A Strong Balanced-Softmax Classifier-Retraining Baseline for Long-Tailed Recognition

# A Strong Balanced-Softmax Classifier-Retraining Baseline for Long-Tailed Recognition
Source: [https://arxiv.org/html/2607.09832](https://arxiv.org/html/2607.09832)
\(July 10, 2026\)

###### Abstract

Long\-tailed recognition methods often modify losses, margins, or representations to reduce the dominance of frequent classes\. We ask whether, after Balanced Softmax training, the remaining tail error can be reduced by retraining only the classifier\. We evaluate BS\-cRT, a two\-stage procedure that trains a backbone and cosine classifier with Balanced Softmax, freezes the backbone, and updates only the classifier on balanced episodic batches\. The second stage keeps the empirical\-prior Balanced Softmax objective and uses raw cosine logits at inference\. Across CIFAR\-100\-LT, CIFAR\-10\-LT, ImageNet\-LT, and Places\-LT, this classifier\-only step consistently improves Few\-shot accuracy over the matched Balanced Softmax checkpoint\. At imbalance factor100100, Few\-shot gains are\+5\.15\+5\.15points on CIFAR\-100\-LT and\+5\.83\+5\.83on CIFAR\-10\-LT; on ImageNet\-LT and Places\-LT, gains are\+6\.92\+6\.92and\+9\.78\+9\.78points, respectively, with a Top\-1/Few\-shot trade\-off on ImageNet\-LT\. We also analyze Counterfactual Boundary Risk Minimization \(CBRM\), a boundary\-probe extension using prototype\-based features near decision boundaries\. CBRM identifies two failure modes: scaled\-logit cosine margins destabilize training, and corrected hardest\-negative probes remain head\-class anchored\. The results support BS\-cRT as a practical classifier\-side baseline and indicate that boundary supervision must account for class frequency\.

*Keywords*Long\-tailed recognition⋅\\cdotImbalanced classification⋅\\cdotClassifier retraining⋅\\cdotBalanced Softmax⋅\\cdotFew\-shot recognition⋅\\cdotDecision boundaries⋅\\cdotClassifier bias

## 1Introduction

Real\-world visual recognition rarely enjoys the balanced label distributions assumed by standard benchmarks\. Instead, categories typically follow a long\-tailed frequency profile: a small number of head classes dominate the training set, while many tail classes appear only a handful of times\. This imbalance is especially problematic because the rare classes are often the most important ones in practice, such as unusual medical findings, rare species, safety\-critical failures, or infrequent industrial defects\. Models trained naively on such data tend to allocate excessive decision space to head classes, yielding deceptively strong overall accuracy while failing on large parts of the label space\(Yang et al\.,[2022](https://arxiv.org/html/2607.09832#bib.bib36)\)\.

A large body of work addresses long\-tailed recognition by modifying the training signal induced by observed examples\. Re\-weighting methods amplify the loss of rare\-class samples\(Cui et al\.,[2019](https://arxiv.org/html/2607.09832#bib.bib5)\); focal\-style objectives reduce the contribution of easy examples\(Lin et al\.,[2017](https://arxiv.org/html/2607.09832#bib.bib18)\); margin and logit\-adjustment methods reshape the classifier geometry according to class frequency\(Cao et al\.,[2019](https://arxiv.org/html/2607.09832#bib.bib2); Ren et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib23)\); and decoupled training recipes separate representation learning from classifier learning, often retraining the classifier under a more balanced sampling distribution\(Kang et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib13)\)\. Among these, Balanced Softmax has become a particularly strong and simple training\-time baseline\. At the same time, classifier retraining \(cRT\) has shown that much of the tail\-class deficit can be addressed after a representation has already been learned\.

This paper revisits the intersection of these two ideas\. We ask a deliberately simple question:*how far can classifier retraining go when the representation is trained with a strong Balanced Softmax anchor?*We instantiate this idea as BS\-cRT, a minimal two\-stage procedure\. In stage 1, we train a backbone and cosine classifier end\-to\-end with Balanced Softmax\. In stage 2, we freeze the backbone and retrain only the cosine classifier, again using Balanced Softmax, but now on class\-balanced episodic batches containingPPclasses andKKsamples per class\. The method has no prototypes, no synthetic samples, no auxiliary objectives, and no additional machinery beyond the usual classifier\-retraining setup\.

Despite its simplicity, BS\-cRT is a strong baseline\. Across CIFAR\-100\-LT and CIFAR\-10\-LT with imbalance factors\{50,100,200\}\\\{50,100,200\\\}, BS\-cRT consistently improves Few\-shot accuracy over the same Balanced Softmax stage\-1 checkpoint, while preserving or improving Top\-1 accuracy\. On CIFAR\-100\-LT at IF=100, BS\-cRT improves Few\-shot accuracy by\+5\.15\+5\.15points over Balanced Softmax; on CIFAR\-10\-LT at IF=100, the gain is\+5\.83\+5\.83points\. The same pattern extends to larger\-scale benchmarks: BS\-cRT improves Few\-shot accuracy by\+6\.92\+6\.92points on ImageNet\-LT and\+9\.78\+9\.78points on Places\-LT\. Overall, the Few\-shot gain is positive in every dataset–imbalance cell we evaluate, suggesting that a carefully matched classifier\-retraining baseline remains a strong reference baseline\.

We then ask a second question:*can tail\-class boundaries be improved further by injecting explicit synthetic supervision near decision boundaries?*This question motivated our initial method, Counterfactual Boundary Risk Minimization \(CBRM\)\. CBRM maintains multi\-prototype class memories, constructs feature\-space probes between a sample’s positive prototype and a negative\-class prototype, approximately bisects the current decision boundary, locally thickens the resulting probe, and penalizes cosine\-margin violations using a class\-balanced top\-kkrisk\. The intuition is natural: tail classes have too few real samples to densely supervise their boundaries, so synthetic boundary\-adjacent probes might provide additional corrective signal\.

The experiments do not support this extension under hardest\-negative probe selection, but they identify two useful failure modes\. First, we identify a margin/scale mismatch that arises when CBRM is combined with cosine classifiers: the original implementation compared a margin parameter in the cosine\-similarity range to logit differences scaled by the classifier temperature, producing gradients roughly30×30\\timeslarger than the Balanced Softmax anchor\. We document this failure mode and fix it by computing CBRM violations directly in cosine\-margin space\. Second, even after this fix, hardest\-negative CBRM probes do not improve over BS\-cRT\. On CIFAR\-100\-LT at IF=100, the corrected hardest\-negative probe variant reduces Few\-shot accuracy by approximately four points relative to BS\-cRT\. Our analysis suggests that this is not merely an optimization artifact: in long\-tailed data, hardest negatives for tail samples are often head classes, so the generated probes are anchored near head\-class regions and can pull tail classifier mass in the wrong direction\.

For long\-tailed recognition, the main implication is methodological\. Balanced\-Softmax classifier retraining provides a low\-complexity way to improve tail accuracy without changing the representation, and it should be treated as a stage\-2 control before tail gains are attributed to more complex mechanisms\. The CBRM diagnostics also show that synthetic boundary supervision is not automatically beneficial: when probes are constructed from hardest negatives, they can be anchored by head\-class competitors and can reinforce the bias that classifier retraining is meant to reduce\. We therefore use BS\-cRT as the main classifier\-side baseline and report CBRM as a controlled negative result for a plausible boundary\-supervision strategy\.

The paper makes three contributions\. First, it provides a matched\-checkpoint evaluation of classifier retraining after Balanced Softmax, isolating the effect of a frozen\-backbone classifier update from representation learning\. Second, it shows that this stage\-2 update consistently improves Few\-shot accuracy across CIFAR\-10/100\-LT, ImageNet\-LT and Places\-LT, with the expected trade\-off between head\-class contraction and tail\-class recovery\. Third, it analyzes a boundary\-probe extension, CBRM, and identifies why hardest\-negative probes can fail in long\-tailed settings even after a cosine\-margin correction\. The code, configurations, and reporting scripts are released to support reproduction\.

## 2Background and Related Work

Long\-tailed recognition lies at the intersection of statistical imbalance, representation learning, and classifier geometry\. The central difficulty is not only that rare classes provide fewer examples, but that empirical risk minimization is driven by a skewed label distribution: head classes appear more often in mini\-batches, contribute more gradient updates, and tend to induce larger and more stable decision regions\(Buda et al\.,[2018](https://arxiv.org/html/2607.09832#bib.bib1); Liu et al\.,[2019](https://arxiv.org/html/2607.09832#bib.bib19); Yang et al\.,[2022](https://arxiv.org/html/2607.09832#bib.bib36)\)\. Tail classes, in contrast, must be recognized from limited evidence while competing against classifier directions learned from substantially richer support\. This imbalance affects the learned features, the final classifier, posterior calibration, and the margins that separate classes in the embedding space\(Cao et al\.,[2019](https://arxiv.org/html/2607.09832#bib.bib2); Kang et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib13); Ren et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib23); Zhong et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib39)\)\.

The literature has addressed this problem from several complementary angles\. Some methods rebalance the contribution of observed examples through resampling, re\-weighting, focal objectives, meta\-learned weights, or effective\-number corrections\(Chawla et al\.,[2002](https://arxiv.org/html/2607.09832#bib.bib3); Lin et al\.,[2017](https://arxiv.org/html/2607.09832#bib.bib18); Ren et al\.,[2018](https://arxiv.org/html/2607.09832#bib.bib22); Shu et al\.,[2019](https://arxiv.org/html/2607.09832#bib.bib25); Cui et al\.,[2019](https://arxiv.org/html/2607.09832#bib.bib5)\)\. Others modify the classifier objective directly through margins, logit adjustment, prior\-aware softmax normalizers, or label\-distribution correction\(Cao et al\.,[2019](https://arxiv.org/html/2607.09832#bib.bib2); Menon et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib21); Ren et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib23); Hong et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib11)\)\. A third family decouples representation learning from classifier learning, showing that a useful feature extractor can still be paired with a biased classifier\(Kang et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib13); Zhou et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib41); Tang et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib28); Zhong et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib39)\)\. Recent work further explores contrastive objectives, prototype\-based representations, multi\-expert models, and synthetic boundary supervision\(Cui et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib6); Zhu et al\.,[2022](https://arxiv.org/html/2607.09832#bib.bib42); Xuan and Zhang,[2024](https://arxiv.org/html/2607.09832#bib.bib35); Snell et al\.,[2017](https://arxiv.org/html/2607.09832#bib.bib26); Jian et al\.,[2025](https://arxiv.org/html/2607.09832#bib.bib12); Yang et al\.,[2026](https://arxiv.org/html/2607.09832#bib.bib37)\)\. These directions provide the context for our study, which revisits a deliberately simple question: after training a representation with Balanced Softmax, how much tail accuracy can be recovered by retraining only the classifier under balanced class exposure?

### 2\.1Long\-tailed recognition

Visual recognition benchmarks have often been designed around balanced evaluation, but visual data collected in the wild rarely follows that pattern\. Large image taxonomies, scene collections, and citizen\-science species observations naturally allocate many samples to a small set of common categories and only a few samples to a large set of rare ones\(Krizhevsky,[2009](https://arxiv.org/html/2607.09832#bib.bib15); Deng et al\.,[2009](https://arxiv.org/html/2607.09832#bib.bib7); Zhou et al\.,[2018](https://arxiv.org/html/2607.09832#bib.bib40); Van Horn et al\.,[2018](https://arxiv.org/html/2607.09832#bib.bib29)\)\. This imbalance is not only a data\-management inconvenience\. Under empirical risk minimization, head classes contribute more images, more mini\-batch appearances, and more gradients\. As a result, the learned posterior, the classifier weights, and often the confidence scores become correlated with the empirical training prior\(Liu et al\.,[2019](https://arxiv.org/html/2607.09832#bib.bib19); Yang et al\.,[2022](https://arxiv.org/html/2607.09832#bib.bib36)\)\. The common Many, Medium, and Few split used in long\-tailed evaluation is therefore more than a reporting convention\. It asks whether a method improves the neglected portion of the label space or merely preserves head\-class dominance\.

This paper follows that question to its smallest useful intervention\. Rather than assuming that tail failure must be solved by a new backbone or a new representation objective, it asks how much rare\-class accuracy remains latent in a representation already trained with a strong long\-tailed loss\. This makes the work close in spirit to the classical long\-tailed recognition literature, but it places the spotlight on classifier geometry after Balanced Softmax training\. The resulting position is deliberately strict: if a method improves Few\-shot accuracy solely by retraining the final layer under more balanced exposure, then that gain should be credited to classifier retraining, not to a more elaborate mechanism\.

### 2\.2Rebalancing observed examples

The most direct family of methods modifies the contribution of observed samples\. Resampling changes which examples the optimizer sees, while re\-weighting changes how strongly each observed loss affects the update\. Systematic studies of class imbalance in convolutional networks showed that imbalance can strongly degrade recognition and that sampling choices are consequential\(Buda et al\.,[2018](https://arxiv.org/html/2607.09832#bib.bib1)\)\. Focal Loss reduces the contribution of easy examples and became influential in dense detection, where the background and foreground imbalance is extreme\(Lin et al\.,[2017](https://arxiv.org/html/2607.09832#bib.bib18)\)\. Class\-Balanced Loss replaces raw inverse\-frequency weights with weights derived from the effective number of samples, which softens the correction when additional samples become redundant\(Cui et al\.,[2019](https://arxiv.org/html/2607.09832#bib.bib5)\)\. Meta\-learning approaches, including Learning to Reweight Examples and Meta\-Weight\-Net, go one step further by learning the weighting rule from validation feedback rather than fixing it by hand\(Ren et al\.,[2018](https://arxiv.org/html/2607.09832#bib.bib22); Shu et al\.,[2019](https://arxiv.org/html/2607.09832#bib.bib25)\)\.

These methods are important because they establish a central theme: long\-tailed learning is not solved by treating every observed example equally\. Yet they also expose a limitation that is central to the present paper\. Re\-weighting and re\-sampling still act through the finite set of observed tail samples, and when used end\-to\-end, they can affect both the representation and the classifier at the same time\. That coupling makes it harder to know whether a tail gain came from better features, from a better decision boundary, or from a changed confidence prior\. BS\-cRT avoids this ambiguity in stage 2\. It freezes the representation and changes only the classifier, so the effect of balanced class exposure can be measured without re\-learning the feature extractor\.

### 2\.3Prior\-aware losses, margins, and logit correction

A second line of work corrects the class\-frequency bias directly in logit or margin space\. LDAM gives rarer classes larger margins and combines this idea with deferred re\-weighting, making the decision boundary itself class\-frequency dependent\(Cao et al\.,[2019](https://arxiv.org/html/2607.09832#bib.bib2)\)\. Logit adjustment provides a statistical view of long\-tail learning by adding a label\-prior correction to logits, either during training or after training\(Menon et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib21)\)\. Balanced Softmax reaches a closely related destination from the softmax normalizer: it incorporates the empirical class prior into the denominator so that optimization better matches a balanced evaluation protocol under label shift\(Ren et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib23)\)\. LADE and PC Softmax further emphasize that the source label distribution can be disentangled from model predictions, and that long\-tailed recognition can be read as a label\-shift problem rather than only as a rare\-sample problem\(Hong et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib11)\)\.

This group of methods is the immediate foundation for the present paper\. Balanced Softmax is a strong anchor because it corrects the training prior without assigning explicit per\-sample weights\. However, a model trained with Balanced Softmax on naturally imbalanced mini\-batches is still exposed to more head\-class gradients than tail\-class gradients\. The loss changes the normalization geometry, but it does not make the stream of classifier updates class\-balanced\. BS\-cRT tests precisely this residual bias: after prior\-aware representation learning, can the classifier directions still be re\-estimated more fairly by balanced episodic exposure?

Margin methods also motivate the diagnostic part of the paper\. LDAM shows that margins should depend on class frequency, but normalized classifiers show that the unit of a margin matters\. NormFace, CosFace, and ArcFace made hyperspherical feature and classifier geometry standard in face recognition by normalizing features and weights, then applying cosine or angular margins with an explicit scale parameter\(F\. Wang et al\.,[2017](https://arxiv.org/html/2607.09832#bib.bib31); H\. Wang et al\.,[2018](https://arxiv.org/html/2607.09832#bib.bib32); Deng et al\.,[2019](https://arxiv.org/html/2607.09832#bib.bib8)\)\. In such models, a margin expressed in cosine similarity is not numerically interchangeable with a margin computed after multiplying logits by the classifier scale\. This is directly relevant to CBRM: a boundary\-risk term must live in the same geometry as the cosine classifier\. Otherwise, the intended margin becomes a scaled\-logit penalty with a very different gradient magnitude\.

### 2\.4Decoupled representation and classifier learning

The closest predecessor of BS\-cRT is the decoupled training paradigm\. Kang et al\. showed that representation learning and classifier learning respond differently to imbalance, and that a representation learned with ordinary instance\-balanced sampling can support strong long\-tailed recognition once the classifier is re\-estimated with class balancing\(Kang et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib13)\)\. Their classifier retraining strategy, cRT, along with weight\-normalization variants such as tau\-normalization and learnable weight scaling, shifted the field away from the assumption that every long\-tail improvement must come from end\-to\-end rebalancing\. BBN reached a similar conclusion through a two\-branch architecture whose cumulative schedule separates broad representation learning from rebalanced classifier learning\(Zhou et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib41)\)\. Causal Normalization interpreted part of the classifier bias through the momentum\-induced confounding effect and removed the harmful head\-class component at inference time\(Tang et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib28)\)\. MiSLAS added another important detail: two\-stage long\-tailed methods can improve accuracy while changing calibration, so classifier retraining should also be viewed as confidence redistribution\(Zhong et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib39)\)\.

Recent work continues to revisit this decoupled view rather than treating it as a solved baseline\. Decoupled Optimization studies parameter groups that contribute differently to Many, Medium, and Few classes\(Cong et al\.,[2024](https://arxiv.org/html/2607.09832#bib.bib4)\)\. Label Over\-Smooth rethinks classifier re\-training under a unified feature representation and argues that simple label smoothing can balance the prediction space in the second stage\(Sun et al\.,[2025](https://arxiv.org/html/2607.09832#bib.bib27)\)\. These studies reinforce the premise that classifier learning is an active research axis, not just a post\-processing footnote\.

BS\-cRT differs from classical cRT in two key ways\. First, its first\-stage representation is trained with Balanced Softmax instead of cross\-entropy, making the stage\-1 checkpoint a stronger, more relevant baseline\. Second, its second\-stage classifier is also trained with Balanced Softmax on balanced episodic batches, rather than switching to cross\-entropy\. This lets BS\-cRT pose a sharper question than cRT: after a prior\-aware loss shapes the representation, does balanced classifier exposure still recover tail accuracy? Empirically, yes—making BS\-cRT a useful reference for evaluating more complex long\-tailed methods\.

### 2\.5Representation learning, contrastive objectives, and prototypes

A large body of complementary literature attempts to improve the representation itself\. OLTR used memory\-augmented features to transfer knowledge from head to tail classes in an open\-world long\-tailed setting\(Liu et al\.,[2019](https://arxiv.org/html/2607.09832#bib.bib19)\)\. LEAP observed that head and tail classes occupy feature space differently, then augmented tail embeddings by transferring intra\-class variation learned from head classes\(Liu et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib20)\)\. MetaSAug used meta\-learning to learn semantic augmentation directions for minority classes, addressing the fact that direct covariance estimation is unreliable when a class has very few samples\(Li et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib16)\)\. DRO\-LT framed the problem as representation bias and optimized a robustness\-inspired loss to improve rare\-class features\(Samuel and Chechik,[2021](https://arxiv.org/html/2607.09832#bib.bib24)\)\. RIDE approached the same imbalance through multiple distribution\-aware experts, reducing variance and bias with routing and diversity\(X\. Wang et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib34)\)\.

Contrastive learning brought another geometric perspective\. Supervised Contrastive Learning pulls same\-class samples together and pushes different\-class samples apart in normalized embedding space\(Khosla et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib14)\)\. In long\-tailed data, however, the number of same\-class positives and negative comparisons is itself imbalanced\. Hybrid contrastive networks, PaCo, BCL, and Decoupled Contrastive Learning each address this issue by changing how positive pairs, class centers, or class\-complement terms contribute to the contrastive objective\(P\. Wang et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib33); Cui et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib6); Zhu et al\.,[2022](https://arxiv.org/html/2607.09832#bib.bib42); Xuan and Zhang,[2024](https://arxiv.org/html/2607.09832#bib.bib35)\)\. Their analyses are highly relevant to this paper because they show that imbalance persists even in normalized representation spaces, where geometry is often easier to reason about\.

Prototype\-based methods connect representation geometry to classifier decisions\. Prototypical Networks established a simple and powerful view of few\-shot learning: classes can be represented by prototypes in an embedding space, and classification can be performed by distances to those prototypes\(Snell et al\.,[2017](https://arxiv.org/html/2607.09832#bib.bib26)\)\. CBRM borrows this geometric instinct but uses prototypes differently\. It does not classify by prototype distance\. Instead, it uses multi\-prototype memories as anchors for synthetic boundary probes\. The negative result in this paper therefore does not reject prototype structure itself\. It shows that when prototypes are used to construct boundary supervision, the choice of negative prototype and the frequency profile of that negative class become part of the learning signal\.

### 2\.6Synthetic samples and boundary supervision

The idea of adding synthetic evidence for minority classes predates deep long\-tailed recognition\. SMOTE creates synthetic minority samples by interpolation, while Borderline\-SMOTE focuses on interpolation near minority examples that are likely to be misclassified\(Chawla et al\.,[2002](https://arxiv.org/html/2607.09832#bib.bib3); Han et al\.,[2005](https://arxiv.org/html/2607.09832#bib.bib10)\)\. In deep networks, mixup and Manifold Mixup generalized interpolation into input and hidden spaces, encouraging smoother behavior between training examples and flatter representations\(Zhang et al\.,[2018](https://arxiv.org/html/2607.09832#bib.bib38); Verma et al\.,[2019](https://arxiv.org/html/2607.09832#bib.bib30)\)\. These methods support the intuition behind CBRM: if tail classes have too few samples to define reliable boundaries, then additional feature\-space supervision near uncertain regions may help\.

Recent long\-tailed work has moved even closer to explicit boundary and decision\-region supervision\. Supervised Exploratory Learning synthesizes exploratory examples to improve class balance, decision regions, and decision boundaries without changing the backbone architecture\(Jian et al\.,[2025](https://arxiv.org/html/2607.09832#bib.bib12)\)\. Decision Boundary\-aware Generation uses generative augmentation to create informative near\-boundary samples and reduce boundary ambiguity in long\-tailed learning\(Yang et al\.,[2026](https://arxiv.org/html/2607.09832#bib.bib37)\)\. These works make boundary supervision a timely direction, but they also clarify why CBRM is a useful diagnostic\. A synthetic boundary sample is not just another training point\. It imposes a directional constraint on the classifier\. If the negative anchor is usually a head class, as can happen with hardest\-negative mining for tail samples, the synthetic probe can become head\-anchored and can pull the tail classifier into the wrong region\. In that regime, more boundary signal is not automatically better\. It must be frequency\-aware, geometrically consistent, and aligned to expand tail decision regions rather than restore head\-class dominance\.

Taken together, these lines of work motivate the design of our study\. Prior\-aware losses and classifier retraining suggest that the final classifier should be examined separately from the representation, while recent boundary\-aware methods show that synthetic near\-boundary samples are a plausible way to target tail regions\. BS\-cRT tests the first idea under a matched Balanced Softmax checkpoint\. CBRM tests the second idea as a controlled extension, and its negative result clarifies a failure mode of hardest\-negative probe construction in long\-tailed data\.

## 3Methods

This section defines BS\-cRT, the proposed classifier\-retraining procedure, and then describes CBRM, the boundary\-supervision objective evaluated as a diagnostic alternative\. The notation is shared across both methods\.

### 3\.1Problem Setup

Let𝒟=\{\(xi,yi\)\}i=1N\\mathcal\{D\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{N\}denote a long\-tailed training set with labelsyi∈\{1,…,C\}y\_\{i\}\\in\\\{1,\\ldots,C\\\}\. The number of training examples in classccis

nc=\|\{i:yi=c\}\|,n\_\{c\}=\\left\|\\\{i:y\_\{i\}=c\\\}\\right\|,\(1\)and the imbalance factor isIF=nmax/nmin\\mathrm\{IF\}=n\_\{\\max\}/n\_\{\\min\}\. A feature extractorfθ:𝒳→ℝdf\_\{\\theta\}:\\mathcal\{X\}\\rightarrow\\mathbb\{R\}^\{d\}is followed by a cosine classifier with class weights\{wc\}c=1C\\\{w\_\{c\}\\\}\_\{c=1\}^\{C\}\. Features and classifier weights are normalized before scoring:

z=fθ​\(x\)∥fθ​\(x\)∥2,w^c=wc∥wc∥2,ℓc​\(z\)=s​w^c⊤​z,z=\\frac\{f\_\{\\theta\}\(x\)\}\{\\lVert f\_\{\\theta\}\(x\)\\rVert\_\{2\}\},\\qquad\\widehat\{w\}\_\{c\}=\\frac\{w\_\{c\}\}\{\\lVert w\_\{c\}\\rVert\_\{2\}\},\\qquad\\ell\_\{c\}\(z\)=s\\,\\widehat\{w\}\_\{c\}^\{\\top\}z,\(2\)wheressis a fixed scale, set to3030in all experiments\.

Balanced Softmax uses the empirical training prior

πc=nc∑c′=1Cnc′\\pi\_\{c\}=\\frac\{n\_\{c\}\}\{\\sum\_\{c^\{\\prime\}=1\}^\{C\}n\_\{c^\{\\prime\}\}\}\(3\)and optimizes

LBS​\(z,y\)=−log⁡exp⁡\(ℓy​\(z\)\+log⁡πy\)∑c=1Cexp⁡\(ℓc​\(z\)\+log⁡πc\)\.L\_\{\\mathrm\{BS\}\}\(z,y\)=\-\\log\\frac\{\\exp\\left\(\\ell\_\{y\}\(z\)\+\\log\\pi\_\{y\}\\right\)\}\{\\sum\_\{c=1\}^\{C\}\\exp\\left\(\\ell\_\{c\}\(z\)\+\\log\\pi\_\{c\}\\right\)\}\.\(4\)At inference, we use the raw cosine logits

ℓc​\(z\)=s​w^c⊤​z\\ell\_\{c\}\(z\)=s\\,\\widehat\{w\}\_\{c\}^\{\\top\}z\(5\)for prediction and confidence estimation\. The log\-prior term in Eq\.[4](https://arxiv.org/html/2607.09832#S3.E4)is used only inside the training loss; it is not added to logits at test time\. Thus, predictions are computed asarg⁡maxc⁡ℓc​\(z\)\\arg\\max\_\{c\}\\ell\_\{c\}\(z\)\.

For reporting, classes are partitioned into Many\-shot, Medium\-shot, and Few\-shot groups following standard long\-tailed recognition protocols\. On CIFAR\-100\-LT, for example, Many\-shot classes satisfync\>100n\_\{c\}\>100, Medium\-shot classes satisfy20<nc≤10020<n\_\{c\}\\leq 100, and Few\-shot classes satisfync≤20n\_\{c\}\\leq 20\.

### 3\.2BS\-cRT: Balanced Softmax Classifier Retraining

BS\-cRT is a two\-stage procedure\. Stage 1 trains the backbonefθf\_\{\\theta\}and cosine classifier end\-to\-end with Balanced Softmax on the original long\-tailed training stream\. We denote the resulting model byfθ\(1\)f\_\{\\theta\}^\{\(1\)\}and\{wc\(1\)\}c=1C\\\{w\_\{c\}^\{\(1\)\}\\\}\_\{c=1\}^\{C\}\.

Stage 2 freezesfθ\(1\)f\_\{\\theta\}^\{\(1\)\}and retrains only the cosine classifier\. The stage\-2 classifier is initialized from\{wc\(1\)\}c=1C\\\{w\_\{c\}^\{\(1\)\}\\\}\_\{c=1\}^\{C\}, so the comparison isolates the effect of balanced classifier optimization from the same representation and checkpoint\. Each stage\-2 batch is sampled episodically\. The sampler is configured with valuesPPandKK, but the effective number of distinct classes per episode isPeff=min⁡\(P,C\)P\_\{\\mathrm\{eff\}\}=\\min\(P,C\)\. We samplePeffP\_\{\\mathrm\{eff\}\}classes uniformly without replacement and then drawKKexamples per selected class, yielding a batch of sizePeff​KP\_\{\\mathrm\{eff\}\}K\. Within a selected class, examples are sampled without replacement when at leastKKexamples are available and with replacement otherwise\. No additional sample weighting is applied\. WhenPeff<CP\_\{\\mathrm\{eff\}\}<C, the episodic batches provide a stochastic approximation to uniform class exposure across updates; whenPeff=CP\_\{\\mathrm\{eff\}\}=C, every class appears in each stage\-2 batch\. Optimization uses SGD with momentum0\.90\.9, weight decay5×10−45\\times 10^\{\-4\}, and a cosine learning\-rate schedule overE2E\_\{2\}epochs\.

The stage\-2 anchor loss remains Balanced Softmax with the empirical training prior\. This point is important because the balanced sampler changes the outer sampling measure but does not redefine the prior inside the loss\. In all BS\-cRT experiments,

πctrain=nc∑c′=1Cnc′\.\\pi\_\{c\}^\{\\mathrm\{train\}\}=\\frac\{n\_\{c\}\}\{\\sum\_\{c^\{\\prime\}=1\}^\{C\}n\_\{c^\{\\prime\}\}\}\.\(6\)Thus, BS\-cRT optimizes the same prior\-aware per\-sample surrogate as stage 1, but under balanced class exposure:

LBS​\-​cRT=1C​∑c=1C𝔼x∼𝒟c​\[−log⁡exp⁡\(ℓc​\(fθ\(1\)​\(x\)\)\+log⁡πctrain\)∑j=1Cexp⁡\(ℓj​\(fθ\(1\)​\(x\)\)\+log⁡πjtrain\)\],L\_\{\\mathrm\{BS\\text\{\-\}cRT\}\}=\\frac\{1\}\{C\}\\sum\_\{c=1\}^\{C\}\\mathbb\{E\}\_\{x\\sim\\mathcal\{D\}\_\{c\}\}\\left\[\-\\log\\frac\{\\exp\\left\(\\ell\_\{c\}\(f\_\{\\theta\}^\{\(1\)\}\(x\)\)\+\\log\\pi\_\{c\}^\{\\mathrm\{train\}\}\\right\)\}\{\\sum\_\{j=1\}^\{C\}\\exp\\left\(\\ell\_\{j\}\(f\_\{\\theta\}^\{\(1\)\}\(x\)\)\+\\log\\pi\_\{j\}^\{\\mathrm\{train\}\}\\right\)\}\\right\],\(7\)where𝒟c\\mathcal\{D\}\_\{c\}denotes the empirical distribution over examples from classcc\. If one instead setπc=1/C\\pi\_\{c\}=1/Cin stage 2, the log\-prior term would be constant across classes and would cancel from the softmax, reducing the anchor to ordinary cross\-entropy under class\-balanced sampling\. That is not the objective evaluated in this work\.

The rationale is that Balanced Softmax improves the stage\-1 representation, but the classifier directions are still updated through an instance\-balanced training stream in which head classes contribute more gradient steps\. Stage 2 keeps the representation fixed and changes only the class exposure seen by the classifier\. The empirical\-prior Balanced Softmax anchor keeps the same prior\-aware training objective as stage 1, while the balanced episodic sampler gives each selected class comparable influence on the retrained classifier directions\. The BS\-cRT objective contains no boundary\-probe loss, no prototype\-dependent gradient, and no auxiliary representation loss; in the implementation it is obtained by settingλcbr=0\\lambda\_\{\\mathrm\{cbr\}\}=0and freezing the backbone\.

### 3\.3Practical Recipe and Cost

Algorithm[1](https://arxiv.org/html/2607.09832#alg1)summarizes the deployment procedure\. The procedure requires only a Balanced Softmax checkpoint, the empirical class counts used by the loss, and the stage\-2 hyperparameters\. The procedure is fixed\-schedule: the evaluation split is reserved for final reporting and is not used for checkpoint selection\. BS\-cRT uses no representation loss, prototype memory, or synthetic probe construction\.

Algorithm 1Practical BS\-cRT procedure\. The loss uses the empirical training priorπc=nc/∑c′nc′\\pi\_\{c\}=n\_\{c\}/\\sum\_\{c^\{\\prime\}\}n\_\{c^\{\\prime\}\}, while evaluation uses raw cosine logits\.1:*Input:*Final scheduled Balanced Softmax checkpoint

\(fθ\(1\),\{wc\(1\)\}\)\(f\_\{\\theta\}^\{\(1\)\},\\\{w\_\{c\}^\{\(1\)\}\\\}\), class counts

\{nc\}\\\{n\_\{c\}\\\},

PP,

KK, learning rate, and number of stage\-2 epochs\.

2:Freeze the backbone

fθ\(1\)f\_\{\\theta\}^\{\(1\)\}\.

3:Initialize the cosine classifier from

\{wc\(1\)\}\\\{w\_\{c\}^\{\(1\)\}\\\}\.

4:Set

Peff=min⁡\(P,C\)P\_\{\\mathrm\{eff\}\}=\\min\(P,C\)\. At each update, sample

PeffP\_\{\\mathrm\{eff\}\}classes uniformly without replacement and

KKimages per selected class, using replacement only when a selected class has fewer than

KKexamples\.

5:Update only the classifier with

LBSL\_\{\\mathrm\{BS\}\}using the empirical training prior\.

6:Run the fixed stage\-2 schedule and keep the final scheduled checkpoint, without selecting by evaluation\-split Top\-1\.

7:Deploy the model with raw cosine logits

ℓc​\(z\)=s​w^c⊤​z\\ell\_\{c\}\(z\)=s\\widehat\{w\}\_\{c\}^\{\\top\}z, without adding

log⁡πc\\log\\pi\_\{c\}at inference\.

The computational footprint is small because only the classifier matrix is updated\. With the normalized linear head used here, stage 2 trainsC​dCdparameters:640640parameters on CIFAR\-10\-LT,6,4006\{,\}400on CIFAR\-100\-LT,2\.052\.05M on ImageNet\-LT, and0\.750\.75M on Places\-LT\. There is no additional inference\-time module, memory, or latency relative to the stage\-1 model\. In our logs, one BS\-cRT stage 2 run takes about0\.060\.06–0\.100\.10hours on CIFAR\-LT,2\.82\.8hours on ImageNet\-LT, and2\.22\.2hours on Places\-LT\. On the two large\-scale benchmarks, this corresponds to roughly77–9%9\\%of the corresponding stage\-1 wall\-clock time\. BS\-cRT therefore adds a short classifier\-only training pass and leaves deployment unchanged\.

Because stage 2 updates only the classifier, BS\-cRT is suitable for deployment settings where the backbone is expensive to retrain or already validated\. The inference graph is unchanged, so the method shifts the operating point toward rare classes without adding runtime modules, prototype memories, or latency\. This makes the procedure especially relevant when rare categories carry operational value, for example, in defect detection, species recognition, or safety monitoring\.

### 3\.4CBRM as a Diagnostic Boundary\-Probe Extension

CBRM is evaluated as a diagnostic extension of BS\-cRT rather than as the main method of the paper\. Its purpose is to test whether synthetic feature\-space probes near current decision boundaries provide useful supervision beyond balanced classifier retraining\. All CBRM variants start from the same Balanced Softmax stage\-1 checkpoint as BS\-cRT\. The stage\-2 objective is

L=LBS\+λcbr​Lcbr\+λcompact​Lcompact\+λsep​Lsep\.L=L\_\{\\mathrm\{BS\}\}\+\\lambda\_\{\\mathrm\{cbr\}\}L\_\{\\mathrm\{cbr\}\}\+\\lambda\_\{\\mathrm\{compact\}\}L\_\{\\mathrm\{compact\}\}\+\\lambda\_\{\\mathrm\{sep\}\}L\_\{\\mathrm\{sep\}\}\.\(8\)BS\-cRT is recovered by settingλcbr=0\\lambda\_\{\\mathrm\{cbr\}\}=0and freezing the backbone\. The CBRM terms are included only to analyze whether boundary probes can improve on this classifier\-only baseline\.

CBRM maintainsMMunit\-norm prototype buffers per class,\{μc,m\}m=1M\\\{\\mu\_\{c,m\}\\\}\_\{m=1\}^\{M\}\. The prototypes are initialized from L2\-normalized features extracted by the stage\-1 checkpoint and are updated during stage 2 by exponential moving average, not by gradient descent\. For a normalized featurezzand classcc, the nearest prototype slot is

m⋆​\(z,c\)=arg⁡maxm⁡z⊤​μc,m\.m^\{\\star\}\(z,c\)=\\arg\\max\_\{m\}z^\{\\top\}\\mu\_\{c,m\}\.\(9\)If no samples are assigned to a prototype in a batch, that buffer is left unchanged\.

For each stage\-2 sample\(zi,yi\)\(z\_\{i\},y\_\{i\}\), CBRM selects a negative classjij\_\{i\}\. The default variant uses the current hardest negative,

ji=arg⁡maxk≠yi⁡ℓk​\(zi\),j\_\{i\}=\\arg\\max\_\{k\\neq y\_\{i\}\}\\ell\_\{k\}\(z\_\{i\}\),\(10\)while the random\-negative diagnostic samplesjij\_\{i\}uniformly from\{1,…,C\}∖\{yi\}\\\{1,\\ldots,C\\\}\\setminus\\\{y\_\{i\}\\\}\. The positive and negative prototype anchors are

μi\+=μyi,m⋆​\(zi,yi\),μi−=μji,m⋆​\(zi,ji\)\.\\mu\_\{i\}^\{\+\}=\\mu\_\{y\_\{i\},m^\{\\star\}\(z\_\{i\},y\_\{i\}\)\},\\qquad\\mu\_\{i\}^\{\-\}=\\mu\_\{j\_\{i\},m^\{\\star\}\(z\_\{i\},j\_\{i\}\)\}\.\(11\)CBRM then searches from the observed feature toward the negative prototype direction:

di=μi−−μi\+‖μi−−μi\+‖2,pi​\(t\)=zi\+t​di‖zi\+t​di‖2,t∈\[0,tmax\]\.d\_\{i\}=\\frac\{\\mu\_\{i\}^\{\-\}\-\\mu\_\{i\}^\{\+\}\}\{\\\|\\mu\_\{i\}^\{\-\}\-\\mu\_\{i\}^\{\+\}\\\|\_\{2\}\},\\qquad p\_\{i\}\(t\)=\\frac\{z\_\{i\}\+td\_\{i\}\}\{\\\|z\_\{i\}\+td\_\{i\}\\\|\_\{2\}\},\\qquad t\\in\[0,t\_\{\\max\}\]\.\(12\)A fixed five\-step bisection search seeks an approximate boundary point whereℓyi​\(pi​\(t\)\)−ℓji​\(pi​\(t\)\)≈0\\ell\_\{y\_\{i\}\}\(p\_\{i\}\(t\)\)\-\\ell\_\{j\_\{i\}\}\(p\_\{i\}\(t\)\)\\approx 0\. If the interval does not bracket a crossing, the implementation keeps the endpoint reached by the search rather than discarding the sample\.

To avoid depending on a single boundary point, the probe is locally thickened by a small normalized perturbation orthogonal todid\_\{i\}\. Withξ∼𝒩​\(0,I\)\\xi\\sim\\mathcal\{N\}\(0,I\),

ξ⟂=ξ−\(ξ⊤​di\)​di,ui=ξ⟂‖ξ⟂‖2\+ε,bi=pi​\(ti⋆\)\+σ​ui‖pi​\(ti⋆\)\+σ​ui‖2\.\\xi\_\{\\perp\}=\\xi\-\(\\xi^\{\\top\}d\_\{i\}\)d\_\{i\},\\qquad u\_\{i\}=\\frac\{\\xi\_\{\\perp\}\}\{\\\|\\xi\_\{\\perp\}\\\|\_\{2\}\+\\varepsilon\},\\qquad b\_\{i\}=\\frac\{p\_\{i\}\(t\_\{i\}^\{\\star\}\)\+\\sigma u\_\{i\}\}\{\\\|p\_\{i\}\(t\_\{i\}^\{\\star\}\)\+\\sigma u\_\{i\}\\\|\_\{2\}\}\.\(13\)
The corrected CBRM violation is computed in cosine\-similarity space rather than on scaled logits:

vi=softplus⁡\(γi−mi​\(bi\)\),mi​\(b\)=w^yi⊤​b−w^ji⊤​b\.v\_\{i\}=\\operatorname\{softplus\}\\\!\\left\(\\gamma\_\{i\}\-m\_\{i\}\(b\_\{i\}\)\\right\),\\qquad m\_\{i\}\(b\)=\\widehat\{w\}\_\{y\_\{i\}\}^\{\\top\}b\-\\widehat\{w\}\_\{j\_\{i\}\}^\{\\top\}b\.\(14\)Heremi​\(b\)∈\[−2,2\]m\_\{i\}\(b\)\\in\[\-2,2\]is an unscaled cosine margin\. The target margin is either fixed or frequency\-adaptive:

γi=γ0\+ρ​log⁡nmaxnyi\.\\gamma\_\{i\}=\\gamma\_\{0\}\+\\rho\\log\\frac\{n\_\{\\max\}\}\{n\_\{y\_\{i\}\}\}\.\(15\)In the reported runs,γ0=0\.2\\gamma\_\{0\}=0\.2andρ=0\.05\\rho=0\.05\. The largest configured value is approximately0\.5450\.545on Places\-LT, which remains inside the feasible cosine\-margin range\. This cosine\-space computation is important because applying the same target margin to scaled logits would multiply the effective gradient with respect to the cosine margin by the classifier scales=30s=30, which caused the loss spike observed in the original CBRM diagnostic\.

CBRM aggregates violations by class rather than by sample\. Letℐc\\mathcal\{I\}\_\{c\}be the set of probes in the batch whose true label iscc\. The classwise upper\-tail risk and boundary loss are

rc\(α\)=1⌈α​\|ℐc\|⌉​∑i∈Topα⁡\(ℐc\)vi,Lcbr=1\|𝒞batch\|​∑c∈𝒞batchrc\(α\)\.r\_\{c\}^\{\(\\alpha\)\}=\\frac\{1\}\{\\lceil\\alpha\|\\mathcal\{I\}\_\{c\}\|\\rceil\}\\sum\_\{i\\in\\operatorname\{Top\}\_\{\\alpha\}\(\\mathcal\{I\}\_\{c\}\)\}v\_\{i\},\\qquad L\_\{\\mathrm\{cbr\}\}=\\frac\{1\}\{\|\\mathcal\{C\}\_\{\\mathrm\{batch\}\}\|\}\\sum\_\{c\\in\\mathcal\{C\}\_\{\\mathrm\{batch\}\}\}r\_\{c\}^\{\(\\alpha\)\}\.\(16\)We useα=0\.25\\alpha=0\.25; settingα=1\\alpha=1recovers the classwise mean violation\.

The compactness and separation terms in Eq\. \([8](https://arxiv.org/html/2607.09832#S3.E8)\) are prototype regularizers retained from the diagnostic implementation\. They are not part of BS\-cRT\. Since prototypes are EMA buffers, the separation term does not backpropagate to the classifier or backbone in the reported implementation\. The compactness term can affect features only when part of the backbone is unfrozen; it is inert in the frozen\-backbone diagnostics\.

Table[1](https://arxiv.org/html/2607.09832#S3.T1)summarizes the stage\-2 variants used in the diagnostic experiments\. The corrected CBRM variants compute violations in the cosine\-margin space of Eq\. \([14](https://arxiv.org/html/2607.09832#S3.E14)\)\. The random\-negative variant replaces hardest\-negative mining with uniform negative\-class sampling to test whether probe failures are tied to systematic head\-class anchoring\.

Table 1:Stage\-2 variants evaluated in the diagnostic experiments\. All variants start from the same Balanced Softmax stage\-1 checkpoint for each seed\.
### 3\.5Implementation Details

CIFAR\-100\-LT and CIFAR\-10\-LT use ResNet\-32 at resolution32×3232\\times 32\. ImageNet\-LT and Places\-LT use ResNet\-50 at resolution224×224224\\times 224\. All experiments use a normalized linear classifier with fixed scales=30s=30\.

Stage 1 uses SGD with learning rate0\.10\.1, momentum0\.90\.9, weight decay5×10−45\\times 10^\{\-4\}, batch size128128, evaluation batch size256256, and cosine learning\-rate decay for200200epochs\. CIFAR\-LT uses no warmup, and AMP is disabled; ImageNet\-LT and Places\-LT use five warmup epochs, and AMP is enabled\. To avoid test\-set\-based model selection, we use the checkpoint at the end of the fixed stage\-1 schedule to initialize stage 2\. The evaluation split is used only for final reporting\.

Stage 2 initializes the classifier from the final scheduled stage\-1 checkpoint, freezes the backbone for BS\-cRT, and trains for 40 epochs with SGD, learning rate 0\.01, momentum 0\.9, weight decay5×10−45\\times 10^\{\-4\}, and cosine decay\. The episodic sampler is configured withP=16P=16andK=8K=8for all reported BS\-cRT runs\. In practice, the effective number of distinct classes per episode isPeff=min⁡\(P,C\)P\_\{\\mathrm\{eff\}\}=\\min\(P,C\)\. Thus CIFAR\-10\-LT usesPeff=10P\_\{\\mathrm\{eff\}\}=10and batch size8080, while CIFAR\-100\-LT, ImageNet\-LT, and Places\-LT usePeff=16P\_\{\\mathrm\{eff\}\}=16and batch size128128\. CIFAR\-LT uses 200 stage\-2 batches per epoch, while ImageNet\-LT and Places\-LT use 500 batches per epoch\. Evaluation uses batch size 256\. The reported stage\-2 model is the checkpoint at the end of this fixed schedule, not a best\-by\-evaluation checkpoint\.

Training augmentations are the standard transforms used by the codebase\. CIFAR\-LT uses random crop with padding44, random horizontal flip, tensor conversion, and CIFAR normalization; evaluation uses tensor conversion and normalization only\. ImageNet\-LT and Places\-LT use random resized crop to224224, random horizontal flip, color jitter with brightness, contrast, and saturation strength0\.40\.4, and ImageNet normalization; evaluation resizes to256256, center\-crops to224224, and applies the same normalization\.

Unless otherwise specified, CBRM usesλcbr=1\.0\\lambda\_\{\\mathrm\{cbr\}\}=1\.0,λcompact=0\.05\\lambda\_\{\\mathrm\{compact\}\}=0\.05, andλsep=0\.01\\lambda\_\{\\mathrm\{sep\}\}=0\.01\. The no\-probe BS\-cRT variants setλcbr=0\\lambda\_\{\\mathrm\{cbr\}\}=0; in frozen\-backbone runs, the compactness and separation terms do not update trainable parameters, as described above\. CBRM also uses five bisection steps,tmax=0\.8t\_\{\\max\}=0\.8,M=2M=2prototypes per class, EMA coefficientαema=0\.99\\alpha\_\{\\mathrm\{ema\}\}=0\.99, tangent noise scaleσ0=0\.02\\sigma\_\{0\}=0\.02, top\-tail fractionα=0\.25\\alpha=0\.25, base marginγ0=0\.2\\gamma\_\{0\}=0\.2, frequency\-adaptive margin coefficientρ=0\.05\\rho=0\.05, and a three\-epoch warmup before enabling the boundary\-risk term\. All main results are reported over seeds\{1,2,3\}\\\{1,2,3\\\}with seeded data loaders and checkpointed configurations\.

All experiments were run on a local workstation equipped with two NVIDIA GeForce RTX 4090 GPUs, an Intel Core i9\-13900K CPU, and 128 GB of system memory\. Each training run used a single GPU unless otherwise stated\. The wall\-clock times reported in Section[3\.3](https://arxiv.org/html/2607.09832#S3.SS3)were measured on this workstation\.

## 4Results

The experiments first measure the effect of BS\-cRT under matched stage\-1 checkpoints, then examine whether the effect persists across imbalance factors and larger benchmarks\. The final part of the section analyzes CBRM to determine whether synthetic boundary probes add useful supervision beyond classifier retraining\. Unless otherwise stated, all values are percentages averaged over three seeds\. Balanced Softmax is the primary reference because BS\-cRT starts from the same stage\-1 checkpoint and changes only the stage\-2 classifier optimization\.

### 4\.1CIFAR\-LT at IF 100

Table[2](https://arxiv.org/html/2607.09832#S4.T2)reports the central CIFAR comparison at an imbalance factor of100100\. On CIFAR\-100\-LT, BS\-cRT approximately preserves Top\-1 accuracy, moving from40\.7340\.73to41\.0741\.07, while increasing Few\-shot accuracy from19\.9119\.91to25\.0625\.06, a gain of\+5\.15\+5\.15points over Balanced Softmax\. On CIFAR\-10\-LT, BS\-cRT improves Top\-1 from75\.6575\.65to77\.8277\.82and Few\-shot accuracy from64\.3464\.34to70\.1770\.17, a gain of\+5\.83\+5\.83points\. These gains are not a consequence of a stronger representation, since the backbone is frozen and shared with the Balanced Softmax stage\-1 model\.

Table 2:Main CIFAR\-LT results at imbalance factor100100\. Top\-1 is reported as mean±\\pmstandard deviation over three seeds\. Many, Medium, and Few are class\-frequency bucket accuracies\. All values are percentages\.Figure[1](https://arxiv.org/html/2607.09832#S4.F1)shows the same result from the frequency\-bucket perspective\. BS\-cRT gives up some Many\-shot accuracy, as expected from class\-balanced classifier retraining, but it recovers substantially more accuracy on the underrepresented classes\. On CIFAR\-100\-LT, Few\-shot accuracy rises by more than five points while Medium accuracy is preserved\. On CIFAR\-10\-LT, the shift is even cleaner: Medium and Few improve together, and the Many\-shot contraction remains small enough that Top\-1 also improves\. This bucket\-level behavior is the core empirical signature of BS\-cRT: it reallocates classifier capacity toward the tail while retaining enough head\-class performance to avoid a global accuracy collapse\.

![Refer to caption](https://arxiv.org/html/2607.09832v1/x9.png)Figure 1:Frequency\-bucket breakdown on CIFAR\-LT at imbalance factor100100\. Panel \(a\) shows CIFAR\-100\-LT, and panel \(b\) shows CIFAR\-10\-LT\. The figure compares Balanced Softmax, classical cRT, and BS\-cRT on Many, Medium, and Few classes\. BS\-cRT trades a moderate reduction in Many\-shot accuracy for a larger gain on Few\-shot classes, which is the desired behavior in balanced long\-tailed evaluation\. Error bars show standard deviation across three seeds\.
### 4\.2Consistency Across Benchmark Cells

The same pattern appears across imbalance factors and datasets\. Figure[2](https://arxiv.org/html/2607.09832#S4.F2)summarizes paired seedwise differences between BS\-cRT and Balanced Softmax across all evaluated benchmark cells, and Table[3](https://arxiv.org/html/2607.09832#S4.T3)reports the corresponding seedwise Few\-shot deltas with paired95%95\\%confidence intervals\. BS\-cRT improves Few\-shot accuracy for every seed in every evaluated benchmark cell: the mean gains are\+5\.18\+5\.18,\+5\.15\+5\.15, and\+5\.10\+5\.10points on CIFAR\-100\-LT for imbalance factors5050,100100, and200200;\+4\.52\+4\.52,\+5\.83\+5\.83, and\+13\.99\+13\.99points on CIFAR\-10\-LT;\+6\.92\+6\.92points on ImageNet\-LT; and\+9\.78\+9\.78points on Places\-LT\. The Top\-1 columns in Table[3](https://arxiv.org/html/2607.09832#S4.T3)clarify the operating point: CIFAR\-100\-LT is best read as Top\-1 preservation with tail recovery, CIFAR\-10\-LT and Places\-LT show Top\-1 gains, and ImageNet\-LT is a clear Top\-1/Few\-shot trade\-off\.

![Refer to caption](https://arxiv.org/html/2607.09832v1/x10.png)Figure 2:Summary of BS\-cRT improvements over Balanced Softmax\. Panel \(a\) shows the paired seedwise Few\-shot gain,ΔFew=FewBS​\-​cRT−FewBS\\Delta\_\{\\mathrm\{Few\}\}=\\mathrm\{Few\}\_\{\\mathrm\{BS\\text\{\-\}cRT\}\}\-\\mathrm\{Few\}\_\{\\mathrm\{BS\}\}, for each benchmark cell\. Error bars denote the standard deviation of paired seed differences\. Panel \(b\) plots the corresponding Top\-1 change against the Few\-shot change\. Labels use compact dataset and imbalance\-factor notation, for example, C100\-50 denotes CIFAR\-100\-LT IF5050\. All evaluated cells lie above zero on the Few\-shot axis, while Top\-1 is preserved or improved in all but ImageNet\-LT\.Table 3:Paired BS\-cRT gains over the matched Balanced Softmax checkpoint\. Seed columns reportΔFew\\Delta\_\{\\mathrm\{Few\}\}in percentage points for seeds11,22, and33\. Confidence intervals are two\-sided pairedttintervals over the three seedwise differences\. Because each interval is computed from three paired seeds, the confidence intervals are descriptive and should be interpreted together with the seedwise differences\. The Top\-1 column is included to make the operating point explicit rather than to claim uniform aggregate improvement\.We additionally report non\-parametric robustness checks over the eight evaluated benchmark cells in Table[4](https://arxiv.org/html/2607.09832#S4.T4)\. These checks use the cell\-level mean deltas from Table[3](https://arxiv.org/html/2607.09832#S4.T3), rather than treating all seed\-level runs as independent observations\. Few\-shot gains are positive in all eight cells, yieldingp=0\.0078p=0\.0078under exact sign, Wilcoxon signed\-rank, and sign\-flip tests\. Top\-1 changes are positive in seven of eight cells but do not reject the null under the same checks, consistent with the ImageNet\-LT Top\-1/Few\-shot trade\-off\.

Table 4:Non\-parametric robustness checks for BS\-cRT gains over the matched Balanced Softmax checkpoint\. Tests are computed over the eight evaluated benchmark cells using the cell\-level mean deltas from Table[3](https://arxiv.org/html/2607.09832#S4.T3)\. Mean and median deltas are reported in percentage points\. All tests are two\-sided and are intended as robustness checks rather than replacements for the seedwise paired deltas\.Figure[3](https://arxiv.org/html/2607.09832#S4.F3)isolates the role of imbalance severity on CIFAR\. On CIFAR\-100\-LT, the Few\-shot gain remains stable as the imbalance factor increases:\+5\.18\+5\.18,\+5\.15\+5\.15, and\+5\.10\+5\.10points for IF5050,100100, and200200\. On CIFAR\-10\-LT, the gain grows with imbalance and reaches\+13\.99\+13\.99points at IF200200\. This supports the mechanism behind BS\-cRT\. When stage\-1 training is increasingly dominated by head\-class gradients, re\-estimating classifier directions under balanced class exposure becomes increasingly valuable\.

![Refer to caption](https://arxiv.org/html/2607.09832v1/x11.png)Figure 3:Few\-shot accuracy as a function of imbalance factor\. Panel \(a\) shows CIFAR\-100\-LT and panel \(b\) shows CIFAR\-10\-LT, using a shared vertical scale from0to100100for direct comparison\. BS\-cRT is compared against the Balanced Softmax checkpoint from which it is initialized\. Error bars show standard deviation across three seeds\. The gap is positive at every imbalance factor and grows sharply for CIFAR\-10\-LT at IF200200\.
### 4\.3Large\-Scale Benchmarks

Table[5](https://arxiv.org/html/2607.09832#S4.T5)reports ImageNet\-LT and Places\-LT\. On ImageNet\-LT, BS\-cRT should be interpreted as a Top\-1/Few\-shot trade\-off: Top\-1 decreases from51\.0151\.01to49\.3749\.37, while Few\-shot accuracy rises substantially from38\.2138\.21to45\.1345\.13\. On Places\-LT, where the natural imbalance is more extreme, the same classifier\-retraining procedure improves both Top\-1 and Few\-shot accuracy\. Top\-1 increases from27\.6427\.64to28\.3828\.38, and Few\-shot accuracy rises from18\.6118\.61to28\.3928\.39\.

Table 5:Large\-scale benchmark results\. Top\-1 is reported as mean±\\pmstandard deviation over three seeds\. Many, Medium, and Few are class\-frequency bucket accuracies\. All values are percentages\.Figure[4](https://arxiv.org/html/2607.09832#S4.F4)makes the large\-scale trade\-off explicit\. On ImageNet\-LT, the Few\-shot gain comes with a visible Many\-shot contraction, so BS\-cRT should be interpreted as a tail\-favoring operating point rather than a universal Top\-1 improvement\. On Places\-LT, the same rebalancing improves Medium and Few classes enough to increase overall Top\-1 despite a reduction in Many\-shot classes\. In both cases, the method behaves consistently with its design: it reduces the classifier’s reliance on head\-class decision regions\.

![Refer to caption](https://arxiv.org/html/2607.09832v1/x12.png)Figure 4:Frequency\-bucket breakdown on ImageNet\-LT and Places\-LT\. Panel \(a\) shows ImageNet\-LT and panel \(b\) shows Places\-LT, using a shared vertical scale from0to7070for direct comparison\. The same classifier\-retraining recipe produces a strong Few\-shot gain on both datasets\. ImageNet\-LT exposes a Top\-1 trade\-off because the Many\-shot contraction is large enough to outweigh the Few\-shot gain in the aggregate\. In contrast, on Places\-LT the Medium\- and Few\-shot gains compensate for the Many\-shot reduction\. Error bars show standard deviation across three seeds\.Table[6](https://arxiv.org/html/2607.09832#S4.T6)places our results next to representative published results under common long\-tailed recognition protocols\. The cited methods differ in backbone, training length, augmentation policy, auxiliary objectives, generative or contrastive branches, and in some cases model multiplicity\. The controlled evidence for BS\-cRT is therefore the matched\-checkpoint comparison in Tables[2](https://arxiv.org/html/2607.09832#S4.T2)and[5](https://arxiv.org/html/2607.09832#S4.T5)\. The role of Table[6](https://arxiv.org/html/2607.09832#S4.T6)is to show where this low\-complexity classifier\-side correction sits relative to stronger published systems\.

Table 6:Published results context under common long\-tailed recognition protocols\. All values are top\-1 accuracy percentages\. “All” denotes overall accuracy, and “Few” denotes the Few\-shot/Tail split reported by the corresponding paper\. The rows are not directly comparable because published methods use different backbones, training schedules, augmentation policies, auxiliary objectives, or model multiplicity\. The table is intended as literature context, not as a controlled comparison\.MethodMain mechanismCIFAR\-100\-LT IF=100ImageNet\-LTPlaces\-LTAllFew/TailAllFewAllFewcRT\(Kang et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib13)\)Classifier retraining43\.318\.149\.627\.336\.724\.9BALMS / Balanced Softmax\(Ren et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib23)\)Prior\-aware softmax50\.8–41\.825\.338\.731\.6BSCE / Balanced Softmax\(Ren et al\.,[2020](https://arxiv.org/html/2607.09832#bib.bib23); Cong et al\.,[2024](https://arxiv.org/html/2607.09832#bib.bib4)\)Prior\-aware loss50\.833\.452\.333\.439\.432\.7MiSLAS\(Zhong et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib39)\)Two\-stage learning and calibration46\.826\.651\.432\.837\.627\.5LADE\(Hong et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib11)\)Label\-distribution disentangling45\.629\.852\.334\.339\.232\.3PaCo\(Cui et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib6)\)Parametric contrastive learning52\.0–57\.039\.141\.235\.3RIDE\(X\. Wang et al\.,[2021](https://arxiv.org/html/2607.09832#bib.bib34)\)Multi\-expert routing48\.023\.755\.736\.040\.333\.0BCL\(Zhu et al\.,[2022](https://arxiv.org/html/2607.09832#bib.bib42)\)Balanced contrastive learning53\.935\.557\.136\.6––GCL\(Li et al\.,[2022](https://arxiv.org/html/2607.09832#bib.bib17)\)Logit/feature rebalancing48\.7–54\.9–––Decoupled Optimisation\(Cong et al\.,[2024](https://arxiv.org/html/2607.09832#bib.bib4)\)Grouped parameter optimization53\.844\.660\.452\.242\.840\.1Label Over\-Smooth\(Sun et al\.,[2025](https://arxiv.org/html/2607.09832#bib.bib27)\)Classifier retraining regularisation54\.9–54\.442\.3––GLMC\+SEL\(Du et al\.,[2023](https://arxiv.org/html/2607.09832#bib.bib9); Jian et al\.,[2025](https://arxiv.org/html/2607.09832#bib.bib12)\)Exploratory synthetic examples56\.48–57\.2438\.28––CBDM\+DBG\(Yang et al\.,[2026](https://arxiv.org/html/2607.09832#bib.bib37)\)Boundary\-aware generation51\.2130\.10––––BS\-cRT \(ours\)Classifier\-only retraining after Balanced Softmax41\.07±1\.0141\.07\\pm 1\.0125\.0649\.37±0\.8049\.37\\pm 0\.8045\.1328\.38±0\.1428\.38\\pm 0\.1428\.39

*Notes\.*Dashes indicate that the corresponding split or dataset was not reported\. BALMS / Balanced Softmax denotes the original BALMS report, whereas BSCE / Balanced Softmax denotes the Balanced Softmax baseline reported in the Decoupled Optimisation comparison\.

This comparison also defines the intended scope of BS\-cRT\. On CIFAR\-100\-LT, modern representation\-learning, multi\-stage, and augmentation\-heavy methods achieve higher overall accuracy\. On ImageNet\-LT, BS\-cRT gives a strong Few\-shot operating point despite updating only the classifier, but this comes with the Top\-1 reduction reported above\. The main point is not that BS\-cRT replaces heavily engineered systems, but that those systems should control for a matched classifier\-retraining stage before attributing tail gains to additional mechanisms\.

### 4\.4CBRM Diagnostics

We next evaluate CBRM as a boundary\-supervision extension of BS\-cRT\. Under the tested hardest\-negative construction, the extension does not improve the classifier\-only baseline\. The original CBRM implementation suffers from a margin/scale mismatch when used with a cosine classifier: the target margin is configured in cosine\-similarity units, while the violation is computed on logits multiplied by the classifier scale\. Figure[5](https://arxiv.org/html/2607.09832#S4.F5)shows the consequence on CIFAR\-100\-LT IF100100, seed11\. When CBRM activates after warmup, training loss jumps by nearly two orders of magnitude, and Top\-1 collapses from roughly40%40\\%to24%24\\%\. Computing the violation in cosine\-margin space removes the loss spike, but it does not make hardest\-negative probes beneficial\.

![Refer to caption](https://arxiv.org/html/2607.09832v1/x13.png)Figure 5:Training dynamics of CBRM and BS\-cRT on CIFAR\-100\-LT IF100100, seed11\. Panel \(a\) shows stage\-2 training loss on a logarithmic scale\. Panel \(b\) shows Top\-1 accuracy during the same run\. The original CBRM formulation exhibits a sharp loss spike and accuracy collapse when the boundary\-risk term activates\. The cosine\-margin fix stabilizes optimization, but BS\-cRT remains the more reliable stage\-2 baseline\.Table[7](https://arxiv.org/html/2607.09832#S4.T7)isolates the probe effect under a frozen backbone\. BS\-cRT, which uses no probes, reaches41\.0741\.07Top\-1 and25\.0625\.06Few\-shot accuracy over three seeds\. Adding corrected hardest\-negative probes gives similar Top\-1,40\.8640\.86, but reduces Few\-shot accuracy to20\.7020\.70, a drop of4\.364\.36points\. The random\-negative row is a one\-seed exploratory diagnostic rather than a full result; it does not show the same tail degradation, suggesting that the failure is tied to the systematic head anchoring induced by hardest\-negative mining\.

Table 7:Frozen\-backbone diagnostic on CIFAR\-100\-LT IF100100\. BS\-cRT disables probes\. The corrected CBRM variants compute violations in cosine\-margin space\. Values are percentages\. Rows with±\\pmreport mean, and standard deviation over three seeds; the random\-negative row is an exploratory one\-seed diagnostic\.Figure[6](https://arxiv.org/html/2607.09832#S4.F6)expands this comparison with additional one\-seed CBRM variants\. Lowering the CBRM weight toλcbr=0\.3\\lambda\_\{\\mathrm\{cbr\}\}=0\.3or reducingtmaxt\_\{\\max\}does not recover the tail accuracy of the no\-probe baseline\. The same pattern appears across these variants: the variant that omits synthetic probes is the most reliable stage\-2 baseline, while hardest\-negative probes tend to move accuracy back toward Many\-shot classes at the expense of the tail\.

![Refer to caption](https://arxiv.org/html/2607.09832v1/x14.png)Figure 6:Probe diagnostics on CIFAR\-100\-LT IF100100\. Panel \(a\) compares Few\-shot accuracy across no\-probe and probe\-based stage\-2 variants\. Panel \(b\) shows the corresponding Top\-1 accuracy\. Error bars appear for variants with three seeds; one\-seed diagnostic variants are shown without error bars\. Lowering the CBRM weight or shortening the probe search interval does not recover the Few\-shot accuracy of BS\-cRT\.These diagnostics identify the failure mode of the tested CBRM construction\. Hardest\-negative probes are intended to strengthen weak boundaries, but in long\-tailed data, the hardest negative for a tail sample is frequently a head class\. The generated probe is therefore anchored near a head\-class region\. Enforcing a positive tail margin on such probes can pull the tail classifier direction toward head\-adjacent feature space, which is precisely the bias that classifier retraining is meant to reduce\. In this setting, balanced classifier retraining without synthetic probes is the more reliable intervention\.

## 5Discussion

The results indicate that part of the residual tail error after Balanced Softmax remains in the classifier\. After stage\-1 training, the representation can still support a classifier whose directions reflect the long\-tailed training stream\. Retraining only the classifier under balanced class exposure recovers Few\-shot accuracy in every evaluated benchmark cell, while aggregate Top\-1 depends on how much Many\-shot accuracy is traded for Medium\- and Few\-shot gains\. This matters when interpreting aggregate accuracy: BS\-cRT is not designed to improve all frequency groups uniformly\. It changes the operating point of the classifier by reallocating decision capacity away from dominant classes and toward underrepresented classes during end\-to\-end training\.

### 5\.1Classifier\-side correction after Balanced Softmax

BS\-cRT is intentionally limited in scope\. It does not introduce a new architecture, representation loss, auxiliary branch, or source of supervision\. Its contribution is the controlled finding that a frozen representation trained with Balanced Softmax still benefits from a separate classifier\-retraining stage\. The use of matched stage\-1 checkpoints is central to this claim: the comparison isolates the effect of the second\-stage classifier update rather than mixing it with representation changes\. In this setting, balancedP×KP\\times Kbatches give each selected class comparable influence on the classifier update, while the empirical\-prior Balanced Softmax loss keeps the same prior\-aware objective used in stage 1\.

This result has a methodological consequence for long\-tailed recognition\. Methods that add prototypes, contrastive heads, generated samples, expert branches, or adaptive margins should be compared not only with end\-to\-end Balanced Softmax, but also with a matched classifier\-retraining stage\. Otherwise, part of the reported tail improvement may reflect the generic effect of re\-estimating the final classifier under balanced exposure\. BS\-cRT is therefore best viewed as a low\-complexity reference baseline rather than as a replacement for representation\-learning or multi\-stage long\-tailed methods\.

### 5\.2Why hardest\-negative boundary probes fail

CBRM was motivated by a plausible hypothesis: tail classes have too few samples to densely constrain their decision boundaries, so synthetic probes near current boundaries might provide useful additional supervision\. The experiments do not support this extension under hardest\-negative mining\. The first failure mode is numerical\. In a normalized classifier, the margin should be measured in cosine\-similarity units\. Computing the violation on scaled logits combines a cosine\-range target margin with the classifier scale, which destabilizes the boundary\-risk term\. Computing violations directly in cosine\-margin space removes the training collapse, but it does not make the probes beneficial\.

The remaining failure mode is geometric\. In long\-tailed data, the hardest negative for a tail sample is often a head class\. A probe constructed between a tail prototype and a head\-class prototype is therefore not a neutral sample near a balanced boundary\. It is anchored near a region already supported by many head\-class examples\. Enforcing a tail margin at that location can pull the tail classifier toward head\-adjacent feature space, partially undoing the correction introduced by balanced classifier retraining\. The random\-negative diagnostic is consistent with this interpretation: when negative classes are sampled uniformly, the same tail degradation is reduced, although this one\-seed variant is not sufficient to define a new method\.

These observations do not rule out boundary\-aware learning for long\-tailed recognition\. They indicate that boundary probes must account for class frequency, class support, and classifier geometry\. In particular, negative\-class selection should avoid conflating “hard” with “head\-dominated,” probe placement should account for asymmetric support between head and tail classes, and the supervised margin should be expressed in the same units as the classifier\.

### 5\.3Limitations and future work

Several aspects of the study delimit the scope of the conclusions\. First, the backbone family is fixed within each benchmark\. We use ResNet\-style architectures, but do not test wider convolutional networks, vision transformers, self\-supervised pretraining, or foundation\-model features\. The strength of BS\-cRT may change when the stage\-1 representation is learned under a substantially different pretraining or architecture regime\.

Second, all main benchmark cells use three seeds\. The Few\-shot gains are large relative to seed variation, but aggregate Top\-1 changes should be interpreted more carefully, especially on ImageNet\-LT, where BS\-cRT trades aggregate accuracy for a stronger Few\-shot operating point\. Third, the stage\-2 anchor loss is not ablated\. All reported BS\-cRT runs use empirical\-prior Balanced Softmax under balanced class exposure\. Replacing this anchor with cross\-entropy, a uniform prior, or another prior choice may lead to different operating points\.

Fourth, the paper focuses on accuracy and frequency\-bucket behavior rather than calibration\. Since classifier retraining changes the distribution of confidence across head and tail classes, per\-bucket reliability and post\-hoc calibration should be studied separately\. Fifth, iNaturalist 2018 is not included in the reported benchmark set, even though it is an important natural long\-tailed dataset\. Finally, the CBRM conclusions are specific to prototype\-to\-prototype interpolation with hardest\-negative selection\. Other forms of synthetic boundary supervision may still be effective if their probes are frequency\-aware and geometrically aligned with tail\-region expansion\.

## 6Conclusion

We studied classifier retraining after Balanced Softmax for long\-tailed recognition\. The results show that a frozen\-backbone stage\-2 update can recover substantial Few\-shot accuracy from the same stage\-1 representation\. BS\-cRT improves Few\-shot accuracy over the matched Balanced Softmax checkpoint across CIFAR\-LT, ImageNet\-LT and Places\-LT, without changing the inference architecture\. The method therefore provides a useful low\-complexity baseline for evaluating more elaborate long\-tailed recognition systems\.

We also evaluated CBRM as a boundary\-probe extension of this baseline\. Correcting the margin computation from scaled\-logit space to cosine\-margin space stabilizes training, but hardest\-negative probes still fail to improve over BS\-cRT and can reduce tail accuracy\. The analysis indicates that head\-class anchoring is a central risk when synthetic boundary probes are generated by hard\-negative mining in long\-tailed data\. Future boundary\-aware methods should therefore control how negative classes are selected, where probes are placed, and how margins are defined relative to the classifier geometry\.

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