The study examines when MAML's objective has a benign landscape.
problem Understanding when MAML's objective landscape is benign.
method Analyzing the landscape of MAML objective on LQR tasks.
result The benign landscape of the MAML objective depends on task similarities.
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.
This work justifies neural collapse under MSE loss and analyzes the optimization landscape.
problem Understanding neural collapse in deep neural networks under MSE loss.
method Global landscape analysis of vanilla nonconvex MSE loss.
result The only global minimizers are neural collapse solutions.
Paper characterizes optimization landscape of Tucker decomposition.
problem Finding exact Tucker decomposition is a nonconvex optimization problem.
method Characterized the optimization landscape and provided a local search algorithm.
result All local minima are globally optimal if tensor has an exact Tucker decomposition.
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.
We propose a general theory for studying the \xl{landscape} of nonconvex \xl{optimization} with underlying symmetric structures \tz{for a class of machine learning problems (e.g., low-rank matrix factorization, phase retrieval, and deep linear neural networks)}. In specific, we characterize the locations of stationary …
We solve the optimization of two-layer ReLU networks using convex math.
problem Optimizing two-layer ReLU neural networks.
method Exact characterization of optimal solutions via convex optimization.
result We prove that all globally optimal solutions can be found via convex optimization.
The past decade has witnessed a successful application of deep learning to solving many challenging problems in machine learning and artificial intelligence. However, the loss functions of deep neural networks (especially nonlinear networks) are still far from being well understood from a theoretical aspect. In this pa…
New method tackles rugged optimization landscapes in contact-rich scenarios.
problem Optimization challenges in dynamic environments with deformable objects.
method Combines Bayesian optimization with semi-local 'leaps' for global search.
result Outperforms gradient-based and gradient-free baselines in simulation and real robot experiments.
A barrier to the wider adoption of neural networks is their lack of interpretability. While local explanation methods exist for one prediction, most global attributions still reduce neural network decisions to a single set of features. In response, we present an approach for generating global attributions called GAM, w…
Hill-ADAM optimizes loss landscapes by exploring state space deterministically.
problem Escaping local minima in loss landscapes.
method Hill-ADAM alternates between minimizing and maximizing error to explore the loss space.
result Hill-ADAM finds the global minimum state in loss landscapes.
SGD with machine learning noise converges to global minimum exponentially fast.
problem Optimizing machine learning models with stochastic gradient descent.
method Analysis of SGD with machine learning noise, focusing on energy landscapes and gradient noise.
result SGD converges to the global minimum exponentially fast under certain conditions.
Due to the success of deep learning to solving a variety of challenging machine learning tasks, there is a rising interest in understanding loss functions for training neural networks from a theoretical aspect. Particularly, the properties of critical points and the landscape around them are of importance to determine …
New findings on optimization landscape of Toeplitz covariance estimation.
problem Understanding the geometry of the Gaussian maximum-likelihood objective for Toeplitz covariance estimation.
method Overparameterized Carathéodory representation of positive definite Toeplitz covariance matrices, focusing on both amplitudes and frequencies.
result Joint optimization of amplitudes and frequencies leads to a benign population landscape, allowing for global recovery of the true Toeplitz covariance.
Optimizes MMD learning for generative models with theoretical guarantees.
problem Theoretical guarantees for optimizing non-convex MMD objectives.
method Analyzes MMD optimization landscape for specific distributions.
result Gradient-based methods globally minimize MMD objective for certain distributions.
SimStock learns stock similarities for better investment management.
problem Challenges in identifying similar stocks due to non-stationary financial markets.
method Temporal self-supervised learning framework combining SSL and temporal domain generalization.
result SimStock outperforms existing methods in finding similar stocks.
We analyze the loss landscape and expressiveness of practical deep convolutional neural networks (CNNs) with shared weights and max pooling layers. We show that such CNNs produce linearly independent features at a "wide" layer which has more neurons than the number of training samples. This condition holds e.g. for the…
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.
Deeper models have a more favorable optimization landscape, making them more robust to noise.
problem Characterizing the effect of depth on the optimization landscape of linear regression models.
method Robust and over-parameterized setting, simple sub-gradient method.
result A simple sub-gradient method converges to a balanced solution that is close to the ground truth and enjoys a flat local landscape.
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…
In this paper we develop a new framework that captures the common landscape underlying the common non-convex low-rank matrix problems including matrix sensing, matrix completion and robust PCA. In particular, we show for all above problems (including asymmetric cases): 1) all local minima are also globally optimal; 2) …
Paper shows no spurious local minima in a specific matrix factorization problem.
problem Optimization of ℓ1-norm rank-one symmetric matrix factorization. method Second-order variational analysis to study the landscape of the problem.
result Any second-order stationary point is globally optimal.
This paper introduces a scalable benchmark for evaluating local posterior sampling in neural networks.
problem Degeneracy in neural network loss landscapes and its impact on SGMCMC algorithms.
method Development of a scalable benchmark for local posterior sampling.
result RMSProp-preconditioned SGLD is most effective at representing the local geometry of the posterior distribution.
Over-parameterization makes optimization easier for simple neural networks, even with minor extra neurons.
problem Understanding the impact of over-parameterization on optimization landscapes of shallow neural networks.
method Analyzing a simple ReLU neural network with Gaussian inputs, focusing on optimization properties and landscape changes.
result Over-parameterization makes the objective function one-point strongly convex in most directions, aiding optimization.
Gradient descent converges to perfect classification in neural nets for non-separable data.
problem Classifying linearly non-separable data using neural networks.
method Analysis of gradient descent dynamics in neural networks with sufficient but not large number of neurons.
result Gradient descent converges to global minima with perfect classification in the landscape of minimization problems.
Embedding principle explains loss landscape of deep neural networks.
problem Understanding the structure of loss landscapes in deep neural networks.
method Proposed an embedding principle that critical points of narrower DNNs can be embedded to critical points of wider DNNs.
result Wide DNNs are often attracted by highly-degenerate critical points embedded from narrower DNNs.
Proves existence of optimal shallow neural networks with ReLU activation.
problem Proving the existence of optimal shallow feedforward networks with ReLU activation.
method Proves existence of global minima in the loss landscape for continuous target functions using shallow feedforward neural networks with ReLU activation.
result Existence of global minima in the loss landscape for shallow feedforward networks with ReLU activation.
Neural collapse occurs in normalized features over a Riemannian manifold.
problem Understanding neural collapse in normalized feature models.
method Simplified multi-class classification task to a nonconvex optimization problem over the Riemannian manifold, analyzing the landscape of critical points.
result The only global minimizers are neural collapse solutions, with all other critical points being strict saddles.
EM algorithm converges to global max in latent Gaussian tree models.
problem Optimizing log-likelihood in latent Gaussian tree models.
method Analyzed the optimization landscape and convergence of EM algorithm.
result EM algorithm converges to global maximum in latent Gaussian tree models.
Analyzes minima of deep linear networks with weight decay.
problem Understanding the loss landscape of deep neural networks.
method Analytical solutions for global minima with weight decay and stochastic neurons.
result The origin is a special point with qualitatively different minima in networks with more than 1 hidden layer.
We explore some mathematical features of the loss landscape of overparameterized neural networks. A priori one might imagine that the loss function looks like a typical function from Rn to R - in particular, nonconvex, with discrete global minima. In this paper, we prove that in at least one impo…
We study nonconvex optimization landscapes for learning overcomplete representations, including learning (i) sparsely used overcomplete dictionaries and (ii) convolutional dictionaries, where these unsupervised learning problems find many applications in high-dimensional data analysis. Despite the empirical success of …
WSFN overcomes saddle points for non-convex functionals in Wasserstein space.
problem Minimizing non-convex functionals over the Wasserstein space with saddle point avoidance.
method WSFN is a second-order method that preconditions the Wasserstein gradient to avoid saddle points.
result WSFN escapes saddle regions and reaches a global minimizer in polynomial time.
The paper analyzes phase retrieval under limited samples, ensuring a benign local landscape for convergence.
problem Ensuring a benign local landscape for phase retrieval under limited samples.
method Fine-grained analysis of local landscape properties under the regime of limited samples.
result Gradient descent can converge to an od(1)-loss solution exponentially fast under certain conditions. Study shows how deep residual networks can be analyzed as shallow network ensembles for optimization.
problem Understanding why deep neural networks can be trained to zero loss despite non-convex optimization landscapes.
method Mean-field analysis of deep residual networks, focusing on their continuum limit as a two-layer network.
result Derives the first global convergence result for multilayer neural networks in the mean-field regime.
SGD vs quasi-Newton optimization in neural networks: different landscapes, different generalizability.
problem Understanding neural network optimization and generalizability.
method Comparison of stochastic gradient descent (SGD) and quasi-Newton optimization methods using computational tools.
result SGD solutions are separated by lower barriers than quasi-Newton solutions, but quasi-Newton solutions are deeper and more isolated.
In this paper we study the problem of learning a shallow artificial neural network that best fits a training data set. We study this problem in the over-parameterized regime where the number of observations are fewer than the number of parameters in the model. We show that with quadratic activations the optimization la…
Despite their practical success, a theoretical understanding of the loss landscape of neural networks has proven challenging due to the high-dimensional, non-convex, and highly nonlinear structure of such models. In this paper, we characterize the training landscape of the mean squared error loss for neural networks wi…
We consider the problem of learning a one-hidden-layer neural network: we assume the input x∈Rd is from Gaussian distribution and the label y=a⊤σ(Bx)+ξ, where a is a nonnegative vector in Rm with m≤d, B∈Rm×d is a full-rank weight matrix, and ξ is a n…
This paper examines how different loss functions affect neural network features and performance.
problem Investigating which loss function is best for deep neural networks.
method Examining last-layer features of deep networks and drawing inspiration from the Neural Collapse phenomenon.
result All relevant loss functions (CE, LS, FL, MSE) produce equivalent features and similar performance.
We analyze neural collapse in neural networks, showing that features collapse to vertices of a Simplex ETF.
problem Understanding and optimizing the features learned in the last layer of neural networks during training.
method Simplified unconstrained feature model, studying the global optimization landscape of cross-entropy loss with weight decay.
result The global minimizers of the loss are Simplex ETFs, and other critical points are strict saddles with negative curvature.
Deep learning dynamics exhibit anomalous superdiffusion initially, aiding escape from local minima.
problem Understanding the dynamics of learning in deep neural networks.
method Novel analysis of SGD dynamics and loss landscape structure.
result SGD exhibits anomalous superdiffusion initially, transitioning to subdiffusion as learning progresses.
We study the error landscape of deep linear and nonlinear neural networks with the squared error loss. Minimizing the loss of a deep linear neural network is a nonconvex problem, and despite recent progress, our understanding of this loss surface is still incomplete. For deep linear networks, we present necessary and s…
The paper reveals surprising star-shaped connectivity in neural networks.
problem Understanding mode connectivity in neural network landscapes.
method Fine-grained analysis of connectivity in overparameterized and finite minima cases.
result Star-shaped connectivity exists in neural network landscapes, suggesting near convexity.
The paper studies the loss landscape of regularized deep matrix factorization, revealing unique and sharp minimizers.
problem Understanding the loss landscape and minimizers of regularized deep matrix factorization problems.
method Theoretical analysis of ℓ2-regularized deep matrix factorization/deep linear network training problems with squared-error loss. result The unique end-to-end minimizer exists for all target matrices except for a set of Lebesgue measure zero.
For one-hidden-layer ReLU networks, we prove that all differentiable local minima are global inside differentiable regions. We give the locations and losses of differentiable local minima, and show that these local minima can be isolated points or continuous hyperplanes, depending on an interplay between data, activati…
Robust estimation methods find global minima efficiently via quasi-gradients.
problem Efficiently solving robust estimation problems with non-convex optimization.
method Identifying generalized quasi-gradients to guarantee low-regret algorithms.
result Generalized quasi-gradients ensure efficient approximation of global minima.
Non-convex optimization with local search heuristics has been widely used in machine learning, achieving many state-of-art results. It becomes increasingly important to understand why they can work for these NP-hard problems on typical data. The landscape of many objective functions in learning has been conjectured to …