Related papers: Beyond first-order methods for non-convex non-conc…
It was recently established that for convex optimization problems with sparse optimal solutions (be it entry-wise sparsity or matrix rank-wise sparsity) it is possible to design first-order methods with linear convergence rates that depend…
This paper studies the complexity of finding approximate stationary points for the smooth nonconvex-strongly-concave (NC-SC) saddle point problem: $\min_x\max_yf(x,y)$. Under the standard first-order smoothness conditions where $f$ is…
In this work, we propose a method for minimizing non-convex functions with Lipschitz continuous $p$th-order derivatives, starting from $p \geq 1$. The method, however, only requires derivative information up to order $(p-1)$, since the…
We describe a novel constructive technique for devising efficient first-order methods for a wide range of large-scale convex minimization settings, including smooth, non-smooth, and strongly convex minimization. The technique builds upon a…
The graduated optimization approach, also known as the continuation method, is a popular heuristic to solving non-convex problems that has received renewed interest over the last decade. Despite its popularity, very little is known in terms…
Second-order optimization methods exhibit fast convergence to critical points, however, in nonconvex optimization, these methods often require restrictive step-sizes to ensure a monotonically decreasing objective function. In the presence…
In this paper, we propose second-order sufficient optimality conditions for a very general nonconvex constrained optimization problem, which covers many prominent mathematical programs.Unlike the existing results in the literature, our…
We study the problem of finding approximate first-order stationary points in optimization problems of the form $\min_{x \in X} \max_{y \in Y} f(x,y)$, where the sets $X,Y$ are convex and $Y$ is compact. The objective function $f$ is smooth,…
Lower-bound analyses for nonconvex strongly-concave minimax optimization problems have shown that stochastic first-order algorithms require at least $\mathcal{O}(\varepsilon^{-4})$ oracle complexity to find an $\varepsilon$-stationary…
In this paper, we propose two novel non-stationary first-order primal-dual algorithms to solve nonsmooth composite convex optimization problems. Unlike existing primal-dual schemes where the parameters are often fixed, our methods use…
On solving a convex-concave bilinear saddle-point problem (SPP), there have been many works studying the complexity results of first-order methods. These results are all about upper complexity bounds, which can determine at most how many…
We study alternating first-order algorithms with no inner loops for solving nonconvex-strongly-concave min-max problems. We show the convergence of the alternating gradient descent--ascent algorithm method by proposing a substantially…
In this work, we consider a connected network of finitely many agents working cooperatively to solve a min-max problem with convex-concave structure. We propose a decentralised first-order algorithm which can be viewed as a non-trivial…
Previous algorithms can solve convex-concave minimax problems $\min_{x \in \mathcal{X}} \max_{y \in \mathcal{Y}} f(x,y)$ with $\mathcal{O}(\epsilon^{-2/3})$ second-order oracle calls using Newton-type methods. This result has been…
We study the smooth minimax optimization problem $\min_{\bf x}\max_{\bf y} f({\bf x},{\bf y})$, where $f$ is $\ell$-smooth, strongly-concave in ${\bf y}$ but possibly nonconvex in ${\bf x}$. Most of existing works focus on finding the…
In this paper, we propose a multilevel stochastic framework for the solution of nonconvex unconstrained optimization problems. The proposed approach uses random regularized first-order models that exploit an available hierarchical…
Modern data science applications demand solving large-scale optimization problems. The prevalent approaches are first-order methods, valued for their scalability. These methods are implemented to tackle highly irregular problems where…
Recent applications that arise in machine learning have surged significant interest in solving min-max saddle point games. This problem has been extensively studied in the convex-concave regime for which a global equilibrium solution can be…
We introduce a class of first-order methods for smooth constrained optimization that are based on an analogy to non-smooth dynamical systems. Two distinctive features of our approach are that (i) projections or optimizations over the entire…
Necessary conditions for high-order optimality in smooth nonlinear constrained optimization are explored and their inherent intricacy discussed. A two-phase minimization algorithm is proposed which can achieve approximate first-, second-…