arXiv research

The study extends Huisken's theorem to nonconvex surfaces that shrink to round points.

New ancient and eternal solutions found for mean curvature flow from minimal surfaces.

Investigates Cheeger sets in rotationally invariant domains and their free boundaries.

OLLA framework efficiently samples from constrained distributions with nonconvex constraints.

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…

Unified framework for constructing nonconvex sparse recovery methods.

New method improves signal reconstruction with nonconvex penalties and parameter control.

We consider a compact, star-shaped, mean convex hypersurface $Σ^2\subset \mathbb{R}^3$. We prove that in some cases the flow exists until it shrinks to a point in a spherical manner, which is very typical for convex surfaces as well (see \cite{An1}). We also prove that in the case we have a surface of revolution which …

Improved SGD methods converge faster for nonconvex optimization.

Schedule-free SGD is optimal for nonconvex optimization problems.

New algorithm for privacy-preserving nonconvex optimization.

We prove that, both in the hyperbolic and spherical 3-spaces, there exist nonconvex compact boundary-free polyhedral surfaces without selfintersections which admit nontrivial continuous deformations preserving all dihedral angles and study properties of such polyhedral surfaces. In particular, we prove that the volume …

PPGD solves nonconvex nonsmooth optimization problems without KL property.

New framework explains why nonconvex methods work well in low-rank matrix estimation.

Develops shuffling gradient-based methods for nonconvex-concave minimax optimization.

New definition of regret for nonconvex online learning models.

In the paper, we study the stochastic alternating direction method of multipliers (ADMM) for the nonconvex optimizations, and propose three classes of the nonconvex stochastic ADMM with variance reduction, based on different reduced variance stochastic gradients. Specifically, the first class called the nonconvex stoch…

This paper analyzes OGDA and EG methods for nonconvex minimax problems.

Simple DP algorithms find approximate solutions for nonconvex ERM.

Unified parametric assumption improves convergence guarantees for nonconvex optimization.

Support vector machines (SVMs) with sparsity-inducing nonconvex penalties have received considerable attentions for the characteristics of automatic classification and variable selection. However, it is quite challenging to solve the nonconvex penalized SVMs due to their nondifferentiability, nonsmoothness and nonconve…

The paper explores nonconvex penalties for deep learning regularization.

In this paper, we study and analyze the mini-batch version of StochAstic Recursive grAdient algoritHm (SARAH), a method employing the stochastic recursive gradient, for solving empirical loss minimization for the case of nonconvex losses. We provide a sublinear convergence rate (to stationary points) for general noncon…

We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum problems, where the nonconvex part is smooth and the nonsmooth part is convex. Surprisingly, unlike the smooth case, our knowledge of this fundamental problem is very limited. For example, it is not known whether the proximal stochastic gra…

The use of convex regularizers allows for easy optimization, though they often produce biased estimation and inferior prediction performance. Recently, nonconvex regularizers have attracted a lot of attention and outperformed convex ones. However, the resultant optimization problem is much harder. In this paper, for a …

PGD algorithm converges to local minima in nonconvex matrix completion.

Paper removes bounded gradient assumption for SGD in nonconvex learning.

This work addresses the issue of large covariance matrix estimation in high-dimensional statistical analysis. Recently, improved iterative algorithms with positive-definite guarantee have been developed. However, these algorithms cannot be directly extended to use a nonconvex penalty for sparsity inducing. Generally, a…

This paper establishes lower bounds for smooth nonconvex finite-sum optimization.

New algorithm tackles nonconvex machine learning problems with adaptive normalization and independent sampling.

As surrogate functions of $L_0$-norm, many nonconvex penalty functions have been proposed to enhance the sparse vector recovery. It is easy to extend these nonconvex penalty functions on singular values of a matrix to enhance low-rank matrix recovery. However, different from convex optimization, solving the nonconvex l…

With the large rising of complex data, the nonconvex models such as nonconvex loss function and nonconvex regularizer are widely used in machine learning and pattern recognition. In this paper, we propose a class of mini-batch stochastic ADMMs (alternating direction method of multipliers) for solving large-scale noncon…

Paper proposes a new method for training nonconvex models.

Two algorithms solve nonconvex minimax problems with linear constraints, achieving complexity guarantees.

Paper proposes an algorithm to solve complex minimax problems efficiently.

PAGE optimizes nonconvex problems with optimal convergence rates.

New algorithm solves nonconvex-convex minimax problems efficiently.

New algorithms solve nonconvex-concave minimax problems without parameter knowledge.

Develops efficient method for nonconvex problems using Regula Falsi.

Safe reinforcement learning with nonconvex constraints using convex approximations.

Paper develops accelerated APCD for nonconvex nonsmooth problems with performance guarantees.

AGDA and variance-reduced methods solve nonconvex-nonconcave minimax problems globally and faster.

Two new algorithms solve nonconvex-strongly concave problems efficiently.

New nonconvex penalty smooths at origin for deep learning.

Unified framework for nonconvex matrix completion with linearly parameterized factors.

Paper analyzes nonconvex bandit problems with improved adaptive methods.

Study uncovers statistical optimality of nonconvex tensor completion methods.

The stochastic gradient descent has been widely used for solving composite optimization problems in big data analyses. Many algorithms and convergence properties have been developed. The composite functions were convex primarily and gradually nonconvex composite functions have been adopted to obtain more desirable prop…