{"id":"e8601b83-1848-42aa-b733-d7c0a522bf4b","arxiv_id":"2608.03846","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A recursive sliding algorithm gives separate oracle complexity bounds for f, g, and bilinear products in Hölder-smooth minimax problems, interpolating from nonsmooth to smooth rates.","lead":"This paper proposes a sliding optimization method for convex-concave saddle-point problems with bilinear coupling, where the two function parts can have different Hölder smoothness levels. It shows how to charge gradient evaluations of each part separately, at rates depending on its own smoothness, plus a coupling term, instead of the worse of the two.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Table 1's strongly convex rates require zero normal vectors at the saddle point; the abstract/table omit this condition and it fails for elementary boundary-constrained instances.","rationale":"The reader's weakest assumption names the same normal-vector boundary condition as the load-bearing gap. My analysis confirms it is not merely a technicality: a simple smooth bilinear problem with active box constraints satisfies every assumption of Theorem 4.4 except ζ=0/ξ=0, and the proof's lower bound breaks in exactly that case. This does not overturn the paper's conditional contribution—the theorem is valid under its stated hypotheses and likely covers unconstrained or interior-saddle applications (X=R^d, or constraints inactive)—but it does mean the headline separated rates in Table 1, as quoted in the strongest_claim, are overbroad. The reader's CONDITIONAL verdict (restate the condition, verify it numerically, or restrict the claim) is therefore appropriate. I also checked the ambient-smoothness condition; it is a genuine extra assumption but it is explicitly stated in the abstract and body, so it is secondary. The proposed test is a minimal 1D instance designed to isolate the normal-vector condition while keeping all other assumptions satisfied.","tokens_in":56082,"tokens_out":26575,"duration_ms":310032,"concrete_test":"Run the restarted sliding method (Algorithm 4 with Theorem 4.4 parameters) on X=Y=[-1,1], B=1, f(x)=-10x, g(y)=0, νx=νy=1, Lxy=1.1, starting at (0,0), with restarts until the original gap is ≤10^-6. Check (a) whether Ψ decreases by 3/4 per restart, and (b) whether total f', g', B/B^T calls match Table 1's O(√κ) counts. Independently, recompute the Appendix H lower bound for this instance with ξ=9, ζ=1: if the residual -βx||ζ||²-βy||ξ||² appears and is not controlled by Ψ(zin), the proof of Theorem 4.4 is specifically reliant on the normal-vector condition, confirming the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The separated rates in Corollary 4.5/Table 1 are proved only under the boundary condition ζ=0 when βx>0 and ξ=0 when βy>0 (Theorem 4.4, Remark 2). The proof in Appendix H uses these identities to replace Bx* by g'(y*) and B^Ty* by -f'(x*) in the p3 anchor quadratics; without them the lower bound on E contains an uncontrolled residual -βx||ζ||² - βy||ξ||² that does not vanish as zin→z* and is not bounded by Ψ(zin). This is not a corner case of the Hölder theory: already in the fully smooth case νx=νy=1, take X=Y=[-1,1], f(x)=-10x, g(y)=0, B=1. The saddle point is (1,1) with ξ=9 and ζ=1, yet δx=δy=1, βx=βy=1/4, and all other assumptions of Theorem 4.4 (including Lxy>µxy) hold. Thus the theorem simply does not apply to this valid instance, even though Table 1 and the abstract present the O(√κx), O(√κy), O(√κxy) rates for the general strongly convex–concave problem on closed convex sets. The condition is stated in Theorem 4.4 but not in Corollary 4.5/Table 1/abstract; if it is essential, the headline claim is narrower than presented.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the bilinearly coupled convex-concave minimax problem min_{x∈X} max_{y∈Y} f(x)+⟨y,Bx⟩−g(y), where f and g have Hölder-continuous (sub)gradients. It develops a recursive sliding method for monotone variational inequalities with finite-sum structure, and specializes it to the bilinear saddle-point problem. The claimed outcome is a componentwise oracle complexity: the numbers of f′-evaluations, g′-evaluations, and B/B^T products are governed separately by the Hölder exponents and constants of f and g and by the spectrum of B, interpolating from nonsmooth O(ε^{-2}) rates to smooth accelerated rates. The paper also treats nonstrongly convex, mixed, and stochastic regimes, and reports synthetic and tomographic experiments supporting the separation of oracle counts across Hölder exponents.","tokens_in":56384,"tokens_out":12460,"duration_ms":152312,"significance":"If the stated results hold, they extend the smooth separated-complexity framework of Borodich et al. (2025) to Hölder-smooth components and provide the first per-oracle separation across the Hölder spectrum for bilinearly coupled minimax problems. The VI-level analysis in Section 3 is self-contained, derives explicit constants without fitted parameters, and correctly recovers known smooth rates at ν=1. The numerical section gives reproducible-seeming experiments and separates the f′ and g′ budgets as predicted. However, the most prominent strongly convex results rest on restrictive boundary conditions that are not reflected in the abstract or Table 1, and the stochastic positive-curvature row is proven only in a narrower setting than the table states. The headline contribution is therefore narrower than presented, though the underlying recursive-sliding idea appears sound under the stated theorem-level assumptions.","major_comments":[{"comment":"The zero-normal condition ζ=0 when βx>0 and ξ=0 when βy>0 is load-bearing and cannot be removed. In the proof of Theorem 4.4, after defining E, the lower bound on the p3 quadratic is converted to a useful term only through the step 'For active coefficients, the normal conditions ... and (28) give Bx*=g′(y*) and B^Ty*=−f′(x*)'. Without this step the bound retains the uncontrolled residual −βx∥Bx*−g′(yin)∥²−βy∥B^Ty*+f′(xin)∥², which at zin=z* equals −βx∥ζ∥²−βy∥ξ∥² and cannot be absorbed into Ψ(zin). This is not a corner case: take X=Y=[−1,1], f(x)=−10x, g(y)=0, B=1, νx=νy=1, Hx=10, Hy=1, Lxy=2, µxy=µyx=1. Then all numerical hypotheses of Theorem 4.4 except the zero-normal condition hold (δx=1, δy=0.1, βx=1/4, βy=1/40, range and ambient-smoothness conditions are satisfied), yet the unique saddle point (1,1) has ξ=9 and ζ=1. Thus Theorem 4.4 does not apply to this valid smooth, strongly-curv","section":"§4, Theorem 4.4, Remark 2, Appendix H; Corollary 4.5 / Table 1"},{"comment":"The stochastic strongly convex row of Table 3 is not proven for the generality that the table suggests. The proof in K.3 assumes X=R^{dx}, Y=R^{dy}, µx>0, µy>0, and explicitly sets µxy=µyx=0, so δx=µx and δy=µy. The table, however, states the δx>0, δy>0 rates without restricting the source of the effective curvature. When the positive curvature comes solely from the coupling, i.e. µxy>0 or µyx>0, the contraction argument in K.3 does not go through: the lower bound used there relies on ambient strong convexity of f and g, not on the coupling-induced regularizer, and the range/normal-cone machinery needed for the coupling-curvature case is not developed for the stochastic setting. The table should be restricted to the case actually proved, or the stochastic proof must be extended to cover coupling-induced effective curvature.","section":"§K.3 and Table 3"},{"comment":"The ambient-smoothness requirement is another load-bearing restriction. Assumptions 4.1 and 4.2 only impose Hölder smoothness on the restriction of f to X and g to Y, but the proof of Theorem 4.4 uses equation (20), which requires the full-space Lipschitz-gradient inequality for every active smooth anchor. The paper does state this in an unnumbered paragraph after Assumption 4.3, and the abstract mentions that a smooth component used in the coupling regularizer must be smooth in the ambient space. Nevertheless, this means the strongly convex smooth rates O(√κx), O(√κy) do not apply to functions that are merely smooth on the feasible set, which is a common situation for constrained problems. This restriction should be promoted into Assumptions 4.1–4.2 (or a single named standing assumption) and should be repeated in the statements of Table 1 and the abstract, since the current presentatio","section":"§4, 'Ambient smoothness for active anchor terms' and Assumptions 4.1–4.2"}],"minor_comments":[{"comment":"The synthetic experiments set µx=µy=µxy=µyx=0 and thus δx=δy=0, so they validate the degenerate rates of Corollary 3.6/Table 4, not the strongly convex rates of Table 1. The abstract's phrase 'numerical experiments reproduce the predicted Hölder exponents' should be qualified to state that the strongly convex regime is not tested.","section":"Section 5 / Appendix L.1"},{"comment":"The statement 'Thus the strongly convex block of Table 3 gives ε-accuracy when Δanc=O(ε)' is accurate only for the full-space, ambiently smooth, µx,µy>0 setting; in the printed text this paragraph appears before the full set of K.3 assumptions and could be misread as applying to the general δx,δy>0 case.","section":"§K.3, paragraph after Eq. (32)"},{"comment":"The relabeling step says 'performing no more than CnRi evaluations' but the construction of the nested budgets uses Ri≥1; this is fine, but the constants κn=2^{2n+2} and Cn=2^{2n+4} differ from the experimental choice κn=1. Appendix L notes this and reports one nonsmooth row 6% above target; the relation between the theorem's sufficient constants and the experimental implementation should be stated explicitly in the main text.","section":"Appendix F, proof of Corollary 3.6"},{"comment":"The definitions βx=1{νy=1}/(4Ly) and βy=1{νx=1}/(4Lx) are asymmetric relative to the notation 'βx' and 'βy'; a reader could expect βx to be associated with the x-block. A sentence explaining that βx scales the x-block quadratic term and uses the smoothness of g (via Ly) would help.","section":"Section 4, Eq. (21)–(22)"}],"recommendation":"major_revision","confidential_remarks":"The paper has a solid core: the VI-level sliding analysis appears coherent, and the degenerate and mixed regimes are handled with a unified framework. The main issue is packaging: the strongly convex headline in the abstract and Table 1 is materially narrower than the theorem-level hypotheses, and the stochastic positive-curvature table is narrower still than its proof. I would ask the authors to either generalize the proofs to cover the advertised settings or substantially narrow the claims, with the boundary-normal condition and the stochastic full-space strong-convexity assumption stated wherever the rates are displayed. If the scope is corrected, the paper is likely publishable; in its current form the central claims overreach."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper extends the recursive sliding framework of Borodich et al. (2025) from smooth to Hölder-smooth components in bilinear minimax, and that extension is real. The per-oracle complexity bounds interpolate cleanly between nonsmooth and smooth, and the construction is not a trivial restatement: the Hölder-to-inexact-oracle reduction is applied componentwise with separated budgets, and the stochastic section adds variance-aware terms rather than just repeating the deterministic argument. The proofs are detailed and the final rates correctly reduce to the known smooth ones at ν=1. No fitted parameters appear. Credit where due: this is a genuine layer on top of prior work, and the experiments (though synthetic) do reproduce the predicted Hölder exponents with tight fits.\n\nThe main caveat is exactly what the stress-test flags. Theorem 4.4 explicitly requires ζ=0 and ξ=0 at the saddle point when the corresponding β terms are active. That condition is not automatic for boundary saddles, and it is easy to construct a valid smooth instance — f,g strongly convex quadratics on [-1,1], B=1 — with the saddle at the boundary and nonzero ξ. The paper itself notes the condition is automatic only at interior saddle points, and it is omitted from the abstract and from Table 1. So the headline strongly-convex rates are narrower than presented. The ambient-smoothness requirement for active smooth anchor terms is a second, lesser restriction, but at least the paper flags it in the body. The tomography experiments do not check whether the boundary conditions hold, which is a missed opportunity. I'd like the authors to either state the condition up front in the abstract and table, or weaken it, and ideally to verify it on their test instances.\n\nMinor: no code is released, and the stochastic strongly-convex section assumes full-space domains and µxy=µyx=0, which is a narrower regime than the deterministic part. Those are stated, so not flaws, just scope.\n\nI think the paper deserves a serious referee. The core contribution is substantive, the proofs are worth checking, and the missing-boundary-condition issue is fixable in revision — it mainly needs honest qualification, not a rework. I would bring it to a reading group and would cite it if I worked on oracle complexity for saddle-point problems.","headline":"Solid Hölder-extension of Borodich et al.'s sliding method with honest per-oracle rates, but the headline strongly-convex table quietly depends on zero-normal-vector boundary conditions that fail on plain box-constrained examples.","tokens_in":56908,"tokens_out":5242,"would_cite":true,"duration_ms":56035,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C47","90C25","90C33","65K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For Hölder-smooth bilinear minimax problems, a recursive sliding method separates oracle costs: each function's gradient budget is set by its own smoothness exponent, and coupling products keep the smooth rate.","keywords":["minimax optimization","Hölder-smooth gradients","bilinear coupling","sliding methods","variational inequalities","oracle complexity","saddle-point problems","separated rates"],"falsifier":"Run the method on a strongly convex instance with νx = 1 whose unique saddle point lies on the boundary of X, for example f(x) = ½(x−c)² − t·x on X = [0,1] with the tilt t chosen so x* = 0, paired with any Hölder g, and measure the per-restart Lyapunov contraction Ψ(z_out)/Ψ(z_in). Theorem 4.4 claims a factor of 3/4 whenever the stated normal-vector condition holds; if the boundary instance does not produce the Table 1 rates, or the contraction fails, the boundary condition is genuinely load-bearing rather than cosmetic.","tokens_in":55951,"feed_emoji":"⚖️","tokens_out":11209,"duration_ms":108658,"temperature":0.7,"pith_summary":"The paper studies convex-concave saddle-point problems with bilinear coupling, where each of the two convex functions may have (sub)gradients of any Hölder smoothness between Lipschitz (exponent 1) and merely bounded-variation nonsmooth (exponent 0). Its central claim is that a recursive sliding method can query the oracles for f, g, and the coupling operator B at different frequencies, so the number of gradient evaluations for each function is governed by that function's own Hölder exponent, while the number of matrix-vector products with B and B^T stays at the smooth rate. In the strongly convex-concave regime the paper derives explicit complexity bounds that interpolate between known nonsmooth and smooth rates, recovering the optimal smooth bilinear rates when both exponents equal 1, and it also covers nonstrongly convex, partially strongly convex, and stochastic settings. This matters in applications where one oracle (say a tomography data gradient or a nonlocal graph gradient) is far more expensive than another: the expensive one pays only for its own regularity.","feed_headline":"Each oracle billed at its own Hölder-smoothness rate","feed_subtitle":"A recursive sliding scheme gives f and g separate gradient budgets, each set by its own Hölder exponent.","key_machinery":"The carrying object is a recursive sliding scheme for finite-sum monotone variational inequalities (Algorithm 1), specialized to a three-component decomposition: p1 = f, p2 = g, and a coupling-induced regularizer p3 = (βx/2)||Bx − g′(y_in)||² + (βy/2)||B^T y + f′(x_in)||² whose anchor gradients are drawn once per restart, with Q3 the skew-symmetric bilinear operator. Three reductions make it run: a Hölder-to-inexact-oracle conversion (Lemma E.1) that turns every (ν, H) function into a (δ, L(δ)) oracle and balances the tolerance against the recursion depth; effective curvatures δx = μx + 4βx μxy², δy = μy + 4βy μyx², condition numbers κx = Lx/δx, κy = Ly/δy, κxy = Lxy²/(δxδy) and scaled const","core_discovery":"The paper claims that when f and g have (νx, Hx)- and (νy, Hy)-Hölder gradients, a restarted three-level sliding method on the associated monotone variational inequality reaches accuracy ε with N_f = O(H̃x^{2/(1+3νx)} ε^{(νx−1)/(1+3νx)}) evaluations of f′ for νx < 1 (Õ(√κx) at νx = 1), the analogue for g′, and N_B = Õ(√κxy) products with B and B^T, independent of the Hölder exponents. So the f-budget is set by f's own smoothness and the coupling spectrum, not by g's; at (νx, νy) = (1,1) the trio collapses to the known optimal smooth separated rates, at ν = 0 to nonsmooth O(ε^{−2})-type scaling, and between them each component interpolates with exponent 2/(1+3ν). The route: convert each Hölde","pith_inferences":["If the separation is robust, the practical scheduling rule in composite applications with heterogeneous oracle costs becomes immediate: allocate recursion depth to the cheapest component and gradient budget to each function by its own Hölder exponent; the paper's own crossover-cost analysis (sliding wins when one gradient call costs more than roughly 20–40 matrix-vector products) suggests where th","The machinery suggests an untested extension the paper only lists as future work: estimating ν and H online and making the (δ, L(δ)) conversion adaptive, which would remove the assumption that the Hölder constants are known to the algorithm.","The boundary normal-vector condition (ζ = 0, ξ = 0 at active smooth anchors) limits the strongly-convex table to saddle points that are interior relative to the anchors; a constraint that binds at the solution would need a correction term, so the most informative next experiment is a boundary-saddle instance.","The coupling budget being independent of νx and νy hints that the bilinear term is always 'as smooth as the smoothest object in the problem'; if that survives a lower-bound analysis, the separation is structural rather than an artifact of the method."],"forward_implications":["When one component is smooth (ν = 1) and the other is genuinely Hölder (ν < 1), the smooth component keeps its logarithmic √κ rate while the other pays only its own ε^{−(1−ν)/(1+3ν)} cost, so the overall rate is not the worse of the two.","In the fully nonsmooth limit νx = νy = 0, the gradient budgets become O(1/ε²)-type nonsmooth rates while the coupling budget remains O(Lxy Ω/ε), matching the optimal rates for first-order convex optimization and for monotone Lipschitz variational inequalities respectively.","For δx = δy = 0 (no effective strong convexity), the regularizer term is dropped and componentwise budgets follow directly from the general VI theorem, so the results cover the nonstrongly convex-concave regime.","In the stochastic setting with bounded-variance gradient estimators, the method gives a uniform expected-gap bound in the degenerate regime, and under positive effective curvature it converges to within an explicit noise floor that mini-batching the anchors reduces to O(ε).","The synthetic experiments reproduce the predicted per-function exponents, with fitted slopes within 4% of 2/(1+3νx), and a tomography benchmark shows runtime gains when the expensive oracle is the genuinely Hölder one."],"supporting_citations":[{"why":"Establishes separated optimal complexities for the smooth bilinear saddle-point problem; this paper's νx = νy = 1 endpoint reduces to it.","marker":"Borodich et al. (2025)"},{"why":"Introduced gradient sliding for composite optimization, the technique this paper extends to inexact oracles and three components.","marker":"Lan (2016)"},{"why":"Mirror-Prox Sliding, the closest smooth two-component predecessor whose rates the paper reproduces and whose cross-component oracle terms it removes.","marker":"Lan and Ouyang (2021)"},{"why":"Universal gradient methods for Hölder-continuous gradients, supplying the ν-parameterized rate that the Hölder-to-inexact-oracle bound builds on.","marker":"Nesterov (2015)"},{"why":"Mirror-Prox for monotone variational inequalities, the standard method whose O(1/t) rate and Lipschitz-operator complexity the coupling budget must match.","marker":"Nemirovski (2004)"},{"why":"Generalized universal Mirror-Prox for Hölder-continuous monotone operators, the main external baseline in the numerical comparisons.","marker":"Stonyakin et al. (2022)"},{"why":"Lower iteration-complexity bounds for bilinear saddle-point problems, used to argue the coupling and degenerate rates are not improvable.","marker":"Zhang et al. (2022)"}],"fun_headline_variants":["Separate oracle budgets for each Hölder-smooth function","Sliding method splits gradient costs by each function's smoothness","Hölder-smooth minimax solved with per-component rates","Adaptive sliding scheme: each oracle pays its own smoothness price","Gradient budget per function, not the worst-case Hölder exponent"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The strongly convex rates rest on the optimality conditions' normal vectors vanishing at the saddle point whenever a smooth anchor term is active, which is automatic at interior saddle points but not at boundary saddle points, together with the requirement that any smooth component's Lipschitz-gradient constant hold on the whole ambient space rather than only on the feasible set.","fun_headline_variants_meta":{"raw":{"variants":["Separate oracle budgets for each Hölder-smooth function","Sliding method splits gradient costs by each function's smoothness","Hölder-smooth minimax solved with per-component rates","Adaptive sliding scheme: each oracle pays its own smoothness price","Gradient budget per function, not the worst-case Hölder exponent"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00117,"raw_usage":{"total_tokens":4763,"prompt_tokens":919,"completion_tokens":3844,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":663,"completion_tokens_details":{"reasoning_tokens":3757}},"tokens_in":663,"tokens_out":3844,"duration_ms":29227,"temperature":1.0,"reasoning_tokens":3757,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T11:07:25.891587+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the method on a strongly convex instance with νx = 1 whose unique saddle point lies on the boundary of X, for example f(x) = ½(x−c)² − t·x on X = [0,1] with the tilt t chosen so x* = 0, paired with any Hölder g, and measure the per-restart Lyapunov contraction Ψ(z_out)/Ψ(z_in). Theorem 4.4 claims a factor of 3/4 whenever the stated normal-vector condition holds; if the boundary instance does not produce the Table 1 rates, or the contraction fails, the boundary condition is genuinely load-bearing rather than cosmetic.","supporting_citations":[],"review_version":1}