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1. A Tale of Tails: Model Collapse as a Change of Scaling Laws

   Feng, Yunzhen; Dohmatob, Elvis; Yang, Pu; Charton, Francois; Kempe, Julia (May 2024, ICLR Workshop on Navigating and Addressing Data Problems for Foundation Models (DPFM))

   As AI model size grows, neural scaling laws have become a crucial tool to predict the improvements of large models when increasing capacity and the size of original (human or natural) training data. Yet, the widespread use of popular models means that the ecosystem of online data and text will co-evolve to progressively contain increased amounts of synthesized data. In this paper we ask: How will the scaling laws change in the inevitable regime where synthetic data makes its way into the training corpus? Will future models, still improve, or be doomed to degenerate up to total (model) collapse? We develop a theoretical framework of model collapse through the lens of scaling laws. We discover a wide range of decay phenomena, analyzing loss of scaling, shifted scaling with number of generations, ``un-learning" of skills, and grokking when mixing human and synthesized data. Our theory is validated by large-scale experiments with a transformer on an arithmetic task and text generation using the large language model Llama2. 

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2. Understanding Scaling Laws with Statistical and Approximation Theory for Transformer Neural Networks on Intrinsically Low-dimensional Data

   Havrilla, Alex; Liao, Wenjing (January 2025, Advances in Neural Information Processing Systems 37 (NeurIPS 2024) Main Conference Track)

   When training deep neural networks, a model's generalization error is often observed to follow a power scaling law dependent both on the model size and the data size. Perhaps the best known example of such scaling laws are for transformer-based large language models (**LLMs**), where networks with billions of parameters are trained on trillions of tokens of text. Yet, despite sustained widespread interest, a rigorous understanding of why transformer scaling laws exist is still missing. To answer this question, we establish novel statistical estimation and mathematical approximation theories for transformers when the input data are concentrated on a low-dimensional manifold. Our theory predicts a power law between the generalization error and both the training data size and the network size for transformers, where the power depends on the intrinsic dimension d of the training data. Notably, the constructed model architecture is shallow, requiring only logarithmic depth in d. By leveraging low-dimensional data structures under a manifold hypothesis, we are able to explain transformer scaling laws in a way which respects the data geometry. Moreover, we test our theory with empirical observation by training LLMs on natural language datasets. We find the observed empirical scaling laws closely agree with our theoretical predictions. Taken together, these results rigorously show the intrinsic dimension of data to be a crucial quantity affecting transformer scaling laws in both theory and practice. 

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3. Scaling Laws for Emulation of Stellar Spectra

   https://doi.org/10.33232/001c.140607  

   Różański, Tomasz; Ting, Yuan-Sen (January 2025, The Open Journal of Astrophysics)

   Neural network-based emulators for the inference of stellar parameters and elemental abundances represent an increasingly popular methodology in modern spectroscopic surveys. However, these approaches are often constrained by their emulation precision and domain transfer capabilities. Greater generalizability has previously been achieved only with significantly larger model architectures, as demonstrated by Transformer-based models in natural language processing. This observation aligns with neural scaling laws, where model performance predictably improves with increased model size, computational resources allocated to model training, and training data volume. In this study, we demonstrate that these scaling laws also apply to Transformer-based spectral emulators in astronomy. Building upon our previous work with TransformerPayne and incorporating Maximum Update Parametrization techniques from natural language models, we provide training guidelines for scaling models to achieve optimal performance. Our results show that within the explored parameter space, clear scaling relationships emerge. These findings suggest that optimal computational resource allocation requires balanced scaling. Specifically, given a tenfold increase in training compute, achieving an optimal seven-fold reduction in mean squared error necessitates an approximately 2.5-fold increase in dataset size and a 3.8-fold increase in model size. This study establishes a foundation for developing spectral foundational models with enhanced domain transfer capabilities. 

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4. Neural Scaling Laws From Large-N Field Theory: Solvable Model Beyond the Ridgeless Limit

   https://doi.org/10.1088/2632-2153/adc872  

   Zhang, Zhengkang (April 2025, Machine Learning: Science and Technology)

   Abstract Many machine learning models based on neural networks exhibit scaling laws: their performance scales as power laws with respect to the sizes of the model and training data set. We use large-N field theory methods to solve a model recently proposed by Maloney, Roberts and Sully which provides a simplified setting to study neural scaling laws. Our solution extends the result in this latter paper to general nonzero values of the ridge parameter, which are essential to regularize the behavior of the model. In addition to obtaining new and more precise scaling laws, we also uncover a duality transformation at the diagrams level which explains the symmetry between model and training data set sizes. The same duality underlies recent efforts to design neural networks to simulate quantum field theories. 

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5. Inverse Scaling: When Bigger Isn’t Better

   McKenzie, IR; Lyzhov, A; Pieler, M; Parrish, A; Mueller, A; Prabhu, A; McLean, E; Kirtland, A; Ross, A; Liu, A; et al (February 2024, Transactions on machine learning research)

   Work on scaling laws has found that large language models (LMs) show predictable improvements to overall loss with increased scale (model size, training data, and compute). Here, we present evidence for the claim that LMs may show inverse scaling, or worse task performance with increased scale, e.g., due to flaws in the training objective and data. We present empirical evidence of inverse scaling on 11 datasets collected by running a public contest, the Inverse Scaling Prize, with a substantial prize pool. Through analysis of the datasets, along with other examples found in the literature, we identify four potential causes of inverse scaling: (i) preference to repeat memorized sequences over following in-context instructions, (ii) imitation of undesirable patterns in the training data, (iii) tasks containing an easy distractor task which LMs could focus on, rather than the harder real task, and (iv) correct but misleading few-shot demonstrations of the task. We release the winning datasets at https://inversescaling.com/data to allow for further investigation of inverse scaling. Our tasks have helped drive the discovery of U-shaped and inverted-U scaling trends, where an initial trend reverses, suggesting that scaling trends are less reliable at predicting the behavior of larger-scale models than previously understood. Overall, our results suggest that there are tasks for which increased model scale alone may not lead to progress, and that more careful thought needs to go into the data and objectives for training language models. 

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