{"id":"878b62d0-4c02-45c5-9680-73b1a1bc335b","arxiv_id":"2506.02257","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"Inflated top-k and inflated full ranking provide assumption-free leave-one-out stability for ranking, with provably minimal ambiguity in the returned set.","lead":"Two new set-valued ranking operators, the inflated top-k and the inflated full ranking, guarantee leave-one-out stability for ranking problems whenever the underlying score estimator is stable. The guarantee is distribution-free, dimension-free, and the returned sets are optimal in a minimax sense.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Full-ranking output can be exponentially large, so the nonempty-intersection stability guarantee can be trivially satisfied and 'informativeness' is not assured for large L; the stability theorems themselves are sound.","rationale":"I checked the main proof steps: property (3) follows from Lemma 13 and the inflated argmax intersection property, and property (5) follows from the iterative construction in the proof of Theorem 8. The proofs are internally consistent, and the transfer from (ε, δ)-stability of A to top-k/full-ranking stability of the composed procedure is correct. The reader's verdict of ACCEPT with high confidence is therefore reasonable. The load-bearing concern is not a mathematical error but a gap between the theorem and the paper's broader claim that the methods provide 'stability without compromising informativeness.' For full ranking, the output can be exponentially large, and the nonempty-intersection stability notion becomes correspondingly weak. This is a limitation the paper partially acknowledges, but it is not resolved by Proposition 10, which only bounds positions. The proposed concrete test would demonstrate the exponential blow-up and make the limitation precise. Since the central stability theorems are sound and the limitation is acknowledged, the reader's verdict should remain unchanged.","tokens_in":18376,"tokens_out":26447,"duration_ms":244574,"concrete_test":"For a synthetic score vector with L = 20 arranged as five blocks of four equal values, with all block values within ε/√2 of each other, compute |ranking(ε)(w)| using the partial-order characterization, or by enumeration for smaller L. If the size grows as the product of block-size factorials (e.g., (4!)^5 ≈ 8 × 10^6) while the outputs on D and D\\i are identical, then the full-ranking stability guarantee is satisfied vacuously and the output set cannot be enumerated for large L.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central conditional stability theorems are correct, but the paper's advertised full-ranking stability is weaker than it appears. For score vectors with entries within ε/√2 of one another, ranking(ε)(w) is the set of all linear extensions of the partial order {i ≺ j : w_i ≥ w_j + ε/√2}; when all scores are tied, this set is S_L and has size L!. Definition 2 only requires f(D) ∩ f(D\\i) ≠ ∅, which is trivially satisfied when both output sets are huge. It does not require the common permutation to be a large fraction of either set, nor does it control the size of the output. Proposition 10 bounds each item's possible position, but the number of permutations consistent with those bounds can still be exponential, and the paper offers no compact representation or efficient enumeration scheme for ranking(ε)(w). The full-ranking experiment uses only L = 5 with average output size 1.76, so the behavior at realistic L is unresolved. This is a practical limitation of the 'informative output' claim, not a defect in Theorems 4 and 8.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops set-valued ranking operators for top-k selection and full ranking, generalizing the inflated argmax of Soloff et al. (2024a). For a score vector w and radius ε, top-k(ε)(w) returns all items that can be brought into the top k by a perturbation of w of size < ε, and ranking(ε)(w) returns all permutations consistent with iteratively applying the inflated argmax. The main results (Theorems 4 and 8) show that if the score-learning algorithm A satisfies the (ε,δ)-stability condition (1), then top-k(ε)∘A has top-k stability δ and ranking(ε)∘A has full ranking stability δ for every dataset and every n ≥ 2, with no distributional assumptions and no dependence on the number L of items. The paper also proves optimality statements (Propositions 5 and 9), efficient-computation results (Propositions 6 and 10), relations between the operators (Propositions 11–13), and reports experiments on Netflix and simulated data.","tokens_in":18547,"tokens_out":15713,"duration_ms":159936,"significance":"The proofs in the appendix are careful and, as far as I can verify, correct; the paper provides a clean finite-sample transfer from score-level stability to ranking-level stability, with uniform guarantees over datasets. The permutation-invariance and the reduction to the inflated argmax are elegant, and the connection to bagging makes the required score-stability condition achievable for a broad class of base learners. The open-source code and reproducible experimental setup are also a strength. The principal caveat, discussed below, is that the full-ranking guarantee relies on a nonempty-intersection condition that can be satisfied trivially when the output set of permutations is exponentially large; this limits the practical force of the 'without compromising informativeness' claim but does not invalidate the theorems.","major_comments":[{"comment":"The full-ranking stability guarantee is weaker than it appears. If the entries of w all lie within ε/√2 of one another, then ranking(ε)(w) = S_L, so the nonempty-intersection condition in Definition 2 is automatically satisfied and conveys no information about whether the two output sets are meaningfully similar. Proposition 10 bounds each item's possible position but does not prevent an exponential number of consistent permutations, and the full-ranking experiment uses only L = 5 with average output size 1.76, leaving the large-L regime unresolved. The claim that the method offers stability 'without compromising informativeness' is therefore not established as stated. The authors should state explicitly that ranking(ε)(w) is the set of linear extensions of the threshold partial order {i ≺ j : w_i ≥ w_j + ε/√2}, which gives a compact representation and makes the exponential-size issue transparent, and they should qualify the informativeness claim or argue why nonempty intersection is the right notion despite this behavior. Relatedly, Proposition 9 does not rule out trivial behavior: the constant function R(w) = S_L satisfies all of its hypotheses, so the optimality statement does not imply that ranking(ε)(w) is small when w has near-ties.","section":"Section 2.3, Definition 2 and Theorem 8"},{"comment":"The word 'assumption-free' overstates the scope of the results. Theorems 4 and 8 are conditional on A satisfying the (ε,δ)-stability inequality (1); although this is not a distributional assumption, it is a genuine condition on the base algorithm. For an arbitrary black-box A the condition must be verified or enforced, for instance by bagging, and the paper should say so in the title or abstract. Qualifying the headline as 'distribution-free' or 'conditional on score stability' would make the actual scope of the guarantee clear to the reader.","section":"Section 2.1, Eq. (1), and the abstract/title"}],"minor_comments":[{"comment":"In Theorems 4 and 8, the references to 'Thm. 1' and 'Thm. 2' should be to Definition 1 and Definition 2, respectively. In the paragraph after Definition 7, the references to 'Thm. 6' and 'Thm. 7' should be to Proposition 6 and Definition 7.","section":"Section 2.2–2.3, theorem/proposition references"},{"comment":"The phrase 'by Thm. 13' should read 'by Lemma 13', since the result is labelled as a lemma.","section":"Appendix A.1, proof of Theorem 4"},{"comment":"The experiment chooses ε = 0.01, but the paper does not verify that the shrinkage-averaged score learner satisfies Eq. (1) with this ε; for the raw vote-counting example the bound would be ε = √2/n ≈ 0.0014. The reported δj values are therefore an illustrative check of the method's behavior rather than a direct verification of Theorem 4.","section":"Section 3.1, Netflix experiment"},{"comment":"The simulated full-ranking experiment uses L = 5, which is too small to demonstrate the behavior of ranking(ε) when L is large; a brief discussion of scalability or the expected output-set size would be helpful.","section":"Section 3.2, full-ranking simulation"}],"recommendation":"major_revision","confidential_remarks":"The technical content is sound and the appendices are credible. My recommendation is driven by the gap between the headline claims (assumption-free, stable without compromising informativeness) and the actual strength of the full-ranking guarantee; both can be addressed by a careful revision. I would be happy to see a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The top-k part of this paper is a clean, correct generalization of the inflated argmax. The operator is natural, the proof of Theorem 4 is right, and Proposition 6 makes it computationally easy. The optimality result in Proposition 5 is also sound and gives a real sense in which the method is minimal. This will be a useful reference for anyone building stable selection procedures without separation assumptions.\n\nThe full-ranking construction is more interesting but also more fragile. The iterative definition in Definition 7 is sensible, and Theorem 8's proof via the shared-permutation induction is correct. Proposition 9 is a nice optimality statement about pairwise orderings. But the stress-test note lands: Definition 2 only asks for a nonempty intersection, and when the score vector has many near-ties, ranking(ε)(w) is the set of all linear extensions of a partial order, which can be L! permutations. Then f(D) ∩ f(D\\i) ≠ ∅ is nearly automatic, and the stability guarantee is trivially satisfied. The paper is transparent about the set-valued nature, and Proposition 10 does bound each item's possible position, but that does not control the total number of permutations. The synthetic experiment uses only L=5 with average output size 1.76, so the behavior at realistic L is genuinely unresolved. This is a limitation in the informativeness of the full-ranking output, not an error in the theorems.\n\nTwo smaller caveats. The title says \"assumption-free,\" but the guarantee is conditional on A satisfying the (ε,δ)-stability inequality (1). The paper says this clearly in Section 2.1, but it is still a conditional guarantee, and \"assumption-free\" refers only to the data distribution. Also, the full-ranking output has no compact representation provided; the position bounds are helpful but not enough to make the output usable for large L.\n\nOverall: the proofs are correct, the writing is honest, and the top-k contribution is solid. The full-ranking stability notion is valid but should be accompanied by a discussion of output set size, or a stronger stability definition that controls informativeness. I would send this to peer review and ask the authors to address the full-ranking output-size issue; the top-k result can be accepted more or less as is.","headline":"A correct and useful top-k stability extension; the full-ranking guarantee is mathematically sound but weaker than advertised because output sets can become exponentially large.","tokens_in":19141,"tokens_out":1558,"would_cite":true,"duration_ms":17117,"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":"If a score learner changes little when one data point is removed, the inflated top-k and inflated full ranking operators make top-k selection and full ranking provably stable, with no distributional assumptions.","keywords":["algorithmic stability","top-k selection","full ranking","set-valued output","inflated argmax","inflated full ranking","bagging","distribution-free guarantee"],"falsifier":"Fix any data set $D$ and any certified $(\\varepsilon,\\delta)$-stable algorithm $A$—for instance vote counting, where $\\varepsilon=\\sqrt{2}/n$ and $\\delta=0$—and compute the empirical fraction of leave-one-out indices $i$ for which $|\\text{top-k}^{(\\varepsilon)}(A(D)) \\cap \\text{top-k}^{(\\varepsilon)}(A(D\\backslash i))| < k$ (or, for full ranking, for which the two output sets are disjoint). The theorem guarantees this fraction is at most $\\delta$, so any data set where the fraction exceeds $\\delta$ refutes the claim; equivalently, one could search directly over score vectors $w,v$ with $\\lVert w-v\\rVert<\\varepsilon$ for a violation of the geometric intersection property (3) or (5).","tokens_in":18140,"feed_emoji":"🏅","tokens_out":13973,"duration_ms":120936,"temperature":0.7,"pith_summary":"This paper asks whether top-$k$ selection and full ranking can be made stable under small perturbations of the training data, without assuming a separation gap between items' scores. It proposes two set-valued operators—the inflated top-$k$ and the inflated full ranking—that return a small set of candidates or a set of possible rankings when the data is ambiguous. The central deliverable is a theorem: if the underlying score-learning algorithm $A$ changes its score vector by less than $\\varepsilon$ in Euclidean norm when one data point is removed, with probability at least $1-\\delta$, then composing either operator with $A$ yields a ranking procedure that is stable with the same $\\delta$, for any sample size $n\\ge 2$, any $k$, and any number of items $L$, with no distributional assumptions. This matters because conventional ranking theory typically requires a separation between true scores, a condition real data such as near-tied movie ratings often violates; here the output grows only slightly—a few extra candidates or a handful of rankings—and the guarantee holds uniformly over every data set.","feed_headline":"Top-k and full rankings get assumption-free stability guarantees","feed_subtitle":"When scores are nearly tied, set-valued outputs keep results stable, adding only a few extra candidates or rankings.","key_machinery":"The load-bearing objects are the two inflated ranking operators, defined from a margin parameter $\\varepsilon$. The inflated top-$k$ is $\\text{top-k}^{(\\varepsilon)}(w) = \\{ j : \\operatorname{dist}(w, C^{\\varepsilon,k}_j) < \\varepsilon \\}$, where $C^{\\varepsilon,k}_j$ is the closed set of score vectors in which item $j$ beats the $(k+1)$-st best item by at least $\\varepsilon/\\sqrt{2}$; for $k=1$ this reduces to the inflated argmax of Soloff et al. (2024a). The inflated full ranking is $\\text{ranking}^{(\\varepsilon)}(w) = \\{ \\pi \\in S_L : 1 \\in \\operatorname{argmax}^{(\\varepsilon)}(w_{\\pi(k)},\\dots,w_{\\pi(L)}) \\text{ for every } k \\}$, an iterative construction that requires each successive item to be an inflated-argmax winner of the remaining subvector. The argument runs on a single geometric transfer property: whenever $\\lVert w-v\\rVert<\\varepsilon$, the two inflated outputs overlap—at least $k$ shared items for top-$k$, and at least one shared permutation for full ranking—so that any $(\\varepsilon,\\delta)$-stable $A$ transfers its stability to the composed ranking procedure, with no distributional assumptions.","core_discovery":"The paper's central claim is that instability in Step 2 of a two-stage ranking pipeline—sorting scores or selecting the top $k$—can be eliminated without any assumptions on the data or on the base score-learning algorithm, at the price of allowing the output to be a set. Concretely, fix any learning algorithm $A$ that maps a data set $D$ to a score vector and satisfies the $(\\varepsilon,\\delta)$-stability inequality: for a uniformly random index $i$, $\\lVert A(D) - A(D\\backslash i)\\rVert < \\varepsilon$ with probability at least $1-\\delta$. Define the $\\varepsilon$-inflated top-$k$ of a score vector $w$ as the set of items $j$ whose score vector lies within distance $\\varepsilon$ of some vector in which $j$ is a top-$k$ item by a margin of $\\varepsilon/\\sqrt{2}$; define the $\\varepsilon$-inflated full ranking as all permutations $\\pi$ such that, at every step $k$, the inflated argmax of the remaining subvector $(w_{\\pi(k)},\\dots,w_{\\pi(L)})$ selects its first entry. Then $\\text{top-k}^{(\\varepsilon)}\\circ A$ has top-$k$ stability $\\delta$ and $\\text{ranking}^{(\\varepsilon)}\\circ A$ has full ranking stability $\\delta$: for every data set $D$, with probability at least $1-\\delta$ over the choice of $i$, the outputs on $D$ and $D\\backslash i$ share at least $k$ items (top-$k$) or share at least one ranking (full ranking). Moreover, both operators are minimal in a precise sense: any permutation-invariant alternative that satisfies the same stability condition returns a set of size $k$ at least as rarely as the inflated top-$k$, and the inflated full ranking preserves every definite pairwise order that any such alternative asserts.","pith_inferences":["The transfer mechanism is purely geometric—a Lipschitz-type overlap property in score space—which suggests a general recipe: any discontinuous downstream operation whose output can be encoded by margin constraints on scores could be stabilized by an analogous inflated, set-valued counterpart.","The full-ranking output set can contain many permutations; the paper bounds each item's possible positions but does not give a compact representation or an efficient sampler, so applying inflated full ranking to very large L would need additional algorithmic work.","A natural extension the paper leaves open is data-driven selection of ε: choosing ε by calibration could let a practitioner control the achieved stability level δ while minimizing the average size of the returned sets.","The paper treats deterministic algorithms; extending the definitions to randomized score learners would require a stability notion that accounts for the algorithm's internal randomness, which the paper only sketches in a footnote."],"forward_implications":["In the preference-voting setting, where A is (√2/n, 0)-stable, the theorems give fully deterministic leave-one-out stability: for every voter i and every vote data set, the inflated top-k outputs on D and D\\i share at least k items, and the inflated full ranking outputs always share a ranking.","For any bounded-output score learner, applying bagging to A makes it (ε,δ)-stable with ε²δ ∝ 1/n, so the guarantee applies to essentially arbitrary learned score functions without distributional requirements on the data.","Because the bound is uniform over data sets and independent of L, the stability guarantee cannot degrade as the candidate pool grows or as the data become more adversarial.","The optimality propositions imply that when the scores are unambiguous the inflated operators return exactly k items or a single permutation, so the price of stability is paid only in genuinely ambiguous cases.","Computing the inflated top-k reduces to computing the inflated argmax of the subvector beginning at the k-th position, so the method remains feasible even with many candidates."],"supporting_citations":[{"why":"supplies the inflated argmax construction and its stability and optimality properties (Thm. 9, Lemma 15, Prop. 20) that the proofs of Thms. 4 and 8 invoke directly.","marker":"Soloff et al. (2024a)"},{"why":"shows bagging makes any bounded-output learning algorithm (ε,δ)-stable with ε²δ ∝ 1/n, so the condition on A is achievable without distributional assumptions.","marker":"Soloff et al. (2024c)"},{"why":"provides the algorithmic-stability formulation to which the (ε,δ)-stability condition (1) is tied.","marker":"Elisseeff et al. (2005)"},{"why":"establishes stability of ℓ²-constrained least squares, the score learner used in the full-ranking simulation.","marker":"Bousquet and Elisseeff (2002)"},{"why":"supplies the Netflix Prize rating data used to measure empirical stability of inflated top-k versus plain top-k.","marker":"Bennett and Lanning (2007)"}],"fun_headline_variants":["Set-valued outputs stabilize rankings without assumptions","Inflated outputs give assumption-free ranking stability","No separation condition needed for ranking stability","Stable top-k and full ranking with no assumptions","Uncertainty in outputs makes ranking stable"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the score-learning algorithm $A$ is $(\\varepsilon,\\delta)$-stable: removing one data point at random changes $A$'s score vector by less than $\\varepsilon$ in Euclidean norm with probability at least $1-\\delta$; if $A$ fails this condition, the inflated operators provide no stability certificate.","fun_headline_variants_meta":{"raw":{"variants":["Set-valued outputs stabilize rankings without assumptions","Inflated outputs give assumption-free ranking stability","No separation condition needed for ranking stability","Stable top-k and full ranking with no assumptions","Uncertainty in outputs makes ranking stable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001575,"raw_usage":{"total_tokens":6412,"prompt_tokens":1195,"completion_tokens":5217,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":811,"completion_tokens_details":{"reasoning_tokens":5151}},"tokens_in":811,"tokens_out":5217,"duration_ms":38144,"temperature":1.0,"reasoning_tokens":5151,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:27:41.236178+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix any data set $D$ and any certified $(\\varepsilon,\\delta)$-stable algorithm $A$—for instance vote counting, where $\\varepsilon=\\sqrt{2}/n$ and $\\delta=0$—and compute the empirical fraction of leave-one-out indices $i$ for which $|\\text{top-k}^{(\\varepsilon)}(A(D)) \\cap \\text{top-k}^{(\\varepsilon)}(A(D\\backslash i))| < k$ (or, for full ranking, for which the two output sets are disjoint). The theorem guarantees this fraction is at most $\\delta$, so any data set where the fraction exceeds $\\delta$ refutes the claim; equivalently, one could search directly over score vectors $w,v$ with $\\lVert w-v\\rVert<\\varepsilon$ for a violation of the geometric intersection property (3) or (5).","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the algorithmic-stability formulation to which the (ε,δ)-stability condition (1) is tied."},{"cited_title":"and Elisseeff, A","cited_arxiv_id":null,"evidence_quote":"establishes stability of ℓ²-constrained least squares, the score learner used in the full-ranking simulation."},{"cited_title":"and Lanning, S","cited_arxiv_id":null,"evidence_quote":"supplies the Netflix Prize rating data used to measure empirical stability of inflated top-k versus plain top-k."}],"review_version":1}