ON NEURAL NETWORK GENERALIZATION VIA PROMOTING WITHIN-LAYER ACTIVATION DIVERSITY
Authors: Firas Laakom
Jenni Raitoharju
Alexandros Iosifidis
Moncef Gabbouj
Venue:  (2021)
Keywords: Deep learning

Abstract: During the last decade, neural networks have been intensively used to tackle various problems and they have often led to state-of-the-art results. These networks are composed of multiple jointly optimized layers arranged in a hierarchical structure. At each layer, the aim is to learn to extract hidden patterns needed to solve the problem at hand and forward it to the next layers. In the standard form, a neural network is trained with gradient-based optimization, where the errors are back-propagated from the last layer back to the first one. Thus at each optimization step, neurons at a given layer receive feedback from neurons belonging to higher layers of the hierarchy. In this paper, we propose to complement this traditional 'between-layer' feedback with additional 'within-layer' feedback to encourage diversity of the activations within the same layer. To this end, we measure the pairwise similarity between the outputs of the neurons and use it to model the layer's overall diversity. By penalizing similarities and promoting diversity, we encourage each neuron to learn a distinctive representation and, thus, to enrich the data representation learned within the layer and to increase the total capacity of the model. We theoretically study how the within-layer activation diversity affects the generalization performance of a neural network in a supervised context and we prove that increasing the diversity of hidden activations reduces the estimation error. In addition to the theoretical guarantees, we present an empirical study confirming that the proposed approach enhances the performance of neural networks.

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ON NEURAL NETWORK GENERALIZATION VIA PROMOTING WITHIN-LAYER ACTIVATION DIVERSITY
Anonymous authors Paper under double-blind review
ABSTRACT
During the last decade, neural networks have been intensively used to tackle var- ious problems and they have often led state-of-the-ar results. These networks are composed of multiple jointly optimized layers arranged in a hierarchical struc- ture. At each layer, the aim is to learn to extract hidden patterns needed to solve the problem at hand and forward it to the next layers. In the standard form, a neural network trained with gradient-based optimization, where the errors are back-propagated from the last layer back to the first one. Thus at each optimiza- tion step, neurons at a given layer receive feedback from neurons belonging to higher layers of the hierarchy. In this paper. propose to complement this tra- ditional "between-layer' feedback with additional 'within-layer' feedback to en- courage diversity of the activations within the same layer. To this end, measure the pairwise similarity between the outputs of the neurons and use it to model the layer's overall diversity. By penalizing similarities and promoting diversity, we encourage each neuron to learn distinctive representation and, thus, to enrich the data representation learned within the layer and to increase the total capacity of the model. We theoretically study how the within-layer activation diversity af- fects the generalization performance of neural network in a supervised context and prove that increasing the diversity of hidden activations reduces the esti- mation error. In addition the theoretical guarantees, we present an empirical study confirming that the proposed approach enhances the performance of neural networks.
INTRODUCTION
Neural networks are a powerful class of non-linear function approximators that have been success- fully used to tackle a wide range of problems. They have enabled breakthroughs in many tasks, such as image classification (Krizhevsky et al. 2012). speech recognition (Hinton et al. 2012a), and anomaly detection (Golan& Yaniv 2018). Formally, the output of neural network consisting of P layers can be defined as follows:
(I) where of(.) is the clement-wise activation function, e.g., ReLU and Sigmoid, of the jth layer and W (Wl ...., WP} are the corresponding weights of the network. The parameters of f(x; W) are optimized by minimizing the empirical loss:
f(x: W) = 6" (w""(o"""
( (W' x)))),
N Σ (f(z.::))))) N
L(f)
(2)
where 1(-) the loss function, and {m;, y/ 1 are the training samples and their associated ground- truth labels. The loss is minimized using the gradient decent-based optimization coupled with back- propagation.
However, neural networks are often over-parameterized, i.e., have more parameters than data. As a result, they tend to overfit to the training samples and not generalize well on unseen examples (Goodfellow et al. 2016). While research Double descent (Belkin et al. 2019: Advani et al.) 2020 Nakkiran et al. 2020) shows that over- parameterization does not necessarily lead to overfit- ting, avoiding overfitting has been extensively studied (Neyshabur al. 2018 Nagarajan & Kolter

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2019
Poggio et al., 2017) and various approaches and strategies have been proposed, such data
augmentation (Goodfellow et al. 2016). regularization (Kukačka et al. 2017
Bietti et al. 2019
Arora et al. 2019), and dropout (Hinton al. 2012b Wang et al. 2019 Lee al.
2019
Li al.
2016). to close the gap between the empirical loss and the expected loss.
Diversity of learners is widely known be important in ensemble learning (Li al. 2012 Yu al.
2011] and, partieularly in deep learning context, diversity of information extracted by the network neurons has been recognized as viable way to improve generalization (Xie et al. 2017a] 2015b).
In most cases, these efforts have focused on making the set of weights more diverse (Yang et al. Malkin & Bilmes| 2009). However, diversity of the activation has not received much attention.
Inspired by the motivation of dropout to co-adapt neuron activation, Cogswell et al. (2016) pro- posed regularize the activations of the network. An additional loss using cross-covariance of hidden activations was proposed, which encourages the neurons to learn diverse or non-redundant representations. The proposed approach, known as Decov, has empirically been proven alleviate overfitting and to improve the generalization ability of neural network, yet theoretical analysis to prove this has so been lacking.
In this work, we propose novel approach to encourage activation diversity within the same layer. We propose complementing 'between-layer" feedback with additional 'within-layer' feedback to pe- nalize similarities between neurons the same layer. Thus, encourage each neuron to learn distinctive representation and to enrich the data representation learned within each layer. Moreover, inspired by Xie al. (2015b), provide theoretical analysis showing that the within-layer acti- vation diversity boosts the generalization performance of neural networks and reduces overfitting.
Our contributions in this paper follows:
approach to encourage the 'diversification' of the layer-wise feature maps' outputs in neural networks. The proposed approach has three variants based how the global diversity is defined. The main intuition is that by pro- moting the within-layer activation diversity, neurons within the same layer learn distinct patterns and, thus, increase the overall capacity of the model.
Methodologically, propose
Theoretically, we analyse the effect the within-layer activation diversity on the generaliza- tion error bound of neural network. The analysis is presented in Section 3 As shown in Theorems 3.7 3.8 3.9 3.10 3.IT 3.12 we express the upper-bound of the estimation error as function of the diversity factor. Thus, we provide theoretical evidence that the layeen-layer activation diversity can help reduce the generalization error.
Empirically, we show that the within-layer activation diversity boosts the performance of neural networks. Experimental results show that the proposed approach outperforms the competing methods.
2 WITHIN-LAYER ACTIVATION DIVERSITY
We propose diversification strategy, where encourage neurons within layer activate in mutually different manner, i.e., to capture different patterns. To this end, we propose an additional within-layer loss which penalizes the neurons that activate similarly. The loss function L() defined in equation is augmented as follows:
P
(
(3)
where J expresses the overall wise similarity of the neurons within the 7th layer and is the penalty coefficient for the diversity loss. As in (Cogswell et al. 2016). proposed diversity loss can be applied to a single layer or multiple layers a network. For simplicity, let us focus on a single layer.
Let oi (xx) and oi (zz)) be the outputs of the th and mh neurons in the 8th layer for the same input sample xj. The similarity 8um between the the th and m neurons can be obtained as the average similarity measure of their outputs for N input samples. We use the radial basis function to
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express the similarity:
N Σ exp( - >x) (a,))))) ==1
8um
(4)
N
where y is hyper-parameter. The similarity 8ron can be computed over the whole dataset or batch- wise. Intuitively, if two neurons and m have similar outputs for many samples, their corresponding similarity 8am will be high. Otherwise, their similarity Smn is small and they are considered "di- verse". Based on these pair-wise similarities, we propose three variants for the global diversity loss Je of the ith layer:
Direct: J = 8nm- In this variant, we model the global layer similarity directly as the sum of the pairwise similarities between the neurons. By minimizing their sum, we encourage the neurons to learn different representations.
Det: J de) where S is similarity matrix defined as Sam 8 This variant is
inspired by the Determinantal Point Process (DPP) (Kulesza & Taskar 2010 (012), as the determinant of S measures the global diversity of the set. Geometrically, det () is the vol- ume of the parallelepiped formed by vectors in the feature space associated with s. Vectors that result in larger volume are considered to be more "diverse". Thus, maximizing det(-) (minimizing -det(-)) encourages the diversity of the learned features.
Logdet: J' -Sogdet(S) I use logdet instead of det as logdet convex function over the positive definite matrix space.
This variant has the same motivation the second one. We
It should be noted here that the first proposed variant, i.e., direct, similar to Decov (Cogswell al., 2016), captures only the pairwise diversity between components and is unable to capture the higher- order "diversity"", whereas the other variants consider the global similarity and are able to mea- sure diversity in a more global manner.
Our newly proposed loss function defined in equation has two terms. The first term is the classic loss function. It computes the loss with respect to the ground-truth. In the back-propagation, this feedback is back-propagated from the last layer the first layer of the network. Thus, it can be considered as between- layer feedback, whereas the second term is computed within a layer. From equation we can see that our proposed approach can be interpreted as regularization scheme. However, regularization in deep learning is usually applied directly on the parameters, i.e., weights (Goodfellow et al., 2016 Kukačka al., (201)), while in our approach, similar to (Cogswell al.) 2016), an additional term is defined over the output maps of the layers. For layer with C neurons and a batch size of N, the additional computational cost is O(C*(N + 1)) for direct variant and O + C)) for both the determinant and log of the determinant variants.
GENERALIZATION ERROR ANALYSIS
In this section, we analyze how the proposed within-layer diversity regularizer affects the general- ization eиror of a neural network. Generalization theory (Zhang et al. 2017 Kawaguchi al.) 2017)
focuses on the relation between the empirical loss, as defined in equation 2 and the expected risk defined as follows:
L(S) = ((()))))(((())...)))))))))))))>)
(5)
where Q is the underlying distribution of the dataset. Let f arg miny L(f) be the expected risk minimizer and f arg min, i.()) be the empirical risk minimizer. We are interested in the estimation error, i.e., -(f*) L()), defined as the gap in the loss between both minimizers (Barron 1994). The estimation error represents how well an algorithm can learn It usually depends on the complexity of the hypothesis class and the number of training samples (Barron 1993 Zhai & Wang 2018).
TTii is defined only if S is positive definite. It can be shown that in our S is positive semi-definite. Thus, in practice regularized version (S + eT) ensure the positive definiteness.
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Several techniques have been used to quantify the estimation error, such as PAC learning (Hanneke) 2016 Arora et al. 2018). VC dimension (Sontag 1998. Harvey et al. 2017 Bartlett et al. 2019).
and the Rademacher complexity (Xie et al. 2015b; Zhai & Wang, 2018
Tang et al. 2020). The
Rademacher complexity has been widely used as if usually leads to tighter generalization error
bound (Sokolic et al., 2 Neyshabur al. 2018 Golowich et al. 2018). The formal definition of the empirical Rademacher complexity is given as follows:
Definition 3.1. (Bartlett Mendelson 2002) For a given dataset with N samples D = (xi, yi li=1
generated by distribution Q @and for model space F X - R with single dimensional output, the empirical Rademacher complexity (F) of the set F is defined as follows:
N
Σ cos((a))
RN(FF) = Eg
sup
(6)
LSEF
N
1=1
where the Rademacher variables a = (0i,".. NNn} are independent uniform random variables in {-1,1}.
In this work, we analyse the estimation error bound of a neural network using the Rademacher complexity and we are interested in the effect of the within-layer diversity on the estimation error. In order to study this effect, inspired by (Xie et al.) 2015b), we assume that with a high probability т, the distance between the output of each pair of neurons, (Ø,, (x) om ( (x))² ,, is lower bounded by
dmin for any input æ. Note that this condition can be expressed in terms of the similarity defined
in equation 4 8mm I e Smin for any two distinct neurons with the probability Our
analysis starts with the following lemma:
Lemma 3.2.
Xie al. 2015b Bartlett & Mendelson 2002
With a probability of at least 6
log (2/6)
I(f) I(M) S 4Rx + BV
N
for B 2 s.P [zf(()) v)l. where (N) is the Rademacher complexity of the loss set A.
It upper-bounds the estimation error using the Rademacher complexity defined over the loss set and supz.y./ (z(f())))) 3)1. Our analysis continues by seeking a tighter upper bound of this error and showing how the within-layer diversity, expressed with dmin affects this upper bound. We start by deriving such an upper-bound for a simple network with one hidden layer trained for regression task and then we extend it for a general multi-layer network and for different losses.
3.1 SINGLE HIDDEN-LAYER NETWORK
Here, we consider a simple neural network with one hidden-layer with M neurons and one- dimensional output trained for regression task. The full characterization of the setup can be sum- marized in the following assumptions: Assumptions 1.
The activation function of the hidden layer, Φ(1), a Lag-Lipschitt continuous function.
The inpul vector x: E PRP satisfies ||a|| S Ci.
The output scalar E E satisfies yl < C2.
The weight matrix W = [wn, W2, (MM E DDM connecting the input to the hidden layer satisfies [|wmlll ll I C3-
The weight vector v E M connecting the hidden-layed to the output neuron satisfies ||u|l < Ct.
The hypothesis class is F = (f]f(x)= ΣM Um em (2:) = EM um
Loss function set A (11((((x).)) = =1(((a)))) - =1
With probability T. for # m, ||@a (x:) om (z) 11 o(( 2 d dmin-
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We recall the following two lemmas related to the estimation error and the Rademacher complexity:
Lemma 3.3. Barrtlett & Mendelson 2002 For F IR*,, assume that R - IR is a Lg Lipschitz continuous function and A 19
E FJ. Then we have
RN S Lg (FF)..
(8)
Lemma 3.4. Xie et al. 2015b) Under Assumptions
7 the Rademacher complexity Rx of the Em=1 Um bmnb( (w w can be upper-bounded
hypothesis class F (J ([))()) as follows:
Em Um Um (2))
2L Cala(O)
Rx(FF) S
VN
VN
where C134 C2Ca and (0)) is the oulput of the activation function at the origin.
Lemma 3.4 provides an upper-bound of the Rademacher complexity for the hypothesis class. In order to find an upper-bound for our estimation error, we start by deriving an upper bound for suPa,f (z=)]:
Lemma 3.5. Under Assumptions with probability least 72, we have
sup æ.
[f(x))) S VJ,
(10)
where Q is equal to the number of neuron pairs defined by M neurons, ie., Q M(M-1) and J = -=](MM3 M(M 1)(C - /2m and Cs CsCiCs + ((0),
The proof can be found in Appendix 7.1 Note that in Lemma 3.5 we have expressed the upper-
bound of sup f [f())) in terms of diminn Using this bound,
find an upper-bound for
su [x(((x)) y) in the following lemma:
Lemma 3.6. Under Assumptions with a probability least 7,, we have
sup [[(((x)))) s (VJ
(11)
w.y.f
The proof can be found in Appendix 7.2 The main goal is to analyze the estimation error bound of the neural network and to see how its upper-bound is linked to the diversity, expressed by dmmins of the different neurons. The main result is presented in Theorem 3.7
Theorem 3.7. Under Assumptions
with probability at least T 2 (1 5), we have
I(f) - 1(T)) ( (40- (23 where C134 - C J C
VM
+ +VJ + +VJ C22 log(22/))
(12)
VN 1)(C - d 2)), and Cs LaCi
The proof can be found in Appendix 7.3
Theorem 3.7 provides an upper-bound for the estimation error. We note that it is decreasing function of dymin- Thus, we say that a higher dmin i.e., more diverse activations, yields a lower estimation error bound. In other words, by promoting the within- layer diversity. we can reduce the generalization error of neural networks. It should be also noted that our Theorem 3.7 has similar form to Theorem difference is that Xie et al. analyse the estimation error with respect to the diversity of the weight vectors. Here, we consider the diversity between the outputs of the activations of the hidden neurons.
(Xie et al., 20155). However, the main
3.2 BINARY CLASSIFICATION
We now extend our analysis of the effect of the within-layer diversity the generalization error in the case of a binary classification task, i.e., y E {11, The extensions of Theorem 3.7 in the case of hinge loss and logistic loss are presented in Theorems (38 3.9 respectively.
Theorem 3.8. Using the hinge loss, we have with probability at least 7 5)
VM
2 2lgg(26))
- < 4 Cia4 Co Co(((1))
+ (11+ VJ)
(13)
VN
N
where C1s4 CCCCC, CM MMM- (lminn2)) and Cs Ci +(0).
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Theorem 3.9. Using the logistic loss 1(f(x).)) v) loog11 + e e-ff())), have with probability at least (1
VM I(F) - L(F)) 1 + ( Cald(0))
4
log vv 21og(2/6)
(14)
VN
where = CiCaCa, J C
e 22), and C5 LaCi 6(0).
The proof's are similar to Lemmas 7 and 8 in (Xie et al. 2015b). As we can see, for the classification task, the error bounds of the estimation error for the hinge and logistic losses are decreasing with respect to dmii. Thus, employing a diversity strategy can improve the generalization also for the binary classification task.
3.3 MULTI-LAYER NETWORKS
Here, we extend our result for networks with P (> 1) hidden layers. We assume that the pair-wise distances between the activations within layer p are lower-bounded by d' with probability T?. In this case, the hypothesis class can be defined recursively. In addition, we replace the fourth assumption in Assumptions with: ||W 1 C for every W ie., the weight matrix of the p-th layer. In this case, the main theorem is extended follows:
Theorem 3.10. With probability of at least II
Ip (TP) (1 8), we have
((f) - I(P) I(F)
s 8(VJ+Cz)
(iC)) P-1
VMC
16(0)
P-1
(2L (2
II VMC j=p
P-1
VN
P-U
+ +VJ++c)) V (og(2/0))
(15)
where QP the number of neuron pairs in the thh layer, defined as QP MP and JP is defined recursively using the following identities: Jo CgC and Jp M (L -00))=
1) Lunn f)r p I.....P.
The proof can be found in Appendix 7.4 In Theorem 3.10 see that P is decreasing with respect to de min. Thus, we see that maximizing the within- layer diversity, we can reduce the estimation error of multi-layer neural network.
3.4 MULTIPLE OUTPUTS
Finally, we consider the case of a neural network with multi culli-diimensonall output, i.e., y E RD. In this case, we can extend Theorem 3.7 decomposing the problem into smaller problems and deriving the global error bound as the sum of the small D bounds. This yields the following two theorems:
Theorem 3.11. For multivariate regression trained with the squared error, we have with proba- bility at least (1 5),
VM + D(VJ +DVJJ+C22) 2 log(2/0) (16) VN
I(f) - I(F) SD(VJ+ C (D00
N
where C134 C,C3CC J CM + M(M L 12)) and Cs LoCiCs +
Theorem 3.12. For a multi-class classification task using the cross-entropy loss, we have with probability at least 7
un- - 1(F" D D-1+ 20J (2L0C104 + C
D(D 1)
VM VN
+ +log (D
2 2log(2/0
(17)
where C134 CCC - CMMM M(M C) d2 d /2) and Cs L Ci C3 (()).
The proofs can be found in Appendix 7.5 Theorems 3.11 and 3. 12 extend our result for the multi-
dimensional regression and classification tasks, respectively. Both bounds inversely proportional to the diversity factor d We note that for the classification task, the upper-bound is exponentially decreasing with respect d

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4 RELATED WORK
Diversity promoting strategies have been widely used in ensemble learning (Li al. 2012 Yu
et al.] 2011), sampling (Derezinski et al.; 2019; Bryık et al.] 2019. Gartrell et al. 2019), ran ranking
Yang et al. Gan et al. 2020), and pruning by reducing redundancy (Kondo & Yamauchi] 2014 He
et al. 2019, Singh et 2020 Lee et al. 2020). In the deep learning context, various approaches have used diversity as direct regularizer on top of the weight parameters. Here, we present brief overview of these regularizers. Based on the way diversity is defined, we can group these approaches into two categories. The first group considers the regularizers that are based on the pairwise dissimilarity of components, i.e., the overall set of weights are diverse if every pair of weights are dissimilar. Given the weight vectors {wm )mnn)mn Yu et al. (201 1) define the regularizer as mm (1-0mn) 0m.), where mm represents the cosine similarity between Wm and w,,, Bao et al. (0133)
log M mm 310mm||3) where 8 is positive hyperparameter. Xie et al. (2015a) 2016) used mean(0mm var(O to regularize Boltzmann machines. They theoretically analyzed its effect on the generalization error bounds in (Xie et al, 20156) and extended it to kernel space in (Xie et al.) (017a)). The second group of regularizers considers a more globalist view of diversity. For example, in (Maikin & Bilmes 2009) 2008 Xie at 7b). weight regularization based on the determinant of the weights covariance is proposed and based on determinantal point process in (Kulesza & Taskar 2012 Kwok Adams 2012).
proposed an incoherence score defined
Unlike the aforementioned methods which promote diversity the weight level and similar to method, method,,Cosswlll et al. (2016) proposed to enforce dissimilarity on the feature map outputs, i.e., on the activations. To this end, they proposed additional loss based on the pairwise covariance of the activation outputs. Their additional loss, Deecv is defined as the squared sum of the non-diagonal elements of the global covariance matrix C:
DeCOv (ICC) |[diag( (]diag(C))lB).
(18)
where ||-[|æ is the Frobenius norm. Their approach, Decov, yielded superior empirical performance; however, it lacks theoretical proof. In this paper, we closed this gap and we showed theoretically how employing diversity strategy on the network activations can indeed decrease the estimation error bound and improve the generalization of the model. Besides, we proposed variants of our approach which consider a global view of diversity.
5 EXPERIMENTAL RESULTS
In this section, we present an empirical study of our approach regression context using Boston Housing price dataset (Dua & Graff, (017) and in classification context using CIFAR10 10 and CI- FART00 datasets (Krizhevsky et al. 2009). We denote as Vanilla the model trained with no diversity protocol and as Decoy the approach proposed in (Cogswell al. 2016).
5.1 REGRESSION
For regression, we use the Boston Housing price dataset (Dua & Graff 2017). It has 404 training samples and 102 test samples with 13 attributes each. We hold the last 100 sam- ple of training as validation set for the hyper-parameter tuning. The loss weight, is chosen from (0.00001, 0.00005, 0.0001, 0.0005, 0.001, 0.001,0.005) for both our approach and De- cov (Cogswell et al., 2016). Parameter ? in the radial basis function is chosen from {0.00001, 0.0001, 0.0T, 1. 10, 100). As a base model, we use a neural network composed of two fully connected hidden layers, each with 64 neurons. The additional loss is applicd on top of both hidden layers.
We train for 80 epochs using stochastic gradient descent with a learning rate of 0.01 and mean square error loss. For hyperparamrter tuning, we keep the model that perform best on the validation and use it in the test phase. We experiment with three different activation functions for the hidden layers: Sigmoid, Rectified Linear Units (ReLU) (airr & Hinton 2010), and LeakyReLU (Maas et al. 2013).
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Table 1: Mean average error of different approaches on Boston Housing price dataset
ReLU
Sigmoid
LeakyReLU
Vanilla
2.97
3.16
2.85
Decov
2.77
2.99
2.80
Ours (direct)
2.72
2.97
2.82
Ours (det)
2.68
2.87
2.83
Ours (logdet)
2.64
2.83
2.77
Table 1 reports the results in terms of the mean average error for the different approaches over the Boston Housing price dataset. First, we note that employing a diversification strategy (ours and Decov) boosts the results compared to the Vanilla approach for all types of activations. The three variants of our approach, i.c., the within-layer approach, consistenly outperform the Decov loss except for the LeakyReLU where the latter outperforms direct variant. Table [ shows that the logdet variant of our approach yields the best performance for all three activation types.
5.2 CLASSIFICATION
For classification, we evaluate the performance of our approach on CIFARIO and CIFAR 100 datasets (Krizhevsky et al. 2009). They contain 60,000 32 x 32 images grouped into 10 and 100 distinct categories, respectively. We train on the 50,000 given training examples and test on the 10,000 specified test samples. We hold the last 10000 of the training set for validation. For the neural network model, we use an architecture composed of 3 convolutional layers. Each convolution layer is composed of 32 3 x filters followed by 2 2 max pooling. The flattened output of the convo- lutional layers is connected to fully connected layer with 128 neurons and a softmax layer. The different additional losses, i.c., ours and Decov, are added only top of the fully connected layer. The models are trained for 150 epochs using stochastic gradient decent with learning rate 0.01 and categorical cross entropy loss. For hyper-paramters tuning, keep the model that performs best on the validation set and use it in the test phase, We experiment with three different activation functions for the hidden layers: sigmoid, Rectified Linear Units (ReLU) (Nair & Hinton 2010), and LeakyReLU (Maas et al., 2013). All reported results are average performance over 4 trials with the standard deviation indicated alongside.
Tables and 3 report the test eтrоr rates of the different approaches for both datasets. Compared the Vanilla network, our within-layer diversity strategies consistently improve the performance of the model. For the CIFAR10, the direct variant yields more than 0.72% improvement for the ReLU and 2% improvement for the sigmoid activation. For the LeakyReLU case, the determinant variant achieves the lowest error rate. This is in accordance with the results on CIFARIO0 Here, we note that our proposed approach outperforms both Vanilla and the Decov models, especially in the sigmoid case. Compared to the Vanilla approach, we note that the model training time cost CIFAR 100 increases by 9% for the direct approach, by 36.1% for the determinant variant, and by 36, 36f2%fo the log of determinant variant.
Table 2: Test error rates CIFARI0
ReLU
Sigmoid
LeakyReLU
Vanilla
32.04 0.57
33.78
0.64
30.99 + 0.27
Decov
30.98 0.25
32.22 = 0.51
30.70 + 0.35
Ours (direct)
31.28 - 0.49
$1.69:0.2
30.86 = 0.75
Ours (det)
31.28 0.60
32.92 0.49
30.93 = 0.44
Ours (logdet)
31.26 0.41
32.61 0.46
30.70 + 0.25
CONCLUSIONS
In this paper, we proposed a new approach to encourage 'diversification' of the layer-wise feature map outputs in neural networks. The main motivation is that by promoting within-layer activation diversity, neurons within the same layer learn to capture mutually distinct patterns. We proposed an additional loss term that be added on top of any fully-connected layer. This term complements
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Table 3: Test emror rates CIFARIOR ReLU
Sigmoid
LeakyReLU
Vanilla
65.81 + 0.42
78.52 0.40
64.90 0.22
Decov
65.26 = 0.21
77.08 = 0.47
64.57 0.23
Ours (direct)
64.95 = 0.32
76,91 = 0.91
64.85 =
Ours (det)
64.90 + 0.40
77.79
0.29
64.46 0.40
Ours (logdet)
64.95 + 0.17
77.70 = 0.61
64.49 + 0.19
the traditional "between-layer' feedback with an additional 'within-layer' feedback encouraging di- versity of the activations. We theoretically proved that the proposed approach decreases the esti- mation error bound, and thus improves the generalization ability of neural networks. This analysis was further supported by experimental results showing that such strategy can indeed improve the performance of neural networks in regression and classification tasks. Our future work includes extensive experimental analysis on the relationship between the distribution of the neurons output and generalization.
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Under review as a conference paper at ICLR 2021
7 APPENDIX
In the following proofs, we use Lipschitz analysis. In particular, a function f: A - IR, C R"
is said to be L-Lipschitz, if there exist constant L 2 0, such that [f(a) f(6) S Llla bl| for every pair points a, E A. Moreover:
SuPzEA f I sup(T]|a<]) + f(0)).
if f continuous and differentiable, == sup (x))
7.1 PROOF OF LEMMA 3.5
Lemma 3.5. Under Assumptions with a probability at least 79, we have
sup [[()) | VJ,
(19)
2.1
where Q is equal to the number of neuron pairs defined by neurons. i.e. Q
M(M
and
2
J = =M + M(M - -C dmnin/2) and Cs +
Proof.
M
M
M
M
(())==
Σ
Um um@m (x))
|10| (m)
Σ
Om (20)
S C
Σ
(a)
C
Σ mm.n
om (x) (2)))
C
Σ om (2o)*
Σ muptir
(20)
We have s (xx) < sup(Lølw' æ| + (()) because is Lipsichchitz Thus, |101 LaCiCs + x(0) Cs. For the first term in equation 20 have
Σn pm (2)2
M(LoCiCs 2(0))²
MC3. The second term, using the identity (2) + (x) (x)² (0m(x) - (n(x))?) can be rewritten
Σ 5m (x) mff
Σ 0m))) + @n(x)) - (((())))((z))
(21)
myf
In addition, we have with a probability t, 110m - (n(21)11 2 dmin for m + n. Thus, we have -
with probability at least 7
1 Σ 22cg ) = M(M - d d
(m)))) (x) S 2 mfn
(22)
Here Q is equal to the number of neuron pairs defined by M neurons, i.e, Q M(M By putting
everything back to equation 20 we have with probability 7
(x S C (MC M(M 1)(C) desin /2) J.
(23)
Thus, with a probability 7
sup [ap]f(})]
S
sup f(x)² s VJ.
(24)
2.1
V
a.
7.2 PROOF OF LEMMA 3.6
Lemma 3.6. Under Assumptions
with a probability at least 79, we have sup [(f()))))] < (VJ C2) m.y.f
(25)
Proof. We have supa..y f()) (f()) vl < 2suP CouPayy.((f((ar (1f(z) + lvl) = 2(VJ + O2) Thus
(upa...l((())) v) (VJ + 22)2
13

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Under review as a conference paper at ICLR 2021
7.3 PROOF OF THEOREM 3.7
Theorem 3.7. Under Assumptions with probability at least (1-1- -6), we have
---) - 1( ( (2a (
VM
+(VJ C2)
2 1lgg(2/8)
(26)
VN
N
where C134 CCCC4,
de min 2)), and C3
+
Proof.
Given that ((-) is K L-iisschitz with a constant K = supe.y. 2 C2), and
using Lemma 3.3 we can show that (AA & KR (F) < 2 + C2) For RN (F),
use the bound found in Lemma 3.4 Using Lemmas 3.2 and 3.6 completes the proof.
7.4 PROOF OF THEOREM 3.10
Theorem 3.10. Under Assumptions
1
with probability of at least
P-1 (TP)O" (1 (1- we have p=
L(f)-LC s 8(VJ+0z)
(2Lo)
Cic
II
P-1
VMrcg
|6(0)
P-1
(2La) (2L0) P-1-p V
p
VN
PMU
+ +VJ++cs)²
2 / (2/(2
(27)
where Q" the number of neuron pairs in the nth layer, defined Q" MP(MPP
and Jo is defined recursively using the following identities: Jo Cg and J M (L
+ (0)))² M(M 1)dmmnn/))) for, p,r
Proof. Lemma 5 in (Xie et al. 2015b) provides upper-bound for the hypothesis class. We denote by o² denote the outputs of the pd hidden layer before applying the activation function:
vd
[w) F,
(28)
M
MP-1
wh (
Σ F
(u }]
(29)
j=l
vº
lwo Φ ,., whe T Φ0),
(30)
where dº = =
06 .).. We have
MP
||0011
Σ
(un m
"22i22
(31)
and uh T "0p S C En oh Thus,
MP
Σ (C)) 20== MPC
62
MMPC2 Σ
S de
(32)
m
mn
We use the same decomposition trick of or, df as in the proof of Lemma 3.5. We need to bound sup, SP:
TT 2 - 6(0) < L W || 000 112 + ((0).
sup S" < sup(Loo
(33)
Thus, we have
| < M M
(00)² - M 1) 2)) = JPP (34)
We found recursive bound for | | we note that for p O, we have 0 < | / C1 5
C&C, C = Jo. Thus,
sup
|f(x) = sup
l VJP.
(35)
&
14

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Under review as a conference paper at ICLR 2021
7.5 PROOFS OF THEOREMS 3.11 AND 3.12
Theorem 3.11. For a multivariate regression trained with the squared error, we have with proba- bility at least 7 (1 6),
VM + +D(VJ +02)²
1 (36)
L(f)-L(f') V (LL
N
where C134= CiCaC,, J= CCC M d 2)), and C3 I. + 600)
Proof. The squared loss [
y| can be decomposed into terms (f(æ)): yk yk)² Using
Theorem 3.7 can derive the bound for each term.
For a multiclass classification task using the cross-entropy loss, we have with probability least Q1 8),
Theorem 3.12.
LÚ) - -0-) D-1+e 1-1+e-203 (2L + Colo(0))
D(D
VM VN
+ log (1 (D (1e1)e2 2 2log(2/8)
N
(37)
where C134 = -C3
d m /)), and Cs L C3 + 6(0).
Proof. Using Lemma 9 in (Xie et al. 2015b), we have supf, I = log (1 + (D - 1)e and /
D-1te-2w7 Lipschitz. Thus, using the decomposition property of the Rademacher complexity, we have
D-1
D(D
2L 2L CM VN
(A) & D D-1+
(38)
VN
15


Reviews:
[{"review_id": "rOJ7gTfoxz9", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "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": "The paper proposed three ways of diversifying outputs of neurons, and the analysis showed that the generalisation bound becomes tighter when the neurons become more diversified. It is an interesting finding, along with theoretical results and empirical results. Although, from a practical perspective, there are still many concerns.\n\nIt is clear that by increasing d_min, the generalisation bound gets tighter. However, it is also obvious that there are other factors that one can control to make the bound tighter, and regularising other factors might be simpler in terms of implementation and optimisation. \n\n1. The constant C_4 in the upper bound of the weight vector connecting the hidden-layer to the output neuron. \\sqrt(J) decays linearly with C_4, and the first term in the generalisation bound for regression tasks decays quadratically w.r.t. C_4. Compared with a linear decay w.r.t. d_min, C_4 seems to be a better option to regularise neural networks. In practice, one can empose an \\ell_2 regularisation on the top linear layer to control the overall norm of the weight matrix so that C_4 is controlled. \n\n2. The constant C_5 = L_\\phi C_1 C_3 + \\phi(0). As we can see in the generalisation bound for regression tasks, \\sqrt(J) decays quadratically w.r.t. C_5, which is even faster than the decay rate w.r.t. C_4. To control C_5, one can choose an activation function that has a small L_\\phi, or to control the weight vectors to the activation function to have a small norm C_3. Both of them can be done relatively easily compared to optimising pair-wise similarity. \n\nOverall, I think there are other regularisations suggested by the bound that could be put into practice, which might also lead to good generalisation, and also simpler optimisation problem. ", "extra_scores": {}, "extra_text": {}}, {"review_id": "PSTTIdq8EOf", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "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 adding regularization terms to encourage diversity of the layer outputs in order to improve the generalization performance. The proposed idea is an extension of Cogswell's work with different regularization terms. In addition, the authors performed detailed generalization analysis based on the Rademacher complexity. The appearance of the term related to the layer output diversity in the generalization bound provides theoretical support for the proposed idea.\n\nThe main weakness of this paper, in my humble opinion, is the lack of important details or rigor in the experiments presented. For example, the authors didn't mention how the hyperparameter selection was conducted, what optimizer (and its parameters) was used, how many runs per result and the confidence interval, whether any test was done to establish statistical significance, why state-of-the-art architecture was not used for the image classification tasks, etc. Without these important details and rigorous comparison, it's hard to have high confidence in the reproducibility of the results.\n\nDetails:\n1) Intro section. The line of work in \"double descent\" shows that overparameterization doesn't necessarily lead to overfitting. For completeness, it'll be good to mention this line of work and qualify the claim on overfitting.\n2) End of section 2. The authors claim that the proposed diversity term induces \"within-layer\" feedback. The regularization term is computed on the outputs of a layer, which do depend on the parameters of the lower layers. So when backpropagation happens, it will affect the parameters of the lower layers. Therefore, \"within-layer\" feedback doesn't sound accurate to me.\n3) Section 3.1, last bullet point. Should $\\tau$ be introduced here?  Otherwise, where does the $\\tau$ later used in Lemma 3.5, Lemma 3.6 and Theorem 3.7 come from?\n4) Section 5. The proposed regularization terms don't seem cheap to compute for large networks with wide layers. It'll be helpful to measure the training cost increase.", "extra_scores": {}, "extra_text": {}}, {"review_id": "G69F_m1IJYG", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "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": "Strong point: the paper addresses an important problem.\n\nThree main weaknesses, which justify the score:\n•\tThe theoretical developments presented in the paper build on the Rademacher complexity, but ignore the conclusions drawn by Zhang et al. in Section 2.2 of their ICLR 2017 paper (Understanding deep learning requires rethinking generalization).\n•\tThe theoretical developments build on the assumption that (i) there exists a lower bound, valid for any input, to the distance between the output of each pair of neurons, and (ii) the proposed diversity loss increases this lower bound. Those two assumptions are central to the theoretical developments, but are quite arguable. For example, a pair of neuron that is not activated by a sample, which is quite common, leads to a zero lower bound.\n•\tExperimental validation are not convincing. Only shallow networks are considered (2 or 3 layers), and the optimization strategy, including the grid search strategy for hyperparameters selection, is not described.\n\nMinor issue: positioning with respect to related works is limited. For example, layer redundancy (which is the opposite of diversity) has been considered in the context of network pruning: https://openaccess.thecvf.com/content_CVPR_2019/papers/He_Filter_Pruning_via_Geometric_Median_for_Deep_Convolutional_Neural_Networks_CVPR_2019_paper.pdf \n", "extra_scores": {}, "extra_text": {}}, {"review_id": "BayzwwLdUuX", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "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": "In this paper, the authors propose a technique to encourage the within-layer\nactivation diversity and therefore improve the model performance.\nSpecifically, they design a within-layer loss that add penalty to the similar\nneurons. They also showed that encouraging the within-layer diversity will\nhelp reduce the generalization error. \n\nThe paper is well-presented and authors provided enough intuition as well as\ntheoretical evidence why the diversity would help. Although I did not check\nall the proofs, the results seem to be right. \n\nThe definition of within-layer diversity seems to be simply the\nconcentration of the values of each individual neuron. How does that affect\nthe distribution of the layer output on the unit ball? Will this lead to a\noutput similar to 'binarized' output?\n\nThe experiment seems insufficient to support the argument. Only very simple\nneural networks on two toy examples are provided. More ablation study of the\nneural/layer output distribution would help better understanding this issue.\n\nOverall I think this paper provides some insights to how the generalization\nerror is related to the neuron outputs and vote for accept.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper looks into generalization performance of NNs in supervised learning setting. The authors propose a regularizer to enhance neuron diversity in each layer(within-layer activation diversity) as a regularizer to improve generalization.  The proposed idea is an extension of Cogswell's work with different regularization terms. The appearance of the term related to the layer output diversity in the generalization bound provides theoretical support for the proposed idea.They use Radamacher complexity as a tool to show this and bound the estimation error.

pros.
-The paper looks into an interesting problem. Designing a regularizer to improve generalization performance of NNs is of huge importance.
-The paper is well-presented and clear.

cons.
-The main drawback of the paper is lack of proper comparison to other regularizers and showing the uniqueness/superiority of this regularizer and how it improves over existing methods either theoretically or with experiments. Without that the significance of this work is limited. 
-The authors response to Reviewer 2's comment was not convincing enough. I encourage the authors to improve this in the next iteration of the paper. 
- i suggest doing a better job in including the related work as also mentioned by the reviewer.
-The experiment section can use more explanation and details on choice of hyper parameters, etc 
- showing performance improvement for a deep architecture would definitely  improve the paper. In the current version only 2 and 3 layer toy examples are shown.
Decision: Reject