Smoothed SGD improves quantile estimation without crossing curves.
problem Estimating quantiles without crossing estimated curves.
method Smoothed SGD algorithm with Bahadur representation and Gaussian approximation.
result Smoothed SGD provides non-asymptotic tail probability bounds and a Gaussian approximation for quantile estimates.
Averaged SGD optimizes a smoothed objective, leading to better generalization.
problem Improving generalization performance in machine learning models.
method Analyzed the smoothed objective function of SGD and proved that averaged SGD can optimize this smoothed function efficiently.
result Averaged SGD can efficiently optimize a smoothed objective, leading to better generalization.
Untuned SGD converges but with an exponential dependence on smoothness, adaptive methods prevent this.
problem The exponential dependence on smoothness in untuned SGD's convergence rate.
method Untuned SGD with arbitrary stepsize η, adaptive methods like NSGD, AMSGrad, and AdaGrad.
result Adaptive methods prevent the exponential dependence on smoothness in SGD.
Paper relaxes SGD privacy and generalization guarantees for non-smooth convex losses.
problem Privacy and generalization in SGD for non-smooth convex losses.
method Relaxes Lipschitz and strong smoothness assumptions to Hölder smoothness, proving ( ε , δ ) (ε,δ) ( ε , δ ) -DP and optimal excess risk. result Noisy SGD with α α α -Hölder smooth losses achieves optimal excess risk with linear gradient complexity for α ≥ 1 / 2 α \geq 1/2 α ≥ 1/2 . This paper improves convergence guarantees for SGD algorithms in non-convex smooth functions.
problem Theoretical convergence properties of SGD algorithms for non-convex smooth functions.
method Analysis of SGD algorithms with arbitrary data ordering for non-convex smooth functions.
result Enhanced convergence guarantees for incremental gradient and single shuffle SGD, improving the optimization term of convergence guarantee.
New proof shows D-SGD and SAM are equivalent, revealing advantages of decentralization.
problem The generalization benefits of decentralized learning.
method Proved D-SGD implicitly minimizes SAM's loss function.
result Decentralized SGD and Average-direction SAM are asymptotically equivalent.
Large-batch stochastic gradient descent (SGD) is widely used for training in distributed deep learning because of its training-time efficiency, however, extremely large-batch SGD leads to poor generalization and easily converges to sharp minima, which prevents naive large-scale data-parallel SGD (DP-SGD) from convergin…
In this paper, we propose a simple variant of the original SVRG, called variance reduced stochastic gradient descent (VR-SGD). Unlike the choices of snapshot and starting points in SVRG and its proximal variant, Prox-SVRG, the two vectors of VR-SGD are set to the average and last iterate of the previous epoch, respecti…
Stochastic Gradient Descent (SGD) is one of the simplest and most popular stochastic optimization methods. While it has already been theoretically studied for decades, the classical analysis usually required non-trivial smoothness assumptions, which do not apply to many modern applications of SGD with non-smooth object…
SGD converges globally to logistic loss minima for two-layer nets.
problem Global convergence of SGD for logistic loss on two-layer neural nets.
method Demonstrates existence of Frobenius norm regularized logistic loss functions as Villani functions, proving convergence and exponential rate.
result SGD converges globally to the global minima of appropriately regularized logistic empirical risk of depth 2 nets.
Adapts SGD to noise and problem specifics for faster convergence.
problem Minimizing smooth, strongly-convex functions with varying noise and problem constants.
method Adaptive SGD with exponentially decreasing step-sizes, Nesterov acceleration, and stochastic line-search.
result Achieves near-optimal convergence rates without knowing noise or problem specifics.
Random Reshuffling outperforms Stochastic Gradient Descent in smooth convex optimization.
problem Theoretical limitations of Random Reshuffling in smooth convex optimization.
method Random Reshuffling (RR) as a variant of Shuffling Stochastic Gradient Descent (Shuffling SGD).
result Random Reshuffling (RR) dominates Stochastic Gradient Descent (SGD) in smooth convex optimization under any reasonable stepsize after any finite number of epochs.
Global convergence of SGD proven for two-layer neural nets with regularization.
problem Proving global convergence of SGD for two-layer neural nets.
method Regularized empirical risk, SGD iterates, Villani functions.
result Global convergence of SGD for a special class of initializations.
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.
In this paper, we propose a simple variant of the original stochastic variance reduction gradient (SVRG), where hereafter we refer to as the variance reduced stochastic gradient descent (VR-SGD). Different from the choices of the snapshot point and starting point in SVRG and its proximal variant, Prox-SVRG, the two vec…
Improved SGD learning for single index models reduces sample complexity.
problem Learning a single index model with optimal sample complexity.
method Using smoothed loss in online SGD to reduce sample complexity.
result Online SGD with smoothed loss achieves optimal sample complexity of d k ⋆ / 2 d^{k^\star/2} d k ⋆ /2 . SGD converges to global minimum for certain non-convex functions.
problem Theoretical challenges in optimizing non-convex functions in machine learning.
method Perturbed SGD on a broad class of non-convex functions.
result SGD converges to global minimum for certain non-convex functions.
IntSGD compresses SGD gradients without floats, converging as SGD.
problem Efficiently compressing stochastic gradients in distributed SGD.
method Adaptive integer compression of gradients, estimating scaling adaptively.
result IntSGD matches SGD's iteration complexity for convex and non-convex functions.
Paper closes convergence gap for SGD without replacement.
problem Establishing convergence rate for SGD without replacement.
method Analyzes convergence rates for strongly convex and smooth functions.
result Achieves a rate of O(1/T^2 + n^2/T^3) for quadratic sums.
New SGD analysis for nonconvex optimization finds optimal rates.
problem Optimizing nonconvex functions in machine learning.
method Introduces a new expected smoothness assumption and analyzes SGD convergence rates.
result Optimal convergence rates for nonconvex smooth functions and global solutions under certain conditions.
SQuARM-SGD improves decentralized SGD efficiency with momentum.
problem Efficient decentralized training of large-scale models over networks.
method Fixed local SGD steps with Nesterov's momentum, sparsified and quantized updates, locally computed triggering criterion.
result Convergence rate matches vanilla SGD, momentum improves test performance.
We design a stochastic algorithm to train any smooth neural network to ε \varepsilon ε -approximate local minima, using O ( ε − 3.25 ) O(\varepsilon^{-3.25}) O ( ε − 3.25 ) backpropagations. The best result was essentially O ( ε − 4 ) O(\varepsilon^{-4}) O ( ε − 4 ) by SGD. More broadly, it finds ε \varepsilon ε -approximate local minima of any smooth nonconvex function in …
Unified analysis for decentralized SGD across various topologies and updates.
problem Analysis of decentralized SGD methods with changing topologies and local updates.
method Unified convergence analysis covering local SGD updates and adaptive network topology.
result Universal convergence rates for smooth problems, interpolating between heterogeneous and iid-data settings.
New stability bounds for SGD on nonsmooth convex losses.
problem Understanding stability of SGD on nonsmooth convex losses.
method Sharp upper and lower bounds for SGD and full-batch GD on nonsmooth convex losses.
result SGD can be less stable but still useful for generalization bounds.
Improved Clipped-SGD achieves near-optimal heavy-tailed statistical estimation in streaming settings.
problem High-dimensional heavy-tailed statistical estimation in streaming with memory constraints.
method Stochastic convex optimization with Clipped-SGD, proving near-optimal sub-Gaussian statistical rates.
result Clipped-SGD achieves an error of T r ( Σ ) + T r ( Σ ) ∥ Σ ∥ 2 log ( log ( T ) δ ) T \sqrt{\frac{\mathsf{Tr}(Σ)+\sqrt{\mathsf{Tr}(Σ)\|Σ\|_2}\log(\frac{\log(T)}δ)}{T}} T Tr ( Σ ) + Tr ( Σ ) ∥Σ ∥ 2 l o g ( δ l o g ( T ) ) with probability 1 − δ 1-δ 1 − δ . Unified analysis of asynchronous-SGD algorithms for distributed learning.
problem Analyzing asynchronous-SGD in heterogeneous settings with varying speeds and data distributions.
method Unified convergence theory for non-convex smooth functions, including pure asynchronous SGD and its modifications.
result Unified convergence rates for various asynchronous algorithms, including novel methods.
Adaptive step-size method improves compressed SGD performance in machine learning.
problem Communication bottleneck in distributed and decentralized optimization.
method Developed an adaptive step-size method for compressed SGD.
result Order-optimal convergence rates for various objective functions.
Paper shows local SGD outperforms mini-batch SGD under certain conditions.
problem Proving local SGD's superiority in distributed learning with heterogeneous data.
method New lower and upper bounds for local SGD under first-order heterogeneity assumptions.
result Local SGD is min-max optimal under certain conditions, resolving understanding of distributed optimization.
CSER improves SGD efficiency by resetting errors and partial synchronization.
problem Limited scalability of Distributed Stochastic Gradient Descent (SGD) due to communication bottlenecks.
method Introduces 'error reset' technique and partial synchronization for gradients and models.
result Proves convergence for smooth non-convex problems and accelerates distributed training significantly.
MindFlayer SGD improves parallel SGD for heterogeneous, random compute times.
problem Minimizing nonconvex functions with heterogeneous, random compute times.
method MindFlayer SGD, designed for stochastic and heterogeneous delays.
result MindFlayer SGD outperforms existing methods in environments with heavy-tailed noise.
Here, we study different update rules in stochastic gradient descent (SGD) for online forecasting problems. The selection of the learning rate parameter is critical in SGD. However, it may not be feasible to tune this parameter in online learning. Therefore, it is necessary to have an update rule that is not sensitive …
We study stochastic gradient descent {\em without replacement} (\sgdwor) for smooth convex functions. \sgdwor is widely observed to converge faster than true \sgd where each sample is drawn independently {\em with replacement} \cite{bottou2009curiously} and hence, is more popular in practice. But it's convergence prope…
This study tightens bounds on how GD and SGD generalize in smooth convex optimization problems.
problem Understanding how GD and SGD generalize in smooth stochastic convex optimization problems.
method Provided tight excess risk lower bounds for GD and SGD under different conditions.
result Lower bounds suggest overfitting occurs and gaps remain in some cases.
Paper explores whether gradient normalization can replace clipping for SGD in heavy-tailed noise.
problem Ensuring convergence of SGD in heavy-tailed noise.
method Revisits gradient clipping and normalization, proving their sufficiency and effectiveness.
result Gradient normalization alone is sufficient for nonconvex SGD convergence under smoothness assumptions.
New bounds show SGD can match deterministic gradient descent's convergence rate.
problem Optimizing SGD convergence rate under strong convexity and smoothness.
method Computer-aided Lyapunov analysis, focusing on bias-optimal bounds.
result SGD achieves optimal convergence rate in bias terms for a wide range of step-sizes.
SGD converges to global minimum for structured non-convex functions.
problem Optimizing non-convex functions using SGD with slow convergence rates.
method Convergence theorems for SGD on structured non-convex functions, including Quasar and PL conditions.
result SGD converges to global minimum for specific non-convex functions under certain conditions.
Paper proves CLT for quantile SGD with constant learning rate.
problem Quantile estimation via SGD with non-smooth, non-strongly convex loss.
method Viewed as a Markov chain, derived stationary distribution, analyzed MGF, proved CLT.
result Centered and standardized stationary distribution converges to Gaussian as η i g h t a r r o w 0 η
ightarrow0 η i g h t a r r o w 0 . Improved analysis for nonconvex SGD methods with flexible sampling.
problem Finding approximately stationary points of nonconvex functions with gradient evaluations.
method Generalized SPIDER and PAGE algorithms with flexible sampling mechanisms.
result Sharper complexity bounds for optimal SGD methods in smooth nonconvex settings.
Improved Local SGD convergence for general convex objectives with bounded second-order heterogeneity.
problem Understanding when and why Local SGD outperforms alternatives in distributed optimization.
method Established improved convergence guarantees for Local SGD on general convex objectives under bounded second-order heterogeneity.
result Upper bounds for Local SGD are nearly tight, providing a sharper convergence theory.
Momentum SGD fails to track nonstationary optima due to drift amplification.
problem Tracking nonstationary optima in stochastic optimization.
method Theoretical analysis of SGD and momentum variants under strong convexity and smoothness.
result Momentum incurs a drift-amplification penalty that diverges as the momentum parameter approaches 1, leading to systematic lag.
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.
Communication bottleneck has been identified as a significant issue in distributed optimization of large-scale learning models. Recently, several approaches to mitigate this problem have been proposed, including different forms of gradient compression or computing local models and mixing them iteratively. In this paper…
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…
Machine learning (ML) models trained by differentially private stochastic gradient descent (DP-SGD) have much lower utility than the non-private ones. To mitigate this degradation, we propose a DP Laplacian smoothing SGD (DP-LSSGD) to train ML models with differential privacy (DP) guarantees. At the core of DP-LSSGD is…
Improved time complexity for parallel stochastic optimization in heterogeneous systems.
problem Time complexity in parallel stochastic optimization for large-scale machine learning models.
method Proposes Rennala MVR, a variance-reduced extension of Rennala SGD based on momentum-based variance reduction.
result Variance reduction improves time complexity in relevant parameter regimes for parallel stochastic optimization in heterogeneous systems.
New algorithm improves gradient-based ERM for smooth convex losses.
problem Empirical risk minimization of smooth, strongly convex loss functions.
method Iterative gradient-based method with local polynomial regression.
result Oracle complexity of O ( ( p ε − 1 ) d / ( 2 η ) ) O((p ε^{-1})^{d/(2η)}) O (( p ε − 1 ) d / ( 2 η ) ) for our algorithm. Paper relaxes stability and generalization assumptions for SGD.
problem Stability and generalization for SGD under restrictive assumptions.
method Introduces on-average model stability and develops novel bounds.
result First-ever-known fast bounds in low-noise setting using stability approach.
Develops anytime-valid stopping rules for SGD based on observed trajectory.
problem Stopping stochastic gradient descent (SGD) based on observed trajectory.
method Develops anytime-valid confidence sequences for stochastic gradient methods.
result Statistically valid, time-uniform stopping rules for SGD across convex and nonconvex settings.