Related papers: Efficiently Escaping Saddle Points in Bilevel Opti…

Adaptive methods such as Adam and RMSProp are widely used in deep learning but are not well understood. In this paper, we seek a crisp, clean and precise characterization of their behavior in nonconvex settings. To this end, we first…

We study the asymptotic behavior of second-order algorithms mixing Newton's method and inertial gradient descent in non-convex landscapes. We show that, despite the Newtonian behavior of these methods, they almost always escape strict…

Bilevel optimization has gained significant attention in recent years due to its broad applications in machine learning. This paper focuses on bilevel optimization in decentralized networks and proposes a novel single-loop algorithm for…

Although gradient descent (GD) almost always escapes saddle points asymptotically [Lee et al., 2016], this paper shows that even with fairly natural random initialization schemes and non-pathological functions, GD can be significantly…

Saddle-point problems have recently gained increased attention from the machine learning community, mainly due to applications in training Generative Adversarial Networks using stochastic gradients. At the same time, in some applications…

Bilevel learning has gained prominence in machine learning, inverse problems, and imaging applications, including hyperparameter optimization, learning data-adaptive regularizers, and optimizing forward operators. The large-scale nature of…

In this paper, we propose a variant of Riemannian stochastic recursive gradient method that can achieve second-order convergence guarantee and escape saddle points using simple perturbation. The idea is to perturb the iterates when gradient…

We consider minimizing a nonconvex, smooth function $f$ on a Riemannian manifold $\mathcal{M}$. We show that a perturbed version of Riemannian gradient descent algorithm converges to a second-order stationary point (and hence is able to…

Algorithms for bilevel optimization often encounter Hessian computations, which are prohibitive in high dimensions. While recent works offer first-order methods for unconstrained bilevel problems, the constrained setting remains relatively…

Recently, saddle point problems have received much attention due to their powerful modeling capability for a lot of problems from diverse domains. Applications of these problems occur in many applied areas, such as robust optimization,…

Nesterov's accelerated gradient descent (AGD), an instance of the general family of "momentum methods", provably achieves faster convergence rate than gradient descent (GD) in the convex setting. However, whether these methods are superior…

Bilevel optimization plays an essential role in many machine learning tasks, ranging from hyperparameter optimization to meta-learning. Existing studies on bilevel optimization, however, focus on either centralized or synchronous…

We provide an overview of primal-dual algorithms for nonsmooth and non-convex-concave saddle-point problems. This flows around a new analysis of such methods, using Bregman divergences to formulate simplified conditions for convergence.

We consider a generic min-max multi-objective bilevel optimization problem with applications in robust machine learning such as representation learning and hyperparameter optimization. We design MORBiT, a novel single-loop gradient…

Many problems of substantial current interest in machine learning, statistics, and data science can be formulated as sparse and low-rank optimization problems. In this paper, we present the nonconvex exterior-point optimization solver NExOS…

Many modern machine learning applications - from online principal component analysis to covariance matrix identification and dictionary learning - can be formulated as minimization problems on Riemannian manifolds, and are typically solved…

Nonconvex optimization underlies many modern machine learning and control tasks, where saddle points pose the dominant obstacle to reliable convergence in high-dimensional settings. Escaping these saddle points deterministically using…

This paper studies first order methods for solving smooth minimax optimization problems $\min_x \max_y g(x,y)$ where $g(\cdot,\cdot)$ is smooth and $g(x,\cdot)$ is concave for each $x$. In terms of $g(\cdot,y)$, we consider two settings --…

Inspired by the Optimistic Gradient Ascent-Proximal Point Algorithm (OGAProx) proposed by Bo{\c{t}}, Csetnek, and Sedlmayer for solving a saddle-point problem associated with a convex-concave function with a nonsmooth coupling function and…

Stochastic nonconvex-concave min-max saddle point problems appear in many machine learning and control problems including distributionally robust optimization, generative adversarial networks, and adversarial learning. In this paper, we…