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REVIEW 3 major objections 4 minor 145 references

ArborEnum: Decision Tree Rashomon Sets over Continuous Features

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read ArborEnum is the first framework that exactly enumerates decision-tree Rashomon sets over continuous features, using ordered-threshold pruning so that no coarse binarization is needed.

desk verdict Novel and mostly sound exact enumeration algorithm; the anytime convergence claim is broken because the root budget is never recomputed after proxy strengthening. read the letter →

arxiv 2608.04310 v1 pith:V3Q4K3Z4 submitted 2026-08-05 cs.LG stat.ML

Zakk Heile , Hayden McTavish , Margo Seltzer , Cynthia Rudin This is my paper · ORCID
classification cs.LGstat.ML
keywords RashomonsetsdecisiontreescontinuousfeaturesexactenumerationanytimealgorithmthresholdpruningAND/ORgraphpredictivemultiplicity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces what it argues is the first framework for exactly enumerating decision-tree Rashomon sets — the collection of all trees whose regularized loss lands within a prescribed budget of the best — directly over continuous features, without pre-binarizing each feature into a few thresholds. The central move is to treat each continuous feature as an ordered block of nested threshold splits and to prune whole intervals of thresholds at once, so the exhaustive set of splits becomes searchable rather than combinatorial. With an optimal proxy the enumeration is exact over all continuous thresholds; with a greedy proxy it becomes a fast relaxation, and an anytime variant starts from a coarse threshold set and progressively refines it, converging to the exact set when run to completion. On 20 datasets the approximate variants keep nearly all trees (worst-case recall at least 94.5%) at a median 270× speedup, and the paper shows coarse binarization hides trees, important features, and predictive multiplicity. If correct, downstream analyses of near-optimal models no longer need to be conditioned on an arbitrary discretization.

What carries the argument

Three mechanisms carry the argument. (1) The proxy-robustness condition: moving $k$ active training samples across a threshold changes the proxy-completed objective by at most $k$; combined with the active-sample distance $\mathrm{dist}_D(s,t)$ (the number of active samples whose branch assignment changes between thresholds $s$ and $t$), it turns one evaluated threshold into a certificate that prunes every threshold closer than the budget deficit $\Delta = P - \varepsilon_{\text{abs}}$ on either side, via binary-search boundary routines. (2) The minimum-objective AND/OR graph, in which each node is a (subproblem bitvector, remaining depth) pair indexed independently of the budget, so a single canonical subgraph is extended in place as budgets grow instead of being rebuilt — eliminating the factorial path duplication of trie-style representations. (3) The LicketySNIP proxy, a relaxation of LicketySPLIT whose greedy completions are memoized, cached across visits to the same subproblem, and reused under different budgets and threshold sets.

What would settle it

On a small dataset, brute-force enumerate every decision tree over all thresholds between consecutive distinct feature values and compare the resulting tree count and objective histogram with ArborEnum's output under an optimal proxy, the same leaf penalty, and the same Rashomon budget — any mismatch refutes the exact-enumeration claim. A sharper test searches for a subproblem where an optimal proxy's completed objective changes by more than the number of active samples moved across a threshold, directly violating the robustness condition on which the interval pruning rests.

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Extended reading notes

Core claim

ArborEnum expands every continuous feature into its full block of nested threshold columns (rules of the form $x_j \le \nu$) and searches these blocks interval by interval. The search is driven by a proxy algorithm: a threshold whose proxy-completed objective $P = P_L + P_R$ exceeds the budget $\varepsilon_{\text{abs}}$ certifies its neighbors, because moving the threshold so that $k$ active samples switch branches can change any proxy-completed objective by at most $k$ (the robustness condition). Any threshold within active-sample distance less than $\Delta = P - \varepsilon_{\text{abs}}$ of the failed threshold must also fail, so the whole interval is pruned by binary search. When the proxy is optimal the pruning removes no feasible split, and the AND/OR graph left behind encodes exactly the continuous-feature Rashomon set; with the LicketySNIP greedy proxy the same machinery is a relaxation, and the anytime variant activates progressively more thresholds and can strengthen the proxy to optimality, converging to the same exact set. The paper thereby claims to close the gap that forced every previous decision-tree Rashomon enumerator to commit in advance to a binarization of the data.

Load-bearing premise

The load-bearing premise is the proxy robustness condition — moving $k$ active samples across a threshold changes the proxy-completed objective by at most $k$ — which the paper asserts holds for optimal proxies and cites to prior work rather than proving in the text; if a proxy violates the condition, the interval pruning can discard feasible thresholds and return a strict subset of the true Rashomon set.

Editorial extensions

If this is right

  • On any dataset where an optimal proxy is affordable, the algorithm returns a certificate that no tree within the Rashomon budget over the exhaustive set of continuous thresholds has been missed — a guarantee no binarization-based enumerator can offer.
  • The approximate variants make near-complete Rashomon sets practical where exact enumeration previously timed out (100 hours) or exhausted memory (128 GB), with worst-case recall of at least 94.5% across the 20 datasets and tree counts recoverable by loosening the root budget.
  • Downstream analyses — feature importance, predictive multiplicity, and the set of high-quality trees — change when continuous thresholds are allowed, so conclusions drawn from a fixed coarse binarization can be incomplete in a way the anytime refinement makes visible.
  • Because subgraphs are budget-independent, a user can start with a conservative Rashomon budget and widen it after the fact without restarting, which also supports incrementally choosing $\varepsilon_{\text{mult}}$ until a target stopping criterion is met.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The interval-pruning rule is effectively a Lipschitz-style sensitivity certificate on the proxy objective, so the same evaluate-one-point-prune-a-neighborhood pattern could transfer to other structured hypothesis classes with an ordering — piecewise-constant regression trees, rule lists, or continuous split search in boosting — the classes the paper names as future work.
  • The anytime algorithm could double as a diagnostic: the rate at which feature counts and predictive-multiplicity counts stabilize across refinement rounds quantifies how much information a chosen binarization was destroying.
  • The near-perfect recall of the greedy proxies suggests the optimal-proxy requirement may matter mainly for the certificate of exactness rather than for the recovered trees; a testable consequence is that budget extension alone, without raising proxy strength, closes most of the remaining recall gap.
  • A more rigorous convergence measure than the paper uses — for example, the symmetric difference or histogram distance between successive anytime graphs — would give an early-stopping rule with a proven error bound.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. ArborEnum introduces a family of algorithms for enumerating decision-tree Rashomon sets over continuous features by exploiting the ordered structure of thresholds. The exact variant (Algorithm 1 with an optimal proxy) uses interval pruning and a budget-independent AND/OR graph to enumerate all trees whose regularized objective is within a multiplicative budget. Approximate variants use greedy proxies such as LicketySNIP, and an anytime variant (Algorithm 4) progressively activates additional thresholds and optionally strengthens the proxy, with the abstract and Section 3 claiming convergence to the exact continuous-feature Rashomon set. Experiments on 20 datasets compare runtimes, memory, and recall against prior binary-feature enumeration methods, reporting large speedups and near-perfect recall.

Significance. If the correctness claims hold, exact enumeration over continuous thresholds is a significant advance over binarization-dependent Rashomon set algorithms, since coarse binarization can miss trees, features, and predictive multiplicity. The paper also contributes theoretical bounds on the discretization optimality gap (Theorems 4–9), a fixed-binarization superset guarantee (Theorem 10), and an empirical demonstration of budget-independent subgraph caching that substantially reduces OR-node counts. The release of code and the detailed pseudocode support reproducibility. However, the anytime algorithm's advertised convergence to the exact continuous-feature Rashomon set is not supported by the presented pseudocode, and the recall metric is weakened by being measured against the best completing method rather than against the exact set. These issues affect major advertised claims and require correction.

major comments (3)
  1. [Algorithm 4, Section 3 (Anytime Algorithm)] The anytime algorithm does not converge to the exact continuous-feature Rashomon set as claimed. In Algorithm 4, line 6 sets ε_abs = (1+ε_mult)·Proxy(D_root,d,γ,S_root) once, using the initial, potentially suboptimal proxy. The subsequent while loop (lines 16–20 of the main-text listing; Algorithm 16 in Appendix A.8) strengthens the proxy but never recomputes ε_abs. Since a suboptimal proxy returns a tree with objective at least the optimal objective, Proxy(D_root) ≥ Optimal(D_root), so the fixed ε_abs is an overestimate of the intended multiplicative budget. RefineGraph (Algorithm 14) only extends the existing graph in place; it never removes or filters splits that were added under the looser budget. Consequently, when the proxy is strengthened to optimality, the returned graph still contains every tree with objective below the stale ε_abs, which is a strict superset of the multiplicative Rashomon set {T : Obj(T) ≤ (1+ε_mult)·Optimal}. This contradicts the abstract and the Section 3 statement that the anytime variant 'converges to the complete continuous-feature Rashomon set.' The central issue is not a minor implementation detail: as written, the algorithm returns a superlevel set of the objective, not the Rashomon set. To fix it, ε_abs must be updated after each proxy-strengthening step, or the graph must be rebuilt/restricted against the new budget.
  2. [Table 2 and Appendix E.2 (Recall metric)] The near-perfect recall claims for the approximate variants are measured relative to the 'best method that finished' using the minimum objective found by any method as the guessed Rashomon bound, rather than against the exact continuous-feature Rashomon set. On datasets where the optimal proxy does not finish (e.g., Credit, Helena, Heloc, Jasmine, Shopping, Spambase at λ=0.02 in Table 5), a recall of 1.000 only certifies agreement with the best approximate method, not recovery of the true Rashomon set. The paper acknowledges this in the table footnotes, but the abstract's statement that approximations 'maintain near-perfect recall' is only established on datasets where an exact reference is available. I recommend reporting recall relative to the exact set for all datasets where the optimal proxy completes, and clearly labeling the other comparisons as agreement rates among approximate methods.
  3. [Section 3 (proxy robustness condition)] The exactness of the continuous-feature pruning rules rests on the proxy robustness condition stated in Section 3: moving k active samples across a threshold changes the proxy-completed objective by at most k. The paper asserts that optimal proxies satisfy this condition but does not provide a proof or a precise reference. This condition is load-bearing: Algorithm 2 lines 16–22 and Algorithm 11 lines 43–51 prune entire ranges of thresholds based on it, and if it fails for the chosen optimal proxy, the algorithm can discard feasible splits and return a strict subset of the true Rashomon set. I believe the condition is true for the optimal objective by a 1-Lipschitz argument applied to the minimum over trees, but the authors should supply a complete, self-contained proof (or an exact statement of where it is proven in the cited literature) in the paper or appendix.
minor comments (4)
  1. [Appendix A.10] In the paragraph beginning 'This does not lose information', the sentence 'we claim that these choices are can be ignored' contains a typing error ('are can' should be 'can').
  2. [Algorithms 1 and 4] The pseudocode for Algorithm 4 in the main text and Algorithm 16 in Appendix A.8 differ slightly in comments and line numbering; please unify them so readers can cross-reference the anytime convergence argument.
  3. [Section 2, related work citation] The citation to 'Brit,a, van der Linden, and Demirović (2025)' contains an encoding artifact in the author name; the correct rendering appears to be 'Briña' and should be fixed throughout the text and references.
  4. [Section 3, anytime description] The text says 'This variant is truly anytime: it can begin with no active continuous thresholds and, as the proxy is strengthened to optimality, converges to the complete continuous-feature Rashomon set.' Given the budget-freezing issue in Algorithm 4, this sentence is misleading; please revise it to describe what the algorithm actually guarantees (e.g., convergence to the set defined by the initial proxy budget) or correct the algorithm.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the exact continuous-feature enumeration is derived from first-principles threshold-interval pruning; self-citations to PRAXIS/Heile et al. are building blocks, not forced inputs.

full rationale

ArborEnum's core exactness claim is grounded in the combinatorial interval-pruning argument (Algorithms 2-3) plus an explicit proxy robustness condition, not in fitting a parameter to the target Rashomon set. The multiplicative Rashomon set is defined with εabs=(1+εmult)Optimal(D,d,γ), and the exact regime uses an optimal proxy; the stated Lipschitz-style robustness condition is an assumption about the objective/proxy, not a definitional rewording of the output. The approximate regime explicitly sets εabs=(1+εmult)Proxy(D,d,γ) and is presented as a relaxation, so its proxy-relative budget is not disguised as an exact prediction. Thresholds chosen by XGBoost for LSR and SNIP+GR are declared heuristics, and recall is measured against exact or best-available baselines, giving external content. Self-citations to Heile et al. (2026) supply shared proxy-optimizer caching and some approximation-side guarantees; these are reused components, while the paper's advertised exact continuous-threshold claim depends on the new interval-pruning derivations and is compared against SORTeD, TreeFARMS, and other external methods. The reader-supplied AnyTimeArborEnum concern (root εabs fixed from the initial proxy and never updated when the proxy is strengthened) is a substantive correctness issue: the final graph can be a stale superlevel set rather than the exact Rashomon set. That is a bug in the convergence claim, not a circularity, because the returned set is not equal to its input by construction. No specific equation reduces to an earlier input, so no circular step is exhibited.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No new physical or model entities are introduced; the AND/OR graph, threshold registry, and exclusion-range tracker are algorithmic data structures. The main load-bearing premise is the proxy robustness condition, which the exact method inherits from optimal-decision-tree bounds. Free parameters are experimental inputs or fitted threshold sets for the approximate variants, not fitted constants in the exact derivation.

free parameters (2)
  • Bproxy (XGBoost-selected threshold set) = 150 depth-2 gradient-boosted estimators with seed 0 plus backward elimination
    Used only in the approximate variants LSR and SNIP+GR and as the initial proxy binarization in the anytime algorithm. It determines which trees the approximations can recover, so the approximate recall and speedup results depend on this fitted choice.
  • Rashomon budget and depth parameters (d, lambda, epsilon_mult) = d=5; lambda in {0.02, 0.01, 0.005}; epsilon_mult = 0.03 in main tables, with additional values in appendices
    These user-chosen experimental inputs define the Rashomon set and runtime regime. They are not fitted constants in the correctness proof, but they shape the empirical claims.
assumptions (4)
  • domain assumption Proxy robustness: moving k active samples across a threshold changes the proxy-completed objective by at most k.
    Stated in Section 3 as the basis for interval pruning; exactness requires it to hold for the optimal proxy, but the paper does not give a full standalone proof. It is plausible for exact objectives because a fixed tree's misclassification changes by at most k.
  • domain assumption Finite threshold representation: each continuous feature is expanded into an ordered block of threshold columns between unique values, and the Rashomon set is over these columns.
    Section 3 states that each continuous feature has been expanded into a contiguous, ordered block of threshold columns. This is standard for finite training data.
  • domain assumption Binarization Hamming snapping condition: every continuous split has a selected binary split within Hamming distance delta, as assumed in Theorem 4.
    Used for the optimality-gap bound; the proof does not address nesting of snapped thresholds on the same feature.
  • domain assumption 64-bit subproblem fingerprints do not collide.
    Appendix A.1 uses non-injective 64-bit fingerprints for subproblem identity; correctness relies on no collision, described as astronomically unlikely.

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Pith. "Pith review of ArborEnum: Decision Tree Rashomon Sets over Continuous Features." pith.science (2026). https://pith.science/paper/V3Q4K3Z4

@misc{pith2026260804310,
  author       = {Pith},
  title        = {Pith review of: ArborEnum: Decision Tree Rashomon Sets over Continuous Features},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V3Q4K3Z4}},
  note         = {Machine review of arXiv:2608.04310}
}
read the original abstract

The Rashomon effect describes the phenomenon that many models can achieve nearly equivalent performance on the same learning task, with significant ramifications for robustness, feature importance, and customizability. These use cases motivate the computation of Rashomon sets: the set of all models whose regularized loss is near-optimal. Decision trees are one of the few model classes for which Rashomon sets can be fully enumerated, but this computation has always been conditional on a binarization of the original data, either restricting which splits each tree is allowed to make or substantially increasing the complexity of an already difficult combinatorial problem. We introduce the first algorithm that exactly enumerates decision-tree Rashomon sets while exploiting the ordered structure of continuous features. We further develop a relaxation for approximate enumeration and an anytime algorithm that progressively refines the set of candidate thresholds, producing increasingly detailed approximations that converge to the continuous-feature Rashomon set. Experiments show that coarse binarization can miss many trees, important features, and predictive multiplicity; our algorithms achieve orders-of-magnitude speedups over existing enumeration methods, with approximations providing further speedups while maintaining near-perfect recall.

Figures

Figures reproduced from arXiv: 2608.04310 by the authors.

Figure 1
Figure 1. InitAndPrune example. Using proxy completions in E for exhaustive thresholds 2 and 18, we prune active thresholds 0, 1, 10, 11, and 12 using active sample distances (for instance, 85−8 > 75, so we prune active threshold 0, whereas 85−12 ≤ 75, so this test cannot prune active threshold 2. For simplicity, we show E storing the sum of proxy completions (not PL and PR); we omit pruning based on γ. From S (not shown), th… view at source ↗
Figure 2
Figure 2. Anytime algorithm on 8 real-world datasets. We display three quantities about the Rashomon set at each stopping [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. An example graph structure for encoding a Rashomon set. The OR nodes at the bottom of this figure have more [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: An example showing a dataset where the subprob [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.