pith. machine review for the scientific record. sign in

arxiv: 2605.07805 · v1 · submitted 2026-05-08 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

Flexible Routing via Uncertainty Decomposition

Charlotte Peale , Siddartha Devic , Parikshit Gopalan , Udi Wieder , Aravind Gollakota
Authors on Pith no claims yet

Pith reviewed 2026-05-11 03:10 UTC · model grok-4.3

classification 💻 cs.LG
keywords model routinguncertainty decompositionabstentionregret boundsclassificationoracle routinghigher-order predictorscost adaptation
0
0 comments X

The pith

Decomposing uncertainty into reducible and irreducible components lets one router choose cheap models, oracles, or abstention and adapt to new costs without retraining.

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

The paper introduces a router that decomposes total uncertainty into reducible and irreducible parts using higher-order predictors to decide routing and abstention in classification tasks. It uses low uncertainty to stay with the cheap model, high reducible uncertainty to call the oracle, and high irreducible uncertainty to abstain. The same trained router adapts to different loss functions and cost ratios simply by changing hyperparameters. It carries theoretical guarantees that bound its regret relative to the best possible task-specific router. Experiments on synthetic and real data show gains in regimes where the two uncertainty types are not strongly correlated.

Core claim

The paper claims that decomposing total uncertainty into its reducible and irreducible components via higher-order predictors yields a unified routing and abstention policy: predict with the weak model on low uncertainty, route to the oracle on high reducible uncertainty, and abstain on high irreducible uncertainty. This policy applies to any classification setting with multiple independent annotations per input, adapts to arbitrary loss and cost parameters through hyperparameter changes alone, and satisfies regret bounds against optimal per-task routers.

What carries the argument

Decomposition of total uncertainty into reducible uncertainty (resolvable by a stronger model) and irreducible uncertainty (inherent ambiguity) performed with higher-order predictors; this split drives the three-way decision and enables the regret analysis.

Load-bearing premise

Multiple independent annotations per input must exist to estimate the uncertainty components, and the reducible and irreducible uncertainties must not be too strongly correlated.

What would settle it

Measure actual regret against the optimal per-task router on a synthetic dataset constructed so that the decomposition is exact and the two uncertainty types are uncorrelated; if regret exceeds the stated bound or if the router fails to improve over a fixed policy, the central claim is false.

Figures

Figures reproduced from arXiv: 2605.07805 by Aravind Gollakota, Charlotte Peale, Parikshit Gopalan, Siddartha Devic, Udi Wieder.

Figure 1
Figure 1. Figure 1: Routing performance of our method on synthetic data and a first-order-calibrated weak model compared to the point-wise optimal routing policy and a router based on total uncertainty. The data generating process is described in Section E.2 and the ground-truth data is visualized in Figure 5a. Irreducible Loss L(f ∗ , f ∗ ) α β (β − α, α) Reducible Loss L(f ∗, f) − L(f ∗, f ∗ ) PREDICT ROUTE ABSTAIN [PITH_F… view at source ↗
Figure 3
Figure 3. Figure 3: Routing performances for different target loss functions for the SNLI dataset. The loss [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Cost-penalized performance of our method on the three-way decision task (Predict vs. [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Examples of synthetic binary data settings used in experiments, as described in Section [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The above routing curve depicts the performance of our routing framework on the synthetic [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The above routing curve compares the performance of two bucketing approaches on [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Routing curves for synthetic data as described in Section [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Routing curves for the CIFAR-10H dataset. The right (b) employs an out-of-the-box [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Routing curves for the SNLI dataset. The right (b) employs an out-of-the-box weak [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Routing curves for the ChaosNLI dataset. The right (b) employs an out-of-the-box [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Routing performances for different target loss functions for synthetic data as described [PITH_FULL_IMAGE:figures/full_fig_p032_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Routing performances for different target loss functions for the CIFAR-10H dataset. [PITH_FULL_IMAGE:figures/full_fig_p033_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Routing performances for different target loss functions for the SNLI dataset. The loss [PITH_FULL_IMAGE:figures/full_fig_p034_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Routing performances for different target loss functions for the ChaosNLI dataset. The loss [PITH_FULL_IMAGE:figures/full_fig_p035_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Depiction of our method’s cost-penalized performance on the three-way decision task [PITH_FULL_IMAGE:figures/full_fig_p036_16.png] view at source ↗
read the original abstract

A key strategy for balancing performance and cost in modern machine learning systems is to dynamically route queries to either a low-cost model or a more expensive oracle (such as a large pretrained model or human expert), an approach known as model routing. In this work we present a new uncertainty-aware router that (1) avoids unnecessary oracle calls on inherently ambiguous queries, and (2) adapts dynamically to different loss functions and cost parameters through simple hyperparameter changes, without retraining. Our method, applicable to any classification setting where multiple independent annotations per input are available, is based on decomposing total uncertainty into irreducible and reducible components using higher-order predictors [Ahdritz et al., 2025]. This enables a unified approach to both routing and abstention: predict with the weak model when uncertainty is low, route to the oracle when reducible uncertainty is high, and abstain when irreducible uncertainty is high. Our router comes with strong theoretical guarantees bounding regret relative to optimal task-specific routers. We conduct experiments on both synthetic and real-world datasets that demonstrate the benefits of our approach in suitable regimes -- in particular, whenever reducible and irreducible uncertainty are not too correlated.

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 / 3 minor

Summary. The paper introduces an uncertainty-aware router for model routing in classification tasks with multiple independent annotations per input. It decomposes total uncertainty into irreducible and reducible components via higher-order predictors (building on Ahdritz et al. 2025), enabling dynamic decisions: use a weak model for low uncertainty, route to an oracle for high reducible uncertainty, and abstain for high irreducible uncertainty. The router adapts to different losses and costs via hyperparameters without retraining and includes theoretical regret bounds relative to optimal task-specific routers. Experiments on synthetic and real data demonstrate benefits when the uncertainty components are not highly correlated.

Significance. If the regret bounds and decomposition hold, the approach offers a flexible, unified framework for routing and abstention that avoids retraining and explicitly handles ambiguity, which could improve cost-performance tradeoffs in deployed ML systems. The explicit conditioning on low correlation between uncertainty components and the dependence on multiple annotations are clearly scoped, strengthening the claims where applicable.

minor comments (3)
  1. The abstract and introduction reference 'strong theoretical guarantees' for regret bounds; ensure the main text explicitly states the assumptions under which these bounds hold (e.g., independence of annotations) and includes a brief proof sketch or reference to the appendix for the key steps.
  2. In the experimental section, clarify how the correlation between reducible and irreducible uncertainty is measured and controlled in the synthetic data generation, as this directly impacts the reported benefits.
  3. Notation for the decomposed uncertainties (e.g., U_r and U_i) should be defined at first use in §3 and used consistently in the regret analysis to avoid ambiguity with total uncertainty U.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on uncertainty-aware routing and for recommending minor revision. We appreciate the recognition of the method's flexibility, theoretical guarantees, and applicability to settings with multiple annotations. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central method decomposes uncertainty via higher-order predictors explicitly cited to Ahdritz et al. 2025 (distinct authors) and applies the decomposition to a routing task with hyperparameter adaptation presented as external to training. Theoretical regret bounds are stated relative to optimal task-specific routers under the decomposition, with no equations or steps in the abstract or description reducing claims to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. Assumptions about low correlation between uncertainty components are explicitly conditioned rather than smuggled in, and the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on the prior higher-order predictor framework for uncertainty decomposition and the domain requirement of multiple annotations; no new free parameters are explicitly introduced beyond tunable hyperparameters for routing thresholds.

free parameters (1)
  • routing hyperparameters
    Simple changes to these allow adaptation to different loss functions and cost parameters without retraining.
axioms (1)
  • domain assumption Multiple independent annotations per input are available
    Required to decompose total uncertainty into irreducible and reducible components using higher-order predictors.

pith-pipeline@v0.9.0 · 5510 in / 1239 out tokens · 40992 ms · 2026-05-11T03:10:26.857358+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 1 internal anchor

  1. [1]

    Integrating expert judgment and algorithmic decision making: An indistinguishability framework

    (document), 1, 4, 4.2, 4.1, 5, 5.1, 7 Rohan Alur, Loren Laine, Darrick K Li, Dennis Shung, Manish Raghavan, and Devavrat Shah. Integrating expert judgment and algorithmic decision making: An indistinguishability framework. arXiv preprint arXiv:2410.08783, 2024a. 2 Rohan Alur, Manish Raghavan, and Devavrat Shah. Human expertise in algorithmic prediction. A...

    work page arXiv

  2. [2]

    A large annotated corpus for learning natural language inference

    2 Samuel Bowman, Gabor Angeli, Christopher Potts, and Christopher D Manning. A large annotated corpus for learning natural language inference. InProceedings of the 2015 conference on empirical methods in natural language processing, pages 632–642,

    work page 2015

  3. [3]

    A unified approach to routing and cascading for llms.arXiv preprint arXiv:2410.10347, 2024

    2 Jasper Dekoninck, Maximilian Baader, and Martin Vechev. A unified approach to routing and cascading for llms.arXiv preprint arXiv:2410.10347,

    work page arXiv

  4. [5]

    BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding

    URL http://arxiv.org/abs/1810.04805. E.1 Dujian Ding, Ankur Mallick, Chi Wang, Robert Sim, Subhabrata Mukherjee, Victor Rühle, Laks VS Lakshmanan, and Ahmed Hassan Awadallah. Hybrid llm: Cost-efficient and quality-aware query routing. InThe Twelfth International Conference on Learning Representations,

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [6]

    Aurélien Garivier and Eric Moulines

    2 Dujian Ding, Ankur Mallick, Shaokun Zhang, Chi Wang, Daniel Madrigal, Mirian Del Car- men Hipolito Garcia, Menglin Xia, Laks VS Lakshmanan, Qingyun Wu, and Victor Rühle. Best-route: Adaptive llm routing with test-time optimal compute.arXiv preprint arXiv:2506.22716,

    work page arXiv

  6. [7]

    Human-algorithm collab- oration: Achieving complementarity and avoiding unfairness

    2 Kate Donahue, Alexandra Chouldechova, and Krishnaram Kenthapadi. Human-algorithm collab- oration: Achieving complementarity and avoiding unfairness. InProceedings of the 2022 ACM Conference on Fairness, Accountability, and Transparency, pages 1639–1656,

    work page 2022

  7. [8]

    When does a predictor know its own loss? In6th Symposium on Foundations of Responsible Computing (FORC 2025), pages 22–1

    1, 2 Aravind Gollakota, Parikshit Gopalan, Aayush Karan, Charlotte Peale, and Udi Wieder. When does a predictor know its own loss? In6th Symposium on Foundations of Responsible Computing (FORC 2025), pages 22–1. Schloss Dagstuhl–Leibniz-Zentrum für Informatik,

    work page 2025

  8. [9]

    P ., Reingold, O., and Wieder, U

    2, 4.2 Parikshit Gopalan, Lunjia Hu, Michael P Kim, Omer Reingold, and Udi Wieder. Loss minimization through the lens of outcome indistinguishability.arXiv preprint arXiv:2210.08649, 2022a. 1, 2, 4.2 Parikshit Gopalan, Adam Tauman Kalai, Omer Reingold, Vatsal Sharan, and Udi Wieder. Om- nipredictors. In13th Innovations in Theoretical Computer Science Conf...

    work page arXiv 2022

  9. [10]

    Levente Kocsis and Csaba Szepesvári

    2, 6.1.2 Wittawat Jitkrittum, Harikrishna Narasimhan, Ankit Singh Rawat, Jeevesh Juneja, Congchao Wang, Zifeng Wang, Alec Go, Chen-Yu Lee, Pradeep Shenoy, Rina Panigrahy, et al. Universal model routing for efficient llm inference.arXiv preprint arXiv:2502.08773,

    work page arXiv

  10. [11]

    Johnson, Daniel Tarlow, David Duvenaud, and Chris J

    2 Daniel D. Johnson, Daniel Tarlow, David Duvenaud, and Chris J. Maddison. Experts don’t cheat: Learning what you don’t know by predicting pairs. In Ruslan Salakhutdinov, Zico Kolter, Katherine A. Heller, Adrian Weller, Nuria Oliver, Jonathan Scarlett, and Felix Berkenkamp, editors, Forty-first International Conference on Machine Learning, ICML 2024, Vien...

    work page 2024

  11. [12]

    Learning to reject with a fixed predictor: Application to decontextualization.arXiv preprint arXiv:2301.09044,

    2 Christopher Mohri, Daniel Andor, Eunsol Choi, and Michael Collins. Learning to reject with a fixed predictor: Application to decontextualization.arXiv preprint arXiv:2301.09044,

    work page arXiv

  12. [13]

    Adver- sarial NLI: A new benchmark for natural language understanding

    6.2 Yixin Nie, Adina Williams, Emily Dinan, Mohit Bansal, Jason Weston, and Douwe Kiela. Adver- sarial NLI: A new benchmark for natural language understanding. InProceedings of the 58th Annual Meeting of the Association for Computational Linguistics. Association for Computational Linguistics, 2020a. E.1 Yixin Nie, Xiang Zhou, and Mohit Bansal. What can we...

    work page 2020

  13. [14]

    Tight pair query lower bounds for matching and earth mover’s distance

    doi: 10.1109/FOCS63196.2025.00084. 2 Isaac Ong, Amjad Almahairi, Vincent Wu, Wei-Lin Chiang, Tianhao Wu, Joseph E Gonzalez, M Waleed Kadous, and Ion Stoica. Routellm: Learning to route llms from preference data. In The Thirteenth International Conference on Learning Representations,

    work page doi:10.1109/focs63196.2025.00084 2025

  14. [15]

    The algorithmic automation problem: Prediction, triage, and human effort

    E.1 Maithra Raghu, Katy Blumer, Greg Corrado, Jon Kleinberg, Ziad Obermeyer, and Sendhil Mul- lainathan. The algorithmic automation problem: Prediction, triage, and human effort.arXiv preprint arXiv:1903.12220,

    work page arXiv 1903

  15. [16]

    Carrot: A cost aware rate optimal router, 2025

    2 15 Seamus Somerstep, Felipe Maia Polo, Allysson Flavio Melo de Oliveira, Prattyush Mangal, Mírian Silva, Onkar Bhardwaj, Mikhail Yurochkin, and Subha Maity. Carrot: A cost aware rate optimal router.arXiv preprint arXiv:2502.03261,

    work page arXiv

  16. [17]

    Doing more with less – implementing routing strategies in large language model-based systems: An extended survey.arXiv preprint arXiv:2502.00409, 2025

    1, 2 Clovis Varangot-Reille, Christophe Bouvard, Antoine Gourru, Mathieu Ciancone, Marion Schaeffer, and François Jacquenet. Doing more with less–implementing routing strategies in large language model-based systems: An extended survey.arXiv preprint arXiv:2502.00409,

    work page arXiv

  17. [18]

    A broad-coverage challenge corpus for sentence understanding through inference

    2 Adina Williams, Nikita Nangia, and Samuel Bowman. A broad-coverage challenge corpus for sentence understanding through inference. InProceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers), pages 1112–1122. Association for Computational Linguistics,

    work page 2018

  18. [19]

    contradiction

    E Additional Experiment Discussion E.1 Dataset Details We detail each dataset and associated weak predictor architecture below. We do not perform any hparam optimization since we are not trying to obtain the best performing model in any setting, we mostly use defaults from other implementations. • Synthetic Data: A set of synthetic data functions with bas...

    work page 2024