{"id":"97235789-4a9e-4d86-b097-3f8b5ba3a707","arxiv_id":"1908.06070","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For two independent symmetric unimodal sources, the optimal energy-harvesting scheduling policy transmits the observation farther from its mean whenever that distance exceeds a time- and energy-dependent threshold.","lead":"This paper finds the best way for a battery-powered scheduler to choose which of two sensors should send its measurement to its estimator over a shared wireless link. The optimal rule is simple: measure how far each observation is from its mean, and transmit the larger one only when that distance clears a threshold that depends on time and battery level.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Missing finite-second-moment condition: as stated, Theorem 1 covers Cauchy-like symmetric unimodal sources where the MSE objective is infinite and Lemma 5's Hardy-Littlewood step lacks integrability.","rationale":"I read the paper in good faith and checked the main line of argument: the MDP reduction in Lemma 1, the information-structure expansion in Section IV, the coordinator POMDP in Section V, the pointwise scheduler optimization in Lemma 4, and the rearrangement step in Lemma 5. The proof is coherent when the sources have finite second moment. The strongest load-bearing concern is not an algebraic slip in the optimization but an omitted regularity condition on the source distributions: Theorem 1 as stated applies to every symmetric unimodal density, including Cauchy-like densities for which the MSE objective is infinite and the Hardy-Littlewood inequality in Appendix B is not an ordering of finite quantities. This directly affects the central claim because Lemma 5 is the step that forces the estimator prescriptions to be the centers a_i, and its proof relies on L¹ integrability of H_t^e. Adding the explicit finite-second-moment assumption repairs the statement without changing the proof's substance. The reader's verdict of CONDITIONAL is appropriate; I also agree that the Section IX claims for N sensors and unequal weights are asserted without proof, but that is secondary to the two-sensor theorem that the paper's central claim rests on. My concern overlaps with the finite-second-moment part of the reader's weakest_assumption, hence partial agreement.","tokens_in":28815,"tokens_out":26855,"duration_ms":279957,"concrete_test":"Take π1 = π2 = Cauchy(0,1), which is symmetric and unimodal in the sense of Definition 1. Evaluate the cost in Eq. (8) under any feasible policy: at every time at most one source is transmitted, so the other source contributes E[‖X_i‖²] = ∞, giving J(f,g1,g2) = ∞ for every strategy profile. This shows the problem is ill-posed on the stated class. If the authors instead add the finite-second-moment assumption E[‖X_i − a_i‖²] < ∞ to Theorem 1 and re-state Lemma 5 with H ∈ L¹, the Cauchy counterexample is excluded and the proof goes through; without such an assumption, Theorem 1 as written is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central two-sensor construction appears internally sound when each source has finite second moment: Lemma 4's pointwise minimization is correct, Lemma 5's rearrangement argument proves x̃=0, and the information-structure expansion is valid. The load-bearing gap is that Theorem 1 states only that π1 and π2 are symmetric and unimodal in the sense of Definition 1, with no finite-second-moment assumption. The objective in Eq. (8) contains E[‖X_i^t − X̂_i^t‖²], so if π_i has infinite second moment, J is not a finite-valued functional. In Lemma 5's proof, H_t^e(x̃; x) = max{0, max{B, κ} − ‖x1 − x̃1‖²} with B = ‖x2 − x̃2‖²; the Hardy-Littlewood inequality used in Eq. (86) is an L¹ rearrangement inequality, and ∫ H π1 can be infinite when E[‖X2‖²] = ∞. A Cauchy source is symmetric and unimodal around its location, satisfies Definition 1, yet makes every feasible strategy have infinite cost: at each time at most one source is transmitted, so the other source contributes an infinite second-moment term. Thus the 'global optimality' statement in Theorem 1 is vacuous for such sources, and the proof's estimates do not establish a finite optimum. The theorem should explicitly assume E[‖X_i − a_i‖²] < ∞, and the Section IX extensions inherit this requirement.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a finite-horizon remote estimation problem with two sensor-estimator pairs sharing a communication channel managed by an energy-harvesting scheduler. The scheduler observes both source realizations and decides, at each time, which (if any) measurement is transmitted, subject to a battery state. The goal is to jointly minimize the sum of mean-squared estimation errors and a transmission cost. The main contribution is a structural result: when the source densities are symmetric and unimodal around their means, the optimal policy is a threshold policy that transmits the source with the largest deviation from its mean only if that deviation exceeds a threshold depending on time and energy; estimators use the received value or the source mean otherwise. The proof uses an expansion of the estimators' information, a common-information-based POMDP formulation, and a rearrangement inequality to show that the optimal estimator prescriptions are the source means. A recursive algorithm computes the thresholds, and numerical examples compare the policy with a blind scheduling benchmark.","tokens_in":29141,"tokens_out":13074,"duration_ms":115535,"significance":"If established, this is a notable exact solution to a non-convex team decision problem with non-classical information structure and energy constraints. The result extends prior one-shot and single-sensor analyses. The proof is largely coherent: Lemma 4's pointwise minimization and Lemma 5's rearrangement argument are valid under the stated assumptions, and the threshold computation is explicit. The paper would be strengthened by stating the necessary finite-second-moment condition, which is currently implicit. The techniques (information structure expansion, common information) are of interest to the networked estimation community.","major_comments":[{"comment":"The theorem assumes only that π1 and π2 are symmetric and unimodal in the sense of Definition 1. This does not guarantee that the cost functional J in Eq. (8) is finite: if a source has infinite second moment (e.g., a Cauchy distribution), then at every time the untransmitted source contributes E[||X_i - a_i||^2] = ∞, making J infinite for every admissible strategy. Under those conditions the global optimality statement is vacuous and the minimization in Eq. (8) is not well-posed as a finite-valued problem. The proof of Lemma 5 in Appendix B, specifically the Hardy-Littlewood step in Eq. (86) and the subsequent integration over the other source, presumes finite second moments. Please add the explicit assumption E[||X_i - a_i||^2] < ∞ for i=1,2 to Theorem 1, and note that Section IX inherits this requirement.","section":"Theorem 1, Eq. (8)"}],"minor_comments":[{"comment":"The claimed global optimality for the unequal-weights and unequal-communication-cost extension is asserted without proof; the paper should either provide a proof or explicitly label this as a conjecture.","section":"Section IX-B, Eqs. (68)-(69)"},{"comment":"The statement that Theorem 1 extends to an arbitrary number of sensors is given without proof; a brief indication of how the proof in Lemmas 4-5 generalizes would be helpful.","section":"Section IX-A"},{"comment":"The definition of the threshold τ*_t(e) in Lemma 4 contains a stray '1' after the square root, and the tie-breaking rule for arg max in Section II-D should be explicitly invoked in Theorem 1 to make (11) well-defined when ‖x1-a1‖ = ‖x2-a2‖.","section":"Section VI, Lemma 4"},{"comment":"The displayed equation for the blind scheduling strategy is missing a closing brace, and the text 'the is given by' should be corrected.","section":"Section VIII, Eq. (60)"},{"comment":"There are typographical errors, including 'transmismitted' and 'accross,' that should be corrected in a revision.","section":"Section I"},{"comment":"The monotonicity of Vπ_t(e) in e is used but not proved; a brief sentence justifying that more energy cannot increase the optimal cost would be useful.","section":"Appendix B, proof of Lemma 5"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope and the central two-sensor proof is convincing once the finite-second-moment assumption is added. The unproved unequal-weight extension is a concern; if it cannot be proved, it should be presented as a conjecture. The numerical examples are illustrative but not necessary for the main claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the two-sensor theorem is the real thing, provided you add the unstated finite-second-moment assumption. The extensions in Section IX are asserted, not proved, and should be downgraded or completed.\n\nWhat's new: this is the first finite-horizon jointly optimal scheduling/estimation policy for two non-collocated estimators with an energy-harvesting scheduler and i.i.d. symmetric unimodal sources. It moves past the one-shot result and the single-sensor battery result by coupling the scheduling decision to both observations and the battery state. The proof is a clean combination of information-structure expansion and common-information POMDP, with a rearrangement inequality to pin the estimator prescription to zero. I checked Lemmas 4 and 5: the pointwise minimization gives the threshold, and the Hardy–Littlewood step is executed correctly for sources with finite second moments. The threshold recursion is well-defined and the numerics illustrate the policy.\n\nSoft spots. First, Theorem 1 states only symmetry and unimodality, but the MSE objective and Lemma 5 need finite second moments. A Cauchy source satisfies the stated assumptions and gives infinite cost under every policy; the theorem becomes vacuous and the rearrangement argument lacks integrability. This is a minor fix—add one line to Theorem 1 and the extensions—but it is a real gap. Second, Section IX-A asserts the N-sensor case without proof, and IX-B gives a formula for unequal weights/costs without derivation. These may be true, but as written they are unsupported claims and should be either proved or explicitly labeled as conjectures. Third, the monotonicity of V_t in e is asserted with an intuitive argument; acceptable, but the proof leans on it.\n\nOverall: the central result holds up under the right assumptions. The paper deserves a serious referee. I'd send it out with a request to add the moment condition and to rewrite Section IX as conjectures if no proofs are provided.","headline":"The two-sensor theorem is the real thing, but the missing finite-second-moment assumption and unproved Section IX extensions need to be addressed before this is taken as fully correct.","tokens_in":29645,"tokens_out":3777,"would_cite":true,"duration_ms":36445,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"For two symmetric unimodal sources, the globally optimal scheduling-estimation strategy is a threshold rule that transmits the observation farthest from its center only when its distance exceeds a threshold depending on time and battery…","keywords":["networked estimation","energy harvesting","sensor scheduling","remote estimation","team decision","common information","threshold policy","symmetric unimodal distributions"],"falsifier":"Run a one-shot version with two independent scalar sources whose densities are not symmetric and unimodal (for example, one exponential and one bimodal mixture), and search numerically over estimation defaults and thresholds for the best achievable cost; if any nonzero default beats the center default when the other sensor's packet is transmitted, Lemma 5 collapses. A sharper test is to evaluate the auxiliary function $J_t^e(\\tilde x_t)$ of Eq. (79) at $\\tilde x_t=0$ and at small perturbations: under the paper's assumption the minimum must occur at zero, so any perturbation that lowers the cost is a counterexample.","tokens_in":1858,"feed_emoji":"🔋","tokens_out":2304,"duration_ms":75010,"temperature":0.7,"pith_summary":"This paper asks how a battery-powered scheduler should decide which of two sensors' measurements to send over a one-packet-per-slot network, while the estimators try to minimize mean-squared error plus transmission cost. The authors claim that, for independent sources with symmetric and unimodal distributions, this joint scheduling-estimation problem has an exact globally optimal solution, despite the non-convexity created by signaling. The optimal scheduler transmits the observation farthest from its distribution center only when that distance clears a threshold set by the remaining time and the battery level; the optimal estimators use the received sample when it arrives and the distribution center otherwise. The paper also gives a backwards recursion to compute the thresholds and extends the result to any number of sensors and to unequal weights and communication costs. If the claim is right, a team problem that is usually intractable becomes a simple offline threshold design.","feed_headline":"Two sensors, one channel: optimal scheduling is a threshold rule","feed_subtitle":"For symmetric unimodal sources, transmit the farthest observation only when its distance passes a battery-and-time threshold.","key_machinery":"The central proof device is the combination of an information-structure expansion with the common information approach. Adding the scheduler's energy history and past transmission outcomes to each estimator's information creates common information among all decision makers, allowing the team problem to be re-expressed as a single coordinator POMDP whose state is $(X_t,E_t)$ and whose belief on $X_t$ is simply the product density $\\pi_1\\pi_2$. The coordinator's dynamic program reduces, for fixed estimation defaults, to a finite-dimensional optimization over those defaults. The Hardy-Littlewood rearrangement inequality then closes the argument: after centering, the error-minimization term is a symmetric decreasing function of the distance from the default, so the zero default—the source center—is a global minimizer. This rearrangement step is what converts an infinite-dimensional team problem into a scalar threshold rule.","core_discovery":"The paper claims that the sequential team problem—two sensors feeding two estimators over a shared network that carries one packet per time slot, with a scheduler whose finite battery harvests random energy—is exactly solvable. Provided each source density is symmetric and unimodal around its center $a_i$ in the radial sense of Definition 1, the strategy profile $(f^\\star, g_1^\\star, g_2^\\star)$ of Theorem 1 is globally optimal for Problem 1. The scheduler's rule is a distance threshold: at time $t$ with energy $e$, it sends nothing if both $\\|x_i-a_i\\| \\le \\tau_t^\\star(e)$, and otherwise it sends the sensor with the largest distance $\\|x_i-a_i\\|$. The estimators use $x_i$ when a packet arrives and $a_i$ when none does. The proof expands the estimators' information so that a coordinator with common information can solve a POMDP, then uses the Hardy-Littlewood rearrangement inequality to show the center default is globally optimal; because the optimal solution is adapted to the original information structure, it is also optimal there.","pith_inferences":["Editorial inference: for correlated or Markov sources, the paper's rearrangement step fails, but the threshold rule remains a natural heuristic, and its suboptimality gap could be quantified against a dynamic-programming lower bound.","Editorial inference: because the value-of-information curves in the examples are nonmonotone in battery capacity, the threshold recursion could be repurposed as a battery-sizing tool that selects the capacity maximizing the benefit of closed-loop scheduling.","Editorial inference: the information-expansion-plus-common-information argument is not tied to the squared-error cost, so similar threshold policies are plausible for other symmetric performance measures, such as $p$-norm errors, with the threshold recursion adjusted accordingly."],"forward_implications":["The optimal thresholds can be computed offline by backward recursion, so the scheduler only needs to compare current distances with a precomputed table indexed by time and battery level.","When the remaining battery energy exceeds the remaining time slots, the optimal threshold is zero and the scheduler always transmits the observation farthest from its center; as the battery depletes, it waits for observations of increasingly large magnitude.","The optimal estimators are memoryless defaults: use the received sample if it arrives, otherwise use the known center; no signaling benefit changes the default estimate.","The same threshold structure holds for any number of sensors: transmit the argmax over all distances if the maximum exceeds the threshold.","With unequal sensor weights and communication costs, the no-transmission region becomes a rectangle defined by two thresholds and the boundary between transmitting sensor 1 and sensor 2 becomes a hyperbola-like curve."],"supporting_citations":[{"why":"Supplies the one-shot version of this problem whose joint optimality result is extended here to the sequential energy-harvesting setting.","marker":"[20]"},{"why":"Defines the networked-estimation setup with a shared medium and the largest-magnitude scheduling heuristic that the optimal policy here generalizes.","marker":"[19]"},{"why":"Shows the common-information equivalence technique for remote estimation with an energy-harvesting sensor, which the proof adapts.","marker":"[10]"},{"why":"Provides the common information approach used to convert the non-classical team problem into a coordinator POMDP.","marker":"[30]"},{"why":"Supplies the proposition that a solution optimal under an expanded information structure and adapted to the original structure is optimal for the original problem, plus the non-classical team framework.","marker":"[5]"},{"why":"Provides the standard MDP results used in Lemma 1 to restrict the scheduler to strategies based on current observations, energy history, and past outputs.","marker":"[31]"},{"why":"Lecture notes on rearrangement inequalities that state the Hardy-Littlewood inequality used in Appendix B.","marker":"[33]"},{"why":"Original source of the Hardy-Littlewood rearrangement inequality used to prove that the zero default is globally optimal.","marker":"[34]"}],"fun_headline_variants":["Two sensors, one channel: optimal schedule is a distance threshold","Energy-harvesting scheduler: send farthest sensor past threshold","Symmetric sources yield an optimal distance-threshold scheduler","Battery-limited scheduler: threshold rule is globally optimal","Optimal scheduling: send only when distance exceeds time-varying threshold"],"cache_read_input_tokens":31744,"weakest_assumption_plain":"The entire result depends on each source distribution being symmetric and unimodal in the strong sense that points closer to the center always have density at least as high as points farther away, together with each source having a finite second moment; if those fail, the threshold policy may stop being globally optimal.","fun_headline_variants_meta":{"raw":{"variants":["Two sensors, one channel: optimal schedule is a distance threshold","Energy-harvesting scheduler: send farthest sensor past threshold","Symmetric sources yield an optimal distance-threshold scheduler","Battery-limited scheduler: threshold rule is globally optimal","Optimal scheduling: send only when distance exceeds time-varying threshold"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000315,"raw_usage":{"total_tokens":1787,"prompt_tokens":950,"completion_tokens":837,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":566,"completion_tokens_details":{"reasoning_tokens":755}},"tokens_in":566,"tokens_out":837,"duration_ms":7839,"temperature":1.0,"reasoning_tokens":755,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:57:06.026201+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a one-shot version with two independent scalar sources whose densities are not symmetric and unimodal (for example, one exponential and one bimodal mixture), and search numerically over estimation defaults and thresholds for the best achievable cost; if any nonzero default beats the center default when the other sensor's packet is transmitted, Lemma 5 collapses. A sharper test is to evaluate the auxiliary function $J_t^e(\\tilde x_t)$ of Eq. (79) at $\\tilde x_t=0$ and at small perturbations: under the paper's assumption the minimum must occur at zero, so any perturbation that lowers the cost is a counterexample.","supporting_citations":[{"cited_title":"Optimal sensor scheduling strategies in networked estimation,","cited_arxiv_id":null,"evidence_quote":"Supplies the one-shot version of this problem whose joint optimality result is extended here to the sequential energy-harvesting setting."},{"cited_title":"Networked state estimation over a shared communication medium,","cited_arxiv_id":null,"evidence_quote":"Defines the networked-estimation setup with a shared medium and the largest-magnitude scheduling heuristic that the optimal policy here generalizes."},{"cited_title":"Optimal strategies for communication and remote estimation with an energy harvesting sensor,","cited_arxiv_id":null,"evidence_quote":"Shows the common-information equivalence technique for remote estimation with an energy-harvesting sensor, which the proof adapts."},{"cited_title":"Decentralized stochastic control with partial history sharing: A common information approach,","cited_arxiv_id":null,"evidence_quote":"Provides the common information approach used to convert the non-classical team problem into a coordinator POMDP."},{"cited_title":"Yuksel and T","cited_arxiv_id":null,"evidence_quote":"Supplies the proposition that a solution optimal under an expanded information structure and adapted to the original structure is optimal for the original problem, plus the non-classical team framework."},{"cited_title":"Kumar and P","cited_arxiv_id":null,"evidence_quote":"Provides the standard MDP results used in Lemma 1 to restrict the scheduler to strategies based on current observations, energy history, and past outputs."},{"cited_title":"A short course on rearrangement inequalities,","cited_arxiv_id":null,"evidence_quote":"Lecture notes on rearrangement inequalities that state the Hardy-Littlewood inequality used in Appendix B."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Original source of the Hardy-Littlewood rearrangement inequality used to prove that the zero default is globally optimal."}],"review_version":1}