Develops a generalized version of Chung's Lemma for stochastic optimization methods.
problem Establishing asymptotic convergence rates for stochastic optimization methods under various step size rules.
method Generalized version of Chung's Lemma for a broader family of step size rules.
result Demonstrates tight non-asymptotic convergence rates for various stochastic methods.
The paper analyzes the risk of bagging regularized M-estimators under proportional asymptotics.
problem Characterizing the risk of ensemble estimators trained with subsamples and regularizers.
method Developed a consistent estimator for the risk of ensemble estimators under proportional asymptotics.
result Optimal subsample size k⋆ tends to be in the overparameterized regime for the full-ensemble estimator. The paper analyzes fixed step-size SA schemes on Riemannian manifolds.
problem Developing efficient algorithms for optimization on curved spaces.
method Fixed step-size stochastic approximation schemes in a Riemannian framework.
result The schemes converge to the solution as the step-size approaches zero.
Large batch sizes reduce gradient variance in DP-SGD, improving privacy.
problem Understanding why large batch sizes work in DP-SGD.
method Decomposed total gradient variance into subsampling and noise-induced variances, proving batch size independence in the limit.
result Large batch sizes reduce effective total gradient variance, improving privacy in DP-SGD.
We analyze SGAs for statistical inference via asymptotics, improving tuning methods.
problem Improper tuning of SGAs for optimization and sampling.
method Characterize large-sample asymptotics of SGAs via step-size and sample-size scaling limits.
result Iterate averaging with large step size is robust and asymptotically has covariance proportional to MLE's.
Directed acyclic graph (DAG) models are popular for capturing causal relationships. From observational and interventional data, a DAG model can only be determined up to its \emph{interventional Markov equivalence class} (I-MEC). We investigate the size of MECs for random DAG models generated by uniformly sampling and o…
This paper analyzes the bias of inexact MCMC methods in high dimensions.
problem Understanding the bias of inexact MCMC methods in high-dimensional spaces.
method Establishing bounds on Wasserstein distances between inexact MCMC methods and target distributions.
result The asymptotic bias of ULA and uHMC depends on key quantities related to the target distribution or the stationary probability measure of the scheme.
This study analyzes LTS in sparse models with finite sample error bounds.
problem Robust regression in high-dimensional sparse models with limited data.
method Non-asymptotic analysis of LTS error bounds.
result Established finite sample error bounds for LTS in sparse models.
Improved analysis for fair federated learning reduces dependence on noise floor.
problem Asymptotic stationarity in group fair federated learning with reduced noise floor dependence.
method DS FedProxGrad framework with inexact local proximal solutions and fairness regularization.
result Algorithm converges asymptotically to stationarity without dependence on a noise floor.
The study finds a trade-off between model size, test loss, and training loss for linear predictors.
problem Finding the optimal balance between model size, test loss, and training loss for linear predictors.
method Established an algorithm and distribution-independent trade-off using non-asymptotic analysis.
result Models with low test loss are either classical (close to noise level training loss) or modern (large number of parameters).
Study ridge ensembles in proportional feature-to-sample size regime, proving risk equivalence and GCV consistency.
problem Characterizing and optimizing ridge ensembles in proportional feature-to-sample size regimes.
method Proportional asymptotics analysis, GCV for tuning, proving risk equivalence.
result Risk of optimal full ridgeless ensemble matches optimal ridge predictor's risk.
Applying standard Markov chain Monte Carlo (MCMC) algorithms to large data sets is computationally infeasible. The recently proposed stochastic gradient Langevin dynamics (SGLD) method circumvents this problem in three ways: it generates proposed moves using only a subset of the data, it skips the Metropolis-Hastings a…
Study SGD dynamics in high-dimensional models, revealing consistent behavior across different batch sizes and learning rates.
problem Understanding SGD dynamics in high-dimensional multi-index models.
method Asymptotic analysis of SGD, developing mean-field equations and Gaussian diffusion approximations.
result Consistent SGD dynamics across different batch sizes and learning rates, distinct from gradient flow and online SGD.
The paper improves methods for estimating set size using samples.
problem Estimating the size of a set from a uniform sample.
method Refines estimators using the birthday problem and maximum of sample.
result Develops a general theory for non-asymptotic error bounds.
Batch normalisation doesn't affect variational inference but fails for larger batch sizes.
problem Failure of Monte Carlo Batch Normalisation (MCBN) for capturing epistemic uncertainty in larger batch sizes.
method Investigated MCBN as an approximate inference technique for Bayesian neural networks, showing its limitations and providing insights for improvement.
result For larger batch sizes, MCBN fails to capture epistemic uncertainty, requiring the batch size to be a variational parameter.
We consider the least-squares regression problem and provide a detailed asymptotic analysis of the performance of averaged constant-step-size stochastic gradient descent (a.k.a. least-mean-squares). In the strongly-convex case, we provide an asymptotic expansion up to explicit exponentially decaying terms. Our analysis…
Study asymptotics of one part monotone Hurwitz numbers in high genus.
problem Asymptotic analysis of one part monotone Hurwitz numbers in high genus.
method Used a linear recurrence and a recent method to extract asymptotics.
result Obtained bivariate asymptotics for one part monotone Hurwitz numbers in high genus.
The paper analyzes the randomized midpoint method for Langevin diffusions, revealing biases and asymptotic properties.
problem Analyzing biases and asymptotic properties of the randomized midpoint method for Langevin diffusions.
method Characterization of stationary distribution and asymptotic normality for numerical integration.
result The step-size needs to go to zero for the method to be asymptotically unbiased.
The paper examines linking numbers in grid models and finds polynomial moments.
problem Analyzing linking numbers in grid models.
method Examined linking numbers as a random variable on isotopy classes of 2-component links, computed moments and limits.
result The uth moment of the linking number is a polynomial in the grid size with degree d≤u, and all odd moments vanish. Determinantal Point Processes (DPPs) are popular models for point processes with repulsion. They appear in numerous contexts, from physics to graph theory, and display appealing theoretical properties. On the more practical side of things, since DPPs tend to select sets of points that are some distance apart (repulsion…
New bounds on efficiency for conformalized regression methods.
problem Efficiency of conformal prediction in regression models.
method Non-asymptotic bounds on prediction set length for conformalized quantile and median regression.
result Identifies phase transitions in convergence rates across different regimes of miscoverage level.
We study the asymptotic properties of the adaptive Lasso in cointegration regressions in the case where all covariates are weakly exogenous. We assume the number of candidate I(1) variables is sub-linear with respect to the sample size (but possibly larger) and the number of candidate I(0) variables is polynomial with …
Random forests remain among the most popular off-the-shelf supervised learning algorithms. Despite their well-documented empirical success, however, until recently, few theoretical results were available to describe their performance and behavior. In this work we push beyond recent work on consistency and asymptotic no…
SGDM accelerates faster than SGD with large batch sizes and permits broader learning rates.
problem Understanding the role of momentum in SGDM and its convergence rates.
method Analysis of SGDM convergence rates under strongly convex settings, including finite-sample rates and asymptotic normality of the averaged estimator.
result SGDM converges faster than SGD with large batch sizes and permits broader learning rates.
We find the entropy's infinite-size behavior in complex manifold sections.
problem Determining entropy behavior in complex manifold sections.
method Analyzing entanglement entropy in tensor powers of hermitian line bundles.
result Asymptotic formula for expected entanglement entropy.
Study on stochastic approximation with Polyak-Ruppert averaging for linear systems.
problem Understanding the asymptotic and non-asymptotic properties of stochastic approximation procedures.
method Detailed analysis of linear stochastic approximation with Polyak-Ruppert averaging, focusing on asymptotic and non-asymptotic properties.
result Proves CLT and non-asymptotic concentration inequality for averaged iterates, providing refined understanding of linear stochastic approximation.
We consider a family of compact, oriented and connected Riemannian manifolds shrinking to a metric graph and describe the asymptotic behaviour of the eigenvalues of the Hodge Laplacian. We apply our results to produce manifolds with spectral gaps of arbitrarily large size in the spectrum of the Hodge Laplacian.
New method for MMD with unequal sample sizes improves test power.
problem Existing MMD methods assume equal sample sizes, discarding valuable data.
method Extended generalized U-statistics to handle unequal sample sizes.
result New asymptotic distributions and power optimization for MMD with unequal sample sizes.
Paper proves robust M-estimators' coordinates' normality in high dimensions.
problem High-dimensional robust M-estimators' asymptotic normality.
method Develops Stein formulae for high-dimensional random vectors on the sphere.
result Asymptotic normality holds for most coordinates of robust M-estimators with convex penalty.
The paper develops a method to create non-asymptotic confidence ellipsoids for linear regression without strong noise distribution assumptions.
problem Constructing reliable confidence regions for linear regression with finite sample sizes and general noise distributions.
method The paper introduces the SPS EOA algorithm to create non-asymptotically guaranteed confidence ellipsoids for linear regression problems.
result The sizes of SPS outer ellipsoids are shown to decrease at the optimal rate for linear regression problems.
Unified analytical tool for non-Markovian jump processes.
problem Analyzing history-dependent jump processes with non-Markovian behavior.
method Developed a standard form of master equations using Laplace-space embedding and asymptotic solution.
result Unified analytical toolset for general non-Markovian processes, leading to the GLE approximation.
This paper introduces time-uniform CLT-based confidence intervals for statistical inference.
problem Developing valid statistical inference methods for sequential data.
method Time-uniform central limit theory and strong invariance principles.
result Asymptotic confidence sequences (CSs) that are uniformly valid over time.
This paper presents compact notations for concentration inequalities and convenient results to streamline probabilistic analysis. The new expressions describe the typical sizes and tails of random variables, allowing for simple operations without heavy use of inessential constants. They bridge classical asymptotic nota…
Paper develops Gaussian approximations and bootstrap for federated LSA with trade-off bounds.
problem Analyzing convergence rates and trade-offs in federated linear stochastic approximation.
method Established Berry-Esseen-type bounds for federated LSA, developed multiplier bootstrap for inference.
result First federated Gaussian approximations with explicit trade-off terms and non-asymptotic validity guarantees.
New mathematical framework proves the effectiveness of reducing neural network sizes.
problem Selecting optimal neural network sizes to avoid overfitting.
method Adaptive group Lasso applied to one-hidden-layer feedforward networks.
result Adaptive group Lasso is consistent and can accurately reconstruct network sizes.
Stochastic gradient descent (SGD) is almost ubiquitously used for training non-convex optimization tasks. Recently, a hypothesis proposed by Keskar et al. [2017] that large batch methods tend to converge to sharp minimizers has received increasing attention. We theoretically justify this hypothesis by providing new pro…
New oracles improve stochastic optimization with noisy or biased measurements.
problem Optimizing functions with noisy or biased measurements.
method Introduced biased gradient oracles for stochastic optimization, analyzed RSG and SGD algorithms with these oracles.
result Derived non-asymptotic bounds for convergence rates of algorithms with biased gradient oracles.
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 …
The paper analyzes bagging in overparameterized learning, deriving risk properties and optimal subsample sizes.
problem Characterizing the risk of bagged predictors in overparameterized settings.
method General strategy using classical results on simple random sampling, specialized for ridge and ridgeless predictors.
result Derives exact asymptotic risk of bagged ridge and ridgeless predictors under various conditions.
We give asymptotically tight estimates of tangent space variation on Riemannian submanifolds of Euclidean space with respect to the local feature size of the submanifolds. We show that the result follows directly from structural properties of local feature size of the Riemannian submanifold and some elementary Euclidea…
We derive formulas for F measures' standard error and confidence intervals.
problem Estimating F measures' accuracy with confidence.
method Analytic formulas based on asymptotic normality.
result Valid formulas for sample size planning.
Implicit Q-learning and SARSA adjust step-sizes automatically, improving stability and performance.
problem Numerical instability and slow progress in Q-learning and SARSA due to step-size calibration.
method Reformulate iterative updates as fixed-point equations, scaling step-sizes inversely with feature norms.
result Implicit methods maintain stability over broader step-size ranges and achieve comparable convergence rates.
Operational risk models commonly employ maximum likelihood estimation (MLE) to fit loss data to heavy-tailed distributions. Yet several desirable properties of MLE (e.g. asymptotic normality) are generally valid only for large sample-sizes, a situation rarely encountered in operational risk. In this paper, we study how…
Study shows how flat flow solutions in 2D converge to disks.
problem Understanding the asymptotics of area-preserving mean curvature flow in 2D.
method Analyzes flat flow solutions starting from bounded sets of finite perimeter.
result Flat flow solutions converge to a union of equally sized disks with exponential rate.
Two novel search strategies reduce complexity for target localization with size-dependent noise.
problem Target localization with varying measurement noise based on query region size.
method Proposes dyaPM and hiePM strategies with low complexity and connected query geometry. result Unified analysis shows dyaPM asymptotically optimal in search time, hiePM near-optimal in rate. Study wSAA for contextual decisions, improving uncertainty quantification under computational constraints.
problem Uncertainty quantification limitations in wSAA for contextual stochastic optimization.
method Establish central limit theorems and asymptotic-normality-based confidence intervals for optimal costs.
result Over-optimizing can mitigate misspecification and preserve asymptotic normality, albeit at a slower convergence rate.
Study bounds variance modulation function for K-spider distributions.
problem Bounding variance modulation function for K-spider distributions.
method Used folded moments and total probabilities of spider legs.
result Gave an interval for the variance modulation function.
SGD's uncertainty quantified in non-convex learning problems.
problem Uncertainty quantification in non-convex learning problems.
method Asymptotic normality of SGD iterates and bias characterization.
result SGD iterates are asymptotically normally distributed around the expected value of the invariant distribution.