Convex Decreasing Algorithms: Distributed Synthesis and Finite-time Termination in Higher Dimension

Recent advancements in the field of intelligent communication technologies and ad-hoc wireless sensor networks for monitoring and control of multi-agent systems have necessitated design of algorithms to assert global information without the complete knowledge of the system. The task of obtaining the global information is often accomplished via a class of algorithms that employ strategies of achieving consensus. In a consensus algorithm, agents iteratively and in a distributed manner agree on a common state. The ideas of distributed consensus algorithms can be traced back to the seminal works, see [1, 2, 3, 4]. Recent works on consensus algorithms are focused on designing protocols to drive agents to the average of their initial states, see [5, 6, 7, 8]. These protocols were designed for cases where the state of each agent is a scalar value. However, increasing storage and computation capabilities of modern day sensor interfacing technologies have motivated large-scale applications, examples of which include distributed machine learning, see [9], multi-agent control and co-ordination, see [10, 11], distributed optimization problems, see [12, 13], distributed sensor localization, see [14]. In order to meet the requirements of such applications there is need of distributed consensus algorithms that allow for vector states. [15] presents such a higher dimensional consensus protocol. The framework in [15] is based on a leader-follower architecture with the agents being partitioned between anchors and sensors. Anchors are agents with fixed states behaving as leaders in the algorithm, while the sensors change their state by taking a convex combination of their state with the neighboring nodes’ states.

Cyber physical systems such as electrical power networks need to accommodate for large number of states for crucial applications such as state-estimations, optimal dispatch, and demand response for ancillary services. [16] formulates the distributed apportioning problem using consensus protocols where only a single state is shared for each protocol. A similar situation involving higher dimensional states arises in distributed resource allocation problems where a fixed amount of resource is required to be apportioned among all participating agents in a network and a convex cost associated with the each resource is to be distributively minimized. The method used to solve the above resource allocation problem involves a higher dimensional consensus protocol (see [17]).

Other applications where accommodating higher dimensional data is required are distributed optimization and their applications to deep neural networks such as in the case of diffusion learning where local agents perform model training using local datasets in parallel, for example see [18], where after the completion of the training process, model parameters are then shared with the neighboring nodes allowing for faster dissemination of the model parameters. This method also allows for asynchronous learning, where different agents learn with models of different parameters at every iteration. Recent frameworks for unsupervised learning such as Generative Adversarial Networks (GANs) where the objective is to train learning models to be robust against adversarial attacks (see [19]). [20] propose methods for scaling adversarial training and learning to large datasets performed on a distributed setup where each agent trains a discriminator model on a local training data and a shared generator model is trained based on the feedback received from the discriminator models of the different agents. These applications need sharing of a high dimensional parameter vector, usually of the order of $32\times 32\times 3\sim 1024\times 1024\times 3$. Applications such as spectrum sensing in Ad hoc cognitive radio networks, see [21], distributed detection of malicious attacks in finite-time consensus networks [22] and control of autonomous agents like unmanned aerial vehicles (UAVs) for search and survey operations, see [10] also depend on implementing higher dimensional consensus protocols.

Termination of consensus algorithms in finite time provides an advantage of getting an approximate consensus while saving valuable computation and communication resources. For the scalar average consensus protocols discussed earlier, the authors in [23, 24] have proposed such a finite time stopping criteria utilizing two additional states namely the global maximum and global minimum over the network. This allows each agent to distributively detect the convergence to the (approximate) average and terminate further computations. The works in [25, 26] generalize this result to the cases of dynamic interconnection topology and communication delays. The authors in [27] present a method based on the minimal polynomial associated with the weight matrix in the state update iterations to achieve the consensus value in a finite number of iterations. However, to calculate the coefficients of the minimal polynomial each node has to run $n$ (total number of agents) different linear iterations each for at least $n+1$ time-steps.

The protocol developed here is based on a new geometric insight into the behavior of push-sum or ratio consensus algorithms that is of independent interest. We prove that many popular consensus algorithms fit into a general class of algorithms which we call “convex decreasing”. We demonstrate that convex decreasing consensus algorithms satisfy monotone convergence properties, which we leverage to give guarantees on network configurations. That is, if it can be determined that all nodes of convex decreasing consensus algorithm belongs to a convex set, then all nodes of the algorithm will remain in said convex set. We develop distributed stopping criteria based on this geometric insight.

Centralized algorithms for finding the convex hull of a finite set of $n$ points in a plane have been long proposed. Such algorithms have the worst-case running time of $\mathcal{O}(n~{}\log n)$, which is also the best achievable performance for obtaining the ordered hull (see [28]). However, with the recent advancements in distributed multi-agent systems, the problem of estimating the convex hull in a distributed manner, in the absence of a centralized entity, has become important. To this regard, distributed algorithms to estimate convex hull have been proposed in the literature for applications such as in classification problems, locational region estimation and formation control to name a few. In classification problems, [29, 30] have proposed an one-class scaled distributed convex hull algorithm where the geometric structure of the convex hull is used to model the boundary of the target class defining the one-class. However an approximation of the convex hull in the original large dimensional space by means of randomly projected decisions on 2-D spaces results in a residual error when applied to finite time applications as analyzed in [31]. In order to obtain the convex hull of the feature space (kernel space) by communicating the extremities, [32] employed a quadratic programming approach. However, the computational complexity of the proposed solution is limiting when extended to higher dimensions. [33] proposed a convex hull algorithm, but require the assumption that the feature space is generated from a Gaussian mixture model and thus is limited to applications of a special class of support vector machines. A difference in paradigm between the convex hull estimation pursued here and the literature on distributed programming for computation of convex hulls, is that the current article constructs a protocol that can be implemented in a plug and play manner in the absence of a central authority or knowledge of the network.

The problem of computing a specified function of the sensor measurements is common in wireless sensor networks, see [34], [35]. The setup in [34] focuses on the problem of determining an arbitrary function of the sensor measurements from a specified sink node. The article studies the maximum rate at which the function can be calculated and communicated to the sink node. The article provides a characterization of the achievable rates for thee different classes of functions. The authors in [35] proposed a method to calculate any arbitrary function in networks utilizing a linear iteration. The authors show that the proposed method can be modeled as a dynamical system. Based on the structured system observability approach it is shown that the linear iterations can determine any specified function of the initial values of the nodes after observing the evolution of the agent values for a large but finite time-steps. However, the method is based on forming observability matrices for all agents in the network; here the method poses limitations as the size of the network increases leading to a large amount of computation and storage requirement. Further, the linear iterations require doubly-stochastic matrices for the agent state updates. The need of doubly-stochastic matrices makes the method in [35] not applicable to directed networks. In this article we propose a distributed algorithm to determine any arbitrary function of the initial values of the agents, applicable to general connected directed graph topologies. Moreover, the proposed method has a fixed storage requirement and the communication overhead of the proposed method compared to the existing methods is insignificant.

In this article, we present a distributed stopping criteria for the higher dimensional consensus problem. Our first progress in this direction can be found in [36], where we investigated distributed stopping algorithm for the special case of “ratio consensus”. Here we present a general theory of convex monotonicity, and demonstrate that one can apply the distributed termination techniques to any algorithm satisfying this criteria. In particular we show that in popular consensus algorithms (ratio consensus and row stochastic for example), the evolution of the convex hull of network states (in any dimension) indexed by time form a nested sequence of convex sets. This motivates an algorithm for distributively computing the convex hull within a time that scales linear with diameter of the network. We further provide a simpler algorithm which guarantees the convergence of consensus algorithm in norm, within any given tolerance.

Statement of contribution:

1. 1.

   This article constructs a general mathematical framework for convergence of network algorithms. In particular a notion of monotoniticity satisfied by important consensus algorithms is introduced. We show ratio consensus [5] and row stochastic updating of a network to be examples of “convex decreasing” algorithms.

2. 2.

   A convex hull algorithm, of independent interest, is developed for distributed determination of the extreme points of a set of vectors in the absence of a central authority. In the context of a distributed convex decreasing algorithm, the hull algorithm can be used by the agents to obtain the convex hull by a fixed time $t$, and thus give guarantees on the state of the convex decreasing algorithm for all times $t^{\prime}\geq t$.

3. 3.

   Feasibility concerns for convex hull computation are addressed for high dimensional data, and an alternative lightweight (in the sense of both computational and communication cost) stopping criterion is given that guarantees finite convergence within an $\varepsilon$-threshold of consensus with respect to an arbitrary norm.

4. 4.

   As application of the theory developed, new stopping criteria are developed for consensus based least square estimation as well as distributed function calculation that give the agents convergence guarantees in a peer to peer network.

The rest of the paper is organized as follows. In Section II, the basic definitions needed for subsequent developments are presented. Further, we discuss the setup for the distributed average consensus in higher dimensions (called the vector consensus problem) using ratio consensus. Sections III presents an analysis on the polytopes of the network states generated in the ratio consensus algorithm. Section V establishes a norm-based finite-time termination criterion for the vector consensus problem. Theoretical findings are validated with simulations presented in Section VII followed by conclusions in Section VIII.

The following theorem shows that consensus algorithms that are convex decreasing converge to the same finite limit at all nodes, and that if one sets an threshold for convergence with an open set about the consensus, the threshold will be met in finite time. Further, when the threshold set is assumed convex, it is proven that if all agents possess a value within the set, their updated values remain within this threshold. In this sense, convex threshold sets provide a guarantee on future behavior of the network.

Theorem III.1 allows us to provide stopping guarantees to convex decreasing consensus algorithms, particularly useful in distributed contexts. We will show that in both consensus frameworks (9) and (10) the network states $\{r^{i}(k)\}_{i=1}^{N}$ and $\{z^{i}(k)\}_{i=1}^{N}$ at time $k$ define a sequence of polytopes $\{r(k)\}_{k=0}^{\infty}$ and $\{z(k)\}_{k=0}^{\infty}$ respectively defined to be

that are convex decreasing.

Below in Figure 1, we briefly illustrate the result in Thoerem III.2 via an example, for the update $z(k+1)={\bf A}z(k)$ with ${\bf A}$ being row stochastic a 30 node Erdos-Rényi graph is initialized with values chosen uniformly at random from $(0,1)^{2}$. The initial values are displayed in red, with the boundary of the convex hull traced in black. The points of $z(0)$ are displayed in red, $z(1)$ in blue, $z(2)$ in green, with the convex hull boundaries of the respective sets traced in black. That row stochastic updating is convex decreasing is instantiated in the nested-ness of the sets $z(i)$.

When $d=1$, the convex hull is simply described, $co(r(k))=[\min_{j}\{r^{j}(k)\}_{j=1}^{n},\max_{j}\{r^{j}(k)\}_{j=1}^{n}]$, so that $co(r(k))\supseteq co(r(k+1))$ gives the monotonicity results from [26], $\min_{j}\{r^{j}(k)\}_{j=1}^{n},\leq\min_{j}\{r^{j}(k+1)\}_{j=1}^{n}$ and $\max_{j}\{r^{j}(k)\}_{j=1}^{n},\geq\max_{j}\{r^{j}(k+1)\}_{j=1}^{n}$. Analogously, applying Theorem III.2 in the one dimensional case delivers the monotonicity of min and max from [23]. In the $d$-dimensional case, taking $\mathcal{F}$ to be the family of all rectangular sets recovers Theorem $4$ of [13].

We will use the monotonicity of convex decreasing consensus algorithms to develop a distributed stopping criteria, guaranteeing convergence of all nodes within an $\epsilon$-ball of the consensus value for a general norm. First we develop a distributed convex hull computation algorithm that is of independent interest. Here, given a function $w:{\mathcal{V}}\to\mathbb{R}^{d}$ the convex hull $co(w)\coloneqq co(\{w(i)\}_{i=1}^{n})$ is to be determined by the network in a distributed fashion. Consider a algorithm where at stage one, agents share their value $w(j)$ with neighbors. The nodes then update their approximation of the convex hull, by determining the extreme points among all values received from their neighbors and their own. This new set of extreme points is communicated to all neighbors and then the process repeats. We show that after $D$ ($D$ being the diameter of the network) iterations, every node will have determined the extreme points of $w$.

In the context of a consensus algorithm protocol $w:{\mathcal{V}}\to\mathbb{R}^{d}$ represents $w(i)=r^{i}(k)$ or $z^{i}(k)$ the value at node $i$ in iteration $k$ of a consensus algorithm, and we implement the following stopping criterion. Given a norm $\|\cdot\|$ and a tolerance $\varepsilon$, implement the convex hull algorithm at time $k$, then at time $k+D$, at a node $j$ if $\max_{e(i),e(j)\in\mathscr{E}(w)}\|e(i)-e(j)\|\leq\varepsilon$, then stop the consensus algorithm. In what follows we demonstrate that upon stopping every node is within $\varepsilon$ of the consensus value in norm.

We now describe a finite time algorithm for distributed convex hull computation. Suppose that $|\mathcal{V}|$ agents indexed by $i$ where each agent has a set $S_{i}$ of elements of $\mathbb{R}^{d}$. The agent can communicate with each other while respecting constraints imposed by a specific communication network. We provide a distributed consensus algorithm through which all agents obtain $\mathscr{E}\left(\cup_{i}S_{i}\right)$ in $D$-iterations of the algorithm. If we let $E$ denote the space of all finite sequences of elements of $\mathbb{R}^{d}$, the convex hull algorithm can be understood as a distributed consensus algorithm.

That is, at iteration $t$, an agent $i$ receives the extreme points known to their “in-neighbors” and forms a new set $s_{i}(t)$ comprised of their previous extreme points and their neighbors. Agent $i$ finds the extreme points of this new set, and then communicates the set to its “out-neighbors” to initiate another iteration.