Unified Neural Scaling Laws

Deep neural networks have achieved remarkable success across domains such as computer vision, natural language processing, mathematical reasoning, and reinforcement learning. This progress has been fueled by exponential growth in model sizes and training data volumes, exemplified by models like GPT series and Vision Transformers. Understanding how model performance scales with parameters, data, and compute is crucial for guiding efficient model development and resource allocation. Early foundational work, such as Kaplan et al.'s power-law scaling laws, revealed simple relationships between single factors (e.g., parameter count) and performance, enabling rough predictions and theoretical insights. However, real-world training involves simultaneous variation of multiple factors, including training steps, inference steps, compute budgets, and hyperparameters, whose interactions are complex and nonlinear. Existing scaling laws predominantly consider single dimensions or rely on empirical fits lacking theoretical unity, limiting their predictive power and generalizability. Addressing this gap requires a unified, multi-dimensional framework that can accurately model and extrapolate performance across diverse tasks and architectures.

The core problem is to develop an accurate, unified model that captures how deep neural network performance varies as multiple scaling dimensions change simultaneously. These dimensions include model parameter count, training dataset size, number of training steps, number of inference steps, compute resources, and hyperparameters. Key challenges include: 1) modeling nonlinear, high-order interactions among these factors; 2) ensuring the model generalizes across different tasks and architectures; 3) achieving robust extrapolation beyond observed training regimes to predict performance at larger scales. Solving this problem is critical for enabling principled design of large-scale models, optimizing training and inference efficiency, and effectively allocating computational resources in both research and industry contexts.

This work introduces several core innovations:

• �� Unified Multi-dimensional Modeling: UNSL integrates multiple scaling factors into a single nonlinear functional form, capturing their joint influence on performance.

• �� Explicit Interaction Terms: By modeling parameter interactions explicitly, UNSL captures complex nonlinear dependencies missed by prior single-variable scaling laws.

• �� Cross-task Generalization: UNSL demonstrates strong predictive accuracy across diverse domains including vision, language, math reasoning, and reinforcement learning, highlighting its broad applicability.

• �� Empirical Validation and Ablation: Extensive experiments validate UNSL's superior performance and identify the critical role of interaction terms through systematic ablation studies.

These innovations collectively advance the theoretical understanding of neural scaling and provide practical tools for efficient model scaling and resource management.

• �� Dataset and Task Selection: Utilize diverse datasets—ImageNet for vision, OpenWebText for language modeling, MATH dataset for mathematical reasoning, and Atari games for reinforcement learning—to cover a broad spectrum of tasks.

• �� Functional Form Design: Define UNSL as a nonlinear function E = a * P^b * D^c * S^d * I^e * C^f * exp(Σ interaction terms), where E is the evaluation metric, P is parameter count, D is data size, S is training steps, I is inference steps, and C is compute budget.

• �� Parameter Estimation: Apply nonlinear least squares optimization to fit model parameters and interaction coefficients using observed performance data.

• �� Cross-validation and Extrapolation Tests: Evaluate model fit on held-out data and test extrapolation to unseen scaling regimes and tasks.

• �� Ablation Studies: Remove interaction terms and power-law components to assess their impact on predictive accuracy.

• �� Comparative Baselines: Benchmark against traditional single-variable scaling laws and recent empirical multi-variable models to demonstrate improvements.

The experimental setup includes:

• �� Datasets: ImageNet (1.2M images, 1000 classes), OpenWebText (large-scale web text corpus), MATH dataset (math problems), and Atari 2600 games.

• �� Baselines: Kaplan et al.'s single-variable power-law scaling laws and recent multi-factor empirical models.

• �� Metrics: Top-1 accuracy for ImageNet, perplexity for language modeling, accuracy for math reasoning, and average game score for Atari.

• �� Hyperparameters: Varied training steps, inference steps, and compute budgets to assess multi-dimensional scaling.

• �� Ablations: Systematic removal of interaction terms and nonlinear components to evaluate their contributions.

• �� Evaluation: Quantitative comparison of prediction errors (e.g., mean squared error) across models and tasks.

Key findings include:

• �� UNSL achieves a 15-17% reduction in prediction error on ImageNet and OpenWebText compared to traditional single-variable scaling laws.

• �� On the MATH dataset, UNSL maintains prediction errors within 5% when extrapolating performance under simultaneous scaling of parameters and data.

• �� In Atari reinforcement learning tasks, UNSL captures complex interactions between training steps and compute resources, reducing prediction error by 12% relative to baselines.

• �� Ablation studies reveal that removing interaction terms increases prediction error by approximately 20%, underscoring their importance.

• �� UNSL demonstrates robust cross-task generalization, accurately modeling scaling behavior across diverse architectures and domains.

UNSL has several practical applications:

• �� Large-scale Model Design: Provides quantitative guidance for selecting model sizes and training data volumes to achieve target performance efficiently.

• �� Compute Resource Optimization: Informs allocation of GPU hours and training steps to balance cost and accuracy.

• �� Inference Efficiency: Models the trade-off between inference steps and accuracy, enabling optimized deployment.

• �� Cross-task Performance Prediction: Supports transfer learning and multi-task learning by predicting performance across varied tasks and architectures.

These applications facilitate more efficient and sustainable development of deep learning systems in both research and industry.

UNSL's limitations include:

• �� Extrapolation to Extremely Large Models: Current validation is limited to small and medium-large models; behavior at ultra-large scales remains uncertain.

• �� Task-specific Nonlinearities: Assumes stable interaction effects across tasks, potentially missing unique nonlinearities in specialized domains.

• �� Hardware Heterogeneity: Compute resource modeling is coarse and does not differentiate hardware architectures, limiting applicability in heterogeneous environments.

• �� Hyperparameter Complexity: Does not fully capture the high-dimensional hyperparameter space's influence on performance.

• �� Dynamic Training Strategies: Lacks modeling of adaptive training schedules and multi-phase training processes common in practice.