FROST – Fast row-stochastic optimization
with uncoordinated step-sizes
In this paper, we discuss distributed optimization over directed graphs, where doubly-stochastic weights cannot be constructed. Most of the existing algorithms overcome this issue by applying push-sum consensus, which utilizes column-stochastic weights. The formulation of column-stochastic weights requires each agent to know (at least) its out-degree, which may be impractical in e.g., broadcast-based communication protocols. In contrast, we describe FROST (Fast Row-stochastic-Optimization with uncoordinated STep-sizes), an optimization algorithm applicable to directed graphs that does not require the knowledge of out-degrees; the implementation of which is straightforward as each agent locally assigns weights to the incoming information and locally chooses a suitable step-size. We show that FROST converges linearly to the optimal solution for smooth and strongly-convex functions given that the largest step-size is positive and sufficiently small.
In this paper, we study distributed optimization, where agents are tasked to solve the following problem:
where each objective, , is private and known only to agent . The goal of the agents is to find the global minimizer of the aggregate cost, , via local communication with their neighbors and without revealing their private objective functions. This formulation has recently received great attention due to its extensive applications in e.g., machine learning [1, 2, 3, 4, 5], control , cognitive networks, [7, 8], and source localization [9, 10].
Early work on this topic includes Distributed Gradient Descent (DGD) [11, 12], which is computationally simple but is slow due to a diminishing step-size. The convergence rates are for general convex functions and for strongly-convex functions, where is the number of iterations. With a constant step-size, DGD converges faster albeit to an inexact solution [13, 14]. Related work also includes methods based on the Lagrangian dual [15, 16, 17, 18]to achieve faster convergence, albeit at the expense of more computation. To achieve both fast convergence and computational simplicity, some fast distributed first-order methods have been proposed. A Nesterov-type approach  achieves for smooth convex functions with bounded gradient assumption. EXTRA  exploits the difference of two consecutive DGD iterates to achieves a linear convergence to the optimal solution. Exact Diffusion [21, 22] applies an Adapt-then-Combine structure  to EXTRA and generalizes the symmetric doubly-stochastic weights required in EXTRA to locally-balanced row-stochastic weights over undirected graphs. Of significant relevance to this paper is a distributed gradient tracking technique built on dynamic consensus , which enables each agent to asymptotically learn the gradient of the global objective function. This technique was first proposed simultaneously in [25, 26]. Refs. [25, 27] combine it with the DGD structure to achieve improved convergence for smooth and convex problems. Refs. [26, 28], on the other hand, propose the NEXT framework for a more general class of non-convex problems.
All of the aforementioned methods assume that the multi-agent network is undirected. In practice, it may not be possible to achieve undirected communication. It is of interest, thus, to develop algorithms that are fast and are applicable to arbitrary directed graphs. The challenge here lies in the fact that doubly-stochastic weights, standard in many distributed optimization algorithms, cannot be constructed over arbitrary directed graphs. In particular, the weight matrices in directed graphs can only be either row-stochastic or column-stochastic, but not both.
We now discuss related work on directed graphs. Early work based on DGD includes subgradient-push [29, 30] and Directed-Distributed Gradient Descent (D-DGD) [31, 32], with a sublinear convergence rate of . Some recent work extends these methods to asynchronous networks [33, 34, 35]. To accelerate the convergence, DEXTRA  combines push-sum  and EXTRA  to achieve linear convergence given that the step-size lies in some non-trivial interval. This restriction on the step-size is later relaxed in ADD-OPT/Push-DIGing [38, 39], which linearly converge for a sufficiently small step-size. Of relevance is also , where distributed non-convex problems are considered with column-stochastic weights. More recent work [41, 42] proposes the and algorithms, which employ both row- and uncoordinated- stochastic weights to achieve (accelerated) linear convergence over arbitrary strongly-connected graphs. Note that although the construction of doubly-stochastic weights is avoided, all of the aforementioned methods require each agent to know its out-degree to formulate doubly- or column-stochastic weights. This requirement may be impractical in situations where the agents use a broadcast-based communication protocol. In contrast, Refs. [43, 44] provide algorithms that only use row-stochastic weights. Row-stochastic weight design is simple and is further applicable to broadcast-based methods.
In this paper, we focus on optimization with row-stochastic weights following the recent work in [43, 44]. We propose a fast optimization algorithm, termed as FROST (Fast Row-stochastic Optimization with uncoordinated STep-sizes), which is applicable to both directed and undirected graphs with uncoordinated step-sizes among the agents. Distributed optimization (based on gradient tracking) with uncoordinated step-sizes has been previously studied in [25, 45, 46], over undirected graphs with doubly-stochastic weights, and in , over directed graphs with column-stochastic weights. These works introduce a notion of heterogeneity among the step-sizes, defined respectively as the relative deviation of the step-sizes from their average in [48, 45], and as the ratio of the largest to the smallest step-size in [46, 47]. It is then shown that when the heterogeneity is small enough, i.e., the step-sizes are very close to each other, and when the largest step-size follows a bound as a function of the heterogeneity, the proposed algorithms linearly converge to the optimal solution. A challenge in this formulation is that choosing a sufficiently small, local step-size does not ensure small heterogeneity, while no step-size can be chosen to be zero. In contrast, a major contribution of this paper is that we establish linear convergence with uncoordinated step-sizes when the upper bound on the step-sizes is independent of any notion of heterogeneity. The implementation of FROST therefore is completely local, since each agent locally chooses a sufficiently small step-size, independent of other step-sizes, and locally assigns row-stochastic weights to the incoming information. In addition, our analysis shows that all step-sizes except one can be zero for the algorithm to work, which is a novel result in distributed optimization. We show that FROST converges linearly to the optimal solution for smooth and strongly-convex functions.
Notation: We use lowercase bold letters to denote vectors and uppercase italic letters to denote matrices. The matrix, , represents the identity, whereas () is the -dimensional uncoordinated vector of all ’s (’s). We further use to denote an -dimensional vector of all ’s except at the th location. For an arbitrary vector, , we denote its th element by and is a diagonal matrix with on its main diagonal. We denote by , the Kronecker product of two matrices, and . For a primitive, row-stochastic matrix, , we denote its left and right Perron eigenvectors by and , respectively, such that ; similarly, for a primitive, column-stochastic matrix, , we denote its left and right Perron eigenvectors by and , respectively, such that . For a matrix, , we denote as its spectral radius and as a diagonal matrix consisting of the corresponding diagonal elements of . The notation denotes the Euclidean norm of vectors and matrices, while denotes the Frobenius norm of matrices. Depending on the argument, we denote either as a particular matrix norm, the choice of which will be clear in Lemma 1, or a vector norm that is compatible with this matrix norm, i.e., for all matrices, , and all vectors, .
We now describe the rest of the paper. Section II states the problem and assumptions. Section III reviews related algorithms that use doubly-stochastic or column-stochastic weights and shows the intuition behind the analysis of these types of algorithms. In Section IV, we provide the main algorithm, FROST, proposed in this paper. In Section V, we develop the convergence properties of FROST. Simulation results are provided in Section VI and Section VII concludes the paper.
Ii Problem Formulation
Consider agents communicating over a strongly-connected network, , where is the set of agents and is the set of edges, , such that agent can send information to agent , i.e., . Define as the collection of in-neighbors, i.e., the set of agents that can send information to agent . Similarly, as the set of out-neighbors of agent . Note that both and include agent . The agents are tasked to solve the following problem:
where is a private cost function only known to agent . We denote the optimal solution of P1 as . We will discuss different distributed algorithms related to this problem under the applicable set of assumptions, described below.
The graph, , is undirected and connected.
The graph, , is directed and strongly-connected.
Each local objective, , is convex with bounded subgradient.
Each local objective, , is smooth and strongly-convex, i.e., and ,
there exists a positive constant such that
there exists a positive constant such that
Clearly, the Lipschitz-continuity and strong-convexity constants for the global objective function, , are and , respectively.
Each agent in the network has and knows its unique identifier, e.g., .
Each agent knows its out-degree in the network, i.e., the number of its out-neighbors.
Iii Related work
In this section, we discuss related distributed first-order methods and provide an intuitive explanation for each one of them.
Iii-a Algorithms using doubly-stochastic weights
A well-known solution to distributed optimization over undirected graphs is Distributed Gradient Descent (DGD) [11, 12], which combines distributed averaging with a local gradient step. Each agent maintains a local estimate, , of the optimal solution, , and implements the following iteration:
where is doubly-stochastic and respects the graph topology. The step-size is diminishing such that and . Under the Assumptions A1, A3, and A6, DGD converges to at the rate of . The convergence rate is slow because of the diminishing step-size. If a constant step-size is used in DGD, i.e., , it converges faster to an error ball, proportional to , around [13, 14]. This is because is not a fixed-point of the above iteration when the step-size is a constant.
To accelerate the convergence, Refs. [25, 27] recently propose a distributed first-order method based on gradient tracking, which uses a constant step-size and replaces the local gradient, at each agent in DGD, with an asymptotic estimator of the global gradient111EXTRA  is another related algorithm, which uses the difference between two consecutive DGD iterates to achieve linear convergence to the optimal solution.. The algorithm is updated as follows [25, 27]:
initialized with and an arbitrary at each agent. The first equation is essentially a descent method, after mixing with neighboring information, where the descent direction is , instead of as was in Eq. (1). The second equation is a global gradient estimator when viewed as dynamic consensus , i.e., asymptotically tracks the average of local gradients: . It is shown in Ref. [27, 45, 39] that converges linearly to under Assumptions A1, A4, A6, with a sufficiently small step-size, . Note that these methods, Eq. (1) and Eqs. (2a)-(2b), are not applicable to directed graphs as they require doubly-stochastic weights.
Iii-B Algorithms using column-stochastic weights
where . From Eq. (3), it is clear that the average of the estimates, , converges to , as Eq. (3) can be viewed as a centralized gradient method if each local estimate converges to . However, since the weight matrix is not row-stochastic, the estimates of agents will not reach an agreement . This discussion motivates combining DGD with an algorithm, called push-sum, briefly discussed next, that enables agreement over directed graphs with column-stochastic weights.
Iii-B1 Push-sum consensus
Push-sum [53, 37] is a technique to achieve average-consensus over arbitrary digraphs. At time , each agent maintains two state vectors, , , and an auxiliary scalar variable, , initialized with . Push-sum performs the following iterations:
where is column-stochastic. Eq. (4a) can be viewed as an independent algorithm to asymptotically learn the right Perron eigenvector of ; recall that the right Perron eigenvector of is not because is not row-stochastic and we denote it by . In fact, it can be verified that and that . Therefore, the limit of , as the ratio of over , is the average of the initial values:
initialized with and an arbitrary at each agent. The step-size, , satisfies the same conditions as in DGD. To understand these iterations, note that Eqs. (5a)-(5c) are nearly the same as Eqs. (4a)-(4c), except that there is an additional gradient term in Eq. (5b), which drives the limit of to . Under the Assumptions A2, A3 and A6, subgradient-push converges to at the rate of . For extensions of subgradient-push to asynchronous networks, see recent work [33, 34, 35]. We next describe an algorithm that significantly improves this convergence rate.
ADD-OPT , extended to time-varying graphs in Push-DIGing , is a fast algorithm over directed graphs, which converges at a linear rate to under the Assumptions A2, A4, and A6, in contrast to the sublinear convergence of subgradient-push. The three vectors, , , , and a scalar maintained at each agent , are updated as follows:
where each agent is initialized with , , and an arbitrary . We note here that ADD-OPT/Push-DIGing essentially applies push-sum to the algorithm in Eqs. (2a)-(2b), when the doubly-stochastic weights therein are replaced by column-stochastic weights.
Iii-B4 The algorithm
As we can see, subgradient-push and ADD-OPT/Push-DIGing, described before, have a nonlinear term that comes from the division by the eigenvector estimation. In contrast, the algorithm, introduced in  and extended to with the addition of a heavy-ball momentum term in  and to time-varying graphs in , removes this nonlinearity and remains applicable to directed graphs by a simultaneous application of row- and column-stochastic weights222See [31, 32] for related work with sublinear rate based on surplus consensus .. Each agent maintains two variables: , , where, as before, is the estimate of , and tracks the average gradient, . The algorithm, initialized with and arbitrary at each agent, performs the following iterations.
where is row-stochastic and is column-stochastic. It is shown that converges linearly to for sufficiently small step-sizes under the Assumptions A2, A4 and A6 . Therefore, can be viewed as a generalization of the algorithm in Eqs. (2a)-(2b) as the doubly-stochastic weights therein are replaced by row- and column-stochastic weights. Furthermore, it is shown in  that ADD-OPT/Push-DIGing in Eqs. (6a)-(6d) in fact can be derived from an equivalent form of after a state transformation on the -update; see  for details. For applications of the algorithm to distributed least squares, see, for instance, .
Iv Algorithms using Row-stochastic Weights
All of the aforementioned methods require at least each agent to know its out-degree in the network in order to construct doubly or column-stochastic weights. This requirement may be infeasible, e.g., when agents use broadcast-based communication protocols. Row-stochastic weights, on the other hand, are easier to implement in a distributed manner as every agent locally assigns an appropriate weight to each incoming variable from its in-neighbors. In the next section, we describe the main contribution of this paper, i.e., a fast optimization algorithm that uses only row-stochastic weights and uncoordinated step-sizes.
To motivate the proposed algorithm, we first consider DGD in Eq. (1) over directed graphs when the weight matrix in DGD is chosen to be row-stochastic, but not column-stochastic. From consensus arguments and the fact that the step-size goes to , it can be verified that the agents achieve agreement. However, this agreement is not on the optimal solution. This can be shown  by defining an accumulation state, , where is the left Perron eigenvector of the row-stochastic weight matrix, to obtain
It can be verified that the agents agree to the limit of the above iteration, which is suboptimal since this iteration minimizes a weighted sum of the objective functions and not the sum. This argument leads to a modification of Eq. (8) that cancels the imbalance in the gradient term caused by the fact that is not a vector of all ’s, a consequence of losing the column-stochasticity in the weight matrix. The modification, introduced in , is implemented as follows:
where is row-stochastic and the algorithm is initialized with and an arbitrary at each agent. Eq. (9a) asymptotically learns the left Perron eigenvector of the row-stochastic weight matrix , i.e., . The above algorithm achieves a sublinear convergence rate of under the Assumptions A2, A3, and A5, see  for details.
Iv-a FROST (Fast Row-stochastic Optimization with uncoordinated STep-sizes)
Based on the insights that gradient tracking and constant step-sizes provide exact and fast linear convergence, we now describe FROST that adds gradient tracking to the algorithm in Eqs. (9a)-(9b) while keeping constant but uncoordinated step-sizes at the agents. Each agent at the th iteration maintains three variables, , and . At -th iteration, agent performs the following update:
where ’s are the uncoordinated step-sizes locally chosen at each agent and the row-stochastic weights, , respect the graph topology such that:
The algorithm is initialized with an arbitrary , , and . We point out that the initial condition for Eq. (10a) and the divisions in Eq. (10c) require each agent to have a unique identifier. Clearly, Assumption A5 is applicable here. Note that Eq. (10c) is a modified gradient tracking update, first applied to optimization with row-stochastic weights in , where the divisions are used to eliminate the imbalance caused by the left Perron eigenvector of the (row-stochastic) weight matrix . We note that the algorithm in  requires identical step-sizes at the agents and thus is a special case of Eqs. (10a)-(10c).
For analysis purposes, we write Eqs. (10a)-(10c) in a compact vector-matrix form. To this aim, we introduce some notation as follows: let , , and collect the local variables , and in a vector in , respectively, and define
Since the weight matrix is primitive with positive diagonals, it is straightforward to verify that is invertible for any . Based on the notation above, Eqs. (10a)-(10c) can be written compactly as follows:
where , and is arbitrary. We emphasize that the implementation of FROST needs no knowledge of agent’s out-degree anywhere in the network in contrast to the earlier related work in [30, 29, 31, 32, 36, 38, 39, 41, 42]. Note that Refs. [21, 22] also use row-stochastic weights but require an additional locally-balanced assumption and are only applicable to undirected graphs.
V Convergence Analysis
Since is primitive and row-stochastic, from the Perron-Frobenius theorem , we note that , where is the left Perron eigenvector of .
V-a Auxiliary relations
We now start the convergence analysis with a key lemma regarding the contraction of the augmented weight matrix under an arbitrary norm.
Let Assumption A2 hold and consider the augmented weight matrix . There exists a vector norm, , such that ,
where is some constant.
As shown above, the existence of a norm in which the consensus process with row-stochastic matrix is a contraction does not follow the standard -norm argument for doubly-stochastic matrices [27, 39]. The ensuing arguments built on this notion of contraction under arbitrary norms were first introduced in  for column-stochastic weights and in  for row-stochastic weights; these arguments are harmonized later to hold simultaneously for both row- and column-stochastic weights in [41, 42]. The next lemma, a direct consequence of the contraction introduced in Lemma 1, is a standard result from consensus and Markov chain theory .
Consider , generated from the weight matrix . We have:
where is some positive constant and is the contraction factor defined in Lemma 1.
As a consequence of Lemma 2, we next establish the linear convergence of the sequences and .
The following inequalities hold : (a) ; (b) .
The proof of (a) is as follows:
where the last inequality uses Lemma 2 and the fact that . The result in (b) is straightforward by applying (a), i.e.,
which completes the proof. ∎
The next lemma presents the dynamics that govern the evolution of the weighted sum of ; recall that , in Eq. (11c), asymptotically tracks the average of local gradients, .
The following equation holds for all :
Recall that . We obtain from Eq. (11c) that
Doing this iteratively, we have that
With the initial conditions that and , we complete the proof. ∎
The next lemma, a standard result in convex optimization theory from , states that the distance to the optimal solution contracts in each step in the centralized gradient method.
Let and be the strong-convexity and Lipschitz-continuity constants for the global objective function, , respectively. Then and , we have
With the help of the previous lemmas, we are ready to derive a crucial contraction relationship in the proposed algorithm.
V-B Contraction relationship
Our strategy to show convergence is to bound , , and as a linear function of their values in the last iteration and ; this approach extends the work in  on doubly-stochastic weights to row-stochastic weights. We will present this relationship in the next lemmas. Before we proceed, we note that since all vector norms are equivalent in , there exist positive constants such that: First, we derive a bound for , the consensus error of the agents.
The following inequality holds, :
where is the equivalence-norm constant such that and is the largest step-size among the agents.
Next, we derive a bound for , i.e., the optimality gap between the accumulation state of the network, , and the optimal solution, .
If , the following inequality holds, :
where and is the equivalence-norm constant such that .
Recalling that and , We have the following:
Since the last term in the inequality above matches the second last term in Eq. (7), we only need to handle the first term. We further note that:
Now, we derive a upper bound for the first term in Eq. (V-B),
If , according to Lemma 5,
where . Next we derive a bound for .
where it is straightforward to bound as
Since and from Lemma 4, we have:
Next, we bound , the error in gradient estimation.
The following inequality holds, :