Probabilistic Explanations for Linear Models

Bernardo Subercaseaux; Kuldeep S Meel; Marcelo Arenas

REVIEW 3 major objections 3 minor 1 cited by

Probabilistic Explanations for Linear Models

T0 review · 3 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper proves that for linear models, a small set of features that determines the model's decision with high probability can be computed in randomized polynomial time, making them the first classifier family with this guarantee.

desk verdict Promising ideas and a likely-true result, but the proof of the main theorem has a real gap that a referee needs to confront. read the letter →

arxiv 2501.00154 v1 pith:HYG4FQHK submitted 2024-12-30 cs.AI cs.CC

Bernardo Subercaseaux , Marcelo Arenas , Kuldeep S Meel This is my paper · ORCID

classification cs.AIcs.CC

keywords formalXAIsufficientreasonsprobabilisticexplanationslinearmodels(δε)-min-SRMonteCarloestimationfeaturescoresexplanationcomplexity

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 tries to establish that linear models are the first class of classifiers for which small probabilistic sufficient reasons can be computed efficiently. It introduces the $(\delta,\varepsilon)$-min-SR relaxation—an explanation that is a minimum $\delta^*$-SR for some $\delta^*$ within $\varepsilon$ of the requested confidence $\delta$—and proves that for any linear model and input, a $(\delta,\varepsilon)$-min-SR can be produced with high probability in time polynomial in the number of features, $1/\varepsilon$, and $1/\delta$. A sympathetic reader should care because the relaxation matches how people actually use confidence thresholds: a user who asks for a 95% explanation would not notice a 94.9997% one, but she would notice an explanation that is suddenly 250 times larger. This positive result contrasts sharply with decision trees, where even approximating small $\delta$-SRs is hard unless SAT can be solved in quasi-polynomial time.

What carries the argument

The machinery has three pieces. First, the feature score $s_i = w_i (2x_i-1)(2\mathcal{L}(x)-1)$ encodes whether feature $i$ helps or hurts the classification and by how much; under the uniform completion distribution, adding features in decreasing score order never decreases the agreement probability (Lemma 1). Second, because of Lemma 1, the minimum $\delta^*$-SR is exactly the shortest score-sorted prefix whose agreement probability reaches $\delta^*$, so the search reduces to a binary search over $k$. Third, the agreement probabilities are estimated by Monte Carlo sampling (Fact 1, via Hoeffding's inequality), and the algorithm randomizes $\delta^*$ uniformly over $[\delta-\varepsilon, \delta+\varepsilon]$ to avoid the finitely many 'hard' thresholds where the probability is within $O(\varepsilon\delta/\log n)$ of $\delta^*$; the proof shows the estimates are accurate enough on the visited prefixes with probability $1-\delta$. The randomization over $\delta^*$ is what breaks the #P-hardness of the exact problem.

What would settle it

Fix a small linear model, say $n=4$ with weights $(1,-2,3,-1)$ and threshold $0$, and an instance $x=(1,1,1,0)$; enumerate all subsets to compute the exact agreement probability of every partial instance under the uniform distribution. If, for some $k$, the top-$k$ prefix by score does not achieve the maximum agreement probability among all $k$-feature subsets, Lemma 10 is false and the algorithm's guarantee collapses. Alternatively, with the same model, draw $\delta^*$ uniformly from $[0.4,0.6]$ and compare the greedy answer to the brute-force minimum $\delta^*$-SR; any size mismatch for a $\delta^*$ more than $\varepsilon$ away from a threshold value would be a counterexample.

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Extended reading notes

Core claim

The paper's central claim is Theorem 2: given a linear model $\mathcal{L}$ and an input $x$, a $(\delta,\varepsilon)$-min-SR can be computed successfully with probability at least $1-\delta$ in $\tilde{O}(n/(\varepsilon^2 \delta^2))$ randomized time. The algorithm samples a target confidence $\delta^*$ uniformly at random from $[\delta-\varepsilon, \delta+\varepsilon]$, sorts features by the score $s_i = w_i (2x_i-1)(2\mathcal{L}(x)-1)$, and uses binary search with Monte Carlo estimates of the agreement probability of the top-$k$ prefixes; Lemma 1 guarantees that the top-$k$ prefix is the minimum $\delta^*$-SR exactly when it is the first prefix whose agreement probability reaches $\delta^*$. The paper also proves (Theorem 3) that under any product distribution every locally minimal $\delta$-SR is subset-minimal for linear models, and (Proposition 2) that approximating $\delta$ does not approximate explanation size: the minimum size can jump by a factor of $n^{1/2-\varepsilon_0}$ when $\delta$ is raised by $\varepsilon$.

Load-bearing premise

The load-bearing premise is that the random completion of an explanation is drawn uniformly over all binary completions (or, more generally, that every feature is an independent Bernoulli variable with the same parameter); the greedy score ordering is provably optimal only under that assumption, and the paper itself notes that extending to arbitrary product distributions is not straightforward.

Editorial extensions

If this is right

Banks and other users of linear scoring models can now ask for a small, high-confidence explanation of a decision—say, 'your loan was rejected because of these three features, with 95% confidence'—and compute it in randomized polynomial time.
Linear models are the first class of classifiers for which a $(\delta,\varepsilon)$-min-SR is efficiently computable; for decision trees the problem remains inapproximable in polynomial time unless SAT is quasi-polynomially solvable.
The relaxation gives both sides what they want: a user receives an explanation no larger than the minimum $(\delta-\varepsilon)$-SR, while a model owner can enforce a strict $\delta$ guarantee by computing a $(\delta+\varepsilon/2,\varepsilon/2)$-min-SR.
For linear models under any product distribution, every locally minimal $\delta$-SR is also subset-minimal, so local-search heuristics that find locally minimal explanations are actually finding globally subset-minimal ones.
Because $\delta$-approximation and size-approximation are decoupled, systems should report the achieved confidence level explicitly rather than silently substituting the requested $\delta$.

Reading between the lines

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

The random-$\delta^*$ trick is transferable: any model class whose score-sorted prefixes have non-decreasing agreement probabilities would inherit the same Monte Carlo plus binary search template; decision trees fail this property, which is exactly why they stay hard.
A testable weakening: replace the uniform distribution with feature-dependent Bernoulli parameters; a majorization or convexity condition on the weights might restore the greedy ordering for some product distributions, a direction the paper leaves open.
The sample count grows like $1/(\varepsilon^2 \delta^2)$, so the authors' own example ($n=500$, $\varepsilon=0.1$, $\delta=0.01$) is expensive; importance sampling or quasi-Monte Carlo could shrink the constant without changing the asymptotic bound.
Because Proposition 2 shows that moving $\delta$ by $\varepsilon$ can enlarge the minimum explanation by a large factor, any deployed system should pair the explanation with its achieved confidence, not just the requested one.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Worth a look, but the main theorem's proof is not sound. The (δ,ε)-SR relaxation is a good idea, and Lemma 1's greedy feature selection for linear models under the uniform distribution is a clean, genuinely useful result. The paper also gives a nice survey of the hardness landscape and an interesting theorem about locally minimal δ-SRs under product distributions. Credit where due: the appendix is serious, and the examples are helpful.

The soft spot is Theorem 2. The correctness proof relies on events E_k: |δ* − p_k| ≥ εδ/log n holding for every queried k, so that Monte Carlo comparisons agree with truth. But there are linear models where this fails deterministically for the final binary-search query. Take n unit weights, threshold n/2 + c√n, and x = 1^n. The probabilities p_k rise in steps of Θ(1/√n). For n large relative to (εδ/log n)^2, the boundary index k* for the sampled δ* has |δ* − p_{k*}| < εδ/log n, so E_{k*} fails with probability 1. The proof's claim that ∩_{k∈I}E_k has probability ≥ (1−δ)^2 is then false for this instance, because I always contains k*. The conditioning argument breaks: conditioning on the binary search path reveals information about δ* that voids the uniform union bound. The algorithm can return a non-minimal explanation for the sampled δ*. The existential (δ,ε)-min-SR claim may still be true—a wrong index in the flat region is usually a minimum δ'-SR for some nearby δ'—but the proof as written does not establish the theorem. This is a load-bearing gap, not a typo.

Other soft spots are minor: the uniform-completion assumption is restrictive and the paper admits it; the runtime dependence on 1/ε^2 δ^2 is heavy. The citation pattern looks fair.

Who is this for? Researchers in formal XAI and anyone who wants a first positive algorithmic result for probabilistic explanations on linear models. I would send it to a serious referee, but the referee should be told to focus on the proof of Theorem 2. If the proof can be patched, the paper is a solid contribution; if not, the main claim needs to be scaled back.

Referee Report

3 major / 3 minor

Summary. The paper introduces (δ, ε)-sufficient reasons, a two-parameter relaxation of the classical probabilistic sufficient reasons of Wäldchen et al., and proves that for linear models a (δ, ε)-min-SR can be computed in polynomial time, specifically in O~(n/(ε^2 δ^2)) time with success probability 1−δ. The algorithm samples a threshold δ* uniformly from [δ−ε, δ+ε], sorts features by a score, and binary-searches over the number of fixed features using Monte Carlo estimates of the acceptance probability. The paper also proves that locally minimal δ-SRs are subset-minimal for linear models under product distributions (Theorem 3), and gives a lower bound showing that a small change in δ can force a very large change in the minimum explanation size (Proposition 2). The structural lemma (Lemma 1) that the top-scoring features are optimal is proved carefully in the appendix.

Significance. If the main algorithmic claim is correct, it is a meaningful positive result in formal XAI: linear models would be the first model class for which approximate minimum probabilistic sufficient reasons can be computed efficiently, in contrast to the strong hardness results for decision trees and neural networks. The structural characterization of feature selection (Lemma 1 and its supporting Lemmas 6–10) is elegant and appears correct. The paper also provides a good summary of the hardness landscape and a useful size-gap lower bound. However, the proof of the central theorem, as written, proves a stronger statement than the theorem needs and that stronger statement is false; this is a load-bearing issue that must be fixed before the main contribution can be accepted.

major comments (3)

[Section 3, proof of Theorem 2 (events E_k and F_k)] The proof requires the event ∩_{k∈I} E_k, where E_k is |δ* − p_k| ≥ εδ/log n and I is the set of indices queried by the binary search. The flaw is that I is not independent of δ*: whenever the search behaves correctly, I contains the boundary index k* = min{k : p_k ≥ δ*}, and for that index p_{k*} is within one step of the sequence {p_k} of δ*. Consider a linear model with n unit weights, threshold n/2 + c√n, and instance x = 1^n. The probabilities p_k have increments Θ(1/√n). For n large enough that 1/√n < εδ/log n, the boundary index satisfies |δ* − p_{k*}| < εδ/log n for every δ*, so E_{k*} fails with probability 1. Hence Pr(∩_{k∈I} E_k) = 0, contradicting the lower bound claimed in the proof. A union bound over a fixed set of size ≤ log n cannot repair this, because I is random and contains exactly the index where the event is most likely to fail. Since the proof's correctness argument depends entirely on this intersection, Theorem 2 is not established as written.
[Section 3, proof of Theorem 2, Eq. (2)–(4)] The proof introduces a set I* defined as the binary-search path conditioned on all events E_k and F_k holding, and then uses Pr(I = I* | ∩ E_k ∩ ∩ F_k) = 1 to lower-bound Pr(∩_{k∈I} E_k). This step is circular: I* is defined by conditioning on the very events whose probability is being estimated, and it is not a fixed set independent of the random estimates and δ*. The transition from Equation (2) to the final bound silently replaces a sum over all possible sets I' by one particular set I*, without a lower bound on the conditional probability of I = I*. Thus the claimed lower bound of (1−δ)^2 is not derived from the preceding inequalities.
[Algorithm 1 and Theorem 2 statement] The output specification of Algorithm 1 and the proof assert that the returned explanation is a minimum δ*-SR for the sampled value δ*. The boundary-index phenomenon described above shows that this stronger assertion is false: in the flat-region example, the Monte Carlo comparison at the final queried index is essentially a coin flip (since the estimator's standard deviation is larger than εδ/log n), so the algorithm can return an index adjacent to k*, which is not the minimum δ*-SR. Theorem 2 as stated only requires the existential condition of Definition 4—that the returned explanation be a minimum δ' -SR for some δ' ∈ [δ−ε, δ+ε]—and that weaker claim may be salvageable. But the current proof establishes only the stronger and false claim, and it provides no separate argument for the existential guarantee. Therefore the main theorem is not proven by the manuscript as it stands.

minor comments (3)

[Section 3 and Theorem 2] The symbol δ is overloaded: it denotes both the probability threshold in (δ, ε)-SR and the failure probability in Theorem 2 and in the sample size ρ in Algorithm 1. This makes the theorem and the algorithm difficult to read; use a separate symbol, e.g., β, for the failure probability.
[Definition 4] Definition 4 should restrict δ* to [0,1], or the theorem should assume ε ≤ min(δ, 1−δ). Otherwise the interval [δ−ε, δ+ε] can contain values outside [0,1], where the notion of a δ*-SR is not defined, and the algorithm samples δ* from that invalid interval.
[Appendix .1 and Section 6] In Appendix .1, the bound '2 · 1014' should read '2 · 10^{-14}'; in Section 6, the phrase 'the feature selection argument of ?? no longer holds' contains an unresolved cross-reference. These should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main derivation is self-contained and the cited prior work is contextual or independently established.

full rationale

The paper's central claim, Theorem 2, is that a (δ,ε)-min-SR for a linear model can be computed in randomized polynomial time. The derivation does not reduce to its inputs by construction. Definition 4 deliberately relaxes the target by allowing any δ* in [δ−ε, δ+ε], and Algorithm 1 samples such a δ* and then searches for a minimum δ*-SR; the substantive content is in the search step, not in the freedom to choose δ*. That search is justified by Lemma 1 and Lemma 10, which prove that the top-score greedy chain contains an optimal partial instance for every δ*, with proofs given in the appendix via Lemmas 6–10. The proof of Theorem 2 uses only Hoeffding's inequality (Fact 1), the random choice of δ*, and monotonicity of the probabilities p_k; it does not presuppose the theorem it is proving. Prior hardness results, including those by the authors, appear as motivation for the relaxation rather than as load-bearing inputs to the positive result. Self-citations such as Arenas et al. (2022) and Barceló et al. (2020) are contextual or are accompanied by self-contained proofs in this paper. The alleged flat-region failure of the events E_k is a correctness critique of the concentration argument, not a circularity: it does not show that the theorem statement is assumed in the proof. No prediction is fitted and then renamed, and no stated definition is equivalent to the claimed conclusion. The paper is therefore self-contained against its own definitions, standard inequalities, and externally checkable facts.

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

The paper introduces no empirical free parameters: ε and δ are inputs to the problem. The central claim rests on standard concentration inequalities and the domain assumption of uniform random completions over binary features. The (δ, ε)-SR is a formal definition, not an empirical entity.

assumptions (5)

standard math Hoeffding's inequality provides a concentration bound for sums of independent bounded random variables.
Used in Fact 1 to bound the error of Monte Carlo probability estimates.
standard math Chernoff bound for sums of independent Bernoulli variables.
Used in Example 2 to show a single feature can be a 0.999999-SR.
standard math Erdős' upper bound on the central binomial coefficient.
Used in Lemma 3 and Lemma 4 to bound binomial tail probabilities in Proposition 2.
domain assumption Completions of partial instances are drawn uniformly at random from the set of all binary completions.
This is the distribution assumption in Definition 2 and the entire algorithm; the greedy score ordering relies on uniform Bernoulli(1/2) features.
domain assumption Models are Boolean functions over binary features (0/1).
The paper restricts to binary features; real-valued or categorical features are left to future work.

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Pith. "Pith review of Probabilistic Explanations for Linear Models." pith.science (2026). https://pith.science/paper/HYG4FQHK

@misc{pith2026250100154,
  author       = {Pith},
  title        = {Pith review of: Probabilistic Explanations for Linear Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HYG4FQHK}},
  note         = {Machine review of arXiv:2501.00154}
}

abstract

Formal XAI is an emerging field that focuses on providing explanations with mathematical guarantees for the decisions made by machine learning models. A significant amount of work in this area is centered on the computation of "sufficient reasons". Given a model $M$ and an input instance $\vec{x}$, a sufficient reason for the decision $M(\vec{x})$ is a subset $S$ of the features of $\vec{x}$ such that for any instance $\vec{z}$ that has the same values as $\vec{x}$ for every feature in $S$, it holds that $M(\vec{x}) = M(\vec{z})$. Intuitively, this means that the features in $S$ are sufficient to fully justify the classification of $\vec{x}$ by $M$. For sufficient reasons to be useful in practice, they should be as small as possible, and a natural way to reduce the size of sufficient reasons is to consider a probabilistic relaxation; the probability of $M(\vec{x}) = M(\vec{z})$ must be at least some value $\delta \in (0,1]$, for a random instance $\vec{z}$ that coincides with $\vec{x}$ on the features in $S$. Computing small $\delta$-sufficient reasons ($\delta$-SRs) is known to be a theoretically hard problem; even over decision trees--traditionally deemed simple and interpretable models--strong inapproximability results make the efficient computation of small $\delta$-SRs unlikely. We propose the notion of $(\delta, \epsilon)$-SR, a simple relaxation of $\delta$-SRs, and show that this kind of explanation can be computed efficiently over linear models.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Explain Yourself, Briefly! Self-Explaining Neural Networks with Concise Sufficient Reasons
cs.LG 2025-02 conditional novelty 5.0 of 10

SST trains models to produce concise sufficient reasons as an extra output, yielding faster and often smaller explanations than post-hoc methods like Anchors and SIS.

Reference graph

Works this paper leans on

25 extracted references · 22 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Arenas, M., B \'a ez, D., Barcel \'o , P., P \'e rez, J., and Subercaseaux, B. (2021). Foundations of Symbolic Languages for Model Interpretability . In Advances in Neural Information Processing Systems , volume 34, pages 11690--11701. Curran Associates, Inc

work page 2021

[2] [2]

Arenas, M., Barcel \'o , P., Romero Orth, M., and Subercaseaux, B. (2022). On computing probabilistic explanations for decision trees. In Koyejo, S., Mohamed, S., Agarwal, A., Belgrave, D., Cho, K., and Oh, A., editors, Advances in Neural Information Processing Systems , volume 35, pages 28695--28707. Curran Associates, Inc

work page 2022

[3] [3]

Arenas, M., Barceló, P., Bustamante, D., Caraball, J., and Subercaseaux, B. (2024). A Uniform Language to Explain Decision Trees . In Proceedings of the 21st International Conference on Principles of Knowledge Representation and Reasoning , pages 60--70

work page 2024

[4] [4]

Barcel\' o , P., Monet, M., P\' e rez, J., and Subercaseaux, B. (2020). Model interpretability through the lens of computational complexity. In Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M., and Lin, H., editors, Advances in Neural Information Processing Systems , volume 33, pages 15487--15498. Curran Associates, Inc

work page 2020

[5] [5]

Barcel \' o , P., P \' e rez, J., and Subercaseaux, B. (2020). Foundations of languages for interpretability and bias detection. AFCI workshop at NeurIPS 2020. Algorithmic Fairness through the Lens of Causality and Interpretability

work page 2020

[6] [6]

Blanc, G., Lange, J., and Tan, L.-Y. (2021). Provably efficient, succinct, and precise explanations. In Beygelzimer, A., Dauphin, Y., Liang, P., and Vaughan, J. W., editors, Advances in Neural Information Processing Systems

work page 2021

[7] [7]

and Koriche, F

Bounia, L. and Koriche, F. (2023). Approximating probabilistic explanations via supermodular minimization. In Proceedings of the Thirty-Ninth Conference on Uncertainty in Artificial Intelligence , UAI '23. JMLR.org

work page 2023

[8] [8]

and Hirth, A

Darwiche, A. and Hirth, A. (2020). On the Reasons Behind Decisions . In ECAI 2020 - 24th European Conference on Artificial Intelligence , 29 August-8 September 2020, Santiago de Compostela , Spain , August 29 - September 8, 2020 - Including 10th Conference on Prestigious Applications of Artificial Intelligence ( PAIS 2020) , pages 712--720

work page 2020