Scaling laws govern predictive uncertainties in deep learning models.
problem Understanding predictive uncertainties in deep learning models.
method Empirical analysis and approximate Bayesian inference on vision and language tasks.
result Scaling laws exist for various measures of predictive uncertainty in deep learning models.
Empirical law predicts accuracy of Google Translate's translation chains.
problem Predicting accuracy in machine translation with multiple hops.
method Empirical testing of Google Translate's sequential translation.
result Accuracy decreases with the number of translating hops, following a power law.
The paper presents a multi-power law for predicting loss curves across different learning rate schedules.
problem Understanding and optimizing the relationship between model performance and hyperparameters, especially learning rates.
method Proposes a multi-power law that combines power laws based on the sum of learning rates and additional laws for loss reduction due to decay.
result The multi-power law accurately predicts loss curves for unseen learning rate schedules and finds a schedule that outperforms cosine learning rate.
The paper predicts loss scaling across different datasets and compute scales.
problem Predicting loss scaling across different datasets and compute scales.
method Derive shifted power law relationships between train and test losses.
result Shifted power law relationships hold for various datasets and tasks, improving prediction accuracy.
Predicts language model performance from public models without training.
problem Understanding how language model performance varies with scale.
method Observational approach building scaling laws from publicly available models.
result Smooth, sigmoidal behavior of emergent phenomena is predictable from small models.
A power-law fit to the empirical inference-compute frontier in LOB prediction suggests a scaling-law-style frontier.
problem Limit order book prediction
method Using a suite of models ranging from small decision trees to neural LOB architectures
result A power-law fit to the low- and mid-compute non-MLPLOB frontier extrapolates across multiple orders of magnitude and attains R2=0.941 on the excluded high-compute MLPLOB target frontier. Sloth predicts LLM performance using latent skills across families.
problem Variations in benchmark performance due to differences in training configurations and data processing across model families.
method Sloth uses publicly available benchmark data and assumes LLM performance is driven by latent skills influenced by model size and training tokens. It exploits correlations across benchmarks to provide accurate predictions.
result Sloth predicts LLM performance accurately and offers insights into scaling behaviors for complex tasks.
The paper improves conformal prediction by analyzing the beta law of conditional coverage.
problem Improving finite-sample marginal coverage guarantees for non-i.i.d. data.
method The method uses Wasserstein distances to quantify deviations from the beta law of conditional coverage.
result The framework provides direct bounds on marginal coverage gaps and bad-calibration probabilities.
The paper develops a method to discover causal relations and predict material laws with uncertainty quantification.
problem Discovering causal relations and predicting material laws with uncertainty in civil engineering applications.
method The paper develops a causal discovery algorithm to infer causal relations among time-history data. It uses a deep neural network with dropout layers for uncertainty quantification and propagates predictions through a causal graph.
result The method accurately predicts material laws and quantifies uncertainty, as demonstrated in two numerical examples.
Unified theory for neural scaling laws in hierarchically compositional data.
problem Understanding neural scaling laws in hierarchically compositional data.
method Probabilistic context-free grammars and power-law distributed production rules.
result Unified learning curve behavior for classification and next-token prediction tasks.
Law explains how LLMs learn to predict next tokens.
problem Understanding how LLMs process input data internally.
method Introduced a precise law governing token embeddings in LLMs.
result Each layer equally contributes to next-token prediction accuracy.
Trimming helps in conformal prediction when it separates anomaly scores.
problem Effectiveness of trimming in conformal prediction under contamination.
method Analyse fixed-threshold trimming as a replacement of the contaminated calibration law with a retained law.
result Trimming helps when it separates anomaly scores, reducing clean-target coverage to a one-dimensional score-CDF transfer problem.
Model predicts neural network performance scaling laws across various factors.
problem Understanding the performance of neural networks across different training factors.
method Random feature model trained with gradient descent, analyzing compute-optimal scaling laws.
result Predicts asymmetric compute-optimal scaling rule and behavior of training and test loss gap.
Theory predicts neural scaling exponents from language statistics.
problem No existing theory could quantitatively predict neural scaling exponents.
method Isolated two key statistical properties of language.
result Derives a simple formula predicting neural scaling exponents.
New method predicts and optimizes test-time scaling for LLMs.
problem Lack of principled guidance on scaling LLMs efficiently.
method Tail-guided search to predict and allocate compute.
result SLG Search achieves higher rewards with less compute.
This paper explains how predictable order flow can lead to Brownian motion in financial prices.
problem Why financial prices exhibit Brownian motion despite predictable order flow.
method Generalized Lillo-Mike-Farmer model to nonlinear price-impact dynamics, mapping to Lévy-walk model.
result Price dynamics remain diffusive under the square-root law, even with persistent order flow.
We provide scientific foundations for athletic performance prediction on an individual level, exposing the phenomenology of individual athletic running performance in the form of a low-rank model dominated by an individual power law. We present, evaluate, and compare a selection of methods for prediction of individual …
Forecasting technological progress is of great interest to engineers, policy makers, and private investors. Several models have been proposed for predicting technological improvement, but how well do these models perform? An early hypothesis made by Theodore Wright in 1936 is that cost decreases as a power law of cumul…
Space exploration technology advances exponentially, consistent with Moore's and Wright's laws.
problem Predicting the advancement of space exploration technology.
method Analysis of Moore's and Wright's laws applied to space exploration technology.
result Spacecraft technology advances exponentially, consistent with Moore's and Wright's laws.
Sharp theory of neural network scaling laws for hierarchical targets.
problem Learning hierarchical multi-index models in neural networks.
method Sharp information-theoretic scaling laws derived for two-layer neural networks.
result Optimal rates achieved by a simple spectral estimator.
This work investigates power laws in deep neural network ensembles and predicts their performance.
problem Understanding the performance of deep neural network ensembles and their optimal structure.
method Investigated the behavior of negative log-likelihood (CNLL) of a deep ensemble as a function of ensemble size and member network size, identifying power law dependencies.
result One large network may perform worse than an ensemble of several medium-size networks, known as a memory split.
We use GANs and signatures to approximate conditional laws in filtering and prediction of diffusion processes.
problem Approximating conditional laws for diffusion processes with noisy observations.
method Conditional GANs combined with signatures for approximation.
result Efficient approximation of conditional laws for diffusion processes.
A new scaling law predicts optimal batch size for training models.
problem Finding the optimal batch size for training models efficiently.
method Proposed a three-term scaling law that considers model size, training data, training steps, and batch size.
result The three-term law accurately recovers the optimal batch size and can be robustly fit with fewer training runs.
New neural scaling law found for simple quadratic function.
problem Neural scaling laws and their predictions for model performance.
method Analysis of neural networks, lottery ticket ensembling, statistical interpretation.
result Found a new scaling law (α=1) for a simple quadratic function, contradicting previous theories. Study reveals neural scaling laws in random graphs and natural language models.
problem Understanding the origin of neural scaling laws in complex systems.
method Examined scaling laws in transformers trained on random walks and simplified natural language models.
result Neural scaling laws emerge in the absence of power law structure in data correlations.
A hierarchical model shows how scaling laws emerge from sequential feature recovery.
problem Emergence of scaling laws from feature learning in multi-layer networks.
method Layer-wise spectral algorithm adapted to compositional structure, sequential feature detection.
result Sequential detection of latent features, leading to explicit power-law decay of prediction error.
Study predicts Gaussian Volterra processes with noisy Brownian motion.
problem Predicting Gaussian Volterra processes with hidden Brownian motion.
method Regular conditional law analysis under model disturbances.
result Developed method for variance reduction in measurement errors.
Study explains Zipf's law using geometric mechanisms from a finite alphabet.
problem Explains Zipf's law in language without relying on linguistic elements.
method Uses the Full Combinatorial Word Model (FCWM) to generate geometric distributions of word lengths.
result Supports predictions of power-law rank-frequency curves, matching various languages.
The study explains transformer scaling laws using statistical and approximation theories.
problem Understanding why transformer scaling laws exist for large models trained on low-dimensional data.
method Established statistical estimation and mathematical approximation theories for transformers on low-dimensional manifolds.
result Predicted a power law between generalization error and model and data sizes, with power depending on intrinsic data dimension.
A simple model explains inference scaling in neural models.
problem Understanding how model performance improves with repeated inference attempts.
method A statistical ansatz based on memorization to study inference scaling laws.
result Inference loss exhibits a power law decay with increasing trials.
Predicts stock market crashes using rational bubble model.
problem Financial market crashes prediction.
method White box model based on rational bubble theory.
result Successfully predicts major crashes in Dow Jones and Bitcoin markets.
Based on empirical financial time-series, we show that the "silence-breaking" probability follows a super-universal power law: the probability of observing a large movement is inversely proportional to the length of the on-going low-variability period. Such a scaling law has been previously predicted theoretically [R. …
New law predicts first extinction in resampling processes.
problem Intractable extinction times in resampling processes.
method Modeling multinomial updates as independent square-root diffusions.
result Closed-form law for first-extinction time with linear cost.
Study on RL on volatility surfaces, proving no free lunch for law-seeking methods.
problem Aligning RL agents with no-arbitrage laws in volatile markets.
method Built a law manifold, defined penalties, and used a Goodhart decomposition.
result No free lunch theorem: Law-seeking RL cannot outperform baselines.
Large models follow power laws in performance with dataset size or parameters.
problem Understanding neural scaling laws in large language models.
method Joint generative data model and random feature model.
result Modeling and solving the dual limit reveals insights into scaling laws.
In this paper, we introduce a physics-driven regularization method for training of deep neural networks (DNNs) for use in engineering design and analysis problems. In particular, we focus on prediction of a physical system, for which in addition to training data, partial or complete information on a set of governing la…
Our purpose is to relate the Fokker-Planck formalism proposed by [Friedrich et al., Phys. Rev. Lett. 84, 5224 (2000)] for the distribution of stock market returns to the empirically well-established power law distribution with an exponent in the range 3-5. We show how to use Friedrich et al.'s formalism to predict that…
This paper addresses the statistical properties of time series driven by rational bubbles a la Blanchard and Watson (1982), corresponding to multiplicative maps, whose study has recently be revived recently in physics as a mechanism of intermittent dynamics generating power law distributions. Using insights on the beha…
We quantify predictive uncertainty using the posterior predictive variance.
problem Quantifying uncertainty in predictive models.
method Using the law of total variance, we generate expansions for the posterior predictive variance.
result Identify the main contributors to prediction intervals and quantify term-wise uncertainty.
Models predict order book recovery from aggressive trading follows a simple t^1/3 scaling.
problem Understanding order book dynamics in financial markets.
method Developed macroscopic models and solved numerically and asymptotically.
result Order book recovery follows a t^1/3 scaling law.
Scaling laws in linear regression explain model performance improvements with size and data.
problem Disagreement between empirical neural scaling laws and conventional wisdom on variance error.
method Infinite dimensional linear regression setup, one-pass SGD, Gaussian prior, power-law spectrum.
result Variance error is dominated by other errors, disappearing from the bound due to SGD's implicit regularization.
Law explains how deep networks separate data for classification.
problem Black-box nature of deep learning limits architecture design and interpretation.
method Studied how deep neural networks process data in intermediate layers.
result Law of geometric data separation emerges in various architectures and datasets.
Study predicts NFT bubbles using LPPL model.
problem Tackles bubble prediction of NFTs.
method Applied logarithmic periodic power law (LPPL) model to NFT price data.
result NFTs, Decentraland, and ArtBlocks are in bubbles, while Ethereum Name Service is in a negative bubble.
Study shows how anisotropic data affects learning dynamics in phase retrieval.
problem Understanding learning dynamics in phase retrieval with anisotropic Gaussian inputs.
method Developed a tractable reduction to reveal a three-phase trajectory and derived scaling laws.
result Found that anisotropy leads to a three-phase trajectory: fast escape, slow convergence, and spectral-tail learning.
Benford's law states that in data sets from different phenomena leading digits tend to be distributed logarithmically such that the numbers beginning with smaller digits occur more often than those with larger ones. Particularly, the law is known to hold for different types of financial data. The Illicit Financial Flow…
SAEs struggle with curved activation manifolds, revealing layer-dependent scaling laws.
problem Sparse autoencoders' reconstruction error varies across layers, not fitting existing scaling laws.
method Cross-layer study of 844 SAE checkpoints, fitting and regressing on manifold geometry.
result Manifold geometry predicts layer-dependent width exponents in SAEs, with transferable coefficients.
Parsimonious neural networks discover interpretable physical laws from data.
problem Discovering interpretable physical laws from data using machine learning.
method Combining neural networks with evolutionary optimization to balance accuracy and parsimony.
result Developed models for classical mechanics and materials melting temperature prediction.
We simplify Volterra process predictions by reducing dimensionality and using a tailored deep learning model.
problem Predicting the conditional law of Volterra processes with stochastic volatility is challenging due to high dimensionality and non-smoothness.
method We developed a stable dimension reduction technique onto a low-dimensional statistical manifold of non-positive curvature and introduced a sequentially deep learning model tailored to this geometry.
result Our model can approximate the conditional law of Volterra processes with approximation rates achievable only with very large networks.