of flatness xf and Kry, are in general not. In principle, variations of our flatness measures can be found that are invariant to such reparameterizations as well (see Appendix but their analysis, except for some empirical evaluations in Appendix E is left for future work.
The second term the generalization bound of Theorem 10 is given by our notion of representa- tiveness. In order to find specific bounds for the e-representativeness of (S, 2s,), distribution over matrices is required that induces a distribution which is similar to localized kernel density estimation (KDE). While notion of representativennss is generalization of classical representativeness, it remains open whether it is efficiently computable. The more feature robust a model is, the more free- dom there is finding specific distributions over matrices that lead 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 eтror. We theoretically investigated this connection by relating this measure to feature robustness. This notion of robustness, together with a novel notion of representativeness provides 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.
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A PROOFS OF MAIN RESULTS
A.1 PROOF OF EQUATION
First note that for ||A|| I 1,
W1
w A A
W2
|
A
Wm
F
Wm
(15)
m
m Σ
|wi AIl v
|w,112
1=1
||w||| F-
From and 5
we get
82 (w.A, HEemp (W.: 5) (w. + O(5))
max ||||S
F(6, S, A)
max
||
15
8² z" H (w.. S)z 0(68))
max
|| w.llr 2
5²
/|wrl1p- zJ S))
max
||
2
8²
|w.ll
> (w.) ) + (
2
where we used the identity that |=X|== Mr XM for any symmetric matrix M.
A.2 PROOF OF THEOREM 5
In this section, we discuss the proof to Theorem 5 Before starting with the formal proof, we discuss the idea simplified setting separate the essential insight from the complicated notation in the setting of neural networks.
Let F, F: Rd + IR denote twice differentiable functions such that F(w) = (Aw) for all w and all X > 0. Later, will correspond to weights of specific layer of the neural network and the functions F' and F will correspond respectively to the neural network functions before and after reparameterizations of possibly all layers of the network. We show that
1
X2 H(F(w)) = HFW())))
Indeed, the second derivative of F Aw with respect coordinates wi, wy given by the differential quotient as
0² F(ww)
F(Jw + he, + hee) F( + hes) - F(w + hee) + F(Aw)
lim h-40
Ow,dwy
h2
F(A(W $ee+ + (ez)))))) F(A(w + (ee))) F(A(10 (ee)) + F()w)
lim h-+0
(4)²
( x2
1 lim
Flw + fez + ke,) - F(w + fes) F(w + (ee))) F(w)
x h0
8² F 2 Ow,dry
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Since this holds for all combinations of coordinates, that HF((w) 1/>²HF(w) for the Hessians of F and F, and hence
1 | (Hol[(HF(Au)) 12110012 HF(() = || (lu
x
Formal Proof Theorem We are given a neural network function s(z; W1: W2..... Wz) parameterized by weights of the i-th layer and positive numbers Ag...., Xy such that f(æ; W1, W2,..., WL) = f(x; A1 A2w2.... ALWL) ) for all w, and all æ. With w defined by (W1, W2....; WL), w Aow and w (w/. w w2..... w2)), we aim show that
k(() n² (w^).
where x² (w) wwll 112 XH (Wr) is the product of the squared norm of vectorized weight matrix wi with the maximal eigenvalue of the Hessian of the empirical error at w with respect to parameters Wi-
Let F(u) Eimees ([((;; W1, w2...., u............................... wL), v) denote the loss Further,
function
the parameters of the I-th layer before reparameterrization Einies (f(2;; w w....., v...., wE).9) y) denote the loss as a function on the parameters of the I-th layer after reparameterization. We define a linear function r by n(u) = uuu. By assumption, we have that F(r(w))) = F(wi) for all wi By the chain rule, we compute for any variable ((.5) of
let F(v)
u,
(F
OF OF(n(u)
Ou(t.))
Oult.)
(F))
((n(u)((.m)
M (k.()
(n())=((w))
Ou((.))
n(u)=n(w))
km
(F
A
On(ta) v-=ww
Similarily, for second derivatives, we get for all 8,
FFF(u)
dF(v)
X
Oult.) (ulla))) n=w;
Ovll)))n(s.0) -=
Consequently, the Hessian H of the empirical error before reparameterizztton and the Hessian H after reparameterizatioo satisfy H (w,, S) Xp H (Aow,, S) and also H.J (wi) = X3 H. (Aww)). Therefore,
re' (w) ||wr|| (H (wi)
||w/
HI. (Aww)) || 2 H. (Arwi) re (w(").
A.3 PROOF OF THROREM 6
Proof. (i) This is just a corollary of Theorem 2 using the trivial bound that the maximal eigenvalue is bounded by the unnormalized trace (sum of eigenvalues) for positive semidefinte matrices (where all eigenvalues are positive).
(ii) We consider the set of orthogonal matrices E Om as equipped with the (unique) normalized Haar measure. (For the definition of the Haar measure, e.g. Krantz & Parks 2008).) We need to
show that E E_ [F(8, S, A)] <
62 /w.ll2 Tr(EEemp (w.)) + O(83) with F(8, S, A) defined
as in Using and we get, similarly to (6)
82 (w.,, FEemp(w... S) (w. + 0169)
EA-O,
F( A)) BA-O..
2
with (,.)) the scalar product with vectorized versions of
WI-AA
W. 1 =
A =
w.. E ERIxm.
W Wdd
W A
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We consider the vectorization of A E Rdm given by (Wix...., Wa.)". By Lemma 11 below, get
8
EAmOm. S, S EamO..
(W) 0
2
4-=1
(16)
8
d
Am (Ww. A)² + 0(69)
2
Here, the notation H Eempl (w.., S) refers to the empirical error w. but the derivatives are only taken over the parameters in the row Wj.-
If W:. o O, then by Proposition 3.2.1 of Krantz & Parks 2008) and the change of variables formula for measures, we get
Ba~Om
(W (w...A))) = | EzeRm ||=||=1 [x" HE (W,, (17)
for all 1 I i.j I d, where the latter expectation is taken over the normalized (uniform) Hausdorff measure over the sphere 5 R Now, using the unnormalized trace Tr([hus)) >3 h4
compute with the help of the so-called Hutchinson's trick:
_==x=",|=]==1 [2_ HEemp(W, 8)z] = B| [T((2 Hempp (Wjm, S)z)
B||+||=1 [Tr(HE (Wj.: 5)22² 2)
(18)
TT((HEemp (W 5) B||=||=1 [2=² "].
Note that 22 = [=:2yla,j and due to symmetry B||=||=1 [ziz]] B|ja||= [2 [2)] for i + j.
hence =||=||=1 [2,2,] 0 whenever i * j. Further B||a||=1 [2]] || Ei=1 2,
Im constant multiple of in |||=|== |||=||]] m for all i. Therefore [|]=||=1 [2z"]
the identity matrix. Putting things together we have
8²
EA-O,, A)
BA-Om. [(win (W) 0
S)(:.. O(59)
2
+/=1
8²
||wia||
2 1 [> ()8 O(5*))
2
11
B
82
||
1
m
1J=1
82
>
d
Σ 1
wioll
Tr Tr(HEemp(W,,..)) + 0060)
2
8²
(1[w-112)-
(-r(HLEemp (W, 0(8))
L
2
m
82
/w. Tr(HHeeppw. (We, 8)) + 0
2m
Lemma 11. (i) Let H = [Hispliss be a positive semidefinite matrix in 122mx22 that consists of E T22m with xa E Rmm we have
submatrices Hi. E Rmmxm S i,j 2. 2. Then for all x =
I1
322
2x H1 S x H2221 + + H1
(ūi) Let d,m E N and H = [H.j be positive definite matrix in ddd that consists of
submatrices Hig E mm 1 5 i,j < d. Then for all x = (Z1...., Id) E ggd with I; E R we have T. Hx < Eij=1 1/j Hr,
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Proof. (i) By definition, H is positive semidefinite if (H is symmetric and) H 2 for all Choosing 2 = (-22, 21) gives z.2 H1 + 2 H2,2 - 2x H1, 20² 2 0, hence 2x 21,242 S
zi 22 + 22 22
(ii) Using that every submatrix Ha,b = (
Haa H
Ha.b Ho,b
)
is positive definite together with (1), we
obtain
x² Hx
Σ
T3 HH, +
Σ 2xl Hag
inj
Σ
ti H4 + Σ
[x] HJ + 2 M( = Σ
1
A.4 PROOF OF THEOREM 10
We are given function f(x) (v (xx) By assumption, f is ((6, S, A), e) e)-fttuuee robust for all matrices |||||
S 1, which implies that
Σ
[e(v(((x;) + &Ab(1:)), e(f()))) w.)] se for all ||||| S 1.
(19)
[S|
ES
Further, we are given that ((S) (e', 21.) repressteeett for hypothesis space H such that H. By Definition 9 (ii) this means that there is some As.A = (Ar, 21 such that (S, is é'reepeseenaatvve for sb. That is, by Definition 9 (i).
B( ((..)~<((((())...))) (([(()) v)
1
Σ EE (((((((())) + Ex), vi)) Sc'
(20)
JS|
(m..:.)E
Since As.A (Ar, do)s Ass there exists probability distribution A of matrices ||A|| S 1 (so that | S 6) such that
Σ
BE,~d, [((((();)) + (x).
EA-A (((b((((r)) + oAb(x:)) 9:)
S
(ESN))S
BANA
Σ
((v(ó(x;) + 6A((2:)).
S
(21)
Putting things together, we get for the generalization error Egen (J, of model f.
Egen (f.)) ED [[(f(z).
Σ
{(f(xz)) y:)
S
(#S.))ES
E(D [e(f(x).
B [e(v(((xx)) 22).
[
+
E EA-A And
1
Σ
{(v(o(a;) + &Ao(a;)), vi)
Σ
e(f(x))
|S|
1St
=D [e(f(z).
EZ (e(v((((z))) + 2x), vs)
S
EA~A
1
Σ
[((p(((z)) + 5Ab(x)) -
S
(xs.8))S
20
é' +e
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B ADDITIONAL MEASURES OF FLATNESS
We present additional measures of flatness we have considered during our study. The original motivation study additional measures was given by the observation that there are other possible reparameterizations of fully connected ReLU network than suitable multiplication of layers by positive scalars: We can use the positive homogeneity and multiply all incoming weights into single neuron by positive number À > O and multiply all outgoing weights of the same neuron by 1/x. Our previous measures of flatness xf and , are in general not independent of the latter reparameterizations. We therefore consider, for a layer I of size n,, feature robustness only for projection matrices F. E IR having zeros everywhere except a one at position (j.j). At local minimum w, of the empirical error, this leads to
5² Cemp(Wr Wix + Ew Eempl (Wi,, S) wI~ (Wem)(Ww))))))))))))))))
O(69)
where (i) denotes the j-th column vector weight matrix wi layer I, and we only consider the Hessian with respect to these weight parameters. We define for each layer and neuron j in that layer flatness measure by
d())(w,)
(E (Wi, (i),(i))
For each I and this measure invariant under all linear reparameterizations that do not change the network function. The proof of the following theorem is given in Section B.I
Theorem 12. Let f = (WW1.W2..... WL) denote a neural network function parameterized by weights We of the i-th layer. Suppose there are positive numbers x(x-))
".... such that the products w² obtained from multiplying weight wi (b.5) at matrix position (i. in layer I by x
satisfy that f(W1, W2,...,Wip WL) = s(w), w for all wi Then pf (j)(w) = =p' (i) for each j and l.
We define measure of flatness for full layer by combinations of the measures of flatness for each individual neuron.
p'(w.) ::y.)=maxx//) (w.)). and (w.)) )) Σd
Since each of the individual expressions is invariant under all linear reparameterizations,, so are the maximum and sum.
2 we get an upper bound for feature robustness for projection matrices Ej- Theorem 13. Let f denote a neural network function of a I--layer fully connected neural network. For each layer I, 1 I L L of size ny let E3 E 1 denote the projection matrix containing only zeros except I at position (j.j). Let Wis denote weights of the I-th layer at a local minimum of the empirical error.
Analogous to Theorem
Then the neural network is ((6, S, E,), 8 (w.)) + O(62)) feature robust for all j at w..
One Value for all layers Our measure of flatness are strongly related to feature robustness, which evaluates the sensitivity toward small changes of features. In good predictor, generalization behavior should correlate with the amount change of the loss under changes of discriminating features. For neural networks, measure is then related by Corollary 13 to changes of the features of the I-th layer. It is however clear that low value of k! for a specific layer alone cannot explain good performance. We therefore specify a common bound for all layers.
consider the output of each layer as feature representation. Each flatness
Denoting by w. the of weights from all layers combined, have l| l|r S ||w./lr for all I. Further, if (l) denotes the Hessian of the loss with respect to only the weights of the 1-th layer, and (t).
H the Hessian with respect the weights of all layers, then Amaz (w!) I XH (w.). (This holds since
X(A) = max TT Au and (v.0)T ||||==
U
= T Av.)
DT
B
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Table Hessian based measures of flatness
Notation
Definition
One value per
Invariance
11122 maz
network
x!
/w/l2 2 H.d (Wi)
layer
layer-wise mult. by pos scalar layer-wise mult. by pos scalar layer-wise mult. by pos scalar layer-wise mult. by pos scalar layer-wise mult. by pos scalar layer-wise mult. by pos scalar all linear reparameterizations all linear reparameterizattios
|w]2
W(,S)) S))
layer
kmaz
maxi (w)
network
k²
Elsr (w)
network
kmaz
maxi wiy (w)
network
RTr
Els (w)
network
((())
wi(s)² HEemp( S)w,(()
neuron
d
max, of (w)
layer
d'
E, E3(((i))))
layer
linear reparameterizations
pma
maxi (w)
network
all linear reparameterizatioos
P
Po
Po
network
all linear reparameterizatiioss
Therefore, matter which layer with activation values o' (zx) for each E S we are perturbing with matrix ||Ai|| S to o' (xx) + dAi o' (x,), we have that
5² F(6,S,A) 2 |/w.112- Aman(w.)) 0
and s² w.) = ||w,|8 113.1
A moo (w.) can be considered as common measure for all layers.
However, x( (W,) not invariant under the reparameterizations considered in Theorem We therefore consider more simple common bounds by combinations of the individual terms k e.g. by taking the maximum of кд over all layers, maa (w.) max; s² (w.). the sum (w., ΣL_ n² (w.)..
Since each of the individual expressions are invariant under linear reparameterizations of full layers, so are the maximum and sum.
Finally, we define m (w.) == max (w.) and p=(ww) = Σt P (w.).
Table T summarizes all measures of flatness, specifying whether each measure is defined per network, layer or neuron, and whether is invariant layer-wise multiplication by a positive scalar (as considered in Theorem or invariant under all linear reparameterization (as considered in Theorem 12)
PROOF OF THEOREM
As in Subsection A.2 first present the idea in simplified setting.
For the proof of Theorem we need to consider the case when we multiply coordinates by different scalars. Let R²2 + IR denote twice differentiable functions such that F(v, w)) = F(v,, pm)) for all E TR, E TR and all X, n 0. In the formal proof, v, w will correspond to outgoing weights for specific neuron, while again F and F correspond network functions before and after reparameterizations of all possibly all weights of the neural network. Then
(v, w) HFF(,, w)
v
(De, H. (AU, (Nw,)w))
Au
w
for all u and all X, p.
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Indeed, the second derivative of F at (Au, pw)) with respect to coordinates v, is given by the differential quotient as
0 F()u, lim F(Au + pw + ke) F(Au + h, pw) F(Au, waw + K) + F(Au, w) 1.
Ovdw
hk
F(M(++ 2), h)) F(A(U+ 2. (.))
F(XU, $)) - F(AU,
lim
h
($)
Ap.
P(u+ x. w -- 4) FF + 4. w) - F(U, w + :.) + w(v,w)
1 lim
Ap h,k-0
h K
1 ((,
Ap
Ovdw
From the calculation above, we also see that
S FA..
FFF(
8 F(v,
1
1.FF(U, w)
and
vvdv
A dvdv
Owdw
112 Owdw
follows that (v, w)
8 F(F FFF(u,w)
O FeF( w)
HF(e, w)
v
2ow
+ w²
w
dvdv
drdw
dwdw
(Au)2 02 F(
OFF(U, w) Ovdw
F(, w) Owdw
+ 2(Au)(ur))
+ (wm)
Oodc
( Av
= (w)) HF((U,
Formal Proof of Theorem 12
We are given a neural network function f(z; W1, W.,..., WL)
parameterized by weights w, of the i-th layer and positive numbers x(x.3) ,..., (6.5) such that the products w obtained from multiplying weight (f.)) at matrix position (i,j) in layer I by x(xa)) satisfies that (a;; W1, W2. WL) f(z; w w.....,,w! .w) for all w, and all æ. We aim to show that
(
each j and where ('))) = wi( HE (j) S)w; wi()) denotes the j-th column of the weight matrix at the [-th layer and HEemp(Wi (J)), denotes the Hessian of the empirical error with respect to the weight parameters in wi()). Similar to the above, denote by (i) the product obtained from multiplying weight wi(f): w/ w((()) at matrix position (i,j) in layer / by (
The proof is very similar to the proof of Theorem only this time have take the different parameters X (t.f) into account. For fixed layer I, we denote the j-th column w/ and wr()).
Let
F(u)= = Σ (ff(; w2, (w), 1), wi + (n)]],
(a.y)es
wz).y)
denote the loss as function on the parameters of the j-th column in the 1-th layer before reparame- terization and
f(v) = Σ
e(ss;; wA,, w "00,..,,(w1... (wi(1))
- 11A,,, v, wi
+ ())))....((()))
WL (C)))))))
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denote the loss function on the parameters of the j-th neuron in the I-th layer after reparameteni zation.
We define linear function 17 by
n "7(11, .nn)) = ru 2/11) (1.9) u2>/2.))
(2,)) ..... Un A ((n.))).
By assumption, we have that F(n((:()))) = F(wi(i)) for all wi()). By the chain rule, we compute for any variable u, of u.
@F(u)
of (F((()))
Ou:
luw((i)
bu
(=
OF (f(())
a(p(u)))
a(n(u)&)
v(()=n((w(1))
Ou,
v(()==((w(6))
aF(v)
Se I))
Similarily, for second derivatives, we get for all i,
F2)(u)
\1 (4-) (aJ) 0F(v) au,āu, v)))(()
Ou,uu, u.)
Consequently, the Hessian H of the empirical error before reparameterization and the Hessian F after reparameterization satisfy that at position (i, s) of the Hessian matrix,
HF(())(..) x/x20 ((.)) (LF(w))(.))
Therefore,
d'()(ww) = wi())² HF(ww) wi())= = Σ
(1:5) (5.5) wi w HF(W))(())
Σ wi (i:1) (5.5) (00r ((5)) HF(w))
(110) ( HF(w))
(Wi()) HF(W?) w((f)= - (1)(
C ADDITIONAL PROPERTIES OF FEATURE ROBUSTNESS
C.1 RELATION TO NOISE INJECTION AT THE FEATURE SPACE
Feature robustness is related to noise injection in the layer of consideration. By defining a probability measure PA matrices A E gmx of norm ||A|| < 1, we can take expectations over matrices. An expectation over such matrices induces for each sample x E X an expectation over probability distribution of vectors ç E R with ||E|| < [| We find the induced probability distribution Pa from the measure Pa defined by Pa (T) = Pftff Ao(x) E T)) for a measurable subset T C R" Then.
EA-PA (F(6,S,
EANPA
Σ ((b(6()) -(
S
(2:2)ES
Σ
Ec_p, [((b(((x)) -(f())))))
(x.N)ES
The latter is robustness to noise injection according to noise distribution P,, for sample x in the feature space defined by d.
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Under review as a conference paper at ICLR 2020
C.2 ADVERSARIAL EXAMPLES
Large changes of loss (adversarial examples) can be hidden in the mean in the definition of feature robustness. We have seen that flatness of the loss curve with respect to some weights is related the mean change loss value when perturbing all data points z; into directions Ar; for some matrix A. For a common bound over different directions governed by the matrix A, we restrict ourselves to matrices ||||| 2 1. One may therefore wonder, what freedom of perturbing individual points do we have?
At first, note that for each fixed sample Tio and direction Eig there is matrix such that Ar = Zio each direction for each datapoint can be considered within bound as above. We get little insight over the change of loss for this perturbation however, since a larger change of the loss may go missing in the mean change of loss over all data points considered in the same bound.
The bound involving k(w,) from above does not directly allow to check the change of the loss when perturbing the samples 2; independently into arbitrary directions... For example, suppose have two samples close to each other and we are interested in the change of loss when perturbing them into directions orthogonal
each other. Specifically, suppose our dataset contains the points (1. O, O.... 0) and (1, ...... 0) for some small 6, and we aim to check how the loss changes when perturbing (1, 0, 0, ..... 0) into direction (1, O, ........ 0) and (1, O, ................................ 0) orthogonally into direction (0, 1, .......... 0). To allow for this simultaneous change, our matrix A has to be of the form
A
0
:
0
Then
0
1
v2
|||| 2 ||A
0
|1
0,
))
Hence, our desired alterations of the input necessarily lead to large matrix norm ||A]| and our attainable bound with []A]])>(w..) becomes almost vacuous.
C.3 CONVOLUTIONAL LAYERS
Feature robustness is not restricted fully connected neural networks. In this section, we briefly consider convolutional layers w * Using linearity. we get w *(x + 8x) = (w + 5w) r. What about changes (w Aw for some matrix A? Since convolution
linear function, there is matrix W such that w * 2 = Wr and there is matrix WA such that w.A * = WAT. We assume that the convolutional layer has full rank, so that there is matrix V with WV Tm Then
caaessiinnalit--eddccin W E Raxm m < n and that the matrix W
(w 5w Wr + SWA A Wx SWVW W(x SVWBz).
As consequence, similar considerations of flatness and feature robustness can be considered for convolutional layers.
'This holds for example for convolutional filter with stride one without padding, in this case W has Toeplitz submatrix of size (m m).
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Under review as a conference paper at ICLR 2020
D A LINK BETWEEN C-REPRESENTATIVENESS AND KERNEL DENSITY ESTIMATION
Compared to classical representativeness, the definition of e-representativeness is far more general, allowing the choice of family of distributions A (A,, 14 V suitable restriction to consider only local distributions X; and V, centered around the origin O. With this, the following connection to kernel density estimation can be established: If a distribution can approximated with emror by kernel density estimation then this sample is representative.
Proposition 14. (i) If a distribution D can be e-approximated by a Kernel Density Estimator with Gaussian kernels and a diagonal bandwidth matrix using sample S of size N E N and if the loss :: y RR is bounded by L, then for any such sample S there some Ax such that (S, Ax) Le-representative for f with respect to 'D and f. (ūi If the probability density function of D lies inside a d-dimensional kernel Hilbert space with Gaussian kernel Kt, i.e. PD(x, 4) E Ha, then (S, Ax) is Le-representatiee with e E EO (N-
Before we proof this proposition it is important to note that this result-in its current form form-cannot cannot
be used obtain generalization bound using Theorem 10 In Proposition 14 A (Az pt) is
chosen such that Pexxxe.) Ka. (2),). where K₁ denotes the Gaussian kernel. Theorem 10 requires
the distribution to be induced by a probability distribution A on feature matrices A with ||| <
However, since Gaussians have support everywhere, the assumption that || < 8 for any finite ō
does not hold. A possible solution would be to use truncated Gaussian kernels, for which ||||| S can be ensured. However, it remains an open question whether there exists probability distribution A over feature matrices A that induces suitable truncated Gaussian distributions which would allow to compute practical bounds on the generalization error.
We now provide the proof to Proposition 14
Proof. Given a sample S ~ with | = N, its representativeness is defined as
Rep(S)- = (a. [e(f(z),
((Er) [e(f(x; + Ez), us + Ey)
N
(RS.N)ES
We can rewrite ((x,)))D (e((2)),)) v)] as
[ ((W))XX (f(x), (oox).)))
/ (())Po(2)dz
B(z (E((())
2EZ
where we abbreviate X' x x-= 2, (z, v). and with slight abuse of notation write {(f(x),)) (x y) = ((z) for fixed f. Furthermore, since xe and v4 are independent we can rewrite
Σ (+
(ss.s (([(= Ex), wi + Ev)l
Σ
I
(((x+++)), Pa.xn) Ev)
N
(a..))s
1
Σ/ E(2 +)) P(axx)((Edg
N SEZ 4,5
By assumption, a Kernel Density Estimator on sample S, i.c.
1 Σ Kn(z 2z)
P(z) = N
Z
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Under review as a conference paper at ICLR 2020
with kernel Kn, approximates Po(z) with approximation error Thus, get that
/ ZE2
(x))PD(2)dz
Σ
{
(z + ()P
N
KEE
/
((=)
Σ
l(zi + E) (xxxx,
+ max (f(=)) .E2
N
=E2
2.E
SEZ
{(2)
1
Σ
Kn(2 21)2
Σ
/
A2 + ((
+ +emax (E(2)) 2E2
JEEZ
2,58
2:5 FEE
1
Σ
(())K<(z - 21)d 21)d2-
J
Ea + E) xv, (E) + tel.
N
RES
2E2
By substituting Ç = = zi and choosing the (A, x u;) such that ((xxxn)(2) = K (z) (which is possible since we assumed the bandwidth matrix to be diagonal), we can further rewrite this as
/
{(z)Kn(z 21)2 [ E(2s + +
SS S N
Σ 2
teb
2EZ
JEEZ
e(s + k))ds() [ E(2 + ()),((x)))()) t
ÇEZ
JEEZ
e(s + 21) (Kr((S) P P(anee) ÇEZ
dc + eL
2:
=0
the probability density function of DD lies inside a d-dimensional kernel Hilbert space with Gaussian kernel Kn, i.e., Pp(x,, v) Ha. then it follows from Theorem 4 in Fasshauer et al. (012) that
V2 (
E c
C c
1+
n n
2
2n
1
E ADDITIONAL EXPERIMENTS
In addition the evaluation on the CIFARIO dataset with LeNet5 network, also conducted experiments on the MNIST dataset. For learning with this data, we employed custom fully connected network with ReLU activations containing 4 hidden layers with 50, 50, 50, and 30 neurons correspondingly. The output layer has 10 neurons with softmax activation. The networks were trained till convergence on the training dataset of MNIST, moreover, the configurations that achieved larger than 0.07 training error were filtered out. All the networks were initialized according to Xavier normal scheme with random seed. For obtaining different convergence minima the batch size was varied between 1000, 2000, 4000, 8000 with learning rate changed from 0.02 to 1.6 correspondingly to keep the ratio constant. All the configurations were trained with SGD. Figure [ shows the correlation between the layer-wise flatness measure based on the trace of the Hessian for the corresponding layer. The values for all four hidden layers are calculated (the trace is not normalized) and aligned with values of generalization error (difference between normalized test error and train error). The observed correlation is strong (with p 2 0.7) and varies slightly for different layers, nevertheless it is hard to identify the most influential layer for identifying generalization properties.
We also calculated neuron-wise flatness measures described in Sec. B for this network configurations. In Figure 6 we depicted correlation between P and generalization loss for each of the layers, and in Figure 7 between p' and generalization loss. The observed correlation is again significant, but compared to the previous measure we can see that it might differ considerably depending on the layer.
The network-wise flatness measures can based both on layer-wise and neuron-wise measures as defined in Sec. B We computed x. x.me,, xif pmaar and p and depicted them in Figure 8 Interesting to note, that each of the network-win wise measures has larger correlation with generalization loss than the original wese and layer-wise measures.
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Under review as a conference paper at ICLR 2020
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uqu squartt wenght norm
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Figure 5: Layer-wiss flatness measure calculated for MNIST trained fully connectdd network. Four plots correspond to four hidden layers the network. For each of the layers a strong correlation with generalization error can be observed.
0.34
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Generalbakrnn
0.06
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wie mossure
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wtweettooonn 6014
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WGS moasure
Figure 6: Neuron-wise flatness measure Pl_ calculated for each of the hidden layers for the fully- connected network trained on MNIST dataset. Each plot corresponds layer.
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Under review as a conference paper at ICLR 2020
0.24
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Ceneratzneto en707
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25
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Nouron wise mezzare
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Genee www.eetttttoon 4004
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(P 0.35
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Figure 7: Wisron-wise flatness measure p' calculated for each of the hidden layers for the fully- connected network trained on MNIST dataset. Each plot corresponds layer.
0.14-
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T7300 x squared weight norm for nobworkk 6T
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Neurnn Milie meature for network gr
Figure 8: Network-wise flatness measures based on various neuron-wise and trace layer-wise mea- sures for the fully-connected network trained MNIST dataset.
24


Reviews:
[{"review_id": "H1e9HffWcH", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes a notion of feature robustness which is invariant with respect to rescaling the weight. The authors discuss the relationship of this notion to generalization.\n\nThe definition of feature robustness is interesting and could potentially be useful. However, the paper has the following major issues:\n\n1- Related work: It seems that authors are unaware of the related work in this area. There are many relevant work in this area that connect feature or weight robustness to generalization look at [1,2,3,4] for some examples. I suggest authors to do a comprehensive literature review.\n\n2- Theoretical results: The theoretical results presented in the paper have very limited value. For example, authors fail to really connect their suggested measure to generalization in any meaningful way. Instead they end up decomposing the test error to the sum of their robustness measure and the gap between robustness and test error which is trivial. I suggest authors to look at the literature on PAC-Bayesian and compression-based bounds to connect their suggested measure to generalization.\n\n3- Experiments: The experiments are not really convincing. The empirical results show that the suggested measure can correlate with generalization when training with different batch-sizes. When varying other things, the measure is not really correlated. Therefore, this is not any better than the version suggested by Keskar et. al. Moreover, the experiments are very limited and I suggest authors to look at more controlled setting to verify the relationship of their measure to generalization. Also, when looking at the generalization, it is important to set the stopping criterion based on the cross-entropy instead of number of epochs.\n\n[1] Dziugaite and Roy. \"Computing nonvacuous generalization bounds for deep (stochastic) neural networks with many more parameters than training data\". AAAI, 2017.\n[2] Neyshabur et. al. \"Exploring Generalization in Deep Learning\", NeurIPS 2017.\n[3] Arora et. al. \"Stronger generalization bounds for deep nets via a compression approach\". ICML 2018.\n[4] Wei and Ma. \"Data-dependent Sample Complexity of Deep Neural Networks via Lipschitz Augmentation\", NeurIPS 2019.\n\n****************************\nAfter author rebuttals:\n\nAuthor have added discussion of related work which was missing in the original submission (thanks!). However, the other two issues are still present. On the theoretical side, I think the major issue is that the paper cannot connect the measure to generalization properly and ends up decomposing the test error to the sum of the robustness measure and the gap between test error and the robustness measure which is not informative. However, this would have been still interesting if the measure could go beyond other empirical measures. Unfortunately, the correlation only happens in case of changing batch size (and learning rate which we know is empirically equivalent to changing batch size) and therefore cannot go beyond what is shown in Keskar et al. 2017. Therefore, my evaluation remains the same.\n\n\n", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "H1e4eAOm9S", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper describes a connection between flatness of minima and generalization in deep neural networks. The authors define a concept called \"feature-robustness\" and show that it is related to flatness. This is derived through a straightforward observation that perturbations in feature space can be recast as perturbations of the model in parameter space. This allows the authors to define a (layerwise) flatness measure for minima in deep networks (this layerwise flatness measure is also invariant to rescalings of the layers in neural networks with positively homogenous activations). The authors combine their notion of feature robustness with epsilon representativeness of a function to connect flatness to generalization. They present a few empirical evaluations on CIFAR10 and MNIST.\n\nI believe this paper is able to once again confirm the relationship between flatness and generalization in an empirical manner with their layerwise measure of flatness. I am not so convinced about the theoretical justification that they claim to provide and thus do not recommend acceptance.\n\nTheory - The key theorem relating generalization and flatness is Theorem 10 which says that if a compositional model is feature robust and the output of the first component is an epsilon-representative for the second component, then the compositional model will generalize. While this is interesting, it is not clear to me that this guarantees generalization for deep neural networks. This result only talks about feature robustness and representativeness for a particular layer. If a deep network has many layers, will the feature robustness layers closer to the input guarantee feature robustness at deeper layers? That might require a further unit operator norm constraint on the layer operator, which is a restriction on the types of weights that can be used. If a sample is epsilon representative at one layer, what is required for the next layer to be epsilon representative for the rest of the deep network? This seems to be a missing step in relating flatness/feature robustness of a layer to the generalization of the whole network.\n\nAnother idea that I think arises from Theorem 10 is that the flatness of loss landscapes is important when you have learning problems where the hypothesis class is compositional. While flatness is only spoken of in the case of deep neural networks, can we identify the same phenomenon in other problems? I would encourage the authors to try and identify another model in which the flatness-generalization relationship exists (even empirical evidence would suffice for now). This would strengthen the case for studying flatness and biasing optimization towards flatter solutions in the case of deep networks.\n\nExperiments - This section seems to be pretty rudimentary, I would like to see more results on different kinds of network architectures (VGG? Inception? AlexNet?), more datasets (KMNIST? Fashion MNIST? SVHN?), and possibly more repetitions. At one point the authors mention that they declare a minimum has been reached if the training loss is < 0.07. Atleast on CIFAR10 and MNIST it is possible to achieve training loss <1e-4 so am not sure if the networks that the authors are testing are minima at all (It is important for them to be minima since the flatness measure is only defined at minima). Can the authors also identify more situations other than large batch vs small batch training that would lead them to obtain flatter/sharper minima?\n\nThe authors also claim that measuring generalization using test error is flawed, but do not provide details about their method of measuring generalization. I would want to see these details and a more thorough discussion of why measuring generalization through test error is flawed.\n\nWhile this is an interesting paper, I do not believe it is ready for acceptance at ICLR 2020.\n\n", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory."}, "extra_text": {}}, {"review_id": "B1x_hzDaYB", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes a flatness measure that is invariant to layer-wise reparametrizations in ReLU networks. The notion of feature robustness, which is a notion the paper proposes, connects the flatness measure to generalization error.\n\nThis paper should be rejected because it is not well-placed in the literature. Similar notions with the proposed flatness measure have repeatedly appeared in the literature. This paper needs to discuss novel insights.\n\nMajor comments:\n1) Many studies proposed or mentioned the flatness measures listed in Table 1 [1, 2, 3, 4]. One of the most relevant work will be [1]. It appears that the Fisher-Rao norm [1] has several advantages over the proposed measure in the submitted paper.\nA) Fisher-Rao norm is invariant to a broader range of linear transformations.\nB) Fisher-Rao norm does not rely on the Hessian, which is more suitable for non-smooth ReLU networks.\nAdditionally and importantly, the Fisher-Rao norm has a direct connection with the size of input gradients, which has a strong relationship with the feature robustness. It is strongly encouraged to discuss the connections and comparisons with the Fisher-Rao norm.\n2) On the connection to the generalization error, Theorem 10 relies on the strong assumption defined in Definition 9. Given the high ability of deep networks to express many functions, assuming that \\phi(S) is epsilon-representative seems difficult to justify. This paper should discuss why the assumption is reasonable. Otherwise, it is hard to claim that this paper connected the modified flatness measure to generalization error.\n\n[1] Liang et al. \"Fisher-Rao Metric, Geometry, and Complexity of Neural Networks.\" AISTATS 2019\n[2] Achille et al. \"Emergence of Invariance and Disentanglement in Deep Representations.\" JMLR 19 (2018)\n[3] Neyshabur et al. \"Exploring Generalization in Deep Learning.\" NeurIPS 2017\n[4] Tsuzuku et al. \"Normalized Flat Minima: Exploring Scale Invariant Definition of Flat Minima for Neural Networks using PAC-Bayesian Analysis.\" arXiv:1901.04653\n\n=====\n\nUpdate:\n\nThank you for the replies and the clarifications. They did address some of my concerns. However, the theoretical result is limited, and I do not think it provides clear connections of flatness and the local loss landscape. I think this paper is not ready for publication, and I keep my score.", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_checking_correctness_of_experiments": "I did not assess the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The authors propose a notion of feature robustness, provide a straightforward decomposition of risk in terms of this robustness measure, and then provide some empirical evidence for their perspective. Across the board, the reviewers raised issues with missing related work, which the authors then addressed. I will point out that some things the authors say about PAC-Bayes are false. E.g., in the rebuttal the authors say that PAC-Bayes is limited to 0-1 error. It is generally trivial to obtain bounds for bounded loss. For unbounded loss functions, there are bounds based on, e.g., sub gaussian assumptions. 

Despite improvements in connections with related work, reviewers continued to find the theoretical contributions to be marginal. Even the empirical contributions were found to be marginal.
Decision: Reject