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arxiv: 2604.07164 · v1 · submitted 2026-04-08 · 🧮 math.OC

Model-Free Aggregative Cooperative Optimization via Randomized Gradient-Free Minimization and Exploration Momentum

Amir Mehrnoosh , Giuseppe Speciale , Riccardo Brumali , Giuseppe Notarstefano , Gianluca Bianchin This is my paper

Pith reviewed 2026-05-10 17:20 UTC · model grok-4.3

classification 🧮 math.OC
keywords aggregative optimizationgradient-free methodsdistributed optimizationcooperative optimizationrandomized algorithmsmomentum methodsmodel-free optimization
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The pith

A randomized gradient-free algorithm solves aggregative cooperative optimization without true gradients.

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

Aggregative cooperative optimization requires agents to minimize objectives that depend on both local decisions and a shared aggregate variable representing global behavior. The paper introduces ARGFree, which performs this task using only function evaluations by combining randomized finite-difference gradient approximations with distributed tracking variables that emulate gradient-based updates. It establishes convergence in expectation to an approximate optimizer, where the gap arises solely from the estimator's randomness. A variant called ARGFree-EM adds momentum to the exploration signals to reduce fluctuations and improve tracking accuracy in high dimensions.

Core claim

ARGFree is the first method capable of solving aggregative cooperative optimization problems without gradient information, by pairing randomized finite-difference approximations with a set of tracking variables that replicate the dynamics of gradient-based distributed algorithms, achieving convergence in expectation to an approximate optimizer; the ARGFree-EM extension incorporates momentum in the exploration signals to smooth fluctuations and thereby tighten the distributed tracking mechanism.

What carries the argument

ARGFree algorithm that emulates gradient descent via randomized finite-difference gradient approximations combined with distributed tracking variables, plus the momentum-enhanced ARGFree-EM variant.

If this is right

  • Agents can reach approximate solutions to aggregative problems using only local function evaluations and neighbor communication.
  • The approximation error is bounded and traceable directly to the finite-difference estimator rather than to communication or distribution.
  • Momentum in exploration signals reduces variance in the tracking variables and improves solution quality as dimension grows.
  • The framework applies to any setting where true gradients are unavailable or costly, such as black-box or sensor-driven systems.

Where Pith is reading between the lines

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

  • The same finite-difference-plus-tracking structure could be tested in non-aggregative distributed problems where only partial gradients are missing.
  • In practice the method suggests a tunable knob between exploration noise level and final accuracy that could be calibrated on real multi-agent hardware.
  • If the momentum term generalizes, similar acceleration might reduce communication rounds needed for convergence in other gradient-free distributed schemes.

Load-bearing premise

Finite-difference approximations and tracking variables can reliably stand in for unavailable true gradients in distributed aggregative problems.

What would settle it

Implement ARGFree on a low-dimensional aggregative benchmark whose optimum is known exactly, then check whether the achieved error remains inside the bound predicted by the randomized estimator or grows without bound.

Figures

Figures reproduced from arXiv: 2604.07164 by Amir Mehrnoosh, Gianluca Bianchin, Giuseppe Notarstefano, Giuseppe Speciale, Riccardo Brumali.

Figure 1
Figure 1. Figure 1: Illustrative application of the aggregative optimization framework [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustrative applications of the aggregative cooperative optimization framework ( [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (Left) Robot trajectories under the algorithm of [7], which relies on exact gradient information. (Center) Trajectories obtained with ARGFree (Algorithm 1). (Right) Trajectories obtained with ARGFree-EM (Algorithm 2). Both proposed methods achieve performance comparable to the gradient￾based approach while requiring no gradient information. Qualitatively, ARGFree-EM produces smoother trajectories that more… view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the relative loss. (a) Noiseless scenario. (b) Noisy [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of the gradient norm ∥∇f(xk)∥ evaluated at the current iterates. (a) Noiseless scenario. (b) Noisy scenario. These results corroborate the relative loss findings, demonstrating the robustness of direct function evaluations against gradient-amplified noise. son, our algorithm relies solely on direct evaluations of the objective function (i.e., measurements of position-dependent signals), which ten… view at source ↗
read the original abstract

Aggregative cooperative optimization problems arise in distributed decision-making settings where each agent's objective depends on its own decision as well as on an aggregate variable capturing global system behavior. Motivated by practical scenarios where gradient information is unavailable, this paper introduces a randomized gradient-free algorithm, named ARGFree, for solving such problems. ARGFree combines finite-difference gradient approximations with a set of tracking variables, emulating the behavior of a gradient-based method. We prove that ARGFree converges in expectation to an approximate optimizer, with the approximation error stemming from the use of a randomized gradient estimator. To enhance performance in high-dimensional settings, we further propose an improved variant, ARGFree-EM, which incorporates momentum in the exploration signals to smooth sudden fluctuations in the gradient exploration signals and thereby improve the accuracy of the underlying distributed tracking mechanism. To the best of our knowledge, the class of ARGFree methods is the first in the literature capable of solving aggregating cooperative optimization problems without gradient information.

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 / 3 minor

Summary. The manuscript introduces ARGFree, a randomized gradient-free algorithm for aggregative cooperative optimization problems. It uses finite-difference estimators combined with distributed tracking variables to emulate gradient-based behavior and proves convergence in expectation to an approximate optimizer, with the error arising from the randomized estimator. An enhanced variant ARGFree-EM adds momentum to the exploration signals to reduce variance and improve tracking accuracy in high dimensions. The authors position the method as the first in the literature capable of solving such problems without gradient information, under standard assumptions including Lipschitz continuity and bounded estimator variance.

Significance. If the stated convergence guarantees hold, the work fills a gap in model-free distributed optimization for aggregative problems common in multi-agent systems. The explicit error analysis separating bias from the one-point/two-point estimator and the demonstration that momentum reduces variance without changing the asymptotic bias term are useful technical contributions. The approach could enable practical deployment in black-box settings such as resource allocation or sensor networks.

minor comments (3)
  1. [§3.2] §3.2, Algorithm 1: the update for the tracking variable x_i^{k+1} is written with a step-size α_k that is not explicitly linked to the conditions in Theorem 1; adding a cross-reference would clarify the parameter schedule.
  2. [§4.3] §4.3, Eq. (18): the variance bound for the momentum-augmented estimator is stated as O(δ^2 + σ^2 / m), but the dependence on the momentum parameter β is only implicit; an explicit expression would strengthen the comparison to the non-momentum case.
  3. [Figure 2] Figure 2: the caption does not indicate the dimension d or the number of agents N used in the numerical example, making it difficult to assess scalability claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition that ARGFree is the first gradient-free method for aggregative cooperative optimization, and the recommendation for minor revision. We appreciate the comments on the significance of the explicit error analysis and the momentum variant.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces the ARGFree algorithm combining finite-difference gradient estimators with distributed tracking variables for aggregative problems, then derives an expectation convergence result to an approximate optimizer. The analysis explicitly invokes standard assumptions (Lipschitz continuity of gradients, bounded variance of the randomized estimator, suitable step-size sequences) to bound tracking error and estimator bias. No step reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation chain; the convergence bound is obtained from the algorithm dynamics and stated hypotheses rather than from renaming or smuggling prior results. The momentum variant ARGFree-EM is analyzed as a variance-reduction extension without altering the asymptotic bias term. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, invented entities, or non-standard axioms are stated. The convergence result implicitly rests on standard assumptions for stochastic approximation and finite-difference estimators in distributed settings.

axioms (1)
  • domain assumption The underlying optimization problem admits convergence of stochastic gradient-free methods with tracking variables
    Invoked to support the stated convergence-in-expectation result.

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