{"id":"994e7b18-9198-4d6b-9965-a3c6124bd605","arxiv_id":"2509.07909","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Proposes using inverse problems to discover LLM scaling laws, but provides no empirical evidence or new results.","lead":"This position paper argues that inverse problems, which infer the ingredients needed to reach a target performance, can uncover cost-saving scaling laws for large language models. It frames data selection, inference optimization, and machine unlearning as inverse problems and calls for new research.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper never specifies a concrete inversion algorithm or well-posedness condition for Eqs. (1a)-(1b); without a controlled demonstration that inversion recovers known scaling laws, the central claim is untestable and the acknowledged ill-posedness remains unresolved.","rationale":"The reader identified the weak assumption as the tractability of ill-posed inverse problems. I agree, and I sharpen the concern by noting that the paper never actually defines an inversion procedure or a regularity condition that would make the problem tractable. This is not an ad hominem or a disagreement with the field's consensus; it is a correctness risk internal to the paper's argument. The central claim depends on the existence of a well-posed inverse, but the paper's own Limitations concede the forward maps are many-to-one, and no remedy is proposed. The concrete test I suggest—recovering the Chinchilla scaling law through the paper's inverse-problem recipe—would settle whether the concern lands. If the framework cannot even be instantiated for this simple case, the programmatic claim is unsupported. I do not recommend rejection, because the paper is a position paper that transparently acknowledges its constraints and the reader's CONDITIONAL verdict already calls for a case study; my concern reinforces that condition but does not shift the verdict.","tokens_in":17325,"tokens_out":3933,"duration_ms":51533,"concrete_test":"Take a known scaling law—e.g., the Chinchilla law C(N,D) from Hoffmann et al. (2022)—and define the forward map from ingredients T=(N,D) to performance C. Using only a sparse set of forward simulations or published data points as observations, attempt to invert the map to recover the scaling-law parameters and to predict C for held-out (N,D) pairs. If the proposed inverse-problem framework cannot be specified for this simple, well-understood case without adding arbitrary regularization, or if it recovers no better than standard least-squares fitting on log C, then the central claim's novelty and tractability collapse. Conversely, if the framework can be precisely defined and outperforms brute-force forward sweeps in sample efficiency, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the inverse problems formed from Eqs. (1a)-(1b) are tractable—i.e., from observations C one can reliably recover the ingredient map T or its scaling-law parameters. The paper's own Limitations section admits that 'many forward problems in LLMs... are inherently many-to-one,' leading to 'potential ambiguity or ill-posedness in their inverse counterparts.' Yet the paper provides no regularization strategy, no identifiability conditions, and no concrete inversion algorithm. It also never distinguishes its 'inverse problem' approach from ordinary parametric fitting of scaling laws from forward experiments (Kaplan et al.; Hoffmann et al.). In the simplest regime where the forward map is a smooth power law, inversion reduces to standard regression—trivially possible but not novel. In the more interesting regime where T includes data composition, architecture, and inference schemes, the map is so high-dimensional and many-to-one that no well-posed inverse is defined. The paper's concrete suggestions (NeuralUCB for prompts, REINFORCE for data selection, watermarking for unlearning verification) are existing optimization/estimation techniques relabeled as inverse problems; they do not demonstrate that scaling laws can be uncovered more efficiently than brute-force forward sweeps. Thus the load-bearing assumption—tractability—is unsupported, and the central claim remains an untested program rather than a demonstrated result.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17663,"tokens_out":4517,"duration_ms":60200,"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":[{"comment":"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.'","section":"Abstract and Eqs. (1a)–(1b)"},{"comment":"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.","section":"Eq. (1b) and Limitations"},{"comment":"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.","section":"§2, §3.1, §4.1"},{"comment":"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.","section":"§2 and §3.3"}],"minor_comments":[{"comment":"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.","section":"Introduction, Eq. (1b)"},{"comment":"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.","section":"Throughout"},{"comment":"The heading says 'Multi-Model LLMs' but the text and acronym MLLM refer to 'Multi-Modal LLMs.' Please fix the heading.","section":"§2.1 heading"},{"comment":"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.","section":"Figure 2"},{"comment":"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.","section":"§3.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on the authors' own prior work for the 'promising directions' (Verma et al. 2025, Lin et al. 2025, Wu et al. 2025, etc.). This is not disqualifying, but without independent benchmarks it is hard to judge whether the inverse-problem framing adds value beyond relabeling existing optimization methods. For a journal venue, the paper is a position/vision piece; if the editors expect empirical validation, the current form is better suited for an opinion/outlook venue or a workshop. I would recommend revision to either add a small proof-of-concept or to explicitly reframe the paper as a set of open problems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis is a position paper, not a results paper. It proposes to treat LLM design—data selection, inference optimization, unlearning—as inverse problems, and advocates that solving these inverses will reveal scaling laws that guide training. If you read it expecting experiments or theorems, you'll be disappointed. But if you read it as a research agenda, it's actually a well-organized map of open problems, with a unifying formalism that I think is useful.\n\nWhat's genuinely new: the paper writes down two forward processes—F(T) → LLM and T(F(T), I) → C—and then asks what it means to invert them. That's a clean way to connect a large literature on influence functions, prompt optimization, and unlearning verification to the mathematical theory of inverse problems. The three case studies (data selection, inference, unlearning) are concrete enough to start work on. I particularly liked the unlearning verification idea: using watermarking as a retraining-free membership check. That's a specific, testable proposal.\n\nWhere the paper is soft: the central claim—that inverse problems can 'efficiently uncover scaling laws'—is asserted rather than demonstrated. There's no toy example, no simulation, no known scaling law recovered by inversion. And the paper itself admits the many-to-one problem, but doesn't offer a way out. The stress-test is right: without a regularization scheme or identifiability condition, the inverse problem may be ill-posed in exactly the regimes that matter. Also, some of the relabeling is cosmetic: prompt optimization as a black-box inverse is essentially existing bandit/BO methodology; data selection via influence functions is already inverse problem thinking. The heavy self-citation (Verma et al., Lin et al., Wu et al.) is not disqualifying, but it does make the paper feel like a group's own agenda rather than an independent synthesis.\n\nThe math itself is just notation—Eqs. (1a)-(1b) are definitional, not derivational. No proofs, no data. That's fine for a position paper, but it means the paper's value is in the questions it poses, not in answers.\n\nBottom line: if a colleague asks 'is there anything here?' I'd say yes, as a framing. It deserves a serious referee if the venue accepts position papers; it shouldn't be sent to a results-oriented track. For me, I won't cite it as evidence for any empirical claim, but I might reference it in related work.\n\nRecommendation: engage with it as a discussion paper, not a contribution to the scaling-laws literature.","headline":"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.","tokens_in":18161,"tokens_out":2620,"would_cite":false,"duration_ms":32133,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["inverse problems","scaling laws","large language models","data selection","inference optimization","machine unlearning","cost-effectiveness","training ingredients"],"falsifier":"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.","tokens_in":17273,"feed_emoji":"📈","tokens_out":7723,"duration_ms":81379,"temperature":0.7,"pith_summary":"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.","feed_headline":"Inverse problems could make LLM scaling laws cheap to find","feed_subtitle":"Instead of training model after model, solve for the data, prompts, and compute that hit a target score.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the classical inverse-problem methodology that the paper says is a tried-and-true route to uncovering scientific laws.","marker":"Groetsch and Groetsch, 1993"},{"why":"Establishes the conventional neural-scaling-law baseline that the inverse framing aims to extend.","marker":"Kaplan et al., 2020"},{"why":"Gives the compute-optimal scaling law that motivates cost-aware ingredient choices.","marker":"Hoffmann et al., 2022"},{"why":"Shows data pruning can beat power-law scaling, grounding the claim that data selection changes scaling behavior.","marker":"Sorscher et al., 2022"},{"why":"Demonstrates a small selected text subset matching full-data performance, the empirical anchor for the data-selection inverse problem.","marker":"Xia et al., 2024"},{"why":"Establishes a test-time compute scaling law that the inference-optimization inverse problem generalizes.","marker":"Chen et al., 2024b"},{"why":"Shows inverse-style optimization already works for speculative decoding configurations, supporting the feasibility of the approach.","marker":"Leviathan et al., 2023"}],"fun_headline_variants":["Inverse problems could reveal LLM scaling laws cheaply","Flip the LLM building problem to cut training costs","Find scaling laws by inverting LLM training","Solve for data and compute that hit target LLM performance","Inverse problems: a cheaper route to LLM scaling laws"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Inverse problems could reveal LLM scaling laws cheaply","Flip the LLM building problem to cut training costs","Find scaling laws by inverting LLM training","Solve for data and compute that hit target LLM performance","Inverse problems: a cheaper route to LLM scaling laws"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000152,"raw_usage":{"total_tokens":969,"prompt_tokens":604,"completion_tokens":365,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":348,"completion_tokens_details":{"reasoning_tokens":286}},"tokens_in":348,"tokens_out":365,"duration_ms":4514,"temperature":1.0,"reasoning_tokens":286,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T21:30:21.305252+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}