{"id":"77153bd3-3683-409d-b86b-6ef37af422a4","arxiv_id":"2204.05551","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"κ-distributed LQR controllers achieve performance exponentially close to the centralized optimum under stabilizability, detectability, and subexponential graph growth.","lead":"The paper shows that for linear-quadratic control of agents on a graph, a controller using only state information from within graph distance κ achieves a performance gap to the centralized optimum that shrinks exponentially in κ. A smart generalist might care because this indicates moderate local communication can deliver near-optimal control in large networked systems without full centralization.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Exponential decay of performance gap with κ relies on subexponentially growing graph condition whose precise use in the proof is the least verified link.","rationale":"The reader's weakest_assumption matches the single assumption required to upgrade the performance gap from sub-exponential to exponential; no other internal inconsistency is visible from the abstract or the stated claim structure. Because the full manuscript is now available, the concrete test above directly checks whether that assumption is doing the claimed work.","tokens_in":1630,"tokens_out":346,"duration_ms":26402,"concrete_test":"Identify the main theorem and the lemma that invokes the subexponentially growing condition; recompute the performance-gap bound after replacing the condition with a polynomially growing graph (e.g., a d-dimensional grid); if the resulting bound on the gap is no longer exponential in κ, the claimed rate depends critically on the stated graph assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim asserts that, under stabilizability, detectability and the subexponentially growing graph condition, the gap between κ-distributed LQR cost and the centralized optimum decays exponentially in κ. The graph-growth condition is explicitly invoked to produce the exponential rate (rather than a slower rate). If the derivation only obtains a sub-exponential or polynomial bound once the condition is relaxed, or if the constants hidden in the exponential depend on graph parameters that grow with system size, the headline statement does not hold at the claimed strength. The abstract alone does not display the intermediate estimates (e.g., decay of the Riccati solution or closed-loop operator norms) that convert the growth condition into an exponential performance gap.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1787,"tokens_out":438,"duration_ms":41028,"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":[{"comment":"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.","section":"Main theorem and its proof (likely §3–4)"},{"comment":"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.","section":"Controller definition and stability argument"}],"minor_comments":[{"comment":"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.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the thoughtful comments and the positive assessment of the contribution. We address each major comment below.","responses":[{"response":"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_made":"partial","referee_comment":"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."},{"response":"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_made":"yes","referee_comment":"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."}],"tokens_in":1296,"tokens_out":490,"duration_ms":45865,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that under stabilizability, detectability, and a subexponentially growing graph condition, the performance gap between a κ-distributed LQR controller and the centralized optimum decays exponentially in κ. This gives a quantitative handle on how much local communication is needed for near-optimal behavior on large networks.","headline":"The paper derives an exponential decay bound on the LQR suboptimality gap for κ-distributed controllers, but the rate depends on a subexponentially growing graph condition whose role in the estimates is not visible from the abstract.","tokens_in":2260,"tokens_out":154,"would_cite":false,"duration_ms":22898,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"under mild assumptions, including stabilizability, detectability, and a subexponentially growing graph condition, the performance difference ... becomes exponentially small in κ"},{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean","rs_theorem":"absolute_floor_iff_bare_distinguishability","paper_passage":"exponential decay property of the optimal gain (Theorem 3.3)"}],"headline":"Distributed LQR truncation analysis on graphs; no RS-shaped cost, ratio symmetry or forcing chain","alignment":"orthogonal","rationale":"The paper's core machinery (exponential decay of Riccati gain entries via graph-induced banded KKT matrices, subexponential node growth, stabilizability/detectability) is standard finite-dimensional linear algebra and perturbation theory. It invokes no J-cost functional equation, no φ-ladder, no 8-tick periodicity, no recognition-cost forcing, and makes no parameter-free derivation of physical constants. The domain (graph-structured optimal control) lies outside the RS forcing chain; the claims neither echo nor contradict any named RS theorem.","tokens_in":61905,"confidence":"high","tokens_out":312,"duration_ms":10034,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The performance difference between κ-distributed LQ control and centralized optimal control decays exponentially with κ.","keywords":["distributed LQ control","networked systems","decentralization","graph distance","exponential decay","optimal control"],"falsifier":"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.","tokens_in":2546,"feed_emoji":"","tokens_out":433,"duration_ms":24384,"temperature":0.7,"pith_summary":"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.","feed_headline":"κ-distributed control nears centralized LQ optimum exponentially","feed_subtitle":"Performance gap shrinks exponentially in κ under stabilizability and subexponential graph growth conditions","key_machinery":"The κ-distributed control, which uses only state information from within distance κ on the underlying graph to make local decisions.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["κ-distributed LQ closes gap to optimum exponentially","Distributed LQR gap decays exponentially in κ on networks","κ-local control nears LQ optimum exponentially","Exponential decay of centralized to distributed LQ gap in κ"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The underlying communication graph satisfies a subexponentially growing condition.","fun_headline_variants_meta":{"raw":{"variants":["κ-distributed LQ closes gap to optimum exponentially","Distributed LQR gap decays exponentially in κ on networks","κ-local control nears LQ optimum exponentially","Exponential decay of centralized to distributed LQ gap in κ"]},"model":"grok-4.3","cost_usd":0.004707,"raw_usage":{"total_tokens":2268,"prompt_tokens":555,"num_sources_used":0,"completion_tokens":61,"cost_in_usd_ticks":47074500,"prompt_tokens_details":{"text_tokens":555,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1652,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":555,"tokens_out":61,"duration_ms":14961,"temperature":1.0,"reasoning_tokens":1652,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-24T12:37:43.623328+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}