Gradient matching method estimates implicit regularization in complex deep learning systems.
problem Estimating implicit regularization in modern deep learning systems with complex modifications.
method Gradient matching methods to empirically estimate implicit regularization.
result Empirical estimation of implicit regularization in arbitrary networks, including dropout.
The paper explains implicit regularization in hierarchical tensor factorization and deep CNNs.
problem Understanding implicit regularization in complex neural network architectures.
method Theoretical analysis using dynamical systems to overcome challenges in hierarchy.
result Established implicit regularization towards low hierarchical tensor rank, equivalent to locality in CNNs.
Deep tensor factorization benefits from implicit regularization with polynomial growth.
problem Tensor factorization's implicit regularization effect in deep networks is not well understood.
method Investigated the implicit regularization in deep tensor factorization, showing polynomial growth.
result Implicit regularization in deep tensor factorization grows polynomially with depth, improving estimation accuracy and convergence.
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.
Gradient descent on ReLU networks with square loss implicitly favors balanced weights.
problem Understanding implicit regularization in nonlinear neural networks with regression losses.
method Analyzing gradient descent dynamics on ReLU networks with square loss.
result It is impossible to characterize the implicit regularization of ReLU networks with square loss by any explicit function of model parameters.
The paper analyzes implicit regularization in tensor factorization using neural networks.
problem Understanding implicit regularization in tensor factorization.
method Dynamical systems perspective and gradient descent analysis.
result Gradient descent induces a form of greedy low tensor rank search.
In this paper, we revisit implicit regularization from the ground up using notions from dynamical systems and invariant subspaces of Morse functions. The key contributions are a new criterion for implicit regularization---a leading contender to explain the generalization power of deep models such as neural networks---a…
AIR-Net adapts low-rank regularization dynamically for better image completion.
problem Fixed low-rank regularization limits adaptability to different images.
method AIR-Net uses adaptive and implicit regularization parameterized by a dynamic Laplacian matrix.
result AIR-Net enhances implicit regularization and outperforms fixed methods in non-uniform missing data scenarios.
Efforts to understand the generalization mystery in deep learning have led to the belief that gradient-based optimization induces a form of implicit regularization, a bias towards models of low "complexity." We study the implicit regularization of gradient descent over deep linear neural networks for matrix completion …
New proof shows norms can't explain deep learning's implicit regularization.
problem Understanding the implicit regularization in deep learning.
method Mathematical proof on matrix factorization problems.
result Implicit regularization drives norms towards infinity, suggesting rank minimization is key.
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 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.
FA algorithm provides convergence guarantees for deep linear networks.
problem Training efficiency and convergence of deep neural networks.
method Theoretical analysis of Feedback Alignment (FA) algorithm for deep linear networks.
result Certain initializations lead to implicit anti-regularization, affecting learning effectiveness.
New insights into network generalization show learning rate affects both norm and sharpness.
problem Understanding the generalization of overparameterized networks.
method Empirical analysis and theoretical proof of the trade-off between norm and sharpness.
result Learning rate influences both norm and sharpness, neither alone minimizes generalization error.
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.
Combining explicit and implicit regularization improves deep learning performance without needing depth.
problem Improving deep learning performance without increasing model complexity.
method Proposes an explicit penalty to mirror implicit regularization bias in adaptive gradient optimizers.
result Single-layer networks can achieve low-rank approximations with similar performance to deep linear networks.
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.
Tuning-free OR-PCA improves scalability for large datasets.
problem Dataset sensitivity of OR-PCA tuning parameters.
method Implicit regularization through modified gradient descents.
result Comparable or better performance on simulated and real-world datasets.
Paper explores how DPP sampling can implicitly regularize kernel regression.
problem Improving kernel regression by reducing redundancy in data.
method Using Determinantal Point Processes (DPPs) to sample subsets implicitly regularizes ridgeless Kernel Regression.
result Ensemble of ridgeless regressors can be effective for datasets with redundant information.
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.
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.
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.
We consider whether algorithmic choices in over-parameterized linear matrix factorization introduce implicit regularization. We focus on noiseless matrix sensing over rank-r positive semi-definite (PSD) matrices in Rn×n, with a sensing mechanism that satisfies restricted isometry properties (RIP)…
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.
New model leads to optimal test loss in sparse linear regression.
problem Sparse linear regression with low test loss despite interpolating training data.
method Developed a new parametrization of the model that combines benefits of ℓ1 and ℓ2 norms.
result Training via gradient descent leads to an interpolator with near-optimal test loss.
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.
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 studies implicit regularization in over-parameterized models for high-dimensional data.
problem Understanding implicit regularization in over-parameterized models for high-dimensional data.
method The paper designs regularization-free algorithms for the high-dimensional single index model and provides theoretical guarantees for the induced implicit regularization phenomenon.
result The proposed methods achieve minimax optimal statistical rates of convergence and outperform classical methods with explicit regularization.
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.
This work improves distribution recovery from sparse data using Random Forest implicit regularization.
problem Distribution recovery from limited statistics.
method Closed-form estimator for scaled beta distributions, using composite quantile and moment matching.
result Improved classification accuracy through closed-form distribution recovery and implicit regularization.
We consider networks, trained via stochastic gradient descent to minimize ℓ2 loss, with the training labels perturbed by independent noise at each iteration. We characterize the behavior of the training dynamics near any parameter vector that achieves zero training error, in terms of an implicit regularization te…
New method for high-fidelity shape representations from raw data.
problem Creating accurate shape representations from raw data.
method A simple loss function encouraging neural network to vanish on input point cloud and have unit norm gradient.
result Our method produces high-fidelity, smooth, and natural zero level set surfaces.
Deep (neural) networks have been applied productively in a wide range of supervised and unsupervised learning tasks. Unlike classical machine learning algorithms, deep networks typically operate in the \emph{overparameterized} regime, where the number of parameters is larger than the number of training data points. Con…
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.
Adam's hyperparameters implicitly regularize solutions, penalizing or impeding loss gradients' norms.
problem Implicit regularization in Adam's hyperparameters and training stage.
method Backward error analysis and ODE approximations to study Adam's behavior.
result Adam's implicit regularization depends on hyperparameters and training stage, involving different norms.
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.
Over-parameterized neural networks generalize well in practice without any explicit regularization. Although it has not been proven yet, empirical evidence suggests that implicit regularization plays a crucial role in deep learning and prevents the network from overfitting. In this work, we introduce the gradient gap d…
We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix X with gradient descent on a factorization of X. We conjecture and provide empirical and theoretical evidence that with small enough step sizes and initialization close enough to the origin, gradient descent on a f…
Many statistical estimators for high-dimensional linear regression are M-estimators, formed through minimizing a data-dependent square loss function plus a regularizer. This work considers a new class of estimators implicitly defined through a discretized gradient dynamic system under overparameterization. We show that…
New algorithm expands FTRL framework with improved worst-case regret bounds.
problem Online learning with improved worst-case regret bounds.
method Generalized implicit Follow-The-Regularized-Leader (FTRL) algorithm.
result Unified framework for designing updates improving worst-case regret bounds.
We introduce Implicit Policy, a general class of expressive policies that can flexibly represent complex action distributions in reinforcement learning, with efficient algorithms to compute entropy regularized policy gradients. We empirically show that, despite its simplicity in implementation, entropy regularization c…
Study shows how networks converge to minimum norm solutions with regularization.
problem Interpolating between known regions in shallow ReLU networks.
method Investigates empirical risk minimizers and weight decay regularizers.
result Empirical risk minimizers converge to minimum norm interpolants under specific conditions.
Study accelerates gradient methods in machine learning, revealing risk and stability connections.
problem Understanding the statistical risk of accelerated gradient methods in machine learning.
method Continuous-time analysis of Nesterov's accelerated gradient method and Polyak's heavy ball method for least squares regression.
result Connections between early stopping, stability, and curvature of loss function are revealed.
We reparametrize ReLU NNs as splines to understand their learning dynamics.
problem Understanding the learning dynamics and inductive bias of neural networks.
method Reparametrize ReLU NNs as continuous piecewise linear splines to study learning dynamics.
result Standard weight initializations yield very flat functions, leading to strength and type of implicit regularization.
Regularization effect found in neural feature alignment.
problem Implicit regularization in deep learning models.
method Geometrical viewpoint and analysis of Rademacher complexity.
result Neural features align along task-relevant directions, leading to regularization.
A tuning-free method recovers jointly sparse signals in MMV using implicit regularization.
problem Recovering jointly sparse signals in MMV with minimal tuning or prior knowledge.
method Reparameterizes MMV estimation matrix into decoupled factors and applies gradient descent to a least-squares objective.
result Gradient descent dynamics exhibit a momentum-like effect, converging towards an idealized row-sparse solution.
The paper analyzes how gradient descent implicitly regularizes solutions in overparameterized neural networks, revealing depth-dependent regularization effects.
problem Understanding implicit regularization in overparameterized linear neural networks for regression problems.
method Analyzing the approximation error between gradient flow limit points and ℓ1-minimization solutions, deriving tight upper and lower bounds. result The approximation error decreases linearly for D≥3 and at a slower rate for D=2, linked to null space property constants. Noise in RNNs promotes flatter minima and more stable dynamics.
problem Understanding and optimizing the training of RNNs with noise.
method Formalizing RNNs as stochastic differential equations and analyzing the effect of noise in the hidden states.
result Noise injection in RNNs leads to flatter minima, more stable dynamics, and improved robustness.