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.
Gradient descent implicitly regularizes neural networks by penalizing large loss gradients.
problem How to optimize deep neural networks without explicit regularization.
method Backward error analysis to calculate implicit gradient regularization and demonstrate its effectiveness empirically.
result Implicit gradient regularization biases gradient descent toward flat minima, improving model robustness and test errors.
Generalizes smoothness conditions for optimization methods.
problem Optimization under non-uniform smoothness conditions.
method Develops a new analysis technique for bounding gradients.
result Obtains convergence rates for gradient descent and Nesterov's method.
This paper presents a stochastic behavior analysis of a kernel-based stochastic restricted-gradient descent method. The restricted gradient gives a steepest ascent direction within the so-called dictionary subspace. The analysis provides the transient and steady state performance in the mean squared error criterion. It…
Multi-subject fMRI data analysis is an interesting and challenging problem in human brain decoding studies. The inherent anatomical and functional variability across subjects make it necessary to do both anatomical and functional alignment before classification analysis. Besides, when it comes to big data, time complex…
Gradient descent learns useful features even in the NTK regime.
problem The ability of neural networks to learn useful features.
method Local convergence analysis of gradient descent with regularization.
result Gradient descent can capture ground-truth directions for feature learning even after the loss threshold is reached.
Functional central limit theorem for kernel gradient flow and infinitesimal gradient boosting
problem Fluctuations of boosting processes around their deterministic limit
method Stochastic perturbation analysis of ODEs in Banach spaces
result Rescaled deviations converge to a Gaussian process
This paper analyzes OGDA and EG methods for nonconvex minimax problems.
problem Theoretical guarantees of OGDA and EG methods in nonconvex settings.
method Unified analysis through single-call extra-gradient methods.
result Established convergence of OGDA and EG methods under NC-SC and NC-C settings.
Algorithm improves online canonical correlation analysis.
problem Online canonical correlation analysis.
method Stochastic Scaled-Gradient Descent (SSGD) for minimizing expectation over Riemannian manifolds.
result Achieved optimal one-time-scale algorithm with explicit rate of local asymptotic convergence.
This paper investigates asymptotic behaviors of gradient descent algorithms (particularly accelerated gradient descent and stochastic gradient descent) in the context of stochastic optimization arising in statistics and machine learning where objective functions are estimated from available data. We show that these alg…
Paper analyzes SGHMC for non-convex optimization with discontinuous gradients.
problem Training neural networks with ReLU activation.
method Non-asymptotic convergence analysis of SGHMC with discontinuous gradients.
result Explicit upper bounds for expected excess risk in non-convex optimization.
Gradient-enhanced GSA uses Poincaré chaos expansions for accurate sensitivity analysis.
problem Accurately estimating Sobol' indices with limited data.
method Integrates sparse, gradient-enhanced regression with Poincaré chaos expansions for derivative-based sensitivity analysis.
result Accurately estimated Sobol' indices using limited data.
The paper analyzes GTD algorithms with finite-sample bounds.
problem Convergence rate analysis of GTD family of algorithms.
method Formulated as stochastic gradient algorithms and analyzed using saddle-point error.
result Obtained finite-sample bounds on GTD performance.
We conduct mathematical analysis on the effect of batch normalization (BN) on gradient backpropogation in residual network training, which is believed to play a critical role in addressing the gradient vanishing/explosion problem, in this work. By analyzing the mean and variance behavior of the input and the gradient i…
Paper analyzes convergence of dynamic policy gradient for MDPs, improving performance in finite-time problems.
problem Optimal policies in finite-time MDPs are not stationary and require epoch-specific training.
method Introduces dynamic policy gradient combining dynamic programming and policy gradient, analyzes convergence for softmax parametrisation.
result Dynamic policy gradient training exploits finite-time structure, leading to better convergence bounds.
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.
The paper analyzes stability and generalization of shallow neural networks using gradient methods.
problem Understanding the generalization of overparameterized shallow neural networks.
method The paper uses gradient descent and stochastic gradient descent to study shallow neural networks, developing consistent excess risk bounds.
result The analysis improves on existing methods by providing a refined estimation of iterates and Hessian eigenvalues, leading to better excess risk bounds.
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.
New analysis improves sample complexity for vanilla policy gradient methods.
problem Improving sample complexity guarantees for vanilla policy gradient methods.
method Adapting tools from SGD analysis to policy gradient methods, with smoothness and gradient approximation assumptions.
result Established improved sample complexity bounds for convergence and global optimum.
In this paper we consider solving saddle point problems using two variants of Gradient Descent-Ascent algorithms, Extra-gradient (EG) and Optimistic Gradient Descent Ascent (OGDA) methods. We show that both of these algorithms admit a unified analysis as approximations of the classical proximal point method for solving…
Proposes HGP to improve survival analysis models.
problem Improving survival analysis models by addressing density function changes.
method Imposes constraints on local data points by regularizing the hazard function gradient.
result HGP enhances survival analysis model performance.
Paper introduces a new reinforcement learning method with improved performance.
problem Designing and analyzing efficient reinforcement learning algorithms.
method Proximal gradient temporal difference learning (GTD) with accelerated algorithm GTD2-MP.
result GTD algorithms have linear complexity and improved convergence rate.
Unified framework for analyzing batch updating methods with noisy gradients.
problem Analyzing convergence of batch updating methods with noisy gradients and approximations.
method Unified framework using convergence of stochastic processes.
result Establishes a general theorem for most known convergence results.
Study on policy gradient for stochastic bandits using diffusion approximation.
problem Improving policy gradient methods for stochastic bandits with optimal regret bounds.
method Continuous-time diffusion approximation of policy gradient with learning rate analysis.
result Proved optimal regret bound of O(klog(k)log(n)/η) for η=O(Δ2/log(n)). The study analyzes implicit biases in neural networks using backward error analysis.
problem Analyzing implicit biases in multitask and continual learning settings.
method Backward error analysis to compute implicit training biases, deriving modified losses with three terms.
result The conflict term, measuring gradient alignment, is a new quantity in continual learning.
We consider the minimization of an objective function given access to unbiased estimates of its gradient through stochastic gradient descent (SGD) with constant step-size. While the detailed analysis was only performed for quadratic functions, we provide an explicit asymptotic expansion of the moments of the averaged S…
New framework improves variational inference with Markov chain methods.
problem Challenges of minimizing KL divergence with stochastic gradient descent.
method Markov chain score ascent (MCSA) methods, including parallel MCSA (pMCSA).
result Improved theoretical and empirical performance of MCSA methods.
Piecewise polynomial interpolation-based gradient descent reduces oracle complexity for smooth loss functions.
problem Optimizing empirical risk minimization loss functions
method Piecewise polynomial interpolation-based gradient descent
result Oracle complexity is reduced for smooth loss functions
New analysis shows GMD can converge linearly under PL-like conditions.
problem Establishing linear convergence for generalized mirror descent.
method PL-based analysis for time-dependent mirrors, Taylor-series approach for stochastic GMD.
result Linear convergence of stochastic GMD under PL-like conditions.
Paper analyzes SVGD algorithm for non-asymptotic convergence.
problem Optimizing a set of particles to approximate a target probability distribution.
method Finite time analysis of SVGD algorithm, providing descent lemma and convergence rates.
result SVGD algorithm decreases the objective at each iteration and converges to the target distribution.
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.
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.
Fine-grained analysis of gradient descent with momentum provides modified loss equations.
problem Understanding the dynamics of gradient descent with momentum.
method Fine-grained analysis and derivation of modified loss equations.
result Global approximation bounds and continuous modified equations for HB.
Despite remarkable empirical success, the training dynamics of generative adversarial networks (GAN), which involves solving a minimax game using stochastic gradients, is still poorly understood. In this work, we analyze last-iterate convergence of simultaneous gradient descent (simGD) and its variants under the assump…
This paper analyzes convergence of FL for neural networks using NTK.
problem Theoretical guarantees of FL for neural networks with explicit forms and multi-step updates are unexplored.
method FL-NTK framework for federated learning of ReLU neural networks trained by gradient descent.
result FL-NTK converges to a global-optimal solution at a linear rate with proper learning parameters.
Unified analysis of SGD variants for nonconvex federated optimization.
problem Performance of stochastic gradient methods in nonconvex optimization.
method Proposed a unified assumption for modeling stochastic gradient second moment, leading to a single convergence analysis for various methods.
result Unified convergence analysis for a wide range of SGD variants and distributed methods.
This paper analyzes implicit bias in Deep Linear Discriminant Analysis.
problem The implicit bias of Deep Linear Discriminant Analysis.
method Analyzing gradient flow on a L-layer diagonal linear network.
result Under balanced initialization, the network transforms additive updates into multiplicative updates, conserving the (2/L) quasi-norm.
Unified analysis of stochastic gradient methods for convex and smooth optimization.
problem Minimizing composite convex and smooth functions.
method Unified convergence analysis of various stochastic gradient methods.
result Unified convergence rates for a variety of methods including proximal SGD, variance reduced methods, quantization, and coordinate descent.
We present a convergence rate analysis for biased stochastic gradient descent (SGD), where individual gradient updates are corrupted by computation errors. We develop stochastic quadratic constraints to formulate a small linear matrix inequality (LMI) whose feasible points lead to convergence bounds of biased SGD. Base…
LDA-GO improves LDA for high-dimensional data via gradient optimization.
problem LDA struggles in high-dimensional settings due to unreliable covariance matrix estimation.
method LDA-GO learns a low-rank precision matrix via gradient optimization, automatically selecting between Gaussian likelihood and cross-entropy loss.
result LDA-GO outperforms other LDA variants in sparse-signal high-dimensional regimes.
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.
Gradient descent finds a global minimum in training deep neural networks despite the objective function being non-convex. The current paper proves gradient descent achieves zero training loss in polynomial time for a deep over-parameterized neural network with residual connections (ResNet). Our analysis relies on the p…
An ADRC-incorporated SGD algorithm improves latent factor analysis speed and accuracy.
problem Slow convergence in standard SGD for HDI matrix analysis.
method Incorporates ADRC principles to refine historical and future learning error states.
result Empirically outperforms state-of-the-art LFA models in HDI matrix prediction.
Recent analyses of certain gradient descent optimization methods have shown that performance can degrade in some settings - such as with stochasticity or implicit momentum. In deep reinforcement learning (Deep RL), such optimization methods are often used for training neural networks via the temporal difference error o…
We provide the first convergence analysis of local gradient descent for minimizing the average of smooth and convex but otherwise arbitrary functions. Problems of this form and local gradient descent as a solution method are of importance in federated learning, where each function is based on private data stored by a u…
Gradient oversmoothing and expansion hinder deep GNN training, solved with normalization.
problem Gradient oversmoothing and expansion prevent deep GNN training.
method Proposed normalization method to constrain the Lipschitz bound of each layer.
result Residual GNNs with hundreds of layers can be efficiently trained with the proposed normalization.
We study the problem of supervised linear dimensionality reduction, taking an information-theoretic viewpoint. The linear projection matrix is designed by maximizing the mutual information between the projected signal and the class label (based on a Shannon entropy measure). By harnessing a recent theoretical result on…
Improves bandit convex optimization with gradient variations.
problem Bandit Convex Optimization with Gradient Variations.
method Refined analysis of non-consecutive gradient variation.
result Improved dimension dependence for convex and strongly convex functions.