Revive Re-weighting in Imbalanced Learning by Density Ratio Estimation
Authors: Jiaan Luo
Feng Hong
Jiangchao Yao
Bo Han
Ya Zhang
Yanfeng Wang
Venue: NeurIPS 2024 poster (2024)
Keywords: machine learning
density ratio estimation
optimization
Primary area: optimization

Abstract: In deep learning, model performance often deteriorates when trained on highly imbalanced datasets, especially when evaluation metrics require robust generalization across underrepresented classes. To address the challenges posed by imbalanced data distributions, this study introduces a novel method utilizing density ratio estimation for dynamic class weight adjustment, termed as Re-weighting with Density Ratio (RDR). Our method adaptively adjusts the importance of each class during training, mitigates overfitting on dominant classes and enhances model adaptability across diverse datasets. Extensive experiments conducted on various large scale benchmark datasets validate the effectiveness of our method. Results demonstrate substantial improvements in generalization capabilities, particularly under severely imbalanced conditions.

Revive Re-weighting in Imbalanced Learning by Density Ratio Estimation
Jiaan Luo 1,3 Feng Hong Jiangchao Yao Bo Han
Ya Zhang2.3 Yanfeng Wang
:Cooperative Medianet Innovation Center, Shanghai Jiao Tong University
Schooo of Artificial Intelligence, Shanghai Jiao Tong University
Shanghai Artificial Intelligence Laboratory
*Hong Kong Baptist University
(luojiaan, feng. hong, Sunarker, 2__zhang, wangyanfeng) jsj jujtu.uuuu.u edu. cn bhanml @comp hkbu.edu.ht.
Abstract
In deep learning, model performance often deteriorates when trained on highly im- balanced datasets, especially when evaluation metrics require robust generalization across underrepresented classes. To address the challenges posed by imbalanced data distributions, this study introduces novel method utilizing density ratio esti- mation for dynamic class weight adjustment, termed as Re-weighting with Density Ratio (RDR). Our method adaptively adjusts the importance of each class during training, mitigates overfitting on dominant classes and enhances model adaptability across diverse datasets. Extensive experiments conducted on various large scale benchmark datasets validate the effectiveness of our method. Results demonstrate substantial improvements generalization capabilities, particularly under severely imbalanced conditions. The code is available here.
Introduction
In recent years, deep learning has made significant strides across various domains by utilizing complex architectures and large-scale datasets, setting new benchmarks for performance. However, these advancements often rely on well-curated datasets that ensure balanced class distributions [Russakovsky et al., 2015]. In contrast, real-world datasets typically exhibit long-tailed distribution, where few classes dominate the majority of samples, while many others are underrepresented [Krizhevsky al., 2009]. This imbalance leads model biases favoring frequent classes, thereby reducing performance on the less common ones. Yet, in many applications -uccc as medical diagnostics and financial analysis- greater emphasis is placed on ensuring strong generalization for underrepresented classes. Addressing this challenge not only reduces data collection costs but also improves the robustness and fairness of the models.
Many excellent methods, such as re-sampling [Bowyer et al., 201 re-weighting [Morik et al., 1999], decoupled learning [Kang al., 2020], margin-based learning [Cao al., 2019, Menon et al., 2020), transfer learning [Yin et al., 2019] and contrastive learning [TTian et al., 2021]. have been proposed to tackle the issue of imbalanced data. Despite the simplicity re-weighting. it falls behind in performance significantly compared with other directions of methods due to the inappropriate weighting coefficients during training. Cui et al. [2019] proposes a method for re-eeighting by effective number, which accounts for potential overlaps among data samples and adjusts the weights for each category based on the actual effective number of samples. Chen al. [2023b] leverages the effective area to re-weight, considering the actual spanned space of each class. However, such subsequent improvements can alleviate but still cannot effectively push that forward. Wang al.
'he first two authors contribute equally. 'The corresponding author is Jiangehao Yao (Sunarkerasjtu edu. en).
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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[2023] obtains fine- aeeeeerrindd generalization bound for re- re-weighting in imbalanced learning through the data-dependent contraction technique. Limited research has focused on the intrinsic limitations of the commonly employed re-weighting-basee loss functions and the corresponding balancing mechanisms designed to enhance parity in class representation.
This study rethinks the characteristics of re-weighted loss and explores the question "Why is re- weighting necessary under conditions of sample imbalance?" Under conditions of sample imbalance, the variation in weights of samples arises due to discrepancies between the distribution of collected data and balanced data distribution. In scenarios where class balance exists, such discrepancies are absent, thus obviating the need for re-weighting. Conversely, in imbalanced settings, re- re-weighting becomes essential to bridge the gap between these distributions. The weights must therefore repre- sent a suitable compromise between balanced and imbalanced distributions and necessarily reflect accurately on each sample. Additionally, as model training progresses dynamically, optimizing the fit to feature distributions, the weights applied to each sample should be continuously updated to maintain robust performance.
This research introduces a novel method, Reweighting with Density Ratio (RDR), designed to mitigate learning disparities in imbalanced distributions. In this method, a feature extractor is employed to discern the features from the training data. A more balanced feature distribution is approximated by continuously updating the momentum on the feature level. This enables real-time density ratio estimation with features learned under imbalanced distributions, thereby obtaining the sample-wise weights. Notably, as the learned features evolve, our method dynamically adjusts weights in response to observed shifts in class density throughout the training cycle, ensuring that the model remains adaptive and effective. This method significantly enhances the robustness and adaptability of the training process. By integrating density ratio estimation to evaluate the difference between the balanced and real data distributions, our approach more accurately reflects the underlying class distribution and improves the model's s generalization capabilities across diverse datasets. The contributions are summarized as follows:
We explore the existing re-weighting techniques, and model the performance of various algorithms during training under different data distributions. This approach offers a novel perspective on understanding re-weighting methods in the scenarios of sample imbalance.
We introduce a novel methodology, Re- Re-weighting with Density Ratio (RDR), which leverages the method of density ratio estimation to dynamically adjust class weights during model training. This approach not only addresses the limitations of prior re-weighting methods but also introduces a mechanism to continuously adapt to the changing importance of classes as learning progresses, thereby enhancing model robustness and adaptability.
We conduct extensive experiments to validate the effectiveness of our proposed RDR method. These experiments are conducted across various large-scale, long-tailed datasets, demonstrating substan- tial improvements in handling class imbalance. Our results illustrate significant enhancements in generalization capabilities, particularly under severely imbalanced scenarios.
2 Related Work
2.1 Re-weighting Based Methods
Re-weighting methods for addressing class imbalance have evolved significantly over the years. Early techniques, such as [Zadrozny et al., 2003), employed inverse frequency techniques address class imbalances but failed to consider deeper data distribution traits, leading to sub-optimal outcomes. Addressing these shortcomings, Huang et al. [2016] introduced cost-sensitive learning framework that, beyond simple frequency adjustments, incorporated misclassification costs to achieve a more nuanced balance. However, this approach still struggled with complexities like class overlap and label noise. To further refine this approach, Lin et al. [2017] developed Focal Loss, which employs a modulation factor based on the prediction probability to adjust the loss function, thereby amplifying the impact of hard-to-clal cacd-so--lassifyy samples while reducing the loss contribution of -acasso-yl samples. Cui et al. [2019] introduced
Maas-BBaaacced Loss which adjusts loss by data overlap, calculating the effective number of each class. Advancements continued with methods based on training gradients, such as (Ren et al., 2020b, Wang et al., 202 Chen et al. [2023b] proposed Adaptive Re-weighting via effective area, which enhances model accuracy by considering the spatial distribution and density

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of data points within classes. Ma et al. [2023] introduced a re-weighting method that adjusts based on semantic richness and visual variability. However, no prior work has tackled the issue of sample imbalance by dynamically re- re-weighting based on the model's performance across training and test sets with differing distributions during the training process.
2.2 Non-re-weighting Based Methods
In addition to re-weighting, many other methods are available to address the issue of sample imbalance. Re-sampling techniques [Kubat and Matwin, 1997, Wallace al., 2011, Han al., 2005, Hong et al., 2024a] mitigate category imbalances by under- under-sampling dominant classes [Buda et al.. 2018] or over-sampling minority classes [Bowyer al., 2011]. However, under-sampling may degrade feature representation by discarding valuable majority class data, whereas over-sampling could cause overfitting by duplicating minority class samples. Decoupled training approaches, such as [Kang et al., 2020], challenge the traditional joint training model by separating representation learning from classification. Margin-based methods such as, LADM [Cao al., 2019], LA [Menon et al., 2020] and VS [Kini et al., 2021], adjust training processes to increase minority class margins, therefore to obtain a more balanced decision boundary. More flexible and robust methods are proposed, including Transfer Learning [Yin et al., 2019, Liu et al., 2019], Contrastive Learning [Li et al., 2022, Chen et al., 2023a], Ensemble Learning [Wang et al., 202 lb, Cai et al., 2021] and Self-supervised Learning [Liu et al., 2022, Zhou al., 2023b]. Please refer to Appendix B for more discussions.
3 Method
3.1 Problem Setup
For a typical classification task in imbalanced learning, suppose given training dataset S (E,,((,,))) 31),. where n is the total number of samples. Denote {π., R.,..., na the sample number of each class. We assume, without loss of generality, that n, ny. when i j. with na typically much larger than nı, reflecting a pronounced imbalance in class distribution. We use a typical loss function like L: W x X x y + R+. Denote { (T: ..., Ta}
as the proportions of each class, such that Σ मं = 1. Define family of deep learning models parameterized by W C T Typically, a model consists of a feature extractor If (x; o) and classifier 1(2;; 0), with Ui 0}. The notations used in this paper are summarized in Appendix A.
3.2 Motivation
Inspired by our review of prior methods, we observe gap in the adaptation of dynamic class-weight adjustments during training phases. Building on the groundwork of static re-weighting strategies, we introduce a novel approach utilizing the method of density ratio estimation dynamically recalibrate class weights. This innovation aims to provide more refined adjustment by estimating real- time class density, thereby promoting an equitable influence of all classes throughout the training.
3.3 Dynamic Re-weighting with Density Ratio
In a typical training optimization problem, our objective is to minimize the empirical risk of the loss function, i.e,, R 17=1 n Σi= (Ix, w). However, in imbalanced datasets, where the frequency of samples across different classes varies, it is necessary to adjust for these gaps by applying different weights for the samples. We assume that the weight of each sample is denoted by a(x, yr w), then the empirical risk can be formulated like π + Σ a(x,, ysi (x,, w)).
In naive re- re-weighting approaches, the weight a of class y is often set to +, on the assumption that the distribution of the training set P and the distribution balanced data set Phat satisfy the equation Pazly: w) = Phot (xly: w). However, in practical training scenarios, both training and test sets are subsets drawn from the actual distribution, leading potential missing of feature patterns. Furthermore, classes with more complex features and lower sample frequencies tend to exhibit more pronounced missing of patterns. Therefore, in the training process, there exists discrepancy between Pa w) and Poot(rlu;: We measure the extent of this discrepancy using the ratio r( r2llp; w) = P(xly; w) Poal(zyy w), incorporating it as correction term into our weighting
1. This setting is based
3

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scheme. Consequently, the empirical risk can be reformulated as follows
R =
(z ys:w)
Ky.
We can explain the rationality of this formula as follows. Considering each class i, where w)) P(yew) and Poat(z:ly(w) = ability formula, we can derive:
with conditional prob-
R = Ep 1 n
(r(s: w(w)) Ep Pyal(): Poas (all(w))))) M2, w:w) P(y:w) P(aly:w)
Poat yiw) (x., y: ())
Ep
= E Poal (:w
P(x, P(x,y:w) w:
Eq. (2) demonstrates that our approach aligns with the balanced risk of the loss function. Consequently, minimizing Eq. (1) also serves to minimize the balanced risk.
Let's take closer look at r(æly; w) w/ (xly; The variable represents the ratio of two different distributions. We approximate this ratio using methods of density ratio estimation. This problem can be solved by first-order moment matching approach. Our goal is to minimize
argmin /
xr (r(xly: Poat (2ll
((rrlg:))dr
(3)
where ||-|| denotes the Euclidean norm. Recall that we capture the features of the input samples by the feature extractor (z; Φ) in our model, and these features are good reflection of what our model learned from the distribution of the input samples. Therefore, in order to capture more complex structures and patterns in raw data, we use fax ø) to obtain a variant of Eq. (3). Our goal can be achieved by obtaining argmin, MM (r), where MM (r) denotes
|/ [x; (rrrll))) w) Posl (xly; / [ax ()
where MM stands for 'moment matching". Let ignore the irrelevant constant in MM (r), and define the rest as MM(r):
|f (aal (1) -2 (aa
/ ((alow)a
(5)
where () denotes the inner product. In practice, as for the real-world imbalanced data distribution P. we denote pr to dynamically reflect the knowledge learned from the distribution P, that is p (f(
f (z.: o)). Remember that the output of feature extractor is z, i.e., 2 = f fx; o) a Z-dimensional vector, then 4 would [Z, n] dimeenssioa vector. Similarly, we denote & for class i, which is 12, n1:] -dimensional vector. As for the balanced data distribution Pout. we design a momentum mechanism to accumulatively estimate the expectation of features learned from balanced data distribution along with the training. Concretely, for each class, maintain prototype feature F' for the entire training progress, using each batch's 's feature expectation for momentum updates. Therefore, define Fr... follows Fr... (Fi,...,
Since the total number of classes Fr... would [Z, d] dimensiioal vector. For each batch, we can obtain z (2..... za), where z, the mean of z of all samples in class i. Then, the momentum updates works as follows
F f- mFFrr Poar + (1-m))
where [0, 1) is momentum coefficient. Back Eq. (5), replace the expectations over Poal and P by 4 and FP.... respectively. Then, take the derivative of MM(r) with respect to r and set it zero. Detailed derivations provided in Appendix C.I. For each class i, we can obtain the estimation of density ratio in imbalanced learning as follows
F = nz /
43
-1
2 F.

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(a) Many Category
(b) Few Category
Figure I: Dynamic trend of the RDR weights well inversely aligns with B' throughout the training
process in different categories. By denotes VTy [L softmax (m)))) where (m))) denotes the minimal prediction the ground-truth class y,, i.e., minges, mm)r) Experiments were conducted
on CIFAR-10-LT dataset with an imbalance factor of 10.
Substitute Eq. (7) into Eq. (1), we can obtain our object to optimize
R
n4
8/))
D F, F.11(I,. U3:w)
T Mj=i
(8)
Φ F. (.-1(.)11;1)) 8
1=
In our implementation, we introduced a warm-up phase to pre-adapt the feature distribution in Fr thereby mitigating excessive oscillations during the initial stages of training. Additionally, employed a temperature coefficient to modulate the influence of weights, which is typically set to 1. When integrating with logit adjustment (LA) [Menon et al., 2020], adhere the same procedures outlined in [Wang et al., 2023] to ensure the fisher consistency. The framework and pseudo-code of our method are shown in Appendix C.2 and Appendix C.3.
3.4 Generalization Bound Analysis
Here, use formal generalization analysis to characterize the interesting point of our method.
Theorem 1. Given model M and the loss function l, for any 8 (0, 1), with probability at least 8 over the training set S, according to [Wang et al., 2023], the following generalization bound holds for the risk the balanced distribution
(m) 2 (1, + Gs(M)
Σ
Wyy Woy softmax (B,(
(9)
di
where & (1. 8) is positively correlated with the empirical re-weighting risk of the training set. (s(M) denotes the empirical complexity of the function M. Bf( denotes the minimal prediction on the ground-truth class in the training set. Wy refers to the weight of class of the re-weighting loss.
Specifically, from the above generalization bound, we can find two inherent requirements for re- weighting methods. 1) Why re-weighting necessary: Wy helps re-balance the imbalanced term /Ty softmax (m))) to get a sharper bound. 2) Why dynamic re-weighting is necessary: The term (m) changes dynamically with model training. Therefore, we need a wy that can adapt dynamically to the changes of (y)m). 3) Why RDR works: From Fig. 1, we observe that the dynamic trend of the RDR weight aligns well with / ,,y [1 softmax (By(m))), denoted as By-- This shows that our RDR can adapt to the dynamics in B', maintaining a sharp bound during training.
3.5 Implementation and Complexity Analysis
At the end of each epoch, a global variable maintained and updated using momentum, as described by Eq. (6). Within each minibatch, the weight of each sample is computed dynamically. Typical
5

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optimization procedures for deep neural networks entail both forward and backward passes per mini- batch, characterized by computational complexity ( where represents the batch size and A denotes the overall parameter size. Within the RDR framework, suppose the feature dimension used as input to the classifier K, and the sample weights are computed according to Eq. (8). This computation for all d classes aggregates to a complexity of O(Ei= (n; x K² + K²)) where n; the sample count of class in minibatch. Given that K generally exceeds n4ç the complexity predominantly stems from the matrix inversion, approximating to (ddk*). The complexity for momentum updates is (BK).. Notice that K and B considerably minor relative the scale of the model parameters, rendering the time overhead of this method manageable.
On the storage front, the memory cost of the RDR primarily arises from the matrix inversion step in Eq. (8), resulting in a space complexity of O(K*). Given the scales of is much lower than A, the extra memory usage is negligible when compared with the memory utilization of the model parameters. To this end, RDR imposes a relatively small computational or space cost, enabling its integration with existing approaches reduced cost. An empirical evaluation of the computational expense is presented in Fig. 2. For more discussions about limitations of RDR. please refer to Appendix E.
240
190
(a) Time Cost on CIFAR-IO-LT
(b) LT
Time Cost on CIFAR-100.
Figure 2: Visualization of the time cost for training 200 epochs using four methods: CE, RDR, SAM and RDR(SAM) on CIFAR-IO-LT and CIFAR-100-LT datasets.
Figure 3: The impact of momen- tum coefficient m in RDR under the measure of top-1 accuracy.
Expriments
4.1 Experimental Setup
Datasets. We conduct experiments on four major long-tailed datasets, CIFAR-IO-LT, CIFAR-100-LT, ImageNet-LT [Liu et al., 2019] and Places-LT [Liu et al., 2019]. CIFAR-IO-LT and 100-LT
two datasets sampled from the original CIFAR [Krizhevsky al., 2009] dataset with total of 10 and 100 classes, respectively. We conduct experiments with different imbalance factors IF Amaz where 7maz and mmmi denotes the number the most and least frequent classes [Kang et al., 2020, Hong et al., 2023, 2024b]. Following the mainstream protocol [Wang al., 2023], set the imbalance factor as 100 and 10 for evaluation. ImageNet-LT has 115.8K training images covering 1000 classes, with imbalance factor being 256. The number of samples per class ranges from 1280 to 5 images. Places-La contains 62.5K training images covering 365 categories, with imbalance factor being 996. The number of samples per class ranges from 4980 to 5 images.
Evaluation Protocol. In the task of long-tailed classification, all classes are treated equally during testing. Following [Rangwani et al., 2022, Zhou et al., 2023c], we also report accuracy on three splits of classes according to the number of training data. Since the number of samples per class increases by its class index, for CIFAR-IO-LE dataset, class[O. 3), class[3, 7) and class[7, 10) reported as Many, Medium and Few classes, respectively. Similarly, CIFAR-IO0-LT is splited as class[O, 35), c3ass|35, 69) and class|69, 100). ImageNet-LT is splited class(O, 390), class|390, 835) and class[835, 1000), while Places-LT is splited as class(O, 131), class[131, 288) and class[288, 365).
Baselines. Our method is combined with existing long-tailed classification methods to demonstrate the efficacy, including the baseline trained by cross-entropy loss (CE), focal loss (Focal) [Lin et al., 2017 class-balanced loss (CB) [Cui et al., 2019] and logit adjustment (LA) [Menon et al., 2020]. Recently, Sharpness-Auu minimization (SAM) [Foret al., 2021] has been proved to be a powerful method in imbalanced learning, therefore we also adopt baseline including SAM [Foret et al., 2021],

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Table 1: Top-1 accuracy (%) (1) results for overall classes on CIFAR-1O-LI CIFAR-100-LT,, ImageNet-LT and Places-LT, CIFAR-IO-LT and CIFAR-100-LT are employed with imbalance factors of 10 and 100, respectively.
CIFAR-
CIFAR-100-14
Dataset
ImageNet-LT Places-LT
IF=10
IF=100
IF=10
IF=100
CE
88.9
75.610.8
59.3
42.7.0.3
43.2.1.1
29.310.2
Focal
8..
76.0.0.1
59.7
43.0.1..6
43.810.8
29.5
CB
89.0=0.4
76.7.0.88
60.4
43.5
43.810.1
32.510.3
LA
89.2=0.3
82.2+0.7
62.3=0.5
48.2+0.4
47.9+0.4
37.5+0.2
RDR+CE
89.9=0.1
81.9
62.3
48.5+10.4
45.2:t0.1
39,4:0.22
RDR+LA
90.2=0.4
83.4
62.9.0.2
49.4
48.1+0.33
39.5+0.1
ImbSAM [Zhou et al., 2023a] and CCSAM [Zhou et al., 2023c], the latter two are also SAM-based methods. The strategies above have been demonstrated superior performance in imbalanced learning,
Implementation details. Our code is implemented with Pytorch 1.12.1. Experiments based on CIFAR- 10-LT and CIFAR-100-LT are carried out on NVIDIA GeForce RTX 3090 GPUs, while experiments based on ImageNet-LT and Places-LT are carried out on NVIDIA A 100 GPUs. For fair comparison, we use ResNet32 on CIFAR-10-LE and CIFAR-100-LT, ResNe150 on ImageNet-LT and pre-trained ResNet-152 on Places-LT. We train each model with batch size of 128 (for CIFAR-IO-LT and CIFAR-100-LT) / 256 (for Places-LT and ImageNet-LT), SGD optimizer with momentum of 0.9, weight decay of 0,0002. The initial learning rate set to 0.1, with cosine learning-rate scheduling along training. The results of ImbSAM and CCSAM are obtained by implementing the official codes.
4.2 Comparison Results
Comparative analyses have been performed evaluate the effectiveness of the proposed RDR. The results are presented in Table 1, Table and Table 4. The metric employed to measure performance the top-1 accuracy on the test sets.
Results on CIFAR-10-LT and CIFAR-100-LT. We first evaluate RDR on CIFAR- and CIFAR- 100-LT. We report the final accuracy of different methods with imbalance factor ratio { {10, 100) in Table 1 and Table 2. We can observe that RDR significantly outperforms all baselines under different imbalance factor ratios across the two datasets. Our observations highlight that RDR consistently outperforms baselines across various class distributions MManti Medium, and Few particularly under severe imbalance (IF=100).
In CIFAR- 10-LT, combined with CE and LA, our method shows substantial improvement in the categories with fewer samples, increasing the accuracy by 19.8% and 9.5% respectively in the Few category under IF=100. This improvement is notable as it effectively addresses the challenge of learning from scarce data. With the inclusion of SAM, the performance of RDR is further enhanced. Under a less severe imbalance (/F=10), where the results show less performance drop-off between categories, RDR combined methods still maintain high performance across all categories, suggesting scalability and reliability of our approach in different imbalance contexts.
In CIFAR-100-LE where the data distributions are more diverse and challenging, RDR also enhances the overall performance, particularly for the Mediam and Few categories. Under the imbalance factor of 10 and 100, RDR increases the accuracy in Few classes by 11.8% and 20.0% respectively, compared to the original CE loss. Furthermore, it is notable that techniques like ImbSAM and CCSAM, which especially focus the Few categories, may heavily sacrifice the performance results in both datasets show that RDR generally outperforms the Many classes compared to the other two variants of SAM, indicating that RDR can efficiently address the overfitting issues for Few classes. For more experimental details, please refer to Appendix D.2 and Appendix D.I.
Many classes. The
Flat minima of loss landscape. Key metrics associated with Eigen Spectral Density, such as the maximum and minimum eigenvalues (Aman and Amin) and the trace of the Hessian matrix (Tr(H)), effectively reflect the smoothness of the loss landscape. Lower values of Amas and (r(H) indicate smoother loss landscape. Rangwani et al. [2022] have demonstrated that smoother loss landscapes correlate with stronger model generalization, which is particularly crucial when dealing
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Table 2: Top-1 accuracy (%) (f) results for Many, Medium, Few and overall classes CIFAR-10: LT, categorized by imbalance factors (IF) 100 and 10. The experiments are employed with the integration of Seappesss-wware-Miimiiaane
methods.
IF=100
IF=10
Dataset
Loss
Method
Many
Med.
Few
Many
Med.
Few
All
SAM
94.8
74.5
60.6
76.4
95.8
85.9
86.5
89.3
ImbSAM
94.6
73.9
67.7
78.3
94.2
84.1
90.1
89.0
CE
CCSAM
85.6
79.2
80.3
81.4
90.6
85.3
90.8
88.5
RDR+SAM
91.8
78.6
81.9
83.6
94.2
86.3
92.5
90.5
CIFAR-10-LT
SAM
90.8
78.1
82.9
83.3
92.6
86.5
91.8
89.9
ImhSAM
85.2
75.1
89.6
82.5
90.8
83.6
94.8
89.1
LA
CCSAM
85.8
77.8
80.5
81.0
90.7
82.4
90.8
87.4
RDR+SAM
88.5
80.0
85.0
84.1
92.5
86.5
93.6
90.4
SAM
72.7
40.3
7.5
41.5
75.3
60.2
45.1
60.8
ImbSAM
71.5
40.6
17.7
44.3
72.5
57.9
53,5
61.7
CCSAM
61.5
50.8
29.7
48.0
62.8
59.3
54.5
59.0
RDR+SAM
63.4
51.9
30.5
49.3
68.0
61.6
58.3
62.8
CIFAR-I00-LT
SAM
64.3
50.5
30.8
49.2
66.1
60.9
59.7
62.3
ImbSAM
58.8
45.4
40.1
48.4
62.9
56.6
64.2
61.2
LA
CCSAM
57.7
48.9
29.0
45.8
61.0
57.8
51.9
57.1
RDR+SAM
63.9
52.4
30.5
49.6
67.6
61.7
60.1
63.2
190
(a) CE
(b) RDR
(c) SAM
(d) RDR(SAM)
Figure 4: Eigen spectral density for the class with the fewest samples across different methods. Experiments conduct on the 1IFAR-10-LT under an imbalance factor of 100. Maximum eigenvalue Amaz (L) minimum eigenvalue Amin (f) in the top right corner of each panel. lower Amaz indicates a smoother loss landscape, while higher Amin suggests conditions more favorable for escaping from saddle points, thereby enhancing the model's generalization capabilities.
with imbalanced data. Amin also serves significant indicator of the loss landscape characteristics. preponderance of negative eigenvalues from the Hessian spectrum, resulting smaller Amin values, empirically suggests convergence to saddle points. Saddle points typically represent regions in the loss landscape characterized by plateau with some negative curvature. In non-convex settings, has been shown that an exponential number of saddle points exist, and convergence to these points is indicative of poor generalization.
Fig. 4 illustrates the Eigen Spectral Density under different loss function training regimes. It is evident that combining our method with the CE technique significantly improves the loss landscape. On one hand, Amaz is substantially reduced, indicating flatter loss landscape. On the other hand, there is an increase in Amins suggesting our method's effectiveness escaping from saddle points.
Table delves deeper into the changes in the loss landscape for all Few classes. It reveals that combining our method with CE and SAM results in average reductions in Amaz by 58.5% and 66. 1%, respectively, and increases in Amin by 49.2% and 7.6%. respectively. Furthermore, our method significantly reduces Tr(H) for minority classes (class7, class8 and class9). The average value of Tr(H) decreases by 55.7% and 66.0% when combined with CE and SAM, respectively. These findings underscore the efficacy of our method improving the loss landscape and enhancing the generalization capability of the model [Dauphin et al., 2014].
Results on ImageNet-LT and Places-LT. Our experiments conducted ImageNet-LT and Places- LT, two large-scale datasets characterized by irregular and complex data distributions, demonstrate notable accuracy improvements through the application of RDR, shown in Table and Table 4.

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Table 3:
Loss landscape metries across different methods on CIFAR-IO-LT, with imbalance factor 100. Average minimum eigenvalues Amin (1), average maximum eigenvalues Amax (L), and the trace Tr (4) of the Hessian matrix for classes with few samples. Tre. Tr7. Trs, and Trg represent the traces of the Hessian matrix for class 6, 7, 8 and 9, respectively, with descending sample quantities. Fer denotes average trace of Hessian matrix over Few classes. Lower Amux and T'r values indicate flatter loss landscape, while higher Amin suggests landscape more conducive to escaping from saddle points, thereby potentially enhancing model generalization.
Method
Amin
Amax
Tre
Tr7
Trs
Tr9
Frwe
SGD RDR
-76.24
110.78
335.64
301.67
234.97
323.87
286.84
-38.73
45.95
290.06
293.04
46.15
41.71
126.96
SAM -14.42
74.82
300.13
314.48
123.10
291.00
242.86
RDR+SAM
-13.33
25.36
179.50
147.11
36.28
64.32
82.57
Specifically, when combined with CE and LA on ImageNet-LT, our method achieves accuracy enhancements of 18.7% and 2% in Few classes, respectively. While the SAM technique combined with CE and LA offers limited accuracy improvements, integration with our approach still results in an overall accuracy increase of approximately 2.3% compared to other methods. Notably, the LA method tends to suppress accuracy Many classes more than the CE method; however, this side effect is effectively mitigated when LA is combined with RDR.
Table 4: Top-1 accuracy (%) (1) results for Many, Medium, Few and overall classes on ImageNet-LT and Places-LT. The experiments are employed with the integration of Aeaennes--ware-innniii based methods.
ImageNet-LT
LL
Loss
Method
Many
Med.
Few
All
Many
Med.
Few
All
SAM
64.6
35.8
10.1
42.8
45.2
27.6
12.1
30.6
ImbSAM
62.5
37.3
13.9
43.3
43.1
26.2
16.1
30.1
CE
CCSAM
54.1
44.1
30.8
45.8
40.4
39.7
31.7
38.2
RDR+SAM
59.1
46.5
26.2
48.1
41.5
41.3
39.0
40.9
SAM
39.6
45.9
39.1
42.3
42.5
41.9
35.5
40.8
ImbSAM
56.1
45.1
37.9
48.2
38.6
35.8
40.4
37.8
LA
CCSAM
52.2
43.9
32.8
45.3
32.1
41.4
40.9
37.9
RDR+SAM
57.5
49.7
35.9
50.5
41.5
42.9
37.8
41.3
In the Places-LT dataset, the performance of RDR is even more pronounced. Combinations of our method with CE and LA result in accuracy gains 10. 1% and 2% in overall classes, respectively. Additionally, integrating SAM with our method also yields incremental improvements of 10.3% and 0.5% under CE and LA conditions, respectively. Our approach not only enhances accuracy in Few classes but also surpasses other methods in Medium classes, indicating its comprehensive efficacy across different categories. This broad applicability is particularly crucial for addressing the challenges of imbalanced learning.
Results on data with label noise. We further investigate the performance of our approach on datasets with label noise. Specifically, we evaluate two datasets: CIFAR-IO-LT-NI and CIFAR-100-LT-N both of which exhibit class imbalance and label noise. Experiments are conducted with noise ratio of 5%, and the results are presented in Fig. 5. As shown, our method demonstrates consistent and significant improvements on more datasets with label noise.
4.3 Ablation Study
In our study, we perform an ablation experiment validate the efficacy of the multiple components that comprise our method. The outcomes from experiments across four datasets are delineated in Table 5. is crucial to note that our weighting definition for each category i follows the formula r r/n..- When
1, our method simplifies to the traditional inverse frequency weighting wt = 1/nz. We explored the differences between our approach with and without the integration of SAM compared to this conventional weighting method.

---

(a) 1F=10 with CE
(b) IF=100 with CE
(c) IF=10 with LA
(d) 1F=100 with LA
Figure 5: Top-1 accuracy (%) (1) results for overall classes on LIFAR-IO-LTE and CUFAR-100-LL
NL with 5% noise ratio, categorized by imbalance factors (IF) of 100 and 10.
Table 5: Top-1 accuracy (%) (1) results from ablation studies across diverse datasets. Experiments conduct on CIFAR- - CIFAR-100-LE ImageNet-LT, and Places-LT, comparing different method combinations. 1/n denote classic inverse frequency weighting method, assigning weights of 1/n; for
class
Method
CIFAR-IO-LT
CIFAR-100-LT
Loss
ImageNet-LT
Places-LT
1/n
RDR
SAM
IF=100
IF=10
IF=100
IF=10
75.61
88.86
42.66
60.17
43.18
29.27
77.78
89.50
44.21
61.89
44.62
34.27
81.87
89.94
48.54
62.28
45.19
35,45
CE
76.44
89.33
41.46
60.82
42.78
30.64
80.98
90.22
44.82
61.43
47.47
38.73
82.88
90.52
49.30
62.80
48.06
48.06
82.15
89.17
48.34
62.27
47.86
37.51
82.27
89.27
48.55
62.26
45.86
37.48
83.44
90.21
49.35
62.88
48.12
37.79
LA
83.37
89.89
49.21
62.34
42.34
40.77
82.71
90.25
47.36
61.88
43.40
39,48
83.56
90.41
49.63
63.06
50.45
41.33
The results depicted in Table reveal that our dynamic weighting approach consistently outperforms the classic method under various scenarios. Without SAM, when combined with CE and LA, our method achieves accuracy improvements ranging from 0.4% 4.3% and 0.3% to 2.3%, respectively. When integrated with SAM, the improvement in accuracy is particularly notable on large datasets. Specifically, the accuracy enhancements on ImageNet-L.T and Places-ITT reach 7.1 % and 1.9% respectively when combined with LA. These results underscore the tangible benefits of our dynamic weighting strategy in enhancing model performance. The impact of momentum coefficient m in RDR shown in Fig. 3. For more experimental details, please refer to Appendix D.3.
5 Conclusion
In this work, we have introduced RDR, a novel approach for mitigating model degradation in imbalanced learning scenarios by dynamically adjusting class weights using density ratio estimation. Our method dynamically adjusts class weights during training based on density ratio estimation, enhancing both model robustness and adaptability. Extensive experiments diverse large-scale datasets demonstrate the effectiveness of RDR, particularly in severely imbalanced settings. Future work will focus on refining the dynamic adjustment mechanisms and exploring broader applicability across various domains and dataset complexities.
Acknowledgement
Jiaan Luo, Feng Hong, Jiangehao Yao, Ya Zhang and Yanfeng Wang are supported by the National Key R&D Program of China (No. 20227DD01 60702), STCSM (No. 2251 I 106101, No. 22D72229005), 111 plan (No. BP071901 19010) and National Natural Science Foundation China (No. 62306178). Bo Han is supported by NSFC General Program No. 62376235, Guangdong Basic and Applied Basic Research Foundation No. 2022A151501 1652 and No. 2024A151501 2399, HKBU Faculty Niche Research Areas No. RC- RC-FNAA-IG/22225 23/SS//00 and HKBU CSD Departmental Incentive Scheme.
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A Notations
In Table 6, we summarize the notations used in this paper.
Table 6: Description of Notations
Category
Notation
Description
S
Training dataset
m
Proportion of class
Data and Sets
72
Sample number of class Number of classes
B
Training batch size
f(;:)
Feature extractor with parameters + Classifier with parameters 0
h(:0)
w U(6
Model parameters consist of and 0 Loss function with input output y and parameters Empirical risk of the loss function Distribution on training set Distribution on balanced data set Density ratio Matrix of knowledge learned from the distribution P Pp for class i Matrix of feature expectation for momentum update Dimension of features extracted from feature extractor
1(x, W:W)
R
P
Model and Functions
bal
p
F Pont
Z
By(-)
Minimal prediction on the ground-truth class
m E M
Model m in function set M
IF
Imbalance factor
Amars Amin
Average maximum and minimum eigenvalues
Others
Tr; Tirew
Trace of Hessian matrix for class i
Average trace of Hessian matrix over Few categories
B More Discussions of Related Work
Beyond imbalanced classification that has discretized label space, an noteworthy area, imbalanced regression that has a continuous label space is also very common in real applications [Yang al., 2021, Gong et al., 2022]. In this direction, the empirical label distribution often does not accurately reflect the true label density in regression tasks, which limits the effectiveness of traditional re-weighting techniques (Yang et al., 2021, Wang and Wang, 2023]. Label Distribution Smoothing (LDS) (Yang al., 2021] ] and Variational Imbalanced Regression (VIR) [Wang and Wang, 2023] propose using kernel smoothing and other techniques to estimate an accurate label density distribution. Ranking Similarity (Ranksim) [Gong et al., 2022] leverages local and global dependencies by encouraging the correspondence between the similarity order of labels and features. Balanced Mean Squared Error (Balanced MSE) [Ren et al., 2022] extends the concept of Balanced Softmax [Ren et al., 2020a] to regression tasks to achieve a balanced predictive distribution. Contrastive Regularizer (ConR) [Keramati et al., 2024] improves contrastive learning techniques to translate label similarities into the feature space.
C Algorithm Details
C.1 Detailed Derivations of Eq. (7)
Back Eq. (5), replace the expectations over Phat and P by upp and PP.... respectively. For each
class i, We can obtain f; = MM(r), where
2
MM(r)
? Ti
p
PT)
- r; F.
Φp
(10)
n4;
15

---

Dynamic Re-weighting
Constant
Weghtt
Peoght
Weoght
Re-weighting
Extracted features
Constant re we ghtng dows match
Step
Im ningng dyramic
Plyl
Feature protolype
Momenturn update
X
Re-weighted Loss
Ordinary Process
Updata
Update
Figure 6:
Framework of RDR
Then, taking the derivative MM(r) with respect to r and setting to zero, we can obtain the estimation of density ratio in imbalanced learning follows
D n
Ppr:
p
F; = 0
(11)
Solving equation above with respect to ri, we can obtain the solution
n4 Φ 1/))
D F.
(12)
C.2 Framework of RDR
We provide the framework of RDR, which is shown in Fig. 6.
C.3 Pseudo-code of RDR
We provide the pseudo-code of RDR to demonstrate the process of implementing method in detail, as shown in Algorithm 1. addition, we also provide pseudo-code that combines our method with the SAM method, as shown in Algorithm 2.
D Supplement for Experiments
D.1 Experiment with More Imbalanced Data
We conduct experiments the more imbalanced CIFAR-10-LE and CIFAR-100-LT datasets, specifi- cally with imbalance factors of 200 and 500. As shown in Table 7, our method consistently achieves significant improvements.
D.2 Dynamically Re-weighting Process
Our approach dynamically adjusts the weights assigned to each category throughout the training process. To gain more insights into RDR, sampled the weights of each category during training. Fig. 7 presents the results of four samplings during the training processes at imbalance factors of 10 and 100, respectively.
The analysis of these results reveals a consistent trend in weight changes across different imbalance factors. For the Many classes, the weights of the categories consistently decrease during training. Specifically, under the highest sample counts, in descending order) decrease by 5.6%, 6.7%, and 8.0%, respectively. Under the IF of 100, these decreases are more pronounced, with reductions of 10.3%, 15.4%, and 14.7%, respectively. For the Medium classes, the weight changes are less marked, with average decrease
10, the weights classO, c, and class2 (the three classes with the
16

---

Algorithm 1 Training Paradigm of RDR.
1: Input: Training dataset S = -.=1 1 (T 81) model Ma with feature extractor fg and classifier
he, loss function I, momentem coefficient m, learning rate a, weight decay coefficient X, batch size b, temperature coefficient
2: Output: Trained parameters x², lr
3: Initialize the model parameters & and ramdomly, Fr = -Fi,....,FA) Fa) t
4: for 1 to T do
5: B +SSampeeMiniBach((, b)
6: I =< (x) output <- 0t)
7:
8:
if < To then
9:
// warm
w+1
11:
else
for class to d do
13:
Φ + (z,) where y; =
compute w, via Eq. (8) and ?
15:
end for
W <- normalize(w")
17:
end if
18:
C(u,, B) + 1/2 Eg w (output, y)
19:
= Wt at [V[ B) + Awa)
20:
Foai f mFP.. + +1-
21:
Optional: anneal the learning rate 04
22:
end for
Algorithm 2 Training Paradigm of RDR combined with SAM.
1: Input: Training dataset S = U
y/) i (I; model M. with feature extractor fe and classifier hg, loss function I, momentem coefficient m, learning rate a, weight decay coefficient A, batch size b, neighborhood size temperature coefficient Y
2: Oufput: Trained parameters Φ², 6
3: Initialize the model parameters ø and 0 ramdomly, Fr... = (Fi,..., Fa) 0
4: for to T do
5: B --SampleMiniBatch((,. b)
6: 2 f( 6.) output +- h(2, 0z)
7:
8:
if < To then
9:
// warm up
10:
w 1
11:
else
12:
for class i to do
13:
T +- (z,) where y/ =
14:
compute w; via Equation and end for
15:
16:
w 4 normalize(w"))
end if
18:
C (w,, B)
+- 5
Eg w po--(outputt y)
V ())
19:
6 + VL, [3r.BB BT
20:
(2(w: 6,, B)
Σ B w I[furtee (1).))
21:
wi = a, V
L W + Gr B)) + Xulel
F f mf Pr.i +
m)
23:
Optional: anneal the learning rate ar
24: end for
17

---

Table 7: Top-1 accuracy (%) (1)
results under more imbalanced conditions.
550500
IF=200
Method
CIFAR- 10-LT CIFAR-I00-LT
CIFAR-IO-LT
CIFAR-100-LT
CE
60.1
34.1
68.6
37.9
RDR+CE
69.7
39.0
76.7
43.1
SAM+CE
60.4
34.5
69.9
39.0
RDR+SAM+CE
71.1
40.0
80.0
44,8
LA
73.9
38,8
76.7
44.0
RDR+LA
75.8
40.1
80.8
45.0
SAM+LA
74.9
39.6
80.2
44.2
RDR+
76.4
40.6
81.6
45.1
0.59
0200
(a) Many
(b) Medium
(c) Few
(d) Many
(e) Medium
Few
Figure 7: Visualization of weight changes across different categories throughout the training process. Experiments conducted on CIFAR-IO-LT. (a), (b), and (c) illustrate the weight changes for Many, Medium, and Few classes, respectively, under an imbalance factor 10. (d), (e), and (f) correspond- ingly show the changes for each class type under imbalance factor of 100.
of 3.7% at IF=10 and 0.3% at IF=100. For the Few classes, there exits a notable increase in weights during training: at /F=10, the weights of class7, class8, and class9 increase by 2.3%, 6.3%, and 9.4%, respectively, while at /F=100, they increase by 6.2%, 6.5%, and 3.2%.
These results suggest that our method increasingly focuses on minority classes as training progresses. Initially, our method effectively learns common features across all categories, while later in training, increasing the weights helps to target learning towards minority samples, thereby enhancing the model's generalizability. Wang al. [2023] also corroborate these findings.
D.3 Ablation Study
We provide more detailed experimental results for each category in ablation study, as illustrated Fig. and Fig. From these figures, we can find that across various datasets and different imbalance factors, our method significantly enhances the generalizability of both Few classes and Medium classes. Moreover, our method maintains superior performance when combined with the SAM method.
18

---

Cangoy
(a) Accuracy under IF=10
(b) Accuracy under IF=10 with SAM
Todan
Accuracy under /F=100
(d) Accuracy under /F=100 with SAM
Figure 8: Visualization of top-1 accuracy (1) across different categories on CIFAR-100-LL, under three different methods: CE, Inverse Frequency (I/n) and RDR. Experiments conducted under /F=10 (Plot (a) and Plot (b)) and 100 (Plot (c) and Plot (d)).
Calepry
Gallglay
(a) Accuracy on ImageNet-LT
(b) Accuracy on ImageNet-LT with SAM
Calorers
Calorte
(c) Accuracy Places-LT
(d) Accuracy Places-LT with SAM
Figure 9: Visualization of t accuracy (1) across different categories on ImageNet-LT (Plot (a) and Plot (b)) and Places-LT (Plot (c) and Plot (d)), under three different methods: CE, Inverse Frequency (1/n) and RDR. Plot (b) and Plot (d) are integrated with SAM while Plot (a) and Plot (c) are not.
19

---

E More Discussions about Limitations
While our approach demonstrates promising results, there are potential challenges that warrant further attention. In particular, as the number of classes increases to very large scale, especially in certain tasks such as face recognition, retail product recommendation, be concerns regarding computational efficiency. It is important to consider lightweight techniques to ensure that scalability does not compromise practical applicability. Additionally, in addressing imbalanced learning, care must be taken to avoid excessive rebalancing toward minority groups, as this could unintentionally affect the learning performance of the majority class, which would not align with the broader goals of fairness. Rebalancing should be conducted within reasonable framework. mindful avoiding misuse or overcompensation that could arise from improper manipulation by any group. Ensuring fairness for all remains critical consideration.
landmark detection, there could
20

---

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Reviews:
[{"review_id": "rfGmNQzVmg", "reviewer": "Reviewer_xrua", "summary": "The paper presents a weighting strategy in order to handle class imbalance. Contrary to existing method, they propose to adapt the weight throughout the training procedure. \n\nTheir method estimates the discrepancy between the sample distribution and the balanced sample distribution for parameterization w and updates the estimate through the training.\n\nThe authors use two resnet architectures to evaluate their contribution on multiple datasets. They also compare to other baselines and show significant gain.", "questions": "* Could you clarify the term, P, etc. You often refer to it as \"real world data distribution\", but the distribution of x does not depends on any other classes (imbalanced or not) ?", "limitations": "The paper would benefit from a clarification in notification (what is the true data distribution, what is the feature distribution, what is an estimate of what quantity, etc.). I believe the contribution is novel and worth it.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "* The paper develops a novel approach for handling class imbalance.\n* The methodology is derived theoretically from the problem formulation\n* The authors propose an analysis of the complexity of the method and empirically evaluate the training time.\n* The methodology is evaluated on multiple datasets and compared to multiple baselines.", "weaknesses": "* The paper is sometimes difficult to read:\n  * Row 125, the authors refer to the distribution of training set, which get parameterized by w. Thus, my understanding is that the authors refer to the distribution of the training set \"captured by the model\". \n  * row 134, P_bal = pi P.. P_bal is the distribution of y in the balanced case ? But should therefore be 1/number classes... and P, should just be the class proportion and we should have P = pi P_bal ?\n  * LDAM and LA terms are not defined at first. First definition of LA is at row 208\n  * row 212 \"trategies\" => strategies", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Row 125, the authors refer to the distribution of training set, which get parameterized by w. Thus, my understanding is that the authors refer to the distribution of the training set \"captured by the model\".\n\nYes, you are right. We use the distribution parameterized by $w$ to represent \"the distribution captured by the model\". We will enrich its description for clarity in the revision. \n\n> row 134, P_bal = pi P.. P_bal is the distribution of y in the balanced case ? But should therefore be 1/number classes... and P, should just be the class proportion and we should have P = pi P_bal ?\n\nThank you for pointing out this oversight: the term  $\\pi_i$  in the equation should indeed be $\\frac{1}{\\pi_i}$. We would like to clarify here that this mistake is limited to this equation and does not affect the correctness of our subsequent derivations. We greatly appreciate the reviewer’s careful review and will make the correction.\n\n> LDAM and LA terms are not defined at first. First definition of LA is at row 208\n\n> row 212 \"trategies\" => strategies\n\nLDAM and LA respectively refer to Label-Distribution-Aware margin loss and Logit Adjustment, two prevalent long-tailed learning methods. We will include their full names at the first mention. For the typo, we have carefully proofread the whole manuscript again and corrected all possible mistakes. These changes will be included in the revision.\n\n> Could you clarify the term, P, etc. You often refer to it as \"real world data distribution\", but the distribution of x does not depends on any other classes (imbalanced or not) ?\n\nSorry for the possible confusion. Here is a clarification of $P$:\n\n$P$ represents the real world data distribution, or more directly, the training set distribution. Specifically, $P(x,y)$ is the joint distribution of the training data (x, y), and the training set $D$ is obtained by IID sampling from $P$. $P(x)$ and $P(y)$ represent the marginal distributions of the training data $x$ and $y$, respectively. $P(x|y)$ and $P(y|x)$ represent the corresponding conditional distributions. For other distribution notations parameterized by $w$, they mean the corresponding distributions captured by the model, as the reivewer's understanding.\n\nWe also place a notation table for clarity in the **attached PDF file**, which will be placed into the manuscript if possible (or in the appendix if not). If you have any further questions or need any clarifications, please let us know. We will carefully improve the manuscript.", "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": "", "extra_scores": {}}, {"review_id": "I52GUULegQ", "reviewer": "Reviewer_MG5X", "summary": "The paper introduces a novel approach called Re-weighting with Density Ratio (RDR) to address the challenges posed by imbalanced data distributions in machine learning.  The RDR approach aims to mitigate overfitting on majority classes and enhance adaptability across diverse datasets by continuously updating the weights in response to observed shifts in class density.  Extensive experiments on various large-scale, long-tailed datasets demonstrate that the RDR method significantly improves the model's generalization capabilities, particularly under severely imbalanced conditions.  The analysis of the weight changes during training reveals that the method increasingly focuses on minority classes as training progresses, initially learning common features across all categories and then targeting learning towards minority samples to enhance generalizability.  The paper also provides an ablation study to further validate the effectiveness of the proposed approach.", "questions": "see above", "limitations": "authors discussed in sec 3.4", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 2, "contribution": 2, "strengths": "1. The paper introduces a novel approach called Re-weighting with Density Ratio (RDR) to address the challenges posed by imbalanced data distributions in machine learning. \n\n2. Extensive experiments on various large-scale, long-tailed datasets demonstrate that the RDR method significantly improves the model's generalization capabilities, particularly under severely imbalanced conditions. \n\n3. The analysis of the weight changes during training reveals that the method increasingly focuses on minority classes as training progresses, initially learning common features across all categories and then targeting learning towards minority samples to enhance generalizability. \n\n4. The paper provides an ablation study to further validate the effectiveness of the proposed approach. \n\n5. The results show that RDR generally outperforms other methods, including Inverse Frequency (1/n) and SAM variants, in both the Many and Few classes, indicating that RDR can efficiently address the overfitting issues for Few classes.", "weaknesses": "- The paper does not provide a detailed theoretical analysis or justification for the proposed Re-weighting with Density Ratio (RDR) method, beyond the intuition that it can mitigate overfitting on majority classes and enhance adaptability across diverse datasets.\n\n- I am interested in how RDR might perform in the presence of extreme imbalance, noisy data, or other challenging scenarios. The current experiment is well-established but dataset itself is relatively simple.\n\n- The paper discusses reweighting/non-reweighting for classification problems. I suggest the authors also briefly discuss reweighting methods in imbalanced regression problems, e.g., VIR [1] for reweighting problems and ConR [2] for non-reweighting problems.\n\n[1] Variational Imbalanced Regression: Fair Uncertainty Quantification via Probabilistic Smoothing, NeurIPS 2023\n\n[2] ConR: Contrastive Regularizer for Deep Imbalanced Regression, ICLR 2024\n\n**Summary** I think the theoretical analysis or at least insights is needed for acceptance, so my suggest score is 5, as the experiment part is excellent in this paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> The paper does not provide a detailed theoretical analysis or justification for the proposed Re-weighting with Density Ratio (RDR) method, beyond the intuition that it can mitigate overfitting on majority classes and enhance adaptability across diverse datasets.\n\nThank you for your suggestion. To address the reviewer's concern, we present a generalization bound sketch for the re-weighting-based methods and provide the possible insights with the empirical verifiction in the following. Note that, about its formal version with the complete hypothesis and the deduction procedure, we will update to the manuscript in the final. \n\nGiven the model $m\\in M$ and a reweighting loss function $L _ {RW}$, for any $\\delta \\in(0,1)$, with probability at least $1-\\delta$ over the training set $\\mathcal{S}$, the following generalization bound holds for the risk on the balanced distribution\n\n$\\mathcal{R} _ {\\text {bal }}^{L}(m) \\precsim \\Phi\\left(L _ {RW}, \\delta\\right)+\\frac{\\mathfrak{S} _ {\\mathcal{S}}(M)}{d \\pi_{1}} \\sum _ {y=1}^{d} w _ y \\sqrt{\\pi _ {y}}\\left[1-\\operatorname{softmax}\\left(B _ {y}(m)\\right)\\right]$\n\nwhere $\\Phi\\left(L _ {RW}, \\delta\\right)$ is positively correlated with the empirical reweighting risk of training set. $\\mathfrak{C} _ {\\mathcal{S}}(M)$ denotes the empirical complexity of the function set $M$. $B _ {y}(f)$ denotes the minimal prediction on the ground-truth class $y$ in the training set. $w_y$ refers to the weight of class $y$ of the reweighting loss $L _ {RW}$. The formal theory and the proof will presented in the revision.\n\nWe can get some insights from the above generalization bound:\n\n1. **Why reweighting is necessary**: $w _ y$ helps to rebalancing the imbalanced term $\\sqrt{\\pi _ {y}}\\left[1-\\operatorname{softmax}\\left(B _ {y}(m)\\right)\\right]$ to get a sharper bound.\n2. **Why dynamic reweighting is necessary**: The term $B _ {y}(m)$ changes dynamically with model training. Therefore, we need a $w _ y$ that can adapt dynamically to the changes of $B _ {y}(m)$.\n3. **Why RDR works**: From Figure 1 in **the attached PDF**, we observe that the dynamic trend of the RDR weight aligns well with $\\sqrt{\\pi _ {y}}\\left[1-\\operatorname{softmax}\\left(B _ {y}(m)\\right)\\right]$, denoted as $B _ y^{\\prime}$. This demonstrates that our RDR can adapt to the dynamic changes in $B _ y^{\\prime}$, maintaining a sharp bound during dynamic training.\n\n> I am interested in how RDR might perform in the presence of extreme imbalance, noisy data, or other challenging scenarios. The current experiment is well-established but dataset itself is relatively simple.\n\nThank you for the suggestion. In the following, we conduct more experiments. The results are shown in Table 2 and Table 3 in the **attached PDF**.\n- We include the results of two new datasets: CIFAR-10-LT-NL and CIFAR-100-LT-NL, with both class imbalance and label noise. We can see that on more complex datasets, our method achieves consistent and significant improvements.\n- Additionally, note that the imbalance factors constructed on ImageNet-LT and Places-LT in the submission are 256 and 996, respectively. These datasets are indeed extremely imbalanced. We will explicitly indicate the imbalance factors of these two datasets in the experiment tables in the revision. Furthermore, we conduct experiments on the more imbalanced CIFAR-10-LT and CIFAR-100-LT datasets, specifically with imbalance factors of 200 and 500. Our method consistently achieves significant improvements.\n\n\n> The paper discusses reweighting/non-reweighting for classification problems. I suggest the authors also briefly discuss reweighting methods in imbalanced regression problems, e.g., VIR [1] for reweighting problems and ConR [2] for non-reweighting problems.\n[1] Variational Imbalanced Regression: Fair Uncertainty Quantification via Probabilistic Smoothing, NeurIPS 2023\n[2] ConR: Contrastive Regularizer for Deep Imbalanced Regression, ICLR 2024\n\nThank you for the valuable suggestion and the recommendation. We will include a brief discussion on imbalanced regression problems in the revision as follows:\n\nWe shall note that the primary focus of this study is on imbalanced classification. Beyond imbalanced classification that has discretized label space, an noteworthy area, imbalanced regression that has a continuous label space is also very common in real applications [a,b]. In this direction, the empirical label distribution often does not accurately reflect the true label density in regression tasks, which limits the effectiveness of traditional reweighting techniques [a,c]. Label Distribution Smoothing (LDS) [a] and Variational Imbalanced Regression (VIR) [c] propose using kernel smoothing and other techniques to estimate an accurate label density distribution. Ranking Similarity (Ranksim) [b] leverages local and global dependencies by encouraging the correspondence between the similarity order of labels and features. Balanced Mean Squared Error (Balanced MSE)[d] extends the concept of Balanced Softmax[e] to regression tasks to achieve a balanced predictive distribution. Contrastive Regularizer (ConR)[f] improves contrastive learning techniques to translate label similarities into the feature space. Considering the different rebalancing paradigms compared with that in imbalanced classification, and the limited space, we will leave the potential extension of our RDR to this area in the future explorations.\n\n[a] Delving into Deep Imbalanced Regression. ICML 2021.\n\n[b] RankSim: Ranking Similarity Regularization for Deep Imbalanced Regression. ICML 2022.\n\n[c] Variational Imbalanced Regression: Fair Uncertainty Quantification via Probabilistic Smoothing. NeurIPS 2023.\n\n[d] Balanced MSE for Imbalanced Visual Regression. CVPR 2022.\n\n[e] Balanced Meta-Softmax for Long-Tailed Visual Recognition. NeurIPS 2020.\n\n[f] ConR: Contrastive Regularizer for Deep Imbalanced Regression. ICLR 2024.", "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": "", "extra_scores": {}}, {"review_id": "c2IewU8rY6", "reviewer": "Reviewer_24eD", "summary": "This paper presents a dynamic re-weighting method for imbalanced learning. The author defines the ratio of the balanced data set distribution to the training set distribution, and tries to estimate it with an iterative update method. The effectiveness of this method is proved by experiments.", "questions": "Please refer to Weaknesses.\n\nBesides, I am also curious about some experimental details. Do the authors use other techniques such as RandAug or mixup?", "limitations": "No. Although the authors say they clarify the limitations in Sec.3.4, I find they mainly highlight the efficiency of the proposed method. More discussion about limitations and potential negative societal impact is recommended.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1.\tThis paper points out a problem with distribution differences, which leads to the potential missing feature patterns in general re-weighting methods.\n2.\tThis paper proposes a new method, which approximates the ratio of the balanced data set distribution to the training set distribution using methods of density ratio estimation. As far as I know, a dynamic re-weighting strategy is novel in this field.\n3.\tThe experimental introduction of this paper is clear, and extensive experiments have been carried out, which validates the effectiveness of the proposed method.", "weaknesses": "1.\tThe formula derivation in Sec. 3.3 can be more detailed. It is suggested to explain how formula (7) is obtained in the appendix.\n2.\tThe introduction may have overlooked some key articles. For example, the article mentions Wang et al. 's article at the end of Sec.3.3, but does not discuss this paper in the introduction section.\n3.\tDoes the new method enjoy the same theoretical boundaries as the general reweighting method? It is recommended to provide more analysis.\n4.\tBesides, there are some typos in the details:\n  - In the experimental section, 'class[390,385] 'may be a typo.\n  - In table 3, the interpretation of Tr_{Few} is supposed to be there.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> The formula derivation in Sec. 3.3 can be more detailed. It is suggested to explain how formula (7) is obtained in the appendix.\n\nThank you for the suggestion. We will include detailed derivations of Eq. (7) in the appendix of revision, which we simply summarize the deduction as follows for clarity:\n\nBack to Eq. (5), replace the expectations over $P _ {bal}$ and $P$ by $\\boldsymbol{\\Phi} _ {P}$ and $F _ {P _ {bal}}$, respectively. For each class $i$, We can obtain\n\n$\\widehat{r _ {i}}=\\widehat{\\mathrm{MM}(r)}$, where $\\widehat{\\mathrm{MM}(r)}=\n\\frac{1}{n _ {i}^{2}}{r _ i}^\\top{\\boldsymbol{\\Phi} _ {P}^{i}}^{\\top} {\\boldsymbol{\\Phi} _ {P}^{i}}{r _ i}-\\frac{2}{n _ {i}}r _ {i}^\\top{\\boldsymbol{\\Phi} _ {P}^{i}}^\\top F _ {i}$\n\nThen, taking the derivative of $\\widehat{\\mathrm{MM}(r)}$ with respect to $r$ and setting it to zero, we can obtain the estimation of density ratio in imbalanced learning as follows\n\n$\\frac{2}{n _ {i}^{2}} {\\boldsymbol{\\Phi} _ {P}^{i}}^{\\top} {\\boldsymbol{\\Phi} _ {P}^{i}} r _ {i}-\\frac{2}{n _ {i}}{\\boldsymbol{\\Phi} _ {P}^{i}}^{\\top}F _ {i}=0$\n\nSolving equation above with respect to $r _ {i}$, we can obtain the solution as \n\n$\\widehat{r _ {i}}=n _ {i}\\left({\\boldsymbol{\\Phi} _ {P}^{i}}^{\\top} \\boldsymbol{\\Phi} _ {P}^{i}\\right)^{-1} {\\boldsymbol{\\Phi} _ {P}^{i}}^{\\top} F _ {i}$\n\n> The introduction may have overlooked some key articles. For example, the article mentions Wang et al. 's article at the end of Sec.3.3, but does not discuss this paper in the introduction section.\n\nWe will take the reviewers' advice to carefully revise the second paragraph of the Introduction (lines 32-34), specially including a discussion of the work by Wang et al. and other related studies. We present the corresponding revision as follows for your reference.\n\nHowever, such subsequent improvements can alleviate but still cannot effectively push that forward. *Wang et al. [2023] obtains a fine-grained generalization bound for re-weighting in imbalanced learning through the data-dependent contraction technique.* Limited research has focused on the intrinsic limitations...\n\n> Does the new method enjoy the same theoretical boundaries as the general reweighting method? It is recommended to provide more analysis.\n\nThank you for your suggestion. To address the reviewer's concern, we present a generalization bound sketch for the re-weighting-based methods and provide the possible insights with the empirical verifiction in the following. Note that, about its formal version with the complete hypothesis and the deduction procedure, we will update to the manuscript in the final. \n\nGiven the model $m\\in M$ and a reweighting loss function $L_{RW}$, for any $\\delta \\in(0,1)$, with probability at least $1-\\delta$ over the training set $\\mathcal{S}$, the following generalization bound holds for the risk on the balanced distribution\n\n$\\mathcal{R} _ {\\text {bal}}^{L}(m) \\precsim \\Phi\\left(L _ {RW}, \\delta\\right)+\\frac{\\mathfrak{S} _ {\\mathcal{S}}(M)}{d \\pi _ {1}} \\sum _ {y=1}^{d} w _ y \\sqrt{\\pi _ {y}}\\left[1-\\operatorname{softmax}\\left(B _ {y}(m)\\right)\\right]$\n\nwhere $\\Phi\\left(L _ {RW}, \\delta\\right)$ is positively correlated with the empirical reweighting risk of training set. $\\mathfrak{C} _ {\\mathcal{S}}(M)$ denotes the empirical complexity of the function set $M$. $B _ {y}(f)$ denotes the minimal prediction on the ground-truth class $y$ in the training set. $w _ y$ refers to the weight of class $y$ of the reweighting loss $L _ {RW}$. The formal theory and the proof will presented in the revision.\n\nWe can get some insights from the above generalization bound:\n\n1. **Why reweighting is necessary**: $w _ y$ helps to rebalancing the imbalanced term $\\sqrt{\\pi_{y}}\\left[1-\\operatorname{softmax}\\left(B _ {y}(m)\\right)\\right]$ to get a sharper bound.\n2. **Why dynamic reweighting is necessary**: The term $B _ {y}(m)$ changes dynamically with model training. Therefore, we need a $w_y$ that can adapt dynamically to the changes of $B _ {y}(m)$.\n3. **Why RDR works**: From Figure 1 in **the attached PDF**, we observe that the dynamic trend of the RDR weight aligns well with $\\sqrt{\\pi _ {y}}\\left[1-\\operatorname{softmax}\\left(B _ {y}(m)\\right)\\right]$, denoted as $B _ y^{\\prime}$. This demonstrates that our RDR can adapt to the dynamic changes in $B _ y^{\\prime}$, maintaining a sharp bound during dynamic training.\n\n> Besides, there are some typos in the details:\n> In the experimental section, 'class[390,385] 'may be a typo.\n> In table 3, the interpretation of Tr_{Few} is supposed to be there.\n> Do the authors use other techniques such as RandAug or mixup?\n\nThank you for pointing it out. We have carefully proofread the whole manuscript, corrected typos, and added a table about necessary explanations of some notations or terms (please refer to the notation table in the **attached PDF file**). $\\overline{\\textit{Tr}}_{Few}$ denotes average trace of Hessian matrix over Few classes. These changes will be included in the revision. We used RandAug in our experiments but did not include mixup, for fair competition.\n\n>  More discussion about limitations and potential negative societal impact is recommended.\n\nThanks for your suggestion. We will enrich the discussion about limitation and social impact in the revision. Specially, we will highlight the potential computational complexity issue when scaling up to a super large number of classes (e.g., in face recognition, retail product recommendation and landmark detections), which should pay additional attention to some corresponding lightweight techniques. Regarding the potential negative societal impact, we should clearly recognize that overly rebalancing to minority groups during training may bring the desctructive effect on the learning of majority groups, which is also not the desirable goal of imbalanced learning. All rebalancing techniques should build on top of a proper range for fairness and we should avoid some improper abuse by the malicious minority groups.", "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": "", "extra_scores": {}}]

Author rebuttal:
We thank reviewers for your valuable feedback, and appreciate the great efforts made by all reviewers, ACs, SACs and PCs. 

Please refer to our detailed responses to each reviewer, where we addressed each question and concern point by point. In the **attached PDF**, we have included a notation summary table for improved readability and clarification, additional tables for experimental results, along with the empirical validation of the theoretical insights presented in the rebuttal.

We appreciate all reviewers’ time again. We are looking forward to your reply!

Meta-review: The authors address the critical and challenging problem of class imbalance in machine learning. They propose a method that adaptively weighs the importance of classes throughout training. The main idea is to use an estimate of the density ratios (existing vs. balanced) to re-weight the samples in the loss. The advantage of the method is demonstrated using several real datasets with imbalance ratios ranging from 10 to 256. Overall, the reviewers are positive about the paper, with the main criticism being the presentation. This work provides an interesting solution to the problem that seems to work well. For these reasons, I recommend acceptance of this paper, and I highly encourage the authors to implement the reviewers' suggestions and work towards improving the writing and presentation.
Decision: Accept (poster)