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REVIEW 3 major objections 4 minor 15 references

Online Feedback Optimization for Constrained Stochastic Problems with Decision-Dependent Distributions: Extended Version

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read A projected primal-dual algorithm tracks the performatively stable saddle points of constrained stochastic optimization problems, with a mean-square error decomposed into four sources.

desk verdict The theory is a real step forward for OFO with decision-dependent distributions, but the numerical section violates the theorem's own contraction condition—worth refereeing after that is fixed. read the letter →

arxiv 2606.19284 v2 pith:EHACE2V2 submitted 2026-06-17 math.OC

Caio Kalil Lauand , Emiliano Dall'Anese This is my paper · ORCID
classification math.OC MSC 90C1590C2590C47
keywords onlinefeedbackoptimizationdecision-dependentdistributionsperformativestabilityprimal-dualalgorithmsurrogatedualconstraintsstochasticapproximationtrackingerrorpowersystems
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

The paper studies online feedback optimization for constrained stochastic problems in which the distribution of random parameters shifts in response to the control input, so the optimization problem itself moves as the controller acts. It proposes a projected primal-dual scheme that replaces the unknown true dual constraint sets with surrogate sets, and proves an upper bound on the mean-square tracking error relative to the sequence of performatively stable saddle points. The error decomposes into four interpretable contributions: stochastic variance, output measurement error, time variability, and surrogate-versus-true dual set mismatch. A distinctive feature is that the step size trades off averaging against tracking: smaller steps reduce stochastic error but amplify time-variation and mismatch terms, so the bound cannot generally be driven to zero by step-size tuning alone. If correct, this gives a principled way to run feedback optimization without knowing the distribution map or the optimal dual variables.

What carries the argument

The load-bearing objects are the performatively stable saddle points z^P_n — fixed points of the arg-min/arg-max map evaluated at the distribution each decision induces — and the projected primal-dual update (5) used to track them. The analysis relies on three quantitative ingredients: the strong monotonicity of the expected regularized Lagrangian gradient with modulus µΨ = min{µ, η}, the Lipschitz constants LΨ and Lν for the gradient map and the decision-dependent distribution shift, and the contraction condition 1/µΨ LΨ Lν < 1, which makes z^P_n unique and gives the one-step mean-square contraction in Lemma 4.3(ii). Surrogate dual sets are handled through ε_H = sup_n max{b_Λ^(n) − b_H^(n),

What would settle it

For a scalar problem with u in a compact interval, objective (1/2)u^2 + λ Φ, constraint E[Φ | u] ≤ 0, and Φ | u ∼ N(u, σ²), compute the unique performatively stable point, run update (5) with a surrogate dual bound b_H smaller than the true bound, and measure the empirical limsup mean-square error over a long horizon while sweeping α over two decades. If the error is not U-shaped in α, or if it does not remain bounded in an instance satisfying 1/µΨ LΨ Lν < 1, the theorem's predictions are contradicted.

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Extended reading notes

Core claim

The paper's central claim is Theorem 2.5: under Assumptions (A1)–(A5), bounded drift of the performatively stable points, and the contraction condition 1/µΨ LΨ Lν < 1, the projected primal-dual scheme (5) — which replaces the unknown true dual constraint sets by surrogate sets H(n) — produces iterates whose mean-square distance to the performatively stable saddle points z^P_n satisfies the bound (6). The bound separates the tracking error into four terms: stochastic variance (ρ_a), output measurement error (ρ_b), time variability (ρ_c), and surrogate-versus-true dual-set mismatch (ρ_d). The paper also establishes that the sequence z^P_n exists and is unique under the same contraction conditi

Load-bearing premise

The load-bearing premise is the contraction condition 1/µΨ LΨ Lν < 1, which the paper assumes rather than derives; if the decision-dependent distribution shift is too strong relative to the regularizer's monotonicity, the performatively stable saddle points may fail to be unique and the contraction that drives the whole bound collapses.

Editorial extensions

If this is right

  • In the static, noiseless setting with known dual constraint sets (ρ_b = ρ_c = ρ_d = 0), the bound becomes O(α), matching the rate expected from stochastic approximation.
  • If the surrogate dual set contains the true dual set, the mismatch term ρ_d is zero; choosing a conservative bound b_H^(n) removes one entire error source.
  • The bound is not monotone in the step size α: decreasing α suppresses stochastic noise but inflates the time-variability and mismatch terms, so optimal tuning requires balancing opposing effects.
  • The algorithm never needs the distribution map D or the optimal dual variables; the distribution shift enters the analysis only through the scalar constant Lν.
  • The numerical power-system example with price-responsive assets exhibits the predicted bounded tracking behavior after a transient.

Reading between the lines

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

  • The paper does not address the case where 1/µΨ LΨ Lν ≥ 1; a natural extension would be to look for weaker guarantees, such as Cesàro-averaged error, that survive without uniqueness of the performatively stable point.
  • Because the four error terms have different scalings in α, the bound could be used as a diagnostic: estimating each term at a given operating point would tell an engineer whether to improve measurements, reduce distribution shift, or enlarge the surrogate dual set.
  • The proof technique appears portable to randomized primal-dual variants or settings with noisy gradient oracles instead of measurement-based gradients; the four-term decomposition would likely persist in those settings.
  • A concrete testable prediction is that, for fixed α, the tracking error should grow with ε_H in the specific quadratic form ρ_d = (1/µeΨ) ε_H(ε_H + 2 b_U,H), so experiments that vary only the surrogate bound b_H could validate or refute the mismatch term directly.
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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 online feedback optimization (OFO) for constrained stochastic optimization problems whose data distribution shifts with the control action (decision-dependent / performative distributions). The proposed algorithm is a projected primal-dual scheme (5) in which the unknown dual constraint sets are replaced by implementable surrogate sets. The main result, Theorem 2.5, gives a steady-state mean-square tracking-error bound (6) relative to the sequence of performatively stable saddle points. The bound is decomposed into four interpretable terms: stochasticity, measurement error, time variability, and surrogate dual-set mismatch, and it exhibits a step-size tradeoff that prevents the error from being made arbitrarily small by step-size alone. The proof builds on a one-step contraction lemma (Lemma 4.4) and a drift bound (Lemma 4.6), and the theory is illustrated with a numerical power-system example in Section 3.

Significance. If the main theorem is correct, the paper makes a useful contribution by extending OFO from deterministic or i.i.d.-stochastic settings to constrained problems with decision-dependent distributions, while simultaneously handling measurement noise, time variation, and unknown dual constraint sets. The bound (6) is interpretable and is a genuine theorem without fitted constants; the proof is mostly self-contained and the algorithm is implementable. The main caveat is the heavy contraction assumption, which is not verified and appears to be violated by the paper's own numerical example. Because this assumption controls all constants in the bound, its practical scope needs to be clarified before the result can be considered broadly applicable.

major comments (3)
  1. [Section 3 and Theorem 2.5] The numerical experiment does not satisfy the key contraction condition 1/µΨ LΨ Lν < 1. In the experiment, Φ_n = E w_n + ξ_n with E = diag(0.4, 0.5, 0.6), so Lν = 0.6. The reported η = 0.02 implies µΨ = min{μ,η} ≤ 0.02 for any μ. The objective gives LΓ ≥ 2, and the formula LΨ = √2√((LΓ + bH√M LΞ + bΞ + μ)² + (bΞ + η)²) from Section 4 yields LΨ ≥ 2.83, so 1/µΨ LΨ Lν ≥ 85 ≫ 1. Even in the limit μ = η → ∞ one has LΨ/µΨ → 2, hence the condition cannot hold for Lν = 0.6. Therefore Figure 1 cannot be cited as 'as expected from Thm. 2.5'; the theorem makes no prediction in that regime. The authors should either revise the experiment to a regime satisfying the condition, or explicitly state that the experiment is outside the theorem's assumptions and provide a verifiable criterion for the contraction condition.
  2. [Lemma 2.2 / Theorem 2.5] The condition 1/µΨ LΨ Lν < 1 is load-bearing: µeΨ = µΨ − LΨLν appears in the denominators of ρa–ρd, in b°, and in the admissible step-size bound. However, it is imported from [7, Prop. 2.11] and is never derived from, or verified against, the problem data. Since LΨ is expressed in terms of constants LΓ, LΞ, bΞ that are themselves not directly computable in applications, the theorem gives no guidance on when it applies. The paper should state the condition as an explicit assumption and discuss its satisfiability, ideally with a concrete check or a discussion of the structural restriction it imposes on the distribution-shift constant Lν.
  3. [Proof of Theorem 2.5, displayed geometric-sum bound] The displayed inequality bounding the geometric sum is typeset ambiguously: it appears to be missing the factor 1/(1−√Υα), which is needed to justify the subsequent bound 2/([µeΨ]² α²). If the fraction is intended, please rewrite the display clearly; if not, the step is unjustified. This is a local issue, but it is important because this step produces the b°ρc/α² and cross terms in (6).
minor comments (4)
  1. [Section 3] The numerical section does not explain how the performatively stable points zP_n were computed. Please provide the computational procedure or solver, and report error bars or multiple runs rather than a single trajectory. Also, the value of μ is not reported, although µΨ = min{μ,η} is needed to assess the assumptions.
  2. [Title/Abstract] The abstract and the first page title disagree: the abstract says 'Online Feedback Optimization...' while the full-text title reads 'Projected Stochastic Gradient Descent with Decision Dependent Distributions'. Please make them consistent.
  3. [Lemmas 2.2 and 2.3] Lemmas 2.2 and 2.3 are stated as imported from [7] and [3] and are not proved. For an extended version, either include their proofs or explicitly mark them as external results; at a minimum, ensure that all notation and assumptions match the present paper, since Lemma 2.2 is central to the existence of {zP_n}.
  4. [After Eq. (6)] The sentence 'if Φ0 ≡ Φn for all n, then ρa = 0' appears to conflate a static deterministic problem with a static distribution. In the stochastic setting, ρa is a gradient-variance term and need not vanish when the distribution is time-invariant. Please rephrase.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional circularity: the MSE bound is a genuine assumption-based theorem; self-citations are supporting prior results, not equivalent to the inputs.

full rationale

The central bound (6) is not fitted or constructed from the data it purports to bound: it follows from an explicit stochastic recursion and from stated assumptions (A1)-(A5) plus the contraction condition 1/µΨ LΨ Lν < 1. The constants ρa-ρd are defined in terms of problem data (σΨ², ε_m, ψ̄, ε_H) and are not tuned to reproduce Fig. 1. The paper relies on two self-citations, Lemma 2.2 from [7] (existence/uniqueness of the performatively stable sequence zP) and Lemma 2.3 from [3] (measurement-error bound), but these are prior published results with stated assumptions and do not, by themselves, imply the paper's tracking bound. Thus they are self-citations rather than equivalence-to-input circularity. The contraction condition is assumed rather than verified, and the numerical experiment may operate in a regime where the theorem's assumptions are violated; this is a correctness/verification concern, not a circularity concern. No step in the derivation reduces Theorem 2.5 to its inputs by construction.

Assumptions & free parameters 3 free parameters · 10 assumptions · 0 invented entities

The theorem's proof rests on standard convex optimization assumptions, a pushforward model of decision-dependent distributions, and a contraction condition that couples regularization to distribution shift. Several key lemmas are imported from prior work by overlapping authors. No parameters are fitted to data; α, μ, η, bH are user-chosen hyperparameters.

free parameters (3)
  • step-size α = 5×10^-3 in experiment (theoretical bound only constrains α)
    Appears explicitly in bound (6); controls the trade-off between stochastic averaging and tracking.
  • regularization μ, η = η=0.02 in experiment; μ not listed
    Set strong-monotonicity μΨ=min{μ,η}; major omission: μ is absent from the numerical parameter list.
  • surrogate dual bound bH = 15 in experiment
    Defines H(n); the mismatch term εH = sup_n max{bΛ−bH,0} depends on the unknown true bound bΛ.
assumptions (10)
  • domain assumption (A1) Convexity and Lipschitz gradients of cost/constraint maps
    Required for strong monotonicity and Lipschitz properties of ∇Ψ; Section 2.1 Preliminaries.
  • domain assumption (A2) Compact convex U(n) and simplex surrogate dual H(n) bounded by bH
    Ensures bounded iterates and well-defined projections; needed for constants bU,H.
  • domain assumption (A3) Φ = ν(u,γ) with u-independent γ~π; mean-square Lipschitz and bounded second moment
    Models decision-dependent distributions; enables coupling between Φn and ΦP_n in the error decomposition.
  • domain assumption (A4) Measurement noise bounded in mean square by εm
    Used in Lemma 2.3 to bound gradient approximation error; standard in OFO.
  • standard math (A5) Slater's constraint qualification
    Ensures existence of bounded dual optimizers.
  • domain assumption Existence of true dual constraint sets Λ(n) with λP_n ∈ Λ(n) and bΛ>0
    Quoted from [14, Sec. 3.1]; load-bearing for the surrogate mismatch analysis but not proved in this paper.
  • domain assumption Contraction condition 1/μΨ LΨ Lν < 1
    Required by Lemma 2.2 and Theorem 2.5; without it μeΨ≤0 and the contraction argument in Lemma 4.3(ii) collapses.
  • domain assumption Bounded drift of performatively stable points: ||zP_{n+1}−zP_n|| ≤ ψ̄ < ∞
    Defines ρc and is needed to bound the time-variability cross-term in Lemma 4.6.
  • standard math Lemma 2.2 (existence/uniqueness of zP_n) from [7, Prop. 2.11]
    Imported; not reproved; establishes the sequence the algorithm tracks.
  • standard math Lemma 2.3 (measurement-error gradient bound) from [3, Lemma 4]
    Imported; bounds En = ∇hatΨ − ∇Ψ in L2.

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Cite this review

Pith. "Pith review of Online Feedback Optimization for Constrained Stochastic Problems with Decision-Dependent Distributions: Extended Version." pith.science (2026). https://pith.science/paper/EHACE2V2

@misc{pith2026260619284,
  author       = {Pith},
  title        = {Pith review of: Online Feedback Optimization for Constrained Stochastic Problems with Decision-Dependent Distributions: Extended Version},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EHACE2V2}},
  note         = {Machine review of arXiv:2606.19284}
}
read the original abstract

Online feedback optimization (OFO) leverages real-time output measurements to optimize the operation of networked systems without requiring full knowledge of system dynamics or disturbances. We develop an OFO approach for constrained stochastic optimization problems in which the distribution of the system's random parameters shifts in response to the control actions. We propose a projected primal-dual algorithm where the true dual constraint sets are replaced by surrogate sets. Our main result is an upper bound on the mean-square tracking error, which decomposes into four interpretable terms reflecting: (i) the stochasticity of the problem, (ii) output measurement errors, (iii) time-variability of the problem, and (iv) the mismatch between surrogate and true dual constraint sets. The theory is illustrated in a numerical experiment for power grids with price-responsive assets.

Figures

Figures reproduced from arXiv: 2606.19284 by the authors.

Figure 1
Figure 1. (a) Norm of the deviation between {un} and {u P n} as a function of n; (b) Evolution of {Γ (n) (un, Φn)}. Transparent curves display instantaneous values, while opaque lines represent a moving average over a window of 200 iterations. 2 Main Results 2.1 Preliminaries Notation: We use ∥ · ∥ to denote the Euclidean norm for vectors and the induced operator norm for matrices. For a random variable X (vector- or matrix-v… view at source ↗

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Reference graph

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