{"id":"518dd710-281b-4d12-b430-8bcf64184d35","arxiv_id":"2608.00686","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under smooth data and Hamiltonian regularity, the value-function error between controlled nonlinear filtering and its N-particle approximation is O(N^{-1/6}) in d=1, O(N^{-1/6}(log N)^{1/3}) in d=2, and O(N^{-1/(3d)}) for d>2.","lead":"This paper proves explicit convergence rates for replacing a controlled nonlinear filtering problem by a centralized N-particle control problem. It extends earlier rate results from a special Gaussian-reducible setting to general nonseparable rewards and controlled drifts, at the cost of slower rates and a strong smoothness assumption on the optimized Hamiltonian.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1 assumes only continuity in t, but the proof (Prop. 4.4 and the start of the proof of Theorem 3.1) requires time-Lipschitz data; Assumption 3.1 does not imply Assumption 4.1, so the claimed uniform rates are unsupported for merely continuous-in-time coefficients.","rationale":"I read the paper as proving a conditional rate theorem: under smooth data, uniform ellipticity, and optimized-Hamiltonian regularity, the N-particle value function converges at the stated rates. I looked for the most load-bearing place where the proof could fail even if all the stated assumptions hold. The reader's choice, Assumption 3.1(iv), is an assumption about scope: it is strong and can fail for natural non-unique-maximizer problems, but the paper is explicit that the result is conditional on it and gives Example 3.1 showing it is satisfiable. That is a limitation, not an internal inconsistency. The more serious issue is an internal mismatch between the assumptions and the proof: Theorem 3.1 is stated under Assumption 3.1 only, but the proof of Prop. 4.4 uses a C|t0−s0|^{1/2} bound for the time-translation error, and the proof of Theorem 3.1 explicitly asserts that Assumption 3.1 implies Assumption 4.1 (Lipschitz data). Since Assumption 3.1(i) only requires continuity in t, this implication is false. With a merely continuous time modulus, Lemma 4.5 gives |t0−s0| ≤ C N^{-1/3}, and the residual can contain a term of size ω(N^{-1/3}), which for slowly varying continuous functions is much larger than N^{-1/6}. The claimed uniform rate is therefore not established under the stated assumptions. The fix is straightforward—add a time-Lipschitz or explicit time-modulus hypothesis—and the overall strategy may be sound, so I do not reject the paper outright. But the theorem as written overreaches, and the reader's conditional verdict should be retained, with the condition now specifically including the missing time regularity. I therefore set verdict_should_be to CONDITIONAL, while noting my load-bearing concern differs from the reader's weakest-assumption identification, hence agreement_with_reader is disagree.","tokens_in":23200,"tokens_out":23171,"duration_ms":302531,"concrete_test":"Check the assertion 'Assumption 3.1 implies Assumption 4.1' at the start of the proof of Theorem 3.1. If it is false, re-run the residual estimate in Prop. 4.4 with coefficients that are C^r in (x,μ,a) but only continuous in t, e.g. ℓ(t,x,m,a)=g(x)h(t) with h(s)-h(t) ~ 1/log(1/|s-t|), and use |t0-s0| ≤ C ε^{2/3} from Lemma 4.5. If the time-translation term in E(N,ε) is not O(α(N)^{1/3}), then Theorem 3.1's rate fails under Assumption 3.1; adding time-Lipschitz to Assumption 3.1 should make the stated rate valid.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The main theorem is stated under Assumption 3.1, whose part (i) requires b and ℓ to be only continuous in t and C^r_b jointly in (x, μ, a) uniformly in t. However, Proposition 4.4 bounds the time-translation part of the residual by C(|t0−s0|^{1/2}+...)(1+r/ε), which requires at least 1/2-Hölder time regularity. More importantly, the proof of Theorem 3.1 begins with: 'Since Assumption 3.1 implies Assumption 4.1, Proposition 4.1 gives comparison...' This implication is false: Assumption 4.1 requires b and ℓ to be Lipschitz continuous on their domains uniformly in a, and continuity in t plus smoothness in (x,μ,a) does not imply Lipschitz dependence on t. Under Lemma 4.5, |t0−s0| ≤ C ε^{2/3}. If the data have only a slow time modulus ω, the time-displacement Hamiltonian error is ω(C N^{-1/3}), not C N^{-1/6}. For example, take h(t)=1/log(e/(T−t)) in a state-dependent term; then ω(|t0−s0|) ~ 1/log N, which dominates the claimed N^{-1/6} rate. Thus the residual estimate in Prop. 4.4 and the comparison sandwich do not establish Theorem 3.1 under the assumptions actually stated. This is an internal gap, not merely an issue of the Hamiltonian regularity assumption being restrictive. Adding time-Lipschitz or an explicit time modulus to Assumption 3.1 would repair the proof; without it, the central claim is not proven for the stated hypothesis.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a partially observed stochastic control problem in which the state is a McKean–Vlasov diffusion on the flat torus and the observation noise is common. The filter serves as the infinite-dimensional state, and the value function is shown to solve a second-order HJB equation on the Wasserstein space. The main result, Theorem 3.1, bounds the uniform error between the value function of a centralized N-particle control problem and the true value function evaluated at the empirical measure, with rates N^{-1/6} in d=1, N^{-1/6}(ln N)^{1/3} in d=2, and N^{-1/(3d)} in d>2, under smoothness of the coefficients, uniform ellipticity, and a strong regularity assumption on the optimized common-control Hamiltonian. The proof combines a translation lift in the common-noise direction, Fourier--Wasserstein inf/sup-convolutions, viscosity comparison, and N-uniform particle derivative estimates.","tokens_in":23588,"tokens_out":6603,"duration_ms":79696,"significance":"If correct, the paper gives the first quantitative value-function convergence rates for particle approximations of controlled nonlinear filtering problems with nonseparable rewards and a common-control Hamiltonian. The proof strategy is technically substantial and builds in an interesting way on recent quantitative mean-field control results, with a clean telescoping residual structure in Proposition 4.4. The authors are transparent about the restrictive nature of Assumption 3.1(iv) and provide a worked example. The comparison with the optimal N^{-1} weak rate in the separable affine case of Bouchard--Tan is clearly explained. However, the central theorem is not proved under the assumptions as stated because the proof requires a time-Lipschitz or Hölder modulus that Assumption 3.1 does not provide.","major_comments":[{"comment":"The proof of Theorem 3.1 begins with the assertion 'Since Assumption 3.1 implies Assumption 4.1'. This is false: Assumption 3.1(i) requires only continuity of b and ℓ in t, while Assumption 4.1 requires these functions to be Lipschitz continuous on their domains uniformly in a. The discrepancy is load-bearing. In Proposition 4.4, the time-translation part of the residual is bounded by C(|t0-s0|^{1/2}+|z0-w0|+r)(1+r/ε). The |t0-s0|^{1/2} term is not available from mere time-continuity; with Lemma 4.5's bound |t0-s0|≤Cε^{2/3}, arbitrary time continuity would give an error of the form ω(C N^{-1/3}) for a generic modulus ω, which can be much larger than the claimed N^{-1/6} rate. Thus Theorem 3.1 is not established for the stated hypotheses. The repair is to add an explicit time-Hölder (e.g., 1/2-Hölder) or Lipschitz condition to Assumption 3.1, or to track the actual time modulus in Proposi","section":"§4, proof of Theorem 3.1; §4.4, Proposition 4.4"},{"comment":"The comparison principle for the limiting equation is central to the sandwich argument in the proof of Theorem 3.1, but Proposition 4.1 is only a 'Proof sketch' and defers to references [20,7,5,6,8]. Since the Hamiltonian H(t,η) here is the optimized common-control Hamiltonian defined by sup after integration, it is not a priori covered by the comparison statements in the cited papers. Please either provide a complete proof of Proposition 4.1, or state precisely which theorem from which reference is being applied and verify all its hypotheses for the present nonseparable, common-control Hamiltonian under Assumptions 3.1 and 4.1.","section":"§4.3, Proposition 4.1"},{"comment":"The Hamiltonian regularity assumption is very strong: it requires the optimized common-control Hamiltonian to be C^r in the measure variable with uniformly continuous Lions derivatives up to total order r. This is not implied by smoothness of the data and will fail in many natural examples where the maximizing control is nonunique. The paper gives one nontrivial example, but no general sufficient condition or discussion of prevalence. Since the N-uniform particle derivative estimates in Proposition 4.2 and the rates in Theorem 3.1 rest directly on this assumption, the scope of the main theorem is considerably narrower than the problem formulation suggests. This is not an internal inconsistency, but it should be stated more prominently and, if possible, supplemented with broader sufficient conditions.","section":"§3.1, Assumption 3.1(iv)"}],"minor_comments":[{"comment":"There is a notation inconsistency between V^N (used for the value function in Sections 2--3 and in Theorem 3.1) and v^N (used in Proposition 4.2 and the proofs). Please unify.","section":"Throughout"},{"comment":"The sentence 'the supremum in (3.1) is exactly H(t, η^{μ,ϕ})' appears to refer to the Hamiltonian definition, but (3.1) is the PDE. The reference should be to (1.1) or the displayed definition in §3.1.","section":"§3.1, text before Example 3.1"},{"comment":"The phrase 'the regularity of ℓ and b in time, state, and measure' is vague. Since the main issue is the time modulus, please state explicitly which regularity (Lipschitz or Hölder) is being used for the time variable at that point.","section":"§4.4, Proposition 4.4 proof"},{"comment":"Lemma 4.1 and Proposition 4.3 are again 'Proof sketch' with references to [8,25]. This is acceptable if the referenced arguments are directly applicable, but the reader should at least see the key step for the Fourier metric and the compact-space empirical W1 estimate.","section":"§4.3, Lemma 4.1 and Proposition 4.3"}],"recommendation":"major_revision","confidential_remarks":"The central claim is likely repairable, but the gap between Assumption 3.1 and Assumption 4.1 is a genuine correctness issue in the main theorem. The paper relies heavily on a chain of the authors' own companion papers for comparison and convolution machinery; I would recommend asking the authors to make the comparison proof self-contained or to state the precise external theorem with verifiable hypotheses. The restrictive Hamiltonian regularity assumption is also worth explicit emphasis, as it may limit the audience."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the paper proves something new—uniform value-function rates for N-particle approximation of a controlled nonlinear filtering problem with a common-control Hamiltonian, nonseparable rewards, and controlled drifts—but the main theorem as stated is not supported by the proof. There is a clean repair, so it deserves a serious referee, not a desk reject.\n\nWhat’s new and good: the Hamiltonian is sup over a of the integrated running cost/drift, not the average of pointwise sups. That changes the particle-derivative scaling, and the paper shows the Fourier–Wasserstein inf/sup-convolution machinery still works. Example 3.1 gives a nontrivial bounded-control model where Assumption 3.1(iv) holds, including the smoothing that makes the sup-value smooth even when the argmax is nonunique. Proposition 4.4’s residual has a clean telescoping structure, and the dimension-dependent rates follow correctly from ε = α(N). The prose is honest about what is deferred.\n\nSoft spots. The reader flagged Assumption 3.1(iv), the C^r Hamiltonian regularity—that is genuinely strong and can fail for smooth data when the optimizer is nonunique. The paper notes the tradeoff and provides a worked example, so it is a limitation, not a hidden flaw.\n\nThe bigger problem is an assumptions-versus-proof mismatch. Theorem 3.1 is stated under Assumption 3.1, whose part (i) only requires b and ℓ continuous in t. But Proposition 4.4 bounds the time-translation part of the residual by |t0−s0|^{1/2}, and the proof of Theorem 3.1 explicitly says “Assumption 3.1 implies Assumption 4.1.” Assumption 4.1 requires Lipschitz continuity in t. Continuity does not imply Lipschitz. With Lemma 4.5 giving |t0−s0| ≤ C ε^{2/3}, Lipschitz data yields C ε^{1/3}; merely continuous data yields only ω(C ε^{2/3}) for the time modulus ω, which can be much larger. Example: h(t)=1/log(e/(T−t)) gives error ~1/log N, dominating the claimed N^{−1/6}. So the theorem’s uniform rate is not established under the hypotheses as written. This is an internal gap, not just a restrictive Hamiltonian condition. The fix is simple: add time-Lipschitz (or an explicit time modulus) to Assumption 3.1 and state Theorem 3.1 accordingly. Then the proof goes through.\n\nAlso check: Proposition 4.1 (comparison for the limiting equation) is a proof sketch deferring entirely to references, including the authors’ own companion papers. That can be acceptable as division of labor, but a referee should verify those results cover the integrated common-control Hamiltonian, not just the pointwise one. Proposition 4.2’s simultaneous induction is compressed but has the right shape.\n\nWho it’s for: people working on quantitative mean-field control, particle approximations, and Wasserstein HJB equations. Worth a careful referee; the gap is fixable and the result matters. I’d send it out with a request to repair the time-regularity assumption.","headline":"A genuinely new rate result for controlled nonlinear filtering with a common-control Hamiltonian, but the main theorem is stated under hypotheses weaker than the proof requires—there's a fix, but the paper needs it.","tokens_in":24116,"tokens_out":3687,"would_cite":false,"duration_ms":39220,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93E11","93E20","49L25","60H30","65C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For controlled nonlinear filtering, an N-particle approximation has explicit value-function error rates.","keywords":["Nonlinear filtering","partially observed control","finite-particle approximation","Wasserstein Hamilton–Jacobi–Bellman equations","common noise","particle filters","viscosity solutions","convergence rate"],"falsifier":"Construct a smooth-data model on T^1 where the supremum defining the Hamiltonian is attained at two distinct actions whose optimizers cross at a measure point, so that H remains continuous but loses C^2 Lions regularity; compute V^N and V for increasing N at a fixed initial measure and check whether the uniform difference still obeys the N^{-1/6} bound. If it does, the Hamiltonian regularity assumption is not necessary; if the observed rate degrades, the assumption is doing the work the proof assigns it.","tokens_in":23024,"feed_emoji":"📊","tokens_out":6330,"duration_ms":71254,"temperature":0.7,"pith_summary":"The paper studies a partially observed control problem in which the state evolves as a McKean–Vlasov diffusion on the torus and the only observations arrive through a common noise. The filter—the conditional law of the state given the observations—becomes the state variable, and the value function solves a second-order Hamilton–Jacobi–Bellman equation on the space of probability measures. The paper shows that this infinite-dimensional problem can be replaced by a centralized N-particle control problem, where one common action is applied to all particles, and that the two value functions differ by at most C N^{-1/6} in dimension one, C N^{-1/6}(log N)^{1/3} in dimension two, and C N^{-1/(3d)} in higher dimensions, uniformly over initial configurations. This matters because it turns a measure-valued stochastic control problem into a finite-particle problem with a guaranteed, explicit error tolerance, without requiring the affine-drift, separable-reward structure of earlier results.","feed_headline":"N particles approximate controlled filtering values to N^{-1/6} error","feed_subtitle":"Centralized N-particle control approximates the true filtering value with explicit N^{-1/6} to N^{-1/(3d)} rates.","key_machinery":"The central object is the optimized common-control Hamiltonian H(t,eta)=sup_a int_{T^d x R^d} [ell(t,x,m_eta,a)+b(t,x,m_eta,a) dot p] eta(dx,dp), where eta is a measure over (x,p) pairs and m_eta is its spatial marginal. Because one action is chosen for all particles, the supremum sits outside the integral; Assumption 3.1(iv) requires this optimized Hamiltonian to be C^r in the measure variable in the Lions sense, with derivatives uniformly continuous and bounded. That regularity implies the empirical derivative scaling |D_{z_{i1}}...D_{z_{iq}} H^N| <= C_R N^{-#{i1,...,iq}}, the key mean-field ingredient for uniform-in-N particle derivative estimates. The other load-bearing device is the tra","core_discovery":"The central claim is Theorem 3.1: under smoothness of the data, uniform ellipticity, and a C^r regularity condition on the optimized common-control Hamiltonian H(t,eta)=sup_a int [ell(t,x,m_eta,a)+b(t,x,m_eta,a) dot p] eta(dx,dp), the value function V^N of the centralized N-particle control problem converges uniformly to V(t, mu^N_x), the value of the true controlled filtering problem evaluated at the empirical measure, at the stated rates. The proof works directly with the Wasserstein HJB equation rather than reducing to a finite-dimensional Gaussian representation. The constant common-noise Hessian is removed by a translation lift that introduces an extra torus variable, and Fourier–Wasser","pith_inferences":["Editorial inference: the stated rates are probably not sharp; under stronger data regularity one would expect rates closer to the N^{-1} weak rate available in the special separable-reward case, but the current proof trades optimality for generality.","Editorial inference: the choice of the Fourier–Wasserstein metric gives convenient Hilbert-space duality for the convolutions; using a sharper metric could improve the dimension dependence but would complicate the derivative identities and residual estimates.","Editorial inference: the comparison machinery should extend to controlled filtering with jumps or path-dependent coefficients if the optimized Hamiltonian keeps analogous Lions regularity and a counterpart of the translation lift exists.","Editorial inference: the uniform particle derivative estimates in v^N suggest that mesh-free or Monte Carlo solvers for the N-particle HJB equation may inherit the N^{-1} mean-field scaling, making the finite-particle problem practically solvable even though its raw dimension grows with N."],"forward_implications":["The centralized N-particle control problem is a numerical proxy for the infinite-dimensional filtering value function with a guaranteed error: N particles give O(N^{-1/6}) accuracy in one dimension, O(N^{-1/6}(log N)^{1/3}) in two dimensions, and O(N^{-1/(3d)}) in higher dimensions.","The error bound is uniform over all initial particle configurations, so it supports worst-case approximation guarantees rather than only average-case ones.","Nonseparable running rewards and controlled drifts are covered, so the rate guarantee does not rely on the Gaussian-convolution reduction used in earlier separable-reward results.","The Hamiltonian regularity condition can hold even when the maximizing action is non-unique, so convergence does not require uniqueness of optimal controls.","In one space dimension, the proof is designed to extend to state- and law-dependent common noise through a measure-dependent flow transformation, broadening the models for which the rates hold."],"supporting_citations":[{"why":"Introduces the centralized finite-population approximation and proves an N^{-1} rate under affine drift and separable rewards; it is the baseline this paper generalizes.","marker":"[10]"},{"why":"Develops particle convergence rates for second-order Wasserstein PDEs using Fourier–Wasserstein techniques; supplies the convolution and residual framework adapted here.","marker":"[8]"},{"why":"Provides algebraic convergence-rate and particle-derivative estimates for McKean–Vlasov control that Proposition 4.2 adapts to the optimized common-control Hamiltonian.","marker":"[12]"},{"why":"Establishes the optimal rate for the mean-field convergence problem; its uniform derivative machinery is the standard this paper modifies.","marker":"[19]"},{"why":"Proves well-posedness and comparison for HJB equations on the Wasserstein space with non-convex Hamiltonians and common noise, giving the limiting comparison principle.","marker":"[20]"},{"why":"Bounds empirical Wasserstein convergence on compact spaces, providing the deterministic empirical approximation and the alpha(N) rates used in localization.","marker":"[25]"},{"why":"Gives a comparison principle for Wasserstein PDEs with state- and law-dependent common noise, supporting the one-dimensional extension described in Remark 3.2.","marker":"[6]"}],"fun_headline_variants":["Controlled filtering via N particles: explicit convergence rates","N-particle control converges to true filtering value","Explicit convergence rates for particle-controlled filtering","Uniform value bounds: N^{-1/6} to N^{-1/(3d)}","Centralized N-particle control: uniform value error bounds"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The whole argument rests on the optimized common-control Hamiltonian being C^r smooth in the measure variable in the Lions sense, with bounded uniformly continuous derivatives; this is the property that forces the N^{-#indices} scaling of empirical derivatives, and smooth data alone does not guarantee it when the maximizing action is non-unique.","fun_headline_variants_meta":{"raw":{"variants":["Controlled filtering via N particles: explicit convergence rates","N-particle control converges to true filtering value","Explicit convergence rates for particle-controlled filtering","Uniform value bounds: N^{-1/6} to N^{-1/(3d)}","Centralized N-particle control: uniform value error bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001044,"raw_usage":{"total_tokens":4248,"prompt_tokens":787,"completion_tokens":3461,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":531,"completion_tokens_details":{"reasoning_tokens":3381}},"tokens_in":531,"tokens_out":3461,"duration_ms":30966,"temperature":1.0,"reasoning_tokens":3381,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T00:31:38.221073+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a smooth-data model on T^1 where the supremum defining the Hamiltonian is attained at two distinct actions whose optimizers cross at a measure point, so that H remains continuous but loses C^2 Lions regularity; compute V^N and V for increasing N at a fixed initial measure and check whether the uniform difference still obeys the N^{-1/6} bound. If it does, the Hamiltonian regularity assumption is not necessary; if the observed rate degrades, the assumption is doing the work the proof assigns it.","supporting_citations":[{"cited_title":"Convergence rate of particle system for second-order pdes on Wasserstein space.SIAM Journal on Control and Optimization, 63(3):1768–1782, 2025","cited_arxiv_id":null,"evidence_quote":"Develops particle convergence rates for second-order Wasserstein PDEs using Fourier–Wasserstein techniques; supplies the convolution and residual framework adapted here."},{"cited_title":"Souganidis","cited_arxiv_id":null,"evidence_quote":"Provides algebraic convergence-rate and particle-derivative estimates for McKean–Vlasov control that Proposition 4.2 adapts to the optimized common-control Hamiltonian."},{"cited_title":"On the optimal rate for the convergence problem in mean field control.Journal of Functional Analysis, 287(12), 2024","cited_arxiv_id":null,"evidence_quote":"Establishes the optimal rate for the mean-field convergence problem; its uniform derivative machinery is the standard this paper modifies."},{"cited_title":"Well-posedness of Hamilton–Jacobi equations in the Wasserstein space: non-convex hamiltonians and common noise.Communications in Partial Differential Equations, 50(1–2):1–52, 2025","cited_arxiv_id":null,"evidence_quote":"Proves well-posedness and comparison for HJB equations on the Wasserstein space with non-convex Hamiltonians and common noise, giving the limiting comparison principle."},{"cited_title":"A comparison principle for Wasserstein PDEs with state- and law-dependent common noise","cited_arxiv_id":"2606.04377","evidence_quote":"Gives a comparison principle for Wasserstein PDEs with state- and law-dependent common noise, supporting the one-dimensional extension described in Remark 3.2."}],"review_version":1}