Tackling benign nonconvexity with smoothing and stochastic gradients
Non-convex optimization problems are ubiquitous in machine learning, especially in Deep Learning. While such complex problems can often be successfully optimized in practice by using stochastic gradient descent (SGD), theoretical analysis cannot adequately explain this success. In particular, the standard analyses do not show global convergence of SGD on non-convex functions, and instead show convergence to stationary points (which can also be local minima or saddle points). We identify a broad class of nonconvex functions for which we can show that perturbed SGD (gradient descent perturbed by stochastic noise -- covering SGD as a special case) converges to a global minimum (or a neighborhood thereof), in contrast to gradient descent without noise that can get stuck in local minima far from a global solution. For example, on non-convex functions that are relatively close to a convex-like (strongly convex or PL) function we show that SGD can converge linearly to a global optimum.
Code (0)
등록된 구현이 없습니다.
Methods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
Nonnegative Low-rank Matrix Recovery Can Have Spurious Local Minima
The classical low-rank matrix recovery problem is well-known to exhibit \emph{benign nonconvexity} under the restricted isometry property (RIP): local optimization is guaranteed to converge to the global optimum, where t…
SSFG: Stochastically Scaling Features and Gradients for Regularizing Graph Convolutional Networks
Graph convolutional networks have been successfully applied in various graph-based tasks. In a typical graph convolutional layer, node features are updated by aggregating neighborhood information. Repeatedly applying gra…
Graph LearningApollo: An Adaptive Parameter-wised Diagonal Quasi-Newton Method for Nonconvex Stochastic Optimization
In this paper, we introduce Apollo, a quasi-newton method for noncovex stochastic optimization, which dynamically incorporates the curvature of the loss function by approximating the Hessian via a diagonal matrix. Algori…
Stochastic OptimizationASDL: A Unified Interface for Gradient Preconditioning in PyTorch
Gradient preconditioning is a key technique to integrate the second-order information into gradients for improving and extending gradient-based learning algorithms. In deep learning, stochasticity, nonconvexity, and high…
EnCAgg: Enhanced Clustering Aggregation for Robust Federated Learning against Dynamic Model Poisoning
Federated learning faces increasing threats from model poisoning attacks, which harms its application to improve privacy. Existing defense methods typically rely on fixed thresholds or perform clustering with a fixed num…
Federated Learning