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arxiv: 2606.21528 · v1 · pith:WD2QCFMWnew · submitted 2026-06-19 · 🧮 math.OC · cs.GT· cs.LG

Accelerated and Stable Convergence with Anchored Optimistic Method

Motahareh Sohrabi , Jianxin You , Simon Lacoste-Julien , Eduard Gorbunov , Gauthier Gidel This is my paper

Pith reviewed 2026-06-26 13:34 UTC · model grok-4.3

classification 🧮 math.OC cs.GTcs.LG
keywords variational inequalitieslast-iterate convergenceoptimistic gradient methodsmin-max optimizationstochastic optimizationanchoring techniquesHalpern iteration
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The pith

Anchored optimistic methods achieve optimal accelerated last-iterate rates for monotone variational inequalities.

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

The paper develops Generalized Optimistic Methods with Anchoring (GOMA) for solving monotone variational inequalities. These methods combine optimistic updates with an anchoring term to reach the optimal last-iterate convergence rate of O(1/k²) on the squared gradient norm in the deterministic case. In the stochastic setting with unbounded variance, a single-call version attains O(1/√k) without variance reduction or batch growth. This provides the first such last-iterate guarantees in the unconstrained stochastic setting and improves practicality for online min-max problems.

Core claim

GOMA achieves the optimal accelerated last-iterate rate O(1/k²) on the squared gradient norm for monotone Lipschitz operators in the deterministic setting. A simplified single-call variant achieves a last-iterate convergence rate of O(1/√k) on the squared gradient norm in the stochastic setting with unbounded variance, marking the first such guarantee for stochastic monotone Lipschitz variational inequalities in the unconstrained setting without variance reduction or growing batches.

What carries the argument

Generalized Optimistic Methods with Anchoring (GOMA) that combine two-time-scale optimistic updates with an anchoring term inspired by Halpern iteration.

If this is right

  • These methods use only one or two gradient evaluations per iteration, enabling use in stochastic and online settings.
  • The convergence is measured on the last iterate rather than averages, which is more relevant for practical applications.
  • The rates hold without additional assumptions like bounded variance or variance reduction techniques.
  • The anchoring improves stability and acceleration compared to standard optimistic methods.

Where Pith is reading between the lines

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

  • If the anchoring term can be generalized, it might apply to non-monotone operators as well.
  • Practical implementations could test whether these rates translate to faster training in adversarial machine learning tasks.
  • The single-call variant might reduce computational cost in high-dimensional problems.

Load-bearing premise

The variational inequality operator is assumed to be monotone and Lipschitz continuous.

What would settle it

A numerical experiment on a simple bilinear saddle-point problem showing that the squared gradient norm fails to decrease at the claimed rate under GOMA would falsify the result.

Figures

Figures reproduced from arXiv: 2606.21528 by Eduard Gorbunov, Gauthier Gidel, Jianxin You, Motahareh Sohrabi, Simon Lacoste-Julien.

Figure 1
Figure 1. Figure 1: Quadratic experiment in the deterministic setting §6.1. Only GOMA and FEG converge. GOMA converges without oscillations, unlike FEG’s dynamics. smaller residuals than FEG by an approximately constant factor, with near-parallel log–log curves indicating the same asymptotic rate but a better constant. 6.2. Stochastic Games We consider two stochastic settings. The first is a low￾dimensional toy problem (d = 2… view at source ↗
Figure 2
Figure 2. Figure 2: Stochastic experiments §6.2. Norm of the gradient operator vs. the number of gradient calls. Left: Stochastic bilinear example with additive noise (σ = 0.5), §6.2.1. Right: Finite-sum saddle-point problem with multiplicative noise, §6.2.2. Stochastic GOMA consistently outperforms the baselines in both tasks, despite using a milder hyperparameter search and simpler setup. the squared operator norm under add… view at source ↗
Figure 3
Figure 3. Figure 3: Monotone QP Lagrangian (n = 200, L = 1). Anchoring-based methods (EAG-C/V, FEG, Anchored Popov, GOMA) achieve the O(1/k2 ) rate on ∥F(zk)∥ 2 , while EG, DSEG, and Nesterov exhibit a substantially slower decay and largely overlap near the top of the plot. GOMA (red) attains the lowest final ∥F(zk)∥ 2 . 36 [PITH_FULL_IMAGE:figures/full_fig_p036_3.png] view at source ↗
read the original abstract

We study first-order methods for solving monotone variational inequalities arising in min-max optimization. Classical approaches such as the extragradient method rely on two gradient queries per iteration, which limits their analysis and applicability in the online and stochastic settings. We propose a family of Generalized Optimistic Methods with Anchoring (GOMA), which combine two-time-scale optimistic updates with an anchoring term inspired by Halpern iteration. In the deterministic setting, GOMA achieves the optimal accelerated last-iterate rate $O(1/k^2)$ on the squared gradient norm for monotone Lipschitz operators. In the stochastic setting with unbounded variance, a simplified single-call variant of GOMA achieves a last-iterate convergence rate of $O(1/\sqrt{k})$ on the squared gradient norm. To the best of our knowledge, this is the first such guarantee for stochastic monotone Lipschitz variational inequalities in the unconstrained setting without variance reduction or growing batches.

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

0 major / 3 minor

Summary. The manuscript introduces the Generalized Optimistic Methods with Anchoring (GOMA), a family of first-order methods that combine two-time-scale optimistic updates with an anchoring term for solving monotone variational inequalities. In the deterministic setting the method is claimed to attain the optimal accelerated last-iterate rate O(1/k²) on the squared gradient norm for monotone Lipschitz operators; in the stochastic setting with unbounded variance a single-call variant is claimed to attain last-iterate convergence O(1/√k) on the same quantity, asserted to be the first such guarantee in the unconstrained setting without variance reduction or growing batch sizes.

Significance. If the stated rates and assumptions are rigorously established, the work supplies the first last-iterate guarantees for stochastic monotone Lipschitz variational inequalities that match known deterministic lower bounds while remaining stable under unbounded variance. The anchoring construction and the single-call stochastic variant constitute a technically interesting synthesis of optimistic and Halpern-style ideas that could influence subsequent analyses of min-max and game-theoretic problems.

minor comments (3)
  1. [§3] §3, Algorithm 1: the two-time-scale parameters α_k and β_k are introduced without an explicit statement of the precise relation required between them for the O(1/k²) proof; a short remark clarifying the admissible range would improve readability.
  2. [Theorem 4.2] Theorem 4.2: the statement of the stochastic rate assumes the operator is monotone and L-Lipschitz but does not restate the precise moment condition on the noise that replaces bounded variance; adding one sentence would make the theorem self-contained.
  3. [Figure 2] Figure 2: the y-axis label “squared gradient norm” is plotted on a log scale without indicating the base or the reference line for the claimed 1/k² slope; this makes visual verification of the rate harder.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The provided abstract and novelty claim present convergence rates for GOMA as independent first-principles results under standard monotonicity and Lipschitz assumptions, with no equations, fitted parameters, self-citations, or ansatzes shown that reduce the claimed O(1/k²) or O(1/√k) rates to prior inputs by construction. The derivation chain cannot be inspected for circular steps because no technical sections, proofs, or load-bearing citations are available in the query; the abstract itself contains no self-referential reductions. This is the expected honest non-finding for an abstract-only view of a paper whose central claims are stated without visible self-definition or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard domain assumptions that the operator is monotone and Lipschitz continuous; no free parameters, new entities, or ad-hoc axioms are visible in the abstract.

axioms (2)
  • domain assumption The operator is monotone
    Required for existence of solutions and for the convergence analysis of variational inequalities.
  • domain assumption The operator is Lipschitz continuous
    Used to control the step-size choices and obtain the explicit rates stated in the abstract.

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Reference graph

Works this paper leans on

18 extracted references · 2 canonical work pages · 1 internal anchor

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    2M η2 k(1−β 2 k)−4M 2η4 k(1−β k)2 −M η 2 kβ2 k 1−2M η 2 k(1−β k)2 −M η 2 kβ2 k # = βk+1 2β k(1−β k)

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    This inequality is equivalent to 0≥ − 1 2 β2 + 3 2 β3 = 1 2 β2(−1 + 3β), which holds forβ≤ 1

    Plugging int= 1 2, we get 4β−5β 2 2β− 3 2 β2 = 4−5β 2− 3 2 β ≥(2 +β)(1−β) = 2−β−β 2. This inequality is equivalent to 0≥ − 1 2 β2 + 3 2 β3 = 1 2 β2(−1 + 3β), which holds forβ≤ 1

  14. [14]

    Sinceβ k ≤ 1 3, condition (4) follows. 17 Accelerated and Stable Convergence with Anchored Optimistic Method Positivity of the potential function decrease.With ck+1 =a k+1, the residual coefficient derived in (29) iseck+1 = akθ 2M η2(1−βk)2 −a k+1, so Vk −V k+1 ≥L 2 akθ 2M η2 −a k+1 ∥xk+1 −y k∥2. Since (1−β k)2 ≤1 gives akθ 2M η2(1−βk)2 ≥ akθ 2M η2 , it s...

  15. [15]

    Thus condition (9) holds for allkwhenevert=M γ 2 ≤ 5 39

    =− 140 81 , so t≤ 3 5 · 140 81 −1 2 5 − 1 9 = 1 27 13 45 = 5 39 . Thus condition (9) holds for allkwhenevert=M γ 2 ≤ 5 39. Positivity of the potential function decrease.We had Vk −V k+1 ≥L 2 akθ 2M γ2 k −a k+1 ∥xk+1 −y k∥2, Recalla k = bkηk 2βk ,b k+1 = bk 1−βk , andη k = (1−β k)γwith constantγ. Then ak+1 ak = bk+1 bk · ηk+1 ηk · βk βk+1 = βk(1−β k+1) βk+...

  16. [16]

    This proves (17) with 66 + 2(1 +e)2 ≤ 94. B.3. Proof of Lemma 4 (stochastic stability) Lemma 4.In the setting of Lemma 3, let bG satisfy Assumption 3, let (xk) be given by (14), and set ek :=x k −¯xk. Then for allN≥0, E∥e N ∥2 ≤ 1√ N+ 1 4σ 2 L2κ + 752κ∥x 0 −x ⋆∥2 .(18) Proof. Step 1: error recursion.Let nk := bG(yk, ξk)−G(y k), so E[nk | F k] = 0 and, by ...

  17. [17]

    It remains to check the determinant conditionS 11 k S22 k ≥(S 12 k )2, that is, ak ≥ (S12 k )2 S22 k = bkc 2√M β k · 1−β k 1−c 2βk

    Since βk = 1 k+2 ≤ 1 2, this gives 1−c 2βk ≥1− 1 4 >0 , hence S22 k >0(andS 11 k =a k ≥0). It remains to check the determinant conditionS 11 k S22 k ≥(S 12 k )2, that is, ak ≥ (S12 k )2 S22 k = bkc 2√M β k · 1−β k 1−c 2βk . 31 Accelerated and Stable Convergence with Anchored Optimistic Method Usinga k = bk−1c 2 √ M βk−1(1−βk−1), cancelingc, M >0, and usin...

    2020

  18. [18]

    We setn= 200

    Therefore this is a 1-smooth saddle problem. We setn= 200. 35 Accelerated and Stable Convergence with Anchored Optimistic Method Methods.We compare the same set of methods as in the negative comonotone experiments. Hyperparameter selection. • EAG-C / EAG-V: Since this experiment is from (Yoon & Ryu, 2021), we use their reported parameters:α= 0.1265/L for ...

    2021