Learned Scaling Factors in Machine Learning

• Learned scaling factors are numerical coefficients that adaptively balance contributions of system components, enhancing model stability and performance.
• They are computed through methods such as least squares minimization, end-to-end learning, and spectral decomposition for precise parameter tuning.
• Applications span structured prediction, generative modeling, physical simulation, and neural network scaling, reducing manual tuning and mitigating interference.

Learned scaling factors are numerical coefficients, either scalar or vector/matrix-valued, that adjust the magnitude or relative contribution of components in a system—such as energy terms, latent bases, frequency calculations, or parameter blocks—where these coefficients are inferred from data or adapted through optimization. Their automated discovery is central to numerous machine learning, signal processing, generative modeling, and physical simulation workflows, replacing ad hoc or manually tuned weighting schemes and supporting robust, generalizable inference, improved disentanglement, and efficient optimization.

1. Formal Definitions and Theoretical Basis

Learned scaling factors parameterize diverse systems by multiplicatively transforming selected inputs, parameters, or outputs. Formally, if a system contains elements $\{\phi_k(\cdot)\}$, the scaled system forms $S(\cdot) = \sum_k \alpha_k \phi_k(\cdot)$, where $\alpha_k$ are the learned scaling coefficients. These coefficients may be derived by minimizing an objective $\mathcal{L}(S(\cdot), Y)$ over $\mathcal{L}$ and/or by imposing secondary constraints (e.g., normalization, regularization). “Uniform” scaling factors apply a single coefficient across all elements, whereas “anisotropic”/“blockwise” scaling allows for structured, component-specific scaling as in aTLAS (Zhang et al., 2024).

In deep structured prediction, scaling factors are especially necessary to balance the contribution of unary and higher-order energy terms in models of the form

$$F(Y|X,\theta) = \sum_i U_i(y_i; X,\theta) + \sum_{\langle i,j\rangle} W_{ij}(y_i,y_j; X,\theta)$$

where naïvely joint training fails absent proper scaling (Shevchenko et al., 2019). In generative modeling, scaling factors emerge as coordinates in bases extracted by spectral decomposition; e.g., StyleGAN latent decomposition $w = E\alpha$ for hypercoordinate basis $E$ yields scaling factor $\alpha$ (Wang et al., 2021). For physical simulation (e.g., vibrational frequency corrections), uniform multiplicative scaling minimizes RMSE between model outputs and experimental measurements, $s^* = \arg\min_s \mathrm{RMSE}(s)$ (Trujillo et al., 2021).

Scaling laws for neural network loss trajectories further instantiate learned scaling factors as the power-law coefficients in

$L(N) = \left(\frac{N_c}{N}\right)^{\alpha_N}$

for model size $N$, with $\alpha_N,N_c$ determined by regression on log-transformed results (Su et al., 2024).

2. Methodologies for Learning Scaling Factors

Approaches to learning scaling factors vary in complexity and domain:

• Least Squares Minimization: Utilized in physical modeling (e.g., harmonic frequency scaling (Trujillo et al., 2021)), the optimal scaling $s$ minimizes $\sum_i [s\,\omega_i^\text{harm} - \nu_i]^2$ yielding

$$s^* = \frac{\sum_i \omega_i^\text{harm} \nu_i}{\sum_i (\omega_i^\text{harm})^2}$$

This is applied globally per model chemistry.

• End-to-End Structured Learning with Scalar or Vector Scaling: In deep structured predictors, balancing terms through learned global scalar $\alpha$ or dynamic epoch-specific scaling restores effective optimization. Offline scaling divides terms by their $\ell_1$-norm and multiplies by $\alpha$, which may be hyperparameter-tuned or learned jointly. Online scaling refits $\alpha$ after every epoch via grid search on validation loss (Shevchenko et al., 2019).
• Spectral Decomposition: Generative modeling applies PCA/eigendecomposition to network weights, forming basis $E$ of principal directions. Latent codes $w$ are projected as $\alpha = E^Tw$, and attribute-conditioned ranges of $\alpha$ are empirically mined to control generation (Wang et al., 2021).
• Blockwise (Anisotropic) Scaling in Task Vector Composition: In transfer and few-shot learning, architectural parameter blocks are individually scaled, $$\Lambda_i = \mathrm{diag}(\lambda_i^{(1)},...,\lambda_i^{(m)})$$, with $\lambda_i^{(j)}$ learned via supervised or self-supervised loss minimization. With low intrinsic dimensionality, only a small number of coefficients per block/task are required (Zhang et al., 2024).
• Scaling Laws for Neural Model Loss: Power-law scaling coefficients $\alpha_N, N_c, \alpha_S, S_c, \alpha_B, B^*$ are fitted using regression against log-transformed observed losses for systematically varied $N, S, B$ (Su et al., 2024).

3. Applications Across Domains

Learned scaling factors are leveraged in structured prediction, generative modeling, physical simulation, and neural model scaling:

• Structured Prediction: In deep energy-based models (e.g., CRFs), correct scaling between potential terms is essential for single-stage joint training. Both offline normalization and online dynamic adjustment recover full accuracy as validated in OCR, chunking, and binary image segmentation tasks (Shevchenko et al., 2019).
• Generative Models: In latent space disentanglement, scaling factors parameterize attribute-controlled sample generation; e.g., FaceCook uses learned hypercoordinate bases and empirical scaling ranges per semantic class, directly synthesizing faces with precise features and elevated diversity, outperforming iterative latent editing (Wang et al., 2021).
• Chemical Physics: Uniform scaling factors for vibrational frequencies correct systematic overestimation by harmonic approximations. Meta-analysis across 1,495 literature instances reveals convergence to $s\approx 0.96$ with hybrid functionals and double/triple-zeta basis sets, subject to a lower error bound from anharmonicity (Trujillo et al., 2021).
• Parameter-Efficient Transfer Learning: aTLAS demonstrates blockwise scaling of task vectors ($\lambda_i^{(j)}$ per block/task) to achieve superior arithmetic and composition properties, reduce interference, and enhance transfer in few-shot or unsupervised scenarios. Blockwise selection and sparsity promote memory efficiency and scalibility (Zhang et al., 2024).
• Scaling Law Prediction for Large Models: Learned scaling factors $\alpha_N,N_c,\alpha_S,S_c,\alpha_B,B^*$ enable accurate prediction of minimum achievable loss, required steps/tokens, optimal batch size, and trajectory for models up to 33B parameters. These coefficients generalize under changes in context length, tokenization, and data, as long as the prefactors are refitted (Su et al., 2024).

4. Quantitative Results and Practical Guidance

Empirical findings validate the efficacy and generality of learned scaling factors:

Domain	Scaling Factor	Result Highlight
Structured Prediction (OCR)	α (global)	E2E w/ scaling: 97.3% (stage: 97.2%)
Deep NER & Segmentation	α (online/offline)	Recovers stage F1/IoU after tuning
Generative Faces (FaceCook)	α ∈ ℝ^k (hypercoord.)	LPIPS diversity +6% (0.681 vs 0.639)
Harmonic Frequency (Chemistry)	s (uniform)	ωB97X-D/6–31G*: s=0.9501, RMSE=25cm⁻¹
Task Vector PEFT (aTLAS)	λ_i^{(j)} (blockwise)	Few-shot acc. +2–4pt vs baseline
LLM Scaling Laws	α_N, N_c, ...	Accurate prediction loss up to 33B

This convergence is robust—e.g., in vibrational spectroscopy, nearly all practical DFT functionals converge to $s \sim 0.96$ and RMSE plateaus once anharmonic errors dominate (Trujillo et al., 2021). In PEFT, aTLAS blockwise scaling reduces pairwise disentanglement error by 2–3% and improves task arithmetic by 16–18% relative (Zhang et al., 2024). In structured prediction, learned scaling halves runtime and matches multi-stage accuracy across tasks (Shevchenko et al., 2019). In scaling laws, fitted coefficients from small models enable extrapolation to large-scale performance (Su et al., 2024).

5. Interference, Disentanglement, and Inductive Bias

Learned scaling factors are foundational in mitigating interference and fostering disentanglement:

• Interference Mitigation: Per-block scaling enables selective suppression or amplification of parameter blocks implicated in conflicting tasks, as shown in aTLAS, where independent $\lambda_i^{(j)}$ reduced interference from 6.7% to 4.3% (Zhang et al., 2024). This mechanism is distinct from uniform scaling and enables finer control in transfer, composition, and test-time adaptation.
• Disentanglement: In latent space generation, properly learned scaling factors align sample diversity with semantic boundaries, as the hypercoordinate ranges in FaceCook yield superior feature accuracy and increased LPIPS diversity (Wang et al., 2021).
• Model Inductive Bias: In structured predictors, learned balancing reduces gradient domination, preserves complementary learning, and sustains stability during joint end-to-end optimization (Shevchenko et al., 2019). In scaling law extrapolation, fitted constants encapsulate inductive biases due to context, dataset, and architecture (Su et al., 2024).

6. Limitations, Assumptions, and Deployment Considerations

Deployment and generalization of learned scaling factors require attention to methodological assumptions:

• Training Regime Constraints: Most scaling law coefficients presume “optimal regime” hyperparameter settings (learning rate, batch size) and access to sufficiently large data (Su et al., 2024).
• Model Family and Data Distribution: Fitted scaling factors must be re-estimated after significant changes in architecture, context length, tokenization, or task distribution; power-law exponents and prefactors are not universal (Su et al., 2024).
• Memory and Computation: Blockwise scaling yields major memory savings, requiring only $mn \ll D$ parameters if the intrinsic dimension is low (e.g., aTLAS with 4.4k coefficients vs millions in LoRA rank 16) (Zhang et al., 2024). Gradient-free optimization further reduces memory at the expense of throughput.
• Empirical Boundaries: In vibrational physics, an anharmonicity-imposed RMSE lower bound ($\sim$25 cm⁻¹) cannot be overcome with improved scaling (Trujillo et al., 2021). In joint energy optimization, failure to maintain scale ratios leads to catastrophic underlearning of energy terms (Shevchenko et al., 2019).

Continued research focuses on automated estimation, transferability, and universal scaling principles that retain empirical reliability across novel domains and architectures.