Disentangling deep neural networks with rectified linear units using duality

1 INTRODUCTION . Despite their success deep neural networks ( DNNs ) are still largely considered as black boxes . The main issue is that in each layer of a DNN , the linear computation , i.e. , multiplication by the weight matrix and the non-linear activations are entangled . Such entanglement has its pros and cons . The commonly held view is that such entanglement is the key to success of DNNs , in that , it allows DNNs to learn sophisticated structures in a layer-by-layer manner . However , in terms of interpretability , such entanglement has an adverse effect : only the final layer is linear and amenable to a feature/weight interpretation , and the hidden layers are non-interpretable due to the non-linearities . Prior works ( Jacot et al. , 2018 ; Arora et al. , 2019 ; Cao & Gu , 2019 ) showed that training an infinite width DNN with gradient descent is equivalent to a kernel method with the so called neural tangent kernel matrix . As a pure kernel method , the NTK matrix performed better than other pure kernel methods . However , in relation to ‘ black box ’ -ness , there are two issues with NTK theory : ( i ) Issue I : Infinite width NTK matrix does not explain fully the success of DNNs because it was observed that finite width DNNs outperform their infinite width NTK counterparts , and it was an open question to understand this performance gap ( Arora et al. , 2019 ) , and ( ii ) Issue II : Since the NTK is based on the gradients , it does not offer further insights about the inner workings of DNNs even for infinite width . A dual view for DNNs with rectified linear units ( ReLUs ) was recently developed by Lakshminarayanan & Singh ( 2020 ) who exploited the gating property ( i.e. , ‘ on/off ’ states ) of the ReLUs . The dual view is essentially linearity in the path space , i.e. , the output is the summation of path contributions . While the weights in a path are the same for each input , whether or not a path contributes to the output is entirely dictated by the gates in the path , which are ‘ on/off ’ based on the input . To understand the role of the gates , a deep gated network ( DGN ) ( see Figure 1 ) was used to disentangle the learning in the gates from the learning in weights . In a DGN , the gates are generated ( and learnt ) in a ‘ gating network ’ which is a DNN with ReLUs and are applied as external signals and the weights are learnt in a ‘ weight network ’ consisting of gated linear units ( GaLUs ) ( Fiat et al. , 2019 ) . Each GaLU multiplies its pre-activation and the external gating signal . Using the DGN , two important insights were provided : ( i ) learning in the gates is the most crucial for finite width networks to outperform infinite width NTK ; this addresses Issue I , and ( ii ) in the limit of infinite width , learning the weights with fixed gates , the NTK is equal to ( but for a scalar ) a so called neural path kernel ( NPK ) which is a kernel solely based on inputs and gates . This shifts Issue II on interpretability to that of interpretability of the gates as opposed to interpretability of the gradients . Our Contribution . We extend the dual view to address ‘ black box ’ -ness by completely disentangling the ‘ gating network ’ and the ‘ weight network ’ . Our contributions are listed below . • Disentangling Gating Network . For this , we propose a novel Deep Linearly Gated Network ( DLGN ) as a mathematically interpretable counterpart of a DNN with ReLUs ( see Figure 1 ) . In a DLGN , the gating network is a deep linear network , i.e. , there is disentanglement because of the absence of non-linear activations . The gating network is mathematically interpretable , because , the transformations from input to the pre-activations are entirely linear ; we call this primal linearity . •Dual View ( Section 3.1 ) . We present an unnoticed insight in prior work on fully connected networks that the NPK is a product kernel and is invariant to layer permutations . We present new results to show that ( i ) the NPK is rotationally invariant for convolutional networks with global average pooling , and ( ii ) the NPK is an ensemble of many kernels in the presence of skip connections . • Disentangling Weight Network . We then argue via theory and experiments that the weight network is disentangled in the path space , i.e. , it learns path-by-path and not layer-by-layer . For this , in Section 3.2 we show via experiments that destroying the layer-by-layer structure by permuting the layers and providing a constant 1 as input ( see DLGN in Figure 1 ) do not degrade performance . These counter intuitive results are difficult to reconcile using the commonly held ‘ sophisticated structures are learnt in layers ’ interpretation . However , these experimental results follow from the theory in Section 3.1 . In other words , it is useful to think that the learning in the weight network happens path-by-path ; we call this dual linearity , which ( for infinite width ) is interpreted via the NPK . Message . The DLGN is not an alternative architecture per se , but a disentanglement and an interpretable re-arrangement of the computations in a DNN with ReLUs . The DLGN disentangles the computations into two ‘ mathematically ’ interpretable linearities ( i ) the ‘ primal ’ linearity and ( ii ) the ‘ dual ’ linearity interpreted via the NPK . Using the facts that the NPK is based on input and the gates , and in a DLGN , the pre-activations in the gating network are ‘ primal ’ linear , we have complete disentanglement . We compare the performance of DNN , DGN and DLGN on CIFAR-10 and CIFAR100 to show that , the DLGN recovers more than 83.5 % of the performance of state-of-the-art DNNs , i.e. , while entanglement in the DNNs enable their improved performance , the ‘ disentangled and interpretable ’ computations in the DLGN recovers most part of the performance . Related Works . We now compare our work with the related works . • Kernels . Several works have examined theoretically as well as empirically two important kernels associated with a DNN namely its NTK based on the correlation of the gradients and the conjugate kernel based on the correlation of the outputs ( Fan & Wang , 2020 ; Geifman et al. , 2020 ; Liu et al. , 2020 ; Chen et al. , 2020 ; Xiao et al. , 2020 ; Jacot et al. , 2018 ; Arora et al. , 2019 ; Novak et al. , 2018 ; Lee et al. , 2017 ; 2020b ) . In contrast , the NPK is based on the correlation of the gates . We do not build pure-kernel method with NPK , but use it as an aid to disentangle finite width DNN with ReLUs . • ReLU , Gating , Dual Linearity . A spline theory based on max-affine linearity was proposed in ( Balestriero et al. , 2018 ; Balestriero & Baraniuk , 2018 ) to show that a DNN with ReLUs performs hierarchical , greedy template matching . In contrast , the dual view exploits the gating property to simplify the NTK into the NPK . Gated linearity was studied in ( Fiat et al. , 2019 ) for single layered networks , along with a non-gradient algorithm to tune the gates . In contrast , we look at networks of any depth , and the gates are tuned via standard optimisers . The main novelty in our work in contrast to the above is that in DLGN the feature generation is linear . The gating in this paper refers to the gating property of the ReLU itself and has no connection to ( Srivastava et al. , 2015 ) where gating is a mechanism to regulate information flow . Also , the soft-gating used in our work and in ( Lakshminarayanan & Singh , 2020 ) enables gradient flow via the gating network and is different from Swish ( Ramachandran et al. , 2018 ) , which is the multiplication of pre-activation and sigmoid . • Finite vs Infinite Width . Lee et al . ( 2020a ) perform an extensive comparison of finite versus infinite width DNNs . An aspect that is absent in their work , but present in the dual view is the disentanglement of gates and weights , and the fact that the learning in gates is crucial for finite width network to outperform infinite width DNNs . In our paper , we make use of theory developed for infinite width DNNs to provide empirical insights into inner workings of finite width networks . • Capacity . Our experiments on destruction of layers , and providing constant 1 input are direct consequences of the insights from dual view theory . These are not explained by mere capacity based studies showing DNNs are powerful to fit even random labelling of datasets ( Zhang et al. , 2016 ) . 2 PRIOR WORK : NEURAL TANGENT KERNEL AND DUAL VIEW . In this section , we will focus on the dual view ( Lakshminarayanan & Singh , 2020 ) and how the dual view helps to address the open question in the NTK theory . We begin with a brief summary of NTK . NTK . An important kernel associated with a DNN is its neural tangent kernel ( NTK ) , which , for a pair of input examples x , x′ ∈ Rdin , and network weights Θ ∈ Rdnet , is given by : NTK ( x , x′ ) = 〈∇Θŷ ( x ) , ∇Θŷ ( x′ ) 〉 , where ŷΘ ( · ) ∈ R is the DNN output . Prior works ( Jacot et al. , 2018 ; Arora et al. , 2019 ; Cao & Gu , 2019 ) have shown that , as the width of the DNN goes to infinity , the NTK matrix converges to a limiting deterministic matrix NTK∞ , and training an infinitely wide DNN is equivalent to a kernel method with NTK∞ . While , as a pure kernel NTK∞ performed better than prior kernels by more than 10 % , Arora et al . ( 2019 ) observed that on CIFAR-10 : CNTK-GAP : 77.43 % ≤ CNN-GAP : 83.30 % where , CNN-GAP is a convolutional neural network with global average pooling and CNTK-GAP is its corresponding NTK∞ matrix . Due to this performance gap of about 5− 6 % , they concluded that NTK∞ does not explain fully the success of DNNs , and explaining this gap was an open question . 2.1 DUAL VIEW FOR DNNS WITH RELUS : CHARACTERISING THE ROLE OF GATES . In the dual view , the computations are broken down path-by-path . The input and the gates ( in each path ) are encoded in a neural path feature vector and the weights ( in each path ) are encoded in a neural path value vector , and the output is the inner product of these two vectors . The learning in the gates and the learning in the weights are separated in a deep gated network ( DGN ) setup , which leads to the two main results of dual view presented in Section 2.1.2 and Section 2.1.3 , wherein , the neural path kernel , the Gram matrix of the neural path features will play a key role .

Summary: The authors of this paper proposes a DLGN (Deep Linear Gated Networks), a novel class of deep networks, inspired by a recent dual view where the computation in DNNs is broken into two parts: learning in the gates and learning in the weights. The DLGN disentangles the computations into 2 parts: (1) "primal" part between input and the pre-activations in the gating network, and (2) "dual" part in the weights network, conditioned on inputs and gates. DLGN’s performance recovers 83.5% of SOTA DNNs. This development may lead to more interpretable deep network models that are also highly performant.