An Asynchronous Approximate Distributed
Alternating Direction Method of Multipliers in Digraphs

In this paper, we are interested in developing a modified ADMM algorithm for solving large-scale optimization problems in digraphs. In that direction, there are two main communication topologies considered: (i) master-workers communication topology [1, 2, 3] and (ii) multi-node communication topology [4, 5, 6, 7]. When the ADMM has a master-worker communication topology, the worker nodes optimize their local objectives and communicate their local variables to the master node which updates the global optimization variable and send it back to the workers. When the ADMM has no master node, the optimization problem is solved over a network of nodes. Herein, we focus on the ADMM realized on multi-node communication topologies.

There are several real-life applications in which synchronous operation is infeasible or costly and complicated. As a result, asynchronous information exchange is necessitated. For example, resource allocation in data centers gives rise to large-scale problems and networks, which naturally call for asynchronous solutions. Recently, different asynchronous distributed ADMM versions are proposed under different assumptions and communication topologies. For example, the version in [1, 2] is based on the master-worker architecture under a star topology. Even though worker updates are parallel and distributed, the master update is centralized as it collects information from workers and broadcasts the updated information to workers for their following iteration. Authors in [4, 5] proposed another version with the almost surely convergence by using information from a randomly selected subset of nodes for ADMM update under the positive possibility selection assumption. Then, authors in [3] combined the above two architecture and designed a master-local processor-link processor architecture with the random node selection. It is worth noting that all the above asynchronous versions are only available for undirected communication graphs, i.e., assuming that every communication link is bidirectional. Very recently, distributed ADMM versions for digraphs are proposed in [6, 7]. However, they are synchronous, which is also the motivation for this work.

In this paper, we propose AsyAD-ADMM, an asynchronous distributed ADMM algorithm for digraphs. First, the optimization problem is restructured such that at every optimization step, each node can solve individually a local convex optimization problem for one of the primal variables, while for the other primal variable the communication among nodes is required, since the cost function is not decomposable. Then, we find the solution for the primal variable whose cost function is non-decomposable by means of a finite-time asynchronous approximate consensus algorithm. More specifically, we adopt the algorithm introduced in [8, Algorithm 2]. The algorithm relies on asynchronous $\min-$consensus and $\max-$consensus protocols to compute the approximate average consensus value considering different communication delays over a finite number of steps. Unlike [8], the consensus algorithm should be repeated at every optimization step, which is also asynchronous. Towards this end, we propose a mechanism in which every next optimization step is initiated after the previous one has finished. The main contributions of the paper are as follows.

• $\bullet$

  We propose an asynchronous distributed ADMM algorithm which can be applied to digraphs.

• $\bullet$

  The convergence analysis shows AsyAD-ADMM converges at a rate of $O(1/k)$, where $k$ is the iteration counter. Factors influencing the algorithm performance are also investigated.

In this work, we consider a strongly connected digraph $\mathcal{G}(\mathcal{V},\mathcal{E})$ in which each node $v_{i}\in\mathcal{V}$ is endowed with a scalar cost function $f_{i}:\mathds{R}^{p}\mapsto\mathds{R}$ assumed to be known to the node only. We assume that each node $v_{i}$ has knowledge of the number of its out-going links, $\mathcal{D}^{+}_{i}$, and has access to local information only via its communication with the in-neighboring nodes, $\mathcal{N}^{-}_{i}$. The only global information available to all the nodes in the network is given in Assumption 1.

While Assumption 1 is limiting, there exist distributed methods for extracting such information; see, e.g., [9, 10].

The objective is to design a discrete-time coordination algorithm that allows every node $v_{i}$ in a digraph to distributively and asynchronously solve the following global optimization problem:

where $x\in\mathds{R}^{p}$ is a global optimization variable (or a common decision variable). In order to solve problem and to enjoy the structure ADMM scheme at the same time, a separate decision variable $x_{i}$ for node $v_{i}$ is introduced and the constraint $\|x_{i}-x_{j}\|\leq\epsilon,\epsilon>0,\forall v_{i},v_{j}\in\mathcal{V}$ is imposed to allow the asynchronous distributed ADMM framework. Thus, problem (6) is reformulated as

where $\epsilon$ is a predefined error tolerance. Define a closed nonempty convex set $\mathcal{C}$ as

By denoting $X\coloneqq\begin{bmatrix}x_{1}^{\mbox{\tiny T}}&x_{2}^{\mbox{\tiny T}}&\ldots&x_{n}^{\mbox{\tiny T}}\end{bmatrix}^{\mbox{\tiny T}}$ and making variable $z\in\mathds{R}^{np}$ as a copy of vector $X$, constraint (8) related to problem (7) becomes

Then, define $g(z)$ as the indicator function of set $\mathcal{C}$ as

Finally, problem (7) with constraint (10) is transformed to

For notational convenience, denote $F(X)\coloneqq\sum_{i=1}^{n}f_{i}(x_{i})$. Thus, denote the Lagrangian function as

where $\lambda\in\mathds{R}^{np}$ is the Lagrange multiplier. The following standard assumptions are required for problem (12).

Assumption 2 allows $f_{i}$ to be non-differentiable [11]. By Assumptions 2-3 and based on the definition of $g(z)$ in (11), $L(X,z,\lambda^{*})$ is convex in $(X,z)$ and $(X^{*},z^{*})$ is a solution to problem (12) [11, 12].

At iteration $k$, the corresponding augmented Lagrangian of optimization problem (12) is written as

where $z_{i}\in\mathds{R}^{p}$ is the $i$-th element of vector $z$. By ignoring terms which are independent of the minimization variables (i.e., $x_{i},z$), for each node $v_{i}$, the standard ADMM updates (3)-(5) change to the following format:

where the last term in (17) comes from the identity $2a^{T}b+b^{2}=(a+b)^{2}-a^{2}$ with $a=\lambda^{k}/\rho$ and $b=X^{k+1}-z$.

Update (16) for $x_{i}^{k+1}$ can be solved by a classical method, e.g., the proximity operator [11, Section 4]. Update (18) for the dual variable $\lambda_{i}^{k+1}$ can be implemented trivially by node $v_{i}$. Note that both updates can be done independently and in parallel by node $v_{i}$.

Since $g$ in (11) is the indicator function of the closed nonempty convex set $\mathcal{C}$, update (17) for $z^{k+1}$ becomes

where $\Pi_{\mathcal{C}}$ denotes the projection (in the Euclidean norm) onto $\mathcal{C}$. Intuitively, from (17) and the definition of $g(z)$ in (11), one can see that the elements of $z$ (i.e., $z_{1},z_{2},\ldots,z_{n}$) should go into $\mathcal{C}$ in finite time. If not, one will have $g(z)=\infty$ and update (17) will never be finished. Then, from the definition of $\mathcal{C}$ in (9), one can see that $z$ going into $\mathcal{C}$ means $\|z_{i}-z_{j}\|\leq\epsilon,\forall v_{i},v_{j}\in\mathcal{V}$.

It is worth noting that if $z_{i}-z_{j}=0,\forall v_{i},v_{j}\in\mathcal{V}$, it means $z_{1}=z_{2}=\ldots=z_{n}$, which is in the mathematical format of consensus. In other words, each node $v_{i}\in\mathcal{V}$ can have $z_{i}$ reach $\frac{1}{n}\sum_{i=1}^{n}z_{i}(0),z_{i}(0)=x_{i}^{k+1}+\lambda_{i}^{k}/\rho$ in a finite number of steps, i.e., finite-time average consensus. Similarly, requiring $\|z_{i}-z_{j}\|\leq\epsilon,\forall v_{i},v_{j}\in\mathcal{V}$ means asking nodes $v_{i},v_{j}\in\mathcal{V}$ to have $z_{i},z_{j}$ with $\epsilon$ closeness to the average consensus, i.e., to have $z_{i},z_{j}$ enter a circle with its center at $\frac{1}{n}\sum_{i=1}^{n}(x_{i}^{k+1}+\lambda_{i}^{k}/\rho)$ and its radius as $\epsilon$. Here we call it finite-time “approximate” ratio consensus.

Before we proceed to the description of our main algorithm, we describe the form of asynchrony that the nodes experience and some consensus algorithms that are key ingredients for the operation of our proposed algorithms.

Our proposed algorithm works on two levels: i) at the optimization level in which the distributed ADMM procedure is described for updating $x_{i}^{k+1}$, $z_{i}^{k+1}$ and $\lambda_{i}^{k+1}$ (Algorithm 1), and ii) at the distributed coordination level for computing $z_{i}^{k+1}$ in an asynchronous manner (Algorithm 2). Algorithm 2 is executed within Algorithm 1.

The distributed least square problem is considered as

where $A_{i}\in\mathds{R}^{q\times p}$ is only known to node $v_{i}$, $b_{i}\in\mathds{R}^{q}$ is the measured data and $x\in\mathds{R}^{p}$ is the common decision variable that needs to be optimized. For the automatic generation of large number of different matrices $A_{i}$, we choose $q=p$ to have the square $A_{i}$. All elements of $A_{i}$ and $b_{i}$ are set from independent and identically distributed (i.i.d.) samples of standard normal distribution $\mathcal{N}(0,1)$.

We choose $p=3$ and $n=600$ to have 600 nodes having a strongly connected digraph. We implement the D-ADMM-FTERC algorithm in [7] as a benchmark to evaluate factors (i.e., $\epsilon,\bar{\tau}$) influencing AsyAD-ADMM performance as D-ADMM-FTERC is developed for calculating the optimal solution to distributed optimization problems for digraphs, i.e., all figures related to D-ADMM-FTERC are optimal. However, D-ADMM-FTERC is synchronous and this is the reason we develop AsyAD-ADMM in this paper.

Fig. 2 demonstrates three points: (i) AsyAD-ADMM solutions are very close to the optimal one from D-ADMM-FTERC. (ii) From Fig. 2 (a) to (b), the smaller the value of $\epsilon$ is, the closer AsyAD-ADMM solutions are to D-ADMM-FTERC one. This is reasonable as smaller value of $\epsilon$ means better and more accurate “approximate” ratio consensus for $z$ update (17) using Algorithm 2.

This can be also validated in Fig. 3 where red and black lines are the $z$ updates using AsyAD-ADMM and D-ADMM-FTERC, respectively. (iii) The values of $\bar{\tau}$ (defined in Sec. IV-A1) do not influence the precision of AsyAD-ADMM, which also makes sense as $\bar{\tau}$ decides the degree of asynchrony, not the asynchronous ADMM precision. Nodes use this information to decide when to update as shown in step 6 of Algorithm 2. The values of $\bar{\tau}$ do influence the iteration steps of Algorithm 2 as described in Table I from which it shows the larger the value of $\bar{\tau}$ is, the more iteration steps Algorithm 2 needs in each AsyAD-ADMM optimization step. For $\epsilon=0.01$, the steps are the same as $1000$ because we set the iteration step upper bound for Algorithm 2 as 1000; it means iteration steps are no less than 1000 when $\epsilon\leq 0.01$ and $\bar{\tau}\geq 3$.

Note that in Fig. 3, we have AsyAD-ADMM with the stopping condition which is for the convenience of figure presentation. The reason that Fig. 3 (a) and (b) have a big difference is $\|z_{i}-z_{j}\|\leq\epsilon$, which means a smaller $\epsilon$ leads to a more accurate $z$ update (17) using Algorithm 2. And the reason is that a smaller $\epsilon$ leads to a larger Algorithm 2 iteration step as one can check it is $9$ and $1000$ from Table I for $\epsilon=0.1$ and $\epsilon=0.01$ in this example. Also from Fig. 3, one can see the ADMM iteration numbers are the same for different values of $\epsilon$ for AsyAD-ADMM with the stopping condition. Furthermore, Table II describes the running time comparison on an Intel Core i5 processor at 2.6 GHz with Matlab R2020b for different values of $\epsilon$ and $\bar{\tau}$ with $k_{\max}=200$. One can see for $\epsilon=0.01$, from $\bar{\tau}=3,5,10$, even though Algorithm 2 iteration steps are the same as 1000, the running time is larger and larger.

This is because Algorithm 2 running time is the time gap between two consecutive iterations multiplies the iteration step while in step 6 of Algorithm 2, $(1+\bar{\tau})D$ is the time gap. Fig. 4 demonstrates two points. (i) It validates Theorem 1 that AsyAD-ADMM has $\mathcal{O}(\frac{1}{k})$ convergence rate. (ii) It is in accordance with Fig. 2 and Fig. 3 that a smaller value of $\epsilon$ has a better solution precision.

What is more, from Fig. 2 and Table II, considering the AsyAD-ADMM solution precision and running time, it is better to have the value of $\epsilon$ neither too large nor too small; for this example, $\epsilon\in[0.01,0.1)$ is a good value range.

An asynchronous approximate distributed alternating direction method of multipliers algorithm is proposed to provide a solution for the distributed optimization problems allowing asynchronous information exchange among nodes in digraphs with assumptions of bounded time-index steps of asynchronous nodes and the known digraph diameter to all nodes. By proposing the finite-time asynchronous approximate ratio consensus algorithm for one ADMM primal variable update, the solution is close to the optimal but acceptable. How to choosing algorithm parameters to have an aggressive performance is also discussed.

Future work will focus on designing a new asynchronous ADMM version without the communication graph diameter information to compute the optimal solution for distributed optimization problems.