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arxiv: 2606.05919 · v2 · pith:Y66G3ZV3 · submitted 2026-06-04 · stat.ML · cs.LG· econ.EM· stat.CO

Finding Most Influential Sets

Lucas D. Konrad , Nikolas Kuschnig This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 reserved 2026-06-27 23:48 UTCgrok-4.3pith:Y66G3ZV3record.jsonopen to challenge →

Figure 1
Figure 1. Figure 1: Greedy failure when finding the 3-MIS for n = 5. Point A is the most influential singleton and is therefore selected first by a greedy procedure. The exact 3-MIS instead consists of the three leftmost points, whose influence is joint and masked in singleton rankings. Enumeration recovers … reproduced from arXiv: 2606.05919
classification stat.ML cs.LGecon.EMstat.CO
keywords most influential setsleave-set-out effectsDinkelbach's methodtop-k problemsorthogonal scorespartial linear modelinfluence analysisstatistical estimands
0
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The pith

For estimands with linear-fractional leave-set-out effects, most influential set selection reduces to a one-parameter sequence of top-k problems.

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

The paper shows that finding size-k subsets whose removal maximally alters a target estimand is normally intractable due to the binomial number of candidates. When the leave-set-out effect takes a linear-fractional form, the search collapses to a one-parameter family of top-k selection tasks. Dinkelbach's method then solves the resulting univariate ratio problem with linear cost per iteration and guaranteed finite termination. A reader would care because the approach yields globally optimal sets for fixed-residual partial linear models and approximates the orthogonal-score objective under estimated nuisances when stability conditions hold. Simulations confirm that previously inaccessible exact recoveries become feasible.

Core claim

For estimands with linear-fractional leave-set-out effects, MIS selection reduces to a one-parameter sequence of top-k problems. Dinkelbach's method yields an algorithm with O(n) cost per iteration and finite termination. For fixed residualized inputs, the algorithm returns a globally optimal set for the univariate ratio objective, including the oracle-residualized partial linear model. With estimated nuisance functions, uniform denominator and generated-score stability imply approximation to the first-order oracle orthogonal-score objective; exact set recovery follows under a separation condition.

What carries the argument

The linear-fractional structure of leave-set-out effects, which reduces MIS selection to a one-parameter sequence of top-k problems solved by Dinkelbach's method.

If this is right

  • The algorithm has O(n) cost per iteration and finite termination.
  • For fixed residualized inputs, it returns a globally optimal set for the univariate ratio objective.
  • With estimated nuisance functions, uniform denominator and generated-score stability imply approximation to the first-order oracle orthogonal-score objective.
  • Exact set recovery follows under a separation condition.
  • Simulations and applications recover exact MIS that were previously computationally inaccessible.

Where Pith is reading between the lines

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

  • The reduction may extend to other fractional objectives in causal inference if they admit an analogous structural property.
  • The approach could scale influence diagnostics to larger n in settings where residualization is cheap.
  • Checking the separation condition on real data would indicate when exact recovery is practically reliable.

Load-bearing premise

The target estimand must possess linear-fractional leave-set-out effects.

What would settle it

On a small dataset where the estimand lacks linear-fractional leave-set-out effects, brute-force enumeration of all k-subsets would yield a different set than the one returned by the algorithm.

Figures

Figures reproduced from arXiv: 2606.05919 by Lucas D. Konrad, Nikolas Kuschnig.

Figure 2
Figure 2. Figure 2: Runtimes of Algorithm 1 in milliseconds (median over 100 runs) for scenarios up to n = 106 and k = 105 (colored tiles), along with an enumeration feasibility frontier (markers) based on n k  candidate sets under an optimistic per-set evaluation cost. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows the Mongolia trial of Attanasio et al. (2015). 3The implementation uses a size-k heap for ordering, and is implemented in R and Rcpp (Eddelbuettel & François, 2011), on a Ryzen AI Max+ Pro 395, using a single thread. We also bench￾marked an implementation of the greedy algorithm by Kuschnig et al. (2021) on a subset of the grid (a runtime of 200 ms is achieved for n = 104 and k = 50), as well as the … view at source ↗
Figure 4
Figure 4. Figure 4: MIS impacts across the remaining six microcredit RCTs (see Meager, 2019). Lines show |∆k| for exact k-MIS (in￾crease/decrease) and greedy; points mark non-nestedness events. dings. For three query pairs with low, medium, and high similarity, we study the local prediction target ϕ(β) = x0β [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

Identifying most influential sets (MIS) - size-$k$ subsets whose removal maximally changes a target estimand - is typically infeasible because it requires searching over $\binom{n}{k}$ subsets. For estimands with linear-fractional leave-set-out effects, we show that MIS selection reduces to a one-parameter sequence of top-$k$ problems. Dinkelbach's method yields an algorithm with $\mathcal{O}(n)$ cost per iteration and finite termination. For fixed residualized inputs, the algorithm returns a globally optimal set for the univariate ratio objective, including the oracle-residualized partial linear model. With estimated nuisance functions, uniform denominator and generated-score stability imply approximation to the first-order oracle orthogonal-score objective; exact set recovery follows under a separation condition. Simulations and applications show that the method recovers exact MIS that were previously computationally inaccessible.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper claims that for estimands with linear-fractional leave-set-out effects, identifying most influential sets (MIS) of size k reduces to a one-parameter sequence of top-k problems. Dinkelbach's method then yields an algorithm with O(n) per-iteration cost and finite termination. For fixed residualized inputs the procedure returns a globally optimal set for the univariate ratio objective (including the oracle-residualized partial-linear model). With estimated nuisance functions, uniform denominator and generated-score stability imply approximation to the first-order oracle orthogonal-score objective, with exact recovery under an additional separation condition. Simulations and applications are reported to recover exact MIS that were previously computationally inaccessible.

Significance. If the central claims hold, the work converts an intractable combinatorial search into a tractable parametric sequence of linear-time selection problems under an explicitly scoped structural hypothesis. The application of Dinkelbach iteration, the finite-termination argument on a finite feasible set, and the explicit stability/separation conditions for the nuisance-estimation case constitute clear algorithmic and theoretical contributions. The extension to oracle-residualized partial-linear models and the empirical demonstration of exact recovery further strengthen the practical utility for influence diagnostics in statistical and machine-learning settings.

minor comments (1)
  1. [Abstract] The abstract refers to 'uniform denominator and generated-score stability' without an inline pointer to the precise definition or assumption statement; a parenthetical reference to the relevant section would improve immediate readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their thorough and positive review. We are grateful for the recommendation to accept and for the recognition of the algorithmic and theoretical contributions, including the reduction to a parametric sequence of top-k problems, the finite-termination argument for Dinkelbach iteration, and the stability conditions for the nuisance-estimated case.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation scopes the main result to estimands whose leave-set-out effects are linear-fractional by assumption, then applies the standard Dinkelbach iteration to convert the ratio objective into a one-parameter family of top-k problems. Finite termination and per-iteration cost follow directly from the finiteness of the feasible set and the monotonic improvement property of the auxiliary-parameter update; neither step reduces to a fitted quantity defined in terms of the output nor to a self-citation chain. Subsequent claims about oracle residuals and nuisance estimation are likewise conditioned on explicitly named stability and separation assumptions that are independent of the algorithm itself. No load-bearing equation or premise collapses to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the estimand admits linear-fractional leave-set-out effects; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Estimands possess linear-fractional leave-set-out effects
    This property is invoked to obtain the reduction to a one-parameter sequence of top-k problems.

pith-pipeline@v0.9.1-grok · 5671 in / 1237 out tokens · 19594 ms · 2026-06-27T23:48:26.510752+00:00 · methodology

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