RESIDUAL NETWORKS CLASSIFY INPUTS BASED ON THEIR NEURAL TRANSIENT DYNAMICS

1 INTRODUCTION . Residual networks ( ResNets ) , first introduced in ( He et al. , 2016 ) , have been more successful in classification tasks in comparison with many other standard methods . This success is attributed to the skip connections between layers that facilitate the propagation of the gradient throughout the network , and in practice allow very deep networks to undergo a successful training . Apart from mitigating the gradient problem in deep networks , the skip connections introduce a dependency between variables in different layers that can be seen as a system state . This novelty provides an opportunity for interesting theoretical analysis of their functioning , and has been the underlying pillar for some interesting analysis of such networks from dynamical system point of view ( Ciccone et al. , 2018 ; Chang et al. , 2018 ; Haber & Ruthotto , 2017 ; Lu et al. , 2017 ; Liao & Poggio , 2016 ; Ruthotto & Haber , 2018 ; Chaudhari et al. , 2017 ) . This study mainly emphasizes the importance of internal transient dynamics in ResNets on classification performance . Using a dynamical system approach , for a general ResNet , we derive dynamics of state evolutions of the residuals in different layers . It is well-known that very deep residual networks with weight sharing are equivalent to shallow recurrent neural networks , with similar performance to ResNets with variable weights between layers ( Liao & Poggio , 2016 ) . Inspired by this work , we study ResNets with shared weights and sigmoid activation functions , which provide a more tractable mathematical analysis . Then , we show empirically that those dynamics are observed in ResNets with variable weights as well . To study the classification performance of ResNets on noisy inputs , we compare the dynamics of noise evolution in these networks with that of Multi-Layer Perceptrons ( MLP ) , and find that for small noise amplitudes , MLP has a higher value of signal to noise ratio . However , for increased noise amplitude , ResNets are more robust . The main contributions of the paper are : 1 . We show that network internal dynamics that are shaped by neural transient dynamics are well separated for each input class . 2 . We compare the transient dynamics of MLP and ResNets , and show how a noise term in the input evolves in the network , and how it affects the classification robustness . 3 . Based on residual dynamics , we develop a new method to obtain an adaptive depth for ResNets , during training , for input classification . 2 RELATED WORK . In ( Haber & Ruthotto , 2017 ) , the authors have studied residual networks using difference equations , and analyzed the stability of the forward propagation of input data , and have linked the inverse problem to the well-posedness of the learning problem . To circumvent the vanishing or exploding gradient problem , it is suggested in ( Haber & Ruthotto , 2017 ) to design the eigenvalues of the feedforward propagation close to the edge of stability , so that the inverse problem is not ill-posed . It is however not clear whether having the eigenvalues set close to the edge of stability is beneficial for the network performance , because it depends on the dynamics required by the actual task . Employing this idea , the authors in ( Chang et al. , 2018 ) suggest a new reversible architecture for neural networks based on Hamiltonian systems . Also , in a recent study , ResNets have been employed as an unrolled non-autonomous time-invariant ( with weight sharing ) system of differential equations ( Ciccone et al. , 2018 ) , wherein , each ResNet block receives an external input , which depends on the previous block . This successive process of feeding the following block by the output of the previous block continues until the latent space converges . Our approach in this paper is similar to the aforementioned studies , however , to understand the classification mechanism in ResNets , we focus on the role of the intrinsic transient dynamics of the residuals over different layers of the network . Some studies on ResNets have focused on tracking the features layer by layer ( Greff et al. , 2017 ; Chu et al. , 2017 ) , and have challenged the idea that deeper layers in neural networks build up abstract features that are different than those formed in lower layers . One supporting evidence for this challenge comes from lesion studies on ResNets ( Veit et al. , 2016 ) and Highway networks ( Srivastava et al. , 2015 ) which show that after the network is trained , perturbing the weights in the deep layers does not have a fundamental effect on the network performance , and therefore , does not bring the performance to chance level . However , changing the weights which are closer to initial layers , have more damaging effect . Empirical studies in ( Greff et al. , 2017 ; Chu et al. , 2017 ) suggest an alternative explanation for feature formation in deep layers ; that is , successive layers estimate the same features which , along the depth of the network , are more refined , and yield an estimate with smaller standard deviation than earlier layers . Our study supports this idea by showing that features in different layers of a ResNet with shared weights are formed by the transient dynamics of residuals that may converge towards their steady state values if they are stable . In attractor networks , perturbing the initial layers changes those dynamics more drastically compared to perturbation of deeper layers , because the residuals in the deep layers are either very close to their stable fixed point , or have already converged . If there are no attractors for the residuals , sensitivity to initial conditions and the internal dynamics of the residuals play an important role in classification . In this case , perturbations of the network at initial layers can potentially change the dynamic evolution of the residuals completely , and this will have a more sever impact on the output . Classification based on unstable internal states is similar to Reservoir networks ( Maass et al. , 2002 ) , where it has been shown that the high dimensionality of the neurons at the readout layer can compensate for the lack of stability of the neural activities . Moreover , one important topic in this domain is the depth of ResNets . On the one hand , the success of ResNets in classification has been attributed to their deep architecture ( He et al. , 2016 ) , on the other hand , there are studies that claim most of the training is accomplished in the initial layers , and having a very deep architecture is not necessary ( Zagoruyko & Komodakis , 2016 ) . Another challenge is to understand the generalization property of ResNets ( which may be related to its depth ) , because their power is correlated with their ability in recognizing unseen data that also belong to the classes that these networks have been trained on . In our analysis of the transient and steady state dynamics of the residuals , we discuss these issues . In fact , an important finding of this paper is that residual networks classify inputs based on summing over all the residual ’ s outputs throughout the network , meaning different transitions of residuals ( convergence to their steady state , or their long wandering trajectories without convergence ) can potentially change the classification result . Interestingly , in biological neuronal networks , it has been suggested that optimal stimulus separation in neurons that encode sensory information occurs during the transients rather than the fixed points of the neuronal activity trajectories ( Mazor & Laurent , 2005 ; Rabinovich et al. , 2008 ) . Also , it has been discussed that spatio-temporal processing in cortical circuits are state dependent , and the role of transients are crucial ( Buonomano & Maass , 2009 ) . In our study we show that also in ResNets , these transients are the decisive factors for classification . Based on this finding , we show how a noise term in the input is mapped to the output , and we develop a new method to control the depth of ResNets . 3 DYNAMICS OF INTERACTIONS BETWEEN RESIDUALS . We consider a dense ResNet with N input dimensions , and arbitrary T layers with exactly N neurons at each layer . A unique property of a ResNet that distinguishes it from conventional feedforward networks ( such as MLP ) is the skip connection between layers . In the ResNet we consider here , the activity of neuron i at layer t , before the output of the previous layer ( t − 1 ) is added to it , is represented as yi ( t ) , and the activity of all neurons in the same layer is represented by the vector y ( t ) . After the integration of the output from layer t − 1 , the output of layer t is represented by x ( t ) . The components of these residuals y ( t ) are calculated based on a linear function of x ( t ) , i.e . zi ( t ) = ∑N i=1 wij ( t ) xj ( t ) followed by a nonlinear function f ( zi ) . Figure 1 illustrates the relation between x and y . Any hidden layer t represents a sample of the dynamical states x after t steps . Input data is considered as the initial condition of the system , and is depicted by x ( 0 ) . Interpreting the network as a dynamical system which evolves throughout the layers , the dynamics of neural activations are x ( t + 1 ) = x ( t ) + y ( t + 1 ) , where y ( t ) is the output of the neurons , and in the rest of the paper , they are called ” residuals ” . This equation implies a difference equation for the variable x ( t ) , that is x ( t+1 ) −x ( t ) = y ( t+1 ) . The left side of this equation resembles the forward Euler method of derivative of a continuous system , when the discretization step is equal to 1 . This approximates a continuous system with dynamics that follow ẋi ( t ) = yi ( t ) . We are interested in understanding how the neural activities evolve over layers in a feedforward fully connected ResNet . The dynamics of x ( t ) and inputs to the residuals , z ( t ) , follow ẋi ( t ) = yi ( t ) = f ( zi ( t ) ) =⇒ xi ( t ) = ∫ t 0 yi ( τ ) dτ + xi ( 0 ) zi ( t ) = N∑ j=1 wij ( t ) xj ( t ) + bi =⇒ żi ( t ) = N∑ j=1 wij ( t ) ẋj ( t ) + ẇij ( t ) xj ( t ) ( 1 ) The first line of equation 1 indicates that x ( t ) stores the sum of the residuals and the input ( x ( 0 ) ) from the input layer up to layer t , meaning that x ( t ) is a cumulative signal for y ( t ) , as well as the input data . For a ResNet with shared weights between layers , ẇij ( t ) = 0 ; because the weights do not change between layers . This constraint makes the analysis simpler , and results in żi ( t ) = ∑N j=1 wij ẋj ( t ) . In this case , after replacing ẋi ( t ) by yi ( t ) in equation 1 , the dynamics of z ( t ) and the residuals y ( t ) will be żi ( t ) = N∑ j=1 wij ( t ) yj ( t ) ẏi ( t ) = ∂yi ( t ) ∂zi ( t ) ∂zi ( t ) ∂t = ∂f ( z ) ∂z ( N∑ j=1 wij ( t ) yj ( t ) ) . ( 2 ) A steady state solution for yi , in a network with a time invariant weight wij ( shared weights across layers ) , is obtained from setting either of the two terms on the right hand side of equation 2 to zero . For a general nonlinear f ( . ) , according to equation 2 , the interactions between the residuals are either competitive or cooperative , depending on the positive or negative influence that they have on each other . For a logistic function f ( . ) , the derivative is yi ( t ) ( 1− yi ( t ) ) , which results in a particular form of predator-prey equation , well-known in studying ecosystems . In this case , equation 2 yields ẏi ( t ) = yi ( t ) ( 1− yi ( t ) ) ( N∑ j=1 wij ( t ) yj ( t ) ) ( 3 ) For N residuals , the number of possible fixed points ( solutions of equation 3 ) is 3N . However , for a given weight W , only some of the fixed points are stable . The initial condition for the residuals is determined by the output of the first layer . After some transients over the next layers , each residual converges to its stable solution , if there is one . Note that the derivatives at yi = 0 or 1 are equal to zero , and the system ’ s trajectories are confined to this space . According to equation 1 and figure 1 , the cumulative of the residuals over the entire network feeds the classifier . Therefore , perturbations at initial layers of the network disrupt the output more severely than perturbations of the deeper layers ( lesion studies have considered weight perturbations , which reflects on neural activity perturbations ) . A discrete-time analysis of the effect of perturbations at different layers on the output value is given in the Appendix . To study the contribution of input noise ( ξ ) in the final hidden layer , we derive the noise evolution for the ResNet considered above ( a similar analysis applies to ResNets with continuous activation functions ) . Represented in vector notations , the signal plus noise evolution is obtained by replacing yi in equation equation 3 by y + ξ. ẏ + ξ̇ = ( y + ξ ) ( 1− y − ξ ) ( W ( y + ξ ) ) = y ( 1− y ) ( Wy ) + y ( 1− y ) ( Wξ ) + ξ ( 1− 2y ) ( W ( y + ξ ) ) − ξ2 ( W ( y + ξ ) ) ( 4 ) Assuming that y ( t ) is the solution of the neural activity at layer t for the input signal without noise ( that is reflected in initial condition y ( 1 ) ) , subtracting equation 3 from equation 4 yields ξ̇ = y ( 1− y ) ( Wξ ) + ξ ( 1− 2y ) ( W ( y + ξ ) ) − ξ2 ( W ( y + ξ ) ) = y ( 1− y ) ( W ( ξ + y − y ) ) + ξ ( 1− 2y − ξ ) ( W ( y + ξ ) ) = y ( 1− y ) ( Wy ) + y ( 1− y ) ( W ( ξ − y ) ) + ξ ( 1− 2y − ξ ) ( W ( y + ξ ) ) = ẏ + y ( 1− y ) ( W ( ξ − y ) ) + ξ ( 1− y − ( y + ξ ) ) ( W ( y + ξ ) ) ( 5 ) In equation 5 and equation 4 , the vectors y and ξ are functions of t. This equation indicates how the growth rate of the noise ( ξ̇ ) differs from the growth rate of the residuals ( ẏ ) .

Summary: In this paper the authors analyse the role of residuals in the performance of residual networks (res-nets) with shared layer weights and propose a method to adapt the depth of such networks during training. They exploit  the fact that res-nets identical blocks / shared layer weights are discrete time dynamical systems to conduct their analysis. Their main contribution seems to be an empirical evaluation of the role of transient dynamics in the performance of such res-nets. Experiments are done on a toy dataset on concentric circles and MNIST data.