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arxiv: 2605.19557 · v1 · pith:BA2UWQGUnew · submitted 2026-05-19 · 📊 stat.ML · cs.LG

Density-Ratio Losses for Post-Hoc Learning to Defer

Alexander Soen , Ragnar Thobaben , Joakim Jald\'en , Richard Nock This is my paper

Pith reviewed 2026-05-20 02:38 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords learning to deferdensity ratio estimationpost-hoc methodsideal distributionsclass probability estimationChow's ruleexpert-tilted posterior
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The pith

Post-hoc learning to defer reduces to estimating the density ratio between a model's and an expert's ideal distributions.

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

The paper establishes that deferral after a model is already trained can be handled by learning a scorer that estimates the density ratio between the ideal distribution for the model and the ideal distribution for the expert. Ideal distributions are reweighted versions of the data under which the model or expert attains low loss through divergence regularization. By reducing density-ratio estimation to class-probability estimation, the authors derive specific loss functions whose outputs can be thresholded to decide when to defer. A sympathetic reader would care because this approach permits changing the deferral rate with a simple threshold adjustment after training, recovers classical optimal rules such as Chow's for KL-based cases, and links deferral to an expert-tilted Bayes posterior that accounts for the expert's performance.

Core claim

We study post-hoc Learning to Defer (L2D) through the lens of ideal distributions: divergence-regularized reweightings of the data distribution under which a model attains low loss. We define deferral via the density-ratio between a model's and an expert's ideals. Using the reduction from density-ratio estimation to class-probability estimation, we derive the DR CPE losses for post-hoc L2D scorers. Deferral decisions are then made by thresholding the scorer, allowing deferral rates to be adjusted without retraining. For KL-based ideal distributions, our deferral rules recovers Chow's rule under the original distribution and a connection to an expert-tilted Bayes posterior -- which incorporat

What carries the argument

The density ratio between a model's ideal distribution and an expert's ideal distribution, obtained via divergence-regularized reweightings and reduced to class-probability estimation losses for training a post-hoc deferral scorer.

Load-bearing premise

The reduction from density-ratio estimation to class-probability estimation holds for the chosen ideal distributions and the resulting scorer can be thresholded to produce valid deferral decisions without additional calibration or assumptions on the joint distribution of model and expert errors.

What would settle it

An experiment in which the DR CPE scorer's thresholded outputs fail to recover Chow's rule under the original distribution or fail to match the expert-tilted Bayes posterior for KL-based ideal distributions would falsify the central derivation.

Figures

Figures reproduced from arXiv: 2605.19557 by Alexander Soen, Joakim Jald\'en, Ragnar Thobaben, Richard Nock.

Figure 1
Figure 1. Figure 1: Post-hoc L2D to learnable density-ratio (DR) deferral. Starting from the post-hoc L2D objective in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Accuracy-deferral trade-off. Titles detail the dataset and ResNet utilized, [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Accuracy-deferral trade-off plots over different [PITH_FULL_IMAGE:figures/full_fig_p035_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Accuracy-deferral trade-off plots over both accuracy and top- [PITH_FULL_IMAGE:figures/full_fig_p036_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Accuracy-deferral trade-off plots over different DR partial losses (with fixed [PITH_FULL_IMAGE:figures/full_fig_p038_5.png] view at source ↗
read the original abstract

We study post-hoc Learning to Defer (L2D) through the lens of ideal distributions: divergence-regularized reweightings of the data distribution under which a model attains low loss. We define deferral via the density-ratio between a model's and an expert's ideals. Using the reduction from density-ratio estimation to class-probability estimation, we derive the DR CPE losses for post-hoc L2D scorers. Deferral decisions are then made by thresholding the scorer, allowing deferral rates to be adjusted without retraining. For KL-based ideal distributions, our deferral rules recovers Chow's rule under the original distribution and a connection to an expert-tilted Bayes posterior -- which incorporates the expert's performance -- depending on if the ideal distributions are joint or marginal distributions. Experimentally, our approach is competitive compared to common baselines and more robust across dataset settings. More broadly, our results cast post-hoc L2D as density-ratio learning between ideal distributions, bridging Chow-style rules, expert comparison, and elucidating connections to related learning settings including anomaly detection.

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

2 major / 2 minor

Summary. The paper proposes framing post-hoc learning to defer (L2D) as density-ratio estimation between 'ideal' distributions, defined as divergence-regularized reweightings of the original data measure under which a model or expert attains low loss. It derives DR-CPE losses by reducing density-ratio estimation to class-probability estimation, obtains deferral scorers that can be thresholded to control deferral rate without retraining, and shows that the KL-based case recovers Chow's rule (under joint ideals) or connects to an expert-tilted Bayes posterior (under marginal ideals). Experiments are reported as competitive with baselines and more robust across settings.

Significance. If the central reduction and thresholding argument hold without additional joint-error assumptions, the work supplies a clean density-ratio perspective that unifies Chow-style rules with post-hoc L2D and links to anomaly detection. The post-hoc, thresholdable nature is practically attractive. The manuscript does not yet provide quantitative tables, error bars, or dataset details in the abstract, so the strength of the empirical claim remains to be verified from the full experiments.

major comments (2)
  1. [§3] §3 (or the derivation following Eq. (3)): the claim that thresholding the DR-CPE scorer produces valid deferral sets appears to rest on the assumption that the class probability is monotonic in the conditional advantage of the expert over the model. The skeptic note correctly flags that marginal ideal distributions alone do not encode instance-level error correlation; the manuscript must explicitly state whether the joint versus marginal choice of ideals is sufficient to guarantee monotonicity, or whether an extra assumption on the joint distribution of model/expert errors is required.
  2. [§4] The recovery of Chow's rule for KL-based joint ideals is stated in the abstract and presumably derived in §4. The derivation must be checked for circularity: if the reweighting factors that define the ideals are realized by a loss on the original data, the resulting ratio estimator must not implicitly presuppose the very deferral decision it is meant to produce. A concrete walk-through of the steps from the ideal densities to the thresholded rule would clarify this.
minor comments (2)
  1. [Abstract] The abstract asserts that experiments are 'competitive and more robust' yet supplies no quantitative results, error bars, or dataset names. These details belong in the abstract or a prominent table in §5.
  2. [Abstract] Notation for the ideal distributions (joint vs. marginal) should be introduced once and used consistently; the current abstract switches between them without a clear forward pointer to the section that defines both.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the recommendation of major revision. We address each major comment point by point below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [§3] §3 (or the derivation following Eq. (3)): the claim that thresholding the DR-CPE scorer produces valid deferral sets appears to rest on the assumption that the class probability is monotonic in the conditional advantage of the expert over the model. The skeptic note correctly flags that marginal ideal distributions alone do not encode instance-level error correlation; the manuscript must explicitly state whether the joint versus marginal choice of ideals is sufficient to guarantee monotonicity, or whether an extra assumption on the joint distribution of model/expert errors is required.

    Authors: We agree that explicit clarification is needed on this point. When joint ideal distributions are used, the density ratio is taken with respect to the joint measure over instances, which directly incorporates the per-instance losses of both the model and the expert. Consequently the class probability obtained from the DR-CPE reduction is monotonic in the conditional advantage of the expert, so that thresholding yields valid deferral sets without further assumptions. When marginal ideal distributions are used, instance-level error correlations are not encoded and monotonicity would indeed require an additional assumption on the joint distribution of model/expert errors. We will revise §3 to state this distinction clearly and to specify the conditions under which the thresholding argument holds. revision: yes

  2. Referee: [§4] The recovery of Chow's rule for KL-based joint ideals is stated in the abstract and presumably derived in §4. The derivation must be checked for circularity: if the reweighting factors that define the ideals are realized by a loss on the original data, the resulting ratio estimator must not implicitly presuppose the very deferral decision it is meant to produce. A concrete walk-through of the steps from the ideal densities to the thresholded rule would clarify this.

    Authors: We thank the referee for raising the possibility of circularity. The ideal distributions are purely theoretical objects: divergence-regularized reweightings of the original measure under which the model or expert attains low loss. The density-ratio estimator is obtained by applying the DR-CPE reduction directly to samples drawn from the original data distribution; the estimator is trained using only the observed per-instance losses of the model and expert and does not involve any deferral decisions or thresholds. The subsequent thresholding step is applied after estimation and does not feed back into the training of the scorer. We will add a concise, numbered walk-through of the steps from the definition of the ideal densities through the DR-CPE reduction to the final thresholded rule in the revised §4. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external definitions and standard reductions

full rationale

The paper starts from the external notion of ideal distributions (divergence-regularized reweightings of the data measure) and defines deferral explicitly as the density ratio between model and expert ideals. It then invokes the known, independently established reduction from density-ratio estimation to class-probability estimation to obtain the DR-CPE losses. Thresholding the resulting scorer is presented as a direct consequence of this construction. For the KL case the paper shows recovery of Chow's rule under the original distribution, which is an external benchmark rather than a self-derived quantity. No equation or step equates a fitted parameter to a prediction by construction, and no load-bearing premise rests solely on self-citation. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of well-defined ideal distributions for both model and expert, the validity of the density-ratio-to-CPE reduction, and the assumption that thresholding the resulting scorer yields useful deferral without further calibration. No explicit free parameters or invented entities are named in the abstract.

axioms (2)
  • domain assumption Ideal distributions exist and can be used to define a meaningful density ratio for deferral decisions.
    Invoked when deferral is defined via the density-ratio between model's and expert's ideals.
  • standard math The standard reduction from density-ratio estimation to class-probability estimation applies directly to the chosen ideal distributions.
    Used to derive the DR CPE losses.

pith-pipeline@v0.9.0 · 5724 in / 1680 out tokens · 37157 ms · 2026-05-20T02:38:52.614854+00:00 · methodology

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