SGD favors flat minima exponentially more than sharp minima in deep learning.
problem Understanding how SGD selects flat minima in deep learning.
method Developed a density diffusion theory (DDT) to analyze minima selection.
result SGD exponentially favors flat minima over sharp minima due to Hessian-dependent noise.
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.
This paper explains why Adam generalizes worse than SGD by analyzing its components.
problem Understanding why Adam generalizes worse than Stochastic Gradient Descent (SGD).
method Diffusion theoretical framework to disentangle the effects of Adaptive Learning Rate and Momentum.
result Adaptive Learning Rate helps escape saddle points but not select flat minima, while Momentum provides a drift effect to help pass through saddle points.
SGD noise helps select flat minima by concentrating in sharp directions and being proportional to loss value.
problem Understanding the implicit regularization of SGD and selecting flat minima in over-parameterized models.
method Relating SGD's linear stability to the Frobenius norm of the Hessian and analyzing the alignment property of SGD noise.
result Flat minima are linearly stable for SGD, and their sharpness is bounded independently of model size and sample size.
Continuous-time analysis shows SGD with noise prefers flat minima.
problem Optimizing neural networks using SGD with noise.
method Continuous-time model for SGD with noise analysis.
result Optimization prefers flat minima in certain noise regimes.
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.
Laplace kernel feature selection offers statistical guarantees for nonparametric models with few samples.
problem Statistical guarantees for kernel-based feature selection in nonconvex optimization problems.
method Sharp characterization of the gradient of the objective function for Laplace kernel feature selection.
result Model-selection consistency for Laplace kernel-based feature selection in nonparametric settings with n∼logp samples. In Deep Learning, Stochastic Gradient Descent (SGD) is usually selected as a training method because of its efficiency; however, recently, a problem in SGD gains research interest: sharp minima in Deep Neural Networks (DNNs) have poor generalization; especially, large-batch SGD tends to converge to sharp minima. It bec…
New rules found to maintain neural network performance in asynchronous training.
problem Asynchronous training leads to degradation in generalization.
method Examined dynamical stability, derived rules for learning rate adjustment.
result Learning rate should be inversely proportional to delay for high delay values.
Study of local optima in neural networks for feature interactions.
problem NNs struggle with local optima in feature interactions for small datasets.
method Proposed a node pruning and feature selection algorithm to improve NN performance.
result NNs have many non-equivalent local optima in XOR-like data with irrelevant variables.
New framework learns sufficient invariant features robustly across distribution shifts.
problem Learning robust models under distribution shifts between training and test datasets.
method Sufficient Invariant Learning (SIL) framework and Adaptive Sharpness-aware Group Distributionally Robust Optimization (ASGDRO) algorithm.
result Empirical evaluations confirm ASGDRO's robustness against distribution shifts.
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.
New method smooths optimization for sparse regularization.
problem Non-smooth, non-convex optimization problems for sparsity.
method Overparameterization and smooth surrogate penalties.
result Surrogate objective has identical global and local minima.
This paper interprets critical scales in persistent homology for compact metric spaces.
problem Understanding critical scales in persistent homology for general compact metric spaces.
method Analyzing local minima of the distance function and their impact on persistence.
result Each decrease in zero-dimensional persistence and increase in one-dimensional persistence is induced by local minima of the distance function.
Study reveals properties of local minima in ReLU networks.
problem Understanding the loss landscape of neural networks.
method Theoretical analysis of one-hidden-layer ReLU networks.
result All differentiable local minima are global within certain regions.
Unified approach to characterize and regularize deep neural network local minima.
problem Characterize and improve generalizability of deep neural network local minima.
method Information-theoretic Fisher information metric for local minima characterization and regularization.
result Unified approach successfully characterizes and improves generalizability of DNNs.
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.
Gradient descent in deep networks tends to find flat minima, which are nearly balanced.
problem Understanding the effect of gradient descent on the structure of minima in deep neural networks.
method Characterized flat minima in linear neural networks trained with a quadratic loss.
result Flat minima correspond to nearly balanced networks where the gain from input to intermediate representations is nearly constant.
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.
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.
New optimization method helps models generalize better after achieving near-perfect training performance.
problem Models can achieve near-perfect training performance but fail to generalize well to unseen examples.
method GROKtimizer combines rapid convergence to interpolation with post-interpolation norm minimization using Critically Damped Momentum.
result GROKtimizer provides a quadratic speedup over classical gradient descent, offering a natural solution for selecting low-norm interpolating solutions.
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. Selecting diverse and important items, called landmarks, from a large set is a problem of interest in machine learning. As a specific example, in order to deal with large training sets, kernel methods often rely on low rank matrix Nyström approximations based on the selection or sampling of landmarks. In this context, …
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.
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.
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.
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…
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.
We present a new approach to learning the structure and parameters of a Bayesian network based on regularized estimation in an exponential family representation. Here we show that, given a fixed variable order, the optimal structure and parameters can be learned efficiently, even without restricting the size of the par…
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.
We continue the comparison between lines of minima and Teichmueller geodesics begun in [CRS1]. We show that in the Teichmueller space of a surface S, lines of minima are quasi-geodesic with respect to the Teichmueller metric. The quasi-geodesic constants depend only on the topological type of S.
We consider deep linear networks with arbitrary convex differentiable loss. We provide a short and elementary proof of the fact that all local minima are global minima if the hidden layers are either 1) at least as wide as the input layer, or 2) at least as wide as the output layer. This result is the strongest possibl…
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.
Deep ReLU networks with extra parameters have mostly good loss landscapes.
problem Finding good local minima in the loss landscape of deep neural networks.
method Analyzing shallow and deep ReLU networks with extra parameters on a generic dataset.
result Most activation patterns correspond to regions with no bad local minima.
In this paper, we theoretically prove that adding one special neuron per output unit eliminates all suboptimal local minima of any deep neural network, for multi-class classification, binary classification, and regression with an arbitrary loss function, under practical assumptions. At every local minimum of any deep n…
New findings suggest non-contrastive learning has many bad minima, not just collapsed ones.
problem The effectiveness of non-contrastive learning in unsupervised feature learning.
method Theoretical analysis and controlled experiments on simple data models.
result Non-contrastive losses have a preponderance of non-collapsed bad minima, and these minima are not avoided during training.
Zeroth-order methods favor flat minima in machine learning.
problem Finding solutions with small Hessian trace in optimization.
method Zeroth-order optimization with two-point estimator.
result Zeroth-order optimization converges to flat minima.
Solves local minima problems on smooth manifolds.
problem Local minima issues on smooth manifolds.
method Introducing valley functions and applying Morse's lemma.
result Eliminates critical points and reduces to 1D.
We define lines of minima in the thick part of Outer space for the free group Fn with n>2 generators. We show that these lines of minima are contracting for the Lipschitz metric. Every fully irreducible outer automorphism of Fn defines such a line a minima. Now let G be a subgroup of the outer automorphism group of Fn …
New bounds link flat minima to good generalisation in overparameterized models.
problem Understanding the relationship between flat minima and generalisation in overparameterized machine learning models.
method Combining PAC-Bayes, Poincaré, and Log-Sobolev inequalities to derive generalisation bounds involving gradient terms.
result Flat minima positively influence generalisation performance, highlighting the benefits of the optimisation phase.
Paper develops a new local convexity condition for non-isolated minima in non-convex optimization.
problem Lack of theory for non-isolated minima in non-convex optimization.
method Formulates a new local convexity condition and studies SGD convergence under this condition.
result Shows SGD can converge locally under the new condition.
Black holes offer insights into machine learning's loss landscapes.
problem Understanding the loss landscape in machine learning.
method Comparing machine learning loss landscapes to black hole entropy.
result Black holes provide an infinite family of potential landscapes with known minima.