Paper proposes a pre-conditioning technique to speed up gradient-descent convergence in distributed linear least-squares problems.
problem Expediting convergence of gradient-descent method for ill-conditioned distributed linear least-squares problems.
method Iterative pre-conditioning technique to improve convergence rate of gradient-descent method.
result Pre-conditioned gradient-descent achieves superlinear convergence for unique solutions and improved linear convergence otherwise.
Researchers find continuous solutions to minimizers in weighted least gradient problems.
problem Existence and regularity of minimizers to weighted least gradient problems.
method Constructing continuous solutions using Sternberg-Williams-Ziemer technique extended to inhomogeneous variations.
result Continuous solutions constructed for minimizers in any dimension n≥2, with level sets being minimal surfaces in a conformal metric.
Gradient flow in least squares regression is at least 1.69 times riskier than ridge regression.
problem Comparing the risk of gradient descent iterates to ridge regression in least squares regression.
method Continuous-time view of gradient descent, proving risk bounds.
result Gradient flow's risk is at least 1.69 times that of ridge regression.
Efficient method for high-dimensional American option pricing and hedging.
problem High-dimensional American option pricing and hedging.
method Gradient-enhanced sparse Hermite polynomial expansions combined with least squares Monte Carlo.
result Outperforms state-of-the-art methods in high dimensions with comparable computational cost.
This work proves convergence of adaptive resampling for random Fourier features.
problem Sampling Fourier frequencies well for high-dimensional data.
method Data adaptive resampling of Fourier frequencies, asymptotically optimal.
result Proves convergence of adaptive resampling method for regression and classification problems.
Extends phase retrieval methods to handle sensing vector errors.
problem Phase retrieval with errors in sensing vectors.
method Total Least Squares (TLS) framework applied to gradient descent.
result Gradient descent can efficiently solve TLS phase retrieval.
New algorithm reduces bias and variance in weighted least-squares solutions.
problem Inconsistent linear least-squares problems with rapidly decaying singular values.
method Regularized block Kaczmarz (ReBlocK) algorithm.
result ReBlocK outperforms RBK and minibatch SGD for inconsistent problems.
Study shows how SGD's implicit regularization relates to ridge regression.
problem Least squares regression optimization with mini-batch SGD.
method Analyzes stochastic gradient flow as a continuous-time model of SGD.
result Bound on excess risk of SGD flow over ridge regression, revealing how parameters drive risk.
Paper accelerates NMF algorithms using extrapolation.
problem Efficiently solving nonnegative matrix factorization problems.
method Inspired by convex optimization, uses extrapolation for two-block exact coordinate descent algorithms.
result Significantly accelerates NMF algorithms on various data sets.
A new method reduces the cost of solving large-scale linear models.
problem Solving large-scale linear statistical models efficiently.
method Sequential Preconditioned Conjugate Gradient Method (SPCG).
result SPCG achieves OLS prediction accuracy with fewer iterations and less time.
LSGANs improve GANs by using least squares loss, leading to better image quality and stability.
problem Vanishing gradients in GANs during training.
method Introducing LSGANs with least squares loss for both discriminator and generator.
result LSGANs generate higher quality images and are more stable during training.
There is widespread sentiment that it is not possible to effectively utilize fast gradient methods (e.g. Nesterov's acceleration, conjugate gradient, heavy ball) for the purposes of stochastic optimization due to their instability and error accumulation, a notion made precise in d'Aspremont 2008 and Devolder, Glineur, …
Improved kernel ridge regression using conjugate gradients.
problem Efficiently solving large-scale kernel ridge regression problems.
method Structured Gaussian regression model with low-rank approximation and conjugate gradients.
result Enhanced approximation of kernel ridge regressor/Gaussian process posterior mean.
Gradient-based sampling improves LS problem solving efficiency.
problem Computational difficulties in large sample size data analysis.
method Adaptive sampling dependent on input and output gradients.
result Gradient-based sampling improves statistical efficiency and computational saving.
Kernel methods with random projections improve least-squares regression efficiency.
problem Efficiently solving least-squares regression problems in high-dimensional spaces.
method Kernel conjugate gradient methods with randomized sketches and Nyström subsampling.
result Optimal generalization and computational advantages with proportional projection dimensions.
Proves a function's locally least gradient property if its level sets are minimal laminations.
problem Understanding the relationship between 1-harmonic functions and minimal laminations.
method Analyzes minimal laminations and their convergence properties, then applies to 1-harmonic functions.
result Proves a function is 1-harmonic if its level sets are minimal laminations.
Maps and measures on surfaces link best Lipschitz and least gradient functions.
problem Analyzing maps between surfaces and their geometric properties.
method Duality between best Lipschitz and least gradient maps, geodesic laminations, and transverse measures.
result The infinity harmonic map defines a geodesic lamination and the least gradient map defines a transverse measure.
Improved SGD for non-strongly-convex regression with faster convergence.
problem Non-strongly-convex least squares regression problems.
method Modified accelerated gradient descent.
result Achieves optimal prediction error rates of O(d/t) and forgets initial conditions faster to O(d/t2). Stochastic gradient descent regularizes least squares problems by smoothing large singular values.
problem Regularization of least squares problems using stochastic gradient descent.
method Analysis of stochastic gradient descent applied to least squares problems, showing a regularization effect.
result Stochastic gradient descent leads to a quick regularization effect, smoothing large singular values.
Simplified proof shows SGD optimality for least squares.
problem Optimizing SGD for least squares efficiency.
method Analyzing SGD as a stochastic process, characterizing stationary covariance matrix.
result Statistical minimax optimality of SGD for least squares.
Least symmetry breaking principle explains SGD's local minima in shallow ReLU networks.
problem Understanding the structure of local minima in two-layer ReLU networks.
method Analyzing the squared loss optimization problem for ReLU networks with Gaussian inputs and applying the principle of least symmetry breaking.
result The principle of least symmetry breaking explains the structure of spurious local minima detected by SGD.
Channel normalization prevents vanishing gradients in convolutional neural networks.
problem Vanishing gradients in convolutional neural networks during optimization.
method Channel normalization, which centers and normalizes each channel individually.
result Channel normalization avoids vanishing gradients, enabling efficient optimization.
We consider the stochastic approximation problem where a convex function has to be minimized, given only the knowledge of unbiased estimates of its gradients at certain points, a framework which includes machine learning methods based on the minimization of the empirical risk. We focus on problems without strong convex…
Gradient-enhanced kriging reduces function evaluations for high-dimensional problems.
problem High-dimensional function evaluations are computationally expensive.
method Developed a new gradient-enhanced surrogate model using partial-least squares to reduce hyperparameters and correlation matrix size.
result Significantly reduces the number of function evaluations required for accurate surrogate models.
New method for unbiased regression reduces excess risk.
problem Least squares regression with optimal solution and Hessian matrix.
method Averaged stochastic gradient descent with time-average estimator.
result Unbiased estimator with O(1/k) expected excess risk.
This paper simplifies finding least favorable priors by reducing dimensionality.
problem Finding least favorable priors is challenging due to infinite-dimensional optimization.
method Develops a dimensionality reduction method using Bregman divergences.
result Allows use of gradient ascent algorithms for finding least favorable priors.
This paper analyzes how BN affects GD convergence and stability.
problem The effect of batch normalization on gradient descent convergence and stability.
method Quantitative analysis of gradient descent with and without batch normalization on ordinary least squares.
result Gradient descent with batch normalization converges for arbitrary learning rates and remains linear under mild conditions.
New method avoids spurious critical points for low-rank matrix recovery.
problem Low-rank matrix recovery problems on Riemannian manifold.
method Riemannian gradient descent with random initialization.
result Riemannian gradient descent avoids spurious critical points and converges nearly linearly.
The paper improves Kaczmarz algorithm with momentum for linear least squares.
problem Improving convergence of the Kaczmarz algorithm for linear least squares.
method Integrates geometrically smoothed momentum into the randomized Kaczmarz algorithm.
result Proves expected error reduction in singular vector directions.
SGD converges to optimal solution in perfect data fitting problem.
problem Noiseless least-squares with perfect data fitting.
method Last iterate of SGD with constant step-size.
result Convergence of SGD final iterate for non-strongly convex problem.
In this paper we study the performance of the Projected Gradient Descent(PGD) algorithm for ℓp-constrained least squares problems that arise in the framework of Compressed Sensing. Relying on the Restricted Isometry Property, we provide convergence guarantees for this algorithm for the entire range of $0\leq p\…
Study on gradient dynamics of shallow ReLU networks for least-squares interpolation.
problem Understanding the gradient dynamics of shallow ReLU networks for interpolation.
method Theoretical and empirical analysis of gradient flow in non-redundant parameterization.
result Identification of two learning regimes: kernel and adaptive, with distinct interpolant shapes.
This work characterizes the benefits of averaging schemes widely used in conjunction with stochastic gradient descent (SGD). In particular, this work provides a sharp analysis of: (1) mini-batching, a method of averaging many samples of a stochastic gradient to both reduce the variance of the stochastic gradient estima…
In this paper, we further study the forward-backward envelope first introduced in [28] and [30] for problems whose objective is the sum of a proper closed convex function and a twice continuously differentiable possibly nonconvex function with Lipschitz continuous gradient. We derive sufficient conditions on the origin…
Paper introduces a new multi-kernel algorithm for better gradient approximation.
problem Improving gradient approximation in high-dimensional problems.
method Develops a multi-kernel passive stochastic gradient algorithm with variance reduction.
result The multi-kernel algorithm performs better in high-dimensional problems.
Study shows how mini-batch GD with random reshuffling affects least squares regression dynamics.
problem Analyzing the error dynamics of mini-batch GD with random reshuffling for least squares regression.
method Represented training and generalization errors through a sample cross-covariance matrix Z, compared with sample covariance matrix of original features X, and used linear scaling rule for analysis.
result Mini-batch GD with random reshuffling exhibits subtle step-size dependence not detectable by gradient flow analysis, converging to a limit dependent on the step size.
Gradient-free learning uses kernel and range space for solving linear equations.
problem Solving linear equations and least squares problems.
method Manipulating kernel and range space to solve linear matrix equations, adapting for neural networks.
result Gradient-free learning framework for neural networks, showing good performance on real-world data.
This paper analyzes multi-pass SGD for least squares, improving generalization bounds.
problem Improving generalization bounds for multi-pass SGD in the least squares problem.
method Develops an instance-dependent excess risk bound for least squares in the interpolation regime.
result SGD performs worse than GD instance-wise but saves computational time.
MGD with early stopping tends to ridge regularization in least squares regression.
problem Characterizing the implicit regularization of MGD with early stopping.
method Continuous-time view of MGD (momentum gradient flow) and comparison with explicit ridge regularization.
result Under optimal tuning, the risk of MGF is no more than 1.54 times that of ridge.
Stochastic gradient descent (SGD) on a low-rank factorization is commonly employed to speed up matrix problems including matrix completion, subspace tracking, and SDP relaxation. In this paper, we exhibit a step size scheme for SGD on a low-rank least-squares problem, and we prove that, under broad sampling conditions,…
New algorithm reduces sketching dimension to effective problem size.
problem Solving L2-regularized least-squares problems efficiently.
method Randomized algorithm using Gaussian and SRHT embeddings.
result Preserves convergence guarantees with reduced embedding dimension.
We introduce a novel semi-supervised version of the least squares classifier. This implicitly constrained least squares (ICLS) classifier minimizes the squared loss on the labeled data among the set of parameters implied by all possible labelings of the unlabeled data. Unlike other discriminative semi-supervised method…
Gradient descent can solve shallow neural network training problems globally.
problem Training shallow linear neural networks.
method Analyzed the geometric properties of the loss landscape.
result Gradient descent can converge globally due to the landscape's properties.
New method uses deep learning to solve linear inverse problems.
problem Solving linear inverse problems with high-dimensional signals.
method Stochastic coarse-to-fine gradient ascent procedure using implicit prior from denoising CNN.
result General algorithm for solving linear inverse problems without additional training.
A new method for estimating large-scale linear models with improved precision.
problem Estimating large-scale linear statistical models efficiently.
method Sequential Least-Squares Estimators with Fast Randomized Sketching (SLSE-FRS), integrating Sketch-and-Solve and Iterative-Sketching methods.
result SLSE-FRS produces high-precision estimators, outperforming state-of-the-art methods.
Algorithm improves stochastic gradient optimization with normalized steps.
problem Stochastic and finite sum minimization problems.
method Trust region algorithm with normalized steps.
result Algorithm converges similarly to traditional stochastic gradient under certain conditions.
We obtain an improved finite-sample guarantee on the linear convergence of stochastic gradient descent for smooth and strongly convex objectives, improving from a quadratic dependence on the conditioning (L/μ)2 (where L is a bound on the smoothness and μ on the strong convexity) to a linear dependence on L/μ. …
Accelerates coordinate descent methods for machine learning problems.
problem Slowness of coordinate descent methods in machine learning.
method Extrapolation-based accelerated coordinate descent.
result Significant speed-up in practice compared to existing methods.