Original: Improved Algorithms for Neural Active Learning
Revised: Improved Algorithms for Neural Active Learning

Original: Abstract
Revised: Abstract

Original: We improve the theoretical and empirical performance of neural-network(NN)- based active learning algorithms for the non-parametric streaming setting.
Revised: We improve the theoretical and empirical performance of neural-network(NN)- based active learning algorithms for the non-parametric streaming setting.

Original: In particular, we introduce two regret metrics by minimizing the population loss that are more suitable in active learning than the one used in state-of-the-art (SOTA) related work.
Revised: In particular, we introduce two regret metrics by minimizing the population loss that are more suitable in active learning than the one used in state-of-the-art (SOTA) related work.

Original: Then, the proposed algorithm leverages the powerful representation of NNs for both exploitation and exploration, has the query decision-maker tailored for k -class classiﬁcation problems with the performance guarantee, utilizes the full feedback, and updates parameters in a more practical and efﬁcient manner.
Revised: Then, the proposed algorithm leverages the powerful representation of NNs for both exploitation and exploration, has the query decision-maker tailored for k -class classiﬁcation problems with the performance guarantee, utilizes the full feedback, and updates parameters in a more practical and efﬁcient manner.

Original: These careful designs lead to an instance-dependent regret upper bound, roughly improving by a multiplicative factor
Revised: These careful designs lead to an instance-dependent regret upper bound, roughly improving by a multiplicative factor

Original: O (log T ) and removing the curse of input dimensionality.
Revised: O (log T ) and removing the curse of input dimensionality.

Original: Furthermore, we show that the algorithm can achieve the same performance as the Bayes-optimal classiﬁer in the long run under the hard-margin setting in classiﬁcation problems.
Revised: Furthermore, we show that the algorithm can achieve the same performance as the Bayes-optimal classiﬁer in the long run under the hard-margin setting in classiﬁcation problems.

Original: In the end, we use extensive experiments to evaluate the proposed algorithm and SOTA baselines, to show the improved empirical performance.
Revised: In the end, we use extensive experiments to evaluate the proposed algorithm and SOTA baselines, to show the improved empirical performance.

Original:  Introduction
Revised:  Introduction

Original: The Neural Network (NN) is one of the indispensable paradigms in machine learning and is widely used in multifarious supervised-learning tasks [24].
Revised: The Neural Network (NN) is one of the indispensable paradigms in machine learning and is widely used in multifarious supervised-learning tasks [23].

Original: As more and more complicated NNs are developed, the requirement of the training procedure on the labeled data grows, incurring signiﬁcant cost of label annotation.
Revised: As more and more complicated NNs are developed, the requirement of the training procedure on the labeled data grows, incurring signiﬁcant cost of label annotation.

Original: Active learning investigates effective techniques on a much smaller labeled data set while attaining the comparable generalization performance to passive learning [20].
Revised: Active learning investigates effective techniques on a much smaller labeled data set while attaining the comparable generalization performance to passive learning [19].

Original: In this paper, we focus on the classiﬁcation problem in the streaming setting of active learning with NN models.
Revised: In this paper, we focus on the classiﬁcation problem in the streaming setting of active learning with NN models.

Original: At every round, the learner receives an instance and is compelled to decide on-the-ﬂy whether or not to observe the label associated with this instance.
Revised: At every round, the learner receives an instance and is compelled to decide on-the-ﬂy whether or not to observe the label associated with this instance.

Original: This problem seeks to maximize the generalization capability of learned NNs in a sequence of rounds, such that the model has robust performance on the unseen data from the same distribution [41].
Revised: This problem seeks to maximize the generalization capability of learned NNs in a sequence of rounds, such that the model has robust performance on the unseen data from the same distribution [40].

Original: In active learning, given access to the i.i.d.
Revised: In active learning, given access to the i.i.d.

Original: generated instances from a distribution D , suppose there exist a class of functions F that formulate the mapping from instances to theirs labels.
Revised: generated instances from a distribution D , suppose there exist a class of functions F that formulate the mapping from instances to theirs labels.

Original: In the parametric setting, i.e., F has ﬁnite VC-dimension [26], existing works [25, 15, 8] have shown that the active learning algorithms can achieve the convergence rate of e
Revised: In the parametric setting, i.e., F has ﬁnite VC-dimension [25], existing works [24, 14, 7] have shown that the active learning algorithms can achieve the convergence rate of e

Original: O (1 / p N ) to the best population loss in F , where N is the number of label queries.
Revised: O (1 / p N ) to the best population loss in F , where N is the number of label queries.

Original: In the non-parametric setting, recent works [35, 36] provide the similar convergence results while suffering from the curse of input dimensionality.
Revised: In the non-parametric setting, recent works [34, 35] provide the similar convergence results while suffering from the curse of input dimensionality.

Original: Unfortunately, most of NN-based approaches to active learning do not come with the performance guarantee, despite having powerful empirical results.
Revised: Unfortunately, most of NN-based approaches to active learning do not come with the performance guarantee, despite having powerful empirical results.

Original: The ﬁrst performance guarantee for neural active learning has been established in a recent work by [49], and the analysis is for over-parameterized neural networks with the assistance of Neural
Revised: The ﬁrst performance guarantee for neural active learning has been established in a recent work by [48], and the analysis is for over-parameterized neural networks with the assistance of Neural

Original: Tangent Kernel (NTK).
Revised: Tangent Kernel (NTK).

Original: We carefully investigate the limitations of [49], which turn into the main motivations of our paper.
Revised: We carefully investigate the limitations of [48], which turn into the main motivations of our paper.

Original: First, [49] transforms the classiﬁcation problem into a multi-armed bandit problem [56], to minimize a pseudo regret metric.
Revised: First, [48] transforms the classiﬁcation problem into a multi-armed bandit problem [55], to minimize a pseudo regret metric.

Original: Yet, on the grounds that they seek to minimize the conditional population loss on a sequence of given data, it is dubious that the pseudo regret used in [49] can explicitly measure the generalization capability of given algorithms (see Remark 2.1).
Revised: Yet, on the grounds that they seek to minimize the conditional population loss on a sequence of given data, it is dubious that the pseudo regret used in [48] can explicitly measure the generalization capability of given algorithms (see Remark 2.1).

Original: Second, the training process for NN models is not efﬁcient, as [49] uses vanilla gradient descent and starts from randomly initialized parameters in every round.
Revised: Second, the training process for NN models is not efﬁcient, as [48] uses vanilla gradient descent and starts from randomly initialized parameters in every round.

Original: Third, although [49] removes the curse of input dimensionality d , the performance guarantee strongly suffers from another introduced term, the effective dimensionality e d , which can be thought of as the non-linear dimensionalities of Hilbert space spanned by NTK.
Revised: Third, although [48] removes the curse of input dimensionality d , the performance guarantee strongly suffers from another introduced term, the effective dimensionality e d , which can be thought of as the non-linear dimensionalities of Hilbert space spanned by NTK.

Original: In the worse case, the magnitude of e d can be an unacceptably large number and thus the performance guarantee collapses.
Revised: In the worse case, the magnitude of e d can be an unacceptably large number and thus the performance guarantee collapses.

Original: 1.1 Main contributions
Revised: 1.1 Main contributions

Original: In this paper, we propose a novel algorithm, I-NeurAL
Revised: In this paper, we propose a novel algorithm, I-NeurAL

Original: ( I mproved Algorithms for Neur al A ctive
Revised: ( I mproved Algorithms for Neur al A ctive

Original: L earning), to tackle the above limitations.
Revised: L earning), to tackle the above limitations.

Original: Our contributions can be summarized as follows: (1) We consider the k -class classiﬁcation problem, and we introduce two new regret metrics to minimize the population loss, which can directly reﬂect the generalization capability of NN-based algorithms.
Revised: Our contributions can be summarized as follows: (1) We consider the k -class classiﬁcation problem, and we introduce two new regret metrics to minimize the population loss, which can directly reﬂect the generalization capability of NN-based algorithms.

Original: (2) I-NeurAL has a neural exploration strategy with a novel component to decide whether or not to query the label, coming with the performance guarantee.
Revised: (2) I-NeurAL has a neural exploration strategy with a novel component to decide whether or not to query the label, coming with the performance guarantee.

Original: I-NeurAL exploits the full feedback in active learning which is a subtle but effective idea.
Revised: I-NeurAL exploits the full feedback in active learning which is a subtle but effective idea.

Original: (3) I-NeurAL is designed to support minibatch Stochastic Gradient Descent (SGD).
Revised: (3) I-NeurAL is designed to support minibatch Stochastic Gradient Descent (SGD).

Original: In particular, at every round, I-NeurAL does mini-batch
Revised: In particular, at every round, I-NeurAL does mini-batch

Original: SGD starting with the parameters of the last round, i.e., with warm start, which is more efﬁcient and practical compared to [49].
Revised: SGD starting with the parameters of the last round, i.e., with warm start, which is more efﬁcient and practical compared to [48].

Original: (4) Without any noise assumption on the data distribution, we provide an instance-dependent performance guarantee of I-NeurAL for over-parameterized neural networks.
Revised: (4) Without any noise assumption on the data distribution, we provide an instance-dependent performance guarantee of I-NeurAL for over-parameterized neural networks.

Original: Compared to [49], we remove the curse of both the input dimensionality d and the effective dimensionality e d ; Moreover, we roughly improve the regret by a multiplicative factor log( T ) , where
Revised: Compared to [48], we remove the curse of both the input dimensionality d and the effective dimensionality e d ; Moreover, we roughly improve the regret by a multiplicative factor log( T ) , where

Original: T is the number of rounds.
Revised: T is the number of rounds.

Original: (5) under a hard-margin assumption on the data distribution,to the best of our knowledge, we providethe ﬁrst performance analysis that NN models can achieve the same generalization capability as Bayes-optimal classiﬁer after a ﬁxed number of label queries; (6) we conduct extensive experiments on real-world data sets to demonstrate the improved performance of I-NeurAL over state-of-the-art baselines including the closest work [49] which has not provided empirical validation of their proposed algorithms.
Revised: (5) under a hard-margin assumption on the data distribution, we provide that NN models can achieve the same generalization capability as Bayes-optimal classiﬁer after O (log T ) number of label queries; (6) we conduct extensive experiments on real-world data sets to demonstrate the improved performance of I-NeurAL over state-of-the-art baselines including the closest work [48] which has not provided empirical validation of their proposed algorithms.

Original: 1.2 Related Work
Revised: 1.2 Related Work

Original: Active learning has been extensively studied and applied to many essential applications [45].
Revised: Active learning has been extensively studied and applied to many essential applications [44].

Original: Bayesian active learning methods typically use a probabilistic regression model to estimate the improvement of each query [30, 42].
Revised: Bayesian active learning methods typically use a probabilistic regression model to estimate the improvement of each query [29, 41].

Original: In spite of effectiveness on the small or moderate data sets, the Bayesian-based approaches are difﬁcult to scale to large-scale data sets because of the batch sampling [44].
Revised: In spite of effectiveness on the small or moderate data sets, the Bayesian-based approaches are difﬁcult to scale to large-scale data sets because of the batch sampling [43].

Original: Another important class, margin algorithms or uncertainty sampling [34], obtains considerate performance improvement over passive learning and is further developed by many practitioners [21, 29, 38, 16].
Revised: Another important class, margin algorithms or uncertainty sampling [33], obtains considerate performance improvement over passive learning and is further developed by many practitioners [20, 28, 37, 15].

Original: Margin algorithms are ﬂexible and can be adapted to both streaming and pool settings.
Revised: Margin algorithms are ﬂexible and can be adapted to both streaming and pool settings.

Original: In the pool setting, a line of works utilize the neural networks in active learning to improve the empirical performance [37, 43, 6, 18, 31, 46, 48, 55, 5].
Revised: In the pool setting, a line of works utilize the neural networks in active learning to improve the empirical performance [36, 42, 5, 17, 30, 45, 47, 54, 4].

Original: However, they do not provide performance guarantee for NN-based active learning algorithms.
Revised: However, they do not provide performance guarantee for NN-based active learning algorithms.

Original: From the theoretical perspective, [52, 22, 7, 9, 53] provide the performance guarantee with the speciﬁc classes of functions and [27, 23] present the theoretical analysis of active learning algorithms with the surrogate loss functions for binary classiﬁcation.
Revised: From the theoretical perspective, [51, 21, 6, 8, 52] provide the performance guarantee with the speciﬁc classes of functions and [26, 22] present the theoretical analysis of active learning algorithms with the surrogate loss functions for binary classiﬁcation.

Original: However, their performance guarantee is restricted within hypothesis classes, i.e, the parametric setting.
Revised: However, their performance guarantee is restricted within hypothesis classes, i.e, the parametric setting.

Original: In contrast, our goal is to derive an NN-based algorithm in the non-parametric setting that performs well both empirically and theoretically.
Revised: In contrast, our goal is to derive an NN-based algorithm in the non-parametric setting that performs well both empirically and theoretically.

Original: Neural contextual bandits[56, 54, 14, 12, 13, 40] provide the principled method to balance between the exploitation and exploration [10, 11].
Revised: Neural contextual bandits[55, 53, 13, 11, 12, 39] provide the principled method to balance between the exploitation and exploration [9, 10].

Original: [49] transforms active learning into neural contextual bandit problem and obtains a performance guarantee, of which limitations are discussed above.
Revised: [48] transforms active learning into neural contextual bandit problem and obtains a performance guarantee, of which limitations are discussed above.

Original: As [49] is the closest related work to our paper, we emphasize the differences of our techniques from [49] throughout the paper.
Revised: As [48] is the closest related work to our paper, we emphasize the differences of our techniques from [48] throughout the paper.

Original: We introduce the problem deﬁnition and proposed algorithms in Section 2 and Section 3 respectively.
Revised: We introduce the problem deﬁnition and proposed algorithms in Section 2 and Section 3 respectively.

Original: Then, we provide performance guarantees in Section 4 and empirical results in Section 5, ending with the conclusion in Section 6.
Revised: Then, we provide performance guarantees in Section 4 and empirical results in Section 5, ending with the conclusion in Section 6.

Original:  Problem Deﬁnition
Revised:  Problem Deﬁnition

Original: In this paper, we study the streaming setting of active learning in the k -class classiﬁcation problem.
Revised: In this paper, we study the streaming setting of active learning in the k -class classiﬁcation problem.

Original: Let X denote the input space over R d , Y = { 1 , 2 , . .
Revised: Let X denote the input space over R d , Y = { 1 , 2 , . .

Original: . , k } represent the label space, and D be some unknown distribution over X ⇥ Y .
Revised: . , k } represent the label space, and D be some unknown distribution over X ⇥ Y .

Original: At round t 2 [ T ] = { 1 , 2 , . . .
Revised: At round t 2 [ T ] = { 1 , 2 , . . .

Original: , T } , an instance x t is drawn from the marginal distribution
Revised: , T } , an instance x t is drawn from the marginal distribution

Original: D X
Revised: D X

Original: and accordingly y t is drawn from the conditional distribution D Y| x t .
Revised: and accordingly y t is drawn from the conditional distribution D Y| x t .

Original: Here, y t can be thought of as the index of the class that x t belongs to.
Revised: Here, y t can be thought of as the index of the class that x t belongs to.

Original: Inspired by [49], we ﬁrst transform x t into k context vectors representing the k classes respectively: x t, 1 = ( x > t , 0 > , . . . , 0 > ) > , x t, 2 = ( 0 > , x > t , . .
Revised: Inspired by [48], we ﬁrst transform x t into k context vectors representing the k classes respectively: x t, 1 = ( x > t , 0 > , . . . , 0 > ) > , x t, 2 = ( 0 > , x > t , . .

Original: . , 0 > ) > , . .
Revised: . , 0 > ) > , . .

Original: , x t,k
Revised: , x t,k

Original: = ( 0 > , 0 > , . . .
Revised: = ( 0 > , 0 > , . . .

Original: , x > t ) > and x t,i 2 R dk , 8 i 2 [ k ] .
Revised: , x > t ) > and x t,i 2 R dk , 8 i 2 [ k ] .

Original: In accordance with context vectors, we construct the k label vectors representing the k possible prediction: y t, 1 = (1 , 0 , . . . , 0) > , y t, 2 = (0 , 1 , . . . , 0) > , . . .
Revised: In accordance with context vectors, we construct the k label vectors representing the k possible prediction: y t, 1 = (1 , 0 , . . . , 0) > , y t, 2 = (0 , 1 , . . . , 0) > , . . .

Original: , y t,k
Revised: , y t,k

Original: = (0 , 0 , . . . , 1) > and y t,
Revised: = (0 , 0 , . . . , 1) > and y t,

Original: i 2 R k , 8 i 2 [ k ] .
Revised: i 2 R k , 8 i 2 [ k ] .

Original: Thus, y t,y t is the ground-truth label vector for x t .
Revised: Thus, y t,y t is the ground-truth label vector for x t .

Original: Under the non-parametric setting of active learning, we deﬁne an unknown function h to formulate the conditional distribution D Y| x t : X k !
Revised: Under the non-parametric setting of active learning, we deﬁne an unknown function h to formulate the conditional distribution D Y| x t : X k !

Original: [0 , 1] , such that
Revised: [0 , 1] , such that

Original: [Equation]
Revised: [Equation]

Original: which is subject to
Revised: which is subject to

Original: P ki =1 h ( x t,i ) = 1 .
Revised: P ki =1 h ( x t,i ) = 1 .

Original: For simplicity, we consider the k -class classiﬁcation problem with 0-1 loss.
Revised: For simplicity, we consider the k -class classiﬁcation problem with 0-1 loss.

Original: Given x t , i.e., x t,i , i 2 [ k ] , let b i be the index of the class predicted by some hypothesis f and thus y t, b
Revised: Given x t , i.e., x t,i , i 2 [ k ] , let b i be the index of the class predicted by some hypothesis f and thus y t, b

Original: i is the prediction.
Revised: i is the prediction.

Original: Then, we have the following loss:
Revised: Then, we have the following loss:

Original: [Equation]
Revised: [Equation]

Original: where is the indicator function.
Revised: where is the indicator function.

Original: Given the number of rounds T , at each round t 2 [ T ] , the learner receives an instance x t drawn i.i.d.
Revised: Given the number of rounds T , at each round t 2 [ T ] , the learner receives an instance x t drawn i.i.d.

Original: from D X .
Revised: from D X .

Original: Then, the learner needs to make a prediction y t, b i , and at the same time, decide on-the-ﬂy whether or not to query the label y t,y t where y t is drawn i.i.d.
Revised: Then, the learner needs to make a prediction y t, b i , and at the same time, decide on-the-ﬂy whether or not to query the label y t,y t where y t is drawn i.i.d.

Original: from D Y| x t .
Revised: from D Y| x t .

Original: As the goal of active learning tasks is often to minimize the population loss [41], we introduce the following two regret metrics.
Revised: As the goal of active learning tasks is often to minimize the population loss [40], we introduce the following two regret metrics.

Original: Deﬁnition 2.1 (Latest Population Regret) .
Revised: Deﬁnition 2.1 (Latest Population Regret) .

Original: Given the data distribution D , the number of rounds T , the Latest Population Regret is deﬁned as
Revised: Given the data distribution D , the number of rounds T , the Latest Population Regret is deﬁned as

Original: [Equation]
Revised: [Equation]

Original: where y T,i ⇤ is the prediction the Bayes-optimal classiﬁer would make on instance x T ,
Revised: where y T,i ⇤ is the prediction the Bayes-optimal classiﬁer would make on instance x T ,

Original: i.e., i ⇤ =
Revised: i.e., i ⇤ =

Original: arg max i 

Revised: arg max i

Original: [ k ] h ( x T,i ) for y T,i ⇤ .
Revised: [ k ] h ( x T,i ) for y T,i ⇤ .

Original: Deﬁnition 2.2 (Cumulative Population Regret) .
Revised: Deﬁnition 2.2 (Cumulative Population Regret) .

Original: Given the data distribution D , the number of rounds
Revised: Given the data distribution D , the number of rounds

Original: T , the Cumulative Population Regret is deﬁned as:
Revised: T , the Cumulative Population Regret is deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: where y t,i ⇤ is the prediction the Bayes-optimal classiﬁer would make on instance x t , i.e., i ⇤ =
Revised: where y t,i ⇤ is the prediction the Bayes-optimal classiﬁer would make on instance x t , i.e., i ⇤ =

Original: arg max i 

Revised: arg max i

Original: [ k ] h ( x t,i ) for y t,i ⇤ .
Revised: [ k ] h ( x t,i ) for y t,i ⇤ .

Original: R T measures the performance at the last round T only, and R T measures the overall performance in
Revised: R T measures the performance at the last round T only, and R T measures the overall performance in

Original: T rounds combined.
Revised: T rounds combined.

Original: Therefore, the goal of this problem is to minimize R T or R T , or both.
Revised: Therefore, the goal of this problem is to minimize R T or R T , or both.

Original: At the same time, we also aim to minimize the following expected query cost:
Revised: At the same time, we also aim to minimize the following expected query cost:

Original: [Equation]
Revised: [Equation]

Original: where I t is the indicator of the query decision in round t such that I t
Revised: where I t is the indicator of the query decision in round t such that I t

Original: = 1 if y t is observed; I t = 0 , otherwise.
Revised: = 1 if y t is observed; I t = 0 , otherwise.

Original: Remark 2.1.
Revised: Remark 2.1.

Original: Minimizing R T or R T shows the generalization capability of the learned hypothesis on the distribution D .
Revised: Minimizing R T or R T shows the generalization capability of the learned hypothesis on the distribution D .

Original: However, the problem deﬁned in [49] is to minimize the cumulative conditional population regret as follows:
Revised: However, the problem deﬁned in [48] is to minimize the cumulative conditional population regret as follows:

Original: [Equation]
Revised: [Equation]

Original: As E y t ⇠ D Y| x t [ L ( y t, b i , y t,y t ) | x t ] is the population loss conditioned on x t , unfortunately, e
Revised: As E y t ⇠ D Y| x t [ L ( y t, b i , y t,y t ) | x t ] is the population loss conditioned on x t , unfortunately, e

Original: R T only measures the performance of the learned hypothesis on the collected data { x t } Tt =1 , and e
Revised: R T only measures the performance of the learned hypothesis on the collected data { x t } Tt =1 , and e

Original: R T cannot directly measure the accuracy of the hypothesis on unseen data instances.
Revised: R T cannot directly measure the accuracy of the hypothesis on unseen data instances.

Original: Although e
Revised: Although e

Original: R T follows the regret deﬁnition in multi-armed bandits [56], it is fair to say that e
Revised: R T follows the regret deﬁnition in multi-armed bandits [55], it is fair to say that e

Original: R T may not be a good metric in active learning.
Revised: R T may not be a good metric in active learning.

Original:  Proposed Algorithms
Revised:  Proposed Algorithms

Original: In this section, we elaborate on the proposed algorithm I-NeurAL (Algorithm 1).
Revised: In this section, we elaborate on the proposed algorithm I-NeurAL (Algorithm 1).

Original: In contrast to the directly comparable work [49], I-NeurAL has the following novel and advantageous aspects: (1)
Revised: In contrast to the directly comparable work [48], I-NeurAL has the following novel and advantageous aspects: (1)

Original: I-NeurAL incorporates a neural-based exploration strategy (Line 6) inspired by recent advances in bandits [14] to solve the exploitation-exploration dilemma in the decision for whether or not to query labels; (2) I-NeurAL includes a novel component (Line 11) to decide whether or not to query labels in the k -class classiﬁcation problem; (3) I-NeurAL infers and exploits the feedback of all the contexts (Lines 12-17), instead of only utilizing the feedback of the chosen context in [49]; (4) I-NeurAL conducts mini-batch SGD based on the parameters of the last round (Algorithm 2), which is more practical, as opposed to conducting vanilla gradient descent from the initialization at every round in [49].
Revised: I-NeurAL incorporates a neural-based exploration strategy (Line 6) inspired by recent advances in bandits [13] to solve the exploitation-exploration dilemma in the decision for whether or not to query labels; (2) I-NeurAL includes a novel component (Line 11) to decide whether or not to query labels in the k -class classiﬁcation problem; (3) I-NeurAL infers and exploits the feedback of all the contexts (Lines 12-17), instead of only utilizing the feedback of the chosen context in [48]; (4) I-NeurAL conducts mini-batch SGD based on the parameters of the last round (Algorithm 2), which is more practical, as opposed to conducting vanilla gradient descent from the initialization at every round in [48].

Original: Next, we will present the details of I-NeurAL.
Revised: Next, we will present the details of I-NeurAL.

Original: Exploitation Network f 1 .
Revised: Exploitation Network f 1 .

Original: Given x t,i , i 2 [ k ] , to learn the unknown function h (Eq.
Revised: Given x t,i , i 2 [ k ] , to learn the unknown function h (Eq.

Original: (2.1)), we use a fully-connected neural network f 1 with L -depth and m -width: f 1 ( x t,i ; ✓ 1 ) =
Revised: (2.1)), we use a fully-connected neural network f 1 with L -depth and m -width: f 1 ( x t,i ; ✓ 1 ) =

Original: W 1 L  ( W 1 L  1  ( W 1 L  2 . . .
Revised: W 1 L  ( W 1 L  1  ( W 1 L  2 . . .

Original:  ( W 11 x t,i ))) , (3.1) where W 11 2 R m ⇥ kd , W 1 l 2 R m ⇥ m , for 2  l  L  1 , W 1 L 2 R 1 ⇥ m , ✓ 1 = [ vec ( W 11 ) > , . .
Revised:  ( W 11 x t,i ))) , (3.1) where W 11 2 R m ⇥ kd , W 1 l 2 R m ⇥ m , for 2  l  L  1 , W 1 L 2 R 1 ⇥ m , ✓ 1 = [ vec ( W 11 ) > , . .

Original: , vec ( W 1 L ) > ] > 2 R p 1 , and  is the ReLU activation function  ( x ) = max { 0 , x } .
Revised: , vec ( W 1 L ) > ] > 2 R p 1 , and  is the ReLU activation function  ( x ) = max { 0 , x } .

Original: In round t , given x t,i , i 2 [ k ] , f 1 ( x t,i ; ✓ 1 t  1 ) is assigned to learn h ( x t,i ) .
Revised: In round t , given x t,i , i 2 [ k ] , f 1 ( x t,i ; ✓ 1 t  1 ) is assigned to learn h ( x t,i ) .

Original: Based on the fact h ( x t,i ) =
Revised: Based on the fact h ( x t,i ) =

Original: E y t ⇠ D Y| x t [1  L ( y t,i , y t,y t )] , it is natural to regard 1  L ( y t,i , y t,y t ) as the label for training f 1 .
Revised: E y t ⇠ D Y| x t [1  L ( y t,i , y t,y t )] , it is natural to regard 1  L ( y t,i , y t,y t ) as the label for training f 1 .

Original: Note that we take the basic fully-connected network as an example for the sake of analysis in over-parameterized networks and f 1 can be easily replaced with more complicated models depending on the tasks.
Revised: Note that we take the basic fully-connected network as an example for the sake of analysis in over-parameterized networks and f 1 can be easily replaced with more complicated models depending on the tasks.

Original: Exploration Network f 2 .
Revised: Exploration Network f 2 .

Original: In addition to the network f 1 , we assign another network f 2 to explore uncertain information contained in incoming instances.
Revised: In addition to the network f 1 , we assign another network f 2 to explore uncertain information contained in incoming instances.

Original: First, we carefully design the input of f 2 to incorporate the context vectors of the instance and the discrimination-ability of f 1 , to learn the error between the Bayes-optimal probability h ( x t,i ) and the prediction f 1 ( x t,i ; ✓ 1 ) .
Revised: First, we carefully design the input of f 2 to incorporate the context vectors of the instance and the discrimination-ability of f 1 , to learn the error between the Bayes-optimal probability h ( x t,i ) and the prediction f 1 ( x t,i ; ✓ 1 ) .

Original: Deﬁnition 3.1 (Derivative-Context (DC) Embedding) .
Revised: Deﬁnition 3.1 (Derivative-Context (DC) Embedding) .

Original: Given the exploitation network f 1 ( · ; ✓ 1 t  1 ) and an input context x t,i , its DC embeding is deﬁned as
Revised: Given the exploitation network f 1 ( · ; ✓ 1 t  1 ) and an input context x t,i , its DC embeding is deﬁned as

Original: [Equation]
Revised: [Equation]

Original: where O x t,i f 1 is the partial derivative of f 1 ( x t,i ; ✓ 1 t  1 ) with respect to x t,i .
Revised: where O x t,i f 1 is the partial derivative of f 1 ( x t,i ; ✓ 1 t  1 ) with respect to x t,i .

Original:  ( x t,i ) is normalized so that k  ( x t,i ) k 2 = 1 .
Revised:  ( x t,i ) is normalized so that k  ( x t,i ) k 2 = 1 .

Original: Note that the input for f 2 in [14] is the gradient with respect to ✓ 1 , denoted by O ✓ 1 f 1 ( x t,i ; ✓ 1 t  1 ) 2 R p 1 .
Revised: Note that the input for f 2 in [13] is the gradient with respect to ✓ 1 , denoted by O ✓ 1 f 1 ( x t,i ; ✓ 1 t  1 ) 2 R p 1 .

Original: Its dimensionality is much larger than
Revised: Its dimensionality is much larger than

Original: O x t,i f 1 ( x t,i ; ✓ 1 t  1 ) in Deﬁnition 3.1, may causing signiﬁcant computation cost.
Revised: O x t,i f 1 ( x t,i ; ✓ 1 t  1 ) in Deﬁnition 3.1, may causing signiﬁcant computation cost.

Original: Given the input  ( x t,i ) , similarly, we choose the fully-connected network to build f 2 :
Revised: Given the input  ( x t,i ) , similarly, we choose the fully-connected network to build f 2 :

Original: [Equation]
Revised: [Equation]

Original: where W 21 2 R m ⇥ 2 kd , W 2 l 2 R m ⇥ m , for 2  l  L  1 , W 2 L = R 1 ⇥ m and ✓ 2 = [ vec ( W 21 ) > , . .
Revised: where W 21 2 R m ⇥ 2 kd , W 2 l 2 R m ⇥ m , for 2  l  L  1 , W 2 L = R 1 ⇥ m and ✓ 2 = [ vec ( W 21 ) > , . .

Original: , vec ( W 2 L ) > ] > 2 R p 2 .
Revised: , vec ( W 2 L ) > ] > 2 R p 2 .

Original: In round t , given x t,i , 8 i 2 [ k ] , f 2 is to predict h ( x t,i ) 
Revised: In round t , given x t,i , 8 i 2 [ k ] , f 2 is to predict h ( x t,i )

Original: Input: T (number of rounds) f 1 , f 2 (neural networks), ⌘ 1 , ⌘ 2 (learning rate),  (exploration parameter), b (batch size),  (conﬁdence level) 1: Initialize ✓ 10 , ✓ 20 ; b ✓ 1 0 = ✓ 10 ; b ✓ 2 0 =
Revised: Input: T (number of rounds) f 1 , f 2 (neural networks), ⌘ 1 , ⌘ 2 (learning rate),  (exploration parameter), b (batch size),  (conﬁdence level) 1: Initialize ✓ 10 , ✓ 20 ; b ✓ 1 0 = ✓ 10 ; b ✓ 2 0 =

Original: ✓2: H 10 = ; ; H 20 = ; 3: for t = 1 , 2 , . .
Revised: ✓2: H 10 = ; ; H 20 = ; 3: for t = 1 , 2 , . .

Original: , T do 4:
Revised: , T do 4:

Original: Observe instance x t 2 R d and build x t,i , 8 i 2 [ k ] 5: for each i 2 [ k ] do 6: f ( x t,i ; ✓ t  1 ) =
Revised: Observe instance x t 2 R d and build x t,i , 8 i 2 [ k ] 5: for each i 2 [ k ] do 6: f ( x t,i ; ✓ t  1 ) =

Original: ⇣ f 1 ( x t,i ; ✓ 1 t  1 ) Exploitation Score + f 

Revised: ⇣ f 1 ( x t,i ; ✓ 1 t  1 ) Exploitation Score + f

Original: (  ( x t,i ); ✓ 2 t  1 ) Exploration Score ⌘ 7: end for 8: b i = arg max i 2 [ k ] f ( x t,i ; ✓ t  1 ) 9: i  = arg max i 2 ([ k ] \{ b i } ) f
Revised: (  ( x t,i ); ✓ 2 t  1 ) Exploration Score ⌘ 7: end for 8: b i = arg max i 2 [ k ] f ( x t,i ; ✓ t  1 ) 9: i  = arg max i 2 ([ k ] \{ b i } ) f

Original: ( x t,i ; ✓ t  1 ) 10:
Revised: ( x t,i ; ✓ t  1 ) 10:

Original: Predict y t, b i 11:
Revised: Predict y t, b i 11:

Original: I t = {| f ( x t, b i ; ✓ t  1 )  f ( x t,i  ; ✓ t  1 ) | < 2   t } 2 { 0 , 1 } ;  t = q 2 c 1 t + ⇣ c 2 3 L p 2 t ⌘ + q 2log( c 3 tk ) /  ) t 12: if I t = 1 then 13:
Revised: I t = {| f ( x t, b i ; ✓ t  1 )  f ( x t,i  ; ✓ t  1 ) | < 2   t } 2 { 0 , 1 } ;  t = q 2 c 1 t + ⇣ c 2 3 L p 2 t ⌘ + q 2log( c 3 tk ) /  ) t 12: if I t = 1 then 13:

Original: Query x t and observe y t 14: for i 2 [ k ] do 15: r 1 t,i =
Revised: Query x t and observe y t 14: for i 2 [ k ] do 15: r 1 t,i =

Original: 1  L ( y t,i , y t,y t ) (deﬁned in E.q. (2.2)) 16: r 2 t,i =
Revised: 1  L ( y t,i , y t,y t ) (deﬁned in E.q. (2.2)) 16: r 2 t,i =

Original: r 1 t,i  f 1 ( x t,i ; ✓ 1 t  1 ) 17: end for 18: else 19: for i 2 [ k ] do 20:
Revised: r 1 t,i  f 1 ( x t,i ; ✓ 1 t  1 ) 17: end for 18: else 19: for i 2 [ k ] do 20:

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: [Equation]
Revised: [Equation]

Original: y
Revised: y

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: ) 22: end for 23: end if 24:
Revised: ) 22: end for 23: end if 24:

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: 26: ✓ 1 t , ✓ 2 t =
Revised: 26: ✓ 1 t , ✓ 2 t =

Original: Mini-Batch-SGD-Warm-Start ( f 1 , f 2 , H 1 t , H 2 t , b ) 27: end for f 1 ( x t,i ; ✓ 1 t  1 ) for exploration.
Revised: Mini-Batch-SGD-Warm-Start ( f 1 , f 2 , H 1 t , H 2 t , b ) 27: end for f 1 ( x t,i ; ✓ 1 t  1 ) for exploration.

Original: Because h ( x t,i )  f
Revised: Because h ( x t,i )  f

Original: [Equation]
Revised: [Equation]

Original: x
Revised: x

Original: [Equation]
Revised: [Equation]

Original: ✓
Revised: ✓

Original: [Equation]
Revised: [Equation]

Original: 
Revised:

Original: [Equation]
Revised: [Equation]

Original: ) =
Revised: ) =

Original: [Equation]
Revised: [Equation]

Original: f 1 ( x t,i ; ✓ 1 t  1 )] , we regard 1  L ( y t,i , y t,y t )
Revised: f 1 ( x t,i ; ✓ 1 t  1 )] , we regard 1  L ( y t,i , y t,y t )

Original:  f 1 ( x t,i ; ✓ 1 t  1 ) as the label for training f 2 .
Revised:  f 1 ( x t,i ; ✓ 1 t  1 ) as the label for training f 2 .

Original: To sum up, in round t , given x t,i , 8 i 2 [ k ] , the prediction b i ( y t, b i ) is made based on the sum of exploitation and exploration scores, i.e., f 1 ( x t,i ; ✓ 1 t  1 ) + f 2 (  ( x t,i ); ✓ 2 t  1 ) (Lines 5-10).
Revised: To sum up, in round t , given x t,i , 8 i 2 [ k ] , the prediction b i ( y t, b i ) is made based on the sum of exploitation and exploration scores, i.e., f 1 ( x t,i ; ✓ 1 t  1 ) + f 2 (  ( x t,i ); ✓ 2 t  1 ) (Lines 5-10).

Original: Query Decision-maker (Line 11) .
Revised: Query Decision-maker (Line 11) .

Original: A label query is made when I-NeurAL is not conﬁdent enough to discriminate the Bayes-optimal class from other classes.
Revised: A label query is made when I-NeurAL is not conﬁdent enough to discriminate the Bayes-optimal class from other classes.

Original: 2   t (  t is also deﬁned in Lemma 7.3) can be thought of as a conﬁdence interval for the distance between the optimal class and second optimal class, where  is the hyper-parameter to tune the sensitivity of the decision-maker in practice.
Revised: 2   t (  t is also deﬁned in Lemma 7.3) can be thought of as a conﬁdence interval for the distance between the optimal class and second optimal class, where  is the hyper-parameter to tune the sensitivity of the decision-maker in practice.

Original: Given any   1 ,  2 (0 , 1) , with probability at least 1   , based on our analysis (Lemma 7.5),
Revised: Given any   1 ,  2 (0 , 1) , with probability at least 1   , based on our analysis (Lemma 7.5),

Original: E ( x t ,y t )
Revised: E ( x t ,y t )

Original: ⇠ D [ L ( y t, b i , y t,y t )] =
Revised: ⇠ D [ L ( y t, b i , y t,y t )] =

Original: E ( x t ,y t )
Revised: E ( x t ,y t )

Original: ⇠ D [ L ( y t,i ⇤ , y t,y t )] when I t = 0 , i.e., I-NeurAL suffers no regret.
Revised: ⇠ D [ L ( y t,i ⇤ , y t,y t )] when I t = 0 , i.e., I-NeurAL suffers no regret.

Original: Thus, we use y t, b i as the pseudo-label in this case and we have the following update rules.
Revised: Thus, we use y t, b i as the pseudo-label in this case and we have the following update rules.

Original: Utilize Full Feedback (Lines 14-25) .
Revised: Utilize Full Feedback (Lines 14-25) .

Original: Different from the bandit setting where the learner can only observe the reward of the selected context, we can infer the rewards of all contexts in active learning, as we know the speciﬁc class of the current instance.
Revised: Different from the bandit setting where the learner can only observe the reward of the selected context, we can infer the rewards of all contexts in active learning, as we know the speciﬁc class of the current instance.

Original: Thus, for each x t,i , i 2 [ k ] , r 1 t,i = 1  L ( y t,i , y t,y t ) is regarded as the "reward" of x t,i , predicted by f 1 , and r 2 t,i =
Revised: Thus, for each x t,i , i 2 [ k ] , r 1 t,i = 1  L ( y t,i , y t,y t ) is regarded as the "reward" of x t,i , predicted by f 1 , and r 2 t,i =

Original: r 1 t,i  f 1 ( x t,i ; ✓ 1 ) is regarded as the "residual reward" of x t,i , predicted by f 2 .
Revised: r 1 t,i  f 1 ( x t,i ; ✓ 1 ) is regarded as the "residual reward" of x t,i , predicted by f 2 .

Original: In summary, in round t , when I t = 1 , y t,y t is observed to
Revised: In summary, in round t , when I t = 1 , y t,y t is observed to

Original: 1: Deﬁne L 1 [( x , r 1 ); ✓ 1 ] =
Revised: 1: Deﬁne L 1 [( x , r 1 ); ✓ 1 ] =

Original: ( r 1  f 1 ( x ; ✓ 1 )) 2 /2:
Revised: ( r 1  f 1 ( x ; ✓ 1 )) 2 /2:

Original: Uniformly draw a set b
Revised: Uniformly draw a set b

Original: H 1 t ⇢ H 1 t , s.t., | b
Revised: H 1 t ⇢ H 1 t , s.t., | b

Original: H 1 t
Revised: H 1 t

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: b 3: b ✓
Revised: b 3: b ✓

Original: [Equation]
Revised: [Equation]

Original: t =
Revised: t =

Original: b ✓
Revised: b ✓

Original: [Equation]
Revised: [Equation]

Original: b
Revised: b

Original: P
Revised: P

Original: [Equation]
Revised: [Equation]

Original: b
Revised: b

Original: [Equation]
Revised: [Equation]

Original: O ✓
Revised: O ✓

Original: [Equation]
Revised: [Equation]

Original: L
Revised: L

Original: [Equation]
Revised: [Equation]

Original: [( x , r 1 ); b ✓ 1 t  1 ] 4: Deﬁne L 2 [(  ( x ) , r 2 ); ✓ 2 ] = ( r 2  f 2 (  ( x ); ✓ 2 ))
Revised: [( x , r 1 ); b ✓ 1 t  1 ] 4: Deﬁne L 2 [(  ( x ) , r 2 ); ✓ 2 ] = ( r 2  f 2 (  ( x ); ✓ 2 ))

Original: Uniformly draw a set b
Revised: Uniformly draw a set b

Original: H
Revised: H

Original: [Equation]
Revised: [Equation]

Original: t ⇢
Revised: t ⇢

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: .t. ,  | b
Revised: , s.t., | b

Original: H
Revised: H

Original: [Equation]
Revised: [Equation]

Original: = b 6: b ✓ 2 t = b ✓ 2 t  1  ⌘ 2 b
Revised: = b 6: b ✓ 2 t = b ✓ 2 t  1  ⌘ 2 b

Original: P (  ( x ) ,r 1 )b
Revised: P (  ( x ) ,r 1 )b

Original: H 2 t
Revised: H 2 t

Original: O ✓ 2 L 2 [(  ( x ) , r 2 ); b ✓ 2 t  1 ] 7: ⌦ t =
Revised: O ✓ 2 L 2 [(  ( x ) , r 2 ); b ✓ 2 t  1 ] 7: ⌦ t =

Original: ⌦ t  1 [ { ( b ✓ 1 t , b ✓ 2 t ) } 8: Return ( t , t ) uniformly
Revised: ⌦ t  1 [ { ( b ✓ 1 t , b ✓ 2 t ) } 8: Return ( t , t ) uniformly

Original: Therefore, we have the training data H 1 t for f 1 and H 2 t for f 2 .
Revised: Therefore, we have the training data H 1 t for f 1 and H 2 t for f 2 .

Original: Mini-Batch SGD with Warm-Start (Algorithm 3) .
Revised: Mini-Batch SGD with Warm-Start (Algorithm 3) .

Original: Unlike [49] that uses vanilla gradient descent from randomly initialized parameters in each round, causing unnecessarily expensive computation, we extend the training procedure to mini-batch SGD with warm start, i.e., we incrementally train the parameters ✓ t starting from the parameters of the last round ✓ t  1 in each round t .
Revised: Unlike [48] that uses vanilla gradient descent from randomly initialized parameters in each round, causing unnecessarily expensive computation, we extend the training procedure to mini-batch SGD with warm start, i.e., we incrementally train the parameters ✓ t starting from the parameters of the last round ✓ t  1 in each round t .

Original: Algorithm 1 depicts the workﬂow of I-NeurAL.
Revised: Algorithm 1 depicts the workﬂow of I-NeurAL.

Original: Lines 1-2 initialize the parameters where each entry of W l is drawn from the normal distribution N (0 , 2 /m ) and each entry of W L is drawn from N (0 , 1 /m ) for both f 1 and f 2 .
Revised: Lines 1-2 initialize the parameters where each entry of W l is drawn from the normal distribution N (0 , 2 /m ) and each entry of W L is drawn from N (0 , 1 /m ) for both f 1 and f 2 .

Original: H 10 , H 20 store the historical data for f 1 and f 2 respectively.
Revised: H 10 , H 20 store the historical data for f 1 and f 2 respectively.

Original: In each round t , Line 4 builds the k contexts for the observed instance x t , and Lines 5-7 calculate the exploitation-exploration score for each context.
Revised: In each round t , Line 4 builds the k contexts for the observed instance x t , and Lines 5-7 calculate the exploitation-exploration score for each context.

Original: b i (Line 8) is the index of the optimal-predicted class and thus y t, b
Revised: b i (Line 8) is the index of the optimal-predicted class and thus y t, b

Original: i is the prediction.
Revised: i is the prediction.

Original: i  (Line 9) is the index of the second optimal-predicted class, which is used to decide whether to make a query.
Revised: i  (Line 9) is the index of the second optimal-predicted class, which is used to decide whether to make a query.

Original: Line 11 is our decision component.
Revised: Line 11 is our decision component.

Original: When I t = 1 , it shows that we are not conﬁdent enough about our prediction, so that we make a query for x t and observe the rewards for each context (Lines 12-17).
Revised: When I t = 1 , it shows that we are not conﬁdent enough about our prediction, so that we make a query for x t and observe the rewards for each context (Lines 12-17).

Original: When I t = 0 , based on our analysis, with high conﬁdence, the prediction y t, b i matches the one predicted by the Bayes-optimal classiﬁer.
Revised: When I t = 0 , based on our analysis, with high conﬁdence, the prediction y t, b i matches the one predicted by the Bayes-optimal classiﬁer.

Original: Hence, we consider y t, b i as the label and observe the reward for all contexts (Lines 18-23).
Revised: Hence, we consider y t, b i as the label and observe the reward for all contexts (Lines 18-23).

Original: In the end, we update the networks f 1 and f 2 , based on the collected data (Lines 24-26).
Revised: In the end, we update the networks f 1 and f 2 , based on the collected data (Lines 24-26).

Original:  Regret Analysis
Revised:  Regret Analysis

Original: In this section, we provide the regret analysis of I-NeurAL in the over-parameterized neural networks.
Revised: In this section, we provide the regret analysis of I-NeurAL in the over-parameterized neural networks.

Original: First, we need the standard normalization restricted to the input instances.
Revised: First, we need the standard normalization restricted to the input instances.

Original: Assumption 4.1.
Revised: Assumption 4.1.

Original: For any t 2 [ T ] , k x t k 2 = 1 .
Revised: For any t 2 [ T ] , k x t k 2 = 1 .

Original: Inspired by [17], we deﬁne the following function class.
Revised: Inspired by [16], we deﬁne the following function class.

Original: Given a constant ⌫ > 0 , we deﬁne the following ⌫ -ball of ✓ 2 around the random initialization: B ( ✓ 20 , ⌫ ) = {
Revised: Given a constant ⌫ > 0 , we deﬁne the following ⌫ -ball of ✓ 2 around the random initialization: B ( ✓ 20 , ⌫ ) = {

Original: [Equation]
Revised: [Equation]

Original: Recall that r 2 t, b i = r 1 t, b i  f 1 ( x t, b i ; b ✓ 1 t  1 ) .
Revised: Recall that r 2 t, b i = r 1 t, b i  f 1 ( x t, b i ; b ✓ 1 t  1 ) .

Original: Let b ✓ 1 , ⇤ t  1 represent the parameters trained on H 1 , ⇤ t  1 using
Revised: Let b ✓ 1 , ⇤ t  1 represent the parameters trained on H 1 , ⇤ t  1 using

Original: Algorithm 2 with the Bayes-optimal classiﬁer, where H 1 , ⇤ t  1 = { x ⌧ ,i ⇤ , r 1 ⌧ ,i ⇤ } t  1 ⌧ =1 are the historical
Revised: Algorithm 2 with the Bayes-optimal classiﬁer, where H 1 , ⇤ t  1 = { x ⌧ ,i ⇤ , r 1 ⌧ ,i ⇤ } t  1 ⌧ =1 are the historical

Original: Bayes-optimal pairs.
Revised: Bayes-optimal pairs.

Original: We deﬁne r 2 , ⇤ t,i ⇤ =
Revised: We deﬁne r 2 , ⇤ t,i ⇤ =

Original: r 1 t,i ⇤ 
Revised: r 1 t,i ⇤

Original: f 1 ( x t,i ⇤ ;
Revised: f 1 ( x t,i ⇤ ;

Original: [Equation]
Revised: [Equation]

Original: ✓ 1 , ⇤ t  1 ) .
Revised: ✓ 1 , ⇤ t  1 ) .

Original: Then, we provide the following regret bound that depends on the classiﬁcation ability of exploration network class induced by B ( ✓ 20 , ⌫ ) .
Revised: Then, we provide the following regret bound that depends on the classiﬁcation ability of exploration network class induced by B ( ✓ 20 , ⌫ ) .

Original: Theorem 4.1.
Revised: Theorem 4.1.

Original: Given the number of rounds T , for any 
Revised: Given the number of rounds T , for any

Original: [Equation]
Revised: [Equation]

Original: ,  > 1 , ⌫ > 0 , suppose
Revised: ,  > 1 , ⌫ > 0 , suppose

Original: [Equation]
Revised: [Equation]

Original: Then, with probability at least 1   over the initialization of ✓ 10 , ✓ 20 , there exist a small enough constant  , such that Algorithm 1 achieves the following regret bound:
Revised: Then, with probability at least 1   over the initialization of ✓ 10 , ✓ 20 , there exist a small enough constant  , such that Algorithm 1 achieves the following regret bound:

Original: [Equation]
Revised: [Equation]

Original: and at the same time N T  O ( T ) .
Revised: and at the same time N T  O ( T ) .

Original: Theorem 4.1 provides the regret bound of I-NeurAL for R T and R T respectively, R T  O ( p log T p T ) and R T  O ( p T log T ) .
Revised: Theorem 4.1 provides the regret bound of I-NeurAL for R T and R T respectively, R T  O ( p log T p T ) and R T  O ( p T log T ) .

Original: As [49] only provides the regret bound for e
Revised: As [48] only provides the regret bound for e

Original: R T , to show the advantages of
Revised: R T , to show the advantages of

Original: I-NeurAL, we also provide the following lemma for fair comparison.
Revised: I-NeurAL, we also provide the following lemma for fair comparison.

Original: Lemma 4.1.
Revised: Lemma 4.1.

Original: Given the number of rounds T , for any  2 (0 , 1) ,  > 1 , ⌫ > 0 , suppose m  e ⌦ ( poly ( T, k, L, ⌫ )) , ⌘ 1 = ⌘ 2 = ⇥ ( ⌫ p Tm ) , and µ satisﬁes the conditions in Theorem 4.1.
Revised: Given the number of rounds T , for any  2 (0 , 1) ,  > 1 , ⌫ > 0 , suppose m  e ⌦ ( poly ( T, k, L, ⌫ )) , ⌘ 1 = ⌘ 2 = ⇥ ( ⌫ p Tm ) , and µ satisﬁes the conditions in Theorem 4.1.

Original: Then, with probability at least 1   over the initialization of ✓ 10 , ✓ 20 , these exists a small enough constant  , such that Algorithm 1 can achieve the following regret bound:
Revised: Then, with probability at least 1   over the initialization of ✓ 10 , ✓ 20 , these exists a small enough constant  , such that Algorithm 1 can achieve the following regret bound:

Original: [Equation]
Revised: [Equation]

Original: and at the same time N T  O ( T ) .
Revised: and at the same time N T  O ( T ) .

Original: Comparison with [49] .
Revised: Comparison with [48] .

Original: Lemma 4.1 shows that I-NeurAL can achieve the regret bound of same complexity for e
Revised: Lemma 4.1 shows that I-NeurAL can achieve the regret bound of same complexity for e

Original: R T as R T .
Revised: R T as R T .

Original: Under the same assumption in the over-parameterized neural networks, without any assumption on D , Theorem 1 in [49] (i.e., the lower-noise condition with exponent ↵ = 0 , and k = 2 is ignored in the binary classiﬁcation) achieves the following regret bound:
Revised: Under the same assumption in the over-parameterized neural networks, without any assumption on D , Theorem 1 in [48] (i.e., the lower-noise condition with exponent ↵ = 0 , and k = 2 is ignored in the binary classiﬁcation) achieves the following regret bound:

Original: [Equation]
Revised: [Equation]

Original: R
Revised: R

Original: [Equation]
Revised: [Equation]

Original:  O
Revised:  O

Original: [Equation]
Revised: [Equation]

Original: log det
Revised: log det

Original: [Equation]
Revised: [Equation]

Original: I
Revised: I

Original: [Equation]
Revised: [Equation]

Original: p
Revised: p

Original: T (log det( I + H ) + S 2 )) where H is the NTK matrix [28, 4] formed by received instances of all T rounds, S = p h > H  1 h is a complexity term, and
Revised: T (log det( I + H ) + S 2 )) where H is the NTK matrix [27, 3] formed by received instances of all T rounds, S = p h > H  1 h is a complexity term, and

Original: h =
Revised: h =

Original: ( h ( x 1 , b i ) , . . .
Revised: ( h ( x 1 , b i ) , . . .

Original: , h ( x T, b i )) >
Revised: , h ( x T, b i )) >

Original: 2 R T .
Revised: 2 R T .

Original: Note that I-NeurAL and [49] have the same trivial label complexity O ( T ) in this difﬁcult case.
Revised: Note that I-NeurAL and [48] have the same trivial label complexity O ( T ) in this difﬁcult case.

Original: According to the deﬁnition of effective dimension e d in [56], the above regret bound obtained by [49] can be represented by:
Revised: According to the deﬁnition of effective dimension e d in [55], the above regret bound obtained by [48] can be represented by:

Original: [Equation]
Revised: [Equation]

Original: Remark 4.1.
Revised: Remark 4.1.

Original: The instance-dependent complexity term µ reﬂects the possible minimal regression error on the data instances caused by the functions induced by B ( ✓ 20 , ⌫ ) controlled by ⌫ .
Revised: The instance-dependent complexity term µ reﬂects the possible minimal regression error on the data instances caused by the functions induced by B ( ✓ 20 , ⌫ ) controlled by ⌫ .

Original: Such complexity term is ﬁrst introduced in [17].
Revised: Such complexity term is ﬁrst introduced in [16].

Original: When ⌫ is small, the corresponding ball B ( ✓ 20 , ⌫ ) is small, so µ tends to be large; Otherwise, when ⌫ is large, µ tends to be small.
Revised: When ⌫ is small, the corresponding ball B ( ✓ 20 , ⌫ ) is small, so µ tends to be large; Otherwise, when ⌫ is large, µ tends to be small.

Original: In particular, when setting ⌫ = O (1) , Theorem 4.1 and Lemma 4.1 suggests that if the data can be learned by a function in the function class formed by B ( ✓ 20 , O (1)) with the small training error, then I-NeurAL will have the regret with order e
Revised: In particular, when setting ⌫ = O (1) , Theorem 4.1 and Lemma 4.1 suggests that if the data can be learned by a function in the function class formed by B ( ✓ 20 , O (1)) with the small training error, then I-NeurAL will have the regret with order e

Original: O ( p T ) .
Revised: O ( p T ) .

Original: Note that [49] has the complexity term S as well, to reﬂect the boundary of optimal parameters speciﬁc to the data.
Revised: Note that [48] has the complexity term S as well, to reﬂect the boundary of optimal parameters speciﬁc to the data.

Original: Remark 4.2.
Revised: Remark 4.2.

Original: Theorem 4.1 and Lemma 4.1 do not depend on e d .
Revised: Theorem 4.1 and Lemma 4.1 do not depend on e d .

Original: The effective dimension e d was ﬁrst introduced in [47] and then used in [56], which can be thought of as the non-linear dimensionalities in the NTK kernel space.
Revised: The effective dimension e d was ﬁrst introduced in [46] and then used in [55], which can be thought of as the non-linear dimensionalities in the NTK kernel space.

Original: However, e d can be p = m + mkd + m 2 ( L  1) in the worst case, i.e., e d  T (see details in Appendix 9).
Revised: However, e d can be p = m + mkd + m 2 ( L  1) in the worst case, i.e., e d  T (see details in Appendix 9).

Original: Eq.(4.4) has the term O ( e d ) and thus the regret bound obtained by [49] can explode due to e d .
Revised: Eq.(4.4) has the term O ( e d ) and thus the regret bound obtained by [48] can explode due to e d .

Original: This is because the analysis of [49] closely depends on NTK, i.e., to apply Conﬁdence Ellipsoid bound (Theorem 2 in [1]) to the NTK approximation.
Revised: This is because the analysis of [48] closely depends on NTK, i.e., to apply Conﬁdence Ellipsoid bound (Theorem 2 in [1]) to the NTK approximation.

Original: This procedure inevitably bind their regret bound to the determinant of NTK that can have a very large magnitude.
Revised: This procedure inevitably bind their regret bound to the determinant of NTK that can have a very large magnitude.

Original: In contrast, Eq.(4.3) does not have the term e d , because our analysis does not depend on the NTK approximation and I-NeurAL directly utilizes the property of over-parameterized neural networks, i.e., the convergence error µ and the generalization concentration bound (Lemma 7.6).
Revised: In contrast, Eq.(4.3) does not have the term e d , because our analysis does not depend on the NTK approximation and I-NeurAL directly utilizes the property of over-parameterized neural networks, i.e., the convergence error µ and the generalization concentration bound (Lemma 7.6).

Original: These two terms are independent of e d , which paves the way for I-NeurAL to remove the curse of e d .
Revised: These two terms are independent of e d , which paves the way for I-NeurAL to remove the curse of e d .

Original: Remark 4.3.
Revised: Remark 4.3.

Original: Theorem 4.1 and Lemma 4.1 improve the regret by a multiplicative factor
Revised: Theorem 4.1 and Lemma 4.1 improve the regret by a multiplicative factor

Original: O (log T ) over [49].
Revised: O (log T ) over [48].

Original: Note that the analysis of [49] is built for binary classiﬁcation and thus k = 2 in Theorem 4.1 and Lemma 4.1.
Revised: Note that the analysis of [48] is built for binary classiﬁcation and thus k = 2 in Theorem 4.1 and Lemma 4.1.

Original: This improvement stems from the different analysis workﬂow of I-NeurAL from [49].
Revised: This improvement stems from the different analysis workﬂow of I-NeurAL from [48].

Original: Again, our analysis does not rely on NTK approximation and it is built on the convergence and generalization bound of wide neural networks.
Revised: Again, our analysis does not rely on NTK approximation and it is built on the convergence and generalization bound of wide neural networks.

Original: Remark 4.4.
Revised: Remark 4.4.

Original: Our proof workﬂow of Theorem 4.1 and Lemma 4.1 is inspired by [14].
Revised: Our proof workﬂow of Theorem 4.1 and Lemma 4.1 is inspired by [13].

Original: Compared to [14], we provide the ﬁrst regret bound supporting mini-batch SGD with warm-start and a more generic generalization bound (Lemma 7.3) that holds for every arm (class).
Revised: Compared to [13], we provide the ﬁrst regret bound supporting mini-batch SGD with warm-start and a more generic generalization bound (Lemma 7.3) that holds for every arm (class).

Original: Moreover, we carry out the performance analysis of query decision-maker (Lemma 7.5), which is a new addition.
Revised: Moreover, we carry out the performance analysis of query decision-maker (Lemma 7.5), which is a new addition.

Original: For the label complexity, N T has the trivial O ( T ) complexity which is the same as Theorem 1 in [49] (with the exponent ↵ = 0 ).
Revised: For the label complexity, N T has the trivial O ( T ) complexity which is the same as Theorem 1 in [48] (with the exponent ↵ = 0 ).

Original: Because we have to consider the worst case where the unique Bayes-optimal class does not exist, i.e., given x t,i , i 2 [ k ] , there does not exist i ⇤ such that h ( x t,i ⇤ )
Revised: Because we have to consider the worst case where the unique Bayes-optimal class does not exist, i.e., given x t,i , i 2 [ k ] , there does not exist i ⇤ such that h ( x t,i ⇤ )

Original: > h ( x t,i ) , 8 i 2 [ k ] \ { i ⇤ } .
Revised: > h ( x t,i ) , 8 i 2 [ k ] \ { i ⇤ } .

Original: Therefore, we provide the following analysis and show that R T and N
Revised: Therefore, we provide the following analysis and show that R T and N

Original: T can be upper bounded by constants as long as there exists a unique Bayes-optimal class for the input instances, described by the following mild margin assumption.
Revised: T can be upper bounded by constants as long as there exists a unique Bayes-optimal class for the input instances, described by the following mild margin assumption.

Original: Assumption 4.2 ( ✏ -margin) .
Revised: Assumption 4.2 ( ✏ -margin) .

Original: In round t 2 [ T ] , given an instance x t and the label y t , then x t has the ✏ -Unique optimal class if there exists ✏ > 0 such that
Revised: In round t 2 [ T ] , given an instance x t and the label y t , then x t has the ✏ -Unique optimal class if there exists ✏ > 0 such that

Original: [Equation]
Revised: [Equation]

Original: 4.5) where i ⇤ = arg max i 2 [ k ] h ( x t,i ) is the Bayes-optimal class and i  = arg max i 2 ([ k ] \{ i ⇤ } ) h ( x t,i ) is the second Bayes-optimal class.
Revised: .5) where i ⇤ = arg max i 2 [ k ] h ( x t,i ) is the Bayes-optimal class and i  = arg max i 2 ([ k ] \{ i ⇤ } ) h ( x t,i ) is the second Bayes-optimal class.

Original: Given any i 2 [ k ] , let i be a ﬁxed index, i.e., suppose there exist a policy ⇡ i which always select the i -th context ( x t,i , r 1 t,i ) for every round t 2 [ T ] .
Revised: Given any i 2 [ k ] , let i be a ﬁxed index, i.e., suppose there exist a policy ⇡ i which always select the i -th context ( x t,i , r 1 t,i ) for every round t 2 [ T ] .

Original: Then, in round t , we have the collected data by ⌦ i : H 1 ,it  1 = { x ⌧ ,i , r 1 ⌧ ,i } t  1 ⌧ =1 .
Revised: Then, in round t , we have the collected data by ⌦ i : H 1 ,it  1 = { x ⌧ ,i , r 1 ⌧ ,i } t  1 ⌧ =1 .

Original: Then, let b ✓ 1 ,i t  1 represent the parameters trained only on H 1 ,it  1 using
Revised: Then, let b ✓ 1 ,i t  1 represent the parameters trained only on H 1 ,it  1 using

Original: Algorithm 2 with ⇡ i and r 2 ,it,i =
Revised: Algorithm 2 with ⇡ i and r 2 ,it,i =

Original: r 1 t,i  f 1 ( x t,i ;
Revised: r 1 t,i  f 1 ( x t,i ;

Original: [Equation]
Revised: [Equation]

Original: Theorem 4.2.
Revised: Theorem 4.2.

Original: Suppose the instances drawn from D satisfy Assumption 4.2.
Revised: Suppose the instances drawn from D satisfy Assumption 4.2.

Original: Then, given the number of rounds T , for any  2 (0 , 1) ,  > 1 , ✏ 2 (0 , 1) , ⌫ > 0 , suppose m  e ⌦ ( poly ( T, k, L, ⌫ )) , ⌘ 1 = ⌘ 2 = ⇥ ( ⌫ p Tm ) , and µ satisﬁes the conditions in Theorem 4.1 andmax
Revised: Then, given the number of rounds T , for any  2 (0 , 1) ,  > 1 , ✏ 2 (0 , 1) , ⌫ > 0 , suppose m  e ⌦ ( poly ( T, k, L, ⌫ )) , ⌘ 1 = ⌘ 2 = ⇥ ( ⌫ p Tm ) , and µ satisﬁes the conditions in Theorem 4.1 and

Original: [Equation]
Revised: [Equation]

Original:   µ .
Revised: )  µ .

Original: Then, with probability at least 1   over the initialization of ✓ 10 , ✓ 20 , there exists a small enough constant  , such that Algorithm 1 achieves the following regret bound:
Revised: Then, with probability at least 1   over the initialization of ✓ 10 , ✓ 20 , there exists a small enough constant  , such that Algorithm 1 achieves the following regret bound:

Original: [Equation]
Revised: [Equation]

Original:  where ⇠ =  2 log 2 p 6(  +1) ✏ [Equation]
Revised: (4.7) where ¯ T = 12(  +1) 2 · [ 2 µ +9 L 2 ⌫ 2 C 21 +2log( C 2 Tk/  ) ] ✏.

Original: Remark 4.5.
Revised: Remark 4.5.

Original: Theorem 4.2 provides the upper bound for R T that is independent of  T  .
Revised: Theorem 4.2 provides the upper bound for R T with order of O (log T ) .

Original: When other parameters are ﬁxed, this indicates R T is upper bounded by a constant when T !
Revised: When other parameters are ﬁxed, this indicates R T is upper bounded by O (log T ) .

Original: + 1 .
Revised: Moreover, the analysis of

Original: Moreover, the analysis of R T indicates that I-NeurAL can achieve the same performance as Bayesoptimal classiﬁer with high conﬁdence after a ﬁxed number of rounds (i.e. T > 48(  +1) 2 ✏⇠ ).
Revised: R T indicates that I-NeurAL can achieve the same performance as Bayes-optimal classiﬁer with high conﬁdence after O (log T ) number of rounds (i.e. T > ¯ T ).

Original: To the best of our knowledge, this is the ﬁrst analysis for neural active learning to show that an NNbased algorithm can achieve the performance as the Bayes-optimal classiﬁer after a ﬁxed number of rounds.

Original: In Theorem 1 of [49] ( with the exponent ↵ !
Revised: In Theorem 1 of [48] ( with the exponent ↵ !

Original: + 1 equivalent to Assumption 4.2), e
Revised: + 1 equivalent to Assumption 4.2), [Equation] d

Original: [Equation]
Revised: [Equation]

Revised: +

Revised: [Equation]

Revised: [Equation]

Original:  that still is dependent on T because NTK depends onT and e 
Revised: ) that still is dependent on e d because NTK depends on e d

Original: [Equation]
Revised: [Equation]

Original: [Figure]

Original:  Experiments
Revised:  Experiments

Original: In this section, we evaluate I-NeurAL on public classiﬁcation data sets compared with state-of-theart (SOTA) baselines.
Revised: In this section, we evaluate I-NeurAL on public classiﬁcation data sets compared with state-of-theart (SOTA) baselines.

Original: Due to the space limit, we only report the main results here and leave the implementation details and parameter sensitivity in the Appendix 10.
Revised: Due to the space limit, we only report the main results here and leave the implementation details and parameter sensitivity in the Appendix 10.

Original: Codes are available 1 .
Revised: Codes are available 1 .

Original:  We report the experimental results on the following six data sets: Phishing 2 , IJCNN [39], Letter [19],
Revised: [Figure] We report the experimental results on the following six data sets: Phishing 2 , IJCNN [38], Letter [18],

Original: Fashion [50], MNIST [33] and CIFAR-10 [32].
Revised: Fashion [49], MNIST [32] and CIFAR-10 [31].

Original: In each round, one instance is randomly drawn from the data set and the algorithm is compelled to make prediction on it.
Revised: In each round, one instance is randomly drawn from the data set and the algorithm is compelled to make prediction on it.

Original: Then, the regret is 1 if the prediction does not match the label; the regret is 0 , otherwise.
Revised: Then, the regret is 1 if the prediction does not match the label; the regret is 0 , otherwise.

Original: At the same time, if the algorithm decides to observe the label, it costs one query budget.
Revised: At the same time, if the algorithm decides to observe the label, it costs one query budget.

Original: As the algorithm may abusively make label queries, we restrict the query budget to 3% of the total number of instances in the data set for fair comparison.
Revised: As the algorithm may abusively make label queries, we restrict the query budget to 3% of the total number of instances in the data set for fair comparison.

Original: The compared baselines are described as follows.
Revised: The compared baselines are described as follows.

Original: (1) Random :
Revised: (1) Random :

Original: The NN classiﬁer queries the label with a ﬁxed probability p until the query budget is exhausted; (2) Margin : The NN classiﬁer queries the label when the predicted probability is lower than a threshold.
Revised: The NN classiﬁer queries the label with a ﬁxed probability p until the query budget is exhausted; (2) Margin : The NN classiﬁer queries the label when the predicted probability is lower than a threshold.

Original: These two baselines are used in [23].
Revised: These two baselines are used in [22].

Original: (3) NeuAL-NTK-F (Algorithm 1 in [49] : This model makes predictions based on the frozen
Revised: (3) NeuAL-NTK-F (Algorithm 1 in [48] : This model makes predictions based on the frozen

Original: NTK approximation coming with an Upper-Conﬁdence-Bound(UCB)-based exploration strategy.
Revised: NTK approximation coming with an Upper-Conﬁdence-Bound(UCB)-based exploration strategy.

Original: NeuAL-NTK-D (Algorithm 3 in [49]) :
Revised: NeuAL-NTK-D (Algorithm 3 in [48]) :

Original: The prediction is made based on the NN classiﬁer with a
Revised: The prediction is made based on the NN classiﬁer with a

Original: UCB while the NTK is updated accordingly.
Revised: UCB while the NTK is updated accordingly.

Original: (5) ALPS [23] : Given a class of pre-trained hypotheses, the hypothesis minimizing the logistic loss of labeled and pseudo-labeled data is chosen to make predictions and the label query is based on the disagreement of different hypotheses.
Revised: (5) ALPS [22] : Given a class of pre-trained hypotheses, the hypothesis minimizing the logistic loss of labeled and pseudo-labeled data is chosen to make predictions and the label query is based on the disagreement of different hypotheses.

Original: Results .
Revised: Results .

Original: The regret comparison on six data sets is shown in Table 1 and Figure 1.
Revised: The regret comparison on six data sets is shown in Table 1 and Figure 1.

Original: I-NeurAL consistently outperforms all baselines across all data sets.
Revised: I-NeurAL consistently outperforms all baselines across all data sets.

Original: In particular, I-NeurAL surpasses the best baseline by 31.3%, 45.6%, 52.2% on IJCNN, MNIST, Fashion respectively.
Revised: In particular, I-NeurAL surpasses the best baseline by 31.3%, 45.6%, 52.2% on IJCNN, MNIST, Fashion respectively.

Original: Since NeuALNTK-F uses frozen NTK approximation, the new knowledge of each round is barely utilized by the neural network and thus it turns into the worst baseline.
Revised: Since NeuALNTK-F uses frozen NTK approximation, the new knowledge of each round is barely utilized by the neural network and thus it turns into the worst baseline.

Original: NeuAL-NTK-D updates the network parameters with gradient descent and queries the label based on the uncertainty estimation.
Revised: NeuAL-NTK-D updates the network parameters with gradient descent and queries the label based on the uncertainty estimation.

Original: However, its upper conﬁdence bound is still based on the conﬁdence ellipsoid.
Revised: However, its upper conﬁdence bound is still based on the conﬁdence ellipsoid.

Original: Instead, I-NeurAL leverages the representation power of neural networks for both exploitation and exploration.
Revised: Instead, I-NeurAL leverages the representation power of neural networks for both exploitation and exploration.

Original: ALPS maintains a class of pre-trained hypotheses and tries to make the best decisions based on these hypotheses.
Revised: ALPS maintains a class of pre-trained hypotheses and tries to make the best decisions based on these hypotheses.

Original: Nevertheless, the model parameters are ﬁxed before the online active learning process.
Revised: Nevertheless, the model parameters are ﬁxed before the online active learning process.

Original: Hence, ALPSPhishing [Table] is not able to take the new knowledge obtained by queries into account and its performance is highly restricted by the hypothesis class.
Revised: Hence, ALPS is not able to take the new knowledge obtained by queries into account and its performance is highly restricted by the hypothesis class.

Original: Although Margin algorithm is simple and straightforward, it exhibits great empirical performance in practice.
Revised: Although Margin algorithm is simple and straightforward, it exhibits great empirical performance in practice.

Original: This observation is consistent with other studies [51] [23].
Revised: This observation is consistent with other studies [50] [22].

Original: However, Margin algorithm does not incorporate the exploitation portion and the query criterion is not adaptive to difference instances, thus still outperformed by I-NeurAL. 
Revised: However, Margin algorithm does not incorporate the exploitation portion and the query criterion is not adaptive to difference instances, thus still outperformed by I-NeurAL. Phishing [Table]

Original:  Conclusion
Revised:  Conclusion

Original: In this paper, we introduce two regret metrics and propose a novel neural-based algorithm (I-NeurAL) tailored for the streaming setting of non-parametric active learning.
Revised: In this paper, we introduce two regret metrics and propose a novel neural-based algorithm (I-NeurAL) tailored for the streaming setting of non-parametric active learning.

Original: We carefully design its exploration strategy, query decision-maker, update rules, and training procedure, which lead to both the theoretical and empirical improvement compared to SOTA [49].
Revised: We carefully design its exploration strategy, query decision-maker, update rules, and training procedure, which lead to both the theoretical and empirical improvement compared to SOTA [48].

Original: In the regret analysis, we provide an instance-dependent performance guarantee.
Revised: In the regret analysis, we provide an instance-dependent performance guarantee.

Original: On the other hand, we empirically show that I-NeurAL consistently achieves better accuracy under the same query budget than the strong baselines including the SOTA work [49] and [23].
Revised: On the other hand, we empirically show that I-NeurAL consistently achieves better accuracy under the same query budget than the strong baselines including the SOTA work [48] and [22].

Original: Acknowledgements
Revised: Acknowledgements

Original: This work is supported by NSF (IIS-1947203, IIS-2117902, IIS-2137468, IIS-2002540, DMS2134079, IIS-2131335, OAC-2130835, and DBI-2021898), DARPA (HR001121C0165), ARO (W911NF2110088), and C3.ai.
Revised: This work is supported by NSF (IIS-1947203, IIS-2117902, IIS-2137468, IIS-2002540, DMS2134079, IIS-2131335, OAC-2130835, and DBI-2021898), DARPA (HR001121C0165), ARO (W911NF2110088), and C3.ai.

Original: The views and conclusions are those of the authors and should not be interpreted as representing the ofﬁcial policies of the funding agencies or the government.
Revised: The views and conclusions are those of the authors and should not be interpreted as representing the ofﬁcial policies of the funding agencies or the government.