Projected Push-Sum Gradient Descent-Ascent for Convex
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Tatiana Zimmermann, Jan NTRODUCTION h transforma- the Hbi and “Hybrid 4 ndation n on ent gents mum sto ss ing ap- ion ng on he ix. he rd or of U x o s f , hl epciglclcntan yadn hmoe a over them adding by constraint constraint global local incorporate to respecting order while in projection wh algorithm, a descent-ascent uses gradient propose distributed push- we novel optimum, the a the on to rely convergence whic not For [15], consensus. do or sum but [4] in updates, ones row-stochastic the use to st also row- closer basic is single algorithm the the push- a of analysis, ture easier into the for algorithm employ matrix-update push-sum we stochastic However, the [13], diffusion. formulating information and by for Simil [5] consensus global. average in and sum approaches local the into constraints to expl the the to exploiting of by commu- ability distinction done directed is the This assuming architectures. while with nication constraints, projection number local the of parameter respect advantage reduced the combine methods’ to seeks work Our have need chosen. sequence they be parameter [13], to penalization no in that global. method advantage constrai projection-free the therefore local the with and agents to the agent Compared of privacy every the by restricts the This known is optimiz are consideration distributed the problem under of constraints DEDP all that the th assumption of to drawback relation structure in major specific methods The projection-based function. mentioned proximal the of convex a projection uses a with that combined consensus push-sum Finally the information. employs matrices gradient projected communication the row-stochastic in diffusing on for contributions rely The [15] and graphs. method [4] undirected proposed to the restri However, communication which matrix-updates, [8]. double-stochastic on in met relies published gradient distributed was projection-based, ods first the [17]. of trivial One non be optimally be to to sequences needs proved dependent parameter those penalization Choosing a determined. for this sequence sequence makes second step-size a the which problems, to optimization next local, However, DEDP. of the range be a including wide to method, a by to tha constraints applicable this is objective method all approach For the this considers of [17]. into it property in One incorporated function. analyzed are penalization published further constraints was and the information, [13] spreading in for respect pu to already algorithm the able uses have also are sum that that publications method methods projection-free of A order constraints. couple first on a focused hand, been other y tracking the gradient in On considered However been a not convergence. the have require for constraints for usually sequence used methods steps-size be order diminishing can first step-size while constant update, gradient a that advantage the Adamy n imization tthe ct [14] , ation ruc- nts. icit ich sh- et. h- ar s, h e s t , , Lagrange multiplier to the local cost functions. known by every agent. 0 Rm Within this article, we make the following contributions: After defining Mi = {µi ∈ |µi 0} and by using T n First, we provide a reformulation of the unconstrained push- µ = (µi )i=1, the dual function of the problem takes the sum consensus, which differs to the one published in e.g. form [5], and provide convergence properties of the update matrix. n T Secondly, we prove convergence of the distributed gradient q(µ) = inf Fi(x)+ µi gi(x) , (2) x∈X descent-ascent method to an optimal solution of the problem, (i=1 ) X respecting privacy of the local constraints. At last, we show that our proposed algorithm is applicable to the DEDP. with which the dual problem The paper is structured as follows: In section II we provide our notation for formulas and graphs. The main part begins max q(µ), (3a) µ in section III with the formulation of the problem class and 0 the results on the reformulation of the push-sum, before s.t. µ ∈M , (3b) our algorithm for solving the defined problems is proposed. 0 Rn×m 0 The subsequent section IV provides the convergence proof with M = {µ ∈ |µi ∈ Mi , i = [n]} can be of the proposed method. In section V we undergird the defined. In accordance to that, the global Lagrangian is the theoretic results by a simulation of a basic DEDP, before sum of the local Lagrangians we summarize and conclude our results in section VI. n n II. NOTATION AND GRAPHS T L(x, µ)= Li(x, µi)= Fi(x)+ µi gi(x). (4) Throughout the paper we use the following notation: The i=1 i=1 X X set of non-negative integers is denoted by Z+. All time indices t in this work belong to this set. To differentiate Before we continue with the analysis of the push-sum between multi-dimensional vectors and scalars we use bold- consensus for information diffusion, we make the following face. || · || denotes the standard euclidean norm. 1, ..., n is assumptions regarding the above problem: denoted by [n]. The element ij of matrix M is denoted Assumption 1. F is strongly convex on X and gi is convex by M . The operation is the projection of onto ij X (u) u for i = [n]. The optimal value F ∗ is finite and there is the convex set X such that (u) = arg min ||x − u||. ∗ ∗ Q X x∈X no duality gap, namely F = q . There exist compact sets Instances of a variable x at time t are denoted by x[t]. 0 Mi ⊂ M ,i = [n] containing the dual optimal vectors The directed graph Q consists of a set of vertices i G = {V, E} V µ∗,i = [n]. and a set of edges E = V×V. Vertex j can send information i to vertex i if (j, i) ∈E. The communication channels can be Assumption 2. X ⊂ Rd is convex and compact. described by the Perron matrix P , where Pij > 0 if (j, i) ∈ E and zero otherwise. This notation includes self-loops such Remark 1. Given the convexity properties of the problem and Slater’s constraint qualification, we have strong duality, that Pii > 0. The set containing the in-neighborhood nodes + − which implies that the duality gap is zero. Furthermore, the is Ni , while Ni is the set of out-neighbors. The out degree − optimal vector for µi is then uniformly bounded in the norm of node i is denoted with di = |Ni |. We say that a directed graph is strongly connected if there exists a path from every (see [2]). Thereby, Assumption 1 is given, for example, if vertex to every other vertex. Slater’s condition holds, F and gi,i = [n], are continuous and the domain X is compact. III. PROJECTED PUSH-SUMGRADIENTDESCENT-ASCENT Assumption 3. The gradients ∇ L (x, µ ), ∇ L (x, µ ) A. Problem formulation and matrix update of the push-sum x i i µi i i exist and are uniformly bounded on X and M , i.e. ∃L < consensus i x ∞,Lµi < ∞ such that ||∇xLi(x, µi)|| ≤ Lx for x ∈ We consider optimization problems of the form X , µi ∈ Mi and ||∇µi Li(x, µi)|| ≤ Lµi for x ∈ X , µi ∈ n Mi. min F (x) = min Fi(x), (1a) x x i=1 Remark 2. Note that this means that for either fixed µi X T or fixed x the Lagrangian L (x, µ ) is Lipschitz-continuous s.t. gi(x) ≤ 0, gi(x) = (gi1(x), ..., gim(x)) , (1b) i i with constant L , L , respectively. x ∈X ⊂ Rd, (1c) x µi Assumption 4. The Graph G = {V, E} is fixed and strongly where functions F : Rd → R and g : Rd → R i ij connected. The associated Perron matrix P is column- are differentiable for i = [n] and for j = [m]1. While we stochastic. consider gi(x) to be local constraint functions of agent i, the constraint set X is assumed to be global and therefore Remark 3. If, for example, agent i weights its messages by 1/(di), with di representing the out degree of i, the resulting 1For the sake of notation, we assume here that all agents have m local constraints. However, the following considerations hold for arbitrary, yet communication matrix contains the elements Pij = 1/di, finite numbers of local constraints that can differ between the agents. which achieves column-stochasticity of P . Recall the push-sum consensus protocol from [1], [3] holds, with w being the right eigenvector of P for the eigenvalue λ =1. Therefore, y[t +1]= Py[t], (5a) 1 −1 1 x[t +1]= Px[t], (5b) lim Φ(t, 0) = (diag[w]) w1T = 11T , t→∞ n n xi[t + 1] zi[t +1]= , (5c) which is a double-stochastic matrix. yi[t + 1] Part c): with initial states x[0] = z[0] = x0 and y[0] = 1. The Using the results from b), we can write for finite s
Following from implication (15), the third line converges t=0 s=0 X X to zero as well, which concludes the proof of part a). Applying this to our case, we conclude Part b): ∞ We have αtbt < ∞. ∞ ∞ ∞ t=0 α ||x [t] − x¯[t]|| ≤ α a + α b X t i t t t t Finally, we have t=0 t=0 t=0 X ∞ X nX ∞ ∞ x 1 x x x + α ||ǫ [t − 1]|| + ||ǫ [t − 1]|| (17) α ||ǫ [t − 1]|| ≤ α 1||ǫ [t − 1]|| < ∞, t i n i t i t− i t=0 i=1 ! t=0 t=0 X X X X with sequences where we used again the implication (16) and the fact that α is non-increasing. n n t 1 Therefore, at = Φ(t − 1, 0)ij − Φ(t − 1, 0)ij ||xj [0]|| ∞ =1 n =1 j i αt||xi[t] − x¯[t]|| < ∞ X X t=0 X and bt = holds, which concludes the proof. t−2 n n 1 Φ(t−1,s+1) − Φ(t−1,s+1) ǫx[s] . ij n ij j s=0 j=1 i=1 The above Lemma is necessary for the convergence proof X X X of the suggested algorithm. Next, we provide an upper bound From Lemma 2 a) we know that the sequence at can be for the algorithm updates in each time step. ′ bounded by a sequence at as follows Proposition 1. Let Assumptions 2, 3 and 4 hold. Then, for n ∗ ∗ t ′ the optimal primal dual pair x ∈ X , µi ∈Mi and t ≥ t0, at ≤ Cλ ||xj [0]|| = at. the following bound holds j=1 X n ∞ ′ ∗ 2 ∗ 2 The series t=0 at converges, as it is a geometric, conver- yi[t + 1]||xi[t + 1] − x || + ||µi[t + 1] − µi || gent series with 0 <λ< 1. By direct comparison test it i=1 ∞ X follows thatP converges as well, because n t=0 at 0 ≤ at ≤ ∗ 2 ∗ 2 ′ ≤ yi[t]||xi[t] − x || + ||µi[t] − µ || at. i Resulting fromP Assumption 5, α is a positive, non- i=1 t X ¯ ∗ ∗ ∗ ∗ ∗ ∗ increasing sequence. Therefore, there exists a 0
In (19d), Li(zi[t], µi[t]) depends affinely on µi[t] with fixed following Lemma, which is the deterministic version of a zi[t] and therefore Theorem in [10]: T ∗ ∞ ∞ ∞ ∞ ∇µi Li(zi[t], µi[t]) (µi[t] − µi ) Lemma 4. Let {vt}t=0, {ut}t=0, {bt}t=0 and {ct}t=0 ∞ ∗ be non-negative sequences such that t=0 bt < ∞ and = Li(zi[t], µi[t]) − Li(zi[t], µi ). ∞ ∗ ∗ ∗ ∗ t=0 ct < ∞ and Combing above results, adding +Li(x , µi ) − Li(x , µi ) P ¯ ∗ ¯ ∗ and +Li(x[t], µi ) − Li(x[t], µi ), as well as summing from P i =0 to n, we receive vt+1 ≤ (1 + bt)vt − ut + ct, ∀t ≥ 0. n ¯ ∗ ∗ ∗ ∞ (19c) + (19d) ≤−2αt L(x[t], µ ) − L(x , µ ) Then vt converges and t=0 ut < ∞. i=1 X ∗ ∗ ∗ This Lemma will beP the key element for proving the + L(x , µ ) − L(x , µ[t]) n following main result: ∗ ∗ − 2αt (Li(zi[t], µ ) − Li(x¯[t], µ )) . i i Theorem 1. Let Assumptions 1-5 hold. Then, xi[t] and µi[t], i=1 X updated by the rules in (11a) - (11d), converge to an optimal ∗ ∗ primal dual pair (x , µi ) ∈X×M for i = [n] as t → ∞. Proof. To apply Lemma 4 let us define Resulting from the strong convexity of F (x) over X , the ∗ ∗ ∗ ∗ n equality L(xˆ, µ ) = L(x , µ ) = minx L(x, µ ) implies ∗ 2 ∗ 2 ˆ ∗ ˆ ˆ ∗ vt = yi[t]||xi[t] − x || + ||µi[t] − µi || , that x = x . Due to dual feasibility of µ, x = x and ∗ ∗ ∗ ∗ i=1 L(x , µˆ) = L(x , µ ) = maxµ≥0 L(x , µ), it is implied X ∗ ∗ ∗ ∗ ∗ ∗ ∗ ut =2αt(L(x¯[t], µ ) − L(x , µ )+ L(x , µ ) that (xˆ, µˆ) = (x , µ ). Next, taking into account Lemma 3a) − L(x∗, µ[t])), and (20), we obtain n n n 2 2 1 ∗ 2 ∗ 2 ct =2αtLxn ||xi[t] − x¯[t]|| + L α δ = lim yi[t]||xi[t] − x || + ||µi[t] − µ || x t w t→∞ i i=1 i=1 i i=1 n X X Xn 2 ∗ 2 ∗ 2 + α Lµ , = lim yi[tl ]||x¯[tl ] − x || + ||µi[tl ] − µ || t i s→∞ s s s i i=1 i=1 X X bt =0. =0.
For showing that ct is sumable, we recall from Lemma 3b) Finally, as y [t] > 0 for all t, we conclude ∞ i that, under given assumptions, =0 αt||xi[t] − x¯[t]|| < ∞. t ∗ Therefore, lim ||xi[t] − x || =0, P t→∞ ∞ n ∗ lim ||µi[t] − µi || =0 for i = [n]. 2Lxn αt ||xi[t] − x¯[t]|| < ∞. t→∞ t=0 i=1 X X By Assumption 5, we directly have 2 n 1 ∞ 2 L + Lµ α < ∞. Together, this V. SIMULATION x i=1 wi i t=0 t results in P ∞P As motivated in the introduction, we consider an economic dispatch problem as an example application. In this problem, ct < ∞. t=0 a group of networked generators seeks to fulfill some pre- X Applying Lemma 4, we can then make the statements defined demand D while minimizing their summed up local n cost functions, which are assumed to take a quadratic form. ∗ 2 ∗ 2 Formally, the problem can be defined by ∃δ, lim vt = lim yi[t]||xi[t]−x || + ||µi[t]−µ || t→∞ t→∞ i i=1 n n X 2 (20) min Ci(pi) = min aipi + bipi + ci, (22a) = δ ≥ 0, p p ∞ ∞ i=1 i=1 ∗ ∗ ∗ Xn X ut = 2αt(L(x¯[t], µ ) − L(x , µ ) t=0 t=0 s. t. pi = D, (22b) X X i=1 + L(x∗, µ∗) − L(x∗, µ[t]) < ∞. (21) X p ≤ p ≤ p , ∀i ∈ [n]. (22c) ∞ i,min i i,max Because the sum of the step-size is infinite, t=0 αt = ∞, by assumption, there need to exist subsequences x[tl], µi[tl], The power balance constraint (22b) consists of the sum of all P such that local decision variables pi and is therefore part of the global ∗ ∗ ∗ ∗ ∗ ∗ constraint set. Furthermore, we add the technical constraint lim L(x¯[tl], µ )−L(x , µ )+L(x , µ )−L(x , µ[tl]) = 0 l→∞ that all power outputs of the generators should be positive ∗ p ≥ 0, defining the global constraint set as P = {p|p ≥ Because L(x[t], µ[t]) is affine for fixed x[t] = x and n ∗ 0, pi = D}. Thereby, we ensure that P is closed and convex for fixed µi[t]= µ , it holds that, ∀tl, i=1 i bounded and therefore compact, satisfying Assumption 2. ¯ ∗ ∗ ∗ ∗ ∗ ∗ L(x[tl], µ )−L(x , µ ) ≥ 0, L(x , µ )−L(x , µ[tl]) ≥ 0. Furthermore,P this provides that the projection onto P binds Therefore, the limit holds, if and only if the generation pi, because, regardless of other pj , j = [n] 6= , it holds that . The lower and upper power ∗ ∗ ∗ i 0 ≤ pi ≤ D lim L(x¯[tl], µ )=L(x , µ ), l→∞ limits of each generator in constraint (22c) allocate the local ∗ ∗ ∗ part of the constraint set and are therefore exclusively known lim L(x , µ[tl])=L(x , µ ). l→∞ by the respective agent i. Following from convergence of vt to some constant δ, the The solution set is non-empty if ¯ subsequences x[tl] and µ[tl] are bounded. With that, we can n n x¯ µ choose convergent subsequences [tls ] and [tls ], such that p ≤ D ≤ p . ¯ ˆ ˆ min,i max,i lims→∞(x[tls ], µ[tls ]) = (x, µ) ∈X×M, since X and M i=1 i=1 are closed. Therefore, it holds that X X Therefore, there exists at least one relative interior point ¯ ∗ ˆ ∗ ∗ ∗ lim L(x[tls ], µ )= L(x, µ )= L(x , µ ) and for which the affine equality and inequality constraints are s→∞ ∗ ∗ ∗ ∗ fulfilled, which satisfies the Slater constraint qualification. lim L(x , µ[tl ]) = L(x , µˆ)= L(x , µ ). s→∞ s Together with the strong convexity of the cost functions, we 1.6 δ1 1.4 δ2 δ3 1.2 δ4 1.0
0.8
0.6 Relative error
1.6 VI. CONCLUSION 0.4 δ1 1.4 δ2 In the work at hand, we tailored a solution method to δ3 a class of distributed, convex optimization problems that 1.2 0.2 δ4 need to respect both global and local constraints. Each agent 1.0 projects its gradient update onto the global set while updating a Lagrange parameter, over which their local constraints 0.8 0 are added to the cost function. Convergence to the optimal 0.6 value was proven and some convergence properties shortly Relative error 0 1,000 2,000discussed by 3an,000 example of the 4 economic,000 dispatch problem. 0.4 IterationFuture work will include augmenting the method for time- 0.2 varying communication architectures in order to make the algorithm more robust against failing communication chan- 0 nels. 0 1,000 2,000 3,000 4,000 Iteration REFERENCES
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