Gradient methods for problems with inexact model of the objective

We consider optimization methods for convex minimization problems under inexact information on the objective function. We introduce inexact model of the objective, which as a particular cases includes inexact oracle [19] and relative smoothness condition [43]. We analyze gradient method which uses this inexact model and obtain convergence rates for convex and strongly convex problems. To show potential applications of our general framework we consider three particular problems. The first one is clustering by electorial model introduced in [49]. The second one is approximating optimal transport distance, for which we propose a Proximal Sinkhorn algorithm. The third one is devoted to approximating optimal transport barycenter and we propose a Proximal Iterative Bregman Projections algorithm. We also illustrate the practical performance of our algorithms by numerical experiments

Gradient methods for problems with inexact model of the objective

We consider optimization methods for convex minimization problems under inexact information on the objective function. We introduce inexact model of the objective, which as a particular cases includes inexact oracle [19] and relative smoothness condition [43]. We analyze gradient method which uses this inexact model and obtain convergence rates for convex and strongly convex problems. To show potential applications of our general framework we consider three particular problems. The first one is clustering by electorial model introduced in [49]. The second one is approximating optimal transport distance, for which we propose a Proximal Sinkhorn algorithm. The third one is devoted to approximating optimal transport barycenter and we propose a Proximal Iterative Bregman Projections algorithm. We also illustrate the practical performance of our algorithms by numerical experiments

Efficient Inexact Proximal Gradient Algorithm for Nonconvex Problems

The proximal gradient algorithm has been popularly used for convex optimization. Recently, it has also been extended for nonconvex problems, and the current state-of-the-art is the nonmonotone accelerated proximal gradient algorithm. However, it typically requires two exact proximal steps in each iteration, and can be inefficient when the proximal step is expensive. In this paper, we propose an efficient proximal gradient algorithm that requires only one inexact (and thus less expensive) proximal step in each iteration. Convergence to a critical point %of the nonconvex problem is still guaranteed and has a $O(1/k)$ convergence rate, which is the best rate for nonconvex problems with first-order methods. Experiments on a number of problems demonstrate that the proposed algorithm has comparable performance as the state-of-the-art, but is much faster

Accelerated Inexact Composite Gradient Methods for Nonconvex Spectral Optimization Problems

This paper presents two inexact composite gradient methods, one inner accelerated and another doubly accelerated, for solving a class of nonconvex spectral composite optimization problems. More specifically, the objective function for these problems is of the form $f_1 + f_2 + h$ where $f_1$ and $f_2$ are differentiable nonconvex matrix functions with Lipschitz continuous gradients, $h$ is a proper closed convex matrix function, and both $f_2$ and $h$ can be expressed as functions that operate on the singular values of their inputs. The methods essentially use an accelerated composite gradient method to solve a sequence of proximal subproblems involving the linear approximation of $f_1$ and the singular value functions underlying $f_2$ and $h$. Unlike other composite gradient-based methods, the proposed methods take advantage of both the composite and spectral structure underlying the objective function in order to efficiently generate their solutions. Numerical experiments are presented to demonstrate the practicality of these methods on a set of real-world and randomly generated spectral optimization problems