Deep Active Learning by Leveraging Training Dynamics
Abstract
Active learning theories and methods have been extensively studied in classicalstatistical learning settings.
However, deep active learning, i.e., active learningwith deep learning models, is usually based on empirical criteria without solidtheoretical justification, thus suffering from heavy doubts when some of those failto provide benefits in real applications.
In this paper, by exploring the connectionbetween the generalization performance and the training dynamics, we propose atheory-driven deep active learning method ( dynamicAL ) which selects samples tomaximize training dynamics.
In particular, we prove that the convergence speedof training and the generalization performance are positively correlated under theultra-wide condition and show that maximizing the training dynamics leads to bettergeneralization performance.
Furthermore, to scale up to large deep neural networksand data sets, we introduce two relaxations for the subset selection problem andreduce the time complexity from polynomial to constant.
Empirical results showthat dynamicAL not only outperforms the other baselines consistently but alsoscales well on large deep learning models.
We hope our work would inspire moreattempts on bridging the theoretical findings of deep networks and practical impactsof deep active learning in real applications.
Introduction
Training deep learning (DL) models usually requires large amount of high-quality labeled data [63]to optimize a model with a massive number of parameters.
The acquisition of such annotated data isusually time-consuming and expensive, making it unaffordable in the fields that require high domainexpertise.
A promising approach for minimizing the labeling effort is active learning (AL), whichaims to identify and label the maximally informative samples, so that a high-performing classifier canbe trained with minimal labeling effort [51].
Under classical statistical learning settings, theories ofactive learning have been extensively studied from the perspective of VC dimension [24].
As a result,a variety of methods have been proposed, such as (i) the version-space-based approaches, whichrequire maintaining a set of models [14, 6], and (ii) the clustering-based approaches, which assumethat the data within the same cluster have pure labels [16].
However, the theoretical analyses for these classical settings may not hold for over-parameterizeddeep neural networks where the traditional wisdom is ineffective [63].
For example, margin-basedmethods select the labeling examples in the vicinity of the learned decision boundary [7, 8].
However,in the over-parameterized regime, every labeled example could potentially be near the learned decisionboundary [31].
As a result, theoretically, such analysis can hardly guide us to design practical activelearning methods.
Besides, empirically, multiple deep active learning works, borrowing observationsand insights from the classical theories and methods, have been observed unable to outperform theirpassive learning counterparts in a few application scenarios [33, 4].
On the other hand, the analysis of neural network’s optimization and generalization performance haswitnessed several exciting developments in recent years in terms of the deep learning theory [29, 3, 37].
It is shown that the training dynamics of deep neural networks using gradient descent can becharacterized by the Neural Tangent Kernel (NTK) of infinite [29] or finite [23]
width networks.
This is further leveraged to characterize the generalization of over-parameterized networks through
Rademacher complexity analysis [3, 11].
We are therefore inspired to ask: How can we design apractical and generic active learning method for deep neural networks with theoretical justifications?
To answer this question, we firstly explore the connection between the model performance on testingdata and the convergence speed on training data for the over-parameterized deep neural networks.
Based on the NTK framework [29, 3], we theoretically show that if a deep neural network convergesfaster (“Train Faster”), then it tends to have better generalization performance (“Generalize Better”),which matches the existing observations [25, 38, 40, 49, 59].
Motivated by the aforementionedconnection, we first introduce Training Dynamics , the derivative of training loss with respect toiteration, as a proxy to quantitatively describe the training process.
On top of it, we formally proposeour generic and theoretically-motivated deep active learning method, dynamicAL , which will querylabels for a subset of unlabeled samples that maximally increase the training dynamics.
In order tocompute the training dynamics by merely using the unlabeled samples, we leverage two relaxations
Pseudo-labeling and Subset Approximation to solve this non-trivial subset selection problem.
Ourrelaxed approaches are capable of effectively estimating the training dynamics as well as efficientlysolving the subset selection problem by reducing the complexity from O ( N b ) to O ( b ) .
In theory, we coin a new term Alignment to measure the length of the label vector’s projection on theneural tangent kernel space.
Then, we demonstrate that higher alignment usually comes with a fasterconvergence speed and a lower generalization bound.
Furthermore, with the help of the maximummean discrepancy [9], we extend the previous analysis to an active learning setting where the i.i.d.assumption may not hold.
Finally, we show that alignment is positively correlated with our activelearning goal, training dynamics, which implies that maximizing training dynamics will lead to bettergeneralization performance.
Regarding experiments, we have empirically verified our theory by conducting extensive experimentson three datasets, CIFAR10 [35], SVHN [43], and Caltech101 [20] using three types of networkstructures: vanilla CNN, ResNet [26], and VGG [54].
We first show that the result of the subsetselection problem delivered by the subset approximation is close to the global optimal solution.
Furthermore, under the active learning setting, our method not only outperforms other baselines butalso scales well on large deep learning models.
The main contributions of our paper can be summarized as follows:
• We propose a theory-driven deep active learning method, dynamicAL , inspired by the observationof “train faster, generalize better”.
To this end, we introduce the Training Dynamics, as a proxy todescribe the training process.•
We demonstrate that the convergence speed of training and the generalization performance isstrongly (positively) correlated under the ultra-wide condition; we also show that maximizing thetraining dynamics will lead to a lower generalization error in the scenario of active learning.•
Our method is easy to implement.
We conduct extensive experiments to evaluate the effectivenessof dynamicAL and empirically show that our method consistently outperforms other methods in awide range of active learning settings.
Background
Notation.
We use the random variable x ∈ X to represent the input data feature and y ∈ Y as thelabel where K is the number of classes and [ K ] := { 1 , 2 , ..., K } .
We are given non-degenerateda data source D with unknown distribution p ( x, y ) .
We further denote the concatenation of x as
X =
[ x 1 , x 2 , ..., x M ] ⊤ and that of y as Y =
[ y 1 , y 2 , ..., y M ] ⊤ .
We consider a deep learning classifierh θ ( x ) = argmax σ ( f ( x ; θ )) :
x → y parameterized by θ ∈ R p , where σ ( · ) is the softmax functionand f is a neural network.
Let ⊗ be the Kronecker Product and I
K ∈ R
K × K be an identity matrix.
Active learning.
The goal of active learning is to improve the learning efficiency of a model with alimited labeling budget.
In this work, we consider the pool-based AL setup, where a finite data set
S = { ( x l , y l ) } Ml =1 with M points are i.i.d.
sampled from p ( x, y ) as the (initial) labeled set.
The AL model receives an unlabeled data set U sampled from p ( x ) and request labels according to p ( y | x )for any x ∈ U in each query round.
There are R rounds in total, and for each round, a query set Qconsisting of b unlabeled samples can be queried.
The total budget size B = b × R .
Neural Tangent Kernel.
The Neural Tangent Kernel [29] has been widely applied to analyze thedynamics of neural networks.
If a neural network is sufficiently wide, properly initialized, andtrained by gradient descent with infinitesimal step size ( i.e. , gradient flow), then the neural network isequivalent to kernel regression predictor with a deterministic kernel Θ ( · , · ) , called Neural Tangent
Kernel (NTK).
When minimizing the mean squared error loss, at the iteration t , the dynamics of theneural network f has a closed-form expression:
[Equation]
where θ ( t ) denotes the parameter of the neural network at iteration t , K t ( X , X ) ∈ R |X|× K ×|X|× Kis called the empirical NTK and K i,jt ( x, x ′ )
=
∇ θ f i ( x ; θ ( t ))
⊤ ∇ θ f j ( x ′ ; θ ( t )) is the inner product ofthe gradient of the i -th class probability and the gradient of the j -th class probability for two samplesx, x ′ ∈ X and i, j ∈ [ K ] .
The time-variant kernel K t ( · , · ) is equivalent to the (time-invariant) NTKwith a high probability, i.e., if the neural network is sufficiently wide and properly initialized, then:
[Equation]
The final learned neural network at iteration t , is equivalent to the kernel regression solution withrespect to the NTK [37].
For any input x and training data { X, Y } we have,
[Equation]
where η is the learning rate, Θ ( x, X ) is the NTK matrix between input x and all samples in trainingdata X .
Method
In section 3.1, we introduce the notion of training dynamics which can be used to describe the trainingprocess.
Then, in section 3.2, based on the training dynamics, we propose dynamicAL .
In section 3.3,we discuss the connection between dynamicAL and existing deep active learning methods.
3.1 Training dynamics
In this section, we introduce the notion of training dynamics.
The cross-entropy loss over the labeledset S is defined as:
[Equation]
where σ i ( f ( x ; θ )) =
exp( f i ( x ; θ )) (cid:80) j exp( f j ( x ; θ )) .
We first analyze the dynamics of the training loss, with respectto iteration t , on one labeled sample (derivation is in Appendix A.1):
[Equation]
For neural networks trained by gradient descent, if the learning rate η is small, then ∇ t θ = θ t +1 − θ t =− η
[Equation]
xl
[Equation]
x l ;
[Equation] [Equation]
∂θ .
Taking the partial derivative of the training loss with respect to theparameters, we have (the derivation of the following equation can be found in Appendix A.2):
[Equation]
Therefore
[Equation]
we can further get the following result for the dynamics of training loss:
[Equation]
Furthermore, we define d i ( X, Y )
=
σ i ( f ( X ; θ ))
− Y i and Y
i is the label vector of all samples fori -th class.
Then, the training dynamics (dynamics of training loss) over training set S , computed withthe empirical NTK K ij ( X, X ) , is denoted by G ( S )
∈ R :
[Equation]
3.2 Active learning by activating training dynamics
Before we present dynamicAL , we state Proposition 1, which serves as the theoretical guidance fordynamicAL and will be proved in Section 4.
Proposition 1.
For deep neural networks, converging faster leads to a lower worst-case generalization error.
Motivated by the connection between convergence speed and generalization performance, we proposethe general-purpose active learning method, dynamicAL , which aims to accelerate the convergence byquerying labels for unlabeled samples.
As we described in the previous section, the training dynamicscan be used to describe the training process.
Therefore, we employ the training dynamics as a proxyto design an active learning method.
Specifically, at each query round, dynamicAL will query labelsfor samples which maximize the training dynamics G ( S ) , i.e. ,
[Equation]
where Q is the corresponding data set for Q with ground-truth labels.
Notice that when applying theabove objective in practice, we are facing two major challenges.
First, G ( S ∪ Q ) cannot be directlycomputed, because the label information of unlabeled examples is not available before the query.
Second, the subset selection problem can be computationally prohibitive if enumerating all possiblesets with size b .
Therefore, we employ the following two relaxations to make this maximizationproblem to be solved with constant time complexity.
Pseudo labeling.
To estimate the training dynamics, we use the predicted label ˆ y u for samplex u in the unlabeled data set U to compute G .
Note, the effectiveness of this adaptation has beendemonstrated in the recent gradient-based methods [4, 42], which compute the gradient as if themodel’s current prediction on the example is the true label.
Therefore, the maximization problem in
Equation (9) is changed to,
[Equation]
where (cid:98) Q is the corresponding data set for Q with pseudo labels (cid:98) Y Q .
Subset approximation.
The subset selection problem of Equation (10) still requires enumerating allpossible subsets of U with size b , which is O ( n b ) .
We simplify the selection problem to the followingproblem without causing any change on the result,
[Equation]
where ∆( (cid:98) Q | S ) =
G ( S ∪ (cid:98) Q ) − G
( S ) is defined as the change of training dynamics.
We approximatethe change of training dynamics caused by query set Q using the summation of the change of trainingdynamics caused by each sample in the query set.
Then the maximization problem can be convertedto Equation (12) which can be solved by a greedy algorithm with O ( b ) .
[Equation]
To further show the approximated result is reasonably good, we decompose the change of trainingdynamics as (derivation in Appendix A.4):
[Equation] [Equation]
where K ij ( x, x ′ ) is the empirical NTK.
The first term in the right hand side is the approximatedchange of training dynamics.
Then, we further define the Approximation Ratio (14) which measuresthe approximation quality,
[Equation]
We empirically measure the expectation of the Approximation Ratio on two data sets with twodifferent neural networks under three different batch sizes.
As shown in Figure 4, the expectation
E Q ∼ U R ( (cid:98) Q | S )
≈ 1 when the model is converged.
Therefore, the approximated result delivered bythe greedy algorithm is close to the global optimal solution of the original maximization problem,
Equation (10), especially when the model is converged.
Based on the above two approximations, we present the proposed method dynamicAL in Algorithm 1.
As described below, the algorithm starts by training a neural network f ( · ; θ ) on the initial labeledset S until convergence.
Then, for every unlabeled sample x u , we compute pseudo label ˆ y u and thechange of training dynamics ∆( { ( x u , (cid:98) y u ) }| S ) .
After that, dynamicAL will query labels for top- bsamples causing the maximal change on training dynamics, train the neural network on the extendedlabeled set, and repeat the process.
Note, to keep close to the theoretical analysis, re-initialization isnot used after each query, which also enables dynamicAL to get rid of the computational overhead ofretraining the deep neural networks every time.
Algorithm
Input: Neural network f ( · ; θ ) , unlabeled sample set U , initial labeled set S , number of query round R , query batch size b .
for r = 1 to R do
Train f ( · ; θ ) on S with cross-entropy loss until convergence.
for x u ∈ U do
Compute its pseudo label ˆ y u = argmax f
( x u ; θ ) .
Compute ∆ ( { ( x u , (cid:98) y u ) }| S ) .
end for
Select b query samples Q with the highest ∆ values, and request their labels from the oracle.
Update the labeled data set S = S ∪ Q .
end for return Final model f ( · ; θ ) .
3.3 Relation to existing methods
Although existing deep active learning methods are usually designed based on heuristic criteria,some of them have empirically shown their effectiveness [4, 39, 28].
We surprisingly found that ourtheoretically-motivated method dynamicAL has some connections with those existing methods fromthe perspective of active learning criterion.
The proposed active learning criterion in Equation (12)can be explicitly written as (derivation in Appendix A.5):
[Equation]
The first term of the right-hand side can be interpreted as the square of gradient length (2-norm)which reflects the uncertainty of the model on the example and has been wildly used as an activelearning criterion in some existing works [28, 4, 53].
The second term can be viewed as the influencefunction [34] with identity hessian matrix.
And recently, [39] has empirically shown that theeffectiveness of using the influence function with identity hessian matrix as active learning criterion.
We hope our theoretical analysis can also shed some light on the interpretation of previous methods.
Theoretical analysis
In this section, we study the correlation between the convergence rate of the training loss and thegeneralization error under the ultra-wide condition [29, 3].
We define a measure named alignmentto quantify the convergence rate and further show its connection with generalization bound.
Theanalysis provides a theoretical guarantee for the phenomenon of “Train Faster, Generalize Better” aswell as our active learning method dynamicAL with a rigorous treatment.
Finally, we show that theactive learning proxy, training dynamics, is correlated with alignment, which indicates that increasingthe training dynamics leads to larger convergence rate and better generalization performance.
Weleave all proofs of theorems and details of verification experiments in Appendix B and D respectively.
4.1 Train faster provably generalize better
Given an ultra-wide neural network, the gradient descent can achieve a near-zero training error [29, 2]and its generalization ability in unseen data can be bounded [3].
It is shown that both the convergenceand generalization of a neural network can be analyzed using the NTK [3].
However, the questionwhat is the relation between the convergence rate and the generalization bound has not been answered.
We formally give a solution by introducing the concept of alignment , which is defined as follows:
Definition 1 (Alignment) .
Given a data set S =
{ X, Y } , the alignment is a measure of correlationbetween X and Y projected in the NTK space.
In particular, the alignment can be computed by
[Equation]
In the following, we will demonstrate why “Train Faster” leads to “Generalize Better” throughalignment.
In particular, the relation of the convergence rate and the generalization bound withalignment is analyzed.
The convergence rate of gradient descent for ultra-wide networks is presentedin following lemma:
Lemma 1 (Convergence Analysis with NTK, Theorem 4.1 of [3]) .
Suppose λ 0 = λ min ( Θ ) > 0 forall subsets of data samples.
For δ ∈ (0 , 1) , if m = Ω( n 7 λ 40 δ 4 ϵ 2 ) and η = O ( λ 0 n 2 ) , with probability atleast 1 − δ , the network can achieve near-zero training error,
[Equation] [Equation]
where n denotes the number of training samples and m denotes the width of hidden layers.
The NTK 209 Θ = V ⊤ Λ V with Λ = { λ i } ni =1 is a diagonal matrix of eigenvalues and V = { ⃗v i } ni =1 is a unitarymatrix.
In this lemma, we take mean square error (MSE) loss as an example for the convenience of illustration.
The conclusion can be extended to other loss functions such as cross-entropy loss (see Appendix B.2in [37]).
From the lemma, we find the convergence rate is governed by the dominant term (16) as
E t ( X, Y ) =
(cid:113)(cid:80)
Kk =1 (cid:80) ni
=1 (1 − ηλ i ) 2 t
( ⃗v ⊤ i Y k ) 2 , which is correlated with the alignment :
Theorem 1 (Relationship between the convergence rate and alignment) .
Under the same assumptionsas in Lemma 1, the convergence rate described by E t satisfies,
[Equation]
Remark 1.
In the above theorem, we demonstrate that the alignment can measure the convergencerate.
Especially, we find that both the upper bound and the lower bound of error E t ( X, Y ) areinversely proportional to the alignment, which implies that higher alignment will lead to achievingfaster convergence.
Now we analyze the generalization performance of the proposed method through complexity analysis.
We demonstrate that the ultra-wide networks can achieve a reasonable generalization bound.
Lemma 2 (Generalization bound with NTK, Theorem 5.1 of [3]) .
Suppose data S
=
( x i , y i ) } ni =1are i.i.d.
samples from a non-degenerate distribution p ( x, y ) , and m ≥ poly( n, λ − 1 0 , δ − 1 ) .
Considerany loss function ℓ : R × R → [0 , 1] that is 1 -Lipschitz, then with probability at least 1 − δ over therandom initialization, the network trained by gradient descent for T ≥ Ω( 1 ηλ 0 log nδ ) iterations haspopulation risk L p =
E ( x,y ) ∼ p ( x,y )
[ ℓ ( f T ( x ; θ ) , y )]
that is bounded as follows:
[Equation]
In this lemma, we show that the dominant term in the generalization upper bound is B ( X, Y )
= 229 (cid:113) 2Tr
[ Y ⊤ Θ − 1 Y ] n .
In the following theorem, we further prove that this bound is inversely proportionalto the alignment
A ( X, Y ) .
Theorem 2 (Relationship between the generalization bound and alignment) .
Under the same assumptions as in Lemma 2, if we define the generalization upper bound as B ( X, Y ) = (cid:113) 2Tr[ Y ⊤ Θ − 1 Y ] n ,then it can be bounded with the alignment as follows:
[Equation] [Equation]
Remark 2.
Theorems 1 and 2 reveal that the cause for the correlated phenomenons “Train Faster”and “Generalize Better” is the projection of label vector on the NTK space (alignment).
4.2 “ Train Faster, Generalize Better ” for active learning
In the NTK framework [3], the empirical average requires datain S is i.i.d.
samples (Lemma 2).
However, this assumptionmay not hold in the active learning setting with multiple queryrounds, because the training data is composed by i.i.d.
sampledinitial label set and samples queried by active learning policy.
To extend the previous analysis principle to active learning, wefollow [58] to reformulate the Lemma 2 as:
[Equation]
where L q = E ( x,y ) ∼ q ( x,y ) [ ℓ ( f ( x ; θ ) , y )] , q ( x, y ) denotes thedata distribution after query, and X, Y includes initial trainingsamples and samples after query.
There is a new term in the upper bound, which is the differencebetween the true risk under different data distributions.
[Equation]
Though in active learning the data distribution for the labeled samples may be different fromthe original distribution, they share the same conditional probability p ( y | x ) .
We define g ( x ) = 250 (cid:82) y ℓ ( f ( x ; θ ) , y ) p ( y | x ) dy , and then we have:
[Equation]
To measure the distance between two distributions, we employ the Maximum Mean Discrepancy(MMD) with neural tangent kernel [30] (derivation in Appendix B.3).
[Equation]
Slightly overloading the notation, we denote the initial labeled set as S 0 , H Θ as the associated Reproducing Kernel Hilbert Space for the NTK Θ , and ∀ x, x ′ ∈ S, Θ ( x, x ′ ) ≤ C .
Note, MMD ( S 0 , S, H Θ )is the empirical measure for MMD ( p ( x ) , q ( x ) , H Θ ) .
We empirically compute MMD and the dominant term of the generalization upper bound B under the active learning setting with our methoddynamicAL .
As shown in Figure 1, on CIFAR10 with a CNN target model (three convolutional layerswith global average pooling), the initial labeled set size | S | = 500 , query round R = 1 and budgetsize b ∈ { 250 , 500 , 1000 }
, we observe that, under different active learning settings, the MMD isalways much smaller than the B .
Besides, we further investigate the MMD and B for R ≥ 2 andobserve the similar results.
Therefore, the lemma 2 still holds for the target model with dynamicAL .
More results and discussions for R ≥ 2 are in Appendix E.4 and the computation details of MMDand NTK are in Appendix D.1.
4.3 Alignment and training dynamics in active learning
[Figure]
In this section, we show the relationship between the alignment and the training dynamics.
To be consistent with theprevious theoretical analysis (Theorem 1 and 2), we use thetraining dynamics with mean square error under the ultrawidth condition, which can be expressed as G MSE ( S ) =
Tr (cid:2) ( f ( X ; θ ) − Y )
⊤ Θ ( X, X )( f ( X ; θ ) − Y ) (cid:3) .
Due to the limited space, we leave the derivation in Appendix A.3.
To furtherquantitatively evaluate the correlation between G MSE ( S ∪ Q )and A ( X ∥ X Q , Y ∥
Y Q ) , we utilize the Kendall
τ coefficient [32]to empirically measure their relation.
As shown in Figure 2, for
CNN on CIFAR10 with active learning setting, where | S | = 500and | Q | = 250 , there is a strong agreement between G MSE ( S ∪ Q ) and A ( X ∥ X Q , Y ∥
Y Q ) , whichfurther indicates that increasing the training dynamics will lead to a faster convergence and bettergeneralization performance.
More details about this verification experiment are in Appendix D.2.
Experiments
5.1 Experiment setup
Baselines.
We compare dynamicAL with the following eight baselines: Random, Corset, Confidence
Sampling (Conf), Margin Sampling (Marg), Entropy, and Active Learning by Learning (ALBL),
Batch Active learning by Diverse Gradient Embeddings (BADGE).
Description of baseline methodsis in Appendix E.1.
Data sets and Target Model.
We evaluate all the methods on three benchmark data sets, namely,
CIFAR10 [35], SVHN [43], and Caltech101 [20].
We use accuracy as the evaluation metric andreport the mean value of 5 runs.
We consider three neural network architectures: vanilla CNN,
ResNet18 [26], and VGG11 [54].
For each model, we keep the hyper-parameters used in their officialimplementations.
More information about the implementation is in Appendix C.1.
Active Learning Protocol.
Following the previous evaluation protocol [4], we compare all thoseactive learning methods in a batch-mode setup with an initial set size M = 500 for all those three datasets, batch size b varying from { 250 , 500 , 1000 } .
For the selection of test set, we use the benchmarksplit of the CIFAR10 [35], SVHN [43] and sample 20% from each class to form the test set for the
Caltech101 [20].
5.2 Results and analysis
The main experimental results have been provided as plots due to the limited space.
We also providetables in which we report the mean and standard deviation for each plot in Appendix E.3.
Overall results.
The average test accuracy at each query round is shown in Figure 3.
Our methoddynamicAL can consistently outperform other methods for all query rounds.
This suggests thatdynamicAL is a good choice regardless of the labeling budget.
And, we notice dynamicAL can workwell on data sets with a large class number, such as Caltech101.
However, the previous state-of-the-artmethod, BADGE, cannot be scaled up to those data sets, because the required memory is linearwith the number of classes.
Besides, because dynamicAL depends on pseudo labeling, a relativelylarge initial labeled set can provide advantages for dynamicAL .
Therefore, it is important to examinewhether dynamicAL can work well with a small initial labeled set.
As shown in Figure 3, dynamicALis able to work well with a relatively small initial labeled set ( M = 500 ).
Due to the limited space,we only show the result under three different settings in Figure 3.
More evaluation results are in
Appendix E.2.
Moreover, although the re-initialization trick makes dynamicAL deviate from thedynamics analysis, we investigate the effect of it to dynamicAL and provide the empirical observationsand analysis in Appendix E.5.
[Figure]
Effect of query size and query round.
Given the total label budget B , the increasing of querysize always leads to the decreasing of query round.
We study the influence of different query sizeand query round on dynamicAL from two perspectives.
First, we study the expected approximationratio with different query batch sizes on different data sets.
As shown in Figure 4, under differentsettings the expected approximation ratio always converges to 1 with the increase of training epochs,which further indicates that the query set selected by using the approximated change of trainingdynamics is a reasonably good result for the query set selection problem.
Second, we study influenceof query round for actual performance of target models.
The performance for different target modelson different data sets with total budge size B = 1000 is shown in Table 1.
For certain query budget,our active learning algorithm can be further improved if more query rounds are allowed.
Comparison with different variants.
The active learning criterion of dynamicAL can be written as 322 (cid:80) ( x,y ) ∈ S ∥∇ θ ℓ ( f ( x ; θ u ) , ˆ y u ) ∥ 2 + γ ∇ θ ℓ ( f ( x u ; θ ) , ˆ y u )
⊤ ∇ θ ℓ ( f ( x ; θ ) , y ) .
We empirically show the
[Figure] [Figure]
[Table]
performance for γ ∈ { 0 , 1 , 2 , ∞} in Figure 5.
With γ = 0 , the criterion is close to the expectedgradient length method [53].
And with γ = ∞ , the selected samples are same with the samplesselected by using the influence function with identity hessian matrix criterion [39].
As shown in
Figure 5, the model achieves the best performance with γ = 2 , which is aligned with the valueindicated by the theoretical analysis (Equation 15).
The result confirms the importance of theoreticalanalysis for the design of deep active learning methods.
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table] [Table]
[Table]
5 # Query Round
T e s t
A cc u r a c y ( % )
SVHN, VGG, Batc Si)e:γ=0 γ=1 γ=∞ γ=      5 # Query Round
T e s t
A cc u r a c y ( % )
CIFAR10, ResNet, Batch Size:γ=0 γ=1 γ=∞ γ=
Related work
Neural Tangent Kernel (NTK): Recent study has shown that under proper conditions, an infinitewidth neural network can be simplified as a linear model with Neural Tangent Kernel (NTK) [29].
Since then, NTK has become a powerful theoretical tool to analyze the behavior of deep learningmodel (CNN, GNN, RNN) from its output dynamics [2, 19, 1] and to characterize the convergenceand generalization error [3].
Besides, [23] studies the finite-width NTK, aiming at making the NTKmore practical.
Active learning: Active learning aims at interactively query labels for unlabeled data points tomaximize model performances [51].
Among others, there are two popular strategies for activelearning, i.e. , diversity sampling [18, 56, 64] and uncertainty sampling [48, 65, 60, 4, 61, 52, 39].
Recently, several papers proposed to use gradient to measure uncertainty [52, 4, 39].
However, thosemethods need to compute gradient for each class, and thus they can hardly be applied on data setswith a large class number.
Besides, one recent work [5] leverages NTK to analyze contextual banditwith streaming data, which is hard to be applied into our pool-based setting.
Conclusion
In this work, we bridge the gap between the theoretic findings of deep neural networks and realworld deep active learning applications.
By exploring the connection between the generalizationperformance and the training dynamics, we propose a theory-driven method, dynamicAL , whichselects samples to maximize training dynamics.
We prove that the convergence speed of training and the generalization performance is (positively) strongly correlated under the ultra-wide condition and we show that maximizing the training dynamics will lead to a lower generalization error.
Empirically our work shows that dynamicAL not only consistently outperforms strong baselines across various setting, but also scales well on large deep learning models
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A APPENDIX: Derivation of Objectives
For the notational convenience, we use f ( x ) to represent f ( x ; θ ) in the Appendix.
A.1 Training Dynamics for Cross-Entropy Loss
The partial derivative for softmax function can be defined with the following,
[Equation]
Then, we have:
[Equation]
A.2 Derivation for Cross-Entropy Loss
[Equation]
A.3 APPENDIX: Training Dynamics for Mean Squared Error
For the labeled data set S , we define the Mean Squared Error(MSE) as:
[Equation]
Then the training loss dynamics for each sample can be defined as:
[Equation]
Because neural networks are optimized by gradient descent, thus:
[Equation]
Therefore
[Equation]
the training dynamics of MSE loss can be expressed as:
[Equation]
A.4 APPENDIX: Decomposition of the Change of Training Dynamics
According to the definition of training dynamics ( Equation (8) ), we have,
[Equation]
The change of training dynamics, ∆( (cid:98) Q | S ) =
G ( S ∪ (cid:98) Q ) − G ( S ) , can be further simplified as:
[Equation]
i
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
A.5 APPENDIX: Simplification of the Change of Training Dynamics
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
The derivative of loss with respect to model parameters can be written as:
[Equation]
Therefore, the change of training dynamics caused by { ( x u , (cid:98) y u ) } can be written as:
[Equation] [Equation]
B APPENDIX: Proofs for Theoretical Analysis
B.1 Proofs for Theorem
Lemma 1 (Convergence Analysis with NTK, Theorem 4.1 of [3]) .
Suppose λ 0 = λ min ( Θ ) > 0 forall subsets of data samples.
For δ ∈ (0 , 1) , if m = Ω( n 7 λ 40 δ 4 ϵ 2 ) and η = O ( λ 0 n 2 ) , with probability atleast 1 − δ , the network can achieve near-zero training error,
[Equation]
where n denotes the number of training samples and m denotes the width of hidden layers.
The NTKΘ =
V ⊤ Λ V with Λ = { λ i } ni =1 is a diagonal matrix of eigenvalues and V = { ⃗v i } ni =1 is a unitarymatrix.
Proof.
According to [3], if m = Ω( n 7 λ 40 δ 4 ϵ 2 ) and learning ratio η = O ( λ 0 n 2 ) , then withprobability at least 1 − δ over the random initialization, we have, ∥ Y l − f t ( X ; θ ( t ))
∥ 2 =(cid:113)(cid:80)
Kk =1 (cid:80) ni
=1 (1 − ηλ i ) 2 t ( v ⊤ i Y kl ) 2 ± ϵ .
We decompose the NTK using Θ = V ⊤ Λ V withΛ =
{ λ i } ni =1 a diagonal matrix of eigenvalues and V = { v i } ni
=1 a unitary matrix.
At each trainingstep in active learning, the labeled samples will be updated by S = S ∪ Q .
We can apply theconvergence result in each of this step and achieve near zero error.
Theorem 1 (Relationship between convergence rate and alignment) .
Under the same assumptions asin Lemma 1, the convergence rate described by E t satisfies,
[Equation] [Equation]
Proof.
We first prove the inequality on the right hand side.
It is easy to see that (1 − ηλ i ) 2 t ≤ (1 − ηλ i ) for each λ
i and t ≥ 1 , based on the fact that ∀ λ i , 0 ≤ 1 − ηλ i ≤ 1 .
Then we can obtain,
[Equation]
Then we use Bernoulli’s inequality to prove the inequality on the left hand side.
Bernoulli’s inequality states that, (1 + x ) r ≥ 1 + rx , for every integer r ≥ 0 and every real number x ≥ − 1 .
It is easy to check that ( − ηλ i ) ≥ − 1 , ∀ λ i .
Therefore,
[Equation]
B.2 Proof for Theorem
Lemma 2 (Generalization bound with NTK, Theorem 5.1 of [3]) .
Suppose data S
=
( x i , y i ) } ni =1are i.i.d.
samples from a non-degenerate distribution p ( x, y ) , and m ≥ poly( n, λ − 1 0 , δ − 1 ) .
Considerany loss function ℓ : R × R → [0 , 1] that is 1 -Lipschitz, with probability at least 1 − δ over therandom initialization, the network trained by gradient descent for T ≥ Ω( 1 ηλ 0 log nδ ) iterations haspopulation risk L p =
E ( x,y ) ∼ p ( x,y )
[ ℓ ( f T ( x ) , y )]
that is bounded as follows:
[Equation]
Proof.
We first show that the generalization bound regrading our method on ultra-wide networks.
The distance between weights of trained networks and their initialization values can be boundedas, ∥ w r ( t ) − w r (0) ∥
=
O ( n √ mλ 0 √ δ ) .
We then give a bound on the ∥ W ( t ) − W (0) ∥ F , where
W = { w 1 , w 2 , . . . } is the set of all parameters.
We definite Z = ∂f ( t ) ∂W ( t ) , then the update functionis given by W ( t + 1) =
W ( t ) − ηZ ( Z ⊤ W ( t ) − Y ) .
Summing over all the time step t = 0 , 1 , . .
,we can obtain that W ( ∞ ) − W (0) =
(cid:80) ∞ t =0 ηZ ( I − η Θ ) y
=
Z Θ − 1 Y .
Thus the distance can bemeasured by ∥ W ( ∞ ) − W (0) ∥ 2 F =
Tr[ Y ⊤ Θ − 1 Y ] .
Then the key step is to apply Rademacher complexity.
Given R > 0 , with probability at least 1 − δ , simultaneously for every B > 0 , the function class F B,R = { f : ∥ w r ( t ) − w r (0) ∥ ≤ R ( ∀ r ∈ m ) , ∥ W
[Equation]
∞ ) − W (0) ∥ 2 F ≤ B } has empirical Rademacher complexity bounded as,
[Equation]
where B = (cid:113)
Tr[ Y ⊤ Θ − 1 ( X, X ) Y ] , and R =
n
[Equation]
Finally, Rademacher complexity directly gives an upper bound on generalization error [41],
[Equation]
, where L S ( f ) ≤ 1 √ n .
Based on this, we apply a union bound over a finite set of different i ’s.
Then with probability at least 1 − δ/ 3 over the sample S , we have sup f ∈F
R,Bi {L p ( f )
[Equation]
L
[Equation]
) , ∀
[Equation]
∈ { 1 , 2 , . . .
, O ( nλ 0 ) } .
Taking a union bound, we know that with probability at least 1 − 23 δ over the sample S , we have, f T ∈ F B ∗ i ,R for some i ∗ ,
R S ( F B ∗ i ,R ) ≤ (cid:113)
Tr[ Y ⊤ Θ − 1 ( X,X ) Y ] 2 n + 2 √ n and
[Equation]
These together can imply,
[Equation]
More proof details can be found in [3].
Theorem 2 (Relationship between the generalization bound and alignment) .
Under the same assumptions as in Lemma (2) , if we define the generalization upper bound as
[Equation]
,then it can be bounded with the alignment as follows,
[Equation]
Proof.
We first expand the following expression:
[Equation]
Then we use this expansion to prove the inequality on the left hand side,
[Equation]
The second line is due to quadratic mean is greater or equal to geometric mean.
Finally, we prove the inequality on the right hand side,
[Equation]
B.3 Derivation for Maximum Mean Discrepancy
The difference between truth risk over p ( x ) and q ( x ) can be defined as,
[Equation]
where g ( x ) = (cid:82) y ℓ ( f ( x ; θ ) , y ) p ( y | x ) dy .
Follow [58], we assume that the prediction functionshave bounded norm ∥ f ∥ F .
Thus, the function g is bounded.
By given the loss function, g is alsomeasurable.
Then, ∃ ˆ g ∈ C ( x ) , such that,
[Equation]
where C ( x ) is the function class of bounded and continuous functions of x .
To make the MMD termbe measurable, we empirically restrict the MMD on a reproducing kernel Hilbert space (RKHS) withthe characteristic kernel H Θ .
Following [22], we know that the relationship between the true MMDand the empirical MMD is,
[Equation]
where MMD ( S 0 , S, H Θ ) is the empirical measure for MMD ( p ( x ) , q ( x ) , H Θ ) .
Slightly overloadingthe notation, we denote S ∼ q ( x ) , which may not be i.i.d., and the initial label set S 0 ∼ p ( x ) .
Then, in the active learning setting, S 0 ⊆ S .
Further, we denote | S 0 | = n 0 , | S | = n and ∀ x, x ′ ∈
S, Θ ( x, x ′ ) ≤ C .
Therefore, we have, (cid:113)
Cn + (cid:113)
Cn 0 ≥ 2 (cid:113)
Cn .
For constant factor γ = MM + B , wehave the following inequality,
[Equation]
Denoting 2 e − γϵ 2 n 4 C
= δ/ 2 , then we have ϵ = (cid:113) 4 C ln(4 /δ ) γn .
Combining all the above results, we showthat with probability at least 1 − δ , the following inequality holds:
[Equation]
Then, we can get,
[Equation]
C APPENDIX: More details of experimental settings
C.1 Implementation Detail
For simple CNN model, we utilize the same architecture used in Pytorch CIFAR10 Image Classification Tutorial 1 .
For ResNet model, we use the Pytorch Offical implementation of ResNet-18 2and set the output dimension to the number of classes.
For VGG model, we use the Pytorch Officalimplementation of VGG-11 3 .
Besides, we leverage the library BackPACK [15] to collect the gradientof samples in batch.
We keep a constant learning rate of 0.001 for all three datasets and all three models.
All the codes
C.2 Computation of Acquisition Function
The acquisition function employed by dynamicAL can be written as the Equation 15.
Furthermore,we simplify it into the following form:
[Equation]
where ∇ θ ℓ ( f ( X S ; θ ) , Y S )
= (cid:80) ( x
,y ) ∈ S ∇ θ ℓ ( f ( x ; θ ) , y ) .
The computational requirement of the
Equation 31 is mainly composed of two parts, the computation of gradient and the computation ofthe inner product.
While PyTorch [45] can compute efficiently batch gradients, BackPACK [15]optimizes the computation of individual gradient and compute the gradient norm, sample per sample,at almost no time overhead.
Thus, the acquisition function can be computed at low computationalcosts.
Note, the efficiency of BackPACK has been verified by several recent works with extensiveexperiments[46, 55].
D APPENDIX: Verification Experiments under Ultra-wide ConditionD.1 Experiment Setting and Computational Detail for the Empirical Comparison betweenNTK and MMD
Experiment Setting For the verification experiment shown in Figure 1, we employ a simple CNNas the target model, in which there are three convolutional layers following with global averagepooling layer, on the CFAIR10 data set.
Note, this CNN architecture is widely used in NTK analysisworks [2, 62].
To make the verification experiment close to the application setting, we keep size ofinitial labeled set and query batch size same as what we used in Section 5.
Computational Detail We follow [30] to compute the MMD with NTK kernel.
The MMD term,
MMD ( p ( x ) , q ( x ) , H Θ ) , can be simplified into the following form:
[Equation]
where x i , x j ∼ p ( x ) and x ′ i , x ′ j ∼ q ( x ) .
Then, we define set S 0 as { x 1 , ..., x n 0 } ∼ p ( x ) andset S as { x ′ 1 , ..., x ′ n } ∼ q ( x ) , where n 0 ≤ n .
The MMD 2 ( S 0 , S ) is an unbiased estimation for
MMD 2 ( p ( x ) , q ( x )) , can we explicitly computed by:
[Equation]
Therefore, the MMD term of Equation (23), MMD ( S 0 , S, H Θ ) , can be empirically approximatedby (cid:113) MMD 2 ( S 0 , S ) .
D.
Experiment for the Correlation Study between Training Dynamics and Alignment
Experiment Setting.
For the verification experiment shown in Figure 2, we also use the simple
CNN on CIFAR10.
And to keep consistent with the application setting, we set | S | = 500 and| Q | = 250 .
The Q is randomly sampled from the unlabeled set and the labeled set S is fixed.
Weindependently sample Q 150 times to compute the correlation between between G MSE ( S ∪ Q ) and
A ( X ∥ X Q , Y ∥
Y Q ) .
Correlation between Training Dynamics computed with pseudo-labels and
In Figure 2, we compute the training dynamics with theground-truth label.
To study the effect of pseudo-labels,we further provide the relation between training dynamicscomputed with pseudo-labels G MSE ( S ∪ Q ) and alignment
A ( X ∥ X Q , Y ∥
Y Q ) , in which we compute the pseudo-labelswith Θ( X Q , X )
⊤ Θ( X, X ) − 1 Y .
The result is shown in the
Figure 6.
Note that we keep hyperparameters the same as previously described.
Compared with Figure 2, we find that thepositive relationship between A and the G computed withground-truth labels is stronger than the G computed withpseudo-labels.
The result is aligned with our expectations,because the extra noise is introduced by the pseudo-labels.
But, the Kendall τ coefficient still achieves 0 .
46 for A andthe G computed with pseudo-labels which indicates the utilityof using G calculating with pseudo-labels as the acquisitionfunction to query samples.
D.3 Correlation Study
[Figure]
between Training Dynamics and Generalization Bound
We present the relation between the training dynamics and thegeneralization bound in Figure 7.
Same as the previous, we set | S | = 500 and | Q | = 250 and the Qis randomly sampled from the unlabeled set.
The result shows that with the increase of G , B willdecrease.
This empirical observation is aligned with our expectation, because Theorem 2 indicatesthat the alignment A is inverse proportional to B and Figure 2 tells us that the G is proportional to
A .
Besides, the τ achieves -0.253 which indicates that the A is moderately inverse proportional to
B [10].
[Figure]
E APPENDIX: More details of experimental results
[Figure]
E.1 Baselines
[Figure]
Random:
Unlabeled data are randomly selected at each round.
[Figure]
Coreset:
This method performs a clustering over the last hidden representations in thenetwork, and calculates the minimum distance between each candidate sample’s embeddingand embeddings of labeled samples.
Then data samples with the maximum distances areselected.
Confidence Sampling (Conf): The method selects b examples with smallest predicted classprobability max Ki f i ( x ; θ ) [57].
Margin Sampling (Marg): The bottom b examples sorted according to the example’s multiclass margin are selected.
The margin is defined as f i ( x ; θ ) − f j ( x ; θ ) , where i and j arethe indices of the largest and second largest entries of f ( x ; θ ) [47].5.
Entropy: Top b samples are selected according to the entropy of the example’s predictiveclass probability distribution, the entropy is defined as H (( f i ( x ; θ ))
Ki =1 ) , where H ( p ) =(cid:80)
Ki p
i ln 1 p i [57].6.
Active Learning by Learning (ALBL):
The bandit-style meta-active learning algorithmcombines Coreset and Conf [27].7.
Batch Active learning by Diverse Gradient Embeddings (BADGE): b samples are selected byusing k-means++ seeding on the gradients of the final layer, in order to query by uncertaintyand diversity.
E.2 Experiment Results
The results for ResNet18, VGG11, and vanilla CNN on CIFAR10, SVHN, and Caltech101 withdifferent batch sizes have been shown in the Figure 3 and 8.
Note, for the large batch size setting( b = 1000 ) on Caltech101, we set the number of query round R = 4 , in which 49.2% images will belabeled after 4 rounds.0 1 2 3 4 5 6 7 8 9 # Query Round
T e s t
A cc u r a c y ( % )
[Table]
Conf ALBL
[Table]
dynamicAL
[Table]
ALBL BADGE dynamicAL 0 1 2 3 4 5 6 7 8 9 # Query Round
T e s t
A cc u r a c y ( % )
CIFAR10, ResNet, Query Batch Size:1000, Initial Set Size:
Random Marg Entropy
Coreset
Conf ALBL BADGE dynamicAL 0 1 2 3 4 5 6 7 8 9 # Query Round
T e s t
A cc u r a c y ( % )
SVHN, VGG, Query Batch Size:1000, Initial Set Size:
Random Marg Entropy
Coreset
Conf ALBL BADGE dynamicAL    4 # Query Round
[Figure]
Coreset
Conf
[Figure]
E.3 Numerical Result of Main Experiments
For the the main experiments, we report the means and standard deviations of active learningperformance under different settings in the the following tables.
[Table] [Table]
[Table] [Table]
[Table] [Table]
[Table] [Table]
E.4 Maximum Mean Discrepancy for Multiple Rounds
As shown in Figure 1, the MMD term is much smaller than the B at the first query round.
To betterunderstand the relation between MMD and B for multiple query setting, we measure the MMD / Bfor R ≥ 2 .
As shown in Figure 9, B is much larger than MMD even multiple query rounds.
Besides,we notice that, for the first round, the larger query batch always leads to larger MMD / B , because thesampling bias introduced by the query policy will be amplified by using large batch size.
9 # Query Round 0.
[Figure]
Furthermore, we measure the MMD / B with a constant total budget size but different query rounds.
The result is shown in Table 10.
As our expectation, spending the total query budget in one queryround will induce the largest MMD / B .
And, with more query rounds, the MMD / B will be lower.
S ETTING
[Table]
E.5 Performance under the Re-initialization Setting
To study the
Following [4], we query samples when training accuracy is greater than 99% and the results aresummarized in Table 11 and 12.
The results show that dynamicAL can still be better than orcompetitive with the commonly used active learning methods.
We notice that the improvement inthe non-retraining setting is more significant.
This is as our expectation.
The dynamic analysis(Equation (8)), that dynamicAL is based on, considers the change of dynamics according to the model’scurrent parameters.
The re-initialization trick will not only causes the computational overhead ofretraining, but also makes dynamicAL deviate from the analysis (Section 4).
E.6 Performance with large query rounds
We provide the experiments with b = 500 , r = 15 on Caltech101 data set with ResNet18 as thebackbone.
We ignore the BACKGROUND Google label and then we have 8677 images in total.
At the last round, we run out of all images in the pool.
As shown in the Figure 10, our methodconsistently outperforms those baselines.
Note, due to the non-retraining setting, the model will havedifferent performance even if all the samples are used for the training.
[Figure]
[Table] [Table]
F Discussion
Limitation and Future Work.
In the work, we study the connection between generalizationperformance and the training dynamics under the NTK regime.
Although the relation between trainingdynamics and generalization performance has been verified by our experiments, the theoreticalanalysis of the relation out of the NTK regime still needs study.
Besides, in the experiments, wemainly focus on the classification problem.
Whether the proposed method is effective for theregression problem is under-explored.
We would like to leave the study of the previously mentionedtwo problems in the future work.
NTK Analysis for the Design of Practical Method.
Although some works [36, 21] discussedthat the NTK assumption is hard to be strictly satisfied in some real-world models, we notice thatsome recent works have shown that the high-level conclusions derived based on NTK is insightfuland useful for the design of practical models.
Some of their applications can achieve SOTA.
Forexample, Park et al.
[44] used the NTK to predict the generalization performance of architectures inthe application of Neural Architecture Search (NAS).
Chen et al.
[12] used the condition number of
NTK to predict a model’s trainability.
Chen et al.
[13] also used the NTK to evaluate the trainabilityof several ImageNet models, such as ResNet.
Deshpande et al.
[17] used the NTK for model selectionin the fine-tuning of pre-trained models on a target task.
In our work, the empirical results in Figure 3and Appendix.
E also show the effectiveness of the high-level conclusions derived from the theorystill hold.
Social Impacts.
In this work, we study the connection between the generalization performanceand the training dynamics and try to bridge the gap between the theoretic findings of deep neuralnetworks and deep active learning applications.
We hope our work would inspire more attempts onthe design of deep active learning algorithms with theoretical justification, which might have positivesocial impacts.
We do not foresee any form of negative social impact induced by our work.
License Privacy Information.
We use the commonly used datasets, CIFAR10 4 , SVHN 5 , Caltech101 6 in the experiments.
Those datasets follow the MIT, CC0 1.0, CC BY 4.0 License respectivelyand are publicly accessible.
No privacy information is included in those datasets.