Cheap methods improve uncertainty in SGD solutions.
problem Uncertainty quantification in SGD solutions.
method Two resampling-based methods: parallel resampling with replacement and online resampling.
result Significantly reduced computation effort in constructing confidence intervals.
SGD improves generalization by using gradient variability as a proxy for data randomness.
problem Improving generalization in machine learning models trained with stochastic gradient descent.
method Bootstrap perspective on SGD, analyzing gradient variability and algorithmic variability.
result SGD avoids spurious solutions and improves generalization by implicitly regularizing the trace of the gradient covariance matrix.
Although stochastic gradient descent (SGD) method and its variants (e.g., stochastic momentum methods, AdaGrad) are the choice of algorithms for solving non-convex problems (especially deep learning), there still remain big gaps between the theory and the practice with many questions unresolved. For example, there is s…
Artificial intelligence is revolutionizing our lives at an ever increasing pace. At the heart of this revolution is the recent advancements in deep neural networks (DNN), learning to perform sophisticated, high-level tasks. However, training DNNs requires massive amounts of data and is very computationally intensive. G…
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.
Study uses SGD to find near-optimal execution cost policies in dynamic markets.
problem Finding optimal execution cost policies in complex markets.
method Stochastic Gradient Descent (SGD) approach to derive near-optimal policies.
result SGD-based policies offer valuable insights and are implementable in volatile markets.
New model shows SGD can prefer sharp or flat solutions based on label noise.
problem Understanding SGD's preference for flat or sharp solutions during training.
method Solved an analytically solvable model to explore SGD behavior.
result Data distribution determines sharpness at convergence; isotropic label noise leads to flat minimum preference.
SGD vs quasi-Newton optimization in neural networks: different landscapes, different generalizability.
problem Understanding neural network optimization and generalizability.
method Comparison of stochastic gradient descent (SGD) and quasi-Newton optimization methods using computational tools.
result SGD solutions are separated by lower barriers than quasi-Newton solutions, but quasi-Newton solutions are deeper and more isolated.
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 balance theory explains SGD's behavior in neural networks.
problem Understanding SGD's navigation in neural network loss landscapes.
method Analyzes minibatch noise and loss function symmetries.
result Derives the stationary distribution of SGD for deep networks.
SGD trains ReLU networks to implement piecewise linear maps with at most 3 knot points.
problem Understanding the training dynamics of neural networks trained via SGD.
method Mean-field analysis of a two-layer ReLU network trained via SGD for a univariate regression problem.
result At convergence, SGD-trained ReLU networks implement piecewise linear maps with at most 3 knot points.
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.
REPAIR mitigates variance collapse to enable linear interpolation between SGD solutions.
problem Linear interpolation between SGD solutions is difficult due to variance collapse in permuted activations.
method REPAIR rescales preactivations of interpolated networks to mitigate variance collapse.
result 60%-100% relative barrier reduction across various architectures and tasks.
FedSel uses local differential privacy to protect data privacy in federated SGD.
problem Privacy leakage from gradients in federated SGD.
method Two-stage framework with top-k dimension selection and gradient accumulation.
result FedSel reduces privacy leakage by privately selecting important dimensions.
This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative eigenval…
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.
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.
Homogenized SGD explains SGD dynamics in high dimensions.
problem Understanding SGD dynamics in high-dimensional settings.
method Developed a homogenized SGD model to analyze high-dimensional SGD.
result Convergent high-dimensional SGD to homogenized SGD for quadratic statistics.
Non-convex SGD learns halfspaces with adversarial label noise efficiently.
problem Agnostically learning halfspaces in adversarial label noise settings.
method Non-convex SGD optimization for halfspace learning.
result Non-convex SGD achieves misclassification error close to optimal with adversarial noise.
New SGD analysis for nonconvex optimization finds optimal rates.
problem Optimizing nonconvex functions in machine learning.
method Introduces a new expected smoothness assumption and analyzes SGD convergence rates.
result Optimal convergence rates for nonconvex smooth functions and global solutions under certain conditions.
Study on adversarial training dynamics in high dimensions using SGD.
problem Analyzing adversarial training of models in high-dimensional settings.
method Deriving deterministic equivalents for SGD iterates under Gaussian mixtures.
result No constant learning rate guarantees monotone descent in adversarial training.
Stochastic Gradient Descent (SGD) based training of neural networks with a large learning rate or a small batch-size typically ends in well-generalizing, flat regions of the weight space, as indicated by small eigenvalues of the Hessian of the training loss. However, the curvature along the SGD trajectory is poorly und…
New insights into SGD and SGD-M in high dimensions.
problem Understanding and comparing SGD and SGD-M in high-dimensional settings.
method Developed high-dimensional scaling limits for SGD-M and online SGD, examining their dynamics and performance.
result SGD-M amplifies high-dimensional effects, potentially degrading performance compared to online SGD.
Adaptive optimization methods, which perform local optimization with a metric constructed from the history of iterates, are becoming increasingly popular for training deep neural networks. Examples include AdaGrad, RMSProp, and Adam. We show that for simple overparameterized problems, adaptive methods often find drasti…
SGD shows distinct phases in learning single-index models, achieving optimal sample complexity and regret.
problem Learning single-index models with SGD in adaptive data settings.
method Stochastic gradient descent (SGD) with an optimal learning rate schedule.
result SGD achieves near-optimal sample complexity and regret guarantees across both burn-in and learning phases.
Paper explores weighted averaging schemes for SGD, achieving asymptotic normality and optimality.
problem Improving convergence of SGD in various settings.
method Develops a general weighted averaging scheme for SGD and establishes asymptotic normality.
result Establishes asymptotic normality and optimality of weighted averaged SGD solutions.
Study on RF regression with SGD shows double descent phenomenon.
problem Understanding generalization in RF models trained with SGD.
method Precise non-asymptotic error bounds derived for RF regression under constant and polynomial-decay step-size SGD.
result RF regression generalizes well for interpolation learning and exhibits double descent behavior.
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.
The optimization of multilayer neural networks typically leads to a solution with zero training error, yet the landscape can exhibit spurious local minima and the minima can be disconnected. In this paper, we shed light on this phenomenon: we show that the combination of stochastic gradient descent (SGD) and over-param…
Paper proposes an online estimator for covariance matrix of SGD iterates.
problem Quantifying variability and randomness of SGD-based estimates in online learning.
method Proposes a fully online estimator for covariance matrix of ASGD using SGD iterates.
result Establishes consistency of the online estimator and shows comparable convergence rate to offline methods.
A long-standing problem in the theory of stochastic gradient descent (SGD) is to prove that its without-replacement version RandomShuffle converges faster than the usual with-replacement version. We present the first (to our knowledge) non-asymptotic solution to this problem, which shows that after a "reasonable" numbe…
SGD converges to global minimum for certain non-convex functions.
problem Theoretical challenges in optimizing non-convex functions in machine learning.
method Perturbed SGD on a broad class of non-convex functions.
result SGD converges to global minimum for certain non-convex functions.
Low precision operations can provide scalability, memory savings, portability, and energy efficiency. This paper proposes SWALP, an approach to low precision training that averages low-precision SGD iterates with a modified learning rate schedule. SWALP is easy to implement and can match the performance of full-precisi…
Adapts SGD to noise and problem specifics for faster convergence.
problem Minimizing smooth, strongly-convex functions with varying noise and problem constants.
method Adaptive SGD with exponentially decreasing step-sizes, Nesterov acceleration, and stochastic line-search.
result Achieves near-optimal convergence rates without knowing noise or problem specifics.
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.
Proposes a new adaptive learning rate for SGD.
problem Finding an efficient learning rate for SGD.
method Stochastic Polyak step-size (SPS).
result SGD with SPS converges faster for over-parameterized models.
Adaptive distributed SGD reduces delay in slow workers.
problem Minimizing delay in distributed SGD with stragglers.
method Adaptive policy for varying k to optimize error-runtime trade-off. result Numerical simulations confirm the effectiveness of the adaptive approach.
Stochastic gradient descent (SGD) forms the core optimization method for deep neural networks. While some theoretical progress has been made, it still remains unclear why SGD leads the learning dynamics in overparameterized networks to solutions that generalize well. Here we show that for overparameterized networks wit…
A new pruning method finds sparse minimizers in flat regions of deep neural networks.
problem Finding sparse minimizers in flat regions of deep neural networks.
method Directional pruning using an ℓ1 proximal gradient algorithm. result Empirical results show promising sparsity (92%) and comparable minima to SGD.
Truncated SGD with heavy-tailed noise eliminates sharp local minima.
problem Avoiding sharp local minima in deep learning models.
method Truncated SGD with heavy-tailed gradient noise.
result Truncated SGD can eliminate sharp local minima entirely from its training trajectory.
Deep neural networks are typically trained by optimizing a loss function with an SGD variant, in conjunction with a decaying learning rate, until convergence. We show that simple averaging of multiple points along the trajectory of SGD, with a cyclical or constant learning rate, leads to better generalization than conv…
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.
We propose a new metaheuristic training scheme that combines Stochastic Gradient Descent (SGD) and Discrete Optimization in an unconventional way. Our idea is to define a discrete neighborhood of the current SGD point containing a number of "potentially good moves" that exploit gradient information, and to search this …
Uniform diffusion approximation for SGD in non-convex settings.
problem Finite-time diffusion approximation for SGD.
method Establishing uniform-in-time diffusion approximation with strong convexity and mild conditions.
result Uniform-in-time diffusion approximation of SGD without convexity of each loss function.
This paper analyzes SGD with increasingly weighted averaging for optimization and generalization.
problem Improving optimization and generalization for non-strongly convex objectives.
method Comprehensive analysis of increasingly weighted averaging schemes for convex, strongly convex, and non-convex objectives.
result The weight α affects both optimization and generalization errors, revealing a trade-off. One-pass SGD dynamics in overparameterized quadratic networks show slow escape from poor solutions.
problem Slow escape from poor generalization solutions in overparameterized neural networks.
method Analysis of one-pass SGD dynamics using ordinary differential equations for overlap matrices.
result Overparameterization only modestly accelerates escape from poor solutions.
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.
Stochastic Gradient Descent underperforms on some problems, contrary to expectations.
problem Understanding the generalization performance of SGD on specific problem instances.
method Analysis of stochastic convex optimization framework, proving empirical and generalization gaps for SGD.
result SGD exhibits both empirical risk and generalization gap of Ω(1) on some problem instances, contradicting its conventional understanding.