Gradient descent near stability threshold exhibits sharpness oscillations.
problem Understanding sharpness behavior near stability threshold in non-Euclidean norms.
method Interpreted EoS through Directional Smoothness and generalized sharpness under arbitrary norms.
result Non-Euclidean GD with generalized sharpness shows sharpness oscillations near 2/η. SoftAD improves classification accuracy with less fine-tuning and fewer computational costs.
problem Improving classification accuracy with less fine-tuning and fewer computational costs.
method SoftAD is a softened, pointwise mechanism that downweights borderline points and limits the effects of outliers.
result SoftAD achieves classification accuracy competitive with flooding and SAM, with a smaller loss generalization gap and model norm.
Improved learning bounds for corrupted data using thresholded gradient descent.
problem Learning from corrupted data with adversarial noise.
method Thresholded gradient descent for sigmoidal, leaky-ReLU, and ReLU activations.
result Improved approximation bounds for various activation functions.
Gradient descent near stability threshold shows sharpness oscillations.
problem Understanding sharpness and stability in non-Euclidean norms during gradient descent.
method Interpreted EoS through Directional Smoothness, defined generalized sharpness for arbitrary norms.
result Non-Euclidean GD exhibits sharpness oscillations around the stability threshold.
Sparse GCA finds linear relationships in multiple datasets, using gradient descent.
problem Finding linear relationships across multiple datasets with sparse loading vectors.
method Formulated as generalized eigenvalue problems, used a thresholded gradient descent algorithm.
result Proposed algorithm yields tight estimation error bounds and demonstrates effectiveness on synthetic datasets.
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.
New method trains neural networks with threshold activation functions efficiently.
problem Training neural networks with threshold activation functions is challenging due to zero gradients.
method We study weight decay regularized training problems of deep neural networks with threshold activations, showing they can be formulated as convex optimization problems.
result Regularized deep threshold network training problems can be formulated as standard convex optimization problems, paralleling the LASSO method.
In this paper we study the performance of the Projected Gradient Descent(PGD) algorithm for ℓp-constrained least squares problems that arise in the framework of Compressed Sensing. Relying on the Restricted Isometry Property, we provide convergence guarantees for this algorithm for the entire range of $0\leq p\…
Gradient descent forces neural network eigenvalues to a specific threshold.
problem Understanding why gradient descent drives eigenvalues to a specific threshold.
method Introduced edge coupling, a functional on consecutive iterate pairs, to explain the trajectory towards the eigenvalue threshold.
result Gradient descent forces the Hessian eigenvalue to the threshold 2/η from arbitrary initialization. Large-scale non-convex sparsity-constrained problems have recently gained extensive attention. Most existing deterministic optimization methods (e.g., GraSP) are not suitable for large-scale and high-dimensional problems, and thus stochastic optimization methods with hard thresholding (e.g., SVRGHT) become more attract…
Noise in linear networks minimizes sharpness and leads to shrinkage-thresholding.
problem Minimizing sharpness in diagonal linear networks.
method Stochastic sharpness-aware minimization (SAM) with isotropic noise.
result Noise forces shrinkage-thresholding of true parameters.
AutoClip automatically adjusts gradient clipping for better audio separation.
problem Improving generalization in audio source separation networks.
method Adaptive gradient clipping based on historical gradient norms.
result Improves generalization performance in audio source separation networks.
Large learning rates cause parameter instability, leading to better generalization.
problem Understanding why deep neural networks perform well despite operating outside the traditional stability regime.
method Analyzing the effect of large learning rates on the orientation of Hessian eigenvectors and parameter exploration.
result Large learning rates induce parameter instability, leading to better generalization through exploration of flatter regions of the loss landscape.
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.
LTP learns per-layer thresholds for efficient pruning of deep networks.
problem Efficiently pruning deep neural networks to reduce computational cost and size.
method LTP learns thresholds via gradient descent, making pruning computationally efficient and scalable.
result LTP achieves competitive compression rates and maintains high accuracy on ImageNet networks.
This paper analyzes convergence of DP-SGD with adaptive quantile clipping.
problem Empirical success of adaptive clipping methods lacks theoretical understanding.
method Comprehensive convergence analysis of SGD with quantile clipping (QC-SGD).
result Establishes theoretical guarantees for DP-QC-SGD, revealing relationships between quantile selection, step size, and convergence.
Momentum affects optimization differently at small vs large batch sizes near instability.
problem Understanding how momentum impacts optimization near the edge of stability.
method Demonstrated through batch-size dependent behavior of SGD with momentum.
result Momentum operates in two distinct regimes: amplifying stochastic fluctuations at small batch sizes and stabilizing at large batch sizes.
A framework for binary classification on top samples.
problem Binary classification problems above/below a threshold.
method General framework for ranking problems, hypothesis testing.
result Theoretical and numerical analysis of methods.
The mushroom body is the key network for the representation of learned olfactory stimuli in Drosophila and insects. The sparse activity of Kenyon cells, the principal neurons in the mushroom body, plays a key role in the learned classification of different odours. In the specific case of the fruit fly, the sparseness o…
This paper solves tensor robust principal component analysis via scaled gradient descent.
problem Extracting useful information from tensor data robust to corruptions and ill-conditioning.
method Directly recovers low-rank tensor factors via scaled gradient descent with adaptive thresholding.
result The proposed algorithm converges linearly to the true low-rank tensor at a constant rate independent of the condition number.
Hard Thresholding Pursuit (HTP) is an iterative greedy selection procedure for finding sparse solutions of underdetermined linear systems. This method has been shown to have strong theoretical guarantee and impressive numerical performance. In this paper, we generalize HTP from compressive sensing to a generic problem …
Random Reshuffling outperforms Stochastic Gradient Descent in smooth convex optimization.
problem Theoretical limitations of Random Reshuffling in smooth convex optimization.
method Random Reshuffling (RR) as a variant of Shuffling Stochastic Gradient Descent (Shuffling SGD).
result Random Reshuffling (RR) dominates Stochastic Gradient Descent (SGD) in smooth convex optimization under any reasonable stepsize after any finite number of epochs.
DeepTopPush improves accuracy at the top for complex classification tasks.
problem Minimizing irrelevant samples above a threshold in binary classification.
method Proposes a new method for end-to-end training of deep networks to minimize loss at the top.
result Demonstrates excellent performance on visual recognition and real-world applications.
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.
Online SGD achieves consistent estimation in high-dimensional non-convex inference tasks.
problem Consistent estimation in high-dimensional non-convex optimization problems.
method Online stochastic gradient descent (SGD) on non-convex losses.
result Nearly sharp thresholds for sample complexity in high-dimensional settings.
We analyse the learning performance of Distributed Gradient Descent in the context of multi-agent decentralised non-parametric regression with the square loss function when i.i.d. samples are assigned to agents. We show that if agents hold sufficiently many samples with respect to the network size, then Distributed Gra…
A new ODE model explains gradient descent dynamics near edge of stability.
problem Understanding gradient-based training over non-convex landscapes.
method Rod Flow, a new ODE approximation of GD dynamics.
result Rod Flow accurately predicts critical sharpness threshold and self-stabilization in quartic potentials.
In many high-dimensional estimation problems the main task consists in minimizing a cost function, which is often strongly non-convex when scanned in the space of parameters to be estimated. A standard solution to flatten the corresponding rough landscape consists in summing the losses associated to different data poin…
In this paper, we study the implicit regularization of the gradient descent algorithm in homogeneous neural networks, including fully-connected and convolutional neural networks with ReLU or LeakyReLU activations. In particular, we study the gradient descent or gradient flow (i.e., gradient descent with infinitesimal s…
Model shows loss curve with two distinct exponents due to sparse activations.
problem Sparse activations impact neural network scaling laws.
method Introduced a model for neural scaling laws under sparse activations, derived asymptotic population loss, and analyzed gradient-descent dynamics.
result Loss curve exhibits double-descent peak near interpolation threshold with two distinct scaling exponents.
New DP-CD method outperforms DP-SGD in solving composite DP-ERM problems.
problem Privacy-preserving machine learning with differential privacy.
method Differentially Private proximal Coordinate Descent (DP-CD) for composite Empirical Risk Minimization (ERM).
result DP-CD outperforms DP-SGD due to larger step sizes and better gradient exploitation.
Sharp pseudospectral bounds prevent transient amplification in coupled gradient descent.
problem Transient amplification in coupled gradient descent systems.
method Developed a sharp pseudospectral theory for block-triangular Jacobians, proving Kreiss constant bounds and matching minimax lower bounds.
result Obtained a finite-horizon iteration-complexity bound of O(K(J)2log(1/δ)) for stochastic coupled descent. We study --both in theory and practice-- the use of momentum motions in classic iterative hard thresholding (IHT) methods. By simply modifying plain IHT, we investigate its convergence behavior on convex optimization criteria with non-convex constraints, under standard assumptions. In diverse scenaria, we observe that …
Randomly sampled interpolators achieve zero generalization error with enough data.
problem Understanding the high generalization ability of machine learning models.
method Algebraic geometry tools to prove zero generalization error for random interpolators.
result Generalization error of randomly sampled interpolators becomes zero once the number of training samples exceeds a geometric threshold.
Binary Iterative Hard Thresholding converges with optimal number of 1-bit measurements.
problem Recovering sparse signals from 1-bit compressed measurements.
method Binary Iterative Hard Thresholding (BIHT) algorithm.
result BIHT converges with only O(k/ε) measurements, optimal for recovery.
Unified analysis of parameter norms in overparameterized linear models, revealing scaling laws and thresholds.
problem Understanding the scaling of parameter norms in overparameterized linear models.
method Simple dual-ray analysis revealing competition between signal spike and bulk of null coordinates.
result Unified closed-form predictions for parameter norm scaling, including elbow and threshold laws.
New insights link no-regret learning to online conformal prediction in adversarial settings.
problem Understanding the relationship between no-regret learning and online conformal prediction in adversarial environments.
method Analysis of existing algorithms and new connections between no-regret learning and conformal prediction.
result No-regret learning algorithms can provide group-conditional coverage guarantees in adversarial settings.
ECI improves time series prediction uncertainty quantification by smoothing miscoverage error.
problem Challenges in uncertainty quantification for time series prediction due to temporal dependence and distribution shift.
method Error-quantified Conformal Inference (ECI) by smoothing quantile loss function and introducing adaptive feedback scale.
result ECI achieves valid miscoverage control and tighter prediction sets than existing methods.
The MBO scheme for data clustering is analyzed in the large data limit, proving convergence to optimal partition problems.
problem Analyzing the MBO scheme for data clustering in the large data limit.
method Implicit gradient descent on the thresholding energy of a similarity graph.
result The MBO scheme outcomes converge to minimizers of a weighted optimal partition problem.
Deep ResNets can achieve zero loss with enough layers and weights.
problem Finding parameters in deep ResNets that fit training data perfectly.
method Mean-field analysis and gradient descent flow to a PDE.
result Gradient descent for ResNet training converges to a zero-loss solution in the large-NN limit.
GD iterates for non-homogeneous deep nets increase margin and converge in direction.
problem Understanding implicit bias in non-homogeneous deep networks.
method Characterization of GD iterates' properties starting from small empirical risk.
result GD iterates converge in direction despite diverging norms, satisfying KKT conditions.
This paper solves quadratic systems with sparse or generative priors.
problem Recovering signals from quadratic systems with full-rank matrices.
method Thresholded Wirtinger flow (TWF) and projected gradient descent (PGD) algorithms.
result The proposed methods significantly outperform existing algorithms in signal recovery.
This paper presents a methodology for selecting the mini-batch size that minimizes Stochastic Gradient Descent (SGD) learning time for single and multiple learner problems. By decoupling algorithmic analysis issues from hardware and software implementation details, we reveal a robust empirical inverse law between mini-…
Iterative hard thresholding (IHT) is a projected gradient descent algorithm, known to achieve state of the art performance for a wide range of structured estimation problems, such as sparse inference. In this work, we consider IHT as a solution to the problem of learning sparse discrete distributions. We study the hard…
Stochastic gradient descent (SGD) is commonly used for optimization in large-scale machine learning problems. Langford et al. (2009) introduce a sparse online learning method to induce sparsity via truncated gradient. With high-dimensional sparse data, however, the method suffers from slow convergence and high variance…
We consider the problem of sparsity-constrained M-estimation when both explanatory and response variables have heavy tails (bounded 4-th moments), or a fraction of arbitrary corruptions. We focus on the k-sparse, high-dimensional regime where the number of variables d and the sample size n are related through $…
SGD recovers multiple signal vectors in noisy tensor PCA.
problem Estimating multiple signal vectors from noisy tensor observations.
method Online stochastic gradient descent (SGD) in high dimensions with detailed analysis of correlations.
result Sequential elimination of correlations allows recovery of all spikes from Np−2 samples. Gradient method converges locally linearly for overparameterized Gaussian mixtures.
problem Learning Gaussian mixtures under overparameterization.
method Gradient-based method alternating short descent steps and long Polyak steps.
result Gradient method converges locally linearly to minimizers.