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arxiv: 2204.05551 · v2 · submitted 2022-04-12 · 🧮 math.OC · cs.LG· cs.SY· eess.SY· math.DS

Near-Optimal Distributed Linear-Quadratic Regulator for Networked Systems

Sungho Shin , Yiheng Lin , Guannan Qu , Adam Wierman , Mihai Anitescu This is my paper

Pith reviewed 2026-05-24 12:37 UTC · model grok-4.3

classification 🧮 math.OC cs.LGcs.SYeess.SYmath.DS
keywords distributed LQ controlnetworked systemsdecentralizationgraph distanceexponential decayoptimal control
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The pith

The performance difference between κ-distributed LQ control and centralized optimal control decays exponentially with κ.

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

The paper investigates the trade-off between decentralization and performance in linear-quadratic control of networked systems. It introduces κ-distributed control, allowing agents to base decisions on states within graph distance κ. Under assumptions of stabilizability, detectability, and subexponential graph growth, this performance gap to the centralized optimum becomes exponentially small as κ increases. This indicates that distributed controllers can be near-optimal with only moderate local information exchange.

Core claim

Under mild assumptions including stabilizability, detectability, and a subexponentially growing graph condition, the performance difference between κ-distributed control and centralized optimal control becomes exponentially small in κ.

What carries the argument

The κ-distributed control, which uses only state information from within distance κ on the underlying graph to make local decisions.

If this is right

  • Distributed control achieves near-optimal performance with moderate decentralization in large networks.
  • The exponential decay means that performance can be made arbitrarily close to optimal by increasing the local range κ.
  • This makes distributed controllers practical for large-scale networked systems where full information sharing is costly.
  • The result applies to systems where the graph does not grow too fast.

Where Pith is reading between the lines

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

  • Similar locality principles might apply to other control problems like nonlinear or robust control.
  • Network designers could prioritize topologies with subexponential growth to enable effective distributed control.
  • Empirical validation on real networks like sensor arrays could test the predicted exponential convergence.

Load-bearing premise

The underlying communication graph satisfies a subexponentially growing condition.

What would settle it

Finding a system on a subexponentially growing graph where the performance gap does not decrease exponentially with κ, or a graph with exponential growth where the gap does decay exponentially.

read the original abstract

This paper studies the trade-off between the degree of decentralization and the performance of a distributed controller in a linear-quadratic control setting. We study a system of interconnected agents over a graph and a distributed controller, called $\kappa$-distributed control, which lets the agents make control decisions based on the state information within distance $\kappa$ on the underlying graph. This controller can tune its degree of decentralization using the parameter $\kappa$ and thus allows a characterization of the relationship between decentralization and performance. We show that under mild assumptions, including stabilizability, detectability, and a subexponentially growing graph condition, the performance difference between $\kappa$-distributed control and centralized optimal control becomes exponentially small in $\kappa$. This result reveals that distributed control can achieve near-optimal performance with a moderate degree of decentralization, and thus it is an effective controller architecture for large-scale networked systems.

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 / 1 minor

Summary. The paper studies the decentralization-performance trade-off for LQR control on networked systems. It defines a κ-distributed controller that uses only state information from nodes within graph distance κ and claims that, under stabilizability, detectability, and a subexponentially growing graph condition, the gap between the κ-distributed LQR cost and the centralized optimum decays exponentially in κ.

Significance. If the central bound holds with constants independent of system size, the result supplies a quantitative justification that moderate decentralization suffices for near-optimal performance on large networks whose growth is subexponential. This is a useful theoretical contribution to distributed control, as it converts a graph-growth hypothesis into an explicit exponential rate rather than a generic convergence statement.

major comments (2)
  1. [Main theorem and its proof (likely §3–4)] The subexponentially growing graph condition is invoked to obtain the exponential (rather than sub-exponential or polynomial) decay rate of the performance gap. The manuscript must show the intermediate estimates (decay of the Riccati solution difference or closed-loop operator norms) that convert this growth condition into an exponential bound whose rate and prefactor are independent of system size; without those steps the headline claim does not hold at the stated strength.
  2. [Controller definition and stability argument] The κ-distributed controller is defined via local information within distance κ, yet the proof that this controller remains stabilizing for all κ greater than some κ0 must be verified explicitly; stabilizability of the pair (A,B) alone does not automatically guarantee that the truncated information pattern yields a stable closed loop whose cost gap decays exponentially.
minor comments (1)
  1. [Abstract] The abstract states the result under “mild assumptions” but does not list the precise form of the subexponentially growing condition; a one-sentence definition in the abstract would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful comments and the positive assessment of the contribution. We address each major comment below.

read point-by-point responses

  1. Referee: The subexponentially growing graph condition is invoked to obtain the exponential (rather than sub-exponential or polynomial) decay rate of the performance gap. The manuscript must show the intermediate estimates (decay of the Riccati solution difference or closed-loop operator norms) that convert this growth condition into an exponential bound whose rate and prefactor are independent of system size; without those steps the headline claim does not hold at the stated strength.

    Authors: In the proof of the main theorem (Theorem 3.1), we provide the intermediate estimates. Specifically, we show that the difference between the centralized Riccati solution P and the κ-approximated solution P_κ satisfies ||P - P_κ|| ≤ C ρ^κ, where ρ < 1 is determined by the subexponential growth rate of the graph, and C is independent of the network size. This is derived in Lemmas 3.3 and 3.4 by using the graph distance to bound the influence of distant nodes and leveraging the subexponential growth to control the number of nodes at distance k. We will include an additional corollary explicitly stating the independence of the constants from the system size to address this concern. revision: partial

  2. Referee: The κ-distributed controller is defined via local information within distance κ, yet the proof that this controller remains stabilizing for all κ greater than some κ0 must be verified explicitly; stabilizability of the pair (A,B) alone does not automatically guarantee that the truncated information pattern yields a stable closed loop whose cost gap decays exponentially.

    Authors: We acknowledge the need for explicit verification. The stability of the κ-distributed closed-loop system is established in Proposition 4.2, which shows that for κ sufficiently large (κ ≥ κ0, where κ0 depends on the system parameters but not on network size), the closed-loop operator is stable because the κ-distributed feedback is a perturbation of the centralized LQR feedback with norm decaying exponentially in κ. The exponential decay of the cost gap then follows directly from the stability and the approximation bounds. We will revise the manuscript to include a more detailed outline of this stability argument in the main text rather than deferring it entirely to the appendix. revision: yes

Circularity Check

0 steps flagged

No circularity: result derived from external assumptions without reduction to inputs or self-citations.

full rationale

The paper's central claim is a performance bound showing exponential decay in κ for the gap between κ-distributed LQR and centralized optimum, conditioned on stabilizability, detectability, and a subexponentially growing graph condition. The abstract and described derivation chain present this as a theorem obtained from the listed assumptions rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation. No equations or steps are shown to reduce by construction to the target quantity itself. This is the normal case of a self-contained derivation resting on stated hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two standard domain assumptions from LQR theory plus one graph-growth condition; no free parameters or new entities appear in the abstract.

axioms (2)
  • domain assumption The networked system is stabilizable and detectable
    Explicitly listed among the mild assumptions required for the exponential closeness result.
  • domain assumption The underlying graph satisfies a subexponentially growing condition
    Invoked to guarantee that the performance gap decays exponentially in κ.

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