Knowledge Scaling Laws

Updated 16 February 2026

Knowledge scaling laws are empirical models that quantify how a system’s ability to store and retrieve factual information scales with model parameters, data volume, and compute.
They extend traditional loss minimization frameworks by distinguishing regimes where model size or data constraints limit the fidelity of learned, rare, or complex knowledge.
These laws inform optimal architectural designs and training strategies by providing clear theoretical bounds and empirical guidance for improved recall, reasoning, and generalization.

Knowledge scaling laws describe how the amount of "knowledge"—task-relevant factual, relational, or semantic information—acquired or expressible by a machine learning system scales as a function of model size, data, and compute. In contrast to traditional capacity scaling laws (which typically address loss or perplexity), knowledge scaling probes a model’s ability to recall, reason with, or generalize factual associations as a function of growth in its parameters, training dataset, or computational footprint.

1. Foundations of Knowledge Scaling Laws

Scaling laws for neural models were first formalized in the context of loss minimization, with power-law relationships established between loss, model size, dataset size, and compute in LLMs and vision architectures. The expansion to "knowledge scaling" concerns the distinct regimes where model size or data quantity constrain the fidelity and breadth of learned factual information, such as entity-relation pairs, long-tail facts, or conceptual taxonomies present in the training distribution.

This domain focuses particularly on how knowledge that is not compressed into model parameters or memorized verbatim requires disproportionately more data and/or model capacity to be faithfully learned and retrieved, especially as one moves from high-frequency (head) to rare (tail) knowledge (Zhou et al., 11 Sep 2025).

2. Mathematical and Empirical Formulation

The canonical form of a knowledge scaling law is the empirical observation:

$\text{Knowledge}(N, D) \propto N^\alpha D^\beta,$

where $\text{Knowledge}(N, D)$ denotes an operational measure of task-relevant knowledge (e.g., successful entity recall, relation completion, or logical reasoning), $N$ the number of model parameters, and $D$ the size of training data. Exponents $\alpha,\ \beta$ are determined empirically.

Simultaneous approximation theory shows that, for deep networks on manifolds, the parameter budget $S$ needed to approximate any function $f$ (and its derivatives up to order $s<k$ ) in Sobolev space $\mathcal{W}_p^k(\mathcal{M}^d)$ to accuracy $\epsilon$ in $\mathcal{W}_p^s$ norm scales as

$S = O\left(\epsilon^{-d/(k-s)}\right),$

where $d$ is the intrinsic manifold dimension (Zhou et al., 11 Sep 2025). In other words, the minimal number of parameters needed to represent "knowledge" at scale $\epsilon$ grows polynomially in $\epsilon^{-1}$ , with an exponent that depends only on the intrinsic complexity (dimension, smoothness, and the required order of derivative accuracy).

3. Regimes of Scaling: Model Size, Data, and Intrinsic Complexity

Distinct scaling regimes emerge:

Model-limited regime ( $N \ll N^*$ ): Knowledge grows as a power of model size; small models can memorize only a subset of the most frequent knowledge statements, or interpolate only the most prominent relations.
Data-limited regime ( $D \ll D^*$ ): Increasing training data yields gains until a saturation point, after which new knowledge is dominated by model capacity.
Intrinsic limit: For tasks involving functions on manifolds, the scaling exponents are dictated by the smoothness $k$ of the function class and the intrinsic manifold dimension $d$ , not the ambient dimension (Zhou et al., 11 Sep 2025).

These observations generalize to many scenarios: LLMs acquiring factual knowledge from text corpora, diffusion models learning physical or behavioral priors, and ML-driven scientific discovery seeking to embed physical laws in a parameter-efficient manner.

4. Optimization Complexity and Knowledge Retrieval

Knowledge scaling interacts crucially with the complexity of recall or retrieval. For constant-depth architectures (e.g., shallow or width-limited ReLU $^{k-1}$ networks), there is a provable lower bound on parameters required to represent all $\epsilon$ -distinct knowledge facts at a specified regularity, showing that approximation efficiency degrades only logarithmically relative to the upper bound (Zhou et al., 11 Sep 2025). This near-optimality suggests that model architecture and parameterization can fundamentally determine the scaling law for knowledge, not just optimization strategy.

Moreover, the sample complexity of acquiring knowledge depends on the volume of the underlying manifold (or conceptual space). For instance, to approximate functions on $d$ -dimensional manifolds from scattered data to error $\epsilon$ , at least $O(\epsilon^{-d/2})$ samples are required (Faigenbaum-Golovin et al., 2020).

5. Knowledge Scaling in Modern Deep Architectures

Recent work has operationalized knowledge scaling in large transformers, diffusion models, and manifold-aware planners:

Neural networks with ReLU $^{k-1}$ activations efficiently approximate both functions and all their derivatives, enabling knowledge-rich representations of PDEs and scientific data on manifolds with parameter budgets matching the theoretical scaling laws (Zhou et al., 11 Sep 2025).
In graph-based or diffusion-based planning, the projection of network outputs onto approximated local tangent spaces of the data manifold ensures that feasible knowledge (trajectories, policy constraints) is preserved even as the ambient dimension or underlying complexity increases, again linking feasible knowledge with local capacity and data density (Lee et al., 1 Jun 2025).
Model order reduction and manifold-valued function approximation methods explicitly blend local tangent-space interpolation with weighted Fréchet means to combine efficient knowledge scaling (parameter/data-wise) with global geometric fidelity (Wang et al., 17 Apr 2025).

6. Implications and Optimality of Knowledge Scaling Laws

The principal insight of knowledge scaling laws is that, unlike raw loss or perplexity scaling—which may benefit from additional heuristics or overparameterization without tight lower bounds—the scaling of acquired or expressible knowledge is sharply characterized by mathematical lower and upper bounds, governed by intrinsic rather than ambient complexity. These findings have central implications:

Curse of Dimensionality Avoidance: Knowledge scaling exponents depend only on intrinsic dimension and regularity of the task, not the raw input dimension, allowing for scalable acquisition and representation of knowledge in high-dimensional settings provided the "knowledge manifold" is low-dimensional (Zhou et al., 11 Sep 2025, Faigenbaum-Golovin et al., 2020).
Architectural Optimality: No constant-depth, bounded-weight network architecture can outperform the optimal scaling by more than logarithmic factors, even for knowledge-rich tasks (Zhou et al., 11 Sep 2025).
Practical Regime: In practical settings, sharp knowledge scaling laws inform the tradeoff between compute, model design, and data curation needed to acquire specific knowledge targets at desired resolutions.

7. Open Challenges and Future Directions

Many open questions remain in the full quantification and operationalization of knowledge scaling:

Identification of precise scaling exponents for diverse model classes on real-world knowledge-rich tasks.
Extension of simultaneous approximation theory to stochastic and continual learning settings.
Characterization of the impact of noise, regularization, and optimization pathologies on attainable knowledge scaling rates.

Recent advances in both the theoretical underpinning and empirical validation of knowledge scaling laws provide a foundation for principled growth of machine learning models targeted at knowledge-rich domains, scientific computing, and autonomous discovery (Zhou et al., 11 Sep 2025, Faigenbaum-Golovin et al., 2020, Wang et al., 17 Apr 2025, Lee et al., 1 Jun 2025).