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REVIEW 3 major objections 5 minor 31 references

A first-order method for constrained nonconvex-nonconcave minimax optimization

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Under a local Kurdyka-Lojasiewicz condition on a lifted inner problem, constrained nonconvex-nonconcave minimax problems can be solved by an inexact proximal gradient method with explicit first-order oracle complexity.

arxiv 2510.01168 v3 pith:G2KQUG7N submitted 2025-10-01 math.OC cs.LGcs.NAmath.NAstat.ML

Zhaosong Lu , Xiangyuan Wang This is my paper
classification math.OCcs.LGcs.NAmath.NAstat.ML MSC 90C2690C3090C4790C9965K05
keywords constrainednonconvex-nonconcaveminimaxoptimizationlocalKurdyka-LojasiewiczconditiongeneralizedHöldersmoothnesssequentialconvexprogramminginexactproximalgradientmethodfirst-orderoraclecomplexity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies minimax optimization problems of the form $\min_x \max_{c(y)\leq 0} [f(x,y)+p(x)-q(y)]$, where the inner problem may be nonconvex in x, nonconcave in y, and constrained by a smooth mapping c. The central claim is that such problems become tractable if a certain 'lifted' reformulation of the inner problem satisfies a local Kurdyka-Lojasiewicz (KL) condition. The authors prove that the maximal function $F^*(x)$ is locally generalized Holder smooth, build a sequential convex programming subroutine that solves the inner maximization with a proven rate, and wrap it in an inexact proximal gradient method. The result is the paper's stated first complexity guarantee for constrained nonconvex-nonconcave minimax: finding an $(r,\epsilon)$-stationary point requires a specific number of iterations and gradient evaluations involving $\epsilon^{-1}$. This matters because many practical minimax formulations—adversarial training, distributionally robust optimization, and penalty reformulations of outer-constrained problems—have exactly this structure.

What carries the argument

The load-bearing mechanism is the lifted minimax reformulation together with the local KL condition imposed on it. The paper replaces the constraint $c(y)\leq 0$ by surrogate constraints $\bar c(y,z)\leq 0$ built from a linearization of c at a reference point z plus a quadratic term, so that the lifted feasible region becomes an intersection of Euclidean balls. Lemma 3 shows the lifted problem is exactly equivalent to the original one. The local KL inequality (Assumption 1(iv)) then forces both a local growth bound on the distance from y to the inner maximizers and a generalized Holder modulus for the maximal function. The SCP subroutine (Algorithm 1) uses these surrogate constraints as convex balls,

What would settle it

Run Algorithm 2 on a deterministic family of instances in which the inner problem has known KL exponent $\theta$ and level-set shrink rate $\sigma$ at the boundary of the allowed range—for example, extend Example 1 so that $\sigma>0$—and compare measured iteration counts against the $\epsilon$ exponents in Theorem 6. Alternatively, construct a problem where Assumption 1(iv) holds with $x$-dependent KL constants and check whether the algorithm still converges at the stated rate; if it does, the uniformity assumption is not necessary.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the constrained inner problem's maximal function $F^*(x) = \max_{c(y)\leq 0} (f(x,y)-q(y))$ inherits a local generalized Holder smoothness from a local KL condition imposed on the lifted function $\bar F(x,y,z) = f(x,y) - q(y) - \delta_{\bar c(y,z)\leq 0} - \delta_Y(z)$, where $\bar c_j(y,z) = c_j(z) + \nabla c_j(z)^T (y-z) + \frac{L_{c_j}}{2} \|y-z\|^2$. Theorem 1 bounds $\nabla F^*$ differences by $L_{\nabla f}\|x-x'\|$ plus $\|x-x'\|^{(1-\theta)/\theta}$ on the region away from stationarity. Using this, Algorithm 2—an inexact proximal gradient method whose inner subproblems are solved by a sequential convex programming routine—outputs an $(\gamma \epsilon^\sigma/(4L_f), \epsilon)$-stationa.

Load-bearing premise

The entire complexity picture rests on Assumption 1(iv): the lifted inner objective must satisfy a local KL inequality with constants $C$ and $\theta$ that do not depend on the outer variable $x$, on level sets that shrink no faster than $\mathrm{dist}(0,\partial \bar\Psi(x))^\sigma$.

Editorial extensions

If this is right

  • Penalty reformulations of fully constrained minimax problems min_{d(x)<=0} max_{c(y)<=0} [f(x,y)+p(x)-q(y)] become solvable by the same method, because the penalty subproblems have exactly the form (1).
  • The local-KL assumption covers problems where global PL or global KL conditions fail; Example 1 demonstrates a concrete instance satisfying the local condition but not the global one.
  • The oracle complexity of Algorithm 2 is explicit: only proximal evaluations of p and q plus gradients of f are needed, no higher-order or stochastic oracles.
  • If the KL exponent is \theta=1/2 (the PL-like case), the worst-case iteration complexity degenerates to O(\epsilon^{-2\max\{1,\sigma\}} \log^2(1/\epsilon)), showing how the level-set shrink rate \sigma enters the final cost.
  • Preliminary numerical experiments on exponentially constrained problems show large objective reductions, indicating the method behaves as predicted in practice.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the uniform-KL assumption fails because the constants C or \theta genuinely depend on x, the stated exponents are not justified; one would expect the effective complexity to degrade by whatever factor the constants grow near stationarity.
  • A natural test is to generate semi-algebraic instances where the KL exponent \theta and shrink parameter \sigma are known exactly, then measure iteration counts and compare them with the predicted \epsilon exponents; this would separate tightness of the analysis from looseness of the hidden constants.
  • A parameter-free variant of Algorithm 2—which the paper notes as worthwhile—could be built by estimating L_f, C, \theta, \gamma, and \sigma on the fly; if such estimates converge, the method would retain its complexity while dropping the input-parameter requirement.
  • The lifted reformulation is the key conceptual export: it converts a nonconvex constraint into a family of convex ball constraints, so the same lifting might extend to other first-order schemes such as accelerated or stochastic variants.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies constrained nonconvex-nonconcave minimax problems of the form (1), where the inner maximization is subject to smooth constraints c(y) ≤ 0. A lifted reformulation (10) is introduced, and under a local Kurdyka–Lojasiewicz condition on the lifted inner problem (Assumption 1(iv)), the authors claim that the maximal function F* is locally generalized Hölder smooth (Theorem 1) and enjoys a local growth property (Theorem 2). They then propose a sequential convex programming method (Algorithm 1) for a constrained KL minimization problem, prove its finite termination and convergence rate (Theorems 4–5), and use it as an inner solver in an inexact proximal gradient method (Algorithm 2). The headline results, Theorems 6–7, state iteration and gradient-evaluation complexity bounds for computing an approximate stationary point of (1) under Assumptions 1–2. Numerical experiments on random instances illustrate the behavior of Algorithm 2.

Significance. If the main results are correct, this is the first complexity guarantee for constrained nonconvex-nonconcave minimax optimization with nonlinear inner constraints. The SCP part of the paper (Theorems 3–5 and Lemma 1) is a genuine contribution and appears, from the arguments given, to be sound; the uniform Slater point obtained from MFCQ in Theorem 3 is a useful device, and the explicit convergence-rate analysis for the SCP subroutine is a strength. The local KL framework is clearly motivated and the paper does not fit parameters to data; the KL constants are assumptions rather than fitted quantities. However, the central complexity claims currently rest on two unproved pillars: the local Hölder smoothness lemma (Lemma 7) and the transfer of the entire complexity analysis from the authors' prior work [17]. In addition, the uniform Lagrange-multiplier bound used in the complexity theorems requires a boundedness condition that is not stated in Assumptions 1–2. These issues must be repaired before the significance claim can be accepted.

major comments (3)
  1. [§4, Lemma 2 and Theorems 6–7] Lemma 2 requires the additional condition G_f := max_{X×Y} ‖∇_y f(x,y)‖ < ∞ (Eq. (40)), but Theorems 6 and 7 are stated under only Assumptions 1–2. This condition is not implied by Assumption 1: Assumption 1(i) bounds ∇_x f through L_f-Lipschitz continuity in x, but ∇_y f can grow with x when X = dom p is unbounded. The uniform bound A_f is then used in the definitions of ω_f, α_f, and K_{f,θ} in Theorem 7, and it affects the SCP termination constants used through Lemma 10 in the proof of Theorem 6. Without a uniform bound, the stated finite complexity does not follow. Please add G_f < ∞ (or a sufficient condition such as boundedness of X) to the assumptions, or prove that Algorithm 2 keeps iterates in a set where such a uniform bound holds.
  2. [§6.1, Lemma 7; §6.3, Lemmas 10–11 and Theorems 6–7] The proof of Lemma 7 is omitted with the explanation that it follows from Lemma 6 and arguments parallel to [17, Theorem 1]. Similarly, Lemmas 10 and 11 are stated as ‘parallel’ to [17, Lemmas 9 and 10], and the proofs of Theorems 6 and 7 are only sketches referring to [17, Theorems 4 and 5]. Lemma 7 is the basis for the local generalized Hölder smoothness of F* (Theorem 1), which in turn justifies the inexact proximal model (38) used throughout the analysis. Because the lifted problem introduces the extra variable z, the term δ_Y(z), and the constraint mapping ¯c, the transfer from the unconstrained setting in [17] is not immediate. A journal submission needs full proofs or a precise statement of which inequalities carry over, with all constants verified. As written, the central complexity claims are not verifiable from the manuscript.
  3. [§3, Algorithm 1, line 4; §4, Theorem 7] Theorem 7 is billed as a first-order oracle complexity bound, but each iteration of Algorithm 1 requires solving the convex subproblem (27), which is a constrained problem with an intersection-of-balls feasible set. The text suggests solving this subproblem with a customized interior-point method (Remark after Algorithm 1). The count bN_ϵ bounds the number of SCP outer iterations, not the number of gradient/prox evaluations needed to solve the subproblems to the required accuracy. Unless the authors provide a first-order algorithm for (27) with an explicit cost, or explicitly declare a different oracle model in which each convex subproblem solve is counted as a unit, the claimed first-order oracle complexity is not an end-to-end guarantee.
minor comments (5)
  1. [§1.1, Theorem 4] The notation ⌈·⌉_+ in the definition of ¯i is not defined. Please define the positive part explicitly.
  2. [§2, Example 1] The verification that the local KL condition holds with C=(3/2)^{1/2}, θ=1/2, γ=1/2, σ=0 is too terse. Please expand the computation of the level set bL_x and the subdifferential distance, as this example is the only concrete illustration of Assumption 1(iv).
  3. [§4, Theorem 6] The constants bC_1, …, bC_8 are defined with no explanation of their roles. A sentence or two motivating each group would greatly improve readability.
  4. [§5, numerical experiments] The choice of KL parameters (C=0.1, θ=0.5, γ=0.01, σ=0.1) is said to be empirical. Since the convergence theory depends on these parameters, the paper should state whether problem (42) is known to satisfy Assumption 1(iv) with these values, or clearly indicate that the experiments are exploratory.
  5. [References] Reference [17] is listed as ‘Accepted by SIAM Journal on Optimization’. If it is now published, please give the full journal reference and year.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main results are genuine derivations from stated KL/MFCQ assumptions; the heavy use of the authors' prior work [17] is reliance on an accepted external result, not an identity or fitted prediction.

full rationale

The paper's derivation chain is not circular. Assumption 1(iv) posits a local KL inequality on the lifted function F-bar; Theorem 1, Corollary 1, Theorem 2 and Lemmas 5-7 then derive generalized Holder smoothness, a descent inequality, and a growth/error bound. These are implications, not restatements: the KL inequality controls the slope of F-bar and yields the asserted bounds through Lemma 4 and elementary calculus, and no parameter appearing in the conclusions is fitted to the quantity being predicted. The SCP convergence theory (Theorems 4-5) is proved from Assumptions 1-3 and Theorem 3, with explicit constants; the termination bound K_theta is a consequence of the descent inequality plus KL, not an input. The complexity claims in Theorems 6-7 are not obtained by fitting: they follow from the stated assumptions, Lemma 2, and Lemmas 10-11, which are stated as consequences of Theorems 1, 2 and 5. The proofs of Theorems 6-7 are sketched by analogy to [17, Theorems 4-5], but [17] is a prior, separately published/accepted work by the same authors, and the current paper introduces genuinely new ingredients for the constrained setting (the lifted problem, MFCQ-based uniform Slater point in Theorem 3, the SCP subroutine, and the multiplier bound in Lemma 2). Citing one's own earlier work as a proof template is not circular unless the cited claim is itself unverified or equivalent to the current claim; that is not the case here. The numerical section sets KL parameters C=0.1, theta=0.5, gamma=0.01, sigma=0.1 by hand, but these are fixed inputs for a demonstration and do not enter the theorem statements as fitted predictions. The one substantive mathematical concern is that Lemma 2 requires G_f = max_{X x Y} ||grad_y f|| < infinity, while this condition is not stated among Assumptions 1-2 and can fail for unbounded X; the skeptic's example illustrates a gap in the hypotheses under which Theorems 6-7 are asserted. That is a missing assumption / correctness risk, not a circularity: it does not make any theorem's conclusion identical to its input. Accordingly, no circular step is identified and the score is 0.

Assumptions & free parameters 5 free parameters · 6 assumptions · 2 invented entities

The central claims depend on the assumed existence and uniformity of KL parameters C,θ,γ,σ, on MFCQ, and on the correctness of the authors' prior work [17] for several key proof steps. The lifted variable z is a mathematical artifact. No data-fitting enters the theory.

free parameters (5)
  • KL constant C
    Assumed to exist in Assumption 1(iv); set to C=0.1 in numerical experiments by hand.
  • KL exponent θ = 0.5
    Assumed θ∈[1/2,1). Experiments chose θ=0.5 because it 'yields the largest reduction'.
  • level-set coefficient γ
    Assumption 1(iv); set to γ=0.01 in experiments by hand.
  • level-set shrink exponent σ
    Assumption 1(iv); set to σ=0.1 in experiments by hand.
  • algorithm parameters L, ρ, β, ε = L=1, ρ=1.25, β=10, ε=10^{-2}
    Chosen empirically in Section 5; they affect the numerical runs but not the theoretical complexity results.
assumptions (6)
  • standard math Standard variational analysis: limiting subdifferential calculus, normal cones, slope/limiting-slope properties, KL theory (Rockafellar-Wets, Drusvyatskiy et al.)
    Used throughout, e.g., Lemma 4 (from [10]) and the chain-rule inclusions in Lemma 8.
  • domain assumption Assumption 1(i)-(iii): f is L_f-Lipschitz in x and L_∇f-smooth on X×Y; p,q simple closed convex with computable prox; Y compact; c_j Lcj-smooth and C² on Y.
    Regularity needed for the value function to be Lipschitz and for the lifted constraint to be smooth.
  • domain assumption Assumption 1(iv): local KL condition on the lifted function ¯F with level sets L(x) = {(y,z): 0<¯F*−¯F ≤ γ dist(0,∂¯Ψ(x))^σ}.
    Core structural assumption; enables the local generalized Hölder smoothing (Theorem 1) and the SCP convergence rates.
  • domain assumption Assumption 2 (MFCQ) at every feasible y∈S.
    Guarantees a uniform Slater point ζ>0 for the lifted constraints (Theorem 3), bounded multipliers (Lemma 1), and well-definedness of Algorithm 1.
  • domain assumption Assumption 3: SCP objective g is L-smooth and its lifted ¯h satisfies a KL condition on the level set {¯h*<¯h≤¯h*+η}.
    Invoked for Algorithm 1 convergence (Theorem 5). In Algorithm 2 this is supplied by Assumption 1(iv) applied at each x_k.
  • ad hoc to paper [17, Theorems 1, 3-5] are correct and directly applicable to the lifted problems with identical constants.
    Lemma 7 and Theorems 6-7 are not proved in this text; the authors assert the proofs 'parallel' their accepted prior paper [17]. This is a load-bearing external reliance.
invented entities (2)
  • Lifted variable z and lifted constraint ¯c(y,z)
    purpose: Reformulate (1) as (10) so the inner feasible set has a quadratic surrogate with a uniform Slater point; used by the SCP subroutine.
    z is an auxiliary copy of y introduced for analysis and algorithm design. A known SCP trick, not an empirical entity.
  • Local KL level-set function L(x) with shrink exponent σ
    purpose: Defines the region where the KL inequality is required; the shrinkage rate controls the complexity exponents.
    A definitional device from [17]; it has no external falsifiable content.

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Pith. "Pith review of A first-order method for constrained nonconvex-nonconcave minimax optimization." pith.science (2026). https://pith.science/paper/G2KQUG7N

@misc{pith2026251001168,
  author       = {Pith},
  title        = {Pith review of: A first-order method for constrained nonconvex-nonconcave minimax optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G2KQUG7N}},
  note         = {Machine review of arXiv:2510.01168}
}
read the original abstract

We study a class of constrained nonconvex-nonconcave minimax optimization problems in which the inner maximization involves potentially complex constraints. Under the assumption that the inner problem of a novel lifted minimax reformulation satisfies a local Kurdyka-Lojasiewicz (KL) condition, we show that the maximal function of the original problem enjoys a local generalized H\"{o}lder smoothness property. We also propose a sequential convex programming (SCP) method for solving constrained optimization problems and establish its convergence rate under a local KL condition. Leveraging these results, we develop an inexact proximal gradient method for the original minimax problem, where the inexact gradient of the maximal function is computed via the SCP method applied to a locally KL-structured subproblem. Finally, we establish complexity guarantees for the proposed method in computing an approximate stationary point of the original minimax problem.

Figures

Figures reproduced from arXiv: 2510.01168 by the authors.

Figure 1
Figure 1. Performance of Algorithm 2 with (n1, n2) = (100, 100) [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗

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