{"id":"c483e48e-52aa-438c-b5dd-f87c862b6d64","arxiv_id":"2606.22688","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":1,"one_line_summary":"GARIP uses a running-average reference to minimize peak lag among causal convex averages, proves local last-iterate convergence at constant anchor strength, and matches R-NaD robustness on matrix, Coin, Connect Four, and Othello games while being easier to tune.","lead":"The paper introduces GARIP, a self-play method that anchors updates to the running average of past policies instead of a fixed or snapshot reference to achieve last-iterate convergence in two-player zero-sum games. A smart generalist might read it to see how the choice of reference policy affects cycling behavior and hyperparameter robustness in competitive AI training.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Local convergence relies on anchor exactly scaling base map rotation by (1-β) to cross stability boundary","rationale":"The reader's weakest_assumption correctly isolates the single algebraic step on which both the local convergence argument and the large-β characterization rest. All other elements (lag-profile comparison, empirical robustness) are downstream of this scaling. Because the full derivation is not supplied, the concern cannot be discharged.","tokens_in":1901,"tokens_out":279,"duration_ms":16676,"concrete_test":"For the 2×2 matching-pennies game, derive the Jacobian of the GARIP update (gradient step plus β-anchor to running average) at the equilibrium; check whether its eigenvalues equal exactly (1-β) times those of the unanchored map for several β in (0,1).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central convergence claim requires that the anchored update map multiplies the rotational eigenvalues of the unanchored base map by precisely (1-β). This is used both to prove local last-iterate convergence at constant β and to characterize large-β consensus failure. If the linearization contains additional cross terms, non-multiplicative effects from the payoff matrix, or higher-order contributions from the running-average reference, the claimed boundary crossing does not follow. The one-line lag theorem is independent of this step.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces GARIP, which anchors self-play updates in two-player zero-sum games to the running average of past policies rather than a fixed reference or periodic snapshot. Its central claims are (i) a one-line peak-lag theorem establishing that, among causal convex averages with fixed mean lag, the running average uniquely minimizes peak lag (flat profile, peak = mean) while a snapshot yields peak = 2× mean, (ii) a local last-iterate convergence proof at constant anchor strength β obtained by showing that the anchor multiplies the rotational eigenvalues of the base map by exactly (1-β), crossing the stability boundary from recurrent to contractive behavior, and (iii) empirical robustness on matrix games, the Coin Game, Connect Four and Othello that matches R-NaD peak performance while being less sensitive to hyperparameter choice.","tokens_in":2024,"tokens_out":645,"duration_ms":14995,"significance":"If the scaling property and lag theorem hold, the work supplies a simple, theoretically grounded default for last-iterate convergence that avoids the need to tune snapshot periods K. The mechanism cleanly separates the effect of reference shape from mean lag and supplies a falsifiable prediction (collapse rate depends on peak rather than mean lag). Experiments are reported as fully reproducible JAX code; the local-convergence result at constant β is a concrete theoretical advance over purely empirical moving-reference methods.","major_comments":[{"comment":"Local convergence argument: the manuscript states that the anchored map multiplies the base map's rotational eigenvalues by precisely (1-β). The linearization must be exhibited (Jacobian of the joint update including the running-average reference) to confirm that no payoff-matrix cross terms or higher-order contributions from the average alter the multiplicative factor; without this explicit calculation the claimed boundary crossing from recurrent to contractive behavior remains unverified and is load-bearing for both the constant-β convergence claim and the large-β consensus-failure characterization.","section":"local convergence argument / peak-lag theorem"},{"comment":"Experimental comparison: the claim that GARIP and R-NaD are statistically indistinguishable over the full grid but that a matched-mean-lag GARIP collapses in 0/40 seeds versus 10/40 for R-NaD at conventional parameterizations requires the exact grid ranges, the definition of 'collapse,' and the statistical test to be stated in a single table or section so that the robustness advantage can be assessed.","section":"experiments"}],"minor_comments":[{"comment":"The abstract and text should clarify whether the peak-lag theorem assumes a fixed mean lag or derives it; the one-line statement is elegant but its precise hypotheses (convexity, causality, stationarity of the mean) need to be listed.","section":"peak-lag theorem"},{"comment":"Notation for the running-average reference (e.g., how the anchor is updated each step) should be introduced once with an equation rather than described only in prose.","section":"method"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"Thank you for the careful review and constructive comments. We address each major point below and will revise the manuscript accordingly to improve clarity and verifiability.","responses":[{"response":"We agree that the explicit Jacobian is required to substantiate the eigenvalue scaling. In the revised manuscript we will derive and display the full linearization of the joint map (policy gradient step together with the running-average update), confirming that the anchor multiplies the rotational eigenvalues of the base map by exactly (1-β) with no additional cross terms arising from the payoff matrix or the averaging operation at linear order. This will directly verify the claimed transition from recurrent to contractive behavior.","revision_made":"yes","referee_comment":"[local convergence argument / peak-lag theorem] Local convergence argument: the manuscript states that the anchored map multiplies the base map's rotational eigenvalues by precisely (1-β). The linearization must be exhibited (Jacobian of the joint update including the running-average reference) to confirm that no payoff-matrix cross terms or higher-order contributions from the average alter the multiplicative factor; without this explicit calculation the claimed boundary crossing from recurrent to contractive behavior remains unverified and is load-bearing for both the constant-β convergence claim and the large-β consensus-failure characterization."},{"response":"We will add a dedicated table (or consolidated subsection) that reports the precise hyperparameter grid ranges, the operational definition of collapse (last-iterate distance exceeding a fixed threshold for a minimum number of iterations), and the statistical procedure used to compare collapse rates. This will allow readers to evaluate the robustness claims directly from the reported data.","revision_made":"yes","referee_comment":"[experiments] Experimental comparison: the claim that GARIP and R-NaD are statistically indistinguishable over the full grid but that a matched-mean-lag GARIP collapses in 0/40 seeds versus 10/40 for R-NaD at conventional parameterizations requires the exact grid ranges, the definition of 'collapse,' and the statistical test to be stated in a single table or section so that the robustness advantage can be assessed."}],"tokens_in":1669,"tokens_out":456,"duration_ms":14578,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing to know is that anchoring self-play to the running average cuts peak lag compared with snapshots and yields local last-iterate convergence at constant anchor strength. The one-line theorem on why a flat average beats a sawtooth for the same mean lag is new relative to MMD and R-NaD, and the experiments show the practical payoff: matched performance with fewer tuning headaches.\n\nThe paper does the mechanism clearly. It isolates how reference choice controls collapse, proves the local result by scaling the base map's rotation by 1-β, and runs reproducible JAX trials on matrix games, Coin Game, Connect Four, and Othello. The grid results are reported both ways—statistically similar collapse rates overall, yet better at conventional settings—which is useful for practitioners.\n\nThe soft spot is the convergence argument. It assumes the anchored map multiplies rotational eigenvalues exactly by 1-β with no extra cross terms from the running average or payoff structure. If that linearization step does not hold precisely, the boundary-crossing claim weakens. Global convergence is only conjectured, and the large-β failure mode is characterized but not fully mapped. The advantage also appears mainly where naive self-play already cycles, so the scope is narrower than the title suggests.\n\nThis is for people working on regularized self-play in zero-sum games. It has enough new math and clean experiments to merit referee time rather than a desk reject, though the proof details will need close checking.","headline":"GARIP gives a clean lag-minimization theorem and local convergence at fixed β via running-average anchoring, but the scaling step in the proof needs verification and global convergence stays conjectural.","tokens_in":2525,"tokens_out":378,"would_cite":false,"duration_ms":16980,"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":"GARIP anchors self-play to a running average to minimize peak lag and restore last-iterate convergence in zero-sum games.","keywords":["self-play","last-iterate convergence","zero-sum games","moving reference","running average","anchor strength","matrix games","board games"],"falsifier":"Run naive self-play and GARIP on the same 2-by-2 matrix game with known cycling behavior and measure whether the distance of the last iterate to equilibrium drops below a fixed threshold within a fixed number of steps only under the running-average anchor.","tokens_in":2776,"feed_emoji":"🎲","tokens_out":786,"duration_ms":13500,"temperature":0.7,"pith_summary":"Self-play with plain gradient ascent produces orbiting last iterates around equilibrium in two-player zero-sum games. Anchoring updates to a moving reference can pull the orbit inward, but the choice of reference determines how much lag is introduced before the pull takes effect. GARIP uses the running average of past policies as that reference. The paper proves that, among all causal convex averages with the same mean lag, the running average is the unique choice that minimizes the maximum lag experienced by any iterate, while a periodic snapshot reference experiences twice that maximum lag. This lag reduction supplies the contraction needed for a local convergence proof at fixed anchor strength and explains why GARIP matches the empirical robustness of snapshot-based methods while requiring less careful hyperparameter tuning.","feed_headline":"Running average reference halves peak lag in self-play","feed_subtitle":"GARIP proves the flat lag profile of the running average is minimal among causal averages and turns recurrent orbits into contractions at fi","key_machinery":"The collapse mechanism, which tracks the peak lag of the reference and converts that lag into a contraction factor of (1-β) on the base map's rotation.","core_discovery":"The central discovery is a one-line comparison of lag profiles: collapse of the last-iterate orbit is governed by the peak lag of the reference; for any fixed mean lag the flat profile of the running average sets peak equal to mean, whereas the sawtooth profile of a snapshot sets peak equal to twice the mean. This identity directly yields local last-iterate convergence because the anchor scales the rotation angle of the base map by exactly (1-β), moving the fixed point from the unit circle into the interior of the unit disk.","pith_inferences":["The same lag-minimization argument could be tested in non-zero-sum or many-player settings where cycling is observed but no equilibrium concept is assumed.","Replacing the running average with other low-pass filters whose lag profile is flatter than a snapshot might yield further gains without changing the mean lag.","The local proof suggests that any base dynamics whose linearization has eigenvalues on the unit circle could be stabilized by a sufficiently strong running-average anchor, offering a general template beyond gradient ascent."],"forward_implications":["Local last-iterate convergence holds at any constant anchor strength β in (0,1) once the rotation scaling crosses the stability boundary.","Global convergence is expected for sufficiently small β; large β produces a consensus failure in which both players lock onto the same suboptimal policy.","On matrix games, the Coin Game, Connect Four and Othello, GARIP reaches the same peak performance as snapshot references but collapses reliably under conventional hyperparameter choices where snapshots require explicit shortening of the snapshot interval.","An anticipatory reference with negative weights further reduces stale-side lag, but only improves outcomes in games where naive self-play already exhibits sustained cycling."],"fun_headline_variants":["Running average minimizes peak lag in self-play","Snapshot doubles peak lag versus running average","Running average turns orbits into contractions","GARIP isolates running average lag minimum"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The anchor multiplies the base map's rotation angle by exactly the factor (1-β).","fun_headline_variants_meta":{"raw":{"variants":["Running average minimizes peak lag in self-play","Snapshot doubles peak lag versus running average","Running average turns orbits into contractions","GARIP isolates running average lag minimum"]},"model":"grok-4.3","cost_usd":0.009212,"raw_usage":{"total_tokens":4127,"prompt_tokens":831,"num_sources_used":0,"completion_tokens":49,"cost_in_usd_ticks":92115500,"prompt_tokens_details":{"text_tokens":831,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3247,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":831,"tokens_out":49,"duration_ms":22693,"temperature":1.0,"reasoning_tokens":3247,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T09:06:04.878002+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Run naive self-play and GARIP on the same 2-by-2 matrix game with known cycling behavior and measure whether the distance of the last iterate to equilibrium drops below a fixed threshold within a fixed number of steps only under the running-average anchor.","supporting_citations":[],"review_version":1}