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REVIEW 3 major objections 4 minor 39 references

Adaptive Stochastic Gradient Descent Ascent Algorithm for Nonconvex Minimax Problems with Decision-Dependent Distributions

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper proposes adaptive gradient descent ascent algorithms that learn the decision-dependent distribution online and reach epsilon-stationary points of nonconvex minimax problems with explicit gradient complexity.

desk verdict A solid first cut at nonconvex SMDD, but the headline rates hide a batch-size floor and an unproven least-squares oracle bound; worth refereeing, not worth quoting as is. read the letter →

arxiv 2509.11018 v1 pith:ZUF6E6YT submitted 2025-09-14 math.OC

Yan Gao , Yongchao Liu This is my paper
classification math.OC MSC 90C2690C1590C47
keywords stochasticminimaxoptimizationdecision-dependentdistributionsnonconvex-concaveadaptivegradientdescentascentdistributionmaplearninglocation-scalemodelPolyak-Lojasiewiczconditionstationarypointcomplexity
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

When the data distribution shifts in response to the decision variables, standard stochastic minimax optimization becomes harder because the distribution is unknown and keeps moving. This paper tries to show that a single-loop adaptive stochastic gradient descent ascent algorithm, which learns the distribution map on the fly, can still find stationary points of the primal function with complexity bounds comparable to settings without distribution shift. For location-scale distribution models, the paper claims an O(epsilon^{-(4+delta)}) complexity in the nonconvex-strongly-concave case and O(epsilon^{-8}) in the nonconvex-concave case, plus an alternating variant that achieves O(kappa_y^4 epsilon^{-4}) when the maximization satisfies the Polyak-Lojasiewicz condition. A sympathetic reader cares because this is the first analysis that handles the biased gradients caused by learning an unknown, decision-dependent distribution within a standard minimax framework.

What carries the argument

The central object is the estimated distribution map psi_t, maintained by an online least squares oracle and used to form stochastic gradients. The bias of these gradients is bounded by L_1 times the Frobenius norm of (nabla psi_t - nabla psi). The proof couples a descent inequality for the primal function (or its Moreau envelope in the concave case) with a recursion controlling the error delta_t = E||y^*(x_t) - y_t||^2. Two-timescale step sizes force the maximizer to move faster than the minimizer. The location-scale assumption reduces the distribution map to linear regression, so the oracle guarantees ||nabla psi_t - nabla psi||_F^2 <= c/(t+d); summing this over T iterations yields O(log T

What would settle it

Run ASGDA on the paper's synthetic example in Section 5.1 (z = 4x - y + xi, xi ~ N(0,1)) and record the online least squares estimate's error ||nabla psi_t - nabla psi||_F at each iteration. If the empirical error does not decay like c/(t+d) (for example, if it plateaus or grows because the regressors (x_t, y_t) keep moving), then the key assumption fails and the O(epsilon^{-(4+delta)}) guarantee would not be observed.

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Extended reading notes

Core claim

The paper studies stochastic minimax problems with decision-dependent distributions (SMDD): min_x max_y E_{z~D(x,y)} l(x,y,z), where D is unknown and changes as the iterates move. The central claim is that the proposed ASGDA algorithm, which maintains an online estimate psi_t of the distribution map and uses two-timescale step sizes, finds an epsilon-stationary point of the primal function Phi(x)=max_y L(x,y). Under a location-scale model D(x,y)=psi(x,y)+xi with xi~P, the complexity is O(epsilon^{-(4+delta)}) for any delta>0 in the nonconvex-strongly-concave case and O(epsilon^{-8}) in the nonconvex-concave case (using the Moreau envelope as the stationarity measure). An alternating variant

Load-bearing premise

The weakest assumption is that the online least squares oracle for the location-scale model satisfies E||nabla psi_t - nabla psi||_F^2 <= c/(t+d) at every iteration, even though the decision variables that generate the data are moving; if the estimation error does not decay this fast along the algorithm's own path, the summed bias terms in Theorems 1-3 fail and the complexity bounds collapse.

Editorial extensions

If this is right

  • For location-scale decision-dependent distributions, provable stationarity comes with essentially the same gradient complexity as standard nonconvex minimax problems without distribution shift: O(epsilon^{-(4+delta)}) versus O(epsilon^{-4}), and O(epsilon^{-8}) in the concave case.
  • The online learning of the distribution map adds only a logarithmic factor to the iteration count, so no separate estimate-then-optimize phase is needed.
  • AASGDA extends the results to the nonconvex-PL-in-y regime, matching the best known single-loop complexity O(kappa_y^4 epsilon^{-4}) for PL minimax problems.
  • The algorithms target stationary points of the primal function, not performative equilibrium points; experiments show that a previously proposed primal-dual method for performative equilibrium diverges on these nonconvex problems.
  • The paper also sketches that an adaptive step size variant can achieve O(epsilon^{-(4+delta)}) with batch size M=O(1) under extra second-order smoothness assumptions.

Reading between the lines

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

  • The proof technique suggests a general recipe: any stochastic gradient method whose bias is controlled by a decaying estimation error can inherit standard complexity bounds, provided the cumulative squared bias is O(log T). This may extend beyond location-scale models to any distribution family with an online estimator achieving 1/t error decay.
  • If the online least squares oracle's 1/t error bound continues to hold along the algorithm's own non-stationary trajectory, then the 'learning while optimizing' paradigm is viable for more general decision-dependent stochastic games. A direct test is to measure the empirical oracle error on the iterates of ASGDA; if the error plateaus, the complexity claim would not hold.
  • The paper does not require strong convexity in x or strong concavity in y for the main results; the Moreau-envelope stationarity measure for the nonconvex-concave case is a natural fit for the weakly-convex primal function induced by the location-scale model.
  • Batch size M enters the bounds through sigma^2/M, so the O(epsilon^{-8}) rate in the concave case could potentially be improved by increasing M, though the paper's analysis fixes M as a constant.
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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 / 4 minor

Summary. The paper studies stochastic minimax problems with decision-dependent distributions (SMDD), where the data distribution depends on the current decision pair (x,y). Under a location-scale model D(x,y)=ψ(x,y)+ξ, the authors propose ASGDA, a single-loop stochastic gradient descent-ascent method that simultaneously learns ψ via an online estimation oracle, and AASGDA, an alternating variant for the nonconvex-PL-in-y case. The main claimed results are: nonconvex-strongly-concave SMDD, O(ε^{-(4+δ)}) stochastic gradient evaluations for any δ>0 (Theorem 1); nonconvex-concave SMDD, O(ε^{-8}) evaluations (Theorem 2); and nonconvex-PL SMDD, O(κ_y^4 ε^{-4}) evaluations (Theorem 3). The proofs are detailed and appear internally consistent given the stated assumptions. Numerical experiments on synthetic and real-world problems compare the proposed algorithms with stochastic primal-dual methods.

Significance. If the claimed rates hold, this would be a meaningful contribution: it extends adaptive stochastic gradient descent-ascent analysis to the decision-dependent distribution setting, a topic of current interest, and gives the first complexity estimates for stationary points of the nonconvex primal function under a location-scale distribution model. The paper is careful to model the bias induced by the estimated distribution map through Lemmas 1 and 2, and the optimization-side arguments follow standard Lyapunov and Moreau-envelope techniques. The reliance on the online least-squares oracle of [27] is explicit and not circular. However, the decisive distribution-estimation rate is imported rather than proved for the adaptive sequence generated by the algorithms, and this is load-bearing for all three theorems.

major comments (3)
  1. [§3.1, after Eq. (19); used in Theorems 1–3] The claimed complexity O(ε^{-(4+δ)}) relies on the assertion, immediately after (19), that under Assumption 5 there is an online least-squares oracle guaranteeing E‖∇ψ_t(x_t,y_t)−∇ψ(x_t,y_t)‖_F² ≤ c/(t+d) at every iterate. This bound is not a consequence of Assumption 5 for the adaptive sequence generated by ASGDA/AASGDA. For ψ(x,y)=Ax+By, the least-squares error is governed by the cumulative design matrix Σ_t = Σ_{s<t} (x_s;y_s)(x_s;y_s)^T. The iterates are not persistently exciting; they may converge to the optimum, in which case Σ_t can grow only logarithmically or remain bounded, and the LS error does not decay as c/t. Consequently the O(log T/T) decay of the last estimation-error terms in (19), (23), and (24) does not follow. This is not a presentation issue: it is the sole mechanism by which the distribution-learning error is absorbed into the optimization rates. The authors need e
  2. [Theorem 1, Eq. (19) and Lemma 1(b)] Inequality (19) contains terms proportional to σ²/M that do not vanish as T→∞. To reach an ε-stationary point one must take M=Ω(σ²/ε²); with M=O(1) the right-hand side has a positive floor, so the stated O(ε^{-(4+δ)}) complexity cannot hold. The sentence after (19) mentioning the adaptive-stepsize variant of [22] is not a proof and does not appear in a theorem. The sample complexity should be stated as a function of M and T, e.g., T=O(ε^{-(4+δ)}) together with M=Ω(ε^{-2}), yielding M·T=O(ε^{-(6+δ)}). As written, the batch-size dependence is hidden and the claimed rate is not accurate.
  3. [Theorems 2 and 3, Eqs. (23) and (24)] The same oracle-rate issue propagates into the nonconvex-concave and nonconvex-PL results. In (23), the terms involving E‖∇yψ_t−∇yψ‖_F and E‖∇xψ_t−∇xψ‖²_F are controlled only by the asserted c/(t+d) bound. In (24), the last two terms are controlled only by the same bound. Without a valid per-iteration oracle error decay, the averaging arguments reduce to O(1) or grow with T, and the stated O(ε^{-8}) and O(κ_y^4 ε^{-4}) rates collapse. This is a load-bearing gap in all three convergence theorems, not a local technical detail.
minor comments (4)
  1. [References, [21] and [22]] References [21] and [22] appear to share the same arXiv identifier 2405.14130 but cite different authors/titles. This is likely a bibliographic error and should be corrected.
  2. [Section 5.2, Figure 4] In the caption and text of Figure 4, the method is referred to as ASGDA, but the algorithm described and implemented is AASGDA. Please fix the typo.
  3. [Section 5, general] The experiments do not report the distribution-map estimation error E‖ψ_t−ψ‖² or the design-matrix condition. Since the oracle rate is the key assumption for the rates, reporting these quantities would help verify the theoretical claim.
  4. [Algorithm 1, Step 4] The dependence of G_x and G_y on D_t(·) is indicated, but the formulas in (13)–(16) use ψ_t(·). It would be clearer to explicitly state the relationship D_t(·)=ψ_t(·)+ξ and the initialization of ψ_0(·).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; convergence proofs are self-contained conditional on an externally cited least-squares oracle.

full rationale

The paper's central claims are convergence-rate bounds for new algorithms (ASGDA/AASGDA) under standard smoothness/convexity/PL assumptions plus a location-scale distribution model. The analysis builds descent lemmas and recursions from these assumptions; the only external input is the online least-squares oracle bound E||∇ψ_t−∇ψ||²_F ≤ c/(t+d), imported from [27] (not the authors' own work). This is independent support, not a circular redefinition: the bound is not derived from the target rates and is not fitted to the algorithms' outputs. The paper's own equations (17)–(19), (21)–(24) do not reduce any 'prediction' to an input. The oracle bound is load-bearing, and its applicability to the moving iterates is a genuine correctness/verification concern: the bound may fail without persistent excitation, as the design matrix of the LS estimator is driven by the algorithm's iterates. Likewise, inequality (19) contains a σ²/M floor, so the stated O(ε^{−(4+δ)}) total-sample claim would require an additional batch-size argument. But these are gaps in external support, not circularity of the derivation. No self-citation chain, no uniqueness theorem imported from the authors, and no fitted parameter renamed as a prediction are present.

Assumptions & free parameters 7 free parameters · 9 assumptions · 0 invented entities

The paper introduces no new entities. It relies on standard smoothness, variance, and concavity/PL assumptions, plus an external oracle error bound from [27]. The only unstated scaling is the batch size M in the strongly-concave case, which the complexity claim silently requires.

free parameters (7)
  • Batch size M (Theorem 1) = O(epsilon^{-2}) required, not stated
    The sigma^2/M constant term in (19) must be at most a fraction of epsilon^2 for the epsilon-stationary claim; the paper's conclusion omits this scaling.
  • eta_x (ASGDA, Theorem 1) = 1/(40(kappa_y+1)^2 ell)
    Chosen to stabilize the primal descent and the delta_t recursion.
  • eta_y (ASGDA, Theorem 1) = 1/(2(ell+mu))
    Chosen for the y-recursion contraction.
  • eta_x (ASGDA, Theorem 2) = 1/(T+1)^(3/4)
    Decaying stepsize for nonconvex-concave setting.
  • eta_y (ASGDA, Theorem 2) = 1/(T+1)^(1/4)
    Decaying stepsize for the maximization step.
  • eta_x (AASGDA, Theorem 3) = min{1/(16 sqrt(T)), 1/(176 ell kappa_y^2)}
    Two-phase stepsize balancing initial transient and asymptotic contraction.
  • eta_y (AASGDA, Theorem 3) = min{11 kappa_y^2 / sqrt(T), 1/ell}
    Two-phase stepsize for the PL maximization step.
assumptions (9)
  • domain assumption Assumption 1: The primal function Phi(x) is bounded below.
    Guarantees well-posedness of minimization; used throughout convergence analysis.
  • domain assumption Assumption 2: L(.) is ell-smooth and L(.,y) is L-Lipschitz continuous.
    Standard smoothness condition for stochastic minimax problems.
  • domain assumption Assumption 3: First moments of gradients bounded by L_1.
    Used to bound the bias from the estimated distribution map.
  • domain assumption Assumption 4: Variance of stochastic gradients bounded by sigma^2.
    Standard bounded-variance condition for stochastic gradients.
  • domain assumption Assumption 5: Location-scale model D(x,y)=psi(x,y)+xi with unknown psi, L_0-Lipschitz and ell_0-smooth.
    Core modeling assumption that enables the learning oracle and bias bounds.
  • domain assumption Assumption 6: Strong concavity in y and compact Y.
    Enables the nonconvex-strongly-concave analysis and the definition of stationarity via the primal gradient.
  • domain assumption Assumption 7: Concavity in y and compact Y.
    Enables the nonconvex-concave analysis using the Moreau envelope gradient.
  • domain assumption Assumption 8: PL condition in y.
    Enables the alternating PL analysis with a global linear-like contraction in y.
  • domain assumption Oracle error bound: E||nabla_psi_t - nabla_psi||_F^2 <= c/(t+d) for the online least squares oracle.
    Imported from [27]; load-bearing for all three convergence theorems because the distribution-map estimation error must vanish at a controlled rate.

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Cite this review

Pith. "Pith review of Adaptive Stochastic Gradient Descent Ascent Algorithm for Nonconvex Minimax Problems with Decision-Dependent Distributions." pith.science (2026). https://pith.science/paper/ZUF6E6YT

@misc{pith2026250911018,
  author       = {Pith},
  title        = {Pith review of: Adaptive Stochastic Gradient Descent Ascent Algorithm for Nonconvex Minimax Problems with Decision-Dependent Distributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZUF6E6YT}},
  note         = {Machine review of arXiv:2509.11018}
}
abstract

In this paper, we study stochastic minimax problems with decision-dependent distributions (SMDD), where the probability distribution of stochastic variable depends on decision variable. For SMDD with nonconvex-(strongly) concave objective function, we propose an adaptive stochastic gradient descent ascent algorithm (ASGDA) to find the stationary points of SMDD, which learns the unknown distribution map dynamically and optimizes the minimax problem simultaneously. When the distribution map follows a location-scale model, we show that ASGDA finds an $\epsilon$-stationary point within $\mathcal{O}\left(\epsilon^{-\left(4+\delta\right)} \right)$ for $\forall\delta>0$, and $\mathcal{O}(\epsilon^{-8})$ stochastic gradient evaluations in nonconvex-strongly concave and nonconvex-concave settings respectively. When the objective function of SMDD is nonconvex in $x$ and satisfies Polyak-{\L}ojasiewicz (P{\L}) inequality in $y$, we propose an alternating adaptive stochastic gradient descent ascent algorithm (AASGDA) and show that AASGDA finds an $\epsilon$-stationary point within $\mathcal{O}(\kappa_y^4\epsilon^{-4})$ stochastic gradient evaluations, where $\kappa_y$ denotes the condition number. We verify the effectiveness of the proposed algorithms through numerical experiments on both synthetic and real-world data.

Figures

Figures reproduced from arXiv: 2509.11018 by the authors.

Figure 1
Figure 1. Performance of ASGDA We record the performance of ASGDA in [PITH_FULL_IMAGE:figures/full_fig_p024_1.png] view at source ↗
Figure 2
Figure 2. Performance of SPD [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Performance of ASGDA and SPD We report the performance of ASGDA and SPD in [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Performance of AASGDA and SPD on synthetic example [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]
Figure 5
Figure 5. Figure 5: Performance of AASGDA and SPD on election prediction [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]

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