{"id":"758cbc62-ef18-4175-b043-7846fded4a20","arxiv_id":"2412.03271","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An input-output variant of Neural Jump ODEs is proven to converge to the L2-optimal conditional expectation for online filtering and classification with irregularly sampled, partially observed data.","lead":"This paper extends Neural Jump ODEs, a neural network model for continuous-time time series, to input-output settings where the observed input differs from the target output. It proves that the model converges to the optimal L2 filter, the conditional expectation, and demonstrates this on filtering and classification examples in finance and health monitoring.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.4's approximation step needs joint continuity of the filter map, which the paper's own threshold-classification example violates; convergence for that central application is therefore not proved as written.","rationale":"The reader's weakest-assumption analysis correctly identified Assumption 3 as the load-bearing condition. My concern sharpens this: the relevant failure is not merely that unsmooth filters fall outside the theorem, but that the proof's approximation argument requires more than the paper verifies for one of its own central examples. Example 5.5 is singled out as a classification application satisfying Assumption 3, yet its jump map is discontinuous in the input path at the threshold. This is an internal tension between the statement of Theorem 4.4 and the example, rather than only a limitation of scope. I do not think this forces rejection: the convergence result may be true and salvageable, for example by replacing the uniform approximation on P_epsilon with an L2 approximation that can handle discontinuities on null sets, or by stating Assumption 3 with explicit joint continuity and removing the classification example from the theorem's coverage. The experiments on the Brownian classification task are encouraging but do not repair the proof gap. Hence the verdict should remain CONDITIONAL, but with the revision requirement made precise.","tokens_in":39793,"tokens_out":35630,"duration_ms":369485,"concrete_test":"Analytic check: in Example 5.5, fix a path U* with U*_tau = alpha and take U^{+,-}_n with U^{+,-}_{tau} = alpha +/- 1/n and t_n = tau + 1/n^3. Then F(t_n, tau, U^{+,-}_n) = 1 - Phi((alpha - U^{+,-}_{tau})/sqrt(t_n - tau)) tends to 1 - Phi(-/+ infinity), giving different limits 1 and 0, while any t-continuous extension at (tau, tau, U*) must assign a single value. This demonstrates that F_j is not jointly continuous on [0,T]^2 x P_epsilon, so Proposition A.5 cannot be applied. A complementary empirical test: train IO NJODE on the Brownian threshold classification task with increasing width/signature order and check whether validation loss actually approaches Psi(V_hat); the current proof gives no guarantee.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 4.4: minimizers of the IO NJODE objective converge to the true conditional expectation. In Step 2 of the proof, the authors invoke Proposition A.5 to uniformly approximate each F_j on [0,T]^2 × P_epsilon. This requires F_j to be jointly continuous in (t, tau, U), not merely differentiable in t for each fixed path. The paper's own online classification example (Example 5.5) has V_t = 1_{W_t >= alpha}, so the jump map is F(t, tau, U) = 1 - Phi((alpha - U_tau)/sqrt(t - tau)) for t > tau, with F(tau, tau, U) = 1_{U_tau >= alpha} a.s. At the point (t, tau, U) with t = tau and U_tau = alpha, the limit as t downarrow tau depends on the rate at which U_tau approaches alpha, so no jointly continuous version exists. Thus Assumption 3, if read as requiring joint continuity, is not satisfied by a headline example; if read only as t-continuity, the proof of Step 2 lacks a necessary hypothesis. Either way, the convergence theorem as proved does not cover the classification application it claims to cover.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends Neural Jump ODEs to input-output systems, where the input process U is observed (possibly irregularly and partially) and the output process V is to be estimated online. The main theoretical result, Theorem 4.4, states that under Assumptions 1–5 the minimizers of the new objective function (4) over growing network classes converge, in the pseudo-metrics d_k, to the true conditional expectation Vhat_t = E[V_t | A_t]; Theorem 4.5 extends this to the Monte Carlo objective. The authors introduce a squared per-term objective to make Vhat the unique minimizer in the input-output setting, prove uniqueness and approximation via truncated signatures and bounded-output neural networks, and provide examples in parameter filtering, stochastic filtering, and online classification. Experiments on synthetic data compare the method with particle filters and classical financial estimators, and Section 7 discusses why the old NJODE objective is inappropriate for input-output filtering.","tokens_in":39961,"tokens_out":14901,"duration_ms":149558,"significance":"If the convergence theorem is valid under the stated assumptions, the paper is a solid contribution: it gives a principled objective for online filtering in a nonparametric, data-driven setting, with a uniqueness result, a layered universal approximation argument, a Monte Carlo convergence theorem, and an open-source implementation with reproducible experiments. The explicit identification of the L2-optimal limit, Vhat, and the comparison with the old objective in Section 7 are valuable. However, the claimed scope currently exceeds what the proofs support: the online classification example in Section 5.3 does not satisfy Assumption 3 as used in the proof, and the uniqueness step in Theorem 4.4 uses an independence property of the observation masks that is not stated among the assumptions. Both issues are local and repairable, but they affect load-bearing parts of the central claim.","major_comments":[{"comment":"The classification example does not satisfy Assumption 3, so the convergence theorem as stated does not cover it. For V_t = 1_{W_t >= alpha}, the natural version of the filter map is F(t,tau,w) = 1 - Phi((alpha - w)/sqrt(t - tau)) for t > tau. This function has no jointly continuous extension to the diagonal: along t downarrow tau with w = alpha it tends to 1 - Phi(0) = 1/2, whereas the diagonal value corresponding to the information at the observation time is 1_{w >= alpha}. The proof of Step 2 requires joint continuity on [0,T]^2 x P_epsilon to apply Proposition A.5. Moreover, the derivative f = partial_t F = phi((alpha - w)/sqrt(t - tau)) (alpha - w) / (2 (t - tau)^{3/2}) satisfies E|f(t)|^2 ~ c (t - tau)^{-3/2} when observation gaps can be arbitrarily small, so the integrability condition in Eq. (1) fails unless one imposes a positive minimum gap. The statement in Section 5.3 that Assumption 3 is easily verified for Example 5.5 is therefore not correct, and the claimed convergence for online classification is not proved as written.","section":"Section 5.3 / Example 5.5 and Theorem 4.4, Step 2"},{"comment":"The proof factors E[1_{k<=n} M_{k,j} (Vhat_{t_k-} - eta_{t_k-})] as E[M_{k,j}] E[1_{k<=n}(Vhat_{t_k-} - eta_{t_k-})], which requires M_k to be independent of A_{t_k-}. Assumption 1 only says that M_k is independent of the observation times and of n; Assumption 6 says that the whole observation framework is independent of Z. Neither condition implies that M_k is independent of the earlier masks M_i (i<k), which are part of A_{t_k-}. If the mask process is serially dependent, the lower bound in Eq. (9) can fail, and the minimizer of the weighted objective Psi need not coincide with Vhat in the unweighted metric d_k. This is load-bearing for the uniqueness argument and for the final d_k-convergence step; please add an explicit independence or conditional independence condition on the masks, or modify the proof accordingly.","section":"Proof of Theorem 4.4, Step 1 (Eq. (9))"}],"minor_comments":[{"comment":"The statement and Step 3 use 'for every 1 <= k <= K', but K is never defined; the corresponding range in Definition 2.4 and Theorem 4.4 is 1 <= k <= bar n.","section":"Theorem 4.5 and its proof"},{"comment":"The push-forward map mu_2 has coordinates (bar t, pi_m0, U_0, U*_t, n_t, delta_t), so its image lies in [0,T] x R^{d'} x R^{dU} x [0,1/epsilon]^2 x [epsilon,T]; the displayed set D_2 = [0,T]^2 x ... has one dimension too many and should be corrected.","section":"Eq. (12) and the surrounding text"},{"comment":"The text refers to 'Assumption 2.1 on delta_min', but the relevant statement is Remark 2.1; also, the phrase '4th (in)equality' in Eq. (15) should be 'fourth equality'.","section":"Proof of Theorem 4.4, after Eq. (14)"}],"recommendation":"major_revision","confidential_remarks":"The paper builds substantially on earlier NJODE work by the same group, and the new material is the input-output objective and its convergence analysis. The two major comments above are, in my view, genuine gaps but not fatal: the mask issue can be fixed by adding a formal independence assumption, and the classification gap can be repaired by either restricting the theorem to filter maps with the needed joint continuity (or to observation schemes with a positive minimum gap) or by proving a separate approximation result for the discontinuous diagonal. I would encourage the editor to request a revision rather than reject, because the core idea and the experimental comparison are valuable and the fixes are localized."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know up front. First, this is a genuine advance in the NJODE line: the paper extends the framework to input-output systems (U ≠ V), introduces a new loss whose unique minimizer is the conditional expectation, and carries the convergence argument through for smooth filtering problems. Second, the central convergence theorem has a real hole: it needs the filter map to be jointly continuous, and the paper's own threshold-classification example is not jointly continuous. The theory as written does not cover one of the two headline applications.\n\nWhat's new and good. The modified objective (4) is the correct fix — Section 7 shows cleanly that under the old loss (28) the minimizer is not the conditional expectation. The compensation-network construction in Theorem 4.4, controlling the error outside the compact path set, is careful, and the Monte Carlo extension (Theorem 4.5) is handled properly. Experiments are honest: the method matches a particle filter when the filter is stable, the CIR case where the particle filter is numerically unstable is reported rather than hidden, and the paper openly acknowledges the implementation/theory mismatch in the shared codebase. The heavy citation of Krach et al. (2022) is legitimate — this is a direct extension of that framework.\n\nWhere it's soft. The stress-test concern lands. Step 2 of Theorem 4.4 invokes Proposition A.5 to approximate F_j uniformly on a compact set, which requires joint continuity in (t, τ, U). Assumption 3 as written says \"continuous and differentiable in their first coordinate t\" — ambiguous, but the proof needs the joint version. In Example 5.5, F(t, τ, U) = 1 − Φ((α − U_τ)/√(t − τ)) for t > τ, with F(τ, τ, U) = 1{U_τ ≥ α}. At the diagonal with U_τ = α, the limit as t ↓ τ depends on the approach direction, so no jointly continuous version exists. The paper's claim that \"the Leibniz integral rule\" verifies Assumption 3 is wrong under the joint reading. Either the assumption is weaker than the proof needs, or the classification example fails it. The fix probably exists — an L2 approximation that ignores the measure-zero discontinuity set would likely work — but it isn't in the paper. Smaller items: no multiple-seed error bars, no deep-filtering baselines (partly excused by the irregular-observation setting).\n\nWho it's for: people working on neural filtering, irregular online classification, and NJODE-type models. It deserves a serious referee; I'd send it to review, not desk-reject, with a request to sharpen Assumption 3 and close the classification gap. If that gets fixed, this is a solid paper.","headline":"The IO NJODE extension is a genuine, mostly careful contribution, but Theorem 4.4 as proved does not cover the paper's own threshold-classification example, so the convergence claims need reworking before I'd rely on them.","tokens_in":40569,"tokens_out":11207,"would_cite":false,"duration_ms":91098,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62M45","60G35","62G05","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Neural jump ODEs provably learn the optimal online filter","keywords":["neural jump ODE","online filtering","input-output systems","conditional expectation","irregular observations","signature transform","L2-optimal filtering","classification"],"falsifier":"Take an input-output system where the optimal filter is discontinuous in time, for example an output that jumps at a random time observed with positive probability, and train a large IO NJODE; if the model's loss stays strictly above the conditional expectation's loss as network size grows, the convergence claim fails because Assumptions 2 and 3 are violated.","tokens_in":1517,"feed_emoji":"📈","tokens_out":4552,"duration_ms":90517,"temperature":0.7,"pith_summary":"This paper extends the Neural Jump ODE framework to input-output systems, where the observed input process and the target output process are distinct. It introduces the IO NJODE model, which evolves a hidden state continuously between observations and jumps at observation times, together with a new objective that squares the pre-jump and post-jump errors separately. The main result is that minimizers of this objective converge to the true conditional expectation as the network size and signature truncation grow, making the model a provably L2-optimal online filter. The practical payoff is that filtering, parameter estimation, and classification become fully data-driven: only sample paths with irregular and partially masked observations are needed, not the underlying distributions.","feed_headline":"Neural jump ODEs provably learn the optimal online filter","feed_subtitle":"A new loss makes them converge to the true conditional expectation from irregular, partial inputs.","key_machinery":"The load-bearing object is the truncated signature of a piecewise-linear interpolation of the observed input path, fed through bounded-output feedforward neural networks inside a jump ODE: the hidden state follows an ODE between observation times and is reset at observation times. The new loss (4), which squares the error just before an observation and the error at the observation separately, is what makes the argument work. Lemma 4.2 decomposes this loss into the intrinsic filter error plus the squared distance from the model to $\\hat V$, proving uniqueness of the conditional expectation as the minimizer; Lemma 4.3 converts sums over random observation times into expectations over a uniform random variable, allowing the proof to treat observation times as an independent source of randomness; and the signature approximation result lets continuous functions of bounded-variation paths be approximated uniformly by functions of the truncated signature.","core_discovery":"The central claim is that, for an observed input process $U$ and a target output process $V$, the IO NJODE trained with the objective function (4) converges to the conditional expectation $\\hat V_t = \\mathbb{E}[V_t\\mid A_t]$ in the pseudo-metric $d_k$ as the neural network size and signature truncation level tend to infinity. Theorem 4.4 establishes this for the theoretical loss, and Theorem 4.5 shows that the same conclusion holds when the loss is replaced by its Monte Carlo approximation on finitely many training paths, with an almost-sure subsequence of trained models converging to the optimal filter. The proof builds on the fact that the new loss has a Pythagorean decomposition around the conditional expectation, so $\\hat V$ is its unique minimizer among adapted processes, and on universal approximation results for truncated signatures and neural networks.","pith_inferences":["As an editorial extension, the same proof strategy should transfer to noisy observations and long-horizon predictions, since the paper states those cases only require replacing the objective and the metric.","A finite-sample trade-off follows: the squared-separately objective is the correct one for L2 optimality, but the old coupled objective learns jumps faster, so practitioners with limited data face a real design choice rather than a free lunch.","If the convergence is correct, online classifiers built by setting the output to class-indicator processes are L2-consistent estimators of class probabilities, a consequence the paper states but does not isolate as a separate theorem."],"forward_implications":["For any input-output system satisfying the assumptions, the trained IO NJODE is an L2-optimal online filter: in the limit its output is indistinguishable from the conditional expectation at observation times.","Because training needs only sample paths rather than the underlying distributions, the method applies where classical particle filters are unusable or numerically unstable, including the CIR examples reported in the paper.","Choosing the output process to be indicator functions of class labels turns the same model into an online classifier of conditional class probabilities.","When all output coordinates also appear as input coordinates, the new loss reproduces the original NJODE setting, and the paper argues the earlier coupled loss is then preferable for learning jumps quickly.","With the Monte Carlo loss, convergence survives finite-sample training: as the sample size and then the model size grow, trained models converge almost surely to the optimal filter."],"supporting_citations":[{"why":"Supplies the original NJODE model, assumptions, and proof strategy that this paper extends to input-output systems.","marker":"Krach et al. (2022)"},{"why":"Introduced Neural Jump ODEs and provides the Black-Scholes dataset used in the loss-function comparison.","marker":"Herrera et al. (2021)"},{"why":"Universal approximation theorem for feedforward networks, used in Step 2 to approximate the conditional-expectation functions on compact sets.","marker":"Hornik et al. (1989)"},{"why":"L2 approximation theorem used to control the network error outside the compact set via push-forward measures.","marker":"Hornik (1991)"},{"why":"Provides the dependent-observation and noisy-observation extensions that the paper states can be incorporated into the framework.","marker":"Andersson et al. (2024)"},{"why":"Particle filter baseline used as a reference approximation of the true conditional expectation in the experiments.","marker":"Djuric et al. (2003)"},{"why":"Gives the conditional multivariate normal formula used to compute true conditional expectations in the Brownian examples.","marker":"Eaton (2007)"}],"fun_headline_variants":["Neural jump ODEs provably achieve optimal online filtering","IO-NJODE: provable convergence to optimal filters","Neural jump ODEs converge asymptotically to L2-optimal filter","Provable filtering guarantees for neural jump ODEs","Neural jump ODEs master irregular-data filtering with proof"],"cache_read_input_tokens":42624,"weakest_assumption_plain":"The optimal filter's conditional-expectation functions must be continuously differentiable in time, so the proof gives no guarantee for filters that jump when information arrives.","fun_headline_variants_meta":{"raw":{"variants":["Neural jump ODEs provably achieve optimal online filtering","IO-NJODE: provable convergence to optimal filters","Neural jump ODEs converge asymptotically to L2-optimal filter","Provable filtering guarantees for neural jump ODEs","Neural jump ODEs master irregular-data filtering with proof"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000875,"raw_usage":{"total_tokens":3723,"prompt_tokens":823,"completion_tokens":2900,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":439,"completion_tokens_details":{"reasoning_tokens":2814}},"tokens_in":439,"tokens_out":2900,"duration_ms":20494,"temperature":1.0,"reasoning_tokens":2814,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:35:03.434692+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an input-output system where the optimal filter is discontinuous in time, for example an output that jumps at a random time observed with positive probability, and train a large IO NJODE; if the model's loss stays strictly above the conditional expectation's loss as network size grows, the convergence claim fails because Assumptions 2 and 3 are violated.","supporting_citations":[{"cited_title":"Neural jump ordinary differential equations: Consistent continuous-time prediction and filtering","cited_arxiv_id":null,"evidence_quote":"Introduced Neural Jump ODEs and provides the Black-Scholes dataset used in the loss-function comparison."},{"cited_title":"Multilayer feedforward networks are universal approximators","cited_arxiv_id":null,"evidence_quote":"Universal approximation theorem for feedforward networks, used in Step 2 to approximate the conditional-expectation functions on compact sets."},{"cited_title":"Extending path-dependent NJ - ODE s to noisy observations and a dependent observation framework","cited_arxiv_id":null,"evidence_quote":"Provides the dependent-observation and noisy-observation extensions that the paper states can be incorporated into the framework."},{"cited_title":"Particle filtering","cited_arxiv_id":null,"evidence_quote":"Particle filter baseline used as a reference approximation of the true conditional expectation in the experiments."},{"cited_title":"Multivariate statistics: A vector space approach","cited_arxiv_id":null,"evidence_quote":"Gives the conditional multivariate normal formula used to compute true conditional expectations in the Brownian examples."}],"review_version":1}