M E -M OMENTUM : E XTRACTING H ARD C ONFIDENT E XAMPLES FROM N OISILY L ABELED D ATA
Anonymous authors
Paper under double-blind review
A BSTRACT
Examples that are close to the decision boundary—that we term hard examples , are essential to shaping accurate classiﬁers.
Extracting conﬁdent examples has been widely studied in the community of learning with noisy labels.
However, it remains elusive how to extract hard conﬁdent examples from the noisy training data.
In this paper, we propose a deep learning paradigm to solve this problem, which is built on the memorization effect of deep neural networks that they would ﬁrst learn simple patterns , i.e., which are deﬁned by these shared by multiple training examples.
To extract hard conﬁdent examples that contain non-simple patterns and are entangled with the inaccurately labeled examples, we borrow the idea of momentum from physics.
Speciﬁcally, we alternately update the conﬁdent examples and reﬁne the classiﬁer.
Note that the extracted conﬁdent examples in the previous round can be exploited to learn a better classiﬁer and that the better classiﬁer will help identify better (and hard) conﬁdent examples.
We call the approach the “ Momentum of Me morization ” (Me-Momentum).
Empirical results on benchmark-simulated and real-world label-noise data illustrate the effectiveness of Me-Momentum for extracting hard conﬁdent examples, leading to better classiﬁcation performance.
I NTRODUCTION
As training datasets are growing big while accurately labeling them is often expensive or sometimes even infeasible, cheap datasets with label noise are ubiquitous in many real-world applications.
Without any care, label noise will degenerate the performance of learning algorithms, especially for these based on deep neural networks (Zhang et al., 2017).
Learning with noisy labels (Angluin &
Laird, 1988) aims to reduce the side-effect of label noise and therefore has become an important topic in machine learning.
Existing methods for learning with noisy labels can be divided into two categories: algorithms that result in statistically consistent or inconsistent classiﬁers.
Methods in the ﬁrst category intent to design classiﬁer-consistent algorithms (Zhang & Sabuncu, 2018; Kremer et al., 2018; Liu & Tao, 2016; Scott, 2015; Natarajan et al., 2013; Goldberger & Ben-Reuven, 2017; Patrini et al., 2017;
Thekumparampil et al., 2018; Yu et al., 2018; Liu & Guo, 2020;
Xu et al., 2019), where classiﬁers learned by using the noisy data will statistically converge to the optimal classiﬁers deﬁned by clean data.
However, these methods rely heavily on the noise transition matrix (Liu & Tao, 2016; Patrini et al., 2017; Xia et al., 2019).
In real-world applications, it is hard to learn the instance-independent noise transition matrix (Cheng et al., 2020).
To be free of estimating the noise transition matrix, methods in the second category employ heuristics to reduce the side-effect of label noise (Yu et al., 2019; Han et al., 2018b; Malach & Shalev-Shwartz, 2017; Ren et al., 2018; Jiang et al., 2018;
Ma et al., 2018; Kremer et al., 2018; Tanaka et al., 2018;
Reed et al., 2015; Han et al., 2018a;
Guo et al., 2018; Veit et al., 2017; Vahdat, 2017; Li et al., 2017; 2020).
These methods were reported to empirically work well, especially in the setting of instance-dependent label noise.
One promising direction in the second category is to extract examples with clean labels— conﬁdent examples — (Northcutt et al., 2017; Cheng et al., 2020;
Chen et al., 2019; Malach & Shalev-Shwartz, 2017; Han et al., 2018b; Yu et al., 2019; Thulasidasan et al., 2019; Nguyen et al., 2020; Dredze et al., 2008; Liu & Tao, 2016;
Wang et al., 2017; Ren et al., 2018; Jiang et al., 2018).
The idea is that compared with the original noisy training data, the extracted examples are less noisy and thus
[Figure]
will lead to a classiﬁer with better performance.
Given only noisy data, state-of-the-art methods exploit the memorization effect (Zhang et al., 2017; Arpit et al., 2017) to extract conﬁdent examples.
The memorization effect will enable deep neural networks to ﬁrst learn patterns that are shared by majority training examples.
As clean labels are of majority in each noisy class (Natarajan et al., 2013; Xiao et al., 2015), deep neural networks would therefore ﬁrst ﬁt training data with clean labels, and then gradually ﬁt the examples with incorrect labels (Chen et al., 2019).
Therefore, early stopping (Li et al., 2020; Song et al., 2019) and the small loss trick (Jiang et al., 2018; Han et al., 2018b; Yu et al., 2019) can be used to exploit conﬁdent examples.
Examples that are close to the decision boundary are called hard examples .
As illustrated in Figure 1, hard (conﬁdent) examples play an important role in shaping the decision boundary.
It has also been widely studied in the traditional classiﬁcation problem that hard examples are essential to train accurate classiﬁers (Vapnik, 2013; Bengio et al., 2009; Huang et al., 2010;
He et al., 2018).
Notwithstanding the importance of hard conﬁdent examples, none of the existing methods studies how to extract hard conﬁdent examples from noisy data.
Note that extracting hard conﬁdent examples is non-effortless.
Since hard examples are often of a small proportion and contain less discriminative information compared with the easy ones (these that are far away from the decision boundary), they are often entangled with inaccurately labeled examples in the procedures of extraction.
In this paper, by alternately updating the conﬁdent examples and reﬁning the classiﬁer, we propose a deep learning paradigm that is able to extract hard conﬁdent examples from the noisy training data, leading to better classiﬁcation performance.
Speciﬁcally, the idea is similar to the usage of momentum from physics.
As stated in the statistical learning theory, with better training data, better classiﬁer can be obtained (Mohri et al., 2018).
We can then think of the classiﬁer as a particle traveling through the hypothesis space, getting acceleration from the conﬁdent data.
Classiﬁers with better performance can be achieved by properly exploiting the previously extracted conﬁdent examples.
This is similar to the momentum trick used in optimization that previous gradient information can be used to escape local minimum and achieve fast convergence rates (Sutskever et al., 2013) 1 .
At a high level, the proposed method is built on the memorization effect of deep neural networks and on the intuition that better conﬁdent examples will result in a better classiﬁer and that a better classiﬁer will identify better conﬁdent examples (and hard conﬁdent examples).
The proposed method is therefore called the Momentum of Memorization (Me-Momentum).
We conduct experiments to show the effectiveness of the proposed Me-Momentum on noisy versions of MNIST , CIFAR10 , CIFAR100 , and a real-world label noise dataset Clothing1M .
Speciﬁcally, on MNIST and CIFAR , we generate class- and instance-dependent label noise and visualize the extracted hard conﬁdent examples, which justiﬁes why Me-Momentum consistently outperforms the baseline methods.
Codes will be available online.
M E -M OMENTUM
In this section, by specifying the proposed method of momentum of memorization (Me-Momentum; summarized in Algorithm 1), we would like to detail how to accomplish extracting hard conﬁdent examples and boosting the classiﬁcation performances.
At the high level, by alternately updating the conﬁdent examples and reﬁning the classiﬁer, Me-Momentum fulﬁls a positive cycle that better conﬁdent examples will result in a better classiﬁer and that a better classiﬁer will identify better conﬁdent examples.
Speciﬁcally, Me-Momentum has two loops, i e., an inner loop and an outer loop.
In the inner loop, Me-Momentum alternates update of the conﬁdent examples and classiﬁer (Steps 2 and 3).
However, the inner loop continually reﬁnes a classiﬁer and thus depends heavily on the initialization of the classiﬁer (Step 1).
It may lead to the memorization of noisy labels and the inferiority of sample-selection bias.
To handle this problem, the outer loop re-initialize the classiﬁer (Step 5) while maintains the previously extracted conﬁdent examples.
There are some points to be clariﬁed for the proposed Algorithm 1:
Q1.
How to initialize a good classiﬁer in Step 1?
Q2.
How to extract conﬁdent examples in Step 2?
Q3.
How to validate the learned classiﬁers in Steps 3 and 5 without a clean validation set?
Q4.
Why the proposed method is called Me-Momentum?
Q5.
Why hard conﬁdent examples can be extracted?
To answer the ﬁrst question , we would like to mention that the aim of the initialization in Step 1 is to initialize a good classiﬁer for the positive cycle: a better classiﬁer will identify better conﬁdent examples and better conﬁdent examples will result in a better classiﬁer.
A good candidate should have a fairly high classiﬁcation accuracy, e.g., higher than random guessing.
Otherwise, the positive cycle cannot be evoked.
Fortunately, the initialization can be made by exploiting the memorization effect of deep neural networks that they would ﬁrst ﬁt clean data (Arpit et al., 2017; Zhang et al., 2017).
Note that this memorization effect is independent of training optimization or network backbones (Arpit et al., 2017).
Speciﬁcally, we use the early stopping trick.
For easy understanding, we would like to introduce a deﬁnition of high-peak .
A noisy validation accuracy at the i -th epoch is called an i -th high-peak if it achieves the highest accuracy in the epoch range { 1 , . .
, i } .
Assume the i -th and j -th high-peaks occur next to each other, having noisy validation accuracies of a and b , respectively.
The training early stops if ( b − a ) /
( j − i ) ≤ τ , where τ is a hyper-parameter.
In the experiments, we set τ = 0 .
1 , which empirically works well across all datasets.
In Appendix A, we compare the difference between the early stopping method and the traditional validation method;
We also study the sensitivity of the hyper-parameter.
The answer to the second question is closely related to the memorization effect.
Note that the classiﬁer initialized in Step 1 would ﬁt the clean data well but not the incorrectly labeled data because of the memorization effect and early stopping.
Therefore, we can treat the training examples whose noisy labels are identical to the ones predicted by the classiﬁer obtained in Step 1 as conﬁdent examples.
This also applies for the classiﬁers in Step 3 to extract conﬁdent examples, which are iteratively trained by employing the updated conﬁdent data.
Note that there are some other feasible methods to extract conﬁdent examples, e.g., extract those who have a large class posterior.
In Step 3, we aim to learn a better classiﬁer compared with the one in the previous round.
This can be achieved because of two reasons: (1) we initialize the network with the parameters of the classiﬁer learned in the previous round; (2) we have a better set of conﬁdent examples as the training sample.
This starts the positive cycle that better conﬁdent examples will result in a better classiﬁer and that a better classiﬁer will identify better conﬁdent examples.
The third question is essential for identifying the classiﬁers in the cycle.
Note that the accurately labeled examples are always assumed to be dominant in each class in the community of learning with noisy labels (Natarajan et al., 2013;
Liu & Tao, 2016; Han et al., 2018b).
Otherwise, the true class label cannot be identiﬁed by only exploiting the noisy data.
This assumption implies that the performance on the noisy validation set (split from the noisy training set) and these on test set are positively correlated.
The noisy validation set could be used as a surrogate to validate the classiﬁers if no clean validation set is available.
We therefore validate the classiﬁers in Steps 3 and 5 with the highest noisy validation accuracies during the training.
Empirical results show that it works well.
To answer the fourth question , we would like to ﬁrst mention that the proposed method heavily relies on the memorization effect of deep neural networks.
Speciﬁcally, in Step 1, a classiﬁer is initialized by exploiting the memorization effect via early stopping, which is used to identity conﬁdent examples.
Later, the classiﬁer and conﬁdent examples are iteratively reﬁned and updated, respectively, which is the positive cycle we have mentioned before.
Note that this cycle also depends on the memorization effect to update conﬁdent examples and reﬁne classiﬁers.
Our method is named as momentum of memorization (Me-Momentum) because it also uses the trick of momentum to better exploit the memorization effect.
Speciﬁcally, we can think of the classiﬁer as a particle traveling through the hypothesis space, getting acceleration from the updated extracted conﬁdent data.
We exploit the previously extracted conﬁdent examples to help learn a better classiﬁer, i.e., initializing the network by using the parameters of the classiﬁer trained with the conﬁdent examples extracted in the previous round.
The impact of the conﬁdent examples in a certain round will decay as we continue updating the classiﬁer.
[Figure]
We intuitively answer the ﬁfth question by employing Figure 2.
Hard conﬁdent examples will be progressively extracted also due to the momentum trick, i.e., previously extracted conﬁdent examples will help identify hard conﬁdent examples.
Speciﬁcally, in Figure 2(a), we show the noisy training examples, where circles and triangles represent two classes respectively and the red ones denote the examples with incorrect labels.
Let the dash line in Figure 2(b) denote a classiﬁer learned by Me-Momentum.
We can use it to extract conﬁdent examples, e.g., the green circles and blue triangles, according to Step 2 of Algorithm 1.
Compared with Figure 1(a), we can see that hard conﬁdent examples close to the optimal boundary, i.e., the circle and triangles with red edges, will be extracted.
Note that as better and better classiﬁers will be learned by Me-Momentum, more and more hard conﬁdent examples will be extracted.
An visualization is shown in Figure 4.
Relation to existing work: The strategy of alternatively optimizing the classiﬁer and updating the training examples is not new for dealing with label noise.
For example, Joint Optim (Tanaka et al., 2018), Co-teaching (Han et al., 2018b; Yu et al., 2019), and SELF (Nguyen et al., 2020) are similar to ours.
Speciﬁcally, Joint Optim and Co-teaching update the classiﬁer with one step of stochastic gradient descent while SELF and the proposed method reﬁne the classiﬁer to be optimal with respect to the extracted conﬁdent examples.
However, existing methods have not focused on extracting hard conﬁdent examples and thus are substantially different from this paper because they neglected the importance of avoiding the accumulated error caused by the single initialization of the classiﬁer and the sample-selection bias.
Experiments (e.g., Figures 3 and 4) show that Me-Momentum with the outer loop part (i.e., re-initialization of the classiﬁer) contributes signiﬁcantly to extracting hard conﬁdent examples and achieving high label precision.
Note that without employing the ensemble technique, SELF is a special case of Me-Momentum with only the inner loop part.
We avoid to integrate the ensemble technique to the proposed method for simplicity and clarity.
Me-Momentum is similar to curriculum learning as it also learns from easy to difﬁcult.
However, curriculum learning needs a predeﬁned curriculum (sample weighting scheme), e.g., assigning big/small wights for conﬁdent/noisy data.
If the curriculum is not available, some clean data is required to learn a mentornet to provide a curriculum (Jiang et al., 2018) or a latent variable could be introduced by self-paced learning (Kumar et al., 2010) to learn a curriculum.
Differently, MeMomentum only based on noisy data and does not explicitly learn a curriculum.
Me-Momentum also has a similar ﬂavor to active learning which tends to choose and label hard examples to learn from at each iteration.
However, for active learning, no label information is available before the data is chosen while Me-Momentum has noisy labels and have to consider the side-effect of label noise.
E XPERIMENTS
Datasets: To verify the effectiveness of the proposed method, we do experiments on datasets with both synthetic and real-world label noise.
Speciﬁcally, we manually corrupt MNIST (LeCun et al., 1998), CIFAR10 , and CIFAR100 (Krizhevsky, 2009) with class-dependent label noise and instancedependent label noise.
We detail how to generate class-dependent and instance-dependent label noise in the supplementary material.
We employ the real-world noisy dataset Clothing1M (Xiao et al., 2015).
These datasets have been widely used in studies with noisy labels (Han et al., 2018b;
Tanaka et al., 2018; Xia et al., 2019).
For MNIST , CIFAR10 , and CIFAR100 , we leave out 10% of the noisy training data as noisy validation data.
Clothing1M contains one million noisy training images, which are crawled from shopping websites, labeling by surrounding text.
Almost all existing work uses the 14 k clean validation data in their experiments.
To verify the robustness of the proposed method, we also employ noisy validation data in our experiments.
Speciﬁcally, 100 k noisy data are randomly left as noisy validation data and the remaining 900 k noisy data as the training data.
Baselines: Me-Momentum is compared against the following state-of-the-art approaches.
(1) Statistically consistent methods: Forward (Patrini et al., 2017), T-revision (Xia et al., 2019), and DMI (Xu et al., 2019); (2) Statistically inconsistent methods: Decoupling (Malach & Shalev-Shwartz, 2017),
MentorNet (Jiang et al., 2018), Co-teaching
Han et al., 2018b), Joint Optim (Tanaka et al., 2018),
SELF (Nguyen et al., 2020) where MentorNet, Co-teaching and SELF use the idea of extracting conﬁdent examples by employing the small loss trick.
Network structure and optimization: All the methods are implemented with default parameters by PyTorch v1.5.
For the experiments on MNIST , CIFAR10 , and CIFAR100 , we set N inner = 20 ,
N outer = 3 , and follow the settings of T-revision (Xia et al., 2019).
Speciﬁcally, LeNet-5, ResNet18 and ResNet-34 networks are used for MNIST , CIFAR10 and CIFAR100 respectively.
We use
SGD with momentum 0 . 9 , weight decay 10 − 4 , batch size 128 , and an initial learning rate of 10 − 2 , divided by 10 after the 40 -th epoch and 80 -th epoch respectively (we ﬁx the learning rate of 10 −for the early stopping method).
Data augmentation is used with horizontal random ﬂips and 32 ×random crops after padding 4 pixels on each side.
For Clothing1M , a ResNet-50 is used.
We set
N inner = 6 and N outer =
To show the effectiveness of the proposed method, we do experiments by randomly initializing the network and pre-training it by employing ImageNet , respectively.
As the noisy training sample contains a large number of examples, we train the network for 5 epochs and choose the best one with the highest classiﬁcation accuracy on the noisy validation set.
We use SGD with momentum 0.9, weight decay 10 − 3 , batch size 32 , with a learning rate of 5 × 10 − 3 , and divided it by 10 at the 3 -rd and 5 -th round in the inner loop.
For each outer loop, the model will be randomly re-initialized (or replaced by a pre-trained one).
The learning rate will be reset to 5 × 10 − 3 .
For data augmentation, all images are resized to 256 × 256 , horizontal random ﬂipped, and cropped in the middle with a size of 224 × 224 .
Note that due to the page limit, some complementary experiments to Sections 3.1 and 3.2 and comparison with SELF are put in the Appendix.
3. V ERIFY M OMENTUM OF M EMORIZATION
In Section 2, we discussed that Me-Momentum is implemented by fulﬁlling the positive cycle that better conﬁdent examples will result in a better classiﬁer and that a better classiﬁer will identify
[Figure]
a
[Figure]
L a b e l P
[Figure]
i s i o n
[Figure]
better conﬁdent examples.
In this subsection, we will empirically verify this positive cycle, which can be done on the synthetic datasets as we have their ground-truth labels.
In Figure 3, we can see that in the inner loops (e.g., rounds 0-5, rounds 6-9, and rounds 10in the ﬁrst column of ﬁgures represent three inner loops respectively), the classiﬁcation accuracy generally increases (note that the ﬁgures are not smooth because the classiﬁers are tested on the unseen test data) and the number of extracted conﬁdent examples clearly increases (although their label precision decreases slightly).
We can also see that in the outer loops (e.g., rounds 0, 6, and 10 in the ﬁrst column of the ﬁgures consist of an outer loop), the classiﬁcation accuracy clearly increases and the label precision of the extracted conﬁdent examples clearly increases (although the number of extracted conﬁdent examples decreases slightly).
This implies that compared with previous classiﬁers and extracted conﬁdent examples, better ones are obtained, which empirically justiﬁes the positive cycle.
Note that the classiﬁcation accuracy in the outer loop is low because the models are re-initialized.
Figure 3 also shows the importance of the outer loop of Me-Momentum.
We can see that the label precision of the extracted conﬁdent examples slightly decreases in the inner loops.
This is because the deep model gradually memorizes the noisy labels as we continually reﬁne it.
This issue can be handled by re-initialize the deep model in the outer loop.
Speciﬁcally, we can see from Figure 3 that by re-initializing the model in the outer loop, the label precision of the extracted conﬁdent examples increases signiﬁcantly.
Although the number of the extracted conﬁdent examples decreases, the overall quality of the extracted conﬁdent examples is increasing as evidenced by the increase of the classiﬁcation accuracy of the classiﬁers trained on the extracted conﬁdent data.
Note that the low data quality in the ﬁrst run of the inner loop also justiﬁes that a single deep model initialization may lead to sample selection bias.
There are interesting observations of the proposed method that the number of extracted conﬁdent examples is close to the number of accurately labeled data in the training set and that the label precision of the extracted conﬁdent examples is quite high, i.e., almost all are above 90%.
This empirically proves that Me-Momentum is powerful in extracting conﬁdent examples.
In the next subsection, we will visualize that Me-Momentum is also good at extracting hard conﬁdent examples.
3. V ISUALIZE HARD CONFIDENT EXAMPLES
To justify that Me-Momentum is able to extract hard conﬁdent examples, we visualize the extracted conﬁdent examples by employing t-SNE (Maaten & Hinton, 2008).
Speciﬁcally, we show how the conﬁdent examples are progressively extracted in the inner and loops.
The results are shown in Figure 4, where green, blue, and red dots represent conﬁdent examples extracted at the beginning, middle, and end rounds of the loops, respectively.
On the datasets of MNIST and CIFAR10 , we can clearly see that the blue and red dots are mostly located at the boundaries of the clusters of green dots.
Although the ﬁgures of CIFAR100 are small, we can also clearly see that there are lots of blue and red dots which are outside of the green clusters in the second and fourth
This supports and justiﬁes our claim that Me-Momentum is able to extract hard conﬁdent examples (those are close to the decision boundary).
Comparing the extracted results of the ﬁrst run of the inner loop (the ﬁrst and third columns) with those of the outer loop (the second and fourth columns), we can ﬁnd that the cluster boundaries in the latter are more clear.
This further justiﬁes that why better classiﬁcation performance can be obtained by re-initialization in the outer loop.
Comparing the conﬁdent examples extracted on the class-dependent label noise datasets with those on the instance-dependent label noise datasets, we can see that the proposed method is not very sensitive to the type of label noise and can work well on the most general instance-dependent label noise cases.
3. C LASSIFICATION ACCURACY
Synthetic data To evaluate the classiﬁcation performance of Me-Momentum, we ﬁrst conduct experiments on MNIST , CIFAR10 , and CIFAR100 with class-dependent and instance-dependent label noise.
Each trial is repeated ﬁve times.
The results are presented in Tables 1, 2, and 3, respectively.
Me-Momentum consistently outperforms the baselines.
Speciﬁcally, CIFAR100 is the most challenging one among the three datasets.
On CIFAR100 , Me-Momentum outperforms the baselines by a clear margin across all the settings as shown in Table 3.
Note that the performance gain in
Me-Momentum is caused by the improvement of the quality of the extracted conﬁdent examples.
In the baselines, Co-teaching, Joint Optim, and T-revision are the representative methods that learn robust classiﬁers by extracting conﬁdent examples, reﬁning the noisy labels, and exploiting the noise transition matrix, respectively.
Note that Co-teaching keeps updating a constant number of conﬁdent
[Table] [Table]
[Table] [Table]
examples from the mini-batches used in SGD.
We therefore do not compare with its extracted conﬁdent examples in Section 3.2 as our method extracts conﬁdent examples from the whole training data at once.
By comparing the classiﬁcation performance
[Table]
we can clearly see that the proposed method is much more powerful in extracting conﬁdent examples.
Note that Joint Optim and Trevision employ all training data to train the classiﬁers; while our method only employs conﬁdent examples and discards the unconﬁdent ones.
The results further justify that Me-Momentum is able to extract high-quality conﬁdent examples.
Note that the performance of Me-Momentum could be further improved by correcting the unconﬁdent data with the idea of Joint Optim
[Table] [Table]
Real-world dataset We compare Me-Momentum with baseline methods on Clothing1M in Table 4, where "pre-trained" and "scratch" mean the network was pre-trained by employing ImageNet and initialized randomly, respectively; "clean" and "noisy" means the validation data is clean and noisy respectively.
First, it is observed that Me-Momentum works well with noisy validation, even surpassing many baselines with clean validation.
For a fair comparison, we also use clean data to validate our method, which achieves the highest test accuracy of 75.18%, better than T-revision by 1% and Joint
Optim by 2.95%.
Note that Forward and T-revision need the 50k clean data for estimating transition matrix, while Me-Momentum does not need any clean data for training.
In addition, to show the robustness of Me-Momentum, we conduct experiments with ResNet-50 from scratch, which achieves the second best accuracy.
C ONCLUSION
In this paper, we propose a method called Me-Momentum that is able to extract hard conﬁdent examples from noisily labeled data by exploiting the memorization effect of deep neural networks.
In a high level, it fulﬁls a positive cycle that better conﬁdent examples will result in a better classiﬁer and that a better classiﬁer will identify better conﬁdent examples.
We have empirically veriﬁed its effectiveness by analyzing the statistics of the extracted examples, visualizing the hard conﬁdent examples, and comparing its classiﬁcation performance with state-of-the-art baselines.
In future, we can extend our work by utilizing and exploiting the unconﬁdent examples.
A EARLY STOPPING
We discuss the early stopping trick used in Step 1 of Algorithm 1.
We repeat explaining the trick here again.
Speciﬁcally, we introduce a deﬁnition of high-peak .
A noisy validation accuracy at the ith epoch is called an i -th high-peak if it achieves the highest accuracy in the epoch range { 1 , . .
, i } .
Assume the i -th and j -th high-peaks occur next to each other, having noisy validation accuracies of a and b , respectively.
The training early stops if ( b − a ) /
( j − i ) ≤ τ , where τ is a hyper-parameter.
In the experiments
[Figure]
we set τ = 0 . 1 , which empirically works well across all datasets.
In Figure 5, we compare the difference between the early stopping method and the traditional validation method
[Figure]
We also study the sensitivity of the hyper-parameter.
Speciﬁcally, we study its sensitivity on CIFAR10 by setting τ to be the values in the range {0.05, 0.07, 0.1, 0.3, 0.5}.
Other settings are the same as those in this paper.
The results are presented in Figure 6.
We can see that the classiﬁcation performance of Me-Momentum is robust and not sensitive to the change of the value of τ .
B G ENERATING LABEL NOISE
Transition matrix
The transition matrix T ( x ) was proposed to explicitly model the generation process of label noise, where T ij ( x ) =
Pr ( ¯ Y =
j | Y =
i, X = x ) , Pr ( A ) denotes as the probability of the event A , X as the random variable for the instance, ¯ Y as the noisy label, and Y as the latent clean label.
In high level, the ij -th entry of the transition matrix denote the probability that the instance will ﬂip from the clean class j to the noisy class i .
Symmetric class-dependent label noise The label noise is generated according to symmetric classdependent and instance-independent noise transition matrices.
If the ﬂip rate is α , the diagonal entries of a symmetric transition matrix are 1 − α and the off-diagonal entries are α/ ( c − 1) .
Instance-dependent label noise We generate the instance-dependent label noise according to the following Algorithm 2 Xia et al.
More details about this algorithm can be found in Xia et al.
Algorithm
C C OMPARE RESULTS WITH SELF
The performance of our re-implemented SELF (Tarvainen & Valpola, 2017) is not as good as that in the original paper.
For a fair comparison, we change our backbone consistently with SELF and compare with the results from the original paper directly (without Mean Teacher (Tarvainen & Valpola, 2017)).
Speciﬁcally, the network is changed to ResNet26 with Shake-shake regularization (Gastaldi, 2017) and the learning rate is changed to 0.05 and weight decay to 2e-4.
Table 5 shows that Me-Momentum outperforms SELF by a large margin in CIFAR10 and CIFAR100 with Symmetric 40% and Symmetric 60% noise.
The gap between the performance of
Me-Momentum and SELF becomes larger in Symmetric 60% in both datasets compared with Symmetric 40% because hard conﬁdent examples play a more important role in more noisy datasets.
Speciﬁcally, both of the methods are based on the same backbone with Cross-Entropy loss, so the improvement of Me-Momentum can only be as a result of the quality of extracted conﬁdent examples.
Therefore, Me-Momentum is able to extract better hard conﬁdent examples than SELF.
[Table]
Furthermore, results in Table 5 shows that the performance of Me-Momentum can be improved by changing a better backbone.
However, to show the effectiveness of the proposed method and avoid complexity, we simply choose in the paper the standard CNN network.
D E XPERIMENTS COMPLEMENTARY TO S ECTION 3.
In Section 3.1, we discuss the statistics of the extracted conﬁdent examples.
However, due the space limit, in the paper, we only provide parts of the empirical results.
In this supplementary material, we provide the statistics on all the employed datasets and settings.
[Figure] [Figure]
[Figure]
E E XPERIMENTS COMPLEMENTARY TO S ECTION 3.
In Section 3.2, we have visualize the extracted conﬁdent examples by using t-SNE.
It veriﬁes that the proposed Me-Momentum is effective to extract hard conﬁdent examples.
However, due the space limit, in the paper, we only provide parts of the empirical results.
In this supplementary material, we provide the visualization on all the employed datasets and settings.
Speciﬁcally, we show how the conﬁdent examples are progressively extracted in the inner and outer loops.
In the ﬁgures, green, blue, and red dots represent conﬁdent examples extracted at the beginning, middle, and end rounds of the loops, respectively.
On the datasets of MNIST and CIFAR10 , we can clearly see that the blue and red dots are mostly located at the boundaries of the clusters of green dots.
On CIFAR100 , we can also clearly see that there are lots of blue and red dots which are outside of the green clusters in the second and fourth ﬁgures.
This supports and justiﬁes our claim that Me-Momentum is able to extract hard conﬁdent examples (those are close to the decision boundary).
The ﬁgures are presented in the next and the following pages.