mlsanityData to 7 Oct 2026

Certified Training: Small Boxes are All You Need

arXiv 2210.04871 ↗First posted 10 Oct 2022cs.LG cs.CRcited by 51

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Abstract (arXiv)

To obtain, deterministic guarantees of adversarial robustness, specialized training methods are used. We propose, SABR, a novel such certified training method, based on the key insight that propagating interval bounds for a small but carefully selected subset of the adversarial input region is sufficient to approximate the worst-case loss over the whole region while significantly reducing approximation errors. We show in an extensive empirical evaluation that SABR outperforms existing certified defenses in terms of both standard and certifiable accuracies across perturbation magnitudes and datasets, pointing to a new class of certified training methods promising to alleviate the robustness-accuracy trade-off.

Key facts. Cited by 51 papers in the map (data through 7 Oct 2026). Builds on 36 of its 56 references that are in the map. First posted on arXiv 10 Oct 2022.

Builds on

36 of 56 references.

References · 36

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  3. 2014Explaining and Harnessing Adversarial Examplescited by 7,287
  4. 2015Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shiftcited by 6,979
  5. 2017Towards Deep Learning Models Resistant to Adversarial Attackscited by 6,326
  6. 2013Intriguing properties of neural networkscited by 5,114
  7. 2016Towards Evaluating the Robustness of Neural Networkscited by 2,976
  8. 2020Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attackscited by 1,442
  9. 2019Certified Adversarial Robustness via Randomized Smoothingcited by 1,324
  10. 2019Adversarial Examples Are Not Bugs, They Are Featurescited by 950
  11. 2017Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networkscited by 739
  12. 2017Evasion Attacks against Machine Learning at Test Timecited by 686
24 more
  1. 2017Provable defenses against adversarial examples via the convex outer adversarial polytopecited by 564
  2. 2020On Adaptive Attacks to Adversarial Example Defensescited by 547
  3. 2018Certified Robustness to Adversarial Examples with Differential Privacycited by 462
  4. 2018Efficient Neural Network Robustness Certification with General Activation Functionscited by 439
  5. 2019Provably Robust Deep Learning via Adversarially Trained Smoothed Classifierscited by 328
  6. 2018On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Modelscited by 322
  7. 2021Beta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Complete and Incomplete Neural Network Robustness Verificationcited by 224
  8. 2018Sorting out Lipschitz function approximationcited by 211
  9. 2019Towards Stable and Efficient Training of Verifiably Robust Neural Networkscited by 204
  10. 2018Certified Adversarial Robustness with Additive Noisecited by 178
  11. 2018Semidefinite relaxations for certifying robustness to adversarial examplescited by 178
  12. 2020Randomized Smoothing of All Shapes and Sizescited by 146
  13. 2018Scaling provable adversarial defensescited by 141
  14. 2019Branch and Bound for Piecewise Linear Neural Network Verificationcited by 138
  15. 2022General Cutting Planes for Bound-Propagation-Based Neural Network Verificationcited by 102
  16. 2020Certified Defenses for Adversarial Patchescited by 100
  17. 2022Complete Verification via Multi-Neuron Relaxation Guided Branch-and-Boundcited by 92
  18. 2021PRIMA: General and Precise Neural Network Certification via Scalable Convex Hull Approximationscited by 77
  19. 2020Enabling certification of verification-agnostic networks via memory-efficient semidefinite programmingcited by 65
  20. 2020Curse of Dimensionality on Randomized Smoothing for Certifiable Robustnesscited by 56
  21. 2022Rethinking Lipschitz Neural Networks and Certified Robustness: A Boolean Function Perspectivecited by 48
  22. 2021Towards Certifying L-infinity Robustness using Neural Networks with L-inf-dist Neuronscited by 40
  23. 2021Boosting the Certified Robustness of L-infinity Distance Netscited by 24
  24. 2022IBP Regularization for Verified Adversarial Robustness via Branch-and-Boundcited by 21

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 1; 2023: 12; 2024: 20; 2025: 10; 2026: 8.

51 papers.

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  1. 2023First Three Years of the International Verification of Neural Networks Competition (VNN-COMP)cited by 58
  2. 2023Understanding Certified Training with Interval Bound Propagationcited by 26
  3. 2023Expressive Losses for Verified Robustness via Convex Combinationscited by 23
  4. 2024Certified Robustness to Data Poisoning in Gradient-Based Trainingcited by 17
  5. 2024CTBENCH: A Library and Benchmark for Certified Trainingcited by 13
  6. 2024Verification of Neural Control Barrier Functions with Symbolic Derivative Bounds Propagationcited by 12
  7. 2025Neural Network Verification is a Programming Language Challengecited by 12
  8. 2023TAPS: Connecting Certified and Adversarial Trainingcited by 11
  9. 2024Certified Robust Accuracy of Neural Networks Are Bounded due to Bayes Errorscited by 7
  10. 2025Adversarial Attacks Leverage Interference Between Features in Superpositioncited by 7
  11. 2024Set-Based Training for Neural Network Verificationcited by 6
  12. 2024Certification for Differentially Private Prediction in Gradient-Based Trainingcited by 6
38 more
  1. 2023PECAN: A Deterministic Certified Defense Against Backdoor Attackscited by 5
  2. 2023DeepBern-Nets: Taming the Complexity of Certifying Neural Networks using Bernstein Polynomial Activations and Precise Bound Propagationcited by 5
  3. 2024Towards Certification of Uncertainty Calibration under Adversarial Attackscited by 5
  4. 2025Enhancing Certified Robustness via Block Reflector Orthogonal Layers and Logit Annealing Losscited by 5
  5. 2024Gaussian Loss Smoothing Enables Certified Training with Tight Convex Relaxationscited by 4
  6. 2022Quantization-aware Interval Bound Propagation for Training Certifiably Robust Quantized Neural Networkscited by 3
  7. 2023Efficient Certified Training and Robustness Verification of Neural ODEscited by 3
  8. 2023Verifiable Learning for Robust Tree Ensemblescited by 3
  9. 2023Raze to the Ground: Query-Efficient Adversarial HTML Attacks on Machine-Learning Phishing Webpage Detectorscited by 3
  10. 2024Data-Driven Lipschitz Continuity: A Cost-Effective Approach to Improve Adversarial Robustnesscited by 3
  11. 2024On Using Certified Training towards Empirical Robustnesscited by 3
  12. 2024Certified Training with Branch-and-Bound for Lyapunov-stable Neural Controlcited by 3
  13. 2026Exact Certification of Data-Poisoning Attacks Using Mixed-Integer Programmingcited by 3
  14. 2023Towards Certified Probabilistic Robustness with High Accuracycited by 2
  15. 2024How Does Bayes Error Limit Probabilistic Robust Accuracycited by 2
  16. 2024Expressiveness of Multi-Neuron Convex Relaxations in Neural Network Certificationcited by 2
  17. 2025Defending against Adversarial Malware Attacks on ML-based Android Malware Detection Systemscited by 2
  18. 2025BEAVER: An Efficient Deterministic LLM Verifiercited by 2
  19. 2023How robust accuracy suffers from certified training with convex relaxationscited by 1
  20. 2024Is Adversarial Training with Compressed Datasets Effective?cited by 1
  21. 2024Towards Precise Observations of Neural Model Robustness in Classificationcited by 1
  22. 2024Cross-Input Certified Training for Universal Perturbationscited by 1
  23. 2024Towards Generalized Certified Robustness with Multi-Norm Trainingcited by 1
  24. 2024Average Certified Radius is a Poor Metric for Randomized Smoothingcited by 1
  25. 2024Integrating uncertainty quantification into randomized smoothing based robustness guaranteescited by 1
  26. 2025A Scalable Approach to Probabilistic Neuro-Symbolic Robustness Verificationcited by 1
  27. 2025Bridging Efficiency and Safety: Formal Verification of Neural Networks with Early Exitscited by 1
  28. 2026Learning Better Certified Models from Empirically-Robust Teacherscited by 1
  29. 2023Adaptive Certified Training: Towards Better Accuracy-Robustness Tradeoffscited by 0
  30. 2024Patch Synthesis for Property Repair of Deep Neural Networkscited by 0
  31. 2025Support is All You Need for Certified VAE Trainingcited by 0
  32. 2025Compression Aware Certified Trainingcited by 0
  33. 2025Dual Randomized Smoothing: Beyond Global Noise Variancecited by 0
  34. 2026On the Extreme Variance of Certified Local Robustness Across Model Seedscited by 0
  35. 2026Rethinking Evaluation Paradigms in IBP-based Certified Trainingcited by 0
  36. 2026Hybrid Robustness Verification for Spatio-Temporal Neural Networkscited by 0
  37. 2026Improving Certified Robustness via Adversarial Distillationcited by 0
  38. 2026Certified Training for Convolutional Perturbationscited by 0

50 most cited of 51 All 51 (JSON)