Data-Dependent Scaling Laws

• Data-dependent scaling is a methodology where performance scaling laws are adjusted based on specific data characteristics such as covariance spectra and compressibility.
• Techniques like spectral analysis, compression metrics, and empirical scaling law fits enable precise estimation of exponents, informing optimal resource split between data and model scaling.
• Practical implications include enhanced transfer learning, improved deduplication strategies, and dynamic sampling, all of which optimize compute usage and performance in complex domains.

Data-dependent scaling encompasses a set of phenomena and methodologies wherein the scaling behavior of machine learning models—how generalization error or other performance metrics decrease with increased resources—depends sensitively and predictively on quantifiable properties of the data distribution, dataset composition, data sampling protocol, or training regime. In contrast to universal or “data-agnostic” scaling laws, data-dependent scaling laws assert that exponents, prefactors, or even the form of the scaling relationship must be adjusted based on structural or statistical features of the data. This principle affects not only core learning-theoretic bounds but also practical choices in transfer learning, large-scale training, synthetic data pipelines, deduplication, and domain-adaptive pretraining.

1. Mathematical Foundations of Data-Dependent Scaling

Data-dependent scaling laws generalize classical power-law learning curves for loss (or error) as a function of dataset size $N$, model size $M$, compute budget $C$, or other resources. The basic form is: $L(N, M) = E + A\,N^{-\alpha_N} + B\,M^{-\alpha_M}$ where $L$ is the expected generalization error, $E$ is the irreducible loss, and $\alpha_N$, $\alpha_M$ are exponents. In data-dependent scaling, these exponents and the coefficients $A$, $B$, and even $M$0 are explicit functions of data properties.

Theoretical advances demonstrate that, under polynomial decay of the data covariance spectrum with index $M$1 and source smoothness exponent $M$2, the excess risk scales as

$M$3

where a “heavier” spectrum (smaller $M$4, i.e., more redundancy) yields a slower rate, and smoother targets (larger $M$5) accelerate convergence. These dependencies hold universally across kernel (linearized/infinite-width) regimes and persist in finite-width neural networks and Transformers (Bi et al., 25 Sep 2025, Brill, 2024, Bahri et al., 2021).

In some contexts, scaling parameters become explicit functions of a data complexity or entropy metric such as gzip-compressibility $M$6, as in

$M$7

where increased data compressibility (lower entropy) sharpens scaling exponents and reduces irreducible loss, fundamentally shifting the optimal allocation of compute resources between model and data scaling (Pandey, 2024).

2. Empirical and Structural Predictors of Scaling Exponents

Key empirical predictors of the scaling law parameters include:

• Covariance spectrum tail ($M$8): Data whose feature covariance matrix has a steeper eigenvalue decay yields higher scaling exponents, accelerating power-law convergence. Empirically-measured spectra allow direct estimation of $M$9, e.g., via log–log fits of sorted eigenvalues (Bi et al., 25 Sep 2025).
• Entropy and compressibility ($C$0): Datasets with higher entropy, as measured by the ratio of gzip-compressed to raw size, exhibit lower scaling exponents and higher baselines. This aligns with source coding theory; more complex data inherently resists compression and learning, requiring more data or capacity for equivalent performance (Pandey, 2024).
• Phase-space dimension: In regression surrogates for physical processes, the scaling exponent is tightly predicted by the intrinsic dimension $C$1, e.g., $C$2 for the number of final-state particles (Bahl et al., 19 Jan 2026).
• Duplication and uniqueness: The effective number of semantically unique samples $C$3 predicts when naive scaling laws break down due to redundancy, particularly for large models whose gradients are aligned across semantic duplicates (Kazdan et al., 18 Feb 2026).

3. Methodologies for Data-Dependent Scaling Analysis

Techniques for quantifying and exploiting data-dependent scaling include:

• Spectral analysis: Estimation of the covariance or kernel spectrum on large data samples to fit the polynomial tail exponent $C$4, which directly determines the learning-curve exponent $C$5 (Bi et al., 25 Sep 2025).
• Compression-based metrics: Application of lightweight algorithms (e.g., gzip) to measure empirical dataset compressibility as a proxy for entropy and complexity, enabling plug-in scaling laws whose coefficients and exponents are explicit (e.g., $C$6) (Pandey, 2024).
• Empirical scaling law fits: Systematic experiments varying data and model size, fit via regression or joint loss minimization (e.g., Huber loss with L-BFGS), to infer the data dependence of the scaling exponents over wide dynamic ranges (Zhang et al., 2024, Yang et al., 17 Apr 2025).
• Data-reuse and active sampling: Analytical demonstration and empirical verification that even when true data size $C$7 is fixed, multiple passes ($C$8) in stochastic optimization can effectively “reuse” data, extending power-law scaling with new exponents up to a computable limit ($C$9 for aligned power-law parameter/prior pairs) (Lin et al., 10 Jun 2025).
• Dynamic, uncertainty-guided sampling: In synthetic data pipelines, iterative estimation of sample informativeness (e.g., predictive entropy) and targeted generation or pruning of new data—approximating optimal pruning in real-time and improving the effective scaling constants and exponents (Askari-Hemmat et al., 21 Feb 2025).
• Scaling law estimation for data-source utility: Running multiple pretraining/annealing runs at varied budgets to empirically fit power-law utility curves for each candidate data source, rather than relying on single-point estimates—crucial for robust compute allocation and avoiding misrankings due to non-invariant scaling (Ostapenko et al., 29 Jul 2025).

4. Case Studies in Data-Dependent Scaling

Several domains have demonstrated the key implications of data-dependent scaling:

Research Area	Data-Dependent Scaling Variable	Impact on Scaling
Amplitude surrogates	Phase-space dimension $L(N, M) = E + A\,N^{-\alpha_N} + B\,M^{-\alpha_M}$0	$L(N, M) = E + A\,N^{-\alpha_N} + B\,M^{-\alpha_M}$1, tighter with lower $L(N, M) = E + A\,N^{-\alpha_N} + B\,M^{-\alpha_M}$2 (Bahl et al., 19 Jan 2026)
Visual transfer learning	Task-specific data/labels $L(N, M) = E + A\,N^{-\alpha_N} + B\,M^{-\alpha_M}$3	Boundaries shift for distillation benefit; threshold analytically computable (Yang et al., 17 Apr 2025)
LLM fine-tuning	Fine-tuning data size, model size, pretraining data	Best method depends on $L(N, M) = E + A\,N^{-\alpha_N} + B\,M^{-\alpha_M}$4; multiplicative scaling law (Zhang et al., 2024)
Synthetic data pipelines	Entropy-guided active selection	Improvement in sample/iteration efficiency via higher scaling exponents (Askari-Hemmat et al., 21 Feb 2025)
Web-scale pretraining	Effective uniqueness $L(N, M) = E + A\,N^{-\alpha_N} + B\,M^{-\alpha_M}$5	Large models face rapid scaling-law breakdowns from semantic collision (Kazdan et al., 18 Feb 2026)
Physical regression	External particle count	Data and compute exponents tightly linked to particle multiplicity (Bahl et al., 19 Jan 2026)

5. Practical Implications and Guidelines

Data-dependent analysis of scaling leads to concrete procedural and allocation recommendations:

• Resource budgeting: For a fixed compute budget, optimal split between model and data size is demanded by their relative exponents, which in turn depend on data complexity metrics such as gzip compressibility. As data becomes harder to compress ($L(N, M) = E + A\,N^{-\alpha_N} + B\,M^{-\alpha_M}$6), optimal policy shifts toward acquiring more data rather than parameter scaling (Pandey, 2024).
• Transfer learning and distillation: There exists a formally computable data threshold $L(N, M) = E + A\,N^{-\alpha_N} + B\,M^{-\alpha_M}$7 below which knowledge distillation is superior, and above which direct large-model fine-tuning dominates. This threshold depends explicitly on the fitted scaling exponents of each regime (Yang et al., 17 Apr 2025).
• Pretraining on limited or redundant datasets: For large models or high redundancy, deduplication should exploit semantic embeddings and estimate $L(N, M) = E + A\,N^{-\alpha_N} + B\,M^{-\alpha_M}$8 rather than rely on surface-form hashes. Data duplication penalties grow rapidly with model scale according to explicit power laws (Kazdan et al., 18 Feb 2026).
• Data source optimization: Before allocating large-scale curation or annealing compute to a new data source, run 3–6 budget-varied experiments to fit scaling exponents. Base allocation on the projected utility at full budget, not on single low-budget runs, to avoid losses from rank flips (Ostapenko et al., 29 Jul 2025).

6. Generalization Across Domains and Theoretical Insights

Theoretical developments establish the universality and explanatory power of data-dependent scaling:

• The scaling exponent is determined by data redundancy via the spectral tail ($L(N, M) = E + A\,N^{-\alpha_N} + B\,M^{-\alpha_M}$9): steeper tails (less redundancy) yield faster power-law improvement and lower sample complexity for a given accuracy (Bi et al., 25 Sep 2025).
• In mixture settings or transfer across domains, the overall exponent is set by the slowest-decaying (“hardest”) component, implying that outlier or minority structures dominate scaling in heterogenous data (Bi et al., 25 Sep 2025).
• Fluctuation-based and derivative-based analysis of physical systems (e.g., glass-forming liquids) reveal that even classical non-ML scaling scenarios display rigorous data-point (state-point) dependence, invalidating naïve universality of exponents (Sanz et al., 2018).
• All scaling law parameters (exponents, constants, irreducible errors) can be considered data functions—either via direct empirical fits, proxy complexity metrics, or structural modeling (covariance, percolation, phase space dimension) (Brill, 2024, Bahl et al., 19 Jan 2026).

7. Limitations and Open Directions

Although data-dependent scaling laws significantly refine model selection and resource allocation, several limitations remain:

• Empirical fits often require operating in the “scaling regime,” i.e., away from plateaus and saturation at small scales (Bahl et al., 19 Jan 2026).
• For heterogenous or evolving datasets, extracting a single meaningful spectral exponent may be nontrivial.
• Many advances rely on synthetic control or domain-specific modeling (e.g., amplitude surrogates, percolation-theory analogs), which may not transfer uncritically to noisy, real-world datasets.

A plausible implication is that for emerging foundation models, future scaling-law–driven development will increasingly intertwine data measurement, curation, embedding analysis, and active data selection—supplanting universalist or purely parameter-centric strategies.

---

References:

• “Scaling Laws are Redundancy Laws” (Bi et al., 25 Sep 2025)
• “gzip Predicts Data-dependent Scaling Laws” (Pandey, 2024)
• “Scale Dependent Data Duplication” (Kazdan et al., 18 Feb 2026)
• “Scaling laws for amplitude surrogates” (Bahl et al., 19 Jan 2026)
• “Scaling Laws for Data-Efficient Visual Transfer Learning” (Yang et al., 17 Apr 2025)
• “When Scaling Meets LLM Finetuning: The Effect of Data, Model and Finetuning Method” (Zhang et al., 2024)
• “Improving the Scaling Laws of Synthetic Data with Deliberate Practice” (Askari-Hemmat et al., 21 Feb 2025)
• “Using Scaling Laws for Data Source Utility Estimation in Domain-Specific Pre-Training” (Ostapenko et al., 29 Jul 2025)
• “Neural Scaling Laws Rooted in the Data Distribution” (Brill, 2024)
• “Improved Scaling Laws in Linear Regression via Data Reuse” (Lin et al., 10 Jun 2025)
• “Experimental evidence of a state-point dependent scaling exponent of liquid dynamics” (Sanz et al., 2018)
• “Explaining Neural Scaling Laws” (Bahri et al., 2021)
• “Adaptive Scaling” (Li et al., 2017)