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arxiv: 2604.25468 · v1 · submitted 2026-04-28 · 📡 eess.SY · cs.SY· math.OC· math.ST· stat.TH

Distributed adaptive estimation for stochastic large regression models

Die Gan , Siyu Xie , Zhixin Liu , Xuebo Zhang This is my paper

Pith reviewed 2026-05-07 15:08 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OCmath.STstat.TH
keywords distributed adaptive estimationrecursive least squaresstochastic regressioncooperative excitationalmost sure convergencemulti-agent systemsprediction regretnon-stationary regressors
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The pith

A distributed recursive least squares algorithm achieves almost sure convergence for stochastic large regression models with infinite parameters under a cooperative excitation condition.

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

This paper develops a distributed recursive least squares algorithm for estimating an unknown infinite-dimensional parameter vector in a stochastic regression model across a network of agents. Each agent constructs a local recursive cost function and updates its estimate using only neighbor information. The analysis proves that the estimates converge almost surely to the true parameters when a cooperative excitation condition holds, blending temporal persistence and spatial cooperation across agents. A separate bound is given on the accumulated prediction regret that requires no excitation assumption at all. The results apply even when regression vectors are correlated and non-stationary, removing common restrictions that often limit practical use in feedback systems.

Core claim

The paper establishes the almost sure convergence of a proposed distributed recursive least squares algorithm for stochastic large regression models under a cooperative excitation condition that incorporates both temporal and spatial information among agents. The growth rate of the regressors' dimension is characterized by a non-decreasing positive function. Additionally, an asymptotic upper bound is derived for the accumulated regret in prediction error without excitation conditions. The analysis handles products of non-independent, non-stationary random matrices with changing dimensions using stochastic Lyapunov functions, double-array martingale theory, and algebraic graph theory, without

What carries the argument

The cooperative excitation condition, which reflects cooperative effects among agents by combining temporal persistence of regressors with spatial information flow across the network.

If this is right

  • The algorithm applies directly to multi-agent systems whose regression signals are correlated through feedback.
  • Prediction performance remains controlled asymptotically even when the cooperative excitation condition is absent.
  • The method accommodates models whose effective parameter dimension grows over time according to any non-decreasing function.
  • Convergence holds without any independence or stationarity requirement on the regression vectors.

Where Pith is reading between the lines

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

  • The regret bound opens a route to online distributed learning in networks whose size or dimension changes dynamically.
  • The same matrix-product techniques may extend to other distributed adaptive filters that must handle time-varying dimension.
  • Practical implementations could relax communication requirements by quantifying how much spatial cooperation is minimally needed for convergence.

Load-bearing premise

The cooperative excitation condition must hold across the network, together with the specified growth rate on regressor dimension.

What would settle it

A simulation or analytic counter-example in which local regressors are persistently exciting at each agent yet the network-wide cooperative excitation condition fails and the parameter estimates diverge almost surely.

read the original abstract

This paper studies the distributed adaptiveestimation problems for stochastic large regression modelswith an infinite number of parameters. By constructing a re-cursive local cost function, we propose a novel distributedrecursive least squares algorithm to estimate the unknownsystem parameters, where the growth rate of regressors'dimension is characterized by a non-decreasing positivefunction. The almost sure convergence of the proposedalgorithm is established under a cooperative excitationcondition, which incorporates the temporal information andthe spatial information to reflect the cooperative effectamong multiple agents. Moreover, we analyze the predic-tion error by establishing the asymptotic upper boundof the accumulated regret without any excitation condi-tions. The main difficulty of theoretical analysis lies in howto analyze properties of the product of non-independentand non-stationary random matrices, whose dimensionschange over time simultaneously. Some techniques, suchas stochastic Lyapunov function, double-array martingaletheory and algebraic graph theory, are employed to dealwith the above issue. Our theoretical results are derivedwithout imposing independence or stationarity assump-tions on the regression vectors, thereby not excluding thecorrelated feedback signals.

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

2 major / 3 minor

Summary. This paper develops a distributed recursive least-squares algorithm for adaptive parameter estimation in stochastic regression models whose regressor dimension grows without bound according to a non-decreasing positive function. Almost-sure convergence of the local estimates is established under a cooperative excitation condition that combines temporal persistence of excitation with spatial cooperation across the network. An asymptotic upper bound on the accumulated regret of the prediction error is derived without any excitation assumption. The analysis relies on stochastic Lyapunov functions, double-array martingale convergence theorems, and algebraic graph theory to control products of non-stationary, non-independent random matrices whose dimensions increase over time, without requiring independence or stationarity of the regression vectors.

Significance. If the stated convergence and regret results hold, the work meaningfully extends distributed adaptive estimation to high-dimensional, non-stationary settings over networks. The cooperative excitation condition and the explicit handling of growing regressor dimension provide a more general framework than prior results that impose stationarity or independence. The regret analysis without excitation conditions is a practical addition. The technical lemmas supplied to close the argument for the time-varying matrix products constitute a solid contribution to the literature on distributed recursive estimation.

major comments (2)
  1. [main convergence theorem and preceding definition] The cooperative excitation condition is load-bearing for the almost-sure convergence claim. The manuscript should explicitly state whether this condition reduces to standard persistence of excitation when the network is fully connected, and whether it can be verified from local data alone (see the statement following the definition of the condition and the hypotheses of the main convergence theorem).
  2. [assumptions on regressor dimension and the regret bound] The growth-rate function that bounds the increase in regressor dimension appears in both the algorithm and the convergence statement. It is not clear whether the derived bounds remain meaningful when this function grows faster than any polynomial; a brief remark on admissible growth rates would strengthen the result.
minor comments (3)
  1. [algorithm description] Notation for the local cost function and the distributed update should be introduced with an explicit equation number at first use to improve readability.
  2. [proof outline] The abstract states that the analysis avoids independence and stationarity assumptions, yet the precise manner in which the double-array martingale lemma is applied to the non-stationary product should be cross-referenced in the main text.
  3. [simulation section] A short numerical example illustrating the cooperative excitation condition on a small network would help readers assess its practical strength.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive comments. We will revise the manuscript to address the points raised and provide clarifications as suggested.

read point-by-point responses

  1. Referee: [main convergence theorem and preceding definition] The cooperative excitation condition is load-bearing for the almost-sure convergence claim. The manuscript should explicitly state whether this condition reduces to standard persistence of excitation when the network is fully connected, and whether it can be verified from local data alone (see the statement following the definition of the condition and the hypotheses of the main convergence theorem).

    Authors: We agree that this clarification would improve the manuscript. In the revised version, we will explicitly state following the definition that when the network is fully connected, the cooperative excitation condition reduces to the standard persistence of excitation condition on the union of the regressors. Furthermore, while the condition involves spatial cooperation, it can be verified using only local and neighbor data exchanges, which are already part of the distributed algorithm. We will add this remark to the text preceding the main convergence theorem. revision: yes

  2. Referee: [assumptions on regressor dimension and the regret bound] The growth-rate function that bounds the increase in regressor dimension appears in both the algorithm and the convergence statement. It is not clear whether the derived bounds remain meaningful when this function grows faster than any polynomial; a brief remark on admissible growth rates would strengthen the result.

    Authors: We thank the referee for this suggestion. The derived bounds are meaningful under the growth rates for which the double-array martingale convergence theorems apply, typically allowing polynomial growth. For faster growth, the accumulated regret bound may lose tightness or require stronger excitation. We will add a brief remark in the manuscript discussing the admissible growth rates to clarify the scope of the results. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes almost-sure convergence of a distributed recursive least-squares estimator for infinite-dimensional regressors by invoking a cooperative excitation condition (temporal persistence plus spatial network cooperation) together with a non-decreasing growth function on regressor dimension. The proof deploys standard external tools—stochastic Lyapunov analysis, double-array martingale convergence theorems, and algebraic graph theory—while supplying the necessary technical lemmas to handle the product of non-stationary, non-independent random matrices whose dimensions increase over time. No step reduces the claimed result to a fitted parameter, a self-defined quantity, or a load-bearing self-citation; the central argument remains independent of the target convergence statement and is closed by externally verifiable probabilistic and graph-theoretic machinery.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard results from stochastic approximation, martingale theory, and algebraic graph theory; no new entities are introduced and no parameters are fitted to data.

axioms (2)
  • standard math Properties of products of non-independent non-stationary random matrices whose dimensions change over time can be analyzed via stochastic Lyapunov functions and double-array martingale theory.
    Invoked to establish almost sure convergence.
  • domain assumption A cooperative excitation condition that mixes temporal and spatial information suffices for convergence in multi-agent systems.
    Central assumption for the main theorem.

pith-pipeline@v0.9.0 · 5498 in / 1277 out tokens · 42286 ms · 2026-05-07T15:08:45.346685+00:00 · methodology

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