How Important is a Neuron

The problem of attributing a deep network ’ s prediction to its input/base features is well-studied ( cf . Simonyan et al . ( 2013 ) ) . We introduce the notion of conductance to extend the notion of attribution to understanding the importance of hidden units . Informally , the conductance of a hidden unit of a deep network is the flow of attribution via this hidden unit . We can use conductance to understand the importance of a hidden unit to the prediction for a specific input , or over a set of inputs . We justify conductance in multiple ways via a qualitative comparison with other methods , via some axiomatic results , and via an empirical evaluation based on a feature selection task . The empirical evaluations are done using the Inception network over ImageNet data , and a convolutional network over text data . In both cases , we demonstrate the effectiveness of conductance in identifying interesting insights about the internal workings of these networks . 1 BACKGROUND AND MOTIVATION . The problem of attributing a deep network ’ s prediction to its input is well-studied ( cf . Baehrens et al . ( 2010 ) ; Simonyan et al . ( 2013 ) ; Shrikumar et al . ( 2017 ) ; Binder et al . ( 2016 ) ; Springenberg et al . ( 2014 ) ; Lundberg & Lee ( 2017 ) ; Sundararajan et al . ( 2017 ) ) . Given a function F : Rn → [ 0 , 1 ] that represents a deep network , and an input x = ( x1 , . . . , xn ) ∈ Rn , an attribution of the prediction at input x is ( a1 , . . . , an ) ∈ Rn where ai can be interpreted as the contribution of xi to the prediction F ( x ) . For instance , in an object recognition network , an attribution method could tell us which pixels of the image were responsible for a prediction . Attributions help determine the influence of base features on the input . Many papers on deep learning speculate about the importance of a hidden unit towards a prediction . They tend to use the activation value of the hidden unit or its product with the gradient as a proxy for feature importance . As we discuss later in the paper , both measures have undesirable behavior . For instance , a ReLU always has positive activation values but could either have positive or negative influence on the prediction ; the fact that the sign of the influence can not be identified is undesirable . In the next subsection , we introduce our approach to the problem of neuron importance . We compare it to the related work in Secion 2 . 1.1 OUR CONTRIBUTION . We introduce the notion of conductance ( see Section 3 ) based on a previously studied attribution technique called Integrated Gradients ( Sundararajan et al . ( 2017 ) ) . Integrated Gradients rely on integrating ( summing ) the gradients ( of the output prediction with respect to the input ) over a series of carefully chosen variants of the input . Informally , the conductance of a hidden unit of a deep network is the flow of Integrated Gradients ’ attribution via this hidden unit . The key idea behind conductance is to decompose the computation of Integrated Gradients via the chain rule ( see Equations 2 and 3 for formal definitions ) . We qualitatively compare conductance to other commonly used measures of neuron importance in Section 4.1 . Conductance takes into account the activation of the neuron , the partial derivative of a the output with respect to the neuron and the partial derivative of the neuron with respect to the input . Intuitively , all three quantities play an important role in judging the importance of a neuron . We find that other measures of neuron importance , like activation values and gradients , ignore one or more of these terms and this results in intuitively undesirable attributions . In Section 4.2 , we provide theoretical justification for conductance . To do this , we first state some desirable properties ( axioms ) of neuron importance . We show that neuron importances that satisfy these very natural axioms must be derived from a “ path method ” , i.e. , as a path integral of the gradients from the output to the hidden layer ( Theorem 1 ) . Furthermore , under an additional condition of consistency to the base feature attributions , we are able to show that the path taken in the hidden layer must depend on the layers below ( Theorem 2 ) . Conductance naturally couples the path at the base features with the path at the hidden layer and we observe that it satisfies all of these axioms ( Theorem 3 ) . We also evaluate the effectiveness of conductance ( against several related methods ) empirically . We do a feature selection study to see if the high-conductance units are also highly predictive . The premise is that if some neurons are consistently important across instances of a certain class , then using these neurons as features , we should be able to classify other instances of the same class . Our empirical evaluations are done over the following two large-scale networks , the Inception architecture ( Szegedy et al . ( 2014 ) ) trained on ImageNet data ( Russakovsky et al . ( 2015 ) ) and a highly-cited convolution network ( Kim ( 2014a ) ) frequently used for text classification tasks . Though all our claims are about importance , and not about interpretability , we acknowledge that the main application of studying neuron importance is to interpret the function of the neuron . In this regard , we demonstrate that conductance can give us intuitive insights about the network . For instance , for the object detection task , we identify filters that are influential across images and labels , which seem to correspond to texture or color features . And for a sentiment analysis task , we use conductance to identify neurons that detect negation . 2 RELATED WORK . A recent paper ( Datta et al . ( 2018 ) ) is closely related to ours in intention and technique . It proposes a notion of influence of an hidden neuron on the output prediction . We explicitly compare our technique to theirs , both qualitatively and quantitatively . We show for instance , that their technique can sometimes result in importance scores with incorrect signs . There is also a large literature on understanding the function/operation of hidden units of a deep network ( cf . Mahendran & Vedaldi ( 2015 ) ; Dosovitskiy & Brox ( 2015 ) ; Yosinski et al . ( 2015 ) ; Mordvintse et al . ( 2015 ; 2017 ) ) . These approaches are usually optimization based , i.e . they explicitly tweak images to optimize activations of a neuron or a group of neurons . The resulting images are an indicator of the function of the hidden unit . However , these works don ’ t identity the importance of a hidden unit ( either to a prediction or to a class of predictions ) . Also , these approaches tend not to work for every hidden unit . Consequently , it is not clear if all the hidden units which are important to the predictive task have been investigated by these approaches . Another approach to identifying the function of hidden units is to build a simple , linear , explanatory model ( cf . Le et al . ( 2012 ) ; Alain & Bengio ( 2016 ) ) in parallel to the original model . But then it is no longer clear if the influence of the hidden units in the new model is the same as what it is in the original model ; e.g . if there are correlated hidden neurons , their weights in the explanatory model may not match those in the original model . Zhou et al . ( 2018 ) approach the interpretability of hidden neurons by aligning human-labeled concepts with the hidden neurons by using neuron activations for each concept . Finally , Kim et al . ( 2018 ) takes a human-defined concept as input , finds hidden neurons that are predictive of this concept , and identifies the influence of these neurons on a certain prediction . Because the network may rely on an inherently different representation from the human , it is not clear how this process identifies all the concepts important to the prediction for a given input or a class of inputs . In contrast , we rely on the network ’ s own abstraction , i.e. , the filters in the hidden layers . This will result in accounting for the whole prediction , but may possibly result in less intuitive explanations . 3 CONDUCTANCE . In this section , we define conductance formally . We start with some notation . Suppose we have a function F : Rn → [ 0 , 1 ] that represents a deep network . Specifically , let x ∈ Rn be the input at hand . We will define attribution methods relative to a baseline input x′ ∈ Rn . 1 For image networks , the baseline could be the black image , while for text models it could be the zero embedding vector . We now recall the definition of Integrated Gradients . We consider the straight-line path ( in Rn ) from the baseline x′ to the input x , and compute the gradients at all points along the path . Integrated gradients are obtained by cumulating these gradients . Specifically , integrated gradients are defined as the path integral of the gradients along the straight-line path from the baseline x′ to the input x . Here is the intuition for why the method works for , say , an object recognition network . The function F varies from a near zero value for the informationless baseline to its final value . The gradients of F with respect to the image pixels explain each step of the variation in the value of F . The integration ( sum ) over the gradients cumulates these micro explanations and accounts for the net difference between the baseline prediction score ( near zero ) and the prediction value at the input x . Formally , the integrated gradient for the ith base feature ( e.g . a pixel ) of an input x and baseline x′ is : IGi ( x ) : := ( xi − x′i ) · ∫ 1 α=0 ∂F ( x′+α ( x−x′ ) ) ∂xi dα ( 1 ) where ∂F ( x ) ∂xi is the gradient of F along the i th dimension at x . Notice that Integrated Gradients produces attributions for base features ( e.g . the pixels of an object recognition network ) . There is a natural way to ‘ lift ’ these attributions to a neuron in a hidden layer . Consider a specific neuron y in a hidden layer of a network . We can define the conductance of neuron y for the attribution to an input variable i as : Condyi ( x ) : := ( xi − x ′ i ) · ∫ 1 α=0 ∂F ( x′+α ( x−x′ ) ) ∂y · ∂y ∂xi dα ( 2 ) We can define the total conductance of the hidden neuron y by summing over the input variables : Condy ( x ) : := ∑ i ( xi − x′i ) · ∫ 1 α=0 ∂F ( x′+α ( x−x′ ) ) ∂y · ∂y ∂xi dα ( 3 ) We will interchangeably use the term “ conductance of neuron j ” to mean either of Equations 2 or 3 . We use the former to explain the function of the neuron in terms of its effect on the base features of the input . We use the latter to discuss the importance of the neuron . Frequently , we will aggregate over a set of logically related neurons that belong to a specific hidden layer . For instance , these could be neurons that belong to a single filter . In this case , we will sum over the conductances of the neurons in the set to define the conductance of the set as whole . Remark 1 . There is a different way to extend Integrated Gradients for neuron importance . The idea is to treat the neuron activations in the hidden layer under consideration as inputs of the network from that layer onwards . Then integrated gradients can be applied at this layer . This idea is described in ( Shrikumar et al . ( 2018 ) ) .

Summary: The authors propose a notion of conductance to attribute the deep neural network’s prediction to its hidden units. The conductance is the flow of attribution via the hidden unit(s) in consideration. The paper proposes using conductance to not only evaluate importance of hidden unit to the prediction for a specific input but also over a set of inputs. The strongest part of the analysis of conductance is that conductance naturally couples  the path at the base features with that of the hidden layer.