{"id":"f34ddd52-0d36-4d2d-be4c-947353161bdc","arxiv_id":"2606.19284","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A projected primal-dual online algorithm is shown to track performatively stable saddle points with bounded mean-square error that decomposes into stochasticity, measurement, time-variation, and dual-set mismatch terms.","lead":"This paper proves a tracking-error bound for an online feedback controller used on constrained stochastic problems where random parameters shift with the control action. It matters because it extends optimization-as-feedback to settings like price-responsive power grids where the controller's own decisions change the statistics of the load.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The contraction condition 1/µΨ LΨ Lν < 1 is not verified and is provably violated by the §3 experiment (Lν=0.6, η=0.02, LΨ≥2.8), so Fig. 1 cannot support the central bound.","rationale":"The reader's weakest assumption is the contraction condition, and I agree it is the load-bearing point. My stress-test sharpens it with a concrete numerical violation: for the experiment's Lν=0.6 and η=0.02, the inequality 1/µΨ LΨ Lν<1 cannot hold for any choice of µ,η (the ratio is at least ~85 with the reported values, and tends to 2Lν=1.2 in the best asymptotic case). This does not invalidate the theorem's conditional statement—the proof appears structurally sound—but it means the numerical section is not a valid demonstration. The paper should either verify the condition for the chosen parameters (e.g., by reducing E or choosing larger equal regularization and Lν<0.5) or explicitly present the experiment as heuristic. Since this is a strengthening of the reader's concern rather than a different verdict, the conditional verdict stands. I am not raising issues about the proof's internal algebra; the bound (6) does match the proof's expansion. The main open risk is that readers will take Fig. 1 as evidence for a theorem whose assumptions are silently violated.","tokens_in":12246,"tokens_out":25087,"duration_ms":212667,"concrete_test":"For the §3 setup, compute the exact constants: LΓ=2, Lν=0.6, bH=15, η=0.02, M=2, and compute LΞ, bΞ from the constraint functions E[y]=P0 and E[(y−P0)²]≤εP. Verify whether 1/µΨ LΨ Lν<1 for any µ>0; it will fail because µΨ≤0.02 and LΨ≥2.83 (or asymptotically 2Lν=1.2). Then re-run the experiment with a smaller shift, e.g., E=diag(0.1,0.2,0.3) so Lν<0.5, and report µ, or state explicitly that the figure is only a qualitative illustration outside the theorem's assumptions. If the tracking behavior is identical outside the assumed regime, the paper should say the bound is conservative rather than claiming Fig. 1 confirms Thm. 2.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.5 and Lemma 4.3 require µeΨ=µΨ−LΨLν>0, i.e. 1/µΨ LΨ Lν<1; if this fails, the contraction in Lemma 4.4 and the bound (6) collapse. The paper assumes the condition without a verifiable criterion. More seriously, the numerical illustration violates it. In §3, Φ=Ew+ξ with E=diag(0.4,0.5,0.6), so Lν=||E||=0.6. The reported η=0.02 implies µΨ=min{µ,η}≤0.02 for any omitted µ. The objective Γ(u,Φ)=||v−P̄||²+w^TΦ+||0.15w||² has LΓ≥2 (the v-gradient is 2(v−P̄)), and the proof's definition gives LΨ=√2√((LΓ+bH√M LΞ+bΞ+µ)²+(bΞ+η)²) ≥2.83. Hence 1/µΨ LΨ Lν ≥ 0.6×2.83/0.02 ≈85≫1. Even taking µ=η=t→∞, LΨ≈2t and µΨ=t, so the ratio tends to 2Lν=1.2>1; for Lν=0.6 the condition is unsatisfiable at any regularization. Thus Figure 1 does not illustrate the theorem: it runs in a regime where the theorem makes no prediction. The condition is load-bearing because it enters every constant (µeΨ in denominators of ρa–ρd, b°, and the step-size bound).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":12705,"tokens_out":12410,"duration_ms":108154,"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":[{"comment":"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.","section":"Section 3 and Theorem 2.5"},{"comment":"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.","section":"Lemma 2.2 / Theorem 2.5"},{"comment":"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).","section":"Proof of Theorem 2.5, displayed geometric-sum bound"}],"minor_comments":[{"comment":"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.","section":"Section 3"},{"comment":"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.","section":"Title/Abstract"},{"comment":"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}.","section":"Lemmas 2.2 and 2.3"},{"comment":"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.","section":"After Eq. (6)"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the authors' own prior work [3], [7], and [10]; the incremental contribution over [3] and [7] should be made more explicit, particularly the treatment of surrogate dual sets. The main reason for my verdict is the mismatch between the theorem's contraction condition and the numerical illustration: if the authors cannot exhibit a parameter regime where the condition holds (or at least discuss its restrictive nature), the practical relevance of the main result is substantially weakened. The proof skeleton appears coherent, but the displayed geometric-sum step should be checked and clarified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick read: this is the first OFO-style bound I know for constrained stochastic problems where the distribution depends on the decision and the dual sets are unknown. The theorem decomposes the tracking error into four interpretable terms—stochasticity, measurement noise, time-variability, surrogate-set mismatch—and shows you can't make the bound arbitrarily small by step-size alone. That's genuinely useful, and the proof of Theorem 2.5 is mostly self-contained and the contraction argument is coherent. The surrogate dual set idea is practically motivated, and the paper positions itself honestly against [3], [7], [10].\n\nThe soft spots are real, though. The contraction condition 1/µΨ LΨ Lν < 1 is load-bearing but never verified, and it's not benign. In the numerical experiment, Lν = ||E|| = 0.6 (from Φ = Ew + ξ), η = 0.02, so µΨ = min{µ, η} ≤ 0.02. The objective's v-gradient alone gives LΓ ≥ 2, and LΨ ≥ √2 LΓ ≥ 2.83 (plus non-negative terms). So LΨ Lν / µΨ ≥ 2.83 × 0.6 / 0.02 ≈ 85 ≫ 1. Even taking µ = η → ∞, the ratio tends to 2Lν = 1.2 > 1, so the condition is unsatisfiable for this Lν at any regularization. The experiment therefore does not illustrate Theorem 2.5; it runs in a regime where the theorem makes no prediction. That's not a fatal flaw in the theory—the theorem is honest about its assumption—but it is a serious gap in the numerical validation. The paper should either pick parameters satisfying the condition (e.g., Lν < 0.5) or explicitly state that the experiment is outside the theorem's scope.\n\nSecondary issues: Lemma 2.2 and 2.3 are imported from overlapping-author papers without reproof, which is acceptable but makes the paper less self-contained. The numerical section reports a single run with no error bars, no explanation of how zP_n was computed, and no value for µ. Also the title in the full text doesn't match the abstract. These are all fixable.\n\nBottom line: the central result is probably correct under its assumptions, and the theorem fills a clear gap. It deserves a serious referee. I'd send it to review with a request to fix the experimental validation and clarify the contraction condition.\n\nBest.","headline":"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.","tokens_in":13132,"tokens_out":6106,"would_cite":true,"duration_ms":49444,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C15","90C25","90C47"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["online feedback optimization","decision-dependent distributions","performative stability","primal-dual algorithm","surrogate dual constraints","stochastic approximation","tracking error","power systems"],"falsifier":"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.","tokens_in":12160,"feed_emoji":"🎯","tokens_out":6360,"duration_ms":59358,"temperature":0.7,"pith_summary":"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.","feed_headline":"Online optimization tracks shifting optima with bounded error","feed_subtitle":"A primal-dual scheme with surrogate constraints keeps mean-square tracking error finite — four error sources isolated.","key_machinery":"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),","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Bounded error in online optimization when distributions follow actions","Surrogate dual sets limit error in shifting stochastic problems","Four error sources isolated for online feedback optimization","Tracking shifting optima with finite mean-square error","Primal-dual scheme with surrogate constraints yields error bound"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Bounded error in online optimization when distributions follow actions","Surrogate dual sets limit error in shifting stochastic problems","Four error sources isolated for online feedback optimization","Tracking shifting optima with finite mean-square error","Primal-dual scheme with surrogate constraints yields error bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000288,"raw_usage":{"total_tokens":1493,"prompt_tokens":678,"completion_tokens":815,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":422,"completion_tokens_details":{"reasoning_tokens":741}},"tokens_in":422,"tokens_out":815,"duration_ms":8571,"temperature":1.0,"reasoning_tokens":741,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T10:53:03.162646+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":2}