We present a new PAC-Bayesian generalization bound. Standard bounds contain a $\sqrt{L_n \cdot \KL/n}$ complexity term which dominates unless , the empirical error of the learning algorithm's randomized predictions, vanishes. We manage to replace by a term which vanishes in many more situations, essentially …
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48 results for “Bernstein bounds”
Sharp inequalities for matrix means with unknown variance.
problem Estimating matrix means with unknown variance.
method Empirical Bernstein inequalities for symmetric random matrices.
result Adapts to unknown variance with tight deviation bounds.
We obtain a Bernstein theorem for special Lagrangian graphs in n-dimensional complex space for arbitrary n only assuming bounded slope, but no quantitative restriction.
Study introduces new Bernstein inequalities for dependent data in Hilbert spaces.
problem Learning from non-independent and non-identically distributed data.
method Data-dependent Bernstein inequalities tailored for vector-valued processes in Hilbert space.
result Achieved novel risk bounds for covariance operator estimation and operator learning.
Improved analysis of UCRL2 with empirical Bernstein inequality reduces exploration-exploitation regret.
problem Exploration-exploitation in communicating Markov Decision Processes.
method Analysis of UCRL2 with Empirical Bernstein inequalities (UCRL2B).
result Regret bound of for UCRL2B.
The paper proves Calabi-Bernstein type results for minimal and maximal surfaces in 3D and 3D-L spacetime.
problem Characterizing minimal and maximal surfaces in 3D and 3D-L spacetime.
method Analyzing surfaces with specific properties and using geometric and functional methods.
result Calabi-Bernstein type results for critical points of a weighted area functional in and .
A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski spacemath.DG
We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hy…
A weighted area estimate for entire graphs with bounded weighted mean curvature in Gauss space is given by a simple proof. Bernstein type theorems for self shrinkers (\cite {wa}) as well as for graphic -hypersurfaces (\cite{ chwe2}) follow immediately as consequences.
New method improves RL in continuous spaces with kernel smoothing.
problem Sample efficiency and structural assumptions in classical RL.
method Kernel smoothing model-based approach with Bernstein-style exploration bonus.
result Achieves improved regret bound in finite-horizon settings.
Bernstein-type Theorems for constant mean curvature surfaces in the three-dimensional light conemath.DG
The study proves that certain constant mean curvature surfaces in a specific cone are either spheres or horospheres.
problem Characterizing constant mean curvature surfaces in a three-dimensional light cone.
method Analyzing entire constant mean curvature graphs in the light cone under the condition of bounded Gaussian curvature.
result Entire constant mean curvature graphs in the light cone are either horospheres or spheres.
New bounds for non-convex estimators without Bernstein condition.
problem Sharp excess risk bounds for non-convex and improper estimators.
method Exponential-tail local Rademacher complexity risk bounds with offset condition.
result Sharp bounds for non-convex and improper estimators without Bernstein condition.
New minimal hypersphere found in 4-sphere solving Bernstein problem.
problem Spherical Bernstein problem in
method Equivariant min-max theory for -invariant minimal hypersurfaces
result Construction of embedded non-equatorial minimal hypersphere
A note on two-dimensional minimal surface graphs in R^n and a theorem of Bernstein-Liouville typemath.DG
Using Schauder's theory for linear elliptic partial differential equations in two independent variables and fundamental estimates for univalent mappings due to E. Heinz we establish an upper bound of the Gaussian curvature of two-dimensional minimal surface graphs in R^n. This leads us to a theorem of Bernstein-Liouvil…
Study improves self-normalized bounds for vector-valued processes beyond sub-Gaussianity.
problem Limited understanding of self-normalized concentration for vector-valued processes outside sub-Gaussian frameworks.
method Developed concentration inequalities for self-normalized processes with light tails (e.g., Bennett, Bernstein bounds) for vector-valued data.
result Provided new insights and bounds for self-normalized processes with non-sub-Gaussian distributions.
New inequality for ternary variables improves on existing measures.
problem Analyzing excess losses and weighted majority votes with ternary random variables.
method Developed a split-kl inequality and its PAC-Bayes extension.
result Outperforms existing inequalities in certain regimes.
This paper improves flow models to better handle perturbations in real-world data.
problem Flow models amplify initial errors in perturbed data, leading to poor generalization.
method Utilizes Bernstein-type polynomials to construct Normalizing Flows (NF) for higher robustness.
result Proposed NF framework provides theoretical upper bounds and practical advantages.
Paper refines PAC-Bayes bounds for bandit problems.
problem Improving probabilistic bounds for off-policy learning.
method Optimizes PAC-Bayesian bounds using a new parameter optimization approach.
result Provides two parameter-free PAC-Bayes bounds that nearly match optimal rates.
We give improved constants for data dependent and variance sensitive confidence bounds, called empirical Bernstein bounds, and extend these inequalities to hold uniformly over classes of functionswhose growth function is polynomial in the sample size n. The bounds lead us to consider sample variance penalization, a nov…
A new method for high-dimensional classification using Bernstein polynomials.
problem Computational difficulties in high-dimensional SVM hinge loss.
method Proposes Bernstein support vector machine (BernSVM) and two efficient algorithms.
result Achieves a prediction accuracy rate of with high probability.
We give tight concentration bounds for mixtures of martingales that are simultaneously uniform over (a) mixture distributions, in a PAC-Bayes sense; and (b) all finite times. These bounds are proved in terms of the martingale variance, extending classical Bernstein inequalities, and sharpening and simplifying prior wor…
Adaptive Bernstein copulas improve risk management by preventing overfitting and reducing simulation effort.
problem Overfitting and high simulation effort in estimating dependence models.
method Constructive approach to Bernstein copulas with an admissible discrete skeleton.
result Comparison of different copula approaches in risk management shows improved accuracy and efficiency.
The betting CI outperforms classical methods in constructing confidence intervals for bounded means.
problem Constructing nonasymptotic confidence intervals for bounded means.
method A betting-based approach to define and time-uniform variants of confidence intervals (CSs).
result The betting CI matches the fundamental limits, outperforming existing empirical Bernstein CIs.
The study proves planes are the only complete uniformly elliptic Weingarten multigraphs.
problem Proving planes are the only complete uniformly elliptic Weingarten multigraphs.
method Proving planes are the only complete multigraphs with quasiconformal Gauss map and bounded second fundamental form.
result Proves planes are the only complete uniformly elliptic Weingarten multigraphs.
Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.
problem Proving theorems for minimal and constant mean curvature graphs in Euclidean and Lorentz-Minkowski spaces.
method Explains several proofs and provides mean curvature estimates for graphs in Euclidean and Lorentz-Minkowski spaces.
result Bernstein-type theorems for constant mean curvature graphs in Euclidean 3-space and space-like graphs in Lorentz-Minkowski 3-space.
In this paper we study nonconvex penalization using Bernstein functions. Since the Bernstein function is concave and nonsmooth at the origin, it can induce a class of nonconvex functions for high-dimensional sparse estimation problems. We derive a threshold function based on the Bernstein penalty and give its mathemati…
We prove semi-empirical concentration inequalities for random variables which are given as possibly nonlinear functions of independent random variables. These inequalities describe concentration of random variable in terms of the data/distribution-dependent Efron-Stein (ES) estimate of its variance and they do not requ…
In this paper we study nonconvex penalization using Bernstein functions whose first-order derivatives are completely monotone. The Bernstein function can induce a class of nonconvex penalty functions for high-dimensional sparse estimation problems. We derive a thresholding function based on the Bernstein penalty and di…
Study Bernstein-Gelfand-Gelfand complexes on Lipschitz domains, computing cohomology and applying to elasticity models.
problem Cohomology of BGG complexes on bounded Lipschitz domains.
method Computes cohomology of conformal deformation and Hessian complexes in Sobolev spaces, allowing multiple input complexes.
result Establishes conformal Korn inequality and proposes generalizations of continuum models with microstructures.
Almost sharp local Bernstein estimates for Laplace eigenfunctions on compact Riemannian manifoldsmath.AP
The study sharpens local Bernstein estimates for Laplace eigenfunctions on compact manifolds.
problem Understanding local growth properties of Laplace eigenfunctions on compact Riemannian manifolds.
method Refined Donnelly-Fefferman method based on --Carleman estimates, combined with elliptic regularity and patching of local Carleman estimates.
result Almost sharp local --Bernstein inequalities for .
New concentration inequality for U-statistics of Markov chains.
problem Proving a concentration inequality for U-statistics of order two in uniformly ergodic Markov chains.
method Inductive analysis using martingale techniques, uniform ergodicity, Nummelin splitting, and Bernstein's inequality.
result Recovery of convergence rate for U-statistics of independent random variables and canonical kernels, with improved results for dependent kernels.
Ancient solutions to mean curvature flow have unique shapes.
problem Understanding unique shapes of ancient solutions.
method Proved a Bernstein theorem for ancient solutions.
result Ancient solutions to mean curvature flow have unique shapes.
We obtain a Bernstein type result for entire two dimensional minimal graphs in , which extends a previous one due to L. Ni. Moreover, we provide a characterization for complex analytic curves.
We improve bounds for stochastic processes, especially those with heavy tails.
problem Bounding the concentration of sub- processes with heavy tails.
method Variational approach to concentration, focusing on sub-Gaussian and other tail conditions.
result First dimension-free self-normalized empirical Bernstein inequality.
Modelling and simulation of dependence structures in nonlife insurance with Bernstein copulasstat.ME
The study models insurance dependence using Bernstein copulas.
problem Modeling dependence structures in nonlife insurance data.
method Review and suggest fitting Bernstein copulas to empirical data.
result Monte Carlo simulation and PML estimation for aggregate losses.
Improved understanding of translating solitons using new techniques.
problem Understanding translating solitons in geometry.
method Using a new test function and gradient estimate technique.
result Better Bernstein type result of translating solitons.
On Bernstein-type theoremsmath.DG
We summarize results concerning the Bernstein property of differential equations.
Explains Bernstein theorems for various geometric PDEs.
problem Bernstein problem for minimal surface, Monge-Ampère, and special Lagrangian equations.
method Expository review of existing theorems and systems.
result Discussion of Bernstein theorems for different geometric PDEs.
Ancient symplectic solutions to mean curvature flow are flat.
problem Understanding ancient solutions to mean curvature flow in symplectic geometry.
method Using a complex phase map to prove a Bernstein theorem.
result Ancient solutions to the symplectic mean curvature flow are flat.
In this paper, we prove some Bernstein type results for -dimensional minimal Lagrangian graphs in quaternion Euclidean space . In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in
The paper develops concentration inequalities for structured random data, extending beyond independent terms.
problem Developing concentration inequalities for structured weighted sums of random data, including tensors and matrix-valued data.
method The paper develops Hoeffding and Bernstein bounds for structured weighted sums under exchangeability, extending beyond the classical framework of independent terms.
result The paper develops a sharper concentration bound for combinatorial sums of matrix arrays.
We present Bernstein-Sato identities for scalar-, spinor- and differential form-valued distribution kernels on Euclidean space associated to conformal symmetry breaking operators. The associated Bernstein-Sato operators lead to partially new formulae for conformal symmetry breaking differential operators on functions, …
Bernstein type gradient estimate for system of weighted local heat equations with potential termmath.AP
Article provides Bernstein gradient estimates for heat equations with potential terms.
problem Gradient estimates for heat equations with potential terms on weighted Riemannian manifolds.
method Derived Bernstein type gradient estimates for two systems of heat equations with linear, exponential, and combined potentials.
result Resolves part of the problem raised by Bhattacharyya et al. in \cite{SB-1}.
The paper solves Bernstein problems for specific submanifolds in high-dimensional spaces.
problem Bernstein problem for smooth maps to lower dimensions forming calibrated submanifolds.
method Established conditions for maps to be affine based on the slope's second elementary symmetric polynomial.
result Conditions ensuring maps are affine for coassociative and Cayley submanifolds in and .
Paper proves generalizations of Bernstein's theorem in higher dimensions.
problem Proving theorems about sets in higher-dimensional spaces.
method Analyzes properties of sets with monotonicity formula and uses geometric and analytic techniques.
result Generalizations of Bernstein's theorem in higher dimensions.
Deep neural networks are optimal for dependent data using PAC-Bayes bounds.
problem Optimizing deep neural networks for dependent data.
method PAC-Bayes oracle inequalities and Bernstein inequality.
result Upper and lower bounds match, proving minimax optimality.
For entire spacelike stationary 2-dimensional graphs in Minkowski spaces, we establish Bernstein type theorems under specific boundedness assumptions either on the W-function or on the total (Gaussian) curvature. These conclusions imply the classical Bernstein theorem for minimal surfaces in 3-dimensional Euclidean spa…
Proof that certain complex equations have only simple solutions.
problem Characterizing solutions to complex equations.
method Analyzing fully nonlinear elliptic operators.
result Entire smooth solutions are quadratic polynomials.
We give concentration bounds for martingales that are uniform over finite times and extend classical Hoeffding and Bernstein inequalities. We also demonstrate our concentration bounds to be optimal with a matching anti-concentration inequality, proved using the same method. Together these constitute a finite-time versi…