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REVIEW 3 major objections 6 minor 300 references

Search order alone can decide whether optimal decision trees finish fast or stall, and two simple priorities beat prior solvers.

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2026-07-31 15:44 UTC pith:UTEXHPUR

load-bearing objection Solid subfield engineering paper: a real unification of ODT search plus a useful 18-way bake-off; SOTA margins are real inside their stack but partly confounded by terminal solvers and continuous-feature handling. the 3 major comments →

arxiv 2607.28170 v1 pith:UTEXHPUR submitted 2026-07-30 cs.LG cs.AI

Search Strategies for Optimal Classification and Regression Trees

Jacobus G. M. van der Linden , Mim van den Bos , Emir Demirovi\'c This is my paper
classification cs.LG cs.AI
keywords optimal decision treessearch strategiesAND-OR searchbranch and boundclassificationregressionanytime performancecontinuous features
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Optimal decision trees are small, globally optimized models that are attractive when interpretability matters, but finding them is hard and different solvers have used different search orders without a shared yardstick. This paper builds one AND-OR search framework that can replay prior strategies and invent new ones, then runs eighteen variants head-to-head on classification and regression. The comparison shows that the order in which subproblems are expanded is a first-order lever: a best-first rule that prefers small-support nodes with strong lower bounds finishes proofs fastest, while a depth-first rule that keeps left and right children balanced delivers the best solution quality over time. Against existing solvers, those two strategies give clearly better anytime curves for classification and more than an order-of-magnitude faster runtimes for regression. A reader who cares about practical ODT solvers gets a concrete ranking of what to implement first, not just another isolated algorithm.

Core claim

Inside a single AND-OR framework for continuous-feature optimal trees, search strategy is a decisive performance factor: among eighteen strategies, best-first search that prioritizes low-support nodes with low lower bounds proves optimality fastest, and balanced depth-first search (expanding the less-explored child of each AND node) yields the strongest anytime performance; together they surpass prior state-of-the-art solvers on anytime classification quality and cut regression runtime by more than ten times.

What carries the argument

An incremental AND-OR search tree whose three hooks—Select (which unexpanded threshold interval and split to open next), Expand (replace an interval with an AND node and residual intervals), and BackPropagate (push bounds and prune)—instantiate DFS, BFS, LDS, and pure AND-OR as different priority and left/right rules while preserving completeness and optimality.

Load-bearing premise

That earlier solvers can be replayed inside this shared framework with only small, practically negligible differences, so measured gaps really come from search order rather than missing engineering.

What would settle it

Re-implement the same two winning priority rules on top of an independent continuous-feature ODT codebase (or re-run the paper’s framework with each baseline’s original cache, terminal solver, and binarization exactly restored) and check whether the anytime and regression-runtime gaps versus the published baselines shrink below an order of magnitude or lose statistical significance on the same UCI suite.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • ODT implementers should default to small-support + low-lower-bound best-first search when the goal is to prove optimality, and to balanced left/right depth-first search when the goal is good trees under a time budget.
  • Anytime performance for deeper trees depends more on balancing left and right expansions than on discrepancy neighborhoods around the greedy tree.
  • The same Select/Expand/BackPropagate skeleton can host new heuristics without rewriting the whole solver, making strategy ablation routine rather than a full redesign.
  • Regression ODT search, previously much slower, becomes practical on the tested continuous datasets once the better strategy is used.
  • Future AND-OR solvers outside trees can test whether “small support first” and “balance AND children” transfer as generic control rules.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If small-support-first works because it quickly produces tight bounds that prune large siblings, similar “solve cheap subproblems first” priorities may help other branch-and-bound ML models (rules, sparse linear models) with additive structure.
  • The weak gains from pure lower-bound AND-OR and from GOSDT-style large-support heuristics suggest that bound quality alone is not enough when the continuous split space is huge; hybrid DFS/BFS switching under memory pressure may become standard.
  • Because the framework deliberately omits caching and still wins on numeric data, the field may have over-weighted DP memoization relative to split-interval pruning and terminal depth-two solvers for continuous features.
  • A natural next measurement is whether the same two strategies keep their ranking when λ > 0, multi-objective losses, or categorical features are required.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proposes CODT, a unified AND-OR search framework for optimal classification and regression trees that incrementally expands UnExp threshold-interval nodes via Select/Expand/BackPropagate (Defs. 1–8, Algs. 1–2). Within this framework the authors instantiate and compare 18 search strategies spanning DFS, BFS, LDS, and AND-OR (Table 2; App. B), prove completeness and optimality preservation of the generic loop (Theorems 1–2, App. C), and add a generalized depth-two terminal solver and similarity/interval pruning for continuous features (App. A). Empirically, BFS-Small-LB is best for proving optimality and DFS-Blossom is best for anytime objective integral (OI); against external baselines, CODT reports better classification anytime performance (Figs. 5–6) and more than an order-of-magnitude regression runtime gains (Fig. 7).

Significance. If the results hold under fair attribution, this is a valuable consolidation paper for the ODT community: it replaces scattered, hard-to-compare solver papers with a common algorithmic lens and the largest head-to-head evaluation of search orders to date (18 strategies). The finding that small-support + low-LB BFS beats the large-support heuristic used by GOSDT, and that balanced left/right DFS (Blossom-style) dominates anytime performance at d≥5, are concrete, transferable design lessons. Strengths include stated completeness/optimality theorems with proofs, a clear OI metric with CDFs and Friedman–Nemenyi testing, multi-class and regression coverage, and a continuous-feature depth-2 subroutine. These make the work more than an incremental solver release.

major comments (3)
  1. [Introduction / Search Framework; App. A–E] Introduction / Search Framework: the central attribution claim—that prior methods are instantiated “exactly or up to small and practically negligible differences”—is load-bearing for both the internal ranking and the SOTA claims (abstract; Figs. 5–7), but is not validated. CODT’s depth-2 terminal solver (App. A.1, Alg. 4), delayed similarity/interval pruning schedule (App. A.2), absence of a cache, root-split multi-threading, and native continuous-feature handling differ from the original ConTree, Quant-BnB, GOSDT, and Branches codebases. Please either (i) ablate these engineering components while holding Select fixed, or (ii) explicitly qualify which gains are attributable to search order versus the shared continuous-feature machinery, and restrict “search strategy” language accordingly.
  2. [Comparison with the Baselines; App. D–E.3; Figs. 5–7, 15] §Comparison with the Baselines and App. E.3: GOSDT and Branches OOM on the full continuous instances and are only compared on 100-sample subsamples (Fig. 15); Quant-BnB is limited to d≤3; CA-ConTree solves no instance to optimality. The abstract’s “compared to the state of the art … order of magnitude for regression” therefore mixes full-scale runs against ConTree/STreeD/Quant-BnB with handicapped or re-hosted baselines. Strengthen the claim by reporting, for each baseline, the exact problem encoding (binarized vs continuous, λ, depth, threads) side-by-side with CODT-ST, and move OOM baselines out of the main SOTA sentence or into a clearly labeled “binarized/subsample” subsection.
  3. [Experiment Setup; App. D.2] Experiments / App. D.2: all runs fix λ=0. With zero complexity penalty the search landscape and the value of lower-bound-guided strategies (AOS, BFS-LB variants) can differ materially from the sparse-ODT regime that motivated GOSDT/OSRT/Blossom. At least a small λ-sweep (e.g., λ∈{0,0.01,0.1}·n or per-dataset defaults from prior work) on a subset of datasets is needed to show that BFS-Small-LB / DFS-Blossom remain preferred when sparsity is enforced; otherwise the strategy recommendations are conditional on λ=0.
minor comments (6)
  1. [Experiments] Code is promised “after acceptance” but is not available for review; a review artifact or anonymous repo would strengthen reproducibility claims.
  2. [Table 1] Table 1 lists “This paper CODT … Many” under search strategy; a pointer to Table 2 / App. B would help readers map the 18 concrete Select definitions.
  3. [Metrics; Fig. 2] Fig. 2 and the OI definition use s̄ from CART and s* as best-found-by-any; state explicitly whether s* is pooled across all strategies/baselines (which can slightly favor methods run in the same bake-off).
  4. [Throughout] Typographical issues: “Brit,a” / “Brit,a” throughout; “andand” / missing spaces in several places; arXiv date “30 Jul 2026” looks like a placeholder.
  5. [App. B.2; Fig. 10] App. B.2: the dynamic BFS→DFS-Prio memory fallback is important for fairness of BFS memory/runtime CDFs (Fig. 10); report how often it triggered.
  6. [App. A.3] Clarify whether multi-threaded CODT shares only root incumbents or also lower bounds across feature threads; this affects reproducibility of the multi-thread curves in Figs. 5–7.

Circularity Check

0 steps flagged

No circularity: empirical runtime/anytime claims measured against external baselines, not quantities forced by construction from fitted inputs or self-citation chains.

full rationale

This is a systems/algorithms paper whose load-bearing claims are empirical CDFs of runtime and objective integral for named search strategies (DFS-Blossom, BFS-Small-LB, etc.) versus external and reimplemented baselines (Quant-BnB, ConTree, CA-ConTree, STreeD, GOSDT, Branches). The AND-OR framework (Defs. 1–8, Alg. 1–2) and the completeness/optimality theorems are ordinary inductive arguments from the stated definitions; they do not define the measured OI or wall-clock times in terms of themselves. Self-citations (MurTree, STreeD, ConTree, Blossom, regression DP) appear as related work and comparison targets, not as uniqueness theorems or ansätze that force the ranking. There is no fitted parameter renamed as a prediction, no self-definitional loop, and no renaming of a known empirical law. Residual concerns about whether prior methods are instantiated only up to negligible engineering differences affect attribution fairness, not circularity under the stated criteria. Score 0 with empty steps is therefore the correct outcome.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 2 invented entities

Load-bearing content is standard branch-and-bound / AND-OR search plus domain modeling of ODTs. No fitted physical constants. Free choices are experimental (timeout, λ, heuristic tie-break ε) and modeling (loss, depth limit, numeric thresholds as midpoints). Invented pieces are the framework packaging and named strategy variants, not new physical entities.

free parameters (4)
  • regularization λ = 0
    Set to 0 in all experiments because not all baselines support it; affects objective and pruning strength.
  • time-out t̄ = 15 min
    15 minutes per run; defines OI normalization and which instances count as solved.
  • BFS tie-break ε = 1e-6
    ε=1e-6 turns support or LB into a pure tie-breaker in several H definitions (Table 3).
  • LDS large/small constants M, m = unspecified large/small
    Used to encode diagonal discrepancy order plus depth tie-break in the LDS heuristic.
axioms (4)
  • domain assumption ODT learning is an AND-OR search over feature tests (OR) and left/right subproblems (AND) with leaf label assignment.
    Preliminaries and Definition 1; standard in Branches and related work.
  • standard math Valid lower/upper bounds never cut an improving root incumbent (optimality preservation).
    Definition 8 and Theorem 2; standard branch-and-bound reasoning.
  • domain assumption Loss is additive over samples (0-1 or SSE) plus λ per branch node; D2 terminal requires element-wise additive cost tuples.
    Problem definition and Appendix A.1 generalization of Solve-D1/D2.
  • ad hoc to paper Prior solvers differ from framework instantiations only by negligible implementation details for comparison purposes.
    Stated in Introduction; required to attribute gains purely to search strategy.
invented entities (2)
  • CODT AND-OR search framework (UnExp threshold intervals + Select/Expand/BackPropagate) no independent evidence
    purpose: Unify and swap search strategies under one complete, optimality-preserving algorithm.
    Core methodological contribution; engineering abstraction rather than a new scientific object.
  • BFS-Small-LB and related support/LB hybrid heuristics no independent evidence
    purpose: New priority rules contrasting GOSDT-style large-support ordering.
    Defined in Table 3 / §Search Strategies; validated only inside this empirical study.

pith-pipeline@v1.2.0-daily-grok45 · 31862 in / 2883 out tokens · 59537 ms · 2026-07-31T15:44:49.169616+00:00 · methodology

0 comments
read the original abstract

Optimal decision trees (ODTs) are compact, interpretable machine learning models that globally optimize a given objective, but their scalability remains challenging. While recent work has proposed a variety of search strategies to improve scalability, the precise contribution of each strategy remains unclear. To address this gap, we introduce a general algorithmic framework for ODTs that instantiates previously used search strategies and enables the definition of new ones. This provides a common lens through which to understand and compare different strategies, which we use to empirically investigate the effect of 18 search strategies. Compared to the state of the art, the best strategy in our evaluation achieves significantly better anytime performance for classification, and improves runtime by more than an order of magnitude for regression.

Figures

Figures reproduced from arXiv: 2607.28170 by Emir Demirovi\'c, Jacobus G. M. van der Linden, Mim van den Bos.

Figure 1
Figure 1. Figure 1: An example of (a) Select and (b) Expand. After expansion BackPropagate propagates the lower and upper bounds on the loss L from the newly expanded nodes back up to the root. Algorithm 1: ODT search defined by the subproce￾dures Select, Expand, and Back-Propagate. Function ODT(D, d) is root ← Create-OR-Node(D, d) while root not completed do node, τi ← Select(root) Expand(node, τi) BackPropagate (node) Expan… view at source ↗
Figure 2
Figure 2. Figure 2: The anytime performance of four search strategy [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Cumulative distribution of the objective integral (OI) as a measure of anytime performance. The lines indicate for a [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Cumulative distribution of runtime (s). and right children, is best for anytime performance. These search strategies also outperform the state of the art. Experiment Setup We ran the experiments on an Intel Xeon Gold 6448Y 32C 2.1GHz with eight cores and 32GB RAM running Linux Red Hat Enterprise 8.10 and repeated the experiments for each dataset five times. For each run, we used a time-out of 15 minutes. W… view at source ↗
Figure 5
Figure 5. Figure 5: Cumulative distribution of the objective integral (OI). For [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The cumulative distribution of runtime (s) for [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of the three terminal cases for CODT (single thread). Both the left-right and our depth-two terminal [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Scalability of solving depth-two trees for all clas [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Cumulative distribution of memory usage (MB) for the four search strategy categories. [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Cumulative distribution of the objective integral [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Results for the best-first search strategies. [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: For LDS, for both runtime and anytime performance (objective integral), picking the midpoint of the threshold [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Cumulative distribution of the objective integral (OI) for all baselines methods including [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Cumulative runtime (s) distribution on the down [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Upper bound (incumbent) and lower bounds for the four search strategy categories for several datasets. The top gray [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Upper bound (incumbent) and lower bounds for the four search strategy categories for several datasets. The top gray [PITH_FULL_IMAGE:figures/full_fig_p021_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Upper bound (incumbent) and lower bounds for the four search strategy categories for several datasets. The top gray [PITH_FULL_IMAGE:figures/full_fig_p022_18.png] view at source ↗

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