A Geometric Perspective towards Neural Calibration via Sensitivity Decomposition

It is well known that vision classification models suffer from poor calibration in the face of data distribution shifts . In this paper , we take a geometric approach to this problem . We propose Geometric Sensitivity Decomposition ( GSD ) which decomposes the norm of a sample feature embedding and the angular similarity to a target classifier into an instance-dependent and an instance-independent component . The instance-dependent component captures the sensitive information about changes in the input while the instance-independent component represents the insensitive information serving solely to minimize the loss on the training dataset . Inspired by the decomposition , we analytically derive a simple extension to current softmax-linear models , which learns to disentangle the two components during training . On several common vision models , the disentangled model outperforms other calibration methods on standard calibration metrics in the face of out-of-distribution ( OOD ) data and corruption with significantly less complexity . Specifically , we surpass the current state of the art by 30.8 % relative improvement on corrupted CIFAR100 in Expected Calibration Error . Code available at https : //github.com/GT-RIPL/Geometric-Sensitivity-Decomposition.git . 1 Introduction . During development , deep learning models are trained and validated on data from the same distribution . However , in the real world sensors degrade and weather conditions change . Similarly , subtle changes in image acquisition and processing can also lead to distribution shift of the input data . This is often known as covariate shift , and will typically decrease the performance ( e.g . classification accuracy ) . However , it has been empirically found that the model ’ s confidence remains high even when accuracy has degraded [ 1 ] . The process of aligning confidence to empirical accuracy is called model calibration . Calibrated probability provides valuable uncertainty information for decision making . For example , knowing when a decision can not be trusted and more data is needed is important for safety and efficiency in real world applications such as self-driving [ 2 ] and active learning [ 3 ] . A comprehensive comparison of calibration methods has been studied for in-distribution ( IND ) data [ 4 ] , However , these methods lead to unsatisfactory performance under distribution shift [ 5 ] . To resolve the problem , high-quality uncertainty estimation [ 6 , 5 ] is required . Principled Bayesian methods [ 7 ] model uncertainty directly but are computationally heavy . Recent deterministic methods [ 8 , 9 ] propose to improve a model ’ s sensitivity to input changes by regularizing the model ’ s intermediate layers . In this context , sensitivity is defined as preserving distance between two different input samples through layers of the model . We would like to utlize the improved sensitivity to better detect 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . Out-of-Distribution ( OOD ) data . However , these methods introduce added architecture changes and large combinatorics of hyperparameters . Unlike existing works , we propose to study sensitivity from a geometric perspective . The last linear layer in a softmax-linear model can be decomposed into the multiplication of a norm and a cosine similarity term [ 10 , 11 , 12 , 13 ] . Geometrically , the angular similarity dictates the membership of an input and the norm only affects the confidence in a softmax-linear model . Counter-intuitively , the norm of a sample ’ s feature embedding exhibits little correlation to the hardness of the input [ 11 ] . Based on this observation , we explore two questions : 1 ) why is a model ’ s confidence insensitive to distribution shift ? 2 ) how do we improve model sensitivity and calibration ? We hypothesize that in part an insensitive norm is responsible for bad calibration especially on shifted data . We observe that the sensitivity of the angular similarity increases with training whereas the sensitivity of the norm remains low . More importantly , calibration worsens during the period when the norm increases while the angular similarity changes slowly . This shows a concrete example of the inability of the norm to adapt when accuracy has dropped . Intuitively , training on clean datasets encourages neural networks to always output increasingly large feature norm to continuously minimize the training loss . Because the probability of the prevalent class of an input is proportional to its norm , larger norms lead to smaller training loss when most training data have been classified correctly ( See Sec . 3.1 ) . This renders the norm insensitive to input differences because the model is trained to always output features with large norm on clean data . While we have put forth that the norm is poorly calibrated , we must emphasize that it can still play an important role in model calibration ( See Sec . 4.1 ) . To encourage sensitivity , we propose to decompose the norm of a sample ’ s feature embedding and the angular similarity into two components : instance-dependent and instance-independent . The instance-dependent component captures the sensitive information about the input while the instanceindependent component represents the insensitive information serving solely to minimize the loss on the training dataset . Inspired by the decomposition , we analytically derive a simple extension to the current softmax-linear model , which learns to disentangle the two components during training . We show that our model outperforms other deterministic methods ( despite their significant complexity ) and is comparable to multi-pass methods with fewer training hyperparameters in Sec . 4.1 . In summary , our contributions are four fold : • We study the problem of calibration geometrically and identify that the insensitive norm is responsible for bad calibration under distribution shift . • We derive a principled but simple geometric decomposition that decomposes the norm into an instance-dependent and instance-independent component . • Based on the decomposition , we propose a simple training and inference scheme to encourage the norm to reflect distribution changes . • We achieve state of the art results in calibration metrics in the face of corruptions while having arguably the simplest calibration method to implement . 2 Related Work . Methods dedicated to strengthening calibration can be divided into two camps : multi-pass models and single-pass deterministic models . The current state-of-the-art multi-pass models are : Bayesian Monte Carlo Drop Out ( MCDO ) [ 7 ] and Deep Ensembles [ 14 ] . Bayesian methods are the most principled way to model uncertainty . Instead of optimizing max likelihood for a single set of parameters , Bayesian methods obtain a posterior distribution over possible parameters given a prior distribution over parameters and calculated data likelihood assuming some process noise . The posterior distribution over parameters captures epistemic uncertainty or uncertainty due to the limits of what the model knows . The final predictive distribution is obtained by marginalizing out model parameters . While Bayesian methods are theoretically sound , they are intractable in practice . Deep Ensembles averages multiple models trained using different random initialization so they learn different classification functions . A recent trend is to use a single-pass deterministic non-Bayesian model to improve uncertainty estimation . Two recent works DUQ [ 8 ] and SNGP [ 9 ] propose to improve uncertainty-awarenesss of deterministic networks by improving the networks ’ sensitivity to input changes . Intuitively , a sensitive model should map samples further from the training data as they become more out-of-distribution . This can be achieved at two levels : feature level and output level . At the feature level , both methods require the feature extractors ( CNNs ) to be regularized to prevent feature collapse , which is the mapping of two different data points to the same embedded vector . This is ensured by having input distance awareness , which is equivalent to ensuring bi-Lipschitz continuity [ 15 ] . In order to achieve this , DUQ [ 8 ] uses a two-sided gradient penalty [ 16 ] and SNGP [ 9 ] uses bounded spectral normalization [ 15 ] . The output level needs to reflect the changes in feature space . This can be done by adopting distance-aware classifiers . DUQ [ 8 ] uses a RBF networks with learned centroids for each class and SNGP [ 9 ] uses an approximate Gaussian Process layer . We were inspired by temperature scaling [ 4 ] , which is another method for bettering calibration , but fails under distribution shift [ 5 ] . Our method does not require input distance awareness and instead leverages the geometric intuitions about the output layer , specifically properties of the norm of the input embedding . 3 Method . Following our hypothesise that the insensitivity of the norm is responsible for bad calibration on distribution shifted data , we propose geometric sensitivity decomposition ( GSD ) for the norm . We first introduce the geometric perspective of the last linear layer in Sec . 3.1 and then derive GSD in Sec . 3.2 . To improve sensitivity of the norm and model calibration on shifted data , we propose a GSD-inspired training and inference procedure in Sec . 3.3 and Sec . 3.4 . 3.1 Norm and Similarity . The output layer of a neural network can be written as a dot-product < x , wy > , where x is the embedded input and wy is the weight vector associated with class y . Though seemingly simple there are strong geometric and calibration related intuitions drawn from this . Several prior works [ 10 , 12 , 11 ] have studied the effects decomposition of the last linear layer in a softmax model can have on classification . The output layer can be decomposed into angular similarity cosφy and norm ‖x‖2 . P ( y|x ) = exp ly∑c j=1 exp lj = exp ( ‖wy‖2‖x‖2 cosφy ) ∑c j=1 exp ( ‖wj‖2‖x‖2 cosφj ) ( 1 ) where ‖wy‖2 is the norm of a specific classifier in the linear layer . We ’ ll use this geometric view of the linear layer instead of the dot-product representation . Based on this perspective , we base the foundation of our work on the following observations from prior works [ 10 , 12 , 11 ] : 1 ) The probability/confidence of the prevalent class of an input is proportional to its norm [ 12 ] . 2 ) While the norm of a feature strongly scales the predictive probability , due to it ’ s unregularized nature the norm is not sensitive to the hardness of the input [ 11 ] . In other words , the norm could be the reason for bad sensitivity of the confidence to input distribution shift . Consequently , the insensitive norm can be causally related to bad calibration . We will examine a strong correlation between the quality of calibration and the magnitude of norm in Sec . 4.2 . 3.2 Geometric Sensitivity Decomposition of Norm and Angular Similarity . To motivate the subsequent geometric decomposition , we can revisit the softmax model , P ( y|x ) ∝ exp ( ‖wy‖2‖x‖2 cosφy ) . There are three terms contributing to the magnitude of the exponential function , ‖wy‖2 , ‖x‖2 and cosφy . Due to weight regularizations , ‖wy‖2 is most likely very small , while cosφy ∈ [ −1 , 1 ] . Therefore , the only way to obtain a high probability/confidence on training data and minimize cross-entropy loss is to 1 ) push the norm ‖x‖2 to a large value and 2 ) keep cos |φy| of the ground truth class close to one , i.e. , |φy| close to zero . This is further supported by [ 17 ] , where it was shown that logits of the ground truth class must diverge to infinity in order to minimize cross-entropy loss under gradient descent . In this process , models tend towards large norms and small angles for all training samples . Therefore , we propose to decompose the norms of features into two components : an instanceindependent scalar offset and an instance-dependent variance factor , which we define in Eq . 2 . The role of the instance-independent offset Cx is to minimize the loss on the entire training set and the instance-dependent component ∆x accounts for differences in samples . Therefore , if we can disentangle the instance-independent component from the instance-dependent component , we can obtain a norm that is sensitive to the hardness of data . Following this logic , we decompose the norm into two components . ‖x‖2 = ‖∆x‖2 + Cx ( 2 ) Similarly , we relax the angles such that the predicted angular similarity does not need to be close to one on the training data , i.e. , making the angles larger . To achieve this , we introduce an instanceindependent relaxation angle Cφ and an instance-dependent angle ∆φy . Analogous to the norm decomposition , the scalar Cφ serves solely to minimize the training loss while the instance-dependent ∆φy accounts for differences in samples . Because we need to account for the sign of the angle , we put an absolute value on it . |φy| = |∆φy| − |Cφ| ( 3 ) The ‖∆x‖2 , |∆φy| are the instance-dependent components and Cx , |Cφ| are the instance-independent components . We can rewrite the pre-softmax logits in Eq . 1 with the decomposed norm and angular similarity . ( Detailed derivation in Sec . A.1 in the Appendix . ) ‖x‖2 cosφy = ‖x‖2 cos |φy| = ( ‖∆x‖2 + Cx ) cos ( |∆φy| − |Cφ| ) ( 4 ) = ( ‖∆x‖2 + Cx ) 1 cos |Cφ| cos |∆φy| ( 1− sin |Cφ|2 ( 1− cos |Cφ| sin |∆φy| sin |Cφ| cos |∆φy| ) ) We can simplify the equation by assuming cos |φy| is close to one , which means |φy| is small . This is due to the fact that |φy| is the angle between the correct class weight and x , which means as training ensues , the angle converges to 0 and thus the cosine similarity converges to 1 . ( Please see Sec . A.2 for empirical support . ) cos |Cφ| sin |∆φy| sin |Cφ| cos |∆φy| = sin ( |∆φy|+ |Cφ| ) + sin |φy| sin ( |∆φy|+ |Cφ| ) − sin |φy| ≈ 1 ( 5 ) Therefore , Eq . 4 , omitting the absolute value on angles because cos is an even function , simplifies : ‖x‖2 cosφy ≈ ( ‖∆x‖2 + Cx ) 1 cos Cφ cos ∆φy ( 6 ) = ( 1 cos Cφ ‖∆x‖2 + 1 cos Cφ Cx ) cos ∆φy = ( 1 α ‖∆x‖2 + β α ) cos ∆φy Because cos Cφ and Cx are instance-independent , we denote them as α and β respectively . This geometric decomposition of norm and cosine similarity inspires us to include α and β as free trainable parameters in a new network and the network can learn to predict the more inputsensitive ‖∆x‖2 and ∆φy instead of the original ‖x‖2 and φy . While both the angle and norm can be decomposed we direct the focus to the norm as the angle is already calibrated to accuracy [ 11 ] . In other words , angles have been shown to be sensitive to input changes in [ 11 ] .

Summary: The paper proposes Geometric Sensitivity Decomposition (GSD) as the new calibration method in the face of out-of-distribution (OOD) data. The paper identifies that the insensitive norm is responsible for bad calibration under distribution shift. Based on GSD, the paper proposes a simple and quick training and inference scheme. The paper achieves SOTA in calibration metrics in the face of corruptions while having arguably the simplest calibration method to implement.