Robust Loss Functions

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Robust loss functions study how to train models when data contain outliers, incorrect labels, heavy-tailed noise, or other unreliable observations. Common approaches include bounded or truncated residual losses, symmetric and generalized cross-entropy losses, divergence-based objectives, and sample reweighting; they are used in settings from regression and image classification to scientific estimation. These methods can improve performance under contamination, but their results depend on the type and level of noise and on choices such as loss shape and scale. A central challenge is gaining robustness without sacrificing clean-data accuracy, calibration, or efficiency, while establishing guarantees that match real-world benchmarks.

Latest papers

50 papers6 papers in the last four weeks, down 14% on the four weeks before. 0.0% of all new papers.

May 19, 2026

Symmetrization of Loss Functions for Robust Training of Neural Networks in the Presence of Noisy Labels

Labeling a training set is often expensive and susceptible to errors, making the design of robust loss functions for label noise an important problem. The symmetry condition provides theoretical guarantees for robustness to such noise. In this work, we study a symmetrization method arising from the unique decomposition of any multi-class loss function into a symmetric component and a class-insensitive term. In particular, symmetrizing the cross-entropy loss leads to a linear multi-class extension of the unhinged loss. Unlike in the binary case, the multi-class version must have specific coefficients in order to satisfy the symmetry condition. Under suitable assumptions, we show that this multi-class unhinged loss is the unique convex multi-class symmetric loss. We also show that it has a fundamental local role: the linear approximation of any symmetric loss around score vectors with equal components is equivalent to the multi-class unhinged loss. We then introduce SGCE and alpha-MAE, two loss functions that interpolate between the multi-class unhinged loss and the Mean Absolute Error while allowing control of the beta-smoothness of the loss. Experiments on standard noisy-label benchmarks show competitive performance compared with existing robust loss functions.

Alexandre Lemire Paquin, Brahim Chaib-Draa, Philippe Giguère

Department of Computer Science and Software Engineering Université Laval, Québec, Canada

Neural Network Robustness Noisy-Label Learning Robust Loss Functions Multiclass Classification Robust Classification

Apr 14, 2026

Risk-Calibrated Learning: Minimizing Fatal Errors in Medical AI

Deep learning models often achieve expert-level accuracy in medical image classification but suffer from a critical flaw: semantic incoherence. These high-confidence mistakes that are semantically incoherent (e.g., classifying a malignant tumor as benign) fundamentally differ from acceptable errors which stem from visual ambiguity. Unlike safe, fine-grained disagreements, these fatal failures erode clinical trust. To address this, we propose Risk-Calibrated Learning, a technique that explicitly distinguishes between visual ambiguity (fine-grained errors) and catastrophic structural errors. By embedding a confusion-aware clinical severity matrix M into the optimization landscape, our method suppresses critical errors (false negatives) without requiring complex architectural changes. We validate our approach in four different imaging modalities: Brain Tumor MRI, ISIC 2018 (Dermoscopy), BreaKHis (Breast Histopathology), and SICAPv2 (Prostate Histopathology). Extensive experiments demonstrate that our Risk-Calibrated Loss consistently reduces the Critical Error Rate (CER) for all four datasets, achieving relative safety improvements ranging from 20.0% (on breast histopathology) to 92.4% (on prostate histopathology) compared to state-of-the-art baselines such as Focal Loss. These results confirm that our method offers a superior safety-accuracy trade-off across both CNN and Transformer architectures.

Abolfazl Mohammadi-Seif, Ricardo Baeza-Yates

Department of Engineering, Universitat Pompeu Fabra, Barcelona, Spain

Medical Image Classification AI Safety Robust Loss Functions Cost-Sensitive Learning

Apr 14, 2026

Socrates Loss: Unifying Confidence Calibration and Classification by Leveraging the Unknown

Deep neural networks, despite their high accuracy, often exhibit poor confidence calibration, limiting their reliability in high-stakes applications. Current ad-hoc confidence calibration methods attempt to fix this during training but face a fundamental trade-off: two-phase training methods achieve strong classification performance at the cost of training instability and poorer confidence calibration, while single-loss methods are stable but underperform in classification. This paper addresses and mitigates this stability-performance trade-off. We propose Socrates Loss, a novel, unified loss function that explicitly leverages uncertainty by incorporating an auxiliary unknown class, whose predictions directly influence the loss function and a dynamic uncertainty penalty. This unified objective allows the model to be optimized for both classification and confidence calibration simultaneously, without the instability of complex, scheduled losses. We provide theoretical guarantees that our method regularizes the model to prevent miscalibration and overfitting. Across four benchmark datasets and multiple architectures, our comprehensive experiments demonstrate that Socrates Loss consistently improves training stability while achieving more favorable accuracy-calibration trade-off, often converging faster than existing methods.

Sandra Gómez-Gálvez, Tobias Olenyi, Gillian Dobbie +1

University of Auckland · Technical University of Munich

Uncertainty Calibration Classification Robust Loss Functions

Apr 9, 2026

Component-Adaptive and Lesion-Level Supervision for Improved Small Structure Segmentation in Brain MRI

Small lesions in brain MRI are hard to segment because they occupy a tiny fraction of the volume and are dominated by background and larger lesions during voxel-wise optimization, so a model can reach a high Dice similarity coefficient (DSC) while missing many of them. We propose CATMIL, a training objective that adds two auxiliary terms to the standard nnU-Net Dice and cross-entropy loss without changing the architecture. The Component-Adaptive Tversky (CAT) term weights lesion voxels by the inverse size of their connected component, so each lesion contributes nearly equally regardless of volume. The lesion-level Multiple Instance Learning (MIL) term treats each lesion as a bag of voxels and penalizes lesions with no detected voxel. For multiple sclerosis lesion segmentation on MSLesSeg, CATMIL achieves the highest small-lesion recall (0.873 vs. 0.796 for Dice+CE; 95% CI of the difference +0.030 to +0.157, higher in all six test patients) and about 48% fewer missed lesions, with comparable DSC and HD95. The gain holds for lesions of at least 3 mm in diameter, the clinical reading size (recall 0.944 vs. 0.870). Standard losses produce no probability response to most small lesions they miss, so no threshold can recover them. The cost is more small false-positive components; a simple component-size filter removes most of them while keeping the sensitivity gain, and at matched lesion-wise precision CATMIL detects more small lesions with higher lesion-wise F1. An ablation attributes the detection gain to the MIL term. On a second dataset, 3D-MR-MS, CATMIL with the same loss weights again improves small-lesion recall, at a larger false-positive cost and slightly lower DSC. Code: https://github.com/luumsk/SmallLesionMRI

Minh Sao Khue Luu, Evgeniy N. Pavlovskiy, Bair N. Tuchinov

The Artificial Intelligence Research Center of Novosibirsk State University, Novosibirsk 630090, Russia

Medical Image Segmentation Robust Loss Functions

Apr 9, 2026

Sparse $ε$ insensitive zone bounded asymmetric elastic net support vector machines for pattern classification

Existing support vector machines(SVM) models are sensitive to noise and lack sparsity, which limits their performance. To address these issues, we combine the elastic net loss with a robust loss framework to construct a sparse

\varepsilon

-insensitive bounded asymmetric elastic net loss, and integrate it with SVM to build

\varepsilon

Insensitive Zone Bounded Asymmetric Elastic Net Loss-based SVM(

\varepsilon

-BAEN-SVM).

\varepsilon

-BAEN-SVM is both sparse and robust. Sparsity is proven by showing that samples inside the

\varepsilon

-insensitive band are not support vectors. Robustness is theoretically guaranteed because the influence function is bounded. To solve the non-convex optimization problem, we design a half-quadratic algorithm based on clipping dual coordinate descent. It transforms the problem into a series of weighted subproblems, improving computational efficiency via the

\varepsilon

parameter. Experiments on simulated and real datasets show that

\varepsilon

-BAEN-SVM outperforms traditional and existing robust SVMs. It balances sparsity and robustness well in noisy environments. Statistical tests confirm its superiority. Under the Gaussian kernel, it achieves better accuracy and noise insensitivity, validating its effectiveness and practical value.

Haiyan Du, Hu Yang

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

Support Vector Machines Robust Loss Functions

Apr 7, 2026

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

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.

Yuanjie Shi, Peihong Li, Zijian Zhang +2

School of EECS, Washington State University, Pullman, WA, USA

Margin-Based Learning Learning with Noisy Labels Robust Classification Robust Loss Functions Statistical Learning Theory

Apr 6, 2026

Outlier-Robust Nonlinear Moving Horizon Estimation using Adaptive Loss Functions

In this work, we propose an adaptive robust loss function framework for MHE, integrating an adaptive robust loss function to reduce the impact of outliers with a regularization term that avoids naive solutions. The proposed approach prioritizes the fitting of uncontaminated data and downweights the contaminated ones. A tuning parameter is incorporated into the framework to control the shape of the loss function for adjusting the estimator's robustness to outliers. The simulation results demonstrate that adaptation occurs in just a few iterations, whereas the traditional behaviour

\mathrm{L_2}

predominates when the measurements are free of outliers.

Nestor Deniz, Guido Sanchez, Fernando Auat Cheein +1

Instituto de Investigación en Senales, Sistemas e Inteligencia Computacional, sınc(i), UNL-CONICET, Santa Fe, Argentina · Harper Adams University, Engineer Department, Edgmond, Newport TF10 8NB

Moving Horizon Estimation Robust Estimation Robust Loss Functions Adaptive Loss Weighting

Mar 20, 2026

Beyond the Mean: Distribution-Aware Loss Functions for Bimodal Regression

Despite the strong predictive performance achieved by machine learning models across many application domains, assessing their trustworthiness through reliable estimates of predictive confidence remains a critical challenge. This issue arises in scenarios where the likelihood of error inferred from learned representations follows a bimodal distribution, resulting from the coexistence of confident and ambiguous predictions. Standard regression approaches often struggle to adequately express this predictive uncertainty, as they implicitly assume unimodal Gaussian noise, leading to mean-collapse behavior in such settings. Although Mixture Density Networks (MDNs) can represent different distributions, they suffer from severe optimization instability. We propose a family of distribution-aware loss functions integrating normalized RMSE with Wasserstein and Cramér distances. When applied to standard deep regression models, our approach recovers bimodal distributions without the volatility of mixture models. Validated across four experimental stages, our results show that the proposed Wasserstein loss establishes a new Pareto efficiency frontier: matching the stability of standard regression losses like MSE in unimodal tasks while reducing Jensen-Shannon Divergence by 45% on complex bimodal datasets. Our framework strictly dominates MDNs in both fidelity and robustness, offering a reliable tool for aleatoric uncertainty estimation in trustworthy AI systems.

Abolfazl Mohammadi-Seif, Carlos Soares, Rita P. Ribeiro +1

Department of Engineering, Universitat Pompeu Fabra, Roc Boronat St., Barcelona, 08018, Barcelona, Spain. · Department of Engineering, University of Porto, Praça de Gomes Teixeira, Porto, 4099-002, Portugal. · Faculty of Sciences, University of Porto, 4169-007 Porto, Portugal. +1

Aleatoric Uncertainty Uncertainty Quantification Robust Loss Functions

Mar 20, 2026

Minimax Generalized Cross-Entropy

Loss functions play a central role in supervised classification. Cross-entropy (CE) is widely used, whereas the mean absolute error (MAE) loss can offer robustness but is difficult to optimize. Interpolating between the CE and MAE losses, generalized cross-entropy (GCE) has recently been introduced to provide a trade-off between optimization difficulty and robustness. Existing formulations of GCE result in a non-convex optimization over classification margins that is prone to underfitting, leading to poor performances with complex datasets. In this paper, we propose a minimax formulation of generalized cross-entropy (MGCE) that results in a convex optimization over classification margins. Moreover, we show that MGCEs can provide an upper bound on the classification error. The proposed bilevel convex optimization can be efficiently implemented using stochastic gradient computed via implicit differentiation. Using benchmark datasets, we show that MGCE achieves strong accuracy, faster convergence, and better calibration, especially in the presence of label noise.

Kartheek Bondugula, Santiago Mazuelas, Aritz Pérez +1

Basque Center for Applied Mathematics, Bilbao, Spain · Ikerbasque, Basque Foundation for Science · Johns Hopkins University, Baltimore, USA

Robust Loss Functions Min-Max Optimization

Mar 18, 2026

rSDNet: Unified Robust Neural Learning against Label Noise and Adversarial Attacks

Neural networks are central to modern artificial intelligence, yet their training remains highly sensitive to data contamination. Standard neural classifiers are trained by minimizing the categorical cross-entropy loss, corresponding to maximum likelihood estimation under a multinomial model. While statistically efficient under ideal conditions, this approach is highly vulnerable to contaminated observations including label noises corrupting supervision in the output space, and adversarial perturbations inducing worst-case deviations in the input space. In this paper, we propose a unified and statistically grounded framework for robust neural classification that addresses both forms of contamination within a single learning objective. We formulate neural network training as a minimum-divergence estimation problem and introduce rSDNet, a robust learning algorithm based on the general class of

S

-divergences. The resulting training objective inherits robustness properties from classical statistical estimation, automatically down-weighting aberrant observations through model probabilities. We establish essential population-level properties of rSDNet, including Fisher consistency, classification calibration implying Bayes optimality, and robustness guarantees under uniform label noise and infinitesimal feature contamination. Experiments on three benchmark image classification datasets show that rSDNet improves robustness to label corruption and adversarial attacks while maintaining competitive accuracy on clean data, Our results highlight minimum-divergence learning as a principled and effective framework for robust neural classification under heterogeneous data contamination.

Suryasis Jana, Abhik Ghosh

Indian Statistical Institute, Kolkata, India

Neural Network Robustness Robust Classification Learning with Noisy Labels Adversarial Robustness Robust Loss Functions

Mar 16, 2026

Global Truncated Loss Minimization for Robust and Threshold-Resilient Geometric Estimation

To achieve outlier-robust geometric estimation, robust objective functions are generally employed to mitigate the influence of outliers. The widely used consensus maximization(CM) is highly robust when paired with global branch-and-bound(BnB) search. However, CM relies solely on inlier counts and is sensitive to the inlier threshold. Besides, the discrete nature of CM leads to loose bounds, necessitating extensive BnB iterations and computation cost. Truncated losses(TL), another continuous alternative, leverage residual information more effectively and could potentially overcome these issues. But to our knowledge, no prior work has systematically explored globally minimizing TL with BnB and its potential for enhanced threshold resilience or search efficiency. In this work, we propose GTM, the first unified BnB-based framework for globally-optimal TL loss minimization across diverse geometric problems. GTM involves a hybrid solving design: given an n-dimensional problem, it performs BnB search over an (n-1)-dimensional subspace while the remaining 1D variable is solved by bounding the objective function. Our hybrid design not only reduces the search space, but also enables us to derive Lipschitz-continuous bounding functions that are general, tight, and can be efficiently solved by a classic global Lipschitz solver named DIRECT, which brings further acceleration. We conduct a systematic evaluation on various BnB-based methods for CM and TL on the robust linear regression problem, showing that GTM enjoys remarkable threshold resilience and the highest efficiency compared to baseline methods. Furthermore, we apply GTM on different geometric estimation problems with diverse residual forms. Extensive experiments demonstrate that GTM achieves state-of-the-art outlier-robustness and threshold-resilience while maintaining high efficiency across these estimation tasks.

Tianyu Huang, Liangzu Peng, Xinyue Zhang +5

Hong Kong Centre For Logistics Robotics, The Chinese University of Hong Kong. · Center for Innovation in Data Engineering and Science (IDEAS), University of Pennsylvania. · Mobile Perception Lab of the School of Information and Technology, ShanghaiTech University. +1

Robust Loss Functions Branch and Bound Robust Regression

Mar 6, 2026

Robust support vector model based on bounded asymmetric elastic net loss for binary classification

In this paper, we propose a novel bounded asymmetric elastic net (

L_{baen}

) loss function and combine it with the support vector machine (SVM), resulting in the BAEN-SVM. The

L_{baen}

is bounded and asymmetric and can degrade to the asymmetric elastic net hinge loss, pinball loss, and asymmetric least squares loss. BAEN-SVM not only effectively handles noise-contaminated data but also addresses the geometric irrationalities in the traditional SVM. By proving the violation tolerance upper bound (VTUB) of BAEN-SVM, we show that the model is geometrically well-defined. Furthermore, we derive that the influence function of BAEN-SVM is bounded, providing a theoretical guarantee of its robustness to noise. The Fisher consistency of the model further ensures its generalization capability. Since the

L_{\text{baen}}

loss is non-convex, we designed a clipping dual coordinate descent-based half-quadratic algorithm to solve the non-convex optimization problem efficiently. Experimental results on artificial and benchmark datasets indicate that the proposed method outperforms classical and advanced SVMs, particularly in noisy environments.

Haiyan Du, Hu Yang

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

Robust Loss Functions Support Vector Machines

Mar 6, 2026

Margin and Consistency Supervision for Calibrated and Robust Vision Models

Deep vision classifiers often achieve high accuracy while remaining poorly calibrated and fragile under small distribution shifts. We present Margin and Consistency Supervision (MaCS), a simple, architecture-agnostic regularization framework that jointly enforces logit-space separation and local prediction stability. MaCS augments cross-entropy with (i) a hinge-squared margin penalty that enforces a target logit gap between the correct class and the strongest competitor, and (ii) a consistency regularizer that minimizes the KL divergence between predictions on clean inputs and mildly perturbed views. We provide a unifying theoretical analysis showing that increasing classification margin while reducing local sensitivity formalized via a Lipschitz-type stability proxy yields improved generalization guarantees and a provable robustness radius bound scaling with the margin-to-sensitivity ratio. Across several image classification benchmarks and several backbones spanning CNNs and Vision Transformers, MaCS consistently improves calibration (lower ECE and NLL) and robustness to common corruptions while preserving or improving top-1 accuracy. Our approach requires no additional data, no architectural changes, and negligible inference overhead, making it an effective drop-in replacement for standard training objectives.

Salim Khazem

Talan Research Center, Paris, France.

Distribution Shift Robustness Model Calibration Robust Loss Functions Image Classification Consistency Regularization

Mar 5, 2026

MCEL: Margin-Based Cross-Entropy Loss for Error-Tolerant Quantized Neural Networks

Robustness to bit errors is a key requirement for the reliable use of neural networks (NNs) on emerging approximate computing platforms and error-prone memory technologies. A common approach to achieve bit error tolerance in NNs is injecting bit flips during training according to a predefined error model. While effective in certain scenarios, training-time bit flip injection introduces substantial computational overhead, often degrades inference accuracy at high error rates, and scales poorly for larger NN architectures. These limitations make error injection an increasingly impractical solution for ensuring robustness on future approximate computing platforms and error-prone memory technologies. In this work, we investigate the mechanisms that enable NNs to tolerate bit errors without relying on error-aware training. We establish a direct connection between bit error tolerance and classification margins at the output layer. Building on this insight, we propose a novel loss function, the Margin Cross-Entropy Loss (MCEL), which explicitly promotes logit-level margin separation while preserving the favorable optimization properties of the standard cross-entropy loss. Furthermore, MCEL introduces an interpretable margin parameter that allows robustness to be tuned in a principled manner. Extensive experimental evaluations across multiple datasets of varying complexity, diverse NN architectures, and a range of quantization schemes demonstrate that MCEL substantially improves bit error tolerance, up to 15 % in accuracy for an error rate of 1 %. Our proposed MCEL method is simple to implement, efficient, and can be integrated as a drop-in replacement for standard CEL. It provides a scalable and principled alternative to training-time bit flip injection, offering new insights into the origins of NN robustness and enabling more efficient deployment on approximate computing and memory systems.

Mikail Yayla, Akash Kumar

Ruhr-Universität Bochum, Germany

Neural Network Robustness Robust Loss Functions

Mar 3, 2026

Functional Properties of the Focal-Entropy

The focal-loss has become a widely used alternative to cross-entropy in class-imbalanced classification problems, particularly in computer vision. Despite its empirical success, a systematic information-theoretic study of the focal-loss remains incomplete. In this work, we adopt a distributional viewpoint and study the focal-entropy, a focal-loss analogue of the cross-entropy. Our analysis establishes conditions for finiteness, convexity, and continuity of the focal-entropy, and provides various asymptotic characterizations. We prove the existence and uniqueness of the focal-entropy minimizer, describe its structure, and show that it can depart significantly from the data distribution. In particular, we rigorously show that the focal-loss amplifies mid-range probabilities, suppresses high-probability outcomes, and, under extreme class imbalance, induces an over-suppression regime in which very small probabilities are further diminished. These results, which are also experimentally validated, offer a theoretical foundation for understanding the focal-loss and clarify the trade-offs that it introduces when applied to imbalanced learning tasks.

Jaimin Shah, Martina Cardone, Alex Dytso

University of Minnesota, USA · Qualcomm Flarion Technology, Inc., USA

Focal Loss Class-Imbalanced Learning Robust Loss Functions

Feb 28, 2026

Heaviside Low-Rank Support Matrix Machine

Support matrix machine (SMM) is an emerging classification framework that directly handles matrix-structured observations, thereby avoiding the spatial correlations destroyed by vectorization. However, most existing SMM variants rely on convex or nonconvex surrogate loss functions, which may lead to high sensitivity to noise. To address this issue, we propose a novel Heaviside low-rank SMM model called HL-SMM, which leverages the Heaviside loss instead of the common hinge or ramp losses for robustness. Moreover, the low-rank constraint is adopted to accurately characterize the inherent global structure. In theory, we analyze the Karush-Kuhn-Tucker (KKT) points and rigorously prove the sufficient and necessary conditions. In algorithms, we develop an effective proximal alternating minimization (PAM) scheme, where all subproblems have closed-form solutions. Extensive experiments on benchmark datasets validate that the proposed HL-SMM achieves superior classification accuracy and robustness compared to state-of-the-art methods.

Xianchao Xiu, Shenghao Sun, Xinrong Li +1

School of Mechatronic Engineering and Automation, Shanghai University, Shanghai, 200400, China. · National Frontiers Science Center for Industrial Intelligence and Systems Optimization, Northeastern University, Shenyang, 110819, China. · Department of Mathematics and Statistics, Loyola University Maryland, Baltimore, MD 21210, USA.

Classification Robust Loss Functions

Feb 25, 2026

Enabling clinical use of foundation models for computational pathology

Foundation models for computational pathology are expected to facilitate the development of high-performing, generalisable deep learning systems. However, in addition to biologically relevant features, current foundation models also capture pre-analytic and scanner-specific variation that bias the predictions made by downstream task-specific models trained on these features. Here we show that introducing novel robustness losses during downstream model training reduces sensitivity to technical variability. A purpose-designed comprehensive experimentation setup with 27,042 whole-slide images from 6,155 patients is used to train thousands of models from the features of eight well-known foundation models for computational pathology. In addition to a substantial improvement in robustness, our approach improves classification accuracy by focusing on biologically relevant features. It mitigates robustness limitations of foundation models for computational pathology without retraining the foundation models themselves, enabling development of models that are more suitable in real-world clinical use.

Audun L Henriksen, Ole-Johan Skrede, Lisa van der Schee +31

Institute for Cancer Genetics and Informatics, Oslo University Hospital, Oslo, Norway · Department of Pathology, University Medical Center Utrecht, Utrecht, The Netherlands · Department of Oncology, University of Oxford, Oxford, UK +20

Computational Pathology Pathology Foundation Models Domain Generalization Histopathology Image Classification Robust Loss Functions

Oct 11, 2025

Wasserstein Robust Classification Error: Surrogate Loss Misalignment and Distributional Attacks

Wasserstein distributionally robust optimization (WDRO) provides a framework for evaluating model robustness through the worst-case expectation in the Wasserstein ball around a reference distribution. We prove that ordering hypotheses by increasing WDRO risk is equivalent to ordering them by decreasing robust accuracy if and only if the loss function is a positive multiple of the zero-one loss plus a label-dependent offset. For a class of loss functions including cross-entropy, we show that a hypothesis with perfect robust accuracy can conversely have a higher WDRO risk than another hypothesis with near zero robust accuracy. To evaluate robust accuracy directly, we develop Wasserstein distributional attacks (WDA), which add a weight generation stage after candidate support search to optimize the allocation of empirical mass under the Wasserstein constraint, rather than assigning each data point's full mass to one returned attack point as existing methods do. We further introduce a support-search algorithm for WDA that finds nearby misclassified inputs. On CIFAR-10, CIFAR-100, and ImageNet, WDA using our search algorithm achieves the lowest adversarial accuracy in most tested settings. Our code is available at https://github.com/OLab-Repo/WDA.

Bach C. Le, Tung V. Dao, Binh T. Nguyen +1

College of Engineering & Computer Science, VinUniversity, Hanoi, Vietnam

Robust Classification Adversarial Robustness Wasserstein Distributionally Robust Optimization Adversarial Attacks Robust Loss Functions

Jun 23, 2025

DPG loss functions for learning parameter-to-solution maps by neural networks

We develop, analyze, and experimentally explore residual-based loss functions for machine learning of parameter-to-solution maps in the context of parameter-dependent families of partial differential equations (PDEs). Our primary concern is on rigorous accuracy certification to enhance the prediction capability of the resulting deep neural network reduced models. This is achieved by the use of variationally correct loss functions. Through one specific example of an elliptic PDE, details for establishing the variational correctness of a loss function from an ultraweak Discontinuous Petrov Galerkin (DPG) discretization are worked out. Despite the focus on the example, the proposed concepts apply to a much wider scope of problems, namely problems for which stable DPG formulations are available. The issue of high-contrast diffusion fields and ensuing difficulties with degrading ellipticity are discussed. Both numerical results and theoretical arguments illustrate that for high-contrast diffusion parameters the proposed DPG loss functions deliver much more robust performance than simpler least-squares losses.

Pablo Cortés Castillo, Wolfgang Dahmen, Jay Gopalakrishnan

PDE Solving PDE Surrogate Modeling Robust Loss Functions PDE Operator Learning Physics-Informed ML

Feb 18, 2025

Fragility-aware Classification for Risk Characterization and Robust Learning

Classification models play a central role in data-driven decision-making applications such as medical diagnosis, recommendation systems, and risk assessment. Traditional performance metrics, such as accuracy and AUC, quantify classification errors and pairwise ranking performance but do not account for the magnitude of errors, i.e., the risk of confident misjudgments. This limitation is particularly consequential in safety-critical and cost-sensitive settings, where overconfident errors can lead to severe outcomes. To address this issue, we propose the Fragility Index (FI), a novel performance metric that evaluates classifiers from a risk-averse perspective by capturing the tail risk of confident misjudgments. We formulate FI within a robust satisficing (RS) framework to quantify classifier sensitivity under distributional uncertainty. Building on this, we develop a tractable surrogate-based training framework and establish provable bounds connecting training fragility to ranking-error FI. We further derive tractable reformulations for a broad class of loss functions, including cross-entropy, hinge-type, and Lipschitz losses, and identify specific conditions under which these reformulations are exact. We also introduce an FI-induced regularizer for deep neural networks. Empirical results on real-world medical diagnosis tasks demonstrate that FI complements existing metrics by revealing error tail risk not reflected in accuracy or AUC. FI-based models achieve competitive predictive performance while generally attaining lower FI. In the simulated human--AI diagnostic workflow, they also achieve lower average operational costs than empirical risk minimization.

Zheng Cui, Daniel Zhuoyu Long, Jin Qi +2

School of Management, Zhejiang University · Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong · Department of Industrial Engineering and Decision Analytics, Hong Kong University of Science and Technology +1

Robust Classification Robust Loss Functions