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SIRUS: Stable and Interpretable RUle Set for Classification

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read SIRUS turns a random forest into a short rule list whose contents converge: with enough data, the same rules are selected on every fresh sample, with accuracy close to the forest.

desk verdict A genuinely useful rule-set method with strong empirical stability, but the headline theorem is proved for a subsampling variant that the released code does not use. read the letter →

arxiv 1908.06852 v5 pith:KCE6LEEN submitted 2019-08-19 stat.ML cs.LGmath.STstat.TH

Clément Bénard (LPSM (UMR_8001)) , Gérard Biau (LPSM (UMR_8001)) , Sébastien da Veiga , Erwan Scornet (CMAP) This is my paper · ORCID
classification stat.MLcs.LGmath.STstat.TH MSC 62H3062G2068T05
keywords classificationinterpretabilityruleensemblesrandomforestsstabilityquantilediscretizationasymptoticconvergence
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

The paper presents SIRUS, a classification algorithm that converts a random forest into a short list of if-then rules, and it argues that interpretability should be judged by the triad of simplicity, stability, and predictive accuracy. Its central claim is that SIRUS is asymptotically stable: given enough data, two independent runs on independent samples select essentially the same rule list, because the selected rule set converges in probability to a deterministic set that depends only on the data distribution. If that claim holds, the traditional weakness of rule models—their tendency to change drastically under small data perturbations—is removed, which matters for applications where a handful of rules must be trusted for critical decisions. The paper proves the convergence under continuous-distribution assumptions and supports it with experiments showing SIRUS's stability exceeds that of existing rule learners while its accuracy remains close to random forests.

What carries the argument

The central object is the path space $\Pi$, made finite by $q$-quantile discretization: a path is a sequence of triplets $(j_k,r_k,s_k)$ encoding the variable, quantile index, and side of each split. Finiteness is what gives each path a nonzero, well-defined occurrence probability in a randomized tree; without discretization, continuous cut points make almost every path unique to a single sample, which is exactly why ordinary rule extraction from trees is unstable. The argument is carried by two convergence results: the empirical CART splitting criterion computed at empirical quantiles converges to the theoretical criterion at population quantiles, and the bagged occurrence-frequency estimates have vanishing variance by a U-statistic bound. Together these show $\hat p_{M,n}(P)\to p^\star(P)$ in probability, which by the threshold selection and deterministic post-treatment yields convergence of the whole rule set.

What would settle it

Simulate data satisfying the continuity assumptions, run SIRUS on two independent samples of increasing size, and measure the Dice–Sorensen index of the two rule lists; if the index does not approach 1 for a threshold away from the tie values, Corollary 1 is false. A sharper boundary check is to engineer a distribution where two splits tie in the population CART criterion and set $p_0$ exactly at their common probability, a case the theorem excludes; the rule lists should flip between samples at that threshold.

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Extended reading notes

Core claim

To make rule extraction stable, SIRUS restricts every split of every tree to a fixed set of empirical $q$-quantiles of each input variable (typically $q=10$), so the universe of possible tree paths is finite and independent of the sample: a path is a sequence of triplets specifying the variable, the quantile index, and the side of each split. The forest then estimates, for each path $P$, the probability $\hat p_{M,n}(P)$ that a randomized tree contains $P$, and the method keeps all paths whose estimated probability exceeds a threshold $p_0$. The paper proves that under assumptions (A1)–(A3) these estimated probabilities converge in probability to theoretical probabilities $p^\star(P)$ defined from population quantiles and the population CART criterion; consequently the selected path set converges to the deterministic set $P^\star_{p_0}$, and the Dice–Sorensen stability index converges to 1 in probability. The post-treatment that removes redundant rules and the final averaging of rules are deterministic functions of the selected paths, so they inherit the convergence. The net claim is that on continuous data the rule list itself is a reproducible estimate of a population-level object, not a sample-dependent artifact.

Load-bearing premise

The stability proof assumes every input variable is continuous, with no ties and no categories; for discrete or categorical data the theorem does not directly apply and the paper relies on practical workarounds.

Editorial extensions

If this is right

  • On data satisfying the assumptions, repeated runs of SIRUS on independent samples converge to the same rule list, so the model's output can be read as a stable statement about the data-generating process.
  • Because deeper paths have exponentially more competitors in the finite path space, the selected rules are almost always one- or two-split rules, which keeps the final model readable.
  • On the UCI and manufacturing datasets tested, SIRUS keeps 1-AUC close to random forests while cutting the model to a handful of rules, so interpretability is not bought with a large accuracy loss.
  • The stability comparison indicates that competing tree-based rule methods such as RuleFit and Node harvest are unstable by design when splits are continuous; the quantile discretization is the step that makes convergence possible.

Reading between the lines

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

  • Beyond the paper: the same quantile-discretization device is portable to regression and to other tree ensembles such as boosting or extremely randomized trees, so a stable rule-surrogate method could be built for any tree black box; regression is flagged as future work in the conclusion.
  • Beyond the paper: because the stability guarantee is proven only for continuous inputs, a practitioner-facing diagnostic suggests itself—plot the Dice–Sorensen index against sample size on real data; if it fails to approach 1, the data are likely outside the continuous regime and the theoretical protection does not apply.
  • Beyond the paper: the theorem excludes thresholds $p_0$ that equal a theoretical path probability, and those boundary values are exactly where finite-sample rule lists should flip; users could avoid unstable outputs by inspecting the histogram of path frequencies and not choosing $p_0$ inside a dense cluster of near-tied probabilities.
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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. The paper introduces SIRUS, a rule-based classification algorithm that extracts a short list of rules from a random forest by selecting paths that occur with frequency above a threshold p0 in a forest grown with split values restricted to empirical quantiles. The selected rules are averaged to produce a class probability estimate. The authors argue that SIRUS is interpretable under a simplicity-stability-predictivity triptych and support this with three types of evidence: (i) a theoretical analysis in Section 4 showing asymptotic stability under assumptions (A1)-(A3), i.e., that the estimated rule set converges in probability to a deterministic set P*_{p0} and that the Dice-Sorensen stability index converges to 1; (ii) extensive experiments on 19 UCI datasets plus the SECOM case study comparing SIRUS with CART, C5.0, RIPPER, PART, FOIL, CBA, BRL, RuleFit, Node harvest, and random forests; and (iii) an open-source R/C++ implementation available on CRAN. The theoretical proofs in Appendix C are detailed for one- and two-split paths, with the extension to arbitrary depth asserted by recursion.

Significance. If the central claims hold, SIRUS would be a valuable contribution to interpretable machine learning: it offers a concrete rule-list model with accuracy close to random forests, markedly higher stability than existing rule learners, and a freely available implementation. The paper is also careful to define a theoretical counterpart of the algorithm and to prove convergence in probability to a distribution-dependent rule set, which is a non-circular route to a stability guarantee. The experimental evaluation is broad, covering multiple rule-learning families and reporting model size, stability, and 1-AUC. The provided software is a practical asset. However, the strength of the contribution is tempered by a mismatch between the theory and the implemented algorithm, as detailed in the major comments.

major comments (3)
  1. [Section 4, Assumption (A1); Algorithm 1; Section 5.1] Theorem 1 and Corollary 1 are proved under Assumption (A1), which requires a_n/n -> 0 and is explicitly tied to subsampling without replacement (Section 4, paragraph before (4.1)). However, Algorithm 1 and the reported experiments use sampling with replacement at the default random-forest settings (a_n = n) via ranger, and the released sirus package inherits this. The proof depends on a_n/n -> 0 in the U-statistic variance bound V[p_n(P)] <= (a_n/n) xi_{a_n,a_n} and in Lemmas 1-2, where sqrt(a_n) times O_p(1/sqrt(n)) vanishes only if a_n/n -> 0; with bootstrap a_n = n these terms are O_p(1) and do not vanish. The central asymptotic stability guarantee therefore applies to a subsampling variant of SIRUS, not to the implemented and benchmarked algorithm, and the manuscript does not state this limitation or provide an alternative analysis.
  2. [Section 3, definition of Pi; Section 4, proof of Corollary 1] The set Pi of all possible paths is defined without a depth bound (Section 3), so for fixed p and q it is infinite. Corollary 1's proof bounds P(hat P_{M_n,n,p0} != P*_{p0}) by a sum over P in Pi, which requires a finite union bound or a uniform convergence result over Pi. Pointwise convergence of hat p_{M_n,n}(P) for each fixed P (Theorem 1) does not control paths of growing depth, and the assertion that the proof 'can be easily extended to a path of any depth d by recursion' does not supply the needed uniformity. The theoretical statement should be restricted to bounded-depth paths (as the software does with max.depth=2) or a uniform argument must be provided.
  3. [Section 5.1 and Corollary 1] The empirical stability metric used in Section 5.1 is the average proportion of shared rules between models built on the 10 folds of a single cross-validation, whose training sets overlap, whereas Corollary 1 and the stability index (3.4) concern two independent samples of size n. These are different quantities, so the reported stability values do not directly evaluate the theoretical guarantee. The paper should either report a stability estimate based on truly independent samples or explain why the overlapping-fold proxy is expected to behave like the independent quantity.
minor comments (4)
  1. [Section 5, Tables 2-4] Standard deviations are omitted from Tables 2-4 with the claim that they are negligible; please report them, at least in an appendix, because several stability differences between SIRUS and its competitors are modest and the reader cannot assess their significance without variability estimates.
  2. [Section 3, paragraph on Pi] The statement that Pi 'is in fact a deterministic quantity, which only depends upon the dimension p and the order q of the quantiles' is inaccurate if paths of unbounded depth are allowed; it should be qualified with a depth bound or a bounded-depth definition.
  3. [Section 5.1 and Section 5.4] There are several typographical errors ('evalute', 'missclassification', '2st or 3nd class') and the notation M_n in the theory is not explicitly reconciled with the experiment-specific M determined by the stopping criterion (5.1); clarifying this would help the reader connect the asymptotic statements to the reported results.
  4. [Section 5.4, stopping criterion] The connection between the stopping criterion (5.1) and Theorem 2 is presented heuristically; a more precise explanation of why the average of epsilon_{M,n,p0} over an estimated set of p0 values controls the stability at the final, data-dependent p0 would be helpful.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the stability theorem is self-contained and the empirical estimator is compared to an independently defined population target.

full rationale

The paper's derivation chain is self-contained. Theorem 1 proves that the empirical path-frequency estimator \hat p_{M_n,n}(P) converges in probability to p*(P), a population quantity defined from the theoretical CART criterion, theoretical quantiles, and the distribution P_{X,Y} alone. The limiting set P*_{p0} is therefore not fitted from the data or defined in terms of the empirical stability index; Corollary 1 is a standard continuous-mapping consequence of this consistency. The Dice-Sorensen index \hat S_{M,n,p0} is the object of study (overlap between two independent fits), not an input to the proof. The only author self-citations are peripheral: Biau and Scornet (2016) is cited for the standard equivalence between the CART criterion and Gini impurity, and Benard and Wright (2020) for the software package itself; neither is load-bearing for the asymptotic result. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled in via citation. The subsampling-without-replacement assumption (A1) being stricter than the bootstrap used in the implementation is a scope or correctness concern, not a circularity, since the theorem explicitly states its assumptions and does not fit constants to the target conclusion.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The theory relies on standard smoothness assumptions for X and on a specific tie-breaking rule for the theoretical tree. The hyperparameters p0, q, depth, and alpha are chosen by defaults or tuning. No new physical or conceptual entities are introduced.

free parameters (4)
  • p0 = tuned per dataset (e.g., 0.04 for SECOM)
    Selection threshold for rule frequency; chosen by cross-validation to maximize accuracy with a small rule set (Section 5.4).
  • q = 10
    Number of quantiles per variable; hand-set default, validated in experiments (Section 5.4).
  • max tree depth = 2
    Default in software sirus; chosen to optimize accuracy and stability (Section 5.4, Table 6).
  • alpha = 0.05
    Threshold in stopping criterion (5.1) to grow enough trees for 95% rule overlap; set by default.
assumptions (5)
  • domain assumption Subsampling rate a_n satisfies a_n -> infinity and a_n/n -> 0 (Assumption A1).
    Used in Theorem 1 proofs to ensure variance of bagged estimates vanishes.
  • domain assumption Number of trees M_n satisfies M_n -> infinity (Assumption A2).
    Needed for Monte-Carlo convergence of the rule frequency estimator.
  • domain assumption X has a strictly positive density with continuous, bounded, strictly positive marginals (Assumption A3).
    Ensures empirical quantiles converge to population quantiles and splits are non-degenerate; central to the stability proof.
  • standard math The sample is i.i.d. from an unknown distribution P_{X,Y}.
    Standard supervised learning assumption stated at the start of Section 3.
  • ad hoc to paper Ties in the theoretical CART criterion are broken according to a limit law defined in Definitions 1 and 2.
    Introduces a specific tie-breaking rule so the theoretical algorithm matches the limiting behavior of the empirical algorithm; this is a modeling choice specific to the proof.

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Pith. "Pith review of SIRUS: Stable and Interpretable RUle Set for Classification." pith.science (2026). https://pith.science/paper/KCE6LEEN

@misc{pith2026190806852,
  author       = {Pith},
  title        = {Pith review of: SIRUS: Stable and Interpretable RUle Set for Classification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KCE6LEEN}},
  note         = {Machine review of arXiv:1908.06852}
}
read the original abstract

State-of-the-art learning algorithms, such as random forests or neural networks, are often qualified as "black-boxes" because of the high number and complexity of operations involved in their prediction mechanism. This lack of interpretability is a strong limitation for applications involving critical decisions, typically the analysis of production processes in the manufacturing industry. In such critical contexts, models have to be interpretable, i.e., simple, stable, and predictive. To address this issue, we design SIRUS (Stable and Interpretable RUle Set), a new classification algorithm based on random forests, which takes the form of a short list of rules. While simple models are usually unstable with respect to data perturbation, SIRUS achieves a remarkable stability improvement over cutting-edge methods. Furthermore, SIRUS inherits a predictive accuracy close to random forests, combined with the simplicity of decision trees. These properties are assessed both from a theoretical and empirical point of view, through extensive numerical experiments based on our R/C++ software implementation sirus available from CRAN.

Figures

Figures reproduced from arXiv: 1908.06852 by the authors.

Figure 1
Figure 1. Example of a root node R2 partitionned by a randomized tree of depth 2: the tree on the right side, the associated paths and hyperrectangles of length d = 2 on the left side. The main step of SIRUS is to extract rules from the modified random forest. The corner￾stone of this extraction mechanism is the notion of path in a decision tree. Indeed, a path describes the sequence of splits to go from the root of the tree … view at source ↗
Figure 2
Figure 2. For the SECOM dataset, error (1-AUC) versus the number of rules when p0 varies, estimated via 10-fold cross-validation (averaged over 10 repetitions of the cross-validation). Errors for CART and random forests are reported for comparisons [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. List of rules output by our software sirus in the R console for the SECOM dataset. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: For the UCI dataset “Credit German”, 1-AUC (on the left) and stability (on the right) versus the number of rules when p0 varies, estimated via 10-fold cross-validation (results are averaged over 10 repetitions) [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 5
Figure 5. Figure 5: For the UCI dataset “Heart Statlog”, 1-AUC (on the left) and stability (on the right) versus the number of rules when p0 varies, estimated via 10-fold cross-validation (results are averaged over 10 repetitions). 20 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: The two lists of rules output by two runs of SIRUS for the SECOM dataset. Dataset C5.0 PART FOIL SIRUS Authentification 11 8 20 13 Breast Wisconsin 5 10 41 24 Credit Approval 9 32 40 16 Credit German 22 68 101 22 Diabetes 12 7 36 8 Haberman 2 2 4 5 Heart C2 10 20 31 20…
Figure 7
Figure 7. Figure 7: The two lists of rules output by two runs of Node harvest for the SECOM dataset. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_7.png]
Figure 8
Figure 8. Figure 8: The two lists of rules output by two runs of RuleFit for the SECOM dataset. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_8.png]
Figure 9
Figure 9. Figure 9: For the UCI dataset “Credit German”, 1-AUC versus the number of rules when p0 varies, estimated via 10-fold cross-validation (repeated 30 times) for two different methods of rule aggregation: the rule average (A.1) in red and a logistic regression (A.2) in blue. To tac…
Figure 10
Figure 10. Figure 10: For the UCI dataset “Credit German”, 1−εM,n,p0 for a sequence of p0 ∈ VˆM,p0 corresponding to final models ranging from 1 to about 25 rules. more demanding than random forests overall. B.2 Theoretical Properties We emphasize that growing more trees does not improve pr…

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