mlsanityData to 14 Oct 2026

Proof-of-Learning: Definitions and Practice

arXiv 2103.05633 ↗First posted 9 Mar 2021cs.LG cs.AI cs.CRcited by 84

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Abstract (arXiv)

Training machine learning (ML) models typically involves expensive iterative optimization. Once the model's final parameters are released, there is currently no mechanism for the entity which trained the model to prove that these parameters were indeed the result of this optimization procedure. Such a mechanism would support security of ML applications in several ways. For instance, it would simplify ownership resolution when multiple parties contest ownership of a specific model. It would also facilitate the distributed training across untrusted workers where Byzantine workers might otherwise mount a denial-of-service by returning incorrect model updates. In this paper, we remediate this problem by introducing the concept of proof-of-learning in ML. Inspired by research on both proof-of-work and verified computations, we observe how a seminal training algorithm, stochastic gradient descent, accumulates secret information due to its stochasticity. This produces a natural construction for a proof-of-learning which demonstrates that a party has expended the compute require to obtain a set of model parameters correctly. In particular, our analyses and experiments show that an adversary seeking to illegitimately manufacture a proof-of-learning needs to perform *at least* as much work than is needed for gradient descent itself. We also instantiate a concrete proof-of-learning mechanism in both of the scenarios described above. In model ownership resolution, it protects the intellectual property of models released publicly. In distributed training, it preserves availability of the training procedure. Our empirical evaluation validates that our proof-of-learning mechanism is robust to variance induced by the hardware (ML accelerators) and software stacks.

Key facts. Cited by 84 papers in the map (data through 14 Oct 2026). Builds on 29 of its 100 references that are in the map. First posted on arXiv 9 Mar 2021.

Builds on

29 of 100 references.

References · 29

  1. 2015Deep Residual Learning for Image Recognitioncited by 52,096
  2. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  3. 2013Intriguing properties of neural networkscited by 5,155
  4. 2015Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classificationcited by 3,611
  5. 2018The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networkscited by 2,262
  6. 2014How transferable are features in deep neural networks?cited by 1,225
  7. 2016Practical Black-Box Attacks against Machine Learningcited by 903
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  9. 2017Evasion Attacks against Machine Learning at Test Timecited by 689
  10. 2016Stealing Machine Learning Models via Prediction APIscited by 619
  11. 2017In-Datacenter Performance Analysis of a Tensor Processing Unitcited by 618
  12. 2017Self-Normalizing Neural Networkscited by 598
17 more
  1. 2018Fine-Pruning: Defending Against Backdooring Attacks on Deep Neural Networkscited by 580
  2. 2017Wild Patterns: Ten Years After the Rise of Adversarial Machine Learningcited by 414
  3. 2018Turning Your Weakness Into a Strength: Watermarking Deep Neural Networks by Backdooringcited by 357
  4. 2018Knockoff Nets: Stealing Functionality of Black-Box Modelscited by 292
  5. 2014cuDNN: Efficient Primitives for Deep Learningcited by 245
  6. 2019High Accuracy and High Fidelity Extraction of Neural Networkscited by 214
  7. 2020Entangled Watermarks as a Defense against Model Extractioncited by 149
  8. 2018How to Start Training: The Effect of Initialization and Architecturecited by 109
  9. 2016Towards the Science of Security and Privacy in Machine Learningcited by 89
  10. 2018NVIDIA Tensor Core Programmability, Performance & Precisioncited by 79
  11. 2018Copycat CNN: Stealing Knowledge by Persuading Confession with Random Non-Labeled Datacited by 70
  12. 2019Robust Watermarking of Neural Network with Exponential Weightingcited by 51
  13. 2017SafetyNets: Verifiable Execution of Deep Neural Networks on an Untrusted Cloudcited by 50
  14. 2019A framework for the extraction of Deep Neural Networks by leveraging public datacited by 22
  15. 2018Defending Against Machine Learning Model Stealing Attacks Using Deceptive Perturbationscited by 11
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 5; 2022: 12; 2023: 19; 2024: 16; 2025: 15; 2026: 17.

84 papers.

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  1. 2022A Survey of Machine Unlearningcited by 312
  2. 2021Unrolling SGD: Understanding Factors Influencing Machine Unlearningcited by 279
  3. 2022The Alignment Problem from a Deep Learning Perspectivecited by 258
  4. 2021On the Necessity of Auditable Algorithmic Definitions for Machine Unlearningcited by 171
  5. 2023Machine Unlearning: Solutions and Challengescited by 93
  6. 2022Trustworthy Graph Neural Networks: Aspects, Methods and Trendscited by 89
  7. 2021Handcrafted Backdoors in Deep Neural Networkscited by 63
  8. 2022Fingerprinting Deep Neural Networks Globally via Universal Adversarial Perturbationscited by 61
  9. 2022Dataset Inference for Self-Supervised Modelscited by 36
  10. 2023HuRef: HUman-REadable Fingerprint for Large Language Modelscited by 34
  11. 2022On the Difficulty of Defending Self-Supervised Learning against Model Extractioncited by 22
  12. 2023Tools for Verifying Neural Models' Training Datacited by 22
38 more
  1. 2022Increasing the Cost of Model Extraction with Calibrated Proof of Workcited by 21
  2. 2023What does it take to catch a Chinchilla? Verifying Rules on Large-Scale Neural Network Training via Compute Monitoringcited by 21
  3. 2021"Adversarial Examples" for Proof-of-Learningcited by 20
  4. 2023zkDL: Efficient Zero-Knowledge Proofs of Deep Learning Trainingcited by 20
  5. 2023False Claims against Model Ownership Resolutioncited by 17
  6. 2023Reconstructing Individual Data Points in Federated Learning Hardened with Differential Privacy and Secure Aggregationcited by 16
  7. 2024Verifiable evaluations of machine learning models using zkSNARKscited by 16
  8. 2022Proof of Unlearning: Definitions and Instantiationcited by 15
  9. 2023IronForge: An Open, Secure, Fair, Decentralized Federated Learningcited by 15
  10. 2022An Introduction to Machine Unlearningcited by 13
  11. 2022Proof-of-Learning is Currently More Broken Than You Thinkcited by 12
  12. 2023Gradients Look Alike: Sensitivity is Often Overestimated in DP-SGDcited by 11
  13. 2025Towards Lifecycle Unlearning Commitment Management: Measuring Sample-level Unlearning Completenesscited by 11
  14. 2021SoK: Machine Learning Governancecited by 10
  15. 2025FIT-Print: Towards False-claim-resistant Model Ownership Verification via Targeted Fingerprintcited by 10
  16. 2022ImpNet: Imperceptible and blackbox-undetectable backdoors in compiled neural networkscited by 9
  17. 2023FedZKP: Federated Model Ownership Verification with Zero-knowledge Proofcited by 9
  18. 2025Verde: Verification via Refereed Delegation for Machine Learning Programscited by 9
  19. 2022On the Robustness of Dataset Inferencecited by 8
  20. 2023Delegating Data Collection in Decentralized Machine Learningcited by 8
  21. 2025Towards Reliable Forgetting: A Survey on Machine Unlearning Verificationcited by 8
  22. 2024Optimistic Verifiable Training by Controlling Hardware Nondeterminismcited by 7
  23. 2024Neural Interactive Proofscited by 7
  24. 2024A Survey on Recommendation Unlearning: Fundamentals, Taxonomy, Evaluation, and Open Questionscited by 7
  25. 2023Machine Learning needs Better Randomness Standards: Randomised Smoothing and PRNG-based attackscited by 6
  26. 2024Towards Lifecycle Unlearning Commitment Management: Measuring Sample-level Approximate Unlearning Completenesscited by 6
  27. 2024SoK: Blockchain-Based Decentralized AI (DeAI)cited by 6
  28. 2023Attesting Distributional Properties of Training Data for Machine Learningcited by 5
  29. 2023ModelGiF: Gradient Fields for Model Functional Distancecited by 5
  30. 2023Can Membership Inferencing be Refuted?cited by 4
  31. 2023On Knowledge Editing in Federated Learning: Perspectives, Challenges, and Future Directionscited by 4
  32. 2023Beyond Labeling Oracles: What does it mean to steal ML models?cited by 4
  33. 2024Is Your AI Truly Yours? Leveraging Blockchain for Copyrights, Provenance, and Lineagecited by 4
  34. 2024Proof-of-Learning with Incentive Securitycited by 4
  35. 2024vTune: Verifiable Fine-Tuning for LLMs Through Backdooringcited by 4
  36. 2024AIArena: A Blockchain-Based Decentralized AI Training Platformcited by 4
  37. 2025Leveraging Per-Instance Privacy for Machine Unlearningcited by 4
  38. 2023Towards Open Federated Learning Platforms: Survey and Vision from Technical and Legal Perspectivescited by 3

50 most cited of 84 All 84 (JSON)