Paper provides uniform deviation bounds for unbounded loss functions, improving k-Means clustering bounds.
problem Uniform deviation bounds for unbounded loss functions, specifically k-Means clustering.
method Novel framework to obtain uniform deviation bounds for unbounded loss functions.
result Improved bounds for k-Means clustering under weak assumptions, achieving $\mathcal{O}\left(m^{-\frac12}
ight)$ rate.
Simple bounds for covariance and Gram matrices across various settings.
problem Capturing the behavior of smaller eigenvalues in covariance and Gram matrices.
method General-purpose theorem converting uniform bounds into relative bounds.
result Sharper control of eigenvalues across the spectrum.
The paper provides a new uniform tail bound for empirical processes.
problem Developing a uniform tail bound for empirical processes indexed by a class of functions.
method Introducing a deflation step to the standard generic chaining argument, and using a natural seminorm based on Cramér functions.
result Established a new uniform tail bound for empirical processes.
Paper proves uniform generalization implies concentration and tight bounds.
problem Mitigating overfitting in learning algorithms.
method Analyzes uniform generalization, proving its equivalence to stability and its relationship to concentration.
result Uniform generalization implies concentration and provides tight bounds.
The paper develops bounds for predictive values in binary classification.
problem Lack of confidence intervals for positive and negative predictive values.
method Bi-criterion framework and distribution-free large deviation and uniform convergence bounds.
result New bounds for predictive values without relying on concentration inequalities.
In this paper, we present the Bennett-type generalization bounds of the learning process for i.i.d. samples, and then show that the generalization bounds have a faster rate of convergence than the traditional results. In particular, we first develop two types of Bennett-type deviation inequality for the i.i.d. learning…
Unified bounds for sketched bilinear forms in machine learning and statistics.
problem Uniform bounds on sketched bilinear forms for modern analyses.
method Generic chaining and new techniques for handling suprema over pairs of sets.
result Improved convergence bounds for sketched Federated Learning and bandit algorithms.
Study on order book dynamics with uniform catastrophes, explaining volatility and trends.
problem Understanding volatility and trends in financial markets with different types of liquidity.
method Stochastic models and population processes with uniform catastrophes.
result Law of large numbers, central limit theorem, and large deviations proved for the model.
Suppose k centers are fit to m points by heuristically minimizing the k-means cost; what is the corresponding fit over the source distribution? This question is resolved here for distributions with p≥4 bounded moments; in particular, the difference between the sample cost and distribution cost decays with $…
Paper extends nonparametric regression bounds for dependent β-mixing samples.
problem Analyzing error in nonparametric regression with dependent data.
method Extends uniform deviation inequalities from independent to dependent β-mixing samples. result Derives generalization bounds for nonparametric regression with dependent data.
Adding noise controls capacity of function compositions.
problem Large capacity of function compositions with bounded capacity classes.
method Adding Gaussian noise to the output of F before composing with H. result Noise effectively controls the capacity of H∘F, offering a general recipe for modular design. Study large deviations in life insurance portfolios without identical distributions.
problem Large deviations in life insurance portfolios with bounded losses and variances.
method Upper bound from standard large deviations, counterexample for full large deviation principle.
result Exponential bound for average loss exceeding a threshold.
New method estimates node community memberships in networks.
problem Estimating community memberships of nodes in networks with overlapping communities.
method Sharp eigenvector deviation bounds for Mixed Membership Stochastic Blockmodel (MMSB).
result Uniform rates of convergence for node community membership vectors.
We consider the problem of predicting as well as the best linear combination of d given functions in least squares regression, and variants of this problem including constraints on the parameters of the linear combination. When the input distribution is known, there already exists an algorithm having an expected excess…
The paper provides a finite-sample deviation bound for stable autoregressive processes.
problem Deviation bounds for least squares estimators in Gaussian AR(n) processes.
method Utilizes martingale concentration inequalities and tail-bound for χ² distributed variables.
result Problem-dependent finite-time bound on the deviation probability of AR(n) process parameters.
Rank-statistic method approximates f-divergences without density-ratio estimation.
problem Approximating f-divergences without explicit density-ratio estimation. method Mapping distribution rank histograms to discrete f-divergence and averaging over random projections. result The rank-statistic estimator is a lower bound of the true f-divergence and converges under mild conditions. Authors derive the first two terms of the Hartman-Watson distribution's expansion for small t.
problem The Hartman-Watson distribution's integral density is difficult to evaluate numerically for small t.
method Saddle point methods and numerical estimates of the integrand.
result Obtained the first two terms of the to0 expansion of the Hartman-Watson distribution. New flexible confidence sequences for robust statistical inference.
problem Creating robust statistical inference methods that work under mild assumptions.
method Proposed a new class of asymptotic time-uniform confidence sequences.
result Sharp asymptotic time-uniform confidence sequences achieved under mild assumptions.
Proposes new deviation measures using Minkowski gauges.
problem Lack of suitable acceptance sets for deviation measures.
method Derives deviation measures through Minkowski gauges of acceptable sets.
result Any positive homogeneous deviation measure can be accommodated in the framework.
Secure gradient descent method for high-dimensional learning with Byzantine workers.
problem Secure training in high-dimensional statistical learning with unreliable workers.
method Proposes a secure variant of gradient descent method that can tolerate up to a constant fraction of Byzantine workers.
result Converges in O(log N) rounds to O(√(q/N) + √(d/N)) error rate, achieving optimal error rate O(√(d/N)) when q=O(d).
Recently theoretical guarantees have been obtained for matrix completion in the non-uniform sampling regime. In particular, if the sampling distribution aligns with the underlying matrix's leverage scores, then with high probability nuclear norm minimization will exactly recover the low rank matrix. In this article, we…
In this paper, we study the risk bounds for samples independently drawn from an infinitely divisible (ID) distribution. In particular, based on a martingale method, we develop two deviation inequalities for a sequence of random variables of an ID distribution with zero Gaussian component. By applying the deviation ineq…
New uniform K-theory and Poincare duality established for manifolds.
problem Developing a new framework for K-theory and K-homology.
method Constructing uniform K-homology, defining external and cap products, proving homotopy invariance and Poincare duality.
result Established Poincare duality between uniform K-theory and uniform K-homology on spin-c manifolds.
Study large deviations rates for SGD with strongly convex functions.
problem High probability metrics with SGD.
method Large deviations theory, generic gradient noise, strongly convex functions.
result Upper large deviations bound for SGD with strongly convex functions.
Improved uniform convergence bound with fat-shattering dimension reduces sample complexity gap.
problem Gap between upper and lower bounds on sample complexity for fat-shattering dimension.
method Provided an improved uniform convergence bound.
result Closed the gap between existing upper and lower bounds on sample complexity.
Uniform entropy bound for Ricci shrinkers with bounded curvature.
problem Bounding entropy for Ricci shrinkers with specific curvature constraints.
method Establishing uniform entropy bounds for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
result Uniform entropy bound for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
Uniform bounds derived for fully non-linear equations.
problem Bounding fully non-linear equations uniformly in background metrics.
method Auxiliary Monge-Ampère equations and entropy-like quantities.
result Uniform L∞ bounds for systems coupling fully non-linear equations to their linearizations. This work certifies non-uniform bounds against adversarial attacks for neural networks.
problem Certifying robust regions around data points against non-uniform adversarial attacks.
method Formulated as an optimization problem with nonlinear constraints, using the augmented Lagrangian method for general feedforward neural networks.
result Non-uniform bounds have larger volumes and better interpretability compared to uniform bounds.
Logistic regression gets a new, simpler uniform bound.
problem Finding a uniform bound for logistic regression's empirical risk.
method PAC-Bayes approach with second-order expansion and Rademacher-complexity bounds.
result Provides a dimension-free uniform concentration bound.
New method improves solving combinatorial optimization problems with smoothed policies.
problem Solving combinatorial optimization problems repeatedly with varying instances.
method Smoothed policies with controlled random perturbations to linear oracle, leading to differentiable surrogate risk.
result Generalization bound decomposes excess risk into bias, estimation, and optimization components.
Researchers introduce a method to assess the safety of interpretable machine learning models.
problem Ensuring safety in machine learning models that are easy to understand.
method Introduce maximum deviation as an optimization problem to find the largest deviation from a safe reference model.
result Interpretability helps in assessing the safety of machine learning models.
Deviation inequalities and limit laws for random walks on metric spaces.
problem Understanding random walks on metric spaces with contracting isometries.
method Adapting Gouëzel's pivotal time construction to establish deviation inequalities.
result Exponential bounds and limit laws for random walks on mapping class groups and CAT(0) spaces.
Study generalizes matrix completion with side info in low noise settings.
problem Matrix completion with side information in low noise conditions.
method Inductive matrix completion with i.i.d. subgaussian noise, uniform sampling, and side information.
result Generalization bounds with noise scaling, convergence to zero, and logarithmic dependence on matrix size.
Uniform bounds derived for nonlinear statistics.
problem Deriving uniform bounds for nonlinear statistics.
method Extended method to Gaussian and Rademacher complexities.
result Tight bounds for U-statistics and error functionals.
This article provides a new toolbox to derive sparse recovery guarantees from small deviations on extreme singular values or extreme eigenvalues obtained in Random Matrix Theory. This work is based on Restricted Isometry Constants (RICs) which are a pivotal notion in Compressed Sensing and High-Dimensional Statistics a…
Study shows gap between uniform convergence and test error in random feature models.
problem Understanding the gap between uniform convergence and test error in random feature models.
method Analytical expressions for uniform convergence over norm balls, interpolators, and minimum norm interpolator risk derived and proved.
result Uniform convergence over interpolators still gives a non-trivial bound of test error even when classical uniform convergence is vacuous.
This paper proves properties of uniformly hyperbolic sets and constructs Markov partitions.
problem Establishing properties of uniformly hyperbolic sets and constructing Markov partitions.
method Backward graph transform, spectral decomposition, shadowing lemma, Markov partitions construction.
result Explicit bounds and Hölder continuity for the coding map.
Paper derives bounds on prediction errors using information theory.
problem Understanding maximum prediction errors in sequential data.
method Information-theoretic approach focusing on conditional entropy.
result Fundamental bounds on prediction errors depend on conditional entropy.
We prove polynomial upper bounds for the deviation of ergodic averages for the straight line flow on every translation surface in almost every direction, in particular for those surfaces arising from rational polygonal billiards.
New tester outperforms existing ones in uniformity testing.
problem Improving uniformity testing accuracy in simulations.
method Introducing a Huber loss-based tester.
result Matches the separation of the collisions tester and has Gaussian-like tails.
Improved median of means estimator with tighter bounds.
problem Improving the efficiency and reliability of median of means estimator.
method Modification of the median of means estimator with sub-Gaussian deviation bounds.
result Achieves nearly optimal constants under minimal assumptions.
The study provides a sample complexity estimate for multi-category classifiers with bounded variation.
problem Controlling the deviation between empirical and generalization performances of multi-category classifiers.
method Using the empirical L1-norm covering number and fat-shattering dimension, the study derives a sample size estimate for classifiers of bounded variation.
result The sample size estimate is sufficient for the performances to be close with high probability, improving the dependency on the number of classes.
A new method achieves optimal uniformity in designs with minimal flexibility.
problem Achieving optimal uniformity in designs with minimal flexibility.
method Derive a lower bound on the uniformity constant and use a greedy construction to achieve this bound, then extend the scheme for more flexibility.
result A simple greedy construction achieves the optimal uniformity constant.
Novel concentration inequalities are obtained for the missing mass, i.e. the total probability mass of the outcomes not observed in the sample. We derive distribution-free deviation bounds with sublinear exponents in deviation size for missing mass and improve the results of Berend and Kontorovich (2013) and Yari Saeed…
Large deviation principle for deep neural networks with ReLU activation.
problem Understanding the behavior of deep neural networks with ReLU activation.
method Proving a large deviation principle for networks with Gaussian weights and ReLU activation functions.
result Simplified expressions and power-series expansions for the ReLU case.
Uniformizes klt pairs using bounded symmetric domains.
problem Characterizing klt pairs uniformizable by bounded symmetric domains.
method Determines conditions for uniformization using Miyaoka-Yau-type inequalities.
result Characterizations of orbifold quotients of polydisc and classical bounded symmetric domains.
Uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures established.
problem Bounding the magnitude of harmonic Beltrami differentials and Weil-Petersson curvatures.
method Using the systole of a hyperbolic surface, the authors derive uniform bounds for the magnitude of harmonic Beltrami differentials and the Weil-Petersson Ricci curvature.
result Uniform bounds on Weil-Petersson curvatures and magnitudes of harmonic Beltrami differentials are established.
Study on U-statistics with heavy-tailed samples, providing tail bounds and LDP.
problem Deviation of U-statistics with heavy-tailed samples.
method Exponential tail bounds and Large Deviation Principle (LDP) for U-statistics.
result Obtained an exponential upper bound for U-statistics tail decay, showing two regions of decay.