Deep networks avoid bad local minima with no bad valleys.
problem Finding sub-optimal local minima in deep neural networks.
method Analyzing over-parameterized networks with specific activation and loss functions.
result No bad local valleys, implying no sub-optimal strict local minima.
Deep learning networks have connected sublevel sets, avoiding local minima.
problem Finding local minima in deep learning networks.
method Analyzing sublevel sets of loss functions in over-parameterized neural nets.
result Sublevel sets are connected and unbounded, ensuring all global minima are accessible.
ResNets with depth and nonlinearity avoid bad local minima.
problem Avoiding bad local minima in deep learning models.
method Proving depth and nonlinearity in ResNets create no bad local minima.
result ResNets with depth and nonlinearity have values no worse than global minimum and can improve further.
Quantum annealer improves classification of handwritten digits.
problem Classical MCMC struggles with missing labels in RBM models.
method Embedding RBM into D-Wave quantum annealer for classification.
result D-Wave QA reduces classification error by more than two times.
The paper shows deep neural networks have no bad local minima and no diverging paths to infinity.
problem The risk of diverging to infinity in deep neural networks.
method Mathematical analysis of regularizers and loss functions.
result For a large class of over-parameterized deep neural networks, the loss function has no bad local minima and no decreasing paths to infinity.
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…
New research finds deep networks often settle in asymmetric valleys, improving generalization.
problem Understanding the nature of local minima in deep neural networks.
method Formal analysis and empirical investigation of SGD trajectories and BN effects.
result SGD solutions biased towards flat sides of asymmetric valleys generalize better.
This paper explores loss landscapes of sparse neural networks, finding unique characteristics compared to dense networks.
problem Understanding the loss landscape of sparse neural networks, especially one-hidden-layer networks.
method Analyzes sparse networks with dense and sparse final layers, focusing on linear and non-linear models.
result Sparse networks can have no spurious valleys under certain conditions, but spurious valleys and minima can exist for wide sparse networks.
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.
Neural networks provide a rich class of high-dimensional, non-convex optimization problems. Despite their non-convexity, gradient-descent methods often successfully optimize these models. This has motivated a recent spur in research attempting to characterize properties of their loss surface that may explain such succe…
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.
Wide networks avoid bad local minima, proving phase transition.
problem Understanding the optimization landscape of neural networks.
method Proving phase transitions in loss surfaces of wide and narrow networks.
result Phase transition from narrow to wide networks: no bad local minima.
Reviews recent findings on neural network landscapes.
problem Non-convexity of loss functions causing bad landscapes.
method Rigorous geometric analysis and empirical exploration.
result Wide neural nets may have sub-optimal local minima.
Removes all bad local minima without adding extra units.
problem Neural network loss landscapes have bad local minima.
method Converts all bad local minima to non-local minima at infinity using auxiliary parameters.
result Removes all bad local minima from any loss landscape.
In this paper, we prove a conjecture published in 1989 and also partially address an open problem announced at the Conference on Learning Theory (COLT) 2015. With no unrealistic assumption, we first prove the following statements for the squared loss function of deep linear neural networks with any depth and any widths…
We investigate the loss surface of neural networks. We prove that even for one-hidden-layer networks with "slightest" nonlinearity, the empirical risks have spurious local minima in most cases. Our results thus indicate that in general "no spurious local minima" is a property limited to deep linear networks, and insigh…
There has been a lot of recent interest in trying to characterize the error surface of deep models. This stems from a long standing question. Given that deep networks are highly nonlinear systems optimized by local gradient methods, why do they not seem to be affected by bad local minima? It is widely believed that tra…
LoRA-Curve connects independent LoRA optima through continuous low-loss valleys, improving Bayesian model averaging.
problem Challenges in estimating epistemic uncertainty in LoRA-based Bayesian inference.
method Introduces LoRA-Curve, a segmented Bézier curve parameterization in the LoRA space, with free and anchored configurations.
result Empirically shows that connecting independent LoRA optima through continuous low-loss valleys improves mutual information of the predictive distribution.
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.
Looped transformers outperform standard transformers in complex reasoning tasks due to a specific loss landscape geometry.
problem Understanding why looped transformers outperform standard transformers in complex reasoning tasks.
method Explained through loss landscape geometry, distinguishing between U-shaped and V-shaped valleys, and proposing SHIFT training strategy.
result Looped transformers' recursive architecture induces a River-V-Valley landscape, leading to better loss convergence and complex pattern learning.
We provide two fundamental results on the population (infinite-sample) likelihood function of Gaussian mixture models with M≥3 components. Our first main result shows that the population likelihood function has bad local maxima even in the special case of equally-weighted mixtures of well-separated and spherical…
This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative eigenval…
D-Wave quantum annealing fails to improve sampling quality from RBMs compared to Gibbs sampling.
problem Improving sampling quality from RBMs using D-Wave quantum annealing.
method Comparison of D-Wave quantum annealing and Gibbs sampling for RBM sampling.
result D-Wave sampling does not significantly improve the number of local valleys compared to Gibbs sampling.
We present novel empirical observations regarding how stochastic gradient descent (SGD) navigates the loss landscape of over-parametrized deep neural networks (DNNs). These observations expose the qualitatively different roles of learning rate and batch-size in DNN optimization and generalization. Specifically we study…
Gradient-based methods find saddle points, not critical points, in neural networks.
problem Gradient-based optimization methods converge to saddle points rather than critical points in deep neural networks.
method Critical point-finding methods used to analyze neural network losses.
result Gradient-based methods often converge to or pass through gradient-flat regions, where gradient norm has a stationary point.
One of the main difficulties in analyzing neural networks is the non-convexity of the loss function which may have many bad local minima. In this paper, we study the landscape of neural networks for binary classification tasks. Under mild assumptions, we prove that after adding one special neuron with a skip connection…
Quantization-aware training can recover accuracy lost by post-training quantization.
problem Post-training quantization (PTQ) can fail sharply at aggressive bitwidths.
method A unified geometric framework that explains PTQ failure and QAT recovery.
result QAT has a useful bias that steers iterates back into the basin.
We use smoothed analysis techniques to provide guarantees on the training loss of Multilayer Neural Networks (MNNs) at differentiable local minima. Specifically, we examine MNNs with piecewise linear activation functions, quadratic loss and a single output, under mild over-parametrization. We prove that for a MNN with …
SGD can converge to bad models if not explicitly regularized.
problem Overparameterized neural networks can converge to poorly-generalizing models.
method Train with random labels first, then correct labels; observe SGD behavior.
result SGD can reach bad global minima if not explicitly regularized.
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.
LLE produces unwanted results without regularization, which can be prevented with regularization.
problem LLE's inherent unwanted results without regularization.
method Mathematical proof and numerical examples of regularization effectiveness.
result Regularization prevents unwanted results in LLE.
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.
New proof shows deep neural nets can have sub-optimal local minima.
problem Can over-parameterization eliminate sub-optimal local minima in deep neural networks?
method Counter-example with generic input data and non-linear activation functions.
result Sub-optimal local minima exist in deep neural networks regardless of width.
A new pruning method reduces neural network computation without retraining.
problem Efficiently reduce neural network computation while maintaining accuracy.
method Structured directional pruning via perturbation orthogonal projection.
result Achieves state-of-the-art pruned accuracy without retraining.
A new Kolmogorov-Arnold network improves function approximation and optimization.
problem Approximating potentially irregular functions in high dimensions.
method Proposes a new Kolmogorov-Arnold network (KAN) and provides error bounds and universal approximation theorems.
result Outperforms multilayer perceptrons in accuracy and convergence speed for irregular functions.
This paper compares EM and GD in two-component mixture models, finding EM escapes bad local optima more reliably.
problem Understanding the convergence of EM and GD in mixture models, especially in regions where one component is missing.
method Analyzing regions called one-cluster regions in two-component mixture models of Gaussians and Bernoullis, comparing the propensity of EM and GD to converge to these regions.
result EM escapes one-cluster regions exponentially fast, while GD escapes them linearly fast, indicating EM is less likely to converge to bad local optima.
We detect and quantify asymmetries in volatility spillovers using the realized semivariances of petroleum commodities: crude oil, gasoline, and heating oil. During the 1987--2014 period we document increasing spillovers from volatility among petroleum commodities that substantially change after the 2008 financial crisi…
Deep networks exhibit permutation saddles and valleys between equivalent minima.
problem Understanding the structure of loss landscapes in deep neural networks.
method Geometric approach to constructing paths between equivalent minima and saddle points.
result Existence of permutation saddles and valleys in deep neural networks.
New technique improves imitation learning by preventing local minima and exploring states.
problem Behavioral cloning gets stuck in local minima and lacks effective exploration.
method Two-phase model with sampling mechanisms and self-attention modules.
result Significantly outperforms previous state-of-the-art in various environments.
This paper proves adding neurons eliminates all bad local minima in deep learning.
problem Eliminating all suboptimal local minima in deep learning models.
method Adding one special neuron per output unit eliminates all suboptimal local minima of deep neural networks.
result At every local minimum, the original neural network parameters are a global minimum.
For the problem of high-dimensional sparse linear regression, it is known that an ℓ0-based estimator can achieve a 1/n "fast" rate on the prediction error without any conditions on the design matrix, whereas in absence of restrictive conditions on the design matrix, popular polynomial-time methods only guarante…
Reservoir computing minimizes prediction error in a spectral radius interval.
problem Lack of guiding principles for neural network parameters.
method Model-free prediction of spatiotemporal dynamical systems using recurrent neural networks.
result A spectral radius interval minimizes prediction error for nonlinear dynamical systems.
Stablecoin liquidity was affected by the SVB collapse, with USDC's transparency leading to market reactions.
problem Impact of stablecoin transparency on liquidity during market turmoil.
method Adapted MCI measure to Uniswap, Difference-in-Differences analysis on MCI and TVL, measured liquidity concentration.
result USDC's transparency led to swift market reactions, while USDT's opacity provided a safety net.
This paper examines SVB's failure and its impact on bank stocks.
problem SVB failure and its contagion effects on bank stocks.
method Analyzed bank-specific vulnerabilities and stock performance.
result Uninsured deposits and unrealized losses were key factors in SVB's impact.
Improves EM algorithm for better local optima in mixture models.
problem EM algorithm's sensitivity to initialization and bad local optima.
method Big Learning principle applied to upgrade EM algorithm.
result BigLearn-EM delivers optimal solution with high probability.
PyBADS optimizes complex functions quickly and reliably.
problem Optimizing rough, noisy, and expensive functions with unknown gradients.
method Bayesian Adaptive Direct Search (BADS) algorithm.
result PyBADS performs well on both artificial and real-world problems.
A simple modification improves GAN performance by discarding bad samples.
problem Improving GAN performance with minimal computational cost.
method Top-k update procedure: zero out gradient contributions from least realistic elements.
result Significant improvement in FID score for conditional generation on CIFAR-10.
Proposes a method to partition univariate data into unimodal subsets.
problem Partitioning univariate multimodal data into unimodal subsets.
method Recursive splitting around valley points of the data density using properties of critical points on the convex hull of the ecdf plot.
result Obtains a hierarchical statistical model of the initial dataset as a mixture of UMMs.