High-probability bound for distributed stochastic approximation tracking error.
problem Analyzing the convergence of distributed stochastic approximation schemes.
method Analysis using ODE approach to stochastic approximation.
result High probability bound for tracking error between iterates and limiting differential equation.
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 ) . New bounds on ReLU networks for low-regular functions.
problem Bounding approximation error for ReLU networks on low-regular functions.
method Complexity analysis of Fourier features residual networks to ReLU networks.
result Approximation error bound proportional to target function norm and inversely proportional to network width and depth.
The paper bounds neural networks' approximation error and applies it to regression and GANs.
problem Bounding the approximation error of norm-constrained neural networks.
method Proved upper and lower bounds on approximation error using Rademacher complexity.
result Obtained convergence rates for over-parameterized neural networks and optimal GAN learning rates.
Algorithm approximates functions into manifolds with curvature bounds.
problem Approximating functions into manifolds with lower curvature bounds.
method Algorithm using manifold exponential and logarithm, with error bounds based on sectional curvature.
result Error bounds for nonnegative sectional curvature are similar to linear space approximations.
Study approximates probability measures using structured classes of functions.
problem Approximating probability measures in Wasserstein- p p p distance. method Structured classes of approximators for functions in L p ( Ω ) L_p(Ω) L p ( Ω ) , transferring to measures in W p ( Ω ) W_p(Ω) W p ( Ω ) . result Linear rate approximation for measures with densities bounded away from zero.
The approximation power of general feedforward neural networks with piecewise linear activation functions is investigated. First, lower bounds on the size of a network are established in terms of the approximation error and network depth and width. These bounds improve upon state-of-the-art bounds for certain classes o…
Paper finds how many neurons are needed to approximate histogram distributions.
problem How many neurons are needed to approximate a target probability distribution?
method Examined for uniform input distribution and histogram target distributions, using efficient neural net construction.
result Obtained a new upper bound on the number of required neurons, strictly better than previous 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.
Sharp bounds on neural network approximation rates and widths.
problem Estimating approximation rates, metric entropy, and n-widths of shallow neural networks.
method Introducing smoothly parameterized dictionaries and providing upper and lower bounds.
result Sharp bounds on approximation rates, metric entropy, and n-widths for neural networks with various activation functions.
Uniform TD(0) bound derived for function approximation with Markov noise.
problem Uniform concentration bound for TD(0) with function approximation.
method Contractive stochastic approximation, martingale and Markov noises, Poisson equation, relaxed concentration inequalities.
result Uniform all-time concentration bound for TD(0) with linear function approximation.
Bayesian bandit algorithms with approximate inference improve regret bounds in stochastic linear bandits.
problem Theoretical justification for Bayesian bandit algorithms with approximate inference in stochastic linear bandits.
method Proposed a theoretical framework to analyze approximate inference impact and conducted frequentist regret analysis on LinTS and LinBUCB.
result LinTS and LinBUCB preserve their original regret upper bounds with larger constant terms in approximate inference settings.
Transformers use ReLUs to approximate softmax efficiently.
problem Analyzing resource usage in softmax transformer models.
method Translating ReLU approximation results to softmax attention mechanisms.
result Economic resource bounds for softmax attention mechanisms.
Stochastic approximation algorithms show exponential progress bounds.
problem Analyzing the convergence of stochastic approximation algorithms.
method Developed geometric ergodicity proofs to establish exponential concentration bounds.
result Proved faster convergence rates for specific algorithms.
Minimum width for ReLU networks to approximate L^p functions is max(d_x+1, d_y).
problem Characterizing the minimum width for ReLU networks to approximate L^p functions.
method Analyzing networks with ReLU activation functions and proving the minimum width required.
result The minimum width required for the universal approximation of L^p functions is exactly max(d_x+1, d_y).
Paper develops bounds for stochastic approximation with averaging.
problem Establish high-probability bounds for averaged stochastic approximation.
method Develops a general framework for non-asymptotic concentration bounds.
result Derives sharp bounds for averaged iterates and tightens existing results.
Theory for deep neural network approximation of score function and its derivatives.
problem Handling data distributions with low-dimensional structure and unbounded support.
method Simultaneous approximation of the score function and its derivatives using deep neural networks.
result Approximation error bounds match literature but relax bounded support requirement.
Paper analyzes normal approximation for two-timescale stochastic algorithms, revealing interaction between fast and slow timescales.
problem Non-asymptotic bounds for accuracy of normal approximation in linear two-timescale stochastic approximation algorithms.
method Established bounds for normal approximation in terms of convex distance, focusing on last iterate and Polyak-Ruppert averaging.
result Normal approximation rate for the last iterate improves with increased timescale separation, while it decreases in the averaged setting.
New bounds for private matrix approximation using Gaussian noise and Dyson Brownian Motion.
problem Private approximation of symmetric matrices with Gaussian noise.
method Viewing Gaussian noise as Dyson Brownian Motion to track eigenvalue and eigenvector evolution.
result Improved bounds on Frobenius-distance utility for private matrix approximation.
Transformers can learn noisy linear systems with depth and IID data.
problem Learning noisy linear dynamical systems with transformers.
method Theoretical analysis of multi-layer and single-layer transformers with respect to L 2 L^2 L 2 -testing loss. result Single-layer transformers have a non-diminishing lower bound on approximation error, suggesting depth separation.
Sharp lower bounds on shallow neural networks' approximation rates are derived.
problem The efficiency of shallow neural networks in approximating functions.
method Lower bounding the L 2 L^2 L 2 -metric entropy and Kolmogorov n n n -widths of the convex hull of neural network basis functions. result Sharp lower bounds on the approximation rates for shallow neural networks are provided.
Transformer networks approximate Hölder and Sobolev functions with fixed-depth networks.
problem Nonparametric regression with dependent observations.
method Established novel upper bounds for Transformer networks approximating Hölder and Sobolev functions under various β β β -mixing data assumptions. result Explicit convergence rates for nonparametric regression problems under β β β -mixing data assumptions. Study on RNNs' ability to approximate past-dependent Hölder functions and their application to regression.
problem Understanding and optimizing the approximation capacity of RNNs for regression tasks.
method Derivation of upper bounds on RNN approximation error for Hölder smooth functions and application to regression.
result Achievement of minimax optimal prediction error bounds for RNNs under various data assumptions.
UCRL-WVTR tackles long-term reinforcement learning with general approximations, achieving horizon-free and instance-dependent regret bounds.
problem Long-term reinforcement learning with general function approximations.
method UCRL-WVTR proposes a novel algorithm, UCRL-WVTR, with weighted value-targeted regression and a high-order moment estimator.
result Achieves horizon-free and instance-dependent regret bounds matching minimax lower bounds up to logarithmic factors.
Derives error bounds for stochastic iterative algorithms using Stein's method.
problem Bounding errors in stochastic iterative algorithms like SGD and SGLD.
method Uses infinite-dimensional Stein's method of exchangeable pairs to derive functional approximation error bounds.
result Establishes non-asymptotic error bounds for algorithm sample paths and variance of iterate averages.
New bound on neural nets complexity for approximating functions.
problem Approximating continuous functions with shallow neural networks.
method Inspired by Stone-Weierstrass theorem, constructive proof.
result General upper bound on neuron count for accuracy.
Paper develops efficient RL algorithm for general value function approximation.
problem Lack of theory for RL with general value function approximation.
method Provable efficient RL algorithm using bounded eluder dimension.
result Achieves a regret bound of O ~ ( p o l y ( d H ) T ) \widetilde{O}(\mathrm{poly}(dH)\sqrt{T}) O ( poly ( d H ) T ) . Paper establishes convergence rates and concentration bounds for stochastic approximation and reinforcement learning with Markovian noise.
problem Analyzing convergence rates and concentration bounds for stochastic approximation and reinforcement learning with Markovian noise.
method Novel discretization of the mean ODE of stochastic approximation algorithms using intervals with diminishing length.
result First almost sure convergence rate and maximal concentration bound with exponential tails for contractive stochastic approximation algorithms with Markovian noise.
The study assesses low-rank approximations in Gaussian Process regression.
problem Improving Gaussian Process regression efficiency with low-rank approximations.
method Analyzes two low-rank approximations: random Fourier features and Mercer expansion truncation.
result Bounds on the divergence and error between exact and approximate GP models.
Variational inference (VI) is a widely used framework in Bayesian estimation. For most of the non-Gaussian statistical models, it is infeasible to find an analytically tractable solution to estimate the posterior distributions of the parameters. Recently, an improved framework, namely the extended variational inference…
EBUCB framework achieves optimal regret with bounded approximate inference error.
problem Theoretical gap between practical performance and theoretical justification of Bayesian bandit algorithms with approximate inference.
method Enhanced Bayesian Upper Confidence Bound (EBUCB) framework that accommodates bandit problems with approximate inference.
result EBUCB achieves optimal regret order O ( log T ) O(\log T) O ( log T ) under certain conditions on inference error. Improved gap-dependent bounds for reinforcement learning with linear approximations.
problem Achieving nearly minimax-optimal performance with linear function approximation.
method Developed and analyzed the LSVI-UCB++ algorithm and its concurrent variant.
result First gap-dependent regret bound for nearly minimax-optimal algorithm LSVI-UCB++.
The study assesses low-rank approximations in Gaussian Process regression.
problem Improving the efficiency of Gaussian Process regression while maintaining accuracy.
method Analyzes two low-rank approximations: random Fourier features and Mercer expansion truncation, and bounds the divergence and error between exact and approximate models.
result Theoretical bounds on the divergence and error between exact and approximate Gaussian Process models are provided.
New method for finite approximations improves inference efficiency and applicability.
problem Challenges in Bayesian nonparametric inference due to infinite dimensionality.
method Automated Independent Finite Approximation (AIFA) for finite-dimensional approximations of CRMs and NCRMs.
result AIFA provides more efficient and straightforward derivations and parallel computing compared to Truncated Finite Approximations (TFAs).
This paper approximates SA iterates using Gaussian distributions for tail bounds.
problem Characterizing the distribution of stochastic approximation iterates in finite time.
method Approximating pre-limit distributions of SA iterates by Gaussian sequences with recursively defined covariances.
result Explicit bounds on the Wasserstein-1 distance between rescaled iterates and Gaussians.
Paper bounds the minimal rank for kernel ridge regression approximations.
problem Efficient memory and computation for kernel ridge regression.
method Lower bound on minimal rank for reliable prediction power.
result Nyström method's computational cost is almost linear in sample size.
Paper improves clustering risk bounds for kernel k-means.
problem Improving clustering risk bounds for kernel k-means.
method Analyzes kernel k-means and Nyström approximation.
result Achieves nearly optimal excess clustering risk bound.
In this paper we derive an easily computed approximation to European basket call prices for a local volatility jump-diffusion model. We apply the asymptotic expansion method to find the approximate value of the lower bound of European basket call prices. If the local volatility function is time independent then there i…
The paper analyzes risk estimation methods and derives bounds for OCE risk.
problem Estimating the Optimized Certainty Equivalent (OCE) risk from samples.
method Derives mean-squared error and concentration bounds for SAA of OCE, and analyzes an efficient stochastic approximation-based estimator.
result Finite sample bounds and mis-identification probability bounds for the efficient estimator.
We introduce the Variational Holder (VH) bound as an alternative to Variational Bayes (VB) for approximate Bayesian inference. Unlike VB which typically involves maximization of a non-convex lower bound with respect to the variational parameters, the VH bound involves minimization of a convex upper bound to the intract…
PCA-Net combines PCA and neural networks for operator approximation, with new bounds on complexity.
problem Developing approximation theory for PCA-Net architecture.
method Combines PCA and neural networks, derives universal approximation results and lower bounds on complexity.
result PCA-Net can overcome the curse of parametric complexity for specific operators.
General lower bounds on neural network approximation in L^p norm.
problem Fundamental limits of neural network expressivity.
method General lower bound proof on approximation in L^p norm, applied to feed-forward neural networks.
result Neural networks can't approximate certain functions as well as previously thought.
Analytic networks with bounded coefficients can't outperform polynomial approximations.
problem Approximation limits of neural networks with analytic activation functions under coefficient constraints.
method Deterministic analysis using comparison argument and Bernstein-type estimates.
result Networks with analytic activation functions and controlled coefficients cannot outperform classical polynomial approximation rates on non-analytic targets.
Improved bounds for function approximation in nonlinear sets.
problem Achieving high probability error with limited samples in nonlinear function approximation.
method Restricting model class to a neighbourhood of the best approximation and estimating sample complexity using tangent and normal spaces' complexities and curvature.
result Improved worst-case bounds for sample complexity in more general sets like tensor networks and neural networks.
The paper provides mean-square error bounds for stochastic approximation algorithms.
problem Error bounds for recursive equations with Markovian disturbances.
method Analysis of mean-square error for stochastic approximation algorithms.
result Mean-square error achieves the optimal rate of O ( 1 / n ) O(1/n) O ( 1/ n ) under certain conditions. Study shows how to approximate and estimate high-dimensional classification functions without the curse of dimensionality.
problem Approximating and estimating classification functions in high-dimensional spaces.
method Modified existing results to show that R B V 2 RBV^2 R B V 2 functions can be approximated by neural networks with bounded weights. Proved the existence of a neural network with bounded weights approximating a classification function. Leveraged these bounds to quantify estimation rates. result Neural networks can approximate R B V 2 RBV^2 R B V 2 functions without the curse of dimensionality, leading to efficient estimation rates. We apply Tian's method in Kahler-Einstein problem to prove that a conic K\''ahler metric with lower Ricci curvature bound can be approximated by smooth K\''ahler metrics with the same lower Ricci curvature bound. Furthermore, conic singularities here can be along a simple normal crossing divisor.
We provide bounds for kernel matrices and new approximations for high-dimensional data.
problem Approximating high-dimensional empirical kernel matrices.
method Decoupling results for U-statistics and non-commutative Khintchine inequality.
result New tighter approximations for inner-product kernel matrices.