{"id":"7f92bfd0-4970-4e79-bf91-3bc97ac1e5d3","arxiv_id":"2607.11056","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"Perturbed S-EG and S-OGDA achieve O(T^{-1/4}) last-iterate restricted primal-dual gap rates when T is known and O(T^{-1/5}) anytime rates under standard stochastic oracles.","lead":"A simple quadratic perturbation turns vanilla stochastic extragradient and optimistic GDA into single-loop methods whose last iterates converge for constrained smooth convex-concave minimax problems under ordinary bounded-variance noise. The rates improve prior last-iterate guarantees and work for both known and unknown horizons, including noncompact domains.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolated the only potential soft spot (the second-moment control that legitimizes the restricted gap) and correctly observed that the same schedules already guarantee the needed summability. The explicit calculation above confirms that the constants remain uniform, so the conversion Lemma 5.5 produces a clean O(T^{-1/4}) without hidden T-dependent factors. All other ingredients (strong monotonicity of the perturbed operator, the refined one-step recursions of Lemmas 5.2–5.3, and the Chung-type lemmas) are standard and carefully tracked. The paper therefore supplies a self-contained, correct last-iterate analysis under the weakest standard oracle; the ACCEPT verdict stands.","tokens_in":34397,"tokens_out":568,"duration_ms":28609,"concrete_test":"Re-derive the uniform bound of Lemma 5.4 for the harmonic schedule alone: substitute \tau = T^{-1/4}, b = \theta(\tau^{-2}), \tau_t \tau = 2/(t+b) into the recursion (7) or (8), verify that \tau^{2} sum \tau_t^{2} remains bounded by an absolute constant independent of T, and confirm that the resulting C enters Lemma 5.5 only through an O(1) factor in front of \tau. If that factor stays O(1) the rate is intact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central O(T^{-1/4}) claim (Theorem 4.1 / Corollary 4.1) rests on the two-stage argument: distance-to-perturbed-saddle (Theorem 5.1) followed by the conversion Lemma 5.5. The only structural point that could have been fragile is the uniform second-moment bound (Lemma 5.4) needed to keep diam(B_x), diam(B_y) and M_R finite. Under both the polynomial and the regularization-aware harmonic schedules the error series that appear in the telescoping argument of Appendix D are summable with sums independent of T and of \tau (for the harmonic case sum \tau^{-2} \tau^{-2} b^{-1} = O(1) because b = \theta(\tau^{-2})). Consequently the constants that multiply \tau and \tau diam^{2} remain O(1) as T\to \tau, the bias term is genuinely O(T^{-1/4}), and the restricted-gap guarantee is non-vacuous. No hidden circularity or schedule-dependent blow-up is present.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies last-iterate convergence of single-loop stochastic first-order methods for smooth convex-concave minimax problems under a standard bounded-variance oracle. Vanilla S-EG and S-OGDA can fail to converge last-iterate even on bilinear problems; the authors stabilize them by adding a quadratic perturbation that makes the problem strongly convex-strongly concave, then run S-EG/S-OGDA on the perturbed operator (PS-EG and PS-OGDA). Analysis proceeds in two stages: non-asymptotic distance bounds to the (possibly time-varying) perturbed saddle, followed by conversion to the restricted primal-dual gap (or gradient norm when unconstrained). With known horizon they obtain O(T^{-1/4}) last-iterate rates (Theorem 4.1, Corollary 4.1); with diminishing schedules they obtain O(T^{-1/5}) anytime rates on general closed convex sets (Theorem 4.2) and a sharper O(T^{-1/4}) anytime gradient-norm rate for unconstrained PS-EG via a reference-tracking argument (Theorem 4.3).","tokens_in":34702,"tokens_out":696,"duration_ms":5612,"significance":"Last-iterate guarantees for stochastic EG/OGDA under the ordinary bounded-variance oracle remain limited, especially with constraints. The paper supplies simple single-loop algorithms that improve the previous best comparable constrained rate (roughly ~O(T^{-1/7})) to O(T^{-1/4}) when T is known, and that also cover non-compact domains and an anytime regime. The two-stage architecture (refined one-step recursions + Chung-type lemmas + conversion lemmas) is fully written out, the uniform second-moment bound needed for the restricted gap is controlled under the stated schedules, and the unconstrained reference-tracking argument for PS-EG is a clean technical contribution. The results are therefore a genuine advance for the standard oracle model.","major_comments":[],"minor_comments":[{"comment":"Table 1 caption and the surrounding text correctly note that G_R coincides with the ordinary gap only when B_x = X and B_y = Y; a short explicit sentence in the introduction or abstract would further reduce the risk that readers over-claim the constrained rate for unbounded domains.","section":null},{"comment":"In the anytime analysis (Theorem 5.2 / Lemma 5.3) the Lyapunov function for PS-OGDA accumulates several (tau_t - tau_{t-1}) terms; a one-line remark that these remain summable under the chosen exponents would make the bookkeeping easier to follow.","section":null},{"comment":"Figure 1 is only described in the text; if the camera-ready version includes the actual plots, ensure the caption states the precise bilinear instance and noise model so the non-convergence of vanilla methods is reproducible.","section":null},{"comment":"A few typographical slips remain (e.g., \"Metho ds\", \"stochas tic\", missing spaces after commas in the arXiv header). They do not affect readability but should be cleaned.","section":null}],"recommendation":"accept","confidential_remarks":"The concurrent works (Ito et al., Sohrabi et al.) are properly distinguished; the manuscript's advantage on non-compact domains and the cleaner O(T^{-1/4}) without logs is real. I see no load-bearing gap that would justify major revision."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The useful fact is simple: under the standard unbiased bounded-variance oracle, both PS-EG and PS-OGDA get last-iterate O(T^{-1/4}) on the restricted primal-dual gap when T is known, and O(T^{-1/5}) anytime on general closed convex sets. On compact domains the restricted gap is the ordinary gap. That improves the previous best comparable single-loop rates (~T^{-1/7} anchoring, T^{-1/6} with Hessian Lipschitz) without mini-batches or variance reduction.\n\nWhat is new is not the quadratic perturbation itself (classical) but the full two-stage analysis that turns distance-to-perturbed-saddle into last-iterate restricted-gap guarantees for both EG and OGDA, covering non-compact domains and anytime schedules. The refined one-step recursions keep the negative quadratic terms that give contraction; the Chung-type lemmas are applied correctly; the conversion lemmas are explicit. The concurrent-work discussion is transparent and fair. The sharper unconstrained anytime gradient-norm rate for PS-EG via reference tracking is a nice extra that avoids the moving-saddle bookkeeping.\n\nThe only structural soft spot is the reliance on uniform mean-square boundedness (Lemma 5.4) to keep the comparison sets and M_R finite. Under the paper’s own schedules the error series are summable with T-independent sums, so the constants stay O(1) and the bias term is genuinely O(T^{-1/4}). No circularity. Rates remain slower than the multi-sample O(T^{-1/3}) unconstrained results, which the authors state clearly. Free parameters are just the usual schedule exponents and offsets; nothing is fitted.\n\nThis is for people who care about last-iterate stochastic VI / minimax under the weakest standard oracle. The math is solid enough that a serious editor should send it to referees. I would cite the known-horizon theorem and the anytime unconstrained corollary.","headline":"Clean single-loop last-iterate rates for S-EG/S-OGDA under the plain bounded-variance oracle; the O(T^{-1/4}) known-horizon result is real and improves the comparable prior art.","tokens_in":35330,"tokens_out":499,"would_cite":true,"duration_ms":5456,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C47","90C15","65K15"],"pacs":[],"model":"grok-4.5","headline":"A simple quadratic perturbation makes the last iterate of stochastic EG and OGDA converge for constrained convex-concave minimax problems.","keywords":["last-iterate convergence","stochastic extragradient","optimistic gradient descent-ascent","convex-concave minimax","quadratic perturbation","restricted primal-dual gap","anytime rates","variational inequalities"],"falsifier":"On a compact bilinear game with additive bounded-variance noise, run PS-EG with the stated horizon-dependent schedule and check whether the empirical last-iterate duality gap decays faster than T to the minus one-fifth and matches the claimed T to the minus one-fourth order; a clear slower decay or divergence falsifies the rate.","tokens_in":35301,"feed_emoji":"📉","tokens_out":677,"duration_ms":6138,"temperature":0.7,"pith_summary":"Vanilla stochastic extragradient and optimistic gradient methods can fail to converge on even simple bilinear games because noise keeps the last iterate from settling. This paper shows that adding a small quadratic regularizer that makes the problem strongly convex-strongly concave, then running the same single-loop updates on the regularized problem, restores last-iterate convergence under ordinary bounded-variance stochastic gradients. When the total number of steps T is known in advance, a fixed perturbation of order T to the minus one-fourth yields an O(T to the minus one-fourth) guarantee on the restricted primal-dual gap; on compact domains that gap is the ordinary duality gap. When T is unknown, slowly diminishing perturbations and stepsizes still give an O(T to the minus one-fifth) rate, and in the unconstrained case the same method recovers an O(T to the minus one-fourth) rate on the gradient norm. The practical payoff is that practitioners can keep the simplest single-loop algorithms and still trust the final output rather than an average.","feed_headline":"Simple noise fix makes last iterate of stochastic EG converge","feed_subtitle":"Quadratic perturbation restores O(T^{-1/4}) last-iterate rates for constrained minimax games","key_machinery":"The quadratic perturbation F_τ = F + (τ/2)‖x‖² - (τ/2)‖y‖², which renders the associated operator strongly monotone; last-iterate distance to the perturbed saddle is first controlled by a contraction-plus-noise recursion, then converted into a restricted primal-dual gap bound via uniform mean-square boundedness of the iterates.","core_discovery":"Under the standard bounded-variance stochastic oracle, both perturbed stochastic extragradient (PS-EG) and perturbed stochastic optimistic GDA (PS-OGDA) produce last iterates whose restricted primal-dual gap converges at rate O(T^{-1/4}) when the horizon is known and the perturbation is held fixed at order T^{-1/4}, and at rate O(T^{-1/5}) under diminishing perturbations when the horizon is unknown; the unrestricted gradient norm inherits the same rates in the unconstrained setting.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Perturbed S-EG restores last-iterate rates for stochastic minimax","Single-loop PS-EG and PS-OGDA hit O(T^{-1/4}) last-iterate gap","Simple quadratic perturbation fixes stochastic EG last-iterate failure","PS-EG, PS-OGDA deliver O(T^{-1/5}) anytime last-iterate convergence","Bounded-noise last iterates via fixed or diminishing perturbation"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The iterates must stay uniformly bounded in mean square so that the restricted gap on a fixed compact set controls the quality of the last iterate; that boundedness is proved only for the stepsize and perturbation schedules already used to obtain contraction.","fun_headline_variants_meta":{"raw":{"variants":["Perturbed S-EG restores last-iterate rates for stochastic minimax","Single-loop PS-EG and PS-OGDA hit O(T^{-1/4}) last-iterate gap","Simple quadratic perturbation fixes stochastic EG last-iterate failure","PS-EG, PS-OGDA deliver O(T^{-1/5}) anytime last-iterate convergence","Bounded-noise last iterates via fixed or diminishing perturbation"]},"model":"grok-4.5","effort":"low","cost_usd":0.004202,"raw_usage":{"total_tokens":1385,"prompt_tokens":931,"num_sources_used":0,"completion_tokens":108,"cost_in_usd_ticks":42020000,"prompt_tokens_details":{"text_tokens":931,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":346,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":931,"tokens_out":108,"duration_ms":3034,"temperature":1.0,"reasoning_tokens":346,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T07:21:11.377617+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"On a compact bilinear game with additive bounded-variance noise, run PS-EG with the stated horizon-dependent schedule and check whether the empirical last-iterate duality gap decays faster than T to the minus one-fifth and matches the claimed T to the minus one-fourth order; a clear slower decay or divergence falsifies the rate.","supporting_citations":[],"review_version":1}