Large batch training with DP-SGD reduces model performance due to implicit bias.
problem Large batch training with DP-SGD reduces model performance.
method The study analyzes the phenomenon of implicit bias in Noisy-SGD (DP-SGD without clipping) and its theoretical solutions for linear models.
result The implicit bias in large batch training with DP-SGD is amplified by additional noise, similar to SGD.
SGD outperforms GD in high dimensions via implicit conditioning, revealed by asymptotic analysis.
problem Understanding why SGD outperforms GD in high-dimensional convex problems.
method Asymptotic analysis of multi-pass SGD on high-dimensional convex quadratics, establishing an equivalence to HSGD.
result SGD's efficiency is explained by implicit conditioning, not regularization.
Study shows SGD's generalization is not explained by implicit bias.
problem Explaining the generalization ability of overparameterized learning algorithms.
method Revisited Stochastic Convex Optimization with SGD, demonstrating limitations of implicit bias.
result No distribution-independent or distribution-dependent implicit regularizer can explain SGD's generalization.
Neural networks learn the support of the target function through SGD's implicit regularization effect.
problem Learning the support of the target function in neural networks.
method Investigation of mini-batch SGD's ability to learn the support in the first layer of a neural network.
result Mini-batch SGD effectively learns the support in the first layer by shrinking irrelevant weights, while vanilla GD requires an explicit regularization term.
SGD noise has no bias advantage in online learning, contrary to offline learning.
problem The role of SGD noise in online learning.
method Extensive empirical analysis of image and language data.
result Small batch sizes do not confer any implicit bias advantages in online learning.
GNIs induce asymmetric heavy-tailed noise in SGD, affecting network performance.
problem The effect of Gaussian noise injections on SGD dynamics and network performance.
method Developed a Langevin-like SDE driven by asymmetric heavy-tailed noise to model the modified SGD dynamics.
result GNIs induce an implicit bias that varies with noise heaviness and asymmetry, affecting network performance.
Study on GD and SGD over diagonal networks, focusing on stepsizes and regularisation.
problem Understanding the impact of stochasticity and large stepsizes on gradient descent and SGD solutions.
method Investigation of GD and SGD over diagonal linear networks with macroscopic stepsizes, proving convergence and characterizing solutions.
result Large stepsizes consistently benefit SGD for sparse regression problems, but can hinder GD recovery of sparse solutions, especially in the edge of stability regime.
Noise in SGD affects overparameterized models, favoring sparse solutions.
problem Understanding and mitigating implicit bias in SGD with parameter-dependent noise.
method Theoretical analysis of a quadratically-parameterized model with label noise and Gaussian noise.
result SGD with label noise recovers sparse ground-truth solutions, while SGD with Gaussian noise overfits dense solutions.
SGD with large learning rates can achieve better test accuracy than expected.
problem SGD with large learning rates often outperforms expected convergence bounds.
method Proved that SGD with small learning rates stays close to gradient flow path on modified loss.
result Explicitly adding an implicit regularizer to the loss improves test accuracy.
The paper analyzes the implicit bias of SGD near loss manifold and provides new insights.
problem Understanding the implicit bias of SGD near loss manifolds in overparametrized models.
method Adapting ideas from Katzenberger (1991) to analyze SGD dynamics using a stochastic differential equation (SDE).
result SGD with label noise locally decreases the sharpness of loss, leading to a global analysis of implicit bias.
SGD implicitly regularizes linear regression problems better than ridge regression for many cases.
problem Understanding implicit regularization in linear regression problems.
method Comparing SGD and ridge regression on a broad class of least squares problems.
result SGD generalizes no worse than ridge regression for many problem instances, sometimes better.
Analysis of SGD+M convergence rates in high dimensions with batch size considerations.
problem Understanding convergence rates of SGD+M in high-dimensional settings.
method Analyzing the dynamics of SGD+M on least squares problems with large batch sizes and dimensions.
result Identifies the implicit conditioning ratio (ICR) that regulates SGD+M's acceleration and convergence rates.
Neural network has attracted great attention for a long time and many researchers are devoted to improve the effectiveness of neural network training algorithms. Though stochastic gradient descent (SGD) and other explicit gradient-based methods are widely adopted, there are still many challenges such as gradient vanish…
The paper analyzes two ISGD modes for statistical inference, deriving error bounds and confidence intervals.
problem Statistical inference with implicit SGD for smooth convex functions.
method Proximal Robbins-Monro (proxRM) and proximal Polyak-Ruppert (proxPR) procedures for ISGD.
result Derives non-asymptotic error bounds and confidence interval estimators for model parameters.
Iterative procedures for parameter estimation based on stochastic gradient descent allow the estimation to scale to massive data sets. However, in both theory and practice, they suffer from numerical instability. Moreover, they are statistically inefficient as estimators of the true parameter value. To address these tw…
This paper proves SGD converges to global minimum for over-parameterized ReLU networks.
problem Theoretical understanding of implicit neural networks is limited.
method Gradient flow analysis of ReLU activated implicit neural networks.
result Randomly initialized gradient descent converges to global minimum at a linear rate for square loss function in over-parameterized ReLU networks.
Paper tackles efficient SGD methods for constrained bilevel optimization.
problem Stochastic bilevel optimization with equality constraints.
method Alternating implicit projected SGD and its variants.
result Achieves sample complexity matching state-of-the-art for unconstrained problems.
We propose graph-dependent implicit regularisation strategies for distributed stochastic subgradient descent (Distributed SGD) for convex problems in multi-agent learning. Under the standard assumptions of convexity, Lipschitz continuity, and smoothness, we establish statistical learning rates that retain, up to logari…
SGD converges to critical points of normalized margin in late-stage training for homogeneous neural networks.
problem Analyzing the implicit bias of SGD on homogeneous neural networks.
method Interpreting SGD dynamics as an Euler-like discretization of a conservative field flow associated with the normalized classification margin.
result Normalized SGD iterates converge to the set of critical points of the normalized margin at late-stage training.
New study shows deep networks generalize well due to loss surface geometry.
problem Why deep networks generalize well despite many parameters.
method Analyzed local geometry of loss surface and its effect on SGD.
result SGD stays close to low-dimensional subspace, leading to better generalization bounds.
Stochastic descent methods (of the gradient and mirror varieties) have become increasingly popular in optimization. In fact, it is now widely recognized that the success of deep learning is not only due to the special deep architecture of the models, but also due to the behavior of the stochastic descent methods used, …
We interpret the variational inference of the Stochastic Gradient Descent (SGD) as minimizing a new potential function named the \textit{quasi-potential}. We analytically construct the quasi-potential function in the case when the loss function is convex and admits only one global minimum point. We show in this case th…
Study on the noise in SGD minibatches near local minima.
problem Understanding the noise in SGD minibatches near local minima.
method Detailed analysis of SGD noise in linear regression and derivation of a general formula for different types of minima.
result Provides insight into the stability of training neural networks and suggests large learning rates can help generalization.
This work shows how exploiting gradient alignment can improve distributed and federated learning performance.
problem Misalignment of gradients across clients in distributed and federated learning.
method Utilizing implicit regularization through a novel GradAlign algorithm that induces gradient alignment with large mini-batches.
result Improvements in test accuracies and generalization performance.
Studied how SGD's stability regularization affects generalization in neural networks.
problem Understanding why SGD often generalizes better than GD in neural networks.
method Analyzed stability of SGD and GD through Frobenius norm and trace of Hessian, and compared their generalization properties.
result Stable minima of SGD generalize well, while GD's stability-induced regularization is too weak.
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.
SGD's escape rate depends on log loss barrier, not linear loss barrier.
problem Understanding the escape rate of SGD from local minima.
method Derived a stochastic differential equation (SDE) with additive noise from SGD's multiplicative noise property.
result The log loss barrier determines the escape rate of SGD, not the linear loss barrier.
New SGD variant makes neural networks compressible without assumptions.
problem Improving neural network compressibility without strong assumptions.
method Introducing heavy-tailed noise to SGD iterates.
result Compressible outputs with high probability for any compression rate.
Label noise in SGD helps converge to flatter minima.
problem Improving generalization in overparametrized models.
method Analyzes SGD with label noise, showing convergence to regularized minima.
result SGD with label noise converges to flatter minima, improving generalization.
Adam avoids simplicity bias in neural networks, leading to better generalization.
problem Simplicity bias in neural networks trained with SGD.
method Comparison of Adam and GD on binary classification tasks with Gaussian data.
result Adam leads to richer and more diverse features, improving generalization.
SGD generalizes better than GD with similar number of iterations.
problem Generalization performance of SGD vs GD in convex optimization.
method Comparison of SGD and GD in stochastic convex optimization model.
result GD requires more iterations to match SGD's generalization performance.
SGD without replacement decouples into curvature-following and flatness-regularizing steps.
problem Theoretical analysis of SGD without replacement for large-scale neural networks.
method Analysis of SGD without replacement in a realistic regime, considering high curvature and flatness.
result Optimizing with SGD without replacement is locally equivalent to an additional regularizer step.
Stochastic gradient descent (SGD) is widely believed to perform implicit regularization when used to train deep neural networks, but the precise manner in which this occurs has thus far been elusive. We prove that SGD minimizes an average potential over the posterior distribution of weights along with an entropic regul…
We study the implicit bias of gradient descent methods in solving a binary classification problem over a linearly separable dataset. The classifier is described by a nonlinear ReLU model and the objective function adopts the exponential loss function. We first characterize the landscape of the loss function and show th…
SGD noise helps select flat minima by concentrating in sharp directions and being proportional to loss value.
problem Understanding the implicit regularization of SGD and selecting flat minima in over-parameterized models.
method Relating SGD's linear stability to the Frobenius norm of the Hessian and analyzing the alignment property of SGD noise.
result Flat minima are linearly stable for SGD, and their sharpness is bounded independently of model size and sample size.
Large SGD step sizes lead to sparse feature learning in neural networks.
problem Sparse feature learning in neural networks with large step sizes.
method Empirical observations and theoretical analysis of SGD dynamics.
result Large step sizes induce implicit regularization leading to sparse predictors.
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 dk⋆/2. In this paper, we propose an implicit gradient descent algorithm for the classic k-means problem. The implicit gradient step or backward Euler is solved via stochastic fixed-point iteration, in which we randomly sample a mini-batch gradient in every iteration. It is the average of the fixed-point trajectory that is c…
Several works have aimed to explain why overparameterized neural networks generalize well when trained by Stochastic Gradient Descent (SGD). The consensus explanation that has emerged credits the randomized nature of SGD for the bias of the training process towards low-complexity models and, thus, for implicit regulari…
SGD can jump from high rank minima to low rank minima in DLNs, but not back.
problem SGD's tendency to get stuck in high rank minima in DLNs.
method Analysis of the L2-regularized loss function of DLNs and the definition of absorbing sets. result SGD has a non-zero probability to jump from high rank minima to low rank minima but zero probability to jump back.
Strictly enforcing orthonormality constraints on parameter matrices has been shown advantageous in deep learning. This amounts to Riemannian optimization on the Stiefel manifold, which, however, is computationally expensive. To address this challenge, we present two main contributions: (1) A new efficient retraction ma…
GD outperforms ridge regression and SGD in linear regression problems.
problem Comparing the risks of GD, ridge regression, and SGD in linear regression problems.
method Instance-wise finite-sample risk analysis of GD, ridge regression, and SGD.
result GD outperforms ridge regression and is incomparable with SGD in some cases.
SGD tends to favor simpler subnetworks, improving generalization.
problem SGD's tendency to favor simpler subnetworks over complex ones.
method Identifying invariant sets and analyzing SGD's behavior around them.
result SGD collapses networks to simpler subnetworks, improving generalization.
Stochastic Gradient Descent shows directional bias with moderate learning rates, impacting optimization outcomes.
problem Understanding the bias of SGD with moderate learning rates in practical scenarios.
method Analyzing SGD and GD on an overparameterized linear regression problem.
result SGD converges along large eigenvalue directions, GD along small ones, affecting early stopping outcomes.
A method for efficient statistical inference from online algorithms.
problem Computational constraints in online algorithms make traditional variance estimation difficult.
method HulC method that wraps around online algorithms to produce valid confidence regions.
result The HulC method produces asymptotically valid confidence regions for online algorithms.
We develop methods for parameter estimation in settings with large-scale data sets, where traditional methods are no longer tenable. Our methods rely on stochastic approximations, which are computationally efficient as they maintain one iterate as a parameter estimate, and successively update that iterate based on a si…
Label noise SGD converges to a simple model with a single linear feature.
problem Understanding the simplicity bias in neural network training.
method Analyzing the convergence of label noise SGD on two-layer neural networks.
result Label noise SGD converges to a model with a single linear feature.
SMD outperforms SGD in over-parametrized linear models for certain data distributions.
problem Understanding the generalization performance of SMD in over-parametrized linear models.
method Analysis of SMD for over-parametrized linear models with binary classification.
result Empirical validation of SMD's generalization performance differing from SGD.