CoLA: Co-Calibrated Logit Adjustment for Long-Tailed Semi-Supervised Learning
Authors: Qian Shao
Qiyuan Chen
Jiahe Chen
Zepeng Li
Qianqian Tang
Hongxia Xu
Jian Wu
Venue: ICLR 2026 Poster (2026)
Keywords: Semi-supervised learning
long-tailed learning
logit adjustment
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: Improve the original logit adjustment for realistic long-tailed semi-supervised learning.

Abstract: Long-tailed semi-supervised learning is hampered by a vicious cycle of confirmation bias, where skewed pseudo-labeling progressively marginalizes tail classes. This challenge is compounded in real-world scenarios by a class distribution mismatch between labeled and unlabeled data, rendering the bias unpredictable and difficult to mitigate. While existing methods adapt Logit Adjustment (LA) using dynamic estimates of the unlabeled distribution, we argue their effectiveness is undermined by two critical limitations stemming from LA's core design, *i.e.*, its class-wise and overall adjustment mechanisms. First, their reliance on simple frequency counting overestimates the prevalence of head classes due to sample redundancy, leading to harmful over-suppression. Second, and more critically, they overlook the interplay between the above two types of adjustment, treating the overall adjustment strength as a fixed hyperparameter. This is a significant oversight, as we empirically find that the optimal strength is highly sensitive to the estimated distribution. To address these limitations, we propose Co-Calibrated Logit Adjustment (CoLA), a framework that co-designs the class-wise and overall LA components. Specifically, CoLA refines the class-wise adjustment by estimating each class's effective sample size via the effective rank of its representations. Subsequently, it formulates the overall adjustment strength as a learnable parameter, which is optimized through a meta-learning procedure on a proxy validation set constructed to mirror the refined distribution. Supported by a theoretical generalization bound, our extensive experiments show that CoLA outperforms existing baselines on $4$ public benchmarks across standard long-tail setups.

Published as a conference paper at ICLR 2026
CoLA: CO-CALIBRATED LOGIT ADJUSTMENT FOR LONG-TAILED SEMI- SUME-UPERVISED LEARNING
Qian Shao' & Qiyuan Chen' & Jiahe Chen' & Zepeng Li & Qianqian Tang & Hongxia Xu²* & Jian Wu
'Transvascular Implantation Devices Research Institute and College of Computer Science & Technology, Zhejiang University, Hangzhou, China *Liangzhu Laboratory and WeDoctor Cloud, Hangzhou, China
The State Key Laboratory of Blockehain and Data Security, Zhejiang University
"Shaoxing Tangtang Technology Co., Ltd.
*Zhejiang Key Laboratory Medical Imaging Artificial Intelligence, Hangzhou, China [qianshao, einstein, wujian2000) )
ABSTRACT
Long-tailed semi-supervised learning is hampered by a vicious cycle of confirma- tion bias, where skewed pseudo-labeling progressively marginalizes tail classes. This challenge is compounded in real-world scenarios by a class distribution mis- match between labeled and unlabeled data, rendering the bias unpredictable and difficult to mitigate. While existing methods adapt Logit Adjustment (LA) us- ing dynamic estimates of the unlabeled distribution, argue their effectiveness is undermined by two critical limitations stemming from LA's core design, i.e. class-wise and overall adjustment mechanisms. First, their reliance on simple frequency counting overestimates the prevalence of head classes due to sample re- dundancy, leading harmful over-suppression. Second, and more critically, they overlook the interplay between the above two types of adjustment, treating the overall adjustment strength as fixed hyperparameter. This a significant over- sight, we empirically find that the optimal strength is highly sensitive the estimated distribution. To address these limitations, we propose Co-Calibrated Logit Adjustment (CoLA), framework that co-designs the class wise and overall LA components. Specifically, CoLAA refines the class-wise adjustment by estimat- ing each class's effective sample size via the effective rank of its representations. Subsequently, it formulates the overall adjustment strength as a learnable param- eter, which is optimized through meta-learning procedure on a proxy validation set constructed to mirror the refined distribution. Supported by a theoretical gener- alization bound, our extensive experiments show that CoLA outperforms existing baselines on 4 public benchmarks across standard long-tat tail setups.
INTRODUCTION
Semi-SSperrvsed Learning (SSL) (Laine & Aila 2017 Sohn et al. 2020 Nguyen 2024 often
struggles in real-world applications due the inherent long-tailed nature data, critical problem known Long-Tailed Semi-Supervised Learning (LTSSL) (Kim al., 2020 Lee et al. 2021 Wei al. 202T). the LTSSL setting, a model initially trained on skewed labeled data develops a strong bias towards the majority classes. This initial bias is then amplified when the model generates large volume of equally biased lseudo-labels from unlabeled data. This process creates a vicious cycle of confirmation bias, causing the model to become increasingly overconfident in majority classes
while progressively neglecting the tail classes (Duan et al. 2022 Fan et al. 2022 Guo Li 2022). Consequently. the ability to produce high-quality, unbiased labuds has become the paramount
challenge in LTSSL (Wei & Gan 2023 Ma et al. 2024 Xing et al. 2025).
Recent advancements in LTSSL predominantly rely on Logit Adjustment (LA) strategy Meeno
et al. 2021) or its variants to enhance the quality of pseudo-labels (Wei & Gan 2023 Ma et al.
"Corresponding author.

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C10 with n 110 C10 with
Head Class
C100 wid 10
C100 with 7 15
Tail Class
Accuracy (ACC)
71.5
Overall adjustment
(a) Long-tailed binary classification
(b) Relations between adjustment and ACC
Figure I: (a) An unadjusted classifier is biased towards head classes (grey solid line), which naively uses class frequency for class-wise adjustment, leading to over-suppression (red dashed line). This is because frequency counting ignores sample redundancy (dark blue spheres), inflating the estimated importance of head classes. De-duplication could mitigate this issue (green dashed line). (b) The optimal overall adjustment strength (four dashed lines) is highly sensitive to the distribution and the number of classes. Counter-intuitively, does not always correlate positively with the imbalance ratio 7/- For instance, on CIFAR-10-L3 (C10), the optimal strength for m = 100 is greater than for n = 150, highlighting the difficulty of relying on a pre-defined strength in complex scenarios.
Hou & Jia 2025). At its core, LA comprises two key components: a class-wise adjustment, which controls the relative adjustment strength of logits for different classes according to the prior probability (i.e., suppressing the head class and encouraging the tail class), and an overall adjust- ment, which uniformly controls the overall scaling magnitude. However, since the true distribution of unlabeled data is unknown, achieving an accurate adjustment is challenging. Consequently, many existing methods (Ma et al., 2024 Hou & Jia 2025) resort to using pre-defined anchor distribu- tion as a proxy for the true class priors. This static assumption proves brittle in more realistic and 2024). where the actual class distribution inevitably deviates from the pre-defined anchor, rendering these methods ineffective.
2024
complex scenarios (Du et al.
More sophisticated works (Wei & Gan 2023] Li et al.] 2024) have pursued dynamically estimat- ing the class distribution of unlabeled data. Despite their conceptual advantages, these approaches encounter a critical two-fold dilemma. First, they typically estimate class distributions via a naive frequency-counting of nigh-confidence predictions. This process often overlooks sample redun- dancy, an issue particularly pronounced in head classes that are saturated with visually similar in- stances (Cui et al. 20199] Zhang al.) 2024). Thus, the estimated prevalence of these classes be- comes inflated, leading to an over-suppression problem that ultimately degrades model performance (Figure [p). Furthermore, these approaches often ignore the interplay between the two adjustments, treating the overall adjustment strength as a fixed hyperparameter. In contrast, we contend that an accurate class-wise adjustment is a prerequisite for determining the optimal overall adjustment. Our empirical analysis reveals that the optimal overall adjustment is highly sensitive to the esti- mated class distribution and the number of classes (Figure [pp). This exposes the second facet of the dilemma: their reliance on a pre-defined overall strength, rendering them incapable of adapting to varying dataset characteristics and ultimately degrading the quality of pseudo-labels. Moreover, most LA-based works lack theoretical grounding to guide the selection of its optimal value.
To address these issues, we propose Co-Calibrated Logit Adjustment (CoLA), which co-designs the class-wise and overall adjustment of LA for any given class distribution. For accurate class- wise adjustment, we introduce De-Duplicated Distribution Estimation (DDDE) to tackle the over- suppression problem. Departing from naive frequency counting, DDDE quantifies sample redun- dancy by calculating the effective rank (Roy & Vetterli 2007) of each class's 's representations, which is inspired by the concept of effective sample size (Cui al. 2019). This process yields a more robust and accurate estimate of class prevalence. Based on the estimated distribution, we design Logit Ceta-Calibration (LMC), which first constructs proxy validation set by resampling labeled data to match the estimated distribution, and then meta-learns the optimal strength on this proxy set.
Theoretically, we establish a generalization bound for our meta-learning procedure and provide convexity analysis of the optimization objective. Empirically, extensive experiments demonstrate that CoLA achieves new state-of-the-arr (SOTA) performance on standard long-tailed benchmarks. Furthermore, comprehensive ablation and visualization experiments validate the individual contri- butions of the key components. The main contributions of this article are as follows:
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We reveal the critical interplay within the LA mechanism, and propose CoLA that adaptively co-designs the class-wise and overall adjustment of LA for LTSSL:
We introduce DDDE, which mitigates the over-suppression problem of head classes by leveraging redundancy-awart estimate of the unlabeled class distribution;
We design LMC, meta-learning strategy that learns the optimal overall adjustment strength proxy set, which mirrors the characteristics of the unlabeled set;
We validate our proposed CoL.A rigorously through theoretical analysis and extensive experiments on 4 benchmarks across 6 different distributions.
RELATED WORK
LTSSL has emerged to address the critical challenge of performance degradation in conventional when confronted with class-imbalanced data. Xing et al. 22255,, loss re-weighting (Lai et al.) 2021; Wei et al. 202T), data augmentation via
SSL methods (Lee al. 2013 Sohn al.
2020)
LTSSL approaches often involve LA (Li et al.
2024
2022
Wang et al.) 2022a), re-sampling (Lee et al.,
2024). distribution alignment (Kim et al. 2220; Lazarow mixing Fann et al.) 2022 Zheng et al.
ct al.
2023 Sanchez Aimar et al. 2024). and others (Kim et al.
2025 Oh et al. (0022).
Due to its efficacy and case of implementation, LA (Menon et al. 2021 is widely used in recent search, including post-hoc correction and/or loss function modification. For instance, CDMAD (Lee
& Kim 2024) employs the post-hoc LA approach, which derives the adjustment strength from the classifier's biased degree by calculating logits on solid color images. LCGC (Xing et al. 2025 fur- ther provides a theoretical foundation for this methodology, enhancing its robustness. In different vein, TCBC (Li et al., (024) corrects the model bias and refines the pseudo-labels by modifying the LA-based loss function. CPE (Ma et al., 2024) designs three distinct classifiers, each tailored to pre-defined anchor distribution through a specialized LA-based loss function. Building on this, Meta-Expert (Hou & Jia 2025) introduces a dynamic gating mechanism to select the most suitable classifier for each unlabeled instance, aiming for more accurate pseudo-label generation. In addi- tion, ACR (Wei & Gan 2023) employs a dual-branch structure that uses these two LA approaches respectively, where the standard branch calculates the distance to three pre-defined anchor distribu- tions to determine the strength of post-hoc LA for accurate pseudo labels, while the balanced branch utilizes an LA-based loss function to generate class-balanced pseudo-labels.
Although the above methods have demonstrated considerable success, most of them lie a shared reliance on small, discrete set of pre-defined anchor distributions, which fail to handle class distri- butions that fall outside these anchors. Consequently, their applicability is limited in more complex and realistic scenarios where distribution shifts can be arbitrary and unpredictable (Du et al. 2024). Related work on other types of LTSSL algorithms can be found in Appendix B
3 PRELIMINARY
Let X' be the input space and y 21, K} be the label space, where K the number of classes. We consider two underlying joint distributions over X' x y: a labeled data distribution Px, (æ, y). and an unlabeled data distribution Px...,. (z, y). abbreviated as Pt and Pa. respectively. The train- ing data consists of two distinct sets: (1) a labeled set D. = {xx, ). drawn i.i.d. from Pt: (2) an unlabeled set D. {ax)) ) where N and M are the numbers of samples in the labeled and unlabeled sets, respectively. For the purpose of analysis, we posit that each sample ay is associated with a true but unobserved label v/a such that the complete pair (o)) is drawn i.i.d. from P,.- Crucially, only
{2e)) M are accessible during the training phase.
For each class E y, let N₂ be the number of samples in D;. We order the classes such that N1 2 N₂ 2 ... 2 NR- The imbalance ratio of the labeled set is defined as " N. assumed that that m > 1. Similarly, let M, represent the true (unobserved) number of samples for class & within the unlabeled dataset D... We denote the unlabeled class indices with the largest and smallest sample numbers, and the imbalance ratio as kmaxx Kmins and Yv = MR /M respectively. These per-class counts (M&) K and the ratio Yu are unknown during training.
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Following the previous works (Du et al. 2024), consider six distributions of unla-
beled data, which cover various real-world situations: (1) consistent long-tailed distribu-
tion, with (Tus &max: kmin)
(?,, K);;; (2) uniform distribution, with %u 1; (3)
(7t, K, 1); (4) middle distribution, with (Yu, kmax, kmin) = (7t- [K] ((K+)) mod 9); (5) head-tail distribution, with (Yu kmax: Kmin) =
reversed distribution, with (Tu, kmax- kmin)
(7t, (K++) mod 2, - 1): (6) unknown distribution, which means that the dataset itself contains some unlabeled data, such as STL-10 (Coates et al. 2011).
The goal of LTSSL is to learn deep model on the train set Danin Dj U Du. that achieves good performances on a class-balanced test set Drest- Let x(x) denote the backbone of the deep model. transforming the sample æ D Durain into the output logits. Following previous works (Wang al. 2022a Wei & Gan 2023), we adopt FixMatch (Sohn et al. 2020) as a backbone SSL algorithm. For labeled sample, FixMatch computes the cross-entrope (CE) loss Ler. between the true label and the model's prediction. For an unlabeled sample, Match first performs strong (Cubuk et al. 2020 DeVries 2017) and weak data augmentation it separately, denoted by A(x=") and (xx"). Then, FixMatch predicts the class probabilities for them as a (=((Axx))) and (=(a(("))) respectively, where (-) denotes the softmax function. The total loss of the backbone L is defined by
N N Σ Len (v((()((()))) + M
M
C
Σ ==1
[[Il(((a((2)))l>>>>> > p) LCE (ūj. (2(A(())))
where II(-) an indicator function, with threshold p. and új arg max, z(a(<)))), denotes the pseudo label. However, training with the objective L on long-tailed set causes the model to develop a significant bias towards majority classes, which directly compromises the quality of pseudo-labels and exacerbates the confirmation bias cycle. To counteract this effect, the post-hoc LA strategy (Menon et al.) 202T) is widely adopted to generate accurate pseudo labels via re-calibrating the predictions unlabeled data:
Úソ arg max (z(a(1;)), - yEY
T log Pr. Py. (u))
where Py, (v) are estimates of the unlabeled class distribution Py, which controls the strength of class-wise adjustment, and r > is hyperparameter that controls the strength of overall adjust- ment. This process effectively penalizes the logits of classes presumed to be frequent under Pv., (v), thereby increasing the likelihood of assigning pseudo-labels to minority classes
4 METHODOLOGY
In this section, we first introduce DDDE method for estimating the class distribution of unlabeled data in Section 4.1 which is specifically designed to handle sample redundancy. With this accurate prior, we then propose LMC in Section 4.2 a data-driven strategy that meta-learns the optimal overall adjustment strength on distribution-matee matched proxy set. Finally, Section 4.3 elaborates on the integration of these components into end-to-end training pipeline.
4.1 DE- DE-DULLICATED DISTRIBUTION ESTIMATION
To solve the over-suppression problem caused by the overestimation of the prevalence of head classes, we draw inspiration from the concept of Effective Number (EN) of samples (Cui et al. 2019), and propose the DDDE method better estimate class frequencies. Specifically, we quantify the EN of samples using the effective rank (erank) (Roy & Vetterli 2007).
For each class y E V, we first gather the representations (z )" E Tdd of samples æ. whose pseudo-label is y and confidence score [=]=(((=))))] is above the threshold p. then forming feature matrix Z, = zy..,, 2,,,) E Rdxmy, where my and d are the number and the dimension of representations, respectively. We assume that Z, is full-rank, since it is exceedingly rare for any two representation vectors to be perfectly linearly dependent, and then compute its singular values, $1, $2, ..., Smy. These values represent the energy of the data along its principal components. To formally apply Shannon entropy, interpret the normalized singular value spectrum within
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probabilistic framework. Specifically, let I. be discrete random variable representing the index of a principal component within the feature space, We define the probability distribution of as p(i) = P(K i) s:/ Σك= 8j. Physically, p(i) represents the probability that unit of feature variance is aligned with the 3-th principal component. The erank is then defined the exponential of the Shannon entropy of this distribution:
my
erank (Z,) exp Σ p(i) logp(i))
The erank effectively measures the uncertainty of the energy distribution across principal directions, serving as a robust proxy for the EN of samples. Finally, the de-duplicated class distribution of the unlabeled data, Py, (y), is obtained by normalizing the effective counts across all classes:
crank Zny erank Zk)
(y) Σ kEY
4.2 LOGIT METAACALLBBATION
As shown in (T) the post-hoc LA technique introduces a hyperparameter r that modulates the overall adjustment strength. While empirical results demonstrate its effectiveness, the optimal choice of data-dependent, and no theoretical guidance is provided for its selection. To address this limitation, we propose a principled, data-driven approach LMC to determine T by making it a learnable param- eter, specifically tailoring it for pseudo-label generation in the context of LTSSL. First, we construct a proxy set De by resampling from the labeled set Dj to match the estimated unlabeled distribution Py, (u). Specifically, for each labeled sample (ar,, y:) E Dts we assign selection probability:
Py, (1:) Ny.
Pr,, (v) Ny
P((Ø), y1) is selected)
max NEY
where the denominator max Prr (v)/N,) serves normalization to ensure all probabilities are at most 1, thus implementing a form of rejection sampling to prevent oversampling. After performing Bernoulli trial for each sample with its corresponding probability, we obtain D. {a=", vt) bY and optimize T by minimizing the CE loss on D,.. The optimal T is found by:
v
T" arg min
Ler ((((((())))) -T-p))
V
where p (Pr. (1), (2),... Pr (K)) is the vector of our estimated de-duplicated class fre- quencies. Notably, we use LA term -T which is motivated by theoretical insights from (Mor & Carmon 2025) and deviates from the original post-hoc LA, which uses the logarithm of the class Trequencies. This linear term, as opposed to the logarithmic one, avoids potential numerical instabil- ity and overly aggressive penalization for classes with very small estimated probabilities, leading to a more stable optimization process. Finally, r" is used to calibrate the logits for generating pseudo- labels on the unlabeled data, thereby producing higher-quality supervisory signals.
4.3 END-TO-END TRAINING
Following the settings of previous researches Wei & Gan 2023 Lee et al.) 2021 Park al. 2024).
we adopt bual-branch architecture comprising balanced branch and standard branch. The bal- anced branch, on which we apply DDDE, is designed to produce class-balanced predictions. Con- currently, the standard branch leverages LMC to generate high-quality pseudo-labels. The overall adjustment parameter 7 is managed in a two-stage process. During an initial warm-up phase, T is configured according ACR (Wei & Gan 2023 Once the model achieves reliable estimate of the class distribution, LMC learns the optimal 7 for the subsequent training process. Additional imple- mentation details and time complexity analysis provided in Appendix G.2 H] respectively.
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5 GENERALIZATION ANALYSIS
We provide a theoretical analysis of the generalization performance classifier parameterized by Our goal is to bound the expected risk the target distribution P. for classifier h, whose controlling parameter T is optimized on the proxy dataset D,- Tighter bounds imply higher quality pseudo-labels. Let H., be a hypothesis space indexed by т, where each h, X + y is classifier. Let E(h, y) y x y - IR+ be a loss function. Then we make four assumptions as follows.
Assumption 1 (Lipschitz and Bounded Loss) The loss function ((h,(x), y) L-Lipschitz with respect to h (x) (the classifier's output) for all E y and is upper-bounded by U.
T is a standard assumption satisfied by many common loss functions like CE loss on a compact domain.
Assumption
Assumption 2. Each sample (xr), V.) ) in Dr is drawn i.i.d. from an underlying joint distribution Px,,y (a. v), abbreviated as P., and its marginal label distribution satisfies (v) = Py, (v).
While De is practically sampled from the finite set Di, for the purpose of theoretical analysis, we model it as being drawn i.i.d. from underlying distribution P... which is standard approach in learning theory. The latter part of Assumption 2 holds by construction within our framework. Specifically, the resampling procedure for constructing D, ensures that the expected marginal label distribution of the resulting dataset is equal to (v).
Assumption 3. The labeled and unlabeled data share the same class-conditional distribution.
Assumption is a central and common assumption in SSL, i.e., Px,IY_ IY_ (xly) = Px,,x (æly). Since the validation set D. is constructed via rejection sampling from the labeled set Dr, they inherently share the same class-conditional distribution, implying Px.ix (acly) = Px.|x; (arly) Following
the literature covariate and distribution shift (Shimodaira 2000), introduce importance
weight function w(x y) Px. Ny (ar.y)
Px Yy (a.2) Under Assumption the weight simplifies to the function
Px. (y.y)-Py..6 (M) Px,m, (elp) Pr, (v) Py. (v)
w(x, y) Px.Ir. (mlu) (v) (x.) (v) Py. (v) denoted as w(y).
Assumption 4. The importance weight w1) is upper- bounded by constant B 2 1.
Assumption holds provided that for all classes y E V, our estimated class frequency Py, (y) not a severe underestimate of its ground-truth value Pv, (v). This assumption highlights the importance of our proposed DDDE, which requires that our method does not produce substantial underestimates of the true class proportions for the bound to hold.
The expected risk of hú, on the target distribution Pu is Pp. (hr) = E(x., (((h Y.)]--
The empirical risk on the proxy dataset D, is Rio (h+) 1. ΣV (x)(())), v.). The weighted empirical risk is defined as (t, = + E w(y; (x))) generalization bound for h, on Pa.
We can now state our
Proposition 1. Under Assumptions I 2 and 4 for any 8 E (0, 1), with probability at least 1 - 8 over the draw of De, the following holds for all Ti, E H,F:
log(1/6)
Rp, (hr) < RD. (hr) + (&D...((h)) RD. (h. ) 2B L (HL)) BV
2V
The first term RD, reflects the performance on the proxy dataset, which we directly minimize when learning Iit..... RD_ measures the divergence between the proxy and target distributions. This theoretically demonstrates that our DDDE method crucial for the success of LMC. A more accurate distribution estimate leads to a smaller discrepancy term, a tighter bound, and thus a more reliable r" found by LMC. This links the two components of our method together. (HL)) is the Rademacher complexity of the hypothesis space, which captures the richness of the classifier family. Due to page limitations, additional analysis the discrepancy term [ŘD,,. w Rp, and the Rademacher complexity (H,,) is presented Appendix F
provides a form is general to many domain adaptation scenarios. To specifically highlight the unique advan- tages of our LMC, we delve into the convexity of the optimization objective in Appendix [F] which guarantees that standard optimization methods (such as gradient descent) can efficiently and reliably converge to the unique global minimum -" limited set D,..
While Proposition D
guarantee for our method, its

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Table 1: Comparison with other LTSSL methods under the various distributions on CIFAR-10/100- LT. Top and second-bes, performances are bolded and underlined, respectively.
CIPAR-IOLT REY
CIFAR-100-L REV
CON
MID
CON
Supersised
63.2788944
1862274
61.7962.4
50.4865.60
50.6425.64
67.12:1.05
47.0061.09
469GGELA
465361138
47,00=1.80
FixMarch
71.931405
71 2682.8
68 68.3023.44
70.102:435
66. 5..58888
G5 65.80 MM
47. 17.-0011.00
50.6622.01
56. 1.05
Mooed
ACR
80 00888822112 12
82 162
63,49EL15
79.7624.76
78.42
J5OSEL44
S1.661844
56. 1..2
CPE
80 28.6 18
82.9hel LB
84 440110011
50.5023.66
73 30.0
0211242 19
4888P1147
58618142
54.42:2.36
ST. 17.
Meta Expert
81 33342 58
83.12e1 09
85 09690.4
50 10.55e13.0
70 70.57
21
47.551115
55.00
35.00=2.00 2.08
S1 17.64:1.70
Sin-Pau Pir
81, 11.57:2222 22:29
8080100
70-781210
68.0011100 190
450661177
$7.56.1.49
58. 1.57
TRAS
05.21
66 35:2-3300
36.16
51 51.3999555
50.57
09.0142144
45.074209
460862445
47.442296
48.18-2.84
Lawy Reveinling Bavest
DeBitaPL
72.6125 41
GA 6.
47.921A1
56.1621.75
16.98
SAW
76.0788
06
7488861366
74.39
70.70
07.095177
48.2041.02
ST.80eL77
36.90:1.95
57 $7.87:1185
Nanoggling Uinse
ABC
1342
1840 85
81 838000
76 28.78:428
30.00
58 44
5F.1-1.70
58 7.557
69
22.57
18.451181
48.181175
56.72-1.01
CReST
70.78:4422
77.71
47.002143
57.4711.99
CReST+
62:224
78.3664400
78.3025.55
47.0241.71
67.7761.79
30.81=1.79
56.58:1142
Minig
19-12-127
97.40=1.70
CLSSEL
60.62112.0
10.1211.4
74.381234
70.3213:44
72.5514:54
67.2811.80
1.00
DARP
11.022272
70.30:4.33
67.8823.40
M673:1
47.8211.80
66.102131
36.82:1.9
56, 1.8
RDA 73.1488.44
71.011105
64.2865.04
70.541:4.00
06.0211.9
47.64:1.81
66 36.1621.46
$6.50-2.31
57 1..00
DASO 70 101188422
74
76.4761775
08.80:3..0
57.02-1
47.8941.52
57 1799
97.27
58. 1.90
COLA (Dus))
$1.87+2.10
93.66#11.9
85.61=1.80
81.80=3.41
80.60++3.22
59.04#1.09
50.26=1.28
00.39:1122
68.71=1.08
09.89:+1.45
6 EXPERIMENT
6.1 DATASET
Following the settings in (Ma al. 2024 Hou & Jia
2025 combine all the LTSSL methods
with the classical SSL algorithm FixMatch (Sohn ct al.
2020) and evaluate the performance on four
widely-used datasets: CIFAR- 100--100100010 (Krizhessyy 2009), STL-10-LT (Coates et al.) (0IT)) and
Small-ImageNet-122 127 (SIN-127)
(Fan et al.) 2022). More details can be found in the Appendix G.1
6.2 COMPARISON WITH OTHER METHODS
We compare CoLA with the following algorithms: supervised learning method, FixMatch (Sohn ct al. 2020), and FixMatch with 6 types of SOTA LTSSL algorithms. LTSSL algorithms include
5 LA algorithms, ie., ACR (Wei & Gan 2023), CPE (Ma et al. 2024). Meta-Expert (Hou &
Jia
2025), Sim-Pro (Du al.) (024). and TRAS (Wei al. (024). 2 re-weiggting algorithms, DeBiasPL (Wang et al. 2022a). and SAW (Lai et al. 2022). re-sampling algorithms, i.e. (Lee al., 2021). ADSH (Guo Li 2022). CReST (Wei et al. (021)) and CReST+ (Weil 2021). 2 data mixing algorithms, i.e. BEM (Zheng et al. (0024), and CoSSL (Fan et al.)
T.e.
ABC
et al.
2022
. distribution alignment algorithm. i.e. DARP (Kim al. 2020), RDA (Duan al. [0222).
UDAL (Lazarow et al. 2023), and ADELLO (Sanchez Aimar et al.) 2024), and I other types of algorithm, i.e. DASO (Oh et al. 2022). The results on CIFAR- 10/100-LT and on STL--10-LTT and SIN-127 are shown in Section 6.2.1 Section 6.2.2, and Section 6.2.3. respectively.
6.2.1 RESULTS ON CIFAR-10/100-E1
We evaluate CoLA against other algorithms on CIFAR-10/100-LT across 5 distributions: consistent (CON), uniform (UNI), reversed (REV), middle (MID), and head-tail (HT). For each distribution, we configure 2 or 4 distinct settings on CIFAR-10-LTT and 2 on CIFAR-100-LT, Each setting is executed 5 times with different random seeds. Due to page limitations, we report the mean and standard deviation aggregated over all runs for each distribution. For instance, the result of the consistent distribution on CIFAR-IO-LT: is the average of 20 individual runs (4 settings details on the experimental setup and results are provided in Appendix G.I and respectively.
seeds). Further
The results are presented Table [ from which we draw several key observations: (1) Our pro- posed CoLA achieves the highest accuracy across all five distributions on both the CIFAR-10-LT and CIFAR-100-LL datasets. This demonstrates its strong capability to handle diverse distributions reflective of complex, real-world scenarios; (2) The performance advantage of CoLA over other LA-based methods is more pronounced on the more challenging CIFAR- 100-LT dataset, where surpasses the runner-up by more than one percentage point in nearly all cases. We hypothesize
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Table Comparison with other LTSSL methods on STL-I0-LT. in the (7t 7a) row indicates that Th is unknown. Top and second-best performances are bolded and underlined, respectively.
(N1, M)
(150,
1004)
(450,
100k)
(1a, (u)
(10,-)
(20,
(10,
[20, -)
Supervised
46.62+1.54
0.66
6..79-1.42
1.42
56.33++1066
FixMatch
61.96:42.85
55.30=0.88
77.10=0.57
69.59:10.95
Logit Adjusiment Baxed
ACR
71.26:+0.75
08.13+1.51 1.51
79.26-0044
75.90++0.68
CPE
71.18:0.000
67 67.23:2.92
18.71=1.10
76.27:10.55
Meta-Expert
71.37:0.577
68.04+1.34
19.07=1.15
76.46:10.89
SimPro
09.24::0.8
63.32:2..00 2.00
77. 07.22=0.87
73.59:10.53
TRAS
50.29:11.50
42.38=1107
64.78:11000
55.78:10.66
Loss Reweighting-Boo Based
DeBiasPL
09.45-1:0.49
65.90+1.00
74.77=0.95
67.83:10.69
SAW
09.26:10.73
65.79=0.95
77.76=0.60
74.57=1.04
Rexampling- Baase
ABC
70.42:10.8
65.74 25.7442.33
78.677 08.67
76. 76.03:1074
ADSH
69.47++0.94
64.95-+0.99
76.03-0.88
72. 07.99
CReST
65.39:10.88
61.32+00955
76. 0.35
73.09
CReST+ 64.4241:.88
56.89=2.28
74.16=0.28
68.66:11.11
Data Mixing-Basse Based
BEM
68.07++0.92
61.48+0.78
75.51+1.38 1.33
65.60=1..2
CoSSL
70.9410.70
1.16
77.46=1.03
68.26:11.13
Distribution Aligionent-Be Basedd
DARP
63.75::2.20
56.08=1.08
75.03=0.68
68.03:20881
RDA
71.1110..2
66.7240.95
78.36=0.73
75.82:1069
UDAL
69.13::0.90
6..26=1142
77.26=1.04
73.74=1.15
Others
DASO
69.25:10.79
6..66=1.43
48-0077
73.80=1.02
CoLA (Ours)
73.32:10.73
68.96-12.55
79.70:10.18
77.53:10.80
that the relative simplicity of CIFAR-IO-LT may not be sufficient to fully distinguish the capabili- ties of highly competitive methods, which turn highlights the efficacy of our approach on more demanding tasks; (3) While CoLA sets
SOTA, other LA-based methods, namely Sim-Pro, Meta-Expert, and ACR, consistently secure the second-best performance across almost all the dis- tributions, outperforming approaches from other types. This validates the general strength the LA paradigm. Our work further advances this line of research by addressing two unique challenges for LA in the LTSSL context, thereby breaking through its performance bottleneck.
6.2.2 RESULTS ON STL-10-LT
We compare CoLA with other a on STL--10LLT, which is more challenging dataset that mirrors the real-world data distribution scenarios: the unlabeled distribution is unknown, and may contain out-of-distribution (OOD) samples The results in Table repbesent mean accuracy and standard deviation over 5 independent runs, from which we observe that CoLA outperforms other LA-based methods in all settings. For instance, given N₁ = 150 with r 10 and N; 450 with n 20, CoLA surpasses the second-best LA-based method Meta-Expert by 1.95% and 1.07%, respectively. Additionally, we compare with ADELLO (Sanchez Aimar et al. (004)). Unlike our work, which focuses on addressing the imbalance problem in SSL, the low-confidence distillation technique in ADELLO emphasizes OOD filtering. Due to page limitations, we present the compar- ison results in Appendix !
6.2.3 RESULTS ON SIN-127
We compare CoLA with several representative methods from other types on SIN-127. Following the settings in (Fan et al. 2022, Park et al. 2024), we conduct comparison experiments where the images down- sampled to the sizes of 32 32 64 x 64. The results shown in Table 3 from which we can observe that CoLA outperforms the other methods. Given that SIN-127 is large-scale dataset, the results demonstrate that CoLAA can also be applied large-scale datasets.
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6.3 ABLATION STUDY
To validate the effectiveness of DDDE and LMC, we con-
Table 3: Comparison on SIN-127.
duct ablation studies on -0110--- across 5 distinct
distributions, where we set (N1, M (1500, 3000) and
Image Size
32 x 32
64 64
(Nr, M (150, 300) on CIFAR-IO-LT and CIFAR-100-
LT, respectively. We evaluate the following variants: (1) w/o
Supervised FixMatch
12.19
21.01
13.30
25.41
D-T: The model without DDDE, where 7 is fixed to 1, 2, 4. (2)
w/o D-L: The model with only LMC, excluding DDDE. (3) D-L (Ours): The full model incorporating both DDDE and LMC. For all variants without DDDE, the unlabeled class dis-
ACR
22.73
36,28
Sim-Pro
21.85
35.89
DeBiasPL
13.49
26.54
ABC
23.66
34.88
tribution is estimated using the empirical sample frequency. From the results of Table 4 have several observations: (1) The optimal fixed r is inconsistent across datasets, for in- stance, on CIFAR-IO-LI
ADSH
17.28
23.26
CReST
12.25
25.59
CoSSL
13.62
27.02
generally performs better than 1, whereas the opposite trend is observed on CIFAR- 1-0- LT. And even the best performance among w/o D-T, r E 1, 2,4 is still lower than w/o D-L, which indicates that using a single fixed causes overall adjustment to conflict with new class-level adjustments, harming final accu- racy. (2) w/o D-L consistently underperforms w/ D-L, which shows that when class-level estimation unreliable, the LMC-llerned t becomes misguided. The above observations indicate that the in- teraction between the two adjustments is bidirectional and also highlight the individual contribution.
DARP
13.51
26.86
DASO
13.47
25.97
CoLA (Ours)
24.18
37.49
Furthermore, we conduct comparative analysis of DDDE against two alternative class distribution estimation methods: the Normalized Weighted Geometric Mean Approximation (NWGMA) (Baldi & Sadowski 2013) and the Monte Carlo Approximation (MCA) (Park et al. 2024). Specifically, we evaluate the average L2 distance between the estimated and true distributions calculated over the final 8 training epochs on CIFAR-10/100-LT across the 5 distributions. As presented in Table(5) our proposed DDDE consistently achieves the minimum L2 distance across all scenarios. This superior performance demonstrates that our method provides a more precise estimation of the unlabeled data distribution, critical factor that significantly improves the accuracy the pseudo-labels.
Table 4: Ablation of DDDE and LMC, ~ in the (kimaxi kiiin) row represents an arbitrary class.
Dalasel
CIFAR 10:LT
CIFAR: 100 LT
(Rmns Kmin)) (x. Tu)
[1, 10)
(-))-)
(10,
10)
(10.5)
(1. 100)
[-.,
(100, I)
(50, 100)
(100, 50)
(100, 100)
(100, 1)
(100, 100)
(100, 100)
(100, 100)
(10, 10)
(10.1)
(10. 10)
(10, 00)
(10, 10)
w/o D-1
83.12
82.03
80.57
82.34
81.30
56,23
18.07
59.33
55.69
57.83
w/o D-2
83.56
82.31
833
83.06
82.11
56.41
47.68
59.02
55.84
57.59
w/o D-1
82.61
81 11
75.41
79.25
53.32
16.05
51.22
56.1
w/o D-L
84.66
83.38
84.77
81.50
83.86
60.16
50.63
60.79
59.64
60.18
D-I. (Ours)
85.04
85.16
84.42
60.42
51.28
61.40
60.10
61.17
Table 5: La distance between the true distribution and the estimated one from different methods.
Dataset
CIFAR-IO-LT
CIFAR-100-LT
(Rmsss Rrutt)
(1. 10)
(-,-~)
(10, 1)
(5. 10)
(10,5)
[1. (1.100)
(~,^<)
(100, 1)
(50. 100)
(100,50)
(Th Tuo
(100,100)
(100,
(100, 200)
(100,100)
(100,100)
[10, 10)
(10,1)
(10, 10)
(10. D0)
(10,10)
MCA
0.0705
0.0464
0.2564
0.1132
0.1
0.0871
0.0431
0.0634
0.05400
0.0499
NWGMA
0.0616
0.0373
0.1495
0.1010
0.1097
0.0386
0.0505
0.0478
DDDE (Ours)
0.0529
0.0305
0.0891
0.0766
0.0585
0.0313
0.0353
0.0482
0.0468
0.0433
6.4 VISUALIZATION
In Figure we visualize the accuracy generated pseudo-labels throughout the training epochs across 5 different distributions on the CIFAR-10/100-LT. We observe distinct trend following the application of the optimal scaling factor т, derived from our LMC (indicated by the gray dashed line). Specifically, the improvement in accuracy is most pronounced on CIFAR- 10-LT under the uniform, middle, and head-tail distributions. Furthermore, a clear, yet modest, enhancement is also observed CIFAR- 100-LT for the consistent and reversed distributions. In the remaining settings,

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Published as a conference paper at ICLR 2026
Accunay po label
0.6
0.6
0.6
0.6
6.4
0.4
%
0.4
15
CIFAR-IBBA
PM
CIFAR-100 10011
700
Epoch
Epoch
Epoch (4) Eledd -ail distrbuttion
(a) Consistentt distribution
(b) Uniloom distribution
(c) Revened distributin
(d) Middle distribution
Figure 2:
Accuracy of pseudo-labels as function of the training epoch under 5 different distribu- tions. The blue and orange eurves represent the mean accuracy over independent runs CIFAR- 10-LT and SITAR-100-LT, respectively. The vertical gray dashed line indicates the epoch at which the optimal derived from our meta-learning applied for all subsequent training.
the accuracy continues to improve at rate comparable to the phase before this application. These results demonstrate that our proposed method not only preserves the baseline pscudo-label accuracy across diverse distributions but also achieves significant improvements in specific scenarios.
7 CONCLUSION
In this paper, identify two critical limitations in the existing LTSSL algorithms stemming from the design of the LA mechanism. First, their reliance on simple frequency counting overestimates the prevalence of head classes due to sample redundancy, resulting in harmful over-suppression of head classes. Second, inaccurate distribution estimation will affect the determination of overall adjustment strength,
empirically find that the optimal overall adjustment is highly sensitive dataset characteristics. Current research regards the overall adjustment as fixed, let alone adaptively adjusting according dataset characteristies. To resolve this, we introduce CoLA, which co- designs the class-wise and overall LA components. CoLA first produces de-duplicated estimate of the unlabeled distribution and then meta-learns the optimal overall adjustment strength on a proxy set that matches the refined distribution. Supported by theoretical analysis and extensive empirical results, CoLA establishes new SOTA the standard long-tail setup, which offers a more principled and adaptive approach to pseudo-labeling in complex, real-world settings.
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8 ETHICS STATEMENT
We confirm that this research strictly adheres to the ICLR Code Ethics. Our work focuses on addressing the problem of algorithmic bias LTSSL, which in itself contributes to improving the fairness of machine learning models. The datasets used in this study are all public computer vi- sion benchmarks (CIFAR- -00AR-00/000-,,, STL-1O-LT, and SIN-127) and do not involve any sensitive or private information, nor do they involve human subjects. Our method is a general algorithmic improvement, and the best our knowledge, there are no potential negative social impacts or risks of misuse. Therefore, we believe that this work does not raise additional ethical concerns.
REPRODUCIBILITY STATEMENT
To ensure the reproducibility of our research, we have made significant efforts. All our experiments were conducted on four public benchmark datasets, and detailed information regarding data pro- cessing, experimental setup, model architecture, and hyperparameters is provided in Appendix G The complete algorithmic details of our proposed CoLA method are described in Section Fur- thermore, have submitted our implementation code supplementary material. We will release our code after acceptance to facilitate the community in reproducing our experimental results and building upon work for future research.
ACKNOWLEDGEMENTS
This research partially supported by National Natural Science Foundation of China under grant No. T2541004, Zhejiang Key R&D Program of China under grants No. 2025C02 120, No. 2024SSYS0026, Zhejiang Key Laboratory of Medical Imaging Artificial Intelligence, and the Transvascular Implantation Devices Research Institute (TIDRI) under Grant No. KY052025003.
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A THE USE OF LLMS
We made light use of a large language model (Gemini 2.5) for grammar checking, spelling correc- tion, and minor language polishing. The model not involved in research ideation, experimental design, analysis, or interpretation. All scientific contributions and conclusions are entirely the re- sponsibility of the authors.
B SUPPLEMENTARY RELATED WORK
This section supplements Section 2 including five other types of LTSSL algorithms:
Loss re-weighting (Ross& & Dollár 2017) increases (decreases) the weight of tail (head) class samples in the training loss, ensuring the model learns better decision boundaries. SAW (Lai
et al. 2022) smoothly re-weights the training loss based on the estimated learning difficulty. De- biasPI. (Wang al. 2022a) designs the adaptive marginal loss to mitigate the pseudo-label bias.
Re-sampling (Chawla et al. 2002 He & Garcia 2009) addresses the bias by constructing bal- anced training batches to mitigate head-class dominance. ABC (Lee et al. (02T) achieves this
through class-aware Bernoulli sampling. paradigm extended by RECD (Park et al. 2024). which
incorporates a Monte Carlo estimation of the unlabeled data distribution for more precise balanc- ing. Other strategies focus re-balancing specific data pools. CReST (Wei al. 2021 prior- itizes the labeled data distribution, augmenting with high-confidence pseudo-labeled samples. Based on this, CReST+ (Wei et al. 2021) adopts a progressive distribution alignment to adap- tively adjust the rebalancing strength. In contrast, ADSH (Guo & Li (022) concentrates on the unlabeled set, generating class-balanees pseudo-labels via an adaptive thresholding mechanism.
Data mixing (Yun et al. 2019 Zhang al. 2018) performs mixing at element-wise or region- wise levels to create diverse training samples, which alleviates the problem of data imbalance. BEM (Zheng et al.) 2024) presents balanced and entropy- based mixing strategy to re-balance the class-wise uncertainty of training samples. COSSI. (Fan et al. (022) proposes a tail-class feature enhancement that expands the dataset by blending unlabeled data features with labeled data features while preserving the label of the labeled samples.
Distribution Alignment (Berthelot et al. 2020) reduces the confirmation bias via regularization added to the pscudo-label during inference. DARP (Kim et al., 2020) models the pseudo-label distribution alignment problem as a convex optimization problem and iteratively soft-corrects pscudo-labels to make their distribution approach the target class prior. RDA (Duan et al. 2022) introduces an auxiliary classifier to mutually align the pseudo-label distribution with the com- plementary label distribution, thereby addressing the challenge of mismatched class distribu- tions between labeled and unlabeled data without requiring any prior distribution information. UDAL (Lazarow al. (2023) unifies distribution alignment and LA. offering a loss-level per- spective. ADELLO (Sanchez Aimar et al. (2044) collaboratively employs dynamic distribution alignment and low-confidence sample distillation alleviate the issue of label distribution shift.
Others. Beyond these core directions, innovative frameworks address label bias through alterna- tive lenses. DASO (Oh et al.) 2022) obtains unbiased pseudo-labels by fusing predictions from similarity-based and linear classifiers. CISSI. (Kim al. 2025) computes independent contrastive losses for labeled and unlabeled samples, respectively, aiming to leverage the ground-truth infor- mation of labeled samples regularize feature representations and mitigate confirmation bias. SoftMatch (Chen et al.) 2023), while designed for standard SSL, addresses the quantity-quality trade-off relevant pseudo-labeling in LTSSL.
C TECHNICAL LEMMAS
Lemma 1 (Theorem 3.3 Mohri al. 2018). Let G be a family of functions mapping from arbitrary input space Z to 10, 1]. Then for any 8 > with probability at least - 8 over the draw of an i...d. sample S of size m, each of the following holds for all g E G:
log
E[g(z)] < m
Σ E((a)
29 (g) +
2m
1=1
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Corollary 1. Let 9 be a family of functions mapping from 2 to (a, b). Then for any > 0, with probability least 8 over the draw of an i.id. sample S of size m, each of the following holds for all g E G:
1 Σ - m
log
Elg(3)1
g(z))
mm (9) + (6 - a)j
2m
Proof. We could straightly follow the proof for Lemma 1 given by (Mohri et al. 2018 to derive Corollary without any other technique. The detailed proof is omitted.
Lemma (Talagrand's lemma (Mohri al. 2018)). Let D1,..., om be to R and 01.-.. ..... om be Rademacher random variables. Then, for any hypothesis set H of real-valued functions, the following inequality holds:
from R
E
sup hEH
0. o (2.)))
S
sup NEN
Σ o,,((z)) = 17))
In particular, if Φ; = t for all i [m], then the following holds:
Rs( H) I {1)s(H
D PROOF OF PROPOSITION
Our proof relies on standard results from statistical learning theory, particularly generalization bounds for importance-weighter risks (Shimodaira 2000).
Proof of Proposition First, we express the target risk Rp, (h,) as an importance-weighted expec- tation the proxy distribution P..
Rp.(h,)
Yu)l
- (() y) (x.
Ixxy (h,(
(o(y) drdy (o(
JXXY ((h,(z)
" ((h v)
Pu Pu(y) dadyy (v(y)
Jr wly) E w
doddy
where Pa y), Pu (arl)), and Pı()) denote the probability density function (PDF) of Pu, class- conditional PDF of Px.,Iy_ (alyy, and class marginal PDF of Py, (v), respectively. While (r y), Pv(y) denote the PDF of P... and class marginal PDF of Py. (y), respectively.
Let us define family of weighted loss functions G {g (x, v) + w(y) (x). v.y) h H, }. By Assumption I (x),)))) E [0, U]. and by Assumption 4 w(y) E B). Therefore, any function g E G maps to the interval [o, U B). We can apply Corollary [ to the family g and the sample D... For any E (0, 1), with probability at least 6, we have for all E g:
log(1/8)
( Y.M < V
Σ g(
29y (9) +U) B
2V
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Substituting the definitions of g, get:
log(1/6)
Rp. (h+) RD 29) U- BV
2V
Next, bound the Rademacher complexity Ry (g). Define Hc the family of loss functions associated to H, mapping from X' V to R: H2 = [he : (ær, y) + (x),,))),)) hy E Hr = = Eo Define mapping by IR R such that (y(a)
w(y) a. The function family 9 can be viewed as the composition of each function E(h, (2), 3) in Hy with the y-dependent function D denoted by fy o He. Since By is linear, and its derivative w(y)) S (Assumption 4). By is B-Lipschitz continuous. Now, we apply Lemma to 9 = B He:
Rv Ry (Φ, o He) S B- Rv,.H.
Although in the canonical form of Lemmaa@the term does not depend on the samples, it is obvious that the result continues to hold when expressed in this multiplicative form that depends subset of the samples (here, y). Since & is I.-Lipschitz (Assumption []. we apply Lemma to Hi = lo H7:
Ry (He)) Ry H.) S L. (H,,).
(4)
Combining and gives:
3
Ry (g) - B Ry (H+).
(5)
Substituting 5 back into (2) we get:
log(1/6)
Rr, (hr) < RD (hr) + 2B L. (yy) + U- BV
2V
Finally, we use the triangle inequality by adding and subtracting the empirical risk:
log(1/8)
Rp,(hr) S Ri (hr) + (p...).) - (h.)) - -2B L.
-L (H4) + B
2V
log log(1/5)
+ +B L. Ry,,,, - BV S RR.. + RD... (h+) - RD. (1r)
2V
This completes the proof of Proposition
E DETAILED ANALYSIS OF PROPOSITION
Discrepancy term |ÂD., Ro.
The discrepancy term RD.,, w (h+)
(h+)
can be con-
trolled via the class- mass-proportion error:
V (w(y²) (
U V
R (h+) Ro. (h+)
vi)
S
wwy(") - 1/.
V
V
i=1
Because w(y) (v) and Py. (v) Pr. (v) by Assumption (8)
taking expectations over D. gives
V
U
ED.
((y))) 11| U Σ Pr. (v) (w) U |Pr. Pv.ll..
Moreover, since 0 w(y) B (Assumption 4 the term |u(v!") -1/ 1/ bounded by max(1, B- 1). V U - 1/. we have that with
By applying Hoeffding's 's inequality to the empirical mean ΣV probability least 1 d:
RD (hr) Rp. (b) SU SU [IBv.
Pr. 11, + max 1)
log(1/8)
2V
Thus, when our estimator Py. is accurate (small L, error), the discrepancy term contracts at the same rate, formally tightening the overall bound and validating the methodology.
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Rademacher complexity Ry (Hr), To simplify the proof, denote the backbone logits (kept fixed when optimizing T) by s(x) E IR and write Sy(x) for the y-th component. Our LMC applies a linear correction -T with (Py, (1)..... Py, (K)). Hence
h (x), s,( T Py:
Py = Pr. (v) 2 0.
The relevant real-valued hypothesis class is therefore
H, (z. v) sy(x) Py r TE.
[0, Tmax] is the search interval used in practice (without bounded interval, the com- plexity would be unbounded).
where T =
For D,, the empirical complexity satisfies
Ri. (H., = E.
sup TET
Σ o.
(sy (2})
po.))
:=1
with o; E [-1,+11, Because the logits are constant with respect T, this splits into
1
RD. (H6+)
E
Σ Jssyt
(a:))
+
B
sup Te(u,
Σ P
V
1
R
sup
P:Pu
V
=E|U,Tmaxi
because E/o; O. Let Sp = Ev= OiPor. S 2 0 the supremum is achieved at 0: otherwise it is achieved at Tmax, yielding muux Spl. Consequently
Ro. (H+)
Tmax
Eg [|sp]]]
(6)
Let By be the count of class y inside D,. Then
Sp
Σ Py
Σ
and
Eg [S =
Σ
nypi
WEY
1:9! my
yEy
By Khintchine's inequality,
E. || S
E. S
pyp
NEy
Combining (6) and 7 yields the finite-sample bound
Rd. (H++)
Tmax
nyp
NEy
Taking expectation draws of D, and applying Jensen's inequality to E[
Evey yields
Ry (7+)
Tmax
Σ
(8)
VV
vEY
Using 8 inside Proposition I
leads to
2BI
log(1
())
Rp, (h,) & RD, (h-) + RD. (kt) )
RID (h+))
+
Σ p3 UB DEY
VV
2V
From the above inequality, can draw the following observations: (1) The third term scales as 1/VV and depends only the search radius Tmax and the geometry of the target class prior, not on the underlying deep backbone. (2) The quantity Σyey P (a third moment the class prior) shrinks when the distribution is closer to uniform and remains bounded even under heavy imbalance. (3) Without [0, Tmax]. Ry would diverge. In practice, the meta-search already constrains т. which simultaneously guarantees a finite complexity term. Consequently, the hypothesis class induced by meta-calibrating single scalar admits the explicit control 8
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F CONVEXITY OF THE OPTIMIZATION OBJECTIVE
significant and unique advantage of our LMC method lies the optimization landscape for the parameter Recall that find the optimal r" by minimizing the CE loss the proxy set D.:
arg min [(r) = arg min
1 V
Σ Las -x 1=1
-T
For this optimization, the logits ((;( and the estimated class distribution vector p are treated as fixed constants. The objective function [(r) is therefore function of single scalar variable We can formally state that [(T) is a convex function with respect to т. This convexity can be established by recognizing that the CE loss is equivalent the negative log- ligllikcl of the softmax function. The log-sum-cxp function is a well-known convex function. Specifically, our objective function for single sample be expressed
K
L((T) = ((r)=-(((((x)))) - .py;) + log
exp (=(a(<))) T .P)).
where the subscripts of =()(()))) and p mean to get the element value corresponding to the respec- tive class indices. The first term is linear in and the second term is log-sum-exp function of an affine transformation of т. Since the composition of a convex function with an affine mapping is convex, and the sum of convex functions remains convex, the total objective [(T) = + Σ, (i(T) convex. The convexity of ((T) is powerful property with profound practical implications:
Guaranteed Global Optimum: Unlike the highly non-convex and complex loss landscape en- countered when fine-tuning a deep neural network, our objective for has no spurious local min- ima. This guarantees that standard optimization methods (like gradient descent) can efficiently and reliably converge to the unique global minimum T".
Efficiency and Robustness: The optimization is performed over single sealar, making it com- putationally trivial and insensitive to initialization.
G DETAILS OF DATASETS AND IMPLEMENTATIONS
G.1 DATASETS
We conduct experiments on CIFAR-I0/100--1 STL-10-LT, and SIN-127. The detailed dataset splits are as follows:
CIFAR-10-LT with 10 classes contains 50, 000 images for training and 10, 000 for testing. We conduct experiments on CIFAR-IO-LT under 5 distributions, including consistent, uniform, reversed, middle, and head-tail distributions. Each one contains 4 settings, shown in Table
CIFAR-100-LY with 100 classes contains 50, 000 images for training and 10, 000 for testing. We conduct experiments on CIFAR-100-LT under distributions, including consistent, uniform, re- versed, middle, and head-tail distributions. Each one contains 2 settings, as shown in Table 7
STL-10-LT with 10 classes contains 000 images for training and 8, 000 images for testing, and 100, 000 unlabeled images as extra training data. The detailed settings are shown in Table g
SIN-127 down-sampled version of ImageNet- 127 (Huh et al. 2016), which is created by con- solidating the 1, 0,00 classes from ImageNet (Russakovsky et al. 2013) into 127 classes. SIN-127 has 281, 167 training samples with an imbalance ratio of approximately 286. Following Fan al. 2022), we randomly select 10% of the samples in each class as labeled samples. The samples are down-sampled sizes of 32 32 and 64 64. The class distribution of the test set of SIN-127 also imbalanced. The detailed settings shown in Table g
G.2 IMPLEMENTATIONS
We provide the pseudocode for CoLA in Algorithm where we use two LA strategies, including the loss modification technique for LA in line 18, and post-hoc LA elsewhere, while CoLA improves
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Table Detailed settings CIFAR-10-LE 10-LT.
No.
Distributions
(kiaaxe kimin)
(N1, Mknax))
(21.7)
1
(100, 100)
(1500,
2
(150, 150)
Consistent
(1,
3
(100, 100)
(500, 4000)
4
(150, 150)
5
(100, 1)
Uniform
(Arbitrary, Arbitrary)
(1500,300)
6
(150, 1)
7
(100, 100)
Reversed
(10,) 1)
(1500, 3000)
8
(150, 150)
9
(100, 100)
(1500, 3000)
10
(150, 150)
Middle
(5, 10)
11
(100, 100)
(500, 4000)
12
(150, 150)
13
(100, 100)
(1500, 3000)
14
(150, 150)
Head-Tail
(10,5)
15
100, 100)
(500, 4000)
16
(150, 150)
Table 7: Detailed settings of CIFAR-I00-LT..
No.
Distributions
(Kmaxe kmin)
(N1, Memse)
(10, 10) (15, 15)
Consistent
(1, 100)
(150, 300)
(10, 1)
Uniform
(Arbitrary, Arbitrary)
(150, 30)
4
(15,1) 1)
(10, 10) (15, 15)
Reversed
(100, 1)
6
7
(150, 300)
(10, 10)
Middle
(50, 100)
(15, 15)
(10, 10)
Head-Tail
(100,50)
10
(15, 15)
Table 8: Detailed settings of STL-10-LT.
denotes unknown.
No.
(N1, M)
(1i, 7w)
(10,
(150, 100k)
2
(20,
3
(10,
(450, 100k)
4
(20,
Table 9: Detailed settings of SIN-127.
No.
Image Size
(N1
(7r, Tw)
x
(27760, 1281167)
(ss 286, 286)
2
64 X 64
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upon post-hoc LA. We combine all the LTSSL methods with the classical FixMatch (Sohn al. 2020). which are trained and evaluated using the codebase Unified SSL Benchmark (USB) (Wang et al. 2022b) on 8XNVIDIA RTX 4090 with 24 GB memory. For training, we use ResNet-50 (He et al. 2016) on the SIN-127 experiment and Wide ResNet-28-2 (Zaoouykoo & Komodakis 2016 on
other datasets. The parameter settings are follows: P 0.95, 0.999, Tu 2. The optimizer for all experiments is SGD SStskever et al. 2013) with an initial learning rate of 0.03. The learning rate decay and momentum are 0.9 and 0.0005, respectively. The training epoch is set to 200 for experiments on SIN-127, and 256 for experiments on other datasets. For our proposed CoLA, the hyperparameter т is set according to ACR (Wei & Gan, 2023) during the initial stage of model training. For the final 16 epochs, its value is then determined using the LMC method. Therefore, we set Eg = 16, E2 = Eg - 32, EDDDE = 5. For evaluation, we calculate the balanced accuracy on the test set and average our results over five independent runs, i.e. seed 0, 0,,,,,3,4
Algorithm Co-Calibrated Logit Adjustment (CoLA)
Require: labeled set Dt unlabeled set Du., confidence P. epochs F., stage cutoffs (EDDDE, Ez, E3), EMA rate B, warm-up Tw: standard branch f. balanced branch f.
1: Initialize: Py <- 1/K, Tmin f- 0, f- Tur 7 <- Tw.
2: for = to F. do
3: if'e < L'DDE then
Warm-up
4: asd - T- log Py-
5: else if EDDDE S < Ey then
b> KL-guided interpolation (Wei & Gan 2023
6:
Compute symmetric KL scores between Pr. and 3 distributions (consistent, uniform and reversed).
7:
f- Tmin + (Tw min) softmax(KL.) aud f- T T-T-log -og Pyi
8:
9:
else
> Meta-learned logit adjustment
10:
add f- 7. P...
11:
end
12:
for
mini-batch (Bi, B.) do
Run f on each labeled sample x! B; to get logits 2(a(=))), Li CE(y, (=)((())))) C +- CE(y,, {(a((=)) Tu log Pvi)-
14:
Run f on unlabeled sample x; B, with weak and strong views to get logits =(a a )), =(A(x)))) {(a(x)))) {(A(x))) and unlabeled features.
9 a (z(a(())) agd), Új & arg max qi- mj < I(max qi 2 p). ā + (=(((((()))) -log Pr,), 0f + arg max q₁. my + I(max (imaxx((((a(aa) (((((<=))) 2 p). if EDDDE S then
17:
DDDE
19:
Within each class i.e., 1 y. gather unlabeled features z with my = 1, then forming feature matrix Z, and compute erank(Zg)
20:
end
21:
if Eg se< Eg then
LMC
22:
Sample proxy set D. by Bernuulii mask p. x Py, (v)) ( ((([(<x))())) - F -r.)), update via SGD on Ly,
23:
L, +
24:
end if
25:
Py. + BP,, + (1- Ener crank Zn)
erank(
26:
C. < CE(Ú,, a((x))))) with mask my V V[(max((((a(()))) 2 p).
27:
C - -EC( {(A(z)))) with mask my V T(max q 2 p).
28:
Update parameters on L Li C L. Cb. end for
29:
end for
31:
return balanced branch f.
H TIME COMPLEXITY ANALYSIS
DDDE Time Complexity. Let = E, my be the total number of high-confidence samples per class in each batch, not exceeding the unlabeled batch size |Bu]- In each training batch, DDDE
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first stacks the high-confidence unlabeled sample representations of each class into matrix Zy Rdxmy, then performs Singular Value Decomposition (SVD) and computes entropy and erank: (1) Constructing the representation matrix by duplicating each high-confidence sample once costs O(dM); (2) The cost of performing full SVD on Zy is ((min ((min(dm. dl myy)) In our experiments, the feature dimension d is typically much larger than the number high-confidene samples my per category in a batch, so the SVD complexity can be approximated as (ddm)); (3) Normalization and entropy computation only require O (my). which is much lower than the SVD cost. Thus, the additional time complexity of DDDE per batch is dominated by K SVD computations. The total additional overhead is approximately (dd EydEy1 m
LMC Time Complexity. Let Cj be the cost of forward pass to compute Logits. The LMC process
involves resampling the labeled set once on the estimated Py, to obtain the proxy set D,, and then optimizing the scalar r using cross-entropy loss: (1) Computing sampling probabilities for each sample and performing one Bernoulli sampling requires constant time, with time complexity O(N);
(2) Recomputing Logits for samples in Dr and adding the linear LA term -r Py. This process includes, for each sample, one forward propagation (O(Cr)) and one K K-dimensional softmax and cross-entropy computation (O(K)), totaling O(V (Cy + K)); (3) Gradient computation for the scalar and SGD update of both require constant time. Therefore, one LMC call only involves performing simple arithmetic operations on K- K-dimensional vectors (i.e., the Logits) over V samples to optimize the scalar, with an overall complexity of O (N + V (Cj- -))). This optimization computationally trivial, consistent with the conclusion at the end of Appendix F
Actual Run-time, We compare the training time of different methods using a single RTX 4090 GPU on the large-scale dataset SIN- 127, including image resolutions of 32 x 32 and 64 x 64. The results are shown in Table TO from which we can observe that the actual runtime of our method is not significantly different from the baseline method without DDDE and LMC ACR)).
Table 10: Actual run-time on SIN-127.
Method
32 x 32
64 x 64
FixMatch
31.12h
84.05h
FixMatch+ACR
38.27h
96.62h
FixMatch+CoLA (Ours)
39.53h
97.63h
I MORE COMPARISON RESULTS ON STL-10-LT
We evaluate the performance of CoLA versus ADELLO. The results are shown in Table 11 from which we can observe that: (1) ADELLO outperforms CoLA under the first, second, and fourth set- tings, which may be because the STL-I0-LT dataset contains a large number of OOD samples in the unlabeled set, and the low-confidence distillation in ADELLO effectively acts threshold-based OOD filtering or regularization mechanism. prevents the model from confidently and incorrectly assigning OOD samples to labeled categories or leveraging them for representation learning with- out hard label assignments, which is particularly advantageous on STL-IO-LT. (2) When ADELLO does not employ distillation techniques (ADELLO"), focusing the comparison on handling imbal- ance, CoLA outperforms ADELLO*, This demonstrates that CoLA offers a superior solution for the core long-tail problem. The contributions of these two approaches are largely orthogonal.
Table 11: Comparison with ADELLO on STL-I0-LT. * represents w/o distillation.
(N1, M)
(150,
100k)
(450, 100k)
(11,7) 7u)
(10,
(20,-)
(10,
(20,-)
ADELLO
71.26:10.4
67.36+1.37
75.17+1.06 79.44=0.46
77.76=0.95
ADELLO
74.2440.65
71.56+0.74
77.95+00.8
CoLA (Ours)
73.32-10.73
68.9642.55
79.70-+0.48
77. 77.33:10.80
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J BREAKDOWN OF RESULTS ON CIFAR-10/100-LI
The comprehensive breakdown of comparison results on CIFAR-101000-L1 under consistent, uni- form, reversed, middle, and head-tail distributions is shown in Table 12 13 14 15 and 16 respec- tively. Our method achieves SOTA performance across almost all data distonbutions.
Comparison with other LTSSL methods under the consistent distribution. Top and second- best performances are bolded and underlined, respectively. Metrics represent mean accuracy and standard deviation over five independent runs.
Table 12:
Dataset
CIFAR-IO-LT
CIFAR-100-EE
(N1, M1)
(1500,
3000)
(500, 4000)
(150, 300)
71 = 7u
100
150
100
150
10
15
Supervised
63.72+1551
5..61=1.58
46.8441.23
42.92:1.14
48.5940.64
45.66:1055
FixMatch
76.71+1.00
72.63+0.80
72.13+2.12
66.25+11.58
57.6240077
54.16+00.65
Logit Adjustment-Bant Based
ACR
84.17=0.50
81.20=0.77
81.31:0.09
76.86:22. 15
39.83:10.66
56. 76:10.8.81
CPE
84.16=0.41
01.28=0.56
79.91+1.05
75.76++099
55. 35.1440..43
51. .1.09+0..59
Mcta-Expert
83.92=00466
8..07=0.34
81.77+1.21
1..54+1.16
56. 1.0101
52.71
Sim-Pro
84.04+0.45
12.21+0.11
81.91
78.1 00.1
39.699+0.51
56.90++0.81
TRAS
66.40==0998
62.23 1.00
56.73++1.00 1.00
51.46-t:1.46
51.48++0.66
47.55-+0.51
Loss Reweighting Baaeeeeinn---eeddd Based
DeBiasPL
17.78=1.00
03
72.71+1.28
66.17+11.70
57.90=0.66
54.8440.77
SAW
80.75=0.69
77.33=0.64
77.13+1.31
71.09++1..68
58.59+00.56
55.59++0.86
Resampling Based
ABC
83.62=0.80
80.62=0.37
79.5810.00
76.78110.0
58.50.10.B
55.63.10.4
ADSH
77.92=0.82
73.20=0.17
75.22410.83
68.04+1:.93
1840044
54.85
CReST
79.04=0.49
74.78=0.61
77.14+:1.88
69.68:12.11
57.76220.38
54.60
CReST+ 79.19=0.44
74.61=0.44
78.66+0044
71.96+1.600
57.51#0.31
54.56:+0.40
Mixing- Based
BEM
80.1210.78
70.71=0.60
75.3240.74
60.9241.50
59.03:10.95
06.13:10.36
CoSSL.
83.41+1.18
81.63+0.26
80.29+0.67
76.74+00.2
58.06++0.65
55.74+00.45
Distribution Alignment-Bant Based
DARP
78.25=0054
74.43=0.41
74..55:11.2
69.00+0.77 76
58.21=0.57
55.2410.61
RDA
77.42+1.32
73.21=1.25
73.48+1.09
68.45410.00
58.13:00.5
54.61
Others
DASO
75.84-0.72
72.21=0.73
69.15111.0
63.4641..00 1.
58.95:1055
55.69-1077
CoLA (Ours)
85.04-t0.32
82.42:t0.54
81.7810.55
78.244-2.09
60.42-00.60
57.67+0.80
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Table 13: Comparison with other LTSSL methods under the uniform distribution. Top and second- best performances are bolded and underlined, respectively. Metrics represent mean accuracy and standard deviation over five independent runs.
Dataset
CIFAR-I06 10-LT
CIFAR ---AT-
(Ni,
(1500, 3000)
(150, 300)
71
100
150
10
15
Supervised
63.46
58.90=1.63
48.61=0.55
45.57=0.36
FixMarch
73.114:125
09.3644..14
48.93:10.51
45.88.10.30
Logit Adjustment-Baute
ACR
83.0411..2
2820.54
50.24=0.69 69
47.72:1.50
CPE
83.45.10.58
81.73-0082
48 13.0.000 6060
45.6411.78
Neta-Expert
84.04++0.48
82.21
49.08=0088
46.61:10.93
Sim-Pro
81.40:::077
50.32:10.54
09.94
46.51:1044
TRAS
60.28:+0.73
56.22:+1.06
46.90=0.54
43.24=00.66
Loss Reweighring-Bian
Baased
DeBiasPL.
74.00
69.29-22.51
49.43=0055
46.42-00.6
SAW
81.84+0.20
80.85=20.62
49.63=0066
46. 16.76:20.59
Resampling- Based
ABC
81.70::0.64
80.47=0.61
50.46 0.333
47.68=0.44 46
ADSH
76.35=20.83
T3.97=0.88
49.74.0.86
46. 066622-0.33 04
CReST
83,26:20.88
80.7320.87
49.03=0.27
45. 15.96100223
CReST+ 83,02++0.77
79.99=4034
49.23=0.34
46 46.02:+0.16
Data Mixing- Based
BEM
81.35+00882
78.7920.72
50.27=0.73
48 48.16:20.58
CoSSL
80.51 0.07
78.33 1.40
50.79:0..00
48.34:10.44
Distribution Alignment-Based
DARP
73.49+t..33 1.39
69. 09:43933339
49. 09.29
46.35=10.30
RDA
78,08+0..00 60
76. 7..95=1.12
49.30=0.64
45.98=0.30
Others
DASO
76.2241.00
73.27=1.22
49.33=0.55
46.4410.45
CoLA (Ours)
84.83.10.34
82.48-L0.39
51.28.10.45
49.23-10.77
24

---

Published as a conference paper at ICLR 2026
Table 14: Comparison with other LTSSL methods under the reversed distribution. Top and second- best performances are bolded and underlined, respectively. Metrics represent mean accuracy and standard deviation over five independent runs.
Dataset
CIFAR- 10-LT
CIFAR-100-LA
(Ni, 711 = Tw
(1500, 3000)
(150, 300)
100
150
10
15
Supervised
63.76
59.71=1.56
48.4820.33
45.4241.21
FixMarch
70.774:2.19
65.82:11466
07.70
53.97:0.59
Logit Adjustment-Based
ACR
84.35:1101
82.62:0.27
60.27=0.16 46
58.16::0.33
CPE
84.46:1:.09
83.5620.44
59.6420.22
57.5810.33
Neta-Expert
85.36++0024
84.69:10.56
60. 0.64
57.04:1066
Sim-Pro
75.83:10.82
73.60=1.33
59.07=0.52
55.66:10.71
TRAS
57.93++0.68
54.399
48.37=0047
43.78=+0.50
Loss Reweighting Baased
DeBiasPL.
71.44++2.06
65.92+
0..46
54.73+0.82
SAW
76.09:20.84
73.56=0.88
59.03=0.63
55.98=10.70
Resampling- Based
ABC
82.56:00.22
81.10=0.30
09.39=0063
37.57=0067
ADSH
69.95-10.88 86
67. 1.::1.145
56.07=0.65
52.64
CReST
84.26:20.70
82.95=33883
59.31=0.58
56.59-10.4 47
CReST+ 72.17+1.28
67. 0..60.63
59.03=0.22
55. 35.700+0.52
Data Mixing- Fixing-Based
BEM
79.27=1.20 1.20
76.27=1.22
59.54=0.79
57.00 57.06=11.8 33
CoSSL
76,20+ 1888
72.5620.69
68:001
55.63
Distribution Alignment-Baan Based
DARP
70,96 47
66. 16.73+1100 1.44
58.04=0.55
54.75=11.00 .04
RDA
73.05+0051
70.56+0.49
57.8 07.5577
54.50=0.77
Others
DASO
77.61+1.05
1.53
59.64=0.71
56.22+1095
CoLA (Ours)
86.84.10.73
84.38+1.08
61.40-10.65
59.38-10.59
25

---

Published as a conference paper at ICLR 2026
Table 15: Comparison with other LTSSL methods under the middle distribution. Top and second- best performances are bolded and underlined, respectively. Metrics represent mean accuracy and standard deviation over five independent runs.
Dataset
LIFAR-I--LT
CIFAR-100-LT
(N1, MPK/a1)
(1500,3000)
(500, 4000)
(150, 300)
2 = Yu
100
150
100
150
10
15
Supervised
63.764 1.38
19.3841.554
46.82442.0
1..94
48.24:10.68
45.41+0.37
FixMatch
75.65+0.71
71.63+0.78
68.76+1.43
64.36+1.38
57.49++033
53.81+0.70
Logit Adjustment-Bont Based
ACR
84.0610.81
81.8611106
00.73:10.68
72. 22.32:12.44
59.13
56. 16.18.10.70
CPR
84.42+00.1
81.98
80.17++1.34
76.65 1.35
56.53++00
52.31 +00988
Meta-Expert
84.3611.22
52.26-40.37
80.71110.70
76.07 +:
57.00+10.51
53. 03.18240..6
Sim-Pro
82.5411044
79.75.70.87
76.75:11.12
71.72:1.38
59.05:10.65
56.05:10.60
TRAS
62.7410.64
57.71110.97
44. 1..75:11.32
40.15:22.12
50.04:10.71
45. 25.24410.22
Loss Reweighting Based
DeBiasPI.
76.83+0.46
+2.27+0033
68.92++0.82
64.68 1.30
57.69++088
54.62++0.40
SAW
80.09+40.56
70.94
73.54:4:.00
67.58422.80
58.66+00.66
35.19440.66
Resampling Based
ABC
83.23:10.54
80.5640.80
78.61:1.61
72.40:12.07
58.81:00.2
35.804:0.
ADSH
72.8410.87
68 0.22
62.51+0.88
18.4111555
56.56:10.48
52. 29:8994-000 67
CReST
83.73+0..2
79. 8620..50 56
81. 0.377
73.8622.92
58.71:0.23
56.22.1-0. 53
CReST+
81.82:00.95
06.76
$1.81+0.80
72.98
58.48:20.39
35.144:0.22
Data Mixing Bixing-Based
BEM
84.35=0.37
82.07+0.422
79.12+1.02
75.62=1120
59.11=0.37
55.80++0.52
CoSSL
83.39+0.37
81.04+0.44
78.17+1.03
74.67+1.29
58.65+0050
55.95++0.77
Distribution Alignment- Based
DARP
76.65-1..00 .0 70.93+4..5
02.34
68.90
1..32:11.35
57.63:10.49
54.02::.77
RDA
66.73:0.668
62.66+11.59
57 37.9841.00 1.52
57.70=20.33
53.37400.6
Others
DASO
75.20+1.05
70.96:0.92
65.93++1.44
61.1841.89
58.9110.74
55.64:0.61
CoLA (Ours)
0..16::0773
83.27:0.37
82.07:0.45
76.9442.81
60.10+0.41
57.32=0.76
Table 16: Comparison with other LTSSL methods under the head-tail distribution. Top and second-
best performances are bolded and underlined, respectively. Metrics represent mean accuracy and standard deviation over five independent runs.
Datasel
CIFAR-IOLA
CIFAR-1004
(N1, MgiRis
(1500, 3000)
(500, 4000)
(150, 300)
100
150
100
150
10
15
Supervised 61:03:11.48
59.73:11.78
47.17:2.001
43.61=11.88
48. 49:10.31
45.52=0.22 20
FixMatch
75.13+11.74
68.99-42.07
63.87:+3.09
58.00+22.37
58 58.25400.7
54.93++1.01
Logit Adjustment-Based
ACR
83.48:20.11
$1.42:20.55
76.13400.98
72.64
59.93:20.55
57.58+1 38
CPE
83 $3.36:1:0221
31.56:10.09
13.63344442
05.06:41.58
58.57440068
55.74:10.17
Meta-Expert
83.70 03.700-0.92
81. 71.79+-0.99 98
78. 894-0.53
75.08::1 1.11
59. 22+-0.52
06.066--.66 63
Sim-Pro
79.722+0.77
77. 77.50:111.0
76, 76.38:222.50
73. 73..444:22.00 86
50.72+0.74
57,000:0.0 63
TRAS
59.09-+0.59
15.11111.17
49.08-40.55
45.47-41.00
50.48-+1.01
45.79-+0.82
Loss Reweighting-Ban
DeBiasPL
75.98=11.67
69.58-+4.04
64.60-33.6
58.9742.78
58.4440.81
55.52++0 69
SAW
79.06:10.90
77.93:1122
76.5440.57
13.28411.13
59. 43.40.88.1
56.32:t0 .2
Resampling- Baaee
ABC
82.98:10.33
81.01:00.66
88.8740.34
74.71:10.95
60.17:40.47
$7.32:10.44
ADSII
75.51+1181
T3.92+11.0 1.07
74.55+22.10
70.68#11.1 1.12
58. 15:+0.39
$4.38:400. 81
CReST
83 22:2900:222 0.28
82,00:10. 49
7..64411667
70,89 61
57. 83:1: 0.23
55.04::0.00
CReST+
50.39++0.887
25.25-42.97
72.44422.3
64+12:00.89
57. 37.99---0.20
55. $5.25-+0.22
Data Mixing-Band Based
BEM
78.45:11.32
75.01
72.08
67.93
59.28
56.83
CoSSL
77.59411.36
74.1742.11
41.4312.008
67.00::1.41
69-10.99
56.63:10.00
Distribution Aligionent BlisedeettBased Based
DARP
75.87:12.02
69.50:12.25
03.66:12.25
59.29113336
58.50-10060
55.35.1 1.02
RDA
75.42++0.85
73.32++
69.00-41.52
65.60++0.76
58.80++0.75
55.31 +-0.74
Others
DASO
75.11+11226
71.1++1.99
66.2141.11
61.70++1.77
59. 86:40.777
56. 45:40.00.0
CoLA (Ours)
84.42-10.45
82.51 1..28
79.80-10.34
75.88:10.59
61.17-10.56
58.61 1..54
26


Reviews:
[{"review_id": "fupnLUGxSn", "reviewer": "Reviewer_nQAr", "summary": "This paper focuses on improving Long-Tailed Semi-Supervised Learning (LTSSL) by employing logit-adjustment methods. The paper identifies two key issues with the logit-adjustment approaches: 1) An over-estimation of popular class probabilities can lead to over-suppression of model predictions for the popular classes; and 2) The overall adjustment factor in the logit-adjustment approach is sensitive to particular dataset and needs to selected carefully. The paper then proposes two solutions to address these two issues, namely *de-duplicated distribution estimation* (DDDE) and *logit meta-calibration* (LMC). The paper provides a generalization analysis for LMC. The paper then provides a comprehensive empirical evidence of the value of the proposed approach while comparing it with existing logit-adjustment-based approaches and other methods beyond logit-adjustment for LTSSL in the literature.", "questions": "- Could you please provide a description/pseudocode of your overall method in the form of an algorithmic block or a figure?\n- In Line 275, the assumption is that $P\\_{X\\_{u} \\mid Y\\_{u}}(\\mathbf{x}|y) =  P\\_{X\\_{l} \\mid Y\\_{l}}(\\mathbf{x}|y)$ (between $u$ and $l$). However, the importance weight in 277 deals with $P\\_{X\\_{u}, Y\\_{u}}$ and  $P\\_{X\\_{v}, Y\\_{v}}$. Are the authors relying on the fact that since $\\mathcal{D}\\_{v}$ is sub-sampled from $\\mathcal{D}\\_{l}$ and thus share the same class conditionals? If yes, please consider making this clearer.\n- Please consider making the notations consistent. E.g., Line 286 uses $R_{P\\_u}$ (with lowercase $u$) while Line 294 uses $R_{P\\_U}$ (with uppercase $U$).\n- Did you use the logit-adjustment form in Line 234 (as opposed to the one in Eq. (1)) for your experiments?\n- Why have you used the form in Line 226 for sampling? Why can one not simply use $\\hat{P}\\_{Y\\_u}(y\\_i)$ as the probability to select $(\\mathbf{x}^l\\_i, y\\_i)$?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "- The paper studies a well motivated problem by identifying key limitations of a widely popular approach in the literature.\n- The empirical results in the paper clearly showcase the improvements compared to competitive baselines from the literature. \n- The paper presents ablation results to establish the value of both DDDE and LMC.", "weaknesses": "- The generalization analysis in Section 5 does not significantly enhance the overall contributions of the paper. Does the theory inspire/motivate the method proposed in the paper? If not, does the theory provide useful guarantees towards the final performance of the proposed solution? \n- The presentation of the theoretical part of the paper can be greatly improved.\n   -  Could the authors expand on the Line 298 (``If our estimation is accurate,...justifying our methodology``) and make it mathematically precise. \n   - If the reviewer understand it correctly, the only optimizing parameter in Section 5 is $\\tau$ and the function class $h_{\\tau}$ is linear with respect to $\\tau$ (since $\\tau$ is the overall logit-adjustment factor). Could the author attempt to provide a more explicit characterization of the Rademarcher complexity in this case?", "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": "", "extra_scores": {}}, {"review_id": "tqtZMkrPFO", "reviewer": "Reviewer_bnRy", "summary": "The authors address in their manuscript the problem of long-tailed semi-supervised learning. They analyze the weaknesses of several methods based on logit adjustment and propose an approach CoLA that is claimed to suffer less from over-suppression. They further propose DDDE to estimate the unlabeled distribution and LMC as a meta-learning strategy. They validate their method on 4 different datasets (CIFAR-10/100-LT, STL-10-LT, and SIN-127).", "questions": "1. (related to weakness 2.): which are the experimental results that clearly show that the _interplay_ of the two types of LA cause the issue? \n2. (related to weakness 4.): what results are obtained with the protocol from Kim et al. 2020 or is there some other way to make the results comparable?\n3. (related to weakness 5./1.): as the statement about state-of-the-art results need to be revised: in which situation does the proposed method shows its main strengths and weaknesses?\n4. (related to weakness 6.): why is the quantification possible and why is $p(i)$ a proper probability?\n5. (related to weakness 7.): what is the overall approach in 4.2?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The paper is well-written (with exception line 090 \"each class's representations\") and structured (DDDE=4.1, LMC=4.2, CoLA=4.3).\n2. The main hypothesis is clearly stated and the approach is presented in a plausible way.\n3. Technical parts are accurately and detailed described.\n4. A good set of datasets has been selected for the experiments.\n5. The results are good (even if not always beating the state of the art)", "weaknesses": "1. The related work is covering a subset of the field, important references (such as [1]) are only used in the appendix, other are completely missing e.g. [2-5].\n2. The premise for the main hypothesis, i.e., the negative effect of overlooking the interplay between the two types of adjustment, has not been clearly supported by experimental results. If the premise is not properly established, all subsequent claims are affected. Figure 1b) only shows that monotonicity is not fulfilled.\n3. Some statements lack evidence or reference, e.g. line 085. \"current work\" embraces all works, see also 1. (this is not saying that any of [2-5] does).\n4. The protocol (for CIFAR) chosen according to Du et al. ICML 2024 (Simpro) deviates from other, previously published protocols, e.g. from the cited paper [6], and makes comparisons difficult, in particular to state-of-the-art methods that remained un-cited and that use those previous protocols.\n5. For experiments that use compatible protocols, e.g. STL-10-LT, the proposed method is inferior to e.g. [4} and rather en par with [3]. \n6. The description of erank is not sufficiently self-contained. In particular, it is not clear why the EN is estimated by erank. Line 197 just says \"we quantify EN ... using ... erank\" and references point to EN and erank, but not why the quantification is possible.  Also, it is not obvious why line 205 $p(i)$ is a probability and not just a point in the $m_y$-simplex.\n7. 4.2 is partly written in a procedural way and the overall approach that leads to the algorithm needs to be stated more clearly.\n\n[1] Zhang et al. Mixup: Beyond empirical risk minimization. ICLR 2018.\n\n[2] Lazarow et al. Unifying distribution alignment as a loss for imbalanced semi-supervised learning. CVPR 2023.\n\n[3] Chen et al. Softmatch: Addressing the quantity-quality tradeoff in semi-supervised learning. ICLR 2023.\n\n[4] Aimar et al. Flexible distribution alignment: Towards long-tailed semi-supervised learning with proper calibration. ECCV 2024.\n\n[5] Kim et al. Separated and Independent Contrastive Semi-Supervised Learning for Imbalanced Datasets. IEEE Access 2025.\n\n[6] Kim et al. Distribution aligning refinery of pseudo-label for imbalanced semi-supervised learning. NeurIPS 2020.", "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": "", "extra_scores": {}}, {"review_id": "JhYxZ526sg", "reviewer": "Reviewer_peuh", "summary": "This paper attempts to tackle an problem in Long-Tailed Semi-Supervised Learning (LTSSL): the confirmation bias driven by biased pseudo-labels. The authors claim that existing methods based on LA suffer from two critical limitations: 1) they rely on naive frequency counting to estimate the unlabeled data distribution, and 2) they treat the overall adjustment strength as a fixed hyperparameter. The authors propose CoLA, which consists of two main components: first, a De-Duplicated Distribution Estimation (DDDE) module that attempts to estimate a more accurate class distribution by calculating the effective rank of class representations to account for sample redundancy. Second, a Logit Meta-Calibration (LMC) procedure that constructs a proxy validation set and uses meta-learning to automatically optimize the overall adjustment strength. The experiment results show that their method establishes a new state-of-the-art on four public benchmarks.", "questions": "See in Weaknesses.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The authors do identify a potentially interesting aspect of the LA mechanism, namely the interplay between the class-wise adjustment and the overall adjustment strength. This is a reasonable observation.\n2. The authors evaluate their method across multiple datasets and distribution mismatch scenarios.", "weaknesses": "1. The first of my concern is novelty. The core idea of this paper is little more than a combination of existing techniques, such as effective number, dual-branch, and meta-learning for hyperparams.\n2. The description of the DDDE module is overly simplistic. The authors propose computing the effective rank for the representation matrix Zy of each class y. They fail to discuss the computational cost of this procedure.\n3. In Figure2 (b,d,e), after applying LMC, the slope of the pseudo-label accuracy improvement barely changes. So I think its contribution is questionable.\n4. The proxy validation set Dv is resampled from the small and imbalanced labeled set Dl. For tail classes, Dl may contain only a few samples. How do you justify that such a severely constrained proxy set can effectively guide the learning of $\\tau$ for a massive and differently distributed unlabeled set Du?", "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": "", "extra_scores": {}}, {"review_id": "iHfiTtpflB", "reviewer": "Reviewer_DkFy", "summary": "This paper addresses the issue of pseudo‑label bias in long‑tailed semi‑supervised learning, and identifies two core limitations in existing Logit Adjustment (LA)–based approaches: distribution estimation distortion caused by sample redundancy, and the use of a fixed overall adjustment strength. To overcome these limitations, the authors propose the CoLA framework, whose main components include: estimating the effective number of samples via the effective rank of the representation matrix to obtain a de‑duplicated class distribution; and, based on the estimated distribution, constructing a proxy validation set and optimizing the overall adjustment strength through meta‑learning so that it adapts to the characteristics of the current distribution. Furthermore, the paper provides a theoretical generalization error bound for the proposed LMC and validates the method through experiments on benchmark datasets.", "questions": "See the weaknesses.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "1. The problems identified in the paper are reasonable and important, and the authors provide a detailed analysis supported by both theoretical justification and experimental evidence.\n2. The paper presents a generalization bound and a convexity analysis for the meta‑learning process, which enhances the theoretical soundness of the proposed method.\n3. The visualization in the ablation study is presented in a clear and comprehensive manner.", "weaknesses": "1. The computation of effective rank and the meta‑learning procedure may introduce substantial computational overhead. It would be beneficial to include an analysis of the time complexity and computational complexity, particularly with respect to runtime performance on large‑scale datasets.\n2. In the downstream experimental evaluation, I notice that the SIN‑127 dataset is a down‑sampled version of ImageNet‑127. Why not perform testing directly on ImageNet‑127? I am curious about the potential results on the full ImageNet‑127 dataset.\n3. The paper lacks a primary diagram illustrating the proposed method. Introducing a main figure would make the methodology clearer and facilitate reader understanding.", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)