Gradient descent with large momentum finds flatter minima.
problem Understanding the effects of momentum in gradient descent.
method Empirical and theoretical analysis of gradient descent with large momentum.
result Large momentum leads to flatter minima than gradient descent.
SAM selects flatter minima late in training, improving generalization.
problem Improving neural network generalization under various settings.
method Sharpness-Aware Minimization (SAM) applied late in training.
result SAM efficiently selects flatter minima late in training, improving generalization.
mSAM improves generalization by making models flatter.
problem Over-parameterized models can have varying generalization performance.
method mSAM modifies loss function to favor flatter minima through adversarial perturbations.
result mSAM achieves flatter minima than SAM and SGD, improving generalization.
GD converges faster to flatter minima than gradient flow in shallow networks.
problem Understanding the dynamics of gradient descent in shallow linear networks.
method Analyzing the convergence rate and solution of gradient descent in depth-2 linear neural networks.
result GD converges linearly to flatter minima than gradient flow, even with large step sizes.
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.
Study shows deep linear networks can converge to flatter minima at large learning rates.
problem Understanding the implicit bias of deep linear networks at large learning rates.
method Characterization of deep linear networks for binary classification using logistic loss in the large learning rate regime.
result Gradient descent iterates converge to a flatter minimum in the catapult phase for certain data separation conditions.
New research shows flat minima in robust loss landscapes correlate with good adversarial robustness.
problem Adversarial training leads to robust overfitting, poor robust generalization.
method Average- and worst-case metrics to measure flatness in robust loss landscapes.
result Flatness in robust loss landscapes correlates with good adversarial robustness.
The paper analyzes how noise geometry influences the performance of SGD in machine learning.
problem Understanding how noise geometry affects the performance of stochastic gradient descent.
method Developed two metrics to quantify noise alignment strength and analyzed their effects on loss and subspace projection dynamics.
result Noise geometry can be used to guarantee alignment under certain conditions, aiding SGD's ability to escape from sharp minima.
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.
Paper confirms winning tickets have sharp minima, useful for generalization.
problem Explaining why over-parameterized models generalize well despite having sharp minima.
method PAC-Bayesian theory applied to analyze winning tickets and their generalization behavior.
result PAC-Bayesian theory confirms winning tickets have relatively sharp minima, a disadvantage for generalization.
Adaptor 'E' extends gradient-based optimizers to explore loss landscapes, improving generalization.
problem Finding lower and better-generalizing minima in deep learning.
method Proposes an adaptor 'E' to extend gradient-based optimizers, encouraging exploration along landscape valleys.
result Adapted optimizers increase test accuracy by an average of 2.5% in large-batch training tasks.
In this work we study generalization of neural networks in gradient-based meta-learning by analyzing various properties of the objective landscapes. We experimentally demonstrate that as meta-training progresses, the meta-test solutions, obtained after adapting the meta-train solution of the model, to new tasks via few…
Function-space MAP estimation leads to better generalization and robustness.
problem The mismatch between parameter posterior and function posterior in model training.
method Directly estimating the most likely function implied by the model and data.
result Function-space MAP estimation can lead to flatter minima, better generalization, and improved robustness.
It was empirically confirmed by Keskar et al.\cite{SharpMinima} that flatter minima generalize better. However, for the popular ReLU network, sharp minimum can also generalize well \cite{SharpMinimacan}. The conclusion demonstrates that the existing definitions of flatness fail to account for the complex geometry of Re…
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.
SGD batch size affects autoencoder global minima sparsity and sharpness.
problem Investigating how batch size impacts autoencoder learning.
method Non-convex autoencoder training with SGD, varying batch sizes.
result SGD batch size influences global minimum sparsity and sharpness.
Transformers are less sensitive to input perturbations compared to other models.
problem Understanding the inductive biases of transformers and distinguishing them from other architectures.
method Identified token-wise sensitivity as a metric to explain transformers' inductive biases across different data modalities.
result Transformers have lower sensitivity than MLPs, CNNs, ConvMixers, and LSTMs, across vision and language tasks.
This research explains why SGD generalizes better than ADAM in deep learning.
problem Understanding the generalization gap between SGD and ADAM in deep learning.
method Analyzing local convergence behaviors through Levy-driven stochastic differential equations (SDEs).
result SGD is more locally unstable and better escapes from sharp minima to flatter ones, leading to better generalization.
New model explains deep learning performance at large learning rates.
problem Understanding deep learning performance at different learning rates.
method Developed neural networks with solvable training dynamics.
result Large learning rates lead to convergence to flatter minima.
Stochastic gradient descent (SGD) is almost ubiquitously used for training non-convex optimization tasks. Recently, a hypothesis proposed by Keskar et al. [2017] that large batch methods tend to converge to sharp minimizers has received increasing attention. We theoretically justify this hypothesis by providing new pro…
Double descent found in DRL, improving generalization with model capacity.
problem Generalization in over-parameterized DRL models.
method Actor-Critic framework, Policy Entropy metric.
result Policy Entropy significantly reduces as model capacity increases, indicating improved generalization.
VL finds flatter solutions at edge of stability, matching theory with practice.
problem Understanding implicit regularization in deep learning.
method Edge of Stability framework, controlling variational posterior shape and sample number.
result VL finds even flatter solutions than gradient descent.
New findings show mini-batch SGD operates in a 'Edge of Stochastic Stability' regime.
problem Understanding the stability and convergence of mini-batch SGD.
method Analyzing the mini-batch Hessian and its directional curvature.
result Mini-batch SGD operates in a different stability regime (Edge of Stochastic Stability) compared to full-batch GD.
Quantum neural networks generalize better due to flatter parameter space.
problem Generalization in quantum neural networks.
method Mapped feature data to a quantum state, applied unitary evolution, and measured for classification.
result Quantum neural networks have better generalization than classical networks.
The pursuit of explaining and improving generalization in deep learning has elicited efforts both in regularization techniques as well as visualization techniques of the loss surface geometry. The latter is related to the intuition prevalent in the community that flatter local optima leads to lower generalization error…
The local geometry of high dimensional neural network loss landscapes can both challenge our cherished theoretical intuitions as well as dramatically impact the practical success of neural network training. Indeed recent works have observed 4 striking local properties of neural loss landscapes on classification tasks: …
Deep neural networks trained on a wide range of datasets demonstrate impressive transferability. Deep features appear general in that they are applicable to many datasets and tasks. Such property is in prevalent use in real-world applications. A neural network pretrained on large datasets, such as ImageNet, can signifi…
X-DER improves continual learning by revising replay memory and learning unseen classes.
problem Deep networks forget when learning new tasks sequentially.
method Combines rehearsal and Knowledge Distillation, revising replay memory and learning unseen classes.
result X-DER outperforms state-of-the-art on various benchmarks.
In several experimental reports on nonconvex optimization problems in machine learning, stochastic gradient descent (SGD) was observed to prefer minimizers with flat basins in comparison to more deterministic methods, yet there is very little rigorous understanding of this phenomenon. In fact, the lack of such work has…
Stochastic Gradient Descent (SGD) and its variants are mainstream methods for training deep networks in practice. SGD is known to find a flat minimum that often generalizes well. However, it is mathematically unclear how deep learning can select a flat minimum among so many minima. To answer the question quantitatively…
Complex-valued neural networks avoid spurious local minima.
problem Finding spurious local minima in neural networks.
method Proved no spurious local minima for shallow complex neural networks with quadratic activations.
result Complex-valued weights eliminate spurious local minima in neural networks.
Expressive quantum circuits are harder to train due to flatter cost landscapes.
problem Designing quantum circuits that are both expressive and trainable.
method Deriving a relationship between expressibility and gradient magnitude, extending barren plateau phenomenon.
result Highly expressive ansätze exhibit flatter cost landscapes, making them harder to train.
Paper finds wide minima are better for generalization and proposes a new learning rate schedule.
problem The challenge of finding optimal learning rates for model training.
method The paper introduces a new hypothesis about the density of wide minima and designs an explore-exploit learning rate schedule.
result The explore-exploit learning rate schedule improves model performance and reduces training time.
Optimizers find approximate global minima in non-convex problems.
problem Understanding why local methods solve non-convex optimization problems.
method Formalizing the hypothesis that many local minima are approximately global minima.
result Most local minima of practical non-convex objectives are approximately global minima.
The study analyzes local minima in ReLU networks and finds low probability of bad local minima.
problem Understanding the existence and probability of local minima in ReLU networks.
method Theoretical analysis combined with linear programming and experiments on MNIST and CIFAR-10 datasets.
result No bad differentiable local minima found almost everywhere in weight space.
In deep learning, \textit{depth}, as well as \textit{nonlinearity}, create non-convex loss surfaces. Then, does depth alone create bad local minima? In this paper, we prove that without nonlinearity, depth alone does not create bad local minima, although it induces non-convex loss surface. Using this insight, we greatl…
Paper proposes faster method to find local minima in nonconvex optimization.
problem Escaping saddle points and finding local minima in nonconvex optimization.
method LENA (Last stEp shriNkAge) framework for faster perturbed stochastic gradient methods.
result LENA finds (ε,εH)-approximate local minima within ildeO(ε−3+εH−6) evaluations. Global minima found for multidimensional scaling with penalties.
problem Finding global minima in multidimensional scaling.
method Combining stress loss function with a quadratic penalty term to find minimizers.
result Trajectory of minimizers leads to global minima.
Recent work has noted that all bad local minima can be removed from neural network loss landscapes, by adding a single unit with a particular parameterization. We show that the core technique from these papers can be used to remove all bad local minima from any loss landscape, so long as the global minimum has a loss o…
Proposes NRS to find flat minima in deep neural networks.
problem Finding optimal solutions in deep neural networks with overparameterization.
method NRS leverages the concept of flat minima and uses Kullback-Leibler divergence to regularize the neighborhood region in weight space.
result NRS drives optimizers towards flat minima, improving generalization ability across various model architectures.
Piecewise linear activations create many spurious local minima in neural networks.
problem Understanding the loss surface of neural networks with piecewise linear activations.
method Proved the existence of infinite spurious local minima and partitioned the loss surface into smooth cells.
result Piecewise linear activations create many spurious local minima that are invariant under a continuous path.
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.
The notion of flat minima has played a key role in the generalization studies of deep learning models. However, existing definitions of the flatness are known to be sensitive to the rescaling of parameters. The issue suggests that the previous definitions of the flatness might not be a good measure of generalization, b…
Study reveals sharp characterisation of local minima in neural network loss landscapes.
problem Characterizing local minima in high-dimensional two-layer ReLU neural networks.
method Exact low-dimensional representation of local minima using summary statistics and link with one-pass SGD dynamics.
result Local minima in overparameterized neural networks form discrete families with varying stability and reachability.
It is well known that (stochastic) gradient descent has an implicit bias towards flat minima. In deep neural network training, this mechanism serves to screen out minima. However, the precise effect that this has on the trained network is not yet fully understood. In this paper, we characterize the flat minima in linea…
New insights into hidden minima in neural networks.
problem Identifying hidden minima in two-layer ReLU networks.
method Analyzing curves along which loss is minimized, focusing on eigenvalue contributions.
result Distinctive structural and symmetry properties of arcs emanating from hidden minima.
In "Width complexes for knots and 3-manifolds," Jennifer Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in the 3-sphere and the isotopies between these positions. She poses several questions about these width complexes; in particular, she asks whether the…
Study of SGD with state-dependent noise, improving escape from local minima.
problem Understanding and improving the dynamics of SGD in non-convex optimization.
method Formal study on SGD with state-dependent noise, proposing power-law dynamic with state-dependent diffusion.
result Power-law dynamic can escape from sharp minima exponentially faster than flat minima.