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1 SIAM J. OPTIM. Vol. 5, No., pp c 05 Society for Industrial and Applied Mathematics EXTRA: AN EXACT FIRSTORDER ALGORITHM FOR DECENTRALIZED CONSENSUS OPTIMIZATION WEI SHI, QING LING, GANG WU, AND WOTAO YIN Abstract. Recently, there has been growing interest in solving consensus optimization problems in a multiagent networ. In this paper, we develop a decentralized algorithm for the consensus optimization problem minimize x R p f(x) = n n i= f i(x), which is defined over a connected networ of n agents, where each function f i is held privately by agent i and encodes the agent s data and objective. All the agents shall collaboratively find the minimizer while each agent can only communicate with its neighbors. Such a computation scheme avoids a data fusion center or longdistance communication and offers better load balance to the networ. This paper proposes a novel decentralized exact firstorder algorithm (abbreviated as EXTRA) to solve the consensus optimization problem. Exact means that it can converge to the exact solution. EXTRA uses a fixed, large step size, which can be determined independently of the networ size or topology. The local variable of every agent i converges uniformly and consensually to an exact minimizer of f. In contrast, the wellnown decentralized gradient descent (DGD) method must use diminishing step sizes in order to converge to an exact minimizer. EXTRA and DGD have the same choice of mixing matrices and similar periteration complexity. EXTRA, however, uses the gradients of the last two iterates, unlie DGD which uses just that of the last iterate. EXTRA has the best nown convergence rates among the existing synchronized firstorder decentralized algorithms for minimizing convex Lipschitz differentiable functions. Specifically, if the f i s are convex and have Lipschitz continuous gradients, EXTRA has an ergodic convergence rate O( ) in terms of the firstorder optimality residual. In addition, as long as f is (restricted) strongly convex (not all individual f i s need to be so), EXTRA converges to an optimal solution at a linear rate O(C )forsomeconstantc>. Key words. consensus optimization, decentralized optimization, gradient method, linear convergence AMS subject classifications. 90C5, 90C30 DOI. 0.37/ X. Introduction. This paper focuses on decentralized consensus optimization, a problem defined on a connected networ and solved by n agents cooperatively, n (.) minimize f(x) = f i (x), x R p n i= over a common variable x R p, and for each agent i, f i : R p R is a convex function privately nown by the agent. We assume that the f i s are continuously differentiable and will introduce a novel firstorder algorithm to solve (.) in a decentralized manner. We stic to the synchronous case in this paper, that is, all the agents carry out their iterations at the same time intervals. Problems of the form (.) that require decentralized computation are found widely in various scientific and engineering areas including sensor networ information processing, multipleagent control and coordination, as well as distributed machine learning. Examples and wors include decentralized averaging [7, 5, 34], learning Received by the editors April 8, 04; accepted for publication (in revised form) January, 05; published electronically May 7, 05. This wor was supported by Chinese Scholarship Council (CSC) grants and , NSFC grant , MOF/MIIT/MOST grant BB000005, and NSF grants DMS and DMS Department of Automation, University of Science and Technology of China, Hefei, China Department of Mathematics, UCLA, Los Angeles, CA
2 EXACT ALGORITHM FOR DECENTRALIZED OPTIMIZATION 945 [9,, 6], estimation [,, 6, 8, 9], sparse optimization [9, 35], and lowran matrix completion [0] problems. Functions f i can tae the form of least squares [7, 5, 34], regularized least squares [,, 9, 8, ], as well as more general ones [6]. The solution x can represent, for example, the average temperature of a room [7, 34], frequencydomain occupancy of spectra [, ], states of a smart grid system [0, 6], sparse vectors [9, 35], a matrix factor [0], and so on. In general, decentralized optimization fits the scenarios in which the data are collected and/or stored in a distributed networ, a fusion center is either infeasible or not economical, and/or computing is required to be performed in a decentralized and collaborative manner by multiple agents... Related methods. Existing firstorder decentralized methods for solving (.) include the (sub)gradient method [, 5, 36], the (sub)gradientpush method [3, 4], the fast (sub)gradient method [5, 4], and the dual averaging method [8]. Compared to classical centralized algorithms, decentralized algorithms encounter more restrictive assumptions and typically worse convergence rates. Most of the above algorithms are analyzed under the assumption of bounded (sub)gradients. The wor [] assumes a bounded Hessian for strongly convex functions. Recent wor [36] relaxes such assumptions for decentralized gradient descent (DGD). When (.) has additional constraints that force x into a bounded set, which also leads to bounded (sub)gradients and a Hessian, projected firstorder algorithms are applicable [7, 37]. When using a fixed step size, these algorithms do not converge to a solution x of problem (.) but a point in its neighborhood regardless of whether the f i s are differentiable or not [36]. This motivates the use of certain diminishing step sizes in [5, 8, 4] to guarantee convergence to x. The rates of convergence are generally weaer than their analogues in centralized computation. For the general convex case and under the bounded (sub)gradient (or Lipschitz continuous objective) assumption, [5] shows that diminishing step sizes α = lead to a convergence rate of O( ln ) in terms of the running best of objective error, and [8] shows that the dual averaging method has a rate of O( ln ) in the ergodic sense in terms of objective error. For the general convex case, under assumptions of fixed step size and Lipschitz continuous, bounded gradient, [4] shows an outer loop convergence rate of O ( ) in terms of objective error, utilizing Nesterov s acceleration, provided that the inner loop performs substantial consensus computation. Without a substantial inner loop, the diminishing step sizes α = lead to a reduced rate of O( ln /3 ). The (sub)gradientpush method [3] can be implemented in a dynamic digraph and, under the bounded (sub)gradient assumption and diminishing step sizes α = O( ), has a rate of O( ln ) in the ergodic sense in terms of objective error. A better rate of O( ln )isproved for the (sub)gradientpush method in [4] under the strong convexity and Lipschitz gradient assumptions, in terms of expected objective error plus squared consensus residual. Some of the other related algorithms are as follows. For general convex functions and assuming closed and bounded feasible sets, the decentralized asynchronous alternating direction nethod of multipliers (ADMM) [3] is proved to have a rate of O( ) in terms of expected objective error and feasibility violation. The augmented Lagrangian based primaldual methods have linear convergence under strong convexity and Lipschitz gradient assumptions [4, 30] or under the positivedefinite bounded Hessian assumption [, 3]. Our proposed algorithm is a synchronous gradientbased algorithm that has an ergodic rate of O( ) (in terms of optimality condition violation) for general convex
3 946 WEI SHI, QING LING, GANG WU, AND WOTAO YIN objectives with Lipschitz differentials and has a linear rate once the sum of, rather than individual, functions f i is also (restricted) strongly convex... Notation. Throughout the paper, we let agent i hold a local copy of the global variable x, which is denoted by x (i) R p ; its value at iteration is denoted by x (i). We introduce an aggregate objective function of the local variables where n f(x) f i (x (i) ), i= x T () x T () x R n p.. x T (n) The gradient of f(x) is defined by T f (x () ) T f (x () ) f(x). T f n (x (n) ) Rn p. Each row i of x and f(x) is associated with agent i. We say that x is consensual if all of its rows are identical, i.e., x () = = x (n). The analysis and results of this paper hold for all p. The reader can assume p = for convenience (so x and f become vectors) without missing any major point. Finally, for given matrix A and symmetric positive semidefinite matrix G, we define the Gmatrix norm A G trace(a T GA). The largest singular value of a matrix A is denoted as σ max (A). The largest and smallest eigenvalues of a symmetric matrix B are denoted as λ max (B) andλ min (B), respectively. The smallest nonzero eigenvalue of a symmetric positive semidefinite matrix B 0 is denoted as λ min (B), which is strictly positive. For a matrix A R m n,null{a} {x R n Ax =0} is the null space of A and span{a} {y R m y = Ax, x R n } is the linear span of all the columns of A..3. Summary of contributions. This paper introduces a novel gradientbased decentralized algorithm EXTRA, establishes its convergence conditions and rates, and presents numerical results in comparison to DGD. EXTRA can use a fixed step size independent of the networ size and quicly converges to the solution to (.). It has a rate of convergence O( ) in terms of best running violation to the firstorder optimality condition when f is Lipschitz differentiable, and has a linear rate of convergence if f is also (restricted) strongly convex. Numerical simulations verify the theoretical results and demonstrate its competitive performance..4. Paper organization. The rest of this paper is organized as follows. Section develops and interprets EXTRA. Section 3 presents its convergence results. Then, section 4 presents three sets of numerical results. Finally, section 5 concludes this paper.
4 EXACT ALGORITHM FOR DECENTRALIZED OPTIMIZATION 947. Algorithm development. This section derives the proposed algorithm EX TRA. We start by briefly reviewing DGD and discussing the dilemma that DGD converges slowly to an exact solution when it uses a sequence of diminishing step sizes, yet it converges faster using a fixed step size but stalls at an inaccurate solution. We then obtain the update formula of EXTRA by taing the difference of two formulas of the DGD update. Provided that the sequence generated by the new update formula with a fixed step size converges to a point, we argue that the point is consensual and optimal. Finally, we briefly discuss the choice of mixing matrices in EXTRA. Formal convergence results and proofs are left to section 3... Review of decentralized gradient descent and its limitation. DGD carries out the following iteration, (.) x + (i) = n w ij x (j) α f i (x (i) ) j= for agent i =,...,n. Recall that x (i) Rp is the local copy of x held by agent i at iteration, W = [w ij ] R n n is a symmetric mixing matrix satisfying null{i W } =span{} and σ max (W n T ) <, and α > 0 is a step size for iteration. If two agents i and j are neither neighbors nor identical, then w ij = 0. This way, the computation of (.) involves only local and neighbor information, and hence the iteration is decentralized. Following our notation, we rewrite (.) for all the agents together as (.) x + = W x α f(x ). With a fixed step size α α, DGD has inexact convergence. For each agent i, x (i) converges to a point in the O(α)neighborhood of a solution to (.), and these points for different agents can be different. On the other hand, properly reducing α enables exact convergence, namely, that each x (i) converges to the same exact solution. However, reducing α causes slower convergence, both in theory and in practice. The wor [36] assumes that the f i s are Lipschitz continuous, and studies DGD with a constant α α. Before the iterates reach the O(α)neighborhood, the objective value reduces at the rate O( ), and this rate improves to linear if the f i s are also (restricted) strongly convex. In comparison, the wor [4] studies DGD with diminishing α = and assumes that the f /3 i s are Lipschitz continuous and bounded. The objective convergence rate slows down to O( ). The wor [5] studies DGD /3 with diminishing α = and assumes that the f / i s are Lipschitz continuous; a slower rate O( ln ) is proved. A simple example of decentralized least squares in section 4. gives a rough comparison of these three schemes (and how they compare to the proposed algorithm). To see the cause of inexact convergence with a fixed step size, letx be the limit of x (assuming the step size is small enough to ensure convergence). Taing the limit over on both sides of iteration (.) gives us x = W x α f(x ). When α is fixed and nonzero, assuming the consensus of x (namely, it has identical rows x (i) ) will mean x = W x,asaresultofw =, and thus f(x )=0, which is equivalent to f i (x (i) )=0 i, i.e., the same point x (i) simultaneously minimizes f i for all agents i. This is impossible in general and is different from our objective to find a point that minimizes n i= f i.
5 948 WEI SHI, QING LING, GANG WU, AND WOTAO YIN.. Development of EXTRA. The next proposition provides simple conditions for the consensus and optimality for problem (.). Proposition.. Assume null{i W } =span{}. If (.3) x x T () x T (). x T (n) satisfies conditions. x = W x (consensus),. T f(x )=0(optimality), then x = x (i), for any i, is a solution to the consensus optimization problem (.). Proof. Since null{i W } =span{}, x is consensual if and only if condition holds, i.e., x = W x.sincex is consensual, we have T f(x )= n i= f i(x ), so condition means optimality. Next, we construct the update formula of EXTRA, following which the iterate sequence will converge to a point satisfying the two conditions in Proposition.. Consider the DGD update (.) written at iterations + and as follows: (.4) (.5) x + = W x + α f(x + ), x + = W x α f(x ), where the former uses the mixing matrix W and the latter uses W = I + W. The choice of W will be generalized later. The update formula of EXTRA is simply their difference, subtracting (.5) from (.4), (.6) x + x + = W x + W x α f(x + )+α f(x ). Given x and x +, the next iterate x + is generated by (.6). Let us assume that {x } converges for now and let x = lim x. Let us also assume that f is continuous. We first establish condition of Proposition.. Taing in (.6) gives us (.7) x x =(W W )x α f(x )+α f(x ) from which it follows that (.8) W x x =(W W )x = 0. Therefore, x is consensual. Provided that T (W W ) = 0, we show that x also satisfies condition of Proposition.. To see this, adding the first update x = W x 0 α f(x 0 )tothe subsequent updates following the formulas of (x x ), (x 3 x ),...,(x + x + ) given by (.6) and then applying telescopic cancellation, we obtain (.9) x + = W + x + α f(x + )+ (W W )x t t=0
6 or, equivalently, EXACT ALGORITHM FOR DECENTRALIZED OPTIMIZATION 949 (.0) x + = W x + α f(x + )+ (W W )x t. Taing,fromx = lim x and x = W x = W x, it follows that (.) α f(x )= (W W )x t. t=0 Left multiplying T on both sides of (.), in light of T (W W ) = 0, we obtain the condition of Proposition.: (.) T f(x )=0. To summarize, provided that null{i W } =span{}, W = I+W, T (W W )= 0, and the continuity of f, if a sequence following EXTRA (.6) converges to a point x, then by Proposition., x is consensual and any of its identical row vectors solves problem (.)..3. The algorithm EXTRA and its assumptions. We present EXTRA an exact firstorder algorithm for decentralized consensus optimization in Algorithm. Algorithm. EXTRA. Choose α>0 and mixing matrices W R n n and W R n n ; Pic any x 0 R n p ;. x W x 0 α f(x 0 );. for =0,,... do x + (I + W )x + W x α [ f(x + ) f(x ) ] ; end for Breaing the computation to the individual agents, step of EXTRA performs updates x (i) = n j= w ij x 0 (j) α f i(x 0 (i) ), t=0 i =,...,n, and step at each iteration performs updates n n [ ] x + (i) = x + (i) + w ij x + (j) w ij x (j) α f i (x + (i) ) f i(x (i) ), i =,...,n. j= j= Each agent computes f i (x (i) )onceforeach and uses it twice for x+ (i) and x + (i). For our recommended choice of W =(W + I)/, each agent computes n j= w ijx (j) once as well. Here we formally give the assumptions on the mixing matrices W and W for EXTRA. All of them will be used in the convergence analysis in the next section. Assumption (mixing matrix). Consider a connected networ G = {V, E} consisting of a set of agents V = {,,...,n} and a set of undirected edges E. The mixing matrices W =[w ij ] R n n and W =[ w ij ] R n n satisfy the following.. (Decentralized property) If i j and (i, j) E,thenw ij = w ij =0.. (Symmetry) W = W T, W = W T.
7 950 WEI SHI, QING LING, GANG WU, AND WOTAO YIN 3. (Null space property) null{w W } =span{}, null{i W } span{}. 4. (Spectral property) W 0and I+W W W. We claim that parts 4 of Assumption imply null{i W } =span{} and the eigenvalues of W lie in (, ], which are commonly assumed for DGD. Therefore, the additional assumptions are merely on W. In fact, EXTRA can use the same W used in DGD and simply tae W = I+W, which satisfies part 4. It is also worth noting that in the recent wor pushdgd [3] relaxes the symmetry condition, yet such relaxation for EXTRA is not trivial and is our future wor. Proposition.. Parts 4 of Assumption imply null{i W } =span{} and that the eigenvalues of W lie in (, ]. Proof. Frompart4,wehave I+W W 0 and thus W I and λ min (W ) >. Also from part 4, we have I+W W and thus I W,whichmeansλ max (W ). Hence, all eigenvalues of W (and those of W ) lie in (, ]. Now, we show null{i W } =span{}. Consideranonzero vector v null{i W }, which satisfies (I W )v =0andthusv T (I W )v =0andv T v = v T W v. From I+W W (part 4), we get v T v = v T ( I+W )v v T W v, while from W W (part 4) we also get v T W v v T W v = v T v. Therefore, we have v T W v = v T v or equivalently ( W I)v = 0, adding which to (I W )v = 0 yields ( W W )v =0. In light of null{w W } = span{} (part 3), we must have v span{} and thus null{i W } =span{}. Note that it is easy to ensure W to be diagonally dominant, so that the range of eigenvalues in Proposition. will lie in [0, ]. Then, the simple choice W = I+W will have λ min ( W ), so that the step size α (given in (3.8) and Theorem 3.3 below) becomes independent of the networ topology..4. Mixing matrices. In EXTRA, the mixing matrices W and W diffuse information throughout the networ. The role of W is similar to that in DGD [5, 3, 36] and average consensus [33]. It has a few common choices, which can significantly affect performance. (i) Symmetric doubly stochastic matrix [5, 3, 36]: W = W T, W =, and w ij 0. Special cases of such matrices include parts (ii) and (iii) below. (ii) Laplacianbased constant edge weight matrix [8, 33], W = I L τ, where L is the Laplacian matrix of the graph G and τ > λ max(l) isa scaling parameter. Denote deg(i) as the degree of agent i. Whenλ max (L) is not available, τ =max i V {deg(i)} + ɛ for some small ɛ>0, say ɛ =,can be used. (iii) Metropolis constant edge weight matrix [3, 34], w ij = max{deg(i),deg(j)}+ɛ if (i, j) E, 0 if (i, j) / E and i j, if i = j w i V for some small positive ɛ>0. (iv) Symmetric fastest distributed linear averaging (FDLA) matrix. It is a symmetric W that achieves fastest information diffusion and can be obtained by a semidefinite program [33].
8 EXACT ALGORITHM FOR DECENTRALIZED OPTIMIZATION 95 It is worth noting that the optimal choice for average consensus, FDLA, no longer appears optimal in decentralized consensus optimization, which is more general. When W is chosen following any strategy above, W = I+W is found to be very efficient..5. EXTRA as corrected DGD. We rewrite (.0) as (.3) x + = W x α f(x ) }{{} DGD (W W )x t, =0,,... + t=0 } {{ } correction An EXTRA update is, therefore, a DGD update with a cumulative correction term. In subsection., we have argued that the DGD update cannot reach consensus asymptotically unless α asymptotically vanishes. Since α f(x )withafixedα>0 cannot vanish in general, it must be corrected, or otherwise x + W x does not vanish, preventing x from being asymptotically consensual. Provided that (.3) converges, the role of the cumulative term t=0 (W W )x t is to neutralize α f(x ) in (span{}), the subspace orthogonal to. If a vector v obeys v T (W W )=0, then the convergence of (.3) means the vanishing of v T f(x ) in the limit. We need T f(x ) = 0 for consensus optimality. The correction term in (.3) is the simplest that we could find so far. In particular, the summation is necessary since each individual term (W W )x t is asymptotically vanishing. The terms must wor cumulatively. 3. Convergence analysis. To establish convergence of EXTRA, this paper maes two additional but common assumptions as follows. Unless otherwise stated, the results in this section are given under Assumptions 3. Assumption (convex objective with Lipschitz continuous gradient). Objective functions f i are proper closed convex and Lipschitz differentiable: f i (x a ) f i (x b ) L fi x a x b x a,x b R p, where L fi 0areconstant. Following Assumption, function f(x) = n i= f i(x (i) )isproperclosedconvex, and f is Lipschitz continuous, f(x a ) f(x b ) F L f x a x b F x a, x b R n p, with constant L f =max i {L fi }. Assumption 3(solution existence). Problem (.) has a nonempty set of optimal solutions: X. 3.. Preliminaries. We first state a lemma that gives the firstorder optimality conditions of (.). Lemma 3. (firstorder optimality conditions). Given mixing matrices W and W, define U =( W W ) / by letting U VS / V T R n n,wherevsv T = W W is the economicalform singular value decomposition. Then, under Assumptions 3, x is consensual and x () x () x (n) is optimal to problem (.) if and only if there exists q = Up for some p R n p such that { (3.) Uq + α f(x )=0, (3.) Ux = 0.
9 95 WEI SHI, QING LING, GANG WU, AND WOTAO YIN Proof. AccordingtoAssumptionandthedefinitionofU, wehave null{u} =null{v T } =null{ W W } =span{}. Hence from Proposition., condition, x is consensual if and only if (3.) holds. Next, following Proposition., condition, x is optimal if and only if T f(x )= 0. Since U is symmetric and U T = 0, (3.) gives T f(x ) = 0. Conversely, if T f(x )=0,then f(x ) span{u} follows from span{u} =(span{}) and thus α f(x )= Uqfor some q. Let q =Proj U q. Then, Uq = Uq and (3.) holds. Let x and q satisfy the optimality conditions (3.) and (3.). Introduce auxiliary sequence q = Ux t and for each, ( (3.3) z q = x t=0 ) ( ) (, z q I 0 = x, G = 0 W The next lemma establishes the relations among x, q, x,andq. Lemma 3.. In EXTRA, the quadruple sequence {x, q, x, q } obeys (I + W (3.4) W )(x + x )+ W (x + x ) = U(q + q ) α[ f(x ) f(x )] for any =0,,... Proof. Similarly to how (.9) is derived, summing EXTRA iterations through +, x = W x 0 α f(x 0 ), x =(I + W )x W x 0 α f(x )+α f(x 0 ),..., x + =(I + W )x W x α f(x )+α f(x ), we get (3.5) x + = W x ( W W )x t α f(x ). t=0 Using q + = + t=0 Uxt and the decomposition W W = U, it follows from (3.5) that (3.6) (I + W W )x + + W (x + x )= Uq + α f(x ). Since (I + W W ) =0,wehave (3.7) (I + W W )x = 0. Subtracting (3.7) from (3.6) and adding 0 = Uq + α f(x ) to (3.6), we obtain (3.4). The convergence analysis is based on the recursion (3.4). Below we will show that x converges to a solution x X and z + z W converges to 0 at a rate of O( ) in an ergodic sense. Further assuming (restricted) strong convexity, we obtain the Qlinear convergence of z z G to 0, which implies the Rlinear convergence of x to x. ).
10 EXACT ALGORITHM FOR DECENTRALIZED OPTIMIZATION Convergence and rate. Let us first interpret the step size condition (3.8) α< λ min( W ) L f, which is assumed by Theorem 3.3 below. First of all, let W satisfy Assumption. It is easy to ensure λ min (W ) 0 since otherwise, we can replace W by I+W. In light of part 4 of Assumption, if we let W = I+W,thenwehaveλ min ( W ),which simplifies the bound (3.8) to α< L f, which is independent of any networ property (size, diameter, etc.). Furthermore, if L fi (i =,...,n) are in the same order, the bound L f has the same order as the bound /( n n i= L f i ), which is used in the (centralized) gradient descent method. In other words, a fixed and rather large step size is permitted by EXTRA. Theorem 3.3. Under Assumptions 3, if α satisfies 0 <α< λmin( W ) L f,then (3.9) z z G z + z G ζ z z + G, =0,,..., where ζ = αl f λ min( W ).Furthermore,z converges to an optimal z. Proof. Following Assumption, f is Lipschitz continuous and thus we have (3.0) α f(x ) f(x ) F L f α x x, f(x ) f(x ) = x + x,α[ f(x ) f(x )] +α x x +, f(x ) f(x ). Substituting (3.4) from Lemma 3. for α[ f(x ) f(x )], it follows from (3.0) that (3.) α f(x ) f(x ) F L f x + x,u(q q + ) + x + x, W (x x + ) x + x I+W W +α x x +, f(x ) f(x ). For the terms on the righthand side of (3.), we have (3.) (3.3) x + x,u(q q + ) = U(x + x ), q q + = Ux +, q q + = q + q, q q +, x + x, W (x x + ) = x + x, W (x x + ), where the second equality follows from Ux = 0 and (3.4) α x x +, f(x ) f(x ) αl f x x + F + α L f f(x ) f(x ) F.
11 954 WEI SHI, QING LING, GANG WU, AND WOTAO YIN Plugging (3.) (3.4) into (3.) and recalling the definitions of z, z,andg, we have (3.5) that is α f(x ) f(x ) F L f q + q, q q + + x + x, W (x x + ) x + x I+W W + αl f x x + F + α f(x ) f(x ) F L, f (3.6) 0 z + z,g(z z + ) x + x I+W W + αl f x x + F. Apply the basic equality z + z,g(z z + ) = z z G z+ z G z z + G to (3.6), we have (3.7) Define 0 z z G z+ z G z z + G x + x I+W W + αl f x x + F. ( I 0 G = 0 W αl f I ). By Assumption, in particular, I + W W 0, we have x + x I+W W 0 and thus z z G z + z G z z + G αl f x x + F = z z + G. Since α< λmin( W ) L f,wehaveg 0and (3.8) z z + G ζ z z + G, which gives (3.9). It shows from (3.9) that for any optimal z, z z G is bounded and contractive, so z z G is converging as z z + G 0. The convergence of z toasolutionz follows from the standard analysis for contraction methods; see, for example, Theorem 3 in []. To estimate the rate of convergence, we need the following result. Proposition 3.4. If a sequence {a } R obeys a 0 and t= a t <, then we have (i) lim a =0, (ii) t= a t = O( ), (iii) min t {a t } = o( ). Proof. Part (i) is obvious. Let b t= a t. By the assumptions, b is uniformly bounded and obeys Part (iii) is due to [6]. lim b <,
12 EXACT ALGORITHM FOR DECENTRALIZED OPTIMIZATION 955 from which part (ii) follows. Since c min {a t} is monotonically nonincreasing, we t have c = min t {a t} t=+ This and the fact that lim t=+ a t 0 give us c = o( ) or part (iii). Theorem 3.5. In the same setting as Theorem 3.3, the following rates hold: () runningaverage progress: ( ) z t z t+ G = O ; t= () runningbest progress: ( ) { min z t z t+ } G = o ; t (3) runningaverage optimality residuals: ( ) Uq t + α f(x t ) W = O t= and a t. ( ) Ux t F = O ; (4) runningbest optimality residuals: ( ) ( ) { min Uq t + α f(x t ) W } { = o and min Ux t } F = o. t t Proof. Parts () and (): Since the individual terms z z G converge to 0, we are able to sum (3.9) in Theorem 3.3 over =0through and apply the telescopic cancellation, i.e., (3.9) z t z t+ G = ( z t z G z t+ z ) z 0 z G = G <. δ δ t=0 t=0 Then, the results follow from Proposition 3.4 immediately. Parts (3) and (4): The progress z z + G can be interpreted as the residual to the firstorder optimality condition. In light of the firstorder optimality conditions (3.) and (3.) in Lemma 3., the optimality residuals are defined as Uq + α f(x ) W and Ux F.Furthermore, t= α T (Uq + α f(x )) = f (x () )+ + f n(x (n) ) is the violation to the firstorder optimality of (.), while Ux F is the violation of consensus. Below we obtain the convergence rates of the optimality residuals. Using the basic inequality a + b F ρ a F ρ b F which holds for any ρ> and any matrices a and b of the same size, it follows that (3.0) z z + G = q q + F + x x + W = x + W W + (I W )x + Uq + α f(x ) W x + W W + ρ Uq + α f(x ) W ρ (I W )x W.
13 956 WEI SHI, QING LING, GANG WU, AND WOTAO YIN Since W W and (I W ) W (I W ) are symmetric and null{ W W } null{(i W ) W (I W )}, there exists a bounded υ > 0 such that (I W )x W = x (I W ) W (I W ) υ x W W. It follows from (3.0) that (3.) z t z t+ G + x W W ( x t+ W W υ ) ρ xt W W + x W W t= + ρ Uqt + α f(x t ) W (Set ρ>υ+) t= = t= t= ( υ ρ + x+ W W + ) Ux t F t= ρ Uqt + α f(x t ) W. As part () shows that t= zt z t+ G = O( ), we have ( ) Uq t + α f(x t ) W = O t= and t= Uxt F = O( ). From (3.) and (3.9), we see that both Uq t + α f(x t ) W and Ux t F are summable. Again, by Proposition 3.4, we have part (4), the o( ) rate of running best firstorder optimality residuals. It is open whether z z + G is monotonic or not. If one can show its monotonicity, then the convergence rates will hold for the last point in the running sequence Linear convergence under restricted strong convexity. In this subsection we prove that EXTRA with a proper step size reaches linear convergence if the original objective f is restricted strongly convex. A convex function h : R p R is strongly convex if there exists μ>0 such that h(x a ) h(x b ),x a x b μ x a x b x a,x b R p. h is restricted strongly convex with respect to point x if there exists μ>0 such that h(x) h( x),x x μ x x x R p. For convenience in the proof, we introduce the function g(x) f(x)+ 4α x W W and claim that f is restricted strongly convex with respect to its solution x if and only if g is so with respect to x = (x ) T. Proposition 3.6. Under Assumptions and, the following two statements are equivalent: (i) The original objective f(x) = n n i= f i(x) is restricted strongly convex with respect to x. There are different definitions of restricted strong convexity. Ours is derived from [7].
14 EXACT ALGORITHM FOR DECENTRALIZED OPTIMIZATION 957 (ii) The penalized function g(x) =f(x)+ 4α x W W is restricted strongly convex with respect to x. In addition, the strong convexity constant of g is no less than that of f. See Appendix A for its proof. Theorem 3.7. If g(x) f(x) + 4α x W W is restricted strongly convex with respect to x with constant μ g > 0, then with proper step size α< μgλmin( W ) L f,there exists δ>0 such that the sequence {z } generated by EXTRA satisfies (3.) z z G ( + δ) z+ z G. That is, z z G converges to 0 at the Qlinear rate O(( + δ) ). Consequently, x x W converges to 0 at the Rlinear rate O(( + δ) ). Proof. Towardalowerboundof z z G z+ z G. From the definition of g and its restricted strong convexity, we have αμ g x + x F α x + x, g(x + ) g(x ) (3.3) = x + x W W +α x + x, f(x + ) f(x ) + x + x,α[ f(x ) f(x )]. Using Lemma 3. for α[ f(x ) f(x )] in (3.3), we get αμ g x + x F x + x W W +α x + x, f(x + ) f(x ) x + x I+W (3.4) W + x+ x,u(q q + ) + x + x, W (x x + ) = x + x ( W W ) (I+W W ) +α x+ x, f(x + ) f(x ) + x + x,u(q q + ) + x + x, W (x x + ). For the last three terms on the righthand side of (3.4), we have from Young s inequality, α x + x, f(x + ) f(x ) αη x + x F + α (3.5) η f(x+ ) f(x ) F αη x + x F + αl f η x+ x F, where η>0 is a tunable parameter and (3.6) x + x,u(q q + ) = q + q, q q + and (3.7) x + x, W (x x + ) = x + x, W (x x + ). Plugging (3.5) (3.7) into (3.4) and recalling the definition of z, z,andg, we obtain αμ g x + x F (3.8) x + x ( W W ) (I+W W ) + αη x+ x F + αl f η x+ x F + z+ z,g(z z + ).
15 958 WEI SHI, QING LING, GANG WU, AND WOTAO YIN By z + z,g(z z + ) = z z G z+ z G z z + G, (3.8) turns into z z G z+ z G x+ x α(μ g η)i ( W W )+(I+W W ) (3.9) + z z + G αl f η x x + F. Acriticalinequality.In order to establish (3.), in light of (3.9), it remains to show (3.30) x + x α(μ g η)i ( W W )+(I+W W ) + z z + G αl f η x x + F δ z + z G. With the terms of z, z in (3.30) expanded and from x + x W W = U(x + x ) F = Ux+ F = q+ q F, (3.30) is equivalent to (3.3) x + x α(μ g η)i+(i+w W ) δ W + x x + W αl f η I δ q+ q F, which is what remains to be shown below. That is, we must find an upper bound for q + q F in terms of x+ x F and x x + F. Establishing (3.3): Step. From Lemma 3. we have (3.3) U(q + q ) F = (I + W W )(x + x )+ W (x + x )+α[ f(x ) f(x )] F = (I + W W )(x + x )+α[ f(x + ) f(x )] + W (x + x )+α[ f(x ) f(x + )] F. From the inequality a+b+c+d F θ( β β a F +β b F )+ θ θ ( γ γ c F +γ d F ), which holds for any θ>, β>, γ>andany matrices a, b, c, d of the same dimensions, it follows that q + q W W ( ) β (3.33) θ β x+ x (I+W W + βα f(x + ) f(x ) ) F + θ ( ) γ θ γ x x + W + γα f(x ) f(x + ) F. By Lemma 3. and the definition of q, all the columns of q and q + lie in the column space of W W. This together with the Lipschitz continuity of f(x) turns (3.33) into (3.34) q + q F θ λ min ( W W ) θ + (θ ) λ min ( W W ) ( βλ max ((I + W W ) ) β ( γλ max ( W ) + γα L f γ ) + βα L f x + x F ) x x + F,
16 EXACT ALGORITHM FOR DECENTRALIZED OPTIMIZATION 959 where λ min ( ) gives the smallest nonzero eigenvalue. To mae a rather tight bound, we choose γ =+ σmax( W ) αl f and β =+ σmax(i+w W ) αl f in (3.34) and obtain q + q F θ(σ max(i + W W )+αl f ) (3.35) λ min ( W x + x F W ) + θ(σ max( W )+αl f ) (θ ) λ min ( W W ) x x + F. Establishing (3.3): Step. In order to establish (3.3), with (3.35), it only remains to show (3.36) (3.37) x + x α(μ g η)i+(i+w W ) δ W + x x + W αl f η ( θ(σmax (I + W δ W )+αl f ) λ min ( W x + x F W ) + θ(σ max( W )+αl f ) ) (θ ) λ min ( W W ) x x + F. To validate (3.36), we need α(μ g η)i +(I + W W ) δ W δθ(σmax(i+w W )+αl f ) λ min( W I, W ) W αl f η I δθ(σmax( W )+αl f ) (θ ) λ min( W W ) I, which holds as long as { α(μ g η) λ min ( δ min W W ) θ(σ max (I + W (3.38) W )+αl f ) + λ max ( W ) λ min ( W W ), (θ )(ηλ min ( W ) αl f ) λ min ( W W ) θη(σ max ( W )+αl f ) To ensure δ>0, the following conditions are what we finally need: ( ) (3.39) η (0, μ g ) and α 0, ηλmin( W ) L f S. Obviously set S is nonempty. Therefore, with a proper step size α S, the sequences z z G are Qlinearly convergent to 0 at the rate O((+δ) ). Since the definition of Gnorm implies x x W z z G, x x W is Rlinearly convergent to 0atthesamerate. Remar (strong convexity condition for linear convergence). The restricted strong convexity assumption in Theorem 3.7 is imposed on g(x) =f(x)+ 4α x W W, not on f(x). In other words, the linear convergence of EXTRA does not require all f i to be individually (restricted) strongly convex. Remar (acceleration by overshooting W ). For conciseness, we used Assumption for both Theorems 3.5 and 3.7. In fact, for Theorem 3.7, the condition I+W W in part 4 of Assumption can be relaxed, thans to μ g in (3.37). Certain W I+W,suchas W =.5I+W.5, can still give linear convergence. In fact, we observed that such an overshot choice of W can slightly accelerate the convergence of EXTRA. Remar 3(step size optimization). We tried deriving an optimal step size and corresponding explicit linear convergence rate by optimizing certain quantities that I }.
17 960 WEI SHI, QING LING, GANG WU, AND WOTAO YIN appear in the proof, but it becomes quite tricy and messy. For the special case W = I+W μg(+λmin(w )), by taing η μ g, we get a satisfactory step size α 4L f. Remar 4(step size for ensuring linear convergence). Interestingly, the critical step size, α< μgλmin( W ) L f = O( μg L f ), in (3.39) for ensuring the linear convergence, and λ the parameter α = min( W W ) μg (+ γ )(μ f L = O( f γ) L f ) in (A.7) for ensuring the restricted strong convexity with O(μ g )=O(μ f ), have the same order. On the other hand, we numerically observed that a step size as large as O( L f ) still leads to linear convergence, and EXTRA becomes faster with this larger step size. It remains an open question to prove linear convergence under this larger step size Decentralized implementation. We shall explain how to perform EX TRA with only local computation and neighbor communication. EXTRA s formula is formed by f(x), W x and W x, andα. By definition f(x) is local computation. Assumption part ensures that W x and W x can be computed with local and neighbor information. Following our convergence theorems above, determining α requires the bounds on L f and λ min ( W ), as well as that of μ g in the (restricted) strongly convex case. As we have argued at the beginning of subsection 3., it is easy to ensure λ min ( W ),soλ min( W ) can be conservatively set as. In many circumstances, the Lipschitz constants L fi and strong convexity constants, or their bounds, are nown a priori and globally. When they are not nown globally, to obtain L f =max i {L fi },a maximum consensus algorithm is needed. For μ f, except in the case that each f i is (restricted) strongly convex, we can conservatively use min i {μ fi }. When no bound μ g is available in the (restricted) strongly convex case, setting α according to the general convex case (subsection 3.) often still leads to linear convergence. 4. Numerical experiments. 4.. Decentralized least squares. Consider a decentralized sensing problem: Each agent i {,...,n} holds its own measurement equation, y (i) = M (i) x + e (i), where y (i) R mi and M (i) R mi p are measured data, x R p is an unnown signal, and e (i) R mi is unnown noise. The goal is to estimate x. We apply the least squares loss and try to solve minimize x f(x) = n n i= M (i)x y (i). The networ in this experiment is randomly generated with connectivity ratio r =0.5, where r is defined as the number of edges divided by L(L ), the number of all possible edges. We set n = 0, m i = i p=5. Datay (i) and M (i),aswellasnoisee (i) i are generated following the standard normal distribution. We normalize the data so that L f =. The algorithm starts from x 0 (i) =0 i and x x 0 (i) = 300. We use the same matrix W by strategy (iv) in section.4 for both DGD and EXTRA. For EXTRA, we simply use the aforementioned matrix W = I+W.Werun α DGD with a fixed step size α, a diminishing one [4], a diminishing one α0 /3 /3 with hand optimized α 0 α, a diminishing one [5], and a diminishing one α0 with / / hand optimized α 0,whereαisthetheoretical critical step size given in [36]. We let EXTRA use the same fixed step size α. The numerical results are illustrated in Figure. In this experiment, we observe that both DGD with the fixed step size and EXTRA show similar linear convergence
18 EXACT ALGORITHM FOR DECENTRALIZED OPTIMIZATION Fig.. Plot of residuals x x F x 0 x F. Constant α =0.576 is the theoretical critical step size given for DGD in [36]. For DGD with diminishing step sizes O(/ /3 ) and O(/ / ), we have hand optimized their initial step sizes as 3α and 5α, respectively. in the first 00 iterations. Then DGD with the fixed step size begins to slow down and eventually stall, and EXTRA continues its progress. 4.. Decentralized robust least squares. Consider the same decentralized sensing setting and networ as in section 4.. In this experiment, we use the Huber loss, which is nown to be robust to outliers, and it allows us to observe both sublinear and linear convergence. We call the problem decentralized robust least squares: minimize x f(x) = n n m i H ξ (M (i)j x y (i)j ), i= where M (i)j is the jth row of matrix M (i) and y (i)j is the jth entry of vector y (i).the Huber loss function H ξ is defined as { H ξ (a) = a for a ξ (l zone), ξ( a ξ) otherwise (l zone). We set ξ =. The optimal solution x is artificially set in the l zone while x0 (i) is set in the l zone at all agents i. Except for new handoptimized initial step sizes for DGD s diminishing step sizes, all other algorithmic parameters remain unchanged from the last test. The numerical results are illustrated in Figure. EXTRA has sublinear convergence for the fist 000 iterations and then begins linear convergence, as x (i) for most i enter the l zone Decentralized logistic regression. Consider the decentralized logistic regression problem minimize x f(x) = n n i= m i m i j= j= ln ( +exp ( )) (M (i)j x)y (i)j, where every agent i holds its training date (M (i)j,y (i)j ) R p {, +}, j =,...,m i, including explanatory/feature variables M (i)j and binary output/outcome y (i)j. To simplify the notation, we set the last entry of every M (i)j to, thus the last entry of x will yield the offset parameter of the logistic regression model.
19 96 WEI SHI, QING LING, GANG WU, AND WOTAO YIN Fig.. Plot of residuals x x F x 0 x F. Constant α =0.576 is the theoretical critical step size given for DGD in [36]. For DGD with diminishing step sizes O(/ /3 ) and O(/ / ), we have hand optimized their initial step sizes as 0α and 0α, respectively. The initial large step sizes have helped them (the red and purple curves) realize faster convergence initially. Fig. 3. Plot of residuals x x F x 0 x F. Constant α = is the theoretical critical step size given for DGD in [36]. For DGD with diminishing step sizes O(/ /3 ) and O(/ / ), we have hand optimized their initial step sizes as 0α and 0α, respectively. We show a decentralized logistic regression problem solved by DGD and EXTRA over a mediumscale networ. The settings are as follows. The connected networ is randomly generated with n = 00 agents and connectivity ratio r =0.. Each agent holds 0 samples, i.e., m i =0 i. The agents shall collaboratively obtain p =0 coefficients via logistic regression. All the 000 samples are randomly generated, and the reference (ground true) logistic classifier x is precomputed with a centralized method. As it is easy to implement in practice, we use the Metropolis constant edge weight matrix W, which is mentioned by strategy (iii) in section.4, with ɛ =, and we use W = I+W. The numerical results are illustrated in Figure 3. EXTRA outperforms DGD, showing linear and exact convergence to the reference logistic classifier x. 5. Conclusion. As one of the fundamental methods, gradient descent has been adapted to decentralized optimization, giving rise to simple and elegant iterations. In this paper, we attempted to address a dilemma or deficiency of the current DGD method: To obtain an accurate solution, it wors slowly as it must use a small step size or iteratively diminish the step size; a large step size will lead to faster convergence
20 EXACT ALGORITHM FOR DECENTRALIZED OPTIMIZATION 963 to, however, an inaccurate solution. Our solution is an exact firstorder algorithm, EXTRA, which uses a fixed large step size and quicly returns an accurate solution. The claim is supported by both theoretical convergence and preliminary numerical results. On the other hand, EXTRA is far from perfect, and more wor is needed to adapt it to the asynchronous and dynamic networ settings. They are interesting open questions for future wor. Appendix A. Proof of Proposition 3.6. Proof. (ii) (i) : By definition of restricted strong convexity, there exists μ g > 0so that for any x, μ g x x F g(x) g(x ), x x (A.) = f(x) f(x ), x x + α x x W W. For any x R p,setx = x T, and from the above inequality, we get μ g x x (A.) n f i (x) f i (x ),x x n i= = f(x) f(x ),x x. Therefore, f(x) is restricted strongly convex with a constant μ f μ g. (i) (ii) : For any x R n p, decompose x = u + v so that every column of u belongs to span{} (i.e., u is consensual) while that of v belongs to span{}. Such an orthogonal decomposition obviously satisfies x F = u F + v F. Since solution x is consensual and thus u x, v = 0, we also have x x F = u x F + v F. In addition, being consensual, u = ut for some u R p. From the inequalities f(u) f(x ), u x = n n f i (u) f i (x ),u x n i= nμ f u x = μ f u x F, f(x) f(u), x u 0, f(u) f(x ), x u L f u x F v F, f(x) f(u), u x L f v F u x F, we get (A.3) f(x) f(x ), x x = f(u) f(x ), u x + f(x) f(u), x u + f(u) f(x ), x u + f(x) f(u), u x μ f u x F L f u x F v F. In addition, from the fact that u x null{ W W } and v span{ W W }, it follows that (A.4) α x x W W = α v W W λ min ( W W ) v α F, where λ min ( ) gives the smallest nonzero eigenvalue of a positive semidefinite matrix.
21 964 WEI SHI, QING LING, GANG WU, AND WOTAO YIN Pic any γ>0. When v F γ u x F, it follows that (A.5) g(x) g(x ), x x = f(x) f(x ), x x + α x x W W μ f u x F L f u x F v F + λ min ( W W ) v F α (μ f L f γ) u x F + λ min ( W W ) v F { α min μ f L f γ, λ min ( W } W ) x x F α. (by (A.3) and (A.4)) When v F γ u x F, it follows that (A.6) g(x) g(x ), x x = f(x) f(x ), x x + α x x W W 0+ λ min ( W W ) v F α λ min ( W W ) α( + γ ) = λ min( W W ) α(+ γ ) x x F. (applied convexity of f and (A.4)) v F + λ min ( W W ) α( + u x F γ ) Finally, in all conditions, (A.7) g(x) g(x ), x x { min μ f L f γ, λ min ( W } W ) α( + x x F γ ) μ g x x F. By, for example, setting γ = μ f 4L f,wehaveμ g > 0. Hence, function g is restricted strongly convex for any α>0aslongasfunction f is restricted strongly convex. In the direction of (ii) (i), we find μ g <μ f, unlie the more pleasant μ f = μ g in the other direction. However, from (A.7), we have α= sup μ g = lim μ g γ,α γ 0 + λmin ( W W ) (+ γ )(μ L f f γ) = μ f, which means that μ g can be arbitrarily close to μ f as α goes to zero. On the other hand, just to have O(μ g )=O(μ f ), we can set γ = O( μ f L f )andα = λ min( W W ) (+ γ )(μ f L f γ) = O( μ f L f )=O( μg L f ). This order of α coincides, in terms of order of magnitude, with the critical step size for ensuring the linear convergence. REFERENCES [] J. Bazerque and G. Giannais, Distributed spectrum sensing for cognitive radio networs by exploiting sparsity, IEEE Trans. Signal Process., 58 (00), pp [] J. Bazerque, G. Mateos, and G. Giannais, GroupLasso on splines for spectrum cartography, IEEE Trans. Signal Process., 59 (0), pp
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