Optimality of Sequential Filtering Under Independent Cost and Selectivity Models

Abhishek Mandal; Hrishikesh Paranjape; Xian Sun

Under an independence model, ordering filters by increasing cost-to-rejection ratio minimizes expected total cost for sequential pipelines.

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2026-06-29 09:07 UTC pith:OHHI6M3T

load-bearing objection Under an independence model the paper proves that sorting filters by increasing cost/rejection ratio minimizes expected total cost via a transitive pairwise swap argument.

arxiv 2606.07589 v1 pith:OHHI6M3T submitted 2026-05-28 cs.LG

Optimality of Sequential Filtering Under Independent Cost and Selectivity Models

Hrishikesh Paranjape , Abhishek Mandal , Xian Sun This is my paper

classification cs.LG

keywords sequential filteringfilter orderingexpected costindependence modelcost-rejection ratiocascaded systemsMonte Carlo evaluation

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.

The paper proves that when filter costs and rejection probabilities are independent, sorting them in increasing order of their cost divided by rejection probability produces the lowest expected cumulative cost. This formal result applies directly to common designs such as cascaded ranking systems, staged machine-learning inference, and fraud-detection pipelines, where stages progressively discard items. Monte Carlo simulations confirm that the ordering rule strictly outperforms standard heuristics in every trial, both on average and across the full distribution of realized costs.

Core claim

Under the independence assumption, the sequence that orders filters by non-decreasing values of cost divided by rejection probability achieves the global minimum expected total cost; any other permutation yields strictly higher expected cost.

What carries the argument

The cost-to-rejection ratio ordering rule, which ranks each filter by c_i / r_i and processes them from smallest to largest ratio.

Load-bearing premise

The costs and rejection probabilities of the filters are statistically independent.

What would settle it

A concrete set of dependent costs and rejection probabilities for which a different ordering yields lower expected cost than the cost-to-rejection ordering.

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

If this is right

The minimal expected cost for any given set of independent filters is obtained by sorting on the ratio.
Common heuristic orderings produce strictly higher expected cost and higher cost in every realization.
The result holds for any number of filters and any positive costs and rejection probabilities between zero and one.

Where Pith is reading between the lines

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

If real-world costs and selectivities exhibit dependence, the optimal ordering may deviate from the ratio rule and would require a joint model.
The same ratio criterion could be tested as a default ordering heuristic in production pipelines even when full independence is only approximate.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Under an independence model the paper proves that sorting filters by increasing cost/rejection ratio minimizes expected total cost via a transitive pairwise swap argument.

read the letter

The key point is that under an independence model, ordering the filters by increasing cost divided by rejection probability minimizes the expected total cost, and this is proven via a simple pairwise swap argument that works for any number of filters.

What stands out as new is the formal optimality proof for this specific ratio under the stated model. The paper does a good job laying out the expected cost objective and showing how the ordering follows directly from comparing pairs, with the transitivity giving the global sort. The Monte Carlo runs are presented as extra evidence that this beats the usual heuristics in practice.

The main soft spot is the independence assumption itself. Costs and selectivities being independent makes the math clean, but real systems often have dependencies, like a cheap filter that rejects based on the same features as an expensive one. The paper is upfront about the model, so that's not hidden. The simulations are only corroborative, and without seeing the exact setup it's hard to judge how thorough the validation is, but they aren't the foundation of the claim.

The math looks solid on the description given, with no circularity or fitting issues. This is the kind of result that could be useful in cascaded inference or fraud detection pipelines.

A reader working on system design for ranking or ML serving would find the ordering rule practical if their filters fit the model. It is worth sending to peer review because the core argument is direct and the application area is active.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that, under an explicit independence model on per-filter costs c_i and rejection probabilities p_i, the ordering that minimizes expected total cost for a sequential pipeline is the one obtained by sorting filters in increasing order of the ratio c_i / p_i. The central argument is a pairwise exchange lemma showing E[AB] < E[BA] precisely when c_A/p_A < c_B/p_B; transitivity then yields the global optimum for any number of filters. Monte Carlo simulations are presented as corroborative evidence that this ordering dominates common heuristics both in expectation and in the full outcome distribution.

Significance. If the result holds, the paper supplies a simple, parameter-free optimality criterion for a ubiquitous design pattern in ranking, cascaded inference, and detection pipelines. The proof is direct (pairwise swap plus transitivity) and does not rely on fitted parameters or simulation; the independence assumption is stated up front, so the claim is conditional but internally consistent. This replaces heuristic ordering with a provably optimal rule under the stated model.

minor comments (2)

[Abstract and §4] The abstract states that simulations show strict dominance 'across all runs,' but does not report the number of Monte Carlo trials, the parameter ranges sampled for c_i and p_i, or the exact heuristics compared; adding these details in §4 would strengthen reproducibility without affecting the proof.
[§3] The reachability probability under independence is written as a product; an explicit one-line expansion of the expected-cost expression immediately after the definition of the ordering would make the dependence on the product form transparent to readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of the direct pairwise-exchange proof and the practical relevance of the c_i/p_i ordering rule under the stated independence model. We appreciate the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a direct mathematical proof of optimality for sequential filter ordering under an explicitly stated independence model on filter costs and rejection probabilities. The derivation relies on a pairwise exchange argument (E_AB < E_BA iff c_A/p_A < c_B/p_B) that is transitive under the product-form reachability probabilities induced by independence, yielding the global optimum without any reduction to fitted parameters, self-definitional constructs, or load-bearing self-citations. Simulations serve only as corroborative evidence and are not the basis for the optimality claim. The result is self-contained against the model's assumptions and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Ledger extracted from abstract only; full paper may contain additional modeling choices.

axioms (1)

domain assumption Independence of cost and selectivity models across filters
The optimality proof is stated to hold under this independence model.

pith-pipeline@v0.9.1-grok · 5629 in / 978 out tokens · 28267 ms · 2026-06-29T09:07:23.649338+00:00 · methodology

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read the original abstract

Sequential filtering pipelines are a common design pattern in large-scale systems, where a large population of items is progressively reduced by a sequence of stages that each incur cost. Despite their prevalence in ranking systems, cascaded machine learning inference, and fraud detection, filter ordering is often determined by heuristics without formal guarantees. We formalize sequential filtering under an expected-cost objective and prove that, under an independence model, ordering filters by increasing ratio of cost to rejection probability minimizes expected total cost. Extensive Monte Carlo simulations show that the optimal ordering strictly dominates common heuristics across all runs, both in expectation and across the full distribution of outcomes.

Figures

Figures reproduced from arXiv: 2606.07589 by Abhishek Mandal, Hrishikesh Paranjape, Xian Sun.

discussion (0)

Forward citations

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cs.LG 2026-07 conditional novelty 6.0

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Reference graph

Works this paper leans on

4 extracted references · cited by 1 Pith paper

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1956

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2001

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Branchynet: Fast inference via early exiting from deep neural networks,

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2016

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Wald,Sequential Analysis

A. Wald,Sequential Analysis. John Wiley and Sons, 1947

1947