{"id":"183e1cf9-c12e-47a7-b12f-e686b9025d4b","arxiv_id":"2606.21528","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"GOMA achieves optimal last-iterate O(1/k²) convergence in deterministic monotone Lipschitz VIs and O(1/√k) in stochastic unbounded-variance settings without variance reduction.","lead":"The paper introduces Generalized Optimistic Methods with Anchoring (GOMA) for solving monotone variational inequalities in min-max problems. These methods could enable faster and more stable training of adversarial machine learning models even with noisy data.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's UNVERDICTED/LOW verdict stems directly from the absence of the full paper. The same information limit prevents any new, technically grounded objection from being raised here.","tokens_in":1685,"tokens_out":191,"duration_ms":10437,"concrete_test":"Load the actual manuscript source referenced in the query and re-check the proof of the deterministic O(1/k²) claim (likely the main theorem) for any implicit bounded-iterate or step-size restriction not stated in the abstract.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The provided query supplies only the abstract plus a placeholder note that full text exists elsewhere; no equations, proofs, or technical sections are available for inspection. Without those details it is impossible to locate a concrete weak assumption, hidden boundedness requirement, or gap in the last-iterate analysis.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1718,"tokens_out":439,"duration_ms":16547,"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.","major_comments":[],"minor_comments":[{"comment":"§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.","section":"§3"},{"comment":"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.","section":"Theorem 4.2"},{"comment":"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.","section":"Figure 2"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were listed in the report.","responses":[],"tokens_in":1191,"tokens_out":47,"duration_ms":11320,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The key takeaway is that the authors present GOMA, which achieves optimal accelerated last-iterate convergence in deterministic monotone variational inequalities and a new O(1/sqrt(k)) rate in the stochastic case with unbounded variance, without needing variance reduction or growing batches.\n\nThey combine two-time-scale optimistic updates with an anchoring term from Halpern iteration. This seems to deliver stable last-iterate behavior where classical extragradient methods struggle, especially in stochastic settings. The deterministic rate of O(1/k^2) on the squared gradient norm matches the optimal known rate, and the stochastic extension is presented as novel for the unconstrained setting.\n\nWhat stands out is the handling of unbounded variance, which is more realistic for some online learning scenarios. The single-call variant for stochastic problems reduces the per-iteration cost, making it more practical.\n\nThe main limitation at this stage is that the abstract alone doesn't let us inspect the proof techniques or any experimental validation. If the derivations rely on specific step-size choices or additional regularity, that could affect how broadly the rates apply. The core assumptions of monotonicity and Lipschitz continuity are explicit but not relaxed, so the result stays within the standard regime for these problems.\n\nThis paper is for optimization researchers focused on min-max problems in machine learning, particularly those needing last-iterate guarantees rather than ergodic averages. Readers interested in practical stochastic methods for adversarial training would find the rates relevant.\n\nThe work engages honestly with the prior literature on optimistic methods and extragradient, so it merits a serious referee. I would send it to peer review to get the technical details checked.","headline":"GOMA gives a claimed first stochastic last-iterate O(1/sqrt(k)) rate without variance reduction for unconstrained monotone Lipschitz VIs, plus optimal deterministic acceleration, but the proofs remain unchecked.","tokens_in":2215,"tokens_out":410,"would_cite":false,"duration_ms":15101,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Anchored optimistic methods achieve optimal accelerated last-iterate rates for monotone variational inequalities.","keywords":["variational inequalities","last-iterate convergence","optimistic gradient methods","min-max optimization","stochastic optimization","anchoring techniques","Halpern iteration"],"falsifier":"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.","tokens_in":2585,"feed_emoji":"","tokens_out":596,"duration_ms":26150,"temperature":0.7,"pith_summary":"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.","feed_headline":"Anchored optimistic method reaches optimal O(1/k²) last-iterate rate","feed_subtitle":"A single-call variant delivers O(1/√k) convergence even with unbounded variance in stochastic monotone variational inequalities.","key_machinery":"Generalized Optimistic Methods with Anchoring (GOMA) that combine two-time-scale optimistic updates with an anchoring term inspired by Halpern iteration.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["GOMA with anchoring hits O(1/k²) last-iterate rate","Single-call GOMA gets O(1/√k) last-iterate in stochastic case","Anchored optimistic method accelerates monotone VI convergence","First stochastic last-iterate rate O(1/√k) for unconstrained VI"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The variational inequality operator is assumed to be monotone and Lipschitz continuous.","fun_headline_variants_meta":{"raw":{"variants":["GOMA with anchoring hits O(1/k²) last-iterate rate","Single-call GOMA gets O(1/√k) last-iterate in stochastic case","Anchored optimistic method accelerates monotone VI convergence","First stochastic last-iterate rate O(1/√k) for unconstrained VI"]},"model":"grok-4.3","cost_usd":0.01109,"raw_usage":{"total_tokens":4852,"prompt_tokens":617,"num_sources_used":0,"completion_tokens":78,"cost_in_usd_ticks":110899500,"prompt_tokens_details":{"text_tokens":617,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4157,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":617,"tokens_out":78,"duration_ms":29099,"temperature":1.0,"reasoning_tokens":4157,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T13:34:28.544228+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}