Last SGD iterate bounds for overparameterized linear regression.
problem Analyzing the last iterate risk bounds of SGD with decaying stepsize for overparameterized linear regression.
method Problem-dependent analysis of last iterate risk bounds of SGD with geometrically decaying stepsize.
result Proved nearly matching upper and lower bounds on the excess risk for last iterate SGD with geometrically decaying stepsize.
This paper provides a framework to analyze stochastic gradient algorithms in a mean squared error (MSE) sense using the asymptotic normality result of the stochastic gradient descent (SGD) iterates. We perform this analysis by taking the asymptotic normality result and applying it to the finite iteration case. Specific…
Paper proposes an online estimator for covariance matrix of SGD iterates.
problem Quantifying variability and randomness of SGD-based estimates in online learning.
method Proposes a fully online estimator for covariance matrix of ASGD using SGD iterates.
result Establishes consistency of the online estimator and shows comparable convergence rate to offline methods.
New study shows faster convergence of SGD and Kaczmarz methods.
problem Improving convergence rates of iterative linear system solvers.
method Last-iterate convergence analysis of SGD with greedy step size over smooth quadratics.
result The t t t -th iterate attains an O ( 1 / t 3 / 4 ) O(1/t^{3/4}) O ( 1/ t 3/4 ) convergence rate. VRL-SGD reduces communication complexity in non-identical data settings.
problem Training machine learning models with non-identical data distribution.
method VRL-SGD, which eliminates gradient variance dependency and achieves linear speedup with lower communication complexity.
result VRL-SGD reduces communication complexity from $O(T^{ rac{3}{4}} N^{ rac{3}{4}})$ to $O(T^{ rac{1}{2}} N^{ rac{3}{2}})$ .
New bounds on SGD's final iterate convergence rate in constant dimension.
problem Characterize the convergence rate of SGD's final iterate in constant dimension.
method Proved lower bounds of Ω ( log d / T ) Ω(\log d/\sqrt{T}) Ω ( log d / T ) and Ω ( log d / T ) Ω(\log d/T) Ω ( log d / T ) for non-smooth Lipschitz convex and strongly convex functions respectively. result First general dimension dependent lower bound on SGD's final iterate convergence rate.
New bounds for SGD show improved performance in various settings.
problem Improving convergence bounds for SGD with random permutations.
method Analyzing convergence of SGD with random reshuffling and arbitrary permutations.
result Tighter lower bounds for weighted average iterates in both convex and strongly-convex cases.
SGD's performance improves with critical batch size, minimizing SFO complexity.
problem Optimizing SGD's performance with batch size and learning rate.
method Analysis of SGD using constant and decaying learning rates, focusing on batch size effects.
result SGD with critical batch size minimizes SFO complexity.
Iterative procedures for parameter estimation based on stochastic gradient descent allow the estimation to scale to massive data sets. However, in both theory and practice, they suffer from numerical instability. Moreover, they are statistically inefficient as estimators of the true parameter value. To address these tw…
SGD in linear regression overfits but performs well due to bias-variance trade-off.
problem Understanding overfitting in SGD for linear regression.
method Constant-stepsize SGD with iterate averaging or tail averaging, analyzing full eigenspectrum of data covariance matrix.
result Sharp excess risk bounds revealing bias-variance decomposition for SGD in linear regression.
SGD generalizes better than GD with similar number of iterations.
problem Generalization performance of SGD vs GD in convex optimization.
method Comparison of SGD and GD in stochastic convex optimization model.
result GD requires more iterations to match SGD's generalization performance.
Mixed-precision CA-SGD for generalized linear models on GPUs
problem SGD communication bottleneck
method Mixed-precision CA-SGD
result Matches FP32 SGD loss within 0.5% on various problems
Minimax optimal convergence rates for classes of stochastic convex optimization problems are well characterized, where the majority of results utilize iterate averaged stochastic gradient descent (SGD) with polynomially decaying step sizes. In contrast, SGD's final iterate behavior has received much less attention desp…
We consider stochastic strongly convex optimization with a complex inequality constraint. This complex inequality constraint may lead to computationally expensive projections in algorithmic iterations of the stochastic gradient descent~(SGD) methods. To reduce the computation costs pertaining to the projections, we pro…
SGD works well with large learning rates at the edge of stability.
problem Stochasticity at the edge of stability in deep learning.
method Sharp convergence guarantees for SGD with multiclass cross-entropy loss.
result SGD self-stabilizes, ensuring convergence with large learning rates.
The paper analyzes SGD with dropout regularization in linear models, proving asymptotic properties and providing inference tools.
problem Analyzing the behavior of SGD with dropout regularization in linear models.
method Establishing geometric-moment contraction (GMC) and proving quenched central limit theorems (CLT).
result The existence of a unique stationary distribution and asymptotic normality results for SGD with dropout.
In this paper we aim to formally explain the phenomenon of fast convergence of SGD observed in modern machine learning. The key observation is that most modern learning architectures are over-parametrized and are trained to interpolate the data by driving the empirical loss (classification and regression) close to zero…
Study on adversarial training dynamics in high dimensions using SGD.
problem Analyzing adversarial training of models in high-dimensional settings.
method Deriving deterministic equivalents for SGD iterates under Gaussian mixtures.
result No constant learning rate guarantees monotone descent in adversarial training.
New method averages SGD iterates to achieve adjustable regularization.
problem Overfitting in machine learning models.
method Averaging SGD iterates for regularized solutions.
result Obtain regularized solutions without tuning parameters.
SGD achieves near optimal convergence rate in smooth interpolation regime.
problem Optimization of smooth convex objectives with zero noise at optimum.
method Stochastic Gradient Descent (SGD) with large stepsize analysis.
result Last iterate of SGD achieves expected excess risk of O(1/T + σ* / √T) with optimal stepsize.
This work addresses the convergence of SGD's final iterate without restrictive assumptions.
problem Prove optimal convergence rate of SGD's final iterate without compact domains or bounded noise.
method Unified proof for general domains, composite objectives, non-Euclidean norms, etc.
result First unified convergence rates in expectation and high probability.
New privacy bounds for DP-SGD's last iterate, even with cyclic sampling.
problem Privacy of the last iterate in DP-SGD with cyclic sampling.
method Established new RDP upper bounds for the last iterate under realistic assumptions.
result Privacy bounds for DP-SGD's last iterate with cyclic sampling and clipping, even for nonconvex losses.
In this note we give a simple proof for the convergence of stochastic gradient (SGD) methods on μ μ μ -convex functions under a (milder than standard) L L L -smoothness assumption. We show that for carefully chosen stepsizes SGD converges after T T T iterations as $O\left( LR^2 \exp \bigl[-\fracμ{4L}T\bigr] + \frac{σ^2}{μT} \r…
SGD generalization bounds derived from information theory.
problem Understanding generalization of SGD for non-convex functions.
method Combining information-theoretic bounds with perturbation analysis.
result Upper bounds on SGD's generalization error based on gradient variance and function smoothness.
New bounds for SGD in high dimensions improve inference efficiency.
problem Quantifying uncertainty in high-dimensional SGD.
method Established non-asymptotic Berry--Esseen bounds for online least-squares SGD.
result Gaussian Central Limit Theorem holds for t ≳ d 1 + δ t \gtrsim d^{1+δ} t ≳ d 1 + δ , extending dimensional scaling. Stochastic Gradient Descent (SGD) has become one of the most popular optimization methods for training machine learning models on massive datasets. However, SGD suffers from two main drawbacks: (i) The noisy gradient updates have high variance, which slows down convergence as the iterates approach the optimum, and (ii)…
Paper proposes a coupling-based diagnostic for SGD stepsize optimization.
problem Optimizing stepsize for SGD convergence.
method Coupling-based convergence diagnostic for monitoring stationarity.
result Proposed stepsize scheme achieves superior performance across convex and non-convex problems.
With the increase in the amount of data and the expansion of model scale, distributed parallel training becomes an important and successful technique to address the optimization challenges. Nevertheless, although distributed stochastic gradient descent (SGD) algorithms can achieve a linear iteration speedup, they are l…
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…
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.
AutoSGD automatically adjusts learning rates for SGD.
problem Inefficient manual learning rate selection for SGD.
method AutoSGD automatically adjusts learning rates based on iteration.
result Empirical results show strong performance across various optimization tasks.
SGD performs optimally on dependent data, achieving both estimation and regret.
problem Performance of SGD on temporally dependent data.
method Stochastic gradient descent with martingale-type and sequential dependence.
result SGD achieves optimal estimation and regret under various dependence structures.
OptEx accelerates first-order optimization with parallelized iterations.
problem Inefficiencies in first-order optimization algorithms for complex tasks.
method Approximately parallelized iterations using kernelized gradient estimation.
result OptEx achieves substantial efficiency improvements with an effective acceleration rate of Ω ( N ) Ω(\sqrt{N}) Ω ( N ) . This paper analyzes Local SGD for federated learning, achieving both statistical and communication efficiency.
problem Statistical estimation and inference in federated learning with decentralized data.
method Local SGD, a multi-round estimation procedure using intermittent communication.
result Local SGD achieves both statistical efficiency and communication efficiency.
New CLT for SGD in high-dimensional regression provides online inference.
problem Quantifying uncertainty in SGD for high-dimensional regression.
method Established a high-dimensional CLT for online SGD iterates.
result Developed an online approach for estimating variance in CLT.
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.
ROOT-SGD solves convex optimization problems with optimal nonasymptotic and near-optimal asymptotic performance.
problem Solving strongly convex and smooth unconstrained optimization problems using stochastic first-order algorithms.
method ROOT-SGD: Recursive One-Over-T SGD, averaging past stochastic gradients.
result Achieves state-of-the-art performance in both nonasymptotic and asymptotic senses.
The paper analyzes convergence rates for SGD and SHB methods.
problem Analyzing convergence rates for stochastic gradient descent and heavy ball methods.
method Stochastic gradient descent and stochastic heavy ball method for general stochastic approximation problems.
result The last iterate of SHB converges almost surely to a minimizer and has faster convergence rates than SGD.
SGD with mini-batches can solve convex low-rank matrix problems efficiently.
problem Solving large-scale convex low-rank matrix problems efficiently.
method Stochastic Gradient Descent with mini-batches and low-rank projections.
result SGD with mini-batches produces low-rank iterates with high probability.
A new algorithm combines SGD and Thompson Sampling for contextual bandits.
problem Finding efficient algorithms for contextual bandits with low time and memory complexity.
method Online Stochastic Gradient Descent (SGD) combined with Thompson Sampling.
result Achieves i l d e O ( T ) ilde{O}(\sqrt{T}) i l d e O ( T ) regret with linear time complexity in T T T and d d d . Adaptive SGD learns optimal batch size for strong convex functions.
problem Finding optimal batch size for SGD in practice.
method Adaptive SGD method that learns optimal batch size.
result Adaptive SGD exhibits nearly optimal performance in experiments.
We present an optimizer which uses Bayesian optimization to tune the system parameters of distributed stochastic gradient descent (SGD). Given a specific context, our goal is to quickly find efficient configurations which appropriately balance the load between the available machines to minimize the average SGD iteratio…
Derives scaling limits and fluctuations for SGD in high dimensions.
problem Understanding SGD behavior in high-dimensional settings with varying noise levels.
method Interacting particle system approach, treating SGD iterates as such, with covariance structure considered.
result Precise three-step phase transition observed in SGD behavior: ballistic, diffusive, then random.
Study on SGD for overparameterized neural networks, focusing on convergence rates.
problem Understanding convergence rates of SGD in overparameterized two-layer neural networks.
method Combines NTK approximation with RKHS analysis to explore SGD dynamics.
result Established sharp convergence rates for SGD in overparameterized two-layer neural networks.
SGD converges to an invariant distribution with sub-Gaussian or sub-exponential properties.
problem Optimizing smooth and strongly convex objectives using SGD.
method Analysis through Markov chains, focusing on convergence and concentration properties.
result SGD iterates and their invariant limit distribution inherit sub-Gaussian or sub-exponential concentration properties.
Study how neural networks optimize to stable linearly connected regions.
problem Understanding how neural networks converge to stable solutions under different training conditions.
method Investigate the stability of neural networks to SGD noise and apply it to iterative magnitude pruning.
result Subnetworks that reach full accuracy must be stable to SGD noise, either at initialization or early in training.
We study the Stochastic Gradient Descent (SGD) method in nonconvex optimization problems from the point of view of approximating diffusion processes. We prove rigorously that the diffusion process can approximate the SGD algorithm weakly using the weak form of master equation for probability evolution. In the small ste…
Improved generalization bounds for SGD in non-convex learning.
problem Understanding generalization properties of SGD in non-convex settings.
method Introducing Type II perturbed SGD (T2pm-SGD) to analyze generalization error bounds.
result Tighter generalization error bounds for SGD in non-convex learning, especially for sub-Gaussian and bounded loss functions.