{"id":"6bb1e374-d67b-4902-ab22-d28275f72c78","arxiv_id":"2502.02516","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Adaptive exploration for multi-reward multi-policy evaluation, with an instance-dependent lower bound and a convex relaxation algorithm.","lead":"This paper derives a sample-complexity bound for evaluating several policies against several reward functions at once in an online Markov decision process, and proposes an adaptive exploration algorithm built on it. The significance is a principled way to decide where to sample so that all policy-reward estimates become accurate with high confidence as quickly as possible.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Optimality claim rests on unproven constant-factor bound between relaxed and exact characteristic times.","rationale":"The reader's weakest_assumption identifies exactly the unproven constant-factor relation between U*_epsilon and T*_epsilon. That is also the most load-bearing concern I see: every other component of the paper -- the lower-bound derivation, the convex relaxation given in Theorem 4.7, the stopping rule, and the numerical comparisons with public code -- is either internally consistent or supported by standard techniques from the best-policy-identification literature. The gap is not a minor technicality: the lower bound in Theorem 4.4 is stated in terms of T*, while the upper bound in Theorem 5.2 is in terms of 4U*. Since the paper's headline claim is asymptotic optimality up to a constant, the ratio U*/T* must be O(1). Section 4.3 explicitly leaves this unquantified, and Section 4.1 only conjectures it. The non-convexity of the alternative set shown in Example 4.6 makes it plausible that the relaxation is loose, and nothing in the appendix closes that gap. A concrete numerical computation of both quantities on a small MDP would settle whether the concern is real for at least one instance; conversely, a proof or a counterexample would resolve the matter. Because the reader already arrived at the same concern and the verdict CONDITIONAL appropriately reflects an unsupported but plausible claim, I recommend no change to the reader's verdict.","tokens_in":42930,"tokens_out":2630,"duration_ms":28890,"concrete_test":"Compute both U*_epsilon(M) and T*_epsilon(M) exactly for a small MDP where the relaxation can be evaluated explicitly. A natural instance is the two-state MDP of Example 4.6 with gamma = 0.9, p2 = 0.5, r2 = 0.5, p3 = 0.01. For T*, enumerate the confusing models by gridding the alternative transition parameter p2' (and the constructed transitions from Proposition 4.1) and evaluate the infimum in Eq. (2); for U*, solve the convex program in Eq. (6). Sweep epsilon over, say, 0.001 to 0.1 and gamma over 0.5 to 0.99, and report sup_epsilon,gamma U*/T*. If this ratio is bounded by a small constant (e.g., < 10) the constant-factor claim survives this instance; if the ratio grows as epsilon decreases or exceeds a reasonable constant, the optimality claim must be weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The advertised 'asymptotic optimality for policy evaluation up to a constant factor' requires an upper bound on E[tau]/log(1/delta) that is a constant multiple of the lower bound. Theorem 5.2 gives limsup E[tau]/log(1/delta) <= 4 U*_epsilon(M), while Theorem 4.4 gives liminf >= T*_epsilon(M). Therefore the claim is valid only if U*_epsilon(M)/T*_epsilon(M) is bounded by a constant independent of epsilon, delta, gamma, and the MDP. The paper does not prove this. Section 4.3 states that 'quantifying the gap |U_epsilon(omega; M) - T_epsilon(omega; M)| remains challenging, and we leave this analysis to future work,' and Section 4.1 only says 'we believe the looseness is at most a constant factor.' The relaxation in Theorem 4.7 also replaces the constrained reward set R^epsilon_pi used in the lower bound by the full set R_pi, and Example 4.6 shows that the set of confusing models can be non-convex, so the relaxation is not a trivial tightening. Without a proof that U*/T* is O(1), the central optimality claim is unsupported; the verified statement is only that MR-NaS achieves the relaxed rate, not that the relaxed rate is within a constant of the true optimal rate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies online (epsilon, delta)-PAC policy evaluation in discounted tabular MDPs when multiple policies must be evaluated over multiple reward sets, with a value-vector error criterion. It first derives an instance-specific lower bound on the sample complexity (Theorem 4.4) expressed through a characteristic time T*_epsilon, defined via an optimization over confusing alternative models. Because this optimization is non-convex (illustrated by Example 4.6), the paper introduces a convex relaxation U_epsilon(omega; M) (Theorem 4.7), and shows how to compute the relaxed optimal allocation for finite and convex reward sets, including a closed form for the reward-free case (Corollary 4.8). The algorithm MR-NaS, adapted from the authors' prior multi-reward best-policy-identification work, is then analyzed: Theorem 5.2 states that MR-NaS is (epsilon, delta)-PAC, stops almost surely, and satisfies limsup E[tau]/log(1/delta) <= 4 U*_epsilon(M). The paper claims in Sections 1 and 7 that this constitutes asymptotic optimality up to a constant factor. Experiments on four tabular environments compare MR-NaS with SF-NR, GVFExplorer, and noisy-policy baselines.","tokens_in":43140,"tokens_out":5242,"duration_ms":54909,"significance":"If the constant-factor optimality claim were fully established, this would be a notable contribution: it would give the first instance-dependent lower bound for multi-reward multi-policy evaluation and a practical algorithm matching it up to a constant. The paper's machinery is substantial: the lower bound construction via confusing models, the relaxation in terms of the one-step value deviation rho, the closed-form solution for reward-free evaluation, and the extension of MR-NaS all appear technically coherent, and the appendix contains detailed proofs. The code is provided, which supports reproducibility. The main reservation is that the headline guarantee is not supported by the stated theorems: Theorem 5.2 bounds the algorithm by the relaxed rate U*, while the lower bound is the exact rate T*, and the paper explicitly leaves the gap U*/T* to future work. The experimental results show good empirical accuracy of MR-NaS relative to baselines, but they do not validate the asymptotic constant-factor claim.","major_comments":[{"comment":"The advertised claim of 'asymptotic optimality for policy evaluation up to a constant factor' is not established by the stated results. Theorem 5.2 gives limsup_{delta->0} E[tau]/log(1/delta) <= 4 U*_epsilon(M), while the lower bound of Theorem 4.4 is liminf >= T*_epsilon(M). For the optimality claim to hold, one needs U*_epsilon(M) <= C T*_epsilon(M) for a constant C independent of epsilon, delta, gamma, and the MDP. Section 4.3 explicitly states that 'quantifying the gap |U_epsilon(omega; M) - T_epsilon(omega; M)| remains challenging, and we leave this analysis to future work', and Section 4.1 only reports a belief that the looseness is at most a constant factor. Without a proof of the constant-factor gap, Theorem 5.2 only shows that MR-NaS attains the relaxed rate, not that the relaxed rate is within a constant of the true optimal rate. This is a load-bearing gap in the central claim of the paper.","section":"Section 4.3, Eq. (6); Theorem 5.2; Section 1"},{"comment":"The relaxation in Theorem 4.7 is not a provably tight surrogate for the true characteristic time. The true rate T_epsilon(omega; M) in Eq. (2) involves an infimum over confusing models in Alt^epsilon_pi,r(M) with rewards restricted to R^epsilon_pi, whereas the relaxed rate U_epsilon in Eq. (6) replaces this by a supremum over the full reward set R_pi and bounds the KL terms using a one-sided necessary condition on the value deviation rho. Example 4.6 shows that the set of confusing parameters can be non-convex, so this is not a benign substitution. Theorem 4.7 provides only T <= U; it gives no quantitative comparison between U*_epsilon and T*_epsilon. Given that Section 4.3 leaves the gap analysis to future work, the paper must either prove the needed bound under its stated assumptions (or a clearly specified subclass, such as reward-free evaluation with the canonical basis, where Corollary 4.8 may make the analysis tractable), or reframe the optimality claim as a conjecture and present Theorem 5.2 as a guarantee relative to the relaxed quantity only.","section":"Theorem 4.7; Example 4.6; Eq. (2)"},{"comment":"The use of the 2epsilon separation also introduces an unquantified constant relative to any 'exact' lower bound for epsilon-accurate evaluation. The text says the 2epsilon weakening 'potentially weakens the resulting lower bound' and states 'we believe the looseness is at most a constant factor (see Theorem 5.2)', but Theorem 5.2 does not connect T*_epsilon (defined with 2epsilon separation) to a rate for algorithms that only need epsilon separation; it merely upper-bounds the algorithm's stopping time by 4U*_epsilon. Even if the relaxation gap were closed, one would still need to justify that the lower bound with a 2epsilon separation is within a constant of the minimal sample complexity for the epsilon-PAC problem as stated in Definition 3.2. Please make this relationship explicit, or restrict the optimality statement accordingly.","section":"Section 4.1 and Section 5; Theorem 5.2"}],"minor_comments":[{"comment":"The claim that the lower bound 'constitutes the first such bound even for the single-policy single-reward case' is too strong as stated; prior single-reward PAC policy-evaluation bounds (e.g., Dann et al., Chen et al.) exist, and the novelty lies in the reward-dependent instance-specific form. Please hedge or qualify this claim with respect to the dependence on the reward value deviation.","section":"Section 4 (introductory paragraph)"},{"comment":"The abbreviation 'sim.' for 'similarly' appears in the notation paragraph; it should be written out for clarity.","section":"Section 3.1, Additional notation paragraph"},{"comment":"The y-axis label '∥VπMr− VπM′r(p2)∥∞' uses M' as a function of p2 but the notation is not defined; please clarify that the alternative model varies with p2 in the caption or in the text.","section":"Figure 2 caption"},{"comment":"The stopping condition uses 'U_epsilon/2(N_t/t; M_t)', but the definition of U_epsilon in Eq. (6) takes an allocation omega; please add an explicit definition of the empirical object U_epsilon(N_t/t; M_t) in the text, or a pointer to the appendix where the empirical stopping rule is formalized.","section":"Algorithm 1, line 1"},{"comment":"The empirical evaluation reports estimation error over time but does not report the stopping time tau or compare the observed stopping time against the theoretical relaxed rate U*_epsilon; reporting this would let the reader assess whether the sample-complexity guarantee has practical relevance beyond the error-vs-time curves.","section":"Section 6 and Figures 3-4"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically careful in its derivations and the relaxed-rate analysis appears sound, but the constant-factor optimality claim is the paper's main selling point and it is not proven. I believe this is fixable within the manuscript's scope: the authors could either prove a bounded U*/T* gap for a natural subclass (e.g., reward-free evaluation or finite reward sets under additional structural assumptions), or substantially weaken the abstract and introduction to say that MR-NaS achieves the relaxed characteristic rate and that constant-factor tightness is conjectured. Given that the lower bound and algorithm are otherwise coherent, I see no reason for rejection, but the overclaim must be addressed before publication. The self-citation density is high but not inappropriate given that the algorithm is a direct extension of the authors' MR-NaS; I would not raise it beyond this remark."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Core issue: the paper's central claim—that MR-NaS is asymptotically optimal up to a constant factor—is not supported by the theorems as stated. Theorem 5.2 gives limsup E[tau]/log(1/delta) <= 4 U*_epsilon, while the lower bound in Theorem 4.4 is T*_epsilon. The gap between U* and T* is explicitly left open in Section 4.3, and Example 4.6 shows the true set of confusing models can be non-convex, so the relaxation is not trivially tight. Without U*/T* = O(1), the upper bound does not match the lower bound. The authors themselves say quantifying the gap is future work; the abstract and intro still advertise constant-factor optimality. That needs to be fixed by proving the gap or softening the claim.\n\nWhat is genuinely new: the multi-reward multi-policy PAC evaluation problem, the instance-dependent lower bound, and the convex relaxation with a closed form for the reward-free case. These go beyond the MR-NaS BPI work, which targets best-policy identification rather than value estimation. The proofs are detailed, the experiments cover four tabular domains with public code, and the empirical comparison against SF-NR and GVFExplorer is honest.\n\nSoft spots, in proportion: the optimality claim is the main one. The 2-epsilon separation and the constant 4 are standard and fine. Assumption 5.1 (unique solution in the open set) is restrictive but not unusual. The lower bound is asymptotic in delta, which matches the BPI literature.\n\nWho this is for: people working on pure exploration, policy evaluation pipelines, and reward modeling. The lower bound and the convex relaxation are the lasting value. This deserves a serious referee, but the referee should insist on either a proof that U*/T* = O(1) or a revised statement that the algorithm achieves the relaxed rate.","headline":"A real contribution to multi-policy evaluation, but the headline optimality claim rests on an unproven constant-factor gap between the relaxed and exact characteristic times.","tokens_in":43679,"tokens_out":2549,"would_cite":true,"duration_ms":24543,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68Q32","68T05","90C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The optimal way to collect data for multi-policy, multi-reward evaluation is to sample in proportion to a one-step value-deviation measure, and MR-NaS provably achieves this rate up to a constant.","keywords":["multi-reward multi-policy evaluation","(epsilon, delta)-PAC sample complexity","adaptive exploration","Markov decision processes","characteristic time","best policy identification","reward-free policy evaluation","value deviation"],"falsifier":"Take any small tabular MDP (for instance the two-state example the paper uses to show non-convexity), fix $\\epsilon$ and a target policy–reward pair, compute the true characteristic time $T^\\star_\\epsilon(M)$ by brute-force enumeration over all alternative transition functions that change the value by more than $2\\epsilon$, and compare it with the relaxed rate $U^\\star_\\epsilon(M)$ from Theorem 4.7; if the ratio $U^\\star_\\epsilon/T^\\star_\\epsilon$ diverges or grows steeply as $\\epsilon \\to 0$ or as the transition probabilities vary, the constant-factor optimality claim fails. Running MR-NaS in the same MDP and measuring $\\mathbb{E}[\\tau]/\\log(1/\\delta)$ against both quantities would show which rate the algorithm actually tracks.","tokens_in":42670,"feed_emoji":"🎯","tokens_out":17206,"duration_ms":130087,"temperature":0.7,"pith_summary":"The paper studies how one run of an agent should gather experience when it must evaluate several policies against several reward functions at once — a task that arises in preference-based fine-tuning, multi-objective planning, and goal-oriented robotics. It establishes an instance-specific $(\\epsilon,\\delta)$-PAC sample-complexity lower bound for this setting, showing that any correct algorithm needs on the order of $T^\\star_\\epsilon(M)\\log(1/\\delta)$ samples, where the difficulty is governed by the hardest policy–reward pair rather than their sum. Because the exact characteristic time is a hard non-convex program, the paper replaces it with a convex relaxation $U_\\epsilon$, whose optimal allocation samples each state–action pair in proportion to a one-step value-deviation measure $\\rho^\\pi_r(s,s')$. Adapting the MR-NaS exploration scheme, it proves an upper bound of $4U^\\star_\\epsilon(M)\\log(1/\\delta)$ samples — optimal data collection up to a constant factor under the stated assumptions — and demonstrates the rule's practicality in tabular experiments. If the paper is right, a single variance-like quantity computed from values and transitions dictates where to spend samples, whether the reward sets are finite, convex, or the full set of all rewards.","feed_headline":"Value-deviation sampling is PAC-optimal for multi-policy evaluation","feed_subtitle":"One run must value many policies and rewards; MR-NaS provably matches the optimal sample count up to a constant.","key_machinery":"The load-bearing object is the one-step value deviation $\\rho^\\pi_r(s,s') := V^\\pi_r(s') - \\mathbb{E}_{\\hat{s}\\sim P(s,\\pi(s))}[V^\\pi_r(\\hat{s})]$, which measures how much the value of policy $\\pi$ under reward $r$ at a next state $s'$ deviates from its expected value after taking $\\pi$'s action in $s$. A state with large $\\|\\rho^\\pi_r(s)\\|_\\infty$ is 'confusing': a statistically near-invisible perturbation of the transition function can change the value by more than $2\\epsilon$, and these alternative models are exactly what the lower bound must rule out. The relaxed characteristic time $U_\\epsilon(\\omega;M) = \\sup_{\\pi\\in\\Pi, r\\in R_\\pi}\\max_s \\gamma^2\\|\\rho^\\pi_r(s)\\|_\\infty^2 / (2\\epsilon^2(1-\\gamma)^2\\omega(s,\\pi(s)))$ packages the deviations into a convex program whose minimizer $\\omega^\\star$ is the optimal data-collection distribution. Writing $\\rho^\\pi_r(s,s') = e_{s'}^\\top\\Gamma^\\pi(s)r$ with $\\Gamma^\\pi(s) = (I - \\mathbf{1}P(s,\\pi(s))^\\top)(I-\\gamma P^\\pi)^{-1}$ turns the maximization over rewards into linear or convex programs, with a closed form over the full reward cube $[0,1]^S$. The machinery converts an abstract infimum over confusing models into a concrete proportional-sampling rule.","core_discovery":"The paper's central claim is that the sample complexity of online $(\\epsilon,\\delta)$-PAC policy evaluation over multiple policies and reward sets is governed by a single instance-dependent quantity — the relaxed characteristic time $U^\\star_\\epsilon(M)$ — and that the exploration rule it induces is asymptotically optimal up to a constant factor, a claim the paper states as the first instance-specific lower bound for this setting. The carrier of the argument is the one-step value deviation $\\rho^\\pi_r(s,s') = V^\\pi_r(s') - \\mathbb{E}_{\\hat{s}\\sim P(s,\\pi(s))}[V^\\pi_r(\\hat{s})]$, a variance-like measure of how much the value of policy $\\pi$ under reward $r$ spreads from the expected next-state value; the optimal allocation samples each state–action pair in proportion to $\\gamma^2\\|\\rho^\\pi_r(s)\\|_\\infty^2 / (2\\epsilon^2(1-\\gamma)^2)$, evaluated at the worst-case policy–reward pair. The paper proves a lower bound of $T^\\star_\\epsilon(M)$ (Theorem 4.4), a convex relaxation $U_\\epsilon(\\omega;M)$ of the potentially non-convex characteristic time that makes the optimal allocation computable for finite and convex reward sets (Theorem 4.7), and a closed-form allocation for the reward-free case computed from the discounted fundamental matrix (Corollary 4.8). For the MR-NaS algorithm — certainty-equivalence planning into the relaxed rate, mixed with a forcing policy, plus a stopping rule — it proves that the returned values are $\\epsilon$-accurate for every policy and reward with probability at least $1-\\delta$, that the algorithm stops almost surely, and that $\\limsup_{\\delta\\to 0} \\mathbb{E}[\\tau]/\\log(1/\\delta) \\le 4U^\\star_\\epsilon(M)$.","pith_inferences":["A direct numerical check, not performed in the paper, would settle the strength of the optimality claim: compute $U^\\star_\\epsilon/T^\\star_\\epsilon$ by exhaustive enumeration on small MDPs (such as the two-state example in the paper) and see whether the unquantified gap stays bounded as $\\epsilon\\to 0$.","The proportional-to-deviation allocation is the PAC analogue of variance-optimal importance sampling; one would expect the same sampling rule to improve finite-sample off-policy evaluation and to extend to stochastic target policies and function approximation, settings this paper does not treat.","A cheap testable signature of the theory is that visit counts under MR-NaS in a fixed tabular environment should grow roughly in proportion to $\\|\\rho^\\pi_r(s)\\|_\\infty^2 / U^\\star_\\epsilon(M)$ over long runs; the paper reports estimation error but not the realized allocation, so this comparison is open."],"forward_implications":["MR-NaS attains the $(\\epsilon,\\delta)$-PAC guarantee with $\\mathbb{E}[\\tau] \\le 4U^\\star_\\epsilon(M)\\log(1/\\delta)$ asymptotically, matching the instance-specific lower bound up to a constant factor.","The optimal exploration rule is concrete: sample each state–action pair $(s,\\pi(s))$ in proportion to the squared one-step value deviation $\\|\\rho^\\pi_r(s)\\|_\\infty^2$ of the hardest policy–reward pair, a quantity computable from value estimates and the transition model.","Sample complexity scales with the worst-case policy–reward pair rather than their sum, so adding policies or rewards to the evaluation set does not compound the data requirement.","For reward-free evaluation over all rewards $[0,1]^S$, the optimal allocation has a closed form in terms of the discounted fundamental matrix $\\Gamma^\\pi(s)$, so no reward-side optimization is needed.","Any behavior policy that renders a high-deviation state transient is provably suboptimal — its information rate collapses to zero — so persistent coverage of high-deviation states is necessary for optimal exploration."],"supporting_citations":[{"why":"Supplies the NaS algorithm and the change-of-measure lower-bound technique that this paper adapts to policy evaluation.","marker":"(Al Marjani et al., 2021)"},{"why":"The original MR-NaS scheme and the proof method for Theorem 3.3 that Theorem 5.2 reuses, plus the multi-reward BPI setting this work extends.","marker":"(Russo & Vannella, 2024)"},{"why":"Provides the change-of-measure inequality, the kl(δ, 1−δ) lower bound, used in the proof of Theorem 4.4.","marker":"(Kaufmann et al., 2016)"},{"why":"The fixed-confidence best-arm identification framework whose information-theoretic lower-bound template the analysis follows.","marker":"(Garivier & Kaufmann, 2016)"},{"why":"Proposition 1 there yields the martingale bound that makes the MR-NaS stopping rule δ-correct.","marker":"(Jonsson et al., 2020)"},{"why":"Source of the variance-driven allocation intuition and of the ForkedRiverswim environment used in the experiments.","marker":"(Russo & Proutiere, 2023a)"},{"why":"Background MDP theory, including the recurrent-chain eigenvector structure used to characterize rewards with zero value deviation.","marker":"(Puterman, 2014)"}],"fun_headline_variants":["PAC-optimal sampling for multi-policy, multi-reward evaluation","Value-deviation bound syncs adaptive exploration to lower bound","One-run evaluation of many policies stays sample-optimal","Adaptive exploration within constant of PAC lower bound","Convex relaxation of hard characteristic time drives sampling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the relaxed characteristic time $U_\\epsilon$, in which the algorithm's upper bound is stated, is within a constant factor of the true characteristic time $T_\\epsilon$, in which the lower bound is stated; Section 4.3 explicitly leaves the gap $|U_\\epsilon - T_\\epsilon|$ unquantified and says quantifying it is future work, so if the ratio $U^\\star_\\epsilon/T^\\star_\\epsilon$ is unbounded or large, the claim of optimality up to a constant factor fails.","fun_headline_variants_meta":{"raw":{"variants":["PAC-optimal sampling for multi-policy, multi-reward evaluation","Value-deviation bound syncs adaptive exploration to lower bound","One-run evaluation of many policies stays sample-optimal","Adaptive exploration within constant of PAC lower bound","Convex relaxation of hard characteristic time drives sampling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000266,"raw_usage":{"total_tokens":1680,"prompt_tokens":1087,"completion_tokens":593,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":703,"completion_tokens_details":{"reasoning_tokens":515}},"tokens_in":703,"tokens_out":593,"duration_ms":6809,"temperature":1.0,"reasoning_tokens":515,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T11:51:49.049063+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any small tabular MDP (for instance the two-state example the paper uses to show non-convexity), fix $\\epsilon$ and a target policy–reward pair, compute the true characteristic time $T^\\star_\\epsilon(M)$ by brute-force enumeration over all alternative transition functions that change the value by more than $2\\epsilon$, and compare it with the relaxed rate $U^\\star_\\epsilon(M)$ from Theorem 4.7; if the ratio $U^\\star_\\epsilon/T^\\star_\\epsilon$ diverges or grows steeply as $\\epsilon \\to 0$ or as the transition probabilities vary, the constant-factor optimality claim fails. Running MR-NaS in the same MDP and measuring $\\mathbb{E}[\\tau]/\\log(1/\\delta)$ against both quantities would show which rate the algorithm actually tracks.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The original MR-NaS scheme and the proof method for Theorem 3.3 that Theorem 5.2 reuses, plus the multi-reward BPI setting this work extends."}],"review_version":1}