KANs with learnable univariate spline activations on edges achieve better accuracy than MLPs with fewer parameters, faster scaling, and direct visualization for scientific discovery.
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Deep Learning Scaling is Predictable, Empirically
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abstract
Deep learning (DL) creates impactful advances following a virtuous recipe: model architecture search, creating large training data sets, and scaling computation. It is widely believed that growing training sets and models should improve accuracy and result in better products. As DL application domains grow, we would like a deeper understanding of the relationships between training set size, computational scale, and model accuracy improvements to advance the state-of-the-art. This paper presents a large scale empirical characterization of generalization error and model size growth as training sets grow. We introduce a methodology for this measurement and test four machine learning domains: machine translation, language modeling, image processing, and speech recognition. Our empirical results show power-law generalization error scaling across a breadth of factors, resulting in power-law exponents---the "steepness" of the learning curve---yet to be explained by theoretical work. Further, model improvements only shift the error but do not appear to affect the power-law exponent. We also show that model size scales sublinearly with data size. These scaling relationships have significant implications on deep learning research, practice, and systems. They can assist model debugging, setting accuracy targets, and decisions about data set growth. They can also guide computing system design and underscore the importance of continued computational scaling.
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- abstract Deep learning (DL) creates impactful advances following a virtuous recipe: model architecture search, creating large training data sets, and scaling computation. It is widely believed that growing training sets and models should improve accuracy and result in better products. As DL application domains grow, we would like a deeper understanding of the relationships between training set size, computational scale, and model accuracy improvements to advance the state-of-the-art. This paper presents a large scale empirical characterization of generalization error and model size growth as training
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representative citing papers
GPT-3 shows that scaling an autoregressive language model to 175 billion parameters enables strong few-shot performance across diverse NLP tasks via in-context prompting without fine-tuning.
IRSL applies IRT to reduce scaling law estimation from O(M×N) to O(M+N) parameters, enabling reliable estimates with only 50 questions per benchmark after calibration and generalizable ability scores across related benchmarks.
LOSCAR-SGD combines local updates, sparse model averaging, and communication-computation overlap with a delay-corrected merge rule, providing convergence rates for smooth non-convex objectives under worker heterogeneity.
Ringmaster LMO extends delay-thresholding from ASGD to LMO-based momentum updates, providing convergence guarantees under (L0, L1)-smoothness and time-complexity bounds that recover optimal rates in the Euclidean case.
PEIRA learns predictive encoders by optimizing the trace of the optimal inter-view linear regressor, with only nontrivial global minimizers as stable equilibria that recover leading nonlinear canonical correlation subspaces.
Olivia harmonizes time series datasets via normalized power spectral density using a Harmonizer module and resonator-based HarmonicAttention, achieving state-of-the-art zero-shot, few-shot, and full-shot forecasting on TSLib, GIFT-Eval, and GluonTS benchmarks.
QuBD extends algorithmic complexity estimation to quantized DNN weights, revealing that complexity decreases during learning, increases with overfitting, follows grokking patterns, and correlates with generalization.
In extensive-width networks, features are recovered sequentially through sharp phase transitions, yielding an effective width k_c that unifies Bayes-optimal generalization error scaling as Θ(k_c d / n).
Inkheart SGD and M4 use bidirectional compression to achieve time complexities in distributed SGD that improve with worker count n and surpass prior lower bounds under a necessary structural assumption.
DBG mitigates boundary overlap in long-tailed learning by generating near-boundary samples, leading to better tail class accuracy and more separable decision spaces.
A cross-population framework for EEG Parkinson's detection using exhaustive 75 directional evaluations and nested validation shows asymmetric transfer and accuracy up to 94.1% when training diversity increases, supported by mixture risk theory.
Contextual entrainment decreases for semantic contexts but increases for non-semantic ones as LLMs scale, following power-law trends with 4x better resistance to misinformation but 2x more copying of arbitrary tokens.
DeepSeek-V2 delivers top-tier open-source LLM performance using only 21B active parameters by compressing the KV cache 93.3% and cutting training costs 42.5% via MLA and DeepSeekMoE.
Autoregressive transformers follow power-law scaling laws for cross-entropy loss with nearly universal exponents relating optimal model size to compute budget across four domains.
T5 casts all NLP tasks as text-to-text generation, systematically explores pre-training choices, and reaches strong performance on summarization, QA, classification and other tasks via large-scale training on the Colossal Clean Crawled Corpus.
SemiPrune uses a small labeled subset and semi-supervised pseudo-labeling to enable supervised dataset pruning methods, achieving state-of-the-art results on domain-specific, image-corrupted, and long-tailed datasets.
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A new scaling law L(N, D, T) = E + (L0 - E) h/(1+h) with h = a/N^α + b/T^β + c N^γ/D^δ that decomposes loss into undercapacity, undertraining, and overfitting terms and saturates between E and L0.
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citing papers explorer
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KAN: Kolmogorov-Arnold Networks
KANs with learnable univariate spline activations on edges achieve better accuracy than MLPs with fewer parameters, faster scaling, and direct visualization for scientific discovery.
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Language Models are Few-Shot Learners
GPT-3 shows that scaling an autoregressive language model to 175 billion parameters enables strong few-shot performance across diverse NLP tasks via in-context prompting without fine-tuning.
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Item Response Scaling Laws: A Measurement Theory Approach for Efficient and Generalizable Neural Scaling Estimation
IRSL applies IRT to reduce scaling law estimation from O(M×N) to O(M+N) parameters, enabling reliable estimates with only 50 questions per benchmark after calibration and generalizable ability scores across related benchmarks.
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LOSCAR-SGD: Local SGD with Communication-Computation Overlap and Delay-Corrected Sparse Model Averaging
LOSCAR-SGD combines local updates, sparse model averaging, and communication-computation overlap with a delay-corrected merge rule, providing convergence rates for smooth non-convex objectives under worker heterogeneity.
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Ringmaster LMO: Asynchronous Linear Minimization Oracle Momentum Method
Ringmaster LMO extends delay-thresholding from ASGD to LMO-based momentum updates, providing convergence guarantees under (L0, L1)-smoothness and time-complexity bounds that recover optimal rates in the Euclidean case.
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PEIRA: Learning Predictive Encoders through Inter-View Regressor Alignment
PEIRA learns predictive encoders by optimizing the trace of the optimal inter-view linear regressor, with only nontrivial global minimizers as stable equilibria that recover leading nonlinear canonical correlation subspaces.
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Olivia: Harmonizing Time Series Foundation Models with Power Spectral Density
Olivia harmonizes time series datasets via normalized power spectral density using a Harmonizer module and resonator-based HarmonicAttention, achieving state-of-the-art zero-shot, few-shot, and full-shot forecasting on TSLib, GIFT-Eval, and GluonTS benchmarks.
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Characterizing Learning in Deep Neural Networks using Tractable Algorithmic Complexity Analysis
QuBD extends algorithmic complexity estimation to quantized DNN weights, revealing that complexity decreases during learning, increases with overfitting, follows grokking patterns, and correlates with generalization.
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Sharp feature-learning transitions and Bayes-optimal neural scaling laws in extensive-width networks
In extensive-width networks, features are recovered sequentially through sharp phase transitions, yielding an effective width k_c that unifies Bayes-optimal generalization error scaling as Θ(k_c d / n).
-
Scalable Distributed Stochastic Optimization via Bidirectional Compression: Beyond Pessimistic Limits
Inkheart SGD and M4 use bidirectional compression to achieve time complexities in distributed SGD that improve with worker count n and surpass prior lower bounds under a necessary structural assumption.
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Decision Boundary-aware Generation for Long-tailed Learning
DBG mitigates boundary overlap in long-tailed learning by generating near-boundary samples, leading to better tail class accuracy and more separable decision spaces.
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Robust and Clinically Reliable EEG Biomarkers: A Cross Population Framework for Generalizable Parkinson's Disease Detection
A cross-population framework for EEG Parkinson's detection using exhaustive 75 directional evaluations and nested validation shows asymmetric transfer and accuracy up to 94.1% when training diversity increases, supported by mixture risk theory.
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Better and Worse with Scale: How Contextual Entrainment Diverges with Model Size
Contextual entrainment decreases for semantic contexts but increases for non-semantic ones as LLMs scale, following power-law trends with 4x better resistance to misinformation but 2x more copying of arbitrary tokens.
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DeepSeek-V2: A Strong, Economical, and Efficient Mixture-of-Experts Language Model
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Scaling Laws for Autoregressive Generative Modeling
Autoregressive transformers follow power-law scaling laws for cross-entropy loss with nearly universal exponents relating optimal model size to compute budget across four domains.
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Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer
T5 casts all NLP tasks as text-to-text generation, systematically explores pre-training choices, and reaches strong performance on summarization, QA, classification and other tasks via large-scale training on the Colossal Clean Crawled Corpus.
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Label-Efficient Dataset Pruning via Semi-Supervised Pseudo-Labeling
SemiPrune uses a small labeled subset and semi-supervised pseudo-labeling to enable supervised dataset pruning methods, achieving state-of-the-art results on domain-specific, image-corrupted, and long-tailed datasets.
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A Boundary-Layer Mechanism for One-Third Scaling in Online Softmax Classification
Derives α^{-1/3} scaling for generalization error in online softmax classification from boundary layers in a teacher-student model.
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Practical Scaling Laws: Converting Compute into Performance in a Data-Constrained World
A new scaling law L(N, D, T) = E + (L0 - E) h/(1+h) with h = a/N^α + b/T^β + c N^γ/D^δ that decomposes loss into undercapacity, undertraining, and overfitting terms and saturates between E and L0.
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AIPO: Learning to Reason from Active Interaction
AIPO adds active multi-agent consultation (Verify, Knowledge, Reasoning agents) plus custom importance sampling to RLVR training so LLMs expand their reasoning boundary and then operate without the agents.
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A Qualitative Test-Risk Mechanism for Scaling Behavior in Normalized Residual Networks
Depth expansion in normalized residual networks yields provable test-risk improvement through representational, optimization, and generalization gains under first-order descent and norm-control conditions.
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Policy-Guided Stepwise Model Routing for Cost-Effective Reasoning
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ZAYA1-8B Technical Report
ZAYA1-8B is a reasoning MoE model with 700M active parameters that matches larger models on math and coding benchmarks and reaches 91.9% on AIME'25 via Markovian RSA test-time compute.
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InfoLaw: Information Scaling Laws for Large Language Models with Quality-Weighted Mixture Data and Repetition
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The Power of Power Law: Asymmetry Enables Compositional Reasoning
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Large language model-enabled automated data extraction for concrete materials informatics
LLM pipeline extracts nearly 9,000 high-quality blended-cement concrete records from over 27,000 publications with F1 scores up to 0.97 and enables ML analyses showing benefits of large diverse datasets.
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Adaptive Test-Time Scaling for Zero-Shot Respiratory Audio Classification
TRIAGE adaptively scales test-time compute via tiered zero-shot stages for respiratory audio classification, reaching mean AUROC 0.744 across nine tasks while outperforming prior zero-shot methods.
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Unsupervised domain adaptation for radioisotope identification in gamma spectroscopy
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Model Merging Scaling Laws in Large Language Models
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Surprisingly High Redundancy in Electronic Structure Data Across Materials Explained by Low Intrinsic Dimensionality
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Does Math Reasoning Improve General LLM Capabilities? Understanding Transferability of LLM Reasoning
Math reasoning gains in LLMs rarely transfer to general domains; RL tuning generalizes while SFT causes forgetting and representation drift.
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Superposition Yields Robust Neural Scaling
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Learning to Reason under Off-Policy Guidance
LUFFY mixes off-policy reasoning traces into RLVR training via Mixed-Policy GRPO and regularized importance sampling, delivering over 6-point gains on math benchmarks and enabling training of weak models where on-policy RLVR fails.
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LLMs on the Line: Data Determines Loss-to-Loss Scaling Laws
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Two-Point Deterministic Equivalence for Stochastic Gradient Dynamics in Linear Models
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Inference Scaling Laws: An Empirical Analysis of Compute-Optimal Inference for Problem-Solving with Language Models
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Large Language Monkeys: Scaling Inference Compute with Repeated Sampling
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The Falcon Series of Open Language Models
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Textbooks Are All You Need
A 1.3B-parameter code model trained on 7B tokens of curated textbook and synthetic data achieves 50.6% on HumanEval, indicating data quality can enable strong performance at small scale.
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SemDeDup: Data-efficient learning at web-scale through semantic deduplication
SemDeDup removes semantic duplicates from datasets like LAION using pre-trained embeddings, cutting data by 50% with minimal performance loss and efficiency gains on C4.
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BLOOM: A 176B-Parameter Open-Access Multilingual Language Model
BLOOM is a 176B-parameter open-access multilingual language model trained on the ROOTS corpus that achieves competitive performance on benchmarks, with improved results after multitask prompted finetuning.
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eDiff-I: Text-to-Image Diffusion Models with an Ensemble of Expert Denoisers
An ensemble of stage-specialized text-to-image diffusion models improves prompt alignment over single shared-parameter models while preserving visual quality and inference speed.
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Language Models (Mostly) Know What They Know
Language models show good calibration when asked to estimate the probability that their own answers are correct, with performance improving as models get larger.
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Scaling Laws and Interpretability of Learning from Repeated Data
Repeating 0.1% of training data 100 times degrades an 800M parameter model's performance to that of a 400M model by damaging copying mechanisms and induction heads associated with generalization.
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A General Language Assistant as a Laboratory for Alignment
Ranked preference modeling outperforms imitation learning for language model alignment and scales more favorably with model size.
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Scaling Laws for Transfer
Effective data transferred from pre-training to fine-tuning is described by a power law in model parameter count and fine-tuning dataset size, acting like a multiplier on the fine-tuning data.
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Asymmetric Scaling Laws from Sparse Features
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Scaling Laws and Tradeoffs in Recurrent Networks of Expressive Neurons
Recurrent networks built from tunable expressive neurons reveal scaling laws with an optimal parameter split that shifts toward higher per-neuron complexity at larger scales.
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Rennala MVR: Improved Time Complexity for Parallel Stochastic Optimization via Momentum-Based Variance Reduction
Rennala MVR improves time complexity over Rennala SGD for smooth nonconvex stochastic optimization in heterogeneous parallel systems under a mean-squared smoothness assumption.
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Physical Foundation Models: Fixed hardware implementations of large-scale neural networks
Physical Foundation Models are fixed physical hardware realizations of foundation-scale neural networks that compute via inherent material dynamics, potentially delivering orders-of-magnitude gains in energy efficiency, speed, and density over digital systems.