Information Plane Analysis of Deep Neural Networks via Matrix--Based Renyi's Entropy and Tensor Kernels

1 INTRODUCTION . Although Deep Neural Networks ( DNNs ) are at the core of most state–of–the art systems in computer vision , the theoretical understanding of such networks is still not at a satisfactory level ( Shwartz-Ziv & Tishby , 2017 ) . In order to provide insight into the inner workings of DNNs , the prospect of utilizing the Mutual Information ( MI ) , a measure of dependency between two random variables , has recently garnered a significant amount of attention ( Cheng et al. , 2018 ; Noshad et al. , 2019 ; Saxe et al. , 2018 ; Shwartz-Ziv & Tishby , 2017 ; Yu et al. , 2018 ; Yu & Principe , 2019 ) . Given the input variable X and the desired output Y for a supervised learning task , a DNN is viewed as a transformation of X into a representation that is favorable for obtaining a good prediction of Y . By treating the output of each hidden layer as a random variable T , one can model the MI I ( X ; T ) between X and T . Likewise , the MI I ( T ; Y ) between T and Y can be modeled . The quantities I ( X ; T ) and I ( T ; Y ) span what is referred to as the Information Plane ( IP ) . Several works have demonstrated that one may unveil interesting properties of the training dynamics by analyzing DNNs in the form of the IP ( Yu & Principe , 2019 ; Goldfeld et al. , 2019 ; Noshad et al. , 2019 ; Chelombiev et al. , 2019 ) . Figure 1 , produced using our proposed estimator , illustrates one such insight that is similar to the observations of Shwartz-Ziv & Tishby ( 2017 ) , where training can be separated into two distinct phases , the fitting phase and the compression phase . This claim has been highly debated as subsequent research has linked the compression phase to saturation of neurons ( Saxe et al. , 2018 ) or clustering of the hidden representations ( Goldfeld et al. , 2019 ) . Contributions We propose a novel approach for estimating MI , wherein a kernel tensor-based estimator of Rényi ’ s entropy allows us to provide the first analysis of large-scale DNNs as commonly found in state-of-the-art methods . We further highlight that the multivariate matrix–based approach , proposed by Yu et al . ( 2019 ) , can be viewed as a special case of our approach . However , our proposed method alleviates numerical instabilities associated with the multivariate matrix– based approach , which enables estimation of entropy for high-dimensional multivariate data . Further , using the proposed estimator , we investigate the claim of Cheng et al . ( 2018 ) that the entropy H ( X ) ≈ I ( T ; X ) and H ( Y ) ≈ I ( T ; Y ) in high dimensions ( in which case MI-based analysis would be meaningless ) and illustrate that this does not hold for our estimator . Finally , our results indicate that the compression phase is apparent mostly for the training data , particularly for more challenging datasets . By utilizing a technique such as early-stopping , a common technique to avoid overfitting , training tends to stop before the compression phase occurs ( see Figure 1 ) . This may indicate that the compression phase is linked to the overfitting phenomena . 2 RELATED WORK . Analyzing DNNs in the IP was first proposed by Tishby & Zaslavsky ( 2015 ) and later demonstrated by Shwartz-Ziv & Tishby ( 2017 ) . Among other results , the authors studied the evolution of the IP during the training process of DNNs and noted that the process was composed of two different phases . First , an initial fitting phase where I ( T ; Y ) increases , followed by a phase of compression where I ( X ; T ) decreases . These results were later questioned by Saxe et al . ( 2018 ) , who argued that the compression phase is not a general property of the DNN training process , but rather an effect of different activation functions . However , a recent study by Noshad et al . ( 2019 ) seems to support the claim of a compression phase , regardless of the activation function . The authors argue that the base estimator of MI utilized in Saxe et al . ( 2018 ) might not be accurate enough and demonstrate that a compression phase does occur , but the amount of compression can vary between different activation functions . Another recent study by ( Chelombiev et al. , 2019 ) also reported a compression phase , but highlighted the importance of adaptive MI estimators . They also showed that when L2-regularization was included in the training compression was observed , regardless of activation function . Also , some recent studies have discussed the limitations of the IP framework for analysis and optimization for particular types of DNN ( Kolchinsky et al . ( 2019 ) ; Amjad & Geiger ( 2019 ) ) . On a different note , Cheng et al . ( 2018 ) proposed an evaluation framework for DNNs based on the IP and demonstrated that MI can be used to infer the capability of DNNs to recognize objects for an image classification task . Furthermore , the authors argue that when the number of neurons in a hidden layer grows large , I ( T ; X ) and I ( Y ; T ) barely change and are , using Cheng et al . ( 2018 ) terminology , approximately deterministic , i.e . I ( T ; X ) ≈ H ( X ) and I ( T ; Y ) ≈ H ( Y ) . Therefore , they only model the MI between X and the last hidden layer , that is the output of the network , and the last hidden layer and Y . Yu & Principe ( 2019 ) initially proposed to utilize empirical estimators for Rényi ’ s MI for investigating different data processing inequalities in stacked autoencoders ( SAEs ) . They also claimed that the compression phase in the IP of SAEs is determined by the values of the SAE bottleneck layer size and the intrinsic dimensionality of the given data . Yu et al . ( 2019 ) also extended the empirical estimators for Rényi ’ s MI to the multivariate scenario and applied the new estimator to simple but realistic Convolutional Neural Networks ( CNNs ) ( Yu et al. , 2018 ) . However , the results so far suffer from high computational burden and are hard to generalize to deep and large-scale CNNs . 3 MATRIX–BASED MUTUAL INFORMATION . Here , we review Rényi ’ s α-order entropy and its multivariate extension proposed by Yu et al . ( 2019 ) . 3.1 MATRIX–BASED RÉNYI ’ S ALPHA-ORDER ENTROPY . Rényi ’ s α-order entropy is a generalization of Shannon ’ s entropy ( Shannon , 1948 ; Renyi , 1961 ) . For a random variable X with probability density function ( PDF ) f ( x ) with support X , Rényi ’ s α-order entropy is defined as : Hα ( f ) = 1 1− α log ∫ X fα ( x ) dx ( 1 ) Equation 1 has been widely applied in machine learning ( Principe , 2010 ) , and the particular case of α = 2 , combined with Parzen window density estimation ( Parzen , 1962 ) , form the basis for Information Theoretic Learning ( Principe , 2010 ) . However , accurately estimating PDFs in highdimensional data , which is typically the case for DNNs , is a challenging task . To avoid the problem of high-dimensional PDF estimation , Giraldo et al . ( 2012 ) proposed a nonparametric framework for estimating entropy directly from data without resorting to kernel density estimation : Definition 3.1 . ( Giraldo et al. , 2012 ) Let xi ∈ X , i = 1 , 2 , . . . , N denote data points and let κ : X × X 7→ R be an infinitely divisible positive definite kernel ( Bhatia , 2006 ) . Given the kernel matrix K ∈ RN×N with elements ( K ) ij = κ ( xi , xj ) and the matrix A , ( A ) ij = 1N ( K ) ij√ ( K ) ii ( K ) jj , the matrix–based Rényi ’ s α–order entropy is given by Sα ( A ) = 1 1− α log2 ( tr ( Aα ) ) = 1 1− α log2 [ N∑ i=1 λi ( A ) α ] . ( 2 ) Here , λi ( A ) denotes the ith eigenvalue of the matrix A . The properties of this quantity was analysed in detail in Giraldo et al . ( 2012 ) . It was shown that the kernel matrix A , obtained from the raw data , acts much as a density matrix in quantum information theory ( Nielsen & Chuang , 2011 ) . It was further shown that the ( Gram ) matrix A is related to an empirical covariance operator on embeddings of probability distributions in a Reproducing Kernel Hilbert Space ( RKHS ) . This is similar to the approach of maximum mean discrepancy and the kernel mean embedding ( Gretton et al. , 2012 ; Muandet et al. , 2017 ) . Moreover , Giraldo et al . ( 2012 ) showed that under certain conditions Equation 2 converges to the trace of the underlying covariance operator , as shown in Proposition B.1 in Appendix B . Notice that the dimensionality of the data does not appear in Proposition B.1 . This means that Sα ( A ) captures properties of the distribution , with a certain robustness with respect to high-dimensional data . This is a beneficial property compared to KNN and KDE based information estimators used in previous works ( Saxe et al. , 2018 ; Chelombiev et al. , 2019 ) , which have difficulties handling high-dimensional data ( Kwak & Chong-Ho Choi , 2002 ) . Not all estimators of entropy have the same property ( Paninski , 2003 ) . Certain approaches developed for estimating the Shannon entropy suffer from the curse of dimensionality ( Kwak & Chong-Ho Choi , 2002 ) . Also , there is no need for any binning procedure utilized in previous works ( Shwartz-Ziv & Tishby , 2017 ) , which are known to struggle with the relu activation function commonly used in DNN ( Saxe et al. , 2018 ) . In Appendix A we have conducted experiments on synthetic data to illustrate the behaviours of these estimators for high-dimensional data . Remark 1 . In the limit when α→ 1 , Equation 2 reduces to the matrix–based von Neumann entropy ( Nielsen & Chuang , 2011 ) that resembles Shannon ’ s definition over probability states , and can be expressed as lim α→1 Sα ( A ) = − N∑ i=1 λi ( A ) log2 [ λi ( A ) ] . ( 3 ) For completeness , the proof of Equation 3 can be found in Appendix C. In addition to the definition of matrix based entropy , Giraldo et al . ( 2012 ) define the joint entropy between x ∈ X and y ∈ Y as Sα ( AX , AY ) = Sα ( AX ◦AY tr ( AX ◦AY ) ) , ( 4 ) where xi and yi are two different representations of the same object and ◦ denotes the Hadamard product . Finally , the MI is , similar to Shannon ’ s formulation , defined as Iα ( AX ; AY ) = Sα ( AX ) + Sα ( AY ) − Sα ( AX , AY ) . ( 5 )

Summary: In this paper, the authors try to resolve the problem of estimating mutual information between high-dimensional layers in neural network. Specifically, the authors try to tackle the convolutional neural networks, by propose a novel tensor-kernel based estimator. They use the proposed estimator to discover the similar trends on the information plane as the previous works.