Decision boundary variability and generalization in neural networks

This paper discovers that the neural network with lower decision boundary ( DB ) variability has better generalizability . Two new notions , algorithm DB variability and ( , η ) -data DB variability , are proposed to measure the decision boundary variability from the algorithm and data perspectives . Extensive experiments show significant negative correlations between the decision boundary variability and the generalizability . From the theoretical view , two lower bounds based on algorithm DB variability are proposed and do not explicity depend on the sample size . By Chebyshev ’ s inequality , we also prove two upper bounds of O ( √ AV ( fQ , D ) /m ( k − 1 ) ) and O ( 1√ m + + η log 1η ) based on algorithm and data DB variability , respectively . The algorithm DB variability upper bound is easier to calculate in practice , while the data DB variability upper bound relies on less assumptions . Moreover , the bounds do not explicitly depend on the network size , which is usually prohibitively large in deep learning . 1 INTRODUCTION . Neural networks ( NNs ) have achieved significant success in vast applications ( Krizhevsky et al. , 2012 ; Vaswani et al. , 2017 ) . However , the advance of NNs is arduous to be characterized by canonical statistical hypothesis complexity , such as VC-dimension ( Vapnik et al. , 1994 ) or Rademacher complexity ( Bartlett & Mendelson , 2002 ) . Considering the intimate connection between low variance and significant performance in learning theory , we investigate decision boundaries ( DBs ) from the perspective of variability in this paper . For neural networks , the decision boundary variability is largely caused by ( 1 ) the randomness from the training algorithm , and ( 2 ) the training data . In this paper , they are measured by two new terms , algorithm DB variability and ( , η ) -data DB variability , and networks with lower DB variability are proved to possess better generalization performance . Algorithm DB variability measures the variability of DBs in different training repeats , and extensive experiments are conducted on CIFAR-10 , and CIFAR-100 ( Krizhevsky & Hinton , 2009 ) datasets to explore the factors influence algorithm DB variability . We visualize the trend of the algorithm DB variability with respect to ( w.r.t . ) different training strategies , training time , sample sizes , and label noise . The empirical results present the negative correlation between algorithm DB variability and the test accuracy in all scenarios , which suggests that the algorithm DB variability largely indicates the generalization of neural networks . From the theoretical view , two lower bounds and an upper bound of the generalization error are proved based on the algorithm DB variability . The upper bound has a rate of O ( √ AV ( fQ , D ) /m ( k − 1 ) ) , where AV ( fQ , D ) is the algorithm DB variability for fQ on data generating distribution D , m is the sample size , and k is the number of classes . To present how the training data influences the decision boundary variability , the ( , η ) -data DB variability is introduced by employing the η-subset to reconstruct decision boundaries with error ( see Definition 5.1 ) . If a decision boundary can be reconstructed by training on a smaller η-subset with a smaller reconstruction error , the decision boundary possesses smaller data DB variability . Moreover , an η- curve can be drawn by varying the value of η and numerically evaluating the label mismatch rate . The area under the η- curve could be a more elaborate indicator for the generalization of NNs because the curve contains richer information than the algorithm DB variability . An O ( 1√ m + + η log 1η ) generalization bound based on the ( , η ) -data DB variability is established theoretically to enhance the relationship between the generalization of NNs and DB variability . Compared to the algorithm DB variability upper bound , the data DB variability bound does not depend on additional Assumption 2 , while the algorithm DB variability is easier to calculate than data DB variability in practice . Moreover , in contrast to many existing generalization bounds based on hypothesis complexity ( Bartlett et al. , 2019 ; Golowich et al. , 2018 ; Bartlett et al. , 2017 ) that require access to the network weight , our generalization bounds based on DB variability only demand the network predictions and thus have advantages in empirically approximating the generalization bound in ( 1 ) black-model settings , where model parameters are unavailable ; and ( 2 ) over-parameterized settings , where calculating the weight norm is of prohibitively high computing burden . 2 RELATED WORKS . Deep learning theory . Due to the bias-variance trade-off , model complexity faces a dilemma in conventional wisdom ( Mohri et al. , 2018 ) . Recently , Zhang et al . ( 2021 ) reveal the surprising ability of neural networks in fitting noise , but the networks still have impressive generalization performance in practice . This gap between the well-known bias-variance trade-off and the universal approximation ability for NNs draws attention to numerous researchers ( Belkin et al. , 2019 ; Nakkiran et al. , 2019 ) . Many works attribute the success of NNs to the effectiveness of the stochastic gradient descent ( SGD ) algorithm ( Bottou , 2010 ; Hardt et al. , 2016 ; Jin et al. , 2017 ) . Some empirical studies also explain the decent performance of networks by uncovering their learning properties ( Nakkiran et al. , 2020 ; Jiang et al. , 2021 ; He et al. , 2021 ) . For instance , neural networks tend to fit the low-frequency information first ( Rahaman et al. , 2019 ; Xu et al. , 2019 ) and gradually learn a more complex function ( Kalimeris et al. , 2019 ) during the training procedure . Decision boundaries and generalization . Recent studies attempt to understand neural networks from the aspect of decision boundaries ( He et al. , 2018 ; Karimi et al. , 2019 ) . Guan & Loew ( 2020 ) empirically show the negative correlation between the complexity of decision boundary and generalization performance . Mickisch et al . ( 2020 ) reveal the phenomenon that the distance from data to decision boundaries continuously decreases during the training procedure . More recently , researchers find that NNs only rely on the most discriminative or simplest features to construct the DBs ( Ortiz-Jimenez et al. , 2020 ; Shah et al. , 2020 ) . Instead , our approach is different from these former methods by considering decision boundary variability , which is empirically and theoretically shown to closely correlate with the generalization in neural networks . Adversarial training and generalization . It has been shown that the adversarial examples , which are created by adding non-perceivable perturbation on the input data , can completely mislead the NNs ( Szegedy et al. , 2013 ; Goodfellow et al. , 2014 ) . To tackle this problem , adversarial training is proposed to improve the robustness of the NNs through continuous training on adversarial examples ( Madry et al. , 2017 ) . Nevertheless , Su et al . ( 2018 ) and Zhang et al . ( 2019 ) show a trade-off between the robustness and the generalization performance of NNs . 3 PRELIMINARIES . We denote the training set by S = { ( xi , yi ) , i = 1 , . . . , m } , where xi ∈ Rn , n is the dimension of input data , yi ∈ { 1 , . . . , k } , k is the number of classes , and m = |S| is the training sample size . We assume that ( xi , yi ) are independent and identically distributed ( i.i.d . ) random variables drawn from the data generating distribution D. Denote the classifier as fθ ( x ) : Rn → Rk , which is a neural network parameterized by θ . The output of fθ ( x ) is a k-dimensional vector and is assumed to be a discrete probability density function . Let f ( i ) θ ( x ) be the i-th component of fθ ( x ) , hence∑k i=1 f ( i ) θ ( x ) = 1 . We define T ( fθ , x ) = { i ∈ { 1 , · · · , k } |f ( i ) θ ( x ) = maxj f ( j ) θ ( x ) } to denote the set of predicted labels by fθ on x . Due to the randomness of the learning algorithm A , let Q ( θ ) = A ( S ) denote the posteriori distribution returned by the learning algorithm A leveraged on the training set S. Hence , we focus on the Gibbs classifier ( a.k.a . random classifier ) fQ = { fθ|θ ∼ Q } . 0− 1 loss is employed in this paper , and the expected risks in terms of θ and Q are defined as : RD ( θ ) = E ( x , y ) ∼D [ I ( y /∈ T ( fθ , x ) ) ] ( 1 ) and RD ( Q ) = E ( x , y ) ∼DEθ∼Q [ I ( y /∈ T ( fθ , x ) ) ] , ( 2 ) respectively . Here I ( · ) is the indicator function . Since the data generating distributionD is unknown , evaluating the expected risk RD is not practical . Therefore , it is a practical way to estimate the expected risk by the empirical riskRS , which is defined as : RS ( θ ) = E ( x , y ) ∼S [ I ( y /∈ T ( fθ , x ) ) ] = 1 m m∑ i=1 I ( yi /∈ T ( fθ , xi ) ) ( 3 ) RS ( Q ) = E ( x , y ) ∼SEθ∼Q [ I ( y /∈ T ( fθ , x ) ) ] = 1 m m∑ i=1 Eθ∼Q [ I ( yi /∈ T ( fθ , xi ) ) ] , ( 4 ) where ( xi , yi ) ∈ S and m = |S| . 3.1 DECISION BOUNDARY . If the output k-dimensional vector fθ ( x ) on the input example x has a tie , i.e. , the maximum value of the vector is not unique , then x is considered to locate on the decision boundary of fθ . With this idea , the decision boundary can be formally defined as below : Definition 3.1 ( decision boundary ) . Let fθ ( x ) : Rn → Rk be a classifier network parameterized by θ . Then the decision boundary of fθ is defined by { x ∈ Rn|∃i , j ∈ { 1 , · · · , k } , i 6= j , f ( i ) θ ( x ) = f ( j ) θ ( x ) = maxq f ( q ) θ ( x ) } ( 5 ) After defining the decision boundary , we have the following remark : Remark 1 . ( 1 ) If an input example ( x , y ) is not located on the decision boundary of fθ , T ( fθ , x ) is a singleton , and we have I ( y /∈ T ( fθ , x ) ) = I ( y 6= T ( fθ , x ) ) . ( 6 ) ( 2 ) If the input x is a boundary point , in practice , we randomly draw a label from the set T ( fθ , x ) as the prediction of fθ on x .

Summary: The authors introduce the notion of decision boundary (DB) variability and develop generalization bounds in terms of this.  They consider two notions, that of 'algorithm DB variability' and 'epsilon-eta DB variability', the latter a generalization of the former.  They show upper and lower bounds for the risk of classifiers in terms of DB variability under a variety of assumptions, as well as experimentally verify some of their claims.