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arxiv: 2605.19452 · v1 · pith:VIQ26PSEnew · submitted 2026-05-19 · 💻 cs.DC · cs.AI

Resilient Byzantine Agreement with Predictions

Julien Dallot , Darya Melnyk , Tijana Milentijevic , Stefan Schmid , Patrik Welters This is my paper

Pith reviewed 2026-05-20 02:36 UTC · model grok-4.3

classification 💻 cs.DC cs.AI
keywords Byzantine agreementpredictor accuracyresilience trade-offsfault toleranceconsistency and robustnessdistributed consensussmoothness
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The pith

Byzantine agreement protocols tolerate αn faulty nodes under accurate predictions but only (1-α)n/2 under arbitrary errors, with tight bounds.

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

This paper characterizes how access to a predictor that flags suspicious nodes shapes the fault tolerance of Byzantine agreement. For n nodes and accuracy parameter α, algorithms exist that handle up to αn faults when predictions match reality and up to (1-α)n/2 -1 faults when predictions are completely wrong in the non-authenticated case, with authentication raising the second bound to (1-α)n -1. These numbers are optimal because one extra fault makes agreement impossible. Resilience falls linearly with each wrong prediction up to a constant fraction of n. Readers care because the work shows how to incorporate imperfect predictors into consensus without requiring them to be perfect.

Core claim

For n nodes and α in [0,1], algorithms tolerate α · n faulty nodes when the predictor is correct and (1-α)/2 · n -1 when arbitrarily wrong in the non-authenticated setting, improving to (1-α) · n -1 with authentication. These trade-offs are exactly tight as one additional faulty node renders the problem impossible. Resilience decreases linearly in the number of wrong predictions within a constant fraction of n, with each wrong prediction losing one unit of resilience non-authenticated and two losing one unit authenticated.

What carries the argument

The scalar α that partitions the predictor's output into a consistency regime of correct suspicions and a robustness regime of arbitrary errors, directly yielding the linear resilience expressions.

If this is right

  • Accurate predictions allow tolerance of a full α fraction of faults.
  • Arbitrary prediction errors reduce tolerance to roughly half the remaining nodes without authentication.
  • Each additional wrong prediction reduces tolerable faults by exactly one without authentication.
  • Authentication halves the impact of each wrong prediction on resilience.
  • The derived bounds cannot be exceeded by any algorithm.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same predictor-based trade-off could be applied to related tasks such as reliable broadcast.
  • If predictor quality improves over successive rounds, overall system resilience could increase dynamically.
  • Practical predictors from anomaly detection might achieve usable α values in real deployments.
  • The linear smoothness result suggests testing whether similar degradation rates appear in other consensus variants.

Load-bearing premise

The predictor's behavior is captured entirely by the single fraction α with no further assumptions about which nodes are misclassified or how errors relate across rounds.

What would settle it

An algorithm that reaches agreement despite one extra faulty node beyond the stated bound for a given α, or an explicit set of faulty nodes and predictions that prevents agreement at that threshold, would disprove the claimed tightness.

read the original abstract

This paper studies the Byzantine Agreement problem where the nodes have access to a predictor that flags nodes for suspicion of faulty (Byzantine) behavior. We focus on algorithmic resilience -- the maximum number of faulty nodes an algorithm can tolerate -- and present algorithms and impossibility results whose resilience depend on the accuracy of the predictor. As our first main result, we bring a complete characterization of the consistency--robustness trade-offs in both the non-authenticated and authenticated settings: for $n$ nodes and a parameter $\alpha \in [0, 1]$, we present algorithms that tolerate up to $\alpha \cdot n$ faulty nodes when the predictor is correct (consistency), and up to $\frac{1-\alpha}{2} \cdot n - 1$ faulty nodes when the predictor is arbitrarily wrong (robustness); in the authenticated setting the robustness bound improves to $(1-\alpha) \cdot n - 1$. These trade-offs are exactly tight as we show that one additional faulty node renders the problem impossible. Our second main result characterizes smoothness: the rate at which resilience degrades as the predictor becomes less accurate. We show that resilience linearly decreases in the number of wrong predictions as long as that number stays within a constant fraction of $n$. Concretely, in the non-authenticated setting each additional wrong prediction loses one unit of resilience, whereas in the authenticated setting the decline is halved since two wrong predictions are needed to lose one unit of resilience.

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

2 major / 2 minor

Summary. The paper studies Byzantine Agreement (BA) augmented by a predictor that flags nodes suspected of faulty behavior. It provides algorithms and matching impossibility results that characterize consistency-robustness trade-offs for a parameter α ∈ [0,1]: algorithms tolerate up to α·n faulty nodes when the predictor is correct, and up to (1-α)/2 ·n -1 faulty nodes (non-authenticated) or (1-α)·n -1 (authenticated) when the predictor is arbitrarily wrong. These bounds are shown to be tight. The paper also characterizes smoothness: resilience degrades linearly with the number of wrong predictions (one unit per wrong prediction in the non-authenticated setting, two in the authenticated setting) as long as the number of errors remains within a constant fraction of n.

Significance. If the central claims hold, the work is significant for distributed computing because it introduces a tunable predictor-based resilience model for BA and derives exact trade-offs together with a smoothness result. The explicit distinction between consistency and robustness regimes, the improvement under authentication, and the linear degradation characterization are all strengths that could inform practical systems with suspicion or ML-based predictors. The tightness via impossibility arguments is a positive feature.

major comments (2)
  1. [§3] §3 (Predictor Model): The predictor is defined solely via the scalar α that partitions correct suspicions from completely arbitrary ones, with no further structure on which nodes are misclassified or whether errors correlate across rounds or rounds. This modeling choice directly determines the exact robustness expressions ((1-α)/2 n−1 and (1-α)n−1) and the linear smoothness claim; if even limited correlation structure were permitted while still satisfying the α bound, an adversary might force a strictly tighter robustness threshold than the stated linear expressions.
  2. [§5] §5 (Impossibility Arguments): The matching impossibility results claim that one additional faulty node beyond the stated robustness bound renders BA impossible. The reduction appears to treat the predictor output as either perfectly aligned with the fault set or else fully adversarial; additional detail is needed on how the predictor's output is incorporated into the indistinguishability argument, especially for the authenticated-channel case where the robustness bound is (1-α)n−1.
minor comments (2)
  1. [Abstract] The abstract states the robustness bounds with “-1” but the smoothness section could more explicitly state the constant-fraction regime in which the linear loss holds.
  2. [§4] Notation for the predictor output (e.g., how the set of suspected nodes is denoted) should be introduced once and used consistently in the algorithm pseudocode and proofs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address each major comment below, providing clarifications on the modeling choices and agreeing to expand the proof details where helpful.

read point-by-point responses

  1. Referee: [§3] §3 (Predictor Model): The predictor is defined solely via the scalar α that partitions correct suspicions from completely arbitrary ones, with no further structure on which nodes are misclassified or whether errors correlate across rounds or rounds. This modeling choice directly determines the exact robustness expressions ((1-α)/2 n−1 and (1-α)n−1) and the linear smoothness claim; if even limited correlation structure were permitted while still satisfying the α bound, an adversary might force a strictly tighter robustness threshold than the stated linear expressions.

    Authors: Our model is intentionally worst-case: the robustness regime allows the predictor output to be chosen completely arbitrarily by the adversary. This includes any pattern of misclassifications or correlations across nodes and rounds, as long as the overall α-bound on the trade-off is respected. The matching impossibility results already establish tightness under fully adversarial error structures. Therefore, the stated linear expressions and smoothness claims hold even when arbitrary correlations are permitted within the model. revision: no

  2. Referee: [§5] §5 (Impossibility Arguments): The matching impossibility results claim that one additional faulty node beyond the stated robustness bound renders BA impossible. The reduction appears to treat the predictor output as either perfectly aligned with the fault set or else fully adversarial; additional detail is needed on how the predictor's output is incorporated into the indistinguishability argument, especially for the authenticated-channel case where the robustness bound is (1-α)n−1.

    Authors: We agree that additional exposition would improve clarity. In the indistinguishability argument, we construct pairs of executions that differ only in the identity of one extra faulty node. In one execution the predictor output is consistent with the actual fault set; in the other it is set adversarially. For the authenticated setting, signatures prevent forgery but the arbitrary predictor still allows the adversary to produce indistinguishable views for correct nodes up to the (1-α)n−1 bound. We will revise Section 5 to spell out these steps explicitly for both the non-authenticated and authenticated cases. revision: yes

Circularity Check

0 steps flagged

Resilience bounds derived from explicit algorithms and impossibility proofs in predictor-augmented BA model

full rationale

The paper's central results consist of new algorithm constructions achieving the stated αn consistency and (1-α)/2 n−1 (or (1-α)n−1 authenticated) robustness bounds, together with matching impossibility arguments showing one additional fault renders the problem impossible. These derivations operate directly in the standard authenticated/non-authenticated BA model with the predictor output modeled by the scalar α; the smoothness claim (linear degradation per misprediction) follows from the same counting analysis. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the model assumptions are stated up front and the bounds are obtained via independent protocol design and adversary arguments that remain falsifiable outside any internal fitting.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the classic Byzantine fault model plus a new predictor whose accuracy is captured by a single parameter α; no additional free parameters or invented entities are introduced.

free parameters (1)
  • α
    Scalar in [0,1] that parameterizes the fraction of correct predictions versus arbitrary ones; chosen to express the consistency-robustness frontier.
axioms (1)
  • domain assumption Standard synchronous or asynchronous message-passing model with n nodes and up to f Byzantine faults, plus authenticated or non-authenticated channels.
    Invoked as the base setting in which the predictor is added.

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Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 1 internal anchor

  1. [1]

    , title =

    Boyar, Joan and Ellen, Faith and Larsen, Kim S. , title =. Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC) , pages =. 2025 , doi =

    work page 2025

  2. [2]

    2025 , url =

    Ben-David, Naama and Dzulfikar, Muhammad Ayaz and Ellen, Faith and Gilbert, Seth , title =. 2025 , url =. doi:10.1145/3732772.3733518 , booktitle =

    work page doi:10.1145/3732772.3733518 2025

  3. [3]

    1996 , volume =

    Chandra, Tushar Deepak and Toueg, Sam , title =. 1996 , volume =. doi:10.1145/226643.226647 , journal =

    work page doi:10.1145/226643.226647 1996

  4. [4]

    Proceedings of the 35th International Conference on Machine Learning,

    Thodoris Lykouris and Sergei Vassilvitskii , title =. Proceedings of the 35th International Conference on Machine Learning,

    work page

  5. [5]

    2018 , url =

    Proceedings of the 35th International Conference on Machine Learning,. 2018 , url =

    work page 2018

  6. [6]

    Alexander Lindermayr and Nicole Megow , note =

    work page

  7. [7]

    and Shostak, R

    Pease, M. and Shostak, R. and Lamport, L. , title =. 1980 , volume =. doi:10.1145/322186.322188 , journal =

    work page doi:10.1145/322186.322188 1980

  8. [8]

    1982 , volume =

    Lamport, Leslie and Shostak, Robert and Pease, Marshall , title =. 1982 , volume =. doi:10.1145/357172.357176 , journal =

    work page doi:10.1145/357172.357176 1982

  9. [9]

    1982 , doi =

    A lower bound for the time to assure interactive consistency , journal =. 1982 , doi =

    work page 1982

  10. [10]

    and Strong, H

    Dolev, D. and Strong, H. R. , title =. SIAM Journal on Computing , volume =. 1983 , doi =

    work page 1983

  11. [11]

    and Micali, S

    Goldreich, O. and Micali, S. and Wigderson, A. , title =. Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing , pages =. 1987 , doi =

    work page 1987

  12. [12]

    and Lynch, Nancy A

    Fischer, Michael J. and Lynch, Nancy A. and Paterson, Michael S. , title =. 1985 , volume =. doi:10.1145/3149.214121 , journal =

    work page doi:10.1145/3149.214121 1985

  13. [13]

    and Garay, J.A

    Berman, P. and Garay, J.A. and Perry, K.J. , booktitle=. Towards optimal distributed consensus , year=

    work page

  14. [14]

    Proceedings of the 10th IEEE Workshop on Computer Security Foundations , pages =

    Malkhi, Dahlia and Reiter, Michael , title =. Proceedings of the 10th IEEE Workshop on Computer Security Foundations , pages =. 1997 , isbn =

    work page 1997

  15. [15]

    and Melliar-Smith, P

    Kihlstrom, Kim Potter and Moser, Louise E. and Melliar-Smith, P. M. , journal=. Byzantine Fault Detectors for Solving Consensus , year=

    work page

  16. [16]

    2003 , note =

    Consensus in Byzantine asynchronous systems , journal =. 2003 , note =. doi:https://doi.org/10.1016/S1570-8667(03)00025-X , author =

    work page doi:10.1016/s1570-8667(03)00025-x 2003

  17. [17]

    Proceedings of the Twenty-Seventh ACM Symposium on Principles of Distributed Computing , pages =

    Druschel, Peter , title =. Proceedings of the Twenty-Seventh ACM Symposium on Principles of Distributed Computing , pages =. 2008 , doi =

    work page 2008

  18. [18]

    Proceedings of the 21st

    Andreas Haeberlen and Petr Kouznetsov and Peter Druschel , title =. Proceedings of the 21st. 2007 , doi =

    work page 2007

  19. [19]

    and Chase, Jeffrey S

    Yumerefendi, Aydan R. and Chase, Jeffrey S. , title =. Proceedings of the First Conference on Hot Topics in System Dependability , pages =. 2005 , publisher =

    work page 2005

  20. [20]

    Proceedings of the Second Conference on Hot Topics in System Dependability , pages =

    Haeberlen, Andreas and Kouznetsov, Petr and Druschel, Peter , title =. Proceedings of the Second Conference on Hot Topics in System Dependability , pages =. 2006 , publisher =

    work page 2006

  21. [21]

    Advances in neural information processing systems , volume=

    Augmenting Online Algorithms with varepsilon -Accurate Predictions , author=. Advances in neural information processing systems , volume=

    work page

  22. [22]

    Advances in Neural Information Processing Systems , volume=

    Learning-augmented priority queues , author=. Advances in Neural Information Processing Systems , volume=

    work page

  23. [23]

    Advances in Neural Information Processing Systems , volume=

    Learning-augmented algorithms with explicit predictors , author=. Advances in Neural Information Processing Systems , volume=

    work page

  24. [24]

    Advances in Neural Information Processing Systems , volume=

    Improving online algorithms via ML predictions , author=. Advances in Neural Information Processing Systems , volume=

    work page

  25. [25]

    Advances in Neural Information Processing Systems , volume=

    Learning-augmented algorithms for the bahncard problem , author=. Advances in Neural Information Processing Systems , volume=

    work page

  26. [26]

    Forty-second International Conference on Machine Learning , year=

    Near-Optimal Consistency-Robustness Trade-Offs for Learning-Augmented Online Knapsack Problems , author=. Forty-second International Conference on Machine Learning , year=

    work page

  27. [27]

    Blockchain Consensus Protocols in the Wild

    Blockchain consensus protocols in the wild , author=. arXiv preprint arXiv:1707.01873 , year=

    work page internal anchor Pith review Pith/arXiv arXiv