Original: B IAS -R EDUCED U NCERTAINTY E STIMATION FOR D EEP N EURAL C LASSIFIERS
Revised: B IAS -R EDUCED U NCERTAINTY E STIMATION FOR D EEP N EURAL C LASSIFIERS

Original: A BSTRACT
Revised: A BSTRACT

Original: We consider the problem of uncertainty estimation in the context of (non-Bayesian) deep neural classiﬁcation.
Revised: We consider the problem of uncertainty estimation in the context of (non-Bayesian) deep neural classiﬁcation.

Original: In this context, all known methods are based on extracting uncertainty signals from a trained network optimized to solve the classiﬁcation problem at hand.
Revised: In this context, all known methods are based on extracting uncertainty signals from a trained network optimized to solve the classiﬁcation problem at hand.

Original: We demonstrate that such techniques tend to introduce biased estimates for instances whose predictions are supposed to be highly conﬁdent.
Revised: We demonstrate that such techniques tend to introduce biased estimates for instances whose predictions are supposed to be highly conﬁdent.

Original: We argue that this deﬁciency is an artifact of the dynamics of training with SGD-like optimizers, and it has some properties similar to overﬁtting.
Revised: We argue that this deﬁciency is an artifact of the dynamics of training with SGD-like optimizers, and it has some properties similar to overﬁtting.

Original: Based on this observation we develop an uncertainty estimation algorithm that selectively estimates the uncertainty of highly conﬁdent points using earlier snapshots of the trained model, before their estimates are jittered (and way before they are ready for actual classiﬁcation).
Revised: Based on this observation we develop an uncertainty estimation algorithm that selectively estimates the uncertainty of highly conﬁdent points using earlier snapshots of the trained model, before their estimates are jittered (and way before they are ready for actual classiﬁcation).

Original: We present extensive experiments indicating that the proposed algorithm provides uncertainty estimates that are consistently better than all known methods.
Revised: We present extensive experiments indicating that the proposed algorithm provides uncertainty estimates that are consistently better than all known methods.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: The deployment of deep learning models in applications with demanding decision-making components such as autonomous driving or medical diagnosis hinges on our ability to monitor and control their statistical uncertainties.
Revised: The deployment of deep learning models in applications with demanding decision-making components such as autonomous driving or medical diagnosis hinges on our ability to monitor and control their statistical uncertainties.

Original: Conceivably, the Bayesian framework offers a principled approach to infer uncertainties from a model; however, there are computational hurdles in implementing it for deep neural networks (Gal & Ghahramani, 2016).
Revised: Conceivably, the Bayesian framework offers a principled approach to infer uncertainties from a model; however, there are computational hurdles in implementing it for deep neural networks (Gal & Ghahramani, 2016).

Original: Presently, practically feasible (say, for image classiﬁcation) uncertainty estimation methods for deep learning are based on signals emerging from standard (non Bayesian) networks that were trained in a standard manner.
Revised: Presently, practically feasible (say, for image classiﬁcation) uncertainty estimation methods for deep learning are based on signals emerging from standard (non Bayesian) networks that were trained in a standard manner.

Original: The most common signals used for uncertainty estimation are the raw softmax response (Cordella et al., 1995), some functions of softmax values (e.g., entropy), signals emerging from embedding layers (Mandelbaum &
Revised: The most common signals used for uncertainty estimation are the raw softmax response (Cordella et al., 1995), some functions of softmax values (e.g., entropy), signals emerging from embedding layers (Mandelbaum &

Original: Weinshall, 2017), and the MC-dropout method (Gal & Ghahramani, 2016) that proxies a Bayesian inference using dropout sampling applied at test time.
Revised: Weinshall, 2017), and the MC-dropout method (Gal & Ghahramani, 2016) that proxies a Bayesian inference using dropout sampling applied at test time.

Original: These methods can be quite effective, but no conclusive evidence on their relative performance has been reported.
Revised: These methods can be quite effective, but no conclusive evidence on their relative performance has been reported.

Original: A recent NIPS paper provides documentation that an ensemble of softmax response values of several networks performs better than the other approaches (Lakshminarayanan et al., 2017).
Revised: A recent NIPS paper provides documentation that an ensemble of softmax response values of several networks performs better than the other approaches (Lakshminarayanan et al., 2017).

Original: In this paper, we present a method of conﬁdence estimation that can consistently improve all the above methods, including the ensemble approach of Lakshminarayanan et al.
Revised: In this paper, we present a method of conﬁdence estimation that can consistently improve all the above methods, including the ensemble approach of Lakshminarayanan et al.

Original: Given a trained classiﬁer and a conﬁdence score function (e.g., generated by softmax response activations), our algorithm will learn an improved conﬁdence score function for the same classiﬁer.
Revised: Given a trained classiﬁer and a conﬁdence score function (e.g., generated by softmax response activations), our algorithm will learn an improved conﬁdence score function for the same classiﬁer.

Original: Our approach is based on the observation that conﬁdence score functions extracted from ordinary deep classiﬁers tend to wrongly estimate conﬁdence, especially for highly conﬁdent instances.
Revised: Our approach is based on the observation that conﬁdence score functions extracted from ordinary deep classiﬁers tend to wrongly estimate conﬁdence, especially for highly conﬁdent instances.

Original: Such erroneous estimates constitute a kind of artifact of the training process with an stochastic gradient descent (SGD) based optimizers.
Revised: Such erroneous estimates constitute a kind of artifact of the training process with an stochastic gradient descent (SGD) based optimizers.

Original: During this process, the conﬁdence in “easy” instances (for which we expect prediction with high conﬁdence) is quickly and reliably assessed during the early SGD epochs.
Revised: During this process, the conﬁdence in “easy” instances (for which we expect prediction with high conﬁdence) is quickly and reliably assessed during the early SGD epochs.

Original: Later on, when the optimization is focused on the “hard” points (whose loss is still large), the conﬁdence estimates of the easy points become impaired.
Revised: Later on, when the optimization is focused on the “hard” points (whose loss is still large), the conﬁdence estimates of the easy points become impaired.

Original: Uncertainty estimates are ultimately provided in terms of probabilities.
Revised: Uncertainty estimates are ultimately provided in terms of probabilities.

Original: Nevertheless, as previously suggested (Geifman & El-Yaniv, 2017; Mandelbaum & Weinshall, 2017; Lakshminarayanan et al., 2017), in a non-Bayesian setting (as we consider here) it is productive to decouple uncertainty estimation into two separate tasks: ordinal ranking according to uncertainty, and probability calibration .
Revised: Nevertheless, as previously suggested (Geifman & El-Yaniv, 2017; Mandelbaum & Weinshall, 2017; Lakshminarayanan et al., 2017), in a non-Bayesian setting (as we consider here) it is productive to decouple uncertainty estimation into two separate tasks: ordinal ranking according to uncertainty, and probability calibration .

Original: Noting that calibration (of ordinal conﬁdence ranking) already has many effective solutions (Naeini et al., 2015; Platt et al., 1999; Zadrozny & Elkan, 2002; Guo et al., 2017), our main focus here is on the core task of ranking uncertainties.
Revised: Noting that calibration (of ordinal conﬁdence ranking) already has many effective solutions (Naeini et al., 2015; Platt et al., 1999; Zadrozny & Elkan, 2002; Guo et al., 2017), our main focus here is on the core task of ranking uncertainties.

Original: We thus adopt the setting of Lakshminarayanan et al.
Revised: We thus adopt the setting of Lakshminarayanan et al.

Original: (2017), and others (Geifman & El-Yaniv, 2017; Mandelbaum & Weinshall, 2017), and
Revised: (2017), and others (Geifman & El-Yaniv, 2017; Mandelbaum & Weinshall, 2017), and

Original: consider uncertainty estimation for classiﬁcation as the following problem.
Revised: consider uncertainty estimation for classiﬁcation as the following problem.

Original: Given labeled data, the goal is to learn a pair ( f, κ ) , where f ( x ) is a classiﬁer and κ ( x ) is a conﬁdence score function.
Revised: Given labeled data, the goal is to learn a pair ( f, κ ) , where f ( x ) is a classiﬁer and κ ( x ) is a conﬁdence score function.

Original: Intuitively, κ should assign lower conﬁdence values to points that are misclassiﬁed by f , relative to correct classiﬁcations (see
Revised: Intuitively, κ should assign lower conﬁdence values to points that are misclassiﬁed by f , relative to correct classiﬁcations (see

Original: Section 2 for details).
Revised: Section 2 for details).

Original: We propose two methods that can boost known conﬁdence scoring functions for deep neural networks (DNNs).
Revised: We propose two methods that can boost known conﬁdence scoring functions for deep neural networks (DNNs).

Original: Our ﬁrst method devises a selection mechanism that assigns for each instance an appropriate early stopped model, which improves that instance’s uncertainty estimation.
Revised: Our ﬁrst method devises a selection mechanism that assigns for each instance an appropriate early stopped model, which improves that instance’s uncertainty estimation.

Original: The mechanism selects the early-stopped model for each individual instance from among snapshots of the network’s weights that were saved during the training process.
Revised: The mechanism selects the early-stopped model for each individual instance from among snapshots of the network’s weights that were saved during the training process.

Original: This method requires an auxiliary training set to train the selection mechanism, and is quite computationally intensive to train.
Revised: This method requires an auxiliary training set to train the selection mechanism, and is quite computationally intensive to train.

Original: The second method approximates the ﬁrst without any additional examples.
Revised: The second method approximates the ﬁrst without any additional examples.

Original: Since there is no consensus on the appropriate performance measure for scoring functions, we formulate such a measure based on concepts from selective prediction (Geifman & El-Yaniv, 2017; Wiener & El-Yaniv, 2015).
Revised: Since there is no consensus on the appropriate performance measure for scoring functions, we formulate such a measure based on concepts from selective prediction (Geifman & El-Yaniv, 2017; Wiener & El-Yaniv, 2015).

Original: We report on extensive experiments with four baseline methods (including all those mentioned above) and four image datasets.
Revised: We report on extensive experiments with four baseline methods (including all those mentioned above) and four image datasets.

Original: The proposed approach consistently improves all baselines, often by a wide margin.
Revised: The proposed approach consistently improves all baselines, often by a wide margin.

Original: For completeness, we also validate our results using probably-calibrated uncertainty estimates of our method that are calibrated with the well-known Platt scaling technique (Platt et al., 1999) and measured with the negative log-likelihood and Brier score Brier (1950).
Revised: For completeness, we also validate our results using probably-calibrated uncertainty estimates of our method that are calibrated with the well-known Platt scaling technique (Platt et al., 1999) and measured with the negative log-likelihood and Brier score Brier (1950).

Original:  P ROBLEM S ETTING
Revised:  P ROBLEM S ETTING

Original: In this work we consider uncertainty estimation for a standard supervised multi-class classiﬁcation problem.
Revised: In this work we consider uncertainty estimation for a standard supervised multi-class classiﬁcation problem.

Original: We note that in our context uncertainty can be viewed as negative conﬁdence and vice versa.
Revised: We note that in our context uncertainty can be viewed as negative conﬁdence and vice versa.

Original: We use these terms interchangeably.
Revised: We use these terms interchangeably.

Original: Let X be some feature space (e.g., raw image pixels) and Y = { 1 , 2 , 3 , . . .
Revised: Let X be some feature space (e.g., raw image pixels) and Y = { 1 , 2 , 3 , . . .

Original: , k } , a label set for the classes.
Revised: , k } , a label set for the classes.

Original: Let P ( X, Y ) be an unknown source distribution over X × Y .
Revised: Let P ( X, Y ) be an unknown source distribution over X × Y .

Original: A classiﬁer f is a function f : X → Y whose true risk w.r.t.
Revised: A classiﬁer f is a function f : X → Y whose true risk w.r.t.

Original: P is R ( f | P ) =
Revised: P is R ( f | P ) =

Original: E P ( X,Y )
Revised: E P ( X,Y )

Original: [ (cid:96) ( f ( x ) , y )] , where (cid:96) :
Revised: [ (cid:96) ( f ( x ) , y )] , where (cid:96) :

Original: Y × Y → R + is a given loss function, for example, the 0/1 error.
Revised: Y × Y → R + is a given loss function, for example, the 0/1 error.

Original: Given a labeled set S m = { ( x i , y i ) } m i =1 ⊆
Revised: Given a labeled set S m = { ( x i , y i ) } m i =1 ⊆

Original: ( X × Y ) sampled i.i.d.
Revised: ( X × Y ) sampled i.i.d.

Original: from P ( X, Y ) , the empirical risk of the classiﬁer f is ˆ r ( f | S m ) =
Revised: from P ( X, Y ) , the empirical risk of the classiﬁer f is ˆ r ( f | S m ) =

Original: ∆ 1 m (cid:80) mi =1 (cid:96) ( f ( x i ) , y i ) .
Revised: ∆ 1 m (cid:80) mi =1 (cid:96) ( f ( x i ) , y i ) .

Original: We consider deep neural classiﬁcation models that utilize a standard softmax (last) layer for multi-class classiﬁcation.
Revised: We consider deep neural classiﬁcation models that utilize a standard softmax (last) layer for multi-class classiﬁcation.

Original: Thus, for each input x ∈ X , the vector f ( x ) =
Revised: Thus, for each input x ∈ X , the vector f ( x ) =

Original: ( f ( x ) 1 , . .
Revised: ( f ( x ) 1 , . .

Original: f ( x ) k ) ∈
Revised: f ( x ) k ) ∈

Original: R k is the softmax activations of the last layer.
Revised: R k is the softmax activations of the last layer.

Original: The model’s predicted class
Revised: The model’s predicted class

Original: ˆ y =
Revised: ˆ y =

Original: ˆ y f ( x ) =
Revised: ˆ y f ( x ) =

Original: argmax i ∈Y f
Revised: argmax i ∈Y f

Original: ( x ) i .
Revised: ( x ) i .

Original: Consider the training process of a deep model f through T epochs using any mini-batch SGD optimization variant.
Revised: Consider the training process of a deep model f through T epochs using any mini-batch SGD optimization variant.

Original: For each 1 ≤ i ≤ T , we denote by f
Revised: For each 1 ≤ i ≤ T , we denote by f

Original: [ i ] a snapshot of the partially trained model immediately after epoch i .
Revised: [ i ] a snapshot of the partially trained model immediately after epoch i .

Original: For a multi-class model f , we would like to deﬁne a conﬁdence score function, κ ( x, i, | f ) , where x ∈ X , and i ∈ Y .
Revised: For a multi-class model f , we would like to deﬁne a conﬁdence score function, κ ( x, i, | f ) , where x ∈ X , and i ∈ Y .

Original: The function κ should quantify conﬁdence in predicting that x is from class i , based on signals extracted from f .
Revised: The function κ should quantify conﬁdence in predicting that x is from class i , based on signals extracted from f .

Original: A κ -score function should induce a partial order over points in X , and thus is not required to distinguish between points with the same score.
Revised: A κ -score function should induce a partial order over points in X , and thus is not required to distinguish between points with the same score.

Original: For example, for any softmax classiﬁer f , the vanilla conﬁdence score function is κ ( x, i | f ) =
Revised: For example, for any softmax classiﬁer f , the vanilla conﬁdence score function is κ ( x, i | f ) =

Original: ∆ f ( x ) i (i.e., the softmax response values themselves).
Revised: ∆ f ( x ) i (i.e., the softmax response values themselves).

Original: Perhaps due to the natural probabilistic interpretation of the softmax function (all values are non-negative and sum to 1), this vanilla κ has long been used as a conﬁdence estimator.
Revised: Perhaps due to the natural probabilistic interpretation of the softmax function (all values are non-negative and sum to 1), this vanilla κ has long been used as a conﬁdence estimator.

Original: Note, however, that we are not concerned with the standard probabilistic interpretation (which needs to be calibrated to properly quantify probabilities (Guo et al., 2017)).
Revised: Note, however, that we are not concerned with the standard probabilistic interpretation (which needs to be calibrated to properly quantify probabilities (Guo et al., 2017)).

Original: An optimal κ (for f ) should reﬂect true loss monotonicity in the sense that for every two labeled instances ( x 1 , y 1 ) ∼ P ( X, Y ) , and ( x 2 , y 2 ) ∼ P ( X, Y ) , κ ( x 1 , ˆ y f ( x ) | f ) ≤ κ ( x 2 , ˆ y f ( x ) | f ) ⇐⇒ Pr P [ˆ y f ( x 1 ) (cid:54)
Revised: An optimal κ (for f ) should reﬂect true loss monotonicity in the sense that for every two labeled instances ( x 1 , y 1 ) ∼ P ( X, Y ) , and ( x 2 , y 2 ) ∼ P ( X, Y ) , κ ( x 1 , ˆ y f ( x ) | f ) ≤ κ ( x 2 , ˆ y f ( x ) | f ) ⇐⇒ Pr P [ˆ y f ( x 1 ) (cid:54)

Original: =
Revised: =

Original: y 1 ] ≥ Pr P [ˆ y f ( x 2 ) (cid:54) =
Revised: y 1 ] ≥ Pr P [ˆ y f ( x 2 ) (cid:54) =

Original: y 2 ] .
Revised: y 2 ] .

Original:  P ERFORMANCE E VALUATION OF C ONFIDENCE S CORES BY S ELECTIVE C LASSIFICATION
Revised:  P ERFORMANCE E VALUATION OF C ONFIDENCE S CORES BY S ELECTIVE C LASSIFICATION

Original: In the domain of (deep) uncertainty estimation there is currently no consensus on how to measure performance (of ordinal estimators).
Revised: In the domain of (deep) uncertainty estimation there is currently no consensus on how to measure performance (of ordinal estimators).

Original: For example, Lakshminarayanan et al.
Revised: For example, Lakshminarayanan et al.

Original: (2017) used the Brier score and the negative-log-likelihood to asses their results, while treating κ values as absolute scores.
Revised: (2017) used the Brier score and the negative-log-likelihood to asses their results, while treating κ values as absolute scores.

Original: In Mandelbaum & Weinshall (2017) the area under the ROC curve was used for measuring performance.
Revised: In Mandelbaum & Weinshall (2017) the area under the ROC curve was used for measuring performance.

Original: In this section we propose a meaningful and unitless performance measure for κ functions, which borrows elements from other known approaches.
Revised: In this section we propose a meaningful and unitless performance measure for κ functions, which borrows elements from other known approaches.

Original: In order to deﬁne a performance measure for κ functions, we require a few concepts from selective classiﬁcation (El-Yaniv & Wiener, 2010; Wiener & El-Yaniv, 2011).
Revised: In order to deﬁne a performance measure for κ functions, we require a few concepts from selective classiﬁcation (El-Yaniv & Wiener, 2010; Wiener & El-Yaniv, 2011).

Original: As noted in (Geifman & ElYaniv, 2017), any κ function can be utilized to construct a selective classiﬁer (i.e., a classiﬁer with a reject option).
Revised: As noted in (Geifman & ElYaniv, 2017), any κ function can be utilized to construct a selective classiﬁer (i.e., a classiﬁer with a reject option).

Original: Thus, selective classiﬁcation is a natural application of conﬁdence score functions based on which it is convenient and meaningful to assess performance.
Revised: Thus, selective classiﬁcation is a natural application of conﬁdence score functions based on which it is convenient and meaningful to assess performance.

Original: The structure of this section is as follows.
Revised: The structure of this section is as follows.

Original: We ﬁrst introduce the (well known) terms selective classiﬁer , selective risk and coverage .
Revised: We ﬁrst introduce the (well known) terms selective classiﬁer , selective risk and coverage .

Original: Then we introduce the risk-coverage curve .
Revised: Then we introduce the risk-coverage curve .

Original: We propose to measure the performance of a κ function as the area under the risk-coverage curve (AURC) of a selective classiﬁer induced by κ .
Revised: We propose to measure the performance of a κ function as the area under the risk-coverage curve (AURC) of a selective classiﬁer induced by κ .

Original: The proposed measure is a normalization of AURC where we subtract the AURC of the best κ in hindsight.
Revised: The proposed measure is a normalization of AURC where we subtract the AURC of the best κ in hindsight.

Original: The beneﬁt of the proposed normalization is that it allows for meaningful comparisons accross problems.
Revised: The beneﬁt of the proposed normalization is that it allows for meaningful comparisons accross problems.

Original: We term the this normalized metric “excess
Revised: We term the this normalized metric “excess

Original: AURC”
Revised: AURC”

Original: (E-ARUC) and it will be used throughout the paper for performance evaluation of κ functions.
Revised: (E-ARUC) and it will be used throughout the paper for performance evaluation of κ functions.

Original: A selective classiﬁer is a pair ( f, g ) , where f is a classiﬁer, and g :
Revised: A selective classiﬁer is a pair ( f, g ) , where f is a classiﬁer, and g :

Original: X → { 0 , 1 } is a selection function , which serves as a binary qualiﬁer for f as follows,
Revised: X → { 0 , 1 } is a selection function , which serves as a binary qualiﬁer for f as follows,

Original: [Equation]
Revised: [Equation]

Original: The performance of a selective classiﬁer is quantiﬁed using coverage and risk .
Revised: The performance of a selective classiﬁer is quantiﬁed using coverage and risk .

Original: Coverage , deﬁned to be φ ( f, g ) =
Revised: Coverage , deﬁned to be φ ( f, g ) =

Original: ∆ E P [ g ( x )] , is the probability mass of the non-rejected region in X .
Revised: ∆ E P [ g ( x )] , is the probability mass of the non-rejected region in X .

Original: The selective risk of ( f, g ) is
Revised: The selective risk of ( f, g ) is

Original: [Equation]
Revised: [Equation]

Original: These two measures can be empirically evaluated over any ﬁnite labeled set S m (not necessarily the training set) in a straightforward manner.
Revised: These two measures can be empirically evaluated over any ﬁnite labeled set S m (not necessarily the training set) in a straightforward manner.

Original: Thus, the empirical selective risk is,
Revised: Thus, the empirical selective risk is,

Original: [Equation]
Revised: [Equation]

Original: where ˆ φ is the empirical coverage , ˆ φ ( f, g | S m )
Revised: where ˆ φ is the empirical coverage , ˆ φ ( f, g | S m )

Original: =
Revised: =

Original: ∆ 1 m (cid:80) mi =1 g ( x i ) .
Revised: ∆ 1 m (cid:80) mi =1 g ( x i ) .

Original: The overall performance proﬁle of a family of selective classiﬁers (optimized for various coverage rates) can be measured using the risk-coverage curve (RC-curve), deﬁned to be the selective risk as a function of coverage.
Revised: The overall performance proﬁle of a family of selective classiﬁers (optimized for various coverage rates) can be measured using the risk-coverage curve (RC-curve), deﬁned to be the selective risk as a function of coverage.

Original: Given a classiﬁer f and conﬁdence score function κ deﬁned for f , we deﬁne an empirical performance measure for κ using an independent set V n of n labeled points.
Revised: Given a classiﬁer f and conﬁdence score function κ deﬁned for f , we deﬁne an empirical performance measure for κ using an independent set V n of n labeled points.

Original: The performance measure is deﬁned in terms of the following selective classiﬁer ( f, g ) (where f is our given classiﬁer), and the selection functions g is deﬁned as a threshold over κ values, g θ ( x | κ, f )
Revised: The performance measure is deﬁned in terms of the following selective classiﬁer ( f, g ) (where f is our given classiﬁer), and the selection functions g is deﬁned as a threshold over κ values, g θ ( x | κ, f )

Original: = 

Revised: =

Original: [ κ ( x, ˆ y f ( x ) | f ) > θ ] .
Revised: [ κ ( x, ˆ y f ( x ) | f ) > θ ] .

Original: Let Θ be the set of all κ values of points in V n , Θ = ∆ { κ ( x, ˆ y f ( x ) | f ) :
Revised: Let Θ be the set of all κ values of points in V n , Θ = ∆ { κ ( x, ˆ y f ( x ) | f ) :

Original: ( x, y ) ∈ V n } ; for now we assume that Θ contains n unique points, and later we note how to deal with duplicate values.
Revised: ( x, y ) ∈ V n } ; for now we assume that Θ contains n unique points, and later we note how to deal with duplicate values.

Original: The performance of κ is deﬁned to be the area under the (empirical) RC-curve (AURC) of the pair ( f, g ) computed over V n ,
Revised: The performance of κ is deﬁned to be the area under the (empirical) RC-curve (AURC) of the pair ( f, g ) computed over V n ,

Original: [Equation]
Revised: [Equation]

Original: Intuitively, a better κ will induce a better selective classiﬁer that will tend to reject ﬁrst the points that are misclassiﬁed by f .
Revised: Intuitively, a better κ will induce a better selective classiﬁer that will tend to reject ﬁrst the points that are misclassiﬁed by f .

Original: Accordingly, the associated RC-curve will decrease faster (with decreasing coverage) and the AURC will be smaller.
Revised: Accordingly, the associated RC-curve will decrease faster (with decreasing coverage) and the AURC will be smaller.

Original: For example, in Figure 1 we show (in blue) the RC-curve of classiﬁer f obtained by training a DNN trained over the CIFAR-100 dataset.
Revised: For example, in Figure 1 we show (in blue) the RC-curve of classiﬁer f obtained by training a DNN trained over the CIFAR-100 dataset.

Original: The κ induced by f is the softmax response conﬁdence score, κ ( x ) = max i f ( x ) i .
Revised: The κ induced by f is the softmax response conﬁdence score, κ ( x ) = max i f ( x ) i .

Original: The RC-curve in the ﬁgure is calculated w.r.t.
Revised: The RC-curve in the ﬁgure is calculated w.r.t.

Original: to an independent labeled set V n of n = 10 , 000 points from CIFAR-100.
Revised: to an independent labeled set V n of n = 10 , 000 points from CIFAR-100.

Original: Each point on the curve is the empirical selective risk (2) of a selective classiﬁer ( f, g θ ) such that θ ∈ Θ .
Revised: Each point on the curve is the empirical selective risk (2) of a selective classiﬁer ( f, g θ ) such that θ ∈ Θ .

Original: As can be seen, the selective risk is monotonically increasing with coverage.
Revised: As can be seen, the selective risk is monotonically increasing with coverage.

Original: For instance, at full coverage = 1, the risk is approximately 0.29.
Revised: For instance, at full coverage = 1, the risk is approximately 0.29.

Original: This risk corresponds to a standard classiﬁer (that always predicts and does not reject anything).
Revised: This risk corresponds to a standard classiﬁer (that always predicts and does not reject anything).

Original: The risk corresponding to coverage = 0.5 is approximately 0.06 and corresponds to a selective classiﬁer that rejects half of the points (those whose conﬁdence is least).
Revised: The risk corresponding to coverage = 0.5 is approximately 0.06 and corresponds to a selective classiﬁer that rejects half of the points (those whose conﬁdence is least).

Original: Not surprisingly, its selective risk is signiﬁcantly lower than the risk obtained at full coverage.
Revised: Not surprisingly, its selective risk is signiﬁcantly lower than the risk obtained at full coverage.

Original: An optimal in hindsight conﬁdence score function for f , denoted by κ ∗ , will yield the optimal risk coverage curve.
Revised: An optimal in hindsight conﬁdence score function for f , denoted by κ ∗ , will yield the optimal risk coverage curve.

Original: This optimal function rates all misclassiﬁed points (by f ) lower than all correctly classiﬁed points.
Revised: This optimal function rates all misclassiﬁed points (by f ) lower than all correctly classiﬁed points.

Original: The selective risk associated with κ ∗ is thus zero at all coverage rates below 1 − ˆ r ( f | V n ) .
Revised: The selective risk associated with κ ∗ is thus zero at all coverage rates below 1 − ˆ r ( f | V n ) .

Original: The reason is that the optimal function rejects all misclassiﬁed points at such rates.
Revised: The reason is that the optimal function rejects all misclassiﬁed points at such rates.

Original: For example, in Figure 1 we show the RC-curve (black) obtained by relying on κ ∗ , which reaches zero at coverage
Revised: For example, in Figure 1 we show the RC-curve (black) obtained by relying on κ ∗ , which reaches zero at coverage

Original: Since the AURC of all RC-curves for f induced by any conﬁdence scoring function will be larger than the AURC of κ ∗ , we normalize by AURC ( κ ∗ ) to obtain a unitless performance measure.
Revised: Since the AURC of all RC-curves for f induced by any conﬁdence scoring function will be larger than the AURC of κ ∗ , we normalize by AURC ( κ ∗ ) to obtain a unitless performance measure.

Original: To compute the AURC of κ ∗ , we compute the discrete integral of ˆ r
Revised: To compute the AURC of κ ∗ , we compute the discrete integral of ˆ r

Original: (w.r.t.
Revised: (w.r.t.

Original: κ ∗ ) from the coverage level of 1 − ˆ r ( f | V n ) (0 errors) to 1 ( n ˆ r errors).
Revised: κ ∗ ) from the coverage level of 1 − ˆ r ( f | V n ) (0 errors) to 1 ( n ˆ r errors).

Original: Thus,
Revised: Thus,

Original: [Equation]
Revised: [Equation]

Original: We approximate (3) using the following integral:
Revised: We approximate (3) using the following integral:

Original: [Equation]
Revised: [Equation]

Original: For example, the gray area in Figure 1 is the AURC of κ ∗ , which equals 0 . 04802 (and approximated by 0 .
Revised: For example, the gray area in Figure 1 is the AURC of κ ∗ , which equals 0 . 04802 (and approximated by 0 .

Original: using the integral).
Revised: using the integral).

Original: To conclude this section, we deﬁne the Excess-AURC (E-AURC)
Revised: To conclude this section, we deﬁne the Excess-AURC (E-AURC)

Original: as E-AURC ( κ, f | V n ) =
Revised: as E-AURC ( κ, f | V n ) =

Original: AURC ( κ, f | V n ) − AURC ( κ ∗ , f | V n ) .
Revised: AURC ( κ, f | V n ) − AURC ( κ ∗ , f | V n ) .

Original: E-AURC is a unitless measure in [0 , 1] , and the optimal κ will have E-AURC =
Revised: E-AURC is a unitless measure in [0 , 1] , and the optimal κ will have E-AURC =

Original: E-AURC is used as our main performance measure.
Revised: E-AURC is used as our main performance measure.

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: The area of uncertainty estimation is huge, and way beyond our scope.
Revised: The area of uncertainty estimation is huge, and way beyond our scope.

Original: Here we focus only on non-Bayesian methods in the context of deep neural classiﬁcation.
Revised: Here we focus only on non-Bayesian methods in the context of deep neural classiﬁcation.

Original: Motivated by a Bayesian approach,
Revised: Motivated by a Bayesian approach,

Original: Gal & Ghahramani (2016) proposed the Monte-Carlo dropout (MC-dropout) technique for
Revised: Gal & Ghahramani (2016) proposed the Monte-Carlo dropout (MC-dropout) technique for

Original: The most common, and well-known approach for obtaining conﬁdence scores for DNNs is by measuring the classiﬁcation margin.
Revised: The most common, and well-known approach for obtaining conﬁdence scores for DNNs is by measuring the classiﬁcation margin.

Original: When softmax is in use at the last layer, its values correspond to the distance from the decision boundary, where large values tend to reﬂect high conﬁdence levels.
Revised: When softmax is in use at the last layer, its values correspond to the distance from the decision boundary, where large values tend to reﬂect high conﬁdence levels.

Original: This concept is widely used in the context of classiﬁcation with a reject option in linear models and in particular, in SVMs (Bartlett & Wegkamp, 2008; Chow, 1970; Fumera & Roli, 2002).
Revised: This concept is widely used in the context of classiﬁcation with a reject option in linear models and in particular, in SVMs (Bartlett & Wegkamp, 2008; Chow, 1970; Fumera & Roli, 2002).

Original: In the context of neural networks, Cordella et al.
Revised: In the context of neural networks, Cordella et al.

Original: De Stefano et al.
Revised: De Stefano et al.

Original: (2000) were the ﬁrst to propose this approach and, for DNNs, it has been recently shown to outperform the MC-dropout on ImageNet (Geifman & El-Yaniv, 2017).
Revised: (2000) were the ﬁrst to propose this approach and, for DNNs, it has been recently shown to outperform the MC-dropout on ImageNet (Geifman & El-Yaniv, 2017).

Original: A K -nearest-neighbors (KNN) algorithm applied in the embedding space of a DNN was recently proposed by Mandelbaum & Weinshall (2017).
Revised: A K -nearest-neighbors (KNN) algorithm applied in the embedding space of a DNN was recently proposed by Mandelbaum & Weinshall (2017).

Original: The KNN-distances are used as a proxy for classconditional probabilities.
Revised: The KNN-distances are used as a proxy for classconditional probabilities.

Original: To the best of our knowledge, this is the ﬁrst non-Bayesian method that estimates neural network uncertainties using activations from non-ﬁnal layers.
Revised: To the best of our knowledge, this is the ﬁrst non-Bayesian method that estimates neural network uncertainties using activations from non-ﬁnal layers.

Original: A new ensemble-based uncertainty score for DNNs was proposed by Lakshminarayanan et al.
Revised: A new ensemble-based uncertainty score for DNNs was proposed by Lakshminarayanan et al.

Original: It is well known that ensemble methods can improve predictive performance (Breiman, 1996).
Revised: It is well known that ensemble methods can improve predictive performance (Breiman, 1996).

Original: Their ensemble consists of several trained DNN models, and conﬁdence estimates were obtained by averaging softmax responses of ensemble members.
Revised: Their ensemble consists of several trained DNN models, and conﬁdence estimates were obtained by averaging softmax responses of ensemble members.

Original: While this method exhibits a signiﬁcant improvement over all known methods (and is presently state-of-the-art), it requires substantially large computing resources for training.
Revised: While this method exhibits a signiﬁcant improvement over all known methods (and is presently state-of-the-art), it requires substantially large computing resources for training.

Original: When considering works that leverage information from the network’s training process, the literature is quite sparse.
Revised: When considering works that leverage information from the network’s training process, the literature is quite sparse.

Original: Huang et al.
Revised: Huang et al.

Original: (2017) proposed to construct an ensemble, composed of several snapshots during training to improve predictive performance with the cost of training only one model.
Revised: (2017) proposed to construct an ensemble, composed of several snapshots during training to improve predictive performance with the cost of training only one model.

Original: However, due to the use of cyclic learning rate schedules, the snapshots that are averaged are fully converged models and produce a result that is both conceptually and quantitatively different from our use of snapshots before convergence.
Revised: However, due to the use of cyclic learning rate schedules, the snapshots that are averaged are fully converged models and produce a result that is both conceptually and quantitatively different from our use of snapshots before convergence.

Original: Izmailov et al.
Revised: Izmailov et al.

Original: (2018) similarly proposed to average the weights across
Revised: (2018) similarly proposed to average the weights across

Original: SGD iterations, but here again the averaging was done on fully converged models that have been only ﬁne-tuned after full training processes.
Revised: SGD iterations, but here again the averaging was done on fully converged models that have been only ﬁne-tuned after full training processes.

Original: Thus both these ensemble methods are superﬁcially similar to our averaging technique but are different than our method that utilizes “premature” ensemble members (in terms of their classiﬁcation performance).
Revised: Thus both these ensemble methods are superﬁcially similar to our averaging technique but are different than our method that utilizes “premature” ensemble members (in terms of their classiﬁcation performance).

Original:  M OTIVATION
Revised:  M OTIVATION

Original: In this section we present an example that motivates our algorithms.
Revised: In this section we present an example that motivates our algorithms.

Original: Consider a deep classiﬁcation model f that has been trained over the set S m through T epochs.
Revised: Consider a deep classiﬁcation model f that has been trained over the set S m through T epochs.

Original: Denote by f [ i ] the model trained at the i th epoch; thus, f = f [ T ] .
Revised: Denote by f [ i ] the model trained at the i th epoch; thus, f = f [ T ] .

Original: Take an independent validation set V n of n labeled points.
Revised: Take an independent validation set V n of n labeled points.

Original: We monitor the quality of the softmax response generated from f (and its intermediate variants f
Revised: We monitor the quality of the softmax response generated from f (and its intermediate variants f

Original: [ i ] ), through the training process, as measured on points in V n .
Revised: [ i ] ), through the training process, as measured on points in V n .

Original: The use of V n allows us to make meaningful statements about the quality of softmax response values (or any other conﬁdence estimation method) for unseen test points.
Revised: The use of V n allows us to make meaningful statements about the quality of softmax response values (or any other conﬁdence estimation method) for unseen test points.

Original: We construct the example by considering two groups of instances in V n deﬁned by conﬁdence assessment assigned using the softmax response values f gives to points in V n .
Revised: We construct the example by considering two groups of instances in V n deﬁned by conﬁdence assessment assigned using the softmax response values f gives to points in V n .

Original: The green group contains the highest (99%-100%) percentile of most conﬁdent points in V n , and the red group contains the lowest (0%-1%) percentile of least conﬁdent points.
Revised: The green group contains the highest (99%-100%) percentile of most conﬁdent points in V n , and the red group contains the lowest (0%-1%) percentile of least conﬁdent points.

Original: Although the softmax response is rightfully criticized in its ability to proxy conﬁdence (Gal & Ghahramani, 2016), it is reasonable to assume it is quite accurate in ranking green vs. red points (i.e., a prediction by f regarding a red point is likely to be less accurate than its prediction about a green point).
Revised: Although the softmax response is rightfully criticized in its ability to proxy conﬁdence (Gal & Ghahramani, 2016), it is reasonable to assume it is quite accurate in ranking green vs. red points (i.e., a prediction by f regarding a red point is likely to be less accurate than its prediction about a green point).

Original: We observe that the prediction of the green points’ labels is learned earlier during training of f , compared to a prediction of any red point.
Revised: We observe that the prediction of the green points’ labels is learned earlier during training of f , compared to a prediction of any red point.

Original: This fact is evident in Figure 2(a) where we see the training of f over CIFAR-100.
Revised: This fact is evident in Figure 2(a) where we see the training of f over CIFAR-100.

Original: Speciﬁcally, we see that the softmax response values of green points stabilize at their maximal values around Epoch 80.
Revised: Speciﬁcally, we see that the softmax response values of green points stabilize at their maximal values around Epoch 80.

Original: We also note that the green points in this top percentile are already correctly classiﬁed very early, near Epoch 25 (not shown in the ﬁgure).
Revised: We also note that the green points in this top percentile are already correctly classiﬁed very early, near Epoch 25 (not shown in the ﬁgure).

Original: In contrast, red points continue to improve their conﬁdence scores throughout.
Revised: In contrast, red points continue to improve their conﬁdence scores throughout.

Original: This observation indicates that green points can be predicted very well by an intermediate model such as f 130 .
Revised: This observation indicates that green points can be predicted very well by an intermediate model such as f 130 .

Original: Can we say that f 130 can estimate the conﬁdence of green points correctly?
Revised: Can we say that f 130 can estimate the conﬁdence of green points correctly?

Original: Recall from Section 3 that a useful method for assessing the quality of a conﬁdence function is the E-AURC measure (applied over an independent validation set).
Revised: Recall from Section 3 that a useful method for assessing the quality of a conﬁdence function is the E-AURC measure (applied over an independent validation set).

Original: We now measure the quality of the softmax response of all intermediate classiﬁers f
Revised: We now measure the quality of the softmax response of all intermediate classiﬁers f

Original: [ i ] ,
Revised: [ i ] ,

Original: [Figure]
Revised: [Figure]

Original: 1 ≤ i ≤ T , over the green points and, separately, over the red points.
Revised: 1 ≤ i ≤ T , over the green points and, separately, over the red points.

Original: Figure 2(b) shows the E-AURC of the green points.
Revised: Figure 2(b) shows the E-AURC of the green points.

Original: The x -axis denotes indices i of the intermediate classiﬁers ( f [ i ] ); the y -axis is
Revised: The x -axis denotes indices i of the intermediate classiﬁers ( f [ i ] ); the y -axis is

Original: E-AURC ( f
Revised: E-AURC ( f

Original: [ i ] , κ |{ green points } ) .
Revised: [ i ] , κ |{ green points } ) .

Original: Similarly, Figure 2(c) shows the E-AURC of the red points.
Revised: Similarly, Figure 2(c) shows the E-AURC of the red points.

Original: We see that for the green points, the conﬁdence estimation quality improves (almost) monotonically and then degrades (almost) monotonically.
Revised: We see that for the green points, the conﬁdence estimation quality improves (almost) monotonically and then degrades (almost) monotonically.

Original: The best conﬁdence estimation is obtained by intermediate classiﬁers such as f 130 .
Revised: The best conﬁdence estimation is obtained by intermediate classiﬁers such as f 130 .

Original: Surprisingly, the ﬁnal model f [ T ] is one of the worst estimators for green points!
Revised: Surprisingly, the ﬁnal model f [ T ] is one of the worst estimators for green points!

Original: In sharp contrast, the conﬁdence estimates for the red points monotonically improves as training continues.
Revised: In sharp contrast, the conﬁdence estimates for the red points monotonically improves as training continues.

Original: The best estimator for red points is the ﬁnal model f
Revised: The best estimator for red points is the ﬁnal model f

Original: [ T ] .
Revised: [ T ] .

Original: This behavior can be observed in all the datasets we considered (not reported).
Revised: This behavior can be observed in all the datasets we considered (not reported).

Original: The above dynamics indicates that the learning of uncertainty estimators for easy instances conceptually resembles overﬁtting in the sense that the assessment of higher conﬁdence points in the test set degrades as training continues after a certain point.
Revised: The above dynamics indicates that the learning of uncertainty estimators for easy instances conceptually resembles overﬁtting in the sense that the assessment of higher conﬁdence points in the test set degrades as training continues after a certain point.

Original: To overcome this deﬁciency we propose an algorithm that uses the concept of early stopping in a pointwise fashion, where for each sample (or set of samples)
Revised: To overcome this deﬁciency we propose an algorithm that uses the concept of early stopping in a pointwise fashion, where for each sample (or set of samples)

Original: we ﬁnd the best intermediate snapshot for uncertainty estimation.
Revised: we ﬁnd the best intermediate snapshot for uncertainty estimation.

Original:  S UPERIOR C ONFIDENCE S CORE BY E ARLY S TOPPING
Revised:  S UPERIOR C ONFIDENCE S CORE BY E ARLY S TOPPING

Original: In this section, ﬁrst we present a supervised algorithm that learns an improved scoring function for a given pair ( f, κ ) , where f is a trained deep neural classiﬁer, and κ is a conﬁdence scoring function for f ’s predictions.
Revised: In this section, ﬁrst we present a supervised algorithm that learns an improved scoring function for a given pair ( f, κ ) , where f is a trained deep neural classiﬁer, and κ is a conﬁdence scoring function for f ’s predictions.

Original: In principle, κ : X → R , where κ ( x ) can be deﬁned as any mapping from the activations of f applied on x to R .
Revised: In principle, κ : X → R , where κ ( x ) can be deﬁned as any mapping from the activations of f applied on x to R .

Original: All the conﬁdence estimation methods we described above comply with this deﬁnition.
Revised: All the conﬁdence estimation methods we described above comply with this deﬁnition.

Original: Our algorithm requires a labeled training sample.
Revised: Our algorithm requires a labeled training sample.

Original: The second algorithm we present is an approximated version of the ﬁrst algorithm, which does not rely on additional training examples.
Revised: The second algorithm we present is an approximated version of the ﬁrst algorithm, which does not rely on additional training examples.

Original: 6. T HE P OINTWISE E ARLY S TOPPING A LGORITHM FOR C ONFIDENCE S CORES
Revised: 6. T HE P OINTWISE E ARLY S TOPPING A LGORITHM FOR C ONFIDENCE S CORES

Original: Let f be a neural classiﬁer that has been trained using any (mini-batch) SGD variant for T epochs, and let F = ∆ { f [ i ] : 1 ≤ i ≤ T } be the set of intermediate models obtained during training ( f [ i ] is the model generated at epoch i ).
Revised: Let f be a neural classiﬁer that has been trained using any (mini-batch) SGD variant for T epochs, and let F = ∆ { f [ i ] : 1 ≤ i ≤ T } be the set of intermediate models obtained during training ( f [ i ] is the model generated at epoch i ).

Original: We assume that f , the snapshots set F , and a conﬁdence score function for f , κ ( · , ·| f ) :
Revised: We assume that f , the snapshots set F , and a conﬁdence score function for f , κ ( · , ·| f ) :

Original: X → (0 , 1]) , are given.
Revised: X → (0 , 1]) , are given.

Original: 2 Let V n be an independent training set.
Revised: 2 Let V n be an independent training set.

Original: The Pointwise Early Stopping (PES) algorithm for conﬁdence scores (see pseudo-code in Algorithm 1) operates as follows.
Revised: The Pointwise Early Stopping (PES) algorithm for conﬁdence scores (see pseudo-code in Algorithm 1) operates as follows.

Original: The pseudo-code contains both the training and inference procedures.
Revised: The pseudo-code contains both the training and inference procedures.

Original: At each iteration of the training main loop (lines 3-11), we extract from V (which is initialized as a clone of the set V n ) a set of the q most uncertain points.
Revised: At each iteration of the training main loop (lines 3-11), we extract from V (which is initialized as a clone of the set V n ) a set of the q most uncertain points.

Original: We abbreviate this set by S (the “layer”).
Revised: We abbreviate this set by S (the “layer”).

Original: The size of the layer is determined by the hyperparameter q .
Revised: The size of the layer is determined by the hyperparameter q .

Original: We then ﬁnd the best model in F using the
Revised: We then ﬁnd the best model in F using the

Original: 1: function T RAIN ( V n , q , κ , F ) 2:
Revised: 1: function T RAIN ( V n , q , κ , F ) 2:

Original: V ← V n ; Θ ← [] ; K ← [] ; T ← | F | ; j ← | F | 3: for i = 0 to (cid:100) n/q (cid:101) do 4: r ← { κ ( x, y f [ T ] ( x ) | f
Revised: V ← V n ; Θ ← [] ; K ← [] ; T ← | F | ; j ← | F | 3: for i = 0 to (cid:100) n/q (cid:101) do 4: r ← { κ ( x, y f [ T ] ( x ) | f

Original: [ j ] ) : ( x, y ) ∈ V } 5: ˜ θ ← r ( q ) (cid:46) r ( q ) indicates the q th order statistic of r 6:
Revised: [ j ] ) : ( x, y ) ∈ V } 5: ˜ θ ← r ( q ) (cid:46) r ( q ) indicates the q th order statistic of r 6:

Original: S ← { ( x, y ) | κ ( x, ˆ y f
Revised: S ← { ( x, y ) | κ ( x, ˆ y f

Original: [ T ] ( x ) | f
Revised: [ T ] ( x ) | f

Original: [ j ] ) <
Revised: [ j ] ) <

Original: ˜ θ } 7: j = argmin 0 <j ≤ T ( E-AURC
Revised: ˜ θ } 7: j = argmin 0 <j ≤ T ( E-AURC

Original: ( κ ( x, ˆ y f
Revised: ( κ ( x, ˆ y f

Original: [ T ] ( x ) | f
Revised: [ T ] ( x ) | f

Original: [ j ] ) , f [ T ] | S ) 8:
Revised: [ j ] ) , f [ T ] | S ) 8:

Original: K [ i ] ← κ ( · , ·| f
Revised: K [ i ] ← κ ( · , ·| f

Original: [ j ] ) 9: Θ[ i ] ← max( { κ ( x, ˆ y f
Revised: [ j ] ) 9: Θ[ i ] ← max( { κ ( x, ˆ y f

Original: [ T ] ( x ) | f
Revised: [ T ] ( x ) | f

Original: [ j ] ) : ( x, y ) ∈ S } ) 10:
Revised: [ j ] ) : ( x, y ) ∈ S } ) 10:

Original: V ← V \ S 11: end for 12: return K, Θ 13: end function 14: function E STIMATE C ONFIDENCE ( x , f , K , Θ ) 15: for i = 0 to | K | − 1 do 16: if κ i ( x, ˆ y f ( x )) ≤ θ i then (cid:46) κ
Revised: V ← V \ S 11: end for 12: return K, Θ 13: end function 14: function E STIMATE C ONFIDENCE ( x , f , K , Θ ) 15: for i = 0 to | K | − 1 do 16: if κ i ( x, ˆ y f ( x )) ≤ θ i then (cid:46) κ

Original: i is the i th element of K 17: return i + κ i ( x, ˆ y f ( x )) 18: end if 19: end for 20: return i + κ i ( x, ˆ y f ( x )) 21: end function
Revised: i is the i th element of K 17: return i + κ i ( x, ˆ y f ( x )) 18: end if 19: end for 20: return i + κ i ( x, ˆ y f ( x )) 21: end function

Original: E-AURC measure with respect to S .
Revised: E-AURC measure with respect to S .

Original: This model, denoted f [ j ] , is found by solving
Revised: This model, denoted f [ j ] , is found by solving

Original: [Equation]
Revised: [Equation]

Original: The best performing conﬁdence score over S , and the threshold over the conﬁdence level, θ , are saved for test time (lines 8-9) and used to associate points with their layers.
Revised: The best performing conﬁdence score over S , and the threshold over the conﬁdence level, θ , are saved for test time (lines 8-9) and used to associate points with their layers.

Original: We iterate and remove layer after layer until V is empty.
Revised: We iterate and remove layer after layer until V is empty.

Original: Our algorithm produces a partition of X comprising layers from least to highest conﬁdence.
Revised: Our algorithm produces a partition of X comprising layers from least to highest conﬁdence.

Original: For each layer we ﬁnd the best performing κ function based on models from F .
Revised: For each layer we ﬁnd the best performing κ function based on models from F .

Original: To infer the conﬁdence rate for given point x at test time, we search for the minimal i that satisﬁes
Revised: To infer the conﬁdence rate for given point x at test time, we search for the minimal i that satisﬁes

Original: [Equation]
Revised: [Equation]

Original: where κ i and θ i are the i ’th elements of K and Θ respectively.
Revised: where κ i and θ i are the i ’th elements of K and Θ respectively.

Original: Thus, we return κ ( x, ˆ y f
Revised: Thus, we return κ ( x, ˆ y f

Original: [ T ] ( x ) | F ) =
Revised: [ T ] ( x ) | F ) =

Original: i + κ i ( x, ˆ y f
Revised: i + κ i ( x, ˆ y f

Original: [ T ] ( x )) , where i is added to enforce full order on the conﬁdence score between layers, recall that κ ∈ (0 , 1] .
Revised: [ T ] ( x )) , where i is added to enforce full order on the conﬁdence score between layers, recall that κ ∈ (0 , 1] .

Original: 6. A VERAGED E ARLY S TOPPING A LGORITHM
Revised: 6. A VERAGED E ARLY S TOPPING A LGORITHM

Original: As we saw in Section 6.1, the computational complexity of the PES algorithm is quite intensive.
Revised: As we saw in Section 6.1, the computational complexity of the PES algorithm is quite intensive.

Original: Moreover, the algorithm requires an additional set of labeled examples, which may not always be available.
Revised: Moreover, the algorithm requires an additional set of labeled examples, which may not always be available.

Original: The Averaged Early Stopping (AES) is a simple approximation of the PES motivated by the observation that “easy” points are learned earlier during training as shown in Figure 2(a).
Revised: The Averaged Early Stopping (AES) is a simple approximation of the PES motivated by the observation that “easy” points are learned earlier during training as shown in Figure 2(a).

Original: By summing the area under the learning curve (a curve that is similar to 2(a))
Revised: By summing the area under the learning curve (a curve that is similar to 2(a))

Original: we leverage this property and avoid some inaccurate conﬁdence assessments generated by the last model alone.
Revised: we leverage this property and avoid some inaccurate conﬁdence assessments generated by the last model alone.

Original: We approximate the area under the curve by averaging k evenly spaced points on that curve.
Revised: We approximate the area under the curve by averaging k evenly spaced points on that curve.

Original: Let F be a set of k intermediate models saved during the training of f ,
Revised: Let F be a set of k intermediate models saved during the training of f ,

Original: [Equation]
Revised: [Equation]

Original: where linspace ( t, T, k ) is a set of k evenly spaced integers between t and T (including t and T ).
Revised: where linspace ( t, T, k ) is a set of k evenly spaced integers between t and T (including t and T ).

Original: We deﬁne the output κ as the average of all κ s associated with models in F , κ ( x, ˆ y f ( x ) | F )
Revised: We deﬁne the output κ as the average of all κ s associated with models in F , κ ( x, ˆ y f ( x ) | F )

Original: [Equation]
Revised: [Equation]

Original: 1 k
Revised: 1 k

Original: [Equation]
Revised: [Equation]

Original: f [ i ]
Revised: f [ i ]

Original: [Equation]
Revised: [Equation]

Original: F
Revised: F

Original: [Equation]
Revised: [Equation]

Original: ( x, ˆ y f ( x ) , f [ i ] ) .
Revised: ( x, ˆ y f ( x ) , f [ i ] ) .

Original: As we show in Section 7, AES works surprisingly well.
Revised: As we show in Section 7, AES works surprisingly well.

Original: In fact, due to the computational burden of running the PES algorithm, we use AES in most of our experiments below.
Revised: In fact, due to the computational burden of running the PES algorithm, we use AES in most of our experiments below.

Original:  E XPERIMENTAL R ESULTS
Revised:  E XPERIMENTAL R ESULTS

Original: We now present results of our AES algorithm applied over the four known conﬁdence scores: softmax response, NN-distance (Mandelbaum & Weinshall, 2017), MC-dropout (Gal & Ghahramani, 2016)
Revised: We now present results of our AES algorithm applied over the four known conﬁdence scores: softmax response, NN-distance (Mandelbaum & Weinshall, 2017), MC-dropout (Gal & Ghahramani, 2016)

Original: ans Ensemble (Lakshminarayanan et al., 2017)
Revised: ans Ensemble (Lakshminarayanan et al., 2017)

Original: (see Section 4).
Revised: (see Section 4).

Original: For implementation details for these methods, see Appendix A.
Revised: For implementation details for these methods, see Appendix A.

Original: We evaluate the performance of these methods and our AES algorithm that uses them as its core κ .
Revised: We evaluate the performance of these methods and our AES algorithm that uses them as its core κ .

Original: In all cases we ran the AES algorithm with k ∈ { 10 , 30 , 50 } , and t = (cid:98) 0 .
Revised: In all cases we ran the AES algorithm with k ∈ { 10 , 30 , 50 } , and t = (cid:98) 0 .

Original: 4 T (cid:99) .
Revised: 4 T (cid:99) .

Original: We experiment with four standard image datasets: CIFAR-10, CIFAR-100, SVHN, and Imagenet (see Appendix A for details).
Revised: We experiment with four standard image datasets: CIFAR-10, CIFAR-100, SVHN, and Imagenet (see Appendix A for details).

Original: Our results are reported in Table 4.
Revised: Our results are reported in Table 4.

Original: The table contains four blocks, one for each dataset.
Revised: The table contains four blocks, one for each dataset.

Original: Within each block we have four rows, one for each baseline method.
Revised: Within each block we have four rows, one for each baseline method.

Original: To explain the structure of this table, consider for example the 4th row, which shows the results corresponding to the softmax response for
Revised: To explain the structure of this table, consider for example the 4th row, which shows the results corresponding to the softmax response for

Original: CIFAR-10.
Revised: CIFAR-10.

Original: In the 2nd column we see the E-AURC ( × 10 3 ) of the softmax response itself (4.78).
Revised: In the 2nd column we see the E-AURC ( × 10 3 ) of the softmax response itself (4.78).

Original: In the 3rd column, the result of AES applied over the softmax response with k = 10 (reaching E-AURC of 4.81).
Revised: In the 3rd column, the result of AES applied over the softmax response with k = 10 (reaching E-AURC of 4.81).

Original: In the 4th column we specify percent of the improvement of AES over the baseline, in this case -0.7% (i.e., in this case AES degraded performance).
Revised: In the 4th column we specify percent of the improvement of AES over the baseline, in this case -0.7% (i.e., in this case AES degraded performance).

Original: For the imagenet dataset, we only present results for the softmax response and ensemble.
Revised: For the imagenet dataset, we only present results for the softmax response and ensemble.

Original: Applying the other methods on this large dataset was computationally prohibitive.
Revised: Applying the other methods on this large dataset was computationally prohibitive.

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: for clarity.
Revised: for clarity.

Original: Let us now analyze these results.
Revised: Let us now analyze these results.

Original: Before considering the relative performance of the baseline methods compares to ours, it is interesting to see that the E-AURC measure nicely quantiﬁes the difﬁculty level of the learning problems.
Revised: Before considering the relative performance of the baseline methods compares to ours, it is interesting to see that the E-AURC measure nicely quantiﬁes the difﬁculty level of the learning problems.

Original: Indeed, CIFAR-10 and SVHN are known as relatively easy problems and the E-AURC ranges we see in the table for most of the methods is quite small and similar.
Revised: Indeed, CIFAR-10 and SVHN are known as relatively easy problems and the E-AURC ranges we see in the table for most of the methods is quite small and similar.

Original: CIFAR-100 is considered harder, which is reﬂected by signiﬁcantly larger E-AURC values recorded for the various methods.
Revised: CIFAR-100 is considered harder, which is reﬂected by signiﬁcantly larger E-AURC values recorded for the various methods.

Original: Finally, Imagenet has the largest E-AURC values and is considered to be the hardest problem.
Revised: Finally, Imagenet has the largest E-AURC values and is considered to be the hardest problem.

Original: This observation supports the usefulness of E-AURC.
Revised: This observation supports the usefulness of E-AURC.

Original: A non-unitless measure such as AUC, the standard measure, would not be useful in such comparisons.
Revised: A non-unitless measure such as AUC, the standard measure, would not be useful in such comparisons.

Original: It is striking that among all 42 experiments, our method improved the baseline method in 39 cases.
Revised: It is striking that among all 42 experiments, our method improved the baseline method in 39 cases.

Original: Moreover, when applying AES with k = 30 , it always reduced the E-AURC of the baseline method.
Revised: Moreover, when applying AES with k = 30 , it always reduced the E-AURC of the baseline method.

Original: For each dataset, the ensemble estimation approach of Lakshminarayanan et al.
Revised: For each dataset, the ensemble estimation approach of Lakshminarayanan et al.

Original: (2017) is the best among the baselines, and is currently state-of-the-art.
Revised: (2017) is the best among the baselines, and is currently state-of-the-art.

Original: It follows that for all of these datasets, the application of AES improves the state-of-the-art.
Revised: It follows that for all of these datasets, the application of AES improves the state-of-the-art.

Original: While the ensemble method (and its improvement by AES) achieve the best results on these datasets, these methods are computationally intensive.
Revised: While the ensemble method (and its improvement by AES) achieve the best results on these datasets, these methods are computationally intensive.

Original: It is, therefore, interesting to identify top performing baselines, which are based on a single classiﬁer.
Revised: It is, therefore, interesting to identify top performing baselines, which are based on a single classiﬁer.

Original: In
Revised: In

Original: CIFAR-10, the best (single-classiﬁer) method is softmax response, whose E-AURC is improved 6% by AES (resulting in the best single-classiﬁer performance).
Revised: CIFAR-10, the best (single-classiﬁer) method is softmax response, whose E-AURC is improved 6% by AES (resulting in the best single-classiﬁer performance).

Original: Interestingly, in this dataset, NN-distance incurs a markedly bad E-AURC (35.1), which is reduced (to 4.58) by AES, making it on par with the best methods for this dataset.
Revised: Interestingly, in this dataset, NN-distance incurs a markedly bad E-AURC (35.1), which is reduced (to 4.58) by AES, making it on par with the best methods for this dataset.

Original: Turning to CIFAR-100, we see that the (single-classiﬁer) top method is
Revised: Turning to CIFAR-100, we see that the (single-classiﬁer) top method is

Original: NN-distance, with an E-AURC of 45.56, which is improved by 22% using AES.
Revised: NN-distance, with an E-AURC of 45.56, which is improved by 22% using AES.

Original: [Table]
Revised: [Table]

Original: Next we examine AES applied together with probability calibration .
Revised: Next we examine AES applied together with probability calibration .

Original: We calibrate the results of the
Revised: We calibrate the results of the

Original: AES algorithm using the Platt scaling technique; see (Platt et al., 1999) for details.
Revised: AES algorithm using the Platt scaling technique; see (Platt et al., 1999) for details.

Original: Platt scaling is applied on the results of the AES algorithm with k = 30 , and compared to the independently scaled underlying measure without AES.
Revised: Platt scaling is applied on the results of the AES algorithm with k = 30 , and compared to the independently scaled underlying measure without AES.

Original: Performance is evaluated using both negative log-likelihood (NLL) and the Brier score (Brier, 1950).
Revised: Performance is evaluated using both negative log-likelihood (NLL) and the Brier score (Brier, 1950).

Original: For further implementation details of this experiment see
Revised: For further implementation details of this experiment see

Original: Appendix A.
Revised: Appendix A.

Original: The results appear in Table 2.
Revised: The results appear in Table 2.

Original: As can easily be seen, the probability scaling results are remarkably consistent with our raw uncertainty estimates (measured with the E-AURC) over all datasets and underlying uncertainty methods.
Revised: As can easily be seen, the probability scaling results are remarkably consistent with our raw uncertainty estimates (measured with the E-AURC) over all datasets and underlying uncertainty methods.

Original: We conclude that AES also improves calibrated probabilities of the underlying uncertainty measures, and the E-AURC can serve as a reliable proxy also for calibrated probabilities.
Revised: We conclude that AES also improves calibrated probabilities of the underlying uncertainty measures, and the E-AURC can serve as a reliable proxy also for calibrated probabilities.

Original: We implemented the PES algorithm only over the softmax response method (SR) for several datasets.
Revised: We implemented the PES algorithm only over the softmax response method (SR) for several datasets.

Original: To generate an independent training set, which is required by PES, we randomly split the original validation set (in each dataset) into two parts and took a random 70% of the set for training our algorithm, using the remaining 30% for validation.
Revised: To generate an independent training set, which is required by PES, we randomly split the original validation set (in each dataset) into two parts and took a random 70% of the set for training our algorithm, using the remaining 30% for validation.

Original: The reason we only applied this algorithm over softmax responses is the excessive computational resources it requires.
Revised: The reason we only applied this algorithm over softmax responses is the excessive computational resources it requires.

Original: For example, when
Revised: For example, when

Original: [Table]
Revised: [Table]

Original: applying PES over NN-distance, the time complexity is nmTk + O
Revised: applying PES over NN-distance, the time complexity is nmTk + O

Original: ( TC f ( S m )
Revised: ( TC f ( S m )

Original: +
Revised: +

Original: TC f
Revised: TC f

Original: ( V n )) , where k is the number of neighbours and C f ( S m ) is the time complexity of running a forward pass of m samples using the classiﬁer f .
Revised: ( V n )) , where k is the number of neighbours and C f ( S m ) is the time complexity of running a forward pass of m samples using the classiﬁer f .

Original: Similarly, the complexity of PES when the underlying scores are from
Revised: Similarly, the complexity of PES when the underlying scores are from

Original: MC-dropout is O ( dTC f ( V n ) where d is the number of dropout iterations (forward passes) of the
Revised: MC-dropout is O ( dTC f ( V n ) where d is the number of dropout iterations (forward passes) of the

Original: MC-dropout algorithm.
Revised: MC-dropout algorithm.

Original: Thus, when n = 7000 , T = 250 (the parameters used for applying PES over CIFAR-100), and with d = 100 (as recommended in Gal & Ghahramani (2016)), this amounts to 175,000,000 forward passes.
Revised: Thus, when n = 7000 , T = 250 (the parameters used for applying PES over CIFAR-100), and with d = 100 (as recommended in Gal & Ghahramani (2016)), this amounts to 175,000,000 forward passes.

Original: We set q = (cid:98) n/ 3 (cid:99) .
Revised: We set q = (cid:98) n/ 3 (cid:99) .

Original: We repeated the experiment over 10 random training–validation splits and report the average results and the standard errors in Table 3.
Revised: We repeated the experiment over 10 random training–validation splits and report the average results and the standard errors in Table 3.

Original: As seen, PES reduced the E-AURC of softmax on all datasets by a signiﬁcant rate.
Revised: As seen, PES reduced the E-AURC of softmax on all datasets by a signiﬁcant rate.

Original: The best improvement was achieved on CIFAR-100 (E-AURC reduced by 18%).
Revised: The best improvement was achieved on CIFAR-100 (E-AURC reduced by 18%).

Original: Our difﬁculties when applying the PES algorithm on many of the underlying conﬁdence methods, and the outstanding results of the AES motivate further research that should lead to improving the algorithm and making it more efﬁcient.
Revised: Our difﬁculties when applying the PES algorithm on many of the underlying conﬁdence methods, and the outstanding results of the AES motivate further research that should lead to improving the algorithm and making it more efﬁcient.

Original:  C ONCLUDING R EMARKS
Revised:  C ONCLUDING R EMARKS

Original: We presented novel uncertainty estimation algorithms, which are motivated by an observation regarding the training process of DNNs using SGD.
Revised: We presented novel uncertainty estimation algorithms, which are motivated by an observation regarding the training process of DNNs using SGD.

Original: In this process, reliable estimates generated in early epochs are later on deformed.
Revised: In this process, reliable estimates generated in early epochs are later on deformed.

Original: This phenomenon somewhat resembles the well-known overﬁtting effect in DNNs.
Revised: This phenomenon somewhat resembles the well-known overﬁtting effect in DNNs.

Original: The PES algorithm we presented requires an additional labeled set and expensive computational resources for training.
Revised: The PES algorithm we presented requires an additional labeled set and expensive computational resources for training.

Original: The approximated version (AES) is simple and scalable.
Revised: The approximated version (AES) is simple and scalable.

Original: The resulting conﬁdence scores our methods generate systematically improve all existing estimation techniques on all the evaluated datasets.
Revised: The resulting conﬁdence scores our methods generate systematically improve all existing estimation techniques on all the evaluated datasets.

Original: Both PES and AES overcome conﬁdence score deformations by utilizing available snapshot models that are generated anyway during training.
Revised: Both PES and AES overcome conﬁdence score deformations by utilizing available snapshot models that are generated anyway during training.

Original: It would be interesting to develop a loss function that will explicitly prevent conﬁdence deformations by design while maintaining high classiﬁcation performance.
Revised: It would be interesting to develop a loss function that will explicitly prevent conﬁdence deformations by design while maintaining high classiﬁcation performance.

Original: In addition, the uncertainty estimation of each instance currently requires several forward passes through the network.
Revised: In addition, the uncertainty estimation of each instance currently requires several forward passes through the network.

Original: Instead it would be interesting to consider incorporating distillation (Hinton et al., 2015) so as to reduce inference time.
Revised: Instead it would be interesting to consider incorporating distillation (Hinton et al., 2015) so as to reduce inference time.

Original: Another direction to mitigate the computational effort at inference time is to approximate PES using a single model per instance based on an early stopping criterion similar to the one proposed by Mahsereci et al. 
Revised: Another direction to mitigate the computational effort at inference time is to approximate PES using a single model per instance based on an early stopping criterion similar to the one proposed by Mahsereci et al. A CKNOWLEDGMENTS

Revised: This research was supported by The Israel Science Foundation (grant No. 81/017).

Original: A I MPLEMENTATION AND E XPERIMENTAL D ETAILS A. D ATASETS
Revised: A I MPLEMENTATION AND E XPERIMENTAL D ETAILS A. D ATASETS

Original: CIFAR-10:
Revised: CIFAR-10:

Original: The CIFAR-10 dataset (Krizhevsky & Hinton, 2009) is an image classiﬁcation dataset containing 50,000 training images and 10,000 test images that are classiﬁed into 10 categories.
Revised: The CIFAR-10 dataset (Krizhevsky & Hinton, 2009) is an image classiﬁcation dataset containing 50,000 training images and 10,000 test images that are classiﬁed into 10 categories.

Original: The image size is 32 × 32 × 3 pixels (RGB images).
Revised: The image size is 32 × 32 × 3 pixels (RGB images).

Original: CIFAR-100: The CIFAR-100 dataset (Krizhevsky & Hinton, 2009) is an image classiﬁcation dataset containing 50,000 training images and 10,000 test images that are classiﬁed into 100 categories.
Revised: CIFAR-100: The CIFAR-100 dataset (Krizhevsky & Hinton, 2009) is an image classiﬁcation dataset containing 50,000 training images and 10,000 test images that are classiﬁed into 100 categories.

Original: The image size is 32 × 32 × 3 pixels (RGB images).
Revised: The image size is 32 × 32 × 3 pixels (RGB images).

Original: Street View House Numbers (SVHN): The SVHN dataset (Netzer et al., 2011) is an image classiﬁcation dataset containing 73,257 training images and 26032 test images classiﬁed into 10 classes representing digits.
Revised: Street View House Numbers (SVHN): The SVHN dataset (Netzer et al., 2011) is an image classiﬁcation dataset containing 73,257 training images and 26032 test images classiﬁed into 10 classes representing digits.

Original: The images are digits of house numbers cropped and aligned, taken from the
Revised: The images are digits of house numbers cropped and aligned, taken from the

Original: Google Street View service.
Revised: Google Street View service.

Original: Image size is 32 × 32 × 3 pixels.
Revised: Image size is 32 × 32 × 3 pixels.

Original: ImageNet: The ImageNet dataset (Deng et al., 2009) is an image classiﬁcation dataset containing 1300000 color training images and another 50,000 test images classiﬁed into 1000 categories.
Revised: ImageNet: The ImageNet dataset (Deng et al., 2009) is an image classiﬁcation dataset containing 1300000 color training images and another 50,000 test images classiﬁed into 1000 categories.

Original: A. A RCHITECTURES AND H YPER P ARAMETERS
Revised: A. A RCHITECTURES AND H YPER P ARAMETERS

Original: VGG-16: For the ﬁrst three datasets (CIFAR10, CIFAR100, and SVHN), we used an architecture inspired by the VGG-16 architecture (Simonyan & Zisserman, 2014).
Revised: VGG-16: For the ﬁrst three datasets (CIFAR10, CIFAR100, and SVHN), we used an architecture inspired by the VGG-16 architecture (Simonyan & Zisserman, 2014).

Original: We adapted the VGGarchitecture to the small images size and a relatively small dataset size based on (Liu & Deng, 2015).
Revised: We adapted the VGGarchitecture to the small images size and a relatively small dataset size based on (Liu & Deng, 2015).

Original: We trained the model for 250 epochs using SGD with a momentum value of 0 .
Revised: We trained the model for 250 epochs using SGD with a momentum value of 0 .

Original: We used an initial learning rate of 0.1, a learning rate multiplicative drop of 0.5 every 20 epochs, and a batch size of 128.
Revised: We used an initial learning rate of 0.1, a learning rate multiplicative drop of 0.5 every 20 epochs, and a batch size of 128.

Original: A standard data augmentation was used including horizontal ﬂips, rotations, and shifts.
Revised: A standard data augmentation was used including horizontal ﬂips, rotations, and shifts.

Original: In this learning regime, we reached a validation error of 6.4% for CIFAR-10, 29.2% for CIFAR-100 and 3.54% for SVHN.
Revised: In this learning regime, we reached a validation error of 6.4% for CIFAR-10, 29.2% for CIFAR-100 and 3.54% for SVHN.

Original: Resnet-18:
Revised: Resnet-18:

Original: For ImageNet dataset, we used the Resnet-18 architecture (He et al., 2016); we trained the model using SGD with a batch size of 256 and momentum of 0.9 for 90 epochs.
Revised: For ImageNet dataset, we used the Resnet-18 architecture (He et al., 2016); we trained the model using SGD with a batch size of 256 and momentum of 0.9 for 90 epochs.

Original: We used a learning rate of 0.1, with a learning rate multiplicative decay of 0.1 every 30 epochs.
Revised: We used a learning rate of 0.1, with a learning rate multiplicative decay of 0.1 every 30 epochs.

Original: The model reached a (single center crop) top 1 validation accuracy of 69.6% and top 5 validation accuracy of 89.1%.
Revised: The model reached a (single center crop) top 1 validation accuracy of 69.6% and top 5 validation accuracy of 89.1%.

Original: A. M ETHODS I MPLEMENTATIONS AND H YPER P ARAMETERS
Revised: A. M ETHODS I MPLEMENTATIONS AND H YPER P ARAMETERS

Original: Softmax Response: For the softmax response method (SR) we simply take the relevant softmax value of the sample, κ ( x, i | f ) =
Revised: Softmax Response: For the softmax response method (SR) we simply take the relevant softmax value of the sample, κ ( x, i | f ) =

Original: f ( x ) i .
Revised: f ( x ) i .

Original: NN-distance: We implemented the NN-distance method using k = 500 for the nearest neighbors parameter.
Revised: NN-distance: We implemented the NN-distance method using k = 500 for the nearest neighbors parameter.

Original: We didn’t implemented the two proposed extensions (embedding regularization, and adversarial training), this add-on will degrade the performance of f for better uncertainty estimation, which we are not interested in.
Revised: We didn’t implemented the two proposed extensions (embedding regularization, and adversarial training), this add-on will degrade the performance of f for better uncertainty estimation, which we are not interested in.

Original: Moreover, running the NN-distance with this add-on will require to add it to all other methods to manage a proper comparison.
Revised: Moreover, running the NN-distance with this add-on will require to add it to all other methods to manage a proper comparison.

Original: MC-dropout: The MC-dropout implemented with p = 0 .
Revised: MC-dropout: The MC-dropout implemented with p = 0 .

Original: 5 for the dropout rate, and 100 feed-forward iterations for each sample.
Revised: 5 for the dropout rate, and 100 feed-forward iterations for each sample.

Original: Ensemble: The Ensemble method is implemented as an average of softmax values across ensemble of 5 DNNs
Revised: Ensemble: The Ensemble method is implemented as an average of softmax values across ensemble of 5 DNNs.

Original: Platt scaling (Platt et al., 1999):
Revised: Platt scaling (Platt et al., 1999):

Original: The Platt scaling is applied as follows.
Revised: The Platt scaling is applied as follows.

Original: Given a conﬁdence measure κ and a validation set V , the scaling is the solution of the logistic regression from κ ( x, ˆ y f ( x ) | f ) to κ ∗ ( x, ˆ y f ( x ) | f ) , where κ ∗ ( x, ˆ y f ( x ) | f ) is deﬁned as 0 when x (cid:54)
Revised: Given a conﬁdence measure κ and a validation set V , the scaling is the solution of the logistic regression from κ ( x, ˆ y f ( x ) | f ) to κ ∗ ( x, ˆ y f ( x ) | f ) , where κ ∗ ( x, ˆ y f ( x ) | f ) is deﬁned as 0 when x (cid:54)

Original: =
Revised: =

Original: ˆ y f ( x ) and 1 otherwise.
Revised: ˆ y f ( x ) and 1 otherwise.

Original: We train the logistic regression models based on all points in V .
Revised: We train the logistic regression models based on all points in V .

Original: To validate the training of this calibration we randomly split (50-50) the original test set to a training and test subsets.
Revised: To validate the training of this calibration we randomly split (50-50) the original test set to a training and test subsets.

Original: The calibration is learned over the training subset and evaluated on the test subset.
Revised: The calibration is learned over the training subset and evaluated on the test subset.

Original: The performance of the resulting scaled probabilities has been evaluated using both negative log likelihood (NLL) and the Brier score (Brier, 1950), which is simply the average L 2 distance between the predicted and the true probabilities.
Revised: The performance of the resulting scaled probabilities has been evaluated using both negative log likelihood (NLL) and the Brier score (Brier, 1950), which is simply the average L 2 distance between the predicted and the true probabilities.

Original: B D ETAILED R ESULTS
Revised: B D ETAILED R ESULTS

Original: We provide here the table of the experiments of AES for softmax response and NN-distance now with standard errors.
Revised: We provide here the table of the experiments of AES for softmax response and NN-distance now with standard errors.

Original: Due to computational complexity the standard error for all other methods has not been computed.Baseline
Revised: Due to computational complexity the standard error for all other methods has not been computed.

Original: AES ( k = 10 )

Original: AES ( k = 30 )

Original: AES ( k = 50 )

Original: E-AURC

Original: E-AURC %

Original: E-AURC %

Original: E-AURC %

Original: CIFARSoftmax 4.78 ± 0.

Original: 4.81 ± 0.

Original: 4.49 ± 0.

Original: 4.49 ± 0.

Original: NN-distance 35.10 ± 6.

Original: [Table]
Revised: [Table]

Original: C M OTIVATION - E XTENDED E XPERIMENTS
Revised: C M OTIVATION - E XTENDED E XPERIMENTS

Original: In Section 5 we motivated our method by dividing the domain X to “easy points” (green) and “hard points” (red).
Revised: In Section 5 we motivated our method by dividing the domain X to “easy points” (green) and “hard points” (red).

Original: We demonstrated that the “easy points” have a phenomenon similar to overﬁtting, where at some point during training the E-AURC measured for “easy points” start degrading.
Revised: We demonstrated that the “easy points” have a phenomenon similar to overﬁtting, where at some point during training the E-AURC measured for “easy points” start degrading.

Original: This observation strongly motivates our strategy that extracts information from early stages of the training process that helps to recover uncertainty estimates of the easy points.
Revised: This observation strongly motivates our strategy that extracts information from early stages of the training process that helps to recover uncertainty estimates of the easy points.

Original: Here, we extend this demonstration that previously was presented done with respect to the softmax κ function.
Revised: Here, we extend this demonstration that previously was presented done with respect to the softmax κ function.

Original: In Figures 3 and 4 we show plots similar to Figure 2(b,c) for the MC-dropout and NN-distance, respectively.
Revised: In Figures 3 and 4 we show plots similar to Figure 2(b,c) for the MC-dropout and NN-distance, respectively.

Original: It is evident that the overﬁtting occurs in all cases.
Revised: It is evident that the overﬁtting occurs in all cases.

Original: but to a much lesser extent in the case of MC-dropout.
Revised: but to a much lesser extent in the case of MC-dropout.

Original: This result is consistent with the results of the AES algorithm where E-AURC improvement over the MC-dropout was smaller compared to the improvements achieved for the other two methods.
Revised: This result is consistent with the results of the AES algorithm where E-AURC improvement over the MC-dropout was smaller compared to the improvements achieved for the other two methods.

Original: In the case of NN-distance a slight overﬁtting also affects the easy points, but the hard instances are affected much more severely.
Revised: In the case of NN-distance a slight overﬁtting also affects the easy points, but the hard instances are affected much more severely.

Original: Thus, from this perspective in all three cases the proposed correction stratgey is potentially useful.
Revised: Thus, from this perspective in all three cases the proposed correction stratgey is potentially useful.

Original: [Figure] [Figure]
Revised: [Figure] [Figure]