Stochastic gradient descent procedures have gained popularity for parameter estimation from large data sets. However, their statistical properties are not well understood, in theory. And in practice, avoiding numerical instability requires careful tuning of key parameters. Here, we introduce implicit stochastic gradien…
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48 results for “Implicit Stochastic Gradient Descent”
Proposes a new algorithm for -means clustering using stochastic backward Euler.
problem Improving -means clustering performance and robustness.
method Implicit gradient descent with stochastic backward Euler iteration.
result The algorithm provides better clustering results compared to traditional -means.
(S)GD over Diagonal Linear Networks: Implicit Regularisation, Large Stepsizes and Edge of Stabilitycs.LG
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.
Study shows how SGD's implicit regularization relates to ridge regression.
problem Least squares regression optimization with mini-batch SGD.
method Analyzes stochastic gradient flow as a continuous-time model of SGD.
result Bound on excess risk of SGD flow over ridge regression, revealing how parameters drive risk.
This work studies the implicit bias of mini-batch SGD in classification.
problem Understanding the implicit bias of mini-batch SGD in multi-class classification.
method Characterizes how batch size, momentum, and variance reduction affect convergence and max-margin behavior under different norms.
result Momentum enables small-batch convergence to an approximate max-margin solution, while variance reduction recovers the exact full-batch bias.
Implicit Regularization or Implicit Conditioning? Exact Risk Trajectories of SGD in High Dimensionsstat.ML
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.
Implicit regularisation in stochastic gradient descent: from single-objective to two-player gamesstat.ML
Novel approach finds implicit regularisation in two-player games using BEA.
problem Understanding implicit regularisation in two-player games.
method Using backward error analysis to construct continuous-time flows with gradient-eligible vector fields.
result Identifies new implicit regularisation effects in two-player games.
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.
VRSMD improves SMD convergence and has implicit regularization.
problem Efficiently estimating models with large datasets.
method Variance reduction in stochastic mirror descent.
result VRSMD converges to the minimum mirror interpolant.
Implicit biases in multitask and continual learning from a backward error analysis perspectivestat.ML
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.
Gradient descent training of neural networks leads to solutions close to natural cubic splines.
problem Understanding the implicit bias of gradient descent in neural networks.
method Analysis of gradient descent training for wide neural networks, focusing on the curvature penalty and initialization schemes.
result The solutions of gradient descent training are polyharmonic splines for certain initialization schemes.
Dropout introduces both explicit and implicit regularization effects.
problem Understanding the full impact of dropout regularization.
method Disentangled explicit and implicit regularization effects through experiments and analytic simplifications.
result Explicit and implicit regularization effects of dropout are distinct and can be characterized analytically.
New implicit regularization drives deep networks towards simple models.
problem Training deep neural networks with noise.
method Stochastic gradient descent with perturbed labels, analyzing dynamics near zero-error parameters.
result The training dynamics are governed by an implicit regularization term, leading to simpler models.
Gradient descent and SGD can converge to max-margin directions in ReLU models.
problem Understanding the implicit bias of gradient methods in ReLU models.
method Characterization of loss function landscape, analysis of GD and SGD convergence, exploration of multi-neuron network learning.
result Gradient descent and SGD can converge to max-margin directions in ReLU models.
Stochastic descent methods achieve optimal performance in certain conditions.
problem Optimizing deep learning models for general loss functions and nonlinear models.
method Revisits minimax properties of stochastic gradient descent and mirror descent.
result Stochastic mirror descent (and gradient descent) is minimax optimal under certain conditions.
Paper introduces robust method for training neural networks.
problem Neural network hyperparameter tuning, particularly learning rate sensitivity.
method Implicit Stochastic Gradient Descent (ISGD) layer-wise approximation for neural networks.
result Our method is more robust to high learning rates and generally outperforms standard backpropagation.
We propose a fast algorithm for spectral embedding using stochastic gradient descent.
problem Scalability issue in spectral embedding due to eigendecomposition bottleneck.
method Reformulate spectral embedding as a stochastic optimization problem, replacing orthogonality constraint with an orthogonalization matrix.
result Efficient algorithm based on mini-batch gradient descent that outperforms existing techniques in execution speed.
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.
Proposes a semi-implicit back propagation method for neural networks.
problem Challenges in training neural networks, especially gradient vanishing and small step sizes.
method Proposes a semi-implicit back propagation method using error back propagation and proximal methods.
result The proposed method leads to better performance in terms of loss decreasing and training/validation accuracy compared to SGD and ProxBP.
Stochastic gradient descent approximates Gaussian process posteriors efficiently.
problem Efficiently sampling from Gaussian process posteriors with limited computational resources.
method Developed stochastic gradient optimization objectives for sampling from Gaussian process posteriors.
result Stochastic gradient descent produces accurate predictive distributions, even in non-convergent cases.
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…
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.
Gradient descent biases towards stable rank networks for nearly-orthogonal data.
problem Understanding implicit bias in non-smooth neural networks trained by gradient descent.
method Analysis of two-layer ReLU and leaky ReLU networks trained by gradient descent on nearly-orthogonal data.
result Gradient descent biases towards networks with stable rank and uniform margin for nearly-orthogonal data.
New variational approach to deep learning via gradient descent.
problem Non-robustness and poor out-of-distribution generalization in deep learning.
method Regularize variational neural networks using gradient descent's implicit bias.
result Strong in- and out-of-distribution performance achieved without additional hyperparameter tuning.
The paper discusses how to improve machine learning models using partial differential equations.
problem Improving the performance and generalization of machine learning models.
method The paper reframes implicit regularization techniques in deep learning as explicit gradient regularization using partial differential equations.
result Explicit regularization using PDEs can lead to better model performance and generalization.
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.
Gradient descent on matrix factorization leads to minimum nuclear norm solution.
problem Optimizing underdetermined quadratic objectives over matrices.
method Gradient descent on a full dimensional factorization of the matrix.
result Gradient descent converges to the minimum nuclear norm solution.
Deep networks generalize well without explicit regularization.
problem Generalization in over-parameterized neural networks.
method Gradient gap deviation and deflection analysis.
result Implicit regularization controls network output to keep interpolating between samples.
Gradient descent converges to a global minimum in nonlinear ReLU implicit networks with linear width.
problem Understanding convergence of gradient methods in nonlinear, infinitely deep ReLU networks.
method Introduced a scaling constant to ensure well-posedness of the equilibrium equation, proving convergence to a global minimum for linear width networks.
result Gradient descent converges to a global minimum at a linear rate for nonlinear ReLU implicit networks with linear width.
Geometric Occam's Razor shapes deep learning solutions.
problem Understanding the regularization in over-parameterized neural networks.
method Analyzing the geometric model complexity and Dirichlet energy in neural networks.
result Over-parameterized neural networks are implicitly regularized by geometric model complexity.
Lectures on deep learning from a learning theory perspective.
problem Understanding how deep learning architectures lead to inductive bias.
method Statistical learning theory and stochastic optimization.
result Gradient descent on linear diagonal networks can lead to various forms of implicit bias.
Gradient descent in tensor factorization favors low-rank solutions.
problem Tackling implicit regularization in tensor factorization problems.
method Gradient descent with small random initialization for overparametrized tensor factorization.
result Gradient descent leads to implicit regularization towards low tubal rank solutions.
Gradient descent implicitly favors group sparsity in neural networks.
problem Understanding implicit regularization in neural networks for structured sparsity.
method Novel neural reparameterization for diagonally grouped linear networks.
result Gradient descent without explicit regularization biases towards group sparsity.
Gradient descent with early stopping achieves optimal sparse recovery.
problem Sparse regression with gradient descent and early stopping.
method Gradient descent on depth-N networks with early stopping.
result Implicit sparse regularization occurs with early stopping for general depth N.
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.
This study investigates how gradient-based methods bias neural networks trained on high-dimensional data.
problem The implicit biases of gradient-based optimization algorithms in neural networks trained on high-dimensional data.
method Investigation of gradient flow and gradient descent in two-layer fully-connected neural networks with leaky ReLU activations.
result Gradient flow and gradient descent lead to neural networks with low-rank solutions and linear decision boundaries.
Quasi-potential as an implicit regularizer for the loss function in the stochastic gradient descentcs.LG
New quasi-potential function helps SGD navigate noisy optimization landscapes.
problem Optimizing noisy loss functions with SGD.
method Interpreted SGD as minimizing quasi-potential function, related to noise covariance structure via PDE.
result Anisotropic noise leads to faster escape from local minima.
Gradient descent in deep matrix factorization favors low-rank solutions, improving recovery accuracy.
problem Understanding the generalization in deep learning models.
method Study of gradient descent over deep linear neural networks for matrix completion and sensing.
result Adding depth enhances an implicit tendency towards low-rank solutions, leading to more accurate recovery.
Implicit Regularization Makes Overparameterized Asymmetric Matrix Sensing Robust to Perturbationscs.LG
Paper shows robustness of gradient descent in matrix sensing despite perturbations.
problem Understanding robustness of gradient descent in matrix sensing.
method Developed perturbed gradient flow to capture noise and improve robustness.
result Gradient descent is robust to perturbations in matrix sensing.
ADSGD method speeds up model identification in sparse optimization.
problem Implicit model identification in sparse optimization problems.
method Accelerated Doubly Stochastic Gradient Method (ADSGD) for faster explicit model identification.
result ADSGD achieves faster explicit model identification and improved algorithm efficiency.
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.
Study on how optimal representations emerge during deep learning training, focusing on the role of implicit regularization.
problem Understanding how optimal representations for tasks are learned during training.
method Investigates the role of implicit regularization in learning minimal sufficient representations, analyzing changes in representation content during training.
result Semantically meaningful but ultimately irrelevant information is encoded in early transient dynamics of training, which is later discarded.
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.
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.
This work investigates implicit bias in multiclass separable data using a novel geometry-aware optimizer.
problem Understanding implicit bias in overparameterized models on multiclass separable data.
method Introduces NucGD, a geometry-aware optimizer enforcing low-rank structures through nuclear norm constraints.
result NucGD enables scalable training and characterizes the impact of stochastic optimization dynamics.
Local Gradient Descent with local steps converges to the centralized model in the interpolation regime.
problem Understanding the implicit bias of Local Gradient Descent in the interpolation regime.
method Analyzing the implicit bias of Local Gradient Descent for classification tasks with linearly separable data.
result The aggregated global model from Local-GD converges exactly to the centralized model in the interpolation regime.
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…
MGD with early stopping tends to ridge regularization in least squares regression.
problem Characterizing the implicit regularization of MGD with early stopping.
method Continuous-time view of MGD (momentum gradient flow) and comparison with explicit ridge regularization.
result Under optimal tuning, the risk of MGF is no more than 1.54 times that of ridge.