Resolving Training Biases via Influence-based Data Relabeling

1 INTRODUCTION . Training data plays an inevitably important role in delivering the model ’ s final performance . It has been well recognized that the training bias issue will compromise model performance to a 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 at possibly every step of the data lifecycle ( Anderson & McGrew , 2017 ; Dolatshah et 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 Ptrain ( x , y ) and Ptest ( x , y ) , but Ptrain is different from Ptest ( Guo et al. , 2020 ; He & Garcia , 2009 ; Zou et al. , 2019 ) . Both corrupted labels and distribution mismatch will hurt the generalization ability of a trained model ( Fang et al. , 2020 ; Zhang et al. , 2017 ) . We generally refer to training samples with corrupted labels or those inducing distribution mismatch as harmful samples . Data resampling is a widely used strategy to deal with harmful training samples . Existing resampling approaches ( Chawla et al. , 2002 ; Mahmoody et al. , 2016 ; Malach & Shalev-Shwartz , 2017 ; Ren et al. , 2018 ) propose to assign different weights to training samples , which aim to mitigate the negative impacts of harmful samples on model ’ s generalization ability . Among them , most resampling approaches ( Arazo et al. , 2019 ; Han et al. , 2018 ; Li et al. , 2020 ; Wang et 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 it is often beneficial to downweight them during the process of model training . However , such loss-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 , one recent work ( Wang et al. , 2020b ) proposes a new resampling scheme based on influence functions ( Cook & Weisberg , 1980 ) . The idea is to estimate the influence of each training sample on model ’ s predictions over the test set . Any training samples that would cause an 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 a specific model or data type ( Koh & Liang , 2017 ; Koh et al. , 2019 ) . Inspired by the success of influence-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 a theoretical guarantee on the model ’ 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 a 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 a 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-based resampling approaches ( Wang et al. , 2018 ; Ting & Brochu , 2018 ; Ren et al. , 2020 ; 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 influencebased 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 applying R over harmful training samples identified by influence functions is able to achieve lower test loss for any classification tasks using cross-entropy loss function . Third , we conduct extensive experiments on real datasets in different domains . The results demonstrate that ( i ) RDIA is effective in reusing harmful training samples towards higher model performance , surpassing the existing influence-based resampling approaches , and ( ii ) RDIA improves model ’ s robustness to label noise , outperforming the current resampling methods by large margins . 2 BACKGROUND . Let D = { ( xi , yi ) ∈ X × Y | 1 ≤ i ≤ N } be the training set which are sampled from Ptrain ( x , y ) . Let zi = ( xi , yi ) where xi ∈ Rd and yi ∈ RK . Let ϕ ( x , θ ) be a model ’ s prediction for x , where θ ∈ Rp is the parameter set of the model . We denote the loss of sample zi by l ( zi , θ ) = L ( yi , ϕ ( xi , θ ) ) and use li ( θ ) to represent l ( zi , θ ) . We consider the standard empirical risk minimization ( ERM ) as the optimization objective . Formally , the empirical risk over D is defined as : L ( D ; θ ) = 1N ∑N i=1 li ( θ ) . 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 . Influence functions . Influence functions , stemming from Robust Statistics ( Huber , 2004 ) , have provided an efficient way to estimate how a small perturbation of a training sample would change the model ’ s predictions ( Koh & Liang , 2017 ; Koh et al. , 2019 ; Yu et al. , 2020 ) . Let θ̂ = arg minθ 1 N ∑N n=1 ln ( θ ) be the optimal model parameters on convergence . When upweighting a training sample zi on its loss term by an infinitesimal step i , we obtain the new optimal parameters θ̂ i on convergence as : θ̂ i = arg min θ 1 N ∑N n=1 ln ( θi ) + ili ( θ ) . Based on influence functions ( Cook & Weisberg , 1980 ; Koh & Liang , 2017 ) , we have the following closed-form expression to estimate the change in model parameters when upweighting zi by i : ψθ ( zi ) , dθ̂ i d i | i=0= −H −1 θ̂ Oθli ( θ̂ ) , ( 1 ) whereHθ̂ , 1 N ∑N n=1 O 2 θln ( θ̂ ) is the Hessian matrix and O 2 θln ( θ ) is the second derivative of the loss at training point zn with respect to θ . Using the chain rule , we can estimate the change of model ’ s prediction at a test data zcj sampled from the given test distribution Ptest ( Koh & Liang , 2017 ) : Φθ ( zi , z c j ) , dlj ( θ̂ i ) d i | i=0= −Oθlj ( θ̂ ) H −1 θ̂ Oθli ( θ̂ ) . ( 2 ) At a fine-grained level , we can measure the influence of perturbing training sample zi from ( xi , yi ) to ( xi , yi + δ ) . Let ziδ = ( xi , yi + δ ) and the new loss li ( ziδ , θ ) = L ( yi + δ , ϕ ( xi , θ ) ) . According to ( Koh & Liang , 2017 ) , the optimal parameters θ̂ iδi after performing perturbation on zi is θ̂ iδi = arg min θ 1 N ∑N n=1 ln ( θ ) + ili ( ziδ , θ ) − ili ( θ ) . This allows us to estimate the change in model parameters after the fine-grained data perturbation using influence functions as : dθ̂ iδi d i | i=0 = ψθ ( ziδ ) − ψθ ( zi ) = −H−1 θ̂ ( Oθli ( ziδ , θ̂ ) − Oθli ( θ̂ ) ) . ( 3 ) Further , the influence of perturbing zi by ziδ on model ’ s prediction at test sample zcj is the following : ηθδ ( zi , z c j ) , dlj ( θ̂ iδi ) d i | i=0 = −Oθlj ( θ̂ ) H−1θ̂ ( Oθli ( ziδ , θ̂ ) − Oθli ( θ̂ ) ) . ( 4 ) It is important to notice that Eq . ( 4 ) holds for arbitrary δ when i 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 et 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 or data structure . Inspired by this , many influence-based 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 . ( 1 ) - ( 2 ) , we perform the fine-grained perturbation on a training sample ’ s label and evaluate its influence using Eq . ( 3 ) - ( 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 a novel viewpoint on handling biased training data .

Summary: The authors present an approach and framework to mitigate training biases by combining influence functions and data relabeling.  The 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.   Other 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.  The current authors combine both approaches, and present a framework that relabels harmful training data based on influence functions (on the test set).   The results of their experiments show that they are able to reduce the test loss compared to other data resampling approaches.