{"id":"fa92bbb9-cc72-4d38-81c3-0e7bae847e74","arxiv_id":"2608.04310","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"ArborEnum exactly and approximately enumerates decision-tree Rashomon sets over continuous thresholds using interval pruning and anytime refinement.","lead":"ArborEnum is a new algorithm for computing Rashomon sets, the full collection of near-optimal decision trees, directly on continuous features instead of on a coarse binarization. Generalists should care because knowing all good models, not just one, exposes predictive multiplicity and robust feature importance for interpretable machine learning.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exactness of continuous-threshold pruning rests on an unproven delegated Lipschitz bound; brute-force small-dataset check would settle whether the exact Rashomon set is truly complete.","rationale":"The paper's strongest claim is exact enumeration over continuous thresholds. For that claim to hold, the interval-pruning logic must never discard a threshold whose optimal completion is within the Rashomon budget. The correctness of that pruning reduces to a Lipschitz property of the optimal completed objective under sample movement. The property is plausible and likely true for an optimal proxy, but the paper does not prove it; it cites prior work on optimal decision trees. My own analysis suggests the bound holds because moving k samples from one child to the other can change each child's optimal objective by at most the number of samples moved, and the leaf-penalty term is unchanged when the same tree structure is applied. However, that reasoning is not spelled out in the paper, and a subtle issue (e.g., empty leaves or structural changes in optimal subtrees) could invalidate it. The reader's weakest assumption identified exactly this delegation, so I agree with that part of the reader's assessment. The other reader concerns (approximate recall not being a certificate, anytime convergence lacking a formal proof) are secondary because they concern the approximate and anytime modes, not the central exactness claim. A brute-force comparison on small datasets directly tests whether the pruning is sound in practice; if it passes, the central claim is supported and only a proof rewrite is needed.","tokens_in":53953,"tokens_out":11460,"duration_ms":108429,"concrete_test":"Run ArborEnum+OPT and a brute-force enumerator on several small datasets (e.g., n ≤ 50, 2–3 continuous features, depth ≤ 3, all thresholds between consecutive unique values) across a grid of γ and ε_mult; verify that the returned Rashomon sets (or AND/OR graphs) are identical. If any instance differs, the interval-pruning exactness claim fails. If all match, the delegated Lipschitz bound is empirically validated and the central claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that ArborEnum with an optimal proxy exactly enumerates the continuous-feature Rashomon set depends on the interval-pruning rules in Algorithms 2 and 3: a threshold whose computed proxy completion exceeds the budget by Δ is used to prune any threshold whose active-sample distance is less than Δ. This pruning is exact only if moving k samples across a threshold changes the proxy-completed objective by at most k. The paper states that this holds for an optimal proxy, but the proof is not given; Theorem 10 merely says \"See Brito et al. 2025 for details of this fact.\" The prior work proves related bounds for finding a single optimal tree, not for Rashomon-set enumeration with a slack budget and a per-leaf penalty. If the Lipschitz property fails in any regime—for example, when a child becomes empty and the leaf-penalty term changes the optimal subtree structure, or when the optimal child subtrees differ between neighboring thresholds—then a feasible threshold could be pruned and the exact enumeration would silently omit trees that are within the Rashomon budget. This is the single most load-bearing assumption for the paper's headline contribution.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces ArborEnum, a framework for enumerating decision-tree Rashomon sets directly over continuous features, without a user-chosen binarization. The method builds on an AND/OR graph representation of the Rashomon set and a proxy-based interval pruning scheme: when a threshold's proxy-completed objective exceeds the budget by Delta, neighboring thresholds whose active-sample Hamming distance is less than Delta are pruned. This pruning is exact when the proxy satisfies a 1-Lipschitz robustness condition, which the paper claims holds for optimal proxies, yielding an exact enumeration algorithm. The paper also proposes an approximate variant using a modified LicketySPLIT proxy (LicketySNIP), an anytime variant that progressively activates more thresholds and optionally strengthens the proxy, and a budget-independent subgraph caching scheme. Experiments on 20 datasets compare runtimes, memory, and recall against existing enumeration methods under exhaustive binarization.","tokens_in":54096,"tokens_out":3851,"duration_ms":36660,"significance":"If the central exactness claim holds, this is a substantial advance: it is the first algorithm that exactly enumerates decision-tree Rashomon sets while exploiting the ordered structure of continuous features, avoiding the information loss and combinatorial blowup of exhaustive binarization. The paper's empirical comparison is extensive, covering 20 datasets, multiple regularization levels, three bootstrap replicates, and existing methods, with clear reporting of timeouts and memory limits. The budget-independent AND/OR graph representation is a meaningful contribution in its own right, with theoretical lower bounds (Theorems 1--3) showing concrete duplication factors avoided. The paper also provides self-contained guarantees for binarization-induced optimality gaps (Theorems 4--9). However, the central exactness proof is delegated to a prior publication in a form that does not clearly cover the Rashomon-set enumeration setting, and the anytime convergence claim is asserted without proof.","major_comments":[{"comment":"The exactness of the interval pruning rules in Algorithms 2 and 3 (lines 16--24 and 43--51) rests on the claim that an optimal proxy's completed objective changes by at most the number of samples moved when a threshold shifts. The manuscript states in Section 3 that 'Optimal proxies satisfy this condition' and Theorem 10's proof says 'See Brita, van der Linden, and Demirovi (2025) for details of this fact,' but the cited work is about finding a single optimal tree, not about Rashomon-set enumeration with a slack budget and a per-leaf penalty. Because the headline contribution---exact enumeration over continuous thresholds---depends entirely on this Lipschitz property, the paper must either prove it self-contained or provide a precise statement and theorem number from the cited work that explicitly covers this setting. As written, the exactness claim is not verifiable from the manuscript.","section":"Section 3, Algorithms 2--3 and Appendix B, Theorem 10"},{"comment":"The claim that the anytime variant is 'truly anytime' and 'converges to the complete continuous-feature Rashomon set' is asserted without a theorem. Specifically, no proof is given that (a) the proxy-strengthening loop can reach an optimal proxy in this setting, and (b) after all thresholds are activated and the proxy is strengthened, RefineGraph produces the same graph as running ArborEnum from scratch with the full threshold set and optimal proxy. Since the anytime convergence is an advertised capability in the abstract, this needs a formal statement with conditions.","section":"Section 3, Anytime Algorithm and Algorithm 4"}],"minor_comments":[{"comment":"The author name 'Brita' appears with a typographical artifact in multiple places (e.g., 'Brit,a'); this should be fixed to 'Brita'.","section":"Throughout (citations)"},{"comment":"The robustness condition is stated only in prose; it would help to state it as a numbered assumption, and to state explicitly that LicketySNIP's greedy completions are not known to satisfy it, so the near-perfect recall results for the approximate variants are empirical rather than guaranteed.","section":"Section 3 and Appendix C.1"},{"comment":"The recall metric is defined relative to the 'best method that finished' using a 'guessed Rashomon bound' from the minimum objective found; this definition should also appear in the main text near Table 2, since it is important for interpreting the reported recall values.","section":"Appendix E.2, Table 5 and Table 10"},{"comment":"The predicate 'notIsProxyOptimal(Proxy)' is used without definition; the paper should specify when the lookahead parameter is considered to make the proxy optimal, e.g., when the lookahead reaches the remaining depth.","section":"Algorithm 4"},{"comment":"Theorem 8 assumes a fixed tie-breaking order, but the main text describes using empirical quantile thresholds with deduplication; the relation between these two settings should be clarified.","section":"Appendix B, Theorem 8"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the delegated proof of the Lipschitz property for optimal proxies. If the authors can supply a self-contained proof or a precise citation covering the Rashomon-budget setting, the central claim is likely sound; otherwise the exactness claim is unsupported. The paper's scope fits the journal, and the empirical work is solid. The repeated self-citations to PRAXIS and LicketySPLIT are relevant and not inappropriate, but the reliance on an inaccessible proof detail is the key concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"ArborEnum is a genuine first: it is the first decision-tree Rashomon set algorithm that operates on continuous thresholds directly instead of requiring a user-chosen binarization. The interval-pruning rules, the budget-independent subgraph caching, and the anytime threshold refinement are real mechanisms, not a relabeled version of the binary case. The exact variant is built on a clean idea: if the proxy-completed objective is 1-Lipschitz in the Hamming distance between threshold columns, then pruning nearby thresholds is valid. The experimental section is substantial and the code is released.\n\nThe soft spots are real but not fatal. The load-bearing Lipschitz property is not proved in this paper. Theorem 10 simply says \"see Brito et al. 2025,\" and that prior work was about finding a single optimal tree, not about Rashomon budgets with a per-leaf penalty. If the leaf-penalty term changes when a child becomes empty, the objective can shift by more than the number of samples moved, and then interval pruning could silently drop feasible thresholds. The paper needs either a direct proof of the robustness condition for this objective and budget, or a brute-force exact check on a small dataset where full enumeration is feasible. To their credit, the appendix tracks the proxy's selected split separately to avoid false pruning, which shows the authors know the approximate proxies can mislead; that makes me trust the engineering even where the theory is thin.\n\nThe empirical claims are slightly oversold. The abstract promises order-of-magnitude speedups over existing enumeration methods, but Table 6 shows PRAXIS finishing faster than exact ArborEnum on Bike, Student, and RL. That claim should be qualified. Also, approximate recall is measured relative to the best method that finished, not against the true continuous-feature Rashomon set; \"near-perfect recall\" is therefore a relative statement. The anytime convergence argument is stated without a formal proof.\n\nWho should read this: anyone working on Rashomon sets, predictive multiplicity, or exact interpretable models. It deserves a serious referee. I would send it out, but ask for a proof of the Lipschitz condition (or a small exact validation), and for the abstract and conclusions to match what Tables 6 and 2 actually show.","headline":"First continuous-threshold Rashomon enumeration, worth reviewing, but exactness rests on a delegated Lipschitz proof and the speedup claims outrun Table 6.","tokens_in":54723,"tokens_out":4007,"would_cite":true,"duration_ms":37080,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes that decision-tree Rashomon sets—all bounded-depth trees within a tolerance of the optimal regularized objective—can be enumerated exactly over continuous thresholds without a user-chosen binarization.","keywords":["decision tree Rashomon set","continuous features","interval pruning","exact enumeration","anytime algorithm","predictive multiplicity","AND/OR graph","threshold binarization"],"falsifier":"Construct a small binary dataset in which shifting a single continuous threshold by one sample changes the optimal subtree cost by more than one, run ArborEnum with the LicketySNIP proxy, and check whether the returned Rashomon set omits a tree that an optimal-proxy run includes; if any such tree is pruned, the Lipschitz robustness condition is violated on that instance and the approximate recall guarantee fails.","tokens_in":53657,"feed_emoji":"🌳","tokens_out":8140,"duration_ms":64379,"temperature":0.7,"pith_summary":"This paper claims that decision-tree Rashomon sets—all bounded-depth trees whose misclassification count plus a per-leaf penalty is within a tolerance of the best achievable value—can be enumerated exactly without first binarizing continuous features. It introduces ArborEnum, which uses the ordering of thresholds on each continuous feature to propagate information from evaluated splits to nearby ones and prune whole ranges of candidate cuts at once. With an optimal cost proxy, the returned AND/OR graph encodes exactly the continuous-feature Rashomon set; with a fast greedy proxy, enumeration is approximate and empirically recovers nearly every tree. The paper also provides an anytime variant that progressively activates more thresholds and converges to the exact set when run to completion, and shows that coarse binarization can miss many trees, important features, and predictive multiplicity.","feed_headline":"First exact tree-Rashomon enumeration over continuous features","feed_subtitle":"ArborEnum finds all near-optimal decision trees over every threshold, where binarization used to hide many of them.","key_machinery":"The load-bearing object is a Lipschitz-style interval-pruning rule for ordered thresholds. For thresholds s and t of one continuous feature, define dist_D(s,t) as the number of active samples whose branch assignment changes when the cut moves from s to t; if the proxy completion at s exceeds the budget by Δ, then any threshold within active-sample distance < Δ cannot enter the budget, provided the proxy robustness condition holds—moving k samples changes the proxy-completed objective by at most k. The algorithm stores a threshold-to-proxy-completion map keyed by exhaustive threshold indices, uses an excluded-range tracker to mark pruned or explored intervals, and encodes recovered subtrees in a budget-independent AND/OR graph with one canonical node per subproblem-depth pair that is extended in place as budgets grow.","core_discovery":"On the paper's own terms, the discovery is that exact Rashomon-set enumeration over continuous thresholds is not a separate, harder combinatorial problem: the nested structure of threshold columns turns candidate splits into an ordered sequence, and bounds on nearby thresholds can be reused. ArborEnum evaluates a threshold, records its left and right proxy completions, and then prunes the interval of thresholds whose active-sample distance from the failed threshold is smaller than the budget excess; when the proxy is optimal this pruning loses no in-budget tree. The same interval machinery powers approximate enumeration with the LicketySNIP greedy proxy and an anytime mode that refines the threshold set. The paper reports that on real datasets the exact variant finishes where prior methods time out, the approximate variants run orders of magnitude faster with near-perfect recall, and the anytime procedure incurs only about 2.7% median overhead.","pith_inferences":["If the exactness claims hold, published tree Rashomon-set analyses could shift to a binarization-free standard, removing a hidden source of researcher discretion in what splits are even allowed.","The anytime refinement view suggests an interactive workflow: inspect coarse summaries (which features appear, where predictions conflict) and spend additional computation only where the coarse view is inconclusive; the paper's observation that downstream properties converge faster than tree counts supports this.","A natural extension is to apply the same active-sample-distance pruning to other ordered model families—e.g., piecewise-constant regression trees or rule lists—where cut points also form nested sample sets; the paper names these as future directions.","A testable extension of the theory: validate Theorem 6 empirically by comparing exact ArborEnum against quantile binarizations at increasing K, to see whether the (2^d − 1)δ gap bound predicts the observed loss in recovered trees and to guide when the anytime algorithm has added enough thresholds."],"forward_implications":["Exact Rashomon sets over continuous thresholds make downstream analyses—feature importance, predictive multiplicity, tree counts—independent of the user's choice of binarization.","Coarse binarization demonstrably under-represents the Rashomon set, missing high-quality trees, important features, and conflicting predictions; ArborEnum recovers the missing structure.","The LicketySNIP approximate variants give near-perfect recall (worst case at least 94.5% over the reported experiment grid) while running orders of magnitude faster, widening the range of datasets on which Rashomon sets can be computed.","The anytime variant lets a practitioner stop early with a partial Rashomon set and later refine it by activating additional thresholds, with a median runtime overhead of about 2.7% relative to running on the final threshold set.","Theoretical bounds on the binarization optimality gap show that a threshold set with covering radius δ gives an objective gap of at most (2^d − 1)δ for depth-d trees, and midpoint refinement halves the covering radius each round."],"supporting_citations":[{"why":"Supplies the threshold-bounds machinery for optimal trees over continuous features that ArborEnum adapts to Rashomon budgets.","marker":"Brit,a, van der Linden, and Demirović 2025"},{"why":"Supplies the proxy-optimizer caching, iterative budget refinement, and AND/OR graph conventions that the new method extends.","marker":"Heile et al. 2026"},{"why":"Prior exact Rashomon-set enumeration over binarized features; establishes the starting point that this paper removes binarization from.","marker":"Xin et al. 2022"},{"why":"Provides the LicketySPLIT proxy that is relaxed into LicketySNIP for continuous thresholds.","marker":"Babbar et al. 2025"},{"why":"SORTeD is the anytime baseline whose budget-dependent representation is compared against budget-independent caching.","marker":"Arslan et al. 2025"},{"why":"Threshold guessing supplies the binarizations used by the restricted proxy variants (LSR and SNIP+GR).","marker":"McTavish et al. 2022"},{"why":"Establishes the scale of the decision-tree search space and the optimal sparse tree framework used for the objective and leaf penalty.","marker":"Hu, Rudin, and Seltzer 2019"},{"why":"Earlier branch-and-bound over continuous feature intervals, limited to depth 2–3, which ArborEnum scales beyond.","marker":"Mazumder, Meng, and Wang 2022"}],"fun_headline_variants":["Exact Rashomon enumeration for continuous-feature decision trees","ArborEnum: all near-optimal trees without binarization","Continuous-feature Rashomon sets, exactly and fast","No more binarization: exact Rashomon sets on raw features"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The pruning is exact only under a Lipschitz condition on the cost proxy—moving a few training samples across a split can change the proxy's estimated objective by at most the number of samples moved—and the fast approximate proxies used in most experiments do not guarantee this condition, so their near-perfect recall is an empirical result rather than a certificate.","fun_headline_variants_meta":{"raw":{"variants":["Exact Rashomon enumeration for continuous-feature decision trees","ArborEnum: all near-optimal trees without binarization","Continuous-feature Rashomon sets, exactly and fast","No more binarization: exact Rashomon sets on raw features"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000546,"raw_usage":{"total_tokens":2593,"prompt_tokens":912,"completion_tokens":1681,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":528,"completion_tokens_details":{"reasoning_tokens":1611}},"tokens_in":528,"tokens_out":1681,"duration_ms":11103,"temperature":1.0,"reasoning_tokens":1611,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T04:13:29.315602+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a small binary dataset in which shifting a single continuous threshold by one sample changes the optimal subtree cost by more than one, run ArborEnum with the LicketySNIP proxy, and check whether the returned Rashomon set omits a tree that an optimal-proxy run includes; if any such tree is pruned, the Lipschitz robustness condition is violated on that instance and the approximate recall guarantee fails.","supporting_citations":[{"cited_title":"From Rashomon Theory to","cited_arxiv_id":null,"evidence_quote":"Supplies the proxy-optimizer caching, iterative budget refinement, and AND/OR graph conventions that the new method extends."}],"review_version":2}