{"id":"66558f10-38fb-40df-b9c6-9412711e516f","arxiv_id":"2506.19375","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Trajectory advantage regression reformulates offline path learning as a regression over path advantages, so that optimizing a path reduces to fitting a least-squares model.","lead":"This paper introduces trajectory advantage regression, a method that converts offline path optimization problems into a simple regression task. The approach could make route planning and path attribution easier for real-world logistics and navigation systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 4.1 depends on an unverified realizability premise: the additive advantage class Eq. 9 may not contain the global minimizer of the penalized V-LP, so optimizing TAR need not recover the optimal path.","rationale":"I read the paper as making a conditional theoretical claim: if the additive advantage parametrization contains a global minimizer of the penalized V-LP, then minimizing TAR and minimizing the penalized V-LP have the same minimizers. The proof of Corollary 4.1 is internally valid under that premise. I do not share the reader's specific concern that the truncation step 'Vθ' ≤ V*' fails: for any V, min(V, V*) never increases the penalty in this reduced MDP, because where V exceeds V* the clipped value satisfies TV* ≤ V*, making the positive-part penalty zero, and elsewhere decreasing the successor value only lowers the penalty. The real load-bearing gap is the realizability premise: the paper never establishes when the global minimizer of L(V) lies in the constrained class Vθ = c + Σ Aθ with Aθ ≤ 0. I constructed a small path-learning instance where the unconstrained minimizer is non-monotone and therefore not representable, so the premise can fail for legitimate choices of P0 and λ. Since the paper also contains no experiments and no discussion of how to choose P0 and λ to avoid this failure, the practical claim that TAR solves offline path learning remains conditional. This matches the reader's CONDITIONAL verdict, so I recommend no change.","tokens_in":5404,"tokens_out":20582,"duration_ms":230204,"concrete_test":"Construct the minimal path MDP described above: actions {a, ⊥}, states {∅, a, a⊥, a⊥⊥} with a⊥⊥ absorbing and V = 0, Ψ = {a⊥}, J(a⊥) = 1. Choose P0 with mass M on a and ε on a⊥, and P1 uniform over all edges. For a range of λ (e.g., λ = 0.01, M = 100), numerically solve the penalized V-LP (Eq. 3) over all V ≥ 0 on this finite state set, and separately solve it over the additive parametrization Vθ = c + Σ Aθ with Aθ ≤ 0. If the unconstrained minimizer has V(a) < V(a⊥) and achieves strictly lower L than every additive minimizer, then the premise of Corollary 4.1 is violated in this instance. Also enumerate Θ* and Θ*_TAR for a linear feature parametrization of Aθ, using the definitions in Eq. 12, to check whether the two argmin sets actually coincide when the premise fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the premise of Corollary 4.1 (stated just after Eq. 12): 'Suppose there exists θ* such that Vθ* ∈ argmin_{V≥0} L(V).' The paper treats this as an expressiveness assumption, but it is not automatic and not characterized. The parametrization Eq. 9 forces Vθ to be nonincreasing along every edge: Vθ(s⊕a) = Vθ(s) + Aθ(s,a) ≤ Vθ(s), because Aθ ≤ 0. The true optimal value V* has this property, but the global minimizer of the penalized loss L(V) for finite λ need not. In a minimal instance with states ∅ → a → a⊥, Ψ = {a⊥}, J(a⊥) = 1, if P0 places large mass M on state a and small mass ε on a⊥, the unconstrained minimizer of Eq. 3 sets V(a) = 0 and V(a⊥) = 1 to avoid the downstream Bellman penalty, paying only the small edge penalty λ. Its cost is about λ, while any monotone V must raise V(a) to at least 1, incurring cost about M. Thus the global minimizer violates monotonicity, no θ* exists, and the corollary's premise fails. The paper gives no condition on P0, λ, or the function class under which the premise holds, so the central conclusion that minimizing the regression loss Eq. 10 yields the optimal path is not established for the general offline PL setting.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes trajectory advantage regression (TAR) for offline path learning. It first reduces offline path learning to offline reinforcement learning (Proposition 3.1), then shows that in the reduced MDP the optimal value function decomposes as a sum of per-action advantages (Lemma 4.1). This motivates a value parametrization in which a path's value is a baseline plus accumulated nonpositive advantages (Eq. (9)). The main theoretical result is Theorem 4.1, which shows that a least-squares objective over this parametrization equals the penalized V-LP objective plus a nonnegative excess term relative to the optimal value function. Corollary 4.1 then claims that the minimizers of the two objectives coincide, so that TAR recovers the optimal path. The paper closes with a brief related-work discussion and a statement that experiments are future work.","tokens_in":5786,"tokens_out":18117,"duration_ms":189757,"significance":"If the claimed equivalence were established, TAR would be a conceptually attractive method: path optimization would reduce to a regression problem, and the learned advantage terms would provide an interpretable decomposition of path yields. Proposition 3.1 and Theorem 4.1 are clean formal statements, and the algebraic decomposition in Theorem 4.1 is a genuine strength. However, the central corollary relies on an unproven pointwise monotonicity claim and an uncharacterized realizability assumption, and the objective in Eq. (10) requires a distribution P0 that is not part of the offline path-learning instance. The paper therefore currently establishes a promising framework rather than a validated algorithm with a correct proof of its main guarantee.","major_comments":[{"comment":"The proof asserts that any minimizer Vθ' of L(V) satisfies Vθ' ≤ V* pointwise, because otherwise the truncation \\tilde V = Vθ' - {Vθ' - V*}_+ would give L(\\tilde V) < L(Vθ'). This step is not justified. Truncation lowers the first term of L, but the squared Bellman penalty term may increase when \\tilde V(s) is set below V(s), and no argument is given to control this increase. A concrete illustration that the interaction is nontrivial is a two-state MDP with states x,y, a transition x→y with reward 1, P0 uniform, and λ = 0.5: the point V(x)=0.8, V(y)=0.1 has L(V) = 0.495 < L(V*) = 0.5, even though V(y) > V*(y) = 0. The proof needs a direct analysis of the penalty term; as written, the equality in Eq. (13) and the inclusion Θ* ⊂ Θ*_TAR are unsupported.","section":"Section 4, Corollary 4.1, proof after Eq. (12)"},{"comment":"The premise that there exists θ* with Vθ* ∈ argmin_{V≥0} L(V) is not automatic and is not characterized. The parametrization Eq. (9) forces Vθ to be nonincreasing along every edge because Aθ ≤ 0. For finite λ, the global minimizer of L need not be monotone. For example, take paths ∅ → a → a⊥ with Ψ = {a⊥}, J(a⊥) = 1, P0 placing large mass M on a and small mass ε on a⊥. The unconstrained minimizer of L sets V(a)=0 and V(a⊥)=1, paying only the small edge penalty, while any monotone V must have V(a) ≥ V(a⊥) = 1, incurring cost proportional to M. Thus no θ* exists and the corollary's premise fails. The paper gives no conditions on P0, λ, or the function class under which the premise holds, so the central claim that minimizing the regression loss Eq. (10) yields the optimal path is not established for the general offline PL setting.","section":"Section 4, Corollary 4.1 premise (sentence after Eq. (12))"},{"comment":"The objective LTAR contains E_{s∼P0}[Vθ(s)], where P0 is introduced in Section 2 as a fixed distribution covering the state space, but P0 is not part of the offline PL instance P = (A, Ψ, PΨ, PY) nor of the reduced offline RL instance R. The paper does not state how P0 is chosen or why a particular choice is legitimate for the regression objective. Similarly, the proof of Theorem 4.1 constructs P1 as (μΨ + \\tilde P1)/2 with an arbitrary \\tilde P1 supported on (S \\ Ψ) × A. These unspecified choices affect the objective and the argmin equivalence, so the statement that TAR 'algorithmically only solves a regression problem' is incomplete without guidance on how P0 and \\tilde P1 are set.","section":"Section 4, Eq. (10) and Theorem 4.1"}],"minor_comments":[{"comment":"The notation PΨY is used but never defined; it should be introduced as the joint distribution of (ψ, y) with ψ ∼ PΨ and y | ψ ∼ PY(ψ).","section":"Section 4, Eq. (10)"},{"comment":"There are several typographical errors, including 'conclude the paper discussion the related work' in the introduction and inconsistent notation for path length (a^t versus aT).","section":"Section 1 and Section 5"},{"comment":"The condition Vθ ≥ 0 in Eq. (12) is not enforced by the least-squares objective in Eq. (10), and the paper does not discuss how the nonnegativity constraint would be handled in practice.","section":"Section 4, Eq. (9) and Eq. (12)"},{"comment":"The comparison with RETRACE would be more useful if the formal relationship between the advantage decomposition and RETRACE's edge-dependent terms were stated explicitly rather than only in prose.","section":"Section 5"},{"comment":"The paper explicitly defers experiments; for a cs.LG methods paper, at least a synthetic validation of the argmin equivalence would considerably strengthen the empirical relevance of the theoretical claims.","section":"Section 6"}],"recommendation":"major_revision","confidential_remarks":"The core idea is interesting and Theorem 4.1 is a solid contribution, but the paper's main guarantee (Corollary 4.1) currently rests on two unproven hinges: the pointwise comparison Vθ' ≤ V* and the realizability of the global minimizer in the advantage parametrization. The second issue is illustrated by a simple counterexample within the paper's own setting, so the claim as stated is not merely missing details. I would encourage the authors to either prove the corollary under explicit, checkable sufficient conditions or reformulate the contribution as a surrogate objective with a bounded suboptimality gap rather than exact equivalence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nRead the Miyaguchi path learning paper. The headline: the reduction of offline path learning to offline RL plus a regression surrogate is a genuinely neat idea, and the algebra in Theorem 4.1 checks out. The real contribution is the advantage parametrization (Eq. 9) with nonpositive Aθ, which makes the Bellman penalty vanish on non-terminal states and turns the whole objective into a least-squares problem, while also giving a natural per-action attribution in terms of drawdowns. That combination is new compared to RETRACE and standard offline RL.\n\nWhat it does well: Lemma 4.1 and the path-scoring identities are correct; the PL-to-RL reduction is standard but cleanly stated. Theorem 4.1’s decomposition is valid. I also think the Corollary 4.1 proof is more robust than the reader’s notes claim. The truncation argument Vθ′ ↦ min(Vθ′,V*) works because V* satisfies TV*≤V*, so truncating above V* zeros the penalty on the truncated region and strictly lowers E[V]. The λ=0 worry is a red herring: with λ=0, L_TAR is exactly L(Vθ), so the argmin equivalence is trivial.\n\nThe real soft spot is the premise of Corollary 4.1: “Suppose there exists θ* such that Vθ* ∈ argmin_{V≥0} L(V).” That premise is load-bearing and never characterized. Because Aθ≤0, every Vθ is nonincreasing along every edge. The global minimizer of the penalized V-LP for finite λ need not be monotone. The stress-test’s two-state example is persuasive: if P0 puts heavy mass on a prefix and the terminal transition carries the yield, the unconstrained minimizer can set the prefix value low and the terminal value high, violating V(a)≥V(a⊥). Then no θ* exists. The paper gives no condition on P0, P1, λ, or the function class that guarantees the premise, so the central claim “minimizing TAR yields the optimal path” is conditional, not general.\n\nAlso worth saying: there are no experiments. The conclusion is honest that implementation is future work, but the abstract’s claim that the method “can be used” is stronger than the theory supports.\n\nWho this is for: a theory-minded reader working on offline RL surrogates or path attribution will get a clear idea worth thinking about. It deserves serious peer review—the counterexample should be fixable by either characterizing the premise or relaxing the monotone parametrization. I’d engage with it, but not cite it in its current form.","headline":"A neat regression surrogate for offline path learning whose main equivalence theorem is conditionally correct—but the load-bearing premise that the penalized V-LP minimizer is representable by the monotone advantage class is never characterized.","tokens_in":6208,"tokens_out":9939,"would_cite":false,"duration_ms":102733,"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":"This paper claims that offline path learning reduces to a regression problem via an advantage decomposition of the optimal value function.","keywords":["offline path learning","trajectory advantage regression","offline reinforcement learning","value linear programming","advantage decomposition","path attribution","least-squares regression","path optimization"],"falsifier":"On a small finite action set with known yields, enumerate all parameters of the advantage form and compare the exact minimizers of $L(V_\\theta)$ and $L_{\\mathrm{TAR}}(V_\\theta)$ for several penalty coefficients $\\lambda$; any parameter that minimizes the regression objective but not the penalized LP disproves Corollary 4.1.","tokens_in":5195,"feed_emoji":"🧭","tokens_out":8038,"duration_ms":79027,"temperature":0.7,"pith_summary":"This paper sets out to show that offline path learning—finding the highest-yield path from noisy observations of path yields—can be reduced to offline reinforcement learning and then solved by a plain least-squares regression. The reduction builds an MDP whose states are path prefixes, whose transitions append actions, and whose rewards are the yields of completed paths. In that MDP the optimal value function decomposes as the best yield plus a sum of per-action advantages, so the paper estimates values in the form of a path-sum of advantages. The paper proves that a squared-error loss over this parametrization has the same minimizers as the penalized value linear program, meaning one regression fit can recover the optimal path and per-action attributions.","feed_headline":"Offline path planning is solved by one regression fit","feed_subtitle":"No full RL solver needed: a single squared-error fit returns the best path and per-action contributions.","key_machinery":"The load-bearing object is the advantage decomposition of the optimal value function in the reduced MDP: for any proper path $a^t$, $V^*(a^t)=J^*+\\sum_{k=1}^t A^*(a^{k-1},a_k)$, where $J^*$ is the optimal policy value and $A^*\\le 0$ is the optimal advantage function. The paper parametrizes value estimates as the analogous path sum, $V_\\theta(a^t)=c_\\theta+\\sum_{k=1}^t A_\\theta(a^{k-1},a_k)$ with $A_\\theta\\le0$ and $V_\\theta\\ge0$. This form makes the regression objective $L_{\\mathrm{TAR}}(V_\\theta)$ in Eq. (10) a simple least-squares problem, and it is the reason the penalty term in the value LP simplifies enough for Corollary 4.1 to hold.","core_discovery":"The central claim is Corollary 4.1: under the constant-plus-advantages parametrization, the minimizers of the trajectory advantage regression objective $L_{\\mathrm{TAR}}(V_\\theta)$ coincide with the minimizers of the penalized value linear program $L(V_\\theta)$. Equivalently, optimizing the regression objective in Eq. (10) recovers the optimal value function $V^*$, and therefore the optimal path, without solving an MDP. The proof relies on an identity relating $L_{\\mathrm{TAR}}$ to $L$ plus a nonnegative penalty for values that exceed $V^*$; when that penalty vanishes, the two objectives have identical minimizers. The same decomposition also gives a path-attribution reading: the predicted yield of a path is the estimated best yield $c_\\theta$ plus the drawdowns introduced action by action.","pith_inferences":["The same regression-as-planning move may extend to other regularized RL objectives—for instance entropy-regularized or distributionally robust value LPs—provided an analogous additive decomposition of $V^*$ exists.","An immediate empirical test would compare the minimizers of $L$ and $L_{\\mathrm{TAR}}$ on small synthetic path-yield datasets across values of the penalty coefficient $\\lambda$; discrepancies would show where the pointwise inequality used in the proof stops holding.","The telescoping structure of the advantage decomposition is not special to routing; any sequential decision problem with additive per-step returns could admit a similar trajectory-advantage regression, from text generation to structured prediction."],"forward_implications":["If the central claim is correct, offline path optimization can be performed with standard regression software rather than a full RL solver.","Learned per-action advantages give a direct explanation of why a path misses the optimal yield, since each $A_\\theta(a^{k-1},a_k)$ measures the drawdown introduced by that step.","Any function approximator suited to least-squares regression becomes available for path value estimation, which may help with large action spaces.","Because the reduction embeds path learning in offline RL, existing offline RL theory on data coverage and distribution shift transfers to path learning."],"supporting_citations":[{"why":"Supplies the MDP formalism, the value linear program, and the existence of a deterministic optimal policy used by the paper's reduction.","marker":"[Put14]"},{"why":"Provides the standard definitions of value functions, advantages, and reinforcement learning that the decomposition builds on.","marker":"[SB+98]"},{"why":"Defines offline reinforcement learning and motivates the fixed-dataset setting that TAR's regression objective targets.","marker":"[LKTF20]"},{"why":"Presents the RETRACE trajectory regression whose path-attribution decomposition is contrasted with TAR's advantage decomposition.","marker":"[IS11]"}],"fun_headline_variants":["One regression fit replaces full RL for path planning","Offline path learning is just a regression problem","Trajectory advantage regression: RL as regression","Path optimization from one squared-error fit","Solving MDPs with a single regression objective"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The equivalence of the two objectives assumes every minimizer of the penalized value LP lies pointwise at or below the true optimal value $V^*$, and that the constant-plus-advantages form is expressive enough to represent such a minimizer.","fun_headline_variants_meta":{"raw":{"variants":["One regression fit replaces full RL for path planning","Offline path learning is just a regression problem","Trajectory advantage regression: RL as regression","Path optimization from one squared-error fit","Solving MDPs with a single regression objective"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001129,"raw_usage":{"total_tokens":4583,"prompt_tokens":723,"completion_tokens":3860,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":339,"completion_tokens_details":{"reasoning_tokens":3791}},"tokens_in":339,"tokens_out":3860,"duration_ms":27437,"temperature":1.0,"reasoning_tokens":3791,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:33:45.565917+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a small finite action set with known yields, enumerate all parameters of the advantage form and compare the exact minimizers of $L(V_\\theta)$ and $L_{\\mathrm{TAR}}(V_\\theta)$ for several penalty coefficients $\\lambda$; any parameter that minimizes the regression objective but not the penalized LP disproves Corollary 4.1.","supporting_citations":[],"review_version":2}