A new method for gradient recovery on manifold data without tangent spaces.
problem Gradient recovery schemes for data on discretized manifolds.
method Parametric Polynomial Preserving Recovery (PPPR) on manifolds.
result Superconvergence and high curvature stability of PPPR.
We revisit resampling procedures for error estimation in binary classification in terms of U-statistics. In particular, we exploit the fact that the error rate estimator involving all learning-testing splits is a U-statistic. Thus, it has minimal variance among all unbiased estimators and is asymptotically normally dis…
Paper introduces a new estimator for Rasch model with exact error analysis.
problem Estimating parameters of the Rasch model with performance guarantees.
method Develops a novel L-MMSE estimator for the Rasch model with nonasymptotic analysis.
result The L-MMSE estimator provides exact error analysis and performs similarly to state-of-the-art estimators.
Study robust linear regression with outliers, providing exact asymptotics for ERM performance.
problem Robust linear regression in high-dimension with outliers.
method Analyzes ℓ2, ℓ1, and Huber losses, providing asymptotic performance metrics. result Optimally-regularised ERM is asymptotically consistent with simple calibration, but Huber loss requires norm calibration.
PPI++ outperforms gold-standard labels only if pseudo-labels are highly correlated.
problem Optimizing statistical estimation using noisy pseudo-labels.
method Exact finite-sample analysis of PPI++ on mean estimation problem.
result PPI++ has provably worse estimation error than gold-standard labels alone in some settings.
The most important aspect of any classifier is its error rate, because this quantifies its predictive capacity. Thus, the accuracy of error estimation is critical. Error estimation is problematic in small-sample classifier design because the error must be estimated using the same data from which the classifier has been…
The paper improves confidence intervals for test error using cross-validation.
problem Improving confidence intervals for test error in machine learning.
method Develops central limit theorems and consistent estimators for cross-validation.
result Provides asymptotically-exact confidence intervals and hypothesis tests.
Exact expressions for double descent and implicit regularization in over-parameterized models.
problem Understanding the generalization error of over-parameterized models like deep neural networks.
method Surrogate random design to replace standard i.i.d. design, leading to exact expressions for mean squared error and implicit regularization.
result Exact non-asymptotic expressions for double descent and implicit regularization in over-parameterized models.
Study on ridge regression in convolutional models shows double descent error behavior.
problem Understanding generalization and estimation error in over-parameterized convolutional models.
method Analysis of ridge estimators for convolutional linear models, derivation of exact error formulae.
result Ridge estimators exhibit double descent error behavior in high-dimensional convolutional models.
We solve matrix denoising with both row and column correlations, setting limits and designing optimal methods.
problem Matrix denoising with doubly heteroscedastic noise (both row and column correlations).
method Established information-theoretic and algorithmic limits, designed a novel spectral estimator with optimality guarantees.
result The novel spectral estimator achieves positive correlation with the signal and Bayes-optimal error under one-sided heteroscedasticity.
Paper calculates the exact error of LDA models.
problem Bayesian generalization error in Latent Dirichlet Allocation (LDA).
method Theoretical analysis of learning coefficient using algebraic geometry.
result Exact asymptotic form of LDA's generalization error.
The study analyzes how covariance estimation errors affect the global minimum-variance portfolio under heavy-tailed distributions.
problem The impact of covariance estimation errors on the global minimum-variance portfolio under heavy-tailed distributions.
method Characterization of covariance-estimation error's effect on GMVP suboptimality, derivation of regret identity and bound, application to heavy-tailed returns.
result The decision geometry of GMVP regret is invariant to a (p-1)-dimensional projection of the error matrix, with invariance to the covariance-scale direction as an exact special case.
Paper connects Sharpe ratio and Student t-statistic, providing exact distribution and asymptotic behavior.
problem Error-prone Sharpe ratio due to statistical estimation of expected returns and volatilities.
method Derive exact distribution of Sharpe ratio for independent normally distributed returns, extend to AR(1) assumptions.
result Empirical Sharpe ratio is asymptotically optimal and achieves Cramer Rao bound.
SS-GEN simulates rare events in heavy and light-tailed data.
problem Estimating probabilities of extreme events in multivariate data.
method Self-Similar Generative Estimation (SS-GEN) decomposes tail distribution into radial and angular components.
result SS-GEN generates representative extreme scenarios and estimates rare-event probabilities beyond observed data.
We study confidence intervals based on hard-thresholding, soft-thresholding, and adaptive soft-thresholding in a linear regression model where the number of regressors k may depend on and diverge with sample size n. In addition to the case of known error variance, we define and study versions of the estimators when…
New method provides exact performance guarantees for MAP inference in deep networks.
problem Analyzing MAP inference in deep networks with complex data.
method Multi-layer vector approximate message passing (ML-VAMP) method.
result Mean squared error of ML-VAMP estimate can be exactly characterized in high-dimensional random limit.
Exact optimality achieved in distributed mean estimation with shared randomness.
problem Achieving optimal communication, privacy, and utility tradeoffs in distributed mean estimation.
method Utilization of a rotationally symmetric shared random codebook and a k-closest encoding mechanism. result Proposed mechanism achieves exact optimality for randomly rotated simplex codebook.
This paper sets fundamental limits for rank-one matrix estimation with varying noise levels.
problem Estimating a rank-one matrix from Gaussian observations with different noise levels across blocks.
method Novel reduction from heterogeneous noise to homogeneous noise, proving asymptotic error bounds.
result Asymptotically exact formulas for minimum mean-squared error in estimating rank-one matrix and factors.
Paper introduces a diagnostic for approximate inference methods.
problem Estimating errors in probabilistic inference algorithms, especially for approximate methods.
method Repeatedly simulate datasets from the prior and perform inference on each, estimating a symmetric KL-divergence.
result A diagnostic for approximate inference methods can be estimated using symmetric KL-divergence.
This paper tackles fast optimization on large data via subsampling with statistical guarantees and MSE approximation.
problem Optimization on large-scale data is computationally difficult.
method Subsampled optimization using a surrogate dataset and solving the subsampled optimization problem.
result Asymptotic properties and MSE approximation of approximate solutions with respect to exact solutions.
Paper proposes a method for estimating sparse and low-rank tensors from sketchings.
problem Estimating sparse and low-rank tensors from limited data.
method Two-stage non-convex implementation using sparse tensor decomposition and thresholded gradient descent.
result Exact and stable recovery of tensors in noisy and noiseless cases with high probability.
New methods for statistical inference in SGD for large-scale data.
problem Statistical inference of true model parameters in SGD.
method Proposed two consistent estimators of asymptotic covariance and debiased estimator for high-dimensional linear regression.
result Construct asymptotically exact confidence intervals and hypothesis tests for true model parameters.
Super-efficient automatic differentiation outperforms analytic methods in min-min optimization.
problem Optimizing functions defined as a minimum using iterative algorithms.
method Comparing automatic differentiation to analytic gradient estimation methods.
result Automatic differentiation yields an asymptotic error close to the square of the optimization error, demonstrating super-efficiency.
Develops asymptotic theory for deep Cox models to enable valid inference.
problem Theoretical gaps in deep neural network estimators for Cox models.
method Asymptotic distribution theory linking in-sample optimization error to population risk.
result Pointwise and multivariate asymptotic normality for subsampled ensemble estimators.
The main goal of the paper is to address the issue of the existence of Kempf's distortion function and the Tian-Yau-Zelditch (TYZ) asymptotic expansion for the Kepler manifold - an important example of non compact manfold. Motivated by the recent results for compact manifolds we construct Kempf's distortion function an…
Exact minibatch MH method improves scalability for large datasets.
problem Inexactness in minibatch MH methods causes inference errors.
method TunaMH proposes an exact minibatch MH method with a tunable batch size.
result TunaMH is asymptotically optimal in terms of batch size.
Max-margin classifiers can overfit without harming performance in high dimensions.
problem Understanding how max-margin classifiers generalize in high-dimensional settings.
method Stylized setting with Gaussian features and labels, proportional asymptotics, and random neural network features.
result Exact expressions for limiting generalization error and conditions for 'benign overfitting'.
Improved computational efficiency for estimating Wasserstein distance.
problem Inefficient computation of Wasserstein distance for large samples.
method Developed Sample-Sketch-Solve paradigm using grid sketches.
result Approximates Wasserstein distance within ε error in ε^(-max(2, (d+1+o(1))/(1+α))) time.
The paper proposes a method to construct well-calibrated prediction sets for correlated target variables.
problem Constructing well-calibrated prediction sets for correlated target variables.
method The method uses vine copulas to estimate the joint cumulative distribution function of non-conformity scores and improves the asymptotic efficiency of the quantile estimate.
result The method guarantees asymptotically exact coverage and competitive efficiency on real-world regression problems.
New method controls gradient error for sparse MRFs.
problem Efficient learning for sparse discrete MRFs with NP-hard inference.
method Stochastic proximal gradient (SPG) with controlled gradient approximation error.
result Novel bounds control gradient approximation quality.
Unified analysis of TD learning using MJLS theory for linear function approximators.
problem Characterizing the exact behaviors of TD learning algorithms with linear function approximators.
method Exploiting connections to Markov jump linear systems (MJLS) theory to analyze TD learning algorithms.
result Closed-form expressions for mean and covariance matrix of TD estimation error at any time step.
Paper finds exact exponent in optimal error rates for crowdsourcing.
problem Determining the optimal number of workers for accurate label aggregation.
method Using the Dawid-Skene model, the paper establishes matching upper and lower bounds with an exact exponent.
result The exact exponent mI(π) is found, allowing precise sample size requirements. Spectral methods improve signal recovery in mixed GLMs with precise asymptotics.
problem Estimating multiple signals from unlabeled observations in mixed GLMs.
method Developed exact asymptotics for spectral methods in a proportional regime.
result Optimized spectral method combined with a linear estimator minimizes estimation error.
Exact learning of tree-structured models with side info and noise.
problem Learning tree-structured graphical models with side information and noise.
method Probabilistic tools from strong large deviations theory.
result Exact asymptotics of structure learning from samples.
Stochastic gradient descent procedures have gained popularity for parameter estimation from large data sets. However, their statistical properties are not well understood, in theory. And in practice, avoiding numerical instability requires careful tuning of key parameters. Here, we introduce implicit stochastic gradien…
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. New method for clustering tasks with heterogeneous data.
problem Clustered multitask learning with semiparametric and heterogeneous nuisances.
method Adaptive fused orthogonal estimator with Neyman-orthogonal losses and data-driven fusion penalties.
result Achieves exact clustering recovery and pooled parametric convergence rates.
Enhances statistical mechanics solving using VANs with MCMC or importance sampling.
problem Sampling error in solving statistical mechanics using VANs.
method Integrates MCMC or importance sampling to correct sampling error in VANs.
result Asymptotically unbiased estimators for physical quantities are achieved.
Study GLS estimator properties in multivariate regression with heteroskedastic and autocorrelated errors.
problem Asymptotic properties of GLS estimator in multivariate regression with specific error structures.
method Derive Wald statistics for linear restrictions and assess their performance.
result Wald statistics remain robust to heteroskedasticity and autocorrelation.
In this paper, we obtain asymptotic formulas with error estimates for the implied volatility associated with a European call pricing function. We show that these formulas imply Lee's moment formulas for the implied volatility and the tail-wing formulas due to Benaim and Friz. In addition, we analyze Pareto-type tails o…
Paper analyzes adversarial training's performance in binary classification.
problem Understanding the generalization performance of adversarial training.
method Derives precise theoretical predictions for adversarial training performance.
result Provides exact asymptotics for test errors of adversarial training.
The paper bounds estimation and prediction errors in time series using entropy.
problem Estimating and predicting errors in time series analysis.
method Information-theoretic approach focusing on conditional entropy.
result Generic bounds on estimation and prediction errors determined by conditional entropy.
New weighted Lasso estimates improve logistic regression performance with measurement error.
problem Improper Lasso estimates in sparse logistic regression with equal penalties.
method Proposed weighted Lasso estimates using McDiarmid inequality for non-asymptotic oracle inequalities.
result Finite sample behavior illustrated by non-asymptotic oracle inequalities for estimation and prediction errors.
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.
Paper analyzes SVM behavior in high dimensions with exact formulas.
problem Characterizing SVM behavior in high-dimensional data with fixed ratio of features to samples.
method Exact asymptotic formulas derived through heuristic leave-one-out calculations.
result Exact formulas for variability of optimal coefficients, support vectors, objective function value, and misclassification error.
Novel method recursively partitions sample space for density estimation.
problem Estimating complex density functions efficiently and accurately.
method Recursive partitioning of the sample space, asymptotically exact.
result Asymptotically exact approximation of any density function.
A common approach to statistical learning with big-data is to randomly split it among m machines and learn the parameter of interest by averaging the m individual estimates. In this paper, focusing on empirical risk minimization, or equivalently M-estimation, we study the statistical error incurred by this strategy…
Estimates and tests treatment effects on entire outcome distributions.
problem Treatment effects on entire outcome distributions, not just averages.
method Proposes a novel estimand and doubly robust estimator, develops a test.
result First test with provably valid type 1 error guarantees in this setting.