Sharp bounds found on expert error in binary advice aggregation.
problem Aggregating binary advice from conditionally independent experts.
method Sharp upper and lower bounds on optimal error probability in asymmetric case.
result Sharp bounds recover and sharpen known results in symmetric case.
Sharp bounds derived for test error of finite-rank kernel ridge regression.
problem Loose bounds on test error for finite-rank kernels in machine learning.
method Sharp non-asymptotic upper and lower bounds for KRR test error.
result Tighter bounds on finite-rank KRR test error, valid for any regularization parameters.
Sharp bounds on uniform generalization errors in binary linear classification.
problem Understanding the uniform generalization errors in binary linear classification.
method Isoperimetric arguments, Poincaré and log-Sobolev inequalities for joint distributions.
result Sharp concentration bounds on uniform generalization errors, almost sure convergence in broad settings.
Sharp bounds on ERM's minimal error in regression.
problem Understanding ERM's performance in regression tasks.
method Sharp lower bounds for ERM in random and fixed design settings.
result ERM's performance depends on the global or local complexity of the model.
Sharp 2-Wasserstein bounds for DDPMs derived from Föllmer process.
problem Sampling error bounds for DDPMs in 2-Wasserstein distance.
method Lipschitz-type conditions on score function, Föllmer process, and log-concave target distributions.
result Sharp upper bounds for DDPMs in 2-Wasserstein distance, optimal in dimension and steps.
Bounds on factual and counterfactual distributions under measurement error in discrete models.
problem Measurement errors in discrete data and their impact on inference.
method Expressing modeling assumptions as linear constraints and using linear programming to derive bounds.
result Sharp bounds on factual and counterfactual distributions for various models, including instrumental variable scenarios.
Efficiently estimates sparse linear regression with heavy-tailed and outlier-contaminated data.
problem Estimating sparse linear regression coefficients with heavy-tailed and outlier-contaminated data.
method Efficient computation of estimators with sharp error bounds.
result Sharp error bounds for efficient estimators.
Sharp analysis of out-of-distribution error in overparameterized models with importance weights.
problem Understanding and quantifying the degradation of performance in overparameterized models when faced with underrepresented data.
method Sharp analysis of an overparameterized Gaussian mixture model with spurious features and cost-sensitive interpolating solutions incorporating importance weights.
result Characterization of a novel tradeoff between worst-case robustness and average accuracy as a function of importance weight magnitude.
Sharp error bounds derived for bidirectional GANs without restrictive assumptions.
problem Estimating the error of bidirectional GANs under various conditions.
method Dudley distance, neural network functions, decomposition of IPM.
result Nearly sharp bounds for bidirectional GAN estimation error.
Improved error estimate for SGLD sampling algorithm.
problem Establishing a precise error bound for SGLD.
method Sharp uniform-in-time error estimate for SGLD under mild assumptions.
result Uniform-in-time O ( η 2 ) O(η^2) O ( η 2 ) bound for KL-divergence between SGLD and Langevin diffusion. Paper analyzes online tensorial ICA convergence with stochastic approximation.
problem Online tensorial ICA convergence analysis.
method Stochastic approximation for nonconvex optimization.
result Sharp finite-sample error bound of O ~ ( d / T ) \tilde{O}(\sqrt{d/T}) O ~ ( d / T ) . Full-batch GD achieves generalization close to any stationary point with fewer assumptions.
problem Generalization and excess risk bounds for smooth losses, including non-Lipschitz and nonconvex cases.
method Path-dependent analysis of GD's generalization error, focusing on optimization error and stability.
result Generalization error is tightly bound in terms of optimization error and iteration count, bypassing common assumptions.
Sharpe ratio is widely used in asset management to compare and benchmark funds and asset managers. It computes the ratio of the excess return over the strategy standard deviation. However, the elements to compute the Sharpe ratio, namely, the expected returns and the volatilities are unknown numbers and need to be esti…
Sharp privacy bounds for sequential analysis of sensitive data.
problem Privacy degradation under sequential analysis of sensitive data.
method Edgeworth expansion in f-differential privacy framework.
result Improved privacy bounds under composition with refined approximation accuracy.
When randomized ensembles such as bagging or random forests are used for binary classification, the prediction error of the ensemble tends to decrease and stabilize as the number of classifiers increases. However, the precise relationship between prediction error and ensemble size is unknown in practice. In the standar…
A new method for streaming PCA provides confidence intervals for eigenvector entries.
problem Uncertainty quantification for individual entries in streaming PCA.
method Oja's algorithm, Bernstein-type concentration bound, Central Limit Theorem, subsampling algorithm.
result Sharp concentration bound and Central Limit Theorem for streaming PCA entries.
Efficiently learns a single neuron with adversarial noise, improving on prior work.
problem Learning a single neuron with adversarial label noise.
method Efficient algorithm using local error bounds from optimization theory.
result Approximates optimal L 2 2 L_2^2 L 2 2 -error within a constant factor. Automated method bounds causal effects in discrete data.
problem Partial identification of causal effects in discrete settings.
method Polynomial programming and dual relaxation for automated bounds.
result Algorithm provides guaranteed non-sharp and ε ε ε -sharp bounds. Deep belief networks can approximate any multivariate density with binary hidden units.
problem Approximating multivariate probability densities with binary hidden units.
method Sharp quantitative bounds on approximation error in terms of hidden units.
result Deep belief networks can approximate any multivariate density with binary hidden units under mild integrability requirements.
The stochastic gradient descent (SGD) optimization algorithm plays a central role in a series of machine learning applications. The scientific literature provides a vast amount of upper error bounds for the SGD method. Much less attention as been paid to proving lower error bounds for the SGD method. It is the key cont…
Forward regression is a statistical model selection and estimation procedure which inductively selects covariates that add predictive power into a working statistical regression model. Once a model is selected, unknown regression parameters are estimated by least squares. This paper analyzes forward regression in high-…
In statistical inference problems, we wish to obtain lower bounds on the minimax risk, that is to bound the performance of any possible estimator. A standard technique to obtain risk lower bounds involves the use of Fano's inequality. In an information-theoretic setting, it is known that Fano's inequality typically doe…
The study establishes minimax bounds for estimating operators from noisy samples.
problem Estimating unknown operators between Hilbert spaces from noisy data.
method Developed a minimax theory for uniformly bounded Lipschitz operators, proving lower and upper bounds.
result Sharp characterizations of minimax risk for generic Lipschitz operators, showing a curse of sample complexity.
Sharp risk bounds for early-stopping in Gaussian linear regression are derived.
problem Minimizing in-sample mean squared error in high-dimensional Gaussian linear regression.
method Early-stopped mirror descent (ESMD) with local Gaussian width bounds.
result Sharp risk bounds extend to early-stopped mirror descent for least squares estimator (LSE).
Sharp Lipschitz bounds for flow-matching and diffusion models with optimal sampling rates.
problem Establishing optimal Lipschitz regularity for flow-matching and diffusion models.
method Sharp Lipschitz regularity theory for flow-matching vector fields and diffusion-model scores.
result Achieves optimal sampling rate of d / N \sqrt{d}/N d / N for Euler-type samplers in dimension d d d . Sharp bounds for approximating Sobolev functions by ridge functions and networks.
problem Approximating Sobolev functions with multivariate ridge functions and networks.
method Proving sharp upper and lower bounds for approximation order.
result Order of approximation asymptotically behaves as n − r / ( d − ℓ ) n^{-r/(d-\ell)} n − r / ( d − ℓ ) . This paper introduces minimum-risk recalibration for probabilistic classifiers, improving their reliability and accuracy.
problem Improving the reliability and accuracy of probabilistic classifiers.
method Minimum-risk recalibration within the MSE decomposition framework, analyzing UMB method and label shift adaptation.
result The optimal number of bins for UMB scales with n 1 / 3 n^{1/3} n 1/3 , resulting in a risk bound of approximately O ( n − 2 / 3 ) O(n^{-2/3}) O ( n − 2/3 ) . Sharp bounds found for minimal surface solutions.
problem Finding bounds for minimal surface solutions.
method Analyzing minimal surface equation with specific boundary conditions.
result Sharp bounds established for solutions over certain domains.
We provide sharp empirical estimates of expectation, variance and normal approximation for a class of statistics whose variation in any argument does not change too much when another argument is modified. Examples of such weak interactions are furnished by U- and V-statistics, Lipschitz L-statistics and various error f…
Unified approach for robust low rank matrix estimation with adversaries.
problem Robust low rank matrix estimation in the presence of adversaries.
method Unified approach combining Huber loss and nuclear norm penalization.
result Sharp estimation error bounds for matrix compressed sensing and completion.
SDP achieves optimal error in noisy phase synchronization.
problem Phase synchronization with noisy measurements.
method SDP relaxation of Maximum Likelihood Estimation (MLE).
result Achieves error bound of ( 1 + o ( 1 ) ) σ 2 2 n p (1+o(1))\frac{σ^2}{2np} ( 1 + o ( 1 )) 2 n p σ 2 under normalized squared ℓ 2 \ell_2 ℓ 2 loss, matching minimax lower bound. When the in-sample Sharpe ratio is obtained by optimizing over a k-dimensional parameter space, it is a biased estimator for what can be expected on unseen data (out-of-sample). We derive (1) an unbiased estimator adjusting for both sources of bias: noise fit and estimation error. We then show (2) how to use the adjust…
Sharp bounds on Alexandrov spaces' boundaries with rigidity analysis.
problem Volume bounds on Alexandrov spaces' boundaries.
method Sharp volume bounds and rigidity analysis of Alexandrov spaces.
result New sharp volume bounds and classification of rigidity cases.
This paper closes the gap on matching pursuit's convergence rate.
problem Improving the understanding of matching pursuit's convergence rate.
method Constructing a worst case dictionary to analyze matching pursuit's performance.
result Sharp characterization of matching pursuit's convergence rate as n − α n^{-α} n − α , with α ≈ 0.182 α \approx 0.182 α ≈ 0.182 . Sharp Gaussian bounds derived for Schrödinger kernel on Ricci solitons.
problem Analyzing Schrödinger heat kernel on gradient shrinking Ricci solitons.
method Deriving sharp Gaussian upper bounds for the Schrödinger heat kernel.
result Sharp upper and lower bounds for eigenvalues of the Schrödinger operator.
The paper extends Weyl's law to CROSSes, showing sharpness and polynomial improvement.
problem Understanding the error term in Weyl's law for different types of manifolds.
method Analyzing the Laplacian eigenvalues on Compact Rank One Symmetric Spaces (CROSSes).
result For CROSSes, the error term in Weyl's law is sharp, and for products of CROSSes, it can be polynomially improved.
Sharp estimates on 2-step nilpotent Lie groups' metrics and cones.
problem Estimating asymptotic metrics in 2-step nilpotent Lie groups.
method Developed a novel technique to perturb rectifiable curves.
result Every 2-step nilpotent Riemannian Lie group is at bounded distance from its asymptotic cone.
Optimizes quickest change detection with bounded means under ARL constraint.
problem Quickest detection of changepoints with bounded means under ARL constraint.
method Derives universal lower and upper bounds for detection delay.
result Achieves universal lower bound in the bounded mean detection setting.
Sharp bounds derived for the first two Steklov eigenvalues of exterior domains.
problem Finding bounds for the first two eigenvalues of Steklov eigenvalue problems on exterior domains.
method Sharp lower and upper bounds derived using the support function and distance function to the origin of the boundary.
result Sharp bounds for the first two eigenvalues of Steklov eigenvalue problems on exterior domains.
This paper improves HNNs by learning optimal curvature for better generalization.
problem Inappropriate curvatures in HNNs lead to suboptimal performance.
method Sharpness-aware curvature learning method to smooth loss landscape.
result Proposed method improves HNNs' generalization across various settings.
Sharp upper bound found for stable minimal surfaces.
problem Bounding the diameter of stable minimal surfaces.
method Analyzing three-dimensional Riemannian manifolds with specific curvature conditions.
result Sharp upper bound for the diameter of stable minimal surfaces.
This paper optimizes portfolio selection by penalizing tracking error, improving Sharpe ratio.
problem Optimizing portfolio allocation with a penalty for deviation from a reference portfolio.
method Formulated as a McKean-Vlasov control problem, provides explicit solutions and asymptotic expansions.
result The penalized portfolio strategy outperforms standard mean-variance and reference portfolios in most cases.
We prove sharp bounds for the growth rate of eigenfunctions of the Ornstein-Uhlenbeck operator and its natural generalizations. The bounds are sharp even up to lower order terms and have important applications to geometric flows.
Sharpe ratio (sometimes also referred to as information ratio) is widely used in asset management to compare and benchmark funds and asset managers. It computes the ratio of the (excess) net return over the strategy standard deviation. However, the elements to compute the Sharpe ratio, namely, the expected returns and …
Sharp bounds for curve isoperimetric deficit derived.
problem Finding sharp bounds for the isoperimetric deficit of curves.
method Fourier analysis applied to derive Wirtinger-type inequalities.
result Sharp lower and upper bounds for the isoperimetric deficit proved.
Sharp upper diameter limit found for Ricci solitons.
problem Bounding the diameter of compact shrinking Ricci solitons.
method Used a sharp logarithmic Sobolev inequality and Vitali-type covering argument.
result Sharp upper diameter bound established in terms of scalar curvature and entropy.
Sharp policy value estimation for contextual bandits with unobserved confounders.
problem Estimating policy value under unobserved confounders with sensitivity analysis.
method Kernel method to approximate conditional moment constraints, leveraging f-divergence.
result Sharp lower bound of policy value, avoiding coarse relaxation of uncertainty set.
This note gives a short, self-contained, proof of a sharp connection between Gittins indices and Bayesian upper confidence bound algorithms. I consider a Gaussian multi-armed bandit problem with discount factor γ γ γ . The Gittins index of an arm is shown to equal the γ γ γ -quantile of the posterior distribution of the arm'…