Stochastic gradient descent on manifolds improves low-rank approximation.
problem Efficiently approximate large matrices with lower rank.
method Stochastic gradient descent on a manifold.
result Algorithm outperforms Euclidean space methods on Netflix Prize data.
A new method improves stochastic gradient descent for faster and more efficient estimation.
problem Efficient and fast parametric estimation methods.
method Projected stochastic gradient descent corrected by Fisher scoring.
result The method is faster and more efficient than traditional methods.
The paper studies stochastic gradient descent with infinite variance gradients.
problem Theoretical properties of SGD with infinite variance gradients.
method Establish asymptotic behavior of SGD with infinite variance gradients.
result Asymptotic distribution of SGD is characterized as a stationary distribution of an Ornstein-Uhlenbeck process driven by a stable Lévy process.
Paper examines the structure of stochastic gradients in deep learning.
problem Exploring the structure and heavy tails of stochastic gradients in deep learning.
method Conducted formal statistical tests on stochastic gradients and gradient noise.
result Stochastic gradients and gradient noise do not exhibit power-law heavy tails, but their covariance spectra do.
Stochastic gradient methods can converge in expectation under heavy-tailed noise.
problem Convergence of stochastic gradient methods under heavy-tailed noise.
method Comprehensive study of stochastic optimization under heavy-tailed noise for extsfSGD, extsfSMD, extsfASMD, extsfSGDM in convex and nonconvex optimization. result Established in-expectation convergence results for various stochastic gradient methods.
New minibatching strategy reduces stochastic gradient bias in optimisation.
problem Reducing bias in stochastic gradient descent.
method Symmetric Minibatching Strategy combined with momentum.
result Reduced stochastic gradient bias from O(h2) to O(h4). New method corrects bias in stochastic gradient samplers.
problem Bias in stochastic gradient samplers.
method Gradient-Guided Monte Carlo with stochastic gradients.
result Corrected sampler yields nonzero acceptance probabilities.
Stochastic gradient descent procedures have gained popularity for parameter estimation from large data sets. However, their statistical properties are not well understood, in theory. And in practice, avoiding numerical instability requires careful tuning of key parameters. Here, we introduce implicit stochastic gradien…
Paper tackles active labeling for partial supervision.
problem Accessing stochastic gradients with partial supervision.
method Streaming technique to minimize generalization error.
result Proves minimization of generalization error ratio.
SGLRW improves robustness of stochastic gradient MCMC methods.
problem Sensitivity to minibatch size and gradient noise in stochastic-gradient MCMC methods.
method Proposes Stochastic Gradient Lattice Random Walk (SGLRW) with lattice-based discretization.
result SGLRW remains stable in regimes where SGLD fails, including heavy-tailed gradient noise.
Study on test risk dynamics in learning theory with stochastic gradient flow.
problem Understanding test risk in stochastic gradient flow dynamics.
method Path integral formulation for small learning rates, explicit computation for weak features.
result Explicit corrections due to stochastic term in dynamics, good agreement with simulations.
Proof given for SGD convergence in a concise manner.
problem Convergence of Stochastic Gradient Descent (SGD)
method Self-contained proof
result SGD convergence proven
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.
The paper analyzes stability and generalization of decentralized SGD.
problem Stability and generalization of decentralized stochastic gradient descent.
method Novel formulation of decentralized stochastic gradient descent combined with non/convex optimization theory.
result First stability and generalization guarantees for decentralized stochastic gradient descent.
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.
Many recent Markov chain Monte Carlo (MCMC) samplers leverage continuous dynamics to define a transition kernel that efficiently explores a target distribution. In tandem, a focus has been on devising scalable variants that subsample the data and use stochastic gradients in place of full-data gradients in the dynamic s…
Stochastic gradient descent approximates Gaussian process posteriors efficiently.
problem Efficiently sampling from Gaussian process posteriors with limited computational resources.
method Developed stochastic gradient optimization objectives for sampling from Gaussian process posteriors.
result Stochastic gradient descent produces accurate predictive distributions, even in non-convergent cases.
SGBD algorithm improves robustness in Bayesian sampling.
problem Inefficiency of existing MCMC algorithms in large datasets.
method Extends Barker MCMC to stochastic gradient framework, introducing bias-corrected version.
result SGBD is more robust to hyperparameter tuning and gradient noise.
(Mini-batch) Stochastic Gradient Descent is a popular optimization method which has been applied to many machine learning applications. But a rather high variance introduced by the stochastic gradient in each step may slow down the convergence. In this paper, we propose the antithetic sampling strategy to reduce the va…
Improved sampling efficiency with Random Reshuffling for Langevin dynamics.
problem Sampling efficiency in stochastic gradient algorithms.
method Random Reshuffling for Stochastic Gradient Langevin Dynamics (SGLD).
result Random Reshuffling leads to improved performance in sampling.
Paper proposes faster method to find local minima in nonconvex optimization.
problem Escaping saddle points and finding local minima in nonconvex optimization.
method LENA (Last stEp shriNkAge) framework for faster perturbed stochastic gradient methods.
result LENA finds (ε,εH)-approximate local minima within ildeO(ε−3+εH−6) evaluations. SGD and stochastic gradient descent converge at optimal rates for certain non-convex functions.
problem Optimal convergence rates for non-convex functions under gradient noise.
method Geometric interpretation of the PL-condition to analyze convergence rates.
result Convergence rates of SGD and stochastic gradient descent match those of strongly convex quadratics.
Unified derivation of high-dimensional linear models using stochastic gradient descent.
problem Performance analysis of high-dimensional linear models trained with stochastic gradient descent.
method Derivation of a deterministic equivalence for the two-point function of a random matrix resolvent.
result Unified understanding of model performance including previously known and novel results.
State space models (SSMs) provide a flexible framework for modeling complex time series via a latent stochastic process. Inference for nonlinear, non-Gaussian SSMs is often tackled with particle methods that do not scale well to long time series. The challenge is two-fold: not only do computations scale linearly with t…
New convergence guarantees for learning with unknown nuisance parameters.
problem Learning problems with unknown nuisance parameters.
method Stochastic gradient optimization with Neyman orthogonality and approximately orthogonalized updates.
result Stochastic gradient algorithms can converge under conditions of nuisance parameters.
Stochastic gradient methods converge for training wide PINNs.
problem Convergence of stochastic gradient descent in training over-parameterized PINNs.
method Established linear convergence of stochastic gradient descent/flow in training over-parameterized two-layer PINNs.
result Linear convergence with high probability for general activation functions.
LASG improves communication efficiency in distributed learning.
problem Efficiently communicating gradients in distributed machine learning.
method Develops a new stochastic gradient descent approach, LASG, that predicts and selects significant communication rounds.
result Achieves communication savings by an order of magnitude in federated learning.
This paper studies the complexity of the stochastic gradient algorithm for PCA when the data are observed in a streaming setting. We also propose an online approach for selecting the learning rate. Simulation experiments confirm the practical relevance of the plain stochastic gradient approach and that drastic improvem…
In this work we introduce a conditional accelerated lazy stochastic gradient descent algorithm with optimal number of calls to a stochastic first-order oracle and convergence rate O(ε21) improving over the projection-free, Online Frank-Wolfe based stochastic gradient descent of Hazan an…
Overview of non-stochastic-gradient SA algorithms in signal processing and ML.
problem Dealing with large data sets and uncertainties in signal processing and machine learning.
method General framework of SA algorithms using Lyapunov functions.
result Unified convergence properties of non-stochastic-gradient algorithms.
Improves posterior approximation speed for Dirichlet process mixture models.
problem Inefficiency of stochastic variational inference in large datasets.
method Uses stochastic gradient ascent with adaptive stepsize optimization.
result Adaptive stepsize improves speed and performance of posterior approximation.
A new tamed stochastic gradient Hamiltonian Monte Carlo algorithm for superlinearly growing stochastic gradients.
problem Sampling and stochastic optimization problems with superlinearly growing stochastic gradients.
method Tamed Stochastic Gradient Hamiltonian Monte Carlo (tSGHMC) algorithm.
result Established a non-asymptotic error bound in Wasserstein-2 distance with a convergence rate of 1/4. We consider the problem of minimizing the sum of two convex functions: one is the average of a large number of smooth component functions, and the other is a general convex function that admits a simple proximal mapping. We assume the whole objective function is strongly convex. Such problems often arise in machine lea…
Stochastic gradient descent outperforms traditional force-directed methods.
problem Improving graph layout quality and efficiency.
method Applying stochastic gradient descent for stress minimization.
result Stochastic gradient descent is simpler and more robust than traditional methods.
Improved convergence for non-log-concave sampling.
problem Sampling from non-log-concave distributions.
method Novel conductance analysis of SGLD with auxiliary Markov Chain.
result SGLD achieves ε-sampling error with fewer evaluations.
Stochastic gradient algorithms estimate the gradient based on only one or a few samples and enjoy low computational cost per iteration. They have been widely used in large-scale optimization problems. However, stochastic gradient algorithms are usually slow to converge and achieve sub-linear convergence rates, due to t…
New dynamics for SGD in small learning rate regime.
problem Improving stochastic gradient descent in small learning rate regime.
method Introducing stochastic modified flows and distribution dependent stochastic modified flows.
result Captures fluctuating dynamics of SGD in small learning rate - infinite width scaling regime.
A new hybrid algorithm reduces stochastic gradient evaluations for nonconvex optimization.
problem Solving stochastic composite nonconvex optimization problems efficiently.
method Proposes a new hybrid variance-reduced proximal gradient method with a stochastic gradient estimator.
result Achieves optimal stochastic oracle complexity bound with one less gradient evaluation.
Stochastic gradient methods are dominant in nonconvex optimization especially for deep models but have low asymptotical convergence due to the fixed smoothness. To address this problem, we propose a simple yet effective method for improving stochastic gradient methods named predictive local smoothness (PLS). First, we …
Despite having various attractive qualities such as high prediction accuracy and the ability to quantify uncertainty and avoid over-fitting, Bayesian Matrix Factorization has not been widely adopted because of the prohibitive cost of inference. In this paper, we propose a scalable distributed Bayesian matrix factorizat…
Paper approximates risk measures using SGD with Langevin dynamics.
problem Approximating arbitrary law invariant risk measures.
method Stochastic Gradient Langevin Dynamics (SGD-Langevin) for general risk measures.
result Non-asymptotic convergence rates of the approximation algorithm.
A large class of machine learning techniques requires the solution of optimization problems involving spectral functions of parametric matrices, e.g. log-determinant and nuclear norm. Unfortunately, computing the gradient of a spectral function is generally of cubic complexity, as such gradient descent methods are rath…
SALSA automatically adjusts learning rates in stochastic gradient methods.
problem Automatic adjustment of learning rates in stochastic gradient methods.
method SALSA uses a line-search procedure to gradually increase the learning rate, then a statistical test to decrease it.
result SALSA matches the performance of best hand-tuned learning rate schedules in deep learning tasks.
Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) is a momentum version of stochastic gradient descent with properly injected Gaussian noise to find a global minimum. In this paper, non-asymptotic convergence analysis of SGHMC is given in the context of non-convex optimization, where subsampling techniques are used o…
SGD handles label noise with bounds improving over SGLD.
problem Label noise in non-convex optimization.
method Stochastic gradient descent with uniform dissipativity and smoothness conditions, using Wasserstein distance and algorithmic stability.
result Generalization error bounds with a rate of n−2/3, better than SGLD's n−1/2. The paper analyzes the variance of different shuffling methods in stochastic gradient descent.
problem Understanding the variance of different shuffling methods in stochastic gradient descent.
method Power spectral density analysis to study the noise sequences of stochastic gradients.
result The stationary variances of iterates decrease in the order of SGD, SGD-RR, and SGD-SO.
SGD fails to converge for deep ReLU networks with limited random initializations.
problem SGD convergence in deep neural networks with limited random initializations.
method Analysis of four discretization parameters: network architecture, training data, gradient steps, and random initializations.
result SGD fails to converge for ReLU networks with depth much larger than width.
Stochastic gradient descent (\textsc{Sgd}) methods are the most powerful optimization tools in training machine learning and deep learning models. Moreover, acceleration (a.k.a. momentum) methods and diagonal scaling (a.k.a. adaptive gradient) methods are the two main techniques to improve the slow convergence of \text…