Unified approach to compute asymptotic constants using optimization.
problem Computing unknown constants in asymptotic expansions.
method Linear Least Squares and Tikhonov Linear Least Squares methods.
result Rigorous asymptotic estimates and convergence-rate guarantees.
Extreme value theory enhances statistical learning extrapolation for rare events.
problem Challenges in traditional machine learning methods for extreme data.
method Asymptotic theory and statistical tools for tail behavior.
result Effective extrapolation methods for extreme quantiles and anomalies.
Overview of high-dimensional time series regression methods.
problem Estimation and inference with high-dimensional time series data.
method Limit theory for high-dimensional dependent data, asymptotic theory for time series regression, statistical learning methods.
result Main limit theory results and asymptotic theory for high-dimensional time series regression.
Theoretical analysis improves understanding of Deep Q-Learning's behavior.
problem Lack of formal guarantees and gaps between theory and practice of Deep Q-Learning.
method Dynamical systems perspective, focusing on realistic assumptions.
result Proves convergence of Deep Q-Learning under specific conditions.
Develops asymptotic theory for adversarial estimators.
problem Estimating unknown functions in machine learning and econometrics.
method Derives convergence rates and normality of A-estimators under various conditions.
result Normality of neural-net M-estimators, overcoming previous technical issues.
The paper develops a learning theory for neural network-based CHARME models.
problem Developing a learning theory for CHARME models using neural networks.
method Proves the stationarity and ergodicity of CHARME models under weak conditions, then applies neural networks to derive strong consistency and asymptotic normality of estimators.
result Strong consistency and asymptotic normality of NN-based estimators of CHARME model weights and biases under weak conditions.
New algorithm achieves instance-optimality in decision making.
problem Develop adaptive algorithms for interactive decision making.
method Introduce Allocation-Estimation Coefficient (AEC) and develop AE2 algorithm. result First non-asymptotic instance-optimal performance guarantees.
Study characterizes training and test risks for MAP regression with Gaussian priors.
problem Understanding high-dimensional behavior of regularized linear regression with informative priors.
method Maximum a posteriori (MAP) regression with Gaussian priors, using random matrix theory.
result Closed-form risk formulas reveal the bias-variance-prior tradeoff and explain double descent.
We find G2-manifolds with specific asymptotic properties.
problem Existence and structure of G2-manifolds with ALC asymptotics.
method Robust Fredholm theory for ALC spaces, proving existence and rigidity results.
result Existence of a G2-analogue of the Atiyah-Hitchin metric and good moduli theory for ALC G2-holonomy metrics.
Survey of spectral theory and dynamics for infinite volume hyperbolic manifolds.
problem Understanding infinite volume asymptotically hyperbolic manifolds.
method Survey of geometry, spectral theory, dynamics, and quantum/classical mechanics.
result Recent results, ideas, and conjectures discussed.
New self-expander found between two given asymptotic ones.
problem Finding new self-expanders between given asymptotic ones.
method Developed a min-max theory for asymptotically conical self-expanders of mean curvature flow.
result Existence of a new asymptotically conical self-expander trapped between two given ones.
Study instantons on asymptotically conical Spin(7)-manifolds, identifying deformation spaces.
problem Deformation theory of instantons on specific Spin(7)-manifolds.
method Relating deformation complex to spinors, identifying kernel of twisted negative Dirac operator.
result Virtual dimension of moduli space calculated using index theorem and Dirac operator spectrum.
This paper provides performance guarantees for neural estimation of statistical distances.
problem Developing performance guarantees for neural estimation of statistical distances.
method Non-asymptotic error bounds using function approximation theorems and empirical process theory.
result Established a fundamental tradeoff between approximation and estimation errors in neural estimation of statistical distances.
The asymptotic dimension theory was founded by Gromov in the early 90s. In this paper we give a survey of its recent history where we emphasize two of its features: an analogy with the dimension theory of compact metric spaces and applications to the theory of discrete groups.
The asymptotic behavior of the stochastic gradient algorithm with a biased gradient estimator is analyzed. Relying on arguments based on the dynamic system theory (chain-recurrence) and the differential geometry (Yomdin theorem and Lojasiewicz inequality), tight bounds on the asymptotic bias of the iterates generated b…
The study provides precise asymptotic theory for in-context learning by Transformers.
problem Understanding the sample complexity, pretraining task diversity, and context length for successful in-context learning.
method An exactly solvable model of linear regression task by linear attention, deriving sharp asymptotics.
result Double-descent learning curve with increasing pretraining examples, phase transition between low and high task diversity regimes.
Study variance-reduced method for estimating fixed points in Banach spaces.
problem Estimating fixed points of contractive operators in Banach spaces with noisy evaluations.
method Variance-reduced stochastic approximation scheme in Banach spaces.
result Establish non-asymptotic bounds for operator defect and estimation error.
Random matrix theory explains how neural networks adapt to data.
problem Understanding how neural networks learn and generalize from data.
method Random matrix analysis of two-layer neural networks.
result Sharp characterization of feature spectrum and generalization error.
New insights into manifold properties using Seiberg-Witten and L2 harmonic theories.
problem Characterizing properties of 4-manifolds with specific geometric conditions.
method Combining Seiberg-Witten theory on compact manifolds and L2 harmonic theory on non-compact manifolds, with a new argument for asymptotic properties. result Found a pair of homeomorphic 4-manifolds with distinct geometric properties under Riemannian metrics.
The paper develops AMP theory for sparse and robust regression with polynomial iterations.
problem Challenges in high-dimensional statistical estimation due to asymptotic theory breakdown.
method Non-asymptotic distributional theory of AMP for sparse and robust regression.
result First finite-sample non-asymptotic distributional theory of AMP for polynomial iterations.
New degree theory proves existence of solitons on 4D manifolds.
problem Existence of gradient expanding solitons on 4D manifolds.
method Developed new degree theory for 4D, asymptotically conical gradient expanding solitons.
result Existence of solitons asymptotic to any cone over S^3 with non-negative scalar curvature.
This study analyzes AdaGrad's stability and convergence in non-convex optimization.
problem Lack of theoretical analysis for AdaGrad in non-convex optimization.
method Novel stopping time-based techniques from probability theory.
result Established stability and derived convergence rates for AdaGrad.
New methods improve temporal difference learning for policy evaluation in Markov decision processes.
problem Improving temporal difference learning for policy evaluation in Markov decision processes.
method Introduced variance-reduced forms of stochastic approximation to achieve non-asymptotic, instance-dependent optimality.
result Temporal difference learning is strictly suboptimal, but variance-reduced forms achieve optimality up to logarithmic factors.
New theory for eigenvectors of generalized Laplacian matrices, addressing dependency issues.
problem Dependency in random matrix theory hinders eigenvector analysis for latent embeddings.
method Introduces generalized Laplacian matrices and a new asymptotic theory framework.
result Established asymptotic normalities for spiked eigenvectors and eigenvalues.
Proves equations for high-dimensional gradient-based methods from Gaussian data.
problem High-dimensional asymptotics of gradient-based learning algorithms.
method Closed-form equations derived from dynamical mean-field theory.
result Equations match those from discretized DMFT for gradient flow.
Establishes scattering theory for de Sitter vacuum solutions in even dimensions.
problem Quantitative nonlinear scattering theory for asymptotically de Sitter vacuum solutions in even spatial dimensions.
method Geometric Littlewood-Paley decomposition of the solution, constructing the scattering map.
result Existence and uniqueness of scattering states, asymptotic completeness, and invertible scattering map with quantitative control.
The EM algorithm's convergence is analyzed using Lyapunov stability theory.
problem Analyzing the convergence of the EM algorithm.
method Reinterpreting the EM algorithm as a dynamical system and applying Lyapunov stability theory.
result Asymptotic stability and convergence of the EM algorithm are established.
The paper develops an asymptotic theory of self-supervised pre-training.
problem Sharpness of current rates in self-supervised pre-training and their accuracy.
method Two-stage M-estimation and tools from Riemannian geometry.
result Characterization of the limiting distribution of the downstream test risk.
A tutorial on non-asymptotic system identification methods.
problem Identifying system parameters in linear models.
method Covering technique, Hanson-Wright Inequality, method of self-normalized martingales.
result Streamlined proofs of least-squares based estimator performance.
Estimates growth of reciprocal classes in Hecke groups.
problem Estimating the growth of reciprocal conjugacy classes in Hecke groups.
method Using free product structure and word lengths of reciprocal elements, with tools from basic probability theory.
result Estimates the asymptotic growth of reciprocal conjugacy classes in Hecke groups.
New theory improves diffusion models' convergence rates.
problem Understanding and optimizing diffusion models for faster data generation.
method Developed non-asymptotic theory for diffusion models with minimal assumptions.
result Established convergence rates for two diffusion models.
We clarify and refine the relation between the asymptotic behavior of the colored Jones polynomial and Chern-Simons gauge theory with complex gauge group SL(2,C). The precise comparison requires a careful understanding of some delicate issues, such as normalization of the colored Jones polynomial and the choice of pola…
The asymptotic lattices and their transformations are studied within the line geometry approach. It is shown that the discrete asymptotic nets are represented by isotropic congruences in the Plucker quadric. On the basis of the Lelieuvre-type representation of asymptotic lattices and of the discrete analog of the Mouta…
Formula derived for special q-hypergeometric series at roots of unity.
problem Asymptotic behavior of special q-hypergeometric series at complex roots of unity.
method Derivation of a formula for the radial asymptotics of Nahm sums at complex roots of unity.
result Formula for the radial asymptotics of Nahm sums at complex roots of unity.
FQE with deep neural networks achieves asymptotic normality and finite-sample bounds.
problem Theoretical understanding of FQE with general differentiable function approximators.
method Z-estimation theory applied to FQE with deep neural networks.
result FQE estimation error is asymptotically normal with explicit variance.
Obtaining labels can be costly and time-consuming. Active learning allows a learning algorithm to intelligently query samples to be labeled for efficient learning. Fisher information ratio (FIR) has been used as an objective for selecting queries in active learning. However, little is known about the theory behind the …
We introduce dynamic asymptotic dimension, a notion of dimension for actions of discrete groups on locally compact spaces, and more generally for locally compact étale groupoids. We study our notion for minimal actions of the integer group, its relation with conditions used by Bartels, Lück, and Reich in the context of…
New theory sharpens Q-learning with LDTZ rate, proving it's best of both worlds.
problem Improving Q-learning's theoretical and practical performance.
method Developed a sharp non-asymptotic error bound and central limit theory for Q-learning with PD2Z-ν schedule.
result Q-learning with LDTZ schedule achieves rapid decay and asymptotic convergence guarantees.
Extends JKO scheme for iterative algorithms with unknown parameters.
problem Computational and statistical analysis of iterative algorithms with unknown parameters.
method Develops statistical methods to estimate unknown parameters and adapts JKO scheme.
result Establishes asymptotic theory for the statistical JKO scheme.
Paper examines risk measure expansions under FGM dependence, improving accuracy at extreme levels.
problem Capturing higher-order tail behavior and dependence effects in risk measures.
method Second-order asymptotic expansions using extreme value theory and regular variation theory.
result Second-order approximations reduce approximation errors, especially at extreme confidence levels.
We review the spectral analysis and the time-dependent approach of scattering theory for manifolds with asymptotically cylindrical ends. For the spectral analysis, higher order resolvent estimates are obtained via Mourre theory for both short-range and long-range behaviors of the metric and the perturbation at infinity…
The paper develops a new algorithm for RBMs using dynamical mean-field theory.
problem Learning in Restricted Boltzmann Machines (RBMs) with complex dependencies.
method Dynamical mean-field theory applied to RBMs with rectangular coupling matrices drawn from a bi-rotation invariant ensemble.
result The algorithm converges globally under a stability criterion, with rates matching numerical simulations.
Study asymptotic properties of generalized shortfall risk measures for heavy-tailed risks.
problem Understanding risk measures for heavy-tailed risks.
method Derive asymptotic expansions for generalized shortfall risk measures.
result Unified theory for risk measures including distortion and utility-based measures.
Study on error probabilities of machine learning classification techniques using large deviations theory.
problem Performance analysis of machine learning binary classification techniques.
method Large deviations theory applied to Data-Driven Decision Function (D3F) for error probability analysis.
result Classification error probabilities vanish exponentially, with an asymptotic formula providing precise error rate estimates.
Proof of Gaussian ML estimator consistency in linear auto-regressive models.
problem Consistency of Gaussian maximum likelihood estimator in linear auto-regressive models.
method Information-theoretic proof without stability assumptions.
result Nearly optimal non-asymptotic rates for parameter recovery.
Extends Arnold's linking theory to higher dimensions and submanifolds.
problem Volume-preserving actions in higher dimensions and submanifolds.
method Generalization of V. Arnold's theory to Rk and Rℓ. result Extension of asymptotic linking to higher dimensions and submanifolds.
Analyzes high-dimensional SGD dynamics using DMFT.
problem Understanding the high-dimensional behavior of multi-pass SGD with small batch sizes.
method Derives DMFT equations for high-dimensional SGD dynamics.
result Proves DMFT equations characterize the asymptotic distribution of SGF parameters.
Study shows how learning and analytical models affect reneging and jockeying in a dual M/M/1 system.
problem How do learning and analytical models affect reneging and jockeying in a dual M/M/1 system?
method Analytical and online trained actor-critic models were used to study reneging and jockeying in a dual M/M/1 system.
result Both analytical and online trained actor-critic models yield the same asymptotic limits for reneging and jockeying, but differ in practical sizes.