Subgradient averaging for multi-agent optimisation with different constraint sets

Abstract

We consider a multi-agent setting with agents exchanging information over a possibly time-varying network, aiming at minimising a separable objective function subject to constraints. To achieve this objective we propose a novel subgradient averaging algorithm that allows for non-differentiable objective functions and different constraint sets per agent. Allowing different constraints per agent simultaneously with a time-varying communication network constitutes a distinctive feature of our approach, extending existing results on distributed subgradient methods. To highlight the necessity of dealing with a different constraint set within a distributed optimisation context, we analyse a problem instance where an existing algorithm does not exhibit a convergent behaviour if adapted to account for different constraint sets. For our proposed iterative scheme we show asymptotic convergence of the iterates to a minimum of the underlying optimisation problem for step sizes of the form $\frac{\eta }{k+1}$, $\eta >0$. We also analyse this scheme under a step size choice of $\frac{\eta }{\sqrt{k+1}}$, $\eta >0$, and establish a convergence rate of $\mathcal{O}\left(\frac{lnk}{\sqrt{k}}\right)$ in objective value. To demonstrate the efficacy of the proposed method, we investigate a robust regression problem and an ${\ell }_2$ regression problem with regularisation.

Introduction

Distributed optimisation deals with multiple agents interacting over a network and has found numerous applications in different domains, such as wireless sensor networks (Baingana et al., 2014, Mateos and Giannakis, 2012), robotics (Martinez, Bullo, Cortes, & Frazzoli, 2007), and power systems (Bolognani, Carli, Cavraro, & Zampieri, 2015), due to its ability to parallelise computation and prevent agents from sharing information considered as private. Typically, distributed algorithms are based on an iterative process in which agents maintain some estimate about the decision vector in an optimisation context, exchange this information with neighbouring agents according to an underlying communication protocol/network, and update their estimate on the basis of the received information.

Despite the intense research activity in this area, only a few algorithms can simultaneously deal with time-varying networks, non-differentiable objective functions and account for the presence of constraints (Liang et al., 2019, Margellos et al., 2018, Nedić and Olshevsky, 2015, Xi and Khan, 2017, Zhu and Martinez, 2012), features that are often treated separately in the literature. Several of the commonly employed methods are based on a projected subgradient or a proximal step and their analysis consists of selecting the step size underlying these algorithms, establishing a convergence rate analysis, and quantifying practical convergence for (near-)real time applications.

In this paper, we study a class of optimisation problems that involves a separable objective function, while the feasible set can be decomposed as an intersection of different compact convex sets. A centralised version of this class of problems has been studied under a stochastic setting in Bianchi (2016) and Patrascu and Necoara (2018). Distributed algorithms for this class have been proposed in Johansson et al., 2008, Lee and Nedić, 2013, Lin et al., 2016, Mai and Abed, 2019, Margellos et al., 2018, Nedic and Ozdaglar, 2009, Nedic et al., 2010 and Zhu and Martinez (2012). References Johansson et al., 2008, Nedic and Ozdaglar, 2009 and Nedic et al. (2010) rely on Bertsekas and Tsitsiklis (1989) and Tsitsiklis, Bertsekas, and Athans (1986) to propose a distributed strategy based on projected sub-gradient methods. These results consist of an averaging step followed by a local sub-gradient projection update. In Margellos et al. (2018) a distributed scheme based on a proximal update is proposed, thus extending Johansson et al. (2008) and Nedic et al. (2010) to the case where different local constraint sets and an arbitrarily time-varying network are considered. The authors in Zhu and Martinez (2012) provide asymptotic convergence for a primal–dual algorithm that allows coupling between agents’ local estimates. We discuss additional related results in Section 4, after the proposed algorithm is presented and some notation introduced.

We motivate our approach by constructing an example showing that extending available algorithms to the case of different constraint sets might not exhibit a convergent behaviour for all problem instances. Hence, a direct adaptation of existing schemes is not always possible when dealing with different constraint sets. Notice also that distributed algorithms developed for the unconstrained case cannot be trivially adapted to our setting, as lifting the constraints in the objective (e.g., via characteristic functions) would violate boundedness of the subgradient, a typical requirement for such algorithms (Duchi et al., 2012, Margellos et al., 2018, Nedić and Olshevsky, 2015, Nedic et al., 2010).

The main contribution of this paper is the introduction and the characterisation of the convergence rate for a new subgradient averaging algorithm. The proposed scheme allows us to account for time-varying networks, non-differentiable objective functions and different constraint sets per agent as in Margellos et al. (2018), while achieving faster practical convergence as it is based on subgradient averaging as in Duchi et al., 2012, Johansson et al., 2008 and Mai and Abed (2019). Note that allowing simultaneously for different constraint sets per agent and time-varying communication network by means of a subgradient averaging scheme is a distinct feature of the algorithm in this paper. Preliminary results related to this paper appeared in Romao, Margellos, Notarstefano, and Papachristodoulou (2019), where several proofs have been omitted. Moreover, the construction of Section 2.2 that motivates the analysis of algorithms with different constraint sets is novel, and offers insight on the limitations of existing algorithms. We also provide detailed numerical examples, not included in the conference version.

The paper is organised as follows. In Section 2 we present the problem statement, the network communication structure, and the main assumptions adopted in this paper, followed by a numerical construction that motivates the algorithm of this paper. In Section 3 we present the proposed scheme and the main convergence results, namely, asymptotic convergence in iterates and a convergence rate as far as the optimal value is concerned. Section 4 provides detailed discussion and comparison of our scheme with other results in the literature. In Section 5 we study the robust linear regression problem and ${\ell }_2$ regression with regularisation to demonstrate the main algorithmic features of our scheme and to compare our strategy against existing methods. Finally, some concluding remarks and future research directions are provided in Section 6. To ease the reader all proofs have been deferred to the Appendix).

Notation: We denote by $\mathbb{R}$ the set of real numbers and $\mathbb{N}$ the set of natural numbers (excluding zero). The symbol ${\mathbb{R}}^n$ stands for the Cartesian product $\mathbb{R}\times \cdots \times \mathbb{R}$ with $n$ terms. A sequence of elements in ${\mathbb{R}}^n$ is denoted by ${\left(x\left(k\right)\right)}_{k\in \mathbb{N}}$. For any set $X\subset {\mathbb{R}}^n$, we denote its interior, relative interior and convex hull by $\mathrm{int}\left(X\right)$, $\mathrm{ri}\left(X\right)$, and $\mathrm{conv}\left(X\right)$, respectively. We also denote by $f\left(X\right)$ as the image of the set $X$ over a function $f$. The subdifferential of $f$ at a point $x\in \mathrm{dom}f$ is denoted by $\partial f\left(x\right)$. For any point $x\in {\mathbb{R}}^n$, ${\Vert x\Vert }_2$ stands for the Euclidean norm of $x$ and ${\Vert x\Vert }_1$ for the ${\ell }_1$ norm of $x\in {\mathbb{R}}^n$, which are reduced to $\left|x\right|$ if $x$ is scalar.