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arxiv: 2604.06468 · v2 · submitted 2026-04-07 · 💻 cs.LG · stat.ML

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Conformal Margin Risk Minimization: An Envelope Framework for Robust Learning under Label Noise

Yuanjie Shi , Peihong Li , Zijian Zhang , Janardhan Rao Doppa , Yan Yan
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Pith reviewed 2026-05-10 19:28 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords label noiserobust classificationconformal predictionmargin regularizationplug-and-play learningnoisy supervision
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The pith

Conformal Margin Risk Minimization wraps any classification loss with one quantile-regularized term to focus training on high-margin samples under arbitrary label noise.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents CMRM as a plug-and-play addition to existing losses that requires no noise transition matrix, clean subset, or pretrained extractor. It computes the margin between the observed label and competing labels, then penalizes samples whose margin falls below a conformal quantile computed on each batch. This directs gradient updates toward likely-correct examples while suppressing probable errors. A generalization bound is derived that holds for any label noise provided the margin distribution satisfies mild regularity. Experiments across five base learners and six datasets with synthetic and real noise show accuracy gains up to 3.39 percent, smaller conformal prediction sets, and no degradation on clean data.

Core claim

CMRM improves robustness to label noise for any base method by adding a single envelope term that thresholds the observed-label margin against a per-batch conformal quantile, thereby reweighting the loss toward high-margin samples without any privileged knowledge of the noise process.

What carries the argument

The conformal quantile threshold on the margin between the given label and alternatives; it supplies a method-agnostic uncertainty signal that regularizes the training loss to down-weight likely mislabeled points.

Load-bearing premise

The distribution of margins is regular enough that an empirical quantile computed on each batch reliably separates clean from noisy examples.

What would settle it

A dataset in which CMRM produces no accuracy gain or larger prediction sets despite the presence of label noise and the same base learner.

Figures

Figures reproduced from arXiv: 2604.06468 by Janardhan Rao Doppa, Peihong Li, Yan Yan, Yuanjie Shi, Zijian Zhang.

Figure 1
Figure 1. Figure 1: Confidence margin distributions for clean (blue) and noisy (orange) samples on CIFAR-100. (a) Class-conditional noise at 20% and (b) human annota￾tion noise at 40%. In both cases, clean samples concentrate on positive margins while noisy samples shift to negative, showing that confidence margins can distinguish clean from noisy labels without assumptions on the noise process. this threshold, a property tha… view at source ↗
Figure 2
Figure 2. Figure 2: Justification experiments for multi-class classification on CIFAR-100 with 20% synthetic label noise. Subfigure (a) shows the training dynamics of total loss (Total), classification loss (cl), and CMRM loss (cr) over epochs. CMRM exhibits stable and monotonic convergence alongside standard loss components. Subfigure (b) reports the ratio of noisy samples among those filtered (with soft weight < 0.5) out by… view at source ↗
Figure 3
Figure 3. Figure 3: Training dynamics of LR+CMRM for bi￾nary classification on the Email dataset. Subfigure (a) Training dynamics of total loss (all loss), classification loss (cl loss), and CMRM loss (cr loss) over epochs. CMRM exhibits stable and monotonic convergence alongside stan￾dard loss components. Subfigure (b) τ − (negative class threshold) and τ + (positive class threshold) of LR+CMRM during training. The separatio… view at source ↗
Figure 4
Figure 4. Figure 4: Histograms of positive confidence distri￾butions for clean (blue) and noisy (orange) samples on the Credit dataset with 20% label noise. The top and bottom rows correspond to samples with observed labels Ye = 0 (negative) and Ye = 1 (positive), respectively. Dis￾tributions are obtained using LR (left) and LR+CMRM (right). CMRM induces a clearer separation between clean and noisy confidence distributions fo… view at source ↗
Figure 5
Figure 5. Figure 5: training dynamics of total loss (Total), classification loss (cl), and CMRM loss (cr) for all base losses over epochs on CIFAR-100 with 20% synthetical label noise. Subfigure (a) CE+CERM; Subfigure (b) Focal+CERM; Subfigure (c) LDAM+CERM; Subfigure (d) GCE+CERM; CMRM exhibits stable and monotonic convergence alongside standard loss components. with its +CMRM variant. Across all objectives, CMRM consistentl… view at source ↗
Figure 6
Figure 6. Figure 6: The ratio of noisy samples among those filtered out by CMRM at each epoch on CIFAR-100 with 20% synthetical label noise. Subfigure (a) CE+CMRM (α = 0.15); Subfigure (b) Focal+CMRM (α = 0.1); Subfigure (c) LDAM+CMRM (α = 0.15); Subfigure (d) GCE+CMRM (α = 0.05); CMRM consistently suppresses noisy examples by excluding low-margin samples during training. (a) CE+CMRM (b) Focal+CMRM (c) LDAM+CMRM (d) GCE+CMRM … view at source ↗
Figure 7
Figure 7. Figure 7: The sensitivity of α of different base losses on CIFAR-100 with 20% synthetical label noise. Subfigure (a) CE+CERM; Subfigure (b) Focal+CERM; Subfigure (c) LDAM+CERM; Subfigure (d) GCE+CERM; CMRM maintains higher accuracy than CE across a range of α values, indicating robustness to hyperparameter α. (a) CE+CMRM (b) Focal+CMRM (c) LDAM+CMRM (d) GCE+CMRM [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The sensitivity of λ of different base losses on CIFAR-100 with 20% synthetical label noise. Subfigure (a) CE+CERM; Subfigure (b) Focal+CERM; Subfigure (c) LDAM+CERM; Subfigure (d) GCE+CERM; CMRM maintains higher accuracy than CE across a range of λ values, indicating robustness to hyperparameter λ. (a) CE+CMRM (b) Focal+CMRM (c) LDAM+CMRM (d) GCE+CMRM [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: the kernel density estimate (KDE) of the margin distribution of different base losses on CIFAR-100 with 20% synthetical label noise. Subfigure (a) CE+CMRM (α = 0.15); Subfigure (b) Focal+CMRM (α = 0.1); Subfigure (c) LDAM+CMRM (α = 0.15); Subfigure (d) GCE+CMRM (α = 0.05); The vertical dashed line indicating the estimated τα(f). The density curve is smooth and strictly positive around τα(f), supporting the… view at source ↗
Figure 10
Figure 10. Figure 10: Dynamics of losses (Top Row) and two thresholds (τ + and τ −, Bottom Row) for binary classification of all base loss (LR, Focal, SVM, GCE) with CMRM on Email datasets. Top row shows the training dynamics of total loss (all loss), classification loss (cl loss), and CMRM regularization loss (CMRM loss) over epochs. CMRM exhibits stable and monotonic convergence alongside standard loss components. Bottom row… view at source ↗
read the original abstract

Most methods for learning with noisy labels require privileged knowledge such as noise transition matrices, clean subsets or pretrained feature extractors, resources typically unavailable when robustness is most needed. We propose Conformal Margin Risk Minimization (CMRM), a plug-and-play envelope framework that improves any classification loss under label noise by adding a single quantile-calibrated regularization term, with no privileged knowledge or training pipeline modification. CMRM measures the confidence margin between the observed label and competing labels, and thresholds it with a conformal quantile estimated per batch to focus training on high-margin samples while suppressing likely mislabeled ones. We derive a learning bound for CMRM under arbitrary label noise requiring only mild regularity of the margin distribution. Across five base methods and six benchmarks with synthetic and real-world noise, CMRM consistently improves accuracy (up to +3.39%), reduces conformal prediction set size (up to -20.44%) and does not hurt under 0% noise, showing that CMRM captures a method-agnostic uncertainty signal that existing mechanisms did not exploit.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes Conformal Margin Risk Minimization (CMRM), a plug-and-play envelope framework that augments any base classification loss with a single quantile-calibrated regularization term derived from the margin between the observed label and competing labels. The quantile is estimated conformally per batch without requiring noise transition matrices, clean subsets, or pretrained extractors. A learning bound is derived for CMRM under arbitrary label noise that requires only mild regularity of the margin distribution. Experiments across five base methods and six benchmarks (synthetic and real-world noise) report consistent accuracy gains (up to +3.39%), smaller conformal prediction sets (up to -20.44%), and no degradation at 0% noise.

Significance. If the bound holds, the work provides a method-agnostic uncertainty signal that existing robust-learning mechanisms have not exploited, with the practical advantage of requiring no pipeline changes or privileged information. The empirical evaluation on multiple bases and both synthetic/real noise is a clear strength, as is the demonstration that performance is not harmed on clean data. The approach correctly leverages established conformal prediction ideas in a novel regularization envelope.

major comments (1)
  1. [Abstract and theoretical section] Abstract and theoretical derivation (bound for CMRM under arbitrary label noise): the central claim that the bound requires 'only mild regularity of the margin distribution' is load-bearing, yet the manuscript does not specify the precise regularity conditions (e.g., continuity of the margin CDF, density bounded away from zero at the quantile level, or Lipschitz continuity). Arbitrary label noise can map clean margins to multimodal or discontinuous observed margins, which would invalidate the conformal quantile calibration guarantee used to control risk; a concrete counter-example or explicit assumption statement is needed to substantiate the bound.
minor comments (2)
  1. [Method description] The per-batch quantile estimation procedure should include a short pseudocode or explicit formula for the conformal score and quantile computation to aid reproducibility.
  2. [Experiments] Table captions and axis labels in the experimental results could more explicitly state the noise rates and base methods for each row/column to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address the major comment below and will revise the manuscript accordingly to strengthen the theoretical presentation.

read point-by-point responses

  1. Referee: [Abstract and theoretical section] Abstract and theoretical derivation (bound for CMRM under arbitrary label noise): the central claim that the bound requires 'only mild regularity of the margin distribution' is load-bearing, yet the manuscript does not specify the precise regularity conditions (e.g., continuity of the margin CDF, density bounded away from zero at the quantile level, or Lipschitz continuity). Arbitrary label noise can map clean margins to multimodal or discontinuous observed margins, which would invalidate the conformal quantile calibration guarantee used to control risk; a concrete counter-example or explicit assumption statement is needed to substantiate the bound.

    Authors: We agree that the precise regularity conditions were not stated explicitly enough in the abstract and theoretical derivation, and that this warrants clarification. In the revised manuscript we will add an explicit assumption statement in the theoretical section: the (observed) margin random variable is assumed to possess a continuous CDF whose density is positive and bounded away from zero in an open neighborhood of the target quantile level. This is the minimal condition needed for the conformal quantile estimator to achieve exact finite-sample marginal coverage and for the subsequent risk bound to hold. We will also include a short remark explaining why this condition is mild and why it is compatible with arbitrary label noise. While label noise can certainly produce multimodal or discontinuous margin distributions, the conformal calibration step itself is distribution-free and relies only on exchangeability within each batch (which is preserved by construction). The regularity assumption rules out only degenerate cases in which the quantile is not uniquely defined; it does not require unimodality or global Lipschitz continuity. We will add a brief discussion of this point and, space permitting, a simple illustrative example showing that the bound continues to hold under moderate multimodality. No counter-example that would invalidate the conformal guarantee under the stated regularity has been identified in our analysis. revision: yes

Circularity Check

0 steps flagged

No significant circularity; learning bound derived from standard conformal prediction theory with independent content

full rationale

The paper's central derivation is a learning bound for CMRM under arbitrary label noise that requires only mild regularity of the margin distribution. This bound is obtained by applying established conformal prediction quantile calibration to the observed margin between the given label and competitors, without reducing the bound to a fitted parameter or self-citation by construction. The per-batch quantile estimation is explicitly data-driven and does not tautologically define the risk control. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear in the derivation chain; the framework is presented as a plug-and-play envelope around any base loss, with empirical gains shown separately on benchmarks. The result therefore remains self-contained against external conformal prediction benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on conformal prediction principles plus one domain assumption; no new entities or fitted constants beyond standard conformal calibration are introduced in the abstract.

axioms (1)
  • domain assumption mild regularity of the margin distribution
    Invoked to derive the learning bound under arbitrary label noise.

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