Pith. sign in

REVIEW 4 major objections 5 minor 13 references

The paper argues that inverse problems—solving for the training and inference ingredients needed to hit a target performance metric—can efficiently uncover LLM scaling laws, and frames data selection, inference optimization, and machine unl

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Proposes using inverse problems to discover LLM scaling laws, but provides no empirical evidence or new results.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection A clean position-paper framing of LLM design as inverse problems, but the central claim is untested and the ill-posedness is unresolved; worth discussing, not citing as a result. the 4 major comments →

arxiv 2509.07909 v1 pith:OXENMCR4 submitted 2025-09-09 cs.LG cs.AIcs.CL

Uncovering Scaling Laws for Large Language Models via Inverse Problems

Arun Verma , Zhaoxuan Wu , Zijian Zhou , Xiaoqiang Lin , Zhiliang Chen , Rachael Hwee Ling Sim , Rui Qiao , Jingtan Wang
show 10 more authors
Nhung Bui Xinyuan Niu Wenyang Hu Gregory Kang Ruey Lau Zi-Yu Khoo Zitong Zhao Xinyi Xu Apivich Hemachandra See-Kiong Ng Bryan Kian Hsiang Low
This is my paper
classification cs.LG cs.AIcs.CL
keywords inverse problemsscaling lawslarge language modelsdata selectioninference optimizationmachine unlearningcost-effectivenesstraining ingredients
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 reading

This position paper argues that the standard forward view of LLM development—choose ingredients, train, evaluate—can be inverted: instead of asking what performance a given model will reach, ask what minimal ingredients are needed to reach a target performance. It proposes that inverse problems can uncover scaling laws that connect data, architecture, training procedures, and inference schemes to performance, with significantly better cost-effectiveness than brute-force trial and error. The paper formalizes two forward processes, then frames data selection, inference optimization, and machine unlearning as inverse problems whose solutions would reveal such scaling laws. The payoff, if right, is a principled way to decide data mixtures, prompts, test-time compute, and unlearning strategies without repeatedly training expensive models.

Core claim

The paper's central claim is that inverse problems—identifying unknown inputs of a process from observed outputs—can efficiently uncover scaling laws that guide building LLMs to a desired performance at lower cost. It formalizes LLM creation and evaluation as two forward processes, F(T)→LLM and T(F(T),I)→C, and advocates inverting them: rather than asking which C a given choice of training data, architecture, procedure, and inference scheme will produce, ask which T and I are needed to achieve a given C. It then argues that data selection, inference optimization, and machine unlearning are best understood as such inverse problems, and that pursuing them will reveal scaling laws for data quan

What carries the argument

The load-bearing object is the pair of forward equations F(T)→LLM and T(F(T),I)→C, together with their inverses. F(T) maps training ingredients T—data, architecture, training procedure—to a trained LLM; T(·,·) maps the model plus inference ingredients I—prompts, exemplars, inference-time compute, model configuration—to a performance metric C. The paper's move is to treat these as forward problems and ask what ingredient sets solve the inverse problem for a target C. This pair of equations is what unifies data selection, inference optimization, and unlearning into a single research program.

Load-bearing premise

The argument depends on the inverse problems in equations (1a) and (1b) being solvable well enough to yield useful scaling laws; the paper itself concedes that many forward LLM processes are many-to-one, so their inverses may be ambiguous or ill-posed.

What would settle it

An experiment on a small task where every ingredient combination can be enumerated: solve the inverse problem to predict the minimum dataset, architecture, and prompt needed to reach a target score, then train all combinations near the predicted frontier. If the target score is reached by many structurally different ingredient sets with no consistent scaling relation, or if the predicted minimal set fails to hit the target, the framework's core premise fails.

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

If this is right

  • If the inverse framing works, a team could specify a target benchmark score and get back the minimal data mixture, model size, and training procedure, rather than training several models to compare.
  • Data selection can be optimized directly for non-differentiable metrics like BLEU or LLM-as-a-judge, using policy-gradient surrogates, so the selected data reflects the actual evaluation criterion.
  • Inference-time scaling laws would extend from single schemes like chain-of-thought to optimal mixes of retrieval, repeated sampling, and ensembles under a fixed compute budget.
  • Machine unlearning can gain retraining-free verification via watermarks and scaling laws that predict how much erased data a model can absorb before retained-data performance drops, guiding whether to unlearn or retrain.
  • Joint optimization across training and inference ingredients could replace separate hand-tuned choices with a single cost-performance frontier.

Where Pith is reading between the lines

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

  • A consequence the paper leaves implicit is that even if inversion is ill-posed, the formalism supplies a useful classification: problems whose inverse is unique enough to yield scaling laws versus problems where only bounds or distributions are possible.
  • The same inverse framing could be applied to data and model valuation—measuring an ingredient's worth by how much the target C moves when it is removed—making attribution a natural subproblem of the inversion.
  • A direct testable extension would be to run the inversion on a small benchmark where the true minimal ingredient set is known by brute force; if the inverted scaling law recovers that set, it can be trusted on larger systems.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. This position paper argues that building LLMs can be guided by 'inverse problems': given a desired performance metric C, one should invert the forward processes F(T)→LLM and T(F(T),I)→C to recover the training/inference ingredients T and I, and thereby uncover scaling laws more efficiently than brute-force forward sweeps. The paper formalizes these two mappings in Eqs. (1a)–(1b), then frames three application areas as inverse problems: data selection (§2), inference optimization (§3), and machine unlearning (§4). For each, it sketches existing or proposed techniques (REINFORCE-based data selection, NeuralUCB prompt optimization, watermarking for unlearning verification) and suggests how scaling laws could be uncovered. The paper concludes with limitations and an ethics statement.

Significance. If the proposed program succeeded, it could offer a principled alternative to the current practice of fitting scaling laws from a small number of expensive forward training runs, and it could unify disparate problems (data selection, prompting, unlearning) under one conceptual umbrella. The historical analogy to Kepler and Newton is evocative, and the paper does identify genuine cost pressures in LLM development. However, the current manuscript is a research agenda rather than a demonstrated result: there is no experiment, simulation, or formal analysis showing that the inverse problems in Eqs. (1a)–(1b) are tractable, that the proposed techniques would recover scaling laws, or that the approach is more cost-effective than standard regression on forward experiments. The paper's own limitations section admits the many-to-one/ill-posedness threat, which is exactly the load-bearing issue that would need to be resolved. The value of the paper lies in its framing and in the concrete open questions it poses, but the central claim is not yet supported.

major comments (4)
  1. [Abstract and Eqs. (1a)–(1b)] The central claim that inverse problems 'can efficiently uncover scaling laws' is asserted rather than demonstrated. The paper contains no experiment, simulation, or proof that inverting either forward map recovers a known scaling law, nor any comparison to direct parametric fitting of scaling laws from forward experiments. As a position paper, this could be acceptable if the claim were explicitly framed as a conjecture or an open challenge; but the abstract states it as a capability. The authors should either provide a proof-of-concept (e.g., recovering a known power-law from synthetic forward evaluations) or temper the claim to 'we conjecture' / 'we advocate exploration.'
  2. [Eq. (1b) and Limitations] The paper never defines a well-posed inverse problem. C is a scalar (or low-dimensional) performance metric, while T and I are high-dimensional; the forward map T(F(T),I)→C is many-to-one, as the Limitations section explicitly acknowledges. The manuscript gives no identifiability conditions, no regularization strategy, no noise model, and no specific inversion algorithm. In classical inverse problems, ill-posedness is addressed precisely by regularization and by a precise statement of the forward operator; none of that is supplied here. This is not a minor omission: the tractability of the inverse problem is the load-bearing assumption of the entire paper.
  3. [§2, §3.1, §4.1] The proposed concrete instantiations are existing optimization/estimation techniques relabeled as inverse problems: REINFORCE for non-differentiable data selection (§2.2), NeuralUCB for prompt optimization (§3.1), and watermarking for unlearning verification (§4.1). The paper does not show how these methods constitute 'solving an inverse problem' in a way distinct from their standard usage, nor how they would reveal scaling laws. Without such a distinction, the central claim reduces to the observation that optimization can be viewed as inversion, which does not establish the proposed efficiency gain.
  4. [§2 and §3.3] The claimed cost-effectiveness is not argued. Section 2 states that solving data selection as an inverse problem 'can significantly reduce computational costs,' but solving an inverse problem typically requires multiple evaluations of the forward process, each of which is a full LLM training run (Eq. (1a)) or a large-scale inference pass (Eq. (1b)). The paper does not quantify how the inverse approach avoids these costs or how it differs from the standard practice of fitting scaling laws from a few forward experiments (Kaplan et al.; Hoffmann et al.). This is a load-bearing assertion that requires at least a rough cost model.
minor comments (5)
  1. [Introduction, Eq. (1b)] The symbol T is used both for training ingredients and for the evaluation mapping T(LLM,I)→C, which is confusing in Eq. (1b) and in the surrounding text. Consider a different symbol, e.g., E or G, for the evaluation operator.
  2. [Throughout] There are several typos and OCR artifacts: 'inputingredients' in the Introduction, 'Antropic' in the references, and 'have lend themselves' in the Limitations section. These should be corrected.
  3. [§2.1 heading] The heading says 'Multi-Model LLMs' but the text and acronym MLLM refer to 'Multi-Modal LLMs.' Please fix the heading.
  4. [Figure 2] The caption describes 'low-quality samples' but the two examples are not clearly annotated as poor image-text pairs; adding a brief explanation of why each is low-quality would improve readability.
  5. [§3.2] The term 'model configuration' is used for both selecting among existing LLMs and choosing the number of active MoE routes; these are distinct problems and should be separated more clearly.

Circularity Check

0 steps flagged

No significant circularity: position paper, not a derivation.

full rationale

This is a position paper that proposes framing LLM data selection, inference optimization, and unlearning as inverse problems (Eqs. 1a–1b). It does not derive or predict any specific scaling law; its central claim is an advocacy that such a framing 'can efficiently uncover scaling laws.' No parameter is fitted and later called a prediction, no ansatz is smuggled in via self-citation, and no uniqueness theorem is imported from the authors' prior work. The many self-citations (e.g., Verma et al. 2025, Lin et al. 2025, Wu et al. 2025, Lau et al. 2024) are used only as pointers to existing optimization or watermarking methods that could be adapted, not as load-bearing evidence for the central claim. The Limitations section is explicit that many forward problems are many-to-one and the resulting inverse problems may be ambiguous or ill-posed, which is a substantive weakness of the research program but not a circularity. Because the paper makes no concrete derivation whose output is equivalent to its input, no circular step can be exhibited. Score 0 is appropriate.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper introduces no free parameters or invented entities. It rests on the domain assumptions that discoverable scaling laws exist and that the inverse problem, despite ill-posedness, can be made tractable. The functional notation is a standard mathematical abstraction.

axioms (3)
  • domain assumption There exist underlying scaling laws governing LLM performance that can be expressed as functions of training and inference ingredients.
    The entire framing assumes discoverable scaling laws exist, as stated in the introduction and conclusion.
  • domain assumption The inverse problems formed from Eq. (1a) and (1b) are tractable despite the acknowledged ill-posedness.
    The paper admits many-to-one mappings cause ill-posedness, yet still advocates the approach without a demonstration of tractability. See Limitations section.
  • standard math The forward process F and evaluation T can be represented as functions amenable to inversion.
    The notation F(T) -> LLM and T(F(T), I) -> C is taken as a given functional abstraction in Section 1.

reviewed 2026-08-04 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Uncovering Scaling Laws for Large Language Models via Inverse Problems." pith.science (2026). https://pith.science/paper/OXENMCR4

@misc{pith2026250907909,
  author       = {Pith},
  title        = {Pith review of: Uncovering Scaling Laws for Large Language Models via Inverse Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OXENMCR4}},
  note         = {Machine review of arXiv:2509.07909}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Large Language Models (LLMs) are large-scale pretrained models that have achieved remarkable success across diverse domains. These successes have been driven by unprecedented complexity and scale in both data and computations. However, due to the high costs of training such models, brute-force trial-and-error approaches to improve LLMs are not feasible. Inspired by the success of inverse problems in uncovering fundamental scientific laws, this position paper advocates that inverse problems can also efficiently uncover scaling laws that guide the building of LLMs to achieve the desirable performance with significantly better cost-effectiveness.

Figures

Figures reproduced from arXiv: 2509.07909 by Apivich Hemachandra, Arun Verma, Bryan Kian Hsiang Low, Gregory Kang Ruey Lau, Jingtan Wang, Nhung Bui, Rachael Hwee Ling Sim, Rui Qiao, See-kiong Ng, Wenyang Hu, Xiaoqiang Lin, Xinyi Xu, Xinyuan Niu, Zhaoxuan Wu, Zhiliang Chen, Zijian Zhou, Zitong Zhao, Zi-Yu Khoo.

Figure 1
Figure 1. Figure 1: Forward processes in large language models. The forward process generates an LLM from key input [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Examples of low-quality samples include instances where the question-answer pair fails to capture the key elements of the image or has limited relevance. Previous efforts have approached the problem as a large-scale data selection challenge, focusing on external evaluators such as established criteria (Wei et al., 2023) or intrinsic features (Chen et al., 2024a; Liu et al., 2024d). For example, Xia et al. … view at source ↗
Figure 3
Figure 3. Figure 3: Watermarking as an unlearning metric. MU metrics can help define scaling laws governing the difficulty of unlearning erased data. Previous work (Zhao et al., 2024) explored how the tug-of-war (ToW) verification metric, which compares the accuracies of the unlearned and retrained models, is influenced by the properties of erased and retained data. It also examined how certain properties of erased data, like… view at source ↗

discussion (0)

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

Reference graph

Works this paper leans on

13 extracted references · 4 canonical work pages · 1 internal anchor

  1. [3]

    arXiv:2001.08361

    Scaling laws for neural language models. arXiv:2001.08361. Zixuan Ke, Yijia Shao, Haowei Lin, Tatsuya Konishi, Gyuhak Kim, and Bing Liu. 2023. Continual pre-training of language models. InProc. ICLR. Will Knight. 2023. OpenAI’s CEO says the age of giant AI models is already over.WIRED. Pang Wei Koh and Percy Liang. 2017. Understanding black-box prediction...

    Pith/arXiv arXiv 2001

  2. [5]

    arXiv:2006.16668

    Gshard: Scaling giant models with conditional computation and automatic sharding. arXiv:2006.16668. Yaniv Leviathan, Matan Kalman, and Yossi Matias

    Pith/arXiv arXiv 2006

  3. [6]

    Fast inference from transformers via speculative decoding. InProc. ICML, pages 19274–19286. Nathaniel Li, Alexander Pan, Anjali Gopal, Summer Yue, Daniel Berrios, Alice Gatti, Justin D. Li, Ann-Kathrin Dombrowski, Shashwat Goel, Long Phan, Gabriel Mukobi, Nathan Helm-Burger, Rassin Lababidi, Lennart Justen, Andrew B. Liu, Michael Chen, Isabelle Barrass, O...

    Pith/arXiv arXiv 2024

  4. [7]

    arXiv:2403.05530

    Gemini 1.5: Unlocking multimodal understanding across millions of tokens of context. arXiv:2403.05530. Laria Reynolds and Kyle McDonell. 2021. Prompt programming for large language models: Beyond the few-shot paradigm. InExtended Abstracts of the 2021 CHI Conference on Human Factors in Computing Systems. Ohad Rubin, Jonathan Herzig, and Jonathan Berant

    Pith/arXiv arXiv 2021

  5. [9]

    BLEURT: Learning robust metrics for text generation. InACL. Rulin Shao, Jacqueline He, Akari Asai, Weijia Shi, Tim Dettmers, Sewon Min, Luke Zettlemoyer, and Pang Wei Koh. 2024. Scaling retrieval-based language models with a trillion-token datastore. arXiv:2407.12854. Rulin Shao, Rui Qiao, Varsha Kishore, Niklas Muennighoff, Xi Victoria Lin, Daniela Rus, ...

    Pith/arXiv arXiv 2024

  6. [11]

    Ronald J Williams

    Instructiongpt-4: A 200-instruction paradigm for fine-tuning minigpt-4.arXiv:2308.12067. Ronald J Williams. 1992. Simple statistical gradient-following algorithms for connectionist reinforcement learning.Machine learning, pages 229–256. Yangzhen Wu, Zhiqing Sun, Shanda Li, Sean Welleck, and Yiming Yang. 2024a. Inference scaling laws: An empirical analysis...

    Pith/arXiv arXiv 1992

  7. [12]

    Judging llm-as-a-judge with mt-bench and chatbot arena. InProc. NeurIPS, volume 36. Dongruo Zhou, Lihong Li, and Quanquan Gu

    work page

  8. [13]

    Neural contextual bandits with UCB-based exploration. InProc. ICML, pages 11492–11502. Xin Zhou, Yi Lu, Ruotian Ma, Tao Gui, Qi Zhang, and Xuanjing Huang. 2024a. Making harmful behaviors unlearnable for large language models. InProc. ACL Findings. Yongchao Zhou, Andrei Ioan Muresanu, Ziwen Han, Keiran Paster, Silviu Pitis, Harris Chan, and Jimmy Ba. 2023a...

    Pith/arXiv arXiv 2025

  9. [2020]

    Explaining black box predictions and unveiling data artifacts through influence functions. InProc. ACL. Apivich Hemachandra, Zhongxiang Dai, Jasraj Singh, See-Kiong Ng, and Bryan Kian Hsiang Low. 2023. Training-free neural active learning with initialization-robustness guarantees. arXiv:2306.04454. Jordan Hoffmann, Sebastian Borgeaud, Arthur Mensch, Elena...

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  10. [2022]

    InNAACL, pages 2655–2671

    Learning to retrieve prompts for in-context learning. InNAACL, pages 2655–2671. Andrea Schioppa, Polina Zablotskaia, David Vilar, and Artem Sokolov. 2022. Scaling up influence functions. InAAAI, pages 8179–8186. Thibault Sellam, Dipanjan Das, and Ankur P Parikh

    work page 2022

  11. [2023]

    Lin Gui, Cristina Gârbacea, and Victor Veitch

    Studying large language model generalization with influence functions.arXiv:2308.03296. Lin Gui, Cristina Gârbacea, and Victor Veitch. 2024. BoNBoN alignment for large language models and the sweetness of best-of-n sampling. InProc. NeurIPS, pages 2851–2885. Qingyan Guo, Rui Wang, Junliang Guo, Bei Li, Kaitao Song, Xu Tan, Guoqing Liu, Jiang Bian, and Yuj...

    Pith/arXiv arXiv 2024

  12. [2024]

    Datainf: Efficiently estimating data influence in loRA-tuned LLMs and diffusion models. InProc. ICLR. Gregory Kang Ruey Lau, Xinyuan Niu, Hieu Dao, Jiangwei Chen, Chuan-Sheng Foo, and Bryan Kian Hsiang Low. 2024. Waterfall: Framework for robust and scalable text watermarking. InICML 2024 Workshop on Foundation Models in the Wild. Dmitry Lepikhin, HyoukJoo...

    work page 2024

  13. [2025]

    NICE data selection for instruction tuning in LLMs with non-differentiable evaluation metric. In Proc. ICML. Ruochen Wang, Minhao Cheng, Xiangning Chen, Xiaocheng Tang, and Cho-Jui Hsieh. 2021. Rethinking architecture selection in differentiable nas.arXiv:2108.04392. Jason Wei, Maarten Bosma, Vincent Zhao, Kelvin Guu, Adams Wei Yu, Brian Lester, Nan Du, A...

    Pith/arXiv arXiv 2021

This paper was first reviewed by deepseek-v4-flash on August 4, 2026.