This paper proves AdaGrad and Adam converge linearly under PL inequality.
problem Understanding the convergence of adaptive gradient methods.
method Unified approach proving AdaGrad and Adam converge linearly under PL inequality.
result AdaGrad and Adam converge linearly when the cost function is smooth and satisfies PL inequality.
Paper improves convergence rates and step sizes for gradient algorithms.
problem Limited convergence rates and step sizes in existing algorithms.
method General proximal incremental aggregated gradient algorithm.
result Better and novel convergence results under general scheme.
This paper improves convergence guarantees for gradient clipping in deep learning.
problem Improving convergence guarantees for gradient clipping in deep learning models.
method Analyzes and provides precise convergence guarantees for arbitrary clipping thresholds.
result Shows tight convergence guarantees for clipped stochastic gradient descent.
Gradient descent achieves exact linear convergence rate for symmetric matrix completion.
problem Low-rank symmetric matrix completion using gradient descent.
method Local analysis of gradient descent for symmetric matrices without additional assumptions.
result Closed-form expression of exact linear convergence rate matches practice.
Adaptive gradient methods are workhorses in deep learning. However, the convergence guarantees of adaptive gradient methods for nonconvex optimization have not been thoroughly studied. In this paper, we provide a fine-grained convergence analysis for a general class of adaptive gradient methods including AMSGrad, RMSPr…
Proof given for SGD convergence in a concise manner.
problem Convergence of Stochastic Gradient Descent (SGD)
method Self-contained proof
result SGD convergence proven
Neural policy gradient methods converge globally and sublinearly.
problem Global optimality and convergence of neural policy gradient methods.
method Actor-critic schemes with neural networks, proving global optimality and sublinear convergence rates.
result Neural natural and vanilla policy gradient methods converge to globally optimal policies and stationary points.
Square-root natural-gradient improves variational inference convergence.
problem Challenges in establishing theoretical convergence guarantees for natural-gradient descent.
method Square-root parameterization for Gaussian covariance.
result Establishes novel convergence guarantees for natural-gradient Gaussian inference.
The paper provides convergence guarantees for multicalibration gradient boosting.
problem Understanding the convergence properties of multicalibration gradient boosting.
method Computational guarantees for multicalibration gradient boosting algorithms, including adaptive variants.
result The magnitude of successive prediction updates decays at O ( 1 / T ) O(1/\sqrt{T}) O ( 1/ T ) , leading to convergence in empirical multicalibration error. Studied SGD convergence under weak conditions.
problem Convergence of SGD in nonconvex optimization.
method Analyzed biased nonconvex SGD under mild conditions.
result Provided convergence rates and complexities.
Policy gradient methods achieve linear convergence in simple MDPs.
problem Analyzing convergence rates of policy gradient methods in finite MDPs.
method Connections with policy iteration to show linear convergence with large step-sizes.
result Policy gradient methods succeed with large step-sizes and achieve linear rate of convergence.
SGD (Stochastic Gradient Descent) is a popular algorithm for large scale optimization problems due to its low iterative cost. However, SGD can not achieve linear convergence rate as FGD (Full Gradient Descent) because of the inherent gradient variance. To attack the problem, mini-batch SGD was proposed to get a trade-o…
New method for natural policy gradients converges linearly.
problem Improving natural policy gradient methods for better convergence.
method Fisher-Rao gradient flow applied to state-action distributions.
result Linear convergence rate with geometry-dependent factor.
Gradient EM converges globally for over-parameterized Gaussian mixtures.
problem Global convergence of gradient EM for Gaussian mixtures with more than 2 components.
method Likelihood-based convergence analysis framework.
result Gradient EM converges globally with a sublinear rate of O(1/√t).
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.
This work analyzes and improves stochastic gradient methods for GAN training.
problem Understanding the training dynamics of GANs, particularly their convergence.
method Continuous-time analysis using differential equations, focusing on simGD and its variants.
result The methods converge under different assumptions, providing new insights into GAN training.
Gradient methods converge better for alternating updates in bilinear zero-sum games.
problem Understanding the dynamics of gradient algorithms for bilinear zero-sum games.
method Systematic analysis of popular gradient updates for simultaneous and alternating versions of bilinear zero-sum games.
result Alternating updates converge better than simultaneous ones, with optimal parameter setup and rates.
GD converges faster to flatter minima than gradient flow in shallow networks.
problem Understanding the dynamics of gradient descent in shallow linear networks.
method Analyzing the convergence rate and solution of gradient descent in depth-2 linear neural networks.
result GD converges linearly to flatter minima than gradient flow, even with large step sizes.
Gradient descent on MMD GAN parameter space converges globally to target distribution.
problem Convergence of gradient descent in Maximum Mean Discrepancy (MMD) GANs.
method Proposes a parametric kernelized gradient flow that mimics the min-max game in gradient regularized MMD GAN.
result Gradient descent on the generator's parameter space in gradient regularized MMD GAN is globally convergent to the target distribution under certain conditions.
Gradient flow of elastic energy converges to elastica.
problem Optimizing closed curves to minimize elastic energy.
method Proving the existence of a unique global solution and convergence via Łojasiewicz--Simon gradient inequality.
result Convergence to elastica established for the H 2 ( d s ) H^2(ds) H 2 ( d s ) -gradient flow of modified elastic energy. Paper removes bounded gradient assumption for SGD in nonconvex learning.
problem Existing theoretical results for SGD in nonconvex learning require uniform boundedness of gradients, which is hard to verify.
method Establishes sufficient conditions for SGD convergence without bounded gradient assumption.
result SGD achieves optimal convergence rates for nonconvex and gradient-dominated objectives.
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.
SGD method converges to minima for non-convex functions.
problem Non-convex optimization problems in machine learning.
method Stochastic gradient descent method for non-convex objective functions.
result Estimates on the rate of convergence to minima.
This work accelerates gradient descent with anytime convergence guarantees.
problem Improving the convergence rate of gradient descent methods.
method Proposes a stepsize schedule for gradient descent that achieves anytime convergence rates.
result Gradient descent can achieve convergence rates of O ( T − 1.119 ) O(T^{-1.119}) O ( T − 1.119 ) for any stopping time T T T . Softmax policy gradient methods converge at O ( 1 / t ) O(1/t) O ( 1/ t ) rate with constants depending on problem and initialization.
problem Understanding convergence rates of softmax policy gradient methods in tabular settings.
method Analysis of softmax policy gradient and entropy regularized policy gradient methods, using Łojasiewicz inequality and lower bounds.
result Entropy regularization improves convergence rate from O ( 1 / t ) O(1/t) O ( 1/ t ) to O ( e − c ⋅ t ) O(e^{-c \cdot t}) O ( e − c ⋅ 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 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 …
Gradient descent proves global convergence for 4-layer matrix factorization.
problem Global convergence of gradient descent on four-layer matrix factorization under random initialization.
method New techniques to show saddle-avoidance properties and extend eigenvalue theories.
result Polynomial-time global convergence guarantee for randomly initialized gradient descent on four-layer matrix factorization.
Policy gradient methods find Nash equilibrium in noisy games.
problem Finding Nash equilibrium in noisy games.
method Policy gradient methods with noise added.
result Policy gradient methods converge to Nash equilibrium in noisy games.
Gradient descent converges globally in deep linear residual networks with ZAS initialization.
problem Optimizing deep linear residual networks for convergence.
method Zero-asymmetric (ZAS) initialization for gradient descent.
result Gradient descent converges to an ε-optimal point in O(L^3 log(1/ε)) iterations.
Improved convergence for nonconvex optimization with dependent data.
problem Constrained smooth nonconvex optimization with dependent data.
method Stochastic projected gradient methods under a general dependent data sampling scheme.
result Achieved worst-case rate of convergence i l d e O ( t − 1 / 4 ) ilde{O}(t^{-1/4}) i l d e O ( t − 1/4 ) and complexity i l d e O ( ε − 4 ) ilde{O}(\varepsilon^{-4}) i l d e O ( ε − 4 ) . 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 ( 1 T ) \mathcal{O}(\frac{1}{T}) O ( T 1 ) non-asymptotic convergence rate for stochastic NGVI in conjugate likelihoods. Gradient descent converges to max-margin solution for hinge loss.
problem Applying gradient descent to the hinge loss for linear classifiers.
method Homotopic gradient descent applied to the hinge loss.
result Explicit convergence rates to max-margin solution for separable data.
Analyze SGD with biased gradients, improving convergence rates and accuracy.
problem Analyzing the convergence of SGD with biased gradients.
method Derive convergence results for smooth non-convex functions and quantify the impact of bias magnitude.
result Improved rates under the Polyak-Lojasiewicz condition and insights into how bias magnitude affects accuracy and convergence.
New method shows AdaGrad converges globally to neural network minima.
problem Convergence of adaptive gradient methods for neural networks.
method Proposed adaptive gradient method for over-parameterized neural networks.
result Converges to global minimum in polynomial time for two-layer networks.
Chebyshev steps improve convergence in deep-unfolded gradient descent.
problem Improving convergence speed in iterative algorithms.
method Introducing Chebyshev steps to bound convergence rate of gradient descent.
result Chebyshev steps lead to asymptotically optimal convergence rate.
We study how gradient convergence speeds up in non-convex learning tasks.
problem Understanding the convergence of gradients in non-convex learning problems.
method We propose vector-valued Rademacher complexities to derive uniform convergence bounds for gradients in non-convex learning problems.
result We show that for non-convex models, gradient convergence can be dimension-independent under certain distributional assumptions.
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 e x t s f S G D extsf{SGD} e x t s f S G D , e x t s f S M D extsf{SMD} e x t s f S M D , e x t s f A S M D extsf{ASMD} e x t s f A S M D , e x t s f S G D M extsf{SGDM} e x t s f S G D M in convex and nonconvex optimization. result Established in-expectation convergence results for various stochastic gradient methods.
Natural gradient descent speeds up convergence in neural networks, especially with overparameterization.
problem Mitigating the effects of curvature in neural network optimization.
method Analysis of natural gradient descent on nonlinear neural networks with stability conditions.
result Natural gradient descent converges efficiently under specific conditions for overparameterized networks.
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.
Gradient clipping helps private SGD converge despite potential bias.
problem Gradient clipping in private SGD can bias convergence.
method Theoretical analysis and empirical evaluation of gradient clipping effects.
result Gradient clipping can prevent convergence to stationary points and introduces bias.
Study shows convergence of stochastic gradient method for unregularized Wasserstein optimization.
problem Wasserstein distributionally robust optimization under potential distribution shifts.
method Regularized approximation with stochastic gradient methods, convergence analysis.
result Stochastic gradient method converges to subgradients of unregularized objective as regularization vanishes.
Gradient methods converge exponentially in concave network games.
problem Finding Nash equilibria in concave network zero-sum games.
method Gradient Ascent and Optimistic Gradient Ascent analyses.
result Exponential convergence rates in various game settings.
Continuous-time SGD converges under certain conditions, useful for deep learning.
problem Minimizing population expected loss in learning problems.
method Continuous-time approximation of stochastic gradient descent.
result Establishes sufficient conditions for convergence, applicable to overparametrized neural networks.
Adaptive algorithm improves convergence rate of Langevin dynamics.
problem Improving convergence rate of Langevin dynamics.
method Adaptive non-reversible stochastic gradient Langevin dynamics algorithm.
result Improved convergence rate of the algorithm.
New analysis shows convergence rate of 1/k for gradient and extra-gradient methods.
problem Finding saddle points in convex-concave problems.
method Interpreted as proximal point method approximations, showing iterates remain bounded.
result Primal dual gap converges at rate O(1/k).
Explains gradient descent methods and their convergence, focusing on simple analysis.
problem Understanding and analyzing gradient descent methods and their variants.
method Elementary mathematical analysis focusing on structures and assumptions of objective functions.
result Unified convergence analysis of various gradient descent methods and variants.
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…