{"id":"9ae7a7ac-2d1e-4be8-8718-d0a7b40a4f31","arxiv_id":"1908.01119","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a scalar Gauss-Markov process watched by constant-gain remote estimators, scheduling the packet with the smallest age-and-precision index minimizes the expected cumulative estimation error.","lead":"This paper finds the best way to choose which sensor packet to send over a lossy link when the goal is accurate remote estimation: send the packet with the lowest combined age-and-noise index. It shows that always sending the freshest packet is not always best, because a slightly older but less noisy measurement can be more useful.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1 uses infinite-horizon sums in Eq. (25) while the claim is finite-horizon optimality; no truncation argument shows the tail beyond T is negligible or sign-definite, so the exchange argument is incomplete.","rationale":"The non-idling concern raised by the reader is real but largely a modeling ambiguity: the action space in Section IV is 'which packet to transmit', so feasible policies are already non-idling. The finite-horizon truncation is a direct gap in the proof of the stated theorem. It is concrete, located in Eq. (25) and Lemma 5, and testable with a small example. The paper has independent merit: the index derivation and the ordering-invariance lemma are sound, and the index itself is a genuine contribution. The appropriate verdict remains CONDITIONAL, since the proof gap is addressable but needs to be fixed before the finite-horizon optimality claim is fully established.","tokens_in":13942,"tokens_out":22862,"duration_ms":215226,"concrete_test":"Set T=3, a=0.5, K=0.5, p=1, σ^2=1, and an initial queue with two packets A and B where W^2_A << W^2_B. Let new packets with indices smaller than B arrive at t=2 and t=3, so the Index Policy never transmits B within the horizon, while the comparison policy transmits B at t=1 and then follows the index rule. Compute exactly (e.g., by direct enumeration of the Gaussian noise variances using the recursion in Eq. (6)) the finite-horizon difference E∑_{t=1}^3(e^2_{πidx}(t)−e^2_{swap}(t)). Compare this value with the RHS of Eq. (25) evaluated with the infinite sums. If the two differ, the proof's use of infinite sums is invalid for the stated finite-horizon objective.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Corollary 1) is that the Index Policy minimizes the finite-horizon objective E(∑_{t=1}^T e^2(t)). In the proof of Theorem 1, the cost difference between the swapped policy and the index policy is written in Eq. (25) as ∆(ψ1,ψ2,1)∑_{t∈[1,T2−1]}α_{1,t} + (∆(ψ2,ψ1,T2)+∆(ψ1,ψ2,1)α_{1,T2})∑_{t≥T2}α_{T2,t}. The second sum is over t≥T2 with no upper limit. For the finite-horizon objective, this expression must be truncated at T. The proof does not condition on T1≤T or argue that the tail beyond T is zero; in fact, after both swapped packets are delivered, the error processes differ because the two noise terms enter with different delays and hence different closed-loop coefficients, so the tail contribution does not vanish. If T1>T (e.g., ψ2 has a very large index and is never served within the horizon because of continuing arrivals), the infinite-sum expression is not the finite-horizon difference. Lemma 5 only establishes positivity of the infinite tail expression, not the truncated one. Thus the proof establishes at best a related infinite-horizon statement, not the finite-horizon optimality claimed in Corollary 1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a discrete-time remote estimation system in which a network node receives noisy observations of a scalar Gauss-Markov process from multiple sources with different sensor precisions, queues the resulting packets, and transmits one packet per slot over an unreliable channel to a remote estimator running a constant-gain linear filter. The objective is to minimize the expected cumulative squared estimation error over a finite horizon T. The authors define an index W^2_psi(t) that combines a packet's age and its source's noise variance, propose the Index Policy that transmits the packet with the smallest index, and claim in Corollary 1 that this policy is optimal under Assumption 1. They also extend the result to N processes whose channel access is chosen i.i.d., showing that the optimal scheduler is a product of single-process index policies (Theorem 2).","tokens_in":14107,"tokens_out":7495,"duration_ms":69898,"significance":"If the optimality proof can be made rigorous, the paper would provide a clean, index-based scheduling rule for remote estimation and would establish the important qualitative point that age-optimal scheduling is not necessarily estimation-optimal when sensor precisions vary. The derivation is principled: the index is computed from the model's error covariance rather than fitted, and the ordering-invariance property in Lemma 2 makes implementation efficient. The significance is conditional, however, because the proof of Theorem 1 as written does not yet establish the finite-horizon claim; the infinite-horizon tail and undefined stopping times are load-bearing issues rather than cosmetic ones.","major_comments":[{"comment":"The proof of Theorem 1 does not establish the finite-horizon claim in Corollary 1. The objective (4) is E(sum_{t=1}^T e^2(t)), but the cost difference in Eq. (25) contains the term (Delta(psi2,psi1,T2)+Delta(psi1,psi2,1)alpha_{1,T2}) sum_{t>=T2} alpha_{T2,t} with no upper limit. Lemma 5 proves positivity of the infinite-tail expression, not of the truncated expression. If T1>T, so that psi2 is never delivered within the horizon, T1 and T2 in Eq. (16) are not even defined on that sample path. Even when T1<=T, the tail t>T does not vanish because the two packets' noise terms enter the error process at different times with different closed-loop coefficients. The proof therefore needs a finite-horizon truncation argument that shows the sign of the cost difference restricted to t<=T, or a reformulation of the theorem as an infinite-horizon result.","section":"Section VI, Theorem 1 and Eq. (25)"},{"comment":"The exchange argument assumes that all compared policies are non-idling: the proof states 'since they are non-idling, the a(t) are the same under all policies.' This restriction is not stated in the problem formulation in Section IV, where the action is only 'which packet to transmit' and there is no explicit idle action. If a policy were allowed to remain idle or transmit a dummy packet while Q(t) is nonempty, the closed-loop gain process a(t) would differ across policies, and the comparison in Eq. (25) would not apply. The authors should either state non-idling as an explicit assumption or extend the argument to allow idling.","section":"Section VI, Theorem 1 proof (non-idling assumption)"},{"comment":"The stopping times T1 and T2 defined in (16) are not guaranteed to exist within the finite horizon. For a sample path on which psi2 is never transmitted successfully under pi_idx by time T, the set {t: psi(t;pi_idx)=psi2} is empty. Lemma 3 and Lemma 4 rely on T1 and T2 being finite and on T2<=T1, but no conditioning on these events is given, and the complement is not analyzed. The proof should define T1 and T2 as hitting times truncated at T (or as infinity) and handle the cases where they exceed T separately.","section":"Eq. (16), Lemma 3"}],"minor_comments":[{"comment":"The definition alpha_{s,t} := product_{m=s}^t a^2(t) appears to have a typo: the product index and the variable inside the product should be m, so that alpha_{s,t} depends on a^2(m) for m=s,...,t.","section":"Eq. (21)"},{"comment":"In the first equality of Eq. (25), the text says 'we used Lemma 2 in order to deduce Delta(psi2,psi1,1) = (a^2)^{T2} Delta(psi2,psi1,1)', but the correct statement is Delta(psi2,psi1,T2) = (a^2)^{T2} Delta(psi2,psi1,1).","section":"Eq. (25)"},{"comment":"In the chain of inequalities in Lemma 5, the step that drops the term -(a_c^2)^T/(1-a^2) reverses the direction of the inequality, because that term is negative; the proof should be corrected or clarified.","section":"Lemma 5 proof"},{"comment":"The paper says sigma_psi is the variance of the observation noise, but the model y_psi = x(t_psi) + sigma_psi w_psi with w_psi of unit variance makes sigma_psi a standard deviation; the notation sigma^2_{s,psi} in Eq. (9) should be aligned with this. There is also a typo in Section VII: 'distribted according to vec p' should be 'distributed according to vec p'.","section":"Section IV and Section VII"},{"comment":"Reference [12] refers to the 'Weiner process'; the correct spelling is 'Wiener process'.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the central index-based result is interesting and the index derivation is elegant, but the proof of Theorem 1 is not yet rigorous for the finite-horizon problem as stated. The authors should be asked to supply a truncation argument or explicitly restrict the claim to an infinite-horizon setting, and to resolve the non-idling and stopping-time issues. These appear to be fixable within the manuscript's scope, so I do not see grounds for rejection, but the current version does not meet the acceptance bar."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: the main contribution is the VoI index that depends on both age and source precision, with a proof that it is optimal for a constant-gain remote estimator. The index and the ordering invariance lemma are genuinely useful, and the paper makes a fair point that age-optimal scheduling need not be estimation-optimal. The multi-process decomposition is a reasonable add-on.\n\nWhat is actually new: prior AoI work mostly treats packets as equally valuable apart from age; this paper shows that when sources have different measurement noise variances, priority should combine age and precision. Lemma 2 gives a clean ordering invariance that makes the index easy to implement, which is a real practical step.\n\nSoft spots, in proportion:\n\n(1) The optimality proof does not actually prove the finite-horizon claim. In the exchange argument, the cost difference in Eq. (25) is written as an infinite sum, but the paper's objective is E from t=1 to T. The tail beyond T is positive by Lemma 5, but positivity of the tail means the finite-horizon difference could be smaller or even negative, so the inequality does not follow. The stopping times T1, T2 may also exceed T, and that case is not handled. This is a real gap, not cosmetic, and needs a truncation argument or a different proof.\n\n(2) The problem statement does not explicitly allow idling, but the proof says \"since they are non-idling\" to equate the closed-loop gain a(t) across policies. If idling were allowed, transmitting a very old, very noisy packet could be worse than not transmitting. Either the model should state that the scheduler must transmit whenever the queue is nonempty, or a no-idling optimality argument should be provided.\n\n(3) Minor: the definitions of W^2_p in the main text and the appendix are inconsistent (one has sigma^2, the other (1+K)^2 sigma^2). The packet ordering is unaffected, but the mismatch should be fixed.\n\nIs the central result likely correct? Probably, and a fix may be straightforward by considering infinite-horizon discounted or average cost, or carefully truncating the tail. But as written, the proof is incomplete. This paper deserves a serious referee; it is a solid extension with a real gap that a careful revision could close. The authors also honestly note the constant-gain and scalar-process limitations, which are real constraints on scope.\n\nFor whom: people working on age of information and remote estimation who care about source quality. I would bring it to a reading group to discuss the proof gap, and I would cite it if I write about remote estimation scheduling.","headline":"A clean, likely-correct value-of-information index for remote estimation with heterogeneous source quality, but the proof has a genuine finite-horizon gap and an unstated non-idling assumption.","tokens_in":14716,"tokens_out":4157,"would_cite":true,"duration_ms":39436,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"For a remote constant-gain estimator of a stable Gauss-Markov process, the paper proves that scheduling the queued packet with the smallest age-and-precision index $W_\\psi^2(t)$ minimizes expected cumulative squared estimation error.","keywords":["age of information","value of information","remote estimation","Gauss-Markov process","constant-gain Kalman filter","index policy","packet scheduling","unreliable channel"],"falsifier":"Brute-force dynamic programming over all non-idling policies on a small instance, say two sources, $T=10$, $a=0.9$, $K=0.05$, and $p=0.5$, should reproduce the index policy's expected cost; if any non-idling policy yields strictly smaller expected cumulative squared error, the paper's optimality claim is false.","tokens_in":13641,"feed_emoji":"📡","tokens_out":11115,"duration_ms":108950,"temperature":0.7,"pith_summary":"The paper asks how a network node sharing one unreliable channel should choose which queued sensor packet to transmit when a remote linear filter with a constant gain uses the packets to estimate a scalar Gauss-Markov process. It claims that the optimal rule is an index policy: compute for each packet a closed-form number that grows with both the packet's age and its source's observation-noise variance, and transmit the packet with the smallest number. This matters because freshness alone can be the wrong objective, since a very fresh but very noisy packet can carry less estimation value than an older, more precise one. Under a stability assumption on the process and filter gain, the paper proves the index policy minimizes the expected cumulative squared estimation error and extends the result to several independent processes sharing a network. A reader should care because the result turns an apparently intractable dynamic scheduling problem into a sorted-queue rule and separates the question of how old the data is from how useful the data is in real-time estimation.","feed_headline":"Optimal updates rank packets by age and noise together","feed_subtitle":"Minimizing age of information alone does not minimize squared estimation error for constant-gain filters.","key_machinery":"The central object is the packet index $W_\\psi^2(t)$, defined as the closed-form contribution of packet $\\psi$ to the one-step expected squared estimation error if it is delivered. Because the term depending on the current error $e(t)$ is common to every choice, minimizing $W^2$ at the last time step is optimal, and an exchange argument shows that swapping packet order never helps at earlier steps either. The time-invariance of the pairwise ordering (Lemma 2) is what makes the policy implementable as a single sorted queue, with a new packet inserted in $O(\\log|Q(t)|)$ time, and it is the property that lets the backward-induction proof go through.","core_discovery":"The paper's central claim is that the single-queue scheduling problem admits an index policy: at each time $t$, assign to each queued packet $\\psi$ the index $$W_\\$psi^{2}$(t) = $a^{2}$$K^{2}$\\left($a^{{2\\tau_\\psi(t)}}$\\sigma_{s,\\psi}^2 + \\$sigma^{2}$\\frac{$a^{{2\\tau_\\psi(t)}}$-1}{$a^{2}$-1}\\right) + \\$sigma^{2}$,$$ where $\\tau_\\psi(t)$ is the packet's age and $\\sigma_{s,\\psi}^2$ is the variance of its source's observation noise, and transmit the packet with the smallest $W_\\psi^2(t)$. Under the stated stability condition on the process $a$, the filter gain $K$, and a non-idling scheduler, this policy minimizes $\\mathbb{E}\\sum_{t=1}^T e^2(t)$ over all scheduling policies. Priority is therefore governed jointly by freshness and source precision, so it is not optimal to order packets by age alone.","pith_inferences":["One extension the paper leaves open is allowing the scheduler to idle: because the optimality proof compares only non-idling policies, a threshold rule that skips transmission when every queued packet's $W_\\psi^2(t)$ exceeds some function of the current error is a natural candidate for the idle-armed problem.","Because the index ordering is time-invariant, the policy may be learnable online: if source precisions or channel reliability are unknown, the scheduler could estimate $\\sigma_{s,\\psi}^2$ from delivered packet innovations and still maintain a sorted queue, though the paper does not address parameter uncertainty.","Since the filter gain $K$ is fixed, the index separates cleanly into sensing-noise and process-noise terms; a plausible next step is to co-optimize $K$ and the scheduling rule, in which case the index would become a function of the chosen gain, but the paper's optimality proof does not cover that joint problem."],"forward_implications":["A system that minimizes age of information alone does not in general minimize expected squared estimation error for a constant-gain filter; both the age and the precision of the source must enter the packet-priority rule.","The optimal scheduler is implementable as a single sorted queue: because the relative order of any two packets is fixed over time, inserting a new packet requires only $O(\\log|Q(t)|)$ comparisons and no re-sorting is ever needed.","For $N$ independent processes sharing a common network node with i.i.d. channel-access probabilities, the joint problem decomposes into $N$ single-process problems, and composing the per-process index policies is optimal with total cost $\\sum_{i=1}^N J_i^*(e_i(0);p p_i;T)$.","The optimal expected cost depends only on the initial error and queue contents, the horizon $T$, and the channel reliability $p$, so the same index rule works unchanged across different arrival processes and time horizons."],"supporting_citations":[{"why":"Defines the constant-gain Kalman filtering update with delayed, time-stamped packets that the paper adopts as its estimator.","marker":"[24]"},{"why":"Supplies the Gauss-Markov process and linear filter/stochastic-control background used to set up the error dynamics.","marker":"[23]"},{"why":"Provides the Markov decision process and dynamic-programming framework used to prove optimality by backward induction.","marker":"[25]"},{"why":"Gives the prior result that age-optimal policies minimize estimation error for a Wiener process, the setting the paper extends and qualifies.","marker":"[12]"}],"fun_headline_variants":["Scheduling packets: Value of information beats age alone","Optimal packet scheduling weighs age and source precision","For constant-gain filters, age-only policies fall short","Priority index: combine packet age with noise variance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes the scheduler is non-idling, meaning whenever the queue is nonempty it must transmit some packet; if deliberate idling were allowed, sending a very old and noisy packet could increase expected error and the index policy would not necessarily remain optimal.","fun_headline_variants_meta":{"raw":{"variants":["Scheduling packets: Value of information beats age alone","Optimal packet scheduling weighs age and source precision","For constant-gain filters, age-only policies fall short","Priority index: combine packet age with noise variance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000225,"raw_usage":{"total_tokens":1454,"prompt_tokens":925,"completion_tokens":529,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":541,"completion_tokens_details":{"reasoning_tokens":467}},"tokens_in":541,"tokens_out":529,"duration_ms":5178,"temperature":1.0,"reasoning_tokens":467,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:23:56.213746+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Brute-force dynamic programming over all non-idling policies on a small instance, say two sources, $T=10$, $a=0.9$, $K=0.05$, and $p=0.5$, should reproduce the index policy's expected cost; if any non-idling policy yields strictly smaller expected cumulative squared error, the paper's optimality claim is false.","supporting_citations":[{"cited_title":"Kalman filtering over a packet-delaying network: A probabilistic approach,","cited_arxiv_id":null,"evidence_quote":"Defines the constant-gain Kalman filtering update with delayed, time-stamped packets that the paper adopts as its estimator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Gauss-Markov process and linear filter/stochastic-control background used to set up the error dynamics."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Markov decision process and dynamic-programming framework used to prove optimality by backward induction."},{"cited_title":"Remote estimation of the wiener process over a channel with random delay ,","cited_arxiv_id":null,"evidence_quote":"Gives the prior result that age-optimal policies minimize estimation error for a Wiener process, the setting the paper extends and qualifies."}],"review_version":1}