Introduction
Regression is an important task in many disciplines such as finance (Zhang et al., 2017b ; Wu et al., 2020b ) , medicine (de Vente et al., 2021 ; Tanaka et al., 2022 ) ,
economics (Zhang et al., 2022 ) , physics (Sia et al., 2020 ; Doi et al., 2022 ) , geography (Liu et al., 2023 ) and more.
However, real-world regression labels are prone to being corrupted with noise, making it an inevitable problem to overcome in practical applications. In previous research, noisy label regression has been studied much in age estimation with noise incurred from Web data crawling (Rothe et al., 2018 ; Yiming et al., 2021 ) .
Beyond that, the issues of continuous label errors have also been reported in the tasks of object detection (Su et al., 2012 ; Ma et al., 2022 ) and pose estimation (Geng and Xia, 2014 ) as well as measurements in hardware systems (Zhou et al., 2012 ; Zang et al., 2019 ) .

The vast amount of noisy label learning research has focused more on classification than regression. Some notable approaches include regularization (Wang et al., 2019 ; Zhang and Sabuncu, 2018 ) ,
data re-weighting (Ren et al., 2018 ; Shen and Sanghavi, 2019 ) , training procedures (Jiang et al., 2018 ) ,
transition matrices (Yao et al., 2020 ; Xia et al., 2020 ) , contrastive learning (Zhang et al., 2021a ; Li et al., 2022b ) ,
refurbishing (Song et al., 2019 ) and sample selection (Lee et al., 2018 ; Ostyakov et al., 2018 ) .
Particularly, sample selection can be further divided into
exploring the memorability of neural networks (Arpit et al., 2017 ; Zhang et al., 2017a ) and delineating samples via the loss magnitude (Wei et al., 2020 ) .

To the best of our knowledge, there have been three works that address the noisy label problem for regression with deep learning. Castells et al. ( 2020 ) propose a weighted loss correction method based on the small loss assumption. Garg and Manwani ( 2020 ) propose an ordinal regression-based loss correction via noise
transition matrix estimation.
However, they assume that accurate noise rates are known in prior (Patrini et al., 2017 ) , which are hard to attain in practice. Yao et al. ( 2022 ) extend MixUp (Zhang et al., 2018 ) for regression to interpolate
the proximal samples in the label space to improve generalization and robustness.
Thanks to its regularizing effect, it can aid the noisy label issue.

In this work, we explore the regression problem with noisy labels, surpassing the scope of previous studies both empirically and methodologically.
For evaluation, we make three notable contributions.
Firstly, recognizing the absence of a standardized benchmark dataset for this task, we take the initiative to curate six balanced real-world datasets.
These datasets span diverse domains, encompassing age estimation (Niu et al., 2016 ; Yiming et al., 2021 ) , music production year estimation (Bertin-Mahieux et al., 2011 ) , and clothing price prediction (Kimura et al., 2021 ) .
Secondly, we conduct a comprehensive empirical benchmark exercise, evaluating the performance of fourteen baselines, which
are carefully selected from various branches of noisy label research extendable to regression tasks.
Lastly, we introduce a performance measure called Error Residual Ratio (ERR), which accounts for the unique property of regression, where labels exhibit varying degrees of noise severity.

Methodologically, we introduce the ConFrag (Contrastive Fragmentation) framework as a novel approach to address label noise in regression.
It is rooted in one fundamental characteristic of regression: the continuous and ordered correlation between the label and feature space.
In other words, data points similar in the feature space are likely to have similar labels.
The framework begins by partitioning the dataset into smaller segments, referred to as fragments, and pairs the most distant fragments in the label space to form what we call contrastive fragment pairs .
Training an expert network on these contrastive fragment pairs aids in generalization due to the distinctive feature matching and conversion of closed-set noise into open-set noise, which is less detrimental for learning.
Next, the framework incorporates neighboring relationships by aggregating and reordering the learned features to detect clean samples.
This is accomplished through the design of Mixture (Jacobs et al., 1991 ) of neighboring fragments.
Furthermore, we enhance our approach with neighborhood jittering regularization, which strengthens the selection process by improving the data coverage of each expert.
This, in turn, leads to improved agreements among neighboring fragments and serves as an effective tool for mitigating overfitting.
Finally, the contributions of this work can be summarized as follows.

We propose a novel method named ConFrag (Contrastive Fragmentation) for noisy labeled regression.
It leverages the inherent orderly relationship within the label and feature space by employing contrastive fragment pairing and constructs a mixture model based on neighborhood agreement.
This is further enhanced by our neighborhood jittering regularization.

We perform one of the most thorough empirical investigations into noisy labeled regression up to date.
We assemble six well-balanced benchmarks using datasets of AFAD (Niu et al., 2016 ) , IMDB-Clean (Yiming et al., 2021 ) , IMDB-WIKI (Rothe et al., 2018 ) , UTKFace (Zhifei et al., 2017 ) , SHIFT15M (Kimura et al., 2021 ) , and MSD (Bertin-Mahieux et al., 2011 ) , on which we evaluate fourteen baselines.
We design a metric termed ERR (Error Residual Ratio), which accounts for the degree of noise severity within the labels, offering a more comprehensive assessment.
Our experiments affirm the superiority of ConFrag over state-of-the-art noisy label learning baselines.