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Corrupted Learning Dynamics in Games

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper shows that optimistic learning dynamics retain near-logarithmic regret in games where players deviate from the prescribed algorithm, with regret degrading only as the square root of opponents' deviations plus a linear penalty…

desk verdict A well-motivated corruption model with a clean external-regret result, but the swap-regret theorem rests on a stability lemma whose proof doesn't close. read the letter →

arxiv 2412.07120 v2 pith:QQPJCXXQ submitted 2024-12-10 cs.GT cs.LGstat.ML

Taira Tsuchiya , Shinji Ito , Haipeng Luo This is my paper · ORCID
classification cs.GTcs.LGstat.ML MSC 91A2691A2091A1091A0568Q32
keywords regretminimizationcorruptedlearningoptimisticfollow-the-regularized-leaderswapNashequilibriumcorrelatedadaptiverateMarkovchainstability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether the fast equilibrium convergence of optimistic learning in games survives when players do not fully follow the prescribed algorithm. It answers yes in a newly defined corrupted regime: OFTRL with adaptive learning rates achieves regret bounds that degrade gracefully with the cumulative strategy deviations and utility corruptions of all players. In two-player zero-sum games, the external regret is bounded by roughly the minimum of a corruption-dependent logarithmic term and a path-length term, plus the player's own deviation; in multi-player general-sum games, swap regret is similarly adaptive. The dynamics do not need to know the corruption levels, and they match the best honest-regime rates when all corruptions are zero. The paper also gives matching lower bounds showing that the square-root dependence on opponents' deviations and the linear dependence on one's own deviation are in general unavoidable.

What carries the argument

The load-bearing object is OFTRL with expert-wise adaptive learning rates, using the negative Shannon entropy regularizer for the two-player case and the log-barrier regularizer for the general-sum swap-regret reduction. The crucial new step is a stability lemma, stated as Eq. (23), which bounds the sum over all experts of the local norms of successive log-barrier iterates by 1/2; this is achieved by capping each learning rate at 1/(256 n $\sqrt$(m)) and requiring T >= 3. That stability condition lets the proof invoke an RVU bound for self-concordant barriers and the Markov-chain tree lemma to control the stationary distributions that define the suggested strategies, thereby avoiding the much worse O($nm^{8}$ log T) honest-regime bound that a naive adaptive analysis would give.

What would settle it

Run the proposed dynamics in a small two-player game (for example m=3, n=2, T=$10^{4}$) while injecting strategy deviations for one player that are large in total but concentrated in bursts, and compute the left-hand side of Eq. (23) over time; if it ever exceeds 1/2 under the stated learning rates, or if the other player's empirical regret exceeds the claimed bound, the proof's key stability lemma is false.

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Extended reading notes

Core claim

The central claim is that corrupted learning can be made robust without sacrificing the fast rates of honest optimistic play. The authors prove that their adaptive OFTRL dynamics give external regret for the x-player in two-player zero-sum games of order min{$\sqrt$((log(m_x m_y)+C_x+C_y) log m_x), $\sqrt$((P_T^inf(~g)+log m_x) log m_x)} + C_x, and similarly for the y-player, where C_i aggregates strategy and utility corruption. For multi-player general-sum games, each player's swap regret is bounded by $nm^{{5/2}}$ log T plus a minimum of an adaptive corruption-dependent term and a sqrt{T log T}-type worst-case term, again plus a linear penalty for that player's own deviation. These are the first external and swap regret bounds for the corrupted regime, and the same dynamics match the best known honest-regime bounds when C_i=0. The argument also extends to corruption in observed expected utilities, and the lower bounds show that the main dependencies on the corruption levels cannot be improved in the worst case.

Load-bearing premise

The whole argument rests on the claimed stability of the Markov chain: the algorithm's consecutive suggested strategies must stay close enough in an adapted metric, and a corruption pattern that violates this would break the regret bound.

Editorial extensions

If this is right

  • If the central claim is correct, then in two-player zero-sum corrupted games the time-averaged joint play is an ~O((C_x+C_y)/T)-approximate Nash equilibrium.
  • In multi-player general-sum games, the time-averaged joint play is an ~O((log T + sqrt(\hat S log T) + max_k \hat C_k)/T)-approximate correlated equilibrium, where \hat S is the total strategy deviation.
  • The bounds create an incentive to follow the prescribed dynamics: one's own deviation enters linearly into regret, while opponents' deviations enter only through a square-root term.
  • The algorithm is agnostic to the corruption levels, so the same dynamics work without knowing \hat C_i or \tilde C_i in advance.
  • The matching lower bounds imply that no learning dynamics can remove the square-root dependence on opponents' deviations or the linear dependence on one's own deviation in the worst case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The utility-corruption setting is structurally similar to time-varying games, so the same adaptive-rates machinery may transfer to payoff or utility functions that drift, a connection the paper leaves mostly implicit.
  • The new RVU bound for log-barrier OFTRL with adaptive learning rates is likely to be useful outside games, for example in online convex optimization with predictable loss sequences, provided the stability condition can be verified in those settings.
  • A practical testable consequence of the stability lemma is that the left-hand side of Eq. (23) can be monitored online; if it ever approaches 1/2 under a given corruption pattern, that is a warning that the fast-rate proof is near its limit.
  • The lower bounds suggest a corruption budget interpretation: total corruption S sets a fundamental floor on equilibrium approximation error, roughly sqrt(S)/T, which could guide how much deviation a system should tolerate before switching to a purely adversarial algorithm.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper introduces a corrupted-regime model for learning in games in which players may deviate from the output of a prescribed OFTRL algorithm and may also observe corrupted utilities. The main claims are corruption-dependent external regret bounds for two-player zero-sum games (Theorem 6) and corruption-dependent swap regret bounds for multi-player general-sum games (Theorem 9), together with lower bounds (Theorem 10). The central technical device for the general-sum result is a stability lemma (Lemma 22) that asserts a uniform bound on the sum, over all experts, of local norms of successive log-barrier OFTRL iterates under a carefully capped adaptive learning rate. This stability bound is used to apply an RVU bound for self-concordant barriers and a Markov-chain stability lemma from prior work.

Significance. If the results are correct, the paper gives the first regret bounds that degrade gracefully with the amount of corruption in strategies and utilities, and it matches the best honest-regime swap regret bound while being agnostic to the corruption levels. The zero-sum external regret analysis and the lower bounds are interesting and appear largely self-contained, with no fitted parameters and no circularity. The general-sum swap regret contribution, however, is the paper's flagship extension, and its correctness rests entirely on the stability of the Markov chain under the adaptive learning rate. That stability proof does not close as written, so the swap-regret half of the paper is not yet supported.

major comments (2)
  1. [E.2, Lemma 22 (Eqs. (27)-(30))] The proof of the key stability condition does not close. After summing the linear lower bound (27) over a in A, the negative contribution is at least of the form (1/2)(3 sqrt(eta/m) + sqrt(16 eta / log T)) up to the simplex-sum bounds, because sum_a sqrt(hat x^{t-1}(a) + hat x^{t-2}(a)) <= sqrt(2m). With the cap eta <= 1/(256 n sqrt m), the second term is of order m^{-1/4}, while the positive quadratic term in (30) is 15/(128m). Thus for large m the negative term dominates the positive term, and the claimed inequality G^{(t+1)}(z) >= G^{(t+1)}(y^{(t)}) is not established. The sentence attributing the final inequality to T >= 3 and eta <= 1/256 is therefore unjustified. Consequently Lemma 20 cannot be invoked to conclude (23).
  2. [5, Theorem 9 and Appendix E.4] Because Lemma 22 is the only supplied mechanism for proving sum_a mu_a^{(t)} <= 1/2 (Lemma 23), the applications of the RVU bound in Lemma 17 and the Markov-chain tree lemma in Lemma 24 are unsupported. The swap regret bounds in Theorem 9, including the honest-regime rate O(n m^{5/2} log T) and the corrupted-regime terms, therefore collapse unless Lemma 22 is repaired. The scaling issue in the proof of Lemma 22 is not cosmetic: with the current cap eta_{i,max} = 1/(256 n sqrt m), the negative term in the stability proof is too large relative to the positive quadratic term, so a substantially different analysis, or a smaller learning-rate cap that would likely worsen the final bound, seems necessary.
minor comments (4)
  1. [Definition 3 vs. Appendix D.1] The definition of C_i is inconsistent: Definition 3 sets C_i = 2 hat C_i + 2 tilde C_i, while Appendix D.1 defines C_x = hat C_x + 2 tilde C_x. Please reconcile this notation.
  2. [Lemma 16 proof] The proof uses the identity bar phi(v) = bar phi(v) - bar phi(x(1)), which requires bar phi(x(1)) = 0. This holds in Algorithm 1 because the first iterate is the uniform distribution, but it is not stated as an assumption in Lemma 16; please add the needed hypothesis or adjust the argument.
  3. [Lemma 22 statement] Equation (23) writes the local norm with respect to F^{(t+1)}_a as (1/sqrt(eta^{(t+1)}_a)) times the norm with respect to phi, but the factor 1/sqrt(eta^{(t+1)}_a) appears outside the sum in an ambiguous way. It should be inside the sum over a.
  4. [Theorem 10 proof] The lower-bound proofs in parts (i) and (ii) assume that hat C_x, tilde C_x, etc. are even integers, but this integrality is not stated in the theorem; please add the standard rounding argument or state the assumption explicitly.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the corruption-adaptive bounds are derived from stated assumptions, and coauthored prior lemmas are used as parameter-free external components rather than as fitted inputs.

full rationale

I walked the derivation chain for Theorems 6 and 9. The external-regret bound (Theorem 6) starts from RVU bound Lemma 12, which is proved in Appendix B from the standard OFTRL regret decomposition (Lemma 11). The corruption levels enter only through triangle-inequality decompositions (Proposition 5 and the argument after Eq. (3)), not through any fitted parameter; the intermediate path-length estimate (5) is obtained by summing the two players' regret bounds and using nonnegativity of the sum of regrets (Lemma 19), which is an independent equilibrium property. For the swap-regret bound (Theorem 9), the paper proves its own stability lemma (Lemma 22), establishing the condition sum_a ||y_a^{t+1}-y_a^t|| <= 1/2 via convex-analysis lemmas (Lemmas 13, 15, 20, 21); it is not assumed as the target conclusion. Lemma 24 from Anagnostides et al. (2022b) is cited for the Markov-chain tree bound connecting expert movements to the stationary distribution; this is a published, parameter-free inequality whose statement does not include the paper's regret bound, so the overlap of author Luo with the present paper does not make the argument circular. The reduction from swap to external regret (Lemma 8) is stated and proved in Appendix E.1, and the final bounds in Theorems 6 and 9 are obtained by algebraically combining these bounds; no quantity is fit to the data being predicted. The skeptic's concern about the size of the negative term in Eq. (27) of Lemma 22, if valid, would be a proof-correctness gap rather than a circularity, because Lemma 22's stability claim is proved rather than assumed or imported as the paper's conclusion.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

There are no free parameters fitted to data; all constants are analytic. The modeling burden is carried by the cumulative corruption budgets and the prior Markov-chain stability lemma, neither of which is invented for this paper. The paper introduces no new entities such as particles, forces, or extra dimensions.

assumptions (6)
  • standard math OFTRL regret decomposition and RVU bounds for Shannon entropy and log-barrier regularizers (Lemmas 11, 12, 17).
    Used to bound external regret of the prescribed dynamics; Lemma 17 additionally assumes the stability condition ||x(t+1)-x(t)||_{x(t),phi} <= 1/2.
  • domain assumption Corrupted regime model: cumulative strategy deviations C_hat_i and utility corruptions C_tilde_i with C_i = 2 C_hat_i + 2 C_tilde_i (Definition 3).
    All regret bounds are defined with respect to these corruption budgets; real-world corruption must be measurable in these norms.
  • domain assumption Markov-chain tree lemma from Anagnostides et al. (2022b): if sum_a mu_a(t) <= 1/2 then ||xhat(t)-xhat(t-1)||_1 <= 8 sum_a mu_a(t) (Lemma 24).
    Borrowed as a black box to convert expert-level movement into stationary-distribution movement; not re-proved here.
  • standard math Self-concordant barrier properties, including Lemmas 13, 14, 15 from Nesterov and Nemirovskii.
    Used in Lemma 16/17 to derive the adaptive-rate RVU bound.
  • standard math Blum-Mansour reduction: swap regret equals sum of external regrets of action-expert minimizers (Lemma 8).
    Proved in Appendix E.1; foundational for the multi-player algorithm.
  • standard math Finite-time lower bounds for online linear optimization over the simplex (Lemma 27 from Orabona-Pal; Lemma 28 re-proved).
    Used to construct matching lower bounds in Theorem 10.

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Pith. "Pith review of Corrupted Learning Dynamics in Games." pith.science (2026). https://pith.science/paper/QQPJCXXQ

@misc{pith2026241207120,
  author       = {Pith},
  title        = {Pith review of: Corrupted Learning Dynamics in Games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QQPJCXXQ}},
  note         = {Machine review of arXiv:2412.07120}
}
abstract

Learning in games refers to scenarios where multiple players interact in a shared environment, each aiming to minimize their regret. An equilibrium can be computed at a fast rate of $O(1/T)$ when all players follow the optimistic follow-the-regularized-leader (OFTRL). However, this acceleration is limited to the honest regime, in which all players adhere to a prescribed algorithm -- a situation that may not be realistic in practice. To address this issue, we present corrupted learning dynamics that adaptively find an equilibrium at a rate that depends on the extent to which each player deviates from the strategy suggested by the prescribed algorithm. First, in two-player zero-sum corrupted games, we provide learning dynamics for which the external regret of $x$-player (and similarly for $y$-player) is roughly bounded by $O(\log (m_x m_y) + \sqrt{\hat{C}_y} + \hat{C}_x)$, where $m_x$ and $m_y$ denote the number of actions of $x$- and $y$-players, respectively, and $\hat{C}_x$ and $\hat{C}_y$ represent their cumulative deviations. We then extend our approach to multi-player general-sum corrupted games, providing learning dynamics for which the swap regret of player $i$ is bounded by $O(\log T + \sqrt{\sum_{k} \hat{C}_k \log T} + \hat{C}_i)$ ignoring dependence on the number of players and actions, where $\hat{C}_i$ is the cumulative deviation of player $i$ from the prescribed algorithm. Our learning dynamics are agnostic to the levels of corruption. A key technical contribution is a new analysis that ensures the stability of a Markov chain under a new adaptive learning rate, thereby allowing us to achieve the desired bound in the corrupted regime while matching the best existing bound in the honest regime. Notably, our framework can be extended to address not only corruption in strategies but also corruption in the observed expected utilities, and we provide several matching lower bounds.

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