Learning with Instance-Dependent Label Noise: A Sample Sieve Approach
Authors: Hao Cheng
Zhaowei Zhu
Xingyu Li
Yifei Gong
Xing Sun
Yang Liu
Venue: ICLR 2021 Poster (2021)
Keywords: Learning with noisy labels
instance-based label noise
deep neural networks.

Abstract: Human-annotated labels are often prone to noise, and the presence of such noise will degrade the performance of the resulting deep neural network (DNN) models. Much of the literature (with several recent exceptions) of learning with noisy labels focuses on the case when the label noise is independent of features. Practically, annotations errors tend to be instance-dependent and often depend on the difficulty levels of recognizing a certain task. Applying existing results from instance-independent settings would require a significant amount of estimation of noise rates. Therefore, providing theoretically rigorous solutions for learning with instance-dependent label noise remains a challenge. In this paper, we propose CORES$^{2}$ (COnfidence REgularized Sample Sieve), which progressively sieves out corrupted examples. The implementation of CORES$^{2}$  does not require specifying noise rates and yet we are able to provide theoretical guarantees of CORES$^{2}$ in filtering out the corrupted examples. This high-quality sample sieve allows us to treat clean examples and the corrupted ones separately in training a DNN solution, and such a separation is shown to be advantageous in the instance-dependent noise setting. We demonstrate the performance of CORES$^{2}$ on CIFAR10 and CIFAR100 datasets with synthetic instance-dependent label noise and Clothing1M with real-world human noise. As of independent interests, our sample sieve provides a generic machinery for anatomizing noisy datasets and provides a flexible interface for various robust training techniques to further improve the performance. Code is available at https://github.com/UCSC-REAL/cores.

Published as a conference paper at ICLR 2021
LEARNING WITH INSTANCE-DEPENDENN LABEL NOISE: A SAMPLE SIEVE APPROACH
Hao Cheng" Zhaowei Zhu"i Xingyu Li", Yifei Gong", Xing Sun and Yang Liu University of California, Santa Cruz, Tennent YouTu Lab (zwzhu,x1l x1i279, yanglin (louischeng, linglluu iu) @ucsc. edu,
winfredsun @tencent.com
ABSTRACT
Human-annotated labels are often prone to noise, and the presence of such noise will degrade the performance of the resulting deep neural network (DNN) mod- els. Much of the literature (with several recent exceptions) of learning with noisy labels focuses on the case when the label noise is independent of features. Practi- cally, annotations errors tend to be destance-deeeneett and often depend the difficulty levels of recognizing a certain task. Applying existing results from instance-independent settings would require a significant amount of estimation of noise rates. Therefore, providing theoretically rigorous solutions for learning with instance-dependent label noise remains a challenge. In this paper, we propose CORES² (COnfidence REgularized Sample Sieve), which progressively sieves out corrupted examples. The implementation of CORES does not require specifying noise rates and yet we are able to provide theoretical guarantees of CORES in filtering out the corrupted examples. This high-quality sample sieve allows us to treat clean examples and the corrupted ones separately in training DNN solu- tion, and such separation is shown to be advantageous in the instance-dependen noise setting. We demonstrate the performance of CORES CIFAR10 10 and CI- FAR100 datasets with synthetic instance- mnstance-dependent label noise and Clothing IM with real-world human noise. As of independent interests, our sample sieve pro- vides a generic machinery for anatomizing noisy datasets and provides a flexi- ble interface for various robust training techniques further improve the perfor- mance. Code is available at
com/ com/UCS /CCc/ REAL/core
INTRODUCTION
Deep neural networks (DNNs) have gained popularity in wide range of applications. The re- markable success of DNNs often relies on the availability of large-scale datasets. However, data annotation inevitably introduces label noise, and it is extremely expensive and time-consuming to clean up the corrupted labels. The existence label noise can weaken the true correlation between features and labels well introducing artificial correlation patterns. Thus, mitigating the effects of noisy labels becomes a critical issue that needs careful treatment.
It is challenging avoid overfitting noisy labels, especially when the noise depends both true labels Y and features X. Unfortunately, this often tends to be the case where human annotations prone different levels of errors for tasks with varying difficulty levels. Recent work has also shown that the presence of distance-dependee cautions training in this scenario (Liu 2021). For such instance-dependent (or feature-dependent, instance-based) label noise settings, theory-supported works usually focus on loss-correction which requires estimating noise rates (Xia et al. 2020 Berthon al., 2020). Recent work by Cheng et al. (2020) addresses the bounded instance- noise by first learning the noisy distribution and then distilling examples according to some thresholds However, with a limited size of datasets, learning an accurate noisy distribution for each example is a non-trivial task. Additionally, the size and the quality distilled examples are sensitive to the thresholds for distillation.
noisy labels imposes additional challenges and
"qqual contributions in alphabetical ordering. Hao leads experiments and Zhaowei leads theories. "orresponning authors: Y. Liu and Z. Zhu [yang] u, zwzhu S @ucacc edu.
'The proposed solution is primarily studied for the binary case in Cheng et al. 2020).

---

Published as a conference paper at ICLR 2021
Departing from the above line of works, we design a sample sieve with theoretical guarantees to provide a high-quality splitting of clean and corrupted examples without the need to estimate noise rates. Instead of learning the noisy distributions noise rates, we focus on learning the underlying clean distribution and design a regularization term to help improve the confidence of the learned classifier, which is proven help safely sieve out corrupted examples. With the division between "clean" and "corrupted" examples, our training enjoys performance improvements by treating the clean examples (using standard loss) and the corrupted ones (using an unsupervised consistency loss) separately.
We summarize our main contributions: 1) We propose to train classifier using a novel confidence regularization (CR) term and theoretically guarantee that, under mild assumptions, minimizing the confidence regularized cross-entropy (CH) loss on the instance-based noisy distribution is equiva- lent to minimizing the pure CE loss on the corresponding "unobservable" clean distribution. This classifier is also shown to be helpful for evaluating each example to build our sample sieve.2) We provide theoretically sound sample sieve that simply compares the example's regularized loss with a closed-form threshold explicitly determined by predictions from the above trained model using confidence regularized loss, without any extra estimates. 3) To the best of our knowledge, the proposed CORES² studied for multi-class classification problem, has theoretical guarantees to avoid overfitting to instance-dependent label noise, and provides high-quality division without knowing or estimating noise rates. 4) By decoupling the regularized loss into separate additive terms, we also provide a novel and promising mechanism for understanding and controlling the effects of general instance- dependent label noise. 5) CORES achieves competitive performance on multiple datasets, includ- ing CIFAR-10, 10, CIFAR-100, and Clothing IM,
CORES" (COnfidence REgularized Sample Sieve) is the first method that is thoroughly
M,, under different label noise settings.
Other related works In addition to recent works by Xia et al. (020)) Berthon et al. (2020), and Cheng et al. 2020), we briefly overview other most relevant references. Detailed related work is left to Appendix A Making the loss function robust to label noise is important for building a robust machine learning model (Zhang et al. 2016). One popular direction to perform loss correction, which first estimates transition matrix (Patrini et al. 2017: Vahdat 2017 Xiao et al. 2015 Zhu jet al., 2021b; Yao et al. (020b), and then performs correction/reweightiim
via forward or back- ward propagation, or further revises the estimated transition matrix with controllable variations (Xia) jet al.] 2019). The other line of work focuses on designing specific losses without estimating transi- tion matrices (Natarajan et al. 2013 Xu et al. 2019: Liu & Guo 2020 Wei & Liu 2021). However, these works assume the label noise is instance-independent
which limits their extension. Another approach is sample selection (Jiang et al. 2017 Han et al. 2018 Yu et al. 2019 Northcutt al., 2019 Yao et al. 2020a Wei et al.] 2020 Zhang et al. (020a), which selects the "small loss" ex- amples as clean ones. However, we find this approach only works well on the instance-independenr label noise. Approaches such as label correction (Veit et al. 2017 Li et al. [017] Han et al.) (019)) or semi-supervised learning (Li et al. 2020 Nguyen et al. 2019) also lack guarantees for the instance- based label noise.
2 CORES CONFIDENCE REGULARIZER SAMPLE SIEVE
Consider a classification problem on a set of N training examples denoted by D {(Tn. Ine]) where [N] == (1, 2, .... N} is the set of example indices. Examples (In. yn) are drawn according to random variables (X, Y) E X x y from a joint distribution D. Let Dx and Dy be the marginal distributions of X and Y. The classification task aims to identify classifier f: X + Y that maps X to Y accurately. One common approach is minimizing the empirical risk using DNNs with respect the cross-entropy loss defined as {(f(x), y) - (n(S21))) y [K), where [zlv] denotes the y-th component of f(x) and K is the number of classes. In real-world appli- cations, such as human- -nnootated images (Krizhevsky al. 2012: Zhang al.) (017) and medical diagnosis (Agarwal et al., 2016), the learner can only observe set of noisy labels. For instance, human annotators may wrongly label some images containing cats as ones that contain dogs acciden- tally or irresponsibly. The label noise of each instance is characterized by noise transition matrix T(X), where each element Ti (X) := IPY j|Y i, X.. The corresponding noisy datase and distribution are denoted by D : (I (n)) InneN and D. Let 1(.) be the indicator function taking
In this paper, the noisy dataset refers to dataset with noisy examples. A noisy example is either clean example (whose label is true) a corrupted example (whose label is wrong).
2

---

Published as a conference paper at ICLR 2021
value 1 when the specified condition is satisfied and 0 otherwise. Similar to the goals in surrogate
loss (Natarajan et al. 2013), LDMI (Xu et al.) 2019) and peer loss (Liu & Guo 2020), we aim to learn a classifier f from the noisy distribution D which also minimizes (X) & (X, D. Beyond their results, we attempt propose a theoretically sound approach addressing a general castance-based noise regime without knowing or estimating noise rates.
2.1 CONFIDENCE REGULARIZATION
In this section, we present a new confidence regularizer (CR). Our design of the CR is mainly motivated by a recently proposed robust loss function called peer loss (Liu & Guo 2020). For each example (In, Gn). peer loss has the following form:
(pr.( ūn) == (z)a), B.) ((J((n)))) Uma);
where (Ina Vo,)) and (Ina Bng) are two randomly sampled and paired peer examples (with replace- ment) for n. Let X,, and Yng be the corresponding random variables. Note Xn, Yng are two independent and uniform random variables being each Uw,' n' E [N] and To', n' E [N] with prob- ability N respectively: P(X,,, Fn D) P(Yn, Yu (D) N Vn' [N]. Let Dj y be the distribution of Yng given dataset D. Peer loss then has the following equivalent form in expectation:
Xn, Kng. 18 1 (L((f( (((())))))))) Yn2)]
N
ME[M]
Σ
((f(ze), Gn) Σ
P(X anl (BlDi)BB 1l n'E(M]
N
ME(M
Σ nE[M]
(e(s(e)). Bw.) i [((f(zn), 811
N
This result characterizes a new loss denoted by lCA:
fex (f( == ((f(f))) - DD (((f(zn). F)).
(1)
Though not studied rigorously by Liu & Guo (022), we show, under condition lca defined in Eqn. 1 encourages confident predictions from by analyzing the gradients:
Theorem 1. For (-)(-), solutions satisfying fx.. [i] > 0, Vi E [K] are not locally optimal at (Ins Un)
See Appendix B.2 for the proof. Partieularly, in binary cases, we have constraint f(xx) [0] + f(x,)[L] = 1. Following Theorem 1 we know minimizing tcs (fx))) Tn) w.r. f under this con- straint leads to either [0x,)[[]] + or [(xx)[[] + I, indicating confident predictions. Therefore, the addition of term -Epy (z.,,, Y)] helps improve the confidence of the learned classifier. Inspired by the above observation, we define the following confidence regularizer:
Confidence Regularizer: (cR(f(xn) == -3- -3-Epp ([[(), Y)],
where 3 is positive and E(-) refers to the CE loss. The prior probability P( Đ) is counted directly from the noisy dataset. In the remaining of this paper, E(-) indicates the CE loss by default.
Why are confident predictions important? Intuitively, when model fits to the label noise, its pre- dictions often become less confident, since the noise usually corrupts the signal encoded in the clean data. From this perspective, encouraging confident predictions plays against fitting to label noise. Compared to iistance-independee
noise, the difficulties in estimating the instance-dependent noise rates largely prevent us from applying existing techniques. In addition, as shown by Manwani & Sastry (2013). the 0-1 loss function is more robust to instance-based noise but hard optimize with. To a certain degree, pushing confident predictions results in differentiable loss function that approximates the 0-1 loss, and therefore restores the robustness property. Besides, as observed by Chatterjee (2200) and Zielinski et al. (2020), gradients from similar examples would reinforce each other. When the overall label information is dominantly informative that (x(X) (X) DNNs
'Detilldd conditions for Theorem 1 are specified at the end of our main contents.
*Our observation can also help partially explain the robustness property of peer loss (Lu & Guo 2020
3

---

Published as a conference paper at ICLR 2021
will receive more correct information statistically. Encouraging confident predictions would discour- age the memorization of the noisy examples (makes hard for noisy labels to reduce the confidence of predictions), and therefore further facilitate DNNs to learn the (clean) dominant information.
fCR NOT the entropy regularization Entropy regularization (ER) is a popular choice for im- proving confidence of the trained classifiers in the literature (Tanaka al. 2018 Yi & Wu 2019). Given a particular prediction probability p for a class, the ER term is based on the function -pn
while our tCR is built on p. Later show fCR offers us favorable theoretical guarantees for training with instance-dependent label noise, while ER does not. In Appendix [C.T] we present both theoretical and experimental evidences that fCR serves as better regularizer compared ER.
2.2 CONFIDENCE REGULARIZED SAMPLE SIEVE
Intuitively, label noise misleads the training thus sieving corrupted examples out datasets is ben- eficial. Furthermore, label noise introduces high variance during training even with the existence of fcR (discussed in Section 3.3). Therefore, rather than accomplishing training solely with fCR, we will first leverage its regularization power to design an efficient sample sieve. Similar to general sieving process physical words that compares the size of particles with the aperture of sieve, evaluate the "size" (quality,
regularized loss) of examples and compare them with some to-be-specified thresholds, therefore the name sample sieve. our formulation, the regularized loss {(f(a, 9n) + (CR( (xx)) ) is employed evaluate examples and a, is used to specify thresholds. Specifically, we aim to solve the sample sieve problem in
Confidence Regularized Sample Sieve
Sample Sieve-O
min wc (0.1)) ME[N]
Σ
Un [e(f(xn), + fca(f(zn)) anl
F1,0
02.3
lex (ex(f(xn) (xx)) == -3 -D El {(f(zx), Y),
Sample Sieve-1
1
Σ
elf(zn) 0) + ten (ex(f((x)) (xx)))
(h.3) =) 1h.3 =>?
= K
Figure Dynamic sample sieves. Green circles are clean examples. Red hexagons are corrupted examples.
K)
(2)
The crucial components in
en E {0, indicates whether example is clean (u,, 1) not (v,, = 0);
(mimicking the aperture of sieve) controls which example should be sieved out;
f a copy of f and does not contribute to the pack-propagation. F the search space f.
Dynamic sample sieve The problem (2) is a combinatorial optimization which is hard to solve directly. A standard solution to is to apply alternate search iteratively as follows:
Starting at = 1, (0) = 1, Vn E [N].
Confidence regularriee model update (at iteration-f):
(()) =arg min feF
Σ
(1-1) [((((=)) fcR (cx()(x))))
en
RE[N]
Sample sieve (at iteration-t):
(0 (((()(((x))))))))))) Tn) (cR(F(( < an.))
(4)
where an,: = ESeen (xF() + lcR( (x,)), f()) and ((0) refer to the specific
classifier and weight at iteration-l. Note the values of fcR (xx)) and (f ( (xr,,, ))
the same. We keep both terms to be consistent with the objective in Eq. (2). In DNNs, we usually update model with one several epochs of data instead of completely solving
Figure | illussrrtes the dynamic sample sieve, where the size of each example corresponds to the regularized loss and the aperture of sieve is determined by an In each iteration-l, sample sieve-
4

---

Published as a conference paper at ICLR 2021
Chis
()) CE Siore (oyme, spoch 20)
(st CE Siove Eoymm.. spach 70)
(3 CORES (ymm, spóci 20)
(3) COREs" lottn, apach 701
10
65.
-10
16
60
-15
-10
10
35
66
(e] Siece (ina,, spach
(0 CE Since [ine...
1g1 CORES" (nntt. opoch 20)
(h) CORES" cpoch 70)
Figure 2: Loss distributions of training on CIFAR-I0 10 with 40% symmetric noise (symm.) or 40%
instance-based noise (inst.). The loss is given by E (x,,) (.n.).))) Gn) + (cu(f()) (z,)) an,t
4 CE
Sieve represents the dynamic sample sieve with standard cross-entropy loss (without CR).
1. "blocks" some corrupted examples by comparing a regularized example loss with a closed-form threshold Ct,,t, which can be immediately obtained given current model f(() and example (Ixx, Mr. (no extra estimation needed). In contrast, most sample selection works (Han al.) 2018 Yu al. 2019 Wei et al. 2020) focus on controlling the number of the selected examples using an intuitive function where the overall noise rate may required, or directly selecting examples by an empiri- cally set threshold (Zhang & Sabuncu 2018). Intuitively, the specially designed thresholds On, for each example should be more accurate than single threshold for the whole dataset. Besides, goal of existing works is often to select clean examples while our sample sieve focuses on removing the corrupted ones. On a high level, we follow a different philosophy from these sample selection works. We coin solution as COnfidence REgularized Sample Sieve (CORES?).
More visualizations of the sample sieve In addition to Figure [ we visualize the superiority of our sample sieve with numerical results Figure 2 The sieved dataset is in the form of two clusters of examples. Particularly, from Figure and Figure ft), we observe that CE suffers from providing good division of clean and corrupted examples due overfitting in the final stage of training. On the other hand, with &CR, there are two distinct clusters and can be separated by the threshold 0 as shown in Figure d) and Figure hh.. Comparing Figure 2 [a) 2 with Figure [[e) we find
the effect of mestance-dependent noise on training is indeed different from the symmetric one, where the instance-dene -eeendeed noise more likely to cause overfitting.
3 THEORETICAL GUARANTEES OF CORES
In this section, we theoretically show the advantages of CORES². The analyses focus on showing
CORES² guarantees quality division, i.e. Y, = (3,,, = Fn,), Vn, with properly set B. To show the effectiveness of this solution, we call model prediction on In is better than random guess Zz, [ym] 1/K, and call it confident if fa., [v] (0. 1}, Vy E [K], where Yn is the clean label and y is an arbitrary label. The quality of sieving out corrupted examples is guaranteed in Theorem 2
Theorem 2. The sample sieve defined in (4) ensures that clean examples (In, Un yn) will not be identified as being corrupted if the model f(()) 's prediction on F, is better than random guess.
Theorem 2 informs us that our sample sieve can progressively and safely filter out corrupted exam- ples, and therefore improves division quality, when the model prediction each F, is better than random guess. The full proof is left to Appendix [B.3] 3] In the next section, we provide evidences that trained model is guaranteed achieve this requirement with sufficient examples.
3.1 DECOUPLING THE CONFIDENCE REGULARIZED LOss
The discussion of performance guarantees the sample sieve focuses on a general instance-based noise transition matrix 7 T(X), which can induce any specific noise regime such as symmetric noise and asymmetric noise (Kim et al.) 2019 Li et al. 2020). Note the feature- ceauure-indeeendency was one critical assumption in state-of ofate-of-the-ar
the-art art theoretically guaranteed noise-resistant literatures Natarajan et al. 2013 Liu & Guo 2020 Xu et al. 2011)) while we do not require. Let Tij == E Vi, J E [K]. Theorem explicitly shows the contributions of clean examples,
corrupted examples, and fcR during training. See Appendix B. for the proof.
Theorem 3. (Main Theorem: Decoupling the Expected Regularized CE Loss) In expectation, the loss with toR can be decoupled three separate additive terms:
5

---

Published as a conference paper at ICLR 2021
Term-1
Term-
B5 + -T-Eo EDA EDA((((()),)))
Σ P(Y ( - -))(((())))))))
(5)
JE[K] SE[K]
Term-3
where T: = min T35 B:= A, = j); A 3= T3p - (x) j. ;x, (X) Tj., and ED ED (T] == 1(A 0) )} 3, 1
Equation (5) provides a generic machinery for anatomizing noisy datasets, where we show the ef- fects of instance-based label noise the lcR regularized loss be decoupled into three additive terms: Term-1 reflects the expectation of CE on clean distribution D, Term-2 shifts the clean distri- bution by changing the prior probability y and Term-3 characterizes how the corrupted examples (represented by Uzg( (X)) might misleah/mii-weightt the loss, as well as the regularization ability of leR (represented by SIP(Y -)). In addition to the design of sample sieve, this additive decou- pling structure also provides a novel and promising perspective for understanding and controlling the effects of generic instance-dependent label noise.
3.2 GUARANTEES OF THE SAMPLE SIEVE
By decoupling the effects of instance-dependent noise into separate additive terms as shown in Theorem 3 we can further study under what conditions, minimizing the confidence regularized CE loss on the (instance-deepnneent) noisy distribution will be equivalent to minimizing the true loss incurred on the clean distribution, which is exactly encoded by Term-1. In other words, we would like to understand when Term-2 and Term-3 in (5) can be controlled not to disrupt the minimization of Term-1. Our next main result establishes this guarantee but will first need the following two assumptions.
Y) Clean labels are Bayes optimal (Y" == arg [ P(Y = ilX)). Assumption 2. (Informative datasets) The noise rate is bounded as Tú (X) (x)(X) 0, Vi E [K],E E [],j i, Dx-
Assumption 1. (Y"=
Feasibility of assumptions: 1) Note for many popular image datasets, e.g. CIFAR, the label of each feature is well-defined and the corresponding distribution is well-separated by human annotation. In this case, each feature X only belongs to one particular class Y. Thus Assumption T is generally held in classification problems (Liu & Tao 2015). Technically, this assumption could be relaxed. We use this assumption for clean presentations. 2) Assumption 2 shows the requirement of noise rates, i.e., for any feature X, a sufficient number of clean examples are necessary for dominant clean information. For example, we require T (X) (X) 0 to ensure examples from class i are informative (Luu & Chen 2017).
Before formally presenting the noise-resistant property of training with fCR, we discuss intuitions here. As discussed earlier in Section 2.1 our tCR regularizes the CE loss to generate/incentivize confident prediction, and thus is able to approximate the 0-1 loss to obtain its robustness property. More explicitly, from 5 fcR affects Term-3 with a scale parameter 3. Recall that Ux)(X) = (X), Vi # j. which exactly the noise transition matrix. Although we have no information about this transition matrix, the confusion brought by (X) can be canceled or reversed by sufficiently large 3 such that (X) :PP( j) < 0. Intuitively, with an appropriate B, all the effects of U2, ( + j can be reversed, and we will get negative loss punishing the classifier for predicting class-j when the clean label is i. Formally, Theorem shows the noise-resistant property
of training with fcR and is proved in Appendix [B.4]
Theorem 4. (Robustness of the Confidence Regularized CE Loss) With Assumption 7 and 2
when P(Ý =) i) - -P(Y =))
(X)
Tx(X)= - (X)
max [JE[K]X>Dx
P(Y 1) <BS
min R
minimizing ()(,) + (x))) is equivalent to minimizing (T)( Y)).
Theorem shows a sufficient condition of 3 for our confidence regularized CE loss to be robust to instance-dependent label noise. The bound on LHS ensures the confusion from label noise could be

---

Published as a conference paper at ICLR 2021
canceled or reversed by the 3 weighted confidence regularizer, and the RHS bound guarantees the model with the minimized regularized loss predicts the most frequent label in each feature w.p. 1.
Theorem 4 alsoo provides guidelines for tuning 3. Although we have no knowledge about Tip (X), we can roughly estimate the range of possible B. One possibly good setting of is linearly increasing with the number of classes, e.g. 3 2 for 10 classes and = 20 for 100 classes.
With infinite model capacity, minimizing Ez (X() Y)] returns the Bayes optimal classifier (since CE is a calibrated loss) which predicts each 2,, better than random guess. Therefore, with suf- ficient number of examples, minimizing E (((((() Y)+ predicts better than random guess, then satisfying the condition required in Theorem 2 to guarantee the quality of sieved examples. Further, since the Bayes optimal classifier always predicts clean la- bels confidently when Assumption hholds, Theorem alsoo guarantees confident predictions. With such predictions, the sample sieve in 4 will achieve 100% precision on both clean and corrupted examples. This guaranteed division is summarized in Corollary[] [
(f()) will also return model that
When conditions in Sholremm@hold, with infinite model capacity and sufficiently many examples, CORES achieves Un = [ynn Un), Vn E [N), i.e. all the sieved clean examples are effectively clean.
Corollary 1.
3.3 TRAINING WITH SIEVED SAMPLES
We discuss the necessity of a dynamic sample sieve in this subsection. Despite the strong guaran- tee in expectation as shown Theorem + performing direct Empirical Risk Minimization (ERM) of the regularized loss is likely to return a sub-optimal solution. Although Theorem 4 guarantees the equivalence of minimizing two first-order statistics, their second-order statistics are also important for estimating the expectation when examples are finite. Intuitively, Term-1 T ED (X([(X())))))))))))))(())) Y) primarily helps distinguish good classifier from a bad one on the clean distribution. The existence of the leading constant T reduces the power of the above discrimination, as effectively the gap be- tween the expected losses become smaller as noise increases (T will decrease). Therefore we would require more examples to recognize the better model. Equivalently, the variance of the selection becomes larger. In Appendix C2.2 we also offer an explanation from the variance's 's perspective. For some instances with extreme label noise, the B satisfying Eqn. (6) in Theorem may not exist. In such case, these instances cannot be properly used and other auxiliary techniques are necessary (e.g., sample pruning).
Sieving out the corrupted examples from the clean ones allows us a couple of better solutions. First, we can focus on performing ERM using these sieved clean examples only. We derive the risk bound for training with these clean examples in Appendix [C.3] Secondly, leveraging the sample sieve distinguish clean examples from corrupted ones provides a flexible interface for various robust training techniques such that the performance can be further improved. For example, semi- supervised learning techniques can be applied (see section for more details).
4 EXPERIMENTS
Now we present experimental evidences of how CORES² works.
Datasets: 2 is evaluated three benchmark datasets: CIFAR-10, 10, CIFAR- 100 (Krizhevsky eet al. 2009) and Clothing IM (Xiao et al., 2015). Following the convention from Xu et al. (2019). we use ResNet34 for CIFAR-10 and CIFAR- 100 and ResNet50 for Clothing IM.
We experiment with three types of label noise: symmetric, asymmetric and instance- dependent label noise. Symmetric noise generated by randomly flipping a true label to the other possible labels w.p. E (Kim et al., 2019). where is called the noise rate. Asymmetric noise is generated by flipping the true label to the next class (i.e., label i+1, mod K) w.p. E. Instance- dependent label noise is a more challenging setting and we generate instance-dependen label noise following the method from Xia et al.] (2020) (See Appendix [D.3 for details). In expectation, the noise rate for all noise regimes is the overall ratio of corrupted examples in the whole dataset.
Noise type:
Consistency training after the sample sieve: Let T be the last iteration of CORES² Define L(r) == {n/n E (f 1}. H() = {n/r E [N], of = 0}. DLC)) == {(En, Dn) n E
'The logarithmic function in fcR is adapted (n( + 10
10-5) -) for numerical stability.
7

---

Published as a conference paper at ICLR 2021
1.0
0.0
0.8
0.8
0.4
0.4
0.4
0.0
20
40
60
80
100
0.0
20
40
60
80
100
0.0
20
40
60
80
100
(a) 20% Symm.
(b) 40% Symm.
(c) 60% Symm.
1.0
0.8
06-
0.4
CORES?
Conterrtnng
04
0.4
Codtartinng
02
0.0
20
40
0.0
600
100
0.0
20
60
100
(d) 20% Inst.
(e) 40% Inst.
60% Inst.
Figure 3: F-score comparisons on CIFARIO under symmetric (Symm.) and instance- (Inst.)
label noise. F-score: = Re where Pre: EnEIN 10,=1,y, =m)) and == ΣnEIni (u., - =..)
Pre+Re
Zne[N]] I(0==1)
2ne[N] [ (Wn=Vn) Vnl
L(T)), DH(-) == (ann E H(r)}. Thus Di(F) is sieved clean examples and D(())) is fil- tered out as corrupted ones. Examples (Ins Fn.) E DL(=) lead the training direction using the CE loss as EnELO) {((xx,) (n). Noting the labels in DH(()) are supposed to be corrupted and can distract the training. we simply drop them. On the other hand, feature information of these examples en- codes useful information that we can further leverage to improve the generalization ability of models. There are different ways to use this unsupervised information, in this paper, we chose to minimize the KL-divergence between predictions on the original feature and the augmented feature to make predictions consistent. This is a common option as chosen by [i et al. (2019). Xie et al. (2019), and Zhang et al. (020b). The consistency loss function in epoch-t is >n H(r) (KL() (xn,)), where f( is copy of the DNN at the beginning of epoch-t but without gradients. Summing the classification and consistency loss yields the total loss. See Appendix [ for an illustration.
Checking the consistency of noisy predictions is only one possible way to leverage the additional information after sample sieves. Our basic idea of first sieving the dataset and then treating corrupted examples differently from clean ones admits other alternatives. There are many other possible designs after sample sieves, e.g., estimating transition matrix using sieved examples then applying loss-correction (Patrini al. 2017: Vahdat Vahdat[2017] [Xioo al.|| 2015). making the consistency loss as another regularization term and retraining the model (Zhang et al. 2020b). correcting the sample selection bias in clean examples and retraining (Cheng et al. 2020 Fang et al.) 2020), or relabeling those corrupted examples and retraining, etc. Additionally, clustering methods on the feature space (Han et al., 2019; Luo et al. 2020) or high-order information (Zhu et al. 202 can also be exploited along with the dynamic sample sieve. Besides, the current structure is ready to include other techniques such as mixup (Zhang et al., 2018).
Other alternatives:
Quality of our sample sieve: Figure shows the F-scores of sieved clean examples with training epochs on the symmetric and the instance-based label noise. F-score quantifies the quality of the sample sieve by the harmonic mean of precision (ratio of actual cleans examples in sieved clean ones) and recall (ratio of sieved cleans examples in actual clean ones). We compare CORES² with Co-teaching and Co-teaching+. Note the F-scores of CORES" and Co-teaching are consistently high on the symmetric noise, while 2 achieves higher performance on the challenging instance- based label noise, especially with the 60% noise rate where the other two methods have low F-scores.
Experiments on CIFAR-10, CIFAR-100 and Clothing 1M: In this section, we compare 2 with several state-of-the-ar methods on CIFAR-10 and CIFAR- 100 under instance-based. symmet- ric and asymmetric label noise settings, which is shown on Table I and Table 2 CORES denotes that we apply consistency training on the corrupted examples after the sample sieve. For a fair com- parison, all the methods use ResNet-34 as the backbone. By comparing the performance of CE on the symmetric and the instance-based label noise, we note the instance-based label noise a more challenging setting. Even though some methods (e.g., LDMI) behaves well on symmetric and asym- metrie label noise, they may reach low test accuracies on the instance- label noise, especially when the noise rate is high the dataset is more complex. However, CORES² consistently works well on the instance- based label noise and adding the consistency training gets better results. Ta- ble 3 verifies CORES² on Clothing M, a dataset with real human label noise. Compared to the other
8

---

Published as a conference paper at ICLR 2021
Table Comparison of test accuracies on clean datasets under instance-bart based label noise.
CIFARTO
Jnst. CIFARTOO
Method
0.4
0.2
0.4
0.6
Cross Entropy
87.16
73.16
44.64
38.72
41.14
25.29
Forward T (Patrani al. 2017
88 $8.08
82.67
41.57
58.95
41.68
22.83
LoMi Xu al
41.77
big (hang & Simmmel 2018
86.45
69.02
32.94
58.18
40.32
23.13
SCE (Wang el ale 2019
89.11
72.04
44.83
59.87
41.76
23.41
Co-teaching
88.66
69.50
34.61
43.03
23.13
7.07
Co-teaching-
Tü ct 8l 2019
89.04
69.15
33.33
41.84
24.40
8.74
JoCoR Wei
Peer Loss Lau & Gio 2020
89.13
81.09
73.73
59.92
45.76
33.61
CORES
89.50
82.84
79.66
61.25
47.81
37.85
CORES
Table 2: Comparison test accuracies on clean datasets under symmerric/asymmetric label noise.
Method
Symm.
CIFARIO
Axymm, CIFRRI
Synom. CIFARIO0
Asymm. CIFARTOO
0.4
0.6
0.2
0.4
c= 0.2
0.3
Cross Enjropy
81.88
74.14
68.39
86.14
20
37.41
59.20
31.40
MAE
Ghush et al. 2017
61.63
41.98
59.67
57.62
7.68
6.45
11.16
8.97
Forward
al
41.59
64.72
Lu
Zhang
Subuncu
87.13
82.54
89.33
85.45
61.77
53.16
66.59
61.45
LoMi
al 2019
83.04
76.51
89.04
87.88
52.32
40.00
60.04
$2.82
NINI
2019
92.43
88.32
93.35
91.80
66.39
56.51
63.12
54.87
SELF
(Nguyen et al. 2019
91.13
93.75
92.42
66.71
70.53
65.09
CORES
93.76
95.18
94.67
72.22
59.16
75.19
73.81
Table 3: The best epoch (clean) test accuracy for each method on Clothing IM.
Method
CE (Baseline)
teaching
JoCoR
LoMi
PVD
CORES
Parrini et al. 2017]
(Han et al. 2018
Wei et al. 2020
(Xu al. 2019
(Nia et al 2020
(our)
Acc.
68.94
70.83
69.21
70.30
72.46
71.67
73.24
approaches, CORES also works fairly well the Clothing IM dataset. See more experiments in We also provide source codes with detailed instructions in supplementary materials.
Appendix D
5 CONCLUSIONS
This paper introduces CORES", a sample sieve that guaranteed to robust to general instance- dependent label noise and sieve corrupted examples, but without using explicit knowledge of the noise rates of labels. The analysis of CORES assumed that the Bayes optimal labels are the same clean labels. Future directions of this work include extensions more general cases where the Bayes optimal labels may differ from clean labels. We are also interested in exploring different possible designs of robust training with sieved examples.
This work is partially supported by the National Science Foundation (NSF) under grant IIS- 2007951 and Office Naval Research under grant N00014-20-1-22.
Acknowledgement
CONDITIONS REQUIRED FOR THEOREM
Theorem holds based on the following three assumptions:
A1. The model capacity is infinite (i.e., it realize arbitrary variation).
A2. The model is updated using the gradient descent algorithm (i.e. updates follow the direction decreasing Ep [X( Y)] - ED, [Exx (f)), Y)D).
A3. The derivative of network function @f(m:)) is smooth (i.e. the network function has singular point), where w,'s 's model parameters.
tws

---

Published as a conference paper at ICLR 2021
REFERENCES
Vibhu Agarwal, Tanya Podchiyska, Juan M Banda, Veena Goel, Tiffany Leung. Evan P Minty, Timothy Sweeney, Elsie Gyang, and Nigam Shah. Learning statistical models of phenotypes using noisy labeled training data. Journal of the American Medical Informatics Association, 23 16):1166-117,, 2016.
Amr M. Alexandari, Anshul Kundaje, and Avanti Shrikumar. corrected calibration hard-to-beat at label shift adaptation. national Conference Machine Learning, ICML '20, 2020.
Maximum likelihood with bias- Proceedings of the 37th Inter-
Ehsan Amid, Manfred KK Warmuth, Rohan Anil, and Tomer Koren. Robust bi-tempered logistic loss based bregman divergences. In Advances in Neural Information Processing Systems, pp. 14987-14996, 2019.
Eric Arazo, Diego Ortego, Paul Albert, Noel E O'Connor, and Kevin McGuinness. Unsupervised label noise modeling and loss correction. arXiv preprint arXiv:1904.11233 2019.
Peter L Bartlett and Shahar Mendelson. Rademacher and gaussian complexities: Risk bounds and structural results. Journal of Machine Learning Research, 3(Nov):4633482, 2002.
Antonin Berthon, Bo Han, Gang Niu, Tongliang Liu, and Masashi Sugiyama. Confidence scores
make instance-dependen label-noise learning possible. arXiv preprint arXiv:2001.03777, 2020.
Satrajit Chatterjee. Coherent gradients: An approach to understanding generalization gradient descent-based optimization. In International Conference on Learning Representations, 2020.
Jiacheng Cheng, Tongliang Liu, Kotagiri Ramamohanarao, and Dacheng Tao. Learning with
bounded instance-and lahel-dependent label noise. In Proceedings of the 37th International Con- ference on Machine Learning, ICML. '20, 2020.
Tongtong Fang, Nan Lu, Gang Niu, and Masashi Sugiyama. Rethinking importance weighting for deep learning under distribution shift. arXiv preprint arXiv:2006.04662. 2020.
Aritra Ghosh, Himanshu Kumar, and PS Sastry. Robust loss functions under label noise for deep neural networks. In Thirty-First AAAI Conference Artificial Intelligence, 2017.
Maoguo Gong, Hao Li, Deyu Meng, Qiguang Miao, and Jia Liu. Decomposition- based evolutionary multiobjective optimization self-paced learning. IEEE Transactions Evolutionary Compu- tation, 23(2):288-33) 2018.
Bo Han, Quanming Yao, Xingrui Yu, Gang Niu, Miao Xu, Weihua Hu, Ivor Tsang, and Masashi Sugiyama. Co-teaching: Robust training of deep neural networks with extremely noisy labels. In Advances in neural information processing systems, pp.
2018.
Jiangfan Han, Ping Luo, and Xiaogang Wang. Deep self-learning from noisy labels. In Proceedings of the IEEE International Conference on Computer Vision, pp. 5138-5147, 2019.
Jiayuan Huang, Arthur Gretton, Karsten Borgwardt, Bernhard Schölkopf, and Alex J Smola. Cor- recting sample selection bias by unlabeled data. In Advances in neural information processing systems, pp. 601-608 2007.
Lu Jiang, Zhengyuan Zhou, Thomas Leung, Li-Jia Li, and Li Fei-Fei. Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels. arXiv preprint arXiv:1712.05005. 2017.
Youngdong Kim, Junho Yim, Juseung Yun, and Junmo Kim. Nini: Negative learning for noisy labels. In Proceedings of the IEEE International Conference on Computer Vision, pp. 101-110, 2019.
Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. Technical report, Citeseer, 2009.
10

---

Published as a conference paper at ICLR 2021
Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convo- lutional neural networks. In Advances in neural information processing systems, pp. 1097-1109 2012.
Junnan Li, Yongkang Wong, Qi Zhao, and Mohan S Kankanhalli. Learning learn from noisy la- beled data. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 50555505155 2019.
Junnan Li, Richard Socher, and Steven C.H. Hoi. Dividemix: Learning with noisy labels as semi- supervised learning. In International Conference on Learning Representations, 2020. URL
https: www.reviewwwweeeeeeoppep net/ www.mmiddruuuaaaaaaa
Yuncheng Li, Jianchao Yang, Yale Song, Liangliang Cao, Jiebo Luo, and Li-Jia Li. Learning from noisy labels with distillation. In Proceedings of the IEEE International Conference on Computer Vision, pp. 1910-1918, 2017.
Tongliang Liu and Dacheng Tao. Classification with noisy labels by importance reweighting. IEEE Transactions on pattern analysis and machine intelligence, 38(3):447-461, 2015.
Yang Liu. The importance of understanding instance- noisy labels, 2021.
Yang Liu and Yiling Chen. Machine-learning aided peer prediction. In Proceedings of the 2017 ACM Conference Economics and Computation, pp. 63-80, 2017.
Yang Liu and Hongyi Guo. Peer loss functions: Learning from noisy labels without knowing noise rates. In Proceedings of the 37th International Conference on Machine Learning. ICML '20, 2020.
Yijing Luo, Bo Han, and Chen Gong. A bi-level formulation for label noise learning with spectral cluster discovery. In International Joint Conference on Artificial Intelligence, pp. (605526611, 2020.
Naresh Manwani and PS Sastry. Noise tolerance under risk minimization. IEEE transactions cybernetics, 43(3): 14411151111111 2013.
Nagarajan Natarajan, Inderjit S Dhillon, Pradeep K Ravikumar, and Ambuj Tewari. Learning with noisy labels. In Advances in neural information processing systems, pp. 1196-1204, 2013.
Duc Tam Nguyen, Chaithanya Kumar Mummadi, Thi Phuong Nhung Ngo. Thi Hoai Phuong Nguyen, Laura Beggel, and Thomas Brox. Self: Learning to filter noisy labels with self- ensembling. arXiv preprint arXiv: 1910.0182... 2019.
Curtis G Northcutt, Lu Jiang, and Isaac L Chuang. Confident learning: Estimating uncertainty in dataset labels. arXiv preprint arXiv: 1911:/99100006 2019.
Giorgio Patrini, Alessandro Rozza, Aditya Krishna Menon, Richard Nock, and Lizhen Qu. Making deep neural networks robust to label noise: A loss correction approach. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1944-1952, 2017.
Scott Reed, Honglak Lee, Dragomir Anguelov, Christian Szegedy, Dumitru Erhan, and Andrew Rabinovich. Training deep neural networks noisy labels with bootstrapping. arXiv preprint arXiv: 1422,556, 2014.
Jun Shu, Qian Zhao, Keyu Chen, Zongben Xu, and Deyu Meng. Learning adaptive loss for robust learning with noisy labels. arXiv preprint arXiv:2002.0642 06482, 2020.
Amos Storkey. When training and test sets different: characterizing learning transfer. Dataset shift in machine learning. pp. 3-28, 2009.
Daiki Tanaka, Daiki Ikami, Toshihiko Yamasaki, and Kiyoharu Aizawa. Joint optimization frame- work for learning with noisy labels. In Proceedings of the IEEE Conference Computer Vision and Pattern Recognition, pp. 5552-5560, 2018
Arash Vahdat. Toward robustness against label noise in training deep discriminative neural networks. In Advances in Neural Information Processing Systems, pp. 5596-56066 2017.
11

---

Published as a conference paper at ICLR 2021
Andreas Veit, Neil Alldrin, Gal Chechik, Ivan Krasin, Abhinav Gupta, and Serge Belongie. Learning from noisy large-scale scale datasets with minimal supervision. In Proceedings of the IEEE Conference Computer Vision and Pattern Recognition, pp. 839-847, 2017.
Yisen Wang, Xingjun Ma, Zaiyi Chen, Yuan Luo, Jinfeng Yi, and James Bailey. Symmetric cross en- tropy for robust learning with noisy labels. In Proceedings of the IEEE International Conference on Computer Vision, pp. 322-330, 2019.
Hongxin Wei, Lei Feng, Xiangyu Chen, and Bo An. Combating noisy labels by agreement: A joint training method with co-regularization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 13726-13735, 2020.
Jiaheng Wei and Yang Liu. When optimizing SfS-divergence is robust with label noise. In Interna- tional Conference on Learning Representations, 2021. URL https://openrevieew net/
Xiaobo Xia, Tongliang Liu, Nannan Wang, Bo Han, Chen Gong, Gang Niu, and Masashi Sugiyama. Are anchor points really indispensable in label-noise learning? In Advances in Neural Information Processing Systems, pp. 6838-6849, 2019.
Xiaobo Xia, Tongliang Liu, Bo Han, Nannan Wang, Mingming Gong, Haifeng Liu, Gang Niu, Dacheng Tao, and Masashi Sugiyama. Parts-dependent label noise: Towards instance-dependent label noise. arXiv preprint arXiv:2006.07836 2020.
Tong Xiao, Tian Xia, Yi Yang, Chang Huang, and Xiaogang Wang. Learning from massive noisy labeled data for image classification. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2691-2699 2015.
Qizhe Xie, Zihang Dai, Eduard Hovy, Minh- Kinh-Thang Luong, and Quoe Le. Unsupervised data augmentation. arXiv preprint Mixiv:/904. 12848, 2019.
Yilun Xu, Peng Cao, Yuqing Kong, and Yizhou Wang. L dmi: An information-theorrtic noise- robust loss function. NeurIPS, 1909.03909 2019.
Quanming Yao, Hansi Yang, Bo Han, Gang Niu, and James T Kwok. Searching to exploit memo- rization effect in learning with noisy labels. In Proceedings of the 37th International Conference on Machine Learning, ICML '20, 2020a.
Yu Yao, Tongliang Liu, Bo Han, Mingming Gong. Jiankang Deng, Gang Niu, and Masashi Sugiyama. Dual T: Reducing estimation error for transition matrix in label-noise learning. In Advances in Neural Information Processing Systems, volume 33, pp. 7260-7271, 2020b.
Kun Yi and Jianxin Wu. Probabilistic end-to-end noise correction for learning with noisy labels. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 7017- 7025, 2019.
Xingrui Yu, Bo Han, Jiangchao Yao, Gang Niu, Ivor W Tsang, and Masashi Sugiyama. How does disagreement help generalization against label corruption? arXiv preprint 190100/90.. 2019.
Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint 1eXiv:1611.03522 2016.
Hongyi Zhang, Moustapha Cisse, Yann N. Dauphin, and David Lopez-Paz. mixup: Beyond em- pirical risk minimization. In International Conference on Learning Representations, 2018. URL https:://penrree net
Rb
Jing Zhang, Victor S Sheng, Tao Li, and Xindong Wu. Improving crowdsourced label quality using noise correction. IEEE transactions on neural networks and learning systems, 2965):1655-1688 2017.
Xuchao Zhang, Xian Wu, Fanglan Chen, Liang Zhao, and Chang-Tien Lu. Self-paced robust learn- ing for leveraging clean labels in noisy data. In AAAI, pp. (863- 2020a.
12

---

Published as a conference paper at ICLR 2021
Zhilu Zhang and Mert Sabuncu. Generalized cross entropy loss for training deep neural networks with noisy labels. In Advances in neural information processing systems. pp. 8778-878888 2018.
Zizhao Zhang, Han Zhang, Sercan O Arik, Honglak Lee, and Tomas Pfister. Distilling effective supervision from severe label noise. In Proceedings of the IEEE/CVF Conference on Computer Vision and Puttern Recognition, pp. 9294-9303, 2020b.
Zhaowei Zhu, Tongliang Liu, and Yang Liu. A second-order approach to learning with instance- dependent label noise. The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 202
Zhaowei Zhu, Yiwen Song, and Yang Liu. Clusterability alternatve anchor points when learning with noisy labels. arXiv preprint arXiv:2/02.05291, 2021b.
Piotr Zielinski, Shankar Krishnan, and Satrajit Chatterjee. Explaining memorization and general- ization: large-scale study with coherent gradients. arXiv preprint arXiv:2003.07422, 2020.
13

---

Published as a conference paper at ICLR 2021
APPENDIX
The appendices are organized as follows. Section A presents the full version of related works. Section B details the proofs for our theorems. Section supplements other necessary evidences to justify CORES?. Section showws more experimental details and results.
A FULL VERSION OF RELATED WORKS
Learning with noisy labels has observed exponentially growing interests. Since the traditional cross- entropy (CE) loss has been proved easily overfit noisy labels (Zhang et al., 2016), researchers try to design different loss functions to handle this problem. There were two main perspectives on designing loss functions. Considering the fact that outputs of logarithm functions in the CE loss grow explosively when the prediction / (x) approaches zero, some researchers tried to design bounded loss functions (Amid et al., 2019; Wang et al. 2019 Gong et al. 2018 Ghosh et 2017). To avoid in loss functions, a meta-lcant meea-learning method was proposed bt Shu al. (2020) to combine the above four loss functions together. However, simply considering loss function values without discussing the noise type and the corresponding statistics could not be noise-tolerant as defined by Manwani & Sastry (2013). As a complementary, others started from noise types and tried to design noise-tolerant loss functions. Based the assumption that label noise only depends on the true class (a.k.a. feature- indeeendent or label-dependent, an unbiased loss function called surrogate loss (Nataraaan et al. 2013), an information bassatdoo---edddddd loss function called LDMI (Xu et al.] 2219),, and a new family of loss functions to punish agreements between classifiers and noisy datasets called peer loss (Liu & Guo 2020) were proposed. They proved theoretically that training DNNs using their loss functions on feature-independent noisy datasets was equivalent training CE on the corresponding unobservable clean datasets. However, surrogate loss focused on the binary classifications and required knowing noise rates. LDMI and peer loss does not require knowing noise rates while LDMI may not be easy for extension and multi-class classification of peer loss requires particular transition matrices.
relying on fine-tuning of byper-parametee
The correction approach is also popular in handling label noise. Previous works (Patrini al. 2017; Vahdat 2017, Xiao et al., (015)) assumed the feature- nnature-ndependent noise transition matrix was given or could be estimated and attempted to use to correct loss functions. For example, Patrini et al. (017)) first estimated the noise transition matrix and then relied on it to correct forward or backward propagation during training. However, without a set clean examples, the noise transition matrix could be hard to estimate correctly, Instead of correcting loss functions, some methods directly corrected labels (Veit et al. 2017] Li et al. 2017 Han et al. 2019), whereas it might introduce extra noise and damage useful information. Recent works (Xia al.] 2020 Berthon et al., (2020) extended loss-correction from the limited ceatrre-ideeennden label noise to part-dependent or a more general instance-dependent noise regime while they relied heavily the noise rate estimation.
Sample selection (Jiang et al.) 2017 Han et al.) 2018 Yu et al. 2019 Yao et al., 2020a Wei et al.) 2020) mainly focused exploiting the memorization of DNNs and treating the "small loss" amples as clean ones, while they only focused on feature-independen label noise. Cheng et al. (2020) tried distill some examples relying on the predictions using the surrogate loss function Natarajan et al.) 2013). Note estimating noise rates are necessary for both applying surrogate loss and determining the threshold for distillation. The sample selection methods could be implemented with some semi-supervised learning techniques to improve the performance, where the corrupted examples were treated as unlabeled data (Li al. 20.0 Nguyen et al., 2019). However, the training mechanisms of these methods were still based on the CE loss, which could not be guaranteed to avoid overfitting to label noise.
B PROOF FOR THEOREMS
In this section, we firstly present the proof for Theorem 3 (our main theorem) in Section B.1 which provides a generic machinery for anatomizing noisy datasets. Then will respectively prove The- orem [ in Section B.2] Theorem in Section B.3 and Theorem in Section B.4 according to the order they appear.
14

---

Published as a conference paper at ICLR 2021
B.1 PROOF FOR THEOREM
Theorem 3. (Main Theorem: Decoupling the Expected Regularized CE Loss) In expectation, the
loss with leu can be decoupled as three separate additive terms:
Term-1
Term
B ((((X))))) (cx
(cx(((())) T.E.E-( V) A ED (ox((((()))))
Σ Σ
P(Y D) BPP( ())((()))))).
JE[K] [E[K]
Term-3
where T minze[a] Tis, A EJEN 4,P(Y =); -,= Ti,x))
J.U (X)(X) Tj.. and ED (()) 1( }}E(K] Eply= -
Proof. The expected form of traditional CE loss on noisy distribution D be written
Ex(e(((X), P)J
= Σ Σ P(Y = (TY=)))))==
jE(K)
Σ Σ
P(Y Ena= +
ΣΣ P(Y (1))) (
(E[K]
The first term could be transformed as:
Σ Σ P(Y =))7)
jE(K)
Σ JEIK]
TijPP(Y = ())((==))))=s/(((x...)) + E TúPP(Y = ()
(E[K],(
= -TED(((f(X), [ED,
Σ
Σ TTP(Y = i) i)
JC(K] iE(K)
where
T min
Tjp A = Σ A,P(Y = 1), A=: -I TI
jE(K]
jE[K]
and
= EjE[K A,P(Y=1)
Ep ((F(),)) if A>0,
Ep. (=))
0 if =
15

---

Published as a conference paper at ICLR 2021
Then Es(e(f(x); K)] - [ED(((((X)))))) AEDA 1U[X))(()) Σ
= (ED-
SE[K] (EK),E)
E P(Y (T, JE(K KE[K]
-TEo((((()) AEDA ((
Σ
= E
JE[N]
Σ
P(Y = 1tw-:://-000-aalffffaa----------
JE[K (
E P(Y = =)) -T))(((())))) -Box=100((x)))1)
jElK)
TEX)((((X), Y) X
Σ P(Y = D) -T)(((((X))))) +
jElK iE())
Σ P(Y j) - - [=xx=,(((())))]] SE(K]
[TEx((((((), Y)] AED [3]()),))]+++++ Σ Σ JE[K] (E(K],
P(Y (,0
Σ = - -73))((((X),1))
JE(K)
[E))((((X) AED (UUX), Σ P 1R
[E[K]
where
(x) (x(x))
=Txx)=T;((]]] T35-
The expected form of CcR on noisy distribution D can be written as
BEp (p..a(((((r)) Yy]] EB ([::)(
- 9 / [(D))E y (x.)), Y)]]
= = Σ P( = j) [x)())))
JE[K]
Σ Σ P(Y i)E = -))((((;)) =
JEIK SE[K]
Thus the expected form of the new regularized loss is
Ep [((f(x)))) + (cx(f(x;)) TED + ((((x), YI
Σ Σ P(Y = E) BP(Ý = -))(((()))))..
(8)
JE[K] [E[]]
B.2 PROOF FOR THEOREM
Theorem 1. For (.a(.), solutions satisfying [1] > 0, Vi E [K] are not locally optimal at (In, (n).
Proof. Let Ec)) be the CE loss. Note this proof does not rely on whether the data distribution is clean not. We use D denote any data distribution and D denote the corresponding dataset. This
16

---

Published as a conference paper at ICLR 2021
notation applies only this proof. For any data distribution D, we have ED [e(f(X),)) - EDvID (((f(a)) Xi]
= =ED (x(((x), Y)] Epy D [ (f(X), Y)I
= -- -Soo
I) + / dx Σ Ia
Σ
Dx yE[K]
yE[k]
S/C dx Σ
In
-(
Dx
уE[K]
The dynamical analyses are based on the following three assumptions:
A1. The model capacity is infinite (i.e., it can realize arbitrary variation).
A2. The model is updated using the gradient descent algorithm (i.e. updates follow the direction decreasing ED [e(f(X),)) Y)] - Epy [EExx [e (f(X), Y)D..
A3. The derivative of network function af(rw) is smooth (i.e. the network function has no singular point), where wt 's are model parameters.
Ow;
Denote the variations of fo [y] during one gradient descent update by (x From Lemma | it can be explicitly written
(2 = fzlvl J dr' Σ
PF - P(2'))(()) Σ
G,(z,
(9)
Dx D'E[K]
kE[K]
where 77 is the learning rate,
G,(,,y) = 0gg (z)
Ogr (x)
Σ
Swlw']
Ow,
Ow,
W'E[K]
and (x) is the network output before the softmax activation. i.e.
exp(g)(*))
[u
Eyeek exp(gy (x))
With (z)), the variation of the regularized loss is
AED [e(f()),))))))) tcR] S dr P(x) Σ (x)
P(ylx) - P()) faly]
(10)
Dx
vE[K]
the training reaches a steady state (a.k.a. local optimum), we have AEp [e(f + tcj = 0. To check the property of this variation, consider the following example. For particular Zg. define
P(y(zo) Ply)
F(xo) = Σ (x))
fz.
yE[R]
Split the labels y into the following two sets (without loss of generality, we ignore the P(ylao) Ply) = 0 cases):
Vzo:- {y: P(ylan) Pyy) < <
and
Vant = {y: Poy(z)) P(y) > 0}.
By assigning (z)) ay V, Y'zo:- and A,, (2o) by O, Vy Vostt finds F(xo) > 0 since Lzo [y] > ΣyEjK] fzoly] = after update. It is easy to check our assigned ay and b, could maintain this constraint by introducing weight Nab to scale b, as follows.
O. Note we have an extra constraint Σ, A, = 0 to ensure
Σ ay Nob Σ b9 0, by Nosb', yE
DEN+
17

---

Published as a conference paper at ICLR 2021
Let (xo) be e-neighbourhood of Tv. Since faly] is continuous, we can set Ax() [1 +
cos all--zzall A (2ro), Vx E (zo) and 0 otherwise. The coefficient {1 + cos الاا~ is added
so that the continuity of [a[y] preserves. This choice will lead AED [e([(X), + [cR] O. Therefore, for any (c yn) with solution fx. i] > 0, vi E [K], we can always find a decreas- ing direction, indicating the solution is not (steady) locally optimal. Note D can be any distribution
in this proof. Thus the result holds for the noisy distribution D.
Lemma 1. (,(x)= = falvl J. dx' Σ Dx
[F(x'. - P(d')P()) E (( iE[K]
y'E[K]
Proof. We need to take into account the actual form of activation function, i.e., the softmax function, as well as the SGD algorithm to demonstrate the correctness of this lemma. The variation Ayo (20) is caused by the change in network parameters {w;}. i.c.,
a5 [vo]
go (2r)
Σ
Sws:
(11)
Ows
#E(K]
where ow, are determined by the SGD algorithm
Sw; = ODD [e(((),, Y) + CCR)
Ow,
P(x, v) Ply) 0fz[v]
fu(vl dw,
J.y
Plugging back to (1I yields
Pl v) P())
ao [yo] [f
yu (rr)
17
Sw,
Ow
SE[K]
2.2
To proceed, we need to expand
@v Ow:
Taking into account the activation function, one has
exp(gr(x))
[x]
Even (x))
where 9y (z) refers to the network output before passed to the activation function. Recall that, by our assumption, derivatives af(w;)) are not singular. Now we have
ow:
0fz
de (x)
EVELK e (y((()) + +e tt)g)
1
Ow;
Ow,
Ow,
EVELKY e (x)((
-e (v(x)
(g)(1)
Σ ())(x) (gr(x)
Ey Eyelk] e (x)
Ow.
(Ewrrtn e eag((t)) 2 (Evretne-aantte v'Elk] Ow;
(gy(x))
fullll (g)
=fa 1y]
Σ
Ow,
Ow,
'' ['E[K]
For simplicity, we can rewrite the above result as
0f-[y]
fa fall)G
Ow:
where
(gy(2) Ow;
Ogy (xx
Gx(z,y)
+ Σ
fall
Own
is a smooth function.
Combining all the above gives Ayo (3ro) follows.
(2o) = [ea[b0] 17 Σ
(Pz. v) - PPx)P(()) Σ (((o
20.0
18

---

Published as a conference paper at ICLR 2021
B.3 PROOF FOR THEOREM
2
Theorem 2. The sample sieve defined in 4 ensures that clean examples (En, In = yn) will not be identified as being corrupted if the model f(() 's prediction on 2n is better than random guess.
Proof. Let yh be the true label corresponding to feature In. For a clean sample, we have Un = yn- Consider arbitrary DNN model f. With the CE loss, we have {(f(r,,, Vn.) --nnn [vnl)) According to Equation (4p in the paper, the necessary and sufficient condition of e, > 0
((f(rn). (n) + leR (fx)) < an + -In(S=,,(uu <
Σ ME(K]
In(f (v))
K
In(fe. < K
Σ
In(Jo..ul)
By Jensen's 's inequality we have
In - Jan, [Mm] In
EvelkI.utu fzalu]
Σ
In(Ja,, (v])
K
K-1
K-1
DE(K],u7u
Therefore, when (sufficient condition)
1 fan [yvl]
- -In(S,, (vm]) < 1-111 In
+ San,
Wn]
K -
K
we have Vi > 0. Inequality Sz,, [yn] > guess.
K indicates the model prediction is better than random
B.4 PROOF FOR THEOREM 4
Before proving Theorem we need to show the effect of adding Term-2 to Term-1 in (5). Let Éx 0.5 be the measure of separation among classes w.r.t feature X distribution D, i.e., P(Y Y*[X) = 1 1 YX D, where Y := arg max P(Y = = is the Bayes optimal label. Let D' be the shifted distribution by adding Term-2 to Term-l and Y' be the shifted label. Then P(X = P(X[Y'), K,KY Y) ~ D, XYY') ~ D' but IP(Y') may be different from P(Y). Lemma 2 shows the invariant property of this label shift.
Lemma 2. Label shift does not change the Bayes optimal label of feature X when cx
minvi,je(k) Tus
Proof. Consider the shifted distribution D'. Let TED(C(((X). Y)] + ÅRD -
where
[b(((( Σ
P(Y' [(U))
JE(K]
and
P(Y" =
C
where
C := EjeiK] Ty j) is constant for normalization. For each possible Y
have P(Y = i|X) [0, Ex] U (1 - ex 0.5. Thus
P(X|Y = i) P(Y = iXX
EXP(
P(X) -x)
E [0,
-
P(Y
P(Y =
19

---

Published as a conference paper at ICLR 2021
Compare D' and D, know there is label shift (Alexandari al. P(X = i) = PXX = i) but IP() and P(Y") may be different.
2020 Storkey 2009) where To ensure the label shift does
not change the Bayes optimal label, we need
P(XY = =))P(Y
max Y arg max = x|x) iE[K]
arg max iE[K]
(X,Y) ~ D.
P(X)
One sufficient condition is
EXP i)
(1-Ex (1-ex)P(Y'
Tj Ta + T3
EX < min Vi
PP(Y i)
P(( =
With Lemma 2 Assumption 1 and Assumption 2
we present the proof for Theorem 4 as follows.
Theorem 4. (Robustness of the Confidence Regularized CE Loss) With Assumption
and when
min Uiy(X) 5B5 (Y==)>(==)))x>>>) [J P(Y j)
Tx(X) -T;((X)
P(Y PP( =)
minimizing Ex(((((X), + lcR (fx)) is equivalent minimizing Eo|((f(X),
Proof. It is easy to check ex = 0, ~ Dx when Assumption hholds Thus adding Term-2 Term-1 (5) does change the Bayes optimal label. With Assumption [ the Bayes optimal classifier on the clean distribution should satisfy f (X())[]] 1, v(X,)) ~ D. On one hand, when
jaX XDDx (X)(X)/P( j), have
- SP = =) s, [K],X~1 ~ Dx.
In this case, minimizing the regularization term results in confident predictions. On the other hand, to make it unbiased to clean results, could not be arbitrarily large. We need to find the upper bound on 8 such that f* also minimizes the loss defined in the latter regularization term. Assume there no loss on confident true predictions and there is one miss-prediction on cxample (In, Yn = j1), i.., the prediction changes from the Bayes optimal prediction fx, [i]] 1 to fz,, [ie] 1, Compared to the optimal one, the first two terms in the right side of (5) is increased by Tj to- where lg > 0 is the regret of one confident wrong prediction. Accordingly, the last term in the right side of (5) is increased by (Bir.i (X) Pin.i (X)) It is supposed that
j1-
T lu (3)..i (I.)) (2n)) 2 0, VJ1.2 E [K],
which is equivalent to
3(P(Ý = ji) -P(( = j2)) s (inii(a)) - Ti (In), Vji, E [K].
Thus
Th (X) (00))TTjjhh (X)
8
min -=>=)>>(>>>">>>x>>>>>>>>>>>>>>>> P(Y = ji) - P(Ý = j2)
By mathematical inductions, it can be generalized to the case with multiple miss-predictions in the
CE term.
C OTHER JUSTIFICATIONS
In this section, we first compare lςR and entropy regularization in Section C.I and highlight our superiority with both theoretical and experimental evidence, then show example for explaining the variances incurred by label noise in Section C.2 and provide the risk bound Section C.3 for training with the sieved examples that satisfy Corollary
20

---

Published as a conference paper at ICLR 2021
Table 4: Comparing foR with ER on CIFAR-10.
Symm 0.4
Asymm
Method
0.2
0.6
0.1
0.2
0.3
Baseline
86.98
81.88
74.14
90.69
88.59
86. 14
Baseline ER
87.61
83.84
80.55
91.36
89.61
87.47
Bascline tCR
90.70
88.29
82.10
92.41
91.02
90.53
C.I COMPARING foR WITH ENTROPY REGULARIZATION
For simplicity, we consider two-class classification problem. Suppose for given feature the probability of æ belonging class is p. The entropy regularization (ER) can be written as:
(p) lp (1 - p) In(1 -p)), p)),
(12)
while our regularization term is written as:
(CR(p) = Imp + In(1 - -p).
(13)
We have the following proposition:
Proposition fcR regularizes models stronger than the entropy regularization terms of gradi- ents.
Proof. First notice that both RER and RcR are symmetric functions around p = 0.5. Thus we can only consider the situation where < p < 0.5. The gradients w.r.t p are:
@RER(P) (Inp - In(1 p)) = In p Öp
1),
and
dRCr (p)
Op
1-p
Now we compare the absolute value of two gradients.
When 0 < < 0.5, it is easy to check
@R)(() In 1)< Op
1 1
aRcu(p)
p
p
p
Op
and both gradients are larger than 0. Therefore, fcR has larger gradients than the entropy regulariza- tion, i.e., fcR has stronger regularization ability than ER.
We can also draw a figure to show this phenomenon. Figure shows the value of RCR and RER with respect to p. We can see the gradient of our regularization larger than entropy regularization, resulting in a more confident prediction. We also perform an experiment to further show the evi- dence. Table records comparison results which show our regularization achieves higher accuracy compared to the entropy term.
C.2 CALCULATING varD((U5(x)))) AND var (( Y) (cR(s
Consider optimal classifier f == arg min, [x)((((X))))))) Y)]. Let lmax be the upper bound of the E(-) loss, and Lmin be the lower bound of the E(.) loss. Denote E by the over noise rate (ratio of corrupted examples in all examples).
For varp varp( (X) ()).), we know the loss E(ft (2n), ya) = lmin for each example. Thus the vari- ance is var (eSs (X), = 0.
21

---

Published as a conference paper at ICLR 2021
-0.5
value
-1.0
-1.5
entropy regularization
-2.0
our regularization
0.0
0.2
0.4
0.6
0.8
1.0
Figure 4: Comparing our regularization with entropy regularization.
For (ff F) + (cR((((())) we know the loss {(fī(zn), Ø yn) = bmins and the loss ((fb(zn), 17 A yn) Lmax. Note
(K 1)m Emin
CC
K
for each example. The expectation is
E))( (cx((((()))))))) =elmax + (1 8)mmin lCR.
Thus the variance is
V)r)( leu (ck(ff(( (X)(
=e(Emax + lcR (élmax (1 eminn + lcRR) (1 e) + lcR (slmax + (1 E) + lcR))2
=E(1 (f m
We know in this example,
vargle(f6(x)),) + (cR(f((X)) = e(1 - 6) -min)) > (eF 0.
C.3 ANALYSIS FOR THE RISK BOUND
Let Dr. and DL- be the set and the distribution of the sieved clean examples according to Corollary [ We know they are supposed to contain only clean examples. Define RD( Ep Exle(f(X)) Y)J, fo = arg wig, (D) R (f) (
E EmneI((((E.)((((a.)) Dn)),
J arg min (f), where 7X) PP (X)/Pp S. (X) stands for the importance of each example correct sample bias such that Rp(s) E, .. (x)))(((X)) The weight can be estimated by kernel mean matching (Huang et al, 20077) and its DNN adaption (Fang et al. 2020 Let D be the marginal distribution of Dz- on X. For example, with particular kernel (XX). the optimization problem is:
xXX
min x(X)
[Exx [x()] ED,, x (X)x(XX
(X) 0 and E,,,, (x)) 1
Note the selection of kernel ((.) is non-trivial, especially for complicated features. See (Fang al. 2020) for a detailed DNN solutions.
Corollary provides risk bound for minimizing CE after sample sieve.
22

---

Published as a conference paper at ICLR 2021
Corollary 2. If 2y.8. is [0, bl-valued, then for any 8 > with probability at least 1 8, we have
log(1/6)
- 20 fo V
2
where the Rademacher complexity R(y o to F) == BD sup
2 [T.1]
Ener, on (fx)) Un))
and {onEL>) } are independent Rademacher variables.
Proof. The sieved clean examples may be biased due to the covariate shift caused by instance- based label noise. One solution to such shift is re-weighting Dr- to match D using importance ve-weighting. Particularly, we need to estimate parameters ((X) such that
RD( = ()) = E (xX))(((X)))))
With the optimal ((X), the ERM should be changed as
fD arg min R RD., (f), jEF
where
1
R Be-a (1)
Σ
(I
[T0]
nEL-
Via Hoeffding's inequality, vf. w.p. at least
8, we have
In(1/5)
R D (f) R (f) R(O 2b
2|4*|
Following the basic Rademacher bound (Bartlett & Mendelson between the expected empirical risks:
2002) on the maximal deviation
Ro(fD.-n) RD(5t)
= =R (B.mm)) R (
( - "b,-,nUBBrr + (Bb -R (B.
- RT (1brr,)))))
+
<0 2 max
(B)
(f) -R (f)l
In(1/8)
229(roloF) 26,
2
where the Rademacher complexity A(y o o F) == Ep,- |sup ser -om Ener. (f, and {onEL=} are independent Rademacher variables. Therefore, we get Corollary
Corollary 2 informs us that, theoretically, the sample sieve is biased and (X) is necessary to correct the selection bias. However, the enror induced by estimating 7(X) may degrade the performance. In addition, it is easy to check the optimal solution of performing direct ERM on the sieved clean examples the same fö in expectation when Assumption hoold.
D MORE DETAILS AND RESULTS FOR EXPERIMENTS
We firstly show training framework in Section D.I then show implementation details and dis- cussions in Section The algorithm for generating the instance- label noise is provided in Section D.3 We show experiments in Section D.4 and the ablation study in Section D.5
23

---

Published as a conference paper at ICLR 2021
D.1 ILLUSTRATION OF THE TRAINING FRAMEWORK
Our experiments follows the framework shown in Figure 5
Model Update
Data Selectian
Low Loss
Diy-1
Dun
Drsei
D
High Loas
Dgo
Dynce
Remove Labal
Reration
Sample sieve
min TEP
Σ 3..
(x..
Duc
An
Dy.Hisi
xx.
(kwe)) (wn)))
Xa2
Augmentatiin
Epoch?
Consistency training
Figure 5: One example of CORES L(t): Indices of sieved clean examples. H(t): Indices of sieved corrupted examples. Dran == (z Un) n L(t)). Dino) = {(Ins Dn) E H(t)), (x,H(+) {In n E HT))).
D.2 IMPLEMENTATION DETAILS AND MORE ANALYSIS
Implementation details on CIFAR-10 and CIFAR-100 with instance-based label noise: The basic hyper-parameters settings for CIFAR- 10 and CIFAR- 100 are listed as follows:
size (64), optimizer (SGD), initial learning rate (0. 1), momentum (0.9), weight decay (0.0005), number of epochs (100) and learning rate decay (0.1 at 50 epochs). Standard data augmentation is applied to each dataset. CORES and bascline share the same hyper-parameters setting except for and in equation 2 When perform CORES We first train network on the dataset for 10 warm-up epochs with only CE (Cross Entropy) loss. Then / is linearly increased from 0 to 2 for next 30 epochs and kept as 2 for the rest of the epochs. The data selection is performed at the 30 epoch and an,t is set to K Σje(k) (x.)) (I.), + fcR (I.,) in epoch-/ as the paper suggests.
When performing * used the sieved result epoch-40. It is worth noting that that time, the sample sieve may not reach the highest test accuracy. However, the division property brought by the confidence regularizer works well that time. We the default setting from UDA Xie et al. 2019) to apply efficient data augmentation.
Implementation details on Clothing-IMs We train the network for 120 epochs on million noisy training images. Batch-size is set to 32. The initial learning rate is set as 0.01 and reduced by factor of 10 at 30, 60, 90 epochs. For each epoch, we sample 1000 mini-batches from the training data while ensuring the (noisy) labels are balanced. Mixup strategy is employed to further avoid the overfitting problem (Zhang al. 2018 Li et al. 2020 is set to 0 first 80 epochs, and linearly increased to 0.4 for next 20 epochs and kept as 0.4 for the rest of the epochs. It is worth noting that Clothing- M actually does not satisfy our Assumption siice the class "Knitwear" (denoted by class-i) and the class "Sweater" (denoted by class-j) can not satisfy (X) (X)(X) Ta TTj- Note consistency training is not implemented on Clothing- IM.
More analysis on 3: The value of 3 mainly affects the sample sieve in CORES From Theorem and Theorem 4 in the paper, when B is set to be small, we do not have the good division property. When 3 is set to be large, the training biased the CE term. Figure visualize this phenomenon. can be seen that in the left and right figure, many clean examples and corrupted examples overlap together located in the left and right clusters, respectively.
24

---

Published as a conference paper at ICLR 2021
lower beta computed samples
proper beta comppted samples
higher beta
2.5
2.00
comgthe samples
1.4
1.75
1.2
1.50
1.0
1.5
camples
2.00
0.6
0.75
0.4
0.2
0.25
0.0
0.0
0.00
loss
Figure 6: Analyzing how the value of /3 influences the division. We set [ =0.5, 2, 10 for lower, proper, and higher beta settings, respectively.
Algorithm Instance- Deennndet Label Noise Generation
Input:
1: Clean examples (zn, ) Noise rate: E; Size of feature: 1 x S; Number of classes: K.
Iteration:
2: Sample instance flip rates qn. from the truncated normal distribution (Ne,, 0.1², [0, 1));
3: Sample W E RSK from the standard normal distribution (O, 12):
for
1 to N do
4:
p == In W
I/ Generate instance dependent flip rates. The size of p is x K. // Only consider entries different from the true label Let On be the probability of getting a wrong label
5:
Py. = -0
6:
p 9n softmax(p)
7:
Pu.. =1 In // Keep clean w.p. 1 On
8:
Randomly choose a label from the label space as noisy label In according to p:
end for
Output:
9: Noisy examples (Ni, Bm)
D.3 GENERATING THE INSTANCE- DEPENDENT LABEL NOISE
In this section, introduce how to generate instance-based label noise which is illustrated in Al- gorithm D Note this algorithm follows the state-of-the-an method (Xia et al. (020). Define the noise rate (the global flipping rate) as E. First, in order to control æ but without constraining all of the instances to have same flip rate, we sample their flip rates from truncated normal distribution N(c, 0.12, [0, 1)), where [o, 1] indicates the range of the truncated normal distribution. Second, we sample parameters W from the standard normal distribution for generating instance-dependent la- bel noise. The size of W is S x K, where S denotes the length of each feature. For each instance (Nn)yn), we Step 5 and Step 6 to ensure that the probability of getting wrong label is q.. Step 7 ensures the sum of all the entries of p is 1.
Suppose there are two features: F; and 2, where F; Iz.- Then the possibility p of these two features, calculated by x W, from the Algorithm I would be exactly the same. Thus the label noise is strongly instance-dependent.
25

---

Published as a conference paper at ICLR 2021
Table 5: Comparison with the results reported by DivideMix (Li et al. 2020 methods use Tre-Re as the backbone. The last epoch test accuracy for each method re- ported. The noise rate is defined as the probability of replacing the label with other labels including the true label.
CIFAR-10. All
Symm
Dataset
Method
0.2
0.5
CE
82.7
57.9
Bootstrap (Reed et al.
2014
82.9
58.4
Forward T (Patrıni
2017
83.1
59.4
Co-teaching+ Yu et al.
2019
88.2
84.1
Mixup (Zhang et 2018)
92.3
77.6
CIFAR-10
P-correction (Y & Wui
2019)
92.0
88.7
Meta- Meta-Learning (Li al.
2019
92.0
88.8
M-correction (Arazo ct al. DivideMix [Li 2020)
(2099)
93.8
91.9
95.7
94.4
CORES
95.9
94.5
Table 6: The best epoch accuracy for each method on Tiny- Ima-emmgeNet.
Symm
Dataset
Model
Method
0.2
0.5
MAE (Ghosh al.
2017)
2.36
1.22
GCE (Zhang & Sabuncu 2018
69.84
66.31
Tiny ImageNet
ResNet18
MentorNet (Jiang et al. CORES
2017)
59.12
53.83
73.47
71.07
Table 7: Comparing CORES (without consistency training) with other noise-robust methods on CIFAR-10.
Symm 0.4
Asymım
Method
0.2
0.6
0.1
0.2
0.3
Cross Entropy
86.98
81.88
74.14
90.69
88.59
86.14
Forward T' (Patrini et al. 2017)
88.11
83,27
75.34
90.11
89.42
88.25
Truncated Ly (Zhang Sabuncu 2018
89.70
87.62
82.70
90.43
89.45
87.10
LDMI [Xu et al. 2019
88.74
83.04
76.51
90.28
89.04
87.88
CORES (without consistency training)
90.70
88.29
82.10
92.41
91.02
90.53
D.4 MORE EXPERIMENTS ON CIFAR- 10 AND TINY-IMAGENET
In this section, we compare C with more methods CIFAR-10 and Tmagemm Table 5 records the comparison results with recent benchmark methods. Table 6 compares CORES? with other methods Tiny-Imas ImageNet. Both tables show that CORES achieves competitive results.
D.5 ABLATION STUDY
2 (without consistency training): By optimizing loss in 2 the model can be forced to concentrate only on clean examples. Thus even without consistency training, the network trained by CORES is also noise-robust. Table compares CORES with other noisc-robust methods which do not apply semi-supervised setting in the framework. We can see CORES still achieves the best performance among all the methods.
2 without confidence regularization or dynamic data selection: The loss in equation consists of data selection strategy and confident regularization term. To see how they influence the final accuracy, we perform the ablation study to show their effect on Table The first row of Table 8 corresponds the traditional CE loss. The second row corresponds to the sample sieve with CE
26

---

Published as a conference paper at ICLR 2021
Table 8: Analysis of each component of CORES CIFAR- 10. All the methods ResNet-34.
Sample Sieve
Symm
Asymm
Consistency training
Data selection
Regularization
0.2
0.4
0.6
0.1
0.2
0.3
X
86.67
81.44
74.63
90.18
88.43
87.27
90.15
86.98
78.36
91.59
90.89
88.51
90.70
88,29
82.10
92.41
91.02
90.53
95.73
93.76
89.78
96.05
95.18
94.67
loss. The third row the typical CORES² The last row is CORES2 We can see both the dynamic sample sieve in (4) and the munfidenc-reegulriiee model update in final accuracy, which suggests the rationality of CORES
show positive effects on the
27


Reviews:
[{"review_id": "zrDwoVgd76Z", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary\n\nThe paper introduces a noise-robust loss function CORES2, motivated by peer loss. The novel loss adds a regularization term that promotes confident prediction and pushes the model prediction away from the prior of the label. Using this loss function, the authors propose a dynamic sample sieve to separate the clean data and corrupted data on-the-fly, by the magnitude of CORES2 loss. The author's approach is to rule out samples whose losses are larger than an adaptive threshold. Importantly, the process of sieving successfully sieves out corrupted samples, both in a theory of 'better than random guess classifier' and in practice. The authors, then show that the proposed CORES2 can be decoupled under the instance-dependent noise setting. Then CORES2 is proved to be noise-robust, which means CORES2 is equivalent to minimizing the original cross-entropy loss. They also show a principle approach for finding the hyperparameters $\\beta$. Further, a consistency loss is adopted after sample sieve on the corrupted samples. The author conducts extensive experiments, including CIFAR10, CIFAR100, and Clothing1M under different settings of noise. CORES2 achieves the SOTA results in all the experiments.\n\nContributions\n\ni) Proposal of a novel confidence regularized loss for image classification and a novel algorithm that dynamically sieves out corrupted samples basing on their loss. \n\nii) The proposed loss is proved to be noise-robust. This theoretical result is valuable since less the instance-based noise is not well studied. A principle way to choose the hyperparameter $\\beta$. \n\niii) Application to CIFAR10, CIFAR100, and Clothing1M (real-world dataset), with the extensive comparison. \n\nIssues:\n\ni) The motivation behind the dynamic sample sieve is unclear to me. One hypothesis is that if we set CORES2 as the objective, the model will fit clean samples much faster than corrupted samples at the beginning of training.  Hence the model can sieve the corrupted samples out. However, Fig2 shows that the cross-entropy can also separate clean/corrupted samples at the early training stage. I am curiously about the performance of cross-entropy loss + dynamic sample sieve. The comparison between CORES2 and cross-entropy in Fig2 is unfair. \n\nTheoretically, the paper only shows that once the model is better than random guess, the dynamic sample sieve will not sieve clean samples out. And the assumption is $ f_{x_n}(y_n) > 1/K$. But $f_{x_n}(\\tilde{y}_n) > 1/K$ would also frequently happen, for $y_n\\not=\\tilde{y}_n$, which means the dynamic sample sieve will also keep these corrupted samples. \nOverall, I think the high-level insight into the sieving process is unclear in the paper.\n\nii) The training stability of the confidence regularizer: The authors show that confidence regularizer can promote confidence prediction. So some entries in $f(x)$ is near 0, and will make the optimization process unstable since the confidence regularizer -> inf in this case.\nThere is another concern related to theorem 4. Since the Bayes optimal classifier is $f^*_x[y]=1$, its CORES2 loss would be -inf, which seems problematic.\n\niii) Selection for $ \\beta$, according to theorem 4: theorem 4 provides a principal way for estimation of \\beta in different datasets. However, to estimate $ \\beta$, we have to estimate the noise transition matrix beforehand. And there may exist some adversary samples that make the rough estimation impossible. In another perspective, we can also give some meaningful upper/lower bound for $\\beta$ in the instance-independent noise setting. So it's unclear to me what makes the difference in the instance-dependent setting if we do not go through all the samples but do a rough estimation.\n\niv) One motivation of Confidence Regularizer is that confident prediction counters the overfitting of noise labels. But if the model capacity is sufficiently large, I think it can both overfit CORES2, and overfit those noise samples. \n\nMinors: \n\na) The assumption shows that CORES2 does not favor the non-diagonal dominant noise rate, which means the diagonal entries in the noise transition matrix T can be smaller than some entries in the same row. Does CORES2 fail in this setting in practice?\n\n\nOverall, the paper's approach is novel and easy to implement. I am willing to raise my score if the authors can address my issues.\n\n#### EDIT------------\n\nI think the authors replied to some of my concerns in a convincing way, hence I raise the score to 6. \n\nUnfortunately, I think the theoretical analysis for the noise-robust loss is orthogonal to their sampling sieving approach. And the analysis for choosing $\\beta$ does not dependent on their instance-dependent noise settings. We can get the same $\\beta$ by very rough estimation(their approach) in instance-independent noise settings. In addition, I guess $f_x^*[y]=1$ is still problematic in the theoretical analysis since CORES2=\\inf given the ideal classifier. \n\nOverall, following author's response, I am leaning towards acceptance, but will let the AC judge the importance of the points above for the final decision.", "extra_scores": {}, "extra_text": {}}, {"review_id": "UQ4bxPhGPDK", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors of the paper propose a new method, the CORES (COnfidence REgularized Sample Sieve), to tackle the important problem of learning under instance dependent label noise. The proposed method, in essence, involves the use of a confidence regularization term that encourages more confident predictions and a sieving process to remove the samples with large losses. Theoretical justification and empirical experiments were conducted to demonstrate the effectiveness of the proposed method. \n\nAll in all, the paper is clearly written and easy to follow. The proposed method seems technically sound and the motivation for the proposal is explained clearly. One major complaint I have for the paper is the lack of novelty of the paper. The two important building blocks of the paper, the confidence regularizer, and the sample sieve are derived from previous papers. Specifically, in my opinion, the confidence regularizer is a marginal extension of the \"peer loss\" [1], and the sample sieve algorithm is essentially the same as that proposed in [2], the only difference being a different choice of loss function for training and sieving, to the best of my knowledge and understanding. I think it is worth commenting on this very relevant line of work in Section 2.2. In addition, it would be interesting if the authors of the paper could offer some insights on why the proposed sieving strategy works better than the one previously proposed [2] based on softmax probability. All in all, with the lack of novelty addressed above, I think the submission is marginally below the acceptance threshold.\n\nOther comments:\n1. I find the intuitive justification for confidence regularization in Section 2.1 to be quite unconvincing. Specifically, it was stated that \"when model overfits to the noise, its predictions often become less confident\". From my understanding, this is not necessarily true at all. In fact, it was previously demonstrated that deep NNs can even perfectly overfit to datasets with randomly assigned labels? From this perspective, wouldn't encouraging confidence make the model overfit harder to the noisy labels? I would appreciate if the authors of the paper could provide further insights and intuition on why the introduced confidence regularization improves noise robustness. \n\n\n[1] Yang Liu and Hongyi Guo. Peer loss functions: Learning from noisy labels without knowing noise rates. In Proceedings of the 37th International Conference on Machine Learning, ICML ’20, 2020.\n\n[2] Zhilu Zhang and Mert Sabuncu. Generalized cross entropy loss for training deep neural networks with noisy labels. In Advances in neural information processing systems, pp. 8778–8788, 2018.\n\n--------------------------------------------------------------------------------------------------------------\nThe authors of the paper addressed carefully the concerns I raised above. As such, I am raising my score to a 6, and would like to recommend accepting this paper. ", "extra_scores": {}, "extra_text": {}}, {"review_id": "OTVC32bqI-a", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "***quality***\nThis paper is quite well-written. The contribution is critical in instance-dependent label noise learning. Moreover, both the theoretical and empirical justifications are convincing.\n\n***clarity***\nAlthough this paper contains heavy mathematics, it is not difficult to understand. I can see that the authors have spent a lot of efforts in paper writing. \n\n***originality***\nIn this paper, the authors proposed a novel sample sieve approach for instance-dependent label noise learning. The proposed model is novel, and the theoretical contributions are also new to the community.\n\n***significance***\nThe proposed algorithm is simple, but the theory behind is rich. I like this kind of work, so I feel that the significance of this paper is high for future research.\n\n***pros and cons***\n\nPros:\n1. The topic is very important for realistic machine learning problems, and is helpful for reducing the human annotation efforts.\n2. The theoretical study of this paper is quite impressive.\n3. The experimental results show that the proposed method achieves SOTA performance.\n\nCons:\n1. The authors claim that their method does not need to estimate the label transition probability or noise rate, which I think is nice! However, it would be better if the authors can explain why the proposed method can avoid this, namely which “component” helps to avoid the estimation for noise rate?\n2. Since this work is an extension of cross-entropy loss to dealing with label noise, I think the comparison with “Symmetric Cross Entropy for Robust Learning with Noisy Labels”(ICCV 2019) is necessary, as this paper also aims to design a robust loss via modifying cross-entropy loss.\n3. I feel that the sample sieve/filtering process (i.e. Eqs. 1-4) looks like self-paced learning (SPL) (see “Self-Paced Robust Learning for Leveraging Clean Labels in Noisy Data”, AAAI 20), as SPL also selects some “important” data for training in each iteration. Maybe the authors can discuss the relationship between these two methods? \n4. The authors misuse the terms “sample” and “example”. Statistically, we say that we have a sample X={x_1,x_2,…,x_n} from some distribution, in which every x_i is an example. \n5. Some recent typical works on label noise learning can be cited, such as “Are Anchor Points Really Indispensable in Label-Noise Learning?”(NeurIPS 19) and “A Bi-level Formulation for Label Noise Learning with Spectral Cluster Discovery” (IJCAI 20).\n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: Dear Authors,

Thank you very much for your very detailed feedback to the reviewers. They have highly contributed to clarifying some of the concerns raised by the reviewers and improved their understanding of this paper.

Overall, all the reviewers acknowledge the merit of this paper and thus I suggest acceptance of this paper.
However, as Reviewer #4 pointed out, there are conceptual and theoretical issues that need to be more carefully addressed.
Please clarify these issues in the final version of the paper.
Decision: Accept (Poster)