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 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.
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.
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.
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 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.
In this paper, we propose a novel technique to implement stochastic gradient methods, which are beneficial for learning from large datasets, through accelerated stochastic dynamics. A stochastic gradient method is based on mini-batch learning for reducing the computational cost when the amount of data is large. The sto…
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…
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.
We develop the mathematical foundations of the stochastic modified equations (SME) framework for analyzing the dynamics of stochastic gradient algorithms, where the latter is approximated by a class of stochastic differential equations with small noise parameters. We prove that this approximation can be understood math…
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.
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.
Riemannian stochastic gradient descent approximates a diffusion process called Riemannian stochastic modified flow.
problem Improving convergence rate of Riemannian stochastic gradient descent.
method Using stochastic differential geometry, the paper shows RSGD can be approximated by the Riemannian stochastic modified flow (RSMF).
result RSGD can be approximated by the solution to the RSMF driven by an infinite-dimensional Wiener process, increasing the order of approximation.
The adjoint sensitivity method scalably computes gradients of solutions to ordinary differential equations. We generalize this method to stochastic differential equations, allowing time-efficient and constant-memory computation of gradients with high-order adaptive solvers. Specifically, we derive a stochastic differen…
This paper analyzes adaptive gradient algorithms for better performance in ill-conditioned problems.
problem Poor performance of standard stochastic gradient algorithms in ill-conditioned problems.
method Non-asymptotic analysis of adaptive gradient algorithms (Adagrad and Stochastic Newton) for strongly convex objectives.
result Theoretical analysis and adaptation to practical applications like linear regression and regularized GLM.
New adaptive step-size method for convex optimization without tuning.
problem Optimizing convex functions efficiently with stochastic gradients.
method Adapted Adaptive Gradient Descent Without Descent to stochastic setting.
result Stochastic gradient descent converges under various assumptions.
Off-policy stochastic actor-critic methods rely on approximating the stochastic policy gradient in order to derive an optimal policy. One may also derive the optimal policy by approximating the action-value gradient. The use of action-value gradients is desirable as policy improvement occurs along the direction of stee…
In this paper, we study and analyze the mini-batch version of StochAstic Recursive grAdient algoritHm (SARAH), a method employing the stochastic recursive gradient, for solving empirical loss minimization for the case of nonconvex losses. We provide a sublinear convergence rate (to stationary points) for general noncon…
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.
Improves understanding of stochastic NGVI convergence rates.
problem Lack of knowledge about non-asymptotic convergence rates in stochastic NGVI.
method Proved non-asymptotic convergence rates for conjugate likelihoods and showed implicit optimization for non-conjugate likelihoods.
result First O(T1) non-asymptotic convergence rate for stochastic NGVI in conjugate likelihoods. FGD reduces noisy gradient variance in SGD for neural networks.
problem Noisy and unreliable gradient estimation in SGD for deep learning.
method Solves an adaptive filtering problem to consistently estimate the local gradient.
result Significantly reduces gradient variance and accelerates convergence.
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.
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). 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.
FSGLD improves federated data sampling by correcting noisy gradients.
problem Noisy gradients and delayed communication in federated data.
method Conductive gradients to correct noisy gradients in distributed SGLD.
result FSGLD converges to true posterior even with delayed communication.
Proposes VSGD optimizer combining probabilistic and gradient-based methods.
problem Uncertainty modeling in deep neural networks.
method Combines probabilistic and gradient-based approaches using SVI.
result VSGD outperforms Adam and SGD on image classification tasks.
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.
Although stochastic gradient descent (SGD) is a driving force behind the recent success of deep learning, our understanding of its dynamics in a high-dimensional parameter space is limited. In recent years, some researchers have used the stochasticity of minibatch gradients, or the signal-to-noise ratio, to better char…
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. 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…
Stochastic variational inference (SVI) lets us scale up Bayesian computation to massive data. It uses stochastic optimization to fit a variational distribution, following easy-to-compute noisy natural gradients. As with most traditional stochastic optimization methods, SVI takes precautions to use unbiased stochastic g…
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, …
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.
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…
Many machine learning, statistical inference, and portfolio optimization problems require minimization of a composition of expected value functions (CEVF). Of particular interest is the finite-sum versions of such compositional optimization problems (FS-CEVF). Compositional stochastic variance reduced gradient (C-SVRG)…
Improved complexity for machine learning optimization methods.
problem Optimizing over-parametrized models in machine learning.
method Stochastic conditional gradient methods with interpolation-like conditions.
result Improved oracle complexities for finding optimal solutions.
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.
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.
Paper addresses fault-tolerance in distributed machine learning with stochastic gradient descent.
problem Fault-tolerance in distributed stochastic gradient descent (D-SGD) for machine learning.
method Proposes norm-based comparative gradient elimination (CGE) to robustify D-SGD against Byzantine faulty agents.
result CGE guarantees fault-tolerance against a bounded fraction of Byzantine agents under standard stochastic assumptions.
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…
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.
We develop the method of stochastic modified equations (SME), in which stochastic gradient algorithms are approximated in the weak sense by continuous-time stochastic differential equations. We exploit the continuous formulation together with optimal control theory to derive novel adaptive hyper-parameter adjustment po…
Proof given for SGD convergence in a concise manner.
problem Convergence of Stochastic Gradient Descent (SGD)
method Self-contained proof
result SGD convergence proven
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.
RES, a regularized stochastic version of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method is proposed to solve convex optimization problems with stochastic objectives. The use of stochastic gradient descent algorithms is widespread, but the number of iterations required to approximate optimal arguments c…
This paper targets solving distributed machine learning problems such as federated learning in a communication-efficient fashion. A class of new stochastic gradient descent (SGD) approaches have been developed, which can be viewed as the stochastic generalization to the recently developed lazily aggregated gradient (LA…
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.