Convergence of the Iterates in Mirror Descent Methods

Abstract

We consider centralized and distributed mirror descent algorithms over a finite-dimensional Hilbert space, and prove that the problem variables converge to an optimizer of a possibly nonsmooth function when the step sizes are square summable but not summable. Prior literature has focused on the convergence of the function value to its optimum. However, applications from distributed optimization and learning in games require the convergence of the variables to an optimizer, which is generally not guaranteed without assuming strong convexity of the objective function. We provide numerical simulations comparing entropic mirror descent and standard subgradient methods for the robust regression problem. √k) for strongly convex or convex objective functions, respectively, [15], [17].

Discounted Mirror Descent Dynamics in Concave Games

“…1 This is an example of the so-called Generative Adversarial Network (GAN), specifically, the Wasserstein GAN (without Lipschitz constraint). Assume ran(A) = R N , then the objective function can be rewritten as [37], [38], and min w∈R M+1 (2,12), (3,15), . .…”

“…Literature review: Mirror descent algorithms have found numerous applications in recent years, e.g. in distributed optimization [2], online learning [4], and variational inequality problems [5]. They fall into the class of so-called primal-dual algorithms; the name mirror descent refers to the two iterative steps: a mapping of the primal variable into a dual space (in the sense of convex conjugate), followed by a mapping of the dual variable, or some post-processing of it, back into the primal space via a mirror map.…”

Mass–spring–damper networks for distributed optimization in non-Euclidean spaces

“…In order to effectively exploit the structure of such non-Euclidean geometries, several attempts have been made to generalize distributed optimization algorithms from Euclidean to non-Euclidean cases. In particular, the distributed mirror descent method [Raginsky and Bouvrie, 2012, Li et al, 2016, Doan et al, 2019 generalizes the distributed subgradient method [Nedic and Ozdaglar, 2009]; the distributed dual averaging algorithm [Duchi et al, 2012] generalizes projected distributed subgradient method [Nedic et al, 2010]; the Bregman parallel direction method of multipliers (PDMM) [Yu et al, 2018] generalizes the proximal distributed alternating direction method of multipliers (ADMM) [Meng et al, 2015]. Compared with their counterparts in Euclidean cases [Nedic and Ozdaglar, 2009, Nedic et al, 2010, Meng et al, 2015, the key feature of these algorithms is to use a Bregman divergence as distance function, which, compared with quadratic function, leads to an improved complexity bound of O(n/ ln n) [Wang andBanerjee, 2014, Yu et al, 2018], where n represents the size of the problem instance.…”

“…Moreover, like other algorithms based on ADMM [Wei and Ozdaglar, 2012, Meng et al, 2015, Yu et al, 2018, Bregman PDMM uses Euler backward method, which solves an optimization problem at each iteration. It is unclear why Euler forward method, which only computes subgradients [Duchi et al, 2012, Li et al, 2016, Doan et al, 2019, cannot achieve similar convergence properties. Motivated by these questions, as well as the connections between algorithm design and physics [Alvarez, 2000, Alvarez et al, 2002, Su et al, 2014, Krichene et al, 2015, Wibisono et al, 2016, we propose a novel algorithm for non-Euclidean distributed optimization with potentially non-smooth convex objective functions over undirected graphs.…”

On the Convergence of Distributed Subgradient Methods under Quantization

“…Finally, there are some related methods based on primal-dual approach for solving problem (1.1), such as, the accelerated primal-dual methods [26,27], the alternating direction method of multipliers (ADMM) [28][29][30][31][32], and the distributed dual methods (mirror descent/dual averaging) [33][34][35]. Our focus in this paper will be on DSG algorithms, as they are both simple and have convergence guarantees that are as strong or stronger than those for dual methods.…”