Resolving Training Biases via Influence-based Data Relabeling
Authors: Shuming Kong
Yanyan Shen
Linpeng Huang
Venue: ICLR 2022 Oral (2022)
Keywords: Training bias
influence functions
data relabeling

Abstract: The performance of supervised learning methods easily suffers from the training bias issue caused by train-test distribution mismatch or label noise. Influence function is a  technique that estimates the impacts of a training sample on the model’s predictions. Recent studies on \emph{data resampling} have employed influence functions to identify \emph{harmful} training samples that will degrade model's test performance. They have shown that discarding or downweighting the identified harmful training samples is an effective way to resolve training biases. In this work, we move one step forward and propose an influence-based relabeling framework named RDIA for reusing harmful training samples toward better model performance. To achieve this, we use influence functions to estimate how relabeling a training sample would affect model's test performance and further develop a novel relabeling function R. We theoretically prove that applying R to relabel harmful training samples allows the model to achieve lower test loss than simply discarding them for any classification tasks using cross-entropy loss. Extensive experiments on ten real-world datasets demonstrate RDIA outperforms the state-of-the-art data resampling methods and improves model's robustness against label noise. 

Published as a conference paper at ICLR 2022
RESOLVING TRAINING BIASES VIA INFLUENCE- BASED DATA RELABELING
Shuming Kong, Yanyan Shen, Linpeng Huang Department of Computer Science and Engineering
Shanghai Jiao Tong University [Leinuo123, shenyy, 1phuang] jtu. edu.
ABSTRACT
The performance of supervised learning methods easily suffers from the training bias issue caused by train-test distribution mismatch or label noise. Influence func- tion is a technique that estimates the impacts of training sample on the model's s predictions. Recent studies on data rexampling have employed influence functions to identify harmful training samples that will degrade model's test performance. They have shown that discarding downweighting the identified harmful train- ing samples is an effective way to resolve training biases. In this work, we move step forward and propose influence-based relabeling framework named RDIA for reusing harmful training samples toward better model performance. To achieve this, use influence functions estimate how relabeling training sam- ple would affect model's test performance and further develop a novel relabeling function R. We theoretically prove that applying samples allows the model to achieve lower test loss than simply discarding them for any classification tasks using cross-entropy loss. Extensive experiments on ten real-wo collworld datasets demonstrate RDIA outperforms the state-of-the- data resampling methods and improves model's robustness against label noise.
relabel harmful training
INTRODUCTION
Training data plays an inevitably important role in delivering the model's final performance. has been well recognized that the training bias issue will compromise model performance to large extent (Arpit et al. 2017). Specifically, there are two major scenarios where training biases show up. The first and most common scenario is that training samples involve corrupted labels that could be originated possibly every step of the data lifecycle (Anderson McGrew 2017:) DDolashah al.] 2018 Pei et al.] 2020: Yu & Qin 2020). The second scenario is that the training and test sets are sampled from the respective distributions Prrain (xr, y) and Ptese( y). but Preain is different from Piest (Guo al. 2020 He & Garcial 2009 Zou et al., 2019). Both corrupted labels and distribution
mismatch will hurt the generalization ability of trained model (Fang et al. [2020] Zhang et al. 2017]
Chen al. 2021). We generally refer training samples with comrupted labels or those inducing distribution mismatch as harmful samples.
Data resampling is widely used strategy deal with harmful training samples. Existing resam- pling approaches (Chawla et al. 2002 Mahmoody et al.) 2016 Malach & Shalev-Shwartz 2017]
Ren al. 2018) propose assign different weights to training samples, which aim to mitigate the negative impacts of harmful samples on model's generalization ability. Among them, most resam- pling approaches (Arazo al. 2019 Han al. 2018 Li al. 2020 Wang al., 2020a) rely on training loss to identify harmful samples from the whole training set. They follow the insight that the samples with higher training losses are very likely to have corrupted labels, and hence of- ten beneficial to downweight them during the process of model training. However, such based resampling methods have two limitations. First, they are only able to deal with the training biases caused by training samples with corrupted labels (aka noisy samples). Second, the small-loss trick typically holds true for deep models but not for any predictive models (Zhang et al. 2017). To address the limitations, based on influence functions (Cook & Weisberg 1980). The idea to estimate the influence each training sample on model's predictions over the test set. Any training samples that would cause
recent work (Wang et al. 2020b) proposes
resampling scheme

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increase in the test loss are considered as harmful and will be downweighted afterwards. It is worth mentioning that the influence functions have been proved to deal with two forms of training biases effectively, and it is agnostic to specific model or data type (Koh & Liang 2017 Koh et al. 2019).
Inspired by the success of influence-basee based data resampling, in this paper, we would like to ask the following question: what would happen if we relabel harmful training data based on influence analysis results? Our motivations on performing data relabeling via influence analysis are twofold. (i) Relabeling noisy samples is able to prevent the model from memorizing the corrupted labels. (ii) Relabeling clean but biased samples is helpful to improve model's robustness to harmful samples. Despite the potential benefits of data relabeling, it is still challenging to develop an influence-based relabeling approach that has theoretical guarantee on the model's 's performance improvement after training with relabeled data.
To answer the question, we first follow (Koh et al. 2019) to measure the influence of each training sample on model's predictions and identify the harmful training samples which would cause an increase in the test loss. Next, we investigate whether relabeling the identified harmful samples rather than discarding them can improve the test performance. To achieve this, we start from binary classification tasks where relabeling training sample is to convert its binary label from y to 1 - y- We theoretically prove that relabeling harmful training data via influence analysis can achieve lower test loss than simply discarding them for binary classification. Furthermore, we design novel relabeling function R for multi-class classification tasks and prove that the advantage of relabeling the identified harmful samples using R. in reducing model's test loss still holds. Following the influence-basee resampling approaches (Wang et al. 2018 Ting & Brochu [018 Ren et al. [2000] Wang et al., 2020b). we only use the test loss for theoretical analysis and empirically calculate influence function with a small but unbiased validation set by assuming the validation set is sampled from the same distribution as the test set. In this way, using the validation loss to calculate the influence function is an unbiased estimation of the true influence function. Otherwise, the problem may lie in the category of transfer learning which is beyond the scope of this work.
To summarize, this work makes the following contributions. First, we propose to combine influence functions with data relabeling for reducing training biases and we develop an end-to-end influence- based relabeling framework named RDIA that reuses harmful training samples toward better model performance. Second, we design a novel relabeling function R and theoretically prove that ap- plying R over harmful training samples identified by influence functions is able to achieve lower test loss for any classification tasks using cross-entropp loss function. Third, we conduct extensive experiments real datasets in different domains. The results demonstrate that (i) RDIA is effec- tive in reusing harmful training samples towards higher model performance, surpassing the existing influence-based resampling approaches, and (ii) RDIA improves model's robustness label noise, outperforming the current resampling methods by large margins.
2 BACKGROUND
N) be the training set which are sampled from (I,, where I2 E Rd and E RK Let yæxx, 0) be a model's predic- tion for x, where A E RR the parameter set of the model. We denote the loss of sample 2; by 1(2,0 0) L(y. p(I;, 0)) and use (() to represent ((z, 0). We consider the standard empirical risk minimization (ERM) as the optimization objective. Formally, the empirical risk over D is defined as: 0) = + ΣN i= ((). Since our relabeling function is dependent to the loss function, we focus on the most effective and versatile loss, i.e., Cross Entropy loss for any classification tasks.
Let D ((I Vi) X x Perain (z. y). Let
Influence functions. Influence functions, stemming from Robust Statistics (Huber) 2004), have provided an efficient way to estimate how a small perturbation of training sample would change the model's predictions (Koh & Liang 2017; Koh et al., 2019 Yu et al., 2020). Let Ø
arg ming 1 >>> (0) be the optimal model parameters on convergence. When upweighting a training sample 2; on its loss term by infinitesimal step e; obtain the new optimal parameters
N
0,, convergence as 0.: arg min 1 Ex= In (0.) (0). Based on influence functions (Cook
& Weisherg 1980
Koh & Liang] 2017). we have the following closed-form expression estimate
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the change in model parameters when upweighting 2, by c;:
(2,)) db,, de, l,==== - -H (o),
where H8 a # (0)n(0) is the Hessian matrix and (0) (f) is the second derivative of the loss at training point En with respect to 8. Using the chain rule, we can estimate the change of model's prediction test data 2j sampled from the given test distribution Ptest (Koh & Liang 2017)
(zis 2j) 2 dlg( h,
b (O) (o)( (V)la))))))))
(2)
de:
At a fine-gratn fine-grained level, we can measure the influence of perturbing training sample from (z,, y:) to (Zss M/4 + 5). Let Ri8 = (It; yı + 5) and the new loss ( (218, 0) = L + 6, ( 0)). According to KKhh & Liang 2017). the optimal parameters de,d, after performing perturbation on 2; is 0e, = arg anginin N En (0) + ecli(zus, 0) e,),(0). This allows to estimate the change in model parameters after the theeggraind data perturbation using influence functions as:
d0.,.
de: 1.=0 vo(2.) via(z))
-V 0) (0).
Further, the influence of perturbing z; by 2i8 on model's prediction at test sample 25 is the following:
1788 (2:, 2J) - dlI (0,()),)))
de, 1,,-0
(4)
(0)
-
(o).
It is important notice that Eq. 44) holds for arbitrary ō when e; is approaching 0. This provides the feasibility of measuring how relabeling a training samples could influence the model's predictions.
Influence-based resampling approaches. Previous researches (Koh & Liang, 2017; Wang al.) 2020b) have shown that influence functions have strong ability to identify harmful samples from the whole training set, which is agnostic to the specific model data structure. Inspired by this, many influence-
resampling approaches (Ting & Brochu 2018 Wang et al. 2018 2020b) proposed to discard or downweight the identified harmful samples to reduce the test loss. However, different from previous works which focus on estimating the influence of each training sample on the test performance using Eq. T- we perform the fine-graines perturbation on a training sample's label and evaluate its influence using Eq. (3)-8 4 Further, our work tries to build an end-to-end influence-based relabeling framework to reuse the harmful samples with a theoretical guarantee on the final model performance for any classification tasks. To be specific, we demonstrate that harmful training instances after being relabeled properly do make contributions to improve the final model performance, which provides novel viewpoint on handling biased training data.
3 METHODOLOGY
Assume we have Q = 1Z) E s sampled from the test distribution Prest and our objective is to minimize the test risk L(Q: = Σ3 (0). Due to the harmful training samples in D, the optimal Ô which minimizes the empirical risk over training set 1 may not be the best risk minimizer over Q. To solve this issue, we propose novel data relabeling framework named RDIA which aims to identify and reuse harmful training instances towards better model performance. We design a relabeling function R that allows the model to achieve lower test risk after being trained with the relabeled harmful instances for any classification tasks. In what follows, first give overview of the RDIA framework. Then describe the details of the major steps in RDIA and provide thcoretical analysis on how the relabeled harmful samples are useful to further reduce the test risk. The algorithm of RDIA could be found in Appendix A
3.1 OVERVIEW OF RDIA
Figure provides an overview of RDIA, which consists of four major steps: Model training, Harm- ful samples identification, Relabeling harmful samples via influence analysis and Model reiraining.
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sel
Traininngset
Training set Đ
Retriming
Treining
Modil
Iniluese function
Refabeling funntinn
Modld
R(-)
Step
Step
Step
Step
Figure 1: The overview of RDIA. We devise relabeling function R. to change the labels of the identified harmful training samples D
Step Model training is to train model based on the training set until convergence and get the model parameters 9.
Step II Harmful samples identification is compute the influence of perturbing the loss term of each training sample 2; E on test risk using Eq. (2) and then use it to identify the harmful training samples from D. We denote the set of identified harmful training samples remaining training instances as D+. The details of this step are provided in Section 3.2
D_ and the set of
Step III: Relabeling harmful samples via influence analysis is apply the relabeling function to modify the label of each identified harmful training sample in D_ and obtain the set of relabeled harmful training samples denoted as D'_. We introduce our relabeling function and theoretically
prove that updating the model's parameters with new training set D = D D can achieve lower test risk over Q than simply discarding downweighting D_ in Section 3.3
Step IV: Model retraining is to retrain the model using / till convergence to get the final optimal parameters BeR-
3.2 HARMFUL SAMPLES IDENTIFICATION
In the second step, we compute D_ S D which contains harmful training samples from the original training set D. Intuitively, training sample is harmful to the model performance if removing it from the training set would reduce the test risk over Q. Based on influence functions, we can measure one sample's 's influence we add a small perturbation e, on the loss term of 2; to change its weight, the change of test loss at test sample 24 E Q can be estimated as follows:
test risk without prohibitive leave-one-out training. According to Eq. -
1(zj.0.) 0,,) - 1( =
= 2.
(5)
where .)) is computed by Eq. test risk follows:
2 We then estimate the influence of perturbing 2; on the whole
M Σ 1=1
1(Q. -1Q. F
00 25).
(6)
Henceforth, we denote by (=,) = Σド M Da(z,, 25) the influence of perturbing the loss term of 2; on the test risk over Q. worth mentioning that given e; E [ ..... Eq. (6) computes the influence of downweighting or discarding the training sample 2;. We denote D {ar E D (2(2)) > 0} as harmful training samples. Similar to (Wang et al. 2020b), we assume that each training sample influences the test risk independently. We derive the Lemma
Lemma 1. Discarding or downweighting the training samples in D_ {2r E D (b(2;) > 0} from D could lead a model with lower test risk over Q:
L(Q, - ((0,
1
Σ
$0(x)) 50,
(
where ê, denotes the optimal model parameters obtained by updating the model's parameters with discarding downweighting samples in D
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Lemma exxlains why influence- based resampling approaches have strong ability to resolve train- ing biases and the proof of Lemma D is provided in Appendix In practice, to further tolerate the estimation error in (z(2)) which may result in the wrong identification of harmful training samples, we select D_ = { E D (0(2;) > a) where the hyperparameter a controls the proportion of harmful samples to be relaheled eventually. We conduct the experiment to show the effects of a and the validation set in Section 5.3
3.3 RELABELING HARMFUT. SAMPLES VIA INFLUENCE ANALYSIS
In the third step, we propose relabeling function and ensure the test risk would be reduced after training with the relabeled harmful samples D'_ To achieve this, we start from a special case (i.e., binary classification) and then extend to any classification tasks.
Relabeling harmful instances on binary classification. We start with binary classification where the relabeling function R is straightforward. Since the label set y is {0, 1}. we have R(2) = 1 y for any (x,, y) E D. Recall that p(v,, 0) denotes the model output for and the training loss of z, = (Ix; yt) E D is: (:(0) = y log(s((s;, 0)) - (1 - y4) log(1 ((*,, 0))
To compute the influence of relabeling training sample 2; in D, we first consider the case that
y/ = 1 and R(z;) = O. The loss & (0) at 2, is changed from - log(vp( 0)) to - log(1 (0x,1_ 0)).
Letting ZRR- (zi, 1 y/) and w: wzz 0) Vol( ZR 0) (0), we have:
w(2,, 0) - Va log(1 1 - ( 0)) + v log(p((;;, 0)) - Vo),
(8)
- (fx., 0)
According to Eq. 29 is:
4 and (8. the influence of relabeling 2; model's prediction at test sample
(on 2ÿ) -ol, (0) "((., 0) -Vo (- 1-110
vol.(0)
Bo( 29)
p(2,, 1
1 p(x,, 0)
Similarly, when the label y/ in 2, is and R(2,) = 1, we can derive the influence of relabeling 2; at 2 as Noze(Zi,2) 2f) 0) to Eq. (6) we could extend the influence of relabeling z, at æf to the whole test risk over Q as:
[=( Let R..n. denote the optimal parameters after relabeling 2, Similar
1(Q, (.) 1(Q,0) = Σ
pon(z, 25).
(10)
According to Eq. (9). the influence of relabeling training samples neR(2,, 2f) is related the influ- ence of perturbing the loss term of training samples, i.e., Fo (2,, (2).))). . In this way, the change of test risk by relabeling (Eq. (TO)) and that by perturbing (Eq. (6)) are interrelated. Then we derive
the Theorem aadd the proof can be found Appendix B
Theorem 1. In binary classification, let be the infinum of _((2.0))
1_p((x.,)) and 1- and D sla..
{ E D bo(2;) > 0}. Relabeling the samples in D_ can achieve lower test risk than discarding or downweighting them from D, because the following inequaaity holds.
L(Q, 0. L(Q,0,)
Σ 4ED
26(2,) S.
(11)
N
Theorem T shows that relabeling the samples in D_ could achieve lower test risk than simply dis- carding or downweighting D_ from the training set for binary classification tasks. We then provide some intuitions on the benefits of relabeling harmful training samples. In the context of binary clas- sification, if training sample z in D_ is noisy, our relabeling method corrects the label noise and improve training data quality; otherwise. 2 is very likely to be biased due to its negative impact on the test risk, and relabeling it might improve model's 's robustness.
Relabeling harmful instances on any classification tasks. We now introduce a relabeling function R. that can be used for any classification tasks. For a classification problem with K class labels (K 2 2), we represent each label y as a K-dimensional one-hot vector. The CE loss at 2 is: (0)) = - Σk= Vi,k (o,(px(1x;,)) 6)). Intuitively, the proposed relabeling function R should satisfy the following principles:
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Consistency: R should produce K -d-dimnnsional label vector: R(I,, Vs) v/ [0, 1]K
Effectiveness: applying R over harmful training samples D_ should guarantee the resul- tant test risk no larger than the one achieved by simply discarding them.
For the consistency principle, we require the new label y/ to be K K-dimensional where V/2 describes the likelihood that 2; takes the k-th class label, & E [1, K]. Here we do not require Σt= V to be one, because we focus on leveraging the identified harmful training samples towards better model performance instead of finding their truth labels.
Consider training sample (Ii, Mi) belonging to the m-th class (n E [1, K]) i.e., 36. = 1. Let Ra,; 36) = V/5 we propose the following relabeling function that fulfills the above two principles:
(0.
=
Vl.x -118pp K K-VT-
(12)
Pm
otherwise
where s0z;; = (PI,,... (Pk) is the probability distribution over K classes produced by the model with parameters 0, i.e. po [0, 1] and Σi=1e: P relabeling function R in Eq. (12) satisfies the first principle. Interestingly, we can verify that for
It is easy to check that our proposed
K 2, we have R(z;) = 1 yi- We further prove the effectiveness of R using the Lemma? 2
Lemma 2. When applying the relabeling function R in Eq. (12) over training sample 2, D with a class label m, the CE loss (0) at 2 is changed from log(<P (I;; 0)) to log(1 Pm (x,, 0))
interesting verify the change in loss (,() acts as an extension of the binary elassification. Similar to Theorem 1, we can drive the following theorem using Eq. (9)4100).
Theorem 2. In multi-class classification, let (y(3,, 0) denote the probability that z; is classified as Us truth class label by the model with the optimal parameters Ô on D, and be the infimum of (y(ee.0) Relabeling the samples in D_ {zz D Φo(zz) > 0} with R leads test risk (.,,))
lower than the one achieved by discarding or downweighting D Formally, we have:
L(Q,0.x)
Σ RED
20(2.) S.
(13)
N
Theorem shows that using our proposed relabeling R can further reduce the test risk than simply discarding or downweighting D_ from the training set for any classification tasks. The detailed proofs of Lemma 2 and Theorem are provided in Appendix B
4 DISCUSSIONS
In this section, we provide numerical analysis on the superior performance of RDIA against other influence-based resampling approaches (Wang al. 2018 2020b). Then we discuss the extension of RDIA in exploiting training loss information.
Numerical analysis. Consider a training point 2; E D_ belonging to class m, where D_ is specified in Section 3.2 According to Eq. (13), if we use R to assign with a new label yf R(2,) instead of discarding or downweighting Fs, the difference in the test risk over Q can be computed as:
(m 0)
g(2.) L(Q, -.(0 N 1 - pml(ss)) 0)
(o(()))
(14)
D_ means the (=(=;) in Fq. [4] is positive, and hence we have g(z)) < O. If the model's prediction for 2, with the optimal parameters i is correct, (m; 0) is the largest component in the vector ((z,, f). We consider 2; as a more harmful sample because it has negative influence on the test loss yet the model has learnt some features from 2; that connects to class m. In practice, 2; very likely be noisy or biased training sample. Interestingly, from Eq. (14), we can see that a small increase in (2, 8) will lead to a rapid increase in g(2). This indicates relabeling such more harmful training samples leads to significant performance gain.
Extension of RDIA. In practice, due to the complexity of calculating the influence functions, iden- tifying harmful samples via influence analysis could incur high computational cost, especially when

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training complex models like deep neural networks. To address the problems, we further extend RDIA by using training loss to identify harmful samples for deep models, and we dub this extension as RDIA-LS. We empirically show that RDIA-LS is effective and efficient to handle training data with corrupted labels for deep learning, which spotlights the great scalability of our approach. The details of RDIA-LS provided in Appendix
5 EXPERIMENTS
In this section, we conduct experiments to evaluate the effectiveness and robustness of our RDIA. We also perform the ablation study to show how hyperparameter a and the size of validation set affect the performance of RDIA. The visualization of identified harmful samples and comparison with other loss-based approaches are provided in Appendix D and Appendix
5.1 EXPERIMENTAL. SETTINGS
Datasets. To verify the effectiveness of RDIA, perform extensive experiments ten public data sets from different domains, including NLP, CV, CTR, etc. Since all the datasets are clean, we build noise transition matrix P = (P.j Y k x to verify the robustness of our proposed approaches for combating noisy labels, where K denotes the class number and Pi denotes the probability of clean label being flipped a noisy label j. In our experiment, we the noise ratio the rate of how many labels are manually corrupted and each clean label has the same probability of being flipped other classes, i.e., Ps. = Rー More details about the statistics of the datasets and Tr-V divisions are provided in Appendix
determine
Comparison methods. We compared our proposed relabeling method RDIA with the following baselines, all of which are agnostic of specific model or data structure.
(1) ERM: means training a model with all the training data with the cross-entropy loss. (2) Random: it is a basic relabel-
ing method that randomly selects and changes the label of training samples. (3) OptLR (Ting & Brochu 2018): it is weighted sampling method which assigns each training sample with a weight proportional its impact on the change in model's parameters v/o Specifically, the weight of 2; is max (a. min (1, { We set a and À to be 1/ max (1x()))((=)) and 1/ (+)))))) respec- tively. (4) Dropout (Wang et al. 20: it is an unweighted subsampling method which simply discards D_ from the training set, i.e., removing all training data with negative influence on the test loss. (5) UIDS (Wang et al., 2020b): it is an unweighted subsampling method which uses Lin- sampling method or Sigmoid sampling method to resample the training data based on influence functions (2)2) It is the best cest--erforming method among all the existing influence-basee methods.
We implemented all the comparison methods by using their published source codes in Pytorch and ran all the experiments on a server with Intel Xeon 1.7GHz CPUs, 128 GB of RAM and a single NVIDIA 2080 Ti GPU. All the baselines mance. To measure the performance of all the approaches, we followed (Wang et al. (2020b) and used the test loss as the metric since we aim to optimize the test loss via influence analysis.
tuned with clean validation data for best model perfor-
Implementation details. For each of the ten datasets, we adopted logistic regression (convex op- timization) as the binary classifier (for MNIST and CIFAR10, we randomly choose two classes to perform binary classification). As for multi-class classification, we implemented two deep models (non-convex optimization), LeNet (2 convolutional layers and fully connected layers) and a CNN with 6 convolutional layers followed by 2 fully connected layers used in (Wang et al., 2019) on MNIST and CIFAR1O. The hyperparameter clean validation set for best performance. More detailed settings are provided in Appendix C
also tuned [0, 0.001, 0.002, ...0 with the
5.2 EXPERIMENTAL RESULTS
Effectiveness of RDIA. To verify the effectiveness of RDIA, we conduct experiments on 10 clean datasets with three different models. The experiments repeated 5 times and the averaged test loss with standard deviation results are reported in Table [ and Table 2 We have the following important observations.
First, our proposed RDIA yields the lowest test loss over 9 out of 10 datasets using logistic re- gression. It outperforms ERM all the datasets, which indicates the effectiveness of relabeling
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Table I: Performance comparison results with logistic regression on binary classification task. Av- erage test loss (#std) over 5 repetitions are reported.
Dataset
ERM
Random
OptLR
Dropout
UIDS
RDIA
Breast-canser
0.0914
0.2619 1 0.0102
0,0934 a 0.0015
0.0731 & 0.0014
0.0786 a 0.00066
0,0649 # 0.0001
Diabetes
0.5170
0.5461 a: 0.0006
0.5232 4 0.0012
0.5083 &: 0.0008
0.5068 a: 0.0004
0.4920 0.0002
News20
0.5157
0.5247 0.0028
0,5253
0.5072 0.0019
0.5075 0.0012
0.5007 0.0015
Adult
0,3383
0.3381 = 0,0001
0,3547 0.0000
0.3383 = 0.0001
0.3383 + 0.0001
0,3383 0,0001
Real-sim
0.2606
0.2638 + 0.0025
0.2884 0.0151
0,2605 0.0024
0.2607 # 0.0031
0,2575 0.0021
Criteol
0.4911
0,4919 0.0011
0.4914 0.0007
0.4995 0.0025
0,4895 + 0.0012
0,4894 0.0010
Covtype
0.6936
0.6906 à: 0.0029
0.6907 0.0026
0.6843 0.0023
0.6784 1: 0.0032
0.6776 0,0024
Avazu
0,3449
0,3449 + 0.0002
0,3450 a 0.0002
0.3576 = 0.0001
0.3447 + 0.0001
0,3447 0.0001
MNIST
0.0245
0.2543 + 0.0005
0.0239 = 0.0004
0.0221 0.0002
0.0238 = 0.0003
0.0207 0.0001
CIFARIO
0.5952
0.6174 2 0.0025
0.6163 0.0021
0.5946 0.0017
0.5845 0.0015
0.5806 0.0012
Table 2: Performance comparison results with deep models on multi-classification task. Average test loss (#std) over 5 repetitions are reported.
Datasel
ERM
Random
OptL.R
Dropout
UDD
RDIA
Neee) 0.0283
0.0407 a 0.0025
0.0756 0.0102
0.0256 a 0.0002
0.0261 a 0.0002
0.0251 2 0.0005
MNISTICNN)
0,0322
0.0385 0,0003
0.0576 0.0042
0.0289 0.0002
0.0302 0.0011
0.0281 0.0006
CIFARIOL
1.1641
1.6247 0,0223
1.8341 + 0.0421
1.1721 0.0019
1.1534 0.0017
1.2631 0.0015
CIFARJO(CNN)
0.7744
0,8303 0.0162
1,2303 # 0.0329
0.7859 # 0.0016
0,7910 = 0.0013
0.6052 0.0029
training samples via influence functions to resolve training biases. Furthermore, RDIA outperforms the state-of-the-art resampling method UIDS on all the datasets except Avazu, which indicates the effectiveness of reusing harmful training samples via relabeling towards higher model performance.
Second, when training deep models, RDIA achieves the best test loss on MNIST+LeNet, MNIST+CNN, and CIFARIO+CNN, where outperforms UIDS by a large margin. We observe LeNet performs much worse than CNN on CIFARIO using the original training set (i.e., the re- sults of ERM) due to its simple architecture. Note that the poor classification results for clean and unbiased training data would interfere the identification of true harmful training samples. Hence, RDIA performs similarly to Random which introduces random noises into the training set and the performance suffers. But we want to emphasize that when training a more suitable model (e.g.. CNN) on CIFARIO, RDIA is more effective to improve model's performance.
Third, Random performs worse than ERM on all the cases except on Adult. This indicates that randomly relabeling harmful training samples may easily inject noisy data that hurt model's perfor- mance significantly. In contrast, our proposed relabeling function is effective to assign appropriate labels to harmful training samples that benefit the test loss.
1.2
1.0
350
Ed4
TEC
1.0
45
48
8 37
XO1
0.5
0.6
0.1
02
3.5
02
Saterets
Mirrrds
(a) Breast-cancer
(b) News20
(c) Real-sim
(d) CIFAR10 (CNN)
Figure 2: Test loss results with different noise ratios. Shaded regions indicate standard deviation.
Robustness to label noises. In order to investigate the robustness of RDIA to noise labels, we set the noise ratio from 0 to 0.8 to manually corrupt the labels in the four datasets from dif- ferent domains, while the results on the other datasets have similar trends. Figure 2 reports the average test loss of all the influence-basee ratios. First, thanks to the high accuracy of estimating the influence functions on logistic regression, all influence-based approaches consistently outperform ERM, which indicates the effectiveness of using influence functions to identify noisy samples. Figure 2(a) 2 and [ho] that RDIA per- forms significantly better than the other influence-based approaches. As the noise ratio becomes larger, the test loss of all the other approaches increases while the test loss reported by RDIA is generally unchanged. This verifies the robustness of RDIAto high noise ratios. We surprisingly find that RDIA at 0.8 noise ratio achieves lower test loss than ERM at zero noise ratio. The reason might be that RDIA could leverage all the training samples and fix noisy labels properly boost the per-
approaches on four noisy datasets with different noise
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Table 3: Effect of hyperparameter RDIA (11684 training samples in total).
Noise ratio
0.2
0.5
0.8
Relabeling number
0
0
0
ERM
Test loss
0.0245
0.2567
0.6942
1,5975
Relabeling number
1439
3626
6140
a 0.01 Test loss
0.0235
0.0443
0.0903
0.1009
Relabeling number
71
1721
4139
6545
0.002
Test loss
0.0207
0.0315
0.0519
0.0465
Relaheling number
530
1804
4193
6589
0.0002
Test loss
0.0903
0.0392
0.0410
0.0405
Table 4: Effect of the number of validation samples used in RDIA (1 (11684 training samples in total).
Number of validation samples
100
200
500
1000
Validation loss
0.5337
0.5309
0.5233
0.5275
ERM
Test loss
0,5219
Validation loss
0.2388
0.2269
0.2417
0.2331
UIDS
Test loss
0.49928
0.3873
0.2783
0.2409
Validation loss
0.0679
0.0583
0.0494
0.0514
RDIA
Test loss
0.3430
0.2080
0.1396
0.0847
formance. Second, Figure 2(d) shows that when combating noisy labels for deep models, RDIA still suffers from the noisy labels like other baselines because the estimation of influence functions with deep models not accurate enough filter out all noisy labels. However, RDIA could still relabel the most negative samples to reduce the test loss.
5.3 ABLATION STUDY
Finally, we investigate the effect of different values of hyperparameters a and size of validation set on the performance of RDIA using MNIST with logistic regression.
Hyparameter a. As discussed in Section 3.3 by varying o, we can derive the percentage of rela- beled training data against the complete training set in RDIA. Table provides the results of how many samples are relabeled and how test loss changed with different values of a under different noise ratios. First, when noise ratio equals to O, there are few biased samples in the training set. In this case, simply relabeling all the identified harmful samples will hurt the performance while using relatively larger could report lower test loss. Second, when noise ratio is 0.8, RDIA achieves better performance with smaller a. This is reasonable since most of training samples involve label noises and increasing a facilitates the relabeling of noisy samples.
Size of the validation set. As discussed in Section [3.]] we use the validation set instead of the test to estimate the influence of each training sample. Table 4 shows the results of how the number of validation samples affects the model performance. We conduct the experiments under 40% noise rates and find the optimal hyperparameter a E [0.0002, 0.01] to get the best results of RDIA.
We have the following observations. 1) Using only 100 validation samples with RDIA achieves 35% lower test loss than ERM. 2) As the number of validation samples increases, RDIA significantly outperforms ERM, achieving up to 90% relative lower in test loss. The reason is that, as the number of validation sets increases, the validation set can gradually reflect the true distribution of test data. In this way, the estimated influence functions are accurate enough to filter out most harmful training samples for the test set. 3) RDIA consistently outperforms UIDS with different sizes of validation set, which empirically shows the effectiveness of our relabeling function R.
6 CONCLUSION
In this paper, propose to perform data relabeling based on influence functions to resolve the train- ing bias issue. We develop a novel relabeling framework named RDIA, which reuses the information of harmful training samples identified by influence analysis towards higher model performance. We theoretically prove that RDIA can further reduce the test loss than simply discarding harmful train- ing samples on any classification tasks using the cross-entropy loss function. Extensive experiments on real datasets verify the effectiveness RDIA in enhancing model's 'ss robustness and final perfor- mance, compared with various resampling and relabeling techniques.
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Reproducibility: We clarify the assumptions in Section and provide the complete proofs of Lemmas, Theorems in Appendix B The statistics of datasets, the data processing, and the de- tails of the experimental settings are described in Appendix Our code could be found in the https://gá t.ub. http://hhhbbbbc///eeccc//////
ACKNOWLEDGMENT
This work is supported by Shanghai Municipal Science and Technology Major Project 10000H22ZX0002), the Tencent Wechat Rhino-Bird Focused Research Program, and SJTU Global Strategic Partnership Fund (2021 SJTU-HKUST). Yanyan Shen is the corresponding author of this paper.
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Appendix
In this appendix, we first provide the algorithm of RDIA (Appendix A) and the complete proofs of the Lemmas and Theorems (Appendix B in the main text. Then give the details of the experimental settings (Appendix C). the extensive analysis of our approach (Appendix D), and the visualization of identified harmful samples (Appendix) E. We then describe RDIA-LS,
extension
of RDIA, to spotlight the scalability of our approach RDIA (Appendix F) and provide empirical results to show that RDIA-LS is effective and efficient handle training data with corrupted labels for deep learning (Appendix G). Finally, we provide additional discussions about the existing data relabeling approaches (Appendix
A RDIA ALGORITHM
Algorithm 1: RDIA
Input:
Training model 0, biased training set D {(x, v:) f= learning rate B, sample selection ratio a such that 0 S S 1, small and unbaised set Q = {(r] k/)
Train the model with D until convergence to get 0;
Initialize D+ for i [1....,[] do
Calculate the influence of the training sample z, (2z, yi) on Q using Eq. 6
(b(2)) > a then
Relabel the identified harmful training samples with z/ <- R(2;);
D_ - D_ U [2
else if (2:) < 0 then
D_ D4 (z}
10
end
11 end
Obtain the new training set < D_UD+
13 Retrain the model with ò till convergence to get the final model parameters O.R
B PROOFS FOR LEMMAS AND THEOREMS
B. PROOF OF LEMMA
Assume the perturbation on 2; is infinitesimal and the influence of each training sample on the test risk is independent.
Lemma 1. Discarding or downweighting the training samples in D_ = {ze E D #0(21) > 0} from D could lead model with lower test risk over Q:
Σ
po(2:) S0.
N
RED
where ô, denotes the optimal model parameters obtained by updating the model's parameters with discarding downweighting samples in D
Proof. Recall that o., arg min N ΣNY (0.) In this way, downweighting the training
sample in D_ means setting E e +, (Noticed that e; = - means discarding training sam- ple z;). For convenience of analysis, set all equal - and have &o(z;) ΣS'n (2, 25). According to Eq. (6), estimate how the test risk is changed by discarding downweighting
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E D_ as follows:
- Σ
Σ
1(2 - 1(zj.0)
SED
Σ
4 X
Do 20 25)
4ED
1
bo(2;) 00
N
2GED
B.2 PROOF OF THEOREM 1
Theorem 1. In binary classification, let be the infinum of 9(2.,0)
1-9(x,,) and 1-pla,0) and D_
p(z..0)
{2, E D bo(2)) 0}. Relabeling the samples in D_ achieve lower test risk than discarding or downweighting them from D, because the following inequality holds.
L(Q, - L(Q,0,) -LQQ N
Σ ALDD
Do(z;) SO.
Proof. Based on Eq. (9), we have:
p0 0)
if if =0
To (Zi; 29)
- 0)
+
2 o zz)
p(x;,0
if =1
pt 0)
It is
worth mentioning that Õ., = arg min 1 N relabeling the training sample 2; in D_ means setting 6
En I, (0) + e (25R, 0) - (.)( In this way,
Similar to the proof of Lemma 1, according to Eq. (6) and Eq. 10 we have:
L(Q.0. I(Q,0.) 0.) = =I( ((Q,0) () I(Q.0.)
Σ
1(2j. 0e. -({=,,)) (1(3), -125-(=330))) Σ
xeED_
M
M
$
NoR (2 27) + Σ
Pe(z,, 29))
RED
j=1
M
(0 25)
1)000(2)2
N
Do 29)
SED
Σ
(2));) 5
N
RED
B.3 PROOF LEMMA
When applying the relabeling function R. in Eq. (12 over training sample 2; E D with a class label m, the CE loss ((8)) at 2; is changed from log( 8)) to log((1 pm (z,, f)).
Lemma 2.
Proof. Recall that the model's prediction o; is sp 0) function is:
(111, (0; -.. PK) and relabeling
0,
if = m otherwise
108px
K-VI - pm,
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Here assume the training example 2; belongs to class which means that Vam = 1 and the other components in the one-hot vector Yi are 0. The prime CE loss is log((m 8)). If we use our relabeling function to change the label of z4, the loss at 2, will be:
i(2s, 8) = - Σ
1og,p
# V
Pm x log(
kmm
log(
VI (mm) log(px))
log(
kmm
log(1 4m)
K
kfm
- looo( Pmm)
In this way, if we use relabeling function R to change the label of example 2,, the loss function will be changed from lo (zy, 0)) log(1 (pm (I..
B.4 PROOF THEOREM 2
Theorem 2. In multi-class classification, let (y(x,,,) 0) denote the probability that is classified as its truth class label by the model with the optimal parameters on D, and be the infimum of sy(x,00) Relabeling the samples in D_ {z+ D Φo(2:) 0} with R leads to test risk --,,(a;,)
lower than the one achieved by discarding or dowmweighting D.. Formally, we have:
I(Q, - T.Q,, 0.)
Σ
(.(2,) 5.
BEDD
Proof. According to Lemma 2 Eq. g and Eq. 10
we can estimate the change of test loss at test
sample zj Q caused by relabeling as follows:
4(2g. ((25.0.,m.) R.) L(25, 0) se e: x (2., 2f)
1 1
Dz 25)
N
py
Further, we can derive the following:
L(Q, (.). L(Q,d.) 0.) =L(Q, L(Q, L(Q, - LAQ, 0c)
Σ
1125 - (1(2) De.) 1(25,0))
2ED
1=1
MoR (zi, (z1,2))
L 29)) N j=1
Σ
N
3ED
j=1
-1
+ ():0
N
(,(x,,0
RED
j=1
Σ
4o(=:) 50
N
2GED_
C EXPERIMENTAL SETTINGS
C.I THE STATISTICS OF THE DATASETS
Table 5 shows the statistics of the datasets. We perform extensive experiments public datasets from different domains to verify the effectiveness and robustness of our approach RDIA. All the datasets could be found in http://www..eemmmmdd.iiiiiiiinne
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Table 5: The statistics of the datasets.
Dataset
#samples
#features
#classes
#domain
Breast-cancer
683
10
Medical
Diabetes
768
2
Medical
News20
19,954
1,355,192
2
Text
Adult
32,561
123
Society
Real-sim
72,309
20,958
Physies
Covtype
581,012
Life
Criteol %
456,674
1,000,000
2
CTR
Avazu
14,596,133 1377
1,000,000
2
CTR
MNIST
70,000
784
2/10
Image
CIFARIO
60,000
3,072
2/10
Image
C.2 TR-VA-TE DIVISIONS
We follow the Ta- (Traiinn/Valldatiinn/Ti set divisions) setting in Wang et al. (2020b) to measure the generalization ability of our approach RDIA. Specifically, the influence of each training instance is estimated with the validation set using the validation loss and the model's performance is tested by additional out-of out-of-sample test set which ensures do not utilize any information of the test data.
When training logistic regression, randomly pick up 30% samples from the training set the validation set. For different influence-basee approaches, the training/validationnees sets are kept the same for fair comparison. Both MNIST and CIFARIO 10 10-classes image classification datasets while logistic regression can only handle binary classification. On MNIST, we select the number and as positive and negative classes, respectively; On CIFARIO, we perform binary classification on cat and dog. For each image, we convert all the pixels into flattened feature vector where each pixel scaled by 1/255.
When training deep models, due to the high time complexity of estimating influence functions, we randomly exclude 100 samples (1%) from the test sets MNIST and CIFARIO as the respective validation sets, and the remaining data is used for testing.
C.3 IMPLEMENTATION DETAILS
We used the Newton-CC algorithm Martens (210)) to calculate the influence functions for the lo- gistic regression model and applied Stochastic estimation Agarwal et al. (2017) for two deep models with 1000 clean data in the validation set. For logistic regression model, we select the regularization term C 0.1 for fair comparison. We adopt the Adam optimizer with the learning rate of 0.001 to train the LeNet on MNIST. After calculating the influence functions and relabeling the identi- fied harmful training samples using R, we reduce the learning rate 10 and update the models until convergence. For CIFAR10, we use the SGD optimizer with the learning rate of 0.01 and the momentum of 0.9 to train the CNN. Then we change the learning rate to 0.001 and update the models based on the relabeled training set. Here we use different optimizers to train the models. This indicates RDIA independent of the update strategy used for model training. The batch size is set to 64 in all the experiments and the hyperparameter is tuned with the validation set for best performance.
D EXTENSIVE ANALYSIS OF RDIA
D.1 COMPLEXITY ANALYSIS
According to Koh & Liang 2017). the time complexity of calculating influence function for one training sample(i.e., Eq. 2 is O(NP), where N and P stand for the sizes of training set and model's parameter set, respectively. Note that the time complexity of relabeling ONN). Considering the complexity calculating influence functions, the time cost of relabeling harmful samples is negligible which means our RDIA is as fast any influence-based approaches.
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Test
Most harmful training samples
Test
Most harmful training samples
0,1033
0.0791
0.0623
0.6224
0.5491
0,4851
1
7
1 0.1215
(
7 0,3006
0.1528
0.1134
0,2993
0.2914
1
1
7 0.0796
7
7
0.0790
0,4522
0.3781
0.3743
cat
cat
0.0483
0,0441
0.0425
0,5728
0.5545
0.5539
dog
dog
(a) No label flipped
(b) 50% label flipped
Figure 3: Identified harmful examples from MNIST and CIFAR10. each test example, three harmful training samples with the highest influence estimates (above the images) are provided.
D.2 RELATIONSHIP WITH PROPENSITY SCORE
Propensity score [Rosenhaum & Rubin 1983: Bickel et al. 2009) is a well-studied technique to solve the distribution mismatch (also called covariate shift) problem where training and testing sets are sampled from two different distributions Pravin (I, 3) and (aest(z, y). respectively. Its basic idea assign the propensity score each training sample make the risk unbiased. Unlike the influence function calculated by measuring the change of test loss, propensity score is calculated directly by estimating the probability of each training sample belonging the test distribution. If we could estimate the training and test distribution accurately, we could also use the propensity score to replace the influence function for identifying whether the training sample is harmful. We leave for the future work.
E VISUALIZATION OF IDENTIFIED HARMFUL SAMPLES
We provide examples of harmful samples identified by influence functions to illustrate the effective- ness of influence analysis. We apply the logistic regression on MNIST (class and 7) and CIFARIO (class cat and dog). The influence functions are estimated by Newton-CCG algorithm (Martens) 2010). We provide three most harmful images which have the highest influence scores and share same label with the test sample.
Figure 3(a) shows three identified harmful training images for each test image when there are no flipped Tabels in the training set. We can see that the identified harmful training samples are visually different from the original pictures, which disturbs the model's prediction on the test image. That is, the presence of clean but harmful training images would damage the model's performance.
Figure 3(b) shows the identified harmful training images when 50% labels of training data have that the harmful images have corrupted labels, which confirms the
been flipped. It is easy effectiveness of applying influence analysis to locate noisy samples.
F RDIA-LS: A LOSS-BASED RELABELING APPROACH
LIMITATIONS OF RDIA
In the main paper, have discussed novel data relabeling framework RDIA via influence analysis. Based on the advantages of influence functions, RDIA able to handle different types of training biases and is agnostic specifie model or data type. However, the time complexity of estimating
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Algorithm 2: RDIA-LS
Input: Deep neural network 0, learning rate B. trainig D. training epoch T,, iteration N. sample
selection ratio p. underweight hyperparameter y such that S 7 S 1.
for E [1..... ,T do
Shuffle training set D: for n [1,...,.. NV] do
Fetch r--th mini-batch from D; Identify harmful samples using training loss:
//Step I
D+ + arg minp:|P|2210) L(D, 0);
D- < D D+:
Relabel the identified harmful training samples:
//ttp II
D'_ < R(D-);
10
Obtain the loss as: LR = -==>>LLD_000 0) + (1 +))((D+,0))
11
Update the model: 0 + 0 (V...
//Stpp III
12
end
13 end
the influence of one training sample is O(NP), where N and P stand for the sizes of training set and model's parameter set, respectively. This is relatively high for deep models which have thousands of parameters. Moreover, according to (Koh & Liang, 2017). the approximate estimation of influence functions on deep models may not be accurate and hence the second step of RDIA suffers from false positives and false negatives. When harmful samples account for the majority of the training set, e.g., high noise rates, is difficult to filter most of harmful samples using the estimated influence.
F.2 RDIA-LS
To address the aforementioned limitations, we aim to extend RDIA to solve the specific problem. Here we focus on combating noisy labels with deep models since label noise usually primary root cause of training bias. We notice that training loss has been used to filter out training samples with corrupted labels in many previous works (Arpit et al. 2017 Han et al. 2018 Wei et al. 2020) Yu et al. 2019). It is worth mentioning that the noisy training samples identified by training loss not equivalent to the harmful ones identified by influence functions because the latter are evaluated to have negative influence on the test performance. Nevertheless, since the selected high-loss training samples are very likely to involve corrupted labels, applying our relabeling function over them has the potential of correcting corrupted labels and benefiting the test performance. Besides, using training loss to identify harmful samples is more efficient as it avoids the estimation of influence functions. Hence, we propose to use training loss identify noisy samples and develop loss- based data relabeling approach named RDIA-LS, which can be viewed as an extension of RDIA for combating corrupted labels with deep models.
RDIA-LS consists of three steps: noisy samples identification, noisy samples relabeling and model updating. It shares the same last two steps with RDIA. The only difference between RDIA-L-S and RDIA is that A-S uses training loss to identify noisy samples in each training epoch so that it does not need to train the model until convergence first. Specifically, given a mini-batch of training instances D C D. RDIA-LS feeds forward all the samples in D and then sorts them in an ascending order of their training losses. Following the prior works (He & Garcial 2009). we regard the large- loss instances as noisy and the small-loss instances clean. We use the rate of p select the possibly clean training instances in D, i..., D. = arg min D:1p)2nl 5) L(D.)). The remaining high- loss training instances are treated as noisy samples, i.e., D_ D D+. We follow (Han et al. 2020) to determine the value of the selection ratio p. After we have D_, we use our relabeling function R. to relabel the samples in D_ and then update the model with D_. U D' In our implementation, we simply modify the loss of the identified noisy samples based on Lemma 2 without performing actual relabeling. We use the hyperparameter Y [0, 1] to control the model tendency of learning from the clean instances and the relabeled noisy instances. The detailed procedure of RDIA-LS is provided in A Algorihhm 2
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Table 6: Average test accuracy (#std) MNIST over the last 10 epochs.
Noise ratio (T)
0.2
0,4
0.6
0.8
ERM
79,46 0.42
59.12
0.37
41.40
0.05
23.43 # 0.30
S-model
97.46
83.52 2 0.14
60.88 # 0.32
41.63 # 1,42
F-correction
0.11
0.05
74.15 1.09
63.83 = 1.76
Self-teaching
0.13
0.14
8.27
75,95 + 1.03
Co-teaching
97.89
9..05
78.54 9..7
SIGUA
97.94
9.05
9.08
83.75 = 0.15
RDIA-LS
98.12 = 0.02
97.57 #
95.32 0.06
87.85 + 0.21
Table 7: Average test accuracy (#std) on CIFARIO over the last 10 epochs.
Noise ratio (T)
0.2
0.4
0.6
0.8
ERM
71.84 1.07
55.62 # 0.31
35.56
0.22
16.90 + 0.62
S-model
76.83
65.37 # 0.39
4
17.41 + 0.08
F-correction
5.25
19.63
Self-teaching
78.92
0.26
0.07
20.32 0.13
Co-teaching
79.43
7.08
6.32
22.47 0.15
SIGUA
81.58 0.36
0.11
66128
24.26 0.23
RDIA-LS
0.14 0.19
6.25
25.35 + 0.17
We conduct the additional experiments in Appendix to empirically show that RDIA-LS is effective and efficient to handle training data with corrupted labels for deep learning, which spotlights the great scalability of our approach RDIA.
G PERFORMANCE EVALUATION OF RDIA-LS
We now conduct the experiments to evaluate the effectiveness and efficiency of RDIA-LS using DNNs MNIST, CIFAR10, CIFAR100 and Clothing The first three datasets clean and corrupted artificially. ClothingIM is a widely used real-world dataset with noisy labels (Patrini al. 2017)
G.1 IMPLEMENTATION DETAILS
We apply the network structures used in the main paper and use LeNet (2 convolutional layers and I fully connected layer) for MNIST, a CNN with convolutional layers followed by fully connected layers used in (Wang et al. 2019) for CIFAR10 and CIFAR 100, and 18-layer ResNet IM.
for Clothing 1M. We follow the settings in (Han et al. 2018) for all the comparison methods. Specif- ically, for MNIST, CIFARIO and CIFARIO0. we use the Adam optimizer with the momentum of 0.9, initial learning rate of 0.001, and the batch size of 128. We run 200 epochs (T=200) in total and linearly decay the learning rate till zero from 80 to 200 epochs. As for Clothing IM, we use the Adam optimizer with the momentum 0.9 and set the batch size to be 64. We run 15 epochs in total and set learning rate to x 10 5 x 10 and 10 for average five epochs.
We set the ratio of small-loss instances with the current training epoch and TV 5 for Clothing M and T3 10 for the other datasets. In this way, we can determine D_ and D in each training epoch. If the noise ratio 7 is not known in advance, could use the method YYu et al. 2018) to estimate The hyperparameter is tuned in (0.05, 0.10, 0.15, ... 0.953 with the validation set for best performance. If there is no validation set, could use training loss to select clean subset from the training
1 min + T, which is changed dynamically
the validation set. Following bass- approach (Han et al. 2020; 2018 Jiang et al. 2018), we use the test accuracy the metric, i.e., (#correct predictions) (#test instances).
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Published as a conference paper at ICLR 2022
Table 8: Average test accuracy (+std) on CIFAR100 over the last 10 epochs.
Noise ratio (T)
0.2
0.4
0.6
0.8
ERM
35.14 + 0.44
20.58 + 0.23
12.87
0.42
4.41 0.14
S-model
45.71 # 0.15
34.94 + 1.29
19.82 + 0.67
2.61 + 1.18
F-correction
4
22.75 1.47
2.20
Self-teaching
4
30.62 0.04
13,49
Co-teaching
47.15
15.15 3.17 0.62
SIGUA
48.52 0.21
0.15
30.73
14.31 # 0.02
RDIA-LS
50.24 0.15
44.20
32.67 0.17
20.21 + 0.04
Table 9: Average test accuracy (+std) results Clothing
Methods
ERM
F-correction
Co-teaching
SIGUA
RDIA-LS
Accuracy(%)
64.54 = 1.05 6 68.36 = 0.35 69.35 0.41
69.64 0.14
G.2 COMPARISON METHODS
We compare our proposed RDIA-LS with the following baselines. S-model (Goldberger & Ben- Reuven 2017) and F-correction (Patrini et al. (017)) are the existing data relabeling approach which aims to estimate the noisy transition matrix correct the noisy labels. The last three approaches are the state-of-the-we
(1) ERM: it trains one network with all the training data using cross-entropy loss. (2) S-model (Goldberger
loss-based resampling approaches.
Ben-Reuven 2017):
uses an additional softmax layer to model the noise transition matrix to correct the model (3) F- correction (Patrini et al. 2017):
y: it corrects the prediction by the noise transition matrix estimated by the other network. (4) Self-teaching (Jiang et al.] 2018): it trains one network with only the selected small-loss instances D+. (5) Co-teaching (Han et al. 2018); it trains two networks simul- taneously and improves self-teaching by updating the parameters of each network with the small-loss instances D+ selected by the peer network. (6) SIGUA (Han et al. (020): it trains one network with the selected small-loss instances D+ and high-loss instances D_ via gradient descent and gradient ascent, respectively.
G.3 EXPERIMENTAL RESULTS
Comparison with the Baselines.
RDIA-LS is proposed to combat noisy labels for deep learning. order to evaluate how RDIA- LS improves the robustness of deep models, we perform experiments on MNIST+LeNet, CI- FAR1O+CNN and CIFARI00+CNN with different noise ratios and the real-world noisy dataset Clothing BeahingIM+Resii M+Resnet18. The average results of test accuracy are reported in Table Table Table and Table D. We have the following observations. (1) RDÍA-LS achieves the highest test accuracy in all the cases. When noise ratio is 0.2, the improvement of RDIA-LS is relatively small. This is reasonable as the performance gain of RDIA-LS obtained from utilizing noisy data is restricted due the low noise ratio. When the noise ratio exceeds 0.4, RDIA-LS significantly outperforms the ex- isting loss-based approaches, achieving up to 5% relative improvement in test accuracy. It indicates that RDIA-LS can still effectively reuse harmful training instances to improve model's robustness under high noise ratios. (2) RDIA-LS consistently performs better than S-model, F-correction, and SIGUA, which implies that using R to relabel noisy training samples identified by training loss is more effective than modeling the noise transition matrix or performing gradient ascent with iden- tified noisy training instances. (3) RDIA-LS outperforms all the baselines on the world noisy dataset Clothing IM, which demonstrates the effectiveness of applying RDIA-LS in practice.
Comparison with RDIA.
Table 10 reports the running time of harmful/noisy samples identification in RDIA and RDIA-LS. We exclude the results of LS on logistic regression since training loss can only be used to filter out noisy samples for training deep models. From the table, identify harmful samples for logistic regression is efficient. However, when training deep models with millions of parameters, using training loss to filter out noisy samples is much more efficient.
that using influence function to
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Published as a conference paper at ICLR 2022
Table 10: Time cost of identifying harmful samples.
Dataset
Model
RDIA
RDIA-LS
Diabetes
LR
0.03 sec
News20
LR
1.8 sec
Criteo1% %
LR
7.1 sec
Avazu
LR
4.2 min
MNIST
LeNet
4'5 hour
0.1 sec
CIFARIO
CNN
7-9 hour
0.6 sec
RDIA-LS is extension of RDIA to combat noisy samples with deep models. The aforementioned experimental results show that RDIA-LS effective and efficient to handle training data with cor- rupted labels for deep learning. However, is worth noticing that RDIA-LS relies on the small-loss trick that the samples which have the larger training loss may contain the corrupted labels. In this way, RDIA-LS is only suitable for the deep models against the corrupted labels and could fail in the situation where the small-loss trick does not hold while RDIA has no such constraint.
H ADDITIONAL DISCUSSION ON DATA RELABELING
Existing relabeling approaches (Goldberger & Ben-Reuven 2017 Jiang et al. 2018 Lee et al,, 2018) are proposed to combat noisy labels with DNNs. They focus on estimating the noise transition matrix to convert the corrupted labels to clean labels. However, current relabeling methods suffer from two limitations. First, they aim to find the labels of the training samples, which means they can only deal with label noise. Second, they employ some additional structures to correct the labels, which are dependent of the specific model structures. For example, Goldberger al. (Goldberger & Ben-Reuven 2017) added an additional softmax layer to present the noise transition matrix and
CleanNet (Lee et al. (018) used the auto-encoder to update the corrupted labels. contrast, we aim to develop relabeling function based on influence analysis to change the labels of harmful samples towards better model performance. We do not require output labels to be one-hot vector since our objective is not to find the truth labels of training samples. Besides,
extend approach to RDIA-LS to effectively combat noisy samples for training DNNs, which outperforms the existing data relabeling approaches (Goldberger & Ben-Reuven 2017 Jiang et al. 2018).
DUTI (Zhang et al., 2018) is an effective data relabeling approach which could debug and correct the wrong labels in the training set. uses a bi-level optimization scheme to recommend the most influential training samples for cleaning and suggest the possibly cleaned labels. The proposed DUTI is different from our approach. Specifically, the relabeling function in DUTI is trained by bi-level optimization using the gradient of the validation loss while our proposed relabeling function has nothing do with gradient, validation loss.
relabeling function
21


Reviews:
[{"review_id": "wlEpNyse3Ge", "reviewer": "Reviewer_XS8o", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "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": "I believe the paper is very well-written and structured. I appreciate that the authors had taken time and consideration into writing an abstract and introduction that clearly motivates the problem in hand, gives enough background into the problem, and that clearly explains the solutions and the experimental results.\n\nThe paper seems to be solidly based in theoretical proofs of their methods, together with experimental results comparing it to some baselines plus the state-of-the-art approach. I liked that the limitations are clearly explained in the appendix.  \nAll in all I think is a good paper. \n\nRegarding weaknesses of the paper that I believe could help at improve the paper if addressed. \n- First, is not until the reader reaches section three that the reader realizes that the authors use influence analysis on the validation set rather than on the test set. I think that the authors should improve the consistency across the paper of using the term \"test set\". Throughout the introduction, abstract, background, and part of the methodology, the validation set is referred as the test set. Which then is clarified to be the validation set, since the test set is only used for a proper test evaluation (without altering the training set). I would suggest to either clarify that at the beginning, or simply use validation set instead of test set. \n- Algorithm for RDIA such as done for RDIA-LS (Algorithm 1) in the Appendix E. I believe this will help portray more details for further reproducibility.\n- Regarding reproducibility, the authors mention that the code is available in the Appendix. Do the authors mean the programming code? If so, I couldn't find a link there. A link with the source code would be beneficial for the same reasons as the point above. \n- My last and probably the most relevant concern is the following. Imagine we have an imbalanced dataset, and that the algorithm is not able to classify well that portion of the dataset with fewer examples. The problem in this case is that we don't have enough data of some particular group of instances, needing some sort of solution like data augmentation (for example). \nHowever, if we would apply the authors solution (or the other authors solution for that matter) we would be treating these examples as incorrect. In the authors case the positive aspect is that the examples won't be removed (as with some of the others solutions). But, how would relabeling those examples help the model learn about this specific category of instances? I am thinking on the lines of a dataset where we have sensitive attributes e.g race, gender. Where some of the people are less represented. From my perspective this solution might have a negative impact in the generalization aspect since the model might not really learn those people that might look different from the rest. Or it could be the case where it helps at mitigating this bias. \nThis comment is more of an opening for a discussion with the authors, rather than \"something to fix\" . It would be great if the authors could just comment what they think on this matter during the discussion period. Since I am sure they have thought about this aspect, and I wouldn't want to have missed or misinterpreted some part of the paper. \nJust to clarify, my scoring hasn't been influenced by this last comment. ", "summary_of_the_paper": "The authors present an approach and framework to mitigate training biases by combining influence functions and data relabeling. \nThe idea behind training biases is that part of the data that is used to train the model does not accurately represent the real data distribution seen in the test set. Thus, having a mismatch between training and test data. This creates a generalization problem for the machine learning model. \n\nOther authors have used different resampling approaches to try and address this problem, relying on the training loss and then relabeling the data; or by using influence functions and changing the weight of the harmful examples so that the effect on the test loss is lower. \nThe current authors combine both approaches, and present a framework that relabels harmful training data based on influence functions (on the test set). \n\nThe results of their experiments show that they are able to reduce the test loss compared to other data resampling approaches.", "summary_of_the_review": "I think this is a good paper with merits to be included as part of the conference proceedings. \nI believe that the paper is well-written, the claims are well-supported, and the experiments seem correct. \n\n\nI believe a few things can be improved (as mentioned above)\nThe reason I don't give a higher score is because I believe that the methodology does not seem to be too novel and the results are marginally better, except for one or two datasets. \n\n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "RetFUFYELw", "reviewer": "Reviewer_136B", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "The paper uses  influence functions from robust statistics to first identify harmful instances and then relabel them based on a novel relabeling function using influence approximation. Influence estimations have been widely used to identify harmful instances or understand the impact of training samples. This paper goes one step further and uses this analysis to relabel the instances in order to achieve better generalization and lower test error. While the technical novelty is limited as the proposed formulations are extensions of existing influence functions and application is interesting. The authors use their relabeling strategy on multiple datasets and models (including deep models).     \n\nPros:\n\n(1) The paper does a focused job of using influence function for identifying harmful examples and fixing them by relabeling. While in the recent times there are papers on influence functions to identify harmful instances, this paper does a very good comprehensive and focused study compared to others.\n\n(2) The paper is well-written and easy to follow. Related works is well laid out and well covered.\n\n(3) Experimental section is complete with experiments on deep models and the proposed RDIA-LS.  The authors acknowledge the limitations of RDIA on deep models due to erroneous influence estimates for deep models and provide a workaround for it in the Appendix. This is an improvement from the earlier versions of the paper (from a previous conference). I would definitely like to highlight this section in the main paper as it’s important for all practical purposes. \n\n\nCons:\n\n(1) The technique of using influence function for identifying harmful instances is not new and well known and applied in the recent times. Hence I feel the technical novelty is not solid and limited in some aspects. However saying that, applications of existing influence functions are not straightforward and hence that’s one point to be noted. \n\n\nIn general, the paper is well-organized, a good study on influence based relabeling and has sufficient empirical studies to backup it’s proposed formulation. ", "summary_of_the_paper": "The paper uses influence functions from robust statistics to perform two tasks in order to mitigate label noise and generalize better: (i) Identify harmful instances via IF; (ii) Relabel the harmful instances to have better generalization. \n", "summary_of_the_review": "The paper does an end to end study of using influence functions for relabeling instances to achieve a lower test-loss. The technical novelty is slightly limited as the formulations are extensions of current influence functions, however the paper is supported by strong empirical studies (including limitations about RDIA in deep models).", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "NKEpRVS6qxq", "reviewer": "Reviewer_FfyB", "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": "## Strength\nThe authors empirically demonstrated that the label correction is much more effective than the standard sampling/data reweighting-based approach for model improvement.\nThis result is coherent with the one previously reported in [Ref1] where a very similar data relabeling approach was studied.\nThe results on DNNs (on MNIST and CIFAR10) are new where [Ref1] considered only kernel-based models.\n\n* [Ref1] Training Set Debugging Using Trusted Items, AAAI 2018.\n    * *I am not the author of [Ref1].*\n\n## Weakness\nThe proposed method is very similar to the one proposed in [Ref1], while the current paper misses this important prior work.\n[Ref1] formulated the data relabeling problem as the bi-level optimization, and proposed an algorithm to optimize the amount of relabeling using gradient descent.\nAn important point here is that the gradient of the amount of relabeling considered in [Ref1] is essentially identical to the relabeling criteria proposed in (10) in this paper (see below for the detail).\nIn addition, once we interpret the proposed label correction criteria as the gradient descent update, Lemma1 and Theorem1 confirming the decrease of the validation loss are more or less trivial (unless the amount of correction is too large).\nIn summary, although the ways the amount of relabeling is optimized are different (using gradient descent or by one step update), I believe the close connection between [Ref1] and the proposed method needs to be described appropriately in the paper, e.g., through detailed discussions and experimental comparisons.\n\n*Close connection to [Ref1]*\n\nAs the special case of the formulation of [Ref1], we can obtain the following bi-level optimization (here, I removed some terms from the original formulation in [Ref1] for simplicity, and used the notation of the current paper).\n\n$$\n\\min\\_{\\delta \\in \\mathbb{R}^{n}} L(Q; \\hat{\\theta}\\_\\delta) := \\frac{1}{M} \\sum\\_{j=1}^{M} l\\_j^c(\\hat{\\theta}\\_\\delta) , \\\\: \\text { s.t. } \\\\: \\hat{\\theta}\\_\\delta = \\underset{\\theta \\in \\mathbb{R}^{p}}{\\operatorname{argmin}} \\frac{1}{N} \\sum\\_{i=1}^{N} l\\_i(z\\_{i\\delta\\_i}, \\theta) ,\n$$\n\nwhere $\\delta$ is the amount of relabeling.\nThis is the optimization of the amount of relabeling so that the the validation loss to be minimized.\nFollowing [Ref1], we can derive the gradient of the validation loss $L(Q; \\hat{\\theta}_\\delta)$ with respect to $\\delta_i$ as\n\n$$\n\\frac{dL(Q; \\hat{\\theta}\\_\\delta)}{d\\delta\\_i} = \\nabla\\_\\theta L(Q; \\hat{\\theta}\\_\\delta) \\frac{d\\hat{\\theta}\\_\\delta}{d\\delta\\_i} = -\\frac{1}{N}\\nabla\\_\\theta L(Q; \\hat{\\theta}\\_\\delta)^\\top H\\_{\\hat{\\theta}\\_\\delta}^{-1} \\frac{\\partial l\\_i(z_{i\\delta_i}, \\hat{\\theta}\\_\\delta)}{\\partial \\delta\\_i} .\n$$\n\nWhen there is no label correction, $\\delta = 0$ and we have $\\hat{\\theta}_0$.\n\nThen, if we consider relabeling the $i$-th training instance with the gradient step size $\\epsilon$, we have the update\n\n$$\n\\delta\\_i \\leftarrow 0 - \\epsilon \\left. \\frac{dL(Q; \\hat{\\theta}\\_\\delta)}{d\\delta\\_i} \\right|\\_{\\delta\\_i = 0}\n$$\n\nand the decrease of the validation loss induced by this update is \n\n$$\nL(Q; \\hat{\\theta}\\_{\\delta}) - L(Q; \\hat{\\theta}\\_0) \\approx \\epsilon \\left. \\frac{dL(Q; \\hat{\\theta}\\_\\delta)}{d\\delta\\_i} \\right|\\_{\\delta\\_i = 0} \n$$\n$$\n= \\frac{\\epsilon}{N} \\nabla\\_\\theta L(Q; \\hat{\\theta}\\_0)^\\top H\\_{\\hat{\\theta}\\_0}^{-1} \\frac{\\partial l\\_i(z\\_{i\\delta\\_i}, \\theta_0)}{\\partial \\delta\\_i}\n$$\n$$\n\\approx \\frac{1}{N}\\nabla\\_\\theta L(Q; \\hat{\\theta}\\_0)^\\top H\\_{\\hat{\\theta}\\_0}^{-1} (l\\_i(z\\_{i \\epsilon}, \\theta) - l\\_i(\\theta)) .\n$$\nThe last line is nothing but the criteria (10) proposed in this paper.\n\n---\n## After Discussion with Authors\n\nI conducted an experiment on breast cancer dataset by myself (see below).\nThere, I confirmed that RDIA does better than the One-Step GD update of [Ref1].\nI therefore decided to increase my score with a strong expectation to the authors for **referring [Ref1] appropriately in the main body of the paper**.\nAs I mentioned in my original review, this is not the first study considering relabeling based on the influence function-like technique.\nRecalling that influence function is one specific example of implicit gradient, [Ref1] is the first work in this direction to my knowledge (even if [Ref1] does not describe the fact explicitly).\nAs I demonstrated in my code, the simplest version of [Ref1] with only one-step update (and without any human intervention) does good job.\n**I strongly expect the authors to pay certain respect to [Ref1] and do not underestimate their technical contribution**.\nFor example, the current writing in Appendix H seems to be not appropriate.\nEven without human intervention, DUTI [Ref1] can do the same (i.e., relabel the training bias towards better model performance) as the proposed approach (but with slightly inferior performance).\n> (2) The learning task of DUTI is to debug the training instances which may contain the wrong labels and predict the true labels. While the target of our approach is to relabel the training bias towards better model performance. In this way, only our approach could relabel the biased training samples with correct labels towards better model performance.\n\n### The results on breast cancer dataset\n\n*Test Accuracy*\n\n|     |   ERM |   One-Step GD |   RDIA |\n|----:|------:|--------------:|-------:|\n| 0   | 0.959 |         0.96  |  0.961 |\n| 0.1 | 0.96  |         0.968 |  0.962 |\n| 0.2 | 0.953 |         0.951 |  0.959 |\n| 0.3 | 0.931 |         0.944 |  0.957 |\n| 0.4 | 0.842 |         0.897 |  0.958 |\n| 0.5 | 0.563 |         0.762 |  0.958 |\n| 0.6 | 0.178 |         0.782 |  0.953 |\n| 0.7 | 0.075 |         0.875 |  0.953 |\n| 0.8 | 0.05  |         0.931 |  0.949 |\n| 0.9 | 0.037 |         0.95  |  0.94  |\n\n*Test Loss*\n\n|     |      ERM |   One-Step GD |     RDIA |\n|----:|---------:|--------------:|---------:|\n| 0   | 0.141025 |      0.161033 | 0.13819  |\n| 0.1 | 0.237151 |      0.24697  | 0.146115 |\n| 0.2 | 0.345921 |      0.347718 | 0.157343 |\n| 0.3 | 0.453018 |      0.451272 | 0.171787 |\n| 0.4 | 0.56437  |      0.557865 | 0.185989 |\n| 0.5 | 0.698697 |      0.63033  | 0.197167 |\n| 0.6 | 0.876071 |      0.590184 | 0.203124 |\n| 0.7 | 1.08732  |      0.492453 | 0.196512 |\n| 0.8 | 1.44151  |      0.362804 | 0.197389 |\n| 0.9 | 2.0108   |      0.265325 | 0.241273 |\n\n```\nimport numpy as np\nimport pandas as pd\nfrom sklearn import datasets\nfrom sklearn.model_selection import train_test_split\nfrom sklearn.preprocessing import StandardScaler\nfrom IPython.display import display\n\ndef sigmoid(u):\n    return 1.0 / (1.0 + np.exp(-u))\n\ndef logloss(w, b, x, y):\n    p = sigmoid(x.dot(w) + b)\n    return - (y.dot(np.log(p)) + (1 - y).dot(np.log(1 - p))) / y.size\n\ndef logreg(x, y, lam=0.1, lr=0.01, max_itr=100):\n    num, dim = x.shape\n    xo = np.concatenate([x, np.ones((num, 1))], axis=1)\n    w = np.zeros(dim+1)\n    for itr in range(max_itr):\n        p = sigmoid(xo.dot(w))\n        g = (p - y).dot(xo) / num\n        g[:-1] = g[:-1] + lam * w[:-1]\n        w = w - lr * g\n    return w[:-1], w[-1]\n\n# experiment\nlam = 1e-2\nacc, loss = [], []\nnr = np.linspace(0, 0.9, 10)\nfor noise_rate in nr:\n    acc_n, loss_n = [], []\n    for seed in range(10):\n\n        # breast cancer data\n        # train / val / test = 300 / 169 / 100\n        x, y = datasets.load_breast_cancer(return_X_y=True, as_frame=False)\n        x, xte, y, yte = train_test_split(x, y, test_size=269, random_state=seed)\n        xval, xte, yval, yte = train_test_split(xte, yte, test_size=100, random_state=seed)\n\n        # normalize features\n        scaler = StandardScaler().fit(x)\n        x = scaler.transform(x)\n        xval = scaler.transform(xval)\n        xte = scaler.transform(xte)\n\n        # noisy label in training\n        np.random.seed(seed)\n        flip = np.random.rand(y.size) < noise_rate\n        y = np.logical_xor(y, flip).astype(int)\n\n        # fit logreg\n        w, b = logreg(x, y, max_itr=500, lam=lam)\n\n        # test accuracy / loss\n        zte = (xte.dot(w) + b > 0).astype(int)\n        acc_te1 = np.mean(yte == zte)\n        loss_te1 = logloss(w, b, xte, yte)\n\n        # influence function\n        num, dim = x.shape\n        xo = np.concatenate([x, np.ones((num, 1))], axis=1)\n        p = sigmoid(x.dot(w) + b)\n        H = xo.T.dot((p * (1 - p))[:, np.newaxis] * xo) / num\n        H = 0.5 * (H + H.T) + np.diag([lam] * dim + [0]) # Hessian\n        g = ((p - y)[:, np.newaxis] * xo) / num # gradient\n        f = - np.linalg.solve(H, g.T).T # influence function\n\n        # label gradient over validation loss\n        xv = np.concatenate([xval, np.ones((xval.shape[0], 1))], axis=1)\n        q = sigmoid(xval.dot(w) + b)\n        gv = (q - yval).dot(xv) / xval.shape[0] # gradient of validation loss\n        gy = f.dot(gv) * (np.log(p) - np.log(1 - p)) # label gradient\n\n        # label correction by one-step gradient descent\n        ynew = y - 1e+3 * gy # step size of GD = 1e+3\n        ynew = np.minimum(1, np.maximum(0, ynew)) # clip label\n        wnew, bnew = logreg(x, ynew, max_itr=500, lam=lam)\n\n        # test accuracy / loss after label correction\n        zte = (xte.dot(wnew) + bnew > 0).astype(int)\n        acc_te2 = np.mean(yte == zte)\n        loss_te2 = logloss(wnew, bnew, xte, yte)\n\n        # label correction by RDIA\n        ynew = y.copy()\n        #ynew[f.dot(gv) > 0] = 1 - ynew[f.dot(gv) > 0] # flip label\n        ynew[f.dot(gv) > 1e-3] = 1 - ynew[f.dot(gv) > 1e-3] # flip label\n        wnew, bnew = logreg(x, ynew, max_itr=500, lam=lam)\n\n        # test accuracy / loss after label correction\n        zte = (xte.dot(wnew) + bnew > 0).astype(int)\n        acc_te3 = np.mean(yte == zte)\n        loss_te3 = logloss(wnew, bnew, xte, yte)\n\n        acc_n.append((acc_te1, acc_te2, acc_te3))\n        loss_n.append((loss_te1, loss_te2, loss_te3))\n    \n    acc.append(acc_n)\n    loss.append(loss_n)\nacc = np.array(acc)\nloss = np.array(loss)\n\nacc = pd.DataFrame(np.mean(acc, axis=1))\nacc.columns = ['ERM', 'One-Step GD', 'RDIA']\nacc.index = nr\nloss = pd.DataFrame(np.mean(loss, axis=1))\nloss.columns = ['ERM', 'One-Step GD', 'RDIA']\nloss.index = nr\nprint('Test Accuracy')\ndisplay(acc)\nprint('Test Loss')\ndisplay(loss)\n```", "summary_of_the_paper": "This paper proposes a data relabeling technique using influence function.\nThe authors first derived the influence function for data relabeling as follows.\n\n$$\n\\eta_{\\theta \\delta}\\left(z\\_i, z\\_j^{c}\\right) \\left. \\triangleq \\frac{d l\\_j^c \\left(\\hat{\\theta}\\_{\\epsilon\\_i \\delta\\_j}\\right)}{d \\epsilon\\_i}\\right|_{c\\_i=0} =-\\nabla\\_{\\theta} l\\_j^c(\\hat{\\theta}) H\\_{\\hat{\\theta}}^{-1}\\left(\\nabla\\_\\theta l\\_i\\left(z\\_{i \\delta}, \\hat{\\theta}\\right)-\\nabla\\_\\theta l\\_i(\\hat{\\theta})\\right)\n$$\n\nThe authors then derived the specific expression for the cross-entropy loss:\n$$\n\\eta_{\\theta R}\\left(z_{i}, z_{j}^{c}\\right)=-\\nabla_{\\theta} l_{j}(\\hat{\\theta}) H_{\\hat{\\theta}}^{-1} w\\left(z_{i}, \\hat{\\theta}\\right)=-\\nabla_{\\theta} l_{j}(\\hat{\\theta}) H_{\\hat{\\theta}}^{-1}\\left(-\\frac{\\nabla_{\\theta} l_{i}(\\hat{\\theta})}{1-\\varphi\\left(x_{i}, \\hat{\\theta}\\right)}\\right)=\\frac{-\\Phi_{\\theta}\\left(z_{i}, z_{j}^{c}\\right)}{1-\\varphi\\left(x_{i}, \\hat{\\theta}\\right)}\n$$\nThe authors experimentally demonstrated that data relabeling is more effective than the standard sampling/data reweighting-based approach for model improvement.", "summary_of_the_review": "The proposed method is very similar to the one proposed in [Ref1], while the current paper misses this important prior work.\nAlthough the ways the amount of relabeling is optimized are different (using gradient descent or by one step update), I believe the close connection between [Ref1] and the proposed method needs to be described appropriately in the paper, e.g., through detailed discussions and experimental comparisons.\n\n* [Ref1] Training Set Debugging Using Trusted Items, AAAI 2018.\n\nThe strong aspect of this paper over [Ref1] is the experimental results where the results on DNNs (on MNIST and CIFAR10) are reported, while [Ref1] considered only kernel-based models.\n\n---\n## After Discussion with Authors\n\nI conducted an experiment on breast cancer dataset by myself (see below).\nThere, I confirmed that RDIA does better than the One-Step GD update of [Ref1].\nI therefore decided to increase my score with a strong expectation to the authors for **referring [Ref1] appropriately in the main body of the paper**.\nAs I mentioned in my original review, this is not the first study considering relabeling based on the influence function-like technique.\nRecalling that influence function is one specific example of implicit gradient, [Ref1] is the first work in this direction to my knowledge (even if [Ref1] does not describe the fact explicitly).\nAs I demonstrated in my code, the simplest version of [Ref1] with only one-step update (and without any human intervention) does good job.\n**I strongly expect the authors to pay certain respect to [Ref1] and do not underestimate their technical contribution**.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "8JfkDCTKdT", "reviewer": "Reviewer_UYUF", "summary": "", "questions": "", "limitations": "", "rating": 8, "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": "The manuscript is well-written and clear. The proposed solution is\nsimple and clean with a sound theoretical justification.\n\nThe practical utility of the approach is unfortunately limited given\nthe cost of estimating influence functions on non-linear models. This\nis a well-known problem, especially when applying influence function\nto deep architectures. The authors address the problem by proposing to\nreplace influence functions with training loss while retaining their\nrelabeling scheme. I wonder how this simple strategy works in\ncomparison to their RDIA approach (the appendix does not report\ncomparisons in terms of accuracy), and under which conditions it could\nfail (e.g. distribution shift rather than noise). Otherwise one has\nthe (probably false?) impression that all you need is training loss +\nrelabeling. ", "summary_of_the_paper": "The paper presents a relabeling scheme for binary and multiclass\nclassification tasks in which harmful training examples identified by\ninfluence functions are relabeled to improve performance on a hold-out\nset. The authors formally prove that the relabeling scheme determines\na reduction in loss wrt down-weighting or discarding examples, and\nreport experimental comparisons confirming the advantage of the\nproposed strategy.", "summary_of_the_review": "A simple and clean solution for influence-based data relabeling.\n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: All reviewers are very positive about this paper. The reviewer with the lowest score did independent experiments that show that the authors' method works well, and has had an extensive discussion with the authors that justifies a higher score. The paper is potentially very valuable to practitioners, since it shows how to compensate for a training set that is not representative of the test data.

Suggestion from the area chair to the authors: Briefly discuss the relationship between influence scores and propensity scores, which are standard in the literature on causal modeling and on sample selection bias, as in https://jmlr.csail.mit.edu/papers/volume10/bickel09a/bickel09a.pdf for example.
Decision: Accept (Oral)