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arxiv: 2606.19284 · v1 · pith:EHACE2V2new · submitted 2026-06-17 · 🧮 math.OC

Projected Stochastic Gradient Descent with Decision Dependent Distributions: Extended Version

Caio Kalil Lauand , Emiliano Dall'Anese This is my paper

Pith reviewed 2026-06-26 19:45 UTC · model grok-4.3

classification 🧮 math.OC
keywords online feedback optimizationstochastic optimizationdecision-dependent distributionsprimal-dual algorithmtracking error boundsurrogate constraint setspower grid control
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The pith

Projected primal-dual algorithm with surrogate sets bounds mean-square tracking error for decision-dependent stochastic optimization.

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

This paper develops an online feedback optimization approach for constrained stochastic problems whose random-parameter distributions shift in response to control actions. It replaces the true dual constraint sets with surrogate sets inside a projected primal-dual algorithm. The central result is an explicit upper bound on mean-square tracking error that splits into four additive terms: stochastic noise, measurement errors, time variation of the problem, and mismatch between surrogate and true sets. The bound holds without requiring full knowledge of system dynamics or disturbances. The claims are illustrated on a power-grid example with price-responsive loads.

Core claim

The paper establishes an upper bound on the mean-square tracking error for a projected primal-dual algorithm that uses surrogate dual constraint sets in place of the true ones, for constrained stochastic optimization problems with decision-dependent distributions. The bound decomposes into interpretable terms reflecting the stochasticity of the problem, output measurement errors, time-variability of the problem, and the mismatch between surrogate and true dual constraint sets.

What carries the argument

Projected primal-dual algorithm that substitutes surrogate dual constraint sets for the true sets, with the resulting mean-square tracking-error bound decomposed into four terms.

If this is right

  • Tracking error stays bounded despite the distribution shifting with each decision.
  • Each of the four sources contributes additively and separately to the total error.
  • The method requires only output measurements, not a full dynamic model.
  • The mismatch term quantifies the price paid for using computationally simpler surrogate sets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Choosing surrogates close to the true sets would drive the mismatch term toward zero and recover tighter bounds.
  • The four-term split could guide algorithm tuning by revealing which error source dominates in practice.
  • The same surrogate-substitution idea might extend to other feedback-based methods for time-varying problems.
  • Numerical validation on power grids leaves open whether the bound remains useful in higher-dimensional or more rapidly varying settings.

Load-bearing premise

Surrogate dual constraint sets can be substituted for the true sets while preserving the stated tracking-error bound, with no quantitative closeness condition supplied.

What would settle it

Run the algorithm on a problem where the surrogate sets differ markedly from the true sets and check whether the observed mean-square tracking error remains below the sum of the four explicit terms.

Figures

Figures reproduced from arXiv: 2606.19284 by Caio Kalil Lauand, Emiliano Dall'Anese.

Figure 1
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- o… view at source ↗
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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper develops a projected primal-dual algorithm for online feedback optimization of constrained stochastic problems in which the distribution of random parameters depends on the control actions. True dual constraint sets are replaced by surrogate sets. The central result is an upper bound on mean-square tracking error that decomposes into four terms reflecting stochasticity of the problem, output measurement errors, time-variability, and mismatch between surrogate and true dual sets. The theory is illustrated via a numerical experiment on power grids with price-responsive assets.

Significance. If the bound derivation holds, the work supplies an interpretable performance guarantee for OFO under decision-dependent distributions, with the explicit mismatch term addressing a practical modeling choice. The decomposition into four distinct sources of error is a clear strength for analysis and design. The power-grid numerical example provides concrete illustration. No machine-checked proofs or parameter-free derivations are claimed.

minor comments (2)
  1. [Abstract] Abstract: the existence of the bound is asserted without stating its explicit form, key assumptions, or a one-line proof sketch; adding these would improve readability without lengthening the abstract substantially.
  2. The title refers to 'Projected Stochastic Gradient Descent' while the abstract and contribution describe a 'projected primal-dual algorithm'; aligning the title with the algorithm class used would reduce potential confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work, as well as the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper presents a derived upper bound on mean-square tracking error for a projected primal-dual algorithm in online feedback optimization, with the bound explicitly decomposed into four additive terms that include the surrogate-true mismatch as an independent contribution. No equations or fitted quantities are defined in terms of the target tracking error itself, and the abstract states the result as an upper bound obtained from the algorithm rather than by construction or renaming. The derivation chain is therefore self-contained against external benchmarks, with no load-bearing self-citation, self-definitional steps, or fitted-input-called-prediction patterns exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.1-grok · 5662 in / 1035 out tokens · 18621 ms · 2026-06-26T19:45:31.214869+00:00 · methodology

discussion (0)

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Reference graph

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