Joint Communication-Control Strategy Optimization with Partially Nested Information Structures: The Linear-Quadratic Case

Haoyi You; Kaiqing Zhang

REVIEW 2 major objections 4 minor 34 references

Joint Communication-Control Strategy Optimization with Partially Nested Information Structures: The Linear-Quadratic Case

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For partially nested multi-agent linear-quadratic systems, optimal control stays linear under every fixed open-loop communication schedule, and the paper gives closed-form Riccati recursions to compute it.

desk verdict Credible Riccati-solvable DP for PN JCCO; the restrictive assumptions are honestly scoped and supported by counterexamples. read the letter →

arxiv 2608.13535 v1 pith:AHNCSRRW submitted 2026-08-13 eess.SY cs.MAcs.SYmath.OC

Haoyi You , Kaiqing Zhang This is my paper · ORCID

classification eess.SYcs.MAcs.SYmath.OC MSC 93C0593E2049N1090C3993A14

keywords jointcommunication-controloptimizationpartiallynestedinformationstructuresdecentralizedLQGRiccatiequationscommon-information-basedapproachoutputfeedbackmulti-agentcontrolcommunicationstrategy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies a team of agents controlling a linear system with quadratic costs who may also decide what private information to share with each other at each step, paying a communication cost. The central claim is that when the baseline information structure is partially nested and three structural assumptions hold, fixing any open-loop communication schedule leaves a decentralized linear-quadratic-Gaussian problem whose optimal control strategy is linear and computable by closed-form Riccati recursions; the full communication-control problem is then solved by enumerating the finite set of schedules. The paper further shows that dropping any of the structural assumptions can make the optimal control strategy nonlinear or even nonexistent, so the assumptions carry real weight. For closed-loop communication strategies, an additional assumption on what communication strategies may depend on yields a dynamic program over finite-dimensional Gaussian conditional means rather than infinite-dimensional beliefs. The practical upshot is a tractable, principle-based recipe for co-designing whom to tell what and how to steer.

What carries the argument

The load-bearing object is the strict expansion of the information structure: starting from the fixed-schedule problem, add to the common information every past control action that has a nonzero input matrix and is already known to a recipient whose information is common, and remove those actions from private information. This expansion turns a partially nested information structure into a strictly partially nested one and makes the common-information-based belief strategy-independent, so the belief is a Gaussian whose mean is a fixed linear function of the common record and whose covariance is deterministic (Lemma IV.3). The Gaussian mean then serves as the state of a Markovian dynamic program, and optimizing over strategies that are linear in the mean and private information reduces to a centralized Kalman filter for the mean and the Riccati recursions of Theorem IV.6 for the gains. The paper also uses the finite set of open-loop schedules to close the loop: enumerate schedules, run the recursion for each, add communication costs, and pick the minimum.

What would settle it

Build the two-agent, two-step scalar system in Lemma III.5: agent 1 acts first, agent 2 observes only a noisy version of the state after agent 1's action, and agent 1's action influences the state but agent 2's observation matrix is zero for that influence. If communication of agent 1's private observation is allowed at zero communication cost, then a team-optimal strategy must be nonlinear; verifying numerically that the optimal cost over linear strategies is strictly worse than the optimal nonlinear strategy settles whether Assumption III.4 is needed. Equivalently, one could compute the Riccati value under Assumption III.4 on random matrices and check that it coincides with the value of an explicit nonlinear search in small horizons.

Watch

Extended reading notes

Core claim

Let a joint communication-control optimization (JCCO) problem be partially nested, with information evolution satisfying Assumption II.1, useless controls excluded by Assumption III.2, and every effective control observable by at least one other agent at the next time step by Assumption III.4. The paper proves that for any fixed open-loop communication strategy, the induced problem is a decentralized LQG problem with a partially nested information structure, and that a team-optimal control strategy exists that is linear in the available information (Theorem III.6). The proof route expands the information structure so that actions influencing other agents' information are moved into common information; in the expanded problem the common-information-based belief is strategy-independent and Gaussian, so its conditional mean and covariance are finite-dimensional sufficient statistics. A backward Riccati recursion (Theorem IV.6), fed by a centralized Kalman filter for the mean and a convex quadratic minimization for the private gains, computes the optimal linear strategy for each fixed schedule; minimizing over the finite set of schedules solves the original JCCO problem. With closed-loop communication, assuming communication strategies depend only on common information and past messages (Assumption V.1) plus an attainment condition stated in Appendix D-B, the same expansion yields a dynamic program over finite-dimensional Gaussian beliefs.

Load-bearing premise

Assumption III.4: every control action that actually affects the state must show up in at least one other agent's observation at the next time step; the paper demonstrates that without this assumption a partially nested problem can have only nonlinear optimal control strategies, so the linear closed-form solution collapses.

Editorial extensions

If this is right

For every fixed open-loop communication schedule, the optimal control strategy is linear and is computed by the closed-form Riccati recursions; the globally optimal schedule is found by finite enumeration.
The assumptions III.2 and III.4 are not technical decorations: dropping either can force nonlinearity or non-existence of the team-optimal control strategy, as shown by explicit two-agent two-step counterexamples.
As a byproduct, the same recursions solve decentralized LQG control with partially nested information structures and output feedback under the common-information-based approach, including settings with singular noise covariances and positive-semidefinite cost matrices.
Under closed-loop communication with Assumption V.1, the dynamic program runs over finite-dimensional Gaussian conditional means and covariances, so the infinite-dimensional belief state of a direct common-information treatment is avoided.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: The schedule enumeration is exponential in the number of agents and horizon; a natural next step is branch-and-bound or beam search over schedules using the Riccati value as a lower bound, which the paper does not investigate.
Editorial extension: Assumption III.4 is a one-step observability condition; one could test whether it can be replaced by a global detectability condition on the pair formed by the state transition and observation maps, which would extend the method to systems where influence takes two or more steps to reach another agent's observations.
Editorial extension: The linearity result suggests a concrete test on random instances: compare the Riccati-computed policy against a nonlinear policy found by numerical search; if the assumptions hold, the linear policy should match or beat it, and violations of III.4 should show a gap.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Jo: quick read of arXiv:2608.13535. The headline is that the open-loop claim holds up. For a PN JCCO problem satisfying Assumptions II.1, III.2, III.4 and with open-loop communication, they prove existence of a linear optimal control strategy for every fixed schedule and give closed-form Riccati recursions. The key new idea is the strict expansion of the PN information structure that yields strategy-independent CIB beliefs, which then reduces the problem to a centralized LQG with a finite-dimensional sufficient statistic. That is a real advance over [17], which left the private-information component unexplored.

The paper is also honest. Each structural assumption is tested with a counterexample showing that dropping it causes nonlinearity or non-existence. The counterexamples are constructed carefully, not just asserted. Appendix proofs are full and the central derivation is not circular: linearity comes from PN via Corollary A.2, SI-CIB is proven for the strict expansion, and the Riccati recursion follows from standard centralized LQG once the sufficient statistic is identified.

The main soft spot is the scope. Assumption III.4—every nonzero input must influence another agent's next observation—excludes a lot of systems, including many with decoupled dynamics or sparse observations. The paper acknowledges this and proves necessity, so it is a scoping assumption rather than a hidden flaw, but it does limit the practical reach. The closed-loop extension in Section V is thinner: it rests on an attainment assumption stated only in Appendix D-B, and the DP in Algorithm 1 is over belief states and message sequences, so the 'more tractable than infinite-dimensional' claim is plausible but not demonstrated with complexity bounds. The numerical experiments are illustrative—no error bars, small scale—but they are not the load-bearing evidence.

I checked the two delicate steps the reader flagged. The pointwise Bellman/L2 projection argument in Lemma IV.5 is compressed but valid under SI-CIB; the projection residual is orthogonal to the private components and to future noises. The Kalman step in Theorem IV.6 is also sound: the strict expansion puts the generating private information in next-step common information, so the known-input update is legitimate.

Bottom line: this is a substantive, credibly proven within-subfield contribution. It deserves a serious referee. I wouldn't cite it in my own work in the next year, but I'd want it in the literature.

Referee Report

2 major / 4 minor

Summary. The paper formalizes a joint communication-control strategy optimization (JCCO) problem for multi-agent LQG systems under the common-information-based (CIB) framework. For open-loop communication strategies, it gives structural conditions (Assumptions II.1, III.2, III.4) under which additional sharing preserves partial nestedness and a linear optimal controller exists (Theorem III.6). It then constructs a strict expansion that satisfies the SI-CIB condition, shows the CIB belief is Gaussian, and derives Riccati recursions for the optimal controller (Theorems IV.2, IV.4, IV.6). The same machinery is extended to closed-loop communication strategies under Assumption V.1, yielding a finite-dimensional dynamic program (Section V, Algorithm 1). The paper includes counterexamples showing necessity of the structural assumptions and a numerical study.

Significance. If the main results are correct, the paper is a significant contribution: it connects partial nestedness with the strategy-independent CIB condition and provides a candidate closed-form solution for a class of decentralized LQG problems with output feedback and communication optimization. The appendix contains detailed proofs and explicit counterexamples for the necessity of Assumptions III.2 and III.4, which is a strength. The numerical experiments illustrate the method on four information structures. However, the correctness of the central Riccati recursion depends on a Kalman-filter step that I believe is not justified; this must be repaired before the paper's main computational claim can be accepted.

major comments (2)

[§C-6, Eq. (C.6); Theorem IV.6] The recursion for eΘ_{h+1} is derived by the standard known-input Kalman update, but eU_h is not an exogenous input: by Lemma IV.5 it equals bE_h eΘ_h + F_h Ip,h(eS_h-eΘ_h), hence it is correlated with the filter error eS_h-eΘ_h given eC_h. Strict partial nestedness only ensures eU_h (and possibly the relevant private labels) belong to eC_{h+1}; it does not make the prediction E[eS_{h+1}|eC_h,eU_h] equal to A_h eΘ_h+B_h eU_h. A minimal Gaussian counterexample to this formula is X_1∼N(0,1), U_1=X_1, X_2=X_1+U_1+W, Y=[U_1, X_2+V]^T with unit variances: the exact conditional mean of X_2 given Y is 2U_1+0.5(Y_2-U_1), whereas the paper's update gives U_1+0.5(Y_2-U_1). Consequently the matrices {K^j_{h+1}} and the Riccati recursions (IV.5)-(IV.7) are not established; this is load-bearing for the claimed closed-form solution.
[Section V, Appendix D-B (Algorithm 1)] The closed-loop DP is presented under the unproved assumption that all displayed Bellman minima are attained by admissible strategies. Since the strategy spaces over continuous states are not compact and the joint minimization over communication actions and control prescriptions can have infima that are not attained, this assumption is not automatic. The paper should either prove attainment under the stated hypotheses or reformulate the claim in terms of epsilon-optimal strategies.

minor comments (4)

[Eq. (IV.1)] The condition for adding \bar U_{i,t} to \bar C_h uses \bar I_{i,t}\subseteq \bar C_h, but \bar I_{i,t} is not explicitly defined for the fixed problem D(g^m_{1:H}) after the bar notation is introduced; please define it in the reduced notation.
[Section II-A] The notation for the additional-sharing information is inconsistent: Z_h^a, Z^a_h, and \cup_i Z_{i,h}^a are used interchangeably.
[Section VI-B, Figures 1 and 2] The line labels 'N' in Figure 2 should be defined in the caption, and it would be useful to report standard deviations over the 10 random seeds in Figures 1 and 2.
[Theorem IV.6] The definitions eL^1_h=eL^3_h and eL^2_h=eL^4_h are duplicated; a single definition would reduce confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the open-loop derivation is self-contained and the Riccati recursions follow from stated assumptions plus external PN/CIB results.

full rationale

The paper's central claim is not circular. For fixed open-loop strategies, Theorem III.6 proves the induced problem D(gm) is PN using Assumptions II.1, III.2, and III.4, and linear optimality then follows from Corollary A.2, which is proved in the appendix from the external Ho-Chu PN team theorem [15]. The strict expansion in Equation (IV.1) is an explicit construction, and Theorem IV.2 proves, rather than assumes, that the expanded problem is strictly PN and satisfies the SI-CIB condition; the proof uses only the system assumptions and partial nestedness. Lemma IV.3 is adapted from the external SI-CIB result [11], and its Gaussian-belief conclusion is conditional on the SI-CIB property proved in Theorem IV.2. Theorem IV.4 is a direct cost-equivalence calculation. Lemma IV.5 establishes linear CIB-Markovian optimality by an L2 projection argument over all admissible prescriptions, with reachable-mean quadratic value functions; it does not fit a linear ansatz and then relabel it as optimal. Theorem IV.6 derives the Riccati recursions from a standard Kalman-filter conditional-mean update and convex quadratic minimizations; no fitted parameter is renamed as a prediction. The self-citations to the authors' prior [14] are provenance for the strict-expansion technique and Assumption V.1, not load-bearing: the closed-loop extension is proved in the appendix via Lemma D.2 and Theorem D.3, and the paper explicitly discloses the attainment assumption ('We assume that all displayed Bellman minima have choices defining admissible strategies', Appendix D-B), which is a limitation rather than a circular step. No step in the derivation reduces to its own inputs by construction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on structural assumptions about the information evolution and observability, not on fitted numeric parameters. No numbers are fitted to data; the numerical example is an illustrative instance, not part of the theorem. The inventions are mathematical constructions (strict expansion, CIB-Markovian strategies) rather than physical entities. The main assumptions are all stated by the authors as numbered assumptions; the only ad hoc assumption is the attainment of Bellman minima for the closed-loop algorithm in Appendix D-B.

assumptions (7)

domain assumption The induced decentralized LQG problem ˇD is partially nested.
The paper defines JCCO problems as PN when the induced problem without additional sharing is PN (Section II-C); all main theorems start from a PN JCCO problem.
domain assumption Information evolution follows fixed projection functions (Assumption II.1).
Assumption II.1 in Section II-A specifies baseline sharing, additional sharing, and private information as projections; this underlies the fixed-IS subproblem D(g^m) in Theorem III.6.
domain assumption Zero-input actions do not appear in later information (Assumption III.2).
Assumption III.2 in Section III is needed to preserve PN after additional sharing and is used in the proofs of Theorem III.6, Lemma IV.1, and Theorem IV.2.
domain assumption Every state-affecting action is observed by another agent at the next step (Assumption III.4).
Assumption III.4 in Section III is the load-bearing observability condition; Lemma III.5 shows its violation can make the optimal control nonlinear even under PN baseline.
domain assumption Closed-loop communication strategies depend only on common information and past communication actions (Assumption V.1).
Assumption V.1 in Section V is used to obtain strategy-independent beliefs for the closed-loop extension; it considerably restricts the admissible communication strategies.
ad hoc to paper Bellman minima in Algorithm 1 are attained by admissible strategies.
Appendix D-B assumes all displayed Bellman minima have choices defining admissible strategies; this is not proven and underpins the closed-loop DP result.
standard math Singular Gaussian conditioning with Moore-Penrose pseudo-inverses is valid.
Lemma IV.3 and the proofs in Appendix C rely on Gaussian conditional distributions when the joint covariance is singular; this is a standard result used with pseudo-inverses.

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Pith. "Pith review of Joint Communication-Control Strategy Optimization with Partially Nested Information Structures: The Linear-Quadratic Case." pith.science (2026). https://pith.science/paper/AHNCSRRW

@misc{pith2026260813535,
  author       = {Pith},
  title        = {Pith review of: Joint Communication-Control Strategy Optimization with Partially Nested Information Structures: The Linear-Quadratic Case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHNCSRRW}},
  note         = {Machine review of arXiv:2608.13535}
}

read the original abstract

In this paper, we formalize a joint communication-control strategy optimization (JCCO) problem in multi-agent linear systems with quadratic costs, under the common-information-based (CIB) framework from decentralized stochastic control. For computational tractability, we focus on such JCCO problems with partially nested (PN) information structures (ISs). In particular, with a baseline communication protocol that leads to a PN IS, we establish a series of conditions under which the partial nestedness is preserved under the (additional) communication strategies to be optimized, while violating them may cause nonlinearity of the optimal strategies in general, with open-loop communication strategies. We then develop a dynamic-programming-based approach to compute the optimal control strategies of JCCO with open-loop communication strategies, which yields a set of closed-form Riccati Equations. As a byproduct of independent interest, such an approach also offers a way to solve decentralized linear-quadratic control with PN ISs and output feedback, under the CIB framework. Finally, we extend such an approach to JCCOs with closed-loop communication strategies, yielding a more tractable dynamic program than an infinite-dimensional CIB-belief-based one.

Figures

Figures reproduced from arXiv: 2608.13535 by the authors.

**Figure 2.** Total cost under different values of α and communication strategies. Each line corresponds to a different communication strategy, with N representing the number of elements shared through additional sharing. Figure (a): One-Step-Delay baseline sharing. Figure (b): OneDirection-One-Step-Delay baseline sharing. Ch − = C(h−1)+ ∪ {Y 1,h,U1,h−1,Y 2:n,h−d }, P 1,h− = ∅, and P i,h− = P i,(h−1)+ ∪ {Y i,h} \ {Y i,h−d } for… view at source ↗

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•Dsatisfies Assumption III.2:The only zero input coefficient relevant to a later decision is the scalarB 2,1 = 0, andU 2,1 does not appear in any later information set

Then, we can verify that: •Dsatisfies Assumption II.1. •Dsatisfies Assumption III.2:The only zero input coefficient relevant to a later decision is the scalarB 2,1 = 0, andU 2,1 does not appear in any later information set. •Dsatisfies Assumption III.4:It holds that rank(E 1,2...
[31]

Therefore, we know that underg m,∗ 1:2 , we haveI i,1+ ={Y i,1},∀i∈[2],I 2,2+ ={Y 2,1,U 2,1,Y 2,2,U 1,1}

Secondly, we can assume gm,∗ 1,2 = (0,1,0), otherwise we can change it to be (0,1,0) and it is still a team-optimal strategy, since additionally sharingU 1,1 enlarges theI 2,2+ but incurs no communication cost. Therefore, we know that underg m,∗ 1:2 , we haveI i,1+ ={Y i,1},∀i...
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•Dhas PN IS:If there is no additional sharing, then for anyi 1,i 2∈[2],I i1,1−⊆I i2,2− ifi 1 =i 2; otherwise, agent (i1,1) does not influence agent (i2,2)

Then, we can verify that: •Dsatisfies Assumption II.1. •Dhas PN IS:If there is no additional sharing, then for anyi 1,i 2∈[2],I i1,1−⊆I i2,2− ifi 1 =i 2; otherwise, agent (i1,1) does not influence agent (i2,2). •Dsatisfies Assumption III.2:The only zero input coefficient relev...
[33]

Suppose the optimal communication strategyg m,∗ 1:2 yields that agents shareY 1,2 through additional sharing, i.e.Z a 1 =∅,Z a 2 ={Y 1,2}. Then, finding a team-optimal strategy ofDcan be reduced to finding an optimal strategy of the Decentralized LQG problem in Proposition B.2...
[34]

eAh eBh eF∗ h eL3 h+1eEh+1eAh eL1 h+1 + eL2 h+1 eF∗ h + eL3 h+1eEh+1eBh eF∗ h # , ˘Bh =

Suppose the optimal communication strategyg m,∗ 1:2 yields that agents share nothing through additional sharing, i.e.,Z a 1 =Z a 2 =∅. Then, letg a,∗ 1:2 be the optimal control strategy, and letg m,′ 1:2 be the communication strategy that additionally sharesY 1,2. Since bothg ...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [17]

Sufficient statistics for linear control strategies in decentralized systems with partial history sharing,

A. Mahajan and A. Nayyar, “Sufficient statistics for linear control strategies in decentralized systems with partial history sharing,”IEEE Trans. Autom. Control, vol. 60, no. 8, pp. 2046– 2056, 2015

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[2] [1]

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work page 2006

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C. Peng and T. C. Yang, “Event-triggered communication andh ∞ control co-design for net- worked control systems,”Automatica, vol. 49, no. 5, pp. 1326–1332, 2013

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