Stochastic gradient descent converges to universal limits in high dimensions.
problem Statistical tasks in high dimensions with specific data projections.
method Stochastic gradient descent applied to mixture distributions, proving universality of limits.
result The ODE limits are universal for mixtures of arbitrary product distributions.
The paper proves universality in optimization problems with i.i.d. random vectors.
problem Optimization problems with i.i.d. random vectors and their projections.
method Proves universality of empirical risk minimization under specific conditions.
result The minimum value of the optimization problem is universal and depends only on the mean and covariance of the random vectors.
Study reveals universal statistics of Fisher information in deep neural networks.
problem Characterizing Fisher information in deep neural networks.
method Used mean field theories with random weights and large width limits.
result Most eigenvalues of Fisher information matrix are close to zero, while the maximum eigenvalue is large.
Modified relative universality for unbiasedness and consistency in dimension reduction.
problem Gap in proof of unbiasedness and Fisher consistency in relative universality.
method Modified definition of relative universality using ǫ-measurability.
result Established unbiasedness and Fisher consistency rigorously.
Study shows perceptrons with random labels perform similarly to Gaussian data.
problem The assumption of Gaussian input data is often seen as a limitation in machine learning.
method Analyzed generalized linear classification (perceptron model) with random labels.
result Minimum training loss is independent of data covariance for high-dimensional input data.
The study uncovers universality laws for Gaussian mixtures in generalized linear models.
problem Understanding the asymptotic behavior of estimators in Gaussian mixture models.
method Investigates the asymptotic joint statistics of generalized linear estimators from empirical risk minimization and Gibbs sampling.
result Characterizes conditions under which the joint statistics depend only on means and covariances of class conditional features.
FedRec learns universal receivers for fading channels without channel statistics.
problem Training neural network-based receivers for diverse fading channels without accurate statistics.
method Federated learning of a MAP detector for downlink fading channels.
result Performance approaches MAP without channel statistics, reduced communication overhead.
The massive amount of available data potentially used to discover patters in machine learning is a challenge for kernel based algorithms with respect to runtime and storage capacities. Local approaches might help to relieve these issues. From a statistical point of view local approaches allow additionally to deal with …
Survey of universal portfolio techniques for minimizing investment regret.
problem Minimizing investment regret in algorithmic trading.
method Explains various universal portfolio techniques and their proofs.
result Coverage of fundamental concepts and algorithms in regret minimization.
Softmax attention approximates complex functions and subsumes many known universal approximators.
problem Universal approximation of continuous sequence-to-sequence functions.
method Interpolation-based analysis of attention's internal mechanism, showing its ability to approximate ReLU functions.
result Softmax attention is a universal approximator for continuous sequence-to-sequence functions.
A universal method for hypothesis tests and confidence sets without regularity conditions.
problem Difficult inference in irregular statistical models.
method Modified likelihood ratio statistic (split LRT).
result Works for any parametric and some nonparametric models.
We investigate the statistics of records in a random sequence {xB(0)=0,xB(1),⋯,xB(n)=xB(0)=0} of n time steps. The sequence xB(k)'s represents the position at step k of a random walk `bridge' of n steps that starts and ends at the origin. At each step, the increment of the position is a random ju…
Paper proves existence of a universal codebook for low-precision quantization.
problem Optimizing low-precision approximation of matrix products in machine learning.
method Develops a universal codebook that is near-optimal for all possible statistics of input data.
result Proves existence of a universal codebook with a 0.11 bit per dimension reduction in rate.
Statistical machine learning plays an important role in modern statistics and computer science. One main goal of statistical machine learning is to provide universally consistent algorithms, i.e., the estimator converges in probability or in some stronger sense to the Bayes risk or to the Bayes decision function. Kerne…
Paper shows how online betting algorithms' regret can be used to create tight confidence sequences.
problem Estimating the expectation of random variables from samples and creating time-uniform confidence sequences.
method Converts the regret guarantee of universal portfolio algorithms into time-uniform concentration inequalities and confidence sequences.
result Numerically obtained confidence sequences are never vacuous and satisfy the law of iterated logarithm.
The common assumption of universal behavior in stock market data can sometimes lead to false conclusions. In statistical physics, the Hurst exponents characterizing long-range correlations are often closely related to universal exponents. We show, that in the case of time series of the traded value, these Hurst exponen…
This paper explores the limits of deep learning in poly-time.
problem Characterizing function distributions that deep learning can or cannot learn efficiently.
method Analysis of SGD and GD-based deep learning approaches, proving universality and non-universality results.
result SGD-based deep learning is efficiently universal, while GD-based is not, especially with large batches.
A plethora of natural, artificial and social systems exist which do not belong to the Boltzmann-Gibbs (BG) statistical-mechanical world, based on the standard additive entropy SBG and its associated exponential BG factor. Frequent behaviors in such complex systems have been shown to be closely related to q-stati…
We uncover scaling laws and statistical structure in complex datasets.
problem Understanding universal traits in complex datasets.
method Analogizing data to physical systems, using statistical physics and RMT.
result Real-world datasets and Gaussian data with long-range correlations share the same RMT universality class.
Deep neural networks near edge of chaos show universal scaling laws.
problem Understanding the behavior of deep neural networks near critical points.
method Analogy to absorbing phase transitions in statistical mechanics, deterministic propagation dynamics, mean-field and directed percolation universality classes.
result Deep neural networks exhibit universal scaling laws near the edge of chaos.
Simple construction for universal quantum gates.
problem Designing efficient quantum gates for topological computers.
method Demonstrated a simple construction for unitary solutions of the braided Yang-Baxter equation in any dimension.
result Proved the existence of universal quantum gates in any dimension.
The goal of this paper is to show that there exists a simple, yet universal statistical logic of spectral graph analysis by recasting it into a nonparametric function estimation problem. The prescribed viewpoint appears to be good enough to accommodate most of the existing spectral graph techniques as a consequence of …
The study uncovers the breakdown of Gaussian universality in high-dimensional empirical risk minimization.
problem Understanding the breakdown of Gaussian universality in high-dimensional empirical risk minimization.
method Extending the Convex Gaussian Min-Max Theorem to non-Gaussian settings, deriving asymptotic min-max characterizations, and proving asymptotic equivalence of regularizers.
result The projection of the ERM estimator onto a test covariate approximately follows a Gaussian convolution under certain conditions.
Single-head transformers with a single self-attention layer can approximate any sequence-to-sequence function and are efficient under certain conditions.
problem Statistical and computational limits of prompt tuning for transformer-based models.
method Investigation of single-head transformers with a single self-attention layer, proving universality and efficiency under SETH.
result Existence of almost-linear time prompt tuning inference algorithms under certain conditions.
Transformers enable in-context learning with guarantees for a wide range of tasks.
problem How to enable in-context learning with transformers for various tasks.
method Developed a universal approximation theory integrating Barron's function approximation with transformer capabilities.
result Transformers can approximate any target function with vanishingly small risk using a few in-context examples.
We use methods of random matrix theory to analyze the cross-correlation matrix C of price changes of the largest 1000 US stocks for the 2-year period 1994-95. We find that the statistics of most of the eigenvalues in the spectrum of C agree with the predictions of random matrix theory, but there are deviations for a fe…
Using the tools developed for statistical physics, we simultaneously analyze statistical properties of the Jakarta and Kuala Lumpur Stock Exchange indices. In spite of the small number of data used in the analysis, the result shows the universal behavior of complex systems previously found in the leading stock indices.…
Unified framework for understanding GRPO as U-statistic.
problem Theoretical properties of GRPO remain less studied.
method Unified framework through classical U-statistics.
result GRPO is asymptotically equivalent to an oracle policy gradient algorithm.
This study reveals statistical patterns in ERC20 token transactions on Ethereum blockchain.
problem Understanding transactional dynamics in decentralized systems.
method Examined over 44 million ERC20 token transfers, categorized by address type (EOA or SC), and analyzed using scaling laws.
result EOA-driven transactions exhibit consistent statistical behavior, while SC-driven activity displays sublinear scaling and bursty activity.
New theorem for generalized group sparsity improves consistency and convergence rates.
problem Improving statistical inference in high-dimensional data with element-wise and group-wise sparsity.
method Developed a generalized version of Sparse-Group Lasso and proved a universal theorem for consistency and convergence rates.
result Obtained results on consistency and convergence rates for different forms of double sparsity regularization.
Generative diffusion models exhibit phase transitions in statistical mechanics, impacting their performance.
problem Understanding the performance and capabilities of generative diffusion models.
method Reformulating generative diffusion models using statistical mechanics, focusing on phase transitions and symmetry breaking.
result Generative diffusion models undergo second-order phase transitions with mean-field universality, critical instability, and mean-field critical exponents.
The Bayesian framework is a well-studied and successful framework for inductive reasoning, which includes hypothesis testing and confirmation, parameter estimation, sequence prediction, classification, and regression. But standard statistical guidelines for choosing the model class and prior are not always available or…
New approach to linear regression using universal learning.
problem Applying universal learning to linear regression.
method Using a Gaussian error hypothesis class and Predictive Normalized Maximum Likelihood (pNML) solution.
result Linear regression can generalize even with over-parametrized models under certain conditions.
Paper shows statistical-computational gaps in learning sparse mixtures and robust estimation.
problem Statistical-computational gaps in learning sparse mixtures and robust estimation.
method Average-case reduction techniques, Imbalanced Sparse Gaussian Mixtures, and algorithmic change of measure.
result New hardness results for robust sparse mean estimation, semirandom planted dense subgraph, and universality principle for sparse mixture problems.
Squared families are a new model class derived from linear transformations, offering convenient properties and universal approximation.
problem Developing a new class of probability models that are easier to handle and have useful properties.
method Introducing squared families as families of probability densities obtained by squaring a linear transformation of a statistic, and showing their properties and applications.
result Squared families have convenient properties and can approximate target densities well.
The paper demonstrates that falsifiability is fundamental to learning. We prove the following theorem for statistical learning and sequential prediction: If a theory is falsifiable then it is learnable -- i.e. admits a strategy that predicts optimally. An analogous result is shown for universal induction.
Diffusion models generate data with Gaussian Universality, matching linear model test errors.
problem Analyzing the performance of models trained on synthetic data generated by diffusion models.
method Investigates Gaussian Universality for data distributions generated via diffusion models, matching test errors of linear models trained on synthetic data to Gaussian Mixture models.
result The test error of a linear model trained on diffusion-generated data matches the test error of a linear model trained on Gaussian Mixture data with matching means and covariances per class.
Study exact limits of matrix reconstruction from noisy projections.
problem Reconstructing matrices from linear projections with high-dimensional data.
method Asymptotic analysis, universality properties, and generalized linear models.
result Exact asymptotic equations for optimal learning performance.
The paper addresses the gap between theoretical and practical confidence set widths in universal inference.
problem Inference procedures can be overly conservative, leading to wider confidence sets than expected.
method The authors identify the source of asymptotic conservativeness and propose a remedy based on studentization and bias correction.
result The proposed method achieves exact asymptotic coverage at the nominal 1−α level, even under model misspecification. Superposition accelerates training to a universal power-law exponent.
problem Training dynamics in neural networks.
method Teacher-student framework and analytic theory.
result Superposition leads to a universal power-law exponent of ~1, independent of data and channel statistics.
This article is written for the online newspaper "The Photon" published by the Department of Physics, University of Maryland. The article describes econophysics research done in the group of Victor Yakovenko. It briefly surveys the subjects "Statistical Mechanics of Money, Income, and Wealth" and "Probability Distribut…
Persistence diagrams from random matrices follow RMT universality, offering a new spectral diagnostic.
problem Understanding spectral properties of random matrices using topological data analysis.
method Applying Morse theory to persistence diagrams of quadratic forms restricted to unit spheres.
result Persistence entropy outperforms traditional level spacing ratios in discriminating random matrix ensembles.
Empirical study on trends reversion in financial markets.
problem Understanding when trends in financial markets revert.
method Polynomial regression and bootstrapping on 30 years of daily futures prices.
result Trends revert when they reach a critical level of statistical significance.
Using detailed statistical analyses of the size distribution of a universe of equity exchange-traded funds (ETFs), we discover a discrete hierarchy of sizes, which imprints a log-periodic structure on the probability distribution of ETF sizes that dominates the details of the asymptotic tail. This allows us to propose …
Develops a universal test for assessing dynamic network models.
problem Determine if observed networks match a candidate dynamic random graph model.
method Formulates and analyzes a universal test for graph-valued, infinite-state Markov processes.
result Exhibits and analyzes a universal test for a natural class of models.
The book explores universal time-series forecasting using mixture predictors.
problem Sequential probability forecasting in a general setting.
method Mixture predictors combining multiple predictors.
result Universality of mixture predictors in a general probabilistic setting.
In this article, a large data set containing every course taken by every undergraduate student in a major university in Canada over 10 years is analysed. Modern machine learning algorithms can use large data sets to build useful tools for the data provider, in this case, the university. In this article, two classifiers…
This work initiates a general study of learning and generalization without the i.i.d. assumption, starting from first principles. While the traditional approach to statistical learning theory typically relies on standard assumptions from probability theory (e.g., i.i.d. or stationary ergodic), in this work we are inter…